- Email: [email protected]
Extremal black holes and elementary string states
UCLEAR PHYSICS
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
Extremal Black Holes and Elementary String States Ashoke Sen a aTata Institute of Fundamental Research Homi Bhabha Road, Bombay 400005, India We study the relationship between entropy of extremal black holes and degeneracy of Bogomol'nyi saturated elementary string states in heterotic string theory compactified on a six dimensionM torus.
In this talk I shall explore the relationship between extremal black holes and Bogomol'nyi saturated elementary string states. Let me begin by giving some motivation for this study. . There have been suggestions from diverse points of view that black holes should be treated as elementary particles[I-7]. Furthermore, many of the recent results on duality in string theory are based on the assumption that extremal black holes can be regarded as elementary particle like states in string theory[8-11]. One of the important problem is to count the degeneracy of such states. Although we'll not be able to say something directly about the degeneracy of a generic extreme black hole state, our analysis will throw some light on a related question. . Among the many different kinds of extremal black holes in string theory, there exist extremal black holes which carry the same quantum numbers as elementary string states. One of the predictions of duality symmetry is that these should not be counted as different states, but must be identified with elementary string states. Thus the question arises: do these black holes have the same properties as elementary string states? In particular, do they have the same degeneracy as the elementary string states? This is the question that we shall address in this talk. We shall restrict our analysis to heterotic string theory compactified on a six dimensional torus
T6[12]. (Generalization of this result to heterotic string theory compactified on T l°-D for 5 _< D _< 9 has been carried out in ref.[13].) For a generic string state of mass m, the degeneracy dE.5;, calculated from tree level string theory increases as exp(m). However for a generic state quantum corrections renormalize m, and hence d~.s.(m) is also renormalized. In order to avoid this problem, we can focus on states saturating Bogomol'nyi bound[14]. For such states there is no renormalization of mass[15], and hence dE.s.(m) calculated at the string tree level is expected to be exact. It turns out that for Bogomol'nyi saturated states dE.s. is still exponential in m. (The exact formula will be given later). On the other hand, for a black hole, the degeneracy (dB.H. -- e x p ( S B . H . ) ) of states is expected to be given by the Bekenstein-Hawking formula. For a generic black hole of mass m, the entropy SB.H. grows quadratically with its mass, and hence dB.H. "~ exp(m2). This seems to be in contradiction with the formula for the degeneracy of elementary string states, b u t we should recall that there can be large renormalization effects which have not been taken into account[4,5]. We can avoid this problem again by focussing our attention to black hole states which saturate Bogomol'nyi bound, and are not expected to receive any mass renormalization[16]. These usually correspond to extremal black holes in string theory whose entropy vanishes. Thus we expect dB.H. "~ 1. This again seems to contradict the string theory expression which grows exponentially with m. If this was the end of the story, then we could conclude that the straightforward identification of
0920-5632/96/$15.00e 1996 Elsevier Science B.V. All rights reserved. PlI: S0920-5632(96)00022-9
199
A. Sen~Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
extremal black holes with elementary string states does not work; and that would be the end of the story. However, we should recall that for extremal black holes of the type we are discussing, the curvature becomes strong near the event horizon, and hence the lowest order approximation that is used to compute the entropy, breaks down. In other words, stringy effects must be taken into account in computing SB.H.. The question that we shall address in this talk is: Is it possible to compute stringy modification of dB.H. and compare it with dE.s. 9. We begin by writing down the effective action describing the dynamics of the massless fields in four dimensions: S-
327rl/d4xx/-'--Ge-~[ RG+Gut]Ou~Ot]¢ i
s
s
tion: M -~ [2Mfl T, Guy + Gut],
¢ + (I,,
Buy + But] {5)
where ~2 is a 28 x 28 matrix satisfying, (6)
i2L~ T = L .
This remains a valid symmetry of the action even after we include the higher derivative terms in the action originating from the higher order corrections in the string world-sheet theory. We also define the canonical Einstein matric gliv as follows: (7)
guy = e - ¢ G l i v .
