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The Riemann surface in the target space and vice versa
Volume 220, number 4
PHYSICS LETTERS B
13 April 1989
T H E R I E M A N N SURFACE I N T H E T A R G E T SPACE A N D VICE VERSA ~ A. G I V E O N , N. M A L K I N and E. R A B I N O V I C I Racah Institute of Physics, Hebrew University, 91904Jerusalem, Israel
Received 20 January 1989
It is observed that the classical part of the partition function associated with the mappings from a genus-g Riemann surface Eg to an "almost complex" target space T2a is equal to that related to the mappings from Ed to T2g. The classical part related to the mappings from a genus-2g Riemann surface E2g,described by a "real" period matrix, to a target space Td is equal to the classical part related to mappings from E2dto Tg. Some physical consequencesof these mathematical identities are discusses.
One imagination capturing feature of string theory is that a string moving in a target space which contains one circular dimension cannot distinguish if the radius of that circle is R or 1/2R [ 1 ]. This duality generalizes to both bosonic and heterotic strings in any d-dimensional toroidal compactification [ 2 ]. For the closed oriented bosonic string it takes, geometrically, the following form. Given the pair (G, B), where G is a positive definite symmetric constant background metric, and B is an antisymmetric constant background, one defines the "parameter matrix" D = G + B . Under the duality transformation D is transformed to 1/D. Some mathematical properties and physical consequences of this transformation were discussed in ref. [ 3 ]. In ref. [ 4 ] a different type of transformation, termed triality, was found in the c = 2 case. It related a string projected from a certain world-sheet torus into a certain 2-dimensional toroidal background to a string projected from a different world-sheet torus into a different background. In this note it is shown that there exists a more general relation between pairs (Z, T), where E denotes the world-sheet and T denotes the target space. It is shown that the classical part o f the partition function describing pairs (Z, T) is related to the classical part of the pair (T, ~). The classical part of the pairs (Z, T) and (T, Z) is manifestly identical if the following conditions are satisfied. Either the target space is "alWork supported in part by the Israeli Academy of Science and by the BSF - American Israeli Bi-National Science Foundation. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
most complex" or the period matrix z describing the Riemann surface is "real". The terms "almost complex" and "real" in this context will be defined below. This symmetry relates the dimension of the target space of a string theory to the order of interaction, namely, the genus of the Riemann surface. Duality relations between spaces of different dimensions were discussed and utilized in the study of statistical mechanics systems [ 5-7 ]. In this letter we present the mathematical identities in an attempt to learn about the symmetries o f string theories in general, and the proposed phase of topological strings [8,9] in particular. We consider here only toroidal bosonic compactifications. The treatment o f superstring and orbifold-like compactifications will be studied separately. Let D be a 2 d × 2d "parameter matrix" [ 3 ] o f the form D=(_
or2 or,
cr2°'l),
(1)
where a~ and a2 are real symmetric d x d matrices, a2 is positive definite. The metric G and antisymmetric background B are, therefore,
Denote by I, the n-dimensional identity matrix, and by J, the standard 2n-dimensional antisymmetric matrix: 551
Volume 220, number 4
J"=
-I,,
PHYSICS LETTERS B
T':
a~a+b,
S':
a - - . - lla,
A':
a~mtam,
' , J~=-I2,.
Define the complex d × d matrix a
13 April 1989
b symmetric, integral,
m s G L ( d , 7/).
(7)
In the loop expansion of string theory one sums over all mappings from a genus-g Riemann surface Z~ to the target space T2d. In a canonical basis of cycles a;, b,, where i = 1..... g (see fig. 1 ), the Riemann surface is described by the period matrix r = r~ +it2. r~ and r2 are real symmetric g X g matrices, r2 is positive definite. The classical piece of the multi-loop path integral will be denoted by Zdas~(a, r). In what follows it is observed that
Invariance under T' reflects the fact that the B terms serve as 2-dimensional 0 parameters [11]. Invariance under S' is the duality symmetry of the closed string with 2d compactified dimensions. The action A' on the background a is equivalent to a change of the basis of the lattice defining the target-space toms. The intimate connection between duality and modular invariance, mentioned in ref. [3] is now better understood. The duality symmetry D ~ 1/D (a--, - 1/ a) is a simple consequence of the symmetry z~a. In fact it is another way to prove that duality is an exact symmetry to all orders in loop expansion. To prove eq. (5) we work in the notations of ref. [ 3 ]. The closed bosonic string action for the compactiffed dimensions is
Zclass(O~, r)~-- Zclass(r , 19") .
s= ~ f d2~r(.,fh h"~a,,. O.X"OpX~
a=aj +ia2.
(3)
The specific form ( 1 ) of D means that the target space is "almost complex" in the sense that D - - i a , i.e.,
G=(ReD)®I2,
B=(-ImD)®J.
