Black holes and solitons in string theory

D. Youm/Physics Reports 316 (1999) 1}232

BLACK HOLES AND SOLITONS IN STRING THEORY

Donam YOUM School of Natural Sciences, Institute for Advanced Study Olden Lane, Princeton, NJ 08540, USA

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Black holes and solitons in string theory Donam Youm School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA Received October 1997; editor: A. Schwimmer Contents 1. Outline of the review 2. Soliton and BPS state 2.1. Physical parameters of solitons 2.2. Supermultiplets of extended supersymmetry 2.3. Positive energy theorem and Nester's formalism 3. Duality symmetries 3.1. Electric-magnetic duality 3.2. Target space and strong-weak coupling dualities of heterotic string on a torus 3.3. String}string duality in six dimensions 3.4. ;-duality and eleven-dimensional supergravity 3.5. S-duality of type-IIB string 3.6. ¹-duality of toroidally compacti"ed strings 3.7. M-theory 4. Black holes in heterotic string on tori 4.1. Solution generating procedure 4.2. Static, spherically symmetric solutions in four dimensions 4.3. Rotating black holes in four dimensions 4.4. General rotating "ve-dimensional solution 4.5. Rotating black holes in higher dimensions

4 5 7 11 20 24 26

37 43 47 49 50 56 62 62 65 80 83 89

5. Black holes in N"2 supergravity theories 5.1. N"2 supergravity theory 5.2. Supersymmetric attractor and black hole entropy 5.3. Explicit solutions 5.4. Principle of a minimal central charge 5.5. Double extreme black holes 5.6. Quantum aspects of N"2 black holes 6. p-branes 6.1. Single-charged p-branes 6.2. Multi-charged p-branes 6.3. Dimensional reduction and higher dimensional embeddings 7. Entropy of black holes and perturbative string states 7.1. Black holes as string states 7.2. BPS, purely electric black holes and perturbative string states 7.3. Near-extreme black holes as string states 7.4. Black holes and fundamental strings 7.5. Dyonic black holes and chiral null model 8. D-branes and entropy of black holes 8.1. Introduction to D-branes 8.2. D-brane as black holes 8.3. D-brane counting argument Acknowledgements References

E-mail address: [email protected] (D. Youm) 0370-1573/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 3 7 - X

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Abstract We review various aspects of classical solutions in string theories. Emphasis is placed on their supersymmetry properties, their special roles in string dualities and microscopic interpretations. Topics include black hole solutions in string theories on tori and N"2 supergravity theories; p-branes; microscopic interpretation of black hole entropy. We also review aspects of dualities and BPS states.  1999 Elsevier Science B.V. All rights reserved. PACS: 11.27.#d

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1. Outline of the review It is a purpose of this review to discuss recent development in black hole and soliton physics in string theories. Recent rapid and exciting development in string dualities over the last couple of years changed our view on string theories. Namely, branes and other types of classical solutions that were previously regarded as irrelevant to string theories are now understood as playing important roles in non-perturbative aspects of string theories; these solutions are required to exist within string spectrum by recently conjectured string dualities. Particularly, D-branes which are identi"ed as non-perturbative string states that carry charges in R-R sector have classical p-brane solutions in string e!ective "eld theories as their long-distance limit description. p-branes, other types of classical stringy solutions and fundamental strings are interrelated via web of recently conjectured string dualities. Much of progress has been made in constructing various p-brane and other classical solutions in string theories in an attempt to understand conjectured (non-perturbative) string dualities. We review such progress in this paper. In particular, we discuss black hole solutions in string e!ective "eld theories in details. Recent years have been active period for constructing black hole solutions in string theories. Construction of black hole solutions in heterotic string on tori with the most general charge con"gurations is close to completion. (As for rotating black holes in heterotic string on ¹, one charge degree of freedom is missing for describing the most general charge con"guration.) Also, signi"cant work has been done on a special class of black holes in N"2 supergravity theories. These solutions, called double extreme black holes, are characterized by constant scalars and correspond to the minimum energy con"gurations among extreme solutions. Among other things, study of black holes and other classical solutions in string theories is of particular interest since these allow to address long-standing problems in quantum gravity such as microscopic interpretation of black hole thermodynamics within the framework of superstring theory. In this review, we concentrate on recent remarkable progress in understanding microscopic origin of black hole entropy. Such exciting developments were prompted by construction of general class of solutions in string theories and realization that non-perturbative R-R charges are carried by D-branes. Within subset of solutions with restricted range of parameters, the Bekenstein} Hawking entropy has been successfully reproduced by stringy microscopic calculations. Since the subject reviewed in this paper is broad and rapidly developing, it would be a di$cult task to survey every aspect given limited time and space. The author made an e!ort to cover as many aspects as possible, especially emphasizing aspects of supergravity solutions, but there are still many issues missing in this paper such as stringy microscopic interpretation of black hole radiation, M(atrix) theory description of black holes and the most recent developments in N"2 black holes and p-branes. The author hopes that some of missing issues will be covered by other forthcoming review paper by Maldacena [470]. The review is organized as follows. Sections 2 and 3 are introductory sections where we discuss basic facts on solitons and string dualities which are necessary for understanding the remaining sections. In these two chapters, we especially illuminate relations between BPS solutions and string dualities. In Section 4, we summarize recently constructed general class of black hole solutions in heterotic string on tori. We show explicit generating solutions in each spacetime dimensions and discuss their properties. In Section 5, we review aspects of black holes in N"2 supergravity theories. We discuss principle of a minimal central charge, double extreme solutions and quantum corrections. In Section 6, we

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summarize recent development in p-branes. Here, we show how p-branes and other related solutions "t into string spectrum and discuss their symmetry properties under string dualities. We systematically study various single-charged p-branes and multi-charged p-branes (dyonic p-branes and intersecting p-branes) in di!erent spacetime dimensions. We also discuss black holes in type-II string on tori as special cases and their embedding to p-branes in higher dimensions. In Sections 7 and 8, we summarize the recent exciting development in microscopic interpretation of black hole entropy within the framework of string theories. We discuss Sen's calculation of statistical entropy of electrically charged black holes, Tseytlin's work on statistical entropy of dyonic black holes within chiral null model and D-brane interpretation of black hole entropy.

2. Soliton and BPS state Solitons are de"ned as time-independent, non-singular, localized solutions of classical equations of motion with "nite energy (density) in a "eld theory [106,512]. Such solutions in D spacetime dimensions are alternatively called p-branes [228,234] if they are localized in D!1!p spatial coordinates and independent of the other p spatial coordinates, where p(D!1. For example, the p"0 case (0-brane) has a characteristic of point particles and is also called a black hole; p"1 case is called a string; p"2 case is a membrane. The main concern of this paper is on the p"0 case, but we discuss the extended objects (p51) in higher dimensions as embeddings of black holes and in relations to string dualities. As non-perturbative solutions of "eld theories, solitons have properties di!erent from perturbative solutions in "eld theories. First, the mass of solitons is inversely proportional to some powers of dimensionless coupling constants in "eld theories. So, in the regime where the perturbative approximations are valid (i.e. weak-coupling limit), the mass of solitons is arbitrarily large and the soliton states decouple from the low energy e!ective theories. So, their contributions to quantum e!ects are negligible. Their contribution to full dynamics becomes signi"cant in the strong coupling regime. Second, solitons are characterized by `topological chargesa, rather than by `Noether chargesa. Whereas the Noether charges are associated with the conservation laws associated with continuous symmetry of the theory, the topological conservation laws are consequence of topological properties of the space of non-singular "nite-energy solutions. The space of non-singular "nite energy solutions is divided into several disconnected parts. It takes in"nite amount of energy to make a transition from one sector to another, i.e. it is not possible to make a transition to the other sector through continuous deformation. Third, the solitons with "xed topological charges are additionally parameterized by a "nite set of numbers called `modulia. Moduli or alternatively called collective coordinates are parameters labeling di!erent degenerate solutions with the same energy. The space of solutions of "xed energy is called moduli space. The moduli of solitons are associated with symmetries of the solutions. For example, due to the translational invariance of the Yang}Mills}Higgs Lagrangian, the monopole solution sitting at the origin has the same energy as the one at an arbitrary point in R; the associated collective coordinates are the center of mass coordinates of the monopole. In addition, there are collective coordinates associated with the gauge invariance of the theory. Note, monopole carries charge of the ;(1) gauge group which is broken from the non-Abelian (S;(2)) one at in"nity (where the Higgs "eld takes its value at the gauge symmetry breaking vacuum). Thus, only relevant gauge

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transformations of non-Abelian gauge group that relate di!erent points in moduli space are those that do not approach identity at in"nity, i.e. those that reduce to non-trivial ;(1) gauge transformations at in"nity. Another important characteristic of solitons is that they are the minimum energy con"gurations for given topological charges, i.e. the energy of solitons saturates the Bogomol'nyi bound [94,510]. The lower bound is determined by the topological charges, e.g. the winding number for strings and ;(1) gauge charge carried by black holes. The original calculation [94] of the energy bound for a soliton in #at spacetime involves taking complete square of the energy density ¹ ; the minimum RR energy is saturated if the complete square terms are zero. Solitons therefore satisfy the "rst-order di!erential equations (`complete square termsa"0), the so-called Bogomol'nyi or self-dual equations. An example is the (anti) self-dual condition F "$夹F for Yang}Mills instantons IJ IJ [68]. Another example is the magnetic monopoles [507,592] in an S;(2) Yang}Mills theory, which satisfy the "rst-order di!erential equation BG"$DGU relating the magnetic "eld BG to the Higgs "eld U. Here, the Higgs "eld takes its values at the minimum of the potential <(U), where the non-Abelian gauge group S;(2) is spontaneously broken down to the Abelian ;(1) gauge group. The energy of solitons in asymptotically #at curved spacetime is given by the ADM mass [1,20,349], i.e. a PoincareH invariant conserved energy of gravitating systems. The ADM mass is de"ned in terms of a surface integral of the conserved current JI"¹IJK over a space-like J hypersurface at spatial in"nity. Here, ¹IJ is the energy-momentum tensor density and K is J a time-like Killing vector of the asymptotic spacetime. The so-called positive-energy theorem [146,295,379,387,483,531}534,631] proves that the ADM mass of gravitating systems is always positive. In such proofs, one calculates the energy associated with a small deviation around the background spacetime and "nds it always positive, implying that the background spacetime (Minkowski or anti-De Sitter space-time) is a stable vacuum con"guration. The proof of the positive energy theorem, "rst given in [631] and re"ned covariantly in [483], involves the volume and the surface integrals (related through the Stokes theorem) of Nester's 2-form, which is de"ned in terms of a spinor and its gravitational covariant derivative. Such proofs have an advantage of being easily generalized to supergravity theories. The positive energy theorem proves that the ADM mass of gravitating systems is always positive, provided the spinor satis"es the Witten's condition and the matter stress energy tensor, if any, satis"es the dominant energy condition. One way of proving positivity of the energy of solitons in curved spacetime is by embedding the solutions into (extended) supergravity theories [295,296] as solutions to equations of motion. In this case, the Nester's form is de"ned in terms of the supersymmetry parameters and their supercovariant derivatives. Then, the surface integral yields the supercharge anticommutation relations of extended supersymmetry, i.e. the 4-momentum term plus the central charge term. The 4-momentum in the surface integral is the ADM 4-momentum [1,20] of the soliton and the central charge corresponds to the topological charge carried by the soliton [201,485,489]; the soliton behaves as if a particle carrying the corresponding 4-momentum and quantum numbers. This is a reminiscent of BPS states in extended supergravities. One can think of solitons as realizations of states in supermultiplet carrying central charges of extended supersymmetry [299,399]. In fact, for

 Note, the stress-energy tensor is second order in derivatives of spacetime coordinates.

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each Killing spinor, de"ned as a spinor "eld which is covariant with respect to the supercovariant derivative, one can de"ne a conserved anticommuting supercharge, whose anticommutation relation is just the surface integral of the Nester's 2-form. The integrand of the volume integral of the Nester's 2-form yields sum of terms bilinear in supersymmetry variations of the fermionic "elds in the supergravity theory. Since such terms are positive semide"nite operators, provided the (generalized) Witten's condition [631] and the dominant energy condition for the matter stress-energy tensor are satis"ed, the terms in the surface integral have to be non-negative, leading to the inequality `(ADM mass)5(the maximum eigenvalue of the topological charge term)a. Again a reminiscent of the mass bound for the states in the BPS supermultiplet. This bound is saturated i! the supersymmetry variations of fermions are all zero. The equations obtained by setting the supersymmetry variations of fermions equal to zero are called the Killing spinor equations. These are a system of "rst-order di!erential equations satis"ed by the minimum energy con"guration among solutions with the same topological charge. Such a con"guration is a bosonic con"guration which is invariant under supersymmetry transformations and therefore is called supersymmetric. The necessary and su$cient condition for the existence of supersymmetric solution is the existence of `non-zeroa superconvariantly constant spinors, i.e. Killing spinors. Note, such Killing spinors de"ne supercharges, which act on the lowest spin state to build up supermultiplets of superalgebra. Killing spinors are Goldstone modes of broken supersymmetries; for each supersymmetry preserved, the corresponding supercharge is projected onto zero norm states, and the rest of supercharges are associated with Goldstone spinor degrees of freedom originated from broken supersymmetries. The number of supercharges which are projected onto the zero norm states is determined by the number of distinct eigenvalues of the central charge matrix. In the language of solitons, such central charge matrix is determined by the charge con"gurations of solitons. Alternatively, one can determine the number of supersymmetries preserved by the solitons from the spinor constraints, which are byproducts of the Killing spinor equations along with self-dual or the "rst-order di!erential equations. The number of constraints on the Killing spinors are again determined by the charge con"guration of the solitons. These constraints determine the number of independent spinor degrees of freedom, i.e. the number of supersymmetries preserved by the soliton. Thus, the number of supersymmetries preserved by solitons is intrinsically related to the topological charge con"gurations of solitons through either the number of eigenvalues of central matrix or the number of constraints on the Killing spinors. In the following, we elaborate on ideas discussed in the above in a more precise and concrete way, by quantifying ideas and giving some examples. First, we discuss how the physical parameters (mass, angular momenta, etc.) are de"ned from solitons. Then, we discuss the BPS multiplets of extended supersymmetry theories. Finally, we discuss positive energy theorem of general relativity and extended supergravity theories. 2.1. Physical parameters of solitons We discuss how to de"ne physical parameters (e.g. the ADM mass, angular momenta, ;(1) charges) of gravitating systems. This serves to "x our conventions for de"ning parameters of solitons. The classical solutions near the space-like in"nity can be regarded as the `imprintsa of the ADM mass, angular momenta and electric/magnetic charges of the source.

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First, we discuss the parameters of spacetime metric. The physical parameters are de"ned with reference to the background (asymptotic) spacetime. We assume that the spacetime is asymptotically Minkowski at space-like in"nity, since the solitons under consideration in this review satisfy this condition. We consider the following general form of action in D spacetime dimensions:








1 R#L , (1)

 16pG" , where G" is the D-dimensional Newton's constant (related to the Plank constant i as i "8pG") , " " , and L is the matter Lagrangian density. For the signature of the metric g , we take the mostly

 IJ positive convention (!#2#). From (1), one obtains the following Einstein "eld equations for gravitation: (2) G "R !g R"8pG,¹  , " IJ IJ IJ  IJ where the matter stress-energy tensor ¹  is de"ned as IJ 2 R((!gL )

 , ¹ , (3) IJ RgIJ (!g S" (!g d"x

where ¹  are stresses, ¹  are momentum densities and ¹  is the mass-energy density GH G  (i, j"1,2, D!1). In order to measure the mass, the momenta and the angular momenta of gravitating systems, one usually goes to the external spacetime far away from the source. In this region, the gravitational "eld is weak and, therefore, the Einstein's "eld equations (2) take the form linear in the deviation h of the metric g from the #at one g (g "g #h , "h ";1). This linearize "eld equations IJ IJ IJ II IJ IJ IJ have the invariance under the in"nitesimal coordinate transformations (xIPxI#mI) h P IJ h !R m !R m , which resembles the gauge transformation of ;(1) gauge "elds. (The linearized IJ J I I J Riemann tensor, Einstein tensor, etc., are examples of invariants under this transformation.) By using this gauge invariance, one can "x the gauge by imposing the `Lorentz gaugea condition R (hIJ!gIJh? )"0. This gauge condition is left invariant under the gauge transformations ? J  satisfying m?@ "0. In this gauge, the Einstein's equations take the form, which resembles the @ Maxwell's equations: 1

h "!16pG" ¹ ! g ¹  ,!16pG"¹M  , (4) IJ , IJ , IJ D!2 IJ






where "R R? is the #at (D!1)-dimensional space Laplacian and ¹,¹I . ? I The linearized Einstein's equations (4) have the following general solution that resembles the retarded wave solution of the Maxwell's equations:





¹M (t!"x!y", y) 1 16pG" 16pG" IJ , , h (xG)" d"\y" ¹M d"\y IJ "x!y""\ r"\ IJ (D!3)X (D!3)X "\ "\ 16pG" xI , yI¹M d"\y#2 , # IJ r"\ X "\



(5)

 In the linearized "eld theory, the spacetime vector indices are raised and lowered by the Minkowski metric g . IJ

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where 2pL> X" L n#1 C 2




is the area of SL and i, k"1,2, D!1, are spatial indices. Note, the ADM D-momentum vector PI and angular momentum tensor JIJ of the gravitating system are de"ned as





PI" ¹I d"\x, JIJ" (xI¹J!xJ¹I) d"\x .

(6)

In particular, the time component of PI is the ADM mass M, i.e. M"P. With a suitable choice of coordinate basis, one can put the spatial components JGH (i, j"1,2, D!1) of JIJ in the following form expressed in terms of the angular momenta J (k"1,2, ["\]) in each rotational plane: I  0 J  !J 0  [JGH]" , (7) 0 J  !J 0  \






where for the even D the last row and column have zero entries. In obtaining the general leading order expression for the metric, one chooses the rest frame (PG"0) with the origin of coordinates at the center of mass of the system (xG¹ d"\x"0). In this frame, JG"0, JGH"2xG¹H d"\x and g takes the form: IJ M 1 16pG" xI 1 16pG" , , #O dt! JIG#O dt dxG ds"! 1! r"\ r"\ X r"\ r"\ (D!2)X "\ "\ M 16pG" , d #(gravitational radiation terms) dxG dxH . # 1# r"\ GH (D!2)(D!3)X "\ (8)

 



 




 

Note, the leading order terms of the asymptotically Minkowski metric is time independent and is determined uniquely by the ADM mass M and the intrinsic angular momenta JGH of the source. The general action (1) contains the following kinetic term for a d-form potential 1 A " A 2 B dxI2 dxIB B d! I I with "eld strength F

"dA : B> B








1 1 d"x(!g F . S " BU 16pG" 2(d#1)! B> ,  We omit the dilaton factor in the kinetic term for the sake of the argument.

(9)

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Note, in this kinetic term, G" is absorbed in the action in contrast to the form of the matter term in , (1). The "eld equations and Bianchi identities of A are B (10) d夹F "2i (!)B夹J , dF "0 , B B> B> " where J is the rank d source current and 夹 denotes the Hodge-dual transformation in D spacetime B dimensions, i.e. 1 (夹A )I2I"\B, eI2I"(A ) "\B>2 " with e2"\"1. I BI B d! The soliton that carries the `Noethera electric charge Q under A is an elementary extended B B object with d-dimensional worldvolume, called (d!1)-brane, and has the electric source J coming from the p-model action of the (d!1)-brane. The `topologicala magnetic charge P I of A is carried B B by a solitonic (i.e. singularity and source free) object with dI -dimensional worldvolume, called (dI !1)-brane, where dI ,D!d!2. The `Noethera electric and the `topologicala magnetic charges of A are de"ned as B 1 (!)B夹J " 夹F , Q ,(2i " M B (2i "\B\ B> B "\B 1 " 1 PI, F . (11) B (2i B> B> 1 " These charges obey the Dirac quantization condition [482,591]:

 

Q PI n B B" , n3Z . 4p 2



(12)

The electric and magnetic charges of A have dimensions [Q ]"¸\"\B\ and B B [P I ]"¸"\B\, respectively. Electric/magnetic charges are dimensionless when D"2(d#1). B Examples are point-like particles (d"1) in D"4, strings (d"2) in D"6 [311,542] and membranes (d"3) in D"8 [311,388]. From (11), one sees that the AnsaK tze for F for the soliton that carries electric or magnetic B> charge of A are respectively given by B , (13) 夹F "(2i Q e I /X I , F "(2i P I e /X B> " B B> B> " B B> B> B> where e denotes the volume form on SL, and the electric and magnetic charges of A are de"ned L B from the asymptotic behaviors: Q PI X u B , F & B> B , (14) A& B B> (2i rB> B (2i r"\B\ " " where r is the transverse distance from the (d!1)-brane, u is the volume form for the (d!1)B brane worldvolume and X is the volume form of SB> surrounding the brane. B> From the elementary (d!1)-brane, one "nds that the electric charge Q is related to the tension B ¹ of the (d!1)-brane in the following way: B Q "(2i ¹ (!)"\BB> . (15) B " B

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Here, ¹ has dimensions [¹ ]"M¸B\ in the unit c"1 and therefore is interpreted as mass per B B unit (d!1)-brane volume. In particular, for a 0-brane (d"1) the tension ¹ is the mass. The Dirac  quantization condition (12), together with (15), yields the following form of the magnetic charge P I of A : B B 2pn PI" (!)"\BB>, n3Z . (16) B (2i ¹ " B We comment on the ADM mass of (d!1)-branes. Note, in deriving (8) we assumed that the metric g depends on all the spatial coordinates. So, (8) applies only to the 0-brane type soliton (or IJ black holes). The (d!1)-branes do not depend on the (d!1) longitudinal coordinates internal to the (d!1)-brane and therefore the Laplacian in (4) is replaced by the #at (D!d!1)-dimensional one. As a consequence, in particular, the (t, t)-component of the metric has the asymptotic behavior:






16pG" M 1 , B g &! 1! . (17) RR (D!2)X < r"\B\ "\B\ B\ Here, the ADM mass M of the (d!1)-brane is de"ned as M ,¹ d"\x"< ¹ d"\Bx, B B B\ where < is the volume of the (d!1)-dimensional space internal to the (d!1)-brane. So, for B\ (d!1)-branes it is the ADM mass `densitya



M o , B " ¹ d"\Bx B < B\ that has the well-de"ned meaning. As an example, we consider the elementary BPS (d!1)-brane in D dimensions. The leading order asymptotic behavior of the (t, t)-component of the metric of (d!1)-brane carrying one unit of the d-form electric charge is






D!d!2 c" B g &! 1! , dI '0 , RR D!2 r"\B\

(18)

where c",2i ¹ /dI X I is the unit (d!1)-brane electric charge and r,(x#2#x ) B " B B>  "\B\ is the radial coordinate of the transverse space. For (d!1)-branes carrying m units of the basic electric charge, c" in (18) is replaced by mc". From (17) and (18), one obtains the following ADM B B mass density of the (d!1)-brane carrying one unit of electric charge: o "¹ "(1/(2i ) "Q " . " B B B

(19)

2.2. Supermultiplets of extended supersymmetry 2.2.1. Spinors in various dimensions Before we discuss the BPS states in extended supersymmetry theories, we summarize the basic properties of spinors for each spacetime dimensions D. More details can be found, for example, in

 When dI "0, e.g. a string in D"4, the metric is asymptotically logarithmically divergent. In this case, the ADM mass density is determined from volume integral of the (t, t)-component of the gravitational energy-momentum pseudo-tensor [440].

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[438,575,628,629]. We assume that there is only one time-like coordinate. The types of superPoincareH algebra satis"ed by supercharges depend on D. The superalgebra is classi"ed according to the fundamental spinor representations of the homogeneous group SO(1, D!1) and the vector representation of the automorphism group that supercharges belong to. The pattern of superalgebra repeats with D mod 8. In even D, one can de"ne c-like matrix c,gcc2c"\ which anticommutes with cI and has the property c"1 (implying g"(!1)"\), required for constructing a projection operator. So, the 2 " complex component Dirac spinor t, which is de"ned to transform as dt"!e RIJt (RIJ,![cI, cJ]) under the in"nitesimal Lorentz transformation, in even D   IJ is decomposed into 2 inequivalent Weyl spinors t "(1#c)t and t "(1!c)t with \  >  2"\ complex components each. We discuss the reality properties of spinors. One can always "nd a matrix B satisfying RIJH"BRIJB\. B de"nes the charge conjugation operation: tPtA"Ct,B\tH .

(20)

By de"nition, the charge conjugation operator C commutes with the Lorentz generators RIJ, implying that t and tA have the same Lorentz transformation properties. If C"1, or equivalently BBH"1, the Dirac spinor t can be reduced to a pair of Majorana spinors (i.e. eigenstates of C) t "(1#C)t and t "(1!C)t. This is possible in D"2, 3, 4, 8, 9 mod 8. First, in odd D,    Majorana spinors are necessarily self-conjugate under C and are always real. So, the Dirac spinors in odd D are real [pseudoreal] in D"1, 3 mod 8 [D"5, 7 mod 8]. In even D, Majorana spinors can be either complex or real depending on whether t and tA have the same or opposite helicity. Namely, since Cc"(!1)"\cC, t and tA have the same (opposite) helicity for even (odd) (D!2)/2, i.e. D"2 mod 8 [D"4, 8 mod 8]. So, in even D, the Dirac spinors are real (complex) or pseudoreal for D"2 mod 8 (D"4, 8 mod 8) or D"6 mod 8, respectively. In particular, in D"2 mod 8, both the Weyl and the Majorana conditions are satis"ed, and therefore in this case the Dirac spinor t is called Majorana}Weyl. We saw that supercharges Q (i"1,2, N), transforming as spinors under SO(1, D!1), have G di!erent chirality and reality properties depending on D. The set +Q ,, furthermore, transforms as a G vector under an automorphism group, with i acting as a vector index. The automorphism group depends on the reality properties of +Q ,. The automorphism group is SO(N), ;Sp(N) or S;(N);;(1) for G real, pseudoreal or complex case, respectively. In D"2 mod 8 and D"6 mod 8, the pair of Weyl spinors with opposite chiralities are not related via C and therefore are independent: the automorphism groups are SO(N );SO(N ) and ;Sp(N );;Sp(N ) in D"2 mod 8 and D"6 mod 8, > > > > respectively, where N #N "N. The central charge Z'( transforms as a rank 2 tensor under the > \ automorphism group with (I, J) acting as tensor indices. In D"0, 1, 7 mod 8 [D"3, 4, 5 mod 8], the central charge has the symmetry property Z'("Z(' [Z'("!Z(']. The number N of supercharges Q' in each D is restricted by the physical requirement that particle helicities should not exceed 2 when compacti"ed to D"4 [17,69,169,274,479]. This limits the maximum D with 1 time-like coordinate and consistent supersymmetric theory to be 11 with N"1 supersymmetry, i.e. 32 supercharge degrees of freedom. This corresponds to N"8 supersymmetry in D"4 when compacti"ed on ¹. In D(11, the number of spinor degrees of freedom cannot exceed that of N"1, D"11 theory. For the pseudoreal cases, i.e. D"5, 6, 7 mod 8, only even N are possible.

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2.2.2. Central charges and super-PoincareH algebra We discuss types of central charge Z'( one would expect in the super PoincareH algebra. According to a theorem by Haag et al. [330], within a unitary theory of point-like particle interactions in D"4 the central charge can be only Lorentz scalar. However, in the presence of p-branes (p51), central charges Z'(2 N transforming as Lorentz tensors can be present in the I I superalgebra without violating unitarity of interactions [201]. In fact, as will be shown, it is the Lorentz tensor type central charges in higher dimensions that are responsible for the missing central charge degrees of freedom in lower dimensions when the higher-dimensional superalgebra is compacti"ed with an assumption that no Lorentz tensor type central charges are present [603]. The Lorentz tensor type central charges appear in the supersymmetry algebra schematically in the form: (21) +Q', Q(,"d'((CcI) P # (CcI2IN) Z'(2 N , ?@ I I ? @ ?@ I N2 where P is D-dimensional momentum, I, J"1,2, N label supersymmetries and a, b are spinor I indices in D dimensions. Here, (CcI) in (21) is replaced by (CcIP!) for positive or negative chiral ?@ ?@ Majorana spinors Q' (e.g. type-IIB theory), where P projects on the positive or negative ! ! chirality subspace, and also similarly for (CcI2IN) . Note, Z'(2 N commute with Q' and P , but ?@ I I ? I transform as second rank tensors under the Lorentz transformation, and therefore are central with respect to supertranslation algebra, only. The number of central charge degrees of freedom is determined by the number of all the possible (I, J) in (21) [41}44]. In the sum term in (21), one has to take into account the overcounting due to the Hodge-duality between p and D!p forms (Z'(2 N&Z'(2 "\N). When p"D!p, Z'(2 N are I I I I I I self-dual or anti-self dual. (For this case, the degrees of freedom are halved.) (I, J) on Z'(2 N are I I de"ned to have the same permutation symmetry as (a, b) in cI2IN so that cI2INZ'(2 N is symmetric ?@ ?@ I I under the simultaneous exchanges of indices in the pairs (I, J) and (a, b) so that they have the same symmetry property (under the exchange of the indices) as the left-hand side of (21). Namely, only terms associated with cI or cI2IN that are either symmetric or antisymmetric under the exchange ?@ ?@ of a and b can be present on the right-hand side of (21). 2.2.3. Central charges and i-symmetry The p-form central charge Z'(2 N in (21) arises from the surface term of the Wess}Zumino (WZ) I I term in the p-brane worldvolume action [201]. Before we discuss this point, we summarize how WZ term emerges in the p-brane worldvolume action [599,600]. In the Green}Schwarz (GS) formalism [86,313,315,351] of the supersymmetric p-brane worldvolume action, one achieves manifest spacetime supersymmetry by generalizing spacetime with bosonic coordinates XI (k"0, 1,2, D!1) and global Lorentz symmetry to superspace R with coordinates Z+"(XI, h?) and super-PoincareH invariance. Here, a is a Ddimensional spacetime spinor index and the spacetime spinor h? takes an additional index I (I"1,2, N) for N-extended supersymmetry theories, i.e. h'?. Fields in the GS action are

 A Majorana spinor Q is de"ned as QM "Q2C, where the bar denotes the Dirac conjugate. The positive or negative chiral spinor Q is de"ned as cQ "$Q . ! ! !

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regarded as maps from the worldvolume = to R. The worldvolume = of a p-brane has coordinates mG"(q, p ,2, p ) with worldvolume vector index i"0, 1,2, p. We denote an immersion from  N = to R as : =PR. The pullback H of a form in R by induces a form in =. To generalize the bosonic p-brane worldvolume Lagrangian density L "¹ [!det(R XI(m)R XJ(m)g )] to be
 N G H IJ invariant under the supersymmetry transformation as well as local reparameterization and global PoincareH transformations, one introduces a supertranslation invariant D-vector-valued 1-form PI,dXI!ihM cI dh. This corresponds to the spacetime component of the left-invariant 1-form P"(PI, P?"dh?) on R. The simplest and straightforward supersymmetric generalization of the bosonic worldvolume p-model action for a p-brane is [2,85}87,201,313,315,351,377] S "¹  N



dN>m(!det(PI(m)PJ(m)g ) , (22) G H IJ 5 where ¹ is the p-brane tension and PI is the mG-component of the pullback of the 1-form PI in R by N G

, i.e. ( H P)(m)"P(m) dmG with PI"R XI!ihM cIR h and P?"R h? (R ,R/RmG). (22) is manifestly G G G G G G G invariant under the global super-PoincareH and local reparameterization transformations, but is not invariant under a local fermionic symmetry, called `i-symmetrya [3,88,131,313,377], which is essential for equivalence of the GS and NSR formalisms of the worldvolume action. To make (22) invariant under the i-symmetry, one introduces an additional term S , called `Wess}Zumino 58 (WZ) actiona, into (22). To construct the Wess}Zumino (WZ) action for a p-brane, one introduces the super-PoincareH invariant closed (p#2)-form h on R. Such closed (p#2)-forms exist only N> for restricted values of D and p. The complete listing of the values of (D, p) are found in [2]. The maximum values of allowed D and p are D "11 and p "5, which can also be determined



 by the worldvolume bose-fermionic degrees of freedom matching condition discussed in the next paragraph. The super-PoincareH invariant closed (p#2)-form, in general, has the form is closed, one can locally write h in terms of h "PI2PIN> dhM c 2 N> dh. Since h I I N> N> N> a (p#1)-form b (on R) as h " db . Then, a super-PoincareH invariant WZ action for N> N> N> over = [2,85}87,201,313,315,351,377]: a p-brane is obtained by integrating b N>



. S "¹ dN>m H b N> 58 N

(23)

Note, whereas S and S are individually invariant under the local reparameterization and global  58 super-PoincareH transformations, the i-symmetry is preserved only in the complete action S "S #S . The i-symmetry gauges way the half of the degrees of freedom of the spinor h, N  58 thereby only 1/2 of spacetime supersymmetry is linearly realized as worldvolume supersymmetry [378]. To summarize, the invariance under the i-symmetry necessitates the introduction of b (on R) via the WZ term; b couples to the worldvolume of the p-branes and becomes the N> N> origin of the central charge term in the supersymmetry algebra. We comment on the allowed values of p and the number N of spacetime supersymmetry for each D. This is determined [241] by matching the worldvolume bosonic degrees N and fermionic degrees N of freedom. First, we consider the case where the worldvolume theory corresponds to $ scalar supermultiplet (with components given by scalars and spinors). By choosing the static gauge (de"ned by XI(m)"(XG(m),>K(m))"(mG,>K(m)), with i"0, 1,2, p and m"p#1,2, D!1), one "nds that the number of on-shell bosonic degrees of freedom is N "D!p!1. We denote the

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number of supersymmetries and the number of real components of the minimal spinor in D-dimensional spacetime [(p#1)-dimensional worldvolume] as N and M [n and m], respectively. Then, since the i-symmetry and the on-shell condition each halves the number of fermionic degrees of freedom, the number of on-shell fermionic degrees of freedom is N "mn"MN. The allowed  $  values of N and p for each D is determined by the worldvolume supersymmetry condition N "N , $ i.e. D!p!1"mn"MN. The complete listing of values of N and p are found in [234,241]. The   maximum number of D in which this condition can be satis"ed is D "11 (p"2) with M"32

 and N"1. So, for other cases (D(11), MN432. Similarly, the maximum value of p for which this condition can be satis"ed is p "5. The `fundamentala super p-branes [2] that satisfy

 this condition are (D, p)"(11, 2), (10, 5), (6, 3), (4, 2). The 4 sets of p-branes obtained from these `fundamentala super p-branes through double-dimensional reduction are named the octonionic, quaternionic, complex and real sequences. Note, in addition to scalars and spinors, there are also higher spin "elds on the worldvolume [241]: vectors or antisymmetric tensors. First, we consider vector supermultiplets. Since a worldvolume vector has (p!1) degrees of freedom, the worldvolume supersymmetry condition N "N becomes D!2"mn"MN. This condition intro $  duces additional points in the brane-scan. Vector supermultiplets exist only for 34p49 and the bose}fermi matching condition can be satis"ed in D"4, 6, 10, only. Second, we consider tensor worldvolume supermultiplets. In p#1"6 worldvolume dimensions, there exists a chiral (n , n )"(2, 0) tensor supermultiplet (B\ , j', '( ), I, J"1,2, 4, with a self-dual 3-form "eld > \ IJ strength, corresponding to the D"11 5-brane. The decomposition of this (2, 0) supermultiplet under (1, 0) into a tensor multiplet with 1 scalar and a hypermultiplet with 4 scalars, followed by truncation to just the tensor multiplet, leads to worldvolume theory of 5-brane in D"7. 2.2.4. Central charges and topological charges We illustrate how Lorentz tensor type central charges (associated with p-branes) arise in the supersymmetry algebra [201]. Since the action S has the manifest super-PoincareH invariance, N one can construct supercharges QG from the conserved Noether currents j associated with ? ? super-PoincareH symmetry. Whereas (22) is invariant under the super-PoincareH variation, i.e. d S "0, the integrand of the WZ action (23) is only quasi-invariant. Namely, since d b "dD 1  1 N> N for some p-form D , the integrand of S transforms by total spatial derivative: N 58 ¹ d ( H b )"d( H D ),R DG dqdp2 dpN. It is D that induces `topological N 1 N> N G N N chargea which becomes central charge in the super-PoincareH algebra. Generally, when a Lagrangian density L is quasi-invariant under some transformation, i.e. d L"R DG , the associated  G  Noether current jG contains an `anomalousa term DG :   RL !DG . jG "d Z+    R(R Z+) G Such an anomalous term modi"es the algebra of the conserved charges Q"dNp jO to include  a topological (or central) terms A .  For a p-brane, the WZ action (23) gives rise to the central term in the supersymmetry algebra of the form:



A "¹ (CcI2IN) dNp jO I2IN , ?@ 2 ?@ N

(24)

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where jO I2IN is the (worldvolume) time component of the topological current density 2 jG I2IN"eGH2HNR XI(m)2R NXIN(m). So, p-form central charges in supersymmetry algebra (21) has 2 H H the following general form [44] given by the surface integral of a (p#1)-form local current J'( 2 N(x) over a space-like surface embedded in D-dimensional spacetime: II I



Z'(2 N" d"\RI J'( 2 N(x) . II I I I

(25)

Here, the (p#1)-form local current J'( 2 N(x) has contributions from individual p-branes with the II I coordinates X? (q, p ,2, p ) and charges z'( (index a labeling each p-brane): I  N ?



J'( 2 N(x)" dq dp 2 dp z'(d"(x!X?(q, p ,2, p ))  N ?  N II I ? (26) ;R X? 2R NX? N (q, p ,2, p ) .  N N I

O I This (p#1)-form current is coupled to a (p#1)-form gauge potential A'( 2 N(x) of the low energy II I e!ective supergravity in the following way:



S& d"x AII2IN(x)J'( 2 N(x) '( II I



(27) " dq dp 2 dp AII2IN(X?)R X? 2R NX? N z'( , O I  N '( N I ? ? where z'( are the charges of A'( 2 N(x) carried by the ath p-brane with the coordinates X? . The ? II I I "eld equation for A'( 2 N(x) is II I (28) RHR A'( 2 N (x)"J'( 2 N(x) . II I H I I I

So, one can think of Z'(2 N as being related to charges of A'( 2 N(x) with the charge source given II I I I by p-branes with their worldvolumes coupled to A'( 2 N(x). There is a one-to-one correspondence II I between A'( 2 N(x) in the e!ective supergravity theory and Z'( 2 N in the superalgebra, i.e. there II I II I are as many central extensions as form "elds in the e!ective supergravity. 2.2.5. S-theory The maximally extended superalgebra has 32 real degrees of freedom in the set +Q', of ? supercharges, i.e. N"1 supersymmetry in D"11 or N"8 supersymmetry in D"4. So, the right-hand side of (21) has at most (32;33)/2"528 degrees of freedom; the sum of D degrees of freedom of the momentum operator PI and the degrees of freedom of central charges Z'(2 N in (21) I I has to be 528. This is the main reason for the necessity of existence of p-branes in higher dimensions [603]; N"1 supersymmetry in D"11 without central charge has only 11 degrees of freedom on the right-hand side of (21).

 In the supersymmetry algebra (21), the p-brane tension ¹ is set equal to 1. N  So, a p-form central charge is related to boundaries of the p-brane. For example for a string ( p"1), Z & I X (0)!X (p). I I

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The extended superalgebra (21) can be derived by compactifying either a type-A superalgebra in D"2#10 or a type-B superalgebra in D"3#10 (the so-called `S-theoriesa) [41}44], with the Lorentz symmetries SO(10, 2) and SO(9, 1)SO(2, 1), respectively. The superalgebra of the typeIIA(B) theory is obtained by compactifying the type-A(B) algebra. First, the type-A algebra has the form: (29) +Q , Q ,"c++Z  #c+2+Z>2  , ?@ ++ ?@ + + ? @ where M "0, 0, 1,2, 10 is a D"12 vector index with 2 time-like indices 0, 0. Note, in D"12 G with 2 time-like coordinates, only gamma matrices which are (anti)symmetric are c++ and c+2+, with c+2+ being self-dual. So, one has terms involving 2-form central charge Z   and ++ self-dual 6-form central charge Z>2 , without momentum operator P+ term, in (29). Generally, + + in the D(12 supersymmetry algebra compacti"ed from the type-A algebra, the spinor indices a and a of 32 spinors Q? are regarded as those of SO(c#1, 1) and SO(D!1, 1) Lorentz groups, ? respectively, where c is the number of compacti"ed dimensions from the point of view of D"10 string theory. Second, the type-B algebra has the form: (30) +Q? , Q@M M ,"cI M (cq )? @M PG #cI M IIc? @M >   #cI M 2I(cq )? @M XG 2  , ? @ ?@ ?@ ?@ G I III G I I where the indices are divided into the D"10 ones a, bM "1, 2,2,16 and k"0, 1,2, 9 with the Lorentz group SO(1, 9), and the D"3 ones a , bM "1, 2 and i"0, 1, 2 with the Lorentz group SO(1, 2). (The barred [unbarred] indices are spinor [spacetime vector] indices.) Here, cI [q ] are G gamma matrices of the SO(1, 9) [SO(1, 2)] Cli!ord algebra and c? @M "ip? @M "e? @M .  We discuss the maximal extended superalgebras that follow from the type-A algebra. First, the following N"1, D"11 superalgebra is obtained from (29) by compactifying the 0-coordinate: +Q , Q ,"(CIC) P #(CIIC) Z  #(CI2IC) X 2  , ?@ I I ?@ I I ? @ ?@ I where each term on the right-hand side emerges from the terms in (29) as

(31)

Z  PP Z   66"11#55 ++ I II (32) 462"462 . Z>2 PX 2  I I + + The central charges Z   and Z 2  are associated with the M2 and M5 branes, respectively. The I I II maximal extended superalgebras in D(11 are obtained by compactifying the D"11 supertranslation algebra (31) on tori. The central charge degrees of freedom in lower dimensions are counted by adding the contribution from the internal momentum P (m"1,2, 11!D) and the number of K ways of wrapping M2 and M5 branes on cycles of ¹\" in obtaining various p-branes in lower dimensions. Schematically, decompositions of the terms on the right-hand side of (31) are P "P P , Z  "Z  ZKZKK , II I I I K II (33) X 2 "X 2 XK2 XKK XKKKXK2KXK2K . I I III II I I I I I The N(N!1) Lorentz scalar central charges of the (maximal) N-extended D(11 superalgebras originate from the Lorentz scalar type terms (under the SO(D!1, 1) group) on the right-hand sides of (33), i.e. P , ZKK and XK2K. In this consideration, one has to take into account equivalence K under the Hodge duality in D dimensions. N(N!1)"56 Lorentz scalar central charges of N"8 superalgebra in D"4 originate from the 7 components of P , (7;6)/(2;1)"21 terms in ZKK, K

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(7;6;5;6;5)/(5;4;3;2;1)"21 terms in XK2K and 7 terms in the Hodge dual of XK2 . I I The similar argument regarding the supertranslational algebras of type-IIB and heterotic theories can be made, and details are found, for example, in [44,603]. We saw that one has to take into account the Lorentz tensor type central charges in higher dimensions to trace the higher-dimensional origin of N(N!1) 0-form central charge degrees of freedom in N-extended supersymmetry in D(11. This supports the idea that in order for the conjectured string dualities (which mix all the electric and magnetic charges associated with N(N!1) 0-form central charges in D(10 among themselves) to be valid, one has to include not only perturbative string states but also the non-perturbative branes within the full string spectrum. In lower dimensions, these central charges are carried by 0-branes (black holes). It is a purpose of Section 4 to construct the most general black hole solutions in string theories carrying all of 0-form central charges. In Section 6.3.3, we identify the (intersecting) p-branes in higher dimensions which reduce to these black holes after dimensional reduction.

2.2.6. Central charges and moduli xelds We comment on relation of central charges to ;(1) charges and moduli "elds [7,8]. Except for special cases of D"4, N"1, 2 and D"5, N"2, scalar kinetic terms in supergravity theories are described by p-model with target space manifold given by coset space G/H. Here, a non-compact continuous group G is the duality group that acts linearly on "eld strengths HK2 N> and is on-shell I I and/or o!-shell symmetry of the action. The isotropy subgroup HLG is decomposed into the automorphism group H of the superalgebra and the group H related to the matter 

 multiplets: H"H H . (Note, the matter multiplets do not exist for N'4 in D"4, 5 and in 

 maximally extended supergravities.) The properties of supergravity theories are "xed by the coset representatives ¸ of G/H. ¸ is a function of the coordinates of G/H (i.e. scalars) and is decomposed as: ¸\"(¸ K, ¸ K) , (34) ¸"(¸KR)"(¸K , ¸K),  '  ' where (A, B) and I respectively correspond to 2-fold tensor representation of H and the  fundamental representation of H . Here, K runs over the dimensions of a representation of G.

 The (p#2)-form strength HK kinetic terms are given in terms of the following kinetic matrix determined by ¸: . (35) NKR"¸ K¸ R !¸ K¸'  ' R So, the `physicala "eld strengths of (p#1)-form potentials in supergravity theories are `dresseda with scalars through the coset representative and the (p#2)-form "eld strengths appear in the supersymmetry transformation laws dressed with the scalars. The central charges of extended superalgebra, which is encoded in the supergravity transformations rules, are expressed in terms of electric QK, "\N\GK and magnetic PK, N>HK charges of (p#2)-form "eld strengths 1 1 HK"dAK (GK"RL/RHK) and the asymptotic values of scalars in the form of the coset representative, manifesting the geometric structure of moduli space. These central charges satisfy the di!erential equations that follow from the Maurer}Cartan equations satis"ed by the coset representative. One of the consequences of these di!erential equations is that the vanishing of a subset of central charges (resulting from the requirement of

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supersymmetry preserving bosonic background) forces the covariant derivative of some other central charges to vanish, i.e. `principle of minimal central chargea [258,259]. Furthermore, from de"ning relations of the kinetic matrix of (p#2)-form "eld strengths and the symmetry properties of the symplectic section under the group G, one obtains the sum rules satis"ed by the central and matter charges. For other cases, in which the scalar manifold cannot be expressed as coset space, one can apply the similar analysis as above by using techniques of special geometry [132,270,576]. For this case, the roles of the coset representative and the Maurer}Cartan equations are played respectively by the symplectic sections and the Picard}Fuchs equations [134]. 2.2.7. BPS supermultiplets We discuss massive representations of extended superalgebras with non-zero central charges [264,330,485,489,575], i.e. the BPS states. It is convenient to go to the rest frame of states de"ned by P "(M, 0,2, 0), where M is the rest mass of the state. The little group, de"ned as a set of I transformations that leave this P invariant, consists of SO(D!1), the automorphism group and I the supertranslations. Since central charges Z'( transform as a second rank tensor under the automorphism group, only the subset of automorphism group that leaves Z'( invariant should be included in the little group. Central charges inactivate some of supercharges, reducing the size of supermultiplets. In the following, we illustrate properties of the BPS states for the D"4 case with an arbitrary number N of supersymmetries. In the Majorana representation, the central charges ;'( and <'( (I, J"1,2, N) appear in the N-extended superalgebra in D"4 in the form [264]: +Q', Q(,"(cIC) P d'(#C ;'(#(c C) <'( , (36) ? @ ?@ I ?@  ?@ where a, b"1,2, 4 are indices of Majorana spinors, C is the charge conjugation matrix C"!C2, and the supercharges Q' are in the Majorana representation. In the Majorana ? representation, ;'( and <'( are hermitian operators and antisymmetric. Alternatively, one can express the superalgebra (36) in the Weyl basis. In this basis, a 4component spinor Q (a"1,2, 4) is decomposed into left- and right-handed 2-component Weyl ? spinors: "(ip Q*')? , (37) Q'"(Q' ) , (Q' )? "e? @Q Q*'  * ? *? 0 @Q  Q where a, b"1, 2 and a, bQ "1, 2 are Weyl spinor indices, and e?@"e?@"(ip ) "!e "!e  Q is  ?@ ?@ ?@ the 2-dimensional Levi}Civita symbol. Namely, the lower and upper components of a 4-component spinor Q (a"1,2, 4) are (Q ) and e? @Q Q* Q , respectively. In this 2-component Weyl basis *@ ? *? representation, anticommutations (36) become +Q', Q*( ,"(p ) Q PId'(, ? @Q I ?@ (38) +Q', Q(,"e Z'(, +Q*'  , Q*( Q ,"e  Q Z'( , @ ?@ ? @ ?@ ? where Z'(,!<'(#i;'(. The central charge matrix Z'( can be brought to the block diagonal form by applying the ;(N) automorphism group: Z'("diag(z eGH, z eGH,2, z eGH)"ip ZK ,   ,

 ,

(39)

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where z (m"1,2, [N/2]) are eigenvalues of Z'( and ZK ,diag(z ,2, z ). There are extra K ,

 ,

0 entries in the Nth row and column in Z'( for an odd N. Further rede"ning the supercharges and making use of the reality condition satis"ed by the supercharges, one can simplify supersymmetry algebra: +SK , S*L ,"d dKLd (M!(!)@z ) , ?? @@ ?@ ?@ L +SK , SL ,"+S*K , S*L ,"0 , (40) ?? @@ ?? @@ where a, b"1, 2, a, b"1, 2, and m, n"1,2, N/2. For odd N, there are extra anticommutation relations associated with the extra 0 entries in Z'(. Since the left-hand sides of (40) are positive semide"nite operators, the rest mass M of the particles in the supermultiplet is always greater than or equal to all the eigenvalue of the central charge matrix and therefore M5max+"z ", . (41) K The state that saturates the bound (41) is called the BPS state. The supermultiplet that do not saturate (41) is called the long multiplet and is the same as that of extended superalgebra without central charges. The BPS supermultiplet is called short multiplet, since there are fewer supercharges (or raising operators) available for building up supermultiplet (since the supercharges that annihilate the supersymmetric vacuum get projected onto zero norm states). The type of supermultiplet that BPS states belong to depends on the number of distinct eigenvalues of the central charge matrix Z'(. In the following, we give examples on all the possible BPS multiplets of N"4 supersymmetry algebra. N"4 superalgebra has [N/2]"2 eigenvalues z and z . There are two types of BPS   supermultiplets in the N"4 superalgebra depending on whether z are the same or di!erent.  When z "z , two raising operators S and S in (40) are projected onto the zero-norm   ? ? states; SK (m"1, 2) annihilate the supersymmetric vacuum state and become zero. So,  of ?  supersymmetry is preserved and the remaining raising operators S and S act on the lowest ? ? helicity state to generate the highest spin 1 multiplet. When z Oz , say, z 'z , the raising operator S is projected onto the zero-norm states.     ? Hence,  of supersymmetry is preserved and the remaining raising operators S , S and S act ? ? ?  on the lowest helicity state to generate the highest spin 3/2 multiplet. 2.3. Positive energy theorem and Nester's formalism We discuss the positive mass theorem [146,295,379,483,531}534,631] of general relativity. The positive mass theorem says that the total energy, i.e. the rest mass plus potential energy plus kinetic energy, of the gravitating system is always positive with a unique zero-energy con"guration with appropriate boundary conditions at in"nity, provided that the matter stress-energy tensor ¹ satis"es the dominant energy condition IJ ¹ ;I
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asymptotically at in"nity. The reason why one cannot make gravitational energy arbitrarily negative by shrinking the size of the gravitating object (as in the case of Newtonian gravity) is that when a system collapses beyond certain size, it forms an event horizon, which hides the inside region that has singularities and negative energy: the system appears to have positive energy to an outside observer. In general relativity, there is no intrinsic de"nition of local energy density due to the equivalence principle. So, one has to de"ne energy of a gravitating system as a global quantity de"ned in background (or asymptotic) region [1,20]. In general, conserved charges of general relativity are associated with generators of symmetries of the asymptotic region. Namely, the conserved charge is de"ned as a surface integral (with the integration surface located at in"nity) of the time component of the conserved current of the symmetry in the asymptotic region. Here, the integration surface is taken to be space-like in this section and thereby gravitational energy is of the ADM type. For a set +k, of vectors that approach the Killing vectors of the asymptotic background, one I can de"ne the conserved charge K through:







1 HIJk dR " I J 4 R

dNOH CJ? kMeIe@ dR , (43) IJM @ ? H NO 1.R where H is the total stress-energy tensor [440] including the pure gravity contributions and IJ C? "C? dxI is the connection 1-form for the metric g . For a time-like Killing vector k, K is @ I@ IJ I the ADM energy relative to the zero-energy background state. For a generic asymptotically #at spacetime, the set of conserved charges K consists of the ADM 4-momentum PI and the angularmomentum tensor JIJ, which satisfy the PoincareH algebra. (The ADM mass M is the norm of the ADM 4-momentum: M"(!PIP .) For a supersymmetric bosonic background, one can de"ne I the (conserved) supercharge [206,590] QK for the conserved current JK"aKRI (R (eaKRI)"0), I I which is de"ned in terms of a Killing spinor aK and a gravitino t satisfying the Rarita}Schwinger I equation: dx eHIk" I R

K"







1 aKeIJMNc t dR " J M N I 2 R

aKeIJMNc c t dR (44)  J N IM 1.R where RI"eIJMNcc t and pIJ,[cI, cJ], etc. The generators KI and QI of the asymptotic J M N   K spacetime symmetries and the supersymmetry transformation satisfy the following algebras: QK"

R

dx eaKR"

[KI , KI ]"C! KI , +QI , QI ,"f +KI , (45)   ! K L KL + where C! are the same structure constants that appear in the commutator of the Killing vectors  k and f  are the same constants in the following relation between the Killing vectors kI and the  KL  Killing spinors a : K a cIa "f  kI . (46) K L KL   At in"nity, the current J"e H kJ is conserved, i.e. RIJ"0. Here, e is the determinant of the background metric I IJ I and kJ is now Killing vectors of the background spacetime.  Here, the integration measures for the surface and volume integrals are respectively de"ned as dR " e dxM dxN and dR "e dxJ dxM dxN. IJ  IJMN I  IJMN  For #at spacetime, one can choose the basis of aK so that f  "c . KL KL

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So, these conserved charges satisfy the supersymmetry algebras [JIJ, QK]" pIJKQL, +QK, QL," cKLPI . (47) L I The third equation in (47) leads to the simple proof [207] of the positivity of energy in quantum supergravity. Since the left-hand side of this equation is a positive semide"nite operator, one has [PI, QK]"0,

\1s"+QK, QLR,"s2"(cIc)KL1s"P "s250 , (48) I where "s2 is a physical state vector. For this inequality to be satis"ed, the eigenvalues P$"P" of cIcP have to be non-negative, leading to proof of the positivity of gravitational energy. Rigorous I proof of the positive energy theorem based on original Witten's proof [631] involves the following antisymmetric tensor (the Nester's 2-form [483]): EIJ"eIJMN(e cc e! e cc e) , (49) M N N M where a Dirac spinor e is assumed to approach a constant spinor e asymptotically  (ePe #O(r\)) and is the gravitational covariant derivative on a spinor. The ADM 4 I momentum P is related to the surface integral of EIJ over the surface S at the space-like in"nity in I the following way:



1 (50) P e cIe " EIJ dR . IJ I  2 1 Proof of the positive energy theorem involves the surface and the volume integrals of the Nester's 2-form, which are related by the Gauss divergence theorem: Z REIJ dS "R EIJ dR . This IJ I J 1.  leads to





P e cIe " G e cIe dRJ# e (cJpIM#pIMcJ) e dR , M J I  R IJ R I

(51)

where G is the Einstein tensor of the metric g . From the Einstein's equations G "¹ , one sees IJ IJ IJ IJ that the "rst term on the right-hand side of (51) is positive if ¹ satis"es the dominant energy IJ condition (42). In the coordinate system in which the x-direction is normal to R, the integrand of the second term on the right-hand side of (51) simpli"es to





  2 e cpGH e"2" e"!2 cG e . G G H H G So, if the spinor e satis"es the `Witten conditiona [631]:

(52)

 cG e"0 , (53) G G the second term on the right-hand side of (51) is also positive. Thus, the left-hand side of (51) is always non-negative for any e , leading to proof that energy P of a gravitating system is  non-negative.

 This is a necessary condition [145,494,513] for a spinor e to satisfy the Witten's condition.

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23

The above proof of the positive energy theorem based on the Nester's formalism can be readily generalized to gravitating con"gurations in extended supergravities. In this case, the gravitational covariant derivative on spinors in the Nester's 2-form (49) is replaced by the super-covariant I derivative K on spinors. So, the Nester's 2-form generalizes to I (54) EK IJ,2( K eCIJMe!e CIJM K e)"EIJ#HIJ , M M where EIJ is the original Nester's 2-form (49) and HIJ denotes the remaining terms, which are usually expressed in terms of gauge "elds of extended supergravities. Here, the supercovariant derivative on a supersymmetry parameter e is given by the gravitino supersymmetry transformation in bosonic background, i.e. dt " K e. The lower bound for mass is given in terms of central I I charges (coming from the HIJ term in (54)) of the extended supergravity and the bound is saturated when the gravitating con"guration is a bosonic con"guration which preserves some of supersymmetry. In the following, we discuss proof of the positive mass theorem in pure N"2, D"4 supergravity as an example [296]. The N"2, D"4 supergravity is "rst obtained in [265] by coupling the (2,3/2) gauge action to the (3/2,1) matter multiplet by means of the Noether procedure. The theory uni"es electromagnetism (spin 1 "eld) with gravity (spin 2 "eld). The theory has a manifest invariance under the O(2) symmetry, which rotates 2 gravitino into each other. In the bosonic background, the supergravity transformation of the gravitino, which de"ne the supercovariant derivative K , is I 1 (G 1 ,F cJcMc e .

K e"

! (55) dt " I JM I I (2pG I 4 (2pG , , Substituting the supercovariant derivative K e from (55) into the Nester's two-form (54), one has I the following expression for HIJ:





HIJ,4e (FIJ#ic 夹FIJ)e .  The integrand of the volume integral takes the form:

(56)

(57)

EK IJ"16pG ¹ I;M#16p(G e (JI#ic JI I)e#4 K eCIJM K e ,  J M J , M , where ;I,e cIe is a non-spacelike 4-vector, provided e is the Killing spinor, and the electric and magnetic 4-vector currents JI and JI I are de"ned through the Maxwell's equations and the Bianchi identities as FIJ"4pJI and 夹FIJ"4pJI I. J J If we assume that the Killing spinor e approaches a constant value e as rPR, then the surface  and the volume integrals are 1 e [!P cI#(G (Q#ic P)]e " [¹ I;J# e (JI#ic JI I)e]dR   J  I  I , G R , 1 #

K eCIJM K e dR , (58) M I 4pG R J , where Q and P are the electric and magnetic charges of the gauge "eld A . I





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The "rst term on the right-hand side of (58) is always non-negative, if ¹  satis"es the IJ generalized dominant energy condition: ¹ ;I
(61)

3. Duality symmetries Past few years have been an active period for studying string dualities [227,500,543,622,635,636]. Five di!erent string theories (E ;E and SO(32) heterotic strings, type-IIA and type-IIB strings,   and type-I string with SO(32) symmetry), which were previously regarded as independent perturbative theories, are now understood as being related via web of dualities. String dualities are classi"ed into T-duality, S-duality and U-duality. T-duality (or target space duality) [306] is a perturbative duality (i.e. duality that relates theories with the same string coupling) that transforms the theory with large (small) target space volume to one with small (large) target space volume [216,425,525] or connects di!erent points in (target-space) moduli space. Under T-duality, type-IIA and type-IIB strings [193,216], and E ;E and SO(32) heterotic strings   [303,480,481] are interchanged. Another examples of T-duality are the O(10!D, 26!D, Z) symmetry [308] of heterotic string on ¹\" and the O(10!D, 10!D, Z) symmetry [304,305,308,570] of type-II string on ¹\". S-duality (or strong-weak coupling duality) is a non-perturbative duality that transforms string coupling to its inverse (while moduli "elds remain "xed) and interchanges perturbative string states and non-perturbative branes. Duality that relates Type-I string and SO(32) heterotic string [505,635] is an example of S-duality. Another examples are (i) the S¸(2, Z) symmetry of type-IIB string [72,376,536,538}540]; (ii) the D"6 string}string duality between the heterotic string on ¹ [on K3] and the type-II string on K3 [on a Calabi} Yau-threefold] [52,93,226,233,235,244,257,397,435,565,624,635]; (iii) the S¸(2, Z) symmetry of N"4 heterotic string in D"4 [268,339,537,544,556}560]. U-duality [39,40,45,381], which is closely related to the D"11 theory (M-theory), is regarded as a consequence of the S¸(2, Z) S-duality of type-IIB string and T-dualities of type-II strings on a torus. Thus, U-duality is a non-perturbative duality of type-II strings which necessarily interchanges NS-NS charged state and R-R charged state.

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String dualities require existence of non-perturbative states within string spectrum, as well as the well-understood perturbative states. These non-perturbative states include smooth solitons and new types of topological objects called D-branes [498]. Such non-perturbative states are extended objects, which in a low energy limit correspond to p-branes of the e!ective "eld theories. So, string theories, which are previously known as theories of (perturbative) strings, are no longer theories of strings only, but contain objects of higher/lower spatial extends. These perturbative and nonperturbative states are interrelated via string dualities. One of important discoveries of string dualities is the conjecture that there exists more fundamental theory in higher dimensions (D'10), which reduces to all of 5 perturbative string theories in di!erent limits in moduli space when the theory is compacti"ed to lower dimensions (D410). Such fundamental theories include M-theory [227,540,543,635] in D"11, F-theory in D"12 [621], and S-theories in D"12, 13 [41}44]. M-theory is de"ned as an unknown theory in D"11 (with 1 time-like coordinate) whose low energy e!ective theory is the D"11 supergravity [158] and which becomes type-IIA theory when the extra 1 spatial coordinate is compacti"ed on S of radius R. Since the radius R of S is related to the string coupling j of type-IIA theory as R"j, the strong coupling limit (j<1) [635] of type-IIA theory is M-theory, i.e. the decompacti"cation limit (RPR) of M-theory on S. Furthermore, the evidence was given in [358] for the conjecture that M-theory compacti"ed on S/Z is E ;E heterotic string. We mentioned in the previous paragraph that type-IIA and    type-IIB strings, and E ;E and SO(32) heterotic strings are related via T-duality, and SO(32)   heterotic string and type-I string are related via S-duality. Thus, all of the 5 di!erent perturbative string theories are obtained from M-theory by compactifying on S or S/Z , and applying  dualities. F-theory is a conjectured theory in D"12 (with 2 time-like coordinates) which is proposed in an attempt to "nd geometric interpretation of the S¸(2, Z) S-duality of type-IIB theory. Namely, the complex scalar formed by the dilaton and the R-R 0-form transforms linear-fractionally under the S¸(2, Z) transformation, just like the transformation of modulus of ¹ under the T-duality of a string theory compacti"ed on ¹. F-theory is, therefore, roughly de"ned as a D"12 theory which reduces to type-IIB theory upon compacti"cation on ¹, with the modulus of ¹ given by the complex scalar. Note, since type-IIB theory on S is equivalent to M-theory on ¹, F-theory on ¹;S is the same as M-theory on ¹. The essence of string dualities is that strong coupling limit of one theory is dual to weak coupling limit of another theory with the strongly coupled string states (dual to perturbative string states) identi"ed with branes. So, branes play an important role in understanding non-perturbative aspects of and dualities in string theories. It is one of purposes of this review to summarize the recent development in solitons and black holes in string theories. The purpose of this section is to give basic facts on dualities in supersymmetric "eld theories and superstring theories that are necessary in understanding the rests of chapters of this review. Therefore, readers are referred to other literatures, e.g. [23,227,228,317,498,499,501,504,539,540,543,605,635,636], for complete understanding of the subject. In the "rst section, we discuss the symplectic transformations in extended supergravities and moduli spaces spanned by scalars in the supermultiplets. In Section 3.2, we summarize T-duality and S-duality of heterotic string on tori. In this section, we also discuss solution generating transformations that induce electric/magnetic charges of ;(1) gauge "elds of heterotic string on tori

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when applied to a charge neutral solution. These dualities are basic transformations for constructing most general black hole solutions in heterotic string on tori. In Section 3.3, we discuss string}string duality between heterotic string on ¹ and type-II string on K3, and string}string} string triality among type-IIA, type-IIB and heterotic strings in D"6. These dualities transform black hole solutions discussed in Section 4 to type-II black holes which carry R-R charges [49,56], thereby enabling interpretation of our black hole solutions in terms of D-brane picture. In the "nal section, we summarize some aspects of M-theory and ;-duality. The generating black hole solutions of heterotic string on tori are the generating solutions for type-II string on tori, as well. Namely, when the generating solutions of heterotic strings are embedded to type-II theories (note, such generating solutions carry only charges of NS-NS sector, which is common to both heterotic and type-II theories), subsets of ;-dualities of type-II theories on tori induce the remaining ;(1) charges of type-II theories on tori [171]. 3.1. Electric-magnetic duality The electric-magnetic duality in electromagnetism was conjectured by Dirac [217] based on the observation that when electric charge and current are non-zero the Maxwell's equations lack symmetry under the exchange of the electric and magnetic "elds, or in other words under the Hodge-duality transformation of the electromagnetic "eld strength. Dirac conjectured the existence of magnetic charges [218] to remedy the situation. Magnetic charges are due to the topological defect of spacetime and are given by the "rst Chern class of the ;(1) principal "ber bundle with the base manifold given by S surrounding the monopole. The requirement of the continuity of the transition function that patches the 2 covers of the northern and southern hemispheres of S or the requirement of the unobservability of Dirac string singularity restricts magnetic charge q to be quantized [546,547,630,640}643] relative to electric charge q in the

 following way through the Dirac}Schwinger}Zwanzinger (DSZ) quantization rule: n qq  " (n3Z) . 4p 2

(62)

Under the duality transformation, electric and magnetic charges are interchanged and correspondingly the coupling of the electromagnetic interactions is inverted due to the relation (62). So, the weak (strong) coupling limit of one theory is described by the strong (weak) coupling limit of its dual theory. The extension of the duality idea to non-Abelian gauge theories was made possible by the discovery of the 't Hooft}Polyakov monopole solution [507,592] in non-Abelian gauge theory. In the 't Hooft}Polyakov monopole con"guration, the non-Abelian gauge group is spontaneously broken down to the Abelian one at spatial in"nity by Higgs "elds that transform as the adjoint representation of the gauge group. The magnetic charge of the 't Hooft}Polyakov monopole is determined by the second homotopy group of S formed by the symmetry breaking Higgs vacuum, i.e. the winding number around S as one wraps around S surrounding the monopole. Within this  context, Montonen and Olive [476] conjectured that the spontaneously broken electric nonAbelian gauge group is dual to the spontaneously broken magnetic non-Abelian gauge group. Under this duality, the gauge coupling of one theory is inverted in its dual theory, leading to the

D. Youm / Physics Reports 316 (1999) 1}232

27

prediction that the strong coupling limit of a gauge theory is the weak coupling limit of its dual theory [549}551]. Note, the hub of the Montonen}Olive conjecture lies in the existence of symmetry breaking Higgs "elds which transform in the adjoint representation of the non-Abelian gauge group. It is a generic feature of extended supersymmetries that scalars live in the same supermultiplet as vector "elds. So, the scalars in vector supermultiplet transform as the adjoint representation under the non-Abelian gauge group. Furthermore, supersymmetric theories obey the well-known non-renormalization theorem (See for example [37,192,267,321,375,473,485,573]). The extended supersymmetry theories have another nice feature that the states preserving some of supersymmetry, i.e. BPS states, are determined entirely by their charges and moduli. These are (degenerate) ground states of the theories (parameterized by moduli). Such BPS mass formula is invariant under dualities and degeneracy of BPS states remains unchanged under dualities. For example, electrically charged BPS states at coupling g have the same mass and degeneracy as magnetically charged states BPS states at coupling 1/g. Furthermore, the supersymmetry algebra prevents the number of degeneracy of BPS states from changing as the coupling constant is varied. Thus, it is BPS states that are suitable for testing ideas on duality symmetries. In the following, we discuss generalization of electric-magnetic duality of Maxwell's equations to N-extended supersymmetry theories and study moduli spaces spanned by scalars. 3.1.1. Symplectic transformations in extended supersymmetries In supersymmetry theories, scalars ' are taken as coordinates of the target space manifold M of non-linear p-model, which we write in general in the form [270]:   L "g ( )gIJ ' ( , (63)   '( I J where the covariant derivative on ' is with respect to the gauge group G that the vector I   "elds AK in the theory belong to: I

'"R '#gAKk'K( ) , (64) I I I of G : with Killing vector "elds kK,k'K( ) R/R ' satisfying the Lie algebra g     D kD . (65) [kK, kR]"fKR Note, g is a subalgebra of the isometry algebra of M .     Here, g is the metric of M . In other words, a scalar is regarded as a map from the spacetime '(   manifold to M . It turns out that the types of allowed target space manifolds formed by scalars   are "xed by the number m of supercharge degrees of freedom in N-extended superalgebra [527]. When m exceeds 8, the target space manifold is "xed as a symmetric space speci"ed by the number n of vector multiplets. For example, D"4, N"8 supergravity, for which m"32, has the target space manifold E /S;(8) [159,160]; the D"4, N"4 theory, for which m"16, has target space  manifold SO(6,n) S;(1,1)  SO(6)SO(n) ;(1) [205,544,545,555,557]. A special case is the e!ective action of the heterotic string on ¹, which is described by the N"4 supergravity coupled to the N"4 super-Yang}Mills theory with the gauge

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group ;(1) (n"22 case). Here, SO(6,12)/SO(6)SO(12) describes (classical) moduli space of Narain torus [480,481], and S;(1,1)/;(1) is parameterized by the dilaton-axion "eld. (In Section 3.2, we discuss the Sen's approach [560] of realizing such target space manifolds within the e!ective supergravity through the dimensional reduction of the heterotic string e!ective action.) For m48, the target space manifolds are less restrictive. For a D"4, N"2 theory, corresponding to m"8, the scalar manifold is factorized into a quaternionic one and a special KaK hler manifold [576,581], which are, respectively, spanned by the scalars in the hypermultiplets and the vector multiplets. For m"4 case, e.g. D"4, N"1 theory, the target space manifold is the KaK hler manifold. Within the extended supersymmetry theories described above, one can generalize [276] the electric-magnetic duality transformations, which preserve equations of motion for the ;(1) "eld strengths. For this purpose, only relevant part of scalars is from vector supermultiplets. Such generalized electric-magnetic duality transformation is realized as follows. We consider the general form of the di!eomorphism of the scalar manifold: t: MT PMT : ' C t'( ) . (66)     The map that corresponds to the isometry of the scalar manifold, i.e. t*g "g , becomes the '( '( candidate for the symmetry of the theory. General form of kinetic term for vector "elds in vector supermultiplets is [270] (67) L "cKR( )FK夹FR#hKR( )FKFR ,    where the "eld strengths FK of gauge "elds AK are IJ I K K K K R D (68) F ,(R A !R A #gfRDA A ) , IJ  I J J I I J and 夹FK ,e FKMN is the Hodge-dual of FK . Here, the n;n matrix c ( ) generalizes the IJ  IJMN IJ '( coupling constant of the conventional gauge theory and the antisymmetric matrix h ( ) is the '( generalization of the h-term [630]. The transformation properties of the gauge "elds and the complex symmetric matrix NKR( ),hKR( )!icKR( ) are determined by the symplectic embedding of the isometry group G of the scalar manifold MT as follows.    We consider the following homomorphism from the group Di!(M ) of di!eomorphisms   t: MT PMT to the general linear group G¸(2n, R):     n : Di!(MT )PG¸(2n, R) . (69) B   One introduces a 2n;1 matrix <"(夹F, 夹G)2, where 夹G,!RL/RF2. Then, the map n in (69) B is de"ned by assigning, for each element m of Di!(MT ), a 2n;2n matrix   A B K n (m)" K B C D K K in GL(2n, R) which transforms < as






夹F  A " R 夹G C R




夹F 夹G

C




B R D R

夹F 夹G

,

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29

or F>  A B R " R G> C D R R




F> G>

C




F>



NF>

,

(70)

where F>,F!i夹F and G>,NF> [212,213]. The transformation law (70) on < is dictated by the requirement that the Bianchi identities and the "eld equations for vector "elds remain invariant. Under the action of the di!eomorphism m on ', N( ) also transforms. If we further require that transformed "eld G to be de"ned as 夹G"!RL/RF2, the transformation property of NKR( ) under the di!eomorphism t on ' is "xed as the fractional linear form: N( ) C N(t( ))"[C #D N( )][A #B N( )]\ , R R R R

(71)

with n (m) now restricted to belong to Sp(2n, R)LG¸(2n, R). B When B O0OC , it is a symmetry of equations of motion only. When B "0OC , the R R R R Lagrangian is invariant up to four-divergence. When B "0"C , the Lagrangian is invariant. In R R particular, the symplectic transformations (70) and (71) with B O0 are non-perturbative, since they R necessarily induce magnetic charge from purely electric con"guration and invert N, which plays the role of the gauge coupling constant. When electric/magnetic charges are quantized, Sp(2n, R) gets broken down to Sp(2n, Z) so that the charge lattice spanned by the quantized electric and magnetic charges is preserved under the transformation (70). This is the generalization of the electric-magnetic duality symmetry to the case of D"4 supersymmetry theory with n vector "elds. 3.1.2. Symplectic embedding of homogeneous spaces When the number of supercharge degrees of freedom exceeds 8, e.g. N53 in D"4, M is   a homogeneous space G/H with the isometry group G. The supersymmetry restricts the dimension of M and the number n of vector multiplets to be related to each other and, therefore, M is     determined uniquely by n. In the following, we discuss the symplectic embedding of the homogeneous space and show how the gauge kinetic matrix NKR is determined. We consider the following embedding of the isometry group G of G/H into Sp(2n, R): n : GPSp(2n, R): ¸( ) C n (¸( )) . B B

(72)

Applying the following isomorphism from the real symplectic group Sp(2n, R) to the complex symplectic group ;sp(n, n),Sp(2n, C)5;(n, n): k:



A B

C D

C



¹ <* < ¹*

,

(73)

where ¹,(A!iB)#(C#iD), <,(A!iB)!(C#iD) ,    

(74)

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one can de"ne the complex symplectic matrix O( ) (3;sp(n, n)), for each coset representative ¸( ) of G/H, as follows: ; ( ) ;*( )  , (75) k ) n : GP;sp(n, n): ¸( ) C O( )"  B ; ( ) ;*( )   where






; ( )R; ( )!; ( )R; ( )"1, ; ( )R; ( )*!; ( )R; ( )*"0 . (76)         From O( ) in (75), which is de"ned for each coset representative ¸( ) of G/H, one can de"ne the following scalar matrix [276] which has all the right properties for the gauge kinetic matrix NKR"hKR!icKR: N,i[;R #;R ]\[;R !;R ] ,     namely, N2"N and N transforms fractional linearly under Sp(2n, R). Speci"cally, we consider the homogeneous space of the form

(77)

SO(m, n) S;(1, 1)  , ST[m, n], SO(m)SO(n) ;(1) where m is the number of graviphotons and n the number of vector multiplets. Here, S;(1, 1)/;(1) is parameterized by the axion-dilaton "eld S and SO(m, n)/[SO(m)SO(n)] is parameterized by a m;n real matrix X. In the real basis, the SO(m, n) T-duality and S¸(2, R) S-duality groups of ST[m, n] are respectively embedded into the symplectic group as:






¸ O 3Sp(2m#2n, R) n : ¸3SO(m, n) C B O (¸2)\









a b a1 bg n: 3S¸(2, R) C 3Sp(2m#2n, R) , B c d cg d1

(78)

where g is an SO(m, n) invariant metric, 1 is the (m#n);(m#n) identity matrix, and a, b, c, d3R satisfy ad!bc"1. In the complex basis, the embeddings are






(¸#g¸g) (¸!g¸g)  3;sp(m#n, m#n), n : ¸3SO(m, n) C  B (¸!g¸g) (¸#g¸g)   t v* Re t 1#i Im t g Re v 1!i Im v g n: 3S;(1, 1) C 3;sp(m#n, m#n) . B v t* Re v 1#i Im v g Re t 1!i Im t g









(79)

The symplectic embeddings (79) make it possible to express the gauge kinetic matrix N in terms of the scalars S and X, which parameterize ST[m, n], as follows. The coset representatives of

 ST[2, n] is the only special KaK hler manifold with direct product structure [266] of this form.

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S;(1, 1)/;(1) and SO(m, n)/[SO(m);SO(n)] are respectively






1 1 \1 >1 , ¸(S), n(S) >1MM 1 \1 (1#XX2) X , ¸(X), X2 (1#X2X)






(80)

where



n(S),

4 Im S . 1#"S"#2Im S

Note, M,¸(X)¸2(X) is a symmetric SO(m, n) matrix, studied by Sen [560], which will be discussed in Section 3.2. Applying transformations (79), one obtains the following symplectic embedding of the coset representations of ST[m, n]:






; (S, X) ;*(S, X)  n (¸(S))
n (¸(X))"  3;sp(n#m, n#m) , (81) B B ; (S, X) ;*(S, X)   where the explicit expressions for n (¸(S)) and n (¸(X)) are obtained by applying the transformations B B (79). Substituting this expression into the general formula in (77), one obtains the following gauge kinetic matrix: N"i Im Sg¸(X)¸2(X)g#Re Sg"i Im SgMg#Re Sg .

(82)

Then, the Lagrangian L #L (cf. see (63) and (67)) takes the following form that corres   ponds to ST[m, n]:



1 1 L"(!g R # R SRISM ! Tr(R MRIM) E 4(Im S) I I 4



1 1 ! Im SF' (g¸g) F(IJ# Re SF' g F( eIJMN . IJ '( IJ '( MN 4 8(!g

(83)

The T-duality and S-duality of the heterotic string on ¹ [465,537,544,560] are special cases of the symplectic transformations (78) with (m, n)"(6, 22). In general, under the SO(m, n) and S¸(2, R) transformations (78), the gauge "elds and the gauge kinetic matrix transform, respectively, as (cf. (70) and (71)) F> C F>"¸F>, N C N"(¸2)\N¸\, F> C F>"aF>#bgNF>, N C N"(dN#cg)(bgN#a)\ .

(84)

Note, O(m, n) [S¸(2, R)] is a perturbative (non-perturbative) symmetry, since N is not inverted (gets inverted). S¸(2, R) is the symmetry of the equations of motion only, since electric and magnetic charges get mixed, and since this corresponds to the transformations (70) and (71) with B O0OC . R R

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3.1.3. Target space manifolds of N"2 theories Contrary to D"4, N53 theories, the scalar manifolds of N"2 theories are not necessarily expressed as homogeneous symmetric coset manifolds. Scalar manifold of the D"4, N"2 theory has the generic form M "SM HM , (85)   L K where SM is a special KaK hler manifold of the complex dimension n" `the number of the vector L supermultipletsa, and the manifold HM spanned by the scalars in the hypermultiplets has the K dimension 4m, where m" `the number of the hypermultipletsa. So, the metric g ( ) of M has '(   the form (86) g ( )d 'd ("g *dz?dz @*#h dqSdqT . ST '( ?@ Here, g * [h ] is the special KaK hler metric on SM [the quaternionic metric on HM ]. ST L L ?@ 3.1.3.1. Special KaK hler manifolds. N"2 super-Yang-Mills theory is described by a chiral super"eld U, which is de"ned by DM ? G U"0 (like chiral super"eld in N"1 theory), with an additional constraint: (87) D? D@ Ue "e e lDM ? IDM l@Q UM e  Q , ?@ G H ?@ GI H where i"1, 2 labels supercharges of N"2 theory, a, a"1, 2 are indices of chiral spinors, and DM ? is a covariant chiral superspace derivative. The component "elds of a N"2 chiral super"eld U are a scalar X, spinors jG, ;(1) gauge "eld strength F , and auxiliary scalars > satisfying a reality IJ GH constraint > "e e l>M Il (due to the constraint (87)). The action of N"2 chiral super"elds U is GH GI H determined by an arbitrary holomorphic function F(U) of U as dxdh F(U)#c.c., and is given, in terms of the component "elds, by (88) L"g M R XRIXM #g M jM cIR j M #Im(F F\F\ IJ)#2 ,  IJ  I G  I where the dots denote the interaction terms involving fermions, g M "R R M K is a KaK hler metric,   and F ,R R F. Note, this action is a special case of the most general coupling of N"1 chiral   super"elds to N"1 Abelian vector super"elds in which the KaK hler potential K and the holomorphic kinetic term function F take the following special forms [283,572]:  K(X, XM )"i[FM (XM )X!F (X)XM ] (F ,R F), F "R R F . (89)       The KaK hler manifold with the KaK hler potential K(X, XM ) determined by the prepotential F(X) [161,211,212] through (89) is called the special Ka( hler manifold [113,128,198,210,212,222,576].  But there is a subclass of homogeneous special manifolds, which are classi"ed in [166]. These are S;(1,1) S;(1,n ) S;(1,1) SO(2,n ) Sp(6,R) S;(3,3) T T , ,  , , , S;(n );;(1) ;(1) SO(2);SO(n ) S;(3);;(1) S;(3);S;(3) ;(1) T T SOH(12) E \ , and S;(6);;(1) E ;SO(2)  with the corresponding symplectic groups Sp(2n #2) respectively given by Sp(4), Sp(2n #2), Sp(2n #4), Sp(14), Sp(20), T T T Sp(32), and Sp(56).