In terms of gut], the action takes the form: S
t
A (a) ---+12~bA? ),
-
1 32~r
/
it]
d4 zv/-2-~ [Rg - ~gU 0u(I)0v(I)
12 GlIu Gt]t] G pp Hlit]pHu,t], p, - -f1~ e - 2 ¢ g Izli ~g t]v ~g p p ' +IGIIVTr(OuMLOvML)]
.
rr
rr
1-1li t] p l Tl p q # p '
,~-¢ ,,lili',,,'," t'(~,)t r. aJr r,~ , p(b)
(1)
+~gUVTr(OliMLOt]ML)] .
Here Gut], Bliv and A (a) (0 < # , u <_ 3, 1 < a < 28) are the string metric, anti-symmetric tensor fields, and U(1) 2s gauge fields respectively, (I) is the dilaton, R e is the scalar curvature associated with the metric Gilt], and, =
H#up
We use the normalization a ' = 16. This corresponds to a string world-sheet action of the form: 64~r
/ d~ali.(X)a.XliO"X
(OuBu p + 2A(a)L (b)~I U a o' F t]p + cyclic permutations of/1, u, p (2)
(g~v) = ~liv, are the field strengths associated with A (~) and But]. M is a 28 × 28 matrix valued scalar field, satisfying,
and,
M T = M,
0)
/6
'
v +...
(9)
where - - • denotes terms involving fermionic fields on the world sheet, as well as the target space fields But], A (~), ¢ and M. We shall restrict ourselves to backgrounds characterized by the foilwing asymptotic forms of various fields:
O,A( o) _ OvA(O),.
M L M T = L,
(8)
(3)
(4)
where In denotes n x n identity matrix. The action (1) is invariant under an 0(6,22) transforma-
(Bliv) = O,
( e-'~ ) = ~-~2, (A(a)) = 0.
(i0)
This gives
(11) From eqs.(8), (9) and (11) we see that the background we are using corresponds to the following values of the Newton's constant G N and string tension T measured by an asymptotic observer: g2 GN = 2, T - 32~r" (12)
200
A. Sen~Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
Note that in eq.(10) we have not specified the asymptotic condition on M. The choice of (M) in fact corresponds to the choice of the Narain lattice[17]. It is well known that the string tree level effective field theory involving massless neutral fields is insensitive to this choice. Thus as long as our computation does not involve string loop effects, or charged fields, the results will be insensitive to the choice of (M). We shall use this freedom to choose a convenient (M) later. (It is precisely this freedom that is reflected in the 0(6,22) transformation given in eq.(5).) We shall first compute the degeneracy of Bogomol'nyisaturated elementary string states. In the normalization convention we are using, the tree level mass formula for generic elementary string states is given by (we have given the mass formula for the Neveu-Schwarz sector states, but a similar formula exists for the Ramond sector states as well) rn I
_-- gS 02 ~ g2 ~2 -~ ( ~ - ~ + 2 N R - - 1) -~- ( - ~ + 2 N L - - 2 )
charged rotating black hole solutions in the theory described by the action (8) were constructed in ref.[7]. We shall specialize on the non-rotating extremal black holes saturating the Bogomol'nyi bound. For an appropriate choice of (M), the most general black hole solution of this type is given by, ds 2
~
gu~dx~dx ~
=
-K-1/2pdtS
+ Ka/Sp-ldp 2
+ K U S p ( d O 2 + sin s OdeS), J~ltV -~ 0 ,
(18)
e '~ = K - 1 / S p g 2 ,
(19)
=
A~~)
-g-~-n(a)~ ° (psinh 21 + m0 sinh ~) for
1 < a < 22,
rnK ( p cosh 2c~+ m 0 c o s h a ) , - g p(a-22) ~
= ,(13)
for where NR and NL are the total oscillator contribution to the squared mass from the left and the ri~ght moving oscillators respectively, and QR and QL denote the left and right handed electric charges carried by the state. In order to saturate the Bogomol'nyi bound NR must be 1/2, so the mass formula takes the form m s = (~2R = g~_~((22L + 2NL - 2) , 8g 2 8 \ g4
(14)
The degeneracy of such states arises due to the many different ways the left moving oscillators make up the total number NL. This degeneracy has been calculated many times (for a recent calculation, see [5]) and is given by, dE.s. ~-- exp(4~rv/-~L) •
(17)
(15)
Thus, the entropy, calculated from the elementary string spectrum, is given by,
23 < a < 28,
(20) ( PnnT M = K -1 \ QpnT
Qnp T) pppT _ ,
where m0 and a are two real numbers, ff is a 22 dimensional unit vector, i6 is a 6 dimensional unit vector, and,
K = (p2+ 2m0p cosh + ml),
(22)
+ p2 cosh2a,
P
=
rn g + 2rnopcosh a
Q
=
2mopsinha+p2sinh2a.