(4)
(5)
The classical piece of the mapping from a Riemann surface Z~.(r) to a target space T2a(a) is equivalent to the mapping from a Riemann surface Ya(a) to a target space T2e(r). This result was observed by Dijkgraaf, Verlinde and Verlinde [4] in the d = 1, g = 1 case. The result (5) can be useful in understanding the properties of moduli space of string theory. For example, there is an isomorphism between Sp(2g, 2[) the group of modular transformations of the Riemann surface generated by [ 10 ] T:
r--,r+b,
S:
r~-
A:
z~m'rm,
1
+ e"~B;,, O,~X"OaX"),
where a, fl=0, 1;/t, u= 1..... 2d; a0, a~ are the worldsheet parameters; h uB is the world-sheet metric; h = d e t h , p ; e'~a is the standard 2-dimensional antisymmetric tensor. The compact space is a 2d-dimensional toms T2a= R Z a / n A 2< The lattice A 2a is spanned by the basis {e~} i= 1, ..., 2d. Thus X;' is identified with XU+nn;e~, where n; are integers. Define the metric G;j and the constant antisymmetric tensor B 0 by
G;,,, =2G~j(e*),(eT),,
b symmetric, integral,
B;,, =2Bo(e*);,(eT),,
l/z, m e G L ( g , 7/),
(6)
and the symmetry transformations on the target's moduli space generated by
a2
a3
ag
Fig. 1. The 2g canonical basis of cycles on a genus-g Riemann surface.
552
(9)
{e,* }, i= 1, ..., 2d, is the basis dual to {e;}. In the conformal gauge the action may be written as
S = 1 f D;j OX;~Xj 7~ d
al
(8)
( 1O)
where D = G + B ; X;-X;,(eT)~'; 0=0o+0~; 0= 0o-0~. The classical piece of the genus-g partition function gives a contribution [ 12-14 ]
Volume 220, number 4
PHYSICS LETTERS B P ~ = ( k " + m " D ) e *,
Zclass(G, B, r) = (det G) g/2
n, IH
+ mia('C2 + "~1Z21 "fl ).:,Goms:,
Zclass(O" , r ) =
-2m,,(ZlZS*),bG,jnjt,+2in,aBom/a]},
(11)
a, b = 1, ..., g; n and rn are 2dg-dimensional vectors with integer components. The normalization is such that Zc*ass= tr [ exp ( - z2H+ iz~P) ] in the one-loop case ( g = 1 ) ( H is the hamiltonian and P is the world-sheet momentum). In the path integral formulation one gets (det G) ~/2 integrating the zero modes Xo. It is the volume of the target space. For G and B of the form (2) Z~l~(a, z) = (det a2) g ~. ii11
,IH2.tl I .H2
exp{--n[nlr21a2nl-I-
P~=(U'-m'Dt)e
*.
(15)
Identifying (k~, k2, m~, m2) in the sum (13) with (nu //2, - m 2 , ml ) in the sum (12) we get
X ~ e x p { - n [ n g . ( z ~ -l)~t,Gonjb
X
13 April 1989
Zclass('t', o').
We recall that eq. (5) is proved for any element r of the Siegel upper-half-space (symmetric complex g × g matrices with positive definite imaginary part). Not all of them correspond to period matrices of Riemann surfaces. The fact that eq. ( 5 ) is correct for any t suggests that there might be a physical meaning to the partition function of a fiat target space for any t. The target-space structure in eq. (5) is restricted to a parameter matrix o f the form a, eqs. (2) and (4). However, one can work with a general d × d background D = G + B and a subset of the 2g-dimensional Siegel upper-half-space of the form
n2"t'2- I 02 n2
T| T2
"(1
J ,
+ m l (z2 + z~ r~-1 r~ )a2m~ +m2 (r2 +r~ r~- ~zj )a2rn2 - 2m ~r~ r~- I a2 n ~- 2m2 r~ z g- ~a2 n2 +2inl al m2 - 2 i n 2 at ml] } ,
(12)
where n~, nz, m,, m2 are dg-dimensional vectors of integer components. A Poisson resummation on m leads to
~ I~ll
exp{--7~[kl 02 1T2kl-l- k2 0"2- I T2 k2
,tH2,kl , k 2
+ m~ (0"2 + cr~ cry- I a~ )z2m~
+m2(~2 +~r~ a~-l~r~ )z2m2
(13)
After the resummation the partition function can be written in a more compact form: Z c ~ ( a , "r) = (det z2) a × ~, exp[in(P~za:,U~ --PR'fabPR)], ~~' nix
where
, D) .
(16)
Zclass(Og, "t'4a)=Zclass(O2d, ~'2g)
+ 2 m l 0"~ a~- ~"r2 k2 - 2m~ a~ cry- ~z2 k~ - 2 i k , "r~ m~ - 2 i k 2 ~1 m 2 ] } •
Zclass(O, T ) = Z d a s s ( T
The classical partition function of a fiat background described by D, and a period matrix r described by T, is the same as that of a background described by T on a Riemann surface described by D. If the target space is "almost complex" and the period matrix is "real" then there is a triality relation
Z,.us~(a, z) = (det ~'2) d X
where T2 is a g X g positive definite symmetric matrix and Tj is a g X g antisymmetric matrix. This subset is isomorphic to O(g, g ) / O ( g ) × O ( g ) . In this case r~ + it2 is equal to J ® ( T2 + T~ ), where J stands for i. ( T2 + T~ ) is real and in that sense r is "real". Define T = 7"2+ T ~one finds
=Zclass(D4g, ~a) ,
(17)
where r, and Db stand for the relevant a X a period matrix and b × b parameter matrix, respectively. It is generalized to a multiality symmetry if the period matrix and the parameter matrix have a higher "complex-real" structure. For example, let
(14) D2.a=y®I2.,
z2,.g=it®I2m,
where y(t) are d x d
(18)
( g × g ) positive-definite sym553
Volume 220, number 4
PHYSICS LETTERS B
metric matrices. Then there is an ( m + n + 1 )-iality symmetry
.... Z(t®I2 ...... ~®I2 .... ) =Z(7®I20, t®I2,.) =Z(t®12 ...... y®I: .... ) =Z(~®I2.+2, t®I2m-z) .....