D. Youm / Physics Reports 316 (1999) 1}232

33

When N"2 chiral "elds UK (K"0, 1,2, n ) are coupled to the Weyl multiplet (with compoT nents given by vierbein, 2 gravitinos and auxiliary "elds) [210,212], the invariance under the dilatation requires F(X) to be a homogeneous function of degree 2 (so that F(X) has Weyl weight 2) [210,212]. Furthermore, the requirement of canonical gravitino kinetic term imposes one constraint on the set of scalars XK as i(XM KFK!FM RXR)"1 ,

(90)

leading to gauge "xing for dilations and the special KaK hler manifold of the dimension n . Note, the T extra chiral super"eld U is introduced to "x the dilatation gauge, to break the S-supersymmetry, and to introduce the physical ;(1) gauge "eld in the N"2 supergravity multiplet (the scalar and the spinor components of the super"eld U do not become additional physical particles). The "nal form of bosonic action describing n numbers of N"2 vector multiplets coupled to N"2 T supergravity is (91) e\L"!R#g *R z?RIz @*!Im(NKR(z, z )F>KF>RIJ) , IJ ?@ I  where z? (a"1,2, n ) are the coordinates of a KaK hler space spanned by the scalars T XK (K"0, 1,2, n ) which satisfy one constraint (90) (therefore, the manifold spanned by XK has T n complex dimensions). A convenient choice for z? is the inhomogeneous coordinates called the T special coordinates: z?"X?(z)/X(z), a"1,2, n . (Note, X?(z)"z? in special coordinates in which T R(X?/X)/Rz@"d? [113,128,134,576].) Here, K and NKR (cf. the scalar matrix N in (71)) are @ determined by F(X) to be of the forms [130,161,198,211,212]: e\)XX "iZM K(z )FK(Z(z))!iZR(z)FM R(ZM (z )) , Im(FK )Im(FRP)X XP , (92) Im(F P)X XP where ZK(z)"e\)XK and ZM K(z )"e\)XM K (K"0, 1,2, n ) are holomorphic sections of the  projective space PC L> [128,129,198], and FKR,RKFR(X). We give some examples of the holomorphic function F(X) of N"2 theories and the corresponding special KaK hler manifold target spaces: NKR"FM KR#2i

F"iXX,

S;(1, 1) , ;(1)

F"(X)/X,

S;(1, 1) , ;(1)

F"!4(X(X),

S;(1, 1) , ;(1)

S;(1, n) F"iX gKRX , , S;(n);(1) K

R

dK RXKX XR F" , X




Calabi}Yau .



XQ L SO(2, 1) SO(2, n) F"!i (X)! (X?) , ; . X SO(2) SO(2);SO(n) ?

(93)

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D. Youm / Physics Reports 316 (1999) 1}232

So far, we de"ned the special Ka( hler manifold as the KaK hler manifold with the special form of the KaK hler metric given by (92), which depends on the holomorphic prepotential F. Now, we discuss the symplectic formalism of the special KaK hler manifolds of N"2 supergravity coupled to n vector supermultiplets.  For the symplectic formalism [132,133,214] of the special KaK lher manifold M, one considers the tensor bundle of the type H"SVL. Here, SVPM denotes a holomorphic #at vector bundle of rank 2n #2 with structural group Sp(2n #2, R), and LPM denotes the complex T T line bundle whose "rst Chern class equals the KaK hler form of the n -dimensional Hodge}KaK hler T manifold M. A holomorphic section of the bundle H has the form [113}115,128,134,198]:


 XK

X"

K, R"0, 1,2, n , T

FR

(94)

which is de"ned for each coordinate patch ; LM of the (Hodge}KaK hler) manifold M and G transforms as a vector under the symplectic transformation Sp(2n #2, R). The Hodge}KaK hler T manifold M with a bundle H described above is called special Ka( hler, if the KaK hler potential is expressed in terms of the holomorphic section X as K"!log(i1X"XM 2)"!log[i(XM KFK!FM RXR)] ,

(95)

where




1X"XM 2,!XR

0

I

!I 0



X

denotes a symplectic inner product. One further introduces the symplectic section of the bundle H according to


 ¸K

<"

MR

,e)X2 .

(96)

Then, by de"nition, < satis"es the constraint [128,130,161,211,212]: 1"i1<"
(97)

and is covariantly holomorphic:

*<"(R *!R *K)<"0 , ? ?  ? where R ,R/Rz? and R *,R/Rz ?*. ? ?

(98)

 The additional two dimensions in SV come from a vector "eld in the supergravity multiplet.  Alternatively, one can de"ne special Ka( hler manifold by introducing the symplectic section < (96) satisfying the constraint (97). Then, the KaK hler potential is determined in terms of the holomorphic section X as in (95) through the constraint (97) with (96) substituted.

D. Youm / Physics Reports 316 (1999) 1}232

35

One further introduces the matrix of the following form:




fK ; " <"(R #R K)<, ? (a"1,2, n ) . (99) ? ? ?  ? T hR ? Then, period matrix NKR (which corresponds to the gauge kinetic matrix in the N"2 theory) is de"ned via the relations M KR f R . MM K"N M KR¸M R, hK "N ? ? Therefore, the period matrix has the following explicit form:

(100)

N M KR"hK
( f \)'R , ' where the (n #1);(n #1) matrices f K and hK in the above are de"ned as T  ' ' K f h K f K" ?K , hK " ? . ' ' ¸M M M K

(101)







(102)

As a consequence, under the di!eomorphism of the base manifold M, NKR transforms fractional linearly, like the gauge kinetic matrix (cf. (71)). Note, in the above symplectic formalism of the special KaK hler manifold no reference was made on the prepotential. In fact, for some cases, the existence of the prepotential is not even guaranteed [136]. We now discuss how the concept of the prepotential emerges within the framework of symplectic formalism. Under the coordinate transformations of M, X transforms as XPX"e\DMX ,

(103)

where the factor e\D corresponds to a ;(1) KaK hler transformation (i.e. K transforms as KPK#Re f (z)) and M3Sp(2n#2, R), and NKR transforms fractional linearly just like a gauge kinetic matrix (cf. (71)). From the transformation law (103) with M"I, one can infer that XK can be regarded as homogeneous coordinates of a (n #1)-dimensional projective space at least locally T [210,212,283], since XK and e\DXK are identi"ed under the KaK hler transformations. This is possible provided the Jacobian matrix R (X@/X) (a, b"1,2, n ) is invertible [132]. In this case, ? T due to the integrability condition following from (97) and (98), the lower components FR of X are expressed as R FR" RF , RX

(104)

in terms of a homogeneous function F(X)"XKFK of degree 2 in XK. Then, one can use  z?,X?/X (a"1,2, n ) as the special coordinates and the holomorphic prepotential is T F(z),(X)\F(X). In terms of F and z?, the KaK hler potential K is expressed as [210] K(z, z )"!log i[2(F!F M )!(R F#R *FM )(z?!z ?*)] . ? ?  For electrically neutral theories, one can always rotate to bases where a prepotential exists [155].

(105)

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3.1.3.2. Hypergeometry. N"2 hypermultiplet consists of a doublet of 0-form spinors with left and right chiralities, and 4 real scalars, which can be locally regarded as the 4 components of a quaternion. The scalars qT (v"1,2, 4n ) in n hypermultiplets form a 4n -dimensional real & & & manifold HM [30,129,198,211,277,353] with a metric ds"h (q)dqSdqT . (106) ST This manifold is endowed with 3 complex structures JV: ¹(HM)P¹(HM) (x"1, 2, 3) satisfying the quaternionic algebra JVJW"!dVW1#eVWXJX. The metric h (q) is hermitian with respect to JV: ST h(JVX, JVY)"h(X, Y), X, Y3¹ HM . (107) From JV, one can de"ne triplet of S;(2) Lie-algebra valued HyperKaK hler forms as KV"KV dqSdqT , (108) ST where KV "h (JV)U. Supersymmetry requires the existence of a principal S;(2)-bundle ST SU T SUPHM with a connection uV. The manifold HM is de"ned by requiring that KV is covariantly closed with respect to the connection uV:

KV,dKV#eVWXuWKX"0 .

(109)

There are two types of hypergeometry: rigid (local) hypergeometry corresponding to global (local) N"2 supersymmetry is called HyperKa( hler (quaternionic). The only di!erence between the two manifolds are the structure of the SU-bundle. A HyperKa( hler manifold has the #at SU-bundle, and a quaternionic manifold has the curvature of the SU-bundle proportional to the HyperKaK hler 2-form. Here, the SU-curvature is de"ned as XV,duV#eVWXuWuX .  In the quaternionic case, the curvature is XV"(1/j)KV ,

(110)

(111)

where j is a real number related to the scale of the quaternionic manifold. In the limit jPR, quaternionic manifold becomes HyperKaK hler manifold [6]. The manifold HM has the following holonomy group: Hol(HM)"S;(2)H for quaternionic manifold , Hol(HM)"1H for HyperKaK hler manifold ,

(112)

where HLSp(2n , R). We denote the #at indices that run in the fundamental representation of & S;(2) [Sp(2n , R)] as i, j"1, 2 [a, b"1,2, 2n ]. Then, the metric of the quaternionic manifold is & & expressed in terms of the vielbein 1-form UG?"UG?(q) dqS as S h "UG?UH@C e , (113) ST S T ?@ GH where C "!C [e "!e ] is the #at Sp(2n ) [Sp(2)&S;(2)] invariant metric. The vielbein ?@ @? GH HG & UG? is covariantly constant with respect to the S;(2)-connection uV and some Sp(2n , R) Lie &

D. Youm / Physics Reports 316 (1999) 1}232

37

algebra valued connection *?@"*@?:

UG?,dUG?#uV(ep e\)G UH?#*?@UGAC "0 ,  V H @A where pV (x"1, 2, 3) are the Pauli spin matrices. Also, UG? satis"es the reality condition:

(114)

U ,(UG?)H"e C UH@ . G? GH ?@ The curvature 2-form XV forms the representation of the quaternionic algebra:

(115)

hQRXV XW "!jdVWh #jeVWXXX , SQ RU SU SU and can be written in terms of the vielbein UG? as

(116)

XV"ijC (pVe\) U?GU@H . ?@ GH

(117)

3.2. Target space and strong-weak coupling dualities of heterotic string on a torus 3.2.1. Ewective xeld theory of heterotic string The e!ective "eld theory of massless states in heterotic string is D"10 N"1 supergravity coupled to N"1 super-Maxwell theory [74,137,142]. The massless bosonic "elds of heterotic string at a generic point of Narain lattice [480,481] are metric GK , 2-form "eld BK , gauge "elds +, +, AK ' of ;(1) and dilaton "eld U, where 04M, N49 and 14I416. The "eld strengths of + AK ' and BK are de"ned as FK ' "R AK ' !R AK ' and HK "R BK !AK ' FK ' #cyc. perms., + +, +, + , , + +,. + ,.  + ,. respectively. The D"10 e!ective action [74,111,137,142] of these massless bosonic modes is HK +,.!FK ' FK ' +,] , (118) L"(1/16pG )(!GK [R K #GK +,R UR U!  HK % + ,  +,.  +,  where GK ,det GK , R is the Ricci scalar of GK , and G is the D"10 gravitational constant. +, %K +,  We choose the mostly positive signature convention (!##2#) for the metric GK . For the +, spacetime vector index convention, the characters (A, B,2) and (M, N,2) denote #at and curved indices, respectively. The supersymmetry transformations of the fermionic "elds, i.e. gravitino t , dilatino j and + gaugini s', are CK ,.e , dt " e!HK + +  +,. dj"(CK +R UK )e!HK CK +,.e , (119) +  +,. ds'"FK ' CK +,e , +, where e"R e#X CK  e is the gravitational covariant derivative on a spinor e. Here, + +  + ): X is the spin-connection de"ned in terms of a Zehnbein EK K (de"ned as EK  g EK "GK +  , +, + X ,!XI #XI !XI , where XI ,EK + EK , R EK , and curved indices are obtained  !  ! ! !  !  , +! by contracting with Zehnbein. CK  are gamma matrices of SO(1, 9) Cli!ord algebra +CK , CK ,"2g (those with several indices are de"ned as the antisymmetrized products of gamma matrices, e.g. CK  ,(CK CK !CK CK )).   In this section, we "x the D"10 gravitational constant to be G "8p. 

38

D. Youm / Physics Reports 316 (1999) 1}232

3.2.2. Kaluza}Klein reduction and moduli space The e!ective "eld theory of massless bosonic "elds in heterotic string on a Narain torus [480,481] at a generic point of moduli space is obtained by compactifying D"10 e!ective "eld discussed in the previous section on ¹\" [465,560]. Before we discuss the compacti"cation Ansatz, we "x our notation for indices. General indices running over D"10 are denoted by upper-case letters (A, B,2; M, N,2). The lower-case Greek letters (a,2, b; k, l,2) are for D(10 spacetime coordinates and the lower-case Latin letters (a, b,2; m, n,2) are for the internal coordinates. The #at indices are denoted by the letters in the beginning of alphabets (A, B,2; a, b,2; a, b,2) and curved indices are denoted by the letters at the latter parts of alphabets (M, N,2; k, l,2; m, n,2). The compacti"cation [143,225,247,416,434,465,528,529] on ¹\" is achieved by choosing the following Abelian Kaluza}Klein (KK) Ansatz for the D"10 metric






e?Pg #G A KA L A KG IJ KL I J I KL , (120) A LG G J KL KL where A K (k"0, 1,2, D!1; m"1,2, 10!D) are KK ;(1) gauge "elds, u,UK !ln det G KL I  is the D-dimensional dilaton and a,2/(D!2). Then, the a!ective action is speci"ed by the following massless bosonic "elds: the (Einstein-frame) graviton g , the dilaton u, (36!2D) ;(1) IJ gauge "elds AG ,(AK, A, A') de"ned as A,BK #BK AL#AK ' A' and A', I I IK I IK IK KL I  K I I AK ' !AK ' AK, the 2-form "eld B with the "eld strength H "R B !AG ¸ FH #cyc.perms., I K I IJ IJM I JM  I GH JM and the following symmetric O(10!D, 26!D) matrix of scalars (moduli) [465,560]: GK " +,




G\

!G\C

!G\a2



M" !C2G\ G#C2G\C#a2a C2G\a2#a2 , aG\C#a

!aG\

I#aG\a2

(121)

where G,[GK ], C,[AK 'AK '#BK ] and a,[AK ' ] are de"ned in terms of the internal parts of KL K KL  K L D"10 "elds. M can be expressed in terms of the following O(10!D, 26!D) matrix as M"<2< [465]:






E\ !E\C !E\a2

<"

0

E

,

(122)

I  where E,[e? ], C,[AK ' AK '#BK ] and a,[AK ' ]. < plays a role of a Vielbein in the K  K L KL K O(10!D, 26!D) target space. Note, M parameterizes the quotient space O(10!D, 26!D)/ [O(10!D);O(26!D)] with dimensions 26!36D#D. The dimensionality precisely matches the number of scalar "elds in the matrix M: (11!D)(10!D)/2 scalars GK , (10!D)(9!D)/2 KL scalars BK , and 16(10!D) scalars AK ' . KL K The resulting theory in D(10 corresponds to 26!D vector multiplets coupled to D(10, N-extended supergravity. The supergravity multiplet consists of graviton g , 2-form potential IJ B , 10!D graviphotons A0? (a"1,2, 10!D), dilaton u, gravitinos t? (a"1,2, N) IJ I I and dilatinos j?. The "eld content in the 26!D vector multiplets is 26!D vector "elds A*' (I"1,2, 26!D), (10!D);(26!D) scalars ?' parameterizing the coset I 0

a

0

D. Youm / Physics Reports 316 (1999) 1}232

39

O(10!D, 26!D)/[O(10!D);O(26!D)] and gauginos s?'. At the string level, the 10!D graviphotons originate from the right moving sector of the heterotic string and the 26!D photons in the vector multiplets originate from the left moving sector. In terms of the "eld strengths FG of I the ;(1)\" gauge group, the graviphoton "eld strengths F0? and matter photon "eld strengths IJ F*' are expressed as IJ F0"< ¸F , F*"< ¸F , (123) IJ 0 IJ IJ * IJ where <"(< , < )2 and the O(10!D, 26!D) invariant metric ¸ is de"ned in (127). 0 * Then, the e!ective D(10 action (in the Einstein frame) takes the form [465,560]



1 1 1 (!g R ! gIJR uR u# gIJ Tr(R M¸R M¸) L" E I J I J (D!2) 8 16pG " 1 1 ! e\?PgIIYgJJYgMMYH H ! e\?PgIIYgJJYFG (¸M¸) FH , IJM IYJYMY 4 IJ GH IYJY 12



(124)

where g,det g , R is the Ricci scalar of g , and FG "R AG !R AG are the ;(1)\" gauge IJ E IJ IJ I J J I "eld strengths. Here, the D(10 gravitational constant G is de"ned in terms of the D"10 one " G as G "(2p(a)\"G , where (a is the radius of internal circles. The Einstein-frame   " metric g is related to the string-frame metric g through Weyl rescaling g"e?Pg . In terms of IJ IJ IJ IJ the graviphotons A0? and photons A0' in vector multiplets, the gauge kinetic terms take the form: I I F (¸M¸)FIJ"F02F0IJ#F*2F*IJ, due to the relation M"<2<"<2< #<2< . IJ IJ IJ 0 0 * * In particular for D"4, the supersymmetry transformations (119) of fermionic "elds in the bosonic background take the following simpli"ed form in terms of D"4 "elds [236]:





1 RS 1 1 (S F0?c?@c C?# Q?@C?@ e , dt " ! ic I ! I 4  ?@ I I S 4 I 8(2  R (S !icS ) 1 1 ! cI I  (S F0?cIJC? e , dj"  IJ S 2(2 4(2  1 1 cI< ¸R <2 ) C! (S F*cIJ e , ds" * I 0  IJ 2(2 (2

 





(125)

where Q?@"(< ¸R <2)?@ is the composite SO(6) connection and S is the axion-dilaton "eld de"ned I 0 I 0 in Section 3.2.3. Here, the D"10 gamma matrices C (A"0, 1,2, 9) are decomposed into the D(10 spacetime parts cI (k"0, 1,2, D!1) and the internal space parts C? (a"1,2, 10!D). 3.2.3. Duality symmetries The D(10 e!ective action (124) is invariant under the O(10!D, 26!D) transformations (T-duality) [465,560]: MPXMX2, AG PX AH , g Pg , uPu, B PB , I GH I IJ IJ IJ IJ  We choose a"1 in most of cases.

(126)

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D. Youm / Physics Reports 316 (1999) 1}232

where X3O(10!D, 26!D), i.e. with the property: X2¸X"¸,




¸" I

0 \" 0

I

0

\" 0 0

0 I



,

(127)

\"

where I denotes the n;n identity matrix. L When electric/magnetic charges are quantized according to the Dirac}Schwinger}Zwanzinger}Witten (DSZW) quantization rule [218,546,547,630,640}643], the quantized, conserved electric a and magnetic b charge vectors live on the even, self-dual, Lorentzian lattice K [410,557]. The subset of O(10!D, 26!D, R) symmetry that preserves the lattice K is O(10!D, 26!D, Z), the so-called T-duality group of heterotic string on a torus. T-duality symmetry is a perturbative symmetry, which is proven to be exact to order by order in string coupling [307]. Under the T-duality, the charge lattice vectors transform as aP¸X¸a, bP¸X¸b .

(128)

In addition, the e!ective "eld theory has an on-shell symmetry called strong-weak coupling duality (S-duality) [268,537,544,545,556}560]. The equations of motion for (124) are invariant under the SO(1, 1) [S¸(2, R)] transformation for 54D410 [D"4]. Such transformations mix electric and magnetic charges, while transforming the dilaton in a nontrivial way. When the DSZW quantization is taken into account, such duality groups break down to integer-valued subgroups Z and S¸(2, Z) for 54D410 and D"4, respectively. These are the conjectured S-dualities in  heterotic string. As an example, we discuss the D"4 S¸(2, R) symmetry. D"4 case is special for the following reason. Since the 2-form "eld strengths are self-dual under the Hodge-duality, ;(1) gauge "elds obey electric-magnetic duality transformations, which leave the Maxwell's equations and Bianchi identities invariant. Also, the "eld strength H of the 2-form potential B is Hodge-dualized to IJM IJ a pseudo-scalar W (axion): eP eIJMNR W , HIJM"! N (!g forming a complex scalar S"W#ie\P with dilaton u. The (Einstein-frame) D"4 theory has the on-shell symmetry under the S¸(2, R) transformations (S-duality) [163,165,560]: aS#b SP , MPM, g Pg , IJ IJ cS#d FG P(cW#d)FG #ce\P(M¸) 夹FH , IJ IJ GH IJ where 1 eIJMNFG , 夹FGIJ" MN 2(!g and a, b, c, d3R satisfy ad!bc"1.

(129)

D. Youm / Physics Reports 316 (1999) 1}232

41

The instanton e!ect breaks the S¸(2, R) symmetry down to S¸(2, Z) [557,569]. The electric and magnetic `lattice charge vectorsa [560] a and b that live on an even, self-dual, Lorentzian lattice K with signature (6, 22) are given in terms of the physical electric and magnetic charges Q and P (de"ned as FG +QG/r and 夹FG +PG/r) as b,¸P and a,e\(M\Q!W b [557]. Under   RP RP the S-duality, a and b transform as [410,557]



a

b




a

!b

a

!c

d

b

P

,

(130)

where a, b, c, d are integers satisfying ad!bc"1. 3.2.4. Solution generating symmetries For stationary solutions, which have the Killing time coordinate, one can further perform Abelian KK compacti"cation of the time coordinate on S. The T-duality transformation of such (D!1)-dimensional action can be applied to a known D-dimensional solution to generate new types of solutions in D dimensions with di!erent spacetime structure [99,144,290,147,148,223,251,278,336,341,393,446,466,552}554,562]. The basic idea on solution generating symmetry is as follows. If the background con"guration is time independent, then under the (time-independent) general coordinate transformations, G y and RI B y transform as vectors, where ky "1,2, D!1. In addition, B y transforms as a vector under the RI RI (time-independent) gauge transformation of the 2-form "eld. So, one can add 2 new ;(1) gauge "elds associate with G y and B y to the existing D-dimensional 36!2D ;(1) gauge "elds, forming RI RI a new multiplet of vectors Ax Gx y (ix "1,2, 38!2D) [562]. In addition, since G and AG transform as I RR R scalars under the transformations mentioned above, the scalar matrix of moduli is enlarged to a (38!2D);(38!2D) matrix [562]. Under the T-duality of the (D!1)-dimensional e!ective action, the (t, t)-component of the D-dimensional metric g mixes with scalars in the moduli matrix M and the t-component of the IJ ;(1)\" gauge "elds AG , and the (t, ky )-components of g mix with the AG and the (t, ky ) I IJ I components of B . So, unlike the D-dimensional T-duality transformation, which leaves the DIJ dimensional spacetime intact, the (D!1)-dimensional T-duality transformation can be applied to a known D-dimensional solution to generate new solutions with di!erent spacetime structure. In particular, such transformations can be imposed on charge neutral solutions to generate electrically charged (under the D-dimensional ;(1)\" gauge group) solutions: 36!2D SO(1, 1) boosts in the (D!1)-dimensional T-duality group generate electric charges of ;(1)\" gauge "eld when acted on charge neutral solutions. The (D!1)-dimensional e!ective action is [495,561,562]: x ¸x R y M x ¸x ) L"(gy e\Py [R y #gy Iy Jy R y uy R y uy #gy Iy Jy Tr(R y M E I J  I J !  gy Iy Iy Ygy Jy Jy Ygy My My YHx y y y Hx y y y !gy Iy Iy Ygy Jy Jy YF x Gx y y (¸x M x ¸x ) x x F x Hx ] ,  IJ IJM IYJYMY GH Iy YJy Y

(131)

 When acted on magnetically charged solutions, such transformations induce unphysical Taub-NUT charge [562]. This is due to the singularity of Dirac monopole.

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where ;(1) gauge "elds A x Gx y (ix "1,2, 38!2D), dilaton uy , 2-form "eld Bx y y and the metric gy y y are I IJ IJ de"ned as Ax Gy ,AGy !(g )\g y AG, 14i436!2D, 14ky 4D!1 , I RR RI R I ,(g )\g y , Ax \" ,B y #AG¸ Ax Hy , Ax \"  RR RI Iy  RI R GH I Iy uy ,u!ln(!g ), gy y y "g y y !(g )\g y g y , RR IJ IJ RR RI RJ  Bx y y ,B y y #(g )\(g y AGy !g y AGy )¸ AH#(g )\(B y g y !B y g y ) , IJ RR RI J RJ I GH R  RR RI RJ RJ RI IJ and the symmetric O(11!D, 27!D) moduli matrix is given by




(132)



M#4(g )\A A2 !2(g )A 2M¸A #4(g )\A (A ¸A ) RR R R RR R R RR R R R !2(g )\A2 (g )\ !2(g )\A2¸A RR R RR RR R R Mx " . (133) 2A2¸M !2(g )\AA2¸A g #4A2¸M¸A R RR R R RR R R #4(g )\A2(A2¸A ) #4(g )\(A2¸A ) RR R R R RR R R Here,


 ¸ 0 0

¸x , 0

0 1

0

1 0

is an O(11!D, 27!D) invariant matrix and A ,[AG]. R R This action has invariance under the O(11!D, 27!D) T-duality [495,562]: x Hx , uy Puy , gy y y Pgy y y , Bx y y PBx y y , (134) Mx PXx M x Xx 2, A x Gx y PXx x x A I GH Iy IJ IJ IJ IJ where Xx 3O(11!D, 27!D), i.e. Xx ¸x Xx 2"¸x . D"3 case is special since a 2-form "eld strength is dual to a scalar. So, the scalar moduli space is enlarged from O(7, 23, Z)!O(7, 23)/[O(7);O(23)] to O(8, 24, Z)!O(8, 24)/[O(8);O(24)] [561]. The O(8, 24, Z) duality symmetry are generated by D"4 S¸(2, Z) S-duality and D"3 O(7, 23, Z) T-duality [561], just as U-duality in type-II string is generated by S-duality in D"10 and T-duality in D(10 [381]. The O(8, 24, Z) symmetry transformation puts the axion-dilaton "eld on the same putting as the other moduli "elds and, therefore, is non-perturbative in nature. Since we consider stationary solution, i.e. a solution with isometry in the time direction, we compactify the time coordinate as well as other internal space coordinates on ¹ to obtain D"3 e!ective action ((124) with D"3 and now k"r, h, ; m"t, 1,2, 6). Such action has an o!-shell symmetry under the O(7, 23) transformation (126). The DSZW quantization condition breaks this symmetry to integer-valued O(7, 23, Z) subset. In D"3, one can perform the following Hodge-duality transformations to trade the D"3 ;(1) "elds AG with a set of scalars t,[tG] [561]: I (135) (!he\PhI I YhJ J Y(M¸) FH  "eI J M R  tG , GH IYJY M where h   is the D"3 space metric and i, j"1,2, 30. Here, M is a symmetric O(7, 23) matrix IJ de"ned as in (121) but now the time component is included, and ¸ is an O(7, 23) invariant metric de"ned in (127). So, the D"3 e!ective theory is described only in terms of graviton and scalars.

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43

The D"3 e!ective action has the form [561]: (136) L"(!h[R #hI J Tr(R  MLR  ML)] ,  F  I J where h,det h   , R is the Ricci scalar of h   . M is a symmetric O(8, 24) matrix of D"3 scalars IJ F IJ de"ned as [561]




M!ePtt2



M¸t!ePt(t2¸t)  ePt2 !eP ePt2¸t  . M" M ¸M t t2¸M ePt2¸t !e\P#t2¸M M  !eP(t2¸t) !ePt2(t2¸t)   The action is manifestly invariant under the O(8, 24) transformations [561]: ePt

MPXMX2, h   Ph   , IJ IJ where X3O(8, 24), i.e.




(137)

(138)

¸M 0 0

XLX2"L,

L" 0 0

0 1 .

(139)

1 0

When electric and magnetic charges are quantized according to the DSZW quantization condition, the O(8, 24) is broken down to O(8, 24, Z). Since O(8, 24, Z) is generated by the conjectured S-duality in D"4 and T-duality in D"3 (which is proven to hold order by order in string coupling), the establishing O(8, 24, Z) invariance of the full string theory is equivalent to proving the S-duality in D"4 [561]. 3.3. String-string duality in six dimensions Six dimensions is special in the duality of (d!1)-branes [234]. A (d!1)-brane in D dimensions is dual to a (dI !1)-brane (dI ,D!d!2) under the Hodge-dual transformation of "eld strengths. So, in particular the heterotic string (1-brane) in D"6 is dual to another string (dI "6!2!2"2) [226]. In fact, it was found out by Du! et al. [246,598] that the type-IIA string compacti"ed on K3 surface has the same moduli space as that of the heterotic string compacti"ed on ¹, i.e. O(4, 20, Z)/O(4, 20, R)![O(4, R);O(20, R)]. Based on these observations, it is conjectured [635] that the heterotic string on ¹ is dual to the Type-IIA string on K3 surface, the so-called string-string duality in D"6 [226,229,235,340,381,635]. The e!ective action of heterotic string compacti"ed on ¹ in the string-frame is [565,635]



1 dx (!Ge\U[R GI J R  UR  U!  GI I YGJ J YGM M YH    H    S" % I J  IJM IYJYMY 16pG  !GI I YGJ J YFG  (¸M¸) FH  #GI J Tr(R  M¸R  M¸)] , (140) IJ GH IYJY  I J where k"0,2, 5, i"1,2, 24, ¸ is an O(4, 20) invariant metric and M is a symmetric O(4, 20) matrix, i.e. M2"M and M¸M2"¸. (De"nitions of D"6 "elds in terms of the D"10 "elds are

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D. Youm / Physics Reports 316 (1999) 1}232

given in Section 3.2.2.) The "eld strengths of the ;(1) gauge "elds and the 2-form potential are FG  "R  AG !R  AG , H    "(R  B   #2AG ¸ FH  )#cyc. perms. . IJ I J J I IJM I JM I GH JM The e!ective action for type-IIA string compacti"ed on a K3 surface is [565,635]

(141)



1 dx((!G[e\UY+R #G I J R  UR  U S" %Y I J 16pG  !  G I I YG J J YG M M YH    H    #G I J Tr(R  M¸R  M¸), IJM IYJYMY  I J            (!G IIYG JJYFG  (¸M¸) FH  ]!eIJMNOCB  FG  ¸ FH  ) , (142)  IJ GH IYJY IJ MN GH OC where now the corresponding D"6 "elds in type-IIA theory are denoted with primes. Note, the "eld strength of B  is de"ned without Chern}Simmons term involving ;(1) gauge "elds IJ H   "R  B  #cyc. perms. . (143) IJM I JM The scalars and metric are de"ned similarly as those in the e!ective action (140) of heterotic string on ¹. But since the K3 surface does not have a continuous isometry, there are no KK ;(1) gauge "elds, instead there are additional ;(1) gauge "elds arising from the 1-form A and the 3-form + A in the R-R sector. +,. These two string-frame e!ective actions are described by the same "eld degrees of freedom and have the same modular space. So, they can be identi"ed as the same action, provided we perform the following conformal transformation of the metric and the Hodge-duality transformation of the 2-form "eld [565,635]: , U"!U, G  "e\UG   , M"M, A ?"A? IJ I I IJ (144) U          (!Ge\ HIJM"eIJMNOCH   .  NOC Under the string}string duality, the dilaton changes its sign, indicating that the string coupling j"e\6U7 of the dual theory is inverse of the original theory. So, a perturbative string state (weak string coupling j;1) in one theory is mapped to a non-perturbative string state (strong string coupling j<1) under the string}string duality. For example, perturbative, singular, fundamental string in one theory is mapped to non-perturbative soliton string in the other theory [565]. 3.3.1. String}string}string triality Upon toroidal compacti"cation to D"4, the D"6 string}string duality (144) interchanges the D"4 S-duality and the (¹ part of) T-duality [226], while the dilaton-axion "eld and the KaK hler structure of ¹ are interchanged. So, the axion-dilaton "eld of the string}string duality transformed theory is given by the KaK hler structure of the original theory. Note, the ¹ part of the full D"4 T-duality group, i.e. the O(2, 2, Z) S¸(2, Z);S¸(2, Z) subgroup, contains not only the S¸(2, Z) factor parameterized by the KaK hler structure of ¹ but also the other S¸(2, Z) factor parameterized by the complex structure of ¹ [215,229,570]. Namely, the e!ective D"4 theory has the S¸(2, Z);S¸(2, Z);S¸(2, Z) symmetry with each S¸(2, Z) factor respectively parameterized by the dilaton-axion "eld, the KaK hler structure and the complex structure. So, on the ground of symmetry argument, one expects another string theory whose axion-dilaton "eld is given by the complex structure of the original theory [236]. In fact, mirror symmetry [24,317,319,320,372] exchanges the

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45

complex structure and the KaK hler structures of an internal manifold. In particular, the mirror symmetry exchanges the type-IIA and type-IIB strings, and transforms heterotic string into itself. Thus, combining the D"6 string}string duality (which interchanges the dilaton-axion "eld and the KaK hler structure) and the Mirror symmetry (which interchanges the KaK hler structure and the complex structure), we establish the `trialitya [236] among the heterotic string on K3;¹ and the type-IIA and type-IIB strings on the Calabi}Yau-threefold. For the purpose of illustrating the triality among these three theories, we consider only the ¹ part and the NS-NS sector (which is common to the three theories) described by the following D"6 e!ective action: (145) L"(1/16pG )(!Ge\U[R #GI J R  UR  U!  GI I YGJ J YGM M YH    H    ] . % I J  IJM IYJYMY  All the three theories with such truncation have the e!ective actions of this form. We label these three D"4 theories as F , where F"H, A, B respectively denoting the heterotic theory, the 678 type-IIA theory and the type-IIB theory, and the subscripts X, >, Z respectively are the axiondilaton "eld, the KaK hler structure and the complex structure of the theory. We can take any of these three theories as the starting point, but for the purpose of de"niteness we start with the heterotic string and impose the string}string duality and the Mirror symmetry to obtain all other "ve theories. Compacti"cation on ¹ is achieved by the following KK Ansatz for the D"6 metric:






eEg #AKALG AKG IJ I J KL I KL , (146) ALG G J KL KL where k, l"0,2, 3 are the D"4 spacetime indices and m, n"1, 2 are the internal space indices. The D"6 2-form "eld is decomposed as: G " IJ






B #(AKB !B AL) B #AKB IJ  I KJ IL J IL I KL . (147) B #B AL B KJ KL J KL Here, g, g , AK, B and G are respectively the D"4 dilaton (de"ned below), Einstein-frame IJ I IJ KL metric, the KK ;(1) gauge "eld, the 2-form "eld and the internal metric. To express the D"4 e!ective action in an S¸(2, Z);S¸(2, Z) T-duality invariant form, we parameterize the internal metric and the 2-form "eld as: B " IJ




G "eM\N KL



e\M#c !c !c

1

,

B "be , KL KL

(148)

and de"ne the D"4 dilaton g and axion a as (149) e\E"e\U(det G "e\U>N, eIJMNR a"(!ge\EgINgJHgMOH , KL N NHO where H is the "eld strength of B . Then, from the above real scalars we de"ne the following IJM IJ complex scalars [215] S"S #iS ,a#ie\E ,   ¹"¹ #i¹ ,b#ie\N ,   ;"; #i; ,c#ie\M ,  

(150)

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which (within the framework of the heterotic string) are respectively the dilaton}axion "eld, the KaK hler structure and the complex structure. Then, the "nal form of the D"4 e!ective action is [236]:



1 (!g R !S gIIYgJJYF2 (M M )F #gIJ Tr(R M LR M L) L" E   IJ 2 3 IYJY  I 2 J 2 16pG  1 gIJR SR SM , (151) # gIJTr(R M LR M L)! I 3 J 3 I I  2(S )  where M , M 3S¸(2, R) are de"ned as 2 3 ¹ ; 1 1 1 1  ,  , M , (152) M , 3 ; ; ";" 2 ¹ ¹ "¹"     and the ;(1) gauge "elds AG (i"1,2, 4) are given by A"B , A"B , A"A, A"A. I I I I I I I I I Here, L is an S¸(2, Z) invariant metric. The action is manifestly invariant under the S¸(2, R);S¸(2, R) T-duality:













M Pu2 M u , M Pu2 M u , F P(u\u\)F , (153) 2 2 2 2 3 3 3 3 IJ 2 3 IJ where u 3S¸(2, R) and the rest of the "elds are inert. In addition, the theory has an on-shell 23 S-duality symmetry:










FG FG a b IJ Pu\ IJ ; u" , (154) 1 夹FG 夹FG c d IJ IJ where a, b, c, d3Z satisfy ad!bc"1, and 夹FG is the Hodge-dual (de"ned from the action (151)) of IJ the "eld strength FG . IJ We denote the theory described by (151) as H , meaning the heterotic theory with the 123 dilaton-axion "eld, the KaK hler structure and the complex structure given respectively by S, ¹, ; de"ned in (150). Under the Mirror symmetry, the KaK hler structure and the complex structure are interchanged, and therefore we obtain H theory, i.e. the heterotic string with the KaK hler 132 structure and the complex structure now respectively given by ; and ¹ de"ned in (150); the e!ective action is (151) with ¹ and ; "elds interchanged. We call H and H as the S-strings, 123 132 meaning the string theories with the dilaton}axion "eld given by S de"ned in (150). Under the D"6 string}string duality (144), the H is transformed to A . Under the Mirror 123 213 symmetry, A is transformed to B . So, the A and B are the T-strings. 213 231 213 231 We apply string}string duality to B to obtain B . Under the Mirror symmetry, B is 231 321 321 transformed to A . Therefore, we have the ;-strings given by B and A . 312 321 312 We comment on relation of the D"4 S-duality to the D"6 string}string duality. Since e!ect of the D"6 string}string duality on the D"4 theory is to interchange the complex structure and the dilaton}axion "eld, the S¸(2, Z) subset T-duality of one theory accounts for the S-duality of the string}string duality transformed theory [226,233]. Namely, the large}small radius T-duality (RPa/R) of one theory corresponds to the strong}weak coupling duality (g/2pP2p/g) of the dual theory. In terms of transformation of ;(1) gauge "elds, one can understand this as follows [236]. Under the string}string duality, electric [magnetic] charges of 2-form ;(1) gauge "elds of aS#b SP , cS#d

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47

one theory is transformed to magnetic [electric] charges of 2-form ;(1) gauge "elds of the dual theory, while those of KK ;(1) "elds remain inert. Since the T-duality interchanges KK ;(1) gauge "elds and 2-form ;(1) gauge "elds (associated with the same internal coordinates), under the combined action of the D"6 string}string duality and the D"4 T-duality, electric [magnetic] charges of KK ;(1) gauge "elds and magnetic [electric] charges of 2-form ;(1) gauge "elds are exchanged, which is exactly the D"4 S-duality. Thus, since the T-duality is proven to hold order by order in string coupling, proof of the conjectured D"4 S-duality amounts to proof of the D"6 string}string duality, and vice versa. Since the string}string duality interchanges the dilaton}axion "eld with the KaK hler structure, the string coupling g/2p of one theory is transformed to the worldsheet coupling a/R of the dual theory [226]. So, string quantum corrections controlled by the string coupling in one theory correspond to the stringy classical corrections (a corrections) controlled by the worldsheet coupling in the dual theory; the a [quantum] corrections in one theory can be understood in terms of the quantum [a] corrections of the dual theory. 3.4. U-duality and eleven-dimensional supergravity Hull and Townsend [381] conjectured that the type-II superstring theories on a torus has full symmetry of low-energy e!ective "eld theory, which is larger than the direct product of the D"10 S¸(2, Z) S-duality and the O(10!d, 10!d, Z) T-duality in D"d(10 [306]. For example, the e!ective action of type-II string on ¹, which is D"4, N"8 supergravity [159,160], has an on-shell E symmetry [160], which contains S¸(2, R);O(6, 6) as the maximal subgroup. Hull and  Townsend [381] conjectured that the subgroup E (Z) (broken due to the DSZW charge quantiz ations) extends to the full string theory as a new uni"ed duality group, called U-duality. U-duality uni"es the S and ¹ dualities and mixes p-model and string coupling constants. The discrete subgroup E (Z) is the intersection of the continuous E group and the discrete   symplectic Sp(28, Z) group, which transforms 28 ;(1) gauge "elds of the e!ective theory linearly: E (Z)"E 5 Sp(28, Z) [381]. Under U-duality, a set of 28;2 electric and magnetic charges   transform as a vector, and all the scalars in the theory mix among themselves. Unlike other types of duality, which assigns a special role to the dilaton, under U-duality the dilaton is on the same footing as moduli. Thus, unlike T-duality which is perturbative in nature, U-duality, like S-duality [560], is non-perturbative in nature, so cannot be tested within perturbative spectrum of string theories. We list the conjectured U-duality groups in various dimensions [381]. Note, the duality group in higher dimensions is a subgroup of lower dimensional duality group, since it survives compacti"cation (cf. The duality group does not act on the Einstein-frame metric). The SO(10!d, 10!d, Z) T-duality in D"d(10 and the S¸(2, Z) S-duality in D"10 are uni"ed to U-duality for d(8. The U-duality groups are, in the descending order starting from D"7: SO(5, 5, Z), S¸(5, Z), E (Z), E (Z), E (Z), E (Z), E (Z). These U-duality groups in D"d are generated      by the T-duality in D"d and n numbers of U-duality groups in D"d#1 [381], where n is the possible numbers of ways in which one can compactify from D"10 to D"d#1 and then down to D"d. In comparison, the heterotic string on ¹\B with d'3 maintains the duality group in the form (T-duality group);(S-duality group), i.e. SO(10!d, 26!d, Z);Z for 105d55 or 

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SO(10!d, 26!d, Z);S¸(2, Z) for d"4. For d"3 [561] and d"2 [563], the T- and S-dualities are uni"ed to U-duality given by SO(8, 24, Z) and SO(8, 24)(Z), respectively. As we will see in Section 7.1, BPS electric states are within perturbative spectrum of heterotic string [248]; all the 28 electric charges in the heterotic string on ¹ are related through the `perturbativea O(6, 22, Z) T-duality. Also, there is non-perturbative spectrum carrying the remaining 28 magnetic charges, related to perturbative spectrum via the Z subset of the `non-pertur bativea S¸(2, Z) S-duality [486,544,558]. These magnetic charges are carried by solitons. The mass of a state in heterotic string on ¹ carrying electric [magnetic] charges of the ;(1) gauge group behaves as &1 [&1/g] in the string frame, as expected for a fundamental string [a soliton]. Q For the type-II string, only 12 of the 28 electric charges couple to perturbative string states, since the remaining 16 electric charges are R-R charges, which cannot be coupled to perturbative string states. The Z subgroup of the S¸(2, Z) S-duality group relates these perturbative states to solitonic  states carrying 12 magnetic charges of the same 12 ;(1) gauge "elds. The remaining 16 electric and 16 magnetic charges of the ;(1) gauge group are carried by another type of non-perturbative states, whose mass behaves as &1/g . Thus, under the (T-duality);(S-duality) subgroup, i.e. Q SO(6, 6, Z);S¸(2, Z)LE (Z), the fundamental representation 56 (representing 28 electric and 28  magnetic charges of the D"4 ;(1) gauge group) is decomposed as 56"(12, 2);(32, 1). Here, the "rst factor (12, 2) corresponds to the 12 numbers of S¸(2, Z) doublets of perturbative and solitonic states in the NS-NS sector and the second factor (32, 1) denotes the remaining non-perturbative states, which are singlets under the S¸(2, Z) group and carries 16 electric and 16 magnetic charges of 16 ;(1) gauge "elds in the R-R sector. Since the O(6, 6, Z) T-duality group of the type-II string on ¹ mixes only NS-NS charges among themselves, it is not inconsistent that string states carry only NS-NS charges. However, it is not consistent with the conjectured U-duality, since U-duality puts all the 28;2 electric and magnetic charges of D"4 ;(1) gauge group on the same putting. In addition, as we saw in the decomposition of the 56 representation, the U-duality requires existence of additional 16#16 electric and magnetic charges in the RR-sector that transform irreducibly under the O(6, 6, Z) T-duality group. Hence, one leads to the conclusion that all the RR charges, which cannot be carried by perturbative string states or solitons, should be carried by another type of nonperturbative states. The low-energy or long-distance description of these non-perturbative string states is R-R p-branes or black holes. In the original work by Hull and Townsend [381], they show that all the R-R charged black holes can arise from extreme p-branes of the D"10 e!ective supergravity via dimensional reduction. In [498], Polchinski shows that the states carrying R-R charges can be realized within string theories without introducing exotic extended objects like p-branes. Such objects are D-branes [193,445], boundaries to which the ends of open strings (with Dirichlet boundary condition) are attached. D-branes carry one unit of R-R charges. D-branes are dynamic objects and the open string states describe their #uctuations. In the strong coupling limit, D-branes become black holes. Such identi"cation made it possible to give precise statistical explanation of black hole entropy. In Section 8, we will summarize aspects of D-branes and the recent development in D-brane calculation of black hole entropy. In the following sections, we discuss the S-duality of the type-IIB string and the T-duality of type-II string on a torus. The U-duality of type-II string on a torus is generated by these two dualities. In particular, starting from generating black holes of type-II theories on a torus, one

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obtains black holes with the general charge con"guration by applying the S-duality and the T-duality transformations. For p-branes in type-II theories, one can generate p-branes of the general charge con"guration by "rst imposing SO(1, 1) boost on a charge neutral solution to induce KK electric charge and then applying the ¹- and the S-duality transformations and/or another SO(1, 1) boosts sequentially. 3.5. S-duality of type-IIB string The type-IIB string [312,536] has the S¸(2, Z) symmetry [381]. The Z LS¸(2, Z) trans formation exchanges the NS-NS 2-form potential BK  (coupled to a perturbative string state) and the R-R 2-form potential BK  (coupled to a non-perturbative D-brane) while changing the sign of the dilaton (or inverting the string coupling). So, the S¸(2, Z) symmetry is a strong}weak coupling duality. It is well-known that a covariant e!ective action for type-IIB string does not exist, while only the "eld equations [376,536] exist. The only problem with construction of the covariant e!ective action is the R-R 4-form potential DK (with the self-dual 5-form "eld strength FK ), whose equation of motion FK "夹FK cannot be derived from the covariant action. So, to construct the covariant e!ective action for type-IIB string, one is forced to set DK to zero. However, it is found in [72] that one can construct the type-IIB e!ective action which gives rise to the correct "eld equations and compacti"es to the correct action for the dimensionally reduced type-IIB theories without setting DK to zero. In this approach, one keeps FK di!erent from zero in the e!ective action but eliminates the self-duality constraint. After the "eld equations are obtained from this e!ective action, the self-duality constraint is imposed in order to get the correct "eld equations for the type-IIB theory. In the string-frame, such e!ective action for type-IIB string has the form [72]:



 



3 1 S " dx (!GK  e\U !RK #4(RU)! (HK ) '' 4 2



1 3 5 1 ! (Rs)! (HK !sHK )! FK ! seGHeDK HK GHK H , 2 4 6 96(!G

(155)

The "eld strengths of the 2-form potentials BK G and the 3-form potential DK , and their gauge transformation rules are HK G"RBK G,

dBK G"RRK G ,

FK "RDK #eGHBK GBK H, dDK "RKK !eGHRRK GBK H ,  

(156)

where RK G and KK are in"nitesimal gauge transformation parameters. The S¸(2, Z) symmetry of the type-IIB theory is manifest in the e!ective action in the Einsteinframe. To go to the Einstein-frame, one Weyl-rescales the metric GK "e\UGK  . +, +,

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The resulting Einstein-frame action has the form:





1 1 3 S#  " dx (!GK !RK # Tr(R MK R+MK \)! HK GMK HK H '' + GH 2 4 4



5 1 ! FK ! eGHeDK HK GHK H , 6 96(!GK

(157)

where MK is a 2;2 real matrix formed by the complex scalar jK "s#ie\U:






"jK " !Re jK 1 . MK " Im jK !Re jK 1

(158)

(157) is manifestly invariant under the S¸(2, R) transformation [376,536]:



 HK  HK 

Pu

HK  HK 

, MK P(u\)2MK u\,

u3S¸(2, R) .

(159)

The S¸(2, R) transformation on jK has the usual fractional-linear form jK P(ajK #b)/(cjK #d). When the DSZW type charge quantization is taken into account, the S¸(2, R) symmetry breaks down to the S¸(2, Z) subset. 3.6. ¹-duality of toroidally compactixed strings Closed strings in D dimensions in target space background with d commuting isometries have O(d, d, Z) T-duality symmetry [304,305,308,570]. The O(d, d,Z) symmetry is a perturbative symmetry proven to hold order by order in string coupling. For heterotic string, the symmetry is enlarged to O(d, d#16, Z) due to the additional rank 16 background gauge "elds. Under the Z subset that inverts the radius of S (i.e. R Pa/R ) and interchanges winding and momentum  G G modes (i.e. m  n ), the type-IIA and the type-IIB theories are interchanged if odd number of circles G G are acted on by the Z transformations, while heterotic string transforms to itself. The  Z transformation between the type-IIA and the type-IIB strings at the e!ective "eld theory level is  understood as "eld rede"nition between type-IIA and type-IIB theories, since the compacti"cation of the type-IIA and the type-IIB supergravities leads to the same supergravity theory. We consider the bosonic string worldsheet p-model with only NS-NS sector "elds (target space metric G (X), 2-form potential B (X) and dilaton U(X)), which are common to both type-II and IJ IJ heterotic strings, turned on. The action with the (curved) D-dimensional target space has form



1 dz[(G (X)#B (X))RXIRXJ! (X)R] . S" IJ IJ  2p

(160)

Let us assume that (160) is invariant under the d commuting, compact Abelian isometries [309] dXI"ekI (i"1,2, d) along the XG-direction, where [k , k ]"0. Then, the background "elds G G H become independent of XG. First, we consider the case where the background "elds have only one isometry (d"1) along, say, the direction h"X. The T-dual pair p-model actions are obtained by gauging the Abelian isometry. Following the procedures discussed in [80,82,100}102], one obtains the action dual to

D. Youm / Physics Reports 316 (1999) 1}232

51

(160) with the dual background "elds (with primes) related to the original ones (without primes) as 1 B G G #B B ? @, G " , G " ? , G "G ! ? @  G ? G ?@ ?@ G   ?@ (161) G B #B G ? @, U"U#log G , B "G G , B "B ! ? @  ? ?  ?@ ?@ G  where XI"(h, X?) (a"1,2, D!1). This is the curved background generalization of the RPa/R T-duality of closed strings on S. Alternatively, the dual pair p-models are obtained by the method of chiral currents [516]. One starts with a (D#d)-dimensional p-model with d Abelian (left-handed) chiral currents JG and (right-handed) anti-chiral currents JM G [309]: S "S #S #S[X] , ">B  ? 1 S " dz[RhG RM hG #RhG RM hG #2R (X)RhG RM hH #C* (X)RX?RM hG #C0 (X)RhG RM X?] ,  2p * * 0 0 GH 0 * ?G * G? 0

 

1 dz[RhG RM hG !RhG RM hG ] , S" * 0 0 * ? 2p

(162)



1 dz[C (X)RX?RM X@!U(X)R] , S[X]" ?@  2p where i, j"1,2, d and a, b"d#1,2, D. Here, chiral and anti-chiral currents, corresponding to the ;(1)B ;;(1)B a$ne symmetries dhG "aG (z), are * 0 *0 *0 JM G"RM hG #R hM H #C0 RM X? . (163) JG"RhG #R hH #C* RX?, 0 GH *  G? * HG 0  ?G By gauging either a vector or an axial subgroup of ;(1)B ;;(1)B , one has the following * 0 D-dimensional dual pair p-model actions:

   



1 1 S!" dz E! (X?)RXI RM XJ ! !(X?)R " 2p IJ ! ! 4

1 " dz E!(X?)RhG RM hH #F0!(X?)RhG RM X?#F*!(X?)RX?RM hG GH ! ! G? ! ?G ! 2p



1 #F!(X?)RX?RM X@! !(X?)R , ?@ 4

(164)

where (XI )"(hG , X?) with k, l"1,2, D, i"1,2, d, a"d#1,2, D. Here, hG ,hG $hG ! ! ! 0 * and the upper (lower) signs in $ and G correspond to the axial (vector) gauged p-model. The

 More general transformations with non-zero R-R "elds are derived in [81].

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background "elds are






E! F0! G@ , !"U#log det(I$R) , E! "G! #B! " GH IJ IJ IJ F*! F! ?H ?@ 1 E!"(I$R) (IGR)\, F!"C $ C* (IGR)\C0 , GH GI IH ?@ ?@ 2 ?G GH H@

(165)

F0!"(IGR)\C0 , F*!"$C* (I$R)\ , G? GH H? ?G ?H HG with G! [B!] denoting the symmetric (the antisymmetric) part of E!. The action S! has an " isometry under the translation in the hG -direction. 8 The dual pair actions (164) are the most general p-model with d commuting compact Abelian symmetries. S> and S\ are related under the combined operations of the sign reversal of R and C*, " " and the coordinate transformation hG P!hG (i.e. hG  hG ), implying equivalence of two gaugings * * > \ at least locally. To achieve global equivalence of the two gaugings, one has to impose the same periodicity conditions on both hG and hG , i.e. hG ,hG #2p. Then, one establishes the equivalence > \ ! ! of vector and axial gauged p-model actions S! (the so-called `vector-axial dualitya [426]). " One can relate S! to the bosonic string p-model action S in (160) by identifying XI"XI . Then, " ! the background "elds in S! are related to those in S in the following way: " G "(E!#E!), B "(E!!E!) , HG GH  GH HG GH  GH G "(F0!#F*!), B "(F0!!F*!) , ?G G?  G? ?G G?  G? G "(F!#F!), B "(F!!F!) . @? ?@  ?@ @? ?@  ?@

(166)

From this, transformation rule of background "elds under T-duality that relates the dual pair bosonic string p-model actions (one related to S> and the other related to S\) follows. When " " S! have isometry along only one coordinate direction (d"1), one recovers the factorized duality " transformation (161). We discuss transformations [305,309] that relate the di!erent backgrounds (of the same action) describing the equivalent conformal "eld theory. S! have the manifest invariance under the " following O(d, d, Z) transformation E!PE!"(a( E!#bK )(c( E!#dK )\




E!



(a!E!c)F0!

"

F*!(cE!#d)\ F!!F*!(cE!#d)\cF0!









det G! 1 ,

!P !" !# log det G! Y 2

g"

a b c

d

3O(d, d, Z) ,

,

(167)

where D;D blocks a( , bK , c( and dK of the O(D, D, Z) matrix are




a( "

a

0



0 I "\B

,




bK "

b 0 0 0




, c( "

c

0

0 0

,




dK "

d

0



0 I "\B

.

(168)

D. Youm / Physics Reports 316 (1999) 1}232

53

Here, I is the n;n identity matrix. The d;d matrices E! transform fractional linearly under L O(d, d, Z): E!PE!"(aE!#b)(cE!#d)\ .

(169)

The constant E! case corresponds to the transformation in the toroidal background. The O(d, d, Z) transformation is generated by the following transformations. E Integer `Ha-parameter shift of E, i.e. E PE #H (H "!H ): GH GH GH GH HG a b I H " B s.t. H"!H2 . c d 0 I B E Homogeneous transformations of E!, F*!, F0! under G¸(d, Z), F*!PA2F*!, F0!PF0!A (A3GL(2, Z)):





a b



A2

0

0

A\

"

c

d



s.t. A3GL(2, Z) .

(170) i.e. E!PA2E!A,

(171)

E Factorized dualities D , corresponding to the inversion of the radius R of the ith circle, i.e. G G R P1/R : G G G\ B\G\ CDE CDE a b I!e e G G " s.t. e "diag(0, 2, 0, 1, 0, 2, 0). (172) G c d e I!e GF F FHF F FI G G







The maximal compact subgroup of O(d, d, Z) is O(d, Z);O(d, Z) with a group element having the form:







hG PO hG , * * *

as ">B h O #O O !O h \ P1 * 0 * 0 \ , 2 O !O O #O h h > * 0 * 0 > C0PO C0, CPC , 0

a b

1 O*#O0 O*!O0 " s.t. O , O 3O(d,Z) . * 0 2 O !O O #O c d * 0 * 0 The O(d, Z);O(d, Z) transformation naturally acts on the action S hG PO hG , 0 0 0





(173)




RPO RO2, C*PC*O2, (174) 0 * * meanwhile on the actions S! as " E!PE!"[(O #O )E!#(O !O )][(O !O )E!#(O #O )]\ , * 0 * 0 * 0 * 0 F*!PF*!"2F*![(O !O )E!#(O #O )]\ , * 0 * 0 (175) F0!PF0! "[(O #O )!E! (O !O )]F0! ,  * 0 * 0 F!PF! "F!!F*![(O !O )E!#(O #O )]\(O !O )F0! . * 0 * 0 * 0 We now specialize to strings in #at background (i.e. toroidal compacti"cation) to understand properties of perturbative string spectrum under T-duality. The relevant part of the worldsheet

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action is the toroidal (¹B) part:

 



1 1 p (176) S" dp dq (gg?@G R XGR XH#e?@B R XGR XH! (g R , GH ? @ GH ? @ 2 4p  where XG&XG#2pmG and i, j"1,2, d. Here, mG is a string `winding numbera along the XGdirection. G and B can be collected into the `background matrixa E"G#B. The matrix E is GH GH a special case of E! in (165) where E! do not depend on X?. The lattice KB, which de"nes ¹B"RB/(pKB), is spanned by basis vectors e satisfying B e?e?"2G . ? G H GH G The mode expansions of XG and conjugate momenta 2pP "G XQ H#B XH  are G GH GH i 1 XG(p, q)"xG#mGp#qGGH(p !B mI)# [aG (E)e\ O\N#aG (E)e\ O>N] , H HI L (2 L$ n L 1 [E2 aH (E)e\ O\N#E aH (E)e\ O>N] , (177) 2pP (p, q)"p # GH L GH L G G (2 L$ where we made analytic continuation qP!iq and the momentum zero modes p take integer G values, i.e. p "n 3Z. The equal-time canonical commutation relations [XG(p, 0), P (p, 0)]" G G H idG d(p!p) lead to commutation relations among the oscillator modes: H [xG, p ]"idG , [aG (E), aH (E)]"[aG (E), aH (E)]"mGGHd . (178) H H L K L K K>L The Hamiltonian takes the form



1 1 p dp(P#P)" Z2M(E)Z#N#NI , H"¸ #¸I " * 0   4p 2  G!BG\B BG\ m , Z" ? , M(E)" !G\B G\ n ? where P are the left- and the right-moving momenta de"ned as *0 P "[2pP #(G!B) XHY]eGH, P "[2pP !(G#B) XHY]eGH , *? G GH ? 0? G GH ? and the number operators of the left- and the right-moving modes are









(179)

(180)

N " aG (E)G aH (E), N " aG (E)G aH (E) . (181) * \L GH L 0 \L GH L L L Here, eGH are dual basis vectors, satisfying B e?eHH"dH and B eGHeHH"(G\)GH. The left- and ? G ? G ? ? ?  the right-moving momenta zero modes p "[n2#m2(B!G)]eH, p "[n2#m2(B#G)]eH , (182) 0 * transforming as a vector under O(d, d, R), form an even self-dual Lorentzian lattice CBB [480,481], i.e. p!p"2mGn 32Z. * 0 G While CBB is preserved under O(d, d, R), the Hamiltonian zero mode H "(p#p) is invariant 0   * only under its maximal compact subgroup O(d, R);O(d, R). So, the zero-mode spectrum is

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unchanged under the O(d, R);O(d, R) subgroup, only. Note, from (182) one sees that (p , p ) and * 0 hence CBB are in one-to-one correspondence with a particular background E"G#B. Thus, the moduli space is isomorphic to O(d, d, R)/[O(d, R);O(d, R)]. Under the O(d, d, Z) transformation (169) [439], M(E)PgM(E)g2 ,

(183) a (E)P(d!cE2)\a (E), a (E)P(d#cE)\a (E) . L L L L So, N are manifestly invariant under O(d, d, Z); the spectrum is O(d, d, Z) invariant. The O(d, d, Z) *0 transformation is generated [304,305,308,570] by integer H-parameter shift of E, the G¸(2, Z) transformation and the factorized duality D , as discussed above. Particularly, under G¸(d, Z), G which changes the basis of KB, EPAEA2 and (m, n)P(A2m, A\n) (A3G¸(d, Z)). In addition, the spectrum is invariant under the worldsheet parity [305] pP!p, which acts on E as BP!B. The e!ect of the worldsheet parity on the spectrum is to interchange the left-handed and the right-handed modes: p  p and a  a . The above transformations generate the full spectrum * 0 L L preserving symmetry group G . B A particular element g of O(d, d, Z) with a"d"0 and b"c"I, i.e. EPE\ [308,570], corresponds to vector-axial duality symmetry (165). Under this transformation, n  m and the Hamiltonian (179) is manifestly invariant. When B"0, the transformation becomes GPG\, i.e. the volume inversion of ¹B. A signi"cance of EPE\ is that the gauge symmetry is enhanced to the a$ne algebra S;(2)B ;S;(2)B at a single "xed point G"I and B"0. The gauge symmetry is * 0 maximally enhanced [308] at "xed points under EPE\ modulo S¸(d, Z) and H(Z) transformations, i.e. E such that E\"M2(E#H)M (M3S¸(d, Z)). Hence, an enhanced symmetry point corresponds to an orbifold singularity point [220,221] in the moduli space under some non-trivial O(d, d, Z) transformation. At the "xed point, E takes the following form in terms of the Cartan matrix C of the rank d, semi-simple, simply laced symmetry group [252]: GH E "0 (i(j) . (184) E "C (i'j), E "C , GH GH GH GG  GG Non-maximally enhanced symmetry points correspond to "xed points under factorized dualities D instead of the full inversion EPE\. A simplest but non-trivial example is the d"2 case G (i.e. compacti"cation on ¹), which we discuss in Section 4.2.2. At the "xed point E"I under EPE\, the gauge symmetry is enhanced to (S;(2);S;(2)) ;(S;(2);S;(2)) . The gauge * 0 symmetry is maximally enhanced to S;(3) ;S;(3) at the point * 0 1 1 . E" 0 1




These "xed points correspond to orbifold singularities [570] in the fundamental domain of the moduli space (parameterized by two complex coordinates of the moduli space S¸(2, R)/;(1); S¸(2, R)/;(1)).

 Note, O(2, 2, R) S¸(2, R);S¸(2, R). Therefore, E, which parameterizes the moduli space O(2, 2, R)/[O(2, R) ;O(2, R)], is reparameterized by the complex coordinates o and q, each parameterizing S¸(2, R)/;(1).

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3.7. M-theory In this section, we discuss some aspects of M-theory. We illustrate how di!erent superstring theories emerge from di!erent moduli space of compacti"ed M-theory and discuss the M-theory origin of string dualities. In this picture, each of "ve di!erent string theories represents a perturbative expansion about di!erent points in moduli space of the compacti"ed M-theory. Namely, 5 perturbative string theories and uncompacti"ed M-theory are located at di!erent subsets of moduli space, and it is dualities that map one subset of moduli space to another, thereby making transition between di!erent theories. In the following, we illustrate this idea by showing how di!erent theories are achieved by taking di!erent limits of parameters of moduli space and how dualities are realized as transformations of parameters of moduli space. First, we discuss the connection between the type-II theories and M-theory. Type-IIA theory is obtained from M-theory by compactifying the extra 1 spatial coordinate on S of radius R  [112,232,383,601,635]. The type-IIA and the type-IIB theories are related via T-duality [193,216]. Namely, the type-IIA theory on S of radius R is perturbatively equivalent (under T-duality) to  the type-IIB theory on S of radius R "1/R . So, one can think of the S-compacti"ed type-IIB  theory as ¹ compacti"ed M-theory. To understand the connection between type-II theories and M-theory, one has to compactify M-theory on ¹"S;S (with the radii of each circle given by R and R from the D"11 point   of view), and compactify the type-IIA and the type-IIB theories on circles of radii R and R ,  respectively. Here, the radius R [R ] is measured with D"10 string frame metric of the type-IIA  [the type-IIB] theory. We "rst relate parameters of the type-II theories (i.e. the radii R and the string couplings  g  in the type-IIA/B theories) to parameters R and R of ¹ moduli space before we Q   understand the various limits in the moduli space. R is related to g as R "(g). As for the  Q  Q second circle of ¹"S;S, which is also the circle upon which the type-IIA theory is compacti"ed, the radius is measured di!erently depending on the dimensionality of spacetime. Note, we denoted the radius measured in D"11 [in D"10 by the type-IIA string-frame metric] as R [R ]. Namely, since the string-frame metric g'' (k, l"0, 1,2, 9) of the type-IIA theory is   IJ related to the D"11 metric G (M, N"0, 1,2, 10) as G&e\(g'', where is the IJ  +, IJ dilaton of the type-IIA theory, one sees that R "R /(g). Furthermore, one can relate the   Q string coupling g  of the type-IIB theory to R and R as follows. Under the T-duality between Q   the type-IIA and the type-IIB theories, the string couplings are related as g "g/R . By using Q Q  other relations among parameters, one "nds that g "R /R . Q  

 Note, due to the no-go theorem for KK compacti"cation of the D"11 supergravity [633], it might be impossible to obtain a chiral theory like type-IIB supergravity through dimensional reduction. This no-go theorem can be circumvented to obtain the (chiral) type-IIB theory by compactifying on orbifolds (rather than manifolds) [197,358,568,639]. In the case of compacti"cation of M-theory on ¹ (which is relevant to our discussion), when the size of ¹ goes to zero at the "xed shape, one obtains `chirala type-IIB theory, due to additional massive &wrapping' modes (of membrane) which become massless [22,81,540,541].  The string coupling g is de"ned as g "e6(7. Q Q

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We discuss the various limits in the M-theory moduli space of ¹ in terms of R and R [22]:   M-theory and the type-IIA, B theories are located at various limiting points in the ¹-moduli space. First, M-theory is located at (R , R )"(R,R) (i.e. the decompacti"cation limit), which is   also the strong coupling limit (g"(R )PR) of the type-IIA theory [601,635]. The (uncomQ  pacti"ed) type-IIA theory, de"ned as R PR and "nite string coupling g, is located at  Q (R , R )"(R, xnite), i.e. M-theory on S of radius R (R. The (uncompacti"ed) type-IIB    theory can be de"ned as the limit R PR, R P0 and "nite string coupling g . In this limit,  Q R "R /(g)"R /(R g )"R(g )\"R\(g )\P0. So, in terms of para    Q  Q Q meters of ¹, the (uncompacti"ed) type-IIB theory corresponds to the limit in which (R , R )"(0, 0) while keeping the ratio g "R /R "nite. The value of g  depends on how   Q   Q the limit (R , R )P(0, 0) is taken and, therefore, (R , R )"(0, 0) is not really a point in the     moduli space. Note, when 2 circles in ¹"S;S are exchanged (i.e. R  R ), g "R /R is inverted:   Q   g P1/g . Such interchange of 2 circles is a subset of more general SL(2, Z) reparameterization of Q Q ¹, which acts on the complex modulus q of ¹ fractional linearly. Thus, the reparameterization symmetry of ¹, upon which M-theory is compacti"ed, manifests in the type-IIB theory as the S¸(2, Z) S-duality [381], which acts on the complex scalar o"s#ie\( (formed by 0-form s and the dilaton ) fractional linearly. Next, we discuss string theories with N"1 supersymmetry, i.e. the E ;E and SO(32) heterotic   strings and type-I string. To understand the connection between M-theory and these N"1 string theories, one has to consider the moduli space of M-theory compacti"ed on S/Z ;S, i.e.  a cylinder of length ¸ and radius R. We "rst comment on the relation of type-I string theory to M-theory. One can think of the type-I theory as an &orientifold' of the type-IIB theory, namely a theory of unoriented closed string (gauged under the worldsheet parity transformation X [355,356,509]) and open string with SO(32) Chan-Paton factor [314]. To see the direct relation to M-theory, it is convenient to "rst compactify one spatial coordinate, which we call >, of the type-I theory on S and then T-dualize along the S-direction, inverting the radius of S. We call such theory as the type-I theory [193,505]. Since the dual coordinate >I is pseudo-scalar under the worldsheet parity transformation (i.e. X[>I ](q, p)"!>I (q,!p)), S is mapped under this T-duality to the orbifold S/Z with "xed  points at >I "0, p. So, the type-I theory is e!ectively described by the type-IIA theory on S/Z ;  closed strings wrapped around S/Z look like open string stretched between two 8-plane  boundaries located at "xed points of S/Z . (These (parallel) boundaries corresponds to D 8 branes.) Second, the E ;E heterotic string theory is obtained by compactifying M-theory on the   orbifold S/Z [357,358]. Namely, M-theory on S/Z of length ¸ gives rise to spacetime with two   D"10 faces (the so-called `end-of-the-world 9-branesa) which are separated by a distance ¸. Each of the two faces carries an E gauge "eld of the E ;E heterotic string [357,358]. In this picture,    a fundamental string of the E ;E theory is interpreted as a cylindrical M 2-brane attached   between the two faces. (So, the intersection of the cylindrical M 2-brane with the faces is a circle.) The string coupling is g#"¸ and, therefore, in the strong coupling limit (g#<1) the two faces Q Q move apart far away from each other, revealing the extra 11th space dimension [357,358,601,635]. When the separation is very small (¸+0), the cylindrical M 2-brane is well approximated by a closed string in D"10. (This is the weak coupling limit g#"¸+0 of the E ;E theory.) Q  

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Finally, the SO(32) heterotic theory is related to the E ;E heterotic theory via T-duality   [303,480,481], and to the type-I theory via S-duality [187,380,505,635]. A corollary of these dualities is the duality between M-theory on a cylinder S/Z ;S and the SO(32) theories on S  [541]. Now, we discuss the various limits in the moduli space of M-theory on S/Z ;S in terms of  parameters ¸ and R [357,358]. An obvious limit in the moduli space is the small ¸ and RPR limit, which is the uncompacti"ed E ;E heterotic string. Here, ¸ is the size of S/Z upon which    M-theory is compacti"ed to lead to the E ;E heterotic theory. The string coupling of the E ;E     heterotic string is g#"¸. The second obvious limit is RP0. In this limit, the M 2-brane Q wrapped around the cylinder looks like open string stretched between the interval S/Z of the  length ¸, i.e. the (uncompacti"ed) type-I open string with SO(16) Chan}Paton factor attached at each end located at the 9-plane boundary. In general, a point in the moduli space with small ¸ and "nite R corresponds to the E ;E   heterotic string on S of radius R. A non-vanishing &Wilson line' around S breaks E ;E down to   subgroups [634], e.g. ;(1) or SO(16);SO(16) depending on the choice of the Wilson line. In particular, the E ;E heterotic string on S with gauge group SO(16);SO(16) is obtained from the   SO(32) heterotic string on S with gauge group SO(16);SO(16) by inverting the radius of S. As R is decreased to a small value, one can switch to the SO(32) heterotic string on S of inverse radius 1/R by using the T-duality between the E ;E and SO(32) heterotic strings. For this case, the   string coupling of the SO(32) heterotic string is g"¸/R, which is small as long as R<¸. As Q the radius R approaches smaller value so that R becomes much smaller than ¸, one can switch to the type-I theory by applying the S-duality between the SO(32) heterotic string and the type-I string. The string coupling of the type-I theory is then given by g'"1/g"R/¸. Note, under the Q Q Z transformation that exchanges two moduli R and ¸ of the cylinder, the string couplings of the  type-I and the SO(32) heterotic theories are inverted, thereby manifesting as the non-perturbative type-I/heterotic duality. In the limit (R, ¸)P(0, 0) with "xed small g'"R/¸, one has the uncompacti"ed type-I theory. Q This is understood as follows. We saw that the type-I theory, which is obtained from the type-I theory on S by inverting the radius, is the RP0 limit of M-theory on the cylinder. If we further let the length ¸ of the cylinder approach zero, then in the type-I side the radius of S, upon which the type-I theory is compacti"ed, approach in"nity (i.e. the decompacti"cation limit of the type-I theory). So far, we discussed connections among M-theory and string theories with either N"1 or N"2 supersymmetry. One can further relate N"1 and N"2 theories. For this purpose, one breaks 1/2 of supersymmetry in N"2 theories by compactifying on a manifold with non-trivial holonomy. An obvious example is the D"6 string}string duality between type-IIA theory on K3 and heterotic string on ¹ [382,635]. Both of the D"6 theories have (non-chiral) N"2 supersymmetry. A corollary of this duality is that M-theory on K3 is equivalent to heterotic string on ¹ [635], since type-IIA theory is M-theory on S. The fundamental string in heterotic string on ¹ is nothing but M 5-brane wrapped around a 4-cycle of K3 surface [226,340,602]. Furthermore, K3-compacti"ed M2-brane through direct dimensional reduction is solitonic 5-brane in the heterotic theory wrapped around a 3-cycle of ¹. Thus, it leads to the conjecture that the strong coupling limit of heterotic string on ¹ is K3-compacti"ed supermembrane in D"11.

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We point out that M-theory and string theories that are connected within the moduli space (of either ¹ for the N"2 string theories or S;S/Z for the N"1 string theories) are on  an equal putting if one includes `non-perturbativea branes, as well as perturbative string states, within the spectra of the 5 superstring theories. Namely, a brane that appears in one theory is necessarily related to branes of the other theories through the dimensional reduction and/or dualities [540,541]. In particular, all the branes in the 5 string theories should have interpretations in terms of M-branes through dimensional reductions and string dualities. It turns out that p-branes obtained in this way have the right property as p-branes of string theories [601] (e.g. the tension T of branes in string-frame depend on the string coupling g as &1, &1/g Q Q and &1/g for a fundamental string, Dp-branes and solitonic 5-brane, respectively) and Q the worldvolume actions (derived from those of M-theory [77,79,86,87]) have the right forms [88,241,530,604]. In the following, we discuss the M-theory origin of branes in string theories. First, we discuss branes in the type-IIA theory. Since the type-IIA theory is M-theory compacti"ed on S, all of p-branes in type-IIA theory (i.e. a fundamental string and a solitonic 5-brane in the NS-NS sector, and Dp-branes with p"0, 2, 4, 6, 8 in the R-R sector) should be obtained in this way. In D"11, there are M 2-brane [86,250] and M 5-brane [327] which are elementary and solitonic branes carrying electric and magnetic charges of the 3-form potential, respectively. Starting from M 2-brane [86,87], one obtains either fundamental string [231,232,250] in the NS-NS sector or the D 2-brane in the R-R sector, through double or direct dimensional reduction. The fundamental string and D 4-brane obtained, respectively, from the M 2- and M 5-branes via double dimensional reduction have the right dependence of the tensions on g , i.e. &1 and &1/g , Q Q respectively. Next, a D 0-brane can be thought of as the KK momentum mode of the D"11 theory on S [601]. This state with the momentum number n along the S-direction has mass (measured in the string-frame) given by M"n/g , which is the right dependence of the mass on the string coupling Q for BPS states carrying R-R charges [635]. The integer n is the electric charge of the KK ;(1) gauge "eld associated with the S-direction, indicating that such KK state is electrically charged under the 1-form potential in the R-R sector. The n"1 KK state is interpreted as a single D0-brane and the n'1 case corresponds to the (marginal) bound state of n D 0-branes. The D 6-brane is regarded as the KK monopole [601], which is magnetically charged under the 1-form potential in the R-R sector. For the D8-brane, currently there is no interpretation in terms of the D"11 theory available yet. Second, we discuss branes in type-IIB theory. In general, branes in type-IIB theory can be obtained from those in type-IIA theory by applying the T-duality between type-IIA and type-IIB theories [29,73,81,199,310]. For example, starting from D 2-brane of type-IIA theory, one obtains D 1-brane of type-IIB theory by T-dualizing along one of coordinates with the Neumann boundary condition (i.e. along a longitudinal direction of the D 2-brane). The fundamental string and the solitonic 5-brane in the NS-NS sector are obtained from the M 2- and the M 5-branes by dimensional reduction similarly as in the type-IIA case. On the other hand, one can directly relate branes in the compacti"ed type-IIB theory to those in compacti"ed M-theory by applying equivalence between type-IIB theory on S and M-theory on ¹. In this relation, one identi"es complex modulus q of ¹ with the complex scalar o of

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(uncompacti"ed) type-IIB theory, i.e. q"o [22,538]. (This is motivated by the observation that the non-perturbative S¸(2, Z) symmetry of type-IIB theory is interpreted as the ¹ moduli group of M-theory on ¹.) First, we discuss 1-branes in (uncompacti"ed) type-IIB theory. These carry (integer valued) electric charges q and q of 2-form potentials in the NS-NS and the R-R sectors [538],   respectively, and are bound states of q fundamental strings and q D-strings. This bound state   is absolutely stable against decay into individual strings i! the integers q and q are relatively   prime [638], due to the `tension gapa and charge conservation. In the string-frame with the vacuum expectation value 1o2"i(g )\, the tension of the (q , q ) string [538] is ¹   " Q   O O  (q#(g )\q¹ , where ¹  is the tension of the fundamental string. This tension formula has   Q   the right limiting behavior: ¹ &1 for a fundamental string and ¹ &(g )\ for a D-string   Q [489]. When these 1-branes are wrapped around S of radius R , one has 0-branes in D"9 with the momentum mode m and the winding mode n around S. This 0-brane of the D"9 type-IIB theory is identi"ed with the M 2-brane wrapped around ¹. Namely, the momentum mode m [winding mode n] of the type-IIB string is interpreted in the M 2-brane language as the wrapping [the KK modes] of the M 2-brane on ¹. Through these identi"cations, one has relations between the tension of the fundamental string of (uncompacti"ed) type-IIB theory and the tension ¹+ of  M 2-brane: (¹ ¸)\"(1/(2p))¹+A, which is consistent with string dualities. Here,   + ¸ "2pR is the circumference of S, upon which type-IIB theory is compacti"ed, and A is the + area of ¹ measured in the D"11 metric. Direct dimensional reduction of 1-branes in type-IIB theory gives rise to strings with charges (q , q ) and the tension ¹   in D"9. This type-IIB string in D"9 is identi"ed with M 2-brane   O O  wrapped around a (q , q ) homology cycle of ¹ with the minimal length ¸   "2pR "q !q q".      O O  Such string of the compacti"ed M-theory has the tension (measured by the D"11 metric) ¹ "¸   ¹+, which is consistent with relation between ¹  and ¹+ in the previous OO O ?     paragraph. Second, we discuss the D"9 p-branes related to D 3-brane [238] of (uncompacti"ed) type-IIB theory. By wrapping D 3-brane around S, one obtains 2-brane with tension ¸ ¹  in D"9. Here,  ¹  is the tension of D 3-brane in D"10. This 2-brane of type-IIB theory is identi"ed with 2-brane  in the ¹-compacti"ed M-theory obtained by direct dimensional reduction of M 2-brane. Such identi"cation of the two 2-branes of type-IIB and M-theory leads to relation between the tensions ¹  and ¹  of fundamental string and D 3-brane:   1 ¹ " (¹ ) .  2p  When D 3-brane of type-IIB theory is compacti"ed via direct dimensional reduction on S, one has 3-brane in D"9. This 3-brane is identi"ed with M 5-brane wrapped around ¹. Such an

 These electric charges (q , q ) transform linearly under SL(2, Z), while the complex scalar o transforms fractional   linearly, i.e. oP(ao#b)/(co#d)(ad!bc"1).

