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Microscopic Entropy of N=2 Extremal Black Holes


CALT-68-2076 RU-96-88 hep-th/9609204

arXiv:hep-th/9609204v1 25 Sep 1996

Microscopic Entropy of N=2 Extremal Black Holes
David M. Kaplan?1 , David A. Lowe?2 , Juan M

. Maldacena?3 and Andrew Strominger?4 Department of Physics University of California Santa Barbara, CA 93106-9530, USA
? ?

California Institute of Technology Pasadena, CA 91125

?

Department of Physics and Astronomy Rutgers University Piscataway, NJ 08855, USA

Abstract String theory is used to compute the microscopic entropy for several examples of black holes in compacti?cations with N = 2 supersymmetry. Agreement with the Bekenstein-Hawking entropy and the moduli-independent N = 2 area formula is found in all cases.

September, 1996
1 2 3 4

dmk@cosmic1.physics.ucsb.edu lowe@theory.caltech.edu malda@physics.rutgers.edu andy@denali.physics.ucsb.edu

1. Introduction More than two decades after its discovery, our understanding of string theory has ?nally developed to the point where it can be used to provide, in special cases, a precise statistical derivation of the thermodynamic Bekenstein-Hawking area law for the entropy. While some de?nite relations between the laws of black hole thermodynamics and the statistical mechanics of stringy microstates have been clearly established, much more remains to be understood. For example a universal derivation of the area law for all types of black holes remains elusive. For these reasons it is important to understand as many cases as possible. In [1] the entropy was microscopically computed for the simplest case of a ?vedimensional extremal black hole. Supersymmetry makes it possible to count the microstates at weak coupling and then extrapolate into the strongly-coupled black hole region. This result was extended to include rotation in [2]. The four dimensional case with N = 8 and N = 4 supersymmetry was microscopically computed in [3] and [4]. The result agreed with the formulae for the area derived in [5,6]. Given the fact that the area is independent of moduli [7,5], these formulae are ?xed up to a few constants by symmetries of N = 8 and N = 4 supergravity. In this paper we analyze several examples with N = 2 supersymmetry, which has not been previously considered. The N = 2 area formula was derived for the pure electric case in [7], for the general case with electric and magnetic charges in [8] and elegantly related to central charge minima in [9]. No symmetries are available here, and the formulae have a rather di?erent character involving rational ?xed points in the special geometry moduli space. Agreement is again found in the examples considered herein.

2. Type IIA Orientifold Example ? First we consider the Type IIA theory on K3 × S 1 × S 1 . This theory has N = 4 supersymmetry in d = 4. In order to construct an N = 2 theory, we orientifold this model by a geometric Z 2 symmetry combined with reversal of worldsheet orientation. Black hole Z solutions of the N = 4 theory invariant under this action will also be solutions of the orientifold model. In this manner we construct black hole solutions of an N = 2 Type IIA orientifold model and compute their macroscopic and microscopic entropy. We consider the Z 2 orientifold which combines the Enriques involution on K3, re?ecZ 1 ? tion on S , and translation by π on S 1 , together with left-right exchange on the worldsheet. 1 ? In M-theory language, this is a purely geometric orbifold of K3 × S 1 × S 1 × S11 , which 1 ? acts as Enriques, combined with re?ection on S 1 × S11 and translation by π on S 1 . This 1 model was discussed in a di?erent construction in [10] (with the translation on S11 rather ?1 ) and with this construction in [11]. It has N=2 supersymmetry in d=4 with 11 than S vector multiplets and 12 hypermultiplets. Now we wish to construct a four-dimensional black hole solution in this N = 2 theory as a collection of intersecting branes [12]. Such a solution is obtained from a set of intersecting branes in the original N = 4 theory as follows. First, Q5 symmetric 5-branes ? wrap K3 × S 1 . We will consider 5-branes not centered at the ?xed points of the re?ection on S 1 . Since we wish to construct a con?guration invariant under the Z 2 symmetry each Z 1