(19)
The reduction in the number o f handles on the world-sheet gives birth to copies of the target-space 7, and vice versa. In the case m = 1, n = 0 or m = 0, n = 1 we get the two possibilities of duality symmetry mentioned before, eq. ( 5 ) and (16). If rn = n = 1 then we get the triality symmetry of eq. ( 17 ). Additional observations on the target-space-worldsheet couplings can be obtained by the following considerations. The exponent in eq. ( 11 ) can be rewritten as
(n-zm)tz~G(n-zm)
+ 2intBm
=Nt(H~ G+iH2B)N
(20)
where we group the n and m into a column vector N = ( mn )
(21,
and the matrices Ht and 112 are defined to be
H, = (
z~'
-z~'z'
- - T I "~9~ I
Z 2 "~- Z l Z 2
) l Zl
'
13 April 1989
P) of the compactified string theory (see section 5 o f ref. [3 ] ). Thus, a close relation between a genus-g Riemann surface and a 2g-dimensional toroidal compactification is suggested. It is interesting that the metric of the target space, G, "feels" the symmetric piece of the jacobian torus hermitian-form, H~. However, the antisymmetric background, B, "feels" only the antisymmetric piece, //2. The constant dilaton background 0 is coupled to the two-dimensional scalar curvature R ~2~. The 2D integration of R ~2) leads to g - 1 where g is the genus. There exists a functional of 0, leading to d - 2 (or d - 26 with the ghost sector), that is the beta function of the dilaton. The full set of constant parameters we are dealing with is (G, B, 0, z~, r2). The parameter 0 is "neutral" under the change of D with z. However, 0 might transform to some 0' in order to keep the overall factor correct (this is also true for the duality symmetry on target space, D--, 1/D, in higher genus). So far we have studied the classical piece of the multi-loop path integral. However, a clue to the symmetry a,---,z could be learned looking at the quantum piece o f a 2d-dimensional string's target space on the genus-g Riemann surface described in fig. 2. The intermediate states connecting the genus- 1 surfaces are the identity operators (very thin and long tubes). Each torus is parametrized by the same modular parameter to, i.e., r=diag(zo, ..., Zo), where roeC, Im r > 0. In this case the quantum piece of the path integral is factorized to a product over the g identical tori: Zquantum(O', diag(ro, ..., Zo) )
The complex matrix H=HI+iH2 is the natural hermitian form o f the jacobian t o m s [ 10]. A Riemann surface with g handles is described by a g-dimensional complex torus (called the jacobian torus) J ~=Cg/A,
A=Z.~NzZ g
(23)
A is the lattice with vectors nR = n--Tin, where n, m are g-vectors with integer components. The hermitian form ~R Z~- ~7~R ( 7~R = n -- "~m ) is related to H as ~Rr~IgR
=N'HN.
(25)
where q = e x p ( 2 i m o ) and q(q) is the Dedekind eta function. The appearance of (det or2) -g is due to the normalization we choose for the classical partition function in eq. (11). The product Z=ZcZq is the Polyakov path-integrand [ 15] divided by the volume of the target space. Under the change tr,--, z for the background a = d i a g ( z o ..... Zo), the genus is
(24)
The vectors ~zR ( 7tL = n + rm) are "equivalent" to the " m o m e n t a " PR (PL) ofeq. ( 15 ) (z is replaced by D and J stands for i). Changing z to tr in eq. (22), H~ is nothing but the metric on the phase space Z = (X', 554
= Irl(q) l-adg(det Z2)-d(det 0"2)-g ,
To
To
To
Fig. 2. A genus-g Riemann surface described by the period matrix r=diag(ro ..... to).