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identi"cation leads to the correct DSZ quantization relation 1 ¹+" (¹+)  2p  on tensions of M 2- and M 5-branes. Third, 5-branes in type-IIB theory carry magnetic charges (p , p ) of 2-form potentials in the R-R   and the NS-NS sectors, with the tension ¹    given similarly as that of 1-branes. 4-branes with N  N  the tension ¸ ¹    are obtained by wrapping this 5-branes around S. The corresponding N N  4-branes in the M-theory side is M 5-brane wrapped around a (p , p ) cycle of ¹. This identi"ca  tion leads to the correct expression for the tension of the type-IIB 5-brane (in the string-frame) given by ¹    "((g )\/(2p))"p !p 1o2"(¹ ). In the limit where the 5-brane carries only Q    N N  either the R-R or the NS-NS magnetic charge, the tension behaves as ¹  &(g )\ and  Q ¹  &(g )\, as expected for solitons and D-branes.  Q 5-branes in D"9 have di!erent interpretations. First, a singlet 5-brane of the type-IIB theory on S is magnetically charged under the KK ;(1) gauge "eld associated with the S-direction. Second, the S¸(2, Z) family of 5-branes of the type-IIB theory on S is charged under the doublet of 2-form ;(1) gauge "elds. The corresponding singlet 5-brane in the ¹-compacti"ed M-theory is magnetically charged under the 3-form ;(1) gauge "eld. The S¸(2, Z) multiplet of 5-branes that are matched with those of the type-IIB theory on S is magnetically charged under the doublet of KK ;(1) "elds of M-theory on ¹. As for the D"9 branes associated with D 7-brane (magnetically charged under the 0-form potential), the M-theory interpretation is not well-understood yet. In [540], it is argued that p-branes with p"7, 8, 9 in M-theory that would give rise to 7-brane in D"9 do not exist, and 6-brane in D"9 cannot be obtained from D 7-brane of type-IIB theory by the periodic array along the compact direction and also is not consistent with the D"9 type-IIB theory. Finally, we comment on branes in the SO(32) theories, i.e. the type-I and the SO(32) heterotic strings. These 2 theories are related by S-duality, which inverts the string couplings (i.e. g"1/g') Q Q and exchanges the 2-form potentials of the 2 theories (the 2-form potential is in the NS-NS sector [the R-R sector] for the SO(32) heterotic theory [the type-I theory]). The electric [magnetic] charge of the 2-form potential is carried by 1-branes [5-branes]. When this 1-brane [5-brane] is compacti"ed on S, one has either 0-brane or 1-brane [4-brane or 5-brane] in D"9 depending on whether or not these branes are wrapped around S. The M-theory origin of these D"9 branes is understood from the observation that M-theory on S/Z ;S with length ¸ and radius R is  related to SO(32) theory on S. Note, while M 2-brane can wrap on S/Z , M 5-brane can wrap  around S, only. So, M 5-brane compacti"ed on S/Z ;S give rise to either 4-brane or 5-brane in  D"9, depending on whether M 5-brane is wrapped around S or not. These branes are identi"ed with those of the SO(32) theory. Similarly, 0-brane and 1-brane of the SO(32) theories on S are identi"ed with the M 2-brane which is "rst wrapped around S/Z and then either wrapped around  S or not. Note, under the exchange of parameters ¸ and R of S/Z ;S, the SO(32) heterotic theory  and the type-I theory is exchanged, while the string couplings are inverted (i.e. g"1/g'"¸/R). Q Q Thus, the roles of R and ¸ are interchanged when one identi"es p-branes of the S-dual theory on S with those of the M-theory on S/Z ;S. These identi"cations yield the tensions for 1-brane  and 5-brane of the SO(32) theories with consistent limiting behavior, i.e. ¹&(g)\ and  Q ¹&1 for the heterotic theory, and ¹' &(g')\ and ¹' &(g')\ for the type-I theory.   Q  Q

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4. Black holes in heterotic string on tori 4.1. Solution generating procedure The primary goal of this section is to generate the general black hole solutions in the e!ective theories of the heterotic string on tori by applying the solution generating transformations described in Section 3.2. In principle, D(10 black hole solutions which have the most general electric/magnetic charge con"gurations and are compatible with the conjectured no-hair theorem [125,343,344,385,386] can be constructed by imposing the SO(1, 1) boosts on charge neutral solutions, i.e. Schwarzschield or Kerr solution. Here, the SO(1, 1) boosts that generate electric charges of ;\"(1) gauge group in D(10 are contained in the O(11!D, 27!D) symmetry group (134) of the (D!1)-dimensional Lagrangian. Meanwhile, magnetic charges of the ;\"(1) gauge group and the D"5 NS-NS 2-form potential are generated by the SO(1, 1) boosts in the O(8, 24) symmetry group (138) of the D"3 action. However, it is not necessary to generate all the electric/magnetic charges by applying the SO(1,1) boosts, as we explain in the following. The D-dimensional dualities, which leave the (Einsteinframe) metric intact, can be used to remove some of charge degrees of freedom (associated with the ;\"(1) gauge group in D(10) of black holes. The general black hole solution with all the redundant charge degrees of freedom removed by the D-dimensional dualities is called the `generating solutiona, since the most general solution in the class is obtained by applying the D-dimensional dualities. Thus, one only needs to generate electric [and magnetic] charges of the generating solutions by applying the SO(1, 1) boosts in O(11!D, 27!D) [O(8, 24)] duality group on the Schwarzschield or the Kerr solution [562]. The generating solution is equivalent to the solution with the most general charge con"guration due to the conjectured string dualities. This is a reminiscence of automorphism transformations of N-extended superalgebra discussed in Section 2.2, which brings the algebra in a simple form in which only [N/2] eigenvalues of central charge matrix appear in the algebra rather than whole N(N!1) central charges. The charge assignments for the generating solution for each dimensions are: E dyonic black holes in D"4 [181]: 5 charge degrees of freedom associated with gauge "elds in the ¹ part. E black holes in D"5 [182]: a magnetic charge of the NS-NS 3-form "eld strength (or an electric charge of its Hodge-dual), and 2 electric charges of KK and 2-form ;(1) gauge "elds associated with the same internal coordinate. E black holes in D56 [184]: 2 electric charges of KK and 2-form ;(1) gauge "elds associated with the same internal coordinate. For the purpose of constructing solutions, it is convenient to choose scalar asymptotic values in the `canonical formsa [562]: M "I and u "0 [and W "0 for the D"4 case]. This  \"\"   is not an arbitrary choice since one can bring arbitrary scalar asymptotic values to the canonical forms by applying the following O(10!D, 26!D, R) transformation: M PM K "XM X2"I , X3O(10!D, 26!D, R) ,    \"\"

(185)

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and the S-duality, for example for the D"4 case, given by the S¸(2, R) transformation S PSK "(aS #b)/d"i, ad"1 . (186)    The D"3 O(8, 24, R) transformation that brings an asymptotic value of modulus M (137) to the form M "I is equivalent [561] to the D"4 S¸(2, R) transformation plus the D"3 O(7, 23, R)   transformation. Furthermore, one brings asymptotic values of ;(1) gauge "elds AG to zero I by applying global ;(1) gauge transformations. Then, the subset of O(11!D, 27!D) [O(8, 24)] that preserves the canonical asymptotic value M "I [M "I ] is SO(26!D, 1);SO(10!D, 1) [SO(22, 2);SO(6, 2)] [562].  \"\"   There are 36!2D[2;28]SO(1, 1) boosts in SO(26!D, 1);SO(10!D, 1) [SO(22, 2);SO(6, 2)]. When applied to a charge neutral solution, these boosts in SO(26!D, 1);SO(10!D, 1) [SO(22, 2) ;SO(6, 2)] induce electric charges of the ;(1)\" gauge group in D(10 [electric and magnetic charges of the ;(1) gauge group in D"4]. The starting point of constructing the generating solution is the D-dimensional Kerr solution, parameterized by the ADM mass and ["\] angular momenta. The solution in the `Boyer}  Lindquista coordinate has the form [478]: (D!2N) D ds"! dt# "\ dr#(r#l cos h#K sin h) dh   D “  (r#l)!2N G NG #(r#l cos t #K sin t ) cos h cos t 2 cos t dt G> G G> G  G\ G !2(l !K )cos h sin h cos t 2 cos t cos t sin t dh dt G> G>  G\ G G G !2 (l!K )cos h cos t 2 cos t H H  G\ GH ;cos t sin t 2 cos t cos t sin t dt dt G G H\ H H G H 4l l kkN k 4l kN # G [(r#l)D#2lkN] d ! G G dt d # G H G H d d , G G G G G G H D D D GH

(187)

where for E Even dimensions: "\ "\ D,a “ (r#l)#r k(r#l)2(r#l ) G G  G\ G G ;(r#l )2(r#l ), G> "\ K ,l sin t #2#l cos t 2 cos t sin t , G G> G "\ G "\ "\ N"mr ,

(188)

 While the SO(1, 1) boosts in SO(22, 1);SO(6, 1)LSO(22, 2);SO(6, 2) induce electric charges of the D"4 ;(1) gauge group, the remaining SO(1, 1) boosts in SO(22, 2);SO(6, 2)!SO(26!D, 1);SO(10!D, 1) induce magnetic charges of the ;(1) gauge group.

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and k ,sin h, k ,cos h sin t ,2 ,    k ,cos h cos t 2 cos t sin t , "\  "\ "\ a,cos h cos t 2 cos t ,  "\

(189)

E Odd dimensions: "\ D,r k(r#l)2(r#l )(r#l )2(r#l ), G  G\ G> "\ G K ,l sin t #2#l cos t 2cos t sin t G G> G "\ G "\ "\ #l cos t 2cos t , "\ G "\ N"mr ,

(190)

and k ,sin h, k ,cos h sin t ,2 ,    k ,cos h cos t 2 cos t sin t , (191) "\  "\ "\ k ,cos h cos t 2 cos t . "\  "\ Here, the repeated indices are summed over: i, j in t [ ] run from 1 to ["\] [from 1 to ["\]].   The ADM mass and the angular momenta J are G X 2 (D!2)X "\m, J " "\ml " Ml , (192) M" G 4pG G D!2 G 8pG " " where G is the D-dimensional Newton's constant and " 2p"\ X " "\ C("\)  is the area of S"\. When compacti"ed to D!1 dimensions [3 dimensions (for the 4-dimensional Kerr solution)], the transformation that generates inequivalent solutions from the Kerr solution is (SO(26!D, 1);SO(10!D, 1))/(SO(26!D);SO(10!D)) [(SO(22, 2);SO(6, 22))/(SO(22);SO(6) ;SO(2))], which has (9!D)#(25!D) [2;28#1] parameters; these parameters are interpreted as (9!D)#(25!D) electric charge degrees of freedom [2;28 electric and magnetic charge degrees of freedom plus unphysical Taub-NUT charge] introduced to the Kerr solution [562]. For the D"5 case, an additional charge associated with the NS-NS 2-form "eld is generated by an SO(1, 1) boost in O(8, 24) [182]. After the generating solutions are constructed from the SO(1, 1) boosts, the remaining charge degrees of freedom are induced (without changing the Einstein-frame spacetime) from subsets of D-dimensional (continuous) duality transformations that generate new charge con"gurations from the generating ones while keeping the canonical scalar asymptotic values intact. This is

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[SO(10!D);SO(26!D)]/[SO(9!D);SO(25!D)], which introduces (9!D)#(25!D) new charge degrees of freedom, and, for the D"4 case, SO(1)LS¸(2, R), which introduces one more charge degree of freedom. Subsequently, to obtain solutions with arbitrary scalar asymptotic values, one has to undo the transformations (185) and (186). In the following sections, we discuss the generating solutions in each dimensions. Note, due to the conjectured string}string duality between heterotic string on ¹ and type-II string on K3, these also correspond to the generating solutions of general black holes in type-II strings on K3;¹L for n"6!D"0, 1, 2. For this case, some of charges of the generating solutions can be dualized to R-R charges, rendering interpretation in terms of D-branes. It turns out that by applying ;-dualities of type-II string on tori to such generating solutions, one can generate the general class of solutions of the e!ective type-II string on tori as well [171] (see Section 6.3.2 for discussions). 4.2. Static, spherically symmetric solutions in four dimensions 4.2.1. Supersymmetric solutions In this section, we derive a general BPS spherically symmetric solution with a diagonal moduli [178]. Such a solution, after subsets of O(6, 22) and S¸(2, R) transformations are applied, satis"es one ;(1) charge constraint, missing one parameter to describe the most general BPS solutions. The solution generalizes the previously known black hole solutions in heterotic string on tori as special cases, and are shown to be exact to all orders in expansions of a [173]. At particular points in moduli space, such a solution becomes massless, enhancing not only gauge symmetry but also supersymmetry [179]. 4.2.1.1. Generating solutions. A general BPS non-rotating black hole solution with a diagonal moduli matrix is obtained by solving the Killing spinor equations. With spherically symmetric AnsaK tze for "elds and a diagonal form of moduli M, the Killing spinor equations dt "0, dj"0 + and ds'"0 (cf. (119)) are satis"ed by restricted charge con"gurations (see [178] for details on allowed charge con"gurations), which we choose without loss of generality to be P, P, Q, Q.     The explicit BPS non-rotating solution with such a charge con"guration has the form [178] j"r/[(r!g P)(r!g P)(r!g Q)(r!g Q)] , .  .  /  /  R"[(r!g P)(r!g P)(r!g Q)(r!g Q)] , .  .  /  /  (r!g P)(r!g P)  .  .  , eP" (r!g Q)(r!g Q) /  / 






r!g P .  , g "  r!g P . 






r!g Q /  , g "  r!g Q / 

(193)

g "1 (mO1, 2) , KK

where j and R are components of the metric g dxI dxJ"!j dt#j\ dr# IJ R(dh#sin h d ), g "$1 and the radial coordinate is chosen so that the horizon is at r"0. ./ The requirement that the ADM mass saturates the Bogomol'nyi bound restricts choice of g ./ such that g sign(P#P)"!1 and g sign(Q#Q)"!1, thus yielding non-negative .   /  

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ADM mass of the form ""P#P"#"Q#Q" . (194) .1     Note, the relative signs for the pairs (Q, Q) and (P, P) are not restricted but in this section     we consider the case where both of pairs have the same relative signs so that the solution (193) has regular horizon. The Killing spinor e of the above BPS solution satis"es the following constraints: M

E PO0 and/or PO0: CK CK ?e"ig e,   . E QO0 and/or QO0: CK CK ?e"g e.   / From these constraints, one sees that purely electric (or magnetic) solutions preserve 1/2, while dyonic solutions preserve 1/4 of N"4 supersymmetry. The former and the latter con"gurations fall into vector- and hyper-supermultiplets, respectively. Since the Killing spinor equations are invariant under the O(6, 22) and S¸(2, R) transformations, one can generate new BPS solutions by applying the O(6, 22) and S¸(2, R) transformations to a known BPS solution. The [SO(6)/SO(4)];[SO(22)/SO(20)] transformation with 6 ) 5!4 ) 3 22 ) 21!20 ) 19 # "50 2 2 parameters to the solution (193) leads to a general solution with zero axion and 4#50" 54"56!2 charges. 28 electric Q and 28 magnetic P charges of such a solution satisfy the two constraints P2M Q"0 (M ,(¸M¸) $¸) . (195) ! !  The subsequent SO(2)LS¸(2, R) transformation introduces one more parameter (along with a non-trivial axion), which replaces the two constraints (195) with the following one S¸(2, R) and O(6, 22) invariant constraint on charges: P2M Q [Q2M Q!P2M P]!(#!)"0 . (196) \ > > Thus, general solution in this class has 4#50#1"55"2 ) 28!1 charge degrees of freedom. By applying the O(6, 22) and S¸(2, R) transformations to (194), one obtains the following ADM mass for general solutions preserving 1/4 of supersymmetry: M "e\P+P2M P#Q2M Q#2[(P2M P)(Q2M Q)!(P2M Q)], . (197) .1 > > > > > This agrees with the expression (213) obtained [236] by the Nester's procedure. When magnetic P and electric Q charges are parallel in the SO(6, 22) sense, i.e. P2M Q"0, (197) becomes the > ADM mass of con"gurations preserving 1/2 of N"4 supersymmetry [34,299,337, 420}422,495,560,564]: M "e\((P2M P#Q2M Q) , > > .1 whose corresponding generating solution is purely electric or magnetic subset of (193).

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4.2.1.2. General supersymmetric solution with xve charges. The BPS solution (193) has a charge con"guration satisfying one constraint (196) when further acted on by the [SO(6)/SO(4)]; [SO(22)/SO(20)] and SO(2) transformations. So, to construct the generating solution for the most general BPS solution, which conforms to the conjectured classical `no-haira theorem, one has to introduce one more charge degree of freedom into (193). Such a generating solution was constructed in [174] by using the chiral null model approach, and has the following charge con"guration: (Q, P)"(q, P ), (Q, P)"(Q , 0) ,       (Q, P)"(!q, P ), (Q, P)"(Q , 0) .       Explicitly, the solution has the form r j" , [(r#Q )(r#Q )(r#P )(r#P )!q[r#(P #P )]]        (r#P )(r#P )   , eP" [(r#Q )(r#Q )(r#P )(r#P )!q[r#(P #P )]]        q(P !P )   W" , 2(r#P )(r#P )   r#P r#Q q[r#(P #P )] ,  . ,   G " G " G "!B "  r#P  r#Q   (r#Q )(r#P )     For this solution to have a regular horizon, the charges have to satisfy the constraints P '0, P '0, Q '0, Q '0 ,     Q Q !q'0, (Q Q !q)P P !q(P !P )'0 .          The ADM mass has the same form as that of the 4-parameter solution (193):

(199)

(200)

(201)

M

"Q #Q #P #P , (202) "+     independent of the additional parameter q. Meanwhile, the horizon area, i.e. A,4p(j\r) , is P modi"ed in the following way due to q: A"4p[(Q Q !q)P P !q(P !P )] . (203)        The following ADM mass and horizon area of BPS non-rotating black hole with general charge con"guration are obtained by applying the [SO(6)/SO(4)];[SO(22)/SO(20)] and SO(2)LS¸(2, R) transformations to (202) and (203): M "a 2k a #e\Pb2k b#e\P[(b2k b)(a2k a)!(b2k a)] , "+  >  > > > > A"peP[(b2¸b)(a2¸a)!(b2¸a)] ,

(204)

where the charge lattice vectors a and b live on the even self-dual Lorentzian lattice K with  signature (6, 22), a ,a#W b and k ,M $¸. Here, a and b are related to the physical ;(1)  ! 

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charges Q and P as: (2P "¸ b , (205) (2Q "ePM (a #W b ), G GH H  H G GH H where we assumed that a"2, G "aeP"eP.  ,  The ADM mass and horizon area in (204) can be put in the S¸(2, Z) S-duality, as well as the O(6,22) T-duality, invariant forms by expressing them with new charge lattice vector *2"(*2, *2),(a2, b2) and by introducing the following S¸(2, R) invariant matrices:




M"eP

W

1

W W#e\P






, L"

0



1

!1 0

.

(206)

The "nal forms are M "(8G )\(M (*?2k *@)#[2L L (*?2k *@)(*A2k *B)]) , "+ , ?@ > ?A @B > > (207) 1  A . "p L L (*?2¸*@)(*A2¸*B) S" 2 ?A @B 4G , These are manifestly S¸(2, R) invariant, since M and * transform under SL(2, R) as [560]



MPuMu2,



*PLuL2*,

u3S¸(2, R) .

(208)

An important observation is that while for "xed values of a and b, mass changes under the variation of moduli and string coupling, entropy remains the same as one moves in the moduli and coupling space [258}260]. The fact that entropy is independent of coupling constants and moduli is consistent with the expectation that degeneracy of BPS states is a topological quantity which is independent of vacuum scalar expectation values and the fact that entropy measures the number of generate microscopic states, which should be independent of continuous parameters. 4.2.1.3. Bogomol'nyi bound. We derive the Bogomol'nyi bound on the ADM mass of asymptotically #at con"gurations within the e!ective theory of heterotic string on ¹ [236]. For this purpose, we introduce the Nester-like 2-form [483]: EK ,e cIJMd tI , (209) IJ C M where d tI is the supersymmetry transformation of physical gravitino in D"4. Given supersymC I metry transformations (119) of fermionic "elds expressed in terms of D"4 "elds [178], the Nester's 2-form reduces to the form: 1 EK IJ"e IJMd e# e\Pe (< ¸(F!icF I )IJ)?C?e#2 , M 0 2(2

(210)

where < is a vielbein de"ned in (122) and ¸ is an invariant metric of O(6, 22) given in (127). Derivation of the Bogomol'nyi bound consists of evaluating the surface integral of the Nester's 2-form (210), which is related through the Stokes theorem to the volume integral of its covariant derivative. The surface integral yields







1 1 dS eEK IJ"e P"+cI# e\P+< ¸(Q!icP),?C? e , IJ  I 0  4pG R 2(2G  . 

(211)

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69

where P"+ is the ADM 4-momentum [20] of the con"guration, and Q and P are physical electric I and magnetic charges of ;(1) gauge group. The integrand of the volume integral is a positive semide"nite operator, provided spinors e satisfy the (modi"ed) Witten's condition [631] n ) K e"0 (n is the 4-vector normal to a space-like hypersurface R). Thus, the bilinear form on the right-hand side of (211) is positive semide"nite, which requires that the ADM mass M has to be greater than or equal to the largest of the following 2 eigenvalues of central charge matrix: 1 e\P #P $2+Q P !(Q2 P ),] , "Z "" 0 0 0 0 0 0  (4G ) 

(212)

where Q ,(2(< ¸Q) and similarly for P . This yields the Bogomol'nyi bound: 0 0 0 1 M 5 e\P #P #2+Q P !(Q2 P ),] . "+ (4G ) 0 0 0 0 0 0 

(213)

This bound is saturated i! supersymmetric variations of fermionic "elds are zero, i.e. BPS con"gurations. The Bogomol'nyi bound (213) can be expressed explicitly in terms of electric Q and magnetic P charges, and asymptotic values M and u of scalars, by using the identity:   ¸<2 < ¸"[¸(M#¸)¸]. For example, Q "Q2¸(M #¸)¸Q. 0  0 0  4.2.2. Singular black holes and enhancement of symmetry In perturbative heterotic string theories, gauge symmetry is enhanced to non-Abelian ones through the Halpern}Frenkel}Kac\ (HFK) mechanism [127,253,273,322}324,480]. The HFK mechanism is due to extra spin one string states which are normally massive at generic points in moduli space but become massless at particular points. These points are the "xed points under discrete subgroups of T-duality of the worldsheet theory. As shown in [382], BPS states in N"4 theories become massless at particular points in the moduli space. Since BPS multiplets in N"4 theories generically contain massive spin one states, gauge symmetry is enhanced to non-Abelian ones when the BPS states become massless. Note, the BPS states carry magnetic, as well as electric, charges and, therefore, are non-perturbative in character. When BPS multiplets with highest spin 3/2 state become massless, supersymmetry as well as gauge symmetry is enhanced, a phenomenon that is never observed within perturbative string theories. In this section, we illustrate the enhancement of symmetries in the BPS states of N"4 theories by studying massless black holes in heterotic string on ¹ [119,141,179]. 4.2.2.1. Massless black holes and symmetry enhancement. First, we consider the subset of BPS states with diagonal M and purely imaginary S [179]. We rewrite the corresponding generating   solution (193) with explicit dependence on scalar asymptotic values j"r/[(r!g P )(r!g P )(r!g Q )(r!g Q )] , .  .  /  /  R"[(r!g P )(r!g P )(r!g Q )(r!g Q )] , .  .  /  / 

(214)

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where the solution now depends on the following `screeneda charges: (P , P , Q , Q ),e\P(g P, g\P, g\Q, g Q)             (215) "(e\Pg b, e\Pg\b, ePg a, ePg\a) .         Here, the quantized charge lattice vectors a and b live on the even, self-dual, Lorentzian lattice K [557]. When g are chosen to satisfy g sign(P#P)"!1 and g sign(Q#Q)"!1, the ./ .   /   ADM mass takes the following form that saturates the BPS bound: "e\P"g b#g\b"#eP"g a#g\a" . (216) .1         Only electric states (b"0) with a2¸a52 are matched onto perturbative string states [248]. As for dyonic states that break 1/4 of supersymmetry (i.e. those with non-parallel electric and magnetic charge vectors, i.e. Q2M PO0, and therefore cannot be related to electric solution via the S¸(2, Z) > transformations), the consistency with the S¸(2, Z) symmetry and consistent electric limit require that the electric and the magnetic lattice vectors separately satisfy the constraints a2¸a5!2 and b2¸b5!2. These subsets of BPS states (214) become massless [48,179,406] at the "xed points under T-duality RP1/R (i.e. at the ¹ self-dual point g "1 [g "1"g ]),    when a"!a"$1 [a"!a"$1 and b"!b"$1] for the case       b"0 [aO0Ob]. There are also additional in"nite number of S¸(2, Z) related massless BPS states. The extra massless spin 1 states associated with a"!a"$1 [a"!a"$1 and     b"!b"$1] at the self-dual points of ¹ contribute to enhancement of Abelian gauge   symmetry to S;(2)[S;(2);S;(2)] non-Abelian symmetry. The extra massless spin 1 states together with generic massless ;(1) gauge "elds form the adjoint representations of the enhanced non-Abelian gauge groups. The BPS multiplet which preserves 1/4 of N"4 supersymmetry contains spin 3/2 state. Thus, additional 4 massless gravitino associated with a"!a"$1 and b"!b"$1     contribute to enhancement of supersymmetry from N"4 to N"8. Note, the in"nite number of S¸(2, Z) related massless states and enhancement of supersymmetry are not realized within perturbative string theories. These are new non-perturbative phase of string theories that are required by non-perturbative string dualities. M

4.2.2.2. Maximal gauge symmetry enhancement in the moduli space of two-torus. In this subsection, we study maximal symmetry enhancement in full moduli space of ¹ parameterized by arbitrary scalar asymptotic values [119,141]. We consider the general BPS mass formula (197). The moduli space of ¹ is parameterized by the following real matrix:




M"

G\

!G\B



!B2G\ G#B2G\B

,

(217)

where G,[G ] and B,[B ] with (m, n)"1, 2 being indices associated with ¹. Thus, the KL KL e!ective theory has the O(2, 2, R) T-duality symmetry. Since O(2, 2, R) S¸(2, R);S¸(2, R) [215,570], M is reparameterized [215] by the following 2 complex scalars, which separately

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parameterize each S¸(2, R) moduli space: (G G q"q #iq , !i , o"o #io ,(G#iB , (218)   G    G   where G,G G !G . Here, q [o] is the complex [KaK hler] structure of ¹. Each S¸(2, Z)    factor [215,305,570] is generated by two transformations, which are given, for the S¸(2, Z) factor associated with q, by 1 S : qP , O q

¹ : qPq#i , O

(219)

where o remains intact, and similarly for the S¸(2, Z) factor associated with o. In addition, the p-model corresponding to ¹ is symmetric under the `mirror symmetrya (S : qo), the world  sheet parity symmetry (q, o)P(q,!o) corresponding to pP!p, and the symmetry (q, o)P (!q,!o) associated with the re#ection XP!X. In accordance with the above reparameterization of moduli space, one can express the central charges (212) of the N"4 theory in the following S¸(2, R) ;S¸(2, R) ;ZO@M invariant form [119] O M  (the subscript R is omitted in ADM mass and central charges) 1 "M " , "Z ""  4(S#SM )(q#q)(o#o) 

(220) M ,(a( #iSbK )PK H, M ,(a( !iSM bK )PK H ,  H H  H H where a( ,(a, a, a,!a)2, bK ,(b, b, b,!b)2, and PK ,(1,!qo, iq, io)2. Here, the         charge lattice vectors a( and bK transform under S¸(2, R) ;S¸(2, R) ;ZO@M as O M  a( a( a( bK bK  Pu  , similary for  ,  ,  , S¸(2, R) : O O a( a( a( bK bK      a( a( a( bK bK  Pu  , similary for  ,  ,  , (221) S¸(2, R) : M M a( a( a( bK bK      ZO@M: a(  a( , bK  bK .      In addition, the central charges (220) are invariant under the S¸(2, R) S-duality: 1 a( a c a( P . (222) S¸(2, R) : 1 bK b d bK




















Note, the ADM mass of BPS states is given by the largest of "Z " and "Z ".   First, we consider the short multiplet, i.e. the BPS multiplet with 2 central charges Z equal in  magnitude. It has the highest spin 1 state and preserves 1/2 of the N"4 supersymmetry. The charge lattice vectors a and b of the short multiplet live on the S-orbit satisfying pa( "sbK (s, p3Z). G G Explicitly, we write a( and bK in the S-orbit as a( "sm ,   bK "pm ,  

a( "sn ,   bK "pn ,  

a( "sn ,   bK "pn ,  

a( "!sm ,   bK "!pm ,  

(223)

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where m [n ] corresponds to momentum [winding] number of perturbative string states when G G p"0 and s [p] denotes electric [magnetic] quantum number associated with the S-modulus. Then, the central charge (220) becomes M"(s#ipS)(m !im q#in o!n qo) . (224)     In the complex moduli space parameterized by (q, o), M can vanish at "xed lines .1 [121,216,384] under the Weyl re#ections w ,S , w ,S S S , w ,¹ S ¹\ and w , 

  M  M  M  M  w\= w in the T-duality group. Along these lines, the Abelian gauge group ;(1) ;;(1) ;    ? @ ;(1) ;;(1) ,;(1);;(1);;(1);;(1) is enhanced to the non-Abelian ;(1);S;(2) A B     group. The BPS states (labeled by a) which become massless along the "xed lines L under w are as G G follows [121,122]: E L "+q"o,: a"k "$(1, 0,!1, 0) contributing to ;(1) ;;(1) ;;(1) ;S;(2)  ! @ B ?>A ?\A E L "+q"o\,: a"k "$(0, 1, 0,!1) contributing to ;(1) ;;(1) ;;(1) ;S;(2)  ! ? A @>B @\B E L "+q"o!i,: a"k "$(1, 1,!1, 0) contributing to ;(1) ;;(1) ;;(1) ;  ! B ?>A ?\@\A S;(2) ?>@\A E L "+q"o/(io#1),: a"k "$(1, 0,!1, 1) contributing to ;(1) ;;(1) ;;(1)  ! @ ?>A ?\A\B ;S;(2) . ?\A>B At points where the lines L intersect [120,122], there are additional massless states, resulting in G the maximal enhancements of gauge symmetries: E L 5L "+q"o"1,: a"k or k contributing to ;(1) ;;(1) ;S;(2) ;S;(2)   ! ! ?>A @>B ?\A @\B E L 5L 5L "+q"o"e p,: a"k or k or k contributing to ;(1) ;;(1) ;    ! ! ! ?>A ?\@\A\B S;(3) . @\B?>@\A>B Along with the above perturbative massless states, there are accompanying in"nite massless dyonic states, so-called S-orbit pa( "sbK , related via S¸(2, Z) S duality. G G 1 Second, we consider the intermediate multiplets, i.e. the BPS multiplets with "Z "O"Z ". They   have the highest spin 3/2 states and preserve 1/4 of the N"4 supersymmetry. In this case, a( and bK are not proportional, i.e. a( bK !a( bK O0. The requirement that the ADM mass is zero, i.e. G H H G "Z ""0 and *Z""Z "!"Z ""0, leads to the relations [119]:    a!aqo#iao!ia"0, b!bqo#ibo!ib"0 . (225)         These relations are satis"ed by the following "xed points [119,141]: E q"o"i: (a, b)"(k , k ), ! ! E q"o"e p: (a, b)"(k ,k ), 24i(j44. ! H! In addition to the above massless dyonic states, there are in"nite number of S¸(2, Z) related 1 dyonic states. Since these additional massless states belong to the highest spin  supermultiplet,  supersymmetry as well as gauge symmetry are enhanced [179]. 4.2.2.3. Properties of massless black holes. When both of the pairs (Q, Q) and (P, P) have     the same relative signs [178], the singularity of the solution (214) is always behind or located at the event horizon at r"0, corresponding to the Reissner-NordstroK m-type horizon or null singularity,

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respectively. However, the moment at least one of the pairs has the opposite relative signs [48,179,406], there is a singularity outside of the event horizon, i.e. naked singularity r '0.   Explicitly, the curvature singularity is at r"r ,max+min["P ", "P "], min["Q ", "Q "],'0.       These singular solutions have an unusual property of repelling massive test particles [406]. (Note, the BPS black holes need not be massless to be able to repel massive test particles [179].) There is a stable gravitational equilibrium point for a test particle at r"r where the graviA tational force is attractive for r'r and repulsive for r(r [406]. This can also be seen by A A calculating the traversal time of the geodesic motion for a test particle with energy E, mass m and zero angular momentum along the radial coordinate r, as measured by an asymptotic observer [179]:



t(r)"

P

E dr

P j(r)(E!mj(r)

.

(226)

The minimum radius that can be reached by a test particle corresponds to r 'r for which

   j(r"r )"E/m, since it takes in"nite amount of time to go beyond r"r . Here, r"r is



   the singularity. Massive test particles cannot reach the singularity of singular black holes in "nite time and are re#ected back. On the other hand, classical massless particles with zero angular momentum do not feel the repulsive gravitational potential due to increasing j(r), and they reach the singularity in a "nite time. Note, for regular solutions, studied in Section 4.2.1, j41, while for singular solutions studied in this section, j51 for r small enough. Thus, for regular solutions, particles are always attracted toward the singularity. When only one charge is non-zero, the regular solution has a naked singularity at r"0; t(r"r "0) is "nite.   4.2.3. Non-extreme solutions The following non-extreme generalization [180] of the BPS solution (193) is obtained by solving the Einstein and Euler}Lagrange equations: j"r(r#2b)/[(r#P)(r#P)(r#Q)(r#Q)] ,     R(r)"[(r#P)(r#P)(r#Q)(r#Q)] ,    



eP"



(r#P)(r#P)    , (r#Q)(r#Q)  

r#P  , g "  r#P 

W"0 ,

r#Q  , g "  r#Q 

g "B "0 (mOn), KL KL

(227)

g "1 (mO1, 2) , KK

a' "0 , K

where b'0 measures deviation from the corresponding BPS solution and P,  b$((P)#b, etc. The ADM mass is  M"P#P#Q#Q!4b .    

(228)

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The signs $ in the expressions for P, etc. should be chosen so that MPM as bP0. To  .1 have a regular horizon, one has to choose the relative signs of both pairs (Q, Q) and (P, P) to     be the same [178]. In this case, the non-extreme solution is (227) with positive signs in the expressions for P, etc. and have the ADM mass  M"((P)#b#((P)#b#((Q)#b#((Q)#b ,     which is always compatible with the Bogomol'nyi bound:

(229)

""P"#"P"#"Q"#"Q" . .1     Such solutions always have nonzero mass.

(230)

M

4.2.3.1. Space}time structure and thermal properties. We now study spacetime properties [178,180] of the regular D"4 4-charged black hole discussed in the previous subsections. There is a spacetime singularity, i.e. the Ricci scalar R blows up, at the point r"r where   R"0. The event horizon, de"ned as a location where the r"constant surface is null, is at r"r & where gPP"j"0. The horizon(s) forms, provided r 'r (time-like singularity). In some cases, &   singularity and the event horizon coincide: r "r . In this case, the singularity is (i) naked   & (space-like singularity) when the singularity is reachable by an outside observer (at r"r 'r ) in  & a "nite a$ne time q"P  dr(EPP"P& dr j\(r) and (ii) also an event horizon (null singularity) ERR P P when q"R. Thermal properties of the solution (193) are speci"ed by spacetime at the event horizon. The Hawking temperature [345,346] is de"ned by the surface gravity i at the event horizon: ¹ "i/(2p)""R j(r"r )"/(4p) . & P & Entropy S is given by the Bekenstein's formula [38,64,65,343,347]:

(231)

S";(the surface area of the event horizon)"pR(r"r ) . (232)  & We classify thermal and spacetime properties according to the number of non-zero charges: E All the 4 charges non-zero: There are 2 horizons at r"0,!2b and a time-like singularity is hidden behind the inner horizon, i.e. the global spacetime is that of the non-extreme Reissner} NordstroK m black hole. The Hawking temperature is ¹ "b/(p(PPQQ) and the &     entropy is S"p(PPQ. When bP0, spacetime is that of extreme Reissner}    ,MPBQRPMK m black holes. E 3 non-zero charges: A space-like singularity is located at the inner horizon (r"!2b). For example when P"0, ¹ "b/(p(2PQQ) and S"p(2bPQQ. When bP0,     &    the singularity coincides with the horizon at r"0. E 2 non-zero charges: A space-like singularity is at r"!2b. For example when PO0OP, ¹ "1/(2p(PP) and S"p(4bPP. As bP0, the singularity co    &   incides with the horizon at r"0. E 1 non-zero charge: A space-like singularity is at r"!2b. For example when PO0, ¹ "1/(2p(2bP) and S"p(8bP. As bP0, the singularity becomes naked.   & 

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4.2.4. General static spherically symmetric black holes in heterotic string on a six-torus In Section 4.2.1, we saw that a general solution obtained by applying subsets of O(6,22) and S¸(2,R) transformations on the 4-charged solution has 1 charge degree of freedom missing for describing non-rotating black holes with the most general charge con"guration. It is a purpose of this section to introduce such 1 missing charge degree of freedom by applying 2 SO(1,1) boosts (in the D"3 O(8,24) duality group) along a ¹ direction (associated with 4 non-zero charges) with a zero-Taub-NUT constraint to construct the `generating solutiona for non-extreme, nonrotating black holes in heterotic string on ¹ with the most general charge con"guration of ;(1) gauge group [181]. (See [390] for an another attempt.) So, the generating solution is parameterized by the non-extremality parameter (or the ADM mass) and 6 ;(1) charges with one zero-TaubNUT constraint. 4.2.4.1. Explicit form of the generating solution. For the purpose of constructing the generating solution in a simplest possible form, it is convenient to "rst generate the 4-charged non-extreme solution (227) with the following non-zero charges by applying 4 SO(1, 1) boosts to the Schwarzschield solution: P"2m sinh d cosh d ,P , P"2m sinh d cosh d ,P ,  N N   N N  Q"2m cosh d sinh d ,Q , Q"2m cosh d sinh d ,Q .  O O   O O 

(233)

Only non-extreme solutions compatible with the Bogomol'nyi bound and, therefore, within spectrum of states, are those with the same relative signs for both pairs (Q , Q ) and (P , P ). For     this case, PK ,2m cosh d !m"$((P )#m, etc. are given with plus signs.  N  As the next step, one introduces one missing charge degree of freedom by applying 2 SO(1, 1)LO(8, 24) boosts with parameters d  satisfying the zero Taub-NUT condition:  P tanh d !Q tan d "0 .    

(234)

Assuming, without loss of generality, that Q 5P , one has, from (234), d in terms of the other    parameters: cosh d "Q cosh d /D,   

sinh d "P sinh d /D ,   

(235)

where D,sign(Q )((Q ) cosh d !(P ) sinh d .     

 An additional SO(1, 1) boost along a ¹ direction on the 4-charged black hole solution necessarily induces Taub-NUT term, since the metric components g get mixed with the -component of the ;(1) gauge potential, which is IJ singular [562].  One can induce any 2 of the remaining charges in the ;(1);;(1);;(1);;(1) gauge group. But we here     choose to induce P and Q.    For the case Q 4P , the role of d and d are interchanged.    

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The "nal form of the generating solution [181] (with zero Taub-NUT charge) is = (r#m)(r!m) , R"(X>!Z), eP" , j" X>!Z (X>!Z) 1 R W" [D(P Q #P Q )#P Q [(P )(r#QK )!(Q )(r#PK )] P           D=



;

with



P P (r!QK ) sinh d #Q Q (r#PK ) cosh d         sinh d cosh d ,   X>!Z

X > G " , G " ,  (r#PK )(r#QK )  (r#PK )(r#QK )     Z G "! ,  (r#PK )(r#QK )   [(Q )(r#PK )!(P )(r#QK )] cosh d sinh d     , B "!   D(r#PK )(r#QK )   G "d , B "0 (i, jO1, 2), a' "0 GH GH GH K

(236)

X"r#[(PK #QK ) cosh d #(QK !PK ) sinh d ]r#(PK QK sinh d #QK PK cosh d ) ,             1 >"r# [(P )(PK !QK ) sinh d #(Q )(PK #QK ) cosh d ]r        D  1 # [(P )QK PK sinh d #(Q )QK PK cosh d ] ,      D    1 Z" [(P P #Q Q )r#(PK Q Q #QK P P )] cosh d sinh d ,           D  

(237)

1 ="r# [(Q )(PK #PK ) cosh d #(P )(QK !QK ) sinh d ]r        D  1 # [(Q )PK PK cosh d #(P )QK QK sinh d ] .      D    For the sake of simpli"cation, the coordinate is chosen so that the outer horizon is at r"m. This solution has the following non-zero charges: P"P Q /D, Q"(PK !PK !QK !QK ) cosh d sinh d ,           P"0, Q"(Q Q cosh d #P P sinh d )/D ,         P"(Q P cosh d #Q P sinh d )/D, Q"0 ,         P"P Q (Q !Q !P !P ) sinh d cosh d /D, Q"D ,          

(238)

 The BPS limit (m"0 and d PR) of this solution is related to the solution (200) via subsets of  SO(2);SO(2)LO(2, 2) (¹ ¹-duality) and SO(2)LS¸(2, R) transformations.

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77

and the ADM mass, compatible with the BPS bound [178,236], is M

1 " [(P )(PK !QK ) sinh d #(Q )(PK #QK ) cosh d ] "+ D         #(PK #QK ) cosh d #(QK !PK ) sinh d .      

(239)

4.2.4.2. S- and T-duality transformations. The additional 51 charge degrees of freedom needed for parameterizing the most general ;(1) charge con"guration are introduced by [O(6);O(22)]/ [O(4);O(20)] and SO(2) transformations. The resulting general solution has the charge con"guration ; (e !e ) ; (m !m ) 1  S B  S B , P"( U2 , (240) Q" U2 e #e m #m S B S B  (2 ; ;  0  0  





















 CDE e2,(Q cos c#P sin c, Q cos c#P sin c, 0, 2, 0) , S    

where

 CDE e2,(P sin c, Q cos c, 0, 2, 0) , B  

(241)

 CDE m2,(P cos c, Q sin c, 0, 2, 0) , S    CDE m2,(P cos c#Q sin c, P cos c#Q sin c, 0, 2, 0) , B    

c is the SO(2)LS¸(2, R) rotational angle, ; 3SO(6), ; 3SO(22), 0 is a (16;1)-matrix with    zero entries and U3O(6, 22, R) brings to the basis where the O(6, 22) invariant metric (127) is diagonal. And the complex scalar S and the moduli M transform to (W cos c!sin c)#ie\P cos c S" , (W sin c#cos c)#ie\P sin c

(242) ;2 0 UMU2  U, ; 0 ;2   where W, e\P and M are the axion, the dilaton and the moduli of the generating solution (236). The `Einstein-framea metric g in (236) remains unchanged, but the `string-framea metric g  is IJ IJ transformed to the most general form g "g /Im (S). IJ IJ




; M"U2  0

0








4.2.4.3. Special cases of the general solution. The generating solution (236), when supplemented by appropriate subsets of S- and T-dualities, reproduces all the previously known spherically symmetric black holes in heterotic string on ¹. Here, we give some examples.

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E Non-rotating black holes in Einstein}Maxwell-dilaton system with the gauge kinetic term e\?PF FIJ [245,281,298,354]: IJ  (1) P "P "Q "Q O0 case: the Reissner}NordstroK m black hole, i.e. a"0     (2) any of 3 charges non-zero and equal: the a"1/(3 case [236] (3) only 2 magnetic (or electric) charges non-zero and equal: the a"1 case [409] (4) only 1 charge non-zero: the a"(3 case, which contains in the extreme limit the followings (i) P O0 case: KK monopole [506,299,325,574], and (ii) P O0 case: H-monopole   [57,286,420}422]. E P "P and Q "Q solution with subsets of S- and T-dualities applied becomes general     axion}dilaton black holes found in [83,404,410,487,486]. E The solution with Q O0OQ , when supplemented by S- and T-dualities, is the general electric   black hole in heterotic string [562,564]. The S-dual counterpart is the general magnetic solution [58]. E The non-BPS extreme solution (i.e. mP0, P "Q "0, "Q "!"P "P0 and d PR, while      keeping meB and ("Q "!"P ")eB as "nite constants) is related by S- and T-dualities to the   non-BPS extreme KK black hole studied in [301]. 4.2.4.4. Global space-time structure and thermal properties. We classify all the possible spacetime and thermal properties of non-rotating black holes in heterotic string on ¹. These properties are determined by the 6 parameters P , Q , d and m of the generating solution (236), since the    D"4 T- and S-dualities, which introduce the remaining charge degrees of freedom, do not a!ect the `Einstein-framea spacetime. We separate the solutions into non-extreme (m'0) and extreme (m"0) ones. Within each class, we analyze their properties according to the range of the other 5 parameters P , Q and d .    4.2.4.4.1. Global space-time structure. There is a spacetime singularity at r"r where   R(r)"0. The event horizon(s) is located at r"r where j"0, provided r 5r .  !  !   (A) Non-extreme solutions (m'0). By analyzing roots of X>!Z, one sees that a singularity is always at r 4!m. Thus, global spacetime is either that of non-extreme Reissner}NordstroK m   black hole when r (!m (case with 2 horizons at r"$m) or that of Schwarzschield black   hole when r "!m.   X>!Z has a single root at r "!m, in which case a singularity and the inner horizon   coincide at r"!m, when (a) d O0 and P "0, or (b) d "0 and at least one of P and Q      is zero. X>!Z has a double root at r "!m, in which case the inner horizon disappears   and a singularity forms at r"!m, when (a) d O0 and only Q is non-zero, or (b) d "0 and at    least 2 of P , Q are zero.  

 In the following analysis, it is understood that Q O0 when d O0. When Q "0, P "0 due to initial assumption     "Q "5"P ". Then, d are not constrained by (234). In this case, we have a non-extreme 4-charged solution with charges    P, P, Q and Q. Such a solution is related to (236) through subsets of SO(2);SO(2)LO(2, 2)LO(6, 22) and     SO(2)LS¸(2, R) transformations.

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(B) Extreme solutions (mP0). When d is "nite, the ADM mass of the generating solution always  saturates the Bogomol'nyi bound as mP0, i.e. becomes BPS extreme solution. When both pairs (P , P ) and (Q , Q ) have the same relative signs, the singularity is always at     r 40. Global spacetime is, therefore, that of the extreme Reissner}NordstroK m black hole when   r (0 (time-like singularity), or the singularity and the horizon coincide (null singularity) when   r "r "0. The latter case happens when at least one out of P , Q (and Q ) is zero with    !    d O0 (with d "0). The horizon at r "0 disappears (naked-singularity) when (i) only Q is     non-zero with d O0, or (ii) only one out of P , Q is non-zero with d "0.     When at least one of the pairs (P , P ) and (Q , Q ) has the opposite relative sign, the singularity     is outside of the horizon, i.e. r '0 (singularity is naked) [48,179,400,406].   In the case of non-BPS extreme solutions [177,181], the singularity is always behind the event horizon (r (r "0), i.e. the global spacetime of the extreme Reissner}NordstroK m black hole    (time-like singularity). 4.2.4.4.2. Thermal properties. Thermal properties are speci"ed by spacetime at the (outer) horizon. So, we consider only regular solutions, which include non-extreme solutions compatible with the Bogomol'nyi bound and extreme solutions with the same relative sign for both pairs (P , P ) and (Q , Q ).     The entropy S is of the form p S" "[(QK #m)(PK #m) cosh d #(PK #m)(QK !m) sinh d ]       "D" ;[(Q )(QK #m)(PK #m) cosh d #(P )(QK #m)(PK !m) sinh d ]         ![P P (QK #m)#Q Q (PK #m)] cosh d sinh d " ,        

(243)

where PK "#(P#m, etc. Entropy increases with d , approaching in"nity ["nite value] as    d PR [non-BPS extreme limit is reached]. For BPS extreme solutions, entropy is (a) non-zero  and "nite, approaching in"nity as d PR, when P and Q are non-zero, and (b) always zero    when at least one of P , Q (and Q ) is zero with d O0 (with d "0).      The Hawking temperature ¹ ""R j(r"m)"/4p is & P m S\ . ¹ " & (2

(244)

As d increases, ¹ decreases, approaching zero. In the BPS extreme limit with at least 3 of  & P ,Q non-zero, ¹ is always zero. With 2 of them non-zero, ¹ is non-zero and "nite,   & & approaching zero as d PR. When only one of them (only Q ) is non-zero (for the case d O0),    ¹ becomes in"nite. In the non-BPS extreme limit, ¹ is zero. & & 4.2.4.5. Duality invariant entropy. We discuss the duality invariant form of entropy of nearextreme, non-rotating black hole in heterotic string on ¹ [170].

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The (t,t)-component of metric for general N"4 spherically symmetric solutions has the form g "p(r#m)(r!m)S\(r) with S(r) given by RR  S(r)"p “ ((r#j ) . G G

(245)

Entropy S of non-extreme solutions is given by S(r) at the outer horizon, i.e. S,S(m). Generally, j are functions of 28#28 electric and magnetic charges (240) and m (through m), G and their duality invariant forms are hard to obtain. However, for the near-extreme case, in which j are expressed to leading order in m around their BPS values j, one can obtain the T- and G G S-duality invariant entropy expression, which reads p   S"p “ (j  # m “ (1/j  j  j  j  #O(m) . G G H I G 2 G GHI G

(246)

Here, the T- and S-duality invariants are  “ j  ,S /p"F(¸, C)F(¸,!C) , .1  G G  j  ,M "(F(M , C)#(F(M ,!C) , G .1 > > G 1 j  j  " (Q2¸Q#P2¸P)#(F(M ,C)F(M ,!C) , > > G H 2g Q GH 1 j  j  j  " +M (Q2¸Q#P2¸P)!(Q2M Q!P2M P) .1 > > G H I 4gM Q .1 GHI ;(Q2¸Q!P2¸P)!4(Q2¸M ¸P)(Q2M P), ,  >

(247)

where 1 F(M ,$C)" (Q2M Q#P2M P$C(M )) > > > > 2g Q C(M )"(4(P2M Q)#(Q2M Q!P2M P) . > > > >

(248)

4.3. Rotating black holes in four dimensions We generalize the 4-charged non-extreme solution (193) to include an angular momentum [183]. (For an another attempt, see [389]. But this solution has only 3 charge degrees of freedom and is a special case of a general solution to be discussed in this section.)

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4.3.1. Explicit solution By applying the solution generating technique discussed in the beginning of this section, one obtains the following D"4, non-extreme, rotating black hole solution [183]: (r#2m sinh d )(r#2m sinh d )#l cos h N C , g "  (r#2m sinh d )(r#2m sinh d )#l cos h N C 2ml cos h( sinh d cosh d sinh d cosh d ! cosh d sinh d cosh d sinh d ) N N C C N N C C , g "  (r#2m sinh d )(r#2m sinh d )#l cos h N C (r#2m sinh d )(r#2m sinh d )#l cos h N C , g "  (r#2m sinh d )(r#2m sinh d )#l cos h N C 2ml cos h( sinh d cosh d cosh d sinh d ! cosh d sinh d sinh d cosh d ) N N C C N N C C , B "!  (r#2m sinh d )(r#2m sinhd )#l cos h N C (r#2m sinh d )(r#2m sinh d )#l cos h N N eP" , D



r!2mr#l cos h dr ds"D ! dt# #dh # D r!2mr#l #

sin h +(r#2m sinh d )(r#2m sinh d )(r#2m sinh d ) N N C D

(r#2m sinh d )#l(1# cos h)r#=#2mlr sin h,d  C 4ml ! +( cosh d cosh d cosh d cosh d ! sinh d sinh d sinh d sinh d )r N N C C N N C C D



#2m sinh d sinh d sinh d sinh d , sin h dt d , N N C C

(249)

where D,(r#2m sinh d )(r#2m sinh d )(r#2m sinh d )(r#2m sinh d ) N N C C #(2lr#=) cos h , =,2ml( sinh d # sinh d # sinh d # sinh d )r N N C C #4ml(2 cosh d cosh d cosh d cosh d sinh d sinh d sinh d sinh d N N C C N N C C !2 sinh d sinh d sinh d sinh d ! sinh d sinh d sinh d N N C C N C C !sinh d sinh d sinh d ! sinh d sinh d sinh d N C C N N C !sinh d sinh d sinh d )#l cos h . (250) N N C The axion W also varies with spatial coordinates, but since its expression turns out to be cumbersome, we shall not write here explicitly.

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The ADM mass, ;(1) charges, and angular momentum are M"2m(cosh 2d # cosh 2d #cosh 2d #cosh 2d ) , C C N N Q"2m sinh 2d , Q"2m sinh 2d ,  C  C (251) P"2m sinh 2d , P"2m sinh 2d ,  N  N J"8lm(cosh d cosh d cosh d cosh d !sinh d sinh d sinh d sinh d ) , C C N N C C N N where G"" and the convention of [478] is followed. ,  When Q"Q"P"P, all the scalars are constant, and thus the solution becomes the     Kerr}Newman solution. The d "d "0 case is the generating solution of a general electric N N rotating solution [562]. The case with Q"P and Q"P is constructed in [389].     The solution (249) has the inner r and the outer r horizons at \ > (252) r "m$(m!l , ! provided m5"l". In this case, the solution has the global spacetime of the Kerr}Newman black hole with the ring singularity at r"min+Q, Q, P, P, and h"p/2.     The extreme solution (r "r ) is obtained by taking the limit mP"l">. In this case, the global > \ spacetime is that of the extreme Kerr}Newman solution. The BPS limit is reached by taking mP0 and d PR while keeping meBCCNN as "nite CCNN constants so that the charges remain non-zero. When J is non-zero, i.e. lO0, the singularity is naked since the condition m5"l" for existence of regular horizon (252) is not satis"ed. To have a BPS solution with regular horizon, one has to take lP0, leading to a solution with J"0. Thus, the only regular BPS solution in D"4 is the non-rotating solution, with global spacetime of the extreme Reissner}NordstroK m black hole. This is in contrast with the D"5 3-charged solution [98,182], where one can take l to zero (so that the BPS solution has regular horizon) but the  angular momenta J can be non-zero. For D'5, the regular BPS limit with non-zero angular  momentum is achieved without taking l to zero if only one angular momentum is non-zero [366]. G 4.3.2. Entropy of general solution The thermal entropy of the solution (249) is [183]




  







    1 A"16p m “ cosh d # “ sinh d #m(m!l “ cosh d ! “ sinh d S" G G G G 4G , G G G G      "16p m “ cosh d # “ sinh d # m “ cosh d ! “ sinh d !J , (253) G G G G G G G G where d ,d and A" dh d (g g " > is the outer-horizon area.  CCNN FF (( PP Note, the thermal entropy has the form which is sum of `left-movinga and `right-movinga contributions. Each term is symmetric in d , i.e. in the 4 charges, manifesting U-duality symmetry G [381]. On the other hand, (253) is asymmetric in J: only the right-moving term has J, which reduces




 See [610] for the same result from the conformal p-model perspective.





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the right-moving contribution to the entropy. This re#ects right-moving worldsheet supersymmetry of the corresponding p-model. When J"0, the entropy becomes [180]:  S"32pm “ cosh d , G G

(254)

which again has U-duality symmetry under the exchange of 4 charges. In the regular BPS limit as well as the extreme limit, the `right-movinga term in (253) becomes zero, however entropy has di!erent form in each case. In the regular BPS limit (J"0) [178]:  S"32pm “ cosh d "2p(PPQQ , G     G

(255)

while in the extreme limit:






  S"16pm “ cosh d # “ sinh d "2p(J#PPQQ . G G     G G

(256)

Entropy of a black hole with general charge con"guration in the class and with arbitrary scalar asymptotic values is independent of scalar asymptotic values when expressed in terms of the charge lattice vectors a and b, and has the S- and T-duality invariant form [183]: S"2p(J#+(a2¸a)(b2¸b)!(a2¸b), .

(257)

4.4. General rotating xve-dimensional solution We construct the most general rotating black hole in heterotic string on ¹ [182]. In D"5, black holes carry only electric charges of ;(1) gauge "elds. Since the NS-NS 3-form "eld strength H is Hodge-dual to a 2-form "eld strength in D"5 in the following way: IJM eP HIJM"! eIJMHNF , HN 2!(!g

(258)

where F is the "eld strength of a new ;(1) gauge "eld A , black holes in D"5 carry an additional IJ I charge associated with the NS-NS 2-form "eld B as well as 26 electric charges of the ;(1) gauge IJ group. Thus, the most general black hole in heterotic string on ¹, compatible with the conjectured `no-hair theorema [125,343,344,385,386], is parameterized by 27 electric charges, 2 angular momenta and the non-extremality parameter. 4.4.1. Generating solution We choose to parameterize the `generating solutiona in terms of electric charges Q, Q and  Q associated with H , A  and A , respectively. These charges are induced through solution  IJM I I generating procedure described in Section 4.1.

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The "nal form of the generating solution is [182] r#2m sinh d #l cos h#l sin h C   g " ,  r#2m sinh d #l cos h#l sin h C   (r#2m sinh d #l cos h#l sin h) C   eP" , “ (r#2m sinh d #l cosh#l sinh) CG   G m cosh d sinh d C C A" , R r#2m sinh d #l cos h#l sin h C   l sinh d sinh d cosh d !l cosh d cosh d sinhd C C C  C C C, A "m sin h  ( r#2m sinh d #l cos h#l sin h C   l cosh d sinh d sinh d !l sinh d cosh d cosh d C C C  C C C, A "m cos h  (  r#2m sinh d #l cos h#l sin h C   m cosh d sinh d C C A" , R r#2m sinh d #l cos h#l sin h C   l cosh d sinh d coshd !l sinh d cosh d sinh d C C C  C C C, A "m sin h  ( r#2m sinh d #l cos h#l sin h C   l sinh d cosh d sinh d !l cosh d sinh d cosh d C C C  C C C, A "m cos h  ( r#2m sinh d #l cos h#l sin h C   l sinh d sinh d cosh d !l cosh d cosh d sinh d C C C  C C C, BK  "!2m sin h  R( r#l cos h#l sin h#2m sinh d   C l sinh d sinh d cosh d !l cosh d cosh d sinh d C C C  C C C, "!2m cos h  BK  R( r#l cos h#l sin h#2m sinh d   C 2m cosh d sinh d cos h(r#l#2m cosh d ) C C  C , BK  "! (( r#l cos h#l sin h#2m sinh d   C (r#l cos h#l sin h)(r#l cos h#l sin h!2m)     ds"DM  ! dt # DM



r 4m cosh sinh # dr#dh# [l l +(r#l cosh#l sinh)    (r#l)(r#l)!2mr DM   !2m( sinh d sinh d # sinh d sinh d #sinh d sinh d ), C C C C C C #2m+(l#l) cosh d cosh d cosh d sinh d sinh d sinh d   C C C C C C !2l l sinh d sinh d sinh d ,]d d

 C C C   4m sin h [(r#l cos h#l sin h)(l cosh d cosh d cosh d !    C C C DM

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!l sinh d sinh d sinh d )#2ml sinh d sinh d sinh d ]d dt  C C C  C C C  4m cos h ! [(r#l cos h#l sin h)(l cosh d cosh d cosh d    C C C DM !l sinh d sinh d sinh d )#2ml sinh d sinh d sinh d ] d dt  C C C  C C C  sin h  # (r#2m sinh d #l) “ (r#2m sinh d #l cos h#l sin h) C  CG   DM G #2m sin h+(l cosh d !l sinh d )(r#l cos h#l sinh)  C  C   #4ml l cosh d cosh d cosh d sinh d sinh d sinh d  C C C C C C !2m sinh d sinh d (l cosh d #l sinhd ) C C  C  C





!2ml sinh d (sinh d #sinh d ), d   C C C 



cos h  # (r#2m sinh d #l) “ (r#2m sinh d #l cos h#l sin h) C  CG   DM G #2m cos h+(l cosh d !l sinh d )(r#l cos h#l sin h)  C  C   #4ml l cosh d cosh d cosh d sinh d sinh d sinh d  C C C C C C !2m sinh d sinh d (l sinh d #l cosh d ) C C  C  C



!2ml sinh d (sinhd #sinh d ), d  ,   C C C

(259)

where DM ,(r#2m sinh d #l cos h#l sin h)(r#2m sinh d #l cos h#l sin h) C   C   ;(r#2m sinh d #l cos h#l sin h) , (260) C   and the subscript E in the line element denotes the Einstein-frame. The ;(1) charges, the ADM mass and the angular momenta of the generating solution (259) (with G""p/4) are , Q"m sinh 2d , Q"m sinh 2d , Q"m sinh 2d ,  C  C C M"m(cosh 2d #cosh 2d #cosh 2d ) C C C (261) "(m#(Q)#(m#(Q)#(m#Q ,   J "4m(l cosh d cosh d cosh d !l sinh d sinh d sinh d ) ,   C C C  C C C J "4m(l cosh d cosh d cosh d !l sinh d sinh d sinh d ) .   C C C  C C C The solution has the outer and inner horizons at: 1 1 1 r "m! l! l$ ((l!l)#4m(m!l!l) ,     ! 2 2 2 provided m5("l "#"l ").  

(262)

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When Q"Q"Q, the generating solution becomes the D"5 Kerr}Newman solution, since   g and u become constant. The generating solution with Q"Q corresponds to the case where   the D"6 dilaton u "u# log det g is constant. In this case, with a subsequent rescaling of    scalar asymptotic values one obtains the static solution of [368] and rotating solution of [95]. The BPS limit with J O0 and regular event horizon is de"ned as the limit in which  mP0, l P0 and d PR while keeping meBC"Q, meBC"Q, meBC"Q,  CCC      l /m"¸ and l /m"¸ constant. In this limit, the generating solution becomes     Q A"   , R  r#Q 

J sin h J cos h A "  , A "  , (  r#Q (  r#Q  

Q A"   , R  r#Q 

J sin h J cos h A "  , A "  , (  r#Q (  r#Q  

J sin h J cos h BK  "!  , BK  " , BK  "!Q cos h ,  R( R( (( r#Q r#Q   r#Q (r#Q)  , eP" g " ,  r#Q [(r#Q)(r#Q)]   



r dr J cos h sin h ds"DM  ! dt# #dh# d d

#   DM r 2DM 2Jr cos h 2Jr sin h dt d # dt d

!   DM DM

 

  

1 sin h (r#Q)(r#Q)(r#Q)! J sin h d  #    4 DM

1 cos h (r#Q)(r#Q)(r#Q)! J cos h d  , #    4 DM

(263)

where DM ,(r#Q)(r#Q)(r#Q) .   The solution is speci"ed by 3 charges and only 1 angular momentum J:

(264)

J "!J ,J"(2QQQ)(¸ !¸ ) ,       while its ADM mass saturates the Bogomol'nyi bound:

(265)

M

"Q#Q#Q . .1  

 When 1 or 3 boost parameters are negative, one has the BPS limit with J "J .  

(266)

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4.4.2. ¹-duality transformation The remaining 27 electric charges (needed for parameterizing the most general charge con"guration) are introduced by the [SO(5);SO(21)]/[SO(4);SO(20)] transformation on the generating solution (259). The "nal expression for electric charges is




; (e !e )  S B Q" U2 e #e S B (2 ;  0  1








,

(267)

where

  CDE CDE e2,(Q, 0, 2, 0), e2,(Q, 0, 2, 0) , B  S 

(268)

; 3SO(5), ; 3SO(21), 0 is a (16;1)-matrix with zero entries and U3O(5, 21, R) brings to the    basis where the O(5, 21) invariant metric ¸ (127) is diagonal. And the charge Q associated with B remains unchanged. The moduli M is transformed to IJ ; 0 ;2 0 M"U2  UMU2  U, (269) 0 ; 0 ;2   where M is the moduli of the generating solution (259). The subsequent O(5, 21);SO(1, 1) transformation leads to the solution with arbitrary asymptotic values M and u .  











4.4.3. Entropy of general solution The thermal entropy of the generating solution (259) is [183]








  1 A"4p m+2m!(l !l ), “ cosh d # “ sinh d S" G G   4G , G G   #m+2m!(l #l ), “ cosh d ! “ sinh d G G   G G    1 "4p 2m “ cosh d # “ sinh d ! (J !J ) G G  16  G G    1 # 2m “ cosh d ! “ sinh d ! (J #J ) , (270) G G  16  G G where d ,d , G "p/4 and the outer horizon area is de"ned as A"  CCC ,  dh d d (g (g  g  !g )" >. ( ( PP   FF ( ( ( ( Note, each term is symmetric under the permutation of d (i.e. 3 charges), manifesting the G conjectured U-duality symmetry [381]. Again, as in the D"4 case, the entropy (270) is cast in the form as sum of `left-movinga and `right-movinga contributions, hinting at the possibility of statistical interpretation of each term as left- and right-moving (D-brane worldvolume) contributions to microscopic degrees of freedom. Each term now carries left- or right-moving angular momentum that could be interpreted as left- or right-moving ;(1) charge [35,36] of the N"4








 





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superconformal "eld theory when the generating solution (259) is transformed to a solution of type-IIA string on K3;S through the conjectured string}string duality in D"6 [635]. When J "0, the entropy rearranges itself as a single term [183,362]: G  (271) S"8(2pm “ cosh d , G G which again has manifest symmetry under permutation of charges. We discuss duality invariant forms of the entropy and the ADM mass of non-extreme, rotating black hole with general charge con"guration (267). The entropy and the ADM mass are expressed in terms of the following T-duality invariants (obtained by applying T-duality to charges of the generating solution): QPX"(Q2M Q#(Q2M Q , >  \   (272) QP>"(Q2M Q!(Q2M Q ,   >  \ while Q remains intact under T-duality. From these 3 T-duality invariant `coordinatesa X, >, Q, one de"nes the following duality invariant `non-extreme hatteda quantities XK : G XK ,(X#m, X "(X, >, Q) . (273) G G G Duality invariant forms of the entropy and the ADM mass are

 

S"2p

“ XK #m XK #(“ (XK !m)!(J !J ) G G G   G G G



# “ XK #m XK !(“ (XK !m)!(J #J ) , G   G G G G G M"XK #>K #QK . When J "0, the duality invariant expression for entropy is G S"4p((XK #m)(>K #m)(QK #m) .

(274)

(275)

4.4.3.1. BPS limit. In the regular BPS limit, the event horizon area (270) becomes [182] (276) "4p[(QQQ)(1!(¸ !¸ ))]"4p[QQQ!J] .     .1     Entropy of BPS black hole with general charge con"guration (267) and with arbitrary scalar asymptotic values depends only on (quantized) charge lattice vectors a and b [182], being a statistical quantity [174,178,258}260,441,579]: A

"4p(b(a2¸a)!J . .1  Here, a and b are related to the physical charges Q and Q as S

(277)

Q"ePM a, Q"e\Pb . 

(278)

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89

In the regular BPS limit, the ADM mass (261) becomes M

"Q#Q#Q . (279) "+   A subset of the SO(5, 21) T-duality transformation on (279) leads to the following T-duality invariant expression for ADM mass of the general D"5 black hole: "eP[a2(M #¸)a]#e\Pb ,  "+ which has dependence on scalar asymptotic values as well as charge lattice vectors. M

(280)

4.4.3.2. Near-extreme limit. In"nitesimal deviation from the BPS limit is achieved by taking the limit in which m and l are very close to zero, and d's are very large such that charges and  l /m"¸ remain as "nite, non-zero constants, and then keeping only the leading order terms   in m [182]. To the leading order in m, the inner and the outer horizons are located at r +m(1!(¸#¸)$([2!(¸ #¸ )][2!(¸ !¸ )]) . !         The outer horizon area to the leading order in m is [95,182] A+4p[(QQQ)(1!(¸ !¸ )#m(QQ#QQ#QQ)          ;([1!(¸ #¸ )]] .    J and J are no longer equal in magnitude and opposite in sign anymore:   J,(J !J )"(2QQQ)(¸ !¸ )#O(m) ,        1 1 1 *J,(J #J )"m(2QQQ) # # (¸ #¸ )#O(m) ,       Q Q Q    while the ADM mass still has the form






(281)

(282)

(283)

M"(m#(Q)#(m#(Q)#(m#Q . (284)   Note, when one of the charges is taken small, e.g. QP0, as in study of the microscopic entropy  near the BPS limit [95,368], the ADM mass is M"M #O(m), while the area is .1 A"A #O(m). However, when all the charges are non-zero, the deviation from the BPS limit .1 is of the forms M"M #O(m) and A"A #O(m). .1 .1 4.5. Rotating black holes in higher dimensions We discuss rotating black holes in heterotic string on ¹\" (44D49) with general ;(1)\" electric charge con"gurations [184,449]. The generating solution is parameterized by the ADM mass M (or alternatively the non-extremality parameter m), ["\] angular momenta &  J (i"1,2, ["\]), and 2 electric charges of the KK and the 2-form ;(1) gauge "elds associated G  with the same compacti"ed direction, which we choose without loss of generality to be Q and  Q, i.e. those associated with the "rst compacti"ed direction, as well as asymptotic values of  a toroidal modulus G and the dilaton u .  

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The non-trivial "elds of the generating solution are [184] N sinh d cosh d N sinh d cosh d  ,  , A" A" R R 2N sinh d #D 2N sinh d #D   Nl k sinh d cosh d Nl k sinh d cosh d  , A " G G  , A " G G (G (G 2N sinh d #D 2N sinh d #D   2N sinh d #D D  eP" , G " ,  2N sinh d #D =  2Nl k sinh d sinh d [m(sinh d #sinh d )r#D] G G     B G"! , R( = B G H"!4Nl l kk sinh d sinh d cosh d cosh d GH G H     (( ;[N(sinh d #sinh d )#D][2N sinh d sinh d     #ND(sinh d # sinh d !1)#D]/[(D!2N)=] ,   D!2N dr dt# ds"D"\"\="\ ! = “ "\ (r#l)!2N G G r#l cos h#K sin h   # dh D



cos h cos t 2cos t  G\ (r#l cos t #K sin t )dt # G> G G> G G D l!K H cos h cos t 2cos t !2 H  G\ D GH ;cos t sin t 2cos t cos t sin t dt dt G G H\ H H G H l !K G> cos h sin h cos t 2cos t cos t sin t dh dt !2 G>  G\ G G G D k # G [(r#l)(2N sinh d #D)(2N sinh d #D) G   D= 2Nl k cosh d cosh d G G   dt d

#2lN(D!2N sinh d sinh d )] d ! G G   G = # GH



4Nl l kk(D!2N sinh d sinh d ) GH G H   d d , G H D=

(285)

where =,(2N sinh d #D)(2N sinh d #D)   and D, K , N, k , a are de"ned separately for even and odd D in (188)}(191). G G

(286)

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The ADM mass, angular momenta and electric charges of the generating solution are X m M " "\ [(D!3)(cosh 2d #cosh 2d )#2] , & 16pG   " X J " "\ml cosh d cosh d , G   G 4pG " (287) X Q" "\ (D!3)m sinh 2d ,   16pG " X Q" "\ (D!3)m sinh 2d .   16pG " For the canonical choice of asymptotic values G "d , i.e. compacti"cation on (10!D) self-dual GH GH circles with radius R"(a, the D-dimensional gravitational constant is G "G /(2p(a)\". "  Also, the KK and the 2-form "eld ;(1) charges Q and Q are quantized as p/(a and q/(a,   respectively, where p, q3Z. The outer horizon area of the generating solution is [184,366] A "2mr X cosh d cosh d , (288) " > "\   where the outer horizon r is determined by > "\

"0 . (289) “ (r#l)!2N G PP> G The surface gravity i at the (outer) event horizon is de"ned as i"lim > j Ij, where PP I mIm ,!j and m,R/Rt#X R/R . Here, X is the angular velocity at the (outer) horizon and is I G G G de"ned by the condition that m is null on the (outer) horizon. The surface gravity and angular velocity at the outer-horizon of the generating solution are







1 1 R (P!2N) l P G i" , X" , (290) G cosh d cosh d r #l cosh d cosh d 4N   PP>   > G where P,“ "\ (r#l). G G The generating solution has a ring-like singularity at (r, h)"(0, p/2) and thus spacetime is that of the Kerr solution. The BPS limit of (285), where the ADM mass M saturates the Bogomol'nyi bound & M 5"Q#Q" , (291) &   is de"ned as the limits mP0 and d PR such that Q remain as "nite constants. For D56   with only one of l non-zero, the BPS limit is also the extreme limit [366], i.e. all the horizons G collapse to r"0 as mP0. However, with more than one l non-zero, the singularity at r"0 G becomes naked, i.e. horizons disappear.

 We use the convention of [478], keeping in mind that matter Lagrangian in (124) has 1/(16pG ) prefactor. "

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5. Black holes in Nⴝ2 supergravity theories 5.1. N"2 supergravity theory 5.1.1. General matter coupled N"2 supergravity We consider the general N"2 supergravity [5,6,128,198,211,271] coupled to n vector mul tiplets and n hypermultiplets. The "eld contents are as follows. The N"2 supergravity multiplet & contains the graviton, the S;(2) doublet of gravitinos tG (the S;(2) index i"1, 2 labels two I supercharges of N"2 supergravity and k"0, 1, 2, 3 is a spacetime vector index), and the graviphoton. The N"2 vector multiplets contain ;(1) gauge "elds, doublets of gauginos j? and G scalars z? (a"1,2, n ), which span the n -dimensional special KaK hler manifold. The hypermultip  lets consist of hyperinos f , f? (a"1,2, 2n ) with left and right chiralities and real scalars ? & qS (u"1,2, 4n ), which span the 4n -dimensional quaternionic manifold. The general form of the & & bosonic action is [5] L,"(!g[!R#g H(z, z ) Iz? z @H#h (q) IqS qT  ?@ I ST I K R K R #i(N M KRF\ F\ IJ!NKRF> F> IJ)] , (292) IJ IJ where g H"R R HK(z, z ) is the KaK hler metric, h (q) is the quaternionic metric, ?@ ? @ ST F!K,(FK $(i/2)eIJMNFK ) are the (anti-)self-dual parts of the "eld strengths II  IJ MN FK "R AK!R AK#gf KR ARA of the ;(1) gauge "elds AK (K"0, 1,2, n ) in the N"2 IJ I I J I I J I  supergravity and N"2 vector multiplets, and g is the gauge coupling. Here, the gauge covariant di!erentials on the scalars are de"ned as:

z?,R z?#gAKk?K(z) , I I I

z ?H,R z ?H#gAKk?H (z ) , (293) I I I K

qS,R qS#gAKkSK(q) , I I I where k?K(z) [kSK(q)] are the holomorphic [triholomorphic] Killing vectors of the KaK hler [quaternionic] manifold (cf. see (65)). We introduce a symplectic vector of the anti-self-dual "eld strengths:




Z\,

F\K G\ R

,

(294)

where G\ M KRF\R [212]. The symplectic vector Z> of self-dual "eld strengths is the complex K ,N conjugation of (294). It is convenient to rede"ne "eld strengths FK as [93,132] ¹\,1<"Z\2"(MKF\K!¸RG\ R ) , M F\K! H¸M KG\ F\?,g?@H1;M H"Z\2"g?@H( HM K) . @ @ K @

(295)

 The KaK hler potential K(z, z ) and the period matrix NKR are de"ned in terms of the holomorphic prepotential F(X) and the scalar "elds XK as in (92).

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Then, ¹\ and F\? (a"1,2, n ), respectively, correspond to the "eld strengths of the gravi-photon  of the supergravity multiplet and the gauge "elds of n super-Yang-Mills multiplets.  The supersymmetry transformation laws for the gravitinos, the gauginos and hyperinos in the bosonic "eld background are dt "D e #[igS g #e ¹\ ]cJeH , GI IG GH IJ GH IJ (296) dj?G"icI z?eG#eGH(F\?cIJ#k?K¸M K)e , H I IJ df "iUH@ qScIe C eG#gNG e , ? S I GH ?@ ?G where e [C ] is the #at Sp(2) [Sp(2n )] invariant matrix and UH@ is the quaternionic vielbein [30]. GH ?@ & S Here, S and NG are mass-matrices given by GH ? i S " (p )Ie PVK¸K, NG "2UG kSK¸M K , (297) GH 2 V G HI ? ?S where PVK is a triplet of real 0-form prepotentials on the quaternionic manifold. 5.1.2. BPS states The BPS states of the N"2 theory have mass equal to the central charge, which is just the graviphoton charge given by 1 Z,! 2



¹\ .

(298)



1 Thus, central charge Z is characterized by the vacuum expectation of the moduli in the symplectic vector < and the symplectic charge vector given by




Q"

PK

QR



; PK,





Re F\K, QR,

1 1 in the following way [133,136,212,550,551]



Re G\ R ,

Z"1<"Q2"(¸KQK!MRPR)"e)X X (XK(z)QK!FR(z)PR) .

(299)

(300)

Note, since two vectors < and Q transform covariantly under the symplectic transformation Sp(2n #2), the central charge and, therefore, the ADM mass M""Z" have manifest symplectic  covariance. 5.2. Supersymmetric attractor and black hole entropy Since entropy is a statistical quantity de"ned as the degeneracy of microscopic states, the horizon area, which de"nes the thermal entropy, should be independent of continuous quantities like scalar asymptotic values. This is an other illustration of no-hair theorem where properties of black holes are independent of scalar hairs; all the information of scalar asymptotic values get lost at the event horizon. It was discovered in [260] within N"2 theories that this is a generic property of BPS solutions in supersymmetric theory and can be derived from supersymmetry alone [8,258,259].

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To illustrate this idea, we consider general magnetic, spherically symmetric solution in N"2 theory coupled to vector super"elds [260]. Spherically symmetric AnsaK tze are [597]: qK ds"g dxI dxJ"!e3 dt#e\3 dx, FK K"   e3P , P IJ r

(301)

and the scalars (moduli) zK,XK/X (K"0, 1,2, n ) depend on r, only.  With these AnsaK tze, the Killing spinor equations dt "0 and dj?G"0 yield the following GI coupled "rst-order di!erential equations [260]:



4;"!

(z Nq )(zNq )(z Nz)     e3 , (zNz)(z Nz )

(302) (zNz)(z  Nq )(z  Nz) e3   (zKq !qK ) , (zK)"!     (z Nz )(zNq ) 4   where the prime denotes the di!erentiation with respect to o,1/r, and (zNq ),zKNKRqR , etc.     From (302), one obtains the following second-order ordinary di!erential equations for the moduli K "elds z :



(zNq ) ((zK)) 1 (zNz)(z Nq )(z Nz)  K   #q   (z )"0 . # ln (303) K K   z q !q (zNz) 2 (z Nz )(zNq )       Eq. (303) can be viewed as a geodesic equation for moduli "elds zK that determines how and (zK)" for the geodesic zK evolves as o varies from 0 to R. The initial conditions zK" M M K K motion in the phase space (with coordinates z and (z )) are the asymptotic values (r"1/oPR) and qK through the second equation in (302)). of zK and their derivatives (determined by zK"   P K K K Given initial conditions z " and (z )" , z evolve with o, following damped geodesic motion M M in the phase space until they run into an attractive "xed point, i.e. a point where the velocities dzK/do of zK vanish. For the special example under consideration, as we see (302), the "xed point is located at




(zK)!