(23)
The horizon and the singularity of this black hole coincide, both being situated at p = 0. We define the electric charge Q(~) el carried b y the black hole through the equations: Q(~) ~t F~ ) - p2
for large p.
Eq.(20) then gives Let us now turn to the computation of entropy of extremal black holes. Most general electrically
(21)
Q(~) eZ
=
n (~)
g--~mo
sinh2a,
(24)
A. Sen~Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
for
1 < a < 22,
201
From this we note the following properties:
p(a-22)
=
g T m °
1. The solution is singular as/5 ---, O.
cosh 2 a , for
23 < a < 28.
(25)
From eqs.(17), (22) we see that the ADM mass of the black hole is given by, 1 m0cosha= m - - G--N 2
1 ~m0cosha.
(26)
Also here f cosh 2c~ nn T (M) = \ sinh 2 ~ p n T
sinh 2c~ np T cosh 2~ppT j .
(27)
The left and the right moving charges are defined as~
Qn
-
Q(~')(L(M)L + L)~bQ (b) 2
Q2L
=
1 Q ( ~ ) ( L ( M ) L - L)~bQ(b) g2m~ sinh 2 a .
(28) 2 From eq.(26) and (28) we see that here m 2 = Q,~/8g 2. Thus the black hole solution saturates the Bogomol'nyi bound. The independent parameters labelling the solution can be taken to be g, m, QL, ff and ~7. In terms of these parameters, -
m0 = 4 1 m 2
Q2
tanh a -
8g 2 '
QL . (29) 2v~mg
Let us examine the behavior of the solution near p = 0. For this we introduce new coordinates, t ~=
/5 = g p ,
m0
.
(30)
For /5 < < mog, the field configuration takes the form:
dS 2
=
G.vdx~dx v -/52dt2 + d/52 + ~2(d02 + sin 2 0d¢2),
¢
.v
l n - -g + In/5,
"~ --
0
F pt (-a)
M ~ 128,
mo
for l,-p (a-22)
~/2
1
3. The string coupling constant e ~ ~ (g/mo)fi is small near the core. Thus string loop effects can be ignored in studying the physics near the core. 4. Since the curvature and other field strengths become strong near/5 ,,~ 1, string world-sheet effects become i m p o r t a n t near this region.
g2mo2 cosh 2o~
_
2. Except for the additive piece in ~, the solution is completely independent of all parameters. Since the conformal field theory around a given background is independent of the additive constant in ~, we see that the string tree level physics near the core of the black hole is described by a universal conformal field theory.