Volume 220, number 4
PHYSICS LETTERSB
changed to d, and the target space becomes 2g-dimensional. Thus, Zquantum is unchanged. In string theory multi-loop expansion, the partition function is integrated over 3 g - 3 complex modular parameters (for g~> 2, and one modular parameter for g = 1 ). On general surfaces the quantum piece is much more complicated. However, one can learn the modular properties of Zqua, (tr, z) knowing the modular properties of Zquan(r, a). Trying to leave the protected shelter of mathematical identities and venture into the realm of their physical applications we wish to focus on three clues, each of which leads us to the context of topological string theories. The first observation is that the identities we have obtained concern only the classical part of the partition function. We do not have enough information to consider the quantum part. The relations between different pairs of target spaces and world-sheets in the absence of quantum oscillations have a natural setting in the topological string theories [8,16-19]. These theories consist of a string moving in a very special background and possess a very large gauge invariance which eliminates all quantum oscillations from the spectrum. If the theory includes in its definition an integral over all worldsheet metrics, the topological gravity term should be added as well to the action. We do not know if given a set of primary fields, one can find a topological field theory whose partition function would coincide with that of the classical partition function based only on the ground states of each primary field. It is in such a theory that the mathematical identities would have a direct physical implication. The second clue concerns the fact that all the identities we have derived so far involve either a "complex" target space or a "real" period matrix. The third clue is that the duality relations obtained suggest, at least for fiat target spaces, a generalization from world-sheets described by allowed period matrices to objects described by any period matrix. There are indications that string theories possess another phase and it was suggested to relate that phase to topological string models. In such a phase the concept of a world-sheet may require a generalization [9 ]. The one afforded by the duality relation of eq. (5), could be an indication as to the direction of the desired generalization. In a topological field theory the short distance re-
13 April 1989
strictions on the value of the Virasoro central charge of the matter sector are removed. The Weyl anomaly does not dictate the allowed dimensions of the target space. If the identities found have a physical significance, they show that there are string theories in which the string cannot distinguish between specific pairs of world-sheets and target spaces. The target space background and dimension are intermingled with the precise form of the interaction the string undergoes. Indeed they both seem to have a common origin and become distinguished only in the broken phase. We find this idea so attractive that we consider it worthwhile presenting the mathematical relations, even though they are attached to a long string of physical conditionals. We have started this note by stressing the "stringy" properties of the duality transformation on the background fields. We have also found that three symmetry properties of the almost complex target spaces, eq. (7), have three symmetry properties on the worldsheet, eq. (6), as their counter parts. All the worldsheet symmetries and two of the target space ones are accepted as natural 2D field theoretical symmetries, while the duality symmetry is singled out as a "stringy" effect. Does there exist a framework in which all these symmetries are on an equal footing? To answer that consider the following example. Imagine that one was given the 2D gravitational field theory in a gauge fixed form. One would then marvel at the various discrete symmetries, eq. (6), the system has. Only after realizing that the system is a gauge fixed form of a general coordinate invariant system, would one realize that the discrete symmetries are a reflection of the global structure of that gauge group. One could have guessed, given eq. (6), that the theory describes world-sheets which are Riemann surfaces. In an analogous manner, all the symmetries apperent in eqs. ( 5 ) - ( 7 ), and ( 16 ) could be viewed, in a second quantized theory, as the global part of some much larger symmetry which is gauge fixed. For that to be true in such a, yet to be discovered, second quantized theory, a functional integral on the background fields should be performed as well. We would like to thank D. Kazhdan for a very supportive guidance, and L. Baulieu and S. Elitzur for discussions. We also thank R. Dijkgraaf for pointing out the results of ref. [ 4 ]. After completion of this 555
Volume 220, number 4
PHYSICS LETTERS B
w o r k we r e c e i v e d a p a p e r by A. S h a p e r e a n d F. W i l c z e k [ 20 ] w h o s e results o v e r l a p w i t h s o m e o f the results o f ref. [ 3 ] a n d the p r e s e n t paper.
References [ 1] See e.g.N. Sakai and I. Senda, Prog. Theor. Phys. 75 (1986) 357. [ 2 ] V.P. Nair, A. Shapere, A. Strominger and F. Wilczek, Nucl. Phys. B 279 (1987) 369. [3] A. Giveon, E. Rabinovici and G. Veneziano, Duality in string background space, preprint CERN-TH.5106/88 (July 1988). [4] R. Dijkgraaf, E. Verlinde and H. Verlinde, in: Perspectives of string theory., eds. P. Di Vecchia and J.L. Petersen (World Scientific, Singapore, 1988 ). [ 5 ] D.J. Amit, S. Elitzur, E. Rabinovici and R. Savit, Nucl. Phys. B 210 [FS6] (1982) 69. [6] R. Savit, Nucl. Phys. B 200 (1982) 233. [7] E. Domany, Phys. Rev. Lett. 52 (1984) 871. [8] E. Witten, Commun. Math. Phys. 118 (198) 411. [9] J.J. Atick and E. Witten, The Hagedorn transition and the number of degrees of freedom of string theory, present IASSNS-HEP-88/14 (April 1988); O. Alvarez and E. Rabinovici, unpublished.
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[ 10] D. Mumford, Tata lectures on theta (Birkh~iuser, Basel, 1983). [ 11 ] K.S. Narain, M.H. Sarmadi and E. Witten, Nucl. Phys. B 279 (1987) 369. [ 12 ] L. Alvarez-Gaum6, G. Moore and C. Vafa, Commun. Math. Phys. 106 (1986) 1. [ 13] L. Alvarez-Gaum6, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Commun. Math. Phys. 112 (1987) 503. [ 14] P. Ginsparg and C. Vafa, Nucl. Phys. B 289 (1987) 414. [ 15] A.M. Polyakov, Phys. Lett. B 103 ( 1981 ) 207. [16] D. Montano and J. Sonnenschein, Topological strings, preprint SLAC-PUB-4664 (June 1988). [17] L. Baulieu and I.M. Singer, The topological sigma model, preprint. [ 18] J.M.F. Labastida, M. Pernici and E. Witten, Topological gravity in two dimensions, preprint IASSNS-HEP-88/29 (June 1988). [ 19 ] E. Witten, Phys. Rev. lett. 61 (1988) 670. [20] A. Shapere and F. Wilzcek, Self-dual models with theta terms, preprint IASSNS-HEP-88/36 (September 1988 ).