(304) zK "qK /q .       At the "xed point, moduli depend on ;(1) charges only, loosing all their information on the initial conditions at in"nity. From this observation, one arrives at stronger version of no-hair theorem for black holes in supersymmetry theories: black holes lose all their scalar hairs near the horizon and are characterized by discrete ;(1) charges (and angular momenta), only. Nearby the horizon, the black hole approximates to the Bertotti}Robinson geometry [92,447,515] with the topology AdS ;S"\. This geometry is conformally #at and the  graviphoton "eld strength is covariantly constant. Thus, in this region the ADM mass reduces to the Bertotti}Robinson mass and the supersymmetry is completely restored [138,292,300, 398,412,453]. In the asymptotic region (rPR), the spacetime is #at and, therefore, supersymmetry is unbroken. In between these regions, solutions break fraction of supersymmetries, indicating the BPS nature. Note, the supersymmetric con"guration under consideration is a bosonic con"guration, i.e. a solution to supergravity theory with all the fermionic "elds set equal to zero. However, the

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supersymmetry parameters associated with the unbroken supersymmetries, called `anti-Killing spinorsa, generate a whole supermultiplets of solutions, i.e. the superpartners to black hole solution. To generate such solutions, one start with a bosonic con"guration and applies supersymmetry transformations iteratively with the supersymmetry parameters given by the anti-Killing spinors. Such a procedure induces fermionic "elds, as well as corrections to bosonic "elds. It was shown in [407] that when this procedure is performed on double-extreme black solutions, i.e. extreme solutions with constant scalars, in the N"2 supergravity coupled to vector and hyper multiplets, there are no corrections to the "elds in the vector and hyper multiplets. This implies that although the metric, graviphoton and gravitino receive corrections, the moduli at the "xed attractor point as functions of ;(1) charges, only, remain intact under the supersymmetry transformations which generate the fermionic partners of the supersymmetric black holes. 5.3. Explicit solutions 5.3.1. General magnetically charged solutions We discuss the general spherically symmetric, magnetic (q R "0) solutions in N"2 supergravity coupled to n vector multiplets with scalar "elds varying with the radial coordinate r [260]. The  AnsaK tze for the "elds are given in (301) with the scalar "elds depending on the radial coordinate r, only. The scalar "elds and the metric components satisfy the di!erential equations (302)}(303). For the purpose of solving the di!erential equations (302)}(303), we consider the simple case with q "0. In this case, the solutions are given by   e3M"e)XX \) , q? z? # K oe\) for z? "z ? ,    4 z?" (305) q? K  oe\)  for z? "!z ? , z? #i    4



ds"!e)X?X ?X?\) dt#e\)X?X ?X?>) dx . The explicit solutions for N"2 theories with speci"c prepotentials F are obtained by substituting the corresponding KaK hler potential K into the general solution (305) [260]. 5.3.2. Dyonic solutions We generalize the magnetic black holes in Section 5.3.1 to include electric charges as well [579]. Since it would be hard to solve the resulting di!erential equations with non-zero electric and magnetic charges, we take all the moduli "elds z' to be constant. In fact, as we discuss in the subsequent sections, such class of solutions corresponds to the minimum energy con"gurations among extreme solutions and, therefore, is physically interesting. Assuming that the moduli z'"X'/X are constant, from dj?G"0 one obtains the following electric and magnetic charges of dyonic solutions:






i (q' , q)"Re CX',!C F (X ) ,   '  2 ' 

(306)

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where C is a constant and the subscript f denotes the "xed point. With a suitable choice of KaK hler gauge, which eliminates the redundant degrees of freedom in X' by half, one can solve the 2n#2 equations in (306) to "nd the expressions for z'"X'/X in terms of quantized charges q' and q.   ' From dt "0 with ;(1) "eld strengths F K '"CX'e> and G>"(C/2)F e> (e> obeys GI ' ' IJ "ie> and is normalized to give 2p after being integrated over S) substituted, one obtains the *e> IJ IJ following solution for the metric: e\3"1#(CCM /4r .

(307)

This solution has the surface area given by p A" CCM . 4

(308)

5.4. Principle of a minimal central charge At the "xed attractor point in phase space, the central charge eigenvalue is extremized with respect to moduli "elds, so-called `principle of a minimal central chargea [254,258,259,414,415]. For N'2 theories, the largest eigenvalue is extremized and the smaller central charges become zero [259] at the "xed point. Since scalar asymptotic values are expressed only in terms of ;(1) charges at the "xed point, the extremized (largest) central charge depends only on ;(1) charges, thereby becoming a candidate for describing black hole entropy. It turns out that entropy of extreme black holes for each dimension has the following universal dependence on the extremal value Z of the (largest) central charge eigenvalue regardless of the number N of supersymmetries   [258,259]: A S" "p"Z "? ,   4

(309)

where a"2[3/2] for D"4 [D"5]. As an example, we consider the BPS dilatonic dyon [292,409] in D"4 with the mass: ""Z""(e\P"p"#eP"q") . (310) .1  The minimum of the central charge "Z" is located at g "eP""p/q", which leads to the following   correct expression for the entropy which is independent of dilaton asymptotic value: M

S"A/4"p"Z ""p"pq" . (311)   One can prove the principle of a minimal central charge as follows. We consider the ungauged N"2 supergravity coupled to Abelian vector multiplets and hypermultiplets, de"ned by the Lagrangian (292) and the supersymmetry transformations (296) with g"0. Since we are interested in con"gurations at the "xed point, the derivatives of scalars are zero, i.e. R z?"0 and R qS"0, at I I  These are obtained by solving dj?G"0.  Thus, for N54 theories, one can determine moduli "elds (at the "xed point) in terms of ;(1) charges, by minimizing the largest eigenvalue and setting the remaining eigenvalues equal to zero.

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the horizon. So, from dj?G"0, one has F\?"0. Note, the covariant derivative of the central IJ charge de"ned in (300) is 1 Z , Z"! ? ? 2



g HF>@H"(QK ¸K!PR MR)"1; "Q2 . ?@ ? ? ?

(312) 1 Since F\?"0 is equivalent to F>?"0, the central charge Z is covariantly constant at the "xed point of the moduli space: 

Z " Z"1; "Q2"0 . (313) ? ? ? It can be shown [258] that the condition (313) is equivalent to the statement that the central charge takes extremum value at the "xed point: R "Z""0 . (314) ? Thus, within ungauged general Abelian N"2 supergravity we establish that central charge is minimized at the xxed point of geodesic motion of moduli evolving with o"1/r. At the "xed point in the moduli space, the central charge is expressed in terms of the symplectic ;(1) charge vector Q and the moduli as [258]: "Z""!Q2 ) M(N) ) Q ,  Im N#(Re N)(Im N)\(Re N) !(Re N)(Im N)\ , M(N), !(Im N)\(Re N) (Im N)\






(315) (316)

with the moduli in the matrix M(N) taking values at the "xed attractor point. The central charge minimization condition (313) "xes the asymptotic values of the moduli in terms of Q. By using the relations 1; "<2"0"1; "
(317)

or in component form: PK"2 Im(ZM ¸K),

QR"2 Im(ZM MR) .

(318)

Eq. (318) can be solved to express the moduli (at the "xed point) in terms of ;(1) charges:




1 <"! 2ZM

0

I



!I 0

) M(F)#i


 I

0

0

I

)Q ,

(319)

or in terms of components: !2ZM ¸K"[iP!(Im N)\(Re N)P#(Im N)\Q]K , !2ZM MR"[iQ!((Im N)#(Re N)(Im N)\(Re N))P#(Re N)(Im N)\Q]R .

(320)

Here, M(F) is de"ned as in (316) with NKR replaced by FKR,RKFR(X). Alternatively, one can rederive the relations (317) obeyed by the moduli and ;(1) charges by a variational principle [54] associated with a potential VQ(>, >M ),!i1PM "P2!1PM #P"Q2 ,

(321)

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where




P,

>K

FR(>)

;

>K,ZM XK .

(322)

Namely, at the minimum of VQ(>,>M ), relation (317) for the minimal central charge, which can be expressed in terms of P as P!PM "iQ, is satis"ed. In particular, the entropy for D"4 black holes is rewritten as S ""Z ""i1PM "P2"">" exp[!K(z, z )]" .     p

(323)

Many black holes are uplifted to intersecting p-branes. In this case, energy of black holes is sum of energies of the constituent p-branes. The minimal energy of p-branes corresponds to the ADM mass of the corresponding double-extreme black holes in lower dimensions. In taking variation of moduli to "nd the minimum energy con"guration, one has to keep the gravitational constant of lower dimensions as constant. The minimum energy of p-brane is achieved when energy contributions from each constituent p-brane are equal [403]. 5.4.1. Generalization to rotating black holes Generally, rotating black holes have naked singularity in the BPS limit. D"5 rotating black holes with 3 charges has regular BPS limit (thereby the horizon area can be de"ned), if 2 angular momenta have the same absolute values [98,182]. We discuss generalization of the principle of minimal central charge to the rotating black hole case [414]. We consider the following truncated theory of 11-dimensional supergravity compacti"ed on a Calabi}Yau three-fold [103,328,329,491]: L"(!g[!R!ePF FIJ!e\PG GIJ#(R u)]   IJ  IJ  I 1 eIJMNHF F B . ! IJ MN H 4(2

(324)

This corresponds to the N"2 theory with F"CKR XKXRX . The supersymmetry transforma  tions of the gravitino t and the gaugino s in the bosonic background are I 1 1 e\PG )e , dt " e# (KMN!4dMCN)(ePF ! I MN (2 MN I I 12 I (325) 1 1 CIR ue# CMN(ePF #(2e\PG )e . ds"! I MN MN 4(3 2(3

 Note, lower-dimensional gravitational constant is expressed in terms of the D"10 gravitational constant and the volume of the internal space, i.e. a modulus.  It is argued in [288] that singular D"4 heterotic BPS rotating black holes can be described by regular D"5 BPS rotating black holes which are compacti"ed through generalized dimensional reduction including massive Kaluza}Klein modes.

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The model corresponds to the N"2 supergravity with the graviphoton (ePF ! MN (1/(2)e\uG ) coupled to one vector multiplet with the vector "eld component MN (ePF #(2e\PG ). MN MN We consider the following D"5 BPS rotating black hole solution [98,182,609] (cf. (263)) to the above theory:









 r  4J sinh 4J cosh dt! d # dt ds" 1!  r p(r!r) p(r!r)   r \ dr!r(dh#sin h d #cos h dt) . ! 1!  r

(326)

For this solution, the scalar u is constant everywhere (double-extreme): eP"j. So, from ds"0, one sees that the vector "eld in the vector multiplet vanishes, i.e. B"!(j/(2)A, from which one can express u in terms of ;(1) charges as





(2 (2 8Q e\P夹G, Q , eP夹F . (327) eP" $ ,j; Q , & 4p  $ 16p  pQ 1 1 & Furthermore, the entropy is still expressed in terms of the central charge Z at the "xed point, but   modi"ed by the non-zero angular momentum J: S"p(Z !J . (328)   The argument can be extended to more general rotating solutions in the N"2 supergravity coupled to n vector multiplets with the gaugino supersymmetry transformations  1 3  i (329) dj "! g (u)CIR u@ e# tK CIJFK e . ? IJ I ? 4 4 2 ?@




From dj "0, one sees that at the "xed point (R u?"0) FK "0. So, the central charge is ? I IJ extremized at the "xed point: R Z"0. ? We discuss the enhancement of supersymmetry near the horizon [138,412]. Since the vector "eld in the vector multiplet is zero for (326), the solution is e!ectively described by the pure N"2 supergravity [265] with the graviphoton FI ,((3/2)jF. The supersymmetry transformation for the gravitino is 1 dt " K e" e# (CMNC #2CMdN)FI e . I I I I I MN 4(3 The integrability condition for the Killing spinor equation dt "0 is I [ K , K ] e"RK e"0 , ? @ ?@ where the super-curvature RK for solution (326) is de"ned as ?@ (r!r)  X (1#C) . RK " ?@ ?@ r

(330)

(331)

(332)

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Here, the explicit forms of matrices X , which can be found in [414], are unimportant for our ?@ purpose. At the horizon (r"r ) and at in"nity (rPR), RK "0, thereby (331) does not constraint  ?@ the spinor e, i.e. supersymmetry is not broken. However, for "nite values of r outside of the event horizon, for which RK O0, e is constrained by the relation: ?@ (1#C)e"0 , (333) indicating that 1/2 of supersymmetry is broken. 5.4.2. Generalization to N'2 case The principle of minimal central charge is generalized to the N"4, 8 cases by reducing N"4, 8 theories to N"2 theories, and then by applying the formalism of N"2 theories [258]. For N54 theories, there are more than 1 central charge eigenvalues Z (i"1,2, [N/2]). The ADM mass of G the BPS con"guration is given by the max+"Z ",. When the principle of minimal central charge is G applied to this eigenvalue, the smaller eigenvalues vanish and all the scalar asymptotic values are expressed in terms of ;(1) charges, only [259]. So, the extremum of the largest central charge continues to depend on integer-valued ;(1) charges, only. The entropy of extreme black holes in each D has the universal dependence on the extreme value of the largest central charge eigenvalue: S"A/4"p"Z "?, where a"2 [3/2] for D"4 [D"5], regardless of the number N of super  symmetry. 5.4.2.1. Pure N"4 supergravity. Pure N"4 theory can be regarded as N"2 supergravity coupled to one N"2 vector multiplet. This can also be regarded as either S;(2);SO(4) or S;(2); S;(4) invariant truncation of N"8 theory. The former corresponds to the N"2 theory with F(X)"!iXX and the latter has no prepotential. These two theories are related by the symplectic transformation [136]: XK "X, FK "F , XK "!F , FK "X , (334)     where the hat denotes the S;(4) model [165]. The SO(4) version [162,164,194] of the D"4, N"4 supergravity action without axion is



1 dx(!g[!R#2R uRIu!(e\PFIJF #ePGI IJGI )] , I" I  IJ IJ 16p

(335)

where the "eld strength GI of the SO(4) theory is related to that G of the S;(4) theory as IJ IJ i 1 GI IJ" e\PeIJMHG . MH 2 (!g The dilatino supersymmetry transformation is 1 1 dK "!cIR e # pIJ(e\(F a !e(GI b )\e( . I ' (2 IJ '( IJ '( 2 '

(336)

At the "xed point (R "0), the Killing spinor equations dK "0 "x in terms of electric and I ' magnetic charges: e\( ""q"/"p". Then, writing dK "0 at the "xed point in the form (Z ) e("0, ' '(  

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101

one learns [409] that E pq'0 case: e non-vanishing, Z "0 and M ""Z ".  "+  E pq(0 case: e non-vanishing, Z "0 and M ""Z ".  "+  So, smaller eigenvalues, which correspond to broken supersymmetries, vanish and entropy is given by the largest eigenvalue at the "xed point. 5.4.2.2. N"4 supergravity coupled to n vector multiplets. The target space manifold of N"4  supergravity coupled to n vector multiplets is  O(6, n ) S;(1, 1)  ; O(6);O(n ) ;(1)  with the "rst factor parameterized by the axion-dilaton "eld S and the second factor by the coset representatives ¸K "(¸GHK , ¸?K) (i, j"1, 2, 3, 4, K"1,2, 6#n and a"1,2, n ) [84]. The central   charge is Z "e)[¸K qK!S¸ KpK] , GH GH GH where K"!ln i(S!SM ) is the KaK hler potential for S. At the "xed point, the gaugino Killing spinor equations dj?"0 require that G K K S¸?Kp !¸? qK"0 .

(337)

(338)

The dilatino Killing spinor equation dsG"0 requires the following smaller of the central charge eigenvalues to vanish: (339) "Z ""(Z ZM GH!((Z ZM GH)!"eGHIJZ Z ") , GH  GH IJ   GH which "xes S at the "xed point. At the "xed point, the di!erence between two eigenvalues "Z "!"Z ""((Z ZM GH)!"eGHIJZ Z " GH  GH IJ    becomes independent of scalars and gives rise to the horizon area [178,236] A"4p(M ) "4p"Z ""2p(qp!(q ) p) . "+    

(340)

(341)

5.4.2.3. N"8 supergravity. The consistent truncation of N"8 down to N"2 is achieved by choosing HLS;(8) such that 2 residual supersymmetries are H-singlet. Such theory corresponds to N"2 supergravity couple to 15 vector multiplets (n "15) and 10 hypermultiplets (n "10).  & (This is the upper limit on the number of matter multiplets that can be coupled to N"2 supergravity.) Under N"2 reduction of N"8, S;(8) group breaks down to S;(2);S;(6), leading to the decomposition of 26 central charges Z of N"8 into (1, 1)#(2, 6)#(1, 15) under  S;(2);S;(6). The S;(2) invariant part (1, 1)#(1, 15) is (Z, D Z), where Z is the N"2 central G charge. So, the horizon area is again A&"Z". For example, for type-IIA theory on ¹ /Z   truncated so that only 2 electric and 2 magnetic charges are non-zero, the central charge at the "xed point is product of ;(1) charges, which is black hole horizon area.

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We consider the following truncation of N"8 supergravity:






1 1 S" dx(!g R! [(Rg)#(Rp)#(Ro)] 16pG 2




!



eE [eN>M(F )#eN\M(F )#e\N\M(F )#e\N>M(F )] .     4

(342)

This is a special case of S¹; model [236] with the real parts of complex scalars zero e\E"Im S,s,

e\N"Im ¹,t, e\M"Im ;,u .

(343)

The following black hole solution to this model is reparameterization [511] of the general solution obtained in [178]: ds"!e3dt#e\3 dx , e3"t t s s ,     t t t s t s e\E"  , e\N"  , e\M"   , s s s t t s       F "$dt dt, FI "$ds dt ,     F "$dt dt, FI "$ds dt ,     where "q " \ "p " \ t " eE>N>M#  , s " e\E\N>M#  ,   r r





 





 

"q " \ "p " \ t " eE\N\M#  , s " e\E>N\M#  ,   r r

(344)

(345)

and s are magnetic potentials related to FI "eE!N\M夹F . The ADM mass of (344) is    1 s u t M " stu"q "# "q "# "P "# "p " . (346) "+ 4  tu  st  su 






By minimizing (346) with respect to s, t, u, one obtains the ADM mass at the "xed point [178]: (M ) ""q p q p " , (347) "+       and "nds that the smaller central charges are zero at the "xed point. This result is proven in general setting as follows. We consider the N"8 supersymmetry transformations [160] of gravitinos W and fermions s at the "xed point: I  ! dW "D e #Z cJe , ds "Z pIJe , (348) I I  IJ  !  IJ !

where A"1,2,8 labels supercharges of N"8 theory. We truncate the Killing spinors as e "0, e "+e , e O0, e "e "0, , ? G    

(349)

 This model also corresponds to ¹ part of type-IIA theory on K ;¹ or heterotic theory on ¹;¹. See Section  3.2.2 for the explicit action.

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where 6 supersymmetries are projected onto null states. Here, we splitted the index A as A"(i, a) in accordance with the breaking of S;(8) to S;(4);S;(4). By bringing Z to a block diagonal  form [264] through S;(8) transformation (See Section 2.2, for details.), one "nds that the supersymmetry variations of t , s and s vanish due to (349) and the block diagonal I? ?@A ?GH choice of Z . From ds "0, one "nds that Z "0 (i.e. Z "Z "0) and from ds "0, one  G?@ ?@   GHI "nds that Z "0. So, we proved within the class of con"gurations characterized by truncation  (349) that the condition for unbroken supersymmetry requires the smaller central charges to vanish. And the largest central charge at the "xed point gives the ADM mass and the horizon area. 5.4.2.4. Five-dimensional theories. N"1, N"2 and N"4 theories in D"5 [11,27,103,213, 328,329,491] have 1, 2 and 3 central charges, respectively. At the "xed point, the largest central charge is minimized and the smaller central charges vanish. The horizon area is given in terms of the central charge at the "xed point by A"4pZ. The general expressions for the (largest) central   charge at the "xed point for each case are as follows: E N"1 theory: Z "(d (q)\q q , where d (q)\ is the inverse of d "d t!(z) evaluated   !    at the "xed point [258]. E N"2 theory: Z "(Q Q), where Q is a charge of the 2-form potential and Q is the   & $ & $ Lorentzian (5, n ) norm of other 5#n charges [580].   E N"4 theory: Z "(q XHJq XKLq XNG), where q is 27 quantized charges transforming   GH JK LN GH under E (Z) and XGH3Sp(8) is traceless.  5.5. Double extreme black holes We discuss the most general extreme spherically symmetric black holes in N"2 theories in which all the scalars are frozen to be constant all the way from the horizon (r"0) to in"nity (rPR) [415], called double-extreme black holes. For this case, the ADM mass (or the largest central charge) takes the minimum value (related to the horizon area) and, therefore, is equal to the Bertotti}Robinson mass. Whereas all the scalars are restricted to take values determined by ;(1) charges, all the ;(1) charges can take on arbitrary values. Double-extreme black holes are also of interests since they are the minimum-energy extreme con"gurations in a moduli space for given charges. The general double-extreme solution is obtained by starting from the spherically symmetric Ansatz for metric ds"e3 dt!e\3 dx, ;(r)P0,

as rPR ,

(350)

 After the "rst draft of this section is "nished, more general class of N"2 supergravity black hole solutions [60,61,521,522,520], which include general rotating black holes and Eguchi}Hanson instantons, are constructed. These solutions are entirely determined in terms of the Kahler potential and the Kahler connection of the underlying special geometry, where also the holomorphic sections are expressed in terms of harmonic functions. Such general class of solutions turns out to be very important in addressing questions related to the conifold transitions in type II superstrings on Calabi-Yau spaces, when they become massless [60].

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and assuming that all the scalars are constant everywhere (R zG"0 and R qS"0) and that I I consistency condition F\G"0 for unbroken supersymmetry is satis"ed. Since all the scalars are constant, the spacetime is that of extreme Reissner}Nordstrom solution: e\3"1#M/r .

(351)

By solving the equations of motion following from (292), one obtains FK"e3

2QK 2PK dtdr! r dhr sin h d . r r

(352)

From equations of motion along with (350)}(352), one obtains the ADM mass M in terms of the electric QK and magnetic PR charges of FK: M"!2 Im NKR(QKQR#PKPR) .

(353)

The ;(1) charges (PR, QK) are related to the symplectic charges Q"(qK , q )"(FK, GR) as:   R qK 2PK   " . (354) 2(Re NKR)PK!2(Im NKR)QK q R







), M is expressed as In terms of (qK , q   R (Im N#Re N Im N\Re N)KR (!Re N ImN\)KR 1 M"! (qK , q K ) 2   (!Im N\ Re N)KR (Im N\)KR





 qR   q R

""Z"#" Z" . (355) ? From the consistency condition F\?"0 for unbroken supersymmetry, one has Z"0, which is ? with equivalent to R Z"0. So, the ADM mass of double extreme black holes is M""Z" ? ?

8 scalars constrained to take values de"ned by Z"0. By solving Z"0, one obtains the ? ? following relation between (q, q ) and the holomorphic section (¸K, MR):   qK 2iZM ¸K   "Re , (356) 2iZM MR q R







which can be solved to express (¸, M) in terms of (q, q ). Since the ADM mass M for   double-extreme solutions obeys the stabilization equations (356), the entropy is related to the ADM mass as: S"pM? ,

(357)

where a"2[3/2] for the D"4 [D"5] solutions. 5.5.1. Moduli space and critical points We have seen that the BPS condition requires scalars at the event horizon take their "xed point values expressed in terms of quantized electric/magnetic charges and, thereby, the (largest) central

 The other sum rule for Z and Z is "Z"!"Z ""!Q2M(F)Q. ? ? 

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charge at the event horizon is related to the black hole entropy. In this subsection, we point out that such property of extreme black holes at the "xed point can be derived from bosonic "eld equations and regularity requirement of con"gurations near the event horizon without using supersymmetry [254]. For non-extreme con"gurations, the horizon area has non-trivial dependence on (continuous) scalar asymptotic values. We consider the following general form of Bosonic Lagrangian: L"(!g[!R#G R ?R @gIJ!kKRFK FR gIHgJM!lKRFK 夹FR gIHgJM] , (358)   ?@ I J  IJ HM  IJ HM where FK ,R AK!R AK are Abelian "eld strengths with charges (pK, qK)"((1/4p)FK, IJ I J J I (1/4p) [kKR夹FR#lKRFR]) and kKR, lKR are moduli dependent matrices. We restrict our attention to static ansatz for the metric ds"e3 dt!e\3c dxK dxL , KL where for spherically symmetric con"gurations c c dq # (dh#sin h du) , c dxK dxL" KL sinhcq sinhcq

(359)

(360)

where q runs from !R (horizon) to 0 (spatial in"nity). The function ; satis"es the boundary conditions ;Pcq as qP!R and ;(0)"1. The equations of motion for ;(q) and ?(q) can be derived from the following 1-dimensional action L




d;  d ? d @ #G " #e3<( , (p, q)) , ?@ dq dq  dq

(361)

describing geodesic motion in a potential <"(p q)







k#lk\l lk\ k\l

k\

p q

and with the constraint




d;  d ? d @ #G !e3<( , (p, q))"c . ?@ dq dq dq

where a constant c is related to the entropy S and temperature ¹ as c"2S¹. For non-extreme con"gurations, where scalars ? have non-zero scalar R? ( ?& ? #(R?/r)), the "rst law of thermodynamics is modi"ed [297] to  i dA #X dJ#tK dqK#sK dpK!G ( )R@ d ? , dM" ?@  8p

(362)

charges

(363)

whereas the Smarr formula remains in a standard form. R? vanish i! ? take the values which extremize M, i.e. double extreme solutions (i.e. ?(q)" ? ), provided < " < is non-negative  ?@ ? @ (convexity condition). Here, is the Levi}Civita covariant derivative with respect to the scalar ? manifold metric G . (This can also be directly seen from (RM/R ?) "!G ( )R@.) For this ?@ (NO ?@  case, ? have to extremize <, i.e. (R
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"xed values in terms of conserved charges (pK, qK). The bound A44p<(p, q,

), which is   derived from the requirement of "nite event horizon area A together with the constraint (362), is saturated for the (double) extreme case. Since ;PMq as qP0, one obtains the following relation from (362): M#G R?R@!<( ? )"4S¹ , (364) ?@  which states that the total self-force on black holes due to the attractive forces of gravity and scalars is not exceeded by the repulsive self-force due to vectors. The net force on black holes vanishes only in the extreme case (c"0). In the double-extreme case, the ADM mass is given by < at the "xed point, i.e. M"<(p, q, ? ) with the "xed values ? of scalars determined by     (R<( , p, q)/R ?) "0, since c"0"R?. From this, one obtains the bound on the ADM mass   M(S, , (p, q))5M(S, , (p, q)). Note, these results are derived only from the requirement of    regularity of con"gurations near the event horizon without using supersymmetry. We specialize to the case where the target space manifold is a KaK hler manifold spanned by complex scalars zG with KaK hler potential K: RK dzGdz H . G d ?d @" ?@ RzGRz H We consider the bosonic action of N"2 supergravity coupled to vector multiplets "!R#G M R zGR z HM gIJ#Im NKRFK FR gIHgJM# Re NKRFK 夹FR gIHgJM . (365) ,  GH I J IJ HM IJ HM The moduli dependent matrices kKR and lKR in (358) are given by l#ik"!N. So, < in (361) has the form <(p, q, ?)""Z(z, p, q)"#"D Z(z, p, q)", where Z is the central charge and D Z is its G G KaK hler covariant derivative. By applying properties of special geometries, one obtains the following ADM mass and scalar charges L

(366) M""Z"(z , p, q), RG"GGHM DM M ZM . H  By applying the general results in the previous paragraph, one can see that at the critical points of < (R <"0), where RG"0, Z is extremized: D Z"0"DM M ZM . Since the second covariant derivative G G I of "Z" at the critical point coincides with the partial (non-covariant) second derivative, one has (RM M R "Z") "G M "Z" . So, when G M is positive (negative) at the critical point, M at the critical point GH G H   GH  reaches its minimum (maximum). Generally, when G M changes its sign and becomes negative, some GH sort of a phase transition occurs and the e!ective Lagrangian breaks down unless new massless states appear. 5.5.2. Examples In the following, we discuss the explicit expression for M in the metric component (351) with speci"c prepotentials. 5.5.2.1. Axion dilaton black holes. The axion-dilaton black holes in the SO(4) [S;(4)] formulation of pure N"4 supergravity can be regarded as black holes in N"2 supergravity coupled to one vector multiplet with the prepotential F"!iXX (without prepotential). The holomorphic sections and ;(1) charges of S;(4) theory [165] (with hats) are related to those of SO(4) theory

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107

[162,164,194] (without hats) as [133,136,212]: XK "X, FK "F , XK "!F , FK "X ,    (367) q(  "q , q( "q, q(  "!q, q( "q .             First, we discuss the SO(4) case. We choose the gauge X"1. Then, the prepotential F"!iXX yields the KaK hler potential






1 X e)" z, 2(z#z ) X

and (356) can be solved to "x the moduli z in terms charges: q!iq   . z"  q!iq    So, one has central charge in terms of ;(1) charges:






qq#q q      (q#iq ) , Z"¸'q!M q' "      ' '   (q)#(q )    by solving (356). This leads to the ADM of the double-extreme black hole:

(368)

(369)

M""Z"""qq#q q " . (370)       The corresponding expressions in the S;(4) theory are obtained by applying the transformations (367). The moduli "eld and the ADM mass are [401,410] q( #iq(   , M""Z"""q(  q( !q( q(  " . z"        q(  !iq(     

(371)

5.5.2.2. N"2 heterotic vacua. The e!ective "eld theory of N"2 heterotic string is described by the N"2 theory with a prepotential [5,6]





L> X XX! (XG) . F"! X G This prepotential corresponds to the manifold [209,266]

(372)

SO(2,n) S;(1,1) ; SO(2);SO(n) ;(1) with the "rst factor parameterized by the axion}dilaton "eld S"!iX/X"!iz and the second factor being the special KaK hler manifold parameterized by n complex moduli zG"XG/X (i"2,2, n#1). S belongs to a vector multiplet and the remaining vector multiplets with the scalar components zG are associated with the ;(1) gauge "elds in the left moving sector of

 From this expression for K, one sees that the real part of the moduli z has to be positive, leading to the constraint Re z""qq#q q ".      

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heterotic string. In particular, the n"2 case is the S¹;-model [59,119] with the complex scalars S, ¹ and ; parameterizing each S¸(2, R) factor of the moduli group. S, ¹ and ; are related to zG as z"iS, z"i¹, z"i; ,

(373)

and, therefore, the prepotential takes the form: F"!S¹;.

(374)

It is convenient to apply a singular symplectic transformation [136] (de"ned as XP!F and  F P!X) on (XK, FR) to bring it to the form [135]:  XK XK " . (375) FR SgRKXK







In this basis, theory has a uniform weak coupling behavior as Im SPR and the holomorphic section satis"es the constraints 1X " X2"1F " F2"X ) F"0 .

(376)

Here, 1A " B2,AKgKRBR"AKgKRBR and A ) B,AKBK with






L

0

0

gKR" 0

L

0

0

0

!I




; L,

0 1 1 0

.

(377)

By solving (356), along with (375), one "xes S in terms of ;(1) symplectic charges and obtains the ADM mass of double extreme solution [178,236]: (1q "q 21q"q2!(q ) q) q ) q       #i , S"   1q "q 2 1q "q 2         M""Z""(1q "q 21q"q2!(q ) q)"(Im S)1q "q 2 .          

(378)

5.5.2.3. Cubic prepotential. We consider the following general form of cubic prepotential [161]: X?X@XA F"d , a, b, c"1,2, n . ?@A X 

(379)

The n "3 case is the S¹; model [59,208].  By solving (356), along with (379), one obtains the ADM mass at the "xed point in the moduli space [571]:



4 1 (D x ?)!9[q (q ) q)!2D] , (380) M"     3 ? 3q   where D,d q? q@ qA , D ,d q@ qA and D ,3D !q q. Here, x ? in (380) are real ?@A       ? ?@A     ? ?   ? solutions to the system of equations: d x ?x @"D . ?@G G

(381)

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The moduli are "xed in terms of the symplectic ;(1) charges as q? 3 x ? x ? 3 [q (q ) q)!2D]#  !i M . z?"     q 2 (D x A) 2 q (D x A)     A A When q "0, (356) can be solved explicitly to yield the ADM mass:   D (qD?#12q) , M"  3 ?



(382)

(383)

where D ,d qA , D?@,[D\] and D?,D?@q. And the moduli z? take the following "xed ?@ ?@A   ?@ @ point value in terms of the symplectic charges: D? q? z?" !i  DM . 6 2

(384)

When the prepotential (379) has an extra topological term [54,374,451] L c ) J  ?z? , 24 ? one only needs to apply the symplectic transformation [54,136] with the matrix






1

0

= 1

3Sp(2n #2) , 

where the non-zero components of =KR are = "c ) J /24. Then, the ADM mass is given by (380) ?  ? or (383) with the symplectic charges (q , q) replaced by   c J q "q!  ?q? ,   24   R ; q K"q (385) K !=RKq   c J  ? q "q! q . ? ? 24  



5.5.2.4. CP(n!1,1) model. The S;(1, n) CP(n!1, 1), S;(n);;(1) model has the isometry group S;(1, n). The n"1 case is the axion}dilaton black holes [83,404,409,415]. The form of prepotential depends on the way in which S;(1, n, Z) is embedded into Sp(2n#2, Z) [523]. For the Sp(2n#2, Z) embedding




X"

A

B

C D

of M";#i<3S;(1, n) given by A";, C"g<,

B"
(386)

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where g is an S;(1, n) invariant metric, the holomorphic prepotential is i F" XRgX . 2

(387)

For this case, X transforms as a vector under S;(1, n), i.e. XPMX, and the moduli space is parameterized by "( ,2, L>)2 as "1/(> and ?"z?/(> with >"1! z?z ?. In ? general, the ADM mass of the extreme solution has the form [524]: "m !n t!Q AG" A GA M " A , .1 2(1!ttM ! AGA M G) G where

(388)

m ,q#iq , n ,iq !q, Q ,iqG !q, t,X/X, AG,XG/X . A    A    GA   G For the following "xed-point values of the moduli "elds, which satisfy (356),

(389)

tM "n /m , A M G"Q /m , A A GA A the ADM mass M takes the minimum value [524]

(390)





g 0 1 M" ("m "!"n "!"Q ")"p(q q ) A GA   0 g 2 A

q

. (391) q   For other embedding X"SXS\ of S;(1, n) into Sp(2n#2) related via the symplectic transformation S3Sp(2n#2), the ADM mass is given in terms of new symplectic ;(1) charges (qYq )2"(qq )S\ by [524]     g 0 qY S2 . (392) M"p(qYq )S   0 g q  





5.5.2.5. General quadratic prepotential. We discuss the case where the lower-component of < is proportional to the upper component [62]:



¸'

¸'



, (393) M R ¸) ( () where R "a !ib with real matrices a and b . Note, it is su$cient to consider the case '( '( '( '( '( where R "!ib , since the most general case with a O0 is achieved by applying the symplectic '( '( '( transformation "




1



0

3Sp(2n#2) a 1 '( on the con"guration with a "0. '(  The ADM mass and entropy transform under this symplectic transformation similarly as (392).

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By solving (356) with (393), one obtains the ADM mass






 


(RM !R)\ (R!RM )\R i M" (qq )   RM (R!RM )\ RM (RM !R)\R 2 b\ !b\a 1 " (qq )   !ab\ b#ab\a 2

q

q  

q

. (394) q   The case R"!ig, where g is an S;(1, n) invariant metric, is the CP(n!1, 1) model, i.e. (394) reduces to (391). The most general extreme solution to this model has the form [597] ds"!e\3 dt#e3 dx ) dx , F' "e R HI ', G "e R H , IJ IJM M (IJ IJM M ( >,ZM ¸"i(R!RM )\(RM HI !H) ,

(395)

where




e3"i(H2 HI 2)




(RM !R)\

(R!RM )\R




RM (R!RM )\ RM (RM !R)\R






q' q HI '" hI '#   , H " h # ( . ( ( r r

H HI

, (396)

The asymptotically #atness condition leads to the following constraint on hI ' and h in (396): ( (RM !R)\ (R!RM )\R h "!i . (397) (h2 hI 2) RM (R!RM )\ RM (RM !R)\R hI







5.6. Quantum aspects of N"2 black holes Supersymmetric "eld theories respect remarkable perturbative non-renormalization theorems. In N"1 theories, superpotentials are not renormalized in perturbation theory [321,375]. N54 theories are "nite to all orders in perturbative quantum corrections [473,573]. So, the classical BPS solutions in N54 theories are exact to all orders in perturbative corrections. (Cf. Some classical solutions of N"4 theories are also exact solutions [47,48,173,174,369}371,518,607,608,614,615] of conformal p-model of string theory and, therefore, exact to all orders in a-corrections.) For N"2 theories, prepotentials, which determine the Lagrangians, receive perturbative quantum corrections up to one-loop level [9,209,321,548]. Hence, one has to study quantum e!ect on prepotentials for complete understanding of solutions in N"2 theories. In the following, we study the quantum aspects of black holes in the e!ective N"2 theories of compacti"ed superstring theories. It is conjectured that the E ;E heterotic string on K3;¹ and   the type-II string on a Calabi}Yau manifold are a N"2 string dual pair [13,14,25,119, 257,396,397,418]. Since the dilaton}axion "eld S belongs to a vector multiplet (hyper multiplet) of

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the heterotic (type-II) theory, moduli space of hyper multiplet (prepotential for a vector multiplet) is exact at the tree level, due to the absence of neutral perturbative couplings between vector multiplets and hypermultiplets [13,118,167,168,418,435]. Thus, applying the duality between heterotic and type-II strings, one can compute the exact prepotential for vector multiplets (hyper multiplet superpotential) of the heterotic (type-II) theory. The prepotentials of the N"2 e!ective "eld theories of these string theories contain the cubic terms: F(X)"d X?X@XA/X , (398) ?@A plus correction terms that include part of quantum corrections, instanton and topological terms which cannot be included in (398). For the type-IIA string on a Calabi}Yau three-fold, real coe$cients d are the topological intersection numbers d , J J J , where J 3H(>, Z) ?@A ?@A ? @ A ? are the KaK hler cone generators. For the heterotic string on K3;¹, d describe the classical parts ?@A and the non-exponential parts of perturbative corrections to the prepotential. The KaK hler potential associated with (398) is K(z, z )"!log(!id (z!z )?(z!z )@(z!z )A) . (399) ?@A General double-extreme black holes and a special class of extreme black holes with non-constant scalars of the N"2 theory with (398) are discussed in [50]. In the following, we study the e!ect of the quantum correction terms of the prepotential on the classical solutions [50,51,54,123,514]. 5.6.1. E ;E heterotic string on K3;¹   At generic points in moduli space, the E ;E heterotic string on K3;¹ is characterized by 65   gauge-neural hypermultiplets (20 from the K3 surface and 45 from the gauge bundle) and 19 vectors (18 from vector multiplets and 1 from the gravity multiplet). So, the moduli z? (a"1,2, n )  in the vector multiplets consist of the axion}dilaton S, the ¹ moduli ¹ and ;, and Wilson lines "i
(400)

We denote the moduli other than S as ¹K (m"2,2, n ). The number of Wilson lines  with one of ;(1) factors coming from the gravity multiplet) depends on the choice of S;(2) bundles with instanton numbers (d , d )"(12!n, 12#n) [4,116,397]. For example, the S¹; model (i.e. the complete Higgsing of   the D"6 gauge group E ;E ) is possible for n"0, 1, 2.   Since prepotentials of N"2 theories are not renormalized beyond one-loop perturbative levels, the prepotentials are written in the form [9,10,136,209]: F"F#F#F,. ,

(401)

 The scalar components of the remaining hypermultiplets in 56 of E are not spectrum-preserving moduli, since their  non-zero vacuum expectation values induce mass for some of the non-Abelian "elds, resulting in change in the spectrum of light particles in the e!ective theories.

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where F is the tree-level prepotential and F (F,.) is the one-loop (non-perturbative) correction. The classical moduli space of the E ;E heterotic string on K3;¹ is   [132,135,136,255,263,266,272] SO(2, n !1) S;(1, 1)   SO(2);SO(n !1) ;(1)  with S residing in the separate moduli space S;(1, 1) . ;(1) The tree-level prepotential is









X LT F"! XX! (XG) "!S ¹;! (


(402)



K"!log (S#SM )!log (¹#¹M )(;#;M )! (




(404) F"!S ¹;! (
(405)

IJ@

where c"c (0)f(3)/8p and e[x]"exp 2pix. Here, c (4kl!b) are the expansion coe$cients of L L particular Jacobi modular form [117] and p (¹, ;, <) is the one-loop cubic polynomial, which L depends on the particular instanton embedding n, given by [70,117,451] p (¹, ;, <)"!;!(#n)<#(1#n);<#n¹< .    L 

(406)

5.6.1.1. Perturbative corrections. In Section 5.5.2, we obtained the general expression for entropy (or the ADM mass of the double extreme black hole) in the tree level e!ective theory of the

 This is possible for the instanton embedding with n"0, 1, 2.

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heterotic string on K3;¹. (See (378) for the tree level expression.) The entropy depends on the symplectic charge vector Q"(q q) through the full triality invariant form   D"1q "q 21q"q2!(q ) q) [123] and is invariant under T-duality since the dilaton Im S       and the combination 1q "q 2 remain intact under T-duality [123,136,209].     Once perturbative quantum corrections are considered, T-duality transformations get modi"ed due to the one-loop corrections to the prepotential:





F"F#F"!S ¹;! (
(409)

K where F K ,RF/RX . Note, F keeps the classical value.  Whereas XK K transforms exactly the same way as in classical theory, the T-duality transformation rules of FK K get modi"ed at one-loop level due to the modi"cation of prepotential [9,121,209,338]:

XK KP;K KRXK R, FK KP[(;K 2)\]RKFK R#[(;K 2)\C]KRXK R ,

(410)

where ;3SO(2, 2#n, Z) and the symmetric integral matrix C encodes the quantum corrections. From the relation X"!FK "!iSXK , one sees that S is no longer invariant under the  perturbative T-duality transformation (410), but transforms as [209] i[(;K 2)\]R (H #CRDXK D)  RK SPS# . ;K KXK

(411)

Note, 1q "q 2 in the classical expression (378) is still invariant under the perturbative     T-duality, but the dilaton Im S transforms under the perturbative T-duality (411). Since superstring theories are exact under T-duality order by order in perturbative corrections, one expects entropy to be invariant under T-duality. One way of making entropy to be manifestly invariant under T-duality is to introduce the invariant dilaton}axion "eld S [209,338] which do not transform  under T-duality. This motivates the following conjectured expression for entropy at one-loop level [123]: S "p"Z ""p(Im S )1p "p 2 .       

(412)

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The perturbative dilaton Im S is understood as follows. The perturbative prepotential (407)  leads to the following perturbative KaK hler potential [209]: K"!log[(S#SM )#< (¹K,¹M K)]!log[(¹K#¹M K)g (¹L#¹M L)] , (413) %1 KL where 2(h#hM )!(¹K#¹M K)(R Kh#R M KhM ) 2 2 (414) < (¹K, ¹M K)" %1 (¹K#¹M K)g (¹L#¹M L) KL is the Green}Schwarz term [124,203,450] describing the mixing of the dilaton with the other moduli ¹K. From this expression for K, one infers that the true string perturbative coupling constant g is of the modi"ed form:  1 4p " (S#SM #< (¹K, ¹M K)) . (415) %1 2 g  One can prove this conjectured form (412) of the perturbative entropy by solving (356) with (407) substituted. In the following, as examples of quantum corrections to N"2 black holes, we "nd explicit expressions of entropy for the axion-free (Re z?"0) solution with special forms of the perturbative prepotential. We assume that 2>!ip">#>M ,jO0. Note, the symplectic magnetic charge in the perturbative basis de"ned by (408) is expressed in terms of the charges pH and qR in the original basis as q( "(p, q , p,2). By solving the stability equation P!PM "iQ with the    following perturbative prepotential: d >?>@>A #ic(>), >K,ZM XK , F(>)" ?@A >

(416)

one obtains the following expression for the entropy (cf. See (323)):





S (p) "!2(q !2cj) j# .  p j

(417)

Here, j in (417) is obtained by solving the following equation derived from (356): d p?p@pA 3p q " ?@A #2cj, q "! d p@pA .  ? j j ?@A

(418)

For the case cpO0, one can express j in terms of charges as 3pq #p?q  ?, j" 6cp

(419)

from which one sees that charges satisfy the following constraint when cp"0: 3pq #p?q "0 .  ? For the case c"0 and q O0, j is  d p?p@pA q , j"$ ?@A 



(420)

(421)

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with the sign $ determined by the condition q j(0 that the entropy should be positive. As  a special case, when c"p"0 the entropy is S "2(q d p?p@pA ,  ?@A p

(422)

with the solution having only n #1 independent charges, i.e. q "0. In particular, with the cubic  ? prepotential (>) >>> #a , F(>)"!b > > entropy (422) becomes S "2(!q (bppp!a(p)) .  p

(423)

The explicit solution with non-constant scalar "elds is obtained by just substituting the symplectic charges by the associated harmonic functions in the stabilization equations. For the case where the prepotential is the general cubic prepotential, i.e. (416) with c"0, the solution has the form [50]: ds"!e\3 dt#e3 dx ) dx, e3"(H d H?H@HA ,  ?@A F? "e R H?, F "R (H )\, z?"iH H?e\3 , KL KLN N  K K   where the harmonic functions in the solution are









(424)



p? q H?"(2 h?# , H "(2 h #  .   r r

(425)

Here, the constants h's are constrained to satisfy the asymptotically #at spacetime condition: 4h C h?h@hA"1 ,  ?@A and, therefore, the ADM mass of the solution is

(426)

q 1 M"  # p?h C h@hA . (427) 4h 2  ?@A  When h's take the values at the horizon expressed in terms of charges as h "q /c and h?"p?/c,   the ADM mass (427) reduces to the entropy S"pM" in (422).

 As a special case, we consider the following prepotential corresponding to the S¹; model with the cubic quantum correction: (X) XXX #a . F" X X

(428)

For this case, the metric component has the form








q e3"4 h #   r

p h! r







p h! r






p p  h# #a h# . r r

(429)

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Note, the quantum correction term acts as a regulator, smoothing out singularity, for example those of massless black holes [48,49,179,318,406,408,488,578], of classical solutions. 5.6.2. Type-II string on Calabi}Yau manifolds Type-IIA string on a Calabi}Yau three-fold [115,373] gives rise to N"2 theory with h #1  vector "elds and h #1 hypermultiplets (h and h being the Hodge numbers of the three-fold),    with the additional hyper multiplet and vector "eld coming from those associated with the dilaton and the gravity multiplet, respectively. The moduli in the vector multiplets consist of the KaK hlerclass moduli t? (a"1,2, n "h ), where h "dim(H(M, Z)). The general form of type-IIA    prepotential is [115,373]






sf(3) 1 1 # nP ¸i e i d t? , (430) F''"! C t?t@tA! ?@A ? 2(2p) (2p) 2 B2BF  6 B BF ? where nP  2 F are the rational instanton numbers of genus 0 and s is the Euler number. Here, the B  B prepotential is de"ned inside of the KaK hler cone p(K)"+ t?J " t?'0,, where J are the (1, 1)? ? ? forms of the Calabi}Yau 3-fold M. In the large KaK hler class moduli limit (t?PR), only the classical part in the prepotential, which is related to the intersection numbers, survives. To the general form (430) of the prepotential, one can add an extra topological term which is determined by the second Chern class c of the Calabi}Yau 3-fold:  F c )J (431) F'' "  ?t?; c ) J , c J .  ?   ? 24 + ? The e!ect of adding such term to the prepotential is the symplectic transformation corresponding to constant shift of the h-angle [630] (cf. the paragraph below (384)). In particular, the type-IIA model dual to the Heterotic string on K3;¹ with n "4 and n"2  (an S¹;< model) [117] corresponds to the compacti"cation on the Calabi}Yau three-fold P (20) [397] with h "4 and Euler number s"!372 [71]. The transformations be  tween the moduli in the pair of these type-IIA and heterotic theories are



t";!2<, t"S!¹, t"¹!;, t"< ,

(432)

and some of the instanton numbers [71,117] and the Euler number of the threefold are given in terms of the quantities of the heterotic string by: nP "!2c (4kl!b), s"2c (0) . J>IIJ>I>@   The cubic intersection-number part of the prepotential is

(433)

!F'' "t((t)#tt#4tt#2tt#3(t))#(t)#8(t)t#t(t)#2(t)t 
  #8ttt#2(t)t#12t(t)#6t(t)#6(t) . (434) In the limit t"<"0 (i.e. the S¹; model of the heterotic string with n"2), the model reduces to the type-IIA string compacti"ed on P (24) with h "3 and s"!480. The linear   topological term, for this case, takes the form: 23 11  c )J  t" t#t#2t"S#¹# ; . 24 6 6 

(435)

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5.6.2.1. Entropy formula. For black holes in type-IIA string on a Calabi}Yau 3-fold, entropy depends not only on ;(1) charges, but also on the topological quantities of the 3-fold. In the following, we consider the double-extreme black holes in the type-IIA superstring on a Calabi}Yau 3-fold with the following special form of prepotential: C >?>@>A c ) J #  ?>>? , F''(>)"! ?@A 24 6>

(436)

where >?,ZM X?. Note, the prepotential is determined by the classical intersection numbers C "!6d and the expansion coe$cients c ) J "24= of the 2nd Chern class c of the ?@A ?@A  ? ?  3-fold. Note, the above form of prepotential can be obtained by imposing the symplectic transformation of the following form on the prepotential without 2nd Chern class terms:



p K q R

"




0

pK

= 1

qR

1

.

(437)

The general expression for the entropy with p"q "0 and = "0 is obtained in (422). By ? ? imposing the symplectic transformation (437), one obtains the following entropy for the type-IIA theory with the prepotential (436) and the charge con"guration p"q "0: ? S "2((q != p ?)d p @p Ap B .  ? @AB p

(438)

5.6.3. Higher-dimensional embedding The above D"4 black holes in string theories with the cubic prepotential arise from the compacti"cation of the following intersecting M-brane solution





1 1 du dv#H du# C H?H@HA dx#H?u , ds "  ?  (C H?H@HA) 6 ?@A  ?@A

(439)

due to the duality [635] among the heterotic string on K3;¹, the type-II string on a Calabi}Yau 3-fold (C>) and M-theory on C>;S [103,261,262]. This solution corresponds to 3 M 5-branes (with the corresponding harmonic functions H?, a"1, 2, 3) intersecting over a 3-brane (with the spatial coordinates x), and the momentum (parameterized by the harmonic function H ) #owing  along the common string. Here, the intersection of 4-cycles that each M 5-brane wraps around is determined by the parameters C and each pair of such 4-cycles intersect over the 2-dimensional ?@A line element u . ? Compactifying the internal coordinates and the common string direction, one obtains the following D"4 solution with spacetime of the extreme Reissner}Nordstrom black hole:



1 1 dt# ! HC H?H@HA dx . ds"! ?@A  6 (!HC H?H@HA ?@A 

(440)

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6. p-branes The purpose of this section is to review the recent development in p-branes and other higherdimensional con"gurations in string theories. (See also [284] for another review on this subject.) Study of p-branes plays an important part in understanding non-perturbative aspects of string theories in string dualities. The recently conjectured string dualities require the existence of p-branes within string spectrum along with well-understood perturbative string states. Furthermore, microscopic interpretation of entropy, absorption and radiation rates of black holes within string theories involves embedding of black holes in higher dimensions as intersecting p-brane. This section is organized as follows. In Section 6.1, we summarize properties of single-charged p-branes. In Section 6.2, we systematically study multi-charged p-branes, which include dyonic p-branes and intersecting p-branes. In the "nal section, we review the lower-dimensional p-branes and their classi"cation. We also discuss various p-brane embeddings of black holes. 6.1. Single-charged p-branes In this section, we discuss p-branes which carry one type of charge. Such single-charged p-branes are basic constituents from which `bound statea multi-charged p-branes, such as dyonic p-branes and intersecting p-branes, are constructed. p-branes in D dimensions are de"ned as p-dimensional objects which are localized in D!1!p spatial coordinates and independent of the other p spatial coordinates, thereby having p translational spacelike isometries. Note, the allowed values of (D, p) for which supersymmetric p-branes exist are limited and can be determined by the bose-fermi matching condition [2]. (The details are discussed in Section 2.2.3.) The e!ective action for a single-charged p-brane has the form:







1 1 1 d"x (!g R! (R )! e\?N(F , (441) I (p)" N> " 2 2(p#2)! 2i " where i is the D-dimensional gravitational constant, is the D-dimensional dilaton and " F ,dA is the "eld strength of (p#1)-form potential A . Here, the parameter a(p) given N> N> N> below is determined by the requirement that the e!ective action (441) and the p-model action (443) scale in the same way [242] under the rescaling of "elds 2(p#1)(p #1) 2(p#1)(p #1) "4! , a(p)"4! D!2 p#p #2

(442)

where p ,D!p!4 corresponds to spatial dimensions of the dual brane. 6.1.1. Elementary p-branes Electric charge of (p#1)-form potential A in I (p) is carried by the `elementarya p-brane. N> " The `elementarya p-brane has a d-function singularity at the core, requiring existence of singular electric charge source for its support so that equations of motion are satis"ed everywhere. Namely, the electric charge carried by the `elementarya p-brane is a Noether charge with the Noether

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current associated with the p-brane worldvolume p-model action:





1 (p!1) (!c S "¹ dN>m ! (!ccGHR X+R X,g e?N(N># G H +, N N 2 2



1 ! eG2GN>R X+2R N>X+N>A 2 N> , G G + + (p#1)!

(443)

where X+(mG) (M"0, 1,2, D!1; i"0, 1,2, p) is the spacetime trajectory of p-branes, ¹ is N the elementary p-brane tension, c (m) (g (X)) is the worldvolume (spacetime) metric and GH +, in (443) is related to the canonical metric A "A 2 N> dX2dXN>. The metric g +, N> + + g  in the Einstein-frame e!ective action (441) through the Weyl-rescaling g "e?N(N>g  . +, +, +, The `elementarya p-brane is a solution to the equations of motion of the combined action I (p)#S . In particular, "eld equation and Bianchi identity of A are " N N> d夹(e\?N(F )"2i (!1)N> 夹J , dF "0 , (444) N> " N> N> where the electric charge source current J "J+2+N>dX 2dX N> is + + N> d"(x!X) J+2+N>(x)"¹ dN>m eG2GN>R X+2R N>X+N> . (445) N G G (!g



Here, 夹 denotes the Hodge-dual operator in D dimensions, i.e. 1 (夹<)+2+"\B, e+2+"< "\B>2 " + + d! with the alternating symbol e+2+" de"ned as e2"\"1. The Noether electric charge is





1 (夹J) " e\?N( 夹F , (446) "\N\ (2i N > N> +"\N\ 1 "  where SN> surrounds the elementary p-brane. In solving the Euler}Lagrange equations of the combined action I (p)#S to obtain the " N elementary p-brane solution, one assumes the P ;SO(D!p!1) symmetry for the con"guraN> tion. Here, P is the (p#1)-dimensional Poincare group of the p-brane worldvolume and N> SO(D!p!1) is the orthogonal group of the transverse space. Accordingly, the spacetime coordinates are splitted into x+"(xI, yK), where k"0,2, p and m"p#1,2, D!1. Due to the P invariance, "elds are independent of xI, and SO(D!p!1) invariance further requires that N> this dependence is only through y"(d yKyL. In solving the equations, it is convenient to make KL the following static gauge choice for the spacetime bosonic coordinates X+ of the p-brane: Q" N

XI"mI, >K"constant ,

(447)

where k"0, 1,2, p (m"p#1,2, D!1) corresponds to directions internal (transverse) to the p-brane. The general Ansatz for metric with P ;SO(D!p!1) symmetry is N> ds"eWg dxI dxJ#e Wd dyK dyL . (448) IJ KL

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By solving the Euler}Lagrange equations with these AnsaK tze, one obtains the following solution for the elementary p-brane: ds"f (y)\N >N>N >g dxI dxJ#f (y)N>N>N >d dyK dyL , IJ KL e("e(f (y)\?N, A

I2IN>

e?N(Š "! e 2 N> f (y)\ , I I Ng

(449)

where Ng is the determinant of the metric g and f (y) is given by IJ



2e?N(Ši ¹ 1 " N 1# , p '!1 , ( pJ #1)X  yNJ > N > f (y)" e?N(Ši ¹ " N ln y, p "!1 . 1! n

(450)

By solving the Killing spinor equations with the P ;SO(D!p!1) symmetric "eld AnsaK tze, N> one sees that the extreme `elementarya p-brane preserves 1/2 of supersymmetry with the Killing spinors satisfying the constraint: (1!CM )e"0 ,

(451)

where 1 CM , eGG2GNR X+R X+2R NX+NC  2 N , G G G ++ + (p#1)!(!c

(452)

which has properties CM "1 and Tr CM "0 that make (1$CM ) a projection operator. This orig inates from the fermionic i-symmetry of the super-p-brane action. The extreme `elementarya p-brane (449) saturates the following Bogomol'nyi bound for the mass per unit area M " d"\N\y h : N  1 i M 5 "Q "e?N( , " N (2 N

(453)

where h is the total energy-momentum pseudo-tensor of the gravity-matter system. +, 6.1.2. Solitonic p -branes The magnetic charge of A is carried by a solitonic p -brane, which is topological in nature and N> free of spacetime singularity. Since magnetic charges can be supported without source at the core, solitonic p -branes are solutions to the Euler}Lagrange equations of the e!ective action I (p) alone. " is de"ned as The `topologicala magnetic charge P  of A N> N



1 P" F , N (2i N> N> 1 "

(454)

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where SN> surrounds the solitonic p -brane. The magnetic charge P  is quantized relative to the N electric charge Q via a Dirac quantization condition: N n Q P N N" , n3Z . (455) 2 4n Solitonic p -brane solution has the form: ds"g(y)\N>N>N >g dxI dxJ#g(y)N >N>N >d dyK dyL , IJ KL (456)  e("e( g(y)?N, F "(2i P  e /X , N> " N N> N> where e is the volume form on SN> and g(y) is given by N> (2e\N>(Š?NN>N >i P  1 " N . (457) g(y)"1# yN> (p#1)X N> The `solitonica p -brane (456) preserves 1/2 of supersymmetry and saturates the following Bogomol'nyi bound for the ADM mass per unit p -volume: 1 M  5 "P  "e\?N( . N (2 N

(458)

Note, the sign di!erence in dependence of the mass densities (453) and (458) on the dilaton asymptotic value . So, in the limit of large , the mass density M (M  ) is large (small), and vice   N N versa. 6.1.3. Dual theory We consider the theory whose actions II (p ) and SI  are given by (441) and(443) with p replaced by " N p "D!p!4. So, in this new theory, p -branes carry electric charge QI  and p-branes carry N magnetic charge PI of (p #1)-form potential AI  . N N> Since a p-brane from one theory and a p -brane from the other theory are both `elementarya (or `solitonica), it is natural to assume that these branes are dual pair describing the same physics. One assumes that the graviton and the dilaton in the pair actions are the same, but the "eld strengths are related by FI  "e\?N( 夹F . (Thereby, the role of "eld equations and F and FI  N> N> N> N> Bianchi identities are interchanged.) Then, it follows that since Q "PI and P  "QI  the Dirac N N N N quantization conditions for electric/magnetic charge pairs (Q , P  ) and (QI  , PI ) lead to the following N N N N quantization condition for the tensions of the dual pair `elementarya p-brane and p -brane: (459) i ¹ ¹  ""n"p . " N N By performing the Weyl-rescaling of metrics to the string-frame, one sees that the p-brane (p -brane) loop counting parameter g (g  ) is N N g "e"\?N(ŠN>, g  "e"\?N (ŠN > , (460) N N with a(p )"!a(p). It follows that the brane loop counting parameters of the dual pair are related by (461) (g )N>"1/(g  )N > . N N

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Thus, the strongly (weakly) coupled p-branes are the weakly (strongly) coupled p -branes. In particular, a string (p"1) in D"6 are dual to another string (p "6!1!4"1), thereby strongly (weakly) coupled string theory being equivalent to weakly (strongly) coupled dual string theory. Other interesting examples are membrane/"vebrane dual pair in D"11, string/"vebrane dual pair in D"10 [219,239,240], self-dual 3-branes in D"10 and self-dual 0-branes in D"4. Note, in D"11 supergravity strong and weak coupling limits do not have meaning due to absence of the dilaton. 6.1.4. Blackbranes We discuss non-extreme generalization of BPS `solitonica p -brane in Section 6.1.2. Such solution is obtained [464] by solving the Euler}Lagrange equations following from I (p) with " R;SO(p#3);E(p ) symmetric "eld AnsaK tze. The non-extreme p -brane solution is ds"!D D\N >N>N > dt#D\D?NN>\ dr > \ > \ #rD?NN> dX #DN>N>N > dxG dx , \ N> \ G e\("D?N, F "(p#1)(r r )N>e , (462) \ N> >\ N> where D ,1!(r /r)N> and i"1,2, p . The magnetic charge P  and the ADM mass per unit ! ! N p -volume M  of (462) are N X X (463) P  " N> (p#1)(r r )N>, M  " N>[(p#2)rN>!rN>] . >\ N > \ N (2i 2i " " The solution (462) has the event horizon at r"r and the inner horizon at r"r , and therefore is > \ alternatively called `blacka p -brane. The requirement of the regular event horizon, i.e. r 5r , > \ leads to the Bogomol'nyi bound (2M  5"P  "e\?N(. N N In the limit r "0, and A become trivial, i.e. P  "0, and spacetime reduces to the product \ N> N of (D!p )-dimensional Schwarzschield spacetime and #at RN . In the extreme limit (r "r ), the > \ symmetry is enhanced to that of the BPS p -brane, i.e. P  ;SO(p#3), since g "g G G. Such RR VV N> extreme solution is related to the BPS solution (456) through the change of variable yN>"rN>!rN>. In the extreme limit, the event-horizon and the singularity completely disap\ pear, i.e. becomes soliton with geodesically complete spacetime in the region r'r "r . > \ 6.2. Multi-charged p-branes In this section, we discuss p-branes carrying more than one types of charges. These p-branes are `bound statesa of single-charged p-branes. Multi-charged p-branes are classi"ed into two categories, namely `marginala and `non-marginala con"gurations. The `marginala (BPS) bound states have zero binding energy and, therefore, the mass density M is sum of the charge densities Q of the G constituent p-branes, i.e. M" Q . Such bound states with n constituent p-branes preserve at least G G ()L of supersymmetry. The marginal bound states include intersecting and overlapping p-branes.  The `non-marginala (BPS) bound states have non-zero binding energy and the mass density of the

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form M" Q. The quantized charges Q of `non-marginala bound states take relatively prime G G G integer values. In general, non-marginal p-brane bound states are obtained from single-charged p-branes or marginal p-brane bound states by applying the S¸(2, Z) electric/magnetic duality transformations and, therefore, preserve the same amount of supersymmetries as the initial p-brane con"gurations (before the S¸(2, Z) transformations). In particular, intersecting p-branes are further categorized into orthogonally intersecting p-branes and p-branes intersecting at angles. 6.2.1. Dyonic p-branes In D"2p#4, i.e. dimensions for which p"p , p-branes can carry both electric and magnetic charges of A . Examples are dyonic black holes (p"0) in D"4; dyonic strings (p"1) in D"6; N> dyonic membranes (p"2) in D"8. Such dyonic p-branes can be constructed by applying the D"2(p#2) S¸(2, Z) electric/magnetic duality transformations on single-charged p-branes. #+ These dyonic p-branes have P ;SO(D!p!1) symmetry and are characterized by one harN> monic function, just like single-charged p-branes, since the S¸(2, Z) transformations leave the Einstein-frame metric intact. In particular, in D"2 mod 4, the (p#2)-form "eld strengths satisfy a real self-duality condition F "!夹F and, thereby, electric and magnetic charges are N> N> identi"ed, i.e. Q "!P . N N The S¸(2, Z) electric/magnetic duality transformations of 2k-form "eld strength F in D"4k #+ I can generally be understood as the ¹ moduli transformations of D"(4k#2) theory compacti"ed on ¹ [311]. We consider the following D"(4k#2) action





I " dI>x (!gR#a [dCH#H 夹H] ,  I>

(464)

where C is a (4k#2)-form potential with the "eld strength H"dC. We compactify the action (464) on ¹ with the following AnsaK tze for the "elds 1 ("q" dy#2Re q dx dy#dx) , ds(M )"ds(M )# I> I Im q C"B dy#A dx,

H"G dy#F dx ,

(465)

where q is the moduli parameter of ¹, (x, y) are coordinates of ¹ (i.e. x&x#1 and y&y#1), and A, B [F, G] are (2k!1)-forms [2k-forms] in D"4k. By applying the self-duality condition [625] of H, one "nds that 2k-form G is an auxiliary "eld, which can be eliminated by its "eld equation as G"Im q 夹F!Re q F, and one "nds that F" dA. The "nal expression for the D"4k action is





 

1 dq dq #a FG , I " dIx (!g R! I 2 (Im q)

(466)

where a real constant a is, in the convention of [388], given by a"2[(2k)!]\. The complex scalar q, which is expressed in terms of real scalars o and p as q"2o#ie\N, parameterizes the target space manifold M"S¸(2, Z)!S¸(2, R)/;(1), which is the fundamental domain of S¸(2, Z) in the upper half q complex plane. Here, ;(1) is a subgroup of S¸(2, R) which preserves the vacuum

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expectation value 1q2. The "eld equations are invariant under the following S¸(2, R) electric/ #+ magnetic duality transformation of F:




a b aq#b ; A" 3S¸(2, R) . (F, G)P(F, G)A\, qP cq#d c d

(467)

Note, this S¸(2, Z) transformation is not S-duality of string theories, since p is not the D"4k #+ dilaton. (In (464), the dilaton is set to zero.) The following electric Q and magnetic P charge densities form an S¸(2, R) doublet: #+ 1 1 G, P" F. (468) Q" X X I I This S¸(2, R) transformation on single-charged (2k!2)-branes yields (2k!2)-branes which #+ carry both electric and magnetic charges of A . Such dyonic p-branes preserve 1/2 of superI\ symmetry. Charges (Q, P) and (Q, P) of two dyonic (2k!2)-branes satisfy the generalized Nepomechie-Teitelboim quantization condition [482,591]:



QP!QP3Z .



(469)

When such dyonic (2k!2)-branes are uplifted to D"(4k#2) through (465), the solutions become self-dual (2k!1)-branes [230,238]. The electric and magnetic charges of the dyonic (2k!2)-branes are interpreted as winding numbers of the self-dual (2k!1)-branes around the x and y directions of ¹. The dyonic (2k!2)-branes (k"1, 2) uplifted to D"11 describe p-brane which interpolates between the M 2-brane and the M 5-brane, i.e. a membrane within a 5-brane (2"2 , 5 ). + + In the following, we speci"cally discuss the k"2 case [311,388]. The associated action ((466) with k"2) is type-IIB e!ective action (155) consistently truncated and compacti"ed to D"8. The N"2, D"8 supergravity has an S¸(3, R);S¸(2, R) on-shell symmetry, whose S¸(3, Z);S¸(2, Z) subset is the conjectured U-duality symmetry of D"8 type-II string. The R-R 4-form "eld strength and its dual "eld strength transform as (1, 2) under S¸(3, R);S¸(2, R). This U-duality group contains as a subset the SO(2, 2, Z),[S¸(2, Z);S¸(2, Z)]/Z T-duality group. The S¸(2, Z) factor  (in S¸(3, Z);S¸(2, Z)) is the electric/magnetic duality (467), which is a subset of `perturbativea T-duality group. All the `non-perturbativea transformations are contained in the S¸(3, Z) factor. The following dyonic membrane solution with 1q2"i is obtained by applying the ;(1)L S¸(2, R) transformation with a parameter m to purely magnetic membrane: #+ ds "H\ ds(M)#H ds(E) ,  F " cos m(夹 dH)# sin m dH\e(M) ,    sin(2m)(1!H)#2iH , (470) q" 2(sinm#H cosm) where ds(M) [ds(E)] is the metric of D"3 Minkowski space M (the "ve-dimensional Euclidean space E), e(M) is the volume form of M, 夹 is the Hodge-dual operator in E and H is a harmonic function given by (457) with p"2. Here, ;(1) is the subgroup of S¸(2, R) that #+ preserves 1q2"i. In the quantum theory, the ;(1) group breaks down to Z due to Dirac 

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quantization condition, resulting in either electric or magnetic solutions when applied to purely magnetic solution. (But the solution in (470) with an arbitrary m satis"es the Euler}Lagrange equations following from (466) and, therefore, can be taken as an initial solution to which the `integer-valueda duality transformations are applied.) To generate dyonic solutions with an arbitrary 1q2 and are relevant to the quantum theory, one has to apply the full S¸(2, R) transformation to (470). The pair of electric and magnetic charges of #+ such dyonic solutions take co-prime integer values. The ADM mass density of this extreme dyonic membrane saturates the Bogomol'nyi bound: M5[e6N7(Q#21o2P)#e\6N7P] , 

(471)

and therefore 1/2 of supersymmetry is preserved. When uplifted to D"10, this dyonic membrane becomes the following self-dual 3-brane of type-IIB theory [238]: ds "H\[ds(M)#dv]#H[ds(E)#du] ,  (472) F "( 夹 dH) du# dH\e(M) dv ,    where the coordinates (u, v) are related to the coordinates (x, y) in (465) through (y, x)"(v, u)A\ (A3S¸(2, Z)). The electric and magnetic charges of the above D"8 dyonic membrane are respectively interpreted as the winding numbers of this D"10 self-dual 3-brane around x and y directions of ¹. The D"8 dyonic membrane (470) uplifted to D"11 is a special case of orthogonally intersecting M-brane interpreted as a membrane within a 5-brane (2"2 , 5 ): + + ds "H(sinm#H cosm)[H\ ds(M)#(sinm#H cosm)\ ds(E)#ds(E)] ,  1 1 3 sin 2m F" cos m 夹 dH# sin m dH\e(M)# dHe(E) . (473)  2 2 2[sin m#H cos m] This M-brane bound state interpolates between the M 5-brane and the M 2-brane as m is varied from 0 to p/2. As long as magnetic charge is non-zero (mO0), (473) is non-singular, thereby singularity of the D"8 dyonic membrane (470) is resolved by its interpretation in D"11. By compactifying an extra spatial isometry direction of (473) on S, one obtains 3 di!erent types of dyonic p-branes in type-IIA theory: (i) a membrane within a 4-brane (2"2, 4), (ii) a membrane within a 5-brane (2"2, 5) and (iii) a string within a 4-brane (1"1, 4). One can construct dyonic p-branes in DO2(p#2) by compactifying purely electric or purely magnetic p-branes down to D"2(p#2) (or D"2(p #2)), applying electric/magnetic duality transformations of p-branes (or p -branes) in D"2(p#2) (or D"2(p #2)), and then uplifting the

 The above procedure can be applied to intersecting p-branes to generate p-brane bound states which interpolate between di!erent intersecting p-branes, e.g. the interpolation between the intersecting two (p#2)-branes and the intersecting two p-branes; the interpolation between the intersecting p-brane and (p#2)-brane and the intersecting (p#2)-brane and p-brane [151].

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dyonic solution to the original dimensions. In particular, the Z subset of the D"2(p#2) (or  D"2(p #2)) electric/magnetic duality transformations relates electric p-brane and magnetic p-brane. Thus, the elementary p-brane in DO2(p#2) is interpreted as the electrically charged partner of magnetic p-brane, establishing electric/magnetic duality between electric `elementarya p-brane and magnetic `solitonica p-brane in DO2(p#2). To enlarge this Z electric/magnetic  duality symmetry in D"2(p#2) (or D"2(p #2)) to the S¸(2, Z) symmetry so that one can generate dyonic p-branes from single-charged p-branes, one has to turn on a pseudo-scalar "eld. For example, the dyonic 5-brane in D"10 type-IIB theory is constructed in [72] by applying the S¸(2, Z) ;S¸(2, Z) transformation of the truncated type-IIB theory in D"6. (This '' #+ SO(2, 2),S¸(2, Z) ;S¸(2, Z) group is a subgroup of SO(5, 5) U-duality group of type-II string '' #+ on ¹.) There, it is found out that non-zero R-R "elds (which are related to the pseudo-scalar "eld) are needed for the solution to have both electric and magnetic charges. The type-IIB S¸(2, Z) S-duality transformation leads to dyonic p-brane whose electric and '' magnetic charges coming from di!erent sectors (NS-NS/R-R) of string theory. General dyonic solutions where form "elds carry both electric and magnetic charges are generated by additionally applying the S¸(2, Z) electric/magnetic transformation in D"6 [72]. In particular, the elec#+ tric/magnetic duality transformation that relates the solitonic 5-brane and elementary 5-brane is the product of (Z ) and (Z ) transformations. Such dyonic 5-brane solutions preserve 1/2 of  ''  #+ supersymmetry. We comment on generalization of dyonic p-branes discussed in this section. In [230], general dyonic p-branes within consistently truncated heterotic string on ¹, where truncated moduli "elds are parameterized by 3 complex modulus parameters ¹G (i"1, 2, 3) of 3 ¹ in ¹, is constructed. Thereby, the O(6, 22, Z) T-duality symmetry of heterotic string on ¹ is broken down to S¸(2, Z). The general class of multi-charged p-brane solution is then characterized by harmonic functions each associated with ¹G and the dilaton}axion scalar S. Such solutions break more than 1/2 of supersymmetry. With trivial S, ()L of supersymmetry is preserved for n non-trivial ¹G. With  non-trivial S, additional 1/2 of supersymmetries is broken unless all of ¹G are non-trivial. With all ¹G non-trivial, () or none of supersymmetry is preserved, depending on the chirality choices. In  particular, a special case of general class of solution with S and only one of ¹G non-trivial corresponds to generalization of D"6 self-dual dyonic string, when such solution is uplifted to D"6. This dyonic solution is parameterized by 2 harmonic functions, which are respectively associated with electric and magnetic charges of 2-form potential. In the self-dual limit, i.e. when the electric and magnetic charges are equal, the solution becomes the D"6 self-dual dyonic string. Within the context of non self-dual theory, the solution preserve only 1/4 of supersymmetry, whereas as a solution of self-dual theory it preserve 1/2 of supersymmetry. 6.2.2. Intersecting p-branes 6.2.2.1. Constituent p-branes. Before we discuss intersecting p-branes, we summarize various single-charged p-branes in D"10, 11, which are constituents of intersecting and overlapping p-branes. These p-branes are special cases of `elementarya p-branes and `solitonica p -branes discussed in Section 6.1. They are characterized by a harmonic function H(y) in the transverse space (with coordinates y ,2,y ) and break 1/2 of supersymmetry. D"11 supergravity has 3-form N> "\ potential. So, the basic p-branes (called M p-branes) are an electric `elementarya membrane and

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a magnetic `solitonica "vebrane: ds"HN>[H\(!dt#dx#2#dx)#(dy #2#dy )] , N N  N N>  c F   ?"! R ?H\, a"3,2,10 (for p"2) , RV V W 2 W N c F ?2 ?" e 2 R ?H , a "6,2, 10 (for p"5) , W W G 2 ? ? W N

(474)

where c"1(!1) for (anti-) branes and harmonic function cQ N N H "1# N " y!y "\N  is for M p-brane located at the +0, 1,2, p, hyperplane at yG"yG . The Killing spinor e of these  M p-branes satis"es the following constraint: (475) C 2 e"ce ,  N where C 2 ,C C 2C is the product of #at spacetime gamma matrices associated with the   N  N worldvolume directions. In D"10, there are 3 types of p-branes, depending on types of charges that p-branes carry. The electric charge of NS-NS 2-form potential is carried by NS-NS strings (or fundamental strings): ds"H\(!dt#dx)#dy#2#dy, e("H\ . (476)   The magnetic charge of NS-NS 2-form potential is carried by NS-NS 5-branes (or solitons): ds"!dt#dx#2#dx#H(dy#2#dy), e("H .     The charges of R-R (p#1)-form potentials are carried by R-R p-branes:

(477)

ds"H\(!dt#dx#2#dx)#H(dy #2#dy) ,  N N>  (478) e("H\N\, F 2 N G"R GH\ , W RV V W where p"0, 2, 4, 6 (p"1, 3, 5, 7) for R-R p-branes in type-IIA (type-IIB) theory. These R-R p-branes of the e!ective "eld theory are long distance limit of D p-branes in type-II superstring theories [193]: the transverse (longitudinal) directions of R-R p-branes correspond to coordinates with Dirichlet (Neumann) boundary conditions. The Killing spinors of D"10 p-branes satisfy one constraint. The left-moving and the rightmoving Majorana}Weyl spinors e and e have the same (opposite) chirality for the type-IIB * 0 (type-IIA) theory, i.e. C e "e [C e "e and C e "!e ]. So, spinor constraints are  *0 *0  * *  0 0 di!erent for type-IIA/B theories: E E E E

NS-NS strings: e "C e , e "!C e *  * 0  0 IIA NS-NS "vebranes: e "C 2 e , e "C 2 e   0 *   * 0 IIB NS-NS "vebranes: e "C 2 e , e "!C 2 e   0 *   * 0 R-R p-branes: e "C 2 e . *  N 0

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Whereas in type-II superstring theories there are D p-branes with p"!1, 0,2, 9, R-R p-branes in massless e!ective "eld theories cover only range p46. So, there is no place in the massless type-II supergravities for R-R 7- and 8-branes. Although the R-R 7-brane in type-IIB theory can be related to 8-form dual of pseudo-scalar of type-IIB theory [293], it cannot be T-dualized to R-R p-branes in type-IIA theory since it is speci"c to the uncompactixed type IIB theory, only. In [76], it is proposed that D p-branes of type-II superstring theories with p'6 can be realized as R-R p-branes of massive type-II supergravity theories. In the following, we discuss R-R 7- and 8-branes in some detail, since their properties and T-duality transformation rules are di!erent from other R-R p-branes. The R-R 8-brane is coupled to 9-form potential. The introduction of 10-form "eld strength into the theory does not increase the bosonic degrees of freedom (therefore, is not ruled out by supersymmetry consideration), but leads to non-zero cosmological constant. In fact, the massive type-IIA supergravity constructed in [204] contains such cosmological constant term. It is argued [498] that the existence of the massive type-IIA supergravity with the cosmological constant is related to the existence of the 9-form potential of type-IIA theory. In [76], new massive type-IIA supergravity is formulated by introducing 9-form potential A whose 10-form "eld strength  F "10dA is interpreted as the cosmological constant once Hodge-dualized. (Note, the cos  mological constant m in the massive type-IIA supergravity is independent of the dilaton in the string-frame, which is typical for terms in R-R sector.) In this new formulation, the cosmological constant m is promoted to a "eld M(x) by introducing A as a Lagrange multiplier for the  constraint dM"0 that M(x)"m is a constant: the "eld equation for M(x) simply determines the new "eld strength F , while the "eld equation for A implies that M(x)"m. It is conjectured in   [154] that the massive type-IIA supergravity is related to hypothetical H-theory [44] in D"13 through the Scherk}Schwarz type dimensional reduction (see below for the detailed discussion), rather than to M-theory. The conjectured type-IIB Sl(2, Z) duality requires the pseudo-scalar s to be periodically identi"ed (s&s#1), which together with T-duality between massive type-IIA and type-IIB supergravities implies that the cosmological constant m is quantized in unit of the radius of type-IIB compacti"cation circle, i.e. m"n/R (n3Z). The massive type-IIA supergravity admits the following 8-brane (or domain wall) as a natural ground state solution: ds"H\g dxI dxJ#H dy, e\("H , (479) IJ and the Killing spinor e satis"es one constraint CM e"$e. The form of harmonic function H(y), W which is linear in y, depends on M(x). When M(x)"m everywhere, H"m"y!y ", with a kink  singularity at y"y . When M(x) is locally constant, the corresponding solution is interpreted as  a domain wall separating regions with di!erent values of M (or cosmological constant). An example is H"!Q y#b [H"Q y#b] for y(0 [y'0], where 8-brane charges Q are \ > ! de"ned as the values of M as yP$Rand b is related to string coupling constant e( at the 8-brane core. This 8-brane solution is asymptotically left-#at (right-#at) when Q "0 (Q "0). The multi \ > 8-brane generalization can be accomplished by allowing kink singularities of H at ordered points y"y (y (2(y .   L  R-R 9-brane of type-IIB theory is interpreted as D"10 spacetime.