23 < a < 28 (31)
Let us now turn to the computation of entropy of this black hole[18]. To do this we first Euclideanize the solution by replacing t by i~. The entropy associated with this solution is then given by the usual t h e r m o d y n a m i c formula: sBx
= ~E - F,
(32)
where ]~ is the inverse temperature, E is the energy, and F is the free energy of the system. For the classical solution, ~ is to be identified as the periodicity of the Euclidean time ~, E is to be identified as the mass, and F as the classical Euclidean action. For a solution which is regular everywhere, the classical Euclidean action is equal to the product of the energy and the range of ~, and hence the entropy vanishes. For an Euclidean black hole solution, however, constant time surfaces intersect at the horizon, and the right hand side of eq.(32) can receive boundary contribution. This is what we need to estimate for our calculation. In this case, the would be horizon lies inside the strong curvature region/5 .~ 1 where the stringy effects become important, and hence it is not a prior~ clear that we can estimate this boundary contribution unambiguously. If we naively ignore
A. Sen~NuclearPhysicsB (Proc. Suppl.) 46 (1996) 198-203
202
the stringy corrections, and use the lowest order results, the answer that we get is zero, but there is no reason why we should trust the lowest order results near this region. We shall now show that even though we do not have the technology for calculating this boundary contribution exactly, we can determine it up to one overall numerical constant. The key point is that the solution is independent of all the parameters near the region f ~ 1 except for the additive constant in (I). The lagrangian density of the full tree level string theory involving the massless bosonic fields is of the form
e-¢ ~-----GC(Gm,, c9~(~,F(~), M) .
(33)
Although we do not know the explicit functional form of £, or the dependence of the various fields (I), Gu~, ~(~) - u~ or M on f, we do know that except for an additive factor of ln(g/mo) in ~, they are independent of all external parameters. Thus the boundary contribution to the entropy must have the form: SB.H. = C m0 , (34) g where C is an undetermined numerical constant. Using eq.(29) we can rewrite this expression as
A~,I /m2-Q2L
v
(35)
This agrees precisely with eq.(16) up to the unknown multiplicative constant C. This concludes the main part of the talk. In order to make this correspondence more concrete, we need to calculate this constant C and verify that eqs.(35) and (16) agree exactly. Unfortunately at present we do not have the necessary technology to do this computation. This involves finding the form of the solution near the core by taking into account stringy modification to the equations of motion, as well as finding the correction to the boundary contribution to SB.H. due to the higher derivative terms in the effective action. It has been argued in ref.[19] that there exists a particular scheme in which the higher derivative terms in the effective action do not modify the solution; but it is not clear how the modification to the contribution to SB.H. looks like in this scheme.
REFERENCES
1. S. Hawking, Monthly Notices Roy. Astron. Soc. 152 (1971) 75; A. Salam, in Quantum Gravity: an Oxford Symposium (eds. Isham, Penrose and Sciama, O.U.P. 1975). 2. G. 't Hooft, Nucl. Phys. B335 (1990) 138. 3. C. Holzhey and F. Wilczek, Nucl. Phys. B380 (1992) 447 [hep-th/9202014]. 4. L. Susskind, 'Some speculations about black hole entropy in string theory', preprint RU93-44 [hep-th/9309145]; L. Susskind and J. Uglum, Phys. Rev. D50 (1994) 2700 [hep-th/9401070]. 5. J. Russo and L. Susskind, 'Asymptotic level density in heterotic string theory and rotating black holes', preprint UTTG-9-94 [hepth/9405117]. 6. M. Duff, R. Khuri, R. Minasian and J. Rahmfeld, Nucl. Phys. B418 (1994) 195 [hepth/9311120]; M. Duff and J. Rahmfeld, Phys. Lett. B345 (1995) 441 [hep-th/9406105]. 