PHYSICS LETTERS B
13 April 1989
T H E R I E M A N N SURFACE I N T H E T A R G E T SPACE A N D VICE VERSA ~ A. G I V E O N , N. M A L K I N and E. R A B I N O V I C I Racah Institute of Physics, Hebrew University, 91904Jerusalem, Israel
Received 20 January 1989
It is observed that the classical part of the partition function associated with the mappings from a genus-g Riemann surface Eg to an "almost complex" target space T2a is equal to that related to the mappings from Ed to T2g. The classical part related to the mappings from a genus-2g Riemann surface E2g,described by a "real" period matrix, to a target space Td is equal to the classical part related to mappings from E2dto Tg. Some physical consequencesof these mathematical identities are discusses.
One imagination capturing feature of string theory is that a string moving in a target space which contains one circular dimension cannot distinguish if the radius of that circle is R or 1/2R [ 1 ]. This duality generalizes to both bosonic and heterotic strings in any d-dimensional toroidal compactification [ 2 ]. For the closed oriented bosonic string it takes, geometrically, the following form. Given the pair (G, B), where G is a positive definite symmetric constant background metric, and B is an antisymmetric constant background, one defines the "parameter matrix" D = G + B . Under the duality transformation D is transformed to 1/D. Some mathematical properties and physical consequences of this transformation were discussed in ref. [ 3 ]. In ref. [ 4 ] a different type of transformation, termed triality, was found in the c = 2 case. It related a string projected from a certain world-sheet torus into a certain 2-dimensional toroidal background to a string projected from a different world-sheet torus into a different background. In this note it is shown that there exists a more general relation between pairs (Z, T), where E denotes the world-sheet and T denotes the target space. It is shown that the classical part o f the partition function describing pairs (Z, T) is related to the classical part of the pair (T, ~). The classical part of the pairs (Z, T) and (T, Z) is manifestly identical if the following conditions are satisfied. Either the target space is "alWork supported in part by the Israeli Academy of Science and by the BSF - American Israeli Bi-National Science Foundation. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
most complex" or the period matrix z describing the Riemann surface is "real". The terms "almost complex" and "real" in this context will be defined below. This symmetry relates the dimension of the target space of a string theory to the order of interaction, namely, the genus of the Riemann surface. Duality relations between spaces of different dimensions were discussed and utilized in the study of statistical mechanics systems [ 5-7 ]. In this letter we present the mathematical identities in an attempt to learn about the symmetries o f string theories in general, and the proposed phase of topological strings [8,9] in particular. We consider here only toroidal bosonic compactifications. The treatment o f superstring and orbifold-like compactifications will be studied separately. Let D be a 2 d × 2d "parameter matrix" [ 3 ] o f the form D=(_
or2 or,
cr2°'l),
(1)
where a~ and a2 are real symmetric d x d matrices, a2 is positive definite. The metric G and antisymmetric background B are, therefore,
Denote by I, the n-dimensional identity matrix, and by J, the standard 2n-dimensional antisymmetric matrix: 551
Volume 220, number 4
J"=
-I,,
PHYSICS LETTERS B
T':
a~a+b,
S':
a - - . - lla,
A':
a~mtam,
' , J~=-I2,.
Define the complex d × d matrix a
13 April 1989
b symmetric, integral,
m s G L ( d , 7/).
(7)
In the loop expansion of string theory one sums over all mappings from a genus-g Riemann surface Z~ to the target space T2d. In a canonical basis of cycles a;, b,, where i = 1..... g (see fig. 1 ), the Riemann surface is described by the period matrix r = r~ +it2. r~ and r2 are real symmetric g X g matrices, r2 is positive definite. The classical piece of the multi-loop path integral will be denoted by Zdas~(a, r). In what follows it is observed that
Invariance under T' reflects the fact that the B terms serve as 2-dimensional 0 parameters [11]. Invariance under S' is the duality symmetry of the closed string with 2d compactified dimensions. The action A' on the background a is equivalent to a change of the basis of the lattice defining the target-space toms. The intimate connection between duality and modular invariance, mentioned in ref. [3] is now better understood. The duality symmetry D ~ 1/D (a--, - 1/ a) is a simple consequence of the symmetry z~a. In fact it is another way to prove that duality is an exact symmetry to all orders in loop expansion. To prove eq. (5) we work in the notations of ref. [ 3 ]. The closed bosonic string action for the compactiffed dimensions is
Zclass(O~, r)~-- Zclass(r , 19") .
s= ~ f d2~r(.,fh h"~a,,. O.X"OpX~
a=aj +ia2.
(3)
The specific form ( 1 ) of D means that the target space is "almost complex" in the sense that D - - i a , i.e.,
G=(ReD)®I2,
B=(-ImD)®J.