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The R-R 8-brane of massive type-IIA supergravity can be interpreted as the KK 6-brane of D"11 supergravity [76]. Namely, after the R-R 8-brane is compacti"ed to 6-brane in D"8, the 6-brane can be lifted as the R-R 6-brane of massless type-IIA supergravity, which is interpreted as the KK monopole of M-theory on S [601]. Under the T-duality, the R-R 8-brane is expected to transform to R-R 7-brane or 9-brane in type-IIB theory. First, T-duality transformation of massless type-II supergravity along a transverse direction of the R-R 8-brane leads to the product of S and D"9 Minkowski spacetime, which is 9-brane. (Note, the direct dimensional reduction requires H to be constant.) Second, T-duality transformation leading to type-IIB 7-brane is much involved and we discuss in detail in the following. T-duality transformation involving massive type-IIA supergravity requires construction of D"9 massive type-IIB supergravity. D"9 massive type-IIB supergravity is obtained from massless type-IIB supergravity through the Scherk}Schwarz type dimensional reduction procedure [528], i.e. "elds are allowed to depend on internal coordinates. This is motivated by the observation that the `StuK ckelberga type symmetry, which "xes the m-dependence of "eld strengths in type-IIA supergravity, is realized within type-IIB supergravity as a general coordinate transformation in the internal direction, which requires some of R-R "elds to depend on the internal coordinates. Namely, an axionic "eld s(x, z) (i.e. R-R 0-form "eld) is allowed to have an additional linear dependence on the internal coordinate z, i.e. s(x, z)Pmz#s(x), where x is the lowerdimensional coordinates. Since s appears always through ds in the action, the compacti"ed action has no dependence on the internal coordinate z. The result is the massive supergravity with cosmological term. The massive type-IIA supergravity compacti"ed on S through the standard KK procedure is related via T-duality to the massless type-IIB supergravity compacti"ed on S through the Scherk}Schwarz procedure. This massive T-duality transformation generalizes those of massless type-II supergravity in [81]. The explicit expression for m-dependent correction to the T-duality transformation can be found in [76]. Under the massive T-duality, the type-IIA 8-brane transforms to the type-IIB 7-brane, which is the "eld theory realization of D 7-brane of type-IIB superstring theory. By applying the T-duality of massless type-II supergravities to this 7-brane, one obtains a 6-brane of type-IIA theory, whereas transformation to 8-brane of type-IIA theory requires the application of massive T-duality. This generalized compacti"cation Ansatz for s is a special case of the generalized compacti"cation on a d-dimensional manifold M where an (n!1)-form potential A (n4d) for which nth B L\ cohomology class HL(M , R) of M is non-trivial is allowed to have an additional linear dependence B B on the (n!1)-form u (z), i.e. A (x, z)"mu (z)#standard terms [154,443]. (All the other L\ L\ L\ "elds are reduced by the standard KK procedure.) Here, du represents non-trivial nth cohomolL\ ogy of M . (In the case of compacti"cation of type-IIB theory on S, the cohomology dz is the B volume form on S.) Since A appears in the action always through dA , the lower-dimenL\ L\ sional action depends on A only through its zero mode harmonics on M , only. The constant L\ B m manifests in lower dimensions as a cosmological constant and, thereby, the compacti"ed D-dimensional action admits domain wall, i.e. (D!2)-brane, solutions. The general pattern for mass generation is as follows. First, the KK vector potentials always become massive. Second, a "eld that appears in a bilinear term in the Chern}Simons modi"cation of a higher rank "eld strength acquires mass if it is multiplied by A (with general dimensional reduction Ansatz). L\

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Third, when axionic "eld AGHI associated with D"11 3-form potential A is used for the  +,. Scherk}Schwarz reduction, the lower-dimensional theory contains a topological mass term. In these mechanisms, the "elds associated with the StuK ckelburg symmetry (under which the eaten "elds undergo pure non-derivative shift symmetries) get absorbed by other "elds to become mass terms for the potentials that absorb them. The consistency of the theory requires that the "elds that are eaten should not appear in the Lagrangian. Note, whereas the original Scherk}Schwarz mechanism [528] is designed to give a mass to the gravitino, thus breaking supersymmetry, and do not generate scalar potentials, in our case a cosmological constant is generated and the full supersymmetry is preserved. In addition to single-charged p-branes, there are other supersymmetric con"gurations which are basic building blocks of p-brane bound states. These are the gravitational plane fronted wave (denoted 0 ), called `pp-wavea, and the KK monopole (denoted 0 ). Their existence within p-brane U K bound states are required by duality symmetries. First, the KK monopole in D"11 is introduced [601] in an attempt to give D"11 interpretation of type-IIA D 6-brane [367]. The KK monopole is the magnetic dual of the KK modes of D"11 theory on S, which is identi"ed as R-R 0-brane (electrically charged under the KK ;(1) gauge "eld) [635]. The D"11 KK monopole, which preserves 1/2 of supersymmetry, has the form [601]: ds "!dt#dy ) dy# depends on D and the numbers of

 But the Scherk}Schwarz reduced theories have smaller symmetry than those reduced via the standard KK procedure [154].

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isometry directions of the con"guration. More general solution is obtained by replacing ="Q/"x"L by =(u, x). In particular, the choice ="f (u)xG corresponds to wave with the pro"le f propagating G G along y; an asymptotic observer, however, observes pp-wave with ="Q/"x"L and Q&1[du f (u)]2. When the Lorentz boost is imposed on a black p-brane and extreme limit is G taken, one has a bound state of a p-brane and pp-wave. 6.2.2.2. Orthogonally intersecting p-branes. From single-charged p-branes, one can construct `marginala bound states of p-branes by applying general intersection rules. The `marginala bound states are called orthogonally `overlappinga (`intersectinga) p-branes if the constituent p-branes are separated (located at the same point) in a direction transverse to all of the p-branes. We denote the con"guration where a (p#r)-brane intersects with a (p#s)-brane over a p-brane as (p"p#r, p#s) [493]. We add subscripts (NS, R, M) in this notation to specify types of charges that the constituent p-branes carry, e.g. 2 , 1 and 3 respectively denote M 2-brane, NS-NS string (or + ,1 0 fundamental string) and R-R 3-brane. The intersection rules are "rst studied in [492] in an attempt to interpret GuK ven solutions [327], and general harmonic superposition rules (which prescribe how products of powers of the harmonic functions of intersecting p-branes occur) of intersecting p-branes are formulated in [611]. (Such harmonic superposition rules of intersecting M-branes are already manifest in general overlapping M-branes constructed in [287].) Intersection rules can be independently derived from the `no forcea condition on a p-brane probe moving in another p-brane background [613]. Alternatively, one can derive general intersection rules for `marginala bound state p-branes in diverse dimensions from equations of motion, only [18]. Namely, from the equations of motion following from general D-dimensional action with one dilaton and arbitrary numbers of n -form  "eld strengths F  (A"1,2, N) with kinetic terms L 1 e?(F , L 2n !   one obtains the following intersection rule prescribing (p "p , p ):  (p #1)(p #1) 1 ! e a e a , (482) p #1"  2  D!2 where e "#(!) when the brane A is electric (magnetic). It is interesting that the relation (482),  derived from the equations of motion of the e!ective action alone, predicts D-branes (0"1 , p ) and ,1 0 D-branes for higher branes. In the following, we discuss intersection rules for a pair of branes. Intersecting con"gurations with more than two constituents are constructed by applying the intersection rules to all the possible pairs of branes. There are 3 types of intersecting p-branes: self-intersections, branes ending on branes and branes within branes. First, p-branes of the same type intersect only over (p!2)-brane (so-called (p!2) self-intersection rule), denoted as (p!2"p, p). This can be understood [492] from the fact that p-brane worldvolume theory contains a scalar (interpreted as a Goldstone mode of spontaneously broken translational invariance by the p-brane), which is Hodge-dualized to a worldvolume (p!1)-form potential that the (p!2)-brane couples to. The second type, denoted as (p!1"p, q) with q'p, is interpreted as a q-brane ending on a p-brane with (p!1)-branes being the ends of

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q-branes on the p-branes. This type of intersecting p-branes can be constructed by applying S- and T-duality transformations on M 2-brane ending on M 5-brane (1"2 , 5 ) or fundamental string + + ending on R-R p-brane (0"1 , p ). The third type, denoted as (p"p, q) with p(q, is interpreted as ,1 0 p-branes inside of the worldvolume of q-brane [224]. This type of intersecting p-branes preserves fraction of supersymmetry when the projection operators P and P (de"ning spinor constraints N O associated with the constituent D p-branes) either commutes or anticommutes [493]. If P and N P commute, then (p"p, q) preserves 1/4 of supersymmetry. This happens i! p"q mode 4. When O P and P anticommute, one has con"gurations that preserve 1/2 of supersymmetry. An example is N O 0-brane within membrane (0"0 , 2 ), which can be obtained from (2"2 , 5 ) (473) through dimen0 0 + + sional reduction on S and T-duality transformations. The spacetime coordinates of intersecting branes are divided into 3 parts: (i) the overall worldvolume coordinates mI (k"0, 1,2, d!1), which are common to all the constituent branes; (ii) the relative transverse coordinates x? (a"1,2, n), which are transverse to part of the constituent branes; (iii) the overall transverse coordinates yG (i"1,2, l), which are transverse to all of the constituent branes. Since a transverse (longitudinal) coordinate of R-R p-branes corresponds to a coordinate with Dirichlet (Neumann) boundary condition, these 3 types of coordinates respectively correspond to coordinates of open strings of the NN-, ND- (or DN-) and DD-types. Intersecting p-branes are divided into 3 types, according to the dependence of harmonic functions on these coordinates. The "rst type, for which the general intersection rules are formulated in [492,611], has all the harmonic functions depending on the overall transverse coordinates, only. For the second type, one harmonic function depends on the overall transverse coordinates and the other on the relative transverse coordinates. The third type has both harmonic functions depending on the relative transverse coordinates. For the "rst (third) type, constituent p-branes are, therefore, localized in the overall (relative) transverse directions but delocalized in the relative (overall) transverse directions. We will be mainly concerned with intersection rules for the "rst type and later we comment on the rest of types. One can construct supersymmetric (BPS) intersecting p-branes when spinor constraints associated with constituent p-branes (given in the previous paragraphs) are compatible with one another with non-zero Killing spinors. When none of spinor constraint is expressed as a combination of other spinor constraints, intersecting N number of p-branes preserve (), of supersymmetry.  (Note, from now on N stands for the number of constituent p-branes, not the number of extended supersymmetries.) One can introduce an additional p-brane without breaking any more supersymmetry, if some combination of spinor constraints of existing constituent p-branes gives rise to spinor constraint of the added p-brane. First, we discuss intersecting M p-branes. Intersecting rules, studies in [492,611], are: E Two M 2-branes (M 5-branes) intersect over a 0-brane (a 3-brane), i.e. (0"2 , 2 )((3"5 , 5 )). + + + + E M 2-brane and M 5-brane intersect over a string, i.e. (1"2 , 5 ), interpreted as M 2-brane ending + + on M 5-brane over a string. The worldvolume theory is described by D"6 (0, 2) supermultiplet with bosonic "elds given by 5 scalars and a 2-form that has self-dual "eld strength, and M 2-brane charge is carried by a self-dual string inside the worldvolume theory. E One can add momentum along an isometry direction by applying an SO(1, 1) boost transformation.

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All the possible intersecting M-branes are determined by these intersection rules. One can intersect up to 8 M p-branes by applying intersection rules to each pair of constituent M p-branes: the complete classi"cation up to T-duality transformations is given in [75]. Intersecting M-brane solutions can be constructed from the harmonic superposition rules. These are "rst formulated in [611] for BPS M-branes and are generalized to the non-extreme case in [175,243] and to the rotating case in [185]. First, the harmonic superposition rules for the BPS case are as follows: E The overall conformal factor of the metric is the product of the appropriate powers of the harmonic functions associated with constituent M p-branes: (483) ds"“ HNG G>(y)[2] , N G where p "2, 5, and H G(y) are harmonic functions in overall transverse space (with dimension l), G N namely of (i) the form cG N for l'2 , H G(y)"1# N "y!y "l\  (ii) logarithmic form for l"2, and (iii) linear form for l"1. Here, y"(y#2#yl is the  radial coordinate of the overall transverse space. So, the spacetime is asymptotically #at when l'2. E The metric is diagonal (unless there is a momentum along an isometry direction) and each component inside of [2] in (483) is the product of the inverse of harmonic functions associated with the constituent p -branes whose worldvolume coordinates include the corresponding G coordinate. E When momentum is added to an isometry direction, say m-direction, the above rules are modi"ed by the harmonic function K(y) associated with the momentum as follows: !dt#dmP!K\(y)dt#K(y)dmY , dmY ,dm#[K\(y)!1] dt .

(484)

E When l'3, one can add a KK monopole in a 3-dimensional subspace of the overall transverse space. All the harmonic functions depend only on these 3 transverse coordinates and the metric is modi"ed in the overall transverse components as follows dyGdyGPdy#2#dyl #H\(dyl $a cos h d )#H(dr#r dX) , \ \ ))  

(485)

where the harmonic function H"1#a /r is associated with the KK monopole charge )) P "$a X . The 2-form "eld strength associated with the KK monopole has the form )) ))  F "P e .  ))  E Non-zero components of 4-form "eld strength F are given by (474) for each constituent M p-branes. Now, we discuss generalization of harmonic superposition rules to the non-extreme, rotating case. The solutions get modi"ed by the harmonic functions g (y)"1#(l/y) associated with G G

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angular momentum parameters l and the following combinations of g : G G



\ a# l\kg\ for even l, 

G G G Gl, f\ ,G l l“ g . G for odd l l\kg\ G G G G l

(486)

Here, a and k are de"ned in (189) and (191). The harmonic superposition rules are modi"ed in the G following way: E Harmonic functions are modi"ed as 2m sinh d 2m sinh d cosh d PH"1#fl , H"1# y l\ y l\

(487)

where the boost parameter d is associated with electric/magnetic charge or momentum 2m sinh d cosh d. E The metric components get further modi"ed as (488) dtPf dt, d dyG dyHPf \dy#y dXY l , \ GH l where f,1!fl(2m/y l\) and f ,G\ l !fl(2m/y \) are non-extremality functions. Here, dXY l denotes the angular parts of the metric. In the limit l "0, dXY l becomes #at metric \ G \ dXl of Sl\. Generally, dXY l is given by the angular metric components of rotating black \ \ hole in (compacti"ed) D"l#1 and the general rules for constructing these components are unknown yet. (The explicit forms of dXY l up to 3 M p-brane intersections with momentum \ along an isometry direction are given in [185].) E The harmonic superposition rule (484) with a momentum along an isometry direction m is modi"ed as !dt#dmPK\f dt#KdmY , dmY ,dm#[K\!1]dt ,

(489)

where d is a boost parameter associated with momentum 2m sinh d cosh d and K\,1!fl(2m sinh d/rl\)K\. E The non-zero components of the 4-form "eld strength are the same as the BPS case, when l "0. G For l O0, there are additional non-zero components associated with the induced electric/ G magnetic "elds due to rotations. The general construction rules are unknown yet; the explicit expressions up to 3 intersections with momentum along an isometry direction are given in [185]. Next, we discuss intersecting p-branes in D"10. The intersection rules are as follows: E Two fundamental strings can only be parallely oriented, i.e. 1 ""1 . ,1 ,1 E Two solitonic 5-branes orthogonally intersect over 3-spaces, i.e. (3"5 , 5 ). ,1 ,1 E A fundamental string and a solitonic 5-brane can only be parallely oriented, i.e. 1 ""5 or ,1 ,1 (1"1 , 5 ). ,1 ,1 E An R-R p-brane and an R-R q-brane intersect over n-spaces (n"p , q ) such that 00 00 p#q!2n"4.

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E A fundamental string orthogonally intersects an R-R p-brane over a point, i.e. (0"1 , p ). ,1 0 E A solitonic 5-brane intersects an R-R p-brane over n-spaces (n"5 , p ) such that p!n"1. ,1 0 These intersecting p-branes preserve 1/4 of supersymmetry except the multi-centered con"guration 1 ""1 . These intersection rules are derived in [613] by applying `no forcea condition and in [18] ,1 ,1 from the equations of motion. Alternatively, one can derive these rules from the intersection rules of M-branes by applying KK procedure and duality transformations, which we discuss in the following in details. Intersecting p-branes in D"10 can be obtained from intersecting M-branes through compacti"cation on S and duality transformations. We have the following p-branes in type-IIA theory from the compacti"cations R and R of M-branes along a longitudinal and a transverse directions, i.e.  , the double and direct dimensional reductions, respectively: 0, 2 , 0 1 , 2 P 2 P ,1 + 0 +

0 4 , 5 P 0, 5 , 5 P + 0 + ,1

(490)

which can be understood from the relations of type-IIA form "elds to the D"11 3-form "eld under the standard KK procedure. Dimensional reduction of 0 and 0 in D"11 yields the following U K type-IIA con"gurations: 0, 0 , 0 P 0 6 , 0 0 , 0 P 0 P 0 U U K 0 U

0, 0 , 0 P K K

(491)

where R (R ) on 0 denotes the reduction in the direction associated with Taub-NUT term (the  , K other directions) of the metric. Duality transformations further relate di!erent types of p-branes. First, we consider T-duality between type-IIA and type-IIB theories on S. T-duality (161) on type-IIA/B R-R p-branes along a tangent (transverse) direction leads to type-IIB/A R-R (p!1)branes ((p#1)-branes). Note, T-duality on a longitudinal direction introduces an additional overall transverse coordinate that harmonic functions have to depend on, which is not always guaranteed. So, the T-duality on a longitudinal direction is called `dangerousa, whereas the T-duality on a transverse direction is `safea since the resulting con"gurations are guaranteed to be solutions to the equations of motion. T-duality transformation rules of NS-NS p-branes can be inferred from T-duality transformation of "elds [81] (see also (161)) as follows. Among other things, T-duality interchanges o!-diagonal metric components g and the same components B of the NS-NS 2-form potential, where a is the I? I? T-duality transformation direction. So, when p-brane has non-trivial (k"t, a)-component of the metric or NS-NS 2-form potential, T-duality transformation is reminiscent of interchange of momentum mode (electric charge of KK ;(1) "eld g ) and winding mode (electric charge of I? NS-NS 2-form ;(1) "eld B ) under T-duality. So, pp-wave (whose linear momentum is identi"ed I? with KK electric charge) and NS-NS string (carrying electric charge of the NS-NS 2-form "eld, identi"ed as string winding mode) are interchanged when T-duality is performed along the longitudinal direction of string or the direction of pp-wave propagation. T-duality along the other directions yields the same type of solutions. Second, the (k" , a)-component of metric (NS-NS G 2-form potential), where are angular coordinates associated with rotational symmetry, corresG ponds to the Taub-NUT term (is associated with magnetic charge of a solitonic 5-brane 5 ). So, ,1 magnetic monopole (or Taub-NUT solution) and NS-NS 5-brane are interchanged when T-duality

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transformation is applied along the direction transverse to NS-NS 5-brane or along the coordinate associated with Taub-NUT term. T-duality along the other directions yields the same type of solutions. Second, the type-IIB S¸(2, Z) S-duality transformation can be used to relate NS-NS charged solutions and R-R charged solutions which are coupled to 2-form potentials. Under the Z  subset transformation, NS-NS string and NS-NS 5-brane transform to R-R 1-brane and R-R 5-brane, respectively. Full S¸(2, Z) transformations on NS-NS string (NS-NS 5-brane) yields `non-marginala BPS bound states of p NS-NS strings and q R-R strings (p NS-NS 5-brane and q R-R 5-brane) with the pair of integers (p, q) relatively prime. The duality transformation rules are, therefore, summarized as follows: 2, (p#1) , 1 P 2 0 , 1 P 2, 1 , 2 (p!1) , p P p P 0 0 0 ,1 U ,1 ,1 0 2 1 , 0 P 2, 0 , 5 P 2 5 , 0 P U ,1 U U ,1 ,1

2, 0 , 5 P ,1 K

(492)

2 5 , 0 P 2, 0 , 1  1 1 , 5  1 5 . 0 P 0 ,1 K ,1 K K ,1 0 We now discuss various intersecting p-branes in D"10. First, we consider intersecting R-R p-branes in type-II theories. D-brane con"gurations are supersymmetric if the number l of coordinates of DN or ND type is the multiple of 4 [501]. (See Section 8.3.2 for details on this point.) At the level of low-energy intersecting R-R p-branes of the e!ective "eld theories, this means that solutions are supersymmetric when the number of relative transverse coordinates is the multiple of 4, i.e. n"4, 8. This can also be derived from the condition that the Killing spinor constraints e "C 2 e of the constituent R-R p-branes *  N 0 are compatible with one another. When both of the harmonic functions depend only on the relative transverse coordinates, BPS con"gurations are possible for n"8 case only, and otherwise only n"4 con"gurations are BPS. Since our main concern is the intersecting R-R p-branes with all the harmonic functions depending only on the overall transverse coordinates, we concentrate on the n"4 case. Since T-duality preserves the total number n of the relative transverse coordinates, one can obtain all the intersecting 2 R-R p-branes with n"4 by applying `safea T-duality transformations to intersecting R-R 0-brane and R-R 4-brane, i.e. (0"0 , 4 ). One can further add R-R p-branes in such a way that n"4 for each pair of constituent 0 0 R-R p-branes. It is shown [75] that one can intersect up to 8 R-R p-branes which satisfy the n"4 rule for each pair: the complete classi"cation up to T-dualities is given in [75]. The explicit solutions for BPS intersecting R-R p-branes can be constructed by the following harmonic superposition rules: E The metric is diagonal, with each component having the multiplicative contribution of H\ (H) from each constituent R-R p-brane whose worldvolume (transverse) coordinates N N include the associated coordinate. E Dilaton is given by the product of harmonic functions associated with the constituent R-R p-branes: e\("“ HNII\. I N E Non-zero components of (p#2)-form "eld strengths are given by (478) for each constituent R-R p-branes.

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As an example, solution for intersecting R-R (p#r)-brane and R-R (p#s)-brane over a p-brane, i.e. (p"p#r, p#s), is of the form:




H  Q (dx?) ds "(H H )\g dmI dmJ# N>P  N>P N>Q IJ H N>Q ? H  Q>P # N>Q (dx?)#(H H )d dyG dyH , N>P N>Q GH H N>P ?Q> e\("HN>P\HN>Q\ , N>P N>Q F 2 Q G"R GH\ , F Q>2 Q>P G"R GH\ . (493) V W RV V W W N>Q RV W N>P We discuss intersecting p-branes which contain NS-NS p-branes. First, (0"1 , p ) with 04p48, ,1 0 is nothing but open strings that end on D-brane. This type of con"gurations can be obtained by "rst compactifying (0"2 , 2 ) on S along a longitudinal direction of one of M 2-brane (resulting in + + (0"1 , 2 )), and then by sequentially applying T-duality transformations along the directions ,1 0 transverse to the NS-NS string. Second, (p!1"5 , p ) with 14p46, are interpreted as D p-brane ,1 0 ending on NS-NS 5-brane. Namely, NS-NS 5-branes act as a D-brane for D-branes. This interpretation is consistent with the observation [46] that M 5-branes are boundaries of M 2-branes. This type of intersecting branes can be constructed by "rst compactifying (1"2 , 5 ) on S along an + + overall transverse direction (resulting in (1"5 , 2 )), and then by sequentially applying T-duality ,1 0 transformations along the longitudinal directions of the NS-NS 5-brane. Third, intersecting NS-NS p-branes can be obtained in the following ways: (i) compacti"cation of (3"5 , 5 ) along an + + overall transverse direction leads to (2"5 , 5 ), (ii) compacti"cation of (1"2 , 5 ) along a relative ,1 ,1 + + transverse direction which is longitudinal to M 2-brane leads to (1"2 , 5 ), (iii) the type-IIB ,1 ,1 S-duality on (!1"1 , 1 ) yields (!1"1 , 1 ). 0 0 ,1 ,1 We comment on the case some or all of harmonic functions depend on the relative transverse coordinates [75]. These types of intersecting p-branes can be constructed by applying the general harmonic superposition rules, taking into account of dependence of harmonic functions on the relative transverse coordinates. In particular, the metric components associated with the relative coordinates (that harmonic functions depend on) have to be the same so that the equations of motion are satis"ed. First, the second type of intersecting p-branes, i.e. one harmonic function depends on the relative transverse coordinates, can be constructed from the "rst type of intersecting p-branes, i.e. all the harmonic functions depend on the overall transverse coordinates, by letting one of harmonic functions depend on the relative transverse coordinates. Thus, the classi"cation of the second type is the same as that of the "rst type. The third type of p-branes, i.e. all the harmonic functions depend on the relative transverse coordinates, have 8 relative transverse coordinates (n"8) for a pair of p-branes. It is impossible to have more than two p-branes with each pair having n"8. In D"11, the only con"guration of the third type is (1"5 , 5 ) [287]. This M-brane + + preserves () of supersymmetry, since the Killing spinor satis"es two constraints of the form (475),  each corresponding to a constituent M 5-brane. By compactifying an overall transverse direction of




 M 2-brane with its longitudinal coordinates given by the overall longitudinal and overall transverse coordinates of (1"5 , 5 ) can be further added without breaking any more supersymmetry. The added M 2-brane intersects the M 5+ + branes over strings and is interpreted as an M 2-brane stretched between two M 5-branes.

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(1"5 , 5 ) on S, one obtains (1"5 , 5 ), which was "rst constructed in [423]. Further application + + ,1 ,1 of the type-IIB SL(2, Z) transformation leads to (1"5 , 5 ). 0 0 The series of application of T-duality transformations, then, yield a set of overlapping 2 R-R p-branes with n"8. (Complete list can be found in [287].) These overlapping p-branes correspond, at string theory level, to D-brane bound states with 8 ND or DN directions, and therefore should be supersymmetric. When 2 R-R p-branes intersect in a point, one can add a fundamental string without breaking any more supersymmetry. This type of con"gurations is interpreted as a fundamental string stretching between two D-branes. For the case where 2 R-R p-branes intersect in a string, one can add pp-wave along the string intersection without breaking anymore supersymmetry. Another third type of intersecting p-branes in D"10 can be constructed by compactifying (1"5 , 5 ) along a relative transverse direction, resulting in (1"4 , 5 ), followed by series of + + 0 ,1 T-duality transformations along the longitudinal directions of the NS-NS 5-brane, resulting in (p!3"p , 5 ) with 34p48. One can further add R-R (p!2)-branes to these con"gurations; 0 ,1 these con"gurations are interpreted as a D (p!2)-brane stretching between D p-brane and NS-NS 5-brane. 6.2.2.3. Other variations of intersecting p-branes. So far, we discussed intersecting p-branes with p!p"0 mod 4. Existence of such classical intersecting p-branes that preserve fraction of supersymmetry is expected from the perturbative D-brane argument [501,504]. One can construct such solutions by applying the harmonic superposition rules discussed in the above. In this subsection, we discuss another type of p-brane bound states which do not follow the intersection rules discussed in the previous subsection. Such p-brane bound states contain pair of constituent p-branes with p!p"2 and still preserve fraction of supersymmetry. These p-brane con"gurations can be generated by applying dimensional reduction or T-duality along a direction at angle with a transverse and a longitudinal directions of the constituent p-branes [96,519,152]. (Hereafter, we call them as `tilteda reduction and `tilteda T-duality.) These p-brane con"gurations can also be constructed by applying `ordinarya dimensional reduction and sequence of `ordinarya duality transformations on (2"2 , 5 ) (473), which preserves 1/2 of supersymmetry. + + In fact, it con#icts with di!eomorphic invariance of the underlying theory that one has to choose speci"c directions (which are either transverse or longitudinal to the constituent p-branes) for dimensional reduction or T-duality transformations [152]. So, the existence of such new p-brane con"gurations is required by (M-theory/IIA string and IIA/IIB string) duality symmetries and di!eomorphism invariance of the underlying theories. We now discuss the basic rules of `tilteda T-duality and `tilteda dimensional reduction on constituent p-branes. Before one applies `tilteda T-duality and dimensional reduction to p-branes,  One can further add a fundamental string along the string intersection without breaking any more supersymmetry. T-duality along the fundamental string direction leads to type-IIB (1"5 , 5 ) with pp-wave propagating along the string ,1 ,1 intersection. Note, the former con"guration preserves only 1/8 of supersymmetry, rather than 1/4, if regarded as a solution of type-IIB theory.  It is argued [501,504] that D-brane bound state with p!p"6 is not supersymmetric and is unstable due to repulsive force. Although a solution that may be interpreted as (0"0 , 6 ) is constructed in [152], its interpretation is 00 00 ambiguous due to abnormal singularity structure of harmonic functions, and it cannot be derived from (2"2 , 5 ) though + + a chain of T-duality transformations.

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one rotates a pair of a longitudinal x and a transverse y coordinates of a (constituent) p-brane by an angle a (a di!eomorphism that mixes x and y):



x

"

y




cos a !sin a

x

sin a

y

cos a

.

(494)

(Note, x and y have to be di!eomorphic directions so that one can compactify these directions on S.) Then, one compacti"es or applies T-duality along the x-direction. These procedures preserve supersymmetry. When a"0 [a"p/2], such compacti"cation or T-duality transformation is the compacti"cation or T-duality transformation along a longitudinal direction x (a transverse direction y). Thus, as the angle a is varied from 0 to p/2, the resulting bound state interpolates between the corresponding two limiting con"gurations. First, we discuss the `tilteda reduction R of solutions in D"11. The `tilteda reduction on ? M-branes leads to type-IIA p-brane bound states, interpreted as `brane within branea: 0? (4"4 , 5 ) , 0? (1"1 , 2 ) , 5 P (495) 2 P ,1 0  + 0 ,1  + where the subscript A means type-IIA con"guration. From 0 and 0 in D"11, one obtains the K U following type-IIA bound states: 0? (0 "6 ) , 0 P 0? (0 "0 ) , 0 P (496) K U 0 U K 0 where (0 "0 ) is interpreted as a `boosteda D 0-brane. U 0 Second, we discuss `tilteda T-duality transformations ¹ . The o!-diagonal metric component ? g induced by the coordinate rotation (494) is transformed to the same component B of 2-form VYWY VYWY potential under T-duality. So, the T-transformed solution has diagonal metric and non-zero F "B #2paF , where F is the worldvolume gauge "eld strength [445]. For R-R p-brane IJ IJ IJ IJ bound states, the corresponding perturbative D-brane now, therefore, satis"es the modi"ed boundary condition R XI!iFIR XJ"0. The induced #ux F is related to a as F "!tan a. L J J VYWY VYWY The ADM mass of the transformed con"guration is of the form M&Q#Q, a characteristic of   non-threshold bound states. First, the `tilteda T-duality on type-IIA/B p-branes leads to the following type-IIB/A bound states: 2? (0 "5 ), (p#1) P 2? (p"p , (p#2) ) , 2? (0 "1 ), 5 P (497) 1 P U ,1 ,1 K ,1 0 0 0 ,1 where (0 "1 ) is simply a boosted fundamental string. The existence of the D-brane bound states U ,1 (p"p , (p#2) ) preserving 1/2 of supersymmetry is also expected from the perturbative D-brane 0 0 considerations. One can apply `tilteda T-duality transformation more than once to obtain new p-brane bound state con"gurations. For example, by applying `tilteda T-duality transformations to D 2-brane in two di!erent directions, one obtains D (4, 2, 2, 0)-brane bound state [96]. Second, `tilteda T-duality on 0 and 0 in type-IIA/B theory yields the following type-IIB/A bound states: K U 2? (0 "1 ) . 2? (0 "5 ), 0 P (498) 0 P K ,1 U U ,1 K Next, we discuss p-brane bound states obtained by "rst imposing a Lorentz boost along a transverse direction and then applying T-duality transformation or reduction along the direction of the boost: T-duality along a boost and reduction along a boost, respectively denoted as ¹ and R . T T The Lorentz boost yields non-threshold bound state of a p-brane and pp-wave. This bound state

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interpolates between extreme p-brane and plane wave, as the boost angle a (see below for its de"nition) varies from zero (no boost) to p/2 (in"nite boost). The ADM energy E of such non-threshold bound state of extreme p-brane (with mass M) and pp-wave (with momentum p) has the form E"M#p, a reminiscent of relativistic kinematic relation of a particle of the rest mass M with the linear momentum p, rendering the interpretation of such bound state as `boosteda p-brane. The Lorentz boost with velocity v/c"sin a along a transverse direction y has the form [519]: tPt"(cos a)\(t#y sin a), yPy"(cos a)\(y#t sin a) .

(499)

In the above, the angle a is related to the boost parameter b as cosh b"1/cos a. After the Lorentz boost (499), one compacti"es or applies T-duality transformation along y. First, we discuss the reduction R along a boost. Since the momentum of pp-wave manifests as T the KK electric charge after reduction along the direction of momentum #ow, the resulting bound state always involves R-R 0-brane. The following type-IIA bound states are obtained from the compacti"cation with a boost of con"gurations in D"11: 0T (0"0 , 5 ) , 0T (0"0 , 2 ) , 5 P 2 P 0 0 + 0 ,1  + 0T (0 "0 ) , 0 P 0T (0"0 , 6 ) . 0 P U 0 U K 0 0

(500)

Second, we discuss the T-duality ¹ along a boost. Under the T-duality, the linear momentum of T a pp-wave is transformed to the electric charge of the NS-NS 2-form potential [359]. So, the T-dualized con"gurations always involve a fundamental string with non-zero winding mode. We have the following type-IIB/A bound states from type-IIA/B con"gurations: 2T 1 , 1 P ,1 ,1

2T (0 "1 ), p P 2T (1"1 , (p#1) ) , 5 P ,1 K ,1 0 ,1 0

2T (0 "1 ), 0 P 2T (0 "1 , 5 ) . 0 P U U ,1 K K $ ,1

(501)

The di!eomorphic invariance and duality symmetries require new type of bound states in M-theory that preserve 1/2 of supersymmetry. These new M-theory con"gurations can be constructed by uplifting new type-IIA con"guration discussed in this subsection. For example, by uplifting (0 "5 ) or (4"4 , 6 ) , one obtains the M 5-brane and the KK monopole bound state in K ,1  0 0 D"11. Another example is the M 2-brane and the KK monopole bound state uplifted from (1"1 , 6 ) or (0 "1 ) . ,1 0  K ,1   This boost angle a can be identi"ed with the angle a of coordinate rotation in (494). Namely, the non-threshold typeIIA (q , q ) string bound state obtained from D"11 pp-wave through tilted dimensional reduction at an angle a,   followed by T-duality, can also be obtained by reduction along a boost of M 2-brane with the same boost angle a, followed by T-duality transformation.  One can straightforwardly apply this procedure to intersecting M-branes, followed by sequence of T-duality transformations, to construct p-brane bound states that interpolate between those that preserve 1/4 of supersymmetry and those that preserve 1/2 of supersymmetry as a is varied from 0 to p/2 [151].

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Non-threshold type-IIB (q , q ) string, obtained from R-R or NS-NS string through S¸(2, Z)   duality, can be related to D"11 pp-wave compacti"ed at angle [519]. This is understood as follows. When the compacti"cation torus ¹ (parameterized by isometric coordinates (x, y)) is rectangular, the angle a of coordinate rotation de"nes the direction of (q , q ) cycle in ¹, around   which the D"11 pp-wave is wrapped, as cos a"q /(q#q. (So, choice of di!erent angle    a corresponds to di!erent choice of a cycle in ¹.) Starting from pp-wave propagating along x, one performs coordinate rotation (494) in the plane (x, y), where y is an isometric transverse direction of the pp-wave, and then compacti"es along y-direction, which is the direction of (q , q )-cycle of   ¹ with coordinates (x, y). The resulting type-IIA con"guration is a non-threshold bound state of pp-wave and D 0-brane. Subsequent T-duality transformation along y leads to type-IIB (q , q )    string solution. This is related to the fact that the type-IIB S¸(2, Z) symmetry is the modular symmetry of the D"11 supergravity on ¹ [81,538,539] (see Section 3.7 for detailed discussion). Had we started from the bound state of M 2-brane and pp-wave along a longitudinal direction of the M 2-brane, we would end up with boosted type-IIB (q , q ) string that preserves 1/4 of   supersymmetry. (The M 2-brane charge is, therefore, interpreted as a momentum of the type-IIB (q , q ) string.)   On the other hand, the non-threshold type-IIB (q , q ) 5-brane can be related to M 5-brane.   Namely, one "rst compacti"es M 5-brane at an angle to obtain (4"4 , 5 ) and then applies 0 ,1  T-duality transformation along the relative transverse direction to obtain type-IIB (q , q ) 5-brane   bound state. Following the similar procedures, one can obtain the non-threshold type-IIB (q , q )   string from M 2-brane. 6.2.2.4. Branes intersecting at angles. In [89], it is shown that one can construct BPS D-brane bound states where the constituent D-branes intersect at angles other than the right angle. We "rst summarize formalism of [89]. Then, we discuss the corresponding classical solutions [33,55,97,151,285] in the e!ective "eld theory. In the presence of a D p-brane, two spinors e and e (corresponding to N"2 spacetime supersymmetry) of type-II string theory satisfy the constraint: N e "C e " : “ eIC e , (502) N G I G where e is an orthonormal frame spanning D p-brane worldvolume. For N numbers of constituent G D-branes, it is convenient to de"ne the following raising and lowering operators from gamma matrices: !iC ), aI"(C #iC ), k"1,2, N , aR"(C I  I\ I I  I\ which satisfy the anticommutation relations:

(503)

+aH, aR,"dH , +aH, aI,"0"+aR, aR, . (504) I I H I The lowering operators aI de"ne the `vacuuma "02, satisfying aI"02"0. Under an S;(N) rotation zGPRG zH of the complex coordinates zI,x?I#ix@I spanned by D p-branes, the raising and the H lowering operators transform as aIPRIaH, aRPRR HaR . H I I H

(505)

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One can construct intersecting D p-branes at angles in the following way. One starts from two D p-branes oriented, say, along the directions Re zG and applies S;(p) rotation zGPRG zH H to one of D p-branes. For this type of intersecting D p-branes at angles, the spinor e satis"es the constraint: N N “ (aR#aI)e" “ (RR HaR#RIaH)e . I I H H I I

(506)

So, the resulting con"guration has unbroken supersymmetries "02 and “N aR"02. These two I I spinors have the same (opposite) chirality for p even (odd). One can further compactify this intersecting D p-branes at angles on a torus and then apply T-duality transformations to obtain other types of intersecting D p-branes at angles. Alternatively, one can start from (q"(p#q) , (p#q) ) and rotate one D (p#q)-brane relative to the other by applying the SO(2p) 0 0 transformation. The resulting con"guration is supersymmetric when the S;(p)LSO(2p) transformation is applied. When these intersecting D-branes are further compacti"ed on tori, the consistency of toroidal compacti"cation imposes the quantization condition for the intersecting angles h's in relation to the moduli of tori [32,89]. When intersecting D-branes at angles are compacti"ed on a manifold M, the unbroken supersymmetry should commute with the `generalizeda holonomy group (de"ned by a modi"ed connection } with torsion } due to non-zero antisymmetric tensor backgrounds) of M. Here, the spinor constraint g"“ eIC g de"nes an action of discrete generalized holonomy. Generally, G G I starting from an intersecting D p-brane at angles with FK "F#B"0, where F [B] is the worldvolume 2-form "eld strength [the NS-NS 2-form], one obtains a con"guration with FK O0 when T-duality is applied. For intersecting D 2-branes at angles, the necessary and su$cient condition for preserving supersymmetry is that FK is anti-self-dual [89]. Classical solution realization of intersecting D-branes at angles is "rst constructed in [97]. Starting from n parallel D 2-branes (with each constituent D 2-brane located at x"x , a"1,2, n, ? and its charge related to l '0) lying in the (y, y) plane, one rotates each constituent D 2-brane by ? an S;(2) angle a in the (y, y) and (y, y) planes, i.e. (z, z)P(e ??z, e\ ??z) where z"y#iy ? and z"y#iy. The solution in the string frame has the form: ds"(1#X








 L 1 !dt# (dyH)# X +[(R )dyG]#[(R )dyH], ? ?G ?H 1#X H ?



 # (dxG) , G



dt L A"  X (R )dyG(R )dyH ? ?G ?H 1#X ? L ! X X sin(a !a )(dydy!dydy) , ? @ ? @ ?@ e("(1#X;



L L X, X # X X sin(a !a ) , ? ? @ ? @ ? ?@

(507)

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where R? (a"1,2, n) are (block-diagonal) SO(4) matrices that correspond to the above mentioned rotation of constituent D 2-branes and harmonic functions






l  1 ? X (x)" ? 3 "x!x " ? are associated with constituent D 2-branes located at x"x . This con"guration preserves 1/4 of ? supersymmetry and interpolates between previously known con"gurations: (i) a "0, p/2 case is ? orthogonally oriented D 2-branes, (ii) a "a , ∀a, case is parallel n D 2-branes oriented in di!erent ?  direction through S;(2) rotations, etc. The ADM mass density of (507) is the sum of those of constituent D 2-branes, i.e. A L m"  l , ? 2i ? and is independent of the S;(2) rotation angles a . The physical charge density is also simply the ? sum of those of constituent D 2-branes, although the charge densities in di!erent planes (yG, yH) of the intersecting D 2-branes depend on a . ? T-duality on (507) yields other types of D-brane bound states. The T-duality along transverse directions leads to angled D p-branes with p'2. The T-duality along worldvolume directions leads to more exotic bound states of D-branes. Namely, since the constituent D 2-branes intersect with one another at angles, the worldvolume direction that one chooses for T-duality transformation is necessarily at angle with some of constituent D 2-branes. Consequently, the resulting con"guration is exotic bound state of D p-brane (pO2) and bound states of the type studied in [96] (e.g. bound state of D (p#1)-brane and D (p!1)-brane, and D (4, 2, 2, 0)-brane bound state) obtained by applying the `tilteda T-duality transformation(s) on a D p-brane. The above intersecting D-branes at angles and related con"gurations are alternatively derived by (i) applying the `tilteda boost transformation on the orthogonally intersecting two D-branes, followed by the sequence of ¹-S-¹ transformations of type-II strings [55], or (ii) applying `reduction along a boosta followed by T-duality transformations [151]. For the former case [151], the resulting con"gurations are mixed bound states of R-R branes that necessarily involves fundamental string. It is essential that one has to turn on both D-brane charges of original orthogonally intersecting D-branes and apply the S-duality between two T-duality transformations to have con"gurations where the D-branes intersect at angles. Intersecting p-branes at angles in more general setting, starting from M-branes with the #at Euclidean transverse space EL replaced by the toric hyper-KaK hler manifold M , are studied in L [285]. We "rst discuss the general formalism and then specialize to the case of intersecting p-branes in D"10, 11. 4n-dimensional toric hyper-KaK hler manifold M with a tri-holomorphic ¹L isometry has the L following general form of metric: ds "; dxG ) dxH#;GH(du #A )(du #A ) (i, j"1,2, n) , &) GH G G H H

(508)

 The manifold M is tri-holomorphic i! the triplet KaK hler 2-forms X"(du #A ) dxG!; dxG;dxH are indepenL G G  GH dent of u , i.e. L GX"0. G ..P

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where u (which are periodically identi"ed } to remove a coordinate singularity } u &u #2p, G G G thereby parameterizing ¹L) correspond to the ;(1) isometry directions of M and L xG"+xG " r"1, 2, 3, parameterize n copies of Euclidean spaces E. The n"1 case is Taub-NUT P space. The hyper-KaK hler condition relates n 1-forms A "dxH ) x with "eld strengths F "dA to G GH G G ; through ePQJRJ ; "FPQ "RPuQ !RQ uP . (So, a toric hyper-KaK hler metric is speci"ed by GH H IG HIG H IG I HG ; alone.) This implies that ; are harmonic functions on M , i.e. ;GHR ) R ;"0. Generally, GH GH L G H a positive-de"nite symmetric n;n matrix ;(xG) is linear combination ,+N, (509) ; ";# ; [+p,, a (+p,)] GH K GH GH + , N K of the following harmonic functions speci"ed by a set of n real numbers +p "i"1,2, n,, called G a `p-vectora, and an arbitrary 3-vector a: pp G H ; [+p,, a]" . (510) GH 2" p xI!a" I I The vacuum hyper-KaK hler manifold EL;¹L with moduli space Sl(n, Z)!Gl(n, R)/SO(n) has the metric (508) with ; "; (so, A "0). Regular non-vacuum hyper-KaK hler manifold is representGH GH G ed by harmonic functions ; [+p,, a] associated with a 3(n!1)-plane in EL de"ned by 3-vector GH equations L p xI"a. The hyper-KaK hler metric (508) is non-singular, provided +p, are coprime I I integers. The Sl(n, Z) transformation ;PS2;S (S3Sl(n, Z)) on the hyper-KaK hler metric (508) leads to another hyper-KaK hler metric with the Sl(n, Z) transformed p-vector S+p,. The angle h between two 3(n!1)-planes de"ned by two p-vectors +p, and +p, is given by cos h"p ) p/(pp. Here, the inner product is de"ned as p ) q"(;)GHp q , which is invariant under Sl(n, Z). The solution G H (508) is, therefore, speci"ed by angles and distances between mutually intersecting 3(n!1)-planes associated with harmonic functions ; [+p,, a (+p,)]. GH K The special case where *;,;!; is diagonal (i.e. the p-vectors have the form (0,2, 1,2, 0) and *; "d (1/2"xG"): there are only n intersecting 3(n!1)-planes) describes n GH GH fundamental BPS monopoles in maximally broken rank (n#1) gauge theories found in [444], thereby called LWY metric. When additionally ; is diagonal (so that ; is diagonal), n 3(n!1)planes intersect orthogonally (cos h"0) and M "M ;2;M . In this case, one can L   always choose u such that F "dA and ; are related as F "夹d; , where 夹 is the Hodge-dual GH G G GG G GG on E. The hyper-KaK hler manifold M preserves fraction of supersymmetry. It admits (n#1) L covariantly constant SO(4n) spinors (in the decomposition of D-dimensional Lorentz spinor representation under the subgroup Sl(n, R);SO(4n)) if the holonomy of M is strictly Sp(n), which L corresponds to the case where 3(n!1)-planes intersect non-orthogonally, i.e. ; is non-diagonal. These covariantly constant SO(4n) spinors arise as singlets in the decomposition of the spinor representation of SO(4n) into representations of holonomy group of M , i.e. Sp(n) for this case. L The only toric hyper-KaK hler manifolds whose holonomy is a proper subgroup of Sp(n) are those corresponding to the `orthogonallya intersecting or `parallela 3(n!1)-planes. For this case, M "M ;2M (i.e. product of n Taub-NUT space) with Sp(1)L holonomy and diagonal L   ; (thereby, 3(n!1)-planes intersecting orthogonally), and more supersymmetry is preserved GH since the non-singlet spinor representations of Sp(n) are further decomposed under the proper

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subgroup Sp(1)L. A trivial case corresponds to the case ; ";, i.e. the vacuum hyper-KaK hler GH GH manifold: since the holonomy group is trivial, all the supersymmetries are preserved. The starting point of general class of intersecting p-branes is the following D"11 solution, which is the product of the D"3 Minkowski space and M :  ds "ds(E)#; dxG ) dxH#;GH(du #A )(du #A ) , (511)  GH G G H H where i"1, 2. For a general solution with non-diagonal ; , thereby with the Sp(2) holonomy for GH M , (511) admits (n#1)"3 covariantly constant spinors. Namely, 32-component real spinor in  D"11 is decomposed under Sl(2, R);SO(8) as 32P(2, 8 )(2, 8 ). Two SO(8) spinor representaQ A tions 8 and 8 are further, respectively, decomposed under Sp(2)LSO(8) as 8 P5111 and Q A Q 8 P44. So, 3/16 of supersymmetry is preserved. When ; is diagonal (so, M "M ;M and A GH    3-planes orthogonally intersect), the holonomy group is Sp(1);Sp(1). Under the Sp(1);Sp(1) subgroup, non-singlet Sp(2) spinor representations 5 and 4 are, respectively, decomposed as 5P(2, 2)(1, 1) and 4P(2, 1)(1, 2). So, 8/32"1/4 of supersymmetry is preserved. One can generalize the solution (511) to include M-branes without breaking any more supersymmetry, resulting in `generalized M-branesa, where the transverse Euclidean space is replaced by M . The harmonic functions (associated with M-branes) are independent of the ;(1) isometry L coordinates u , thereby M p-branes are delocalized in the u -directions. G G First, one can naturally include an M 2-brane to the solution (511), since the transverse space of M 2-brane has dimensions 8: ds "H\ds(E)#H[; dxG ) dxH#;GH(du #A )(du #A )] ,  GH G G H H F"$u(E)dH\ ,

(512)

where u(E) is the volume form on E, the signs $ are those of M 2-brane charge and H"H(xG) is a harmonic function (associated with M 2-brane) on M , i.e. ;GHR ) R H"0. The  G H SO(1, 10) Killing spinor of this solution is decomposed into the SO(8) spinors of de"nite chiralities 8 and 8 , which are related to the signs $. So, depending on the sign of M 2-brane charge, either A Q all supersymmetries are broken or 3/16 of supersymmetry is preserved. Second, one can add M 5-branes to the solution (511) if M "M ;M , i.e.    ;"diag(; (x), ; (x)). For this purpose, it is convenient to introduce 2 1-form potentials AI   G (i"1, 2) with "eld strengths FI which can be related to the harmonic functions H (x) and H (x) G   (associated with 2 M 5-branes) as dH "夹FI . (This is analogous to the relations d; "夹F G G G G satis"ed by the diagonal components of ; and the "eld strengths F "dA of the solution (511) G G when both *; and ; are diagonal.) Here, 夹 is the Hodge-dual on E. The explicit solution has the form: ds "(H H )[(H H )\ ds(E)#H\[; dx ) dx#;\(du #A )]           # H\[; dx ) dx#;\(du #A )]#dz] ,      F"[FI (du #A )#FI (du #A )]dz .        The subscripts s and c denote two possible SO(8) chiralities.

(513)

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Generally with non-constant ; and H , (513) preserves 1/8 of supersymmetry, provided the proper G G relative sign of M 5-brane charges is chosen. In the following, we discuss the intersecting (overlapping) p-brane interpretation of solutions obtained via dimensional reduction and duality transformations of the D"11 solutions (511)}(513). Due to the triholomorphicity of the Killing vector "elds R/Ru , the Killing spinors G survive in these procedures. As for the p-branes associated with the harmonic functions ; , there is GH a one-to-one correspondence between 3(n!1)-planes and p-branes, and the intersection angle of p-branes is given by the angle between the corresponding p-vectors, which de"ne 3(n!1)-planes. First, we discuss intersecting (overlapping) p-branes related to (511) and (512). Since (511) is a special case of (512) with H"1, we consider p-branes related to (512), and then comment on the H"1 case. First, one compacti"es one of the ;(1) isometry directions of M , say u without loss   of generality, on S, resulting in a type-IIA solution, and then applies the T-duality transformation along the other ;(1) isometry direction, i.e. the u -direction, to obtain the following type-IIB  solution: ds"(det ;)H[H\(det ;)\ds(E)#(det ;)\; dxG ) dxH#H\ dz] , # GH

(514)

(det ; ; , i D"u(E)dH\ , B "A dz, q"! #i I G G ; ;   where is the dilaton, q,l#ie\( (l"R-R 0-form "eld), B (i"1, 2) are 2-form potentials in G the NS-NS and R-R sectors, D is the 4-form potential, and z,u . This solution is interpreted as  D 3-brane (with harmonic function H) stretching between 5-branes along the z-direction. The 5-branes in this type-IIB con"guration are speci"ed by a set of intersecting 3-planes L p xI"a I I in E. From the expression for q in (514), one sees that the Sl(2, R) transformation ;P(S\)2;S\ (S3Sl(2, R)) in M is realized in this type-IIB con"guration as the type-IIB Sl(2, R) symmetry  aq#b qP cq#d of equations of motion, where




S"

a b c

d

.

The condition that Sl(2, R) is broken down to Sl(2, Z) so that M with the coprime integers +p , p ,    remains non-singular after the transformation is translated into the type-IIB language that the Sl(2, R) symmetry of the equations of motion is broken down to the Sl(2, Z) S-duality symmetry of type-IIB string theory. In the following we discuss particular cases of (514). We "rst consider the solution (514) with ;"diag(H (x), H (x)) and H"1. In this case,   M "M ;M with holonomy Sp(1);Sp(1), thereby preserving 1/4 of supersymmetry. The    corresponding solution is `orthogonallya intersecting (2"5 , 5 ): ,1 0 ds"(H H )[(H H )\ds(E)#H\dx ) dx#H\dx ) dx#dz] , (515) #       where harmonic functions H "1#(2"xG")\ (i"1, 2) are respectively associated with NS-NS G 5-brane and R-R 5-brane [285].

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A particular case of (514) with ;"1"H and a single 3-plane in E (de"ned by +p , p ,) is the   bound state of NS-NS 5-brane and R-R 5-brane with charge vector (p , p ). So, the restriction that   M is non-singular, i.e. +p , p , are coprime integer, manifests in the type-IIB theory that the    corresponding p-brane con"guration is a non-marginal bound state of NS-NS 5-brane and R-R 5-brane. There is a correlation between the D"11 Sl(2, Z) transformation, which rotates a 3-plane in E, and the type-IIB Sl(2, Z) transformation, which rotates the charge vectors of the 2-form "eld doublet B . G The general type-IIB solution (514) with non-diagonal ; is interpreted as an arbitrary number of 5-branes intersecting or overlapping at angles. p-vectors and ; specify orientations and charges of 5-branes, and a determines distance of 5-branes from the origin. Since the corresponding M has the Sp(2) holonomy, 3/16 of supersymmetry is preserved.  Next, we discuss the intersecting p-branes in type-IIA string and M-theory related to (512). The intersecting p-branes in type-IIA theory are constructed in the following way. First, one T-dualizes the type-IIB solution (514) along a direction in E to obtain the following type-IIA `generalized fundamental stringa solution, which can also be obtained from (511) by compactifying on a spatial direction in E: ds"H\ds(E)#; dxG ) dxH#;GH(du #A )(du #A ) , GH G G H H (516) B"u(E)H\, "! ln H .   Subsequent T-dualities along u and u lead to the type-IIA solution:   ds"H\ds(E)#; (dxG ) dxH#duGduH) , GH (517) B"A duG#u(E)H\, " ln det ;! ln H , G    where B is the NS-NS 2-form potential and is the dilaton. This solution is interpreted as an  arbitrary number of NS-NS 5-branes intersecting on a fundamental string (with a harmonic function H), generalizing the solutions in [423]. The case with diagonal ; represents orthogonally intersecting NS-NS 5-branes. More general case with non-diagonal ; represents intersecting NS-NS 5-branes at angles and preserves 3/16 of supersymmetry. The following solution in D"11 is obtained by uplifting the type-IIA solution (517): ds "H(det ;)[H\(det ;)\ds(E)  # (det ;)\; (dxG ) dxH#duG duH)#H\dy] , (518) GH F"[F duG#u(E)dH\]dy . G When ; is of LWY type, this solution represents parallel M 2-branes which intersect intersecting 2 M 5-branes (orthogonally when ; is diagonal as well) over a string. For more general form of ;, the solution represents the M 2-branes intersecting arbitrary numbers of M 5-branes at angles and preserves 3/16 of supersymmetry. T-duality on the type-IIA solution (517) along the fundamental string direction leads to intersecting type-IIB NS-NS 5-branes with a pp-wave along the common intersection direction. Further application of Z LS¸(2, Z) S-duality transformation leads to the following solution  involving R-R 5-branes, which preserves 3/16 of supersymmetry when 5-branes intersect at

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angles: ds"(det ;)[dt dp#Hdp#; (dxG ) dxH#duG duH)] , # GH (519) B"A duG, q"i(det ; , G where B is the R-R 2-form potential. The H"1 case is classical solution realization of intersecting D-branes at angles of [89]. In [89], the condition for unbroken supersymmetry is given by the holonomy condition arising in the KK compacti"cations. This corresponds to the holonomy condition on the hyper-KaK hler manifold of [285]. Namely, intersecting R-R p-branes preserve fraction of supersymmetry if orientations of the constituent R-R p-branes are related by rotations in the Sp(2) subgroup of SO(8). This is seen by considering spinor constraints of intersecting two D 5-branes, where one D 5-brane is oriented in the (12345) 5-plane and the other D 5-brane rotated into the (16289) 5-plane by an angle h. For this con"guration, type-IIB chiral spinors e (A"1, 2) satisfy the constraints C e"e and R\(h)C R(h)e"e, where R(h)"   exp+!h(C #C #C #C ), is the SO(1, 9) spinor representation of the above mentioned      SO(8) rotation. (The rotational matrix R(h) is associated with an element of ;(2, H) Sp(2) that commutes with quaternionic conjugation, which is the rotation mentioned above.) It can be shown [285] that (i) for h"0, p, 1/2 of supersymmetry is preserved since the second spinor constraint is trivially satis"ed, (ii) for h"$p/2, 1/4 is preserved, and (iii) for all other values of h, 3/16 is preserved. Finally, we discuss intersecting (overlapping) p-branes related to (513). The compacti"cation on one of the ;(1) isometry directions followed by the T-duality along the other ;(1) isometry direction yields the following type-IIB solution: ds"(H H ; ; )[(; ; H H )\ds(E)#(; H )\dx ) dx #           #(; H )\dx ) dx#(H H )\dz#(; ; )\dy] ,       (520) H ;  . B"A dz#AI dy, B"A dz#AI dy, q"i     H ;   This solution represents 2 NS-NS 5-branes in the planes (1, 2, 3, 4, 5) and (1, 6, 7, 8, 9) and 2 R-R 5-branes in the planes (1, 5, 6, 7, 8) and (1, 2, 3, 4, 9) intersecting orthogonally. Since the spinor constraint associated with one of the constituent p-branes is expressed as a combination of the rest three independent spinor constraints, the solution preserves () of supersymmetry. Related  intersecting p-brane is constructed by applying T-duality, oxidation and dimensional reduction. One can further include additional p-branes without breaking any more supersymmetry, provided the spinor constraints of the added p-branes can be expressed as a combination of spinor constraints of the existing p-branes.



6.3. Dimensional reduction and higher dimensional embeddings The lower-dimensional (D(10) p-branes can be obtained from those in D"10, 11 through dimensional reduction. Reversely, most of lower-dimensional p-branes are related to D"10, 11 p-branes via dimensional reduction and dualities. In particular, many black holes in D(10 originate from D"10,11 p-brane bound states, which makes it possible to "nd microscopic origin

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of black hole entropy. It is purpose of this section to discuss various p-branes in D(10. We also discuss various p-brane embeddings of black holes. There are two ways of compactifying p-branes to lower dimensions. First, one can compactify along a longitudinal direction. It is called the `double dimensional reductiona (since both worldvolume and spacetime dimensions are reduced, bringing a p-brane in D dimensions to (p!1)brane in D!1 dimensions diagonally in the D versus p brane-scan) or `wrappinga of branes (around cycles of compacti"cation manifold). Since target space "elds are independent of longitudinal coordinates, one only needs to require periodicity of "elds in the compacti"cation directions. Second, one can compactify a transverse direction of a p-brane. It is called the `direct dimensional reductiona (since this takes us vertically on the bran-scan, taking a p-brane in D dimensions to a p-brane in D!1 dimensions) or `constructing periodic arraysa of p-branes (along the compacti"ed direction). Since "elds depend on transverse coordinates, direct dimensional reduction is more involved [286,422,461]. For this purpose, one takes periodic array of parallel p-branes (with the period of the size of compact manifold) along the transverse direction. Then, one takes average over the transverse coordinate, integrating over continuum of charges distributed over the transverse direction. The resulting con"guration is independent of the transverse coordinate, making it possible to apply standard Kaluza}Klein dimensional reduction. In the double [direct] dimensional reduction, the values of p [p] and D are preserved; in the direct dimensional reduction, the asymptotic behavior of the "elds (which goes as &1/"y"N ) changes. Conventionally, the direct dimensional reduction uses the zero-force property of BPS p-branes, which allows the construction of multicentered p-branes. Note, however that it is also possible to apply the vertical dimensional reduction even for non-BPS extreme p-branes [461] and non-extreme p-branes [463,442], contrary to the conventional lore. Namely, since the equations of motion (of a non-extreme, axially symmetric black (D!4)-brane in D dimensions, for the non-extreme case) can be reduced to Laplace equations in the transverse space with suitable choice of "eld AnsaK tze, one can still construct multi-center p-branes for non-BPS and non-extreme cases as well. For the non-extreme case, when an inxnite number of non-extreme p-branes are periodically arrayed along a line, the net force on each p-brane is zero and the conical singularities along the axis of periodic array act like `strutsa that hold the constituents in place. Furthermore, since the direction of periodic array is compacti"ed on S with each p-brane precisely separated by the circumference of S, the instability problem of such a con"guration can be overcome. One can also lift p-branes as another p-branes in higher-dimensions, so-called &dimensional oxidation'. First, the oxidation of a p-brane in D dimensions to a p-brane in D#1 dimensions (i.e. the reverse of the direct dimensional reduction) is never possible in the standard KK dimensional reduction, since the oxidized p-brane in D#1 dimensions has to depend on the extra transverse

 Such staking-up procedure breaks down for (D!3)-branes in D dimensions, due to conical asymptotic spacetime [461]. This is also related to the fact that (D!2)-branes reside in massive supergravity, rather than ordinary massless type-II theory.  Or one can promote such isometry directions as the spatial worldvolume of a p-brane, leading to an intersecting p-brane solution [287,424,461,611].

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coordinate introduced by oxidation. (This is analogous to the `dangerousa T-duality transformation.) Second, the oxidation of a p-brane in D dimensions to a (p#1)-brane in D#1 dimensions (i.e. the reverse of the double dimensional reduction) is classi"ed into two groups. A p-brane in D dimensions is called `rustya if it can be oxidized to a (p#1)-brane in D#1 dimensions. Otherwise, it is called `stainlessa. Thus, p-branes in bran scan are KK descendants of stainless p-branes in some higher dimensions. A p-brane in D dimensions is `stainlessa when (i) there is not the antisymmetric tensor in D#1 dimensions that the corresponding (p#1)-brane couples to, or (ii) the exponential prefactors a and a for (p#2)-form "eld strength kinetic terms in D#1 and "> " D dimensions do not satisfy the relation 2(p #1) . a "a ! "> " (D!2)(D!1) (This relation is satis"ed by the expression for a in (531), provided D remains unchanged in the dimensional reduction procedure.) In this section, we focus on p-branes in D(10 with only "eld strengths of the same rank turned on, comprehensively studied in [243,455}463]. A special case is black holes, which are 0-brane bound states. We also discuss their supersymmetry properties and interpretations as bound states of higher-dimensional p-branes. 6.3.1. General solutions We concentrate on p-branes in D"11 supergravity on tori. Bosonic Lagrangian of D"11 supergravity is (521) L "(!GK [R K !  FK ]#FK FK AK , %        where FK "dAK is the "eld strength of the 3-form potential AK . So, such p-branes have interpreta   tion in terms of M-theory or type-II string theory con"gurations. Although one can directly reduce the D"11 action down to D(11 by compactifying on ¹\" in one step, it turns out to be more convenient to reduce the action (521) one dimension at a time iteratively until one reaches D dimensions. Namely, one compacti"es 11!D times on S, making use of the following KK Ansatz: ds "e?Pds #e\"\?P(dz#A ) , "> "  (522) A (x, z)"A (x)#A (x)dz , L L L\ where u is a dilatonic scalar, A "A dxI is a KK 1-form "eld, A is an n-form "eld arising from  I L AK and a,1/(2(D!1)(D!2). The explicit form of resulting D(11 action can be found  elsewhere [456]. The advantage of such compacti"cation procedure is that spin-0 "elds are

 Such a p-brane in D dimensions should rather be viewed as a p-brane in D#1 dimensions whose charge is uniformly distributed along the extra coordinate. This is interpreted as the limit where one of charges of intersecting two p-branes in D#1 dimensions is zero [461].  Contrary to the conventional lore that all the p-branes in D(10 are obtained from those in D"10, 11 through dimensional reductions, there are stainless p-branes in D(10 which cannot be viewed as dimensional reductions of p-branes in D"10, 11. So, the conventional brane scan is modi"ed [459].

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manifestly divided into two classes in the Lagrangian. Namely, only dilatonic scalars

"( ,2,

) appear in the exponential prefactors of the n-form "eld kinetic terms. The  \" dilatonic scalars originate from the diagonal components of the internal metric and are true scalars. The couplings of to n-form potentials A? are characterized by the `dilaton vectorsa a in the L ? following way: 1 (523) e\L "! ea?  (F? ) . L> LU 2n! ? The complete expressions for `dilaton vectorsa, which are expressed as linear combinations of basic constant vectors, are found in [456]. On the other hand, the remaining spin-0 "elds coming from the o!-diagonal components of GK and the internal components of AK are axionic, being +,  associated with constant shift symmetries, and should rather be called 0-form potentials, which couple to solitonic (D!3)-branes. (Note, there are no elementary p-branes for 1-form "eld strengths.) Up to present time, study of p-branes within the above described theory has been mostly concentrated on the case where only n-form potentials of the same rank are turned on, with the restrictions that terms related to the last term in (521) (denoted as L from now on) and the $$ `Chern}Simonsa terms in (n#1)-form "eld strengths are zero. These restrictions place constraints on possible charge con"gurations for p-branes. These constraints become non-trivial when a pbrane involves both undualized and dualized "eld strengths, i.e. when the p-brane has both electric and magnetic charges coming from di!erent "eld strengths. The former [later] type of constraint is satis"ed as long as the dualized and undualized "eld strengths have [do not have] common internal indices i, j, k. 6.3.1.1. Supersymmetry properties. The supersymmetry preserved by p-branes is determined from the Bogomol'nyi matrix M, which is de"ned by the commutator of supercharges Q " Re C !t dR per unit p-volume: !  C .



N dR &eR Me , (524)    .R where N "e C !d t is the Nester's form de"ned from the supersymmetry transformation rule  C ! of D"11 gravitino t :  (525) N "e C !D e #e C!!e F #  e C !2!e F 2  .  !!    ! !  !  The "rst term in N gives rise to the ADM mass density and the last two terms respectively contribute to the electric and magnetic charge density terms in M. The Bogomol'nyi matrix for the 11-dimensional supergravity on (S)\" is in the form of the ADM mass density m term plus the electric and magnetic Page charge density [490] (de"ned respectively as (1/4u ) "\L夹F and L "\L 1 (1/4u ) LF ) terms. L 1 L [Q , Q ]" C C

 In particular, to set the 0-forms AGHI to zero consistently with their equations of motion, the bilinear products of "eld  strengths that occur multiplied by AGHI in L should vanish.  $$

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Since the Bogomol'nyi matrix is obtained from the Hermitian supercharges, its eigenvalues are non-negative. The matrix M has zero eigenvalues for each component of unbroken supersymmetry associated with the Killing spinor e satisfying d t "0. Since D"11 spinor has 32 components, C  the fraction of preserved supersymmetry is k/32, where k is the number of 0 eigenvalues of M (i.e. the nullity of the matrix M) or equivalently the number of linearly independent Killing spinors. The amount of preserved supersymmetry is determined as follows. First, one calculates the ADM mass density m from the p-brane solutions. Then, one plugs m, together with the Page electric and magnetic charge densities of the p-branes, into the Bogomol'nyi matrix. The multiplicity k of 0 eigenvalues of the resulting matrix M determines the fraction of supersymmetry preserved by the corresponding p-branes. 6.3.1.2. Multi-scalar p-branes. General p-branes with more than one non-zero p-brane charges are called `multi-scalar p-branesa, since such p-branes have more than one non-trivial dilatonic scalars. The Lagrangian density has the following truncated form:





1 1 ea?  (F? ) , (526) L"(!g R! (R )! N> 2 2(p#2)! ? where "eld strengths F? "dA? are de"ned without `Chern}Simonsa modi"cations. In this N> N> action, the rank p#2 of "eld strengths is assumed to not exceed D/2, namely those with p#2'D/2 are Hodge-dualized. This is justi"ed by the fact that the dual of "eld strength of an elementary (solitonic) p-brane is identical to the "eld strength of solitonic (elementary) (D!p!2)-brane, with the corresponding dilaton vector di!ering only by sign. We consider extreme p-branes with N non-zero (p#2)-form "eld strengths, each of which is either elementary or solitonic, but not both. For the simplicity of calculations, the SO(1, p!2);SO(D!p#1) symmetric metric Ansatz (448) is assumed to satisfy (p#1)A# (p #1)B"0 [457], so that the "eld equations are linear. The p-brane solutions are then determined completely by the dot products M ,a ) a of the dilaton vectors a associated with non?@ ? @ ? zero (p#1)-form "eld strengths F? (a"1,2, N). In solving the equations, it is assumed that N> M is invertible, which requires the number N of non-trivial F? to be not greater than the num?@ N> ber of the components in , i.e. N411!D. For such p-branes, only N components u ,a ) of ? ?

are non-trivial. If one further takes the Ansatz !eu #2(p#1)AJ (M\) u , then M @ ?@ @ ?@ ? takes the form: 2(p#1)(p #1) M "4d ! . ?@ ?@ D!2

(527)

The conditions on "elds that linearize the "eld equations and lead to M of the form (527) are also ?@ dictated by supersymmetry transformation rules for the BPS con"gurations. Thus, a necessary condition for `multi-scalar p-branesa to be BPS is for the dilaton vectors a associated with ? participating "eld strengths to satisfy (527). The following extreme multi-scalar p-brane solution is

 When M is singular, analysis depends on the number of rescaling parameters [456]. The only new solution is the ?@ case a "0, which yields solutions with a"0 and (F?)"F/N, ∀a. ? ?

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obtained by taking further simplifying Ansatz discussed in [457]: , , eCP?\N>"H , ds" “ H\N >"\ dxI dxJg # “ HN>"\ dyK dyK , (528) ? ? IJ ? ? ? where harmonic functions 1 j H "1# ? (y,(yKyK)  ? p #1 yN> are associated with p-branes carrying the Page charges P "j /4, and the "eld strengths are given, ? ? respectively, for the electric and magnetic cases by F? "dH\dN>x, F? "夹(dH\dN>x) . (529) N> ? N> ? The elementary and solitonic p-branes are related by P! . The ADM mass density is the sum of the mass densities of the constituent p-branes, i.e. m" , P . Multi-center generalization is ? ? achieved by replacing harmonic functions by [424]. j ?G H "1# ? "y!y "N > ?G G 6.3.1.3. Single-scalar p-branes. We discuss the case where the bosonic Lagrangian for 11-dimensional supergravity on (S)\" is consistently truncated to the following form with one dilatonic scalar and one (p#2)-form "eld strength [456,459]:





1 1 e?((F ) , L"(!g R! (R )! N> 2 (p#2)!

(530)

where we parameterize the exponential prefactor a in the form: 2(p#1)(p #1) a"D! . D!2

(531)

This expression for a is motivated from (422), now with an arbitrary parameter D replacing 4. By consistently truncating (526), one has (530) with (F ), (F? ) and a, given by (for the case ? N> N> M is invertible) ?@ \ a" (M\) , "a (M\) a ) . (532) ?@ ? ?@ ?@ ?@ By taking AnsaK tze which reduce equations of motion for (530) to the "rst order, one obtains [459] the `single-scalar p-branea solution with the Page charge density P"j/4:




e("H?CD,



ds"H\N >D"\ dxI dxJ g #HN>D"\ dyK dyK , IJ

where 1 (Dj H"1! .  2(p#1) rN>

(533)

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The mass density is m"j/2(D. Although inequivalent charge con"gurations give rise to the same D, i.e. the same solution, supersymmetry properties depend on charge con"gurations. Note, although one can obtain p-brane solutions (533) for any values of p and a, hence for any values of D, only speci"c values of p and a can occur in supergravity theories. The value of D is preserved in the compacti"cation process, provided no "elds are truncated. For p-branes with 1 constituent, D is always 4, as can be seen from the form of a in (442), determined by the requirement of scaling symmetry, and always 1/2 of supersymmetry is preserved. The value D"4 can also be understood from the facts that D"4 in D"11, since there is no dilaton in D"11, and the value of D is preserved in dimensional reduction not involving "eld truncations. When the "eld strength is a linear combination of more than one original "eld strengths, D(4. With all the Page charges P "j /4 of `multi-scalar p-branea (528) equal, one has `single-scalar ? ? p-branea with the Page charge P"j/4 (j"(Nj ). By substituting M in (527) into (532), one has ? ?@ D"4/N. So, `single-scalar p-branesa with D"4/N (N52) are bound states of N single-charged p-branes (with D"4) with zero binding energy, and preserve the same fraction of supersymmetry as their multi-scalar generalizations. Only `single-scalar p-branesa with D"4/N (N5Z>) and `multi-scalar p-branesa can be supersymmetric. (Non-supersymmetric p-branes in this class is related to supersymmetric ones by reversing the signs of certain charges.) And only single-scalar p-branes with D"4/N (N52) have multi-scalar generalizations. 6.3.1.4. Dyonic p-branes. In D"2(p#2), p-branes can carry both electric and magnetic charges of (p#2)-form "eld strengths. There are two types of dyonic p-branes [456]: (i) the "rst type has electric and magnetic charges coming from di!erent "eld strengths, (ii) the second type has dyonic "eld strengths. As in the multi-scalar p-brane case, the requirements that L "0 and the $$ Chern}Simons terms are zero place constraints on possible dyonic solutions in D"2(p#2)"4, 6, 8. Such restrictions rule out dyonic p-branes of the "rst type in D"6, 8. For the second type, dyonic p-brane in D"8 is special since it has non-zero 0-form potential A [388], thereby requiring non-zero source term FK FK e+,./0123, and can be obtained  +,./ 0123 from purely electric/magnetic membrane by duality rotation, unlike dyonic D"6 string and D"4 0-brane of the second type. Dyonic p-branes of the second type include self-dual 3-branes in D"10 [238,367], self-dual string [242] and dyonic string [230] in D"6, and dyonic black hole in D"4 [456]. There are two possible dyonic p-branes (associate with (530)) of the second type with the Page charge densities j /4 [456]: (1) a"p#1 case (i.e. *"2p#2) with the solution G 1 1 j j , e?(\N>"1#  , e\?(\N>"1#  a(2 rN> a(2 rN>

(534)

 For p-branes with DO4, the scaling symmetry of combined worldvolume and e!ective supergravity actions is broken.  The solutions for dyonic p-branes of the "rst type have the form (528) with Lagrangian (526) containing both Hodgedualized and undualized "eld strengths.

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and (2) a"0 case (i.e. D"p#1) with the solution



1 j#j 1  

"0, e\N>"1# . 2 p#1 rN>

(535)

The ADM mass density for (534) is m"(1/2(D)(j #j ), whereas for (535)   m"(1/2(D)(j#j. The solution (535) is invariant under electric/magnetic duality and, there  fore, is equivalent to the purely elementary (j "0) or solitonic (j "0) case. For (534) with   j "j , the "eld strength is self-dual and "0, thereby (534) and (535) are equivalent, but for (535)   j and j are independent. When j "!j , (534) corresponds to anti-self-dual massless string     with enhanced supersymmetry. Note, the solutions (534) and (535) are not restricted to those obtained from the D"11 supergravity on tori. For D"8, 6 and 4, which are relevant for the D"11 supergravity on tori, D's for (534) and (535) are respectively +6, 3,, +4, 2, and +2, 1,. So, (534) and (534) with p"2 (i.e. D"8) are excluded. 6.3.1.5. Black p-branes. We discuss non-extreme p-branes. Non-extreme p-branes are additionally parameterized by the non-extremality parameter k'0. There are two ways of constructing non-extreme p-branes. The "rst method involves a universal prescription for `blackeninga extreme p-branes, which deforms extreme solutions with [243]: k (r,"y") eD"1!  rN> dtPeD dt, drPe\D dr ,

(536)

while modifying harmonic functions associated with p-branes as H"1#k sin h2d / ? rN >P1#k sinh d /rN >. The resulting non-extreme p-branes, called `type-2 non-extreme p? branesa, have an event horizon at r"r "kN >, which covers the singularity at the core r"0. > The ADM mass density has the generic form m& ((Q )#k, which is always larger than the ? ? extreme counterpart, and all the supersymmetry is broken since the Bogomol'nyi bound is not saturated. For type-2 non-extreme p-branes (with p51), the PoincareH invariance is broken down to R;EN because of the extra factor eD in the (t, t)-component of the metric. For 0-branes, the metric remains isotropic but the quantity (p#1)A#(p #1)B no longer vanishes. In the second method, the metric Ansatz (448) remains intact but instead general solution to the "eld equations is obtained [455,462] without simplifying AnsaK tze, e.g. (p#1)A#(p #1)B"0, that linearize "eld equations. (In solving the "eld equations without simplifying AnsaK tze, one encounters an additional integration constant interpreted as non-extremality parameter.) So, the resulting non-extreme p-branes, called `type-1 non-extreme p-branesa, preserve the full PoincareH invariance (in the worldvolume) of extreme p-branes. So, type-1 non-extreme p'0 solutions do not overlap with the type-2 non-extreme counterparts. But type-1 non-extreme 0-branes contain type-2 non-extreme 0-branes as a subset. The equations of motion for single-scalar p-branes and dyonic p-branes of the second type are, respectively, casted into the forms of the Liouville equation and the Toda-like equations for two

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variables, which are subject to the "rst integral constraint. The equations of motion for dyonic p-branes are solvable when a"p#1 (i.e. D"2(p#1)) or a"3(p#1) (i.e. D"4(p#1)). When a"p#1, the equations of motion are reduced to two independent Liouville equations. Since D44 in supergravity theories, only dyonic strings in D"6 and dyonic black holes in D"4 are relevant, with only dyonic strings having BPS limit. When a"3(p#1), the equations of motion are reduced to S;(3) Toda equations. Only dyonic black holes are possible in supergravity theory for this case. In the extreme limit, such black holes preserve supersymmetry when either electric or magnetic charge is zero. For multi-scalar p-branes with N "eld strengths, the equations of motion are Toda-like in general, but when the extreme limit is BPS (i.e. dilaton vectors satisfy (527)) the equations of motion become N independent Liouville equations. The requirements that non-extreme p-branes are asymptotically Minkowskian and dilatons are "nite at the event horizon (thereby the event horizon is regular) place restrictions on parameters of the solutions. 6.3.1.6. Massless p-branes. For multi-scalar p-branes and a dyonic p-brane (535) of the second type, the ADM mass density has the form m& j . So, they can be massless when some of the ? ? Page charges are negative. In this case, there are additional 0 eigenvalues of the Bogomol'nyi matrix, enhancing supersymmetry. Generally massless p-branes are ruled out if one requires the Bogomol'nyi matrix to have only non-negative eigenvalues, since the Bogomol'nyi matrix is obtained from the commutator of the Hermitian supercharges. Since some of the Page charges are negative, the massless p-branes have naked singularity. On the other hand, if one allows negative eigenvalues, one can have p-branes preserving more than 1/2 of supersymmetry and some of non-BPS multi-scalar p-branes can become supersymmetric due to the appearance of 0 eigenvalues with suitable sign choice of Page charges (but their single-scalar counterparts are non-BPS, since Page charges have to be equal in the single-scalar limit) [457]. 6.3.2. Classixcation of solutions In this subsection, we classify p-branes discussed in the previous subsection according to their supersymmetry properties. Since single-scalar p-branes and their multi-scalar generalizations preserve the same amount of supersymmetry (except for the special case of massless p-branes), the classi"cation of multi-scalar p-branes is along the same line as that of single-scalar p-branes. Single-scalar p-branes are supersymmetric only when D"4/N (N3Z>) and the dilaton vectors a (associated with the participating "eld strengths) of their multi-scalar p-brane counterparts ? satisfy relations (527). Spin-0 "elds, i.e. dilatonic scalars and 0-form "elds, form target space manifold of p-model. The target space manifold has a coset structure G/H, where GKE (R) (n"11!D) is the (realL>L valued) ;-duality group and H is a linearly realized maximal subgroup of G. Under the G-transformations, the equations of motion are invariant. When the DSZ quantization is taken

 In the exceptional case of 4-scalar solution with 2-form "eld strengths in D"4, it is possible to have massless p-branes where the Bogomol'nyi matrices have no negative eigenvalues [457].  For D46, this is the case only when all the "eld strengths are Hodge-dualized to those with rank 4D/2 [458].