7. A. Sen, 'Black hole solutions in heterotic string theory on a torus', preprint TIFR-TH94-47 [hep-th/9411187]. 8. C. Hull and P. Townsend, Nucl. Phys. B438 (1995) 109 [hep-th/9410167]; P. Townsend, 'The eleven dimensional supermembrane revisited', preprint D A M T P / R / 9 5 / 2 [hep-th/9501068]. 9. E. Witten, 'String theory dynamics in various dimensions', preprint IASSNS-HEP-95-18 [hep-th/9503124]. 10. A. Strominger, 'Massless black holes and conifolds in string theory', preprint [hepth/9504090] B. Greene, D. Morrison and A. Strominger, 'Black hole condensation and the unification of string vacua', preprint CLNS-95/1335 [hepth/9504145]. 11. A. Burinskii, 'Some properties of the Kerr solution to the low energy string theory', preprint IBRAE-95-08 [hep-th/9504139]; H. Nishino, 'Stationary axi-symmetric black holes, N=2 superstring, and self-dual gauge or gravity fields', preprint UMDEPP 95-111
A. Sen~Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
[hep-th/9504142]. 12. A. Sen, preprint TIFR/TH-95-19 [hepth/9504147]. 13. A. Peet, preprint PUPT-1548 [hep-th/9506200]. 14. G. Gibbons and C. Hull, Phys. Lett. B109 (1982) 109; J. Harvey and J. Liu, Phys. Lett. B268 (1991) 40; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D46 (1992) 5278 [hep-th/9205027]. 15. E. Witten and D. Olive, Phys. Lett. B78 (1978) 97. 16. R. Kallosh, Phys. Lett. B282 (1992) 80. 17. K. Narain, Phys. Lett. B169 (1986) 41; K. Narain, H. Sarmadi and E. Witten, Nucl. Phys. B279 (1987) 369. 18. G. Gibbons and S. Hawking, Phys. Rev. D15 (1977) 2752~ Comm. Math. Phys. 66 (1979) 291. 19. G. Horowitz and A. Tseytlin, Phys. Rev. Lett. 73 (1994) 3351 [hep-th/9408040].
203
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
Extremal Black Holes and Elementary String States Ashoke Sen a aTata Institute of Fundamental Research Homi Bhabha Road, Bombay 400005, India We study the relationship between entropy of extremal black holes and degeneracy of Bogomol'nyi saturated elementary string states in heterotic string theory compactified on a six dimensionM torus.
In this talk I shall explore the relationship between extremal black holes and Bogomol'nyi saturated elementary string states. Let me begin by giving some motivation for this study. . There have been suggestions from diverse points of view that black holes should be treated as elementary particles[I-7]. Furthermore, many of the recent results on duality in string theory are based on the assumption that extremal black holes can be regarded as elementary particle like states in string theory[8-11]. One of the important problem is to count the degeneracy of such states. Although we'll not be able to say something directly about the degeneracy of a generic extreme black hole state, our analysis will throw some light on a related question. . Among the many different kinds of extremal black holes in string theory, there exist extremal black holes which carry the same quantum numbers as elementary string states. One of the predictions of duality symmetry is that these should not be counted as different states, but must be identified with elementary string states. Thus the question arises: do these black holes have the same properties as elementary string states? In particular, do they have the same degeneracy as the elementary string states? This is the question that we shall address in this talk. We shall restrict our analysis to heterotic string theory compactified on a six dimensional torus