5-brane is accompanied by its Z 2 image, so that Q5 is even. In addition, a 4-brane wraps Z ? the product of a holomorphic 2-cycle, Σ, of K3 with S 1 × S 1 . It is possible to show that for any Σ ∈ H? (K3) with even self-intersection number m = 2Q2 , there is a choice of 4 complex structure of K3 such that the cycle may be realized as a holomorphic curve of ? genus Q2 +1. Since S 1 × S 1 is odd under the Z 2 , we must impose the additional restriction Z 4 that the 2-cycle of K3 be odd under the Enriques involution. In order that the 4-cycle be supersymmetric in the orientifold theory, the 2-cycle must be holomorphic with respect to the odd self-dual two-form of K3. Note also that the symmetric 5-branes cut each of the 4-cycles into Q5 pieces along the S 1 direction. Finally, we include n quanta of momentum ? along the S 1 direction. In terms of the original N = 4 theory, we are considering a set of Q5 symmetric 5-branes, a 4-brane with self-intersection number 2Q2 and total momentum n. This con4 ?guration is related by duality to a con?guration of one 6-brane and Q2 + 1 2-branes, Q5 4 5-branes and momentum n. The statistical entropy of such a con?guration was calculated in [3] and found to be S = 2π Q2 Q5 n , 4 (2.1)

in the limit of large charges. This agrees with the Bekenstein-Hawking entropy of the corresponding black hole solution. The entropy is duality invariant, so (2.1) will also hold in the case at hand. Now let us consider the Bekenstein-Hawking entropy in the orientifold theory. This may be computed by dividing the horizon area A10 of the ten-dimensional solution by the ten-dimensional Newton’s constant G10 . Newton’s constant is una?ected by the orientifolding but the Z2 acts freely on the horizon and hence divides the area in half. The entropy S ′ in the orientifold theory is then S′ = A′ A10 S 10 = , ′ = 8G 4G10 2 10 (2.2)

where the prime denotes quantities in the orientifold theory. It is also easily seen that orientifolding reduces the microscopic entropy by half. The microscopic entropy is carried by Q2 Q5 massless supermultiplets that live on a string 4 ? wrapping S 1 . The orientifold reduces the length of this string by half. Since the entropy is an extensive quantity it is also reduced by half. The orientifold also introduces twisted spatial boundary conditions for the massless supermultiplets. However this a?ects mainly the zero mode structure and not the asymptotic form of the entropy for large n.

3. Type I Example In this section we consider black holes in Type I theory on K3. These theories have N = 2 supersymmetry in four dimensions when we further compactify two more dimensions on a torus. First we describe the classical black hole solutions, then quantize the charges in four and ?ve dimensions and compute the Bekenstein-Hawking entropy. This is found to agree with the number of microscopic con?gurations obtained using D-brane techniques. 2

3.1. Classical Solutions ? For the ?ve(four)-dimensional black holes we consider Type I on K3×S 1 (×S 1 ). The four-dimensional classical solution was found in [5] for the case of toroidal compacti?cation. The solution in the N = 2 case is the same as the N = 4 case as the relevant terms in the low energy supergravity lagrangians involved are identical. In four dimensions the black hole we consider carries charges corresponding to NS solitonic 5-branes wrapping around ? K3×S 1 , Kaluza-Klein monopoles on S 1 , and fundamental string winding and momentum 1 along S . The ?ve-dimensional con?guration is the same, except for the absence of the Kaluza-Klein monopole. The ?ve dimensional solutions are treated in [13]. It is useful to rewrite the entropy formulas in terms of integer quantized charges. The fundamental strings are winding along S 1 so the charge quantization condition will be the same as in N = 4. The quantization for momentum will be the same. The NS 5-brane is ? the Dirac dual to the string winding along S 1 , so it also has the same quantum of charge as in N = 4 case, and the Kaluza-Klein monopole is the Dirac dual to momentum along ? S 1 so again it is the same as in the N = 4 case. Therefore the formula for the entropy is S = 2π Q5 QKK Q1 n , (3.1)