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into account, G and H break down to integer-valued subgroups. The subgroup G(Z) is the conjectured ;-duality group of type-II string on tori. The asymptotic values of spin-0 "elds, called `modulia, de"ne the `scalar vacuuma. The asymptotic values of dilatonic scalars and 0-form "elds are, respectively, interpreted as the `coupling constantsa and `h-anglesa of the theory. One can parameterize spin-0 "elds by a G-valued matrix <(x) which transforms under rigid G-transformation by right multiplication and under local H-transformation by left transformation: <(x)Ph(x)<(x)K\, h(x)3H and K3G. It is convenient to de"ne a new scalar matrix M,<2<, which is inert under H but transforms under G as MPKMK2. So, the ;-duality group G generally changes the `vacuuma of the theory. By applying a G-transformation, one can bring the asymptotic value of M to the canonical form M "1. The subgroup H leaves M "1 intact (i.e. H is the ;-duality little group of the scalar   vacuum), thereby acting as solution classifying isotropy group (of the vacuum) that organizes the distinct solutions of the theory into families of the same energy. The integer-valued subgroup G(Z)5H is identi"ed with the Weyl group of G that transforms the set of dilaton vectors a associated with "eld strengths of the same rank as weight vectors of the irreducible representa? tions of G(Z). The ; Weyl group in D49 contains a subgroup S consisting of the permutations of the \" internal coordinates (i  j), corresponding to the permutations of "eld strengths, and (for D48) the additional discrete symmetries that interchange "eld strengths and the Hodge dualized "eld strengths (namely, the interchange of "eld strength equations of motion and Bianchi identity). At the same time, the associated dilaton vectors transform in such a way that theory is invariant, respectively, by permutation or change of signs, forming an irreducible multiplet under the Weyl group. In particular, M are invariant under the ; Weyl transformations and therefore D is also ?@ preserved. Furthermore, since the Bogomol'nyi matrix M is invariant under the ; Weyl group, p-branes with the same eigenvalues of M and, therefore, the same supersymmetry property are related by the ; Weyl group. Starting with a p-brane with a set of a , one generates a ; Weyl group ? multiplet of p-branes with the same M (or same D) and the same eigenvalues of M. In the case of ?@ multi-scalar or dyonic p-branes, where the N Page charges are independent parameters, the size of the ; Weyl multiplet is larger than that of single-scalar p-brane counterparts, since the participating "eld strengths are now distinguishable. We classify p-branes according to the rank of "eld strengths that p-branes couple to [456,457]. Particularly, BPS p-branes are possible with N"1 4-form/3-form "eld strength, N44 2-form "eld strengths and N47 1-form "eld strengths. BPS p-branes with N participating "eld strengths appear in lower dimensions once they occur in some higher dimensions; the p-branes in those maximal dimensions are `stainless super p-branesa. Generally, BPS p-branes with D"4, 2,   respectively preserve , ,  of supersymmetry, and 0-branes with D", , , which occur only in     D"4, all preserve  . As for the super p-branes with 4 "eld strengths, there are two inequivalent  solutions: (i) those that preserve  of supersymmetry (denoted D"1) and are coupled to 2-form   In the following, we also call the real-valued group G as the ;-duality group, but the distinction between G(R) and G(Z) will be clear from the context.  This permutation is a discrete subset of G(R) which acts on a "eld strength multiplet linearly.

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and 1-form "eld strengths (ii) those that preserve  (denoted D"1) and are coupled to 1-form "eld  strengths, only. Lastly, we show that H-transformations on black holes discussed in Section 4 generate the most general black holes in D"11 supergravity on tori [171]. 6.3.2.1. 4-form xeld strength. There is only one 4-form "eld strength in each dimension, but within the supergravity models under consideration in this section, the 4-form "eld strength exists only in D58, since those in D(8 are Hodge-dualized to lower ranks. So, no multi-scalar generalization is possible. There is a unique single-scalar p-brane, which is either elementary membrane or solitonic (D!6)-brane. In D"8, one can construct dyonic membrane of the second type (534), but it is ruled out by the constraint L "0. $$ 6.3.2.2. 3-form xeld strengths. There are 11!D 3-forms in D56, except in D"7 where there is an extra 3-form coming from the Hodge-dualization of the 4-form. The associated dilaton vectors satisfy M "2d !2(D!6)/(D!2), which are not of the form (527), and, therefore, the multi?@ ?@ scalar generalization is not possible. In fact, this expression for M yields D"2#2/N in the limit ?@ F"F/N, ∀a: supersymmetry is completely broken unless N"1 (i.e. D"4), in which case 1/2 of ? supersymmetry is preserved. In D"6, one can construct dyonic strings. Due to the constraint L "0, only dyonic strings of the second type, which are (534) and (535) with D"4 and 2, $$ respectively, are possible. 6.3.2.3. 2-Form xeld strengths. Two-form "eld strengths couple to elementary 0-branes and solitonic (D!4)-branes. Analysis of 2-form "eld strengths and 1-form "eld strengths is complicated due to their proliferation in lower dimensions. We therefore discuss only supersymmetric cases; complete classi"cation of p-branes including non-supersymmetric ones can be found in [456,457]. The dilaton vectors a associated with N participating 2-form "eld strengths satisfy (527) only for ? N44. The dimensions D in which these BPS p-branes with N"1, 2, 3, 4 2-form "eld strengths appear are, respectively, D410, 9, 5, 4. For N"1, 2, 3, the p-branes preserve 2\, of supersymmetry, and for the N"4 case, the solutions preserve 1/8. Whereas p-branes with N43 can be either purely electric/magnetic or dyonic (of the "rst type), p-branes with N"4 are intrinsically dyonic (of the "rst type). In D"4, there are 4 inequivalent BPS black holes with D"4/N (N"1, 2, 3, 4), corresponding to dilaton couplings a"(3, 1, 1/(3, 0, respectively. These black holes are interpreted as bound states of N D"5 KK black holes with a"(3 [249,511]. In D"4, dyonic 0-branes of both "rst and second types satisfy the constraint L "0. $$ Discussion on the "rst type is along the same line as the multi-scalar 0-branes. As for the second type, we have solutions (534) and (535) with a"1/(3 (i.e. N"2) and a"0 (i.e. N"4), respectively. First, a"0 case is intrinsically dyonic of the "rst type even when j "0 or j "0. Although   the explicit forms of solutions are insensitive to signs of Page charges, their supersymmetry properties depend on their relative signs. Second, the supersymmetry property of a"1/(3 case is insensitive to the signs of Page charges. Supersymmetry is preserved when (i) j "0 or j "0,    Note, only for these values of a, the 0-branes have a regular event horizon.

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corresponding to purely solitonic or elementary solution preserving 1/4 of supersymmetry, or (ii) j "!j , corresponding to a massless black hole preserving 1/2.   6.3.2.4. 1-form xeld strengths. 1-form "eld strengths couple to solitonic (D!3)-branes, only. The dilaton vectors satisfy (527) for N47 participating "eld strengths. N"1, 2, 3 cases occur, respectively, in D49, 8, 6, whereas N"5, 6, 7 cases occur in D"4, only. As for the N"4 case, there are 2 inequivalent BPS solutions: (i) those occurring in D46, denoted N"4 or D"1 and (ii) those occurring only in D"4, denoted N"4 or D"1. For generic values of Page charges, p-branes preserve 2\, [  ] of supersymmetry for N"1, 2, 3, 4 [N"5, 6, 7]. The N"4 case preserves . In   the case D"1, , , , p-branes are BPS or non-BPS, depending on the signs of the Page charges.  However, for D"4, 2, , 1, their supersymmetry properties are independent of the Page charge  signs. 6.3.2.5. Black holes in 44D49. The 0-branes in D"11 supergravity on tori with the most general charge con"gurations can be obtained by applying subsets of ;-duality transformations on the generating solutions. As in the case of black holes in heterotic string theory on tori, the set of transformations that generate the general black holes with the canonical asymptotic value of scalar matrix M "1 from the generating solutions is of the form H/H , where H is the    largest subgroup of H that leaves the generating solutions intact. The H/H transformation  introduces dim(H)!dim(H ) parameters, which together with the parameters of the generating  solutions form the complete parameters of the most general solution. The number of ;(1) charges of the generating solutions are 5, 3, 2 for D"4, 5,56, respectively. The charge con"gurations for these generating solutions are the same as the heterotic case in Section 4, with all the charges coming from the NS-NS sector. To generate solutions with an arbitrary asymptotic value of the scalar matrix M, one additionally imposes a general (real-valued) ;-duality transformation. The `dresseda 0-brane charge ZM "< Z can be rearranged in an N;N anti-symmetric complex  matrix Z (A, B"1,2, N), where N is the number of the maximal supersymmetry " in D dimensions. ZM and Z are invariant under the global G-transformation but transform " under the local H-transformation. Z appears in the supersymmetry algebra in the form " [Q , Q ]"C Z . In general, Z is splitted into blocks of ,;, submatrices. Two diagonal ? @ ?@ "  "   blocks Z correspond to NS-NS charges and two o!-diagonal blocks represent R-R charges. 0* By applying the H-transformation Z PZ "hZ h2 (h3H), one can bring the matrix Z into " " " " the skew-diagonal form with complex skew eigenvalues j (i"1,2, N/2). These eigenvalues j G G are related to charges of the generating solutions in a simple way, which we show in the following.

 For 0-branes in D"4, the matrix Z is de"ned as follows [159,381]. The electric q and magnetic p charges of the  ' ' ;(1) gauge group of the N"8, D"4 supergravity are combined into a 56-vector Z2"(p', q ), which transforms ' under G as ZPKZ. The dressed 0-brane charge ZM "< Z"(p ', q )2 is invariant under G but transforms under local  ' S;(8). The central charge matrix Z , which is the complex antisymmetric representation of S;(8), that appears in the  N"8, D"4 supersymmetry algebra is related to the `dresseda charges q and p ' as Z "(q #ip ')t' , where '  '  t' "!t' are the generators of SO(8).  

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We now discuss the subsets of H that generate 0-branes with the most general charge con"gurations from the generating solutions. E D"4: The most general 0-brane carries 28#28 electric and magnetic charges of the ;(1) gauge group. The U-duality group is G"E with the maximal compact subgroup H"S;(8).  The skew eigenvalues j (i"1,2, 4) are related to the charges Q ,Q $Q , G 0*   P ,P $P and q of the generating solution as 0*   j "Q #P , j "Q !P ,  0 0  0 0 (537) j "Q #P #2iq, j "Q !P !2iq .  * *  * * The subset of H"S;(8) that leaves the generating solution unchanged is SO(4) ;SO(4) . The * 0 63!(6#6)"51 parameters of H/H "S;(8)/[SO(4) ;SO(4) ] are introduced to the generat * 0 ing solution. E D"5: The `dresseda 27 electric charges of the most general 0-brane transform as a 27 of the ;Sp(8) maximal compact subgroup of U-duality group E . The skew eigenvalues j (i"1,2, 4)  G ,Q $Q and Q of the generating with a constraint  j "0 are related to the charges Q G G 0*   solution as

E

E

E

E

j "Q#Q , j "Q!Q , j "!Q#Q , j "!Q!Q . (538)  0  0  *  * The subset SO(4) ;SO(4) L;Sp(8) leaves this charge con"guration intact. The * 0 ;Sp(8)/[SO(4) ;SO(4) ] transformation introduces remaining 36!12"24 charge degrees of * 0 freedom into the generating solution. D"6: The most general 0-brane carries 16 electric charges, which transform as a 16 (spinor) of the SO(5, 5) U-duality group, whereas the `dresseda charges transform as (4, 4) under the maximal compact subgroup SO(5);SO(5). The skew eigenvalues j (i"1, 2) are related to the G charges Q ,Q $Q as 0*   j "Q , j "Q . (539)  0  * The subgroup SO(3) ;SO(3) of the maximal compact subgroup SO(5);SO(5) leaves the * 0 generating solution intact. The transformation [SO(5);SO(5)]/[SO(3) ;SO(3) ] introduces * 0 remaining 2(10!3)"14 charge degrees of freedom. D"7: The most general 0-brane carries 10 electric charges, which transform as a 10 under the S¸(5, R) U-duality group, whereas the `dresseda charges also transform as 10 under SO(5). The skew eigenvalues j (i"1, 2) are related to the charges Q ,Q $Q in the same way as G 0*   the D"6 case. The subgroup SO(2) ;SO(2) of the maximal compact subgroup SO(5) preserves * 0 the generating solution. 10!2"8 parameters of SO(5)/[SO(2) ;SO(2) ] are introduced into * 0 the generating solution. D"8: 6 electric charges of the general 0-brane transform as (3, 2) under the U-duality group S¸(3, R);S¸(2, R). There is no subgroup of the maximal compact subgroup SO(3);;(1) that leaves the generating solution intact. The SO(3);;(1) transformation induces 3#1"4 remaining charge degrees of freedom into the generating solution. D"9: 4 electric charges of the general 0-brane transform as (3, 1) under the U-duality group S¸(2, R);R>. The maximal compact subgroup ;(1) introduces an additional electric charge into the generating solution.

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Since the equations of motion and, especially, the Einstein frame metric are invariant under the U-duality, it is natural to expect that quantities derived from the metric, e.g. the ADM mass and the Bekenstein}Hawking entropy, are U-duality invariant. In fact, one can express the Bekenstein} Hawking entropy in a manifestly U-duality invariant form in terms of unique G invariants of D"11 supergravity on tori. Such manifestly U-duality invariant entropy depends only on `integer-valueda quantized bare charges [260]. In the following, we give the manifestly U-duality invariant form for the Bekenstein}Hawking entropy of black holes with general charge con"guration. E D"4: The quartic E invariant is given in terms of Z as [160]   J "Z ZM !Z ZM "!(Z ZM  )#  (e ZM  ZM !"ZM #$ZM %&   !"#$%&          !"     #e !"#$%&Z Z Z Z ). (540)    !"  #$  %& In terms of the skew-eigenvalues j , J takes the form G    J " "j "!2 "j ""j "#4(jM jM jM jM #j j j j ) . (541)  G G H         G HG By substituting j in (537) into the following E invariant entropy [405,413], one reproduces the G  Bekenstein}Hawking entropy (203) of the generating solution: p S " (J . & 8 

(542)

E D"5: The cubic E invariant has the form [156,157]:   X#$Z , (543) J "! X Z X!"Z  !  "#  $  2  $ which is expressed in terms of the real skew eigenvalues j as G  J "2 j . (544)  G G Here, X is the USp(8) symplectic invariant. The manifestly E invariant expression for the  entropy of general solution is of the form [171]: (545) S "p(  J , &   which reproduces the entropy (276) of the generating solution if the expressions for j in (538) are G substituted. E 64D49: There is no non-trivial U-duality invariant in D56. This is consistent with the fact that the Bekenstein}Hawking entropy of the general BPS black holes in D56 is zero, which is the only U-duality invariant in D56. For near-extreme black holes, which has non-zero Bekenstein}Hawking entropy, the entropy can be expressed in a duality invariant form in terms of `dresseda electric charges and, therefore, has dependence on scalar asymptotic values [171]. The ADM mass M of the BPS solution is given by the largest eigenvalue max+"j ", of Z . The G " U-duality invariant form of the ADM mass can be expressed in terms of the U-duality invariant quantities Tr(>K) (m"1,2, [N/2]!p#1; > ,Z2 Z ) and corresponds to the largest root of " " " "

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a polynomial of degree [N/2]!p#1 in M with coe$cients involving Tr(>K). The BPS solution " preserves p/N of supersymmetry if p of the central charge eigenvalues have the same magnitude, i.e. "j ""2""j ". This depends on charge con"gurations of black holes. As for the generating  N solutions, the number of identical eigenvalues "j " can be determined from (537), (538) and (539). In G the following, we discuss D"4 black holes as an example [171]. E p"4 case: The generating solution preserves 1/2 of supersymmetry when only one charge is non-zero. The U-duality invariant ADM mass is M"!Tr(> ).   E p"3 case: An example is the case where (Q , Q , P "P )O0 with q"((Q #P )!Q .      0 0 * The U-duality invariant mass has the form M"!Tr(> )#(  Tr(>)!  (Tr > ).       E p"2 case: An example is the case where only Q and Q are non-zero. The U-duality invariant   mass M is the largest root of a cubic equation in M with coe$cients involving U-duality invariants Tr(>K) (m"1, 2, 3).  E p"1 case: Examples are the case where only Q , Q and P are non-zero, or the case where all    the "ve charges are non-zero and independent. The largest root of a quartic equation involving invariants Tr(>K) (m"1, 2, 4) corresponds to the ADM mass of the BPS solution.  6.3.3. p-Brane embedding of black holes We discuss the D"10, 11 p-brane embeddings of black holes in D(10. Starting from p-branes in D"10, 11, one obtains 0-branes in D(10 by wrapping all the constituent p-branes around the cycles of the internal manifold. The resulting black hole solution has the following generic form: ds "h"\(r)[!h\(r) f (r) dt#f \(r) dr#r dX ] , " "\ where 2m 2m sinh d ? f"1! H "1# ? r"\ r"\



(546)



is harmonic function associated with non-extremality parameter m [charge Q " ? (D!3)m sinh 2d ] and h(r)"“, H (r). The ADM mass and the Bekenstein}Hawking entropy ? ? ? are D!2 N , , ! k, M "2m (D!3) sinh d #D!2 " (Q#k#2 ? ? "+ D!3 2 ? ? (547) 1 , , S " (2m)"\"\u “ cosh d &k"\"\\, “ ((Q#k#k) , "\ & 4 ? ? ? ?










 The fraction of supersymmetry preserved by N"8, D"4 BPS black holes can also be determined from the Killing spinor equations [140].  Generally, for a multi-charged black p-brane with the Page charges j "(p #1)m sinh 2d (a"1,2, N), ? ? , , p #2 N M "2m (p #1) sinh d #p #2 " (j#k#2 ! k ? ? "+ p #1 2 ? ? 1 , , S " (2m)N >N >u  “ cosh d &kN >N >\,u  “ ((j#k#k), & 4 N> ? N> ? ? ? where k,(p #1)m is the rescaled non-extremality parameter.










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where k,(D!3)m is the rescaled non-extremality parameter and we neglected overall factor related to the gravitational constant, since we are interested only in the dependence on m, d and Q . ? ? As can be seen from (546), dimensional reduction of single-charged p-branes leads to black holes with singular horizon and zero horizon area in the BPS limit. To construct black holes with regular event horizon and non-zero horizon area in the BPS limit, one has to start from multi-charged p-branes in higher dimensions. This is achieved in D"4, 5 with N"4, 3, respectively, which can be seen from the BPS limit of entropy in (547). In fact, it is shown in [75] that the number N of distinct BPS black hole solutions to the equations of motion for the action (530) that have intersecting p-brane origins in D"11, 10 is N"4, 3 and 2 for D"4, 5 and D56, respectively. In the following, we discuss intersecting p-branes which give rise to regular BPS black holes in D"4, 5, as well as black holes with singular BPS limit. We concentrate on intersecting M-branes; intersecting p-branes in D"10 are related to intersecting M-branes through dimensional reduction and dualities. All the possible D"10, 11 BPS, intersecting p-branes that satisfy intersection rules are classi"ed in [75]. Also, there is a M-brane con"guration (473) interpreted as a M 2-brane within a M 5-brane (2L5), which preserves 1/2 of supersymmetry [388]. By including 2L5 con"gurations, one constructs new type of black holes [149,150] which are mixture of marginal and non-marginal bound states. Namely, the ADM mass and horizon area of p-branes that contain 2L5 have the forms M& (a#k#ck and A &kAY“ (k#(a#k) with a " G G G G & G (Q#P for each 2L5 constituent, where Q [P ] is charge of M 2-brane (M 5-brane) in the G G G G 2L5 constituent and c, c are appropriate constants. One can also add KK monopole to intersecting M-branes with overall transverse space dimensions higher than 3. We will not show the explicit intersecting p-brane solutions, since one can straightforwardly construct them applying harmonic superposition rules discussed in the previous section; explicit solutions can be found, for example, in [149,150,175,287,432,611]. 6.3.3.1. Four-dimensional black holes. Intersecting M-branes which reduce to D"4 black holes with 4 charges, i.e. (546) with N"4, should have 4 or 3 (with momentum along an isometry direction) M p-brane constituents and at least 3 overall transverse directions. Such con"gurations are (i) 2N2N5N5 for N"4, and (ii) 5N5N5, 2N5N5 and 2N2N5 for N"3. Additionally, one has the following D"11 con"gurations that reduce to D"4 black holes preserving 1/8 of supersymmetry: (i) 2N2N2#KK monopole, (ii) 2N5#boost#KK monople, (iii) (2L5)N(2L5)N(2L5), (iv) (2L5)N5#boost, (v) (2L5)N2#KK monopole, (vi) (2L5)#boost#KK monopole. Also, the boost#KK monopole con"guration reduces to D"4 black hole that carries KK electric and magnetic charges and preserves 1/4 of supersymmetry. 6.3.3.2. Five-dimensional black holes. Intersecting M-branes with 3 or 2 (with a boost along an isometry direction) M p-brane constituents and at least 4 overall transverse directions can be reduced to D"5 black holes with 3 charges. These are 2N2N2 and 2N5 with a momentum along an isometry direction. An additional M-brane con"guration that reduces to D"5 black hole with 3 charges is (2L5)N2, which preserves 1/4 of supersymmetry. 6.3.3.3. Black holes in D56. 0-branes in D56 can be supersymmetric with up to 2 constituent p-branes. Supersymmetric 2 intersecting M-branes are 5N5, 2N5 and 2N2, which are compacti"ed

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to 2-charged black holes in D"4, 5 and 7, respectively, after wrapping each constituent M p-brane around cycles of a compact manifold. One can compactify overall transverse directions of these D"5, 7 black holes to obtain black holes with 2 charges in D"4 and D46, respectively. One can also construct black holes in D49 and D45 by compactifying M 2-brane and M 5-brane with momentum along a longitudinal direction, respectively. Additionally, the M-brane con"guration (2N5)#boost reduces to D"6 black hole that preserves 1/4 of supersymmetry.

7. Entropy of black holes and perturbative string states One of challenging problems in quantum gravity for past decades is the issues related to black hole thermodynamics. It was early 1970s [38,63}65,126] when it was "rst noticed that the event horizon area A behaves much like entropy S of classical thermodynamics. Namely, it is observed [343,496] that the event horizon area tends to grow (dA50), resembling the second law of thermodynamics (dS50). Furthermore, Bardeen et al. [38] proved that the surface gravity i of a stationary black hole is constant over the event horizon, resembling the zeroth law of thermodynamics, which states that the temperature is uniform over a body in thermal equilibrium. They also realized the following relation between the ADM mass M of black holes and the event horizon area A: 1 i dA , (548) dM" 8pG , which resembles the thermodynamic relation between energy E and entropy S (the "rst law of thermodynamics): dE"¹dS ,

(549)

if one identi"es the energy E with the ADM mass M and the entropy S with the event horizon area A with some unknown constant of proportionally. Such analogy between horizon area and entropy met initially with skepticism, until Hawking discovered [345,346] that black hole is indeed thermal system which radiates (quasi-Planckian black body) thermal spectrum with temperature ¹ " i/2p, due to quantum e!ect. Since then, it is widely accepted that a black hole, as a thermal & system, is endowed with `thermodynamica entropy given by a quarter of the event horizon area in Planck units, the so-called Bekenstein}Hawking entropy [38,63}65,343,347,411,477,627]: S "A/4 G . (550) & , Puzzles on black hole entropy stem from the fact that entropy is a thermodynamic quantity, which arises from the fundamental microscopic dynamics of a large complicated system as

 For the Kerr}Newmann black hole, this relation is generalized to 1 dM" i dA#U dQ#X dJ, 8pG , where U (X) is the potential (angular velocity) at the event horizon and Q [J] is a ;(1) charge (angular momentum).

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a universal macroscopic quantity which does not depend on the details of the underlying microscopic dynamics. So, if the correspondence between laws of black hole mechanics and thermodynamic laws is to be valid, the thermodynamic black hole entropy (550) should have a statistical interpretation in terms of the degeneracy of the corresponding microscopic degrees of freedom. Based upon our knowledge of statistical mechanics, one could guess several possible interpretations of the statistical origin of black hole entropy: (i) internal black hole states associated with a single black hole exterior [64,66,67,347], (ii) the number of di!erent ways the black holes can be formed [64,347], (iii) the number of horizon quantum states [587,594], (iv) missing information during the black hole evolution [302,348]. Another di$culty comes from the fact that contrary to ordinary thermodynamics, understanding of black hole thermodynamics requires the treatment of quantum e!ects, as we noted the crucial role that the Hawking e!ect plays in establishing black hole thermodynamics. Thus, the statistical interpretation of black hole entropy should necessarily entail quantum theory of gravity, of which we have only rudimentary understanding. The early attempts during the 1970s and 1980s were not successful in the sense that the most of approaches either (i) did not touch upon the statistical meaning of entropy, since the calculations were mostly based on thermodynamic relations (e.g. calculating entropy using Clausius's rule S"dM/¹ given that the black hole temperature ¹ is determined by the surface gravity method [38,626]), or (ii) is purely classical (e.g. in Gibbons}Hawking Euclidean (on-shell functional integral) method [294] involving grand partition function, the black hole entropy A/(4 G ) is , reproduced at tree-level of quantum gravity calculation). Another major was the ultraviolet quadratic divergence (related to the divergence of the number of energy levels a particle can occupy in the vicinity of black hole horizon) in black hole entropy when the Euclidean functional formulation of the partition function is evaluated for quantum "elds in the black hole background. To avoid such divergences, 't Hooft [593] introduced `cuto! a at a small distance e just above the event horizon in the path integral of real free scalar "eld , assuming that there are no states at the interval between the event horizon and the cuto! (the so-called brick-wall method). The black hole entropy in "eld theory based on brick-wall method in general depends on the cuto! distance e in the form S &A/4e, re#ecting the quadratic divergence. It was conjectured [594] by 't Hooft ( that such ultraviolet divergence of the statistical entropy might be related to Hawking's information paradox [348], i.e. a black hole is an in"nite sink of information. In [588], it is shown that the divergence associated with the Euclidean functional formulation of the partition function for canonical quantum gravity (of point-like particles) is related to the renormalization of the gravitational coupling G . When the contributions to entropy (obtained , from the partition function) from the pure gravity and matter "elds are added, the entropy takes a suggestive form




c A 1 # S" e 4 G ,



 This is closely related to the fact that quantum gravity in point-like particle "eld theory is non-renormalizable, as we will discuss below.  't Hooft proposed that the entropy of black hole is nothing but the entropy of particles which are in thermal equilibrium with black hole background [592,593,595].

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similar to (550) but the bare gravitational constant G is renormalized. The explicit calculation , of quantum corrections of quantum gravity shows [202,394,395] that the renormalized gravitational constant G takes the same form. Since superstring theory is a promising candidate for , a "nite theory of quantum gravity, the contradiction encountered in the point-like particle "eld theory has to be resolved [583,585,588,589]. It is indeed shown in [186,588] that theory of superstring propagating in a black hole background gives rise to a xnite expression for black hole entropy in the calculation of partition function through Euclidean path integral, with the xnite renormalized gravitational coupling G : the genus zero contribution gives rise to the classical result , (550) with a bare Newton's constant G and the higher genus terms contribute to "nite corrections , to G . , This can be seen intuitively by considering microscopic states near the event horizon [584]. For point-like particles, due to the arbitrarily small longitudinal Lorentz contraction near the event horizon, an arbitrarily large number of particles can be packed close to the event horizon, giving rise to a divergent entropy. However, the Lorentz contraction of strings along the longitudinal direction is eventually halted to a "nite extent and, therefore, only "nite number of strings can be packed near the horizon, leading to "nite entropy. Just as only contribution to the "rst-quantized path integral of the point-like particle "eld theory is from the set of paths that encircle or touch the black hole event horizon, only string graphs which contribute to the entropy through the partition functions are those that are somehow entangled with the event horizon. From the point of view of an external observer, this kind of closed strings, which are partially hidden behind the event horizon, look like open strings frozen to the horizon. Thus, the black hole entropy can somehow be interpreted as being associated with oscillation degrees of freedom of #uctuating open strings whose ends are attached to the black hole horizon [588]. This section is organized as follows. In Section 7.1, we discuss connection between black holes and perturbative string states. Identi"cation of black holes with string states makes it possible to do explicit calculations of statistical entropy of black holes, based on the conjecture that microscopic origin of entropy is from degenerate string states with mass given by the corresponding ADM mass of the black hole. In Section 7.2, we discuss Sen's original calculation of statistical entropy of the BPS static black hole, which was compared to Bekenstein}Hawking entropy evaluated at the stretched horizon. In Section 7.3, we generalize Sen's result to near-extreme rotating black holes. In Section 7.4, we discuss the level matching of black holes to macroscopic string states at the core. This justi"es our working hypothesis that black holes are perturbative string states. In Section 7.5, we summarize Tseytlin's method of chiral null model for calculating statistical entropy of the BPS black holes that carries magnetic charges as well as perturbative NS-NS electric charges.

7.1. Black holes as string states It is not a new idea that elementary particles might behave like black holes [342,526,594]. A particle whose mass exceeds the Plank mass and therefore whose wavelength is less than its Schwarzschield radius exhibits an event horizon, a characteristic property of black holes. Since typical massive excitations of strings have mass of the order of the Planck mass, one would expect that massive string states become black holes when gravitational coupling (or string coupling) is

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large enough. It has been shown that black holes with given charges and angular momenta behave like string states with the corresponding quantum numbers. Before we discuss identi"cation of black holes with string states, we summarize [308] some aspects of perturbative spectrum, moduli space and T-duality of heterotic string on a torus. Heterotic string [322] is a theory of closed string whose left- and right-moving modes are respectively described by bosonic and super string theories. So, the critical dimensions, in which the conformal anomali is absent, for each mode are di!erent: D"26 for the left-movers and D"10 for the right-movers. In compactifying the extra 16 coordinates of the left-movers on ¹, one obtains a rank 16 non-Abelian gauge group which is associated with the even-self-dual lattice of the type E ;E or Spin(32)/Z . Thus, the massless bosonic modes of heterotic string in D"10 at a generic    point of moduli space are ;(1) gauge "elds, as well as graviton, 2-form "eld and dilaton in the NS-NS sector ground state. We compactify the extra d spatial coordinates XI (k"1,2, d) on ¹B to obtain a theory in D"10!d. The toroidal compacti"cation is de"ned by the periodic identi"cation of each internal coordinate, i.e. XI&XI#2pmI, where mI is the integer-valued string `winding modea. Since the holonomy of a torus is trivial, all of supersymmetry is preserved in compacti"cation, i.e. N"4 for the compacti"cation on ¹. The ¹B part of the heterotic string worldsheet action in #at background, including the coupling to gauge "elds A? and a 2-form potential B , is I IJ 1 dz[(G #B )RXIRM XJ#A RXIRM X? S" IJ IJ I? 2p



(551) # (G #B )RX?RM X@]#(ferminionic terms) , ?@ ?@ where k, l"1,2, d (a"1,2, 16) correspond to the coordinates of ¹B (¹ of left-movers), and the complex worldsheet coordinate and derivative are de"ned as 1 1 (q#ip), R" (R !iR ) . z" N (2 O (2

(552)

The internal coordinates X? live on the weight lattice of E ;E or Spin(32)/Z . The background    "elds G , B and A , which parameterize the moduli space O(d#16, d, Z)/[O(d#16, Z) IJ IJ I? ;O(d, Z)] of ¹B;¹, can be organized into the `background matrixa of the form:






(G#B#A)A ) A  ) GH G( , (553) 0 (G#B) '( where B [G] is the antisymmetric (symmetric) part of E. (For the relations between the background "elds in (551) and E, see the next footnote.) Here, the indices i, j [I, J], associated with ¹B [¹], run from 1 to d (from 1 to 16). In particular, the components E "(B#G) of E are related to the '( '( Cartan matrix C of E ;E or Spin(32)/Z as '(    (554) E "C (I'J), E "C , E "0 (I(J) . '( '( '( ''  '' The Narain lattice [480,481] KB>, which de"nes ¹B;¹ is spanned by basis vectors a,(a , a ) of the form: G ' a "(e , A)E ) , a "(0, E ) , (555) G G G ) ' ' E,B#G"

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where the vectors +E? " I"1,2, 16, and +eI " i"1,2, d, are de"ned through ' G E ) E "2G , e ) e "2G , e ) E "0 . (556) ' ( '( G H GH G ' The zero modes (p , p ) of the right- and left-moving momenta form an even self-dual lattice 0 * CBB>"CBBC. Quantized momentum zero modes (p , p ) are embedded into 0 * CB>B> as





p pG 0 " 0 pH p * *

0



p( *

,

(557)

where p "[nR#mR(B!G)]aH, p "[nR#mR(B#G)]aH, m,n3RB> . 0 * Here, aH is the basis vector of the lattice dual to KB>:

(558)

aGH"(eGH, 0), a'H"(!A'eGH, E'H) , (559)  G where E'H [eGH] are dual to E [e ], i.e. B eIeHH"dH [  E?E(H"d( ]. ' G I G I G ? ' ? ' The heterotic string with the action (551) has an O(d#16, d, Z) ¹-duality symmetry. This group is a subgroup of the following O(d#16, d#16, Z) transformation that preserves the triangular form of E in (553): EPE"(aE#b)(cE#d)\,


 a b c

d

3O(d#16, d#16, Z) ,

(560)

and (p , p ) in CBB> transforms as a vector. T-duality is proven [307] to be exact to all orders in 0 * string coupling. The mass of perturbative states for heterotic string on a torus is [322] 1 1 +(p )#2N !1," +(p )#2N !2, , (561) M" 0 * 8j * 8j 0   where N are left- and right-moving oscillator numbers, j is the vacuum expectation value of * 0  the dilaton (or string coupling). We now identify string states with black holes. The mass of the BPS purely electric black holes in heterotic string on ¹, which preserves  of the N"4 supersymmetry, is [337,544,558,560,564]:  1 m" a?(M#¸) a@ , (562) ?@ 16j  where a is the charge lattice vector on an even, self-dual, Lorentzian lattice K with the O(6, 22) metric ¸ and the subscript (0) denotes asymptotic values. Here, M is the moduli matrix of ¹ de"ned in (121). Under the T-duality, the moduli matrix and the charge lattice transform

 By contracting with eGH and E'H, one obtains the background "elds in (551) from E. For example, I ? G?@"2G (E'H)?(E(H)@ and BIJ"2B (eGH)I(eHH)J, etc. '( GH

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as [560] MPXMX2, KP¸X¸K, X3O(6, 22)

(563)

and the BPS mass (562) is invariant. With a choice of the asymptotic values j"1 and  M"I , the mass takes a simple form:  #¸) a@"(a ), a? ,(I $¸) a@ . (564) m"  a?(I  ?@  0 0*   ?@  The string momentum (winding) zero modes are identi"ed with the quantized electric charges of KK (2-form) ;(1) gauge "elds, i.e. a "p . Then, m"M, provided N " [248]: the BPS black 0* 0* 0  holes are identi"ed with the ground states of the right movers. With a further inspection of (561), one "nds N in terms of a [248]: * (565) N !1"((a )!(a ))"a2¸a , *  *  0 leading to a2¸a5!2. So, the various BPS black holes in the heterotic string on a torus are identi"ed [248] as string states with the corresponding value of N [or a2¸a]. * Non-extreme black holes are identi"ed with string states with the right movers excited as well. Identi"cation of black holes in other dimensions and in type-II theories with string states is proceeded similarly as above. For type-II string theories, the mass of perturbative string state is 1 1 M" +(p )#2N !1," +(p )#2N !1, . (566) 0 * 8j 0 8j *   Since the type-II strings have supersymmetry in both the right- and left-moving sectors, perturbative string states can (i) preserve supersymmetry in both sectors (N "N "1/2), leading to 0 * short supermultiplet; (ii) preserve supersymmetry in one sector, only (N 'N "1/2), leading 0* *0 to intermediate supermultiplet; (iii) break supersymmetry in both sectors (N , N '1/2), leading to 0 * long supermultiplet. Further study of equivalence of string states and extreme black holes, including spins of string states and dipole moments of rotating black holes, is carried out in [237,249].

7.2. BPS, purely electric black holes and perturbative string states In the previous section, we showed that BPS electric black holes in the string low energy e!ective actions are identi"ed with perturbative string states. Thus, it is natural to infer that the microscopic degeneracy of black holes originates from the degenerate string states in a corresponding level. In general, non-extreme black holes are also identi"ed with perturbative string states. However, non-extreme solutions are plagued with (unknown gravitational) quantum corrections and, therefore, the ADM mass cannot be trusted. In fact, the number of states in non-extreme black hole grows with the ADM mass M like &e+
 [594], whereas the string state level density grows with
 the string state mass M as &e+ . Thus, if one is to identify string states with black hole states,   one is forced to identify M with M [517,583]. In [583], Susskind attributes the discrepancy to
   the mass renormalization due to unknown quantum corrections. (See also Section 8.2, where it is discussed that the Bekenstein}Hawking entropy of non-extreme black holes has to be evaluated

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at the speci"c string coupling at the black hole and microscopic con"guration (D-branes and fundamental string) transition point.) As "rst pointed out by Vafa [583], the BPS solutions do not receive quantum corrections [485] due to renormalization theorem of supersymmetry. Such class of solutions are, therefore, suitable for testing the hypothesis that the statistical origin of black hole entropy is from the degenerate string states with mass given by the ADM mass of black hole. So, one can calculate the `statisticala entropy by taking logarithm of the string level density. This yields the "nite non-zero entropy &(N . However, the `thermala entropy of the BPS purely * electric black holes in heterotic string is zero. In [564], Sen circumvented with problem by postulating that the `thermala entropy of the BPS black hole is not the event horizon area, but the area of a surface close to the event horizon, a so-called `stretcheda horizon [586,587,596]. Although the BPS electric black hole solutions are free of quantum corrections, they receive (classical) stringy a corrections due to the singularity at the event horizon. This leads to the shift of the event horizon by the amount of an order of a. Originally, the stretched horizon is de"ned [587] as the surface where the local Unruh temperature for an observer, who is stationary in the Schwarzschield coordinate, is of the order of the Hagedorn temperature [331]. Namely, it is a surface where the string interactions become signi"cant. In [419,475,582], it is observed that the transverse size of strings diverges logarithmically and "ll up a region at the stretched horizon, melting to form a single string. Thereby, information in the string states is stored and thermalized with black hole environment in the region near the stretched horizon [452,454,475,582,584], and black hole states are in one-to-one correspondence with single string states. So, the statistical entropy is due to degenerate strings states in equilibrium with the black hole background at the stretched horizon [588]. In this section, we summarize [495,564] to illustrate this idea. The electric black hole considered in [564] is a special case of the general solution [178] discussed in Section 4.2.1. But for the purpose of illustrating the idea of perturbative string state and black hole correspondence, we follow Sen's parameterization of solution in terms of left-moving and right-moving electric charges, rather than in terms of KK and 2-form electric charges. 7.2.1. Black hole solution In Sen's notation, the most general non-rotating, electric black hole solution in the heterotic string on ¹, in the Einstein-frame, is [564] D r(r!2m) dt# dr#D(dh#sinh du) , g dxI dxJ"! IJ r(r!2m) D

(567)

where D,r[r#2mr(cosh a cosh b!1)#m(cosh a!cosh b)].

 It is argued in [350] that the Bekenstein}Hawking formula for entropy, i.e. (entropy)Jhorizon area), ceases to hold for extreme case and entropy of an extreme black hole is always zero, displaying discontinuity in going from non-extreme to extreme case. This is attributed [350] as being due to di!erence in Euclidean topologies for the two cases. The cure for this discontinuity is proposed in [291], where it is suggested that one has to extremize after quantization, rather than quantizing after extremization.

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The ADM mass and the electric charges are 1 m(1#cosh a cosh b) , M " & 2G , m g nG sinh a cosh b for 14i422 , Q (2 QG" m g pG\ sinh b cosh a for 234i428 , (2 Q



(568)

where n [p] is an arbitrary 22 [6] component unit vector. The left- and right-handed charges are de"ned as nG 1 !¸) QH" * g m sinh a cosh b , QG , (I GH * 2  (2 Q nG 1 #¸) QH" 0 g m sinh b cosh a , QG , (I GH 0 2  (2 Q

(569)

and the 28-component left- and right-handed unit vectors n and n are similarly de"ned. The * 0 solution (567) is in the frame where the O(6, 22) invariant metric ¸ (127) is diagonal. This parameterization of black hole solution has a convenient form in which only left (right) handed charges are non-zero when b"0 (a"0) with all the parameters "nite. The solution has 2 horizons at r"r "2m, 0. The event horizon area is >\



A" dh du(g g " >"8pm(cosh a#cosh b) . FF PP PP

(570)

The surface gravity at the event horizon is 1 i" lim (gPPR (!g " " . P RR F 2m(cosh a#cosh b) PP>

(571)

7.2.2. Extreme limit and string states The extreme limit is de"ned as a limit where the inner and outer horizons coincide, i.e. mP0. In taking the `non-extremality parametera m to zero, one has to let one (or both) of the boost parameters a and b go to in"nity so that the electric charges (569) do not vanish. Since we are interested in the BPS solutions, we let the ADM mass depend only on the right-handed electric charge. This is achieved by taking the limit bPR and mP0 such that m( "me@ remains as  a "nite non-zero constant, while a remaining "nite. In this limit, the ADM mass and the electric charges are 1 m( cosh a , " .1 2G , 1 QG " g nG m( sinh a, * (2 Q * M

1 QG " g nG m( cosh a , 0 (2 Q 0

(572)

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thereby the ADM mass depends on the right-handed electric charge, only: 1 Q , M " .1 8g 0 Q where G "2. In this limit, the solution has the form (567) with m"0 and , D"r(r#2m( r cosh a#m( ) .

(573)

(574)

The event horizon area (570) is zero in the BPS limit. However, the string states are degenerate. One can circumvent such problem by calculating entropy at the `stretched horizona right above the event horizon. To "nd a location of the stretched horizon, one considers a region close to the event horizon in the `string framea metric: r dS,g  dxI dxJK! g dt#g dr#gr(dh#sin h du) Q Q IJ m(  Q "!r  dtM #dr #r (dh#sin h du) ,

(575)

where r ,g r and tM ,t/m( . Note, in the frame (tM , r , h, ), all the dependence on the other parameters Q has disappeared. One can show that the other background "elds also become independent of the parameters near the event horizon, if one performs a suitable O(6, 22) transformation. Thus, the location of stretched horizon, i.e. the location where higher-order stringy corrections become important, is unambiguously estimated to be located at r "C, a distance of order 1 (in unit of string scale) from the event horizon. In terms of the original coordinate, the stretched horizon is located at r"C/g ,g. Q The stretched horizon area, calculated from (567) with m"0 and (574), is AK4pgm( "4pm( C/g , Q where only the term leading order in g is kept, and therefore the thermal entropy is

(576)

A p m( C S , " . & 4G 2 g Q , To compare this expression with the statistical entropy, one expresses S in terms of electric & charges by using the relation



Q m( "4 M ! * & 8g Q derived from (572):



Q 2pC M ! * . (577) S " & 8g & g Q Now we compare the thermal entropy (577) with the degeneracy of string states. Since string states identi"ed with BPS black holes have the right movers in ground state (N "), the string 0  state degeneracy is from the left movers with N given in terms of electric charges as (details are *

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along the same line as in Section 7.1):






Q 4 (578) N K M ! * , .1 8g * g Q Q for large Q. So, the statistical entropy associated with the degeneracy of string states is * 8p Q M ! * . (579) S ,ln d(N )K4p(N K 122 * * g & 8g Q Q This entropy expression has the same dependence on M and Q as the thermal entropy (577) .1 * calculated at the stretched horizon, and the two expressions agree if one chooses C"4 in (577). Note, it is crucial that the constant C does not depend on parameters of black holes; otherwise, the dependence of S (579) on Q and M cannot be trusted because of the unknown dependence of 122 * & C on these parameters.



7.3. Near-extreme black holes as string states In the previous section, we saw that thermal entropy of the BPS, non-rotating, electric black holes agrees (up to numerical factor of order one) with statistical entropy associated with the degeneracy of string states, if it is evaluated at the `stretched horizona. However, the rotating black hole case is problematic for the following reasons. Since the electric, rotating black hole (285) in the BPS limit with all the angular momenta non-zero has naked singularity, thermal quantities cannot be de"ned. The BPS limit with a horizon is possible in D56 with at most 1 non-zero angular momentum [366]. Even for this case, not only the event horizon is singular (i.e. the event horizon and the singularity coincide) and has zero surface area, but also the area of the stretched horizon (which is assumed to be independent of parameters of the black hole) is independent of angular momenta. We surmise that this is due to the unknown dependence of the location of the stretched horizon on physical parameters, unlike the non-rotating black hole case. The determination of the stretched horizon location may require understanding of a corrections with rotating black hole as the target space con"guration, which is di$cult to estimate at this point. We propose [184] an alternative way to circumvent the problems of the BPS electric black holes. Instead of de"ning the thermal entropy of the BPS black holes at the stretched horizon, we propose to calculate the thermal entropy of near extreme black holes at the event horizon. Then, the thermal entropy of near-extreme black holes takes suggestive form which can be interpreted in terms of string state degeneracy. We attempt statistical interpretation of such thermal entropy expression by using the conformal "eld theory of p-model with the near-extreme solution as a target space con"guration and with angular momenta identi"ed with [(D!1)/2] ;(1) left-moving world-sheet currents. 7.3.1. Thermal entropy of near-BPS black holes The proper way of taking the near-BPS limit of rotating black holes is to take the limit in such a way that the angular momentum contribution to the thermal entropy is not negligible compared with the contribution of the other terms, while ensuring the regular horizon so that thermal quantities can be evaluated at the event horizon. This is achieved as follows. First of all, the

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near-BPS limit is de"ned as the limit in which the non-extremal parameter m'0 is very small and the boost parameters d are very large such that the combinations meBG (i"1, 2) remain as "nite, G non-zero constants. Then, as long as l are non-zero, J (287) do not vanish. Second, the requirement G G of the regular event horizon restricts the range of the parameters of the solution [478], e.g. m5"l "  for D"4 and m5("l "#"l ") for D"5. For an arbitrary D, we write such a constraint generally   as . QQ
Third, for the thermal entropy to be macroscopically non-negligible, the electric charges have to be very large, i.e. Q
In such a near-BPS limit, the thermal entropy (288) takes the form [184]:





4  2 "\

J . S "2p QQ(2m)"\! G   (D!3)   (D!3) G

(580)

7.3.2. Microscopic interpretation In this section, we calculate the statistical entropy of near-extreme, rotating black holes by counting the degenerate string states with the speci"c angular momenta. In principle, to calculate the statistical entropy of rotating black holes, one has to extract the degenerate string states (in a given level) with the speci"c values of angular momenta. This was "rst attempted in [517]. Note, string states in a given level consist of states with di!erent angular momenta (with the maximum angular momentum determined by the level). Alternatively, one can use the level density formula that has contribution from all the possible angular momenta in a given level and employ the technique of conformal "eld theory to extract the speci"c contribution of states with given angular momenta. The main point is that the ;(1) charges of the left-moving worldsheet currents are interpreted as target space spins of string states. States with non-zero spins are obtained by applying the a$ne ;(1) current operators to spin zero states. The procedures described in the following paragraphs are also applicable to D-brane interpretation of entropy of rotating black holes [95,98,471]. As in the BPS case, one identi"es the KK electric charges Q [the 2-form electric charges Q] G G with the (internal) momentum zero modes [string winding modes]. Then, mass of the perturbative string states takes the form:






1 4 4 M "(Q#Q)# N ! "(Q!Q)# (N !1) , 0       2 a * a

(581)

where each circle in the torus has self-dual radius R"(a. From the second equality in (581), i.e. the Virasoro constraint, one has the following relation between N and N : * 0 N "aQ Q #N # . (582) *   0 

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Note, for the statistical interpretation to be valid, the electric charge (quantized in unit 1/(a) has to be very large, i.e. Q<1/(a.  In the near extreme limit (m+0), the BPS mass (287) takes the form M +(Q#Q)#O(m) . (583) &   By identifying the ADM mass (583) with the mass of string state (581), one "nds that the right movers are barely excited: N +#O(m). So, N is negligible compared to N and N +aQQ 0  0 * *   to a good approximation: N +aQQ#1#O(m)+aQQ


1 ln d(N , N )+2p c N "4p(N , * 0 * 6  *

(584)

with the right-mover contributions neglected, just like BPS black holes. Here, c "26!2"24  since we are considering left moving bosonic string modes, only. Note, this level density contains contribution from all the spin states in the level (N , N ). * 0 To extract the contribution by the states with particular spins, we employ conformal "eld theory technique. Recall that the p-model with the target space con"guration [104,105] given by a rotating black hole is described by the WZNW model [289,437,484,632] with the ;(1) "\ a$ne Lie algebra (i.e. Cartan sub-algebra of the O(D!1) rotational group), or the conformal "eld theory with the ;(1) "\ group manifold. The eigenvalues of the left-moving ;(1) worldsheet currents j "iR  HG (i"1,2, [(D!1)/2]) are G X interpreted as the [(D!1)/2] spins of string states. A general state in this WZNW model is labeled by charges of the a$ne Lie algebra as well as by the oscillator numbers. The conformal "eld can be expressed as U 2 "\ with ;(1) charges J 2   "\

( "\

U 2 "\ " “ eG(G&GU , (585) (  G where U is a conformal "eld without ;(1) charges, or without target space spins. Thus, the  left-moving conformal dimensions hM 's, i.e. the eigenvalues of the left-moving Virasoro generator ¸ ,  of U 2 "\ and U are related as:  ( 1 "\

hM U(2 "\ " J#hM U . (586) G 2 G This implies that the total number N of left moving oscillations of spinless states is reduced by the * amount  "\ J relative to the total number N of left moving oscillations of states with the  G G * speci"c spins J 2 :   "\

1 "\

N PN "N ! J . (587) * * * 2 G G Note, the level density for spinless states in a given level (N , N ) di!ers from the level density * 0 d(N , N ) of all the states in the level (N , N ) by a numerical factor, which can be neglected in the * 0 * 0

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large (N , N ) limit if one takes logarithm of the level densities. So, one can use the formula (584) as * 0 the logarithm of the level density of spinless states to a good approximation. Then, the statistical at the level entropy associated with degenerate string states with particular spins J 2   "\

(N , N ) is * 0  1 "\

J , (588) S ,log d(N , N )+4p(N "4p N ! G * 2   * 0 * G in the limit






1 "\

N < J. * 2 G G For the angular momenta contribution to be statistically non-negligible, "\

J<1 G G has to be satis"ed. In the near-extreme limit, N +aQQ. So, in terms of electric charges and angular momenta, *   the statistical entropy takes the form: (N *






 "\

. S "2p 4aQ Q !2 J   G   G This qualitatively agrees with the thermal entropy (580).

(589)

7.4. Black holes and fundamental strings In the previous sections, we calculated statistical entropy of black holes by assuming that perturbative string states are black holes. Based on this assumption, we equated the mass of string states with the ADM mass of black holes, and identi"ed the left and right moving momentum zero modes of the string states with the left and right handed electric charges of black holes. This "xes N (which determines the microscopic degeneracy of states) in terms of the macroscopic para*0 meters of black holes, making it possible to calculate the statistical entropy of black holes. In this section, we justify [110,189] such identi"cation of perturbative string states with microscopic black hole states. The starting point is the fundamental string in 54D410. Here, the fundamental string is de"ned as a 1-brane solution of the combined action S#S with macroN scopic string (described by S ) as its electric charge source. When the fundamental string is N compacti"ed on S along its longitudinal direction, it asymptotically approaches black hole, as rPR, with its core having milder singularity (than black hole in (D!1) dimensions) of a D-dimensional string source. With this identi"cation, microscopic degrees of freedom of the asymptotic black hole in (D!1) dimensions is interpreted as being due to oscillating macroscopic string at its core. And electric charges and angular momenta of the black hole are determined by momentum and winding modes of the core string along its longitudinal direction and the frequency of string oscillation in the rotational planes.

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7.4.1. Fundamental string in D dimensions We consider the following worldsheet action of macroscopic string moving in a background of the string massless modes:

 



1 1 dp (cc?@R X+R X,GK #e?@R X+R X,BK ! (cUK R . S" ? @ +, ? @ +, 2 N 4pa

(590)

The conformal invariance of S leads to the equations of motion for the massless background "elds N [606,607]. These equations of motion can be reproduced [111] by the Euler}Lagrange equations of the e!ective action:







1 1 d"x (!GK e\UK R K #4R UK R+UK ! HK HK +,. , (591) S" % + 12 +,. 2i " where the D-dimensional gravitational constant i is related to the Newton's constant G  as " " G "i /8p. " " When the target space is #at, i.e. GK "g and BK "0, one can exactly solve the p-model +, +, +, (590) to construct perturbative string states. We concentrate on compacti"cation of the p-model on S of radius R in #at background. The string states in this model are characterized by string winding number n and quantized momentum zero mode m/R along the S-direction. The right- and left-moving momenta along the S-direction are nR m p " ! , 0 2R 2a

m nR p " # . * 2R 2a

(592)

For BPS states in heterotic string, whose supersymmetry is generated by right-moving worldsheet current, all the right movers are in the ground state (N ") and the total number of left-moving 0  oscillations is determined by the Virasoro constraint to be: N "1#a(p!p)"1!mn . (593) * 0 * The mass of BPS states depend p , only: 0 M "4p . (594)   0 From now on, we concentrate on BPS fundamental string solution [187,189}191,380] to this theory. The background "elds of such `straighta fundamental string solution have the form [190]: BK "(eUK !1) , GK dx+ dx,"!eUK du dv#dx ) dx, ST  +, Q i " e\UK "1# , Q" , (595) r"\ pa(D!4)X "\ where u,x!x"\ and v"x#x"\ are the lightcone coordinates along the string worldsheet, and xK (m"1,2, (D!2)) are the transverse coordinates. In deriving this solution, we chose the static gauge for X+:






XI"mI, XK"constant ,  G is related to the quantities in (590) as G "aeUK . " " 

(596)

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where mI"(q, p) is the worldsheet coordinate. This fundamental string solution preserves 1/2 of the spacetime supersymmetry [190]. When the fundamental string is compacti"ed along its longitudinal direction on S of radius R, i.e. x"\"x"\#2pR, one obtains point-like solution in D!1 dimensions with its charge proportional to the winding number n along the S-direction. We now obtain solution that also has an arbitrary left-moving oscillation, which is a source for microscopic degeneracy. The zero-modes of the left-moving oscillation induce momentum m/R in the S-direction. Applying the general prescription of solution generating transformation discussed in [279,280,282,617,618] to the straight fundamental string solution (595), one obtains the following left-moving oscillating fundamental string solution GK dx+ dx,"!eUK (du dv!¹(v, x) dv)#dx ) dx , +, 1 Q BK " (eUK !1), e\UK "1# , ST 2 r"\

(597)

where ¹(v, x) is a solution to Rx ¹(v, x)"0. The general form of ¹(v, x) that can be matched onto the string source at the core is ¹(v, x)"f (v) ) x#p(v)r\"> ,

(598)

where the "rst term corresponds to oscillating string source and the second term corresponds to a momentum without oscillations. One can bring (597) to a manifestly asymptotically #at form by applying the coordinate transformations



u"u!2FQ ) x#2FQ ) F!

v"v,

TY

FQ  dv,

x"x!F ,

(599)

where Q,R/Rv, f (v)"!2F$ and FQ "FQ ) FQ . In this new coordinates, (597) takes the form (with primes suppressed) GK dx+ dx,"!eUK du dv#[eUK p(v)r\">!(eUK !1)FQ ] dv#2(eUK !1)FQ ) dx dv#dx ) dx , +, BK "(eUK !1), BK "FQ (eUK !1) , (600) ST  TG G Q e\UK "1# . "x!F""\ This solution has the ADM mass p and the ADM momentum per unit length pG , and the "+ "+ D-momentum #ow along the string p"\+ given by "+ p+ Q#QFQ #p QFQ G !QFQ !p "+ "(D!4)X"\ . (601) 2i p"\+ !QF Q !p !QF Q G !Q#QF Q #p " "+











7.4.2. Level matching condition In Section 7.4.1, we constructed oscillating fundamental string solution by solving the equations of motion (following from S#S ) of the target space background "elds and imposing solution N generating transformations. Note, all of such solutions do not correspond to the underlying string states. To ensure that these solutions match onto the perturbative string states at the core, one has

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to additionally solve the equations of motion for string coordinates X+ and the Virasoro constraints. From the Virasoro constraints (or the level matching conditions), one extracts the relations between the macroscopic quantities of the oscillating fundamental string (600) and the microscopic quantities of the perturbative string states. This allows to interpret the entropy of the target space solution in terms of the perturbative string state degrees of freedom. We start by choosing the following static gauge for the string coordinates X+ in the coordinate frame corresponding to the solution in (597): ;";(p>, p\),

<"<(p>, p\),

XK"0 ,

(602)

where p!"q$p are the light-cone worldsheet coordinates. Also, we choose the conformal gauge for strings, i.e. c "diag(!1, 1). From the Virasoro constraints ¹ "0"¹ , one has the ?@ >> \\ following form of string coordinates (602): ;"(2Rn#a)p\,

<"2Rnp> ,

(603)

where a,(1/p)p0LFQ  is the zero mode of FQ . And the constant Q in (597) is expressed in terms of  the perturbative string state quantities as ni " Q" . (604) pa(D!4)X "\ More information on matching of the spacetime solution onto states of the core string source is extracted by taking the #at spacetime limit i P0, in which for example the Virasoro relations " (592), (593) and(561) are valid. For this purpose, we go to the frame represented by (600), where the metric is manifestly asymptotically #at, by applying the transformations (599). In this new frame (denoted by primes), X+ take the form:



FQ , X"F(<) , (605) 4Y manifestly showing that the core string is oscillating with pro"le F(<). In the #at spacetime limit, i.e. i P0 or 1UK 2P0, a perturbative string state has the momentum p+ (conjugate to X+, " obtained from S ) and the winding vector n+ given in the coordinates (X, X, X"\) by N n+"(0, 0, n), p+"(2a)\(2nR#a, 0,!a) . (606) <"2Rnp>,

;"(2Rn#a)p\#

This expression for p+ agrees with the i P0 limit of the ADM momentum p+ (601) of the target " "+ space solution (600) with Q given by (604). This con"rms that oscillating fundamental strings are matched onto perturbative states of the core macroscopic string. Since the momentum zero mode of a perturbative string state along the (compacti"ed) X"\direction is m/R and N "!nm ((593) in the large N limit), one can read o! the expression for * * pI in (606) to express m and N in terms of the macroscopic quantities of the fundamental string * solution: m"!Ra/2a, N "nRa/2a . (607) * Thus, we see that the oscillating fundamental string solution (600) with p(v)"0 is matched onto the perturbative string state with n, m and N given in (607). These are a subclass of solutions (600) that * can be matched onto perturbative states of the core source string.

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7.4.3. Black holes as string states When the longitudinal direction of the fundamental string (600) is compacti"ed, one has point-like object in D!1 dimensions. Such a solution approaches Sen's BPS electric black hole [562] as rPR. This allows one to relate the macroscopic quantities, which are de"ned at spatial in"nity, of black holes to the microscopic quantities of perturbative string states. Note, such point-like solutions in D!1 dimensions asymptotically approach only a subset of Sen's black holes that can be matched onto perturbative string states. So, for example, the angular momenta of such solutions follow the Regge bound of perturbative string states, whereas Sen's rotating black holes [562], in general, take arbitrary values of angular momenta, which do not satisfy the Regge bound. In the following, we discuss the D"5 case for the purpose of illustrating basic ideas. The generalization to an arbitrary D is straightforward; one starts from D-dimensional solution (600) with more general pro"le function F(v). One compacti"es the longitudinal direction of the D"5 fundamental string (600) with p(v)"0 on S of radius R to obtain a point-like solution in D"4. To make the resulting D"4 point-like solution approach a `rotatinga black hole asymptotically, one chooses the following form of F that describes rotation in the (x, x)-plane with amplitude A and angular frequency u: F"A(e( cos ut#e( sin ut) , (608)   where e( is a unit vector in the xG-direction. G Since the D"4 point-like solution depends on the compacti"ed coordinate x and the time coordinate t through v"x#t, the compacti"cation on x (i.e. taking average over x so that only the zero modes of "elds are kept) is equivalent to taking the time-average. By taking the time-average of the leading order terms of the "elds at large r, one can read o! the following ADM mass M , angular momentum J, the right- and left-handed electric charges & Q ,($Q#Q)/(2, and the right- and left-handed magnetic moments k of the rotating 0* 0* black hole that the point-like solution approaches asymptotically: Q(1#Au) M " , & 4

QAu J" , 4

Q Q (Au!1), Q "! (1#Au) , Q "! 0 * 2(2 2(2

(609)

k "0, k "!(2J , * 0 with choice of unit in which G "1. With this choice of normalization, Q in (604) becomes " Q"4nR/a, and p and Q are related as Q "2(2p . Furthermore, the periodicity of *0 *0 *0 *0 x requires u"l/(nR) for some integer l. With this identi"cation, one has M "2p , as one would expect from the fact that the target & 0 space solution with the ADM mass M is matched onto the string source with the right moving & momentum p . This proves the assumption that the perturbative string states are black holes. 0  The generalization to a (D!1)-dimensional rotational black hole with [(D!2)/2] angular momenta involves F representing independent rotations in [(D!2)/2] mutually orthogonal planes.

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Furthermore, J in (609), which reduced to the form J"Al/a, satis"es the Regge bound J4 " 2#a(p!p)""Al/a, with J forming the Regge trajectory when l"1. Finally, the 0 * gyromagnetic ratios g , de"ned by k "g Q J/2M , are *0 *0 *0 *0 & g "0, *

g "2 . 0

(610)

This result is consistent with the fact that the BPS black holes correspond to the perturbative string states with only left-mover excited, since the gyromagnetic ratios are related to the left- and right-moving angular momenta J as *0 g

J "2 0* . *0 J #J 0 *

Moreover, the right-moving gyromagnetic ratio is 2 as expected from the fact that the underlying states are fundamental. 7.5. Dyonic black holes and chiral null model So far, we discussed statistical interpretation for entropy of purely electric black holes in terms of microscopic degrees of freedom of perturbative string states. The BPS electric black holes have a nice virtue of being free of quantum corrections, thereby the ADM mass can be trusted. However, due to singularity at the horizon, the horizon gets shifted by &(a through the a-corrections. Thus, entropy is known only up to the order of a. Furthermore, such solutions are not black holes in the conventional sense, since the event horizon coincides with the singularity and has zero area. It is the construction of dyonic solutions [178] in the heterotic string on ¹ that triggered renewed interests in black hole entropy and made the precise calculation of the statistical entropy possible. Such dyonic solutions not only do not receive quantum corrections, but also are free of classical a-corrections [173] since they are described by exact conformal p-model and the event horizon is free of singularity. Such dyonic solutions contain as a subset the Reissner}NordstroK m solution and have non-zero event horizon area. Since the event horizon is regular, the acorrections are under control at the event horizon. And as in the pure electric case, the dilaton is "nite at the event horizon, implying that the string loop corrections are under control. Being free of plagues (i.e. a-corrections of purely electric solutions and the string loop corrections of the purely magnetic solutions) su!ered by the previously known solutions in string theories, the dyonic solution [178] is suitable for studying statistical origin of black hole entropy. Such observation was "rst made in [441] (see also Refs. [363,364]), where it is proposed that the microscopic degrees of freedom of the dyonic black holes are due to the hair associated with the oscillations in the internal dimensions. The "rst attempt to explain the statistical origin of the BPS black holes with non-zero event horizon area is based upon a special class of string worldsheet p-model called `chiral null modela. In this approach, the BPS black holes are embedded as background "elds of the chiral null model and the throat region conformal model is studied for understanding microscopic degeneracy of string states. Remarkably, the throat region conformal theory approximates to the

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WZNW model of perturbative string theory with string tension rescaled by magnetic charges of the black holes. So, the degeneracy of string states carrying magnetic charges is obtained by applying level density formula of perturbative string states. In this section, we summarize the results of [174,609,612], which study chiral null model interpretation of black hole entropy. 7.5.1. Chiral null model String theory is a promising candidate for consistent quantum gravity theory, being free from ultra-violet divergences, which plagued quantum gravity of point-like particles. So, it is useful to study classical string solutions to address problems in quantum gravity. However, it is almost hopeless to obtain exact classical solutions to the equations of motion following from e!ective "eld theory of string massless states, since the e!ective action consists of in"nite series of terms of all derivatives multiplied by powers of a, which are also ambiguous due to the freedom of choosing di!erent renormalization schemes (or "eld rede"nitions). So, the only exact classical string solutions that one can study are those that do not have a-corrections. In fact, there exist classes of string p-models whose background "elds do not receive a-corrections in a special renormalization scheme. One starts from a string p-model which is shown to be conformal to all orders in a and looks for classical solutions to the leading order (in a) e!ective "eld theory which can be embedded as target space background "eld con"gurations of the p-model, or vice versa. This approach of studying classical solutions of string theory is based upon a remarkable relationship between the conformally invariant string p-model and the extremum of the e!ective action. To the leading order in string coupling, the string "eld equations are obtained by the conformal (Weyl) invariance condition of p-model, which are equivalent to the stationary conditions of the e!ective action due to the proportionality between the Weyl anomality coe$cients and derivatives (with respect to "elds) of the e!ective action. Given a conformal p-model, one obtains a string solution not modi"ed by a-corrections. The general bosonic p-model describing string propagation in background of massless "elds G , B and U is +, +, 1 dz[(G #B )(X)RX+RM X,#aRU(X)] . (611) I" +, +, pa



The chiral null model is a special case of (611) with the Lagrangian: ¸"F(x)Ru[RM v#K(x, u)RM u#2A (x, u)RM xG]#(G #B )(x)RxGRM xH#RU(x) , (612) G GH GH where X+ are splitted into `light-conea coordinates u, v and `transversea coordinates xG, i.e. X+"(u, v, xG). Note, F does not depend on u. There exists a special renormalization scheme in which the p-model (612) is conformal to all orders in a, provided (i) the transverse p-model ¸ "(G #B )(x)RxGRM xH#R (x), where , GH GH

,U! ln F, is conformal and (ii) F, K, A and U satisfy conformal invariance conditions, which  G  In di!erent approach [332}335] based upon the Rindler geometry in throat region, black hole in the weak coupling limit is described by closed strings in the background of black hole carrying non-perturbative charges. The e!ect of these non-perturbative charges on closed strings is to rescale the string tension, as in the case of chiral null model approach.

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for the case ¸ has at least (4, 0) supersymmetry take the form [174]: , ! F\#RG R F\"0, ! K#RG R K#R AG"0 ,  G  G S G (613) i.e. (e\(FGH)!e\(F HGIH"0 , ! K FGH#R FGH"0, G G  GI  G where F "R A !R A , H "3R B and the covariant derivative K , (CK ) is de"ned in terms GH G H H G GHI G HI

of the generalized connection CK G ,CG #HG with torsion. HI HI  HI The chiral null model has one null Killing vector which generates shifts of v: the action is invariant under an a$ne symmetry vPv"v#h(q#p). The associated null Killing vector R/Rv gives rise to the conserved current J "F(x)Ru on the string worldsheet. A balance between the T metric and the antisymmetric tensor (G "B ) implies that the conserved current J is chiral, SG SG T which is a crucial condition for the conformal invariance [369,371]. The action (613) is also invariant under the following subgroup of coordinate transformations on v combined with a gauge transformation of K and A : G vPv!2g(x, u), KPK#2R g, A PA #R g . (614) S G G G Unless K, A and U do not depend on u, one can choose a gauge in which K"0 by applying (614). G When the "elds are independent of u, the chiral null model turns out to be self-dual. Namely, a leading-order duality transformation along any non-null direction in the (u, v)-plane (say along the u-direction, where v"v( #au with a constant) leads to a p-model of the same form with duality transformed background "elds given by (615) F"(K#a)\, K"F\, A"A , U"U! ln[F(K#a)] . G G  When background "elds are independent of u, the conformal invariance conditions (613) take the form of the Laplace equations in the transverse space. Background "eld solutions to the conformal invariance conditions are then parameterized by harmonic functions in the transverse space. Since the equations are linear, one can superpose harmonic functions to generate multi-center solutions. One can further generalize background "elds to depend on u in such a way that the conformal invariance conditions (613) are still satis"ed. Such changing of background "elds is viewed as `marginal deformationsa [371] of the conformal "eld theory. In particular, adding zero modes of A has the e!ect of adding a Taub-NUT charge, angular momenta or extra electric/magnetic G charges to the original solutions. The chiral null model (612) generalizes K-model (plane fronted wave solution) and F-model (a generalization of the fundamental string solution). First, in the limit F"1, (612) describes a class of plane fronted wave backgrounds which have a covariantly constant null vector R/Rv (K-model). For this case, one can add another vector coupling 2A M (x, u)RxGRM u to the Lagrangian while still G preserving conformal invariance but breaking chiral structure. Second, when K"0 with background "elds independent of u, (612) reduces to the F-model, which has two null Killing vectors R/Ru and R/Rv associated with a$ne symmetries u"u#f (q!p) and v"v#h(q#p). Since the

 This follows [371] from the standard leading order conformal invariance conditions RK #2DK DK U"0 of \+, \+ \, the general p-model (611). Here, the Ricci tensor RK and covariant derivative DK are de"ned in terms of the \+, \+ generalized connection CK . "C. $H. with torsion H . !+, +,  +, +,.

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coupling to u and v is chiral (G "B ), the associated 2 conserved currents JM "FRM v and J "FRu ST ST S T are chiral. The F model and the K model are related by a duality transformation (615) along u with a"0. The dimensional reduction of chiral null model leads to various charged (under the KK or 2-form gauge "eld) black hole and string solutions, which can also carry angular momenta or the Taub-NUT charge. The chiral coupling leads to a no-force condition (a characteristic of BPS solutions) on the solutions, which allows the construction of multi-centered solutions. The balance between G and B in chiral theories manifests in lower dimensional solutions as the "xed ratio +, +, of mass and charge, i.e. the BPS condition. In the following, we discuss black hole solutions that satisfy the conformal invariance conditions (613) and therefore are exact to all orders in a. For the purpose of obtaining general D"4, 5 black holes, we split the transverse coordinates xG into non-compact ones xQ and compact ones yL, i.e. xG"(xQ, yL), where s"1,2, D!1 and n"1,2, 9!D. We decompose the 8-dimensional transverse space into the direct product M;¹ of some 4-space M and ¹. The chiral null model Lagrangian (612) is accordingly expressed as the sum of two terms associated with each space. Here, M has SO(3) [SO(4)] symmetry for the D"4 [D"5] black holes and, therefore, M is parameterized by (xQ, y) [xK] with background "elds depending on the coordinates only through r"(xQxQ [r"(xKxK]. We consider the case where M has the torsion related to dilaton

in the speci"c way HKLI"!(2/(G)eKLIJR , so that the last conformal invariance condition J in (613) simpli"es to a Laplace-form (e\(FGH )"0, where FGH ,FGH#夹FGH with G > > 夹FGH"(1/2(G)eGHIJF . IJ 7.5.1.1. General four-dimensional, static, BPS black hole. We consider the case where 4-dimensional transverse part of the metric has the form G "f (x)g where f"0, , (g) and g is GH GH GH a hyper-KaK hler metric with a translational isometry in the x-direction. The D"6 part (u, v, x,2, x) of (612) then takes the special form: ¸"F(x)Ru[RM v#K(x)RM u#2A(x)(RM x#a (x)RM xQ)]#R ln F(x)#¸ , Q  , ¸ "f (x)k(x)(Rx#a (x)RxQ)(RM x#a (x)RM xQ)#f (x)k\(x)RxQRM xQ , Q Q #b (x)(RxRM xQ!RM xRxQ)#R (x) , Q

(616)

where xQ"(x, x, x) are non-compact coordinates and compact coordinates are x"y and  u"y . Here, we chose A in (612) to take the form A "A and A "Aa so that the D"4 metric  G  Q Q has no Taub-NUT term. The Lagrangian (616) is invariant under T-duality transformations in the x-direction (P  P   and qP!q): fPk\, kPf \, a  b , AP( f k)\A , Q Q

(617)

and in the u-direction (Q  Q ):   FPK\, KPF\, U(F)PU(K\) .

(618)

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When A"0, the Lagrangian has remarkable manifest invariance under D"4 S-duality, under which (618) transforms as u  x and FPf \, KPk\, fPF\, kPK\ ,

(619)

and under the D"6 string}string duality (GPe\UG, dBPe\U夹dB, UP!U) between the heterotic string on ¹ and the type-II string on K3: FPf \, KPK,

fPF\, kPk .

(620)

Note, the invariance of (616) under the T-duality is manifest only when all the charges associated with 4 harmonic functions F, K, f and k are non-zero. The self-dual case F"K\"f \"k and a "b corresponds to the D"4 Reissner}NordstroK m solution. As expected, the combined Q Q transformation of T-duality and the string}string duality yields the D"4 S-duality (619). One can obtain D"4 black hole solution which is exact to all orders in a by solving (613) with (616) and all the background "elds depending on non-compact transverse coordinates xQ, only. Solutions for background "elds are expressed in terms of harmonic functions f, k, F and K, which satisfy (linear) Laplace equations. Particularly, A is given in terms of harmonic functions by A"q k\#q f k (q " const). If one further assumes the asymptotic #atness condition (i.e.    : q "!q . kP1, fP1, AP0 as r"(xQx PR), coe$cients in A are restricted such that q "    Q The solutions for background "elds are: Q P Q F\"1# , K"1# , f"1# , r r r

P k\"1#  , r

a dxQ"P (1!cos h)du, b dxQ"P (1!cos h)du , Q  Q 

(621)

r#P q r#(P #P )  , eU"Fe("   , A" ) r#P r#Q r   where q,2q (P !P ). Since the resulting (conformal invariance condition) equations are of    Laplace-type, one can superpose harmonic functions to obtain multi-center generalization of the above. The D"4 spherically symmetric solution in (200) is obtained by applying the standard KK procedure with all the background "eld in (612) properly identi"ed with those in (611) and setting u"y , v"2t, x "y .    7.5.1.2. General xve-dimensional, rotating, BPS black hole. We consider the case where M-part is (locally) SO(4)-invariant. The D"6 part of chiral null model Lagrangian is again given by (616) with A "(A , 0) and the transverse part ¸ replaced by G K , ¸ "f (x)RxKRM xK#B (x)RxKRM xK#R (x) , , KL

(622)

where m, n,2"1,2, 4, and the "elds are given in terms of a harmonic function f (x) (Rf"0) R "!e R f. By solving the conformal by " ln f, G "fd and H "!2(GGNJe KL KL KLI KLIN J KLIJ J 

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invariance conditions (613) with above AnsaK tze, one obtains Q P f"1# , F\"1# , r r

Q K"1#  , r

(623)

r#P c c e "Ff" , A " sin h, A " cos h , P P r#Q r r  U

where r,(xKxK and c is related to the angular momenta as J "J "(p/4G )c. Note, the   , conformal invariance condition (e\(FKL)"0 is solved by imposing the #at-space anti-selfK > duality condition F "0 on the "eld strength of the potential A , and as a result the 2 angular >KL K momenta are the same. By superposing harmonic functions, one obtains the multi-center generalization of the above. The dimensional reduction along u leads to D"5, rotating, BPS black hole with charge con"guration (Q "Q, Q "Q, P), where P is a magnetic charge of the NS-NS 3-form "eld     strength (or an electric charge of its Hodge-dual). The Einstein-frame metric is





 c ds"!j(dt#A dxK)#j\ dxK dx "!j dt# (sinh du #cos h du ) # K K   r #j\[dr#r(dh#sin h du#cosh du)]   r j"(F\Kf )\" . [(r#Q )(r#Q )(r#P)]  

(624)

7.5.2. Level matching condition To calculate statistical entropy of black holes, one has to relate macroscopic quantities of black holes to microscopic quantities of perturbative string states through `level matching conditiona [189]. Strictly speaking, level matching process is possible for electric solutions, only, since perturbative string states do not carry magnetic charges and string momentum [winding] modes are matched onto `electrica charges of KK [2-form "eld] ;(1) "elds. Furthermore, magnetic solutions can be supported without source at the core (cf. magnetic solutions are regular everywhere including the core), since they are topological in character. However, it turns out [174] that the dyonic solution found in [178], which is a `bound statea of fundamental string and solitonic 5-brane, still needs a source for its support and satis"es the same form of level matching condition as the fundamental string. The crucial point in the level matching of such dyonic solutions onto the perturbative string spectrum is that as in the purely magnetic case "elds are perfectly regular near the horizon (or the throat region), making it possible to describe the solutions at the throat region in terms of WZNW conformal model [289,437,484,632]. Since the solution is regular and the dilaton is "nite near the event horizon (implying that the classical a and the string loop corrections are under control), such e!ective WZNW model near the horizon can be trusted. For large magnetic charges (or large level), the theory e!ectively looks like a free p-model for perturbative string theory with the string tension rescaled by magnetic charges. Namely, for large magnetic charges, the dyonic

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solutions are matched onto perturbative string states with string tension rescaled by magnetic charges. To match background "eld solutions onto the macroscopic string source at the core, one considers the combined action of string p-model and the e!ective "eld theory. Among the equations of motion of the combined action, the relevant parts are the Einstein equations for target space metric, equations of motion for XI and the Virasoro conditions. Requiring that all the solutions are supported by sources, one obtains the level matching condition:



1 p0 E(u), du E(u)"0, (E(u),[F(x)K(u, x)] "0) . V 2pR 

(625)

This condition is satis"ed without modi"cation even when solutions carry magnetic charges. 7.5.3. Throat region conformal model and magnetic renormalization of string tension 7.5.3.1. Four-dimensional dyonic solutions. The p-model (616) of dyonic black hole (193) with charge con"guration (P, Q, P, Q),(P , Q , P , Q ) takes the following form near the hor        izon (rP0) [174,609,612]:





1 1 dp ¸ " dp(e\XRuRM v#Q Q\RuRM u) I" P pa   pa



P P #   dp(RzRM z#Ry RM y #RuRM u#RhRM h!2 cos hRy RM u) .    pa

(626)

This is the S¸(2, R);S;(2) WZNW model with the level i"(4/a)P P . (Since the level has to be   an integer, one has the quantization condition 4P P /a3Z.) Here, the coordinates are de"ned as   z,ln(Q /r)PR, u ,(Q\Q P P )\u, v ,(Q Q\P P )\v, and y ,P\y #u.             For large P (or i), the transverse (o, y , u, h) part of (626) looks like a free theory of   perturbative string with the string tension ¹"1/(2pa) renormalized by P :  1 P P P P 1 P "  "   , aR a a a , 

(627)

where R "(a is the radius of the internal coordinate associated with P .   7.5.3.2. Five-dimensional dyonic solution. The p-model with the target space con"guration given by the 3-charged, BPS, non-rotating black hole, i.e. (263) with J"0, takes the following form in the limit rP0 [609,612]:





1 1 dp ¸ " dp(e\XRuRM v#Q Q\RuRM u) I" P pa   pa #



P dp(RzRM z#Ru RM u #Ru RM u #RhRM h!2 cos hRu RM u ) ,       4pa

(628)

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where z,ln(Q /r)PR. This is the S¸(2, R);S;(2) WZNW model with the level i,(1/a)P.  In the limit of large P or large i, the transverse part (o, u , u , h) of (628) reduces to free perturbative   string theory with the renormalized string tension: 1 P 1 P " . 4a a a ,

(629)

7.5.4. Marginal deformation The degeneracy of micro-states responsible for statistical entropy is traced to the degrees of freedom associated with oscillations or marginal deformations around the classical solutions. The marginal deformations lead to a family of all the possible solutions (obeying conformal and BPS conditions) with the same values of electric/magnetic charges but di!erent short distance structures that depend on a choice of oscillation pro"le function. Thus, one has to consider the region near the horizon (at r"0) to determine the microscopic degrees of freedom, since in this region degeneracy of solutions is lifted. General chiral null model action which represents deformation from the classical BPS solutions in Section 7.5.1 and preserves the BPS and conformal invariance properties is given by (612) with K and A "(A , A ) having an additional dependence on u, where u,z!t with the longitudinal G K ? direction z satisfying the periodicity condition z,z#2pR. Here, A &q (u)/r and A "q (u)/r K K ? ? (r,x x ) are respectively `deformationsa in the non-compact xK and the compact y? directions. K K On the other hand, the perturbation K(u, x)"h (u)xK#k(u)/r does not contribute to the K degeneracy, since h (u)xK drops out and k(u) has zero mean value. K The perturbations K, A and A represent various `left-movinga waves propagating along the ? K string and are invisible far away from the core. The mean values q (u) and q(u) are related to the ? K oscillation numbers of the macroscopic string at the core as p pa NK" q (u), N?" q(u) , (630) * * 16G K 16G ? , , thereby contributing to the microscopic degrees of freedom. These marginal deformations do not contribute to the microscopic degeneracy of black holes with the same order of magnitude [612]. This can be inferred from the fact that the classical BPS black holes are solutions of both heterotic and type-II string. Namely, although the thermal entropy is the same whether one embeds the solutions within heterotic or type-II string, one faces the discrepancy in factor of 2 in the statistical entropies within the two theories, if one takes the degeneracy contributions of all the oscillators to be of the same order of magnitude. In fact, as can be seen from the conformal invariance condition (e\(FGH )"0, the perturbations A in the G > ? compact directions y are decoupled from the non-trivial non-compact parts of the solution. On ? the other hand, the perturbations A in the non-compact directions x are non-trivially coupled to K K the magnetic harmonic functions, with the net e!ect being the rescaling of q (u) terms by the K  The chiral null model corresponding to rotating black hole (624) also approaches the S¸(2, R);S;(2) WZW model in the throat limit. The requirements that the level i"P/a is an integer and u "u #2cP\Q\u (u ,u #u ) has       the period 4p lead to the quantization of P and J: P"ai and J"iwl with k, l3Z (w" string winding number). The regularity of the underlying conformal model requires that J(imw, i.e. the `Regge bounda.