T6[12]. (Generalization of this result to heterotic string theory compactified on T l°-D for 5 _< D _< 9 has been carried out in ref.[13].) For a generic string state of mass m, the degeneracy dE.5;, calculated from tree level string theory increases as exp(m). However for a generic state quantum corrections renormalize m, and hence d~.s.(m) is also renormalized. In order to avoid this problem, we can focus on states saturating Bogomol'nyi bound[14]. For such states there is no renormalization of mass[15], and hence dE.s.(m) calculated at the string tree level is expected to be exact. It turns out that for Bogomol'nyi saturated states dE.s. is still exponential in m. (The exact formula will be given later). On the other hand, for a black hole, the degeneracy (dB.H. -- e x p ( S B . H . ) ) of states is expected to be given by the Bekenstein-Hawking formula. For a generic black hole of mass m, the entropy SB.H. grows quadratically with its mass, and hence dB.H. "~ exp(m2). This seems to be in contradiction with the formula for the degeneracy of elementary string states, b u t we should recall that there can be large renormalization effects which have not been taken into account[4,5]. We can avoid this problem again by focussing our attention to black hole states which saturate Bogomol'nyi bound, and are not expected to receive any mass renormalization[16]. These usually correspond to extremal black holes in string theory whose entropy vanishes. Thus we expect dB.H. "~ 1. This again seems to contradict the string theory expression which grows exponentially with m. If this was the end of the story, then we could conclude that the straightforward identification of
0920-5632/96/$15.00e 1996 Elsevier Science B.V. All rights reserved. PlI: S0920-5632(96)00022-9
199
A. Sen~Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
extremal black holes with elementary string states does not work; and that would be the end of the story. However, we should recall that for extremal black holes of the type we are discussing, the curvature becomes strong near the event horizon, and hence the lowest order approximation that is used to compute the entropy, breaks down. In other words, stringy effects must be taken into account in computing SB.H.. The question that we shall address in this talk is: Is it possible to compute stringy modification of dB.H. and compare it with dE.s. 9. We begin by writing down the effective action describing the dynamics of the massless fields in four dimensions: S-
327rl/d4xx/-'--Ge-~[ RG+Gut]Ou~Ot]¢ i
s
s
tion: M -~ [2Mfl T, Guy + Gut],
¢ + (I,,
Buy + But] {5)
where ~2 is a 28 x 28 matrix satisfying, (6)
i2L~ T = L .
This remains a valid symmetry of the action even after we include the higher derivative terms in the action originating from the higher order corrections in the string world-sheet theory. We also define the canonical Einstein matric gliv as follows: (7)
guy = e - ¢ G l i v .
In terms of gut], the action takes the form: S
t
A (a) ---+12~bA? ),
-
1 32~r
/
it]
d4 zv/-2-~ [Rg - ~gU 0u(I)0v(I)
12 GlIu Gt]t] G pp Hlit]pHu,t], p, - -f1~ e - 2 ¢ g Izli ~g t]v ~g p p ' +IGIIVTr(OuMLOvML)]
.
rr
rr
1-1li t] p l Tl p q # p '
,~-¢ ,,lili',,,'," t'(~,)t r. aJr r,~ , p(b)
(1)
+~gUVTr(OliMLOt]ML)] .
Here Gut], Bliv and A (a) (0 < # , u <_ 3, 1 < a < 28) are the string metric, anti-symmetric tensor fields, and U(1) 2s gauge fields respectively, (I) is the dilaton, R e is the scalar curvature associated with the metric Gilt], and, =
H#up
We use the normalization a ' = 16. This corresponds to a string world-sheet action of the form: 64~r
/ d~ali.(X)a.XliO"X
(OuBu p + 2A(a)L (b)~I U a o' F t]p + cyclic permutations of/1, u, p (2)
(g~v) = ~liv, are the field strengths associated with A (~) and But]. M is a 28 × 28 matrix valued scalar field, satisfying,
and,
M T = M,
0)
/6
'
v +...
(9)
where - - • denotes terms involving fermionic fields on the world sheet, as well as the target space fields But], A (~), ¢ and M. We shall restrict ourselves to backgrounds characterized by the foilwing asymptotic forms of various fields:
O,A( o) _ OvA(O),.
M L M T = L,
(8)
(3)
(4)
where In denotes n x n identity matrix. The action (1) is invariant under an 0(6,22) transforma-
(Bliv) = O,
( e-'~ ) = ~-~2, (A(a)) = 0.