(4)
(5)
The classical piece of the mapping from a Riemann surface Z~.(r) to a target space T2a(a) is equivalent to the mapping from a Riemann surface Ya(a) to a target space T2e(r). This result was observed by Dijkgraaf, Verlinde and Verlinde [4] in the d = 1, g = 1 case. The result (5) can be useful in understanding the properties of moduli space of string theory. For example, there is an isomorphism between Sp(2g, 2[) the group of modular transformations of the Riemann surface generated by [ 10 ] T:
r--,r+b,
S:
r~-
A:
z~m'rm,
1
+ e"~B;,, O,~X"OaX"),
where a, fl=0, 1;/t, u= 1..... 2d; a0, a~ are the worldsheet parameters; h uB is the world-sheet metric; h = d e t h , p ; e'~a is the standard 2-dimensional antisymmetric tensor. The compact space is a 2d-dimensional toms T2a= R Z a / n A 2< The lattice A 2a is spanned by the basis {e~} i= 1, ..., 2d. Thus X;' is identified with XU+nn;e~, where n; are integers. Define the metric G;j and the constant antisymmetric tensor B 0 by
G;,,, =2G~j(e*),(eT),,
b symmetric, integral,
B;,, =2Bo(e*);,(eT),,
l/z, m e G L ( g , 7/),
(6)
and the symmetry transformations on the target's moduli space generated by
a2
a3
ag
Fig. 1. The 2g canonical basis of cycles on a genus-g Riemann surface.
552
(9)
{e,* }, i= 1, ..., 2d, is the basis dual to {e;}. In the conformal gauge the action may be written as
S = 1 f D;j OX;~Xj 7~ d
al
(8)
( 1O)
where D = G + B ; X;-X;,(eT)~'; 0=0o+0~; 0= 0o-0~. The classical piece of the genus-g partition function gives a contribution [ 12-14 ]
Volume 220, number 4
PHYSICS LETTERS B P ~ = ( k " + m " D ) e *,
Zclass(G, B, r) = (det G) g/2
n, IH
+ mia('C2 + "~1Z21 "fl ).:,Goms:,
Zclass(O" , r ) =
-2m,,(ZlZS*),bG,jnjt,+2in,aBom/a]},
(11)
a, b = 1, ..., g; n and rn are 2dg-dimensional vectors with integer components. The normalization is such that Zc*ass= tr [ exp ( - z2H+ iz~P) ] in the one-loop case ( g = 1 ) ( H is the hamiltonian and P is the world-sheet momentum). In the path integral formulation one gets (det G) ~/2 integrating the zero modes Xo. It is the volume of the target space. For G and B of the form (2) Z~l~(a, z) = (det a2) g ~. ii11
,IH2.tl I .H2
exp{--n[nlr21a2nl-I-
P~=(U'-m'Dt)e
*.
(15)
Identifying (k~, k2, m~, m2) in the sum (13) with (nu //2, - m 2 , ml ) in the sum (12) we get
X ~ e x p { - n [ n g . ( z ~ -l)~t,Gonjb
X
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Zclass('t', o').
We recall that eq. (5) is proved for any element r of the Siegel upper-half-space (symmetric complex g × g matrices with positive definite imaginary part). Not all of them correspond to period matrices of Riemann surfaces. The fact that eq. ( 5 ) is correct for any t suggests that there might be a physical meaning to the partition function of a fiat target space for any t. The target-space structure in eq. (5) is restricted to a parameter matrix o f the form a, eqs. (2) and (4). However, one can work with a general d × d background D = G + B and a subset of the 2g-dimensional Siegel upper-half-space of the form
n2"t'2- I 02 n2
T| T2
"(1
J ,
+ m l (z2 + z~ r~-1 r~ )a2m~ +m2 (r2 +r~ r~- ~zj )a2rn2 - 2m ~r~ r~- I a2 n ~- 2m2 r~ z g- ~a2 n2 +2inl al m2 - 2 i n 2 at ml] } ,
(12)
where n~, nz, m,, m2 are dg-dimensional vectors of integer components. A Poisson resummation on m leads to
~ I~ll
exp{--7~[kl 02 1T2kl-l- k2 0"2- I T2 k2
,tH2,kl , k 2
+ m~ (0"2 + cr~ cry- I a~ )z2m~
+m2(~2 +~r~ a~-l~r~ )z2m2
(13)
After the resummation the partition function can be written in a more compact form: Z c ~ ( a , "r) = (det z2) a × ~, exp[in(P~za:,U~ --PR'fabPR)], ~~' nix
where
, D) .
(16)
Zclass(Og, "t'4a)=Zclass(O2d, ~'2g)
+ 2 m l 0"~ a~- ~"r2 k2 - 2m~ a~ cry- ~z2 k~ - 2 i k , "r~ m~ - 2 i k 2 ~1 m 2 ] } •
Zclass(O, T ) = Z d a s s ( T
The classical partition function of a fiat background described by D, and a period matrix r described by T, is the same as that of a background described by T on a Riemann surface described by D. If the target space is "almost complex" and the period matrix is "real" then there is a triality relation
Z,.us~(a, z) = (det ~'2) d X
where T2 is a g X g positive definite symmetric matrix and Tj is a g X g antisymmetric matrix. This subset is isomorphic to O(g, g ) / O ( g ) × O ( g ) . In this case r~ + it2 is equal to J ® ( T2 + T~ ), where J stands for i. ( T2 + T~ ) is real and in that sense r is "real". Define T = 7"2+ T ~one finds
=Zclass(D4g, ~a) ,
(17)
where r, and Db stand for the relevant a X a period matrix and b × b parameter matrix, respectively. It is generalized to a multiality symmetry if the period matrix and the parameter matrix have a higher "complex-real" structure. For example, let
(14) D2.a=y®I2.,
z2,.g=it®I2m,
where y(t) are d x d
(18)
( g × g ) positive-definite sym553
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metric matrices. Then there is an ( m + n + 1 )-iality symmetry
.... Z(t®I2 ...... ~®I2 .... ) =Z(7®I20, t®I2,.) =Z(t®12 ...... y®I: .... ) =Z(~®I2.+2, t®I2m-z) .....