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magnetic charges. (This is related to the scaling of the string tension in the transverse directions by the magnetic charge(s).) So, the marginal deformation contributions from the compact directions, which are di!erent for the two theories, are suppressed relative to those of non-compact directions by the factor of the inverse of magnetic charge(s), thereby negligible for a large magnetic charge(s) or the large level i. Only the marginal deformations from the non-compact directions and the compact direction associated with non-zero magnetic charge(s), which are common for both theories, have the leading contribution to the degeneracy. Only these 4 string coordinates get their tension e!ectively rescaled by the magnetic charge(s). Furthermore, the marginal deformations on the original black hole solutions have to be only left-moving (i.e. depend only on u, not on v) so that marginally deformed p-models are conformaly invariant. This is related to the `chirala condition on the p-model; only left-moving deformations lead to supersymmetric action. Thus, for large magnetic charge(s), the statistical entropy calculations within heterotic and type-II strings agree. As pointed out, the marginal deformations A (u, x)"q (u)/r in the compact directions ? ? y contribute to the microscopic degeneracy to sub-leading order (suppressed by the inverse of ? magnetic charge(s)), which can be neglected for large value of magnetic charge(s). But the zero modes q of the Fourier expansions of q (u)"q #q (u) (q (u) denoting the oscillating parts) ? ? ? ? ? produce additional left-handed electric charges [174] of D"6 strings. Namely, the internal marginal deformation A (u)"q (u)/r on the p-model associated with D"4 4-charged BPS black ? ? hole (193) leads to 5-charged BPS black hole solution (621) with the zero mode q corresponding to ? an additional charge parameter q. The mean oscillation values q  (u) of the marginal deformations A (u, x)&q (u)/r K K K (q (u)"q #q (u) with q and q (u) respectively denoting the zero modes and oscillating parts) K K K K K in the non-compact directions x contribute to N to the leading order. Meanwhile, the zero modes K * q have an interpretation as angular momenta [610] of black holes. Namely, the rotational K marginal deformation corresponding to S;(2) Cartan current deformation A (u, x) dxK" K (c(u)/r)(sin h du #cos h du ) on the p-model action of the D"5 3-charged non-rotating   solution leads to the rotating solution (624) [609,610]. To calculate the statistical entropy associated with degeneracy of solutions (i.e. all the possible marginal deformations of the original classical solution), one has to determine N of the macro* scopic string at the core. For this purpose, we write the marginally deformed p-model action (612) in the throat region in the form:



1 dp(marginal deformation terms) I"I# P pa



1 dp [e\XRuRM v#E(u)RuRM u]#other terms , " pa

(631)

where I stands for the throat limit WZNW models (626) and (628) of the (undeformed) classical solutions. First, for the D"4 dyon, the marginal deformation (612) gives rise to the following expression for E(u): E(u)"Q Q\!(P P )\Q\q(u)!Q\q(u)      L  ? "(P P )\Q\[Q Q P P !q(u)!P P q(u)] .        L   ?

(632)

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191

Note, since the string tension a (627) of the transverse parts is rescaled by the magnetic charges, , the coe$cient in front of the term q(u) is rescaled by (P P )\. Applying the level matching   L condition E(u)"0 (625), one "nds that q(u)"P P (Q Q !q ), where q denote the zero modes L     ? ? of the oscillations q (u) in the compact directions. Thus, the statistical entropy is [174] ? p (P P (Q Q !q ) , (633) S +2p(N "     ?   * 2G , in agreement with the thermal entropy. Second, we consider D"5 solutions. For the non-rotating solutions, the marginal deformation (612) leads to E(u)"Q Q\#Q\k(u)!P\Q\q (u)#O(P\) . (634)     K Applying the level matching condition E(u)"0, one "nds that q (u)"Q Q P for large P, which K   reproduces the thermal entropy [609,612]: p (Q Q P . (635) S +2p(N "     * 2G , For the rotating solution, one introduces the marginal deformation A in a non-compact K direction. Then, one has E(u)"Q Q\!P\Q\c(u)"P\Q\[Q Q P!c(u)] , (636)       where c(u)"c#c(u) with c and c(u) respectively denoting zero and oscillating modes of c(u). From the level matching condition E(u)"0, one has c(u)"Q Q P!c, reproducing entropy of   the rotating black hole [609,610] in the limit of P +P and Q large:    p (Q Q P!c . (637) S +2p(N "     * 2G , 8. D-branes and entropy of black holes Past year or so has been an active period for investigation on microscopic origin of black hole entropy. The construction of general class of BPS black hole solution in heterotic string on ¹ [178] motivated renewed interest [441] in the study of black hole entropy within perturbative string theory. The explicit calculation of statistical entropy of BPS solution in [178] by the method of WZNW model in the throat region of black hole reproduced the Bekenstein}Hawking entropy. Realization [498] that D-branes in open string theory can carry R-R charges motivated the explicit D-brane calculation of statistical entropy of non-rotating BPS solution in D"5 with 3 charges [580]. This is generalized to rotating black hole [98] in D"5, near extreme black hole [109,368] in D"5 and near extreme rotating black hole [95] in D"5. Meanwhile, the WZNW model approach was generalized to the case of non-rotating BPS D"5 black hole [609] and rotating BPS D"5 black hole [612]. D-brane approach was soon extended to D"4 cases: non-rotating BPS case in [392,471], near extreme case in [361] and extreme rotating case in [361]. Later, it is shown [365] that the microscopic counting argument in string theory can be extended even to

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non-extreme black holes as well, provided the entropy is evaluated at the proper transition point of black hole and D-brane (or perturbative string) descriptions. In this section, we review the recent works on D-brane interpretation of black hole entropy (for other reviews, see Refs. [468,360]). With realization [498] that R-R charges, which were previously known to be decoupled from string states, can couple to D-branes [193], it became possible to do conformal "eld theory of extended objects (p-branes) within string theories and to perform counting [90,91,566,567,619,620] of string states that carries R-R charges as well as NS-NS charges. To apply D-brane techniques to the calculation of microscopic degeneracy of black holes, one has to map non-perturbative NS-NS charges of the generating black hole solutions of heterotic string on tori to R-R charges by applying subsets of U-duality transformations. In D-brane picture of black holes [109], the microscopic degrees of freedom are carried by oscillating open strings which are attached to D-branes. Whereas the e!ect of magnetic charges in the chiral null model and Rindler geometry approaches is to rescale the string tension, the e!ect of R-R charges on open strings in the D-brane description is to alter the central charge (i.e. the bosonic and fermionic degrees of freedom) of open strings from the free open string theory value. In the D-brane picture of [109] (see also Ref. [196]), the number of degrees of freedom of open strings is increased relative to the free open string value because of an additional factor (proportional to the product of D-brane charges) related to all the possible ways of attaching the ends of open strings to di!erent D-branes. So, the net calculation results of statistical entropy in both descriptions are the same. This chapter is organized as follows. In Section 8.1, we summarize the basic facts on D-branes necessary in understanding D-brane description of black holes. In Section 8.2, we discuss the D-brane embeddings of black holes. The D-brane counting arguments for the statistical entropy of black holes are discussed in Section 8.3. 8.1. Introduction to D-branes We discuss basic facts on D-branes necessary in understanding D-brane description of black holes. Comprehensive account of the subject is found, for example, in [193,391,445,498,501,504], which we follow closely. The basic knowledge on string theories is referred to [316,535]. Each end of open strings can satisfy two types of boundary conditions. Namely, from the boundary term (1/2pa) M dp dXIR X , where RM is the boundary of the worldsheet M swept by L I . an open string and dXI[R X ] is the variation (the derivative) of bosonic coordinates XI parallel to L I (normal to) RM, in the variation (with respect to XI) of the worldsheet action, one sees that the ends of the string either can have zero normal derivatives R XI"0 , L called Neumann boundary condition, or have "xed position in target spacetime XI"constant ,

(638)

(639)

called Dirichlet boundary condition. In order for the T-duality to be an exact symmetry of the string theory, open string has to satisfy both the Neumann and Dirichlet boundary conditions [193]. Under the T-duality of open string theory with the coordinate XG"XG (z)#XG (z ) compacti"ed on S of radius R , R PR"1/R and 0 * G G G G

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193

XGP>G"XG (z)!XG (z ). So, the Neumann and the Dirichlet boundary conditions get inter0 * changed under T-duality:













Rz Rz Rz Rz RXG! RM XG" R>G# RM >G"R >G , R XG" O L Rq Rq Rq Rq

(640)

where q is the worldsheet time coordinate, which is tangent to RM. Starting from the D"10 open string with the Neumann boundary conditions and with the coordinates XG (i"p#1,2, 9) compacti"ed on circles of radii RG, one obtains open string theory with the ends of the dual coordinates >G con"ned to the p-dimensional hyperplane in the R P0 limit (or the decompacti"caG tion limit, i.e. 1/R PR, of the dual theory). Such p-dimensional hyperplane is called D-brane G [498]. A further T-duality in the direction tangent (orthogonal) to a Dp-brane results in a D(p!1)-brane (D(p#1)-brane). D-brane is a dynamical surface [498] with the states of open strings (attached to the D-brane) interpreted as excitations of #uctuating D-brane. The massless bosonic excitation mode in the open string spectrum is the photon with the vertex operator < "A R X+, where R is the  + R R derivative tangent to RM. So, the bosonic low energy e!ective action is that of N"1, D"10 Yang-Mills theory [637]:



1 dx Tr[F FIJ] . IJ 2g

(641)

In the T-dual theory, the vertex operator < of the photon is decomposed into  < " N A (XI)R XI and < " u (XI)R XG, corresponding respectively to ;(1) gauge boson GN G N I I R P  and scalars on the p-brane worldvolume. The scalars u are regarded as the collective coordinates G for transverse motions of the p-brane. The bosonic low energy e!ective action of the T-dual theory is, therefore, obtained by compactifying (641) down to p#1 dimensions [637]. In this action, the worldvolume scalars u have potential term <" Tr[u , u ]. GH G H G Note, open strings can have non-dynamic degrees of freedom called Chan-Paton factors (i, j) at both ends of strings [139,316,501,504,637]. The indices (i, j), which label the state at each end of the string, run over the representation of the symmetry group G. When the state "K, ij2 describes a massless vector, (i, j) run over the adjoint representation of G. Each vertex operator of an open string state carries antihermitian matrices j? (a"1,2, dim G) representing the algebra of G and GH j? describe the Chan-Paton degrees of freedom of the open string states. The global symmetry G of GH the worldsheet amplitude manifests as a gauge symmetry in target space. For the oriented open strings, G is ;(N) and each end of the open string is respectively in complex and complex conjugate representations of ;(N). When open string states are invariant under the worldsheet parity transformation X (pPp!p or zP!z ), i.e. the exchange of two ends plus reversal of the orientation of an open string, the open string is called unoriented. For this case, the representations R and RM on both ends of the string are equivalent. For unoriented open strings, the transformation property of the Chan-Paton matrix j under the worldsheet parity X GH determines G [474,535]. Under the worldsheet parity symmetry, the open string state j "K, ij2 GH  The tachyon is removed by the GSO projection of supersymmetric theory.

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transforms to Xj "K, ij2"j "K, ij2, j"Mj2M\ . GH GH

(642)

If M is symmetric, i.e. M"M2"I , then the photon j aI "k2 survives the projection under the , GH \ gauged worldsheet parity and the Chan-Paton factor is antisymmetric (j2"!j), giving rise to SO(N) gauge group. If M is antisymmetric, i.e.




0

M"!M2"i

!I ,



I , , 0

then the gauge group is ;Sp(N), i.e. j"!Mj2M. When the Chan-Paton factors are present at the ends of open string, a Wilson line, say, for the ;(N) oriented open string theory with the coordinate X compacti"ed on S of radius R given by A "diag(h ,2, h )/(2pR)"!iK\R K ,   , 

(643)

where K"diag(e 6Fp0,2, e 6F,p0), receives non-trivial phase factor diag(e\ F,2, e\ F,) under the transformation XPX#2pR. As a result, the momentum number of an open string along S can have a fractional value. So, in the T-dual theory the winding number takes on fractional values [497], meaning that two ends of open strings can live on N di!erent hyperplanes (D-branes) located at >"h R"2paA (k"1,2, N): I II



>(p)!>(0)"

p



dp R >"(2p#h !h )R , N H G

(644)

where R"a/R is the radius of the circle in the dual theory. Note, without the Chan-Paton factors taken into account, both ends of open strings of the dual theory are con"ned to the same hyperplane up to the integer multiple of periodicity 2pR of the dual coordinate >. The similar argument can be made for unoriented open string theories. With SO(N) Chan-Paton symmetry, the Wilson line can be brought to the form: diag(h ,!h ,2, h ,!h ) .   , ,

(645)

Note, in the dual coordinate >K, worldsheet parity reversal symmetry (zP!z ) of the original theory is translated into the product of worldsheet and spacetime parity operations. Since unoriented strings are invariant under the worldsheet parity, the T-dual spacetime is a torus modded by spacetime parity symmetry Z . The "xed planes >K"0, pR under spacetime parity  symmetry are called orientifolds [193]. Away from the orientifold plane, the physics is that of oriented open strings, with a string away from the orientifold "xed plane being related to the string at the image point. Open strings can be attached to the orientifolds, but the orientifolds do not correspond to the dynamic surface since the projection X"#1 [355,356,509] removes the open string states corresponding to collective motion of D-branes away from the orientifold plane. In this T-dual theory, there are N/2 D-branes on the segment 04>K(pR and the remaining N/2 are at the image points under Z . Open strings can stretch between pairs of Z re#ection planes, as   well as between di!erent planes on one side.

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With a single coordinate X compacti"ed on S of radius R, the mass spectrum of the dual open string is




M"



[2pn#(h !h )]R 1 G H # (N!1) . 2pa a

(646)

Thus, the massless states arise in the ground state (N"1) with no winding mode (n"0) and both ends of the string attached to the same hyperplane (h "h ): G H aI "k, ii2, <"A R XI , \ I R (647) a "k, ii2, <"uR X"uR > , \ R L respectively corresponding to D-brane worldvolume photon and scalar. For the case of oriented open string theories, when all the N hyperplanes (located at h R) do not coincide (i.e. I h Oh , ∀i, j), ;(N) is broken down to ;(1),, corresponding to N massless ;(1) gauge "elds at each G H hyperplane located at h R (k"1,2, N). When m hyperplanes (m4N) coincide, say I h "2"h , the additional massless ;(1) gauge "elds (associated with open strings originally  K stretched between these m hyperplanes) contribute to the restoration of the symmetry to ;(m);;(1),\K [497,637]. Furthermore, m massless scalars (interpreted as positions [193,637] of m distinct hyperplanes) are promoted to m;m matrix of m massless scalars when these m hyperplanes coincide [637]. For unoriented open strings with SO(N) symmetry, the generic gauge group with all the N/2 D-branes distinct is ;(1),. When m D-branes coincide, the symmetry is enhanced to ;(m);;(1),\K as in the oriented case. But when m D-branes are located at an orientifold plane, the symmetry is enhanced to SO(2m);;(1),\K, due to additional massless ;(1) gauge bosons arising from open strings that originally stretched between pairs of Z image branes.  In the language of e!ective "eld theory, this symmetry enhancement or reversely symmetry breaking to Abelian group is interpreted as Higgs mechanism with scalars associated with location and separation of D-branes interpreted as Higgs "elds. The fermionic spectrum of open strings is divided into subsectors according to the boundary conditions that fermions tQ I (04p4p, !R(q(R) satisfy at one end of string. There are two types of boundary conditions on the fermion: R: tI(0, q)"tI I(0, q) tI(p, q)"tI I(p, q) , NS: tI(0, q)"!tI I(0, q) tI(p, q)"tI I(p, q) .

(648)

De"ning tI(2p!p, q),tI I(p, q), one sees that the Ramond (R) [Neveu-Schwarz (NS)] boundary condition becomes the periodic (anti-periodic) boundary condition on the rede"ned fermion t(p, q)(04p42p, !R(q(R), leading to integer (half-integer) modded Fourier series decomposition. In the NS sector, the ground state consists of 8 transverse polarizations tI "k2 of massless \ open string photon A . In the R sector, the ground state is degenerate, transforming as 32 spinor I representation of SO(8). The Virasoro conditions pick out 2 irreducible representations 8 and 8 . Q A Here, the subscript s[c] means eigenstates "s , s , s , s , s 2 (s , s "$) with even (odd) number of       G   The same argument can be applied for the case of ;Sp(N) symmetry.

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eigenvalues ! of S "iS and S "SGG>, where SIJ"! t I tJ are the fermionic part of   G  P \P P the D"10 Lorentz generators. These 2 representations are physically equivalent for open strings. The GSO projection picks out 8 and, therefore, the ground state of the open string theory is 8 8 , Q T Q forming a vector multiplet of D"10, N"1 theory. Including the Chan-Paton factors, the gauge group G of the N"1, D"10 super-Yang-Mills theory is ;(N) [SO(N) or ;Sp(N)] for an oriented [an unoriented] open string theory. For an open string theory with 9!p coordinates compacti"ed, the massless spectrum of Dp-brane worldvolume theory of dualized open string is described by the D"10, N"1 supersymmetric gauge theory compacti"ed to D"p#1. We brie#y discuss some aspects of type-II closed string relevant for understanding Dp-branes of open string theory. For type-II string, i.e. the closed string theory with supersymmetry on both leftand right-moving modes, the two choices of the GSO projections in the R sector are not equivalent. So, there are two types of type-II theories de"ned according to the possible inequivalent choices of the GSO projections on the left- and the right-moving modes. The massless sectors of these two type-II theories are Type IIA: (8 8 )(8 8 ) , T Q T A (649) Type IIB: (8 8 )(8 8 ) . T Q T Q The massless modes 8 8 in the NS-NS sector of the both theories are the same: dilaton, gravitino T T and the 2-form "eld. In the R-R sector, the massless modes 8 ;8 [8 ;8 ] of type-IIA [type-IIB] Q A Q Q theory are 1- and 3-form potentials (0-, 2- and self-dual 4-form potentials) [275]. The massless modes in NS-R and R-NS sectors contain 2 spinors and 2 gravitinos of the same (opposite) chirality for the type-IIB (type-IIA) theory. When an oriented type-II theory with a coordinate compacti"ed on S is T-dualized [193,216], the chirality of the right-movers gets reversed. So, when odd (even) number of coordinates in type-IIA/B theory are T-dualized, one ends up with type-IIB/A (type-IIA/B) theory. The e!ect of `odda T-duality, which exchanges type-IIA and type-IIB theories, on massless R-R "elds is to add (remove) the indices (of (p#1)-form potential) corresponding to the T-dualized coordinates, if those indices are absent (present) in the (p#1)-form potential. For example, the T-duality on B [B ] along xM (kOoOl) produces B [B ]. IJ IM IJM I A worldsheet parity symmetry X in a closed string, de"ned as pP!p or zPz , interchanges left- and right-moving oscillators. The unoriented closed string is de"ned by projecting only even parity states, i.e. X"t2"#"t2, as in the open string case. When type-II string is coupled to open superstring (type-I string), the orientation projection of type-I string picks up only one linear combination of 2 gravitinos in type-II theory, resulting in an N"1 theory. The only possible consistent coupling of type-I and closed superstring theories is between (unoriented) SO(32) type-I theory [108,503] and unoriented N"1 type-II theory. But in the T-dual theory, type-II theories without D-branes are invariant under N"2 supersymmetry, with orientation projection relating a gravitino state to the state of the image gravitino. The chiralities of these 2 gravitinos are the same

 T-duality on the type-II theory leads to 16 D-branes on a ¹ /Z orbifold. (The restriction to 16, i.e. SO(32) gauge \N  symmetry, comes from the conservation of R-R charges.) However, in non-compact space, one can have a consistent theory with an arbitrary number of D-branes.

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(opposite) if even (odd) number of directions are T-dualized. In the presence of D-branes, only one linear combination of supercharges in the T-dual type-II theory is conserved, resulting in a theory with 1/2 of supersymmetry broken (N"1 theory), i.e. the BPS state [498]. For this case, the leftand right-moving supersymmetry parameters (of the T-dual type-II theory) are constrained by the relation [326,501,504] e "C2CNe . (650) 0 * The conserved charges carried by D-branes are charges of the antisymmetric tensors in the R-R sector. The worldvolume of a D p-brane naturally couples to a (p#1)-form potential in the R-R sector, with the relevant space-time and D p-brane actions given by



1 G 夹G #ik N> N 2 N>



C , (651) N> NU
  where k"2p(4pa)\N is the D p-brane charge [498]. The worldvolume action is given by the N following Dirac}Born}Infeld type action [269,445] describing interaction of the world-brane ;(1) vector "eld and scalar "elds with the background "elds [445]: !¹



N

dN>m e\P det(G #B #2paF ) , ?@ ?@ ?@

(652)

where ¹ is the D p-brane tension [199,311], and G and B are the pull-back of the spacetime N ?@ ?@ "elds to the brane. In the amplitudes of parallel D-brane interactions, terms involving exchange of the closed string NS-NS states and the closed string R-R states cancel [498], a reminiscence of no-force condition of BPS states. Furthermore, the D-brane tension, which measures the coupling of the closed string states to D-branes, has the g\ behavior [445], a property of R-R p-branes [367,381,601]. The Q "eld strengths which couple to D p- and D(6!p)-branes are Hodge-dual to each other, and the corresponding conserved R-R charges are subject to the Dirac quantization condition [482,498,591] k k "2pn. \N N 8.2. D-brane as black holes In Section 8.1, we observed that D p-branes have all the right properties of the R-R p-branes in the e!ective "eld theories. As solutions of the e!ective "eld theories, which are compacti"ed from the D"10 string e!ective actions, black holes can be embedded in D"10 as bound states [224,448,637] of D p-branes. Here, p takes the even (odd) integer values for the type-IIA [type-IIB] theory. Such p-branes of the e!ective "eld theories correspond to the string background "eld con"gurations [104,105] (with the p-brane worldvolume action being the source of (p#1)-form charges) to the leading order in string length scale l "(a and describe the long range "elds away Q from D p-branes. As long as the spacetime curvature of the soliton solutions (at the event horizon in string frame) is small compared to the string scale 1/l, the e!ective "eld theory solutions can be Q trusted, since the higher-order corrections to the spacetime metric is negligible. The e!ective "eld theory metric description of solitons in superstring theories is valid only for length scales larger than a string. The D-brane picture of black holes, or more generally black p-branes, is as follows.

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The string-frame ADM mass M of p-branes carrying NS-NS electric charge [187,380], NS-NS magnetic charge [577] and R-R charge [381] behaves as &1,&1/g and &1/g (in the unit where Q Q l &1), respectively. Since the gravitational constant G is proportional to g, the gravitational Q , Q "eld strength (JG M) of the NS-NS electric charged and the R-R charged p-branes vanishes as , g P0. Namely, in the limit g P0, strings live in the #at spacetime background. Note, in the limit Q Q g P0, the description of R-R charged con"gurations in terms of black p-branes is not valid, since Q the size of the p-brane horizon (A&g) is smaller than D-brane size; the black hole is surrounded Q by a halo which is large compared to its Schwarzschield radius. In the limit g <1, one can Q integrate out massive string states (with their masses increasing in Planck units, de"ned as l"1,  as g increases) to obtain string e!ective "eld theories. (So, one can trust black hole solutions in the Q e!ective "eld theories in the strong string coupling limit.) In this limit, the massive string states form black holes and these degenerate massive states (whose mass is identi"ed with black hole mass) are degenerate black hole microscopic states, which are origin of statistical entropy. To summarize, the weak string coupling description of R-R charged con"guration is the perturbative D-branes in #at background, and in the strong string coupling limit the horizon size (&G M) , becomes larger than the string scale (with string states undergoing gravitational collapse inside the horizon), thereby, the black p-brane description emerges. The transition point of the two descriptions occurs at the point where the horizon size r is of the order of the string length scale l [365].  Q This occurs when g N&1 or g Q&1, where N is the string excitation level and Q is an R-R Q Q charge. This implies that g is very small for large N or Q. (Note, however that the e!ective string Q coupling of these bound states is g&g N or g Q, which is of order 1.) At this transition point, Q Q Q the mass of a string state M&N/l becomes comparable to black hole mass: M &r/G . Q Q
  , The essence of the D-brane description of black hole entropy is that the number of degenerate BPS states is a topological invariant which is independent of (continuous) moduli "elds, including g [549}551]. Furthermore, mass of the BPS states is not renormalized [485]. It is argued Q [195,496,469] that even for near BPS D-brane states the D-brane counting results can be extrapolated invariantly to strong coupling limit. It is also shown that D-brane approach reproduces entropy of non-supersymmetric extreme black holes [188,361] and even non-extreme case [365] as well. So, the statistical entropy of p-branes can be calculated by counting the number of degenerate perturbative (g P0) string states in D p-brane con"guration. Q Each D p-brane carries one unit of R-R charge and the g PR limit of Q D p-branes is black Q N p-brane carrying R-R (p#1)-form charge Q . R-R p-branes have a p-volume tension behaving as N &g\ [601]. So, although these are non-perturbative, R-R p-branes have singularities, except for Q p"3. The transverse (longitudinal) directions of R-R p-branes correspond to open string coordinates with Dirichlet (Neumann) boundary condition. The same T-duality rules of D-branes hold for R-R p-branes: T-duality on the transverse (longitudinal) directions of p-branes produces (p#1)-[(p!1)-] branes. When longitudinal directions are compacti"ed, single-charged p-branes become black holes having singular horizon with zero surface area and diverging dilaton at the horizon. This is due to

 The gravitational constant in D"d is GB "G/< , where < is the volume of the (10!d)-dimensional , , \B \B internal space and G"8pga&gl. , Q Q Q  The Planck length is de"ned as l "m\"( G /c).   ,

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the brane tension which makes the volume parallel (perpendicular) to the brane shrinks (expands) as one gets close to the brane [109]. Black holes having regular horizon with non-zero area in the BPS limit are constructed from bound states of p-branes (with a momentum) to balance the tension to stabilize the volume internal to all the constituent p-branes. Such regular black holes are obtained with the minimum of 4 [3] p-brane charges for the D"4 [D"5] black holes. The basic constituent of black holes is the R-R p-brane in D"10 [367]: ds "g  dxI dxJ"f \(!dt#dx#2#dx)   IJ N  N #f (dx #2#dx) , N N>  (653) e\P"f N\, A 2 "!( f \!1) ,  N N  N where f "1#Q c/r\N (r,(x #2#x )). Here, c is related to the basic (p#1)-form N N N N>  N potential charge and can be estimated by comparing the ADM mass of (653) to the mass of D-brane state carrying one unit of the (p#1)-form potential charge. The Killing spinors of this solution are constrained by [367]: e "C2CNe , e "!C2CNe , (654) * * 0 0 where e denotes the left/right handed chiral spinor (Ce "e ). One can construct solutions *0 *0 *0 for bound states of p-branes by applying the intersection rules. (See Section 6.2.2 for details on intersection rules.) In particular, the dilaton is the product of individual factors associated with those of the constituent p-branes: e\P"f N \2f NI I\. N N To add a momentum along an isometry direction xG, one oscillates p-brane so that it carries traveling waves along the xG-direction. (See Section 7.4.1 for the detailed discussion on the construction of such solutions.) At a long distance region, the solutions approach the form where all the oscillation pro"le functions are (time or phase) averaged over. So, the long-distance region p-brane solution carrying a momentum along the xG-direction is obtained by just imposing SO(1, 1) boost among the coordinates (x, xG), with the net e!ect on the metric being the following substitution: !dt#dxP!dt#dx#k(dt!dx ) , (655) G G G where k"cN/r\N with N interpreted as a momentum along the xG-direction. The momentum N along the xG-direction adds one more constraint on the Killing spinor: e "CCGe , e "CCGe . (656) 0 0 * * In general, the intersecting n p-branes preserve at least 1/2L of supersymmetry; since a single p-brane breaks 1/2 of supersymmetry with one spinor constraint (654), as one increases the number of constituents more supersymmetry get broken. One obtains BPS con"gurations if spinor constraints of constituent p-branes are compatible with non-zero spinor e . The intersecting D p0* and D p-branes preserve 1/2 of supersymmetry i! p"p mod 4 [224]. All the supersymmetries are broken when the dimension of the relative transverse space is neither 4 nor 8. When there is

 It is shown [431] that the stringy BPS black holes with non-zero horizon area are not possible for D56. This can be seen from the explicit solutions discussed in Section 4.5.

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a momentum in the xG-direction, the additional Killing spinor constraint (656) breaks 1/2 of the remaining supersymmetry. Black holes in lower dimensions are obtained by compactifying (intersecting) p-branes in D"10. In the language of p-branes, this compacti"cation procedure corresponds to wrapping p-branes along the cycles of compact manifold. Since the compacti"ed space is very small, the con"guration looks point-like (0-brane) in lower dimensions. In the following subsections, we discuss various D p-brane embeddings of D"4, 5 black holes having the regular BPS limit with non-zero horizon area. 8.2.1. Five-dimensional black hole We discuss D"5 type-IIB black hole originated from intersecting Q D1-branes (along x) and  Q D5-branes (along x,2, x) with a momentum P #owing in the common string direction [109],  i.e. the x-direction. The 1- and 5-brane charges are electric and magnetic charges of the R-R 2-form "eld, and the momentum corresponds to the KK electric charge associated with the metric component G.  To obtain a black hole in D"5, one wraps Q D1-branes around S (along x) of radius R and  wrap Q D5-branes around ¹"¹;S. Here, ¹ has coordinates (x ,2, x ) and volume <. The    momentum (of open string) P"N/R #ows around S. The resulting D"5 solution has the form [109,182]:



















r \ 1!  dr#r dX ,  r

r ds"!f \(r) 1!  dt#f (r)  r

(657)

where f (r) is given by




r sinh d  f (r)" 1#  r

r sinh d  1#  r



(658)

R
(659)

r sinh d N . 1#  r

The three charges carried by the black hole are:
r Q "  sinh 2d ,  2g  Q

The ADM energy E, entropy S, and the Hawking temperature ¹ of (657) are & R
(660)

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From the asymptotic values of G and G (i"5,2, 8), one obtains the following expressions  GG for pressures in the x- and xG-directions:





1 R
(661)

So, in the xG-directions, which are parallel to 5-brane but perpendicular to 1-brane, shrinking e!ect of 5-brane and expanding e!ect of 1-brane compete and become balanced when d "d . In the   x-direction, which are parallel to both 5- and 1-branes, the shrinking e!ects of 5- and 1-branes are compensated by momentum in the x-direction. The 6 parameters r , d , d , d , < and R of the solution (657) can be traded with the numbers    N of 1-branes, anti-1-branes, 5-branes, anti-5-branes, right-moving momentum, and left-moving momentum, respectively given by [362]

(662)

rR< N " e\BN . * 4g Q

These parameters are related to the charges in (659) as Q "N !N  , Q "N !N  and       N"N !N . In terms of the new parameters, the ADM energy and the Bekenstein}Hawking 0 * entropy take simple and suggestive forms [362]: R< 1 R E" (N #N  )# (N #N  )# (N #N ) ,     * g R 0 g Q Q

(663)

S"2p((N #(N  )((N #(N  )((N #(N ) ,    * 0  where




<"

N N     , N N  




R"



gN N  Q 0 * . N N  

(664)

8.2.2. Four-dimensional black hole There are various ways in which one can construct D"4 black holes having regular BPS-limit with non-zero horizon area. The criteria for such construction are that supersymmetries are preserved and all the contributions of D-brane tensions from constituents compensate for each other so that internal space is stable against the shrinking e!ect near the horizon.

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The "rst type of con"guration is obtained by "rst T-dualizing the type-IIB D-brane bound state discussed in Section 8.2.1 along x, resulting in bound state of Q D 2-branes (along x, x) and Q   D 6-branes (along x,2, x) with the momentum N #owing along the x-direction. This con"guration has a zero horizon area in the BPS limit, since the radii along the directions x, x shrink as one approaches the horizon, due to the tensions of D 2- and D 6-branes and the momentum along the x direction. This is compensated by adding solitonic 5-brane along x,2, x and with magnetic charge of the NS-NS 2-form "eld B , where k"0,2, 3. Since the spinor constraint of I the additional solitonic 5-brane does not further reduce the Killing spinor degrees of freedom, the solution still preserves 1/8 of supersymmetry. D"4 black hole is obtained by compactifying this p-brane bound state on ¹"¹;SY;S [361,471]. Here, ¹ with the coordinates (x,2, x) has  the volume < and SY [S] with the coordinate x [x] has the radius R [R ]. Namely, the Q    D 6-branes wrap around ¹, Q D 2-branes wrap around SY;S, solitonic 5-branes wrap around  ¹;S and the momentum #ows along S. The resulting D"4 solution has the form [178,180,181,361]:




 



 


r ds"!f \(r) 1!  dt#f (r)  r




r sinh d  f (r)" 1#  r

r sinh d  1#  r



r \ dr#r(dh#sin h d ) , 1!  r




r sinh d  1#  r



(665)

r sinh d N . 1#  r

The electric/magnetic charges carried by the black hole are r < Q "  sinh 2d , Q "r R sinh 2d ,       g Q r r
(666)

The pressures along the directions x, xG (i"5,2, 8) and x are r
(667)

r
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One can trade the 8 parameters r , d , d , d , d , <, R , and R with the numbers of right- (left-)     N   moving momentum modes, (anti-) 2-branes, (anti-) 5-branes, and (anti-) 6-branes: r
(668)

r r N "  eB, N  "  e\B .  2g  2g Q Q In terms of these parameters, the ADM mass and entropy take forms [361]: 1 R R




(669)



N N N N gN N  , R "   , R R" Q 0 * . (670)    N N N N gN N      Q   We discuss couple of other ways [392] to construct D"4 black holes. The "rst con"guration is a type-IIB con"guration where Q parallel D 1-branes (along x) and Q parallel D 5-branes (along   x,2, x) intersect in the x-direction, along which momentum P"N/¸ #ows, and magnetic monopole in the subspace (t, r, h, , x). The D"4 black hole is obtained by "rst wrapping Q  D 5-branes around 4-cycles of K3 surface (with the coordinates x,2, x), resulting in a D-string bound state along with Q D 1-branes in D"6. This bound state in D"6 has momentum P along  x and the KK magnetic charge associated with the metric component g . One further compac( ti"es the coordinates (x, x) on ¹ to obtain D"4 black hole. Entropy of this solution is <"

A "2p(Q Q N . S"   4G , The second con"guration is a type-IIA solution obtained by T-dualizing the above con"guration along x. Namely, this is a bound state of Q D 0-branes and Q D 4-branes (along x,2, x) with   open strings wound around the x-direction (with the winding number =) and the KK monopole in the subspace (t, r, h, , x). Similarly as in the "rst case, to have a D"4 black hole, one "rst wrappes Q D 4-branes around 4-cycles of K3 surface (with the coordinates x,2, x), resulting in  a D particle bound state in D"6, and then compacti"es the coordinates (x, x) on ¹. Such a solution has entropy A "2p(Q Q = . S"   4G ,

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8.3. D-brane counting argument In this section, we discuss D-brane interpretation of black hole entropy. The number of degenerate BPS states is calculated in the weak string coupling regime (g +0), in which the Q D-brane picture, rather than black p-brane description, of charge con"gurations is applicable. Since mass of R-R charge carrier behaves as &1/g , D-branes become in"nitely massive in the perturQ bative region. So, the leading contribution to the degeneracy of states is from perturbative states of open strings, which are attached to D-branes. So, perturbative D-brane con"gurations are described by conformal "eld theory of open strings in the target space manifold determined by D-branes and the internal space. In the following, we discuss conformal "eld theories of D-branes which correspond to speci"c D"4, 5 black holes and calculate the number of degenerate perturbative open string states. More intuitive picture of D-brane argument is discussed in the subsequent subsection. 8.3.1. Conformal model for D-brane conxgurations We saw that the worldvolume theory of massless modes in a bound state of N D p-branes is described by D"10 ;(N) super-Yang-Mills theory compacti"ed to D"p#1 [637]. Namely, dynamics of collective coordinates [107] of N parallel D p-branes is described by a supersymmetric ;(N) gauge theory. The BPS states of D-brane con"guration correspond to the supersymmetric vacuum of the corresponding super-Yang-Mills theory. The ;(N) group is decomposed into a ;(1) group, describing the center of mass motion of D-branes, and an S;(N) group [637]. It is the supersymmetric S;(N) vacuum that contains states with a mass gap, which are relevant for degeneracy of D-brane bound states. When there is a mass gap, the number of ground states in the S;(N) super-Yang-Mills theory is the same as the degeneracy of the D-brane bound states. Here, the ground states are identi"ed with the cohomology elements of the S;(N) instanton moduli space M,, implying that degeneracy of D-brane bound states is given by the number of cohomology elements of M, [619,620]. D-brane bound states are e!ectively described by the p-model on the instanton moduli space M,. Generally, the p-model with target space manifold given by hyperKaK hler manifold of dimension 4k has the central charge c"6k. For example, the moduli space M, of S;(N) instantons on K3 with the instanton I number k has the dimension 4[N(k!N)#1]. It has been checked [90,91,566,567,619,620] that calculation of the degeneracy of D-brane bound states based on this idea yields the results which are consistent with conjectured string dualities that relate perturbative string states to D-brane bound states. In particular, P of BPS perturbative string states at the level N "1#P is *  dualized to the intersection number of D-brane bound states [619]. One of setbacks in study of D-brane bound states is that a bound state of m D p-branes is marginally stable, i.e. there is no energy barrier against decay into m D p-branes (each carrying the unit R-R (p#1)-form charge). Such a problem is circumvented [566] by compactifying an extra one direction, say y, on S so that states in the multiplet carry momentum or winding number n along S. If the pair (m, n) is relatively prime, the corresponding states in the D"9!p theory, i.e.

 In particular, the singular points in the moduli space [21,26,635], at which gauge symmetry is enhanced, correspond to the case where D-branes coincide.

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D"10 string bound states compacti"ed on ¹N;S, become absolutely stable against decay. Such worldvolume theory is described by the N"1 ;(m) gauge theory compacti"ed to D"p#1 with states characterized by n units of ;(1)L;(m) electric #ux along y. The non-trivial information on degeneracy of the BPS D-brane bound states is contained in the supersymmetric ground states of the S;(m) part of theory on the base manifold ¹N;R, where R is labeled by the time coordinate. It is shown [619] that for the type-IIA bound state of m 0-branes and 1 4-brane (or the bound state of m 1-branes and 1 5-brane in the T-dual theory) the instanton moduli space M of the corresponding ;(m);;(1) super-Yang-Mills theory is (¹)K/S , where S is the permutation A K K group on m objects. The degeneracy of the cohomology in this space is in one-to-one correspondence with the partition function of left-movers of the superstring. To put it another way, the ground states of this gauge theory are related to the degenerate string states at level m. The quantum numbers (F #F , F !F ) of the cohomology of (¹)K/S , where F [F ] is the [anti-] holomor0 * 0 K * 0  * phic degrees of the homology shifted by half the complex dimension, is mapped to the light-cone helicities of the left oscillator states of type-II superstring [623]. The generalization to D-brane bound states wrapped around K3 surface is carried out in [91,620]. The bound state of N D 4-branes wrapped around K3 is described by a ;(N) gauge theory on K3;R with R parameterized by the time coordinate. For this D-brane con"guration, it is shown [91] that the quantum corrections induce the e!ective D 0-brane charge M"k!N. So, the momentum square P"P!P is P/2"NM"N(k!N). If N and k are relatively prime, 0 * there is a mass gap and the moduli space M, is smooth with a discrete spectrum, with cohomology I of M, being that of N(k!N)#1 unordered points on K3. When N"2 and k is odd, the moduli I space is the symmetric product of K3's: M"(K3)I\/S . The Euler characteristic d(2k!3) of I I\ M is the same as the degeneracy d(N )"d(P#1) of string states at level N , or the degeneracy * I *  of D-brane bound states with charges (N,M). Here, d(n) is de"ned in terms of the Dedekind eta function g(q) as g(q)\"q\ d(n)qL. L 8.3.1.1. Applications to statistical entropy of black holes. We consider a type-IIB D"5 black hole carrying electric charges Q and Q of ;(1) "elds, respectively, originated from R-R 2-form $ & potential AK 00  and NS-NS 2-form potential BK ,1  [95,98,580]. Here, the D"5 ;(1) "eld strength ++ ++ which is associated with R-R 2-form potential in D"10 is Hodge-dual to the "eld strength of R-R in D"5. From the D"10 perspective, Q (Q ) is magnetic (electric) charge of 2-form "eld AK 00 $ & I I AK 00  (BK ,1 ), which couples to the worldvolume of R-R 5-brane [NS-NS string]. The D"5 ;(1) ++ ++ "eld originated from AK 00 , therefore, carries electric charge. The internal index m in the D"5 ++ NS-NS ;(1) "eld BK ,1 is along S, upon which an extra coordinate of the D"6 type-IIB theory is IK compacti"ed. So, this black hole is regarded as a bound state of 5-brane with R-R charge Q and $ string with N-N charge Q with the 5-brane wrapped around a holomorphic 4-cycle of K3 and & partially around S, and the string wound round S. T-duality along S leads to a type-IIA solution corresponding to 4-brane with 5-form charge Q and momentum #owing along S. $ The non-rotating BPS black hole with the above charge con"guration has spacetime of the Reissner}Nordstrom black hole [182,580,609]:









r   r  \ dt# 1!  dr#r dX , ds"! 1!   r r

(671)

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where r "(8Q Q/p). The scalars are constant everywhere and expressed in terms of the  & $ (integer-valued) electric charges Q [260]. The Bekenstein}Hawking entropy is &$ A Q Q & $. S " "2p (672) & 4 2



For D"5 rotating solutions, the regular BPS limit is possible only for the case where 2 angular momenta have the same magnitude, i.e. "J """J ""J. The solution is [98,182,610]    k ku sin h ku cos h dt! d # d

ds"! 1!   r (r!k) (r!k)





 



k \ dr#r(dh#sin h d #cos h d ) , # 1!   r

(673)

with Q "k/j, Q "!(p/2(2)kj and J "!J "(p/4)ku. The Bekenstein}Hawking entropy & $   is



Q Q A & $!J . S " "2p & 4 2

(674)

We discuss the D-brane interpretation of the entropies (672) and (674). In terms of D-brane language, the above black holes are the bound state of Q D 5-branes wrapped around K3;S and $ a fundamental string wound around S with the winding mode Q . The product Q ) Q is the & $ $ intersection number of D 5-branes in the K3 homology. We consider the case where K3 is small compared to the size of S. For this case, the solution looks like a fundamental string in the 5-brane background. The theory is e!ectively described by the conformal "eld theory on S;R with the target space manifold given by the symmetric product of K3 surfaces [580]: (K3)B /$>

. (675) M" S  $ / >

This conformal "eld theory has central charge c"6(Q#1), since the real dimension of this  $ manifold is 2(Q#2). $ First, we consider the non-rotating black hole (671) with the thermal entropy (672) [580]. The statistical entropy of the BPS solutions is given in terms of degeneracy d(n, c) of conformal states with the left-moving oscillators at level ¸ "n (n<1) and the right-moving oscillators in ground  state:



S "ln d(n, c)&2p  

nc . 6

(676)

For the solution (671), n"Q and, therefore, the statistical entropy (676) reproduces the thermal & entropy (672): (677) S &2p(Q (Q#1) .   & $ Next, we consider the thermal entropy (674) of the rotating black hole (673) [98]. The rotation group SO(4) of the D"6 lightcone-frame theory is identi"ed with the S;(2) ;S;(2) symmetry of * 0

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the N"4 superconformal algebra. The charges (F , F ) of ;(1) ;;(1) LS;(2) ;S;(2) are * 0 * 0 * 0 interpreted as spins of string states [98,623], and are related to angular momenta of the black hole (673) as J "(F #F ), J "(F !F ) . (678)   * 0   * 0 Note, the ;(1) current J can be bosonized as J "((c/3)R and a conformal state U * carrying * * * $ ;(1) charge F is obtained by applying an operator exp(iF /(c/3) to the state U without ;(1)  * * * * charge. The net e!ect is to shift the left-moving oscillator level n of string states carrying the ;(1) * charge F with respect to the level n for string states without the ;(1) charge F : n "n!3F/2c. *  * *  * For a large value of n , one can safely use the string level density formula d , which includes  LA contributions of string states with all the possible spin values, in calculating the statistical entropy. Here, c"6(Q#1) and n"Q , as in the non-rotating case. Substituting all of these values into &  $ (676), one obtains the following statistical entropy in agreement with the thermal entropy (674): S &2p  








1 n c 1  &2p Q Q#1 ! ("J "#"J ") , & 2 $  4  6

(679)

where J "!J "J.   8.3.2. Another description We discuss more intuitive D-brane picture of black holes based upon the arguments in [109]. In this picture, one counts all the possible oscillator contributions of open strings attached to D-branes. D-brane con"gurations, which are the weak string coupling limit of black holes, are described by the p-model of open strings with the target space background determined by D-branes, upon which the ends of the open strings live. The statistical entropy of black holes is due to degenerate states in "xed oscillator levels N *0 (or a "xed mass) of open strings attached to D-branes:



c ((N #(N ) , S "ln d(N , N ; c)&2p * 0   * 0 6

(680)

in the limit N <1. The central charge c has an additive contribution of 1 () for each bosonic *0  (fermionic) coordinate. N are determined by the NS-NS electric charges (i.e. momentum and winding modes in the *0 compacti"ed space) from the mass formula of open string states, which is derived from the Virasoro condition ¸ !a"0. Here, a is the zero point energy (or the normal ordering constant of  oscillator modes). The contribution to the zero point energy a is additive with the contribution of !  (  ) for each bosonic coordinate with the periodic (anti-periodic) boundary condition,   i.e. the integer (half-integer) moding of oscillator, and for a fermionic coordinate there is an extra minus sign.

 For detailed discussions on this point, see Section 7.3.2.

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Note, the Virasoro operator ¸ , say of the bosonic coordinates, is de"ned in terms of the  oscillator modes as ¸ "aI a # aI a , where k is a D"10 vector index. Here, aI "pI is L \L LI     I the center-of-mass frame momentum of an open string and N,(a/4) aI a is the oscillator L \L LI number operator. For string theory compacti"ed to D"d (d(10), we divide D"10 momentum pI into the D"d part pI and the internal part pK, i.e. (pI)"(pI , pK). Since mass in D"d is de"ned as M"!p  pI , one obtains the following mass formula from the Virasoro condition ¸ !a"0:  I 4 4 (681) M"(p )# (N #a )"(p )# (N #a ) , * 0 0 * a 0 a * where the subscripts L and R denote left- and right-moving sectors, and p are momenta in the *0 internal directions. Note, this expression for mass has all the contributions from bosonic and fermionic coordinates. In particular, for the compacti"cation on S of radius R, the left- and right-moving momenta in the S-direction is p "n/RGmR/a, where n and m are respectively *0 momentum and winding modes around S. Mass of the BPS states is M "(p ). So, for the BPS states, the right-movers are in ground .1 0 state and only left-movers are excited with their total oscillation numbers determined (from the mass formula (681)) by the left- and right-moving momenta in the internal space: a N "!a , N " [(p )!(p )]!a . * * 0 0 * 4 0 ), the right movers, as well as the left movers, are excited: .1 a N "!a #k, N " [(p )!(p )]!a #k (k3Z) . 0 0 * 4 0 * *

(682)

For non-BPS states (M'M

(683)

Since N are "xed by NS-NS electric charges, the main problem of calculating the statistical *0 entropy in D-brane picture is to determine the value of c, i.e. the total degrees of freedom of bosonic and fermionic coordinates of open strings. There are two types of open strings. The "rst type is open strings that stretch between the same type of D-brane, called the (p, p) type. The second type is open strings that stretch between di!erent types of D-branes, called the (p, p) type, pOp. For the second type, (p, p) and (p, p) are not equivalent, since open strings are oriented. Open strings of the (p, p) type can be ignored in the calculation of the open string state degeneracy, since the excited states of open strings which stretch between D-branes of the same type become very heavy as the relative separation between a pair of D-branes gets large. We brie#y discuss some aspects of open string states of the (p, p) type. The D"10 spacetime coordinates XI for this type of open strings are divided into 3 classes. The "rst (second) type, called the DD (NN) type, corresponds to coordinates for which both ends of strings satisfy the Dirichlet (Neumann) boundary conditions (i.e. coordinates which are transverse (longitudinal) to both branes). For these cases, the bosonic string coordinates are integer modded and, thereby, have the

 The fermionic coordinates have the similar expressions. In the presence of both bosonic and fermionic coordinates, the Virasoro operator is sum of the 2 contributions, i.e. ¸ "¸
#¸ .   

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zero point energy contribution of !  for each bosonic coordinate. The third type, called DN or  ND type, corresponds to coordinates for which each end of open strings satis"es di!erent boundary conditions (i.e. coordinates which are transverse to one brane and longitudinal to the other brane). For this type, the bosonic string coordinates are half-integer modded and, therefore, have the zero point energy contribution of  for each bosonic coordinate. The fermionic coordi nates have the same moding (as the bosonic coordinates) in the R sector and the opposite in the NS sector. So, the total zero point energy is always 0 for the R sector. For the NS sector, the zero point energy depends on the total number l of the ND and DN coordinates, which is always even:









1 1 1 1 1 l #l # "! # . (8!l) ! ! 24 48 24 48 2 8

(684)

Only when l is a multiple of 4, D-brane con"gurations are supersymmetric and degeneracy between the NS and R sectors is possible. When l"4, N"2 supersymmetry is preserved in D"4 and the zero point energy is zero in both the R and NS sectors. An example is the intersecting D string and D 5-brane, corresponding to the D"5 black hole that we consider in the following. For this type of open strings, degrees of freedom of the fermionic and bosonic coordinates are determined as follows. In the NS sector, among 4 fermionic coordinates s in the ND directions only 2 of them survive the GSO projection CGCGCGCGs"s, where i ,2, i correspond to the ND directions. In the R sector, also only 2 of   4 periodic transverse fermions (in the NN and DD directions) survive the GSO projection. Furthermore, there are Q Q ways to attach open strings between D p- and D p-branes. Since open N NY strings are oriented, we have the same numbers of fermionic and bosonic degrees of freedom satisfying the above constraints in the (p, p) and the (p, p) sectors. So, there are 4Q Q bosonic and N NY 4Q Q fermionic degrees freedom for each (p, p) and (p, p) types. Thus, the central charge of open N NY strings of (p, p) and (p, p) types is (685) c"4Q Q (1#)"6Q Q . N NY N NY  We comment on di!erent interpretation of the entropy expression (680). Let us consider a BPS con"guration (N "0 case) with the total momentum number N along S of radius R. When the 0 number of the bosonic (fermionic) degrees of freedom are N (N ), the central charge is $ c"N #N . Then, the statistical entropy (680) becomes the following entropy formula describ $ ing the N bosonic and N fermionic species with energy E"N/R in a 1-dimensional space of $ length ¸"2pR [200,471]: S"(p(2N #N )E¸/6 . (686) $ Note, the entropy formula (680) is derived assuming that N <1, i.e. the NS electric charges are *0 much bigger than the number Q of D p-branes. When Q and the NS electric charges are of the N N same order in magnitude, the string level density formula fails to reproduce the Bekenstein} Hawking entropy [472]. This stems from the fact that (with Q of the order of N) the notion of N extensivity (of entropy, energy, etc.) fails when radius R of S, around which D-branes are wrapped,

 These two types are di!erent since open strings are oriented.

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is smaller than the size of black holes ( fat black hole limit), since the wavelength (&1/¹, where ¹ is the e!ective temperature of the left movers) of a typical quantum exceeds the size ¸"2pR of the system [472]. The remedy to this problem proposed in [472] is as follows. One considers a bound state of Q  D 5-branes and Q D 1-branes which, respectively, wrap around M ;S and S. Here, M is    a 4-dimensional compact manifold and S has radius R. One neglects the directions associated with M and therefore the D 5-branes are regarded as strings wrapped around S. Instead of taking Q  N D p-branes as Q numbers of strings wrapped around S once, one regards it as a single string N wrapped Q times around S, thereby having length 2pQ R [467]. So, a bound state of D 1- and N N D 5-branes is described by a single string of the length 2pQ Q R wrapped Q Q times around S.     This is interpreted as a single species with the energy E"N/R"N/R in a 1-dimensional space of length ¸"2pR (thereby simulating a spectrum of fractional charges), where N,Q Q N and   R,Q Q R. Note, both the oscillator level and the radius of S are multiplied by Q Q . In this     description of D-brane bound states, the size ¸ of systems is always bigger than the wavelength of a typical quantum, thereby the extensivity condition is always satis"ed. Furthermore, since the (e!ective) oscillator level N"Q Q N is always much bigger than the number of D-branes (which   is 1) even when Q +N+Q , one can still use the statistical entropy expression of the type (680).   The statistical entropy is then S"(2pE¸"2p(N, where ¸"Q Q ¸"(N #N )¸ and     $ N"Q Q N. Note, this is the same as (686) when expressed in terms of the original variables, but   with this new description the approximation (of string level density) leading to the statistical entropy is valid. Near-extreme black holes correspond to D-brane con"gurations with a small amount of right moving oscillations excited, i.e. N given by (683) with a small integer n. As long as the string *0 coupling is very small and the density of strings (J1/R) is low, the interaction between the left- and the right-movers is negligible [368]. In this limit, entropy contributions from the left- and the right-movers are additive, leading to [362,368] S "(p(2N #N )E ¸/6#(p(2N #N )E ¸/6 , $ *   $ 0

(687)

where ¸"2pR and E "N /R with N very small. This expression correctly reproduces *0 *0 0 the Bekenstein}Hawking entropy of near extreme black holes. We comment that the above Dbrane interpretation of near extreme black holes is valid when only one of the constituents of the D-brane bound state has energy contribution much smaller than the others. For example, the above calculation is done in the limit where R is larger than the size < of the other internal manifold and, thereby, black holes are e!ectively described by (oscillating) strings in spacetime 1 dimension higher. In this limit, the momentum modes (of open strings) are much lighter than the branes (cf. (663)); the leading order contribution to the black hole entropy is from open string modes. For other limits where one of branes has much smaller energy than the other constituents, one can apply the similar argument in calculating statistical entropy of near extreme black holes [362] and the result is consistent with the conjectured U-duality. (In these cases, the leading contribution to the degeneracy is from D-branes rather than from open strings.) For the case where more than one constituents are light or all the constituents have energies of the same order of magnitude, the statistical interpretation of near extreme black holes is not known yet.

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Generalization to the rotating black hole case [95,98,361] is along the same line described in Section 7.3.2. Note, only the fermionic coordinates of the (p, p) type can carry angular momentum under the spatial rotational group. These are 4 periodic transverse fermions (from the NN and DD directions) in the R sector, which are spinors under the Lorentz group of the external spacetime. When the GSO projection is imposed, only the positive chirality representation of the external Lorentz group survives. We concentrate on the (1, 5) type, which is related to the D"5 (D"4) black holes under consideration in this section (through T-duality). For this case, the D"10 Lorentz group SO(1, 9) is decomposed as: SO(1, 9)MSO(1, 1)SO(4) SO(4) , (688) # ' where the "rst (third) factor acts on the D 1-brane worldsheet (the space internal to D 5-brane) and the second factor on the space external to the con"guration. Worldsheet spinors are decomposed into the 2-dimensional positive chirality representation 2> (with R-type quantization in ND directions of the NS sector) under the internal rotational group SO(4) (thereby a boson under the ' spacetime SO(1, 5) Lorentz group) and the 2-dimensional representation 2> (with NS-type quantiz> ation in ND directions of the R sector) which is positive-chiral under the external group SO(1, 1);SO(4) LSO(1, 5) (thereby a spacetime fermion). # Here, the spatial rotation group SO(4) (external to the D-brane con"guration) is isomorphic to # S;(2) ;S;(2) , which is identi"ed with the symmetry of (4, 4) superconformal theories. This is 0 * related to the fact that worldsheet fermions, which carry angular momenta, manifest themselves as spacetime fermions. The ;(1) LS;(2) charges F are related to angular momenta J of the *0 *0 0*  spatial rotational group SO(4) as in (678). # The e!ect of angular momenta is to reduce the total oscillation numbers [98]: 3F 3F N "N ! *, N "N ! 0 , 0 0 * * 2c 2c

(689)

where N (N ) are the total oscillation numbers of states that do not carry the ;(1) charges *0 *0 *0 (have the ;(1) charges F ). To obtain the statistical entropy of black holes, one plug this *0 *0 expression for N into (680). *0 For non-extreme or near-extreme black holes, which do not saturate the BPS bound, the right moving as well as the left moving supersymmetries are preserved, and thereby F are both 0* non-zero. The statistical entropy of near-extreme black holes is:



c S &2p ((N #(N ) ,   * 0 6

(690)

where N are given in (689). *0 However, for D"5 BPS black holes, F "0 (leading to J "J ":J) since only the left-moving 0   supersymmetry survives (i.e. (0, 4) superconformal theory). So, for BPS black holes, N "N "0 0 0 and N "N !(6/c)J, leading to the statistical entropy [361]: * *





c c S &2p (N "2p N !J . *   6 6 *

(691)

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For the D"5 rotating black hole with charge con"guration described in the above, c"6Q Q   and N "N. So, the statistical entropy (691) correctly reproduces the Bekenstein}Hawking * entropy: S "2p(NQ Q !J.     To obtain D"4 rotating black hole with regular extreme limit, one "rst applies T-duality along the common string direction of the intersecting D 1- and D 5-branes with open strings wound around the common string direction. The resulting con"guration is bound state of D 2- and D 6-branes with momentum #owing along one of the common intersection directions. To have non-zero horizon area in the BPS limit, one adds solitonic 5-brane with charge Q . Here,  D 6-branes, D 2-branes and solitonic 5-brane, respectively, wrap around ¹"¹;S;S, S;S and ¹;S, and momentum #ows along S. Since the right moving supersymmetry breaks in the presence of solitonic 5-branes, the ;(1) charge F in the right moving sector vanishes (i.e. 0 J "J "J). Particularly, D"4 extreme rotating black hole corresponds to the minimum energy   con"guration which is regular with non-zero angular momentum, which happens when N "0 * [361]. So, the left movers are constrained to carry angular momentum, only. In this case, N "N 0 0 and N "6J/c. By plugging these into (683), one obtains * a 6 N " J! [p!p] , * 0 c 4 0 leading to the statistical entropy:





c ac S &2p (N "2p J! [p!p] . 0 *   6 24 0

(692)

For the D-brane bound state corresponding to the D"4 black hole described above, c"6Q Q Q    (since there are 4Q Q Q species of bosons and fermions) and (a/4)[p!p]"N is the total    * 0 momentum mode of open strings around S. So, the statistical entropy S "2p(J#NQ Q Q      agrees with the Bekenstein}Hawking entropy. For further reading, see Refs. [15,16,28,31,53,78,153,172,176,256,402,417,433,502,508,616].

Acknowledgements I would like to thank M. Cvetic\ for suggesting me to write this review paper, for discussion during the initial stage of the work and for proof-reading part of the review. Part of the work was done while the author was at University of Pennsylvania. The work is supported by DOE grant DE-FG02-90ER40542.

 The extra factor of Q comes from the fact that whenever D 2-branes (which intersect the solitonic 5-brane along S)  cross the solitonic 5-brane they can break up and ends separate in di!erent positions in ¹. Thus, Q D 2-branes break up  into Q Q D 2-branes.  

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References [1] L.F. Abbott, S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B 195 (1982) 76}99. [2] A. Achucarro, J.M. Evans, P.K. Townsend, D.L. Wiltshire, Super p-branes, Phys. Lett. B 198 (1987) 441}446. [3] M. Aganagic, C. Popescu, J.H. Schwarz, Gauge invariant and gauge "xed D-brane actions, Nucl. Phys. B 495 (1997) 99}126. [4] G. Aldazabal, A. Font, L.E. Ibanez, F. Quevedo, Chains of N"2, D"4 heterotic type II duals, Nucl. Phys. B 461 (1996) 85}100. [5] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P. Fre, General matter coupled N"2 supergravity, Nucl. Phys. B 476 (1996) 397}417. [6] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P. Fre, T. Magri, N"2 supergravity and N"2 super Yang-Mills theory on general scalar manifolds: symplectic covariance, gaugings and the momentum map, Preprint POLFIS-TH-03-96, hep-th/9605032, 1996. [7] L. Andrianopoli, R. D'Auria, S. Ferrara, Central extension of extended supergravities in diverse dimensions, Int. J. Mod. Phys. A 12 (1997) 3759}3774. [8] L. Andrianopoli, R. D'Auria, S. Ferrara, ; duality and central charges in various dimensions revisited, Prepint CERN-TH-96-348, hep-th/9612105, 1996. [9] I. Antoniadis, S. Ferrara, E. Gava, K.S. Narain, T.R. Taylor, Perturbative prepotential and monodromies in N"2 heterotic superstring, Nucl. Phys. B 447 (1995) 35}61. [10] I. Antoniadis, S. Ferrara, E. Gava, K.S. Narain, T.R. Taylor, Duality symmetries in N"2 heterotic superstring, Talk given at international workshop of supersymmetry and uni"cation of fundamental interactions (SUSY 95), Palaiseau, Cedex, France, 15}19 May 1995, Nucl. Phys. Proc. Suppl. 45 BC (1996) 177}187, Nucl. Phys. Proc. Suppl. 46 (1996) 162}172. [11] I. Antoniadis, S. Ferrara, T.R. Taylor, N"2 heterotic superstring and its dual theory in "ve-dimensions, Nucl. Phys. B 460 (1996) 489}505. [12] I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, Topological amplitudes in string theory, Nucl. Phys. B 413 (1994) 162}184. [13] I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, N"2 Type II heterotic duality and higher derivative F terms, Nucl. Phys. B 455 (1995) 109}130. [14] I. Antoniadis, H. Partouche, Exact monodromy group of N"2 heterotic superstring, Nucl. Phys. B 460 (1996) 470}488. [15] I. Antoniadis, H. Partouche, T.R. Taylor, Spontaneous breaking of N"2 global supersymmetry, Phys. Lett. B 372 (1995) 83}87. [16] I. Antoniadis, T.R. Taylor, Dual N"2 SUSY breaking, Workshop on recent developments in theoretical physics: S¹;-dualities and nonperturbative phenomena in superstrings and supergravity, 27 Nov}1 Dec 1995, Geneva, Switzerland, Fortsch. Phys. 44 (1996) 487}492. [17] C. Aragone, S. Deser, Consistency problems of hypergravity, Phys. Lett. B 86 (1979) 161}163. [18] R. Argurio, F. Englert, L. Houart, Intersection rules for p-branes, Phys. Lett. B 398 (1997) 61}68. [19] P.C. Argyres, A.E. Faraggi, The vacuum structure and spectrum of N"2 supersymmetric S;(N) gauge theory, Phys. Rev. Lett. 74 (1995) 3931}3934. [20] R. Arnowitt, S. Deser, C.W. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. 122 (1961) 997}1006. [21] P.S. Aspinwall, Enhanced gauge symmetries and K3 surfaces, Phys. Lett. B 357 (1995) 329}334. [22] P.S. Aspinwall, Some relationships between dualities in string theory, Presented at ICTP Trieste Conference on Physical and Mathematical Implications of Mirror Symmetry in String Theory, Trieste, Italy, June 5}9, 1995, Nucl. Phys. Proc. Suppl. 46 (1996) 30}38. [23] P.S. Aspinwall, K3 surfaces and string duality, Preprint RU-96-98, hep-th/9611137, 1996. [24] P.S. Aspinwall, B.R. Greene, D.R. Morrison, Calabi-Yau moduli space, mirror manifolds and space-time topology change in string theory, Nucl. Phys. B 416 (1994) 414}480. [25] P.S. Aspinwall, J. Louis, On the ubiquity of K3 "brations in string duality, Phys. Lett. B 369 (1996) 233}242. [26] P.S. Aspinwall, D.R. Morrison, String theory on K3 surfaces, Preprint DUK-TH-94-68, hep-th/9404151, 1994.

214

D. Youm / Physics Reports 316 (1999) 1}232

[27] W. Awada, P.K. Townsend, N"4 Maxwell-Einstein supergravity in "ve-dimensions and its S;(2) gauging, Nucl. Phys. B 255 (1985) 617}632. [28] M. Awada, P.K. Townsend, G. Sierra, Six-dimensional simple and extended chiral supergravity in superspace, Class. Quant. Grav. 2 (1985) L85}L90. [29] C. Bachas, D-brane actions, Phys. Lett. B 374 (1996) 37}42. [30] J. Bagger, E. Witten, Matter couplings in N"2 supergravity, Nucl. Phys. B 222 (1983) 1}10. [31] V. Balasubramanian, F. Larsen, On D-branes and black holes in four-dimensions, Nucl. Phys. B 478 (1996) 199}208. [32] V. Balasubramanian, R.G. Leigh, D-branes, moduli and supersymmetry, Phys. Rev. D 55 (1997) 6415}6422. [33] V. Balasubramanian, F. Larsen, R.G. Leigh, Branes at angles and black holes, Preprint PUPT-1690, hepth/9704143, 1997. [34] T. Banks, M. Dine, H. Dykstra, W. Fischler, Magnetic monopole solutions of string theory, Phys. Lett. B 212 (1988) 45}50. [35] T. Banks, L.J. Dixon, Constraints on string vacua with spacetime supersymmetry, Nucl. Phys. B 307 (1988) 93}108. [36] T. Banks, L.J. Dixon, D. Friedan, E. Martinec, Phenomenology and conformal "eld theory or can string theory predict the weak mixing angle? Nucl. Phys. B 299 (1988) 613}626. [37] R. Barbieri, S. Cecotti, Radiative corrections to the e!ective potential in N"1 supergravity, Z. Phys. C 17 (1983) 183}187. [38] J.M. Bardeen, B. Carter, S.W. Hawking, The four laws of black hole mechanics, Comm. Math. Phys. 31 (1973) 161}170. [39] I. Bars, Stringy evidence for D"11 structure in strongly coupled type IIA superstring, Phys. Rev. D 52 (1995) 3567}3575. [40] I. Bars, Consistency between 11D and ;-duality, Nucl. Phys. Proc. Suppl. 49 (1996) 115}122. [41] I. Bars, Supersymmetry, p-brane duality and hidden space and time dimensions, Phys. Rev. D 54 (1996) 5203}5210. [42] I. Bars, Duality and hidden dimensions, Lectures given at frontiers in quantum "eld theory in honor of the 60th birthday of prof. K. Kikkawa, Toyonaka, Japan, 14}17 Dec 1995, Preprint USC-96-HEP-B3A, hep-th/9604200, 1996. [43] I. Bars, Algebraic structure of S-theory, Lectures given at 2nd international Sakharov conference on physics, Moscow, Russia, 20}23 May 1996, and at Strings 96, Santa Barbara, CA, 15}20 July 1996, Preprint USC-96-HEPB5, hep-th/9608061, 1996. [44] I. Bars, S-theory, Phys. Rev. D 55 (1997) 2373}2381. [45] I. Bars, S. Yankielowicz, ; duality multiplets and nonperturbative superstring states, Phys. Rev. D 53 (1996) 4489}4498. [46] K. Becker, M. Becker, Boundaries in M-Theory, Nucl. Phys. B 472 (1996) 221}230. [47] K. Behrndt, The 10-D Chiral null model and the relation to 4-D string solutions, Phys. Lett. B 348 (1995) 395}401. [48] K. Behrndt, About a class of exact string backgrounds, Nucl. Phys. B 455 (1995) 188}210. [49] K. Behrndt, String duality and massless string states, Preprint SU-ITP-95-17, hep-th/9510080, 1995. [50] K. Behrndt, Quantum corrections for D"4 black holes and D"5 strings, Phys. Lett. B 396 (1997) 77}84. [51] K. Behrndt, Classical and quantum aspects of 4 dimensional black holes, Talk given at 30th Ahrenshoop international symposium on the theory of elementary particles, Buckow, Germany, 27}31 August 1996, Preprint HUB-EP-97-03, hep-th/9701053, 1997. [52] K. Behrndt, E. Bergshoe!, B. Janssen, Type II duality symmetries in six dimensions, Nucl. Phys. B 467 (1996) 100}126. [53] K. Behrndt, E. Bergshoe!, B. Janssen, Intersecting D-branes in ten and six dimensions, Phys. Rev. D 55 (1997) 3785}3792. [54] K. Behrndt, G.L. Cardoso, B. de Wit, R. Kallosh, D. Lust, T. Mohaupt, Classical and quantum N"2 supersymmetric black holes, Nucl. Phys. B 488 (1997) 236}260. [55] K. Behrndt, M. Cvetic\ , BPS saturated bound states of tilted p-branes in type II string theory, Phys. Rev. D 56 (1997) 1188}1193.

D. Youm / Physics Reports 316 (1999) 1}232 [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86]

215

K. Behrndt, H. Dorn, String}string duality for some black hole type solutions, Phys. Lett. B 370 (1996) 45}48. K. Behrndt, R. Kallosh, F and H monopoles, Nucl. Phys. B 468 (1996) 85}112. K. Behrndt, R. Kallosh, O(6, 22) BPS con"gurations of the heterotic string, Phys. Rev. D 53 (1996) 589}592. K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova, W. Wong, S¹; black holes and string triality, Phys. Rev. D 54 (1996) 6293}6301. K. Behrndt, D. Lust, W.A. Sabra, Moving moduli, Calabi-Yau phase transitions and massless BPS con"gurations in type II superstrings, Preprint HUB-EP-97-40, hep-th/9708065, 1997. K. Behrndt, D. Lust, W.A. Sabra, Stationary solutions of N"2 supergravity, Preprint HUB-EP-97-30, hepth/9705169, 1997. K. Behrndt, W.A. Sabra, Static N"2 black holes for quadratic prepotentials, Phys. Lett. B 401 (1997) 258}262. J.D. Bekenstein, Black holes and the second law, Lett. Nuov. Cimento 4 (1972) 737}740. J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333}2346. J.D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974) 3292}3300. J.D. Bekenstein, Statistical black hole thermodynamics, Phys. Rev. D 12 (1975) 3077}3085. J.D. Bekenstein, Black holes and thermodynamics, Phys. Rev. D 13 (1976) 191}197. A.A. Belavin, A.M. Polyakov, A.S. Shvarts, Y.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85}87. F.A. Berends, J.W. van Holten, B. de Wit, P. van Nieuwenhuizen, On spin 5/2 gauge "elds, J. Phys. A 13 (1980) 1643}1649. P. Berglund, S. Katz, A. Klemm, Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties, Nucl. Phys. B 456 (1995) 153}204. P. Berglund, S. Katz, A. Klemm, P. Mayr, New Higgs transitions between dual N"2 string models, Nucl. Phys. B 483 (1997) 209}228. E. Bergshoe!, H.J. Boonstra, T. OrtiD n, S duality and dyonic p-brane solutions in type II string theory, Phys. Rev. D 53 (1996) 7206}7212. E. Bergshoe!, M. de Roo, D-branes and ¹-duality, Phys. Lett. B 380 (1996) 265}272. E. Bergshoe!, M. de Roo, B. de Wit, P. van Nieuwenhuizen, Ten-dimensional Maxwell-Einstein supergravity, its currents, and the issue of its auxiliary "elds, Nucl. Phys. B 195 (1982) 97}136. E. Bergshoe!, M. de Roo, E. Eyras, B. Janssen, J.P. van der Schaar, Multiple intersections of D-branes and M-branes, Nucl. Phys. B 494 (1996) 119}143. E. Bergshoe!, M. de Roo, M.B. Green, G. Papadopoulos, P.K. Townsend, Duality of type II 7-branes and 8-branes, Nucl. Phys. B 470 (1996) 113}135. E. Bergshoe!, M. de Roo, T. OrtiD n, The eleven-dimensional "ve-brane, Phys. Lett. B 386 (1996) 85}90. E. Bergshoe!, M. de Roo, S. Panda, Four-dimensional high-branes as intersecting D-branes, Phys. Lett. B 390 (1997) 143}147. E. Bergshoe!, M.J. Du!, C.N. Pope, E. Sezgin, Compacti"cations of the eleven-dimensional supermembrane, Phys. Lett. B 224 (1989) 71}78. E. Bergshoe!, I. Entrop, R. Kallosh, Exact duality in string e!ective action, Phys. Rev. D 49 (1994) 6663}6673. E. Bergshoe!, C. Hull, T. OrtiD n, Duality in the type-II superstring e!ective action, Nucl. Phys. B 451 (1995) 547}578. E. Bergshoe!, R. Kallosh, T. OrtiD n, Duality versus supersymmetry and compacti"cation, Phys. Rev. D 51 (1995) 3009}3016. E. Bergshoe!, R. Kallosh, T. OrtiD n, Stationary axion/dilaton solutions and supersymmetry, Nucl. Phys. B 478 (1996) 156}180. E. Bergshoe!, I.G. Koh, E. Sezgin, Coupling of Yang-Mills to N"4, D"4 supergravity, Phys. Lett. B 155 (1985) 71}75. E. Bergshoe!, E. Sezgin, Y. Tanii, P.K. Townsend, Super p-branes as gauge theories of volume preserving di!eomorphisms, Ann. Phys. 199 (1990) 340}365. E. Bergshoe!, E. Sezgin, P.K. Townsend, Supermembranes and eleven-dimensional supergravity, Phys. Lett. B 189 (1987) 75}78.

216

D. Youm / Physics Reports 316 (1999) 1}232

[87] E. Bergshoe!, E. Sezgin, P.K. Townsend, Properties of the eleven-dimensional super membrane theory, Ann. Phys. 185 (1988) 330}368. [88] E. Bergshoe!, P.K. Townsend, Super D-branes, Nucl. Phys. B 490 (1997) 145}162. [89] M. Berkooz, M.R. Douglas, R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B 480 (1996) 265}278. [90] M. Bershadsky, V. Sadov, C. Vafa, D-strings on D-manifolds, Nucl. Phys. B 463 (1996) 398}414. [91] M. Bershadsky, V. Sadov, C. Vafa, D-branes and topological "eld theories, Nucl. Phys. B 463 (1996) 420}434. [92] B. Bertotti, Uniform electromagnetic "eld in the theory of general relativity, Phys. Rev. 116 (1959) 1331}1333. [93] M. Billo, A. Ceresole, R. D'Auria, S. Ferrara, P. Fre, T. Regge, P. Soriani, A. Van Proeyen, A search for nonperturbative dualities of local N"2 Yang-Mills theories from Calabi-Yau threefolds, Class. Quant. Grav. 13 (1996) 831}864. [94] E.B. Bogomol'nyi, Stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976) 449}454. [95] J.C. Breckenridge, D.A. Lowe, R.C. Myers, A.W. Peet, A. Strominger, C. Vafa, Macroscopic and microscopic entropy of near-extremal spinning black holes, Phys. Lett. B 381 (1996) 423}426. [96] J.C. Breckenridge, G. Michaud, R.C. Myers, More D-brane bound states, Phys. Rev. D 55 (1997) 6438}6446. [97] J.C. Breckenridge, G. Michaud, R.C. Myers, New angles on D-branes, Preprint MCGILL-97-02, hep-th/9703041, 1997. [98] J.C. Breckenridge, R.C. Myers, A.W. Peet, C. Vafa, D-branes and spinning black holes, Phys. Lett. B 391 (1997) 93}98. [99] P. Breitenlohner, D. Maison, G. Gibbons, Four-dimensional black holes from Kaluza-Klein theories, Comm. Math. Phys. 120 (1988) 295}333. [100] T.H. Buscher, Quantum corrections and extended supersymmetry in new sigma models, Phys. Lett. B 159 (1985) 127}130. [101] T.H. Buscher, A symmetry of the string background "eld equations, Phys. Lett. B 194 (1987) 59}62. [102] T.H. Buscher, Path integral derivation of quantum duality in nonlinear sigma models, Phys. Lett. B 201 (1988) 466}472. [103] A.C. Cadavid, A. Ceresole, R. D'Auria, S. Ferrara, Compacti"cation of D"11 supergravity on spaces of exceptional holonomy, Phys. Lett. B 357 (1995) 300}306. [104] C. Callan, J. Harvey, A. Strominger, World sheet approach to heterotic instantons and solitons, Nucl. Phys. B 359 (1991) 611}634. [105] C. Callan, J. Harvey, A. Strominger, Worldbrane actions for string solitons, Nucl. Phys. B 367 (1991) 60}82. [106] C. Callan, J. Harvey, A. Strominger, Supersymmetric string solitons, in Trieste 1991, Proceedings, String theory and quantum gravity '91 208}244 and Chicago Univ. (91/11,rec.Feb.92), Preprint EFI-91-66, hep-th/9112030, 1991. [107] C. Callan, I.R. Klebanov, D-brane boundary state dynamics, Nucl. Phys. B 465 (1996) 473}486. [108] C.G. Callan, C. Lovelace, C.R. Nappi, S.A. Yost, Loop corrections to superstring equations of motion, Nucl. Phys. B 308 (1988) 221}284. [109] C.G. Callan, J.M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl. Phys. B 472 (1996) 591}610. [110] C.G. Callan, J.M. Maldacena, A.W. Peet, Extremal black holes as fundamental strings, Nucl. Phys. B 475 (1995) 645}678. [111] C.G. Callan, E.J. Martinec, M.J. Perry, D. Friedan, Strings in background "elds, Nucl. Phys. B 262 (1985) 593}609. [112] I.C.G. Campbell, P.C. West, N"2 D"10 nonchiral supergravity and its spontaneous compacti"cation, Nucl. Phys. B 243 (1984) 112}124. [113] P. Candelas, X.C. de la Ossa, Moduli space of Calabi-Yau manifolds, Nucl. Phys. B 355 (1991) 455}481. [114] P. Candelas, X.C. de la Ossa, P.S. Green, L. Parkes, An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds, Phys. Lett. B 258 (1991) 118}126. [115] P. Candelas, X.C. De La Ossa, P.S. Green, L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21}74. [116] P. Candelas, A. Font, Duality between the webs of heterotic and type II vacua, Preprint UTTG-04-96, hepth/9603170, 1996.