(i0)
This gives
(11) From eqs.(8), (9) and (11) we see that the background we are using corresponds to the following values of the Newton's constant G N and string tension T measured by an asymptotic observer: g2 GN = 2, T - 32~r" (12)
200
A. Sen~Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
Note that in eq.(10) we have not specified the asymptotic condition on M. The choice of (M) in fact corresponds to the choice of the Narain lattice[17]. It is well known that the string tree level effective field theory involving massless neutral fields is insensitive to this choice. Thus as long as our computation does not involve string loop effects, or charged fields, the results will be insensitive to the choice of (M). We shall use this freedom to choose a convenient (M) later. (It is precisely this freedom that is reflected in the 0(6,22) transformation given in eq.(5).) We shall first compute the degeneracy of Bogomol'nyisaturated elementary string states. In the normalization convention we are using, the tree level mass formula for generic elementary string states is given by (we have given the mass formula for the Neveu-Schwarz sector states, but a similar formula exists for the Ramond sector states as well) rn I
_-- gS 02 ~ g2 ~2 -~ ( ~ - ~ + 2 N R - - 1) -~- ( - ~ + 2 N L - - 2 )
charged rotating black hole solutions in the theory described by the action (8) were constructed in ref.[7]. We shall specialize on the non-rotating extremal black holes saturating the Bogomol'nyi bound. For an appropriate choice of (M), the most general black hole solution of this type is given by, ds 2
~
gu~dx~dx ~
=
-K-1/2pdtS
+ Ka/Sp-ldp 2
+ K U S p ( d O 2 + sin s OdeS), J~ltV -~ 0 ,
(18)
e '~ = K - 1 / S p g 2 ,
(19)
=
A~~)
-g-~-n(a)~ ° (psinh 21 + m0 sinh ~) for
1 < a < 22,
rnK ( p cosh 2c~+ m 0 c o s h a ) , - g p(a-22) ~
= ,(13)
for where NR and NL are the total oscillator contribution to the squared mass from the left and the ri~ght moving oscillators respectively, and QR and QL denote the left and right handed electric charges carried by the state. In order to saturate the Bogomol'nyi bound NR must be 1/2, so the mass formula takes the form m s = (~2R = g~_~((22L + 2NL - 2) , 8g 2 8 \ g4
(14)
The degeneracy of such states arises due to the many different ways the left moving oscillators make up the total number NL. This degeneracy has been calculated many times (for a recent calculation, see [5]) and is given by, dE.s. ~-- exp(4~rv/-~L) •
(17)
(15)
Thus, the entropy, calculated from the elementary string spectrum, is given by,
23 < a < 28,
(20) ( PnnT M = K -1 \ QpnT
Qnp T) pppT _ ,
where m0 and a are two real numbers, ff is a 22 dimensional unit vector, i6 is a 6 dimensional unit vector, and,
K = (p2+ 2m0p cosh + ml),
(22)
+ p2 cosh2a,
P
=
rn g + 2rnopcosh a
Q
=
2mopsinha+p2sinh2a.
(23)
The horizon and the singularity of this black hole coincide, both being situated at p = 0. We define the electric charge Q(~) el carried b y the black hole through the equations: Q(~) ~t F~ ) - p2
for large p.
Eq.(20) then gives Let us now turn to the computation of entropy of extremal black holes. Most general electrically
(21)
Q(~) eZ
=
n (~)
g--~mo
sinh2a,
(24)
A. Sen~Nuclear Physics B (Proc. Suppl.) 46 (1996) 198-203
for
1 < a < 22,
201
From this we note the following properties:
p(a-22)
=
g T m °
1. The solution is singular as/5 ---, O.
cosh 2 a , for
23 < a < 28.
(25)
From eqs.(17), (22) we see that the ADM mass of the black hole is given by, 1 m0cosha= m - - G--N 2
1 ~m0cosha.
(26)
Also here f cosh 2c~ nn T (M) = \ sinh 2 ~ p n T
sinh 2c~ np T cosh 2~ppT j .