(19)
The reduction in the number o f handles on the world-sheet gives birth to copies of the target-space 7, and vice versa. In the case m = 1, n = 0 or m = 0, n = 1 we get the two possibilities of duality symmetry mentioned before, eq. ( 5 ) and (16). If rn = n = 1 then we get the triality symmetry of eq. ( 17 ). Additional observations on the target-space-worldsheet couplings can be obtained by the following considerations. The exponent in eq. ( 11 ) can be rewritten as
(n-zm)tz~G(n-zm)
+ 2intBm
=Nt(H~ G+iH2B)N
(20)
where we group the n and m into a column vector N = ( mn )
(21,
and the matrices Ht and 112 are defined to be
H, = (
z~'
-z~'z'
- - T I "~9~ I
Z 2 "~- Z l Z 2
) l Zl
'
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P) of the compactified string theory (see section 5 o f ref. [3 ] ). Thus, a close relation between a genus-g Riemann surface and a 2g-dimensional toroidal compactification is suggested. It is interesting that the metric of the target space, G, "feels" the symmetric piece of the jacobian torus hermitian-form, H~. However, the antisymmetric background, B, "feels" only the antisymmetric piece, //2. The constant dilaton background 0 is coupled to the two-dimensional scalar curvature R ~2~. The 2D integration of R ~2) leads to g - 1 where g is the genus. There exists a functional of 0, leading to d - 2 (or d - 26 with the ghost sector), that is the beta function of the dilaton. The full set of constant parameters we are dealing with is (G, B, 0, z~, r2). The parameter 0 is "neutral" under the change of D with z. However, 0 might transform to some 0' in order to keep the overall factor correct (this is also true for the duality symmetry on target space, D--, 1/D, in higher genus). So far we have studied the classical piece of the multi-loop path integral. However, a clue to the symmetry a,---,z could be learned looking at the quantum piece o f a 2d-dimensional string's target space on the genus-g Riemann surface described in fig. 2. The intermediate states connecting the genus- 1 surfaces are the identity operators (very thin and long tubes). Each torus is parametrized by the same modular parameter to, i.e., r=diag(zo, ..., Zo), where roeC, Im r > 0. In this case the quantum piece of the path integral is factorized to a product over the g identical tori: Zquantum(O', diag(ro, ..., Zo) )
The complex matrix H=HI+iH2 is the natural hermitian form o f the jacobian t o m s [ 10]. A Riemann surface with g handles is described by a g-dimensional complex torus (called the jacobian torus) J ~=Cg/A,
A=Z.~NzZ g
(23)
A is the lattice with vectors nR = n--Tin, where n, m are g-vectors with integer components. The hermitian form ~R Z~- ~7~R ( 7~R = n -- "~m ) is related to H as ~Rr~IgR
=N'HN.
(25)
where q = e x p ( 2 i m o ) and q(q) is the Dedekind eta function. The appearance of (det or2) -g is due to the normalization we choose for the classical partition function in eq. (11). The product Z=ZcZq is the Polyakov path-integrand [ 15] divided by the volume of the target space. Under the change tr,--, z for the background a = d i a g ( z o ..... Zo), the genus is
(24)
The vectors ~zR ( 7tL = n + rm) are "equivalent" to the " m o m e n t a " PR (PL) ofeq. ( 15 ) (z is replaced by D and J stands for i). Changing z to tr in eq. (22), H~ is nothing but the metric on the phase space Z = (X', 554
= Irl(q) l-adg(det Z2)-d(det 0"2)-g ,
To
To
To
Fig. 2. A genus-g Riemann surface described by the period matrix r=diag(ro ..... to).