where Q5 , QKK , Q1 , n are the number of NS 5-branes, Kaluza-Klein monopoles, winding strings, and momentum, respectively. The entropy of the ?ve-dimensional black hole is as in (3.1) with the factor QKK set to one. 3.2. D-Brane Counting Consider ?rst the ?ve-dimensional case. The Type I con?guration on K3×S 1 consists of D 5-branes wrapping K3×S 1 , D-strings wrapping S 1 and momentum ?owing along S 1 . Note that these are D-branes of Type I theory so that the D 5-brane has an SU (2) = Sp(1) gauge ?eld living on it. When Q5 D-5-branes coincide we have an Sp(Q5 ) gauge theory on the brane. The D-1-brane charge is carried by instantons of this gauge theory which are self dual gauge connections on the K3. For large Q1 and Q5 the number of bosonic degrees of freedom of the instanton moduli space is 4Q1 Q5 and they come mainly from the di?erent ways of orienting the instantons inside the gauge group. We could also, as in [14] , count the moduli by considering open strings going between the D 1-branes and the D 5-branes. These open strings are unoriented, so there are 2 bosonic ground states for each string, plus 2 possible Chan Paton factors for each 5-brane. This leads to a total of 4Q1 Q5 bosonic states5 . (In the corresponding Type II counting there is a factor of 2 arising from the 2 possible orientations of the open string). There are also 4Q1 Q5 fermionic degrees of freedom. The momentum n along the S1 direction will be carried by oscillations in the instanton moduli space or, equivalently, by the (1,5) strings. In either case the counting is that of a 1+1-dimensional gas with 4Q1 Q5 bosonic and fermionic ?avors6 . We can
The actual state will also have some (1,1) and (5,5) strings excited in such a way that D-terms vanish [15,16]. It should be kept in mind that when n is not much bigger than Q1 Q5 then the e?ects of multiple windings become important and both the number of ?avors and the e?ective length of the circle increase, giving the same result for the entropy [17].
6 5

3

therefore use the standard asymptotic formula for the entropy which yields S = 2π Q1 Q5 n . (3.2)

This agrees with the classical result. Now let us consider the four-dimensional case. We have the same con?guration of branes as in ?ve dimensions plus a Kaluza-Klein monopole. It was shown in [4] for N = 4 (but the arguments also apply to N = 2) for the case of one monopole that (3.2) is unchanged, in agreement with (3.1) for QKK = 1. The general formula then follows from duality. It is nevertheless instructive, as well as useful for following sections, to see how this ? works in detail. We ?rst perform a T-duality along S 1 to a Type I ′ theory. The D 5-branes ? become D 6-branes and the D-strings become D 2-branes wrapped around S 1 × S 1 . The momentum remains momentum and the Kaluza-Klein monopole becomes a NS 5-brane ? wrapping K3×S 1 . This theory also has two orientifold planes perpendicular to S 1 and 16 D-8-branes with 16 images on the other side of the orientifold planes. Locally, there is a Type II theory in between the orientifold planes. We take the NS 5-branes to sit at ? particular points of S 1 . The total number of Type II 5-branes is 2QKK : a 5-brane and an image 5-brane is the minimum that we can have, therefore it is the “unit” of solitonic 5brane charge in this theory. However only QKK of them are in between a pair of orientifold hyperplanes so that the e?ective number of 5-branes in the locally Type II theory between two orientifold hyperplanes is QII = QKK . The number of D-2-branes in the locally Type 5 II theory is QII = Q1 (the same as the number of original one branes), while the number 2 of locally Type II D-6-branes is QII = 2Q5 . The extra factor of two comes from the extra 6 Sp(1) Chan Paton index carried by Type I 5-branes. We can now use the Type II result [3] to count the number of con?gurations between two orientifold planes. The counting in [3] relied on the fact that the solitonic 5-branes slice the 2-branes in QII pieces. Now 5 we must in addition consider the D-8-branes, which also intersect the 2-branes, further dividing up their worldvolumes. However for large charges they will have a subleading e?ect since the number of 8-branes is much smaller than QII in that limit (16 ? QII ). 5 5 Similarly we will not worry about possible slicing of the 6-branes by the 8-branes7 . The fact that the 2-branes can end on 5-branes [18,19] implies that di?erent slices in between di?erent solitonic 5-branes can move independently. The momentum is carried mainly by (2,6) strings living between particular 5-branes. Note however that the orientifold projection will correlate what happens on one side of the orientifold hyperplane with what happens on the other side. In particular when we put a unit of momentum between two 5-branes we also have to put a unit of momentum on the two image 5-branes on other side of the orientifold hyperplane