(27)
The left and the right moving charges are defined as~
Qn
-
Q(~')(L(M)L + L)~bQ (b) 2
Q2L
=
1 Q ( ~ ) ( L ( M ) L - L)~bQ(b) g2m~ sinh 2 a .
(28) 2 From eq.(26) and (28) we see that here m 2 = Q,~/8g 2. Thus the black hole solution saturates the Bogomol'nyi bound. The independent parameters labelling the solution can be taken to be g, m, QL, ff and ~7. In terms of these parameters, -
m0 = 4 1 m 2
Q2
tanh a -
8g 2 '
QL . (29) 2v~mg
Let us examine the behavior of the solution near p = 0. For this we introduce new coordinates, t ~=
/5 = g p ,
m0
.
(30)
For /5 < < mog, the field configuration takes the form:
dS 2
=
G.vdx~dx v -/52dt2 + d/52 + ~2(d02 + sin 2 0d¢2),
¢
.v
l n - -g + In/5,
"~ --
0
F pt (-a)
M ~ 128,
mo
for l,-p (a-22)
~/2
1
3. The string coupling constant e ~ ~ (g/mo)fi is small near the core. Thus string loop effects can be ignored in studying the physics near the core. 4. Since the curvature and other field strengths become strong near/5 ,,~ 1, string world-sheet effects become i m p o r t a n t near this region.
g2mo2 cosh 2o~
_
2. Except for the additive piece in ~, the solution is completely independent of all parameters. Since the conformal field theory around a given background is independent of the additive constant in ~, we see that the string tree level physics near the core of the black hole is described by a universal conformal field theory.
23 < a < 28 (31)
Let us now turn to the computation of entropy of this black hole[18]. To do this we first Euclideanize the solution by replacing t by i~. The entropy associated with this solution is then given by the usual t h e r m o d y n a m i c formula: sBx
= ~E - F,
(32)
where ]~ is the inverse temperature, E is the energy, and F is the free energy of the system. For the classical solution, ~ is to be identified as the periodicity of the Euclidean time ~, E is to be identified as the mass, and F as the classical Euclidean action. For a solution which is regular everywhere, the classical Euclidean action is equal to the product of the energy and the range of ~, and hence the entropy vanishes. For an Euclidean black hole solution, however, constant time surfaces intersect at the horizon, and the right hand side of eq.(32) can receive boundary contribution. This is what we need to estimate for our calculation. In this case, the would be horizon lies inside the strong curvature region/5 .~ 1 where the stringy effects become important, and hence it is not a prior~ clear that we can estimate this boundary contribution unambiguously. If we naively ignore
A. Sen~NuclearPhysicsB (Proc. Suppl.) 46 (1996) 198-203
202
the stringy corrections, and use the lowest order results, the answer that we get is zero, but there is no reason why we should trust the lowest order results near this region. We shall now show that even though we do not have the technology for calculating this boundary contribution exactly, we can determine it up to one overall numerical constant. The key point is that the solution is independent of all the parameters near the region f ~ 1 except for the additive constant in (I). The lagrangian density of the full tree level string theory involving the massless bosonic fields is of the form
e-¢ ~-----GC(Gm,, c9~(~,F(~), M) .
(33)
Although we do not know the explicit functional form of £, or the dependence of the various fields (I), Gu~, ~(~) - u~ or M on f, we do know that except for an additive factor of ln(g/mo) in ~, they are independent of all external parameters. Thus the boundary contribution to the entropy must have the form: SB.H. = C m0 , (34) g where C is an undetermined numerical constant. Using eq.(29) we can rewrite this expression as
A~,I /m2-Q2L
v
(35)
This agrees precisely with eq.(16) up to the unknown multiplicative constant C. This concludes the main part of the talk. In order to make this correspondence more concrete, we need to calculate this constant C and verify that eqs.(35) and (16) agree exactly. Unfortunately at present we do not have the necessary technology to do this computation. This involves finding the form of the solution near the core by taking into account stringy modification to the equations of motion, as well as finding the correction to the boundary contribution to SB.H. due to the higher derivative terms in the effective action. It has been argued in ref.[19] that there exists a particular scheme in which the higher derivative terms in the effective action do not modify the solution; but it is not clear how the modification to the contribution to SB.H. looks like in this scheme.
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