D. Youm / Physics Reports 316 (1999) 1}232

217

[117] C.L. Cardoso, G. Curio, D. Lust, Perturbative couplings and modular forms in N"2 string models with a Wilson line, Nucl. Phys. B 491 (1997) 147}183. [118] C.L. Cardoso, G. Curio, D. Lust, T. Mohaupt, Instanton numbers and exchange symmetries in N"2 dual string pairs, Phys. Lett. B 382 (1996) 241}250. [119] C.L. Cardoso, G. Curio, D. Lust, T. Mohaupt, S.J. Rey, BPS Spectra and nonperturbative gravitational couplings in N"2, N"4 supersymmetric string theories, Nucl. Phys. B 464 (1996) 18}58. [120] C.L. Cardoso, D. Lust, T. Mohaupt, Threshold corrections and symmetry enhancement in string compacti"cations, Nucl. Phys. B 450 (1995) 115}173. [121] C.L. Cardoso, D. Lust, T. Mohaupt, Nonperturbative monodromies in N"2 heterotic string vacua, Nucl. Phys. B 455 (1995) 131}164. [122] C.L. Cardoso, D. Lust, T. Mohaupt, Perturbative and nonperturbative results for N"2 heterotic strings, Nucl. Phys. Proc. Suppl. 45 BC (1995) 167}176. [123] C.L. Cardoso, D. Lust, T. Mohaupt, Modular symmetries of N"2 black holes, Phys. Lett. B 388 (1996) 266}272. [124] G.L. Cardoso, B.A. Ovrut, Coordinate and KaK hler sigma model anomalies and their cancellation in string e!ective "eld theories, Nucl. Phys. B 392 (1993) 315}344. [125] B. Carter, Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26 (1970) 331}333. [126] B. Carter, Rigidity of a black hole, Nature 238 (1972) 71}72. [127] A. Casher, F. Englert, H. Nicolai, A. Taormina, Consistent superstrings as solutions of the D"26 bosonic string theory, Phys. Lett. B 162 (1985) 121}126. [128] L. Castellani, R. D'Auria, S. Ferrara, Special KaK hler geometry: an intrinsic formulation from N"2 space-time supersymmetry, Phys. Lett. B 241 (1990) 57}62. [129] L. Castellani, R. D'Auria, S. Ferrara, Special geometry without special coordinates, Class. Quant. Grav. 7 (1990) 1767}1790. [130] S. Cecotti, S. Ferrara, L. Girardello, Geometry of type-II superstrings and the moduli of superconformal "eld theories, Int. J. Mod. Phys. A 4 (1989) 2475}2529. [131] M. Cederwall, A. von Gussich, B.E.W. Nilsson, P. Sundell, A. Westerberg, The Dirichlet super p-branes in ten-dimensional type IIA and IIB supergravity, Nucl. Phys. B 490 (1997) 179}201. [132] A. Ceresole, R. D'Auria, S. Ferrara, On the geometry of moduli space of vacua in N"2 supersymmetric Yang-Mills theory, Phys. Lett. B 339 (1994) 71}76. [133] A. Ceresole, R. D'Auria, S. Ferrara, The symplectic structure of N"2 supergravity and its central extension, Talk given at ICTP Trieste conference on physical and mathematical implications of mirror symmetry in string theory, Trieste, Italy, 5}9 June 1995, Nucl. Phys. Proc. Suppl. 46 (1996) 67}74. [134] A. Ceresole, R. D'Auria, S. Ferrara, S.W. Lerche, J. Louis, Picard-Fuchs equations and special geometry, Int. J. Mod. Phys. A 8 (1993) 79}114. [135] A. Ceresole, R. D'Auria, S. Ferrara, A. van Proeyen, On electromagnetic duality in locally supersymmetric N"2 Yang-Mills theory, Contributed to joint U.S.-Polish workshop on physics from Planck scale to electro-weak Scale (SUSY 94), Warsaw, Poland, 21}24 September 1994, Preprint CERN-TH-7510-94, hep-th/9412200. [136] A. Ceresole, R. D'Auria, S. Ferrara, A. van Proeyen, Duality transformations in supersymmetric Yang-Mills theories coupled to supergravity, Nucl. Phys. B 444 (1995) 92}124. [137] A.W. Chamseddine, N"4 Supergravity coupled to N"4 matter, Nucl. Phys. B 185 (1981) 403}415. [138] A.W. Chamseddine, S. Ferrara, G. Gibbons, Enhancement of supersymmetry near 5d black hole horizon, Phys. Rev. D 55 (1997) 3647}3653. [139] H. Chan, J.E. Paton, Generalized Veneziano model with isospin, Nucl. Phys. B 10 (1969) 516}520. [140] K.L. Chan, Supersymmetric dyonic black holes of IIA string on six torus, Preprint UPR-0717-T, hep-th/9610005, 1996. [141] K.L. Chan, M. Cvetic, Massless BPS saturated states on the two torus moduli subspace of heterotic string, Phys. Lett. B 375 (1996) 98}102. [142] G.F. Chapline, N.S. Manton, Uni"cation of Yang-Mills theory and supergravity in ten-dimensions, Phys. Lett. B 120 (1983) 105}109. [143] Y.M. Cho, P.G.O. Freund, Nonabelian gauge "elds in Nambu-Goldstone "elds, Phys. Rev. D 12 (1975) 1711}1720.

218

D. Youm / Physics Reports 316 (1999) 1}232

[144] A. Chodos, S. Detweiler, Spherically symmetric solutions in "ve-dimensional general relativity, Gel. Rel. Grav. 14 (1982) 879}890. [145] Y. Choquet-Bruhat, D. Christodoulou, Elliptic systems in H spaces on manifolds which are Euclidean at in"nity, QB Acta Math. 146 (1981) 129}150. [146] Y. Choquet-Bruhat, J. Marsden, Solution of the local mass problem in general relativity, Comm. Math. Phys. 51 (1976) 283}296. [147] G. Clement, Stationary solutions in "ve-dimensional general relativity, Gen. Rel. Grav. 18 (1986) 137}160. [148] G. Clement, Rotating Kaluza-Klein monopoles and dyons, Phys. Lett. A 118 (1986) 11}13. [149] M.S. Costa, Composite M-branes, Nucl. Phys. B 490 (1997) 202}216. [150] M.S. Costa, Black composite M branes, Nucl. Phys. B 495 (1997) 195}205. [151] M.S. Costa, M. Cvetic\ , Nonthreshold D-brane bound states and black holes with nonzero entropy, Preprint DAMTP-R-97-15, hep-th/9703204, 1997. [152] M.S. Costa, G. Papadopoulos, Superstring dualities and p-brane bound states, Preprint DAMTP-R-96-60, hep-th/9612204, 1996. [153] M.S. Costa, M.J. Perry, Kaluza-Klein electrically charged black branes in M theory, Class. Quant. Grav. 14 (1997) 603}614. [154] P.M. Cowdall, H. Lu, C.N. Pope, K.S. Stelle, P.K. Townsend, Domain walls in massive supergravities, Nucl. Phys. B 486 (1997) 49}76. [155] B. Craps, F. Roose, W. Troost, A. van Proeyen, The de"nitions of special geometry, Preprint KUL-TF-96-11, hep-th/9606073, 1996. [156] E. Cremmer, Dimensional reduction in "eld theory and hidden symmetries in extended supergravity, in: S. Ferrara, J.G. Taylor (Eds.), Trieste 1981, Proceedings, Supergravity, Cambridge University Press, Cambridge, 1980. [157] E. Cremmer, Supergravity in 5 dimensions, in: S.W. Hawking, M. Roc\ ek (Eds.), Superspace and Supergravity, Cambridge University Press, Cambridge, 1981. [158] E. Cremmer, B. Julia, Supergravity theory in eleven-dimensions, Phys. Lett. B 76 (1978) 409}412. [159] E. Cremmer, B. Julia, The N"8 supergravity theory. 1. the lagrangian, Phys. Lett. B 80 (1978) 48}51. [160] E. Cremmer, B. Julia, The SO(8) supergravity, Nucl. Phys. B 159 (1979) 141}212. [161] E. Cremmer, C. Kounnas, A. Van Proeyen, J.P. Derendinger, S. Ferrara, B. de Wit, L. Girardello, Vector multiplets coupled to N"2 supergravity: superhiggs e!ect, #at potentials and geometric structure, Nucl. Phys. B 250 (1985) 385}426. [162] E. Cremmer, J. Scherk, Algebraic simpli"cations in supergravity theories, Nucl. Phys. B 127 (1977) 259}268. [163] E. Cremmer, J. Scherk, The supersymmetric nonlinear sigma model in four-dimensions and its coupling to supergravity, Phys. Lett. B 74 (1978) 341}343. [164] E. Cremmer, J. Scherk, S. Ferrara, ;(N) invariance in extended supergravity, Phys. Lett. B 68 (1977) 234}238. [165] E. Cremmer, J. Scherk, S. Ferrara, S;(4) invariant supergravity theory, Phys. Lett. B 74 (1978) 61}64. [166] E. Cremmer, A. van Proeyen, Classi"cation of KaK hler manifolds in N"2 vector multiplet supergravity couplings, Class. Quant. Grav. 2 (1985) 445}454. [167] G. Curio, Topological partition function and string}string duality, Phys. Lett. B 366 (1996) 131}133. [168] G. Curio, String dual F1 function in the three parameter model, Phys. Lett. B 368 (1996) 78}82. [169] T.L. Curtright, Massless "eld supermultiplets with arbitrary spin, Phys. Lett. B 85 (1979) 219}224. [170] M. Cvetic\ , I. Gaida, Duality invariant nonextreme black holes in toroidally compacti"ed string theory, Preprint UPR-0734-T, hep-th/9703134, 1997. [171] M. Cvetic\ , C. Hull, Black holes and ;-duality, Nucl. Phys. B 480 (1996) 296}316. [172] M. Cvetic\ , C. Hull, unpublished. [173] M. Cvetic\ , A.A. Tseytlin, General class of BPS saturated dyonic black holes as exact superstring solutions, Phys. Lett. B 366 (1996) 95}103. [174] M. Cvetic\ , A.A. Tseytlin, Solitonic strings and BPS saturated dyonic black holes, Phys. Rev. D 53 (1996) 5619}5633. [175] M. Cvetic\ , A.A. Tseytlin, Nonextreme black holes from nonextreme intersecting M-branes, Nucl. Phys. B 478 (1996) 181}198.

D. Youm / Physics Reports 316 (1999) 1}232

219

[176] M. Cvetic\ , D. Youm, Supersymmetric dyonic black holes in Kaluza-Klein theory, Nucl. Phys. B 438 (1995) 182}210, Addendum-ibid. B 449 (1995) 146}148. [177] M. Cvetic\ , D. Youm, All the four dimensional static, spherically symmetric solutions of Abelian Kaluza-Klein theory, Phys. Rev. Lett. 75 (1995) 4165}4168. [178] M. Cvetic\ , D. Youm, Dyonic BPS saturated black holes of heterotic string on a six-torus, Phys. Rev. D 53 (1995) 584}588. [179] M. Cvetic\ , D. Youm, Singular BPS saturated states and enhanced symmetries of four-dimensional N"4 supersymmetric string vacua, Phys. Lett. B 359 (1995) 87}92. [180] M. Cvetic\ , D. Youm, BPS saturated and non-extreme states in Abelian Kaluza-Klein theory and e!ective N"4 supersymmetric string vacua, in: Bars et al. (Eds.), The proceedings of STRINGS 95: Future Perspectives in String Theory, I. World Scienti"c, Singapore, 1995, Preprint UPR-675-T, hep-th/9508058, 1995. [181] M. Cvetic\ , D. Youm, General static spherically symmetric black holes of heterotic string on a six torus, Nucl. Phys. B 472 (1996) 249}270. [182] M. Cvetic\ , D. Youm, General rotating "ve dimensional black holes of toroidally compacti"ed heterotic string, Nucl. Phys. B 476 (1996) 118}132. [183] M. Cvetic\ , D. Youm, Entropy of non-extreme charged rotating black holes in string theory, Phys. Rev. D 54 (1996) 2612}2620. [184] M. Cvetic\ , D. Youm, Near-BPS-saturated rotating electrically charged black holes as string states, Nucl. Phys. B 477 (1996) 449}464. [185] M. Cvetic\ , D. Youm, Rotating intersecting M-branes, Preprint IASSNS-HEP-96-123, Nucl. Phys. B 499 (1996) 253}282. [186] A. Dabholkar, Quantum corrections to black hole entropy in string theory, Phys. Lett. B 347 (1995) 222}229. [187] A. Dabholkar, Ten-dimensional heterotic string as a soliton, Phys. Lett. B 357 (1995) 307}312. [188] A. Dabholkar, Microstates of nonsupersymmetric black holes, Phys. Lett. B 402 (1997) 53}58. [189] A. Dabholkar, J.P. Gauntlett, J.A. Harvey, D. Waldram, Strings as solitons & black holes as strings, Nucl. Phys. B 474 (1996) 85}121. [190] A. Dabholkar, G. Gibbons, J.A. Harvey, Superstrings and solitons, Nucl. Phys. B 340 (1990) 33}55. [191] A. Dabholkar, J.A. Harvey, Nonrenormalization of the superstring tension, Phys. Rev. Lett. 63 (1989) 478}481. [192] A. D'Adda, R. Horsley, P. di Vecchia, Supersymmetric magnetic monopoles and dyons, Phys. Lett. B 76 (1978) 298}302. [193] J. Dai, R.G. Leigh, J. Polchinski, New connections between string theories, Mod. Phys. Lett. A 4 (1989) 2073}2083. [194] A. Das, SO(4) invariant extended supergravity, Phys. Rev. D 15 (1977) 2805}2809. [195] S.R. Das, The e!ectiveness of D-branes in the description of near extremal black, Phys. Rev. D 56 (1997) 3582}3590. [196] S.R. Das, S.D. Mathur, Excitations of D-strings, entropy and duality, Phys. Lett. B 375 (1996) 103}110. [197] K. Dasgupta, S. Mukhi, Orbifolds of M theory, Nucl. Phys. B 465 (1996) 399}412. [198] R. D'Auria, S. Ferrara, P. FreH , Special and quaternionic isometries: general couplings in N"2 supergravity and the scalar potential, Nucl. Phys. B 359 (1991) 705}740. [199] S.P. De Alwis, A note on brane tension and M-theory, Phys. Lett. B 388 (1996) 291}295. [200] S.P. De Alwis, K. Sato, Radiation from a class of string theoretic black holes, Phys. Rev. D 55 (1996) 6181}6188. [201] J.A. De Azcarraga, J.P. Gauntlett, J.M. Izquierdo, P.K. Townsend, Topological extensions of the supersymmetry algebra for extended objects, Phys. Rev. Lett. 63 (1989) 2443}2446. [202] J.G. Demers, R. Lafrance, R.C. Myers, Black hole entropy without brick walls, Phys. Rev. D 52 (1995) 2245}2253. [203] J.P. Derendinger, S. Ferrara, C. Kounnas, F. Zwirner, On loop corrections to string e!ective "eld theories: "eld dependent gauge couplings and sigma model anomalies, Nucl. Phys. B 372 (1992) 145}188. [204] L.J. Romans, Massive N"2A supergravity in ten-dimensions, Phys. Lett. B 169 (1986) 374. [205] M. De Roo, Matter coupling in N"4 supergravity, Nucl. Phys. B 255 (1985) 515}531. [206] S. Deser, J.H. Kay, K.S. Stelle, Hamiltonian formulation of supergravity, Phys. Rev. D 16 (1977) 2448}2455. [207] S. Deser, C. Teitelboim, Supergravity has positive energy, Phys. Rev. Lett. 39 (1977) 249}252. [208] B. De Wit, G.L. Cardoso, D. Lust, D. Mohaupt, S.J. Rey, Higher order gravitational couplings and modular forms in N"2, D"4 heterotic string compacti"cations, Nucl. Phys. B 481 (1996) 353}388.

220

D. Youm / Physics Reports 316 (1999) 1}232

[209] B. De Wit, V. Kaplunovsky, J. Louis, D. Lust, Perturbative couplings of vector multiplets in N"2 heterotic string vacua, Nucl. Phys. B 451 (1995) 53}95. [210] B. De Wit, P.G. Lauwers, R. Philippe, S.Q. Su, A. van Proeyen, Gauge and matter "elds coupled to N"2 supergravity, Phys. Lett. B 134 (1984) 37}43. [211] B. De Wit, P.G. Lauwers, A. van Proeyen, Lagrangians of N"2 supergravity-matter systems, Nucl. Phys. B 255 (1985) 569}608. [212] B. De Wit, A. van Proeyen, Potentials and symmetries of general gauged N"2 supergravity-Yang-Mills models, Nucl. Phys. B 245 (1984) 89}117. [213] B. De Wit, A. van Proeyen, Broken sigma model isometries in very special geometry, Phys. Lett. B 293 (1992) 94}99. [214] B. De Wit, A. van Proeyen, Isometries of special manifolds, Presented at meeting on quaternionic Structures, Trieste, Italy, Sep 1994, Preprint THU-95-13, hep-th/9505097, 1995. [215] R. Dijkgraaf, E. Verlinde, H. Verlinde, On moduli spaces of conformal "eld theories with c51, in: P. Di Vecchia, J.L. Petersen (Eds.), Perspectives in String Theory, World Scienti"c, Singapore, 1988. [216] M. Dine, P. Huet, N. Seiberg, Large and small radius in string theory, Nucl. Phys. B 322 (1989) 301}316. [217] P. Dirac, Quantized singularities in the electromagnetic "eld, Proc. Roy. Soc. London A1 33 (1931) 60}72. [218] P. Dirac, The theory of magnetic poles, Phys. Rev. 74 (1948) 817}830. [219] J.A. Dixon, M.J. Du!, J.C. Plefka, Putting string/"ve-brane duality to the test, Phys. Rev. Lett. 69 (1992) 3009}3012. [220] J.A. Dixon, J.A. Harvey, C. Vafa, E. Witten, Strings on orbifolds, Nucl. Phys. B 261 (1985) 678}686. [221] J.A. Dixon, J.A. Harvey, C. Vafa, E. Witten, Strings on orbifolds 2., Nucl. Phys. B 274 (1986) 285}314. [222] J.A. Dixon, V. Kaplunovsky, J. Louis, On e!ective "eld theories describing (2, 2) vacua of the heterotic string, Nucl. Phys. B 329 (1990) 27}82. [223] P. Dobiasch, D. Maison, Stationary, spherically symmetric solutions of Jordan's uni"ed theory of gravity and electromagnetism, Gen. Rel. Grav. 14 (1982) 231}242. [224] M.R. Douglas, Branes within branes, Preprint RU-95-92, hep-th/9512077, 1995. [225] M.J. Du!, Kaluza-Klein theory in perspective, Talk given at the Oskar Klein centenary symposium, Stockholm, Sweden, 19}21 September 1994, Preprint NI-94-015, hep-th/9410046, 1994. [226] M.J. Du!, Strong/weak coupling duality from the dual string, Nucl. Phys. B 442 (1995) 47}63. [227] M.J. Du!, M-Theory (the theory formerly known as strings), Int. J. Mod. Phys. A 11 (1997) 5623}5642. [228] M.J. Du!, Supermembranes, Lectures given at theoretical advanced study institute in elementary particle physics (TASI 96): Fields, strings, and duality, Boulder, CO, 2}28 June 1996, and at 26th British Universities summer school in theoretical elementary particle physics, Swansea, Wales, 3}18 September 1996, Preprint CTP-TAMU61/96, hep-th/9611203, 1997. [229] M.J. Du!, S. Ferrara, R.R. Khuri, Supersymmetry and dual string solitons, Phys. Lett. B 356 (1995) 479}486. [230] M.J. Du!, S. Ferrara, R.R. Khuri, J. Rahmfeld, Supersymmetry and dual string solitons, Phys. Lett. B 356 (1995) 479}486. [231] M.J. Du!, G.W. Gibbons, P.K. Townsend, Macroscopic superstrings as interpolating solitons, Phys. Lett. B 332 (1994) 321}328. [232] M.J. Du!, P.S. Howe, T. Inami, K.S. Stelle, Superstrings in D"10 from supermembranes in D"11, Phys. Lett. B 191 (1987) 70}74. [233] M.J. Du!, R.R. Khuri, Four-dimensional string/string duality, Nucl. Phys. B 411 (1994) 473}486. [234] M.J. Du!, R.R. Khuri, J.X. Lu, String solitons, Phys. Rep. 259 (1995) 213}326. [235] M.J. Du!, J.T. Liu, R. Minasian, Eleven dimensional origin of string/string duality: a one loop test, Nucl. Phys. B 452 (1995) 261}282. [236] M.J. Du!, J.T. Liu, J. Rahmfeld, Four dimensional string/string/string triality, Nucl. Phys. B 459 (1996) 125}159. [237] M.J. Du!, J.T. Liu, J. Rahmfeld, Dipole moments of black holes and string states, Nucl. Phys. B 494 (1996) 161}199. [238] M.J. Du!, H. Lu, The selfdual type IIB superthreebrane, Phys. Lett. B 273 (1991) 409}414. [239] M.J. Du!, J.X. Lu, Remarks on string/"ve brane duality, Nucl. Phys. B 354 (1991) 129}140. [240] M.J. Du!, J.X. Lu, Elementary "ve brane solutions of D"10 supergravity, Nucl. Phys. B 354 (1991) 141}153.

D. Youm / Physics Reports 316 (1999) 1}232 [241] [242] [243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] [255] [256] [257] [258] [259] [260] [261] [262] [263] [264] [265] [266] [267] [268] [269] [270]

[271]

[272]

221

M.J. Du!, H. Lu, Type II p-branes: the brane scan revisited, Nucl. Phys. B 390 (1993) 276}290. M.J. Du!, H. Lu, Black and super p-branes in diverse dimensions, Nucl. Phys. B 416 (1994) 301}334. M.J. Du!, H. Lu, C.N. Pope, The black branes of M theory, Phys. Lett. B 382 (1996) 73}80. M.J. Du!, R. Minasian, Putting string/string duality to the test, Nucl. Phys. B 436 (1995) 507}528. M.J. Du!, R. Minasian, J. Rahmfeld, New black hole, string and membrane solutions of the four-dimensional heterotic string, Nucl. Phys. B 418 (1994) 195}205. M.J. Du!, B.E.W. Nilsson, Four-dimensional string theory from the K3 lattice, Phys. Lett. B 175 (1986) 417}422. M.J. Du!, B.E.W. Nilsson, C.N. Pope, Kaluza-Klein supergravity, Phys. Rep. 130 (1986) 1}142. M.J. Du!, J. Rahmfeld, Massive string states as extreme black holes, Phys. Lett. B 345 (1995) 441}447. M.J. Du!, J. Rahmfeld, Bound states of black holes and other p-branes, Nucl. Phys. B 481 (1996) 332}352. M.J. Du!, K.S. Stelle, Multimembrane solutions of D"11 supergravity, Phys. Lett. B 253 (1991) 113}118. J. Ehlers, in Les theories relativistes de la gravitation, CNRS, Paris, 1959, p. 275. S. Elitzur, E. Gross, E. Rabinovici, N. Seiberg, Aspects of bosonization in string theory, Nucl. Phys. B 283 (1987) 413. F. Englert, A. Neveu, Nonabelian compacti"cation of the interacting bosonic string, Phys. Lett. B 163 (1985) 349}352. S. Ferrara, G.W. Gibbons, R. Kallosh, Black holes and critical points in moduli space, Nucl. Phys. B 500 (1997) 75}93. S. Ferrara, L. Girardello, C. Kounnas, M. Porrati, E!ective lagrangians for four-dimensional superstrings, Phys. Lett. 192 (1987) 368}376. S. Ferrara, L. Girardello, M. Porrati, Spontaneous breaking of N"2 to N"1 in rigid and local supersymmetric theories, Phys. Lett. B 376 (1996) 275}281. S. Ferrara, J.A. Harvey, A. Strominger, C. Vafa, Second quantized mirror symmetry, Phys. Lett. B 361 (1995) 59}65. S. Ferrara, R. Kallosh, Supersymmetry and attractors, Phys. Rev. D 54 (1996) 1514}1524. S. Ferrara, R. Kallosh, Universality of sypersymmetric attractors, Phys. Rev. D 54 (1996) 1525}1534. S. Ferrara, R. Kallosh, A. Strominger, N"2 extremal black holes, Phys. Rev. D 52 (1995) 5412}5416. S. Ferrara, R.R. Khuri, R. Minasian, M-theory on a Calabi-Yau manifold, Phys. Lett. B 375 (1996) 81}88. S. Ferrara, R. Minasian, A. Sagnotti, Low-energy analysis of M and F theories on Calabi-Yau threefolds, Nucl. Phys. B 474 (1996) 323}342. S. Ferrara, M. Porrati, The manifolds of scalar background "elds in Z(N) orbifolds, Phys. Lett. B 216 (1989) 289}296. S. Ferrara, C.A. Savoy, B. Zumino, General massive multiplets in extended supersymmetry, Phys. Lett. B 100 (1981) 393}398. S. Ferrara, P. van Nieuwenhuizen, Consistent supergravity with complex spin 3/2 gauge "elds, Phys. Rev. Lett. 37 (1976) 1669}1671. S. Ferrara, A. van Proeyen, A theorem on N"2 special KaK hler product manifolds, Class. Quant. Grav. 6 (1989) L243}L247. R. Flume, From N"1 to N"4 extended supersymmetric Yang-Mills "elds in a covariant Wess-Zumino gauge, Nucl. Phys. B 217 (1983) 531}543. A. Font, L.E. Ibanez, D. Lust, F. Quevedo, Strong-weak coupling duality and nonperturbative e!ects in string theory, Phys. Lett. B 249 (1990) 35}43. E.S. Fradkin, A.A. Tseytlin, Nonlinear electrodynamics from quantized strings, Phys. Lett. B 163 (1985) 123}130. P. Fre, Lectures on special KaK hler geometry and electric-magnetic duality rotations, Lectures given at ICTP workshop on string theory, gauge theory and quantum gravity, Trieste, Italy, 5}7 April 1995, Nucl. Phys. Proc. Suppl. 45 BC (1996) 59}114. P. Fre, The complete form of N"2 supergravity and its place in the general framework of D"4 N extended supergravities, Presented at spring school and workshop on string theory, gauge theory and quantum gravity, Trieste, Italy, 18}29 March 1996, hep-th/9611182, 1996. P. Fre, P. Soriani, Symplectic embeddings, special KaK hler geometry and automorphic functions: the case of SK(N#1)"S;(1, 1)/;(1);SO(2, N)!SO(2);SO(N), Nucl. Phys. B 371 (1992) 659}682.

222

D. Youm / Physics Reports 316 (1999) 1}232

[273] P.G.O. Freund, Superstrings from twenty six dimensions? Phys. Lett. B 151 (1985) 387}390. [274] P.G.O. Freund, Introduction to Supersymmetry, Cambridge University Press, Cambridge, 1986. [275] D. Friedan, E. Martinec, S. Shenker, Conformal invariance, supersymmetry and string theory, Nucl. Phys. B 271 (1986) 93}165. [276] M.K. Gaillard, B. Zumino, Duality rotations for interacting "elds, Nucl. Phys. B 193 (1981) 221}244. [277] K. Galicki, A generalization of the momentum mapping construction for quaternionic KaK hler manifolds, Comm. Math. Phys. 108 (1987) 117}138. [278] D.V. Gal'tsov, O.V. Kechkin, Ehlers-Harrisson type transformations in dilaton-axion gravity, Phys. Rev. D 50 (1994) 7394}7399. [279] D. Gar"nkle, Traveling waves in strongly gravitating cosmic strings, Phys. Rev. D 41 (1990) 1112}1115. [280] D. Gar"nkle, Black string traveling wave, Phys. Rev. D 46 (1992) 4286}4288. [281] D. Gar"nkle, G. Horowitz, A. Strominger, Charged black holes in string theory, Phys. Rev. D 43 (1991) 3140}3143, erratum, ibid. D 45 (1992) 3888. [282] D. Gar"nkle, T. Vachaspati, Phys. Rev. 42 (1990) 1960}1963. [283] S.J. Gates, Superspace formulation of new nonlinear sigma models, Nucl. Phys. B 238 (1984) 349}366. [284] J.P. Gauntlett, Intersecting branes, Preprint QMW-PH-97-13, hep-th/9705011, 1997. [285] J.P. Gauntlett, G.W. Gibbons, G. Papadopoulos, P.K. Townsend, Hyper-KaK hler manifolds and multiply intersecting branes, Nucl. Phys. B 500 (1997) 133}162. [286] J.P. Gauntlett, J.A. Harvey, J.T. Liu, Magnetic monopoles in string theory, Nucl. Phys. B 409 (1993) 363}381. [287] J.P. Gauntlett, D.A. Kastor, J. Traschen, Overlapping branes in M theory, Nucl. Phys. B 478 (1996) 544}560. [288] J.P. Gauntlett, T. Tada, Entropy of four-dimensional rotating BPS dyons, Phys. Rev. D 55 (1997) 1707}1710. [289] D. Gepner, E. Witten, String theory on group manifolds, Nucl. Phys. B 278 (1986) 493}549. [290] R. Geroch, A method for generating solutions of Einstein's equations, J. Math. Phys. 12 (1971) 918}924. [291] A. Ghosh, P. Mitra, Understanding the area proposal for extremal black hole entropy, Phys. Rev. Lett. 78 (1997) 1858}1860. [292] G.W. Gibbons, Antigravitating black hole solitons with scalar hair in N"4 supergravity, Nucl. Phys. B 207 (1982) 337}349. [293] G.W. Gibbons, M.B. Green, M.J. Perry, Instantons and seven-branes in type IIB superstring theory, Phys. Lett. B 370 (1996) 37}44. [294] G.W. Gibbons, S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752}2756. [295] G.W. Gibbons, S.W. Hawking, G.T. Horowitz, M.J. Perry, Positive mass theorems for black holes, Comm. Math. Phys. 88 (1983) 295}308. [296] G.W. Gibbons, C.M. Hull, A Bogomolny bound for general relativity and solitons in N"2 supergravity, Phys. Lett. B 109 (1982) 190}194. [297] G.W. Gibbons, R. Kallosh, B. Kol, Moduli, scalar charges, and the "rst law of black hole thermodynamics, Phys. Rev. Lett. 77 (1996) 4992}4995. [298] G.W. Gibbons, K. Maeda, Black holes and membranes in higher-dimensional theories with dilaton "elds, Nucl. Phys. B 298 (1988) 741}775. [299] G.W. Gibbons, M.J. Perry, Soliton-supermultiplets and Kaluza-Klein theory, Nucl. Phys. B 248 (1984) 629}646. [300] G.W. Gibbons, P.K. Townsend, Vacuum Interpolation in supergravity via super p-branes, Phys. Rev. Lett. 71 (1993) 3754}3757. [301] G.W. Gibbons, D.L. Wiltshire, Black holes in Kaluza Klein theory, Ann. Phys. 167 (1986) 201}223, erratum, ibid. 176 (1987) 393. [302] S.B. Giddings, Comments on information loss and remnants, Phys. Rev. D 49 (1994) 4078}4088. [303] P. Ginsparg, Comment on toroidal compacti"cation of heterotic superstrings, Phys. Rev. D 35 (1987) 648}654. [304] A. Giveon, N. Malkin, E. Rabinovici, The Riemann surface in the target space and vice versa, Phys. Lett. B 220 (1989) 551}556. [305] A. Giveon, N. Malkin, E. Rabinovici, On discrete symmetries and fundamental domains of target space, Phys. Lett. B 238 (1990) 57}64. [306] A. Giveon, M. Porrati, E. Rabinovici, Target space duality in string theory, Phys. Rep. 244 (1994) 77}202.

D. Youm / Physics Reports 316 (1999) 1}232

223

[307] A. Giveon, E. Rabinovici, A.A. Tseytlin, Heterotic string solutions and coset conformal "eld theories, Nucl. Phys. B 409 (1993) 339}362. [308] A. Giveon, E. Rabinovici, G. Veneziano, Duality in string background space, Nucl. Phys. B 322 (1989) 167}184. [309] A. Giveon, M. Rocek, Generalized duality in curved string backgrounds, Nucl. Phys. B 380 (1992) 128}146. [310] M.B. Green, C.M. Hull, P.K. Townsend, D-brane Wess-Zumino actions, ¹-duality and the cosmological constant, Phys. Lett. B 382 (1996) 65}72. [311] M.B. Green, N.D. Lambert, G. Papadopoulos, P.K. Townsend, Dyonic p-branes from self-dual (p#1)-branes, Phys. Lett. B 384 (1996) 86}92. [312] M.B. Green, J.H. Schwarz, Supersymmetrical string theories, Phys. Lett. B 109 (1982) 444}448. [313] M.B. Green, J.H. Schwarz, Covariant description of superstrings, Phys. Lett. 136 (1984) 367}370. [314] M.B. Green, J.H. Schwarz, Anomaly cancellations in supersymmetric D"10 gauge theory require SO(32), Phys. Lett. B 149 (1984) 117}122. [315] M.B. Green, J.H. Schwarz, Properties of the covariant formulation of superstring theories, Nucl. Phys. B 243 (1984) 285}306. [316] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, Vol. 1 & Vol. 2, Cambridge University Press, Cambridge, 1987. [317] B. Greene, String theory on Calabi-Yau manifolds, June 1996, 150pp, Lectures given at Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality, Boulder, CO, 2}28 June 1996, Preprint CU-TP-812, hep-th/9702155, 1997. [318] B. Greene, D. Morrison, A. Strominger, Black hole condensation and the uni"cation of string vacua, Nucl. Phys. B 451 (1995) 109}120. [319] B. Greene, M.R. Plesser, Duality in Calabi-Yau moduli space, Nucl. Phys. B 338 (1990) 15}37. [320] B. Greene, M.R. Plesser, Mirror Manifolds: a brief review and progress report, Based in part on invited paper given at 2nd International Conference PASCOS'91, Boston, MA, March 25}30, 1991, Preprint CLNS-91-1109, hep-th/9110014. [321] M.T. Grisaru, W. Siegel, M. Rocek, Improved methods for supergraphs, Nucl. Phys. B 159 (1979) 429}450. [322] D.J. Gross, J.A. Harvey, E. Martinec, R. Rohm, The heterotic string, Phys. Rev. Lett. 54 (1985) 502}505. [323] D.J. Gross, J.A. Harvey, E. Martinec, R. Rohm, Heterotic string theory. (I). The free heterotic string, Nucl. Phys. B 256 (1985) 253}284. [324] D.J. Gross, J.A. Harvey, E. Martinec, R. Rohm, Heterotic string theory. (II). The interacting heterotic string, Nucl. Phys. B 267 (1985) 75}124. [325] D.J. Gross, M.J. Perry, Magnetic monopoles in Kaluza-Klein theories, Nucl. Phys. B 226 (1983) 29}48. [326] S.S. Gubser, A. Hashimoto, I.R. Klebanov, J.M. Maldacena, Gravitational lensing by p-branes, Nucl. Phys. B 472 (1996) 231}248. [327] R. Gueven, Black p-brane solutions of D"11 supergravity theory, Phys. Lett. B 276 (1992) 49}55. [328] M. GuK naydin, G. Sierra, P.K. Townsend, The geometry of N"2 Maxwell-Einstein supergravity and Jordan algebras, Nucl. Phys. B 242 (1984) 244}268. [329] M. GuK naydin, G. Sierra, P.K. Townsend, Gauging the D"5 Maxwell-Einstein supergravity theories: more on Jordan algebras, Nucl. Phys. B 253 (1985) 573}608. [330] R. Haag, J.T. Lopuszanski, M. Sohnius, All possible generators of supersymmetries of the S matrix, Nucl. Phys. B 88 (1975) 257}274. [331] R. Hagedorn, Statistical thermodynamics of strong interactions at high energies, Nuovo Cim. Suppl. 3 (1965) 147}186. [332] E. Halyo, Reissner-Nordstrom black holes and strings with rescaled tension, Preprint SU-ITP-42, hepth/9610068, 1996. [333] E. Halyo, Four-dimensional black holes and strings with rescaled tension, Preprint SU-ITP-96-51, hepth/9611175, 1996. [334] E. Halyo, B. Kol, A. Rajaraman, L. Susskind, Counting Schwarzschild and charged black holes, Phys. Lett. B 401 (1997) 15}20. [335] E. Halyo, B. Kol, A. Rajaraman, L. Susskind, Braneless black holes, Phys. Lett. B 392 (1997) 319}322. [336] B.K. Harrison, New solutions of the Einstein-Maxwell equations from old, J. Math. Phys. 9 (1968) 1744}1752.

224

D. Youm / Physics Reports 316 (1999) 1}232

[337] J.A. Harvey, J. Liu, Magnetic monopoles in N"4 supersymmetric low-energy superstring theory, Phys. Lett. B 268 (1991) 40}46. [338] J.A. Harvey, G. Moore, Algebras, BPS states, and strings, Nucl. Phys. B 463 (1996) 315}368. [339] J.A. Harvey, G. Moore, A. Strominger, Reducing S-duality to T-duality, Phys. Rev. D 52 (1995) 7161}7167. [340] J.A. Harvey, A. Strominger, The heterotic string is a soliton. Nucl. Phys. B 449 (1995) 535}552, erratum, ibid. B 458 (1996) 456}473. [341] S.F. Hassan, A. Sen, Twisting classical solutions in heterotic string theory, Nucl. Phys. B 375 (1992) 103}118. [342] S.W. Hawking, Mon. Not. Roy. Astron. Soc. 152 (1971) 75. [343] S.W. Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett. 26 (1971) 1344}1346. [344] S.W. Hawking, Black holes in the Brans-Dicke theory of gravitation, Commun. Math. Phys. 25 (1972) 152}171. [345] S.W. Hawking, Black hole explosions, Nature 248 (1974) 30}31. [346] S.W. Hawking, Particle creation by black holes, Comm. Math. Phys. 43 (1975) 199}220. [347] S.W. Hawking, Black holes and thermodynamics, Phys. Rev. D 13 (1976) 191}197. [348] S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460}2473. [349] S.W. Hawking, G.T. Horowitz, The gravitational Hamiltonian, action, entropy, and surface terms, Class. Quant. Grav. 13 (1996) 1487}1498. [350] S.W. Hawking, G.T. Horowitz, S.F. Ross, Entropy, area, and black hole pairs, Phys. Rev. D 51 (1995) 4302}4314. [351] M. Henneaux, L. Mezincescu, A sigma model interpretation of Green-Schwarz covariant superstring action, Phys. Lett. B 152 (1985) 340}342. [352] M. Henningson, G. Moore, Threshold corrections in K3;¹ heterotic string compacti"cations, Nucl. Phys. B 482 (1996) 187}212. [353] N.J. Hitchin, A. Karlhede, U. Lindstrom, M. Rocek, HyperkaK hler metrics and supersymmetry, Comm. Math. Phys. 108 (1987) 535}589. [354] C.F.E. Holzhey, F. Wilczek, Black holes as elementary particles, Nucl. Phys. B 380 (1992) 447}477. [355] P. Horava, Strings on world sheet orbifolds, Nucl. Phys. B 327 (1989) 461}484. [356] P. Horava, Background duality of open string models, Phys. Lett. B 231 (1989) 251}257. [357] P. Horava, E. Witten, Heterotic and type I string dynamics from eleven dimensions, Nucl. Phys. B 460 (1995) 506}524. [358] P. Horava, E. Witten, Eleven-dimensional supergravity on a manifold with boundary, Nucl. Phys. B 475 (1996) 94}114. [359] J.H. Horne, G.T. Horowitz, A.R. Steif, An equivalence between momentum and charge in string theory, Phys. Rev. Lett. 68 (1992) 568}571. [360] G.T. Horowitz, The origin of black hole entropy in string theory, Talk given at Paci"c Conference on Gravitation and Cosmology, Seoul, Korea, 1}6 February 1996, Preprint UCSBTH-96-07, gr-qc/9604051, 1996. [361] G.T. Horowitz, D.A. Lowe, J.M. Maldacena, Statistical entropy of nonextremal four-dimensional black holes and ;-duality, Phys. Rev. Lett. 77 (1996) 430}433. [362] G.T. Horowitz, J.M. Maldacena, A. Strominger, Nonextremal black hole microstates and ;-duality, Phys. Lett. B 383 (1996) 151}159. [363] G.T. Horowitz, D. Marolf, Counting states of black strings with traveling waves, Phys. Rev. D 55 (1997) 835}845. [364] G.T. Horowitz, D. Marolf, Counting states of black strings with traveling waves II, Phys. Rev. D 55 (1997) 846}852. [365] G.T. Horowitz, J. Polchinski, A correspondence principle for black holes and strings, Preprint NSF-ITP-96-144; hep-th/9612146, 1996. [366] G.T. Horowitz, A. Sen, Rotating black holes which saturate a Bogomol'nyi bound, Phys. Rev. D 53 (1996) 808}815. [367] G.T. Horowitz, A. Strominger, Black strings and p-branes, Nucl. Phys. B 360 (1991) 197}209. [368] G.T. Horowitz, A. Strominger, Counting states of near-extremal black holes, Phys. Rev. Lett. 77 (1996) 2368}2371. [369] G.T. Horowitz, A.A. Tseytlin, On exact solutions and singularities in string theory, Phys. Rev. D 50 (1994) 5204}5224. [370] G.T. Horowitz, A.A. Tseytlin, Extremal black holes as exact string solutions, Phys. Rev. Lett. 73 (1994) 3351}3354. [371] G.T. Horowitz, A.A. Tseytlin, A new class of exact solutions in string theory, Phys. Rev. D 51 (1995) 2896}2917.

D. Youm / Physics Reports 316 (1999) 1}232

225

[372] S. Hosono, A. Klemm, S. Theisen, Lectures on mirror symmetry, Extended version of lecture given at 3rd Baltic Student Seminar, Helsinki, Finland, 1993, Preprint HUTMP-94-01, hep-th/9403096, 1994. [373] S. Hosono, A. Klemm, S. Theisen, S.T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Comm. Math. Phys. 167 (1995) 301}350. [374] S. Hosono, B.H. Lion, S.T. Yau, GKZ generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces, Comm. Math. Phys. 182 (1996) 535}578. [375] P.S. Howe, K.S. Stelle, P.C. West, A class of "nite four-dimensional supersymmetric "eld theories, Phys. Lett. B 124 (1983) 55}63. [376] P.S. Howe, P.C. West, The complete N"2, D"10 supergravity, Nucl. Phys. B 238 (1984) 181}220. [377] J. Hughes, J. Liu, J. Polchinski, Supermembranes, Phys. Lett. B 180 (1986) 370}374. [378] J. Hughes, J. Polchinski, Partially broken global supersymmetry and the superstring, Nucl. Phys. B 278 (1986) 147}169. [379] C.M. Hull, The positivity of gravitational energy and global supersymmetry, Comm. Math. Phys. 90 (1983) 545}561. [380] C.M. Hull, String-string duality in ten-dimensions, Phys. Lett. B 357 (1995) 545}551. [381] C.M. Hull, P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109}137. [382] C.M. Hull, P.K. Townsend, Enhanced gauge symmetries in superstring theories, Nucl. Phys. B 451 (1995) 525}546. [383] M. Huq, M.A. Namazie, Kaluza-Klein supergravity in ten-dimensions, Class. Quant. Grav. 2 (1985) 293}308. [384] L. Ibanez, W. Lerche, D. Lust, S. Theisen, Some considerations about the stringy Higgs e!ect, Nucl. Phys. B 352 (1991) 435}450. [385] W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967) 1776}1779. [386] W. Israel, Event horizon in static electrovac space-times, Comm. Math. Phys. 8 (1968) 245}260. [387] W. Israel, Positivity of the Bondi gravitational mass, Phys. Lett. 85 A (1981) 259}260. [388] J.M. Izquierdo, N.D. Lambert, G. Papadopoulos, P.K. Townsend, Dyonic membranes, Nucl. Phys. B 460 (1996) 560}578. [389] D.P. Jatkar, S. Mukherji, S. Panda, Rotating dyonic black holes in heterotic string theory, Phys. Lett. B 384 (1996) 63}69. [390] D.P. Jatkar, S. Mukherji, S. Panda, Dyonic black hole in heterotic string theory, Nucl. Phys. B 484 (1997) 223}244. [391] C.V. Johnson, Introduction to D-branes, with applications, Nucl. Phys. Proc. Suppl. A 52 (1997) 326}334. [392] C.V. Johnson, R.R. Khuri, R.C. Myers, Entropy of 4D extremal black holes, Phys. Lett. B 378 (1996) 78}86. [393] C.V. Johnson, R.C. Myers, Taub-NUT dyons in heterotic string theory, Phys. Rev. D 50 (1994) 6512}6518. [394] D. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B 453 (1995) 281}302. [395] D. Kabat, S. Shenker, M.J. Strassler, Black hole entropy in the O(N) model, Phys. Rev. D 52 (1995) 7027}7036. [396] S. Kachru, A. Klemm, W. Lerche, P. Mayr, C. Vafa, Nonperturbative results on the point particle limit of N"2 heterotic string compacti"cations, Nucl. Phys. B 459 (1996) 537}558. [397] S. Kachru, C. Vafa, Exact results for N"2 compacti"cations of heterotic strings, Nucl. Phys. B 450 (1995) 69}89. [398] R. Kallosh, Supersymmetric black holes, Phys. Lett. B 282 (1992) 80}88. [399] R. Kallosh, Black hole multiplets and spontaneous breaking of local Supersymmetry, Phys. Rev. D 52 (1995) 1234}1241. [400] R. Kallosh, Duality symmetric quantization of superstring, Phys. Rev. D 52 (1995) 6020}6042. [401] R. Kallosh, Superpotential from black holes, Phys. Rev. D 54 (1996) 4709}4713. [402] R. Kallosh, Wrapped supermembrane, Preprint SU-ITP-96-56, hep-th/9612004, 1996. [403] R. Kallosh, Bound states of branes with minimal energy, Phys. Rev. D 55 (1997) 3241}3245. [404] R. Kallosh, D. Kastor, T. OrtiD n, T. Torma, Supersymmetry and stationary solutions in dilaton-axion gravity, Phys. Rev. D 50 (1994) 6374}6384. [405] R. Kallosh, B. Kol, E(7) symmetric area of the black hole horizon, Phys. Rev. D 5 (1996) 5344}5348. [406] R. Kallosh, A. Linde, Exact supersymmetric massive and massless white holes, Phys. Rev. D 52 (1995) 7137}7145. [407] R. Kallosh, A. Linde, Black hole superpartners and "xed scalars, Phys. Rev. D 56 (1997) 3509}3514. [408] R. Kallosh, A. Linde, Supersymmetric balance of forces and condensation of BPS states, Phys. Rev. D 53 (1996) 5734}5744.

226

D. Youm / Physics Reports 316 (1999) 1}232

[409] R. Kallosh, A. Linde, T. OrtiD n, A. Peet, A. van Proeyen, Supersymmetry as a cosmic censor, Phys. Rev. D 46 (1992) 5278}5302. [410] R. Kallosh, T. OrtiD n, Charge quantization of axion-dilaton black holes, Phys. Rev. D 48 (1993) 742}747. [411] R. Kallosh, T. OrtiD n, A. Peet, Entropy and action of dilaton black holes, Phys. Rev. D 47 (1993) 5400}5407. [412] R. Kallosh, A. Peet, Dilaton black holes near the horizon, Phys. Rev. D 46 (1992) 5223}5227. [413] R. Kallosh, A. Rajaraman, Brane-anti-brane democracy, Phys. Rev. D 54 (1996) 6381}6386. [414] R. Kallosh, A. Rajaraman, W. Wong, Supersymmetric rotating black holes and attractors, Phys. Rev. D 55 (1997) 3246}3249. [415] R. Kallosh, M. Shmakova, W. Wong, Freezing of moduli by N"2 dyons, Phys. Rev. D 54 (1996) 6284}6292. [416] T. Kaluza, On the unity problem of physics, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. K 1 (1921) 966}972. [417] D.M. Kaplan, D.A. Lowe, J.M. Maldacena, A. Strominger, Microscopic entropy of N"2 extremal black holes, Phys. Rev. D 55 (1997) 4898}4902. [418] V. Kaplunovsky, J. Louis, S. Theisen, Aspects of duality in N"2 string vacua, Phys. Lett. B 357 (1995) 71}75. [419] M. Karliner, I. Klebanov, L. Susskind, Size and shape of strings, Int. J. Mod. Phys. A 3 (1988) 1981}1996. [420] R. Khuri, Some instanton solutions in string theory, Phys. Lett. B 259 (1991) 261}266. [421] R. Khuri, A multimonopole solution in string theory, Phys. Lett. B 294 (1992) 325}330. [422] R. Khuri, A heterotic multimonopole solution, Nucl. Phys. B 387 (1992) 315}332. [423] R. Khuri, A comment on string solitons, Preprint CTP-TAMU-18-93, hep-th/9305143, 1993. [424] N. Khviengia, Z. Khviengia, H. Lu, C.N. Pope, Intersecting M-branes and bound states, Phys. Lett. B 388 (1996) 21}28. [425] K. Kikkawa, M. Yamasaki, Casimir e!ects in superstring theories, Phys. Lett. B 149 (1984) 357}360. [426] E.B. Kiritsis, Duality in gauged WZW models, Mod. Phys. Lett. A 6 (1991) 2871}2880. [427] E. Kiritsis, C. Kounnas, Infrared regularization of superstring theory and the one loop calculation of coupling constants, Nucl. Phys. B 442 (1995) 472}493. [428] E. Kiritsis, C. Kounnas, Infrared behavior of closed superstrings in strong magnetic and gravitational "elds, Nucl. Phys. B 456 (1995) 699}731. [429] E. Kiritsis, C. Kounnas, P.M. Petropoulos, J. Rizos, On the heterotic e!ective action at one loop gauge couplings and the gravitational sector, Preprint CERN-TH-96-091, hep-th/9605011, 1996. [430] E. Kiritsis, C. Kounnas, P.M. Petropoulos, J. Rizos, Universality properties of N"2 and N"1 heterotic threshold corrections, Nucl. Phys. B 483 (1997) 141}171. [431] I.R. Klebanov, A.A. Tseytlin, Entropy of near-extremal black p-branes, Nucl. Phys. B 475 (1996) 164}178. [432] I.R. Klebanov, A.A. Tseytlin, Intersecting M-branes as four-dimensional black holes, Nucl. Phys. B 475 (1996) 179}192. [433] I.R. Klebanov, A.A. Tseytlin, Near-extremal black hole entropy and #uctuating 3-branes, Nucl. Phys. B 479 (1996) 319}335. [434] O. Klein, Quantum theory and "ve-dimensional theory of relativity, Z. Phys. 37 (1926) 895}906. [435] A. Klemm, W. Lerche, P. Mayr, K3 "brations and heterotic type II string duality, Phys. Lett. B 357 (1995) 313}322. [436] A. Klemm, W. Lerche, S. Yankielowicz, S. Theisen, Simple singularities and N"2 supersymmetric Yang-Mills theory, Phys. Lett. B 344 (1995) 169}175. [437] V.G. Knizhnik, A.B. Zamolodchikov, Current algebra and Wess-Zumino model in two-dimensions, Nucl. Phys. B 247 (1984) 83}103. [438] T. Kugo, P. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B 221 (1983) 357}380. [439] T. Kugo, B. Zwiebach, Target space duality as a symmetry of string "eld theory, Prog. Theor. Phys. 87 (1992) 801}860. [440] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, 4th Edition, Pergamon Press, Oxford, 1989, Section 96. [441] F. Larsen, F. Wilczek, Internal structure of black holes, Phys. Lett. B 375 (1996) 37}42. [442] I.V. Lavrinenko, H. Lu, C.N. Pope, T.A. Tran, Harmonic superpositions of nonextremal p-branes, Preprint CTP-TAMU-9-97, hep-th/9702058, 1997.

D. Youm / Physics Reports 316 (1999) 1}232

227

[443] I.V. Lavrinenko, H. Lu, C.N. Pope, From topology to generalized dimensional reduction, Nucl. Phys. B 492 (1997) 278}300. [444] K. Lee, E.J. Weinberg, P. Yi, The moduli space of many BPS monopoles for arbitrary gauge groups, Phys. Rev. D 54 (1996) 1633}1643. [445] R.G. Leigh, Dirac-Born-Infeld action from Dirichlet p-model, Mod. Phys. Lett. A 4 (1989) 2767}2772. [446] H. Leutwyler, Arch. Sci. Gene`ve 13 (1960) 549. [447] T. Levi-Civita, R.C. Acad. Lincei 26 (1917) 519. [448] M. Li, Boundary states of D-branes and dy-strings, Nucl. Phys. B 460 (1996) 351}361. [449] P.M. Llatas, Electrically Charged Black-holes for the Heterotic String Compacti"ed on a (10!D)-Torus, Phys. Lett. B 397 (1997) 63}75. [450] J. Louis, Nonharmonic gauge coupling constants in supersymmetry and superstring theory, Talk presented at the 2nd International Symposium on Particles, Strings and Cosmology, Boston, MA, March 25}30, 1991, Preprint SLAC-PUB-5527. [451] J. Louis, J. Sonnenschein, S. Theisen, S. Yankielowicz, Nonperturbative properties of heterotic string vacua compacti"ed on K3;¹, Nucl. Phys. B 480 (1996) 185}212. [452] D.A. Lowe, J. Polchinski, L. Susskind, L. Thorlacius, J. Uglum, Black hole complementarity vs. locality, Phys. Rev. D 52 (1995) 6997}7010. [453] D.A. Lowe, A. Strominger, Exact four-dimensional dyonic black holes and Bertotti-Robinson space-times in string theory, Phys. Rev. Lett. 73 (1994) 1468}1471. [454] D.A. Lowe, L. Susskind, J. Uglum, Information spreading in interacting string "eld theory, Phys. Lett. B 327 (1994) 226}233. [455] H. Lu, S. Mukherji, C.N. Pope, From p-branes to cosmology, Preprint CTP-TAMU-65-96, hep-th/9612224, 1996. [456] H. Lu, C.N. Pope, p-brane solitons in maximal supergravities, Nucl. Phys. B 465 (1996) 127}156. [457] H. Lu, C.N. Pope, Multi-scalar p-brane solitons, Int. J. Mod. Phys. A 12 (1997) 437}450. [458] H. Lu, C.N. Pope, ¹-duality and ;-duality in toroidally compacti"ed strings, Preprint CTP-TAMU-06-97, hep-th/9701177, 1997. [459] H. Lu, C.N. Pope, E. Sezgin, K.S. Stelle, Stainless super p-branes, Nucl. Phys. B 456 (1995) 669}698. [460] H. Lu, C.N. Pope, K.S. Stelle, Weyl group invariance and p-brane multiplets, Nucl. Phys. B 476 (1996) 89}117. [461] H. Lu, C.N. Pope, K.S. Stelle, Vertical versus diagonal dimensional reduction for p-branes, Nucl. Phys. B 481 (1996) 313}331. [462] H. Lu, C.N. Pope, K.W. Xu, Liouville and Toda solutions of M theory, Mod. Phys. Lett. A 11 (1996) 1785}1796. [463] H. Lu, C.N. Pope, K.W. Xu, Black p-branes and their vertical dimensional reduction, Nucl. Phys. B 489 (1997) 264}278. [464] J.X. Lu, ADM masses for black strings and p-branes, Phys. Lett. B 313 (1993) 29}34. [465] J. Maharana, J.H. Schwarz, Noncompact symmetries in string theory, Nucl. Phys. B 390 (1993) 3}32. [466] D. Maison, Ehlers-Harrison type transformations for Jordan's extended theory of gravitation, Gen. Rel. Grav. 10 (1979) 717}723. [467] J.M. Maldacena, Statistical entropy of near extremal "ve-branes, Nucl. Phys. B 477 (1996) 168}174. [468] J.M. Maldacena, Black holes in string theory, Ph.D. Thesis, hep-th/9607235, 1996. [469] J.M. Maldacena, D-branes and near extremal black holes at low-energies, Phys. Rev. D 55 (1997) 7645}7650. [470] J.M. Maldacena, in preparation. [471] J.M. Maldacena, A. Strominger, Statistical entropy of four-dimensional extremal black holes, Phys. Rev. Lett. 77 (1996) 428}429. [472] J.M. Maldacena, L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475 (1996) 679}690. [473] S. Mandelstam, Light cone superspace and the ultraviolet "niteness of the N"4 model, Nucl. Phys. B 213 (1983) 149}168. [474] N. Marcus, A. Sagnotti, Tree level constraints on gauge groups for type-I superstrings, Phys. Lett. B 119 (1982) 97}99. [475] A. Mezhlumian, A. Peet, L. Thorlacious, String thermalization at a black hole horizon, Phys. Rev. D 50 (1994) 2725}2730. [476] C. Montonen, D. Olive, Magnetic monopoles as gauge particles? Phys. Lett. B 72 (1977) 117}120.

228 [477] [478] [479] [480] [481] [482] [483] [484] [485] [486] [487] [488] [489] [490] [491] [492] [493] [494] [495] [496] [497] [498] [499] [500] [501] [502] [503] [504] [505] [506] [507] [508] [509] [510] [511] [512] [513]

D. Youm / Physics Reports 316 (1999) 1}232 I. Moss, Black hole thermodynamics and quantum hair, Phys. Rev. Lett. 69 (1992) 1852}1855. R.C. Myers, M. Perry, Black holes in higher dimensional space-times, Ann. Phys. 172 (1986) 304}347. W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149}166. K. Narain, New heterotic string theories in uncompacti"ed dimensions (10, Phys. Lett. B 169 (1986) 41}46. K. Narain, H. Sarmadi, E. Witten, A note on toroidal compacti"cation of heterotic string theory, Nucl. Phys. B 279 (1987) 369}379. R.I. Nepomechie, Magnetic monopoles from antisymmetric tensor gauge "elds, Phys. Rev. D 31 (1985) 1921}1924. J. Nester, A new gravitational energy expression with a simple positivity proof, Phys. Lett. 83 A (1981) 241}242. S.P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37 (1982) 3}49. D. Olive, E. Witten, Supersymmetry algebras that include topological charges, Phys. Lett. B 78 (1978) 97}101. T. OrtiD n, Electric-magnetic duality and supersymmetry in stringy black holes, Phys. Rev. D 47 (1993) 3136}3143. T. OrtiD n, S¸(2, R) Duality covariance of Killing spinors in axion-dilaton black holes, Phys. Rev. D 51 (1995) 790}794. T. OrtiD n, Massless black holes as black diholes and quadruholes, Phys. Rev. Lett. 76 (1996) 3890}3893. H. Osborn, Topological charges for N"4 supersymmetric gauge theories and monopoles of spin 1, Phys. Lett. B 83 (1979) 321}326. D.N. Page, Classical stability of round and squashed seven spheres in eleven-dimensional supergravity, Phys. Rev. D 28 (1983) 2976}2982. G. Papadopoulos, P.K. Townsend, Compacti"cation of D"11 supergravity on spaces of exceptional holonomy, Phys. Lett. B 357 (1995) 300}306. G. Papadopoulos, P.K. Townsend, Intersecting M-branes, Phys. Lett. B 380 (1996) 273}279. G. Papadopoulos, P.K. Townsend, Kaluza-Klein on the brane, Phys. Lett. B 393 (1997) 59}64. T. Parker, C.H. Taubes, On Witten's proof of the positive energy theorem, Commun. Math. Phys. 84 (1982) 223}238. A. Peet, Entropy and supersymmetry of D dimensional extremal electric black holes versus string states, Nucl. Phys. B 456 (1995) 732}752. R. Penrose, R.M. Floyd, Extraction of rotational energy from a black hole, Nature (Phys. Sci.) (1971) 177}179. J. Polchinski, Combinatorics of boundaries in string theory, Phys. Rev. D 50 (1994) 6041}6045. J. Polchinski, Dirichlet-branes and Ramond-Ramond charges, Phys. Rev. Lett. 75 (1995) 4724}4727. J. Polchinski, Recent results in string duality, Prog. Theor. Phys. Suppl. 123 (1996) 9}18. J. Polchinski, String duality } a colloquium, Rev. Mod. Phys. 68 (1996) 1245}1258. J. Polchinski, TASI lectures on D-branes, Preprint NSF-ITP-96-145, hep-th/9611050, 1996. J. Polchinski, (1996) unpublished. J. Polchinski, Y. Cai, Consistency of open superstring theories, Nucl. Phys. B 296 (1988) 91}128. J. Polchinski, S. Chaudhuri, C.V. Johnson, Notes on D-branes, Preprint NSF-ITP-96-003, hep-th/9602052, 1996. J. Polchinski, E. Witten, Evidence for Heterotic-Type I String Duality, Nucl. Phys. B 460 (1996) 525}540. D. Pollard, Antigravity and Classical Solutions of Five-dimensional Kaluza-Klein theory, J. Phys. A 16 (1983) 565}574. A.M. Polyakov, Particle spectrum in the quantum "eld theory, JETP Lett. 20 (1974) 194}195. C.N. Pope, Fission and fusion bound states of p-brane solitons, Preprint CTP-TAMU-22-96, hep-th/9606047, 1996. G. Pradisi, A. Sagnotti, Open string orbifolds, Phys. Lett. B 216 (1989) 59}67. M.K. Prasad, C.M. Sommer"eld, An exact classical solution for the 't Hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett. 35 (1975) 760}762. J. Rahmfeld, Extremal black holes as bound states, Phys. Lett. B 372 (1996) 198}203. R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, North-Holland, Amsterdam, 1982. O. Reula, Existence theorem for solutions of Witten's equation and nonnegativity of total mass, J. Math. Phys. 23 (1982) 810}814.

D. Youm / Physics Reports 316 (1999) 1}232

229

[514] S.J. Rey, Classical and quantum aspects of BPS black holes in N"2, D"4 heterotic string compacti"cations, Preprint SLAC-PUB-7326, hep-th/9610157, 1996. [515] I. Robinson, A solution of the Maxwell-Einstein equations, Bull. Acad. Polon. Sci. SeH r. Sci. Math. Astr. Phys. 7 (1959) 351}352. [516] M. RocH ek, E. Verlinde, Duality, quotients, and currents, Nucl. Phys. B 373 (1992) 630}646. [517] J.G. Russo, L. Susskind, Asymptotic level density in heterotic string theory and rotating black holes, Nucl. Phys. B 437 (1995) 611}626. [518] J.G. Russo, A.A. Tseytlin, Exactly solvable string models of curved space-time backgrounds, Nucl. Phys. B 449 (1995) 91}145. [519] J.G. Russo, A.A. Tseytlin, Waves, boosted branes and BPS states in M-theory, Nucl. Phys. B 490 (1997) 121}144. [520] W.A. Sabra, General BPS black holes in "ve-dimensions, Preprint HUB-EP-97-47, hep-th/9708103, 1997. [521] W.A. Sabra, Black holes in N"2 supergravity theories and harmonic functions, Preprint HUB-EP-97-26, hep-th/9704147, 1997. [522] W.A. Sabra, General static N"2 black holes, Preprint HUB-EP-97-18, hep-th/9703101, 1997. [523] W.A. Sabra, Symplectic embeddings and special KaK hler geometry of CP(n!1, 1), Nucl. Phys. B 486 (1997) 629}649. [524] W.A. Sabra, Classical entropy of N"2 black holes: the minimal coupling case, Mod. Phys. Lett. A 12 (1997) 789}798. [525] S. Sakai, I. Senda, Vacuum energies of string compacti"ed on torus, Prog. Theor. Phys. 75 (1986) 692}705, erratum, ibid. 77 (1987) 773. [526] A. Salam, in: C.J. Isham, R. Penrose, D.W. Sciama (Eds.), Quantum Gravity; an Oxford Symposium, O.U.P, Oxford, 1975. [527] A. Salam, E. Sezgin, Supergravities in Diverse Dimensions, Vols. 1, 2, World Scienti"c, Singapore, 1989. [528] J. Scherk, J.H. Schwarz, Spontaneous breaking of supersymmetry through dimensional reduction, Phys. Lett. B 82 (1979) 60}64. [529] J. Scherk, J.H. Schwarz, How to get masses from extra dimensions, Nucl. Phys. B 153 (1979) 61}88. [530] C. Schmidhuber, D-brane actions, Nucl. Phys. B 467 (1996) 146}158. [531] R. Schoen, S.T. Yau, Proof of the positive action conjecture in quantum relativity, Phys. Rev. Lett. 42 (1979) 547}548. [532] R. Schoen, S.T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979) 45}76. [533] R. Schoen, S.T. Yau, Proof of the positive mass theorem II, Comm. Math. Phys. 79 (1982) 231}260. [534] R. Schoen, S.T. Yau, Proof that the Bondi mass is positive, Phys. Rev. Lett. 48 (1982) 369}371. [535] J.H. Schwarz, Superstring theory, Phys. Rep. 89 (1982) 223}322. [536] J.H. Schwarz, Covariant "eld equations of chiral N"2 D"10 supergravity, Nucl. Phys. B 226 (1983) 269}288. [537] J.H. Schwarz, Dilaton-axion symmetry, presented at International Workshop on String Theory, Quantum Gravity and the Uni"cation of Fundamental Interactions, Rome, Italy, 21}26 September 1992, Preprint CALT68-1815, hep-th/9209125. [538] J.H. Schwarz, An S¸(2, Z) multiplet of type IIB superstrings, Phys. Lett. B 360 (1996) 13}18, erratum, ibid. B 364 (1995) 252. [539] J.H. Schwarz, Superstring dualities, Nucl. Phys. Proc. Suppl. 49 (1996) 183}190. [540] J.H. Schwarz, The power of M theory, Phys. Lett. B 367 (1996) 97}103. [541] J.H. Schwarz, M theory extensions of ¹ duality, Presented at Workshop Frontiers in Quantum Field Theory, Toyonaka, Japan, 14}17 December 1995, Preprint CALT-68-2034, hep-th/9601077, 1996. [542] J.H. Schwarz, Self-dual superstring in six dimensions, Preprint CALT-68-2046, hep-th/9604171, 1996. [543] J.H. Schwarz, Lectures on superstring and M theory dualities, Lectures given at Spring School and Workshop on String Theory, Gauge Theory and Quantum Gravity, Trieste, Italy, 18}29 March 1996, and at the Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96), Fields, Strings, and Duality, Boulder, CO, 2}28 June 1996, Preprint CALT-68-2065, hep-th/9607201. [544] J.H. Schwarz, A. Sen, Duality symmetries of 4D heterotic strings, Phys. Lett. B 312 (1993) 105}114. [545] J.H. Schwarz, A. Sen, Duality symmetric actions, Nucl. Phys. B 411 (1994) 35}63.

230 [546] [547] [548] [549] [550] [551] [552] [553] [554] [555] [556] [557] [558] [559] [560] [561] [562] [563] [564] [565] [566] [567] [568] [569] [570] [571] [572] [573] [574] [575] [576] [577] [578] [579] [580] [581] [582] [583] [584] [585] [586]

D. Youm / Physics Reports 316 (1999) 1}232 J. Schwinger, Magnetic charge and quantum "eld theory, Phys. Rev. 144 (1966) 1087}1093. J. Schwinger, Sources and magnetic charge, Phys. Rev. 173 (1968) 1536}1544. N. Seiberg, Supersymmetry and nonperturbative beta functions, Phys. Lett. B 206 (1988) 75}80. N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D 49 (1994) 6857}6863. N. Seiberg, E. Witten, Electric-magnetic duality, monopole condensation, and con"nement in N"2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19}52, erratum, ibid. B 430 (1994) 485}486. N. Seiberg, E. Witten, Monopoles, duality and chiral symmetry breaking in N"2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484}550. A. Sen, O(D);O(D) symmetry of the space of cosmological solutions in string theory, scale factor duality and two-dimensional black holes, Phys. Lett. B 271 (1991) 295}300. A. Sen, Twisted black p-brane solutions in string theory, Phys. Lett. B 274 (1992) 34}40. A. Sen, Rotating charged black hole solution in heterotic string theory, Phys. Rev. Lett. 69 (1992) 1006}1009. A. Sen, Macroscopic charged heterotic string, Nucl. Phys. B 388 (1992) 457}473. A. Sen, Electric magnetic duality in string theory, Nucl. Phys. B 404 (1993) 109}126. A. Sen, Quantization of dyon charge and electric magnetic duality in string theory, Phys. Lett. B 303 (1993) 22}26. A. Sen, S¸(2, Z) duality and magnetically charged strings, Int. J. Mod. Phys. A 8 (1993) 5079}5094. A. Sen, Magnetic monopoles, Bogomolny bound and S¸(2, Z) invariance in string theory, Mod. Phys. Lett. A 8 (1993) 2023}2036. A. Sen, Strong-weak coupling duality in four-dimensional string theory, Int. J. Mod. Phys. A 9 (1994) 3707}3750. A. Sen, Strong-weak coupling duality in three-dimensional string theory, Nucl. Phys. B 434 (1995) 179}209. A. Sen, Black hole solutions in heterotic string theory on a torus, Nucl. Phys. B 440 (1995) 421}440. A. Sen, Duality symmetry group of two-dimensional heterotic string theory, Nucl. Phys. B 447 (1995) 62}84. A. Sen, Extremal black holes and elementary string states, Mod. Phys. Lett. A 10 (1995) 2081}2094. A. Sen, String string duality conjecture in six-dimensions and charged solitonic strings, Nucl. Phys. B 450 (1995) 103}114. A. Sen, A note on marginally stable bound states in type II string theory, Phys. Rev. D 54 (1996) 2964}2967. A. Sen, ;-duality and intersecting D-branes, Phys. Rev. D 53 (1996) 2874}2894. A. Sen, Orbifolds of M theory and string theory, Mod. Phys. Lett. A 11 (1996) 1339}1348. A. Shapere, S. Trivedi, F. Wilczek, Dual dilaton dyons, Mod. Phys. Lett. A 6 (1991) 2677}2686. A. Shapere, F. Wilczek, Selfdual models with theta terms, Nucl. Phys. B 320 (1989) 669}695. M. Shmakova, Calabi-Yau black holes, Phys. Rev. D 56 (1997) 540}544. G. Sierra, P.K. Townsend, An introduction to N"2 rigid supersymmetry, Preprint LPTENS-83-26, Lectures given at the 19th Karpacz Winter School on Theoretical Physics, Karpacz, Poland, 14}28 February, 1983. M.F. Sohnius, P.C. West, Conformal invariance in N"4 supersymmetric Yang-Mills theory, Phys. Lett. B 100 (1981) 245}250. R.D. Sorkin, Kaluza-Klein monopole, Phys. Rev. Lett. 51 (1983) 87}90. J. Strathdee, Extended Poincare supersymmetry, Int. J. Mod. Phys. A 2 (1987) 273}300. A. Strominger, Special geometry, Comm. Math. Phys. 133 (1990) 163}180. A. Strominger, Heterotic solitons, Nucl. Phys. B 343 (1990) 167}184, erratum, ibid. B 353 (1991) 565. A. Strominger, Massless black holes and conifolds in string theory, Nucl. Phys. B 451 (1995) 96}108. A. Strominger, Macroscopic entropy of N"2 extremal black holes, Phys. Lett. B 383 (1996) 39}43. A. Strominger, C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99}104. A. Strominger, E. Witten, New manifolds for superstring compacti"cation, Comm. Math. Phys. 101 (1985) 341}361. L. Susskind, String theory and the principle of black hole complementarity, Phys. Rev. Lett. 71 (1993) 2367}2368. L. Susskind, Some speculations about black hole entropy in string theory, Preprint RU-93-44, hep-th/9309145, 1993. L. Susskind, String, black holes and Lorentz contraction, Phys. Rev. D 49 (1994) 6606}6611. L. Susskind, Particle growth and BPS saturated states, Preprint SU-ITP-95-23, hep-th/9511116, 1995. L. Susskind, L. Thorlacius, Gedanken experiments involving black holes, Phys. Rev. D 49 (1994) 966}974.

D. Youm / Physics Reports 316 (1999) 1}232

231

[587] L. Susskind, L. Thorlacius, J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743}3761. [588] L. Susskind, J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700}2711. [589] L. Susskind, J. Uglum, String physics and black holes, Nucl. Phys. Proc. Suppl. 45 BC (1996) 115}134. [590] C. Teitelboim, Surface integrals as symmetry generators in supergravity theory, Phys. Lett. B 69 (1977) 240}244. [591] C. Teitelboim, Monopoles of higher rank, Phys. Lett. B 167 (1986) 69}72. [592] G. 't Hooft, Magnetic monopoles in uni"ed gauge theories, Nucl. Phys. B 79 (1974) 276}284. [593] G. 't Hooft, On the quantum structure of a black hole, Nucl. Phys. B 256 (1985) 727}745. [594] G. 't Hooft, The black hole interpretation of string theory, Nucl. Phys. B 335 (1990) 138}154. [595] G. 't Hooft, The black hole horizon as a quantum surface, Phys. Scripta T 36 (1991) 247}252. [596] K.S. Thorne, R.H. Price, D.A. MacDonald, Black Holes, The Membrane Paradigm, Yale University Press, New Haven, 1986. [597] K.P. Tod, All metrics admitting supercovariantly constant spinors, Phys. Lett. B 121 (1983) 241}244. [598] P.K. Townsend, A new anomaly free chiral supergravity theory from compacti"cation on K3, Phys. Lett. B 139 (1984) 283}287. [599] P.K. Townsend, Three lectures on supermembranes, in: M. Green, M. Grisaru, R. Iengo, E. Sezgin, A. Strominger (Eds.), Superstrings '88, World Scienti"c, Singapore, 1989. [600] P.K. Townsend, Three lectures on supersymmetry and extended objects, in: L.A. Ibort, M.A. Rodriguez (Eds.), Integrable Systems, Quantum Groups and Quantum Field Theory, Kluwer Academic, Dordrecht, 1993. [601] P.K. Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184}187. [602] P.K. Townsend, String-membrane duality in seven dimensions, Phys. Lett. B 354 (1995) 247}255. [603] P.K. Townsend, p-brane democracy, Preprint DAMTP-R-95-34, hep-th/9507048, 1995. [604] P.K. Townsend, D-branes from M-branes, Phys. Lett. B 373 (1996) 68}75. [605] P.K. Townsend, Four lectures on M-theory, Talk given at Summer School in High-energy Physics and Cosmology, Trieste, Italy, 10 June}26 July 1996, Preprint DAMTP-R-96-58, hep-th/9612121. [606] A.A. Tseytlin, String theory e!ective action: string loop corrections, Int. J. Mod. Phys. A 3 (1988) 365}395. [607] A.A. Tseytlin, Sigma model approach to string theory, Int. J. Mod. Phys. A 4 (1989) 1257}1318. [608] A.A. Tseytlin, Exact solutions of closed string theory, Class. Quant. Grav. 12 (1995) 2365}2410. [609] A.A Tseytlin, Extreme dyonic black holes in string theory, Mod. Phys. Lett. A 11 (1996) 689}714. [610] A.A. Tseytlin, Generalized chiral null models and rotating string backgrounds, Phys. Lett. B 381 (1996) 73}80. [611] A.A. Tseytlin, Harmonic superpositions of M-branes, Nucl. Phys. B 475 (1996) 149}163. [612] A.A. Tseytlin, Extremal black hole entropy from conformal string sigma model, Nucl. Phys. B 477 (1996) 431}448. [613] A.A. Tseytlin, No-force condition and BPS combinations of p-branes in 11 and 10 dimensions, Nucl. Phys. B 487 (1997) 141}154. [614] A.A. Tseytlin, On the structure of composite black p-brane con"gurations and related black holes, Phys. Lett. B 395 (1997) 24}27. [615] A.A. Tseytlin, Composite BPS con"gurations of p-branes in 10 and 11 dimensions, Class. Quant. Grav. 14 (1997) 2085}2105. [616] W.G. Unruh, Absorption cross section of small black holes, Phys. Rev. D (1976) 3251}3259. [617] T. Vachaspati, Gravitational e!ects of cosmic strings, Nucl. Phys. B 277 (1986) 593}604. [618] T. Vachaspati, Traveling waves on domain walls and cosmic strings, Phys. Lett. B 238 (1990) 41}44. [619] C. Vafa, Gas of D-Branes and Hagedorn density of BPS states, Nucl. Phys. B 463 (1996) 415}419. [620] C. Vafa, Instantons on D-branes, Nucl. Phys. B 463 (1996) 435}442. [621] C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403}418. [622] C. Vafa, Lectures on strings and dualities, Preprint HUTP-97-A009, hep-th/9702201, 1997. [623] C. Vafa, E. Witten, A strong coupling test of S-duality, Nucl. Phys. B 431 (1994) 3}77. [624] C. Vafa, E. Witten, A one-loop test of string duality, Nucl. Phys. B 447 (1995) 261}270. [625] E. Verlinde, Global aspects of electric-magnetic duality, Nucl. Phys. B 455 (1995) 211}228. [626] M. Visser, Dirty black holes: thermodynamics and horizon structure, Phys. Rev. D 46 (1992) 2445}2451. [627] M. Visser, Dirty black holes: entropy versus area, Phys. Rev. D 48 (1993) 583}591.

232

D. Youm / Physics Reports 316 (1999) 1}232

[628] C. Wetterich, Massless spinors in more than four dimensions, Nucl. Phys. B 211 (1983) 177}188. [629] C. Wetterich, Dimensional reduction of Weyl, Majorana and Majorana-Weyl spinors, Nucl. Phys. B 222 (1983) 20}44. [630] E. Witten, Dyons of charge eh/2n, Phys. Lett. B 86 (1979) 283}287. [631] E. Witten, A simple proof of the positive energy theorem, Comm. Math. Phys. 80 (1981) 381}402. [632] E. Witten, Nonabelian bosonization in two dimensions, Comm. Math. Phys. 92 (1984) 455}472. [633] E. Witten, Fermion quantum numbers in Kaluza-Klein theory, in: R. Jackiw, N. Khuri, S. Weinberg (Eds.), Quantum Field Theory and the Fundamental Problems in Physics, MIT Press, Cambridge, 1985. [634] E. Witten, Symmetry breaking patterns in superstring models, Nucl. Phys. B 258 (1985) 75}100. [635] E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85}126. [636] E. Witten, Some comments on string dynamics, Contributed to STRINGS 95: Future Perspectives in String Theory, Los Angeles, CA, 13}18 March 1995, Preprint IASSNS-HEP-95-63, hep-th/9507121. [637] E. Witten, Bound states of strings and p-branes, Nucl. Phys. B 460 (1996) 335}350. [638] E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541}559. [639] E. Witten, Five-branes and M theory on an orbifold, Nucl. Phys. B 463 (1996) 383}397. [640] T. Wu, C. Yang, Dirac's monopole without strings: classical Lagrangian theory, Phys. Rev. D 14 (1976) 437}445. [641] T. Wu, C. Yang, Dirac monopole without strings: monopole harmonics, Nucl. Phys. B 107 (1976) 365}380. [642] D. Zwanzinger, Exactly soluble nonrelativistic mode of particles with both electric and magnetic charges, Phys. Rev. 176 (1968) 1480}1488. [643] D. Zwanzinger, Quantum "eld theory of particles with both electric and magnetic charges, Phys. Rev. 176 (1968) 1489}1495.