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changed to d, and the target space becomes 2g-dimensional. Thus, Zquantum is unchanged. In string theory multi-loop expansion, the partition function is integrated over 3 g - 3 complex modular parameters (for g~> 2, and one modular parameter for g = 1 ). On general surfaces the quantum piece is much more complicated. However, one can learn the modular properties of Zqua, (tr, z) knowing the modular properties of Zquan(r, a). Trying to leave the protected shelter of mathematical identities and venture into the realm of their physical applications we wish to focus on three clues, each of which leads us to the context of topological string theories. The first observation is that the identities we have obtained concern only the classical part of the partition function. We do not have enough information to consider the quantum part. The relations between different pairs of target spaces and world-sheets in the absence of quantum oscillations have a natural setting in the topological string theories [8,16-19]. These theories consist of a string moving in a very special background and possess a very large gauge invariance which eliminates all quantum oscillations from the spectrum. If the theory includes in its definition an integral over all worldsheet metrics, the topological gravity term should be added as well to the action. We do not know if given a set of primary fields, one can find a topological field theory whose partition function would coincide with that of the classical partition function based only on the ground states of each primary field. It is in such a theory that the mathematical identities would have a direct physical implication. The second clue concerns the fact that all the identities we have derived so far involve either a "complex" target space or a "real" period matrix. The third clue is that the duality relations obtained suggest, at least for fiat target spaces, a generalization from world-sheets described by allowed period matrices to objects described by any period matrix. There are indications that string theories possess another phase and it was suggested to relate that phase to topological string models. In such a phase the concept of a world-sheet may require a generalization [9 ]. The one afforded by the duality relation of eq. (5), could be an indication as to the direction of the desired generalization. In a topological field theory the short distance re-
13 April 1989
strictions on the value of the Virasoro central charge of the matter sector are removed. The Weyl anomaly does not dictate the allowed dimensions of the target space. If the identities found have a physical significance, they show that there are string theories in which the string cannot distinguish between specific pairs of world-sheets and target spaces. The target space background and dimension are intermingled with the precise form of the interaction the string undergoes. Indeed they both seem to have a common origin and become distinguished only in the broken phase. We find this idea so attractive that we consider it worthwhile presenting the mathematical relations, even though they are attached to a long string of physical conditionals. We have started this note by stressing the "stringy" properties of the duality transformation on the background fields. We have also found that three symmetry properties of the almost complex target spaces, eq. (7), have three symmetry properties on the worldsheet, eq. (6), as their counter parts. All the worldsheet symmetries and two of the target space ones are accepted as natural 2D field theoretical symmetries, while the duality symmetry is singled out as a "stringy" effect. Does there exist a framework in which all these symmetries are on an equal footing? To answer that consider the following example. Imagine that one was given the 2D gravitational field theory in a gauge fixed form. One would then marvel at the various discrete symmetries, eq. (6), the system has. Only after realizing that the system is a gauge fixed form of a general coordinate invariant system, would one realize that the discrete symmetries are a reflection of the global structure of that gauge group. One could have guessed, given eq. (6), that the theory describes world-sheets which are Riemann surfaces. In an analogous manner, all the symmetries apperent in eqs. ( 5 ) - ( 7 ), and ( 16 ) could be viewed, in a second quantized theory, as the global part of some much larger symmetry which is gauge fixed. For that to be true in such a, yet to be discovered, second quantized theory, a functional integral on the background fields should be performed as well. We would like to thank D. Kazhdan for a very supportive guidance, and L. Baulieu and S. Elitzur for discussions. We also thank R. Dijkgraaf for pointing out the results of ref. [ 4 ]. After completion of this 555
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w o r k we r e c e i v e d a p a p e r by A. S h a p e r e a n d F. W i l c z e k [ 20 ] w h o s e results o v e r l a p w i t h s o m e o f the results o f ref. [ 3 ] a n d the p r e s e n t paper.
References [ 1] See e.g.N. Sakai and I. Senda, Prog. Theor. Phys. 75 (1986) 357. [ 2 ] V.P. Nair, A. Shapere, A. Strominger and F. Wilczek, Nucl. Phys. B 279 (1987) 369. [3] A. Giveon, E. Rabinovici and G. Veneziano, Duality in string background space, preprint CERN-TH.5106/88 (July 1988). [4] R. Dijkgraaf, E. Verlinde and H. Verlinde, in: Perspectives of string theory., eds. P. Di Vecchia and J.L. Petersen (World Scientific, Singapore, 1988 ). [ 5 ] D.J. Amit, S. Elitzur, E. Rabinovici and R. Savit, Nucl. Phys. B 210 [FS6] (1982) 69. [6] R. Savit, Nucl. Phys. B 200 (1982) 233. [7] E. Domany, Phys. Rev. Lett. 52 (1984) 871. [8] E. Witten, Commun. Math. Phys. 118 (198) 411. [9] J.J. Atick and E. Witten, The Hagedorn transition and the number of degrees of freedom of string theory, present IASSNS-HEP-88/14 (April 1988); O. Alvarez and E. Rabinovici, unpublished.
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[ 10] D. Mumford, Tata lectures on theta (Birkh~iuser, Basel, 1983). [ 11 ] K.S. Narain, M.H. Sarmadi and E. Witten, Nucl. Phys. B 279 (1987) 369. [ 12 ] L. Alvarez-Gaum6, G. Moore and C. Vafa, Commun. Math. Phys. 106 (1986) 1. [ 13] L. Alvarez-Gaum6, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Commun. Math. Phys. 112 (1987) 503. [ 14] P. Ginsparg and C. Vafa, Nucl. Phys. B 289 (1987) 414. [ 15] A.M. Polyakov, Phys. Lett. B 103 ( 1981 ) 207. [16] D. Montano and J. Sonnenschein, Topological strings, preprint SLAC-PUB-4664 (June 1988). [17] L. Baulieu and I.M. Singer, The topological sigma model, preprint. [ 18] J.M.F. Labastida, M. Pernici and E. Witten, Topological gravity in two dimensions, preprint IASSNS-HEP-88/29 (June 1988). [ 19 ] E. Witten, Phys. Rev. lett. 61 (1988) 670. [20] A. Shapere and F. Wilzcek, Self-dual models with theta terms, preprint IASSNS-HEP-88/36 (September 1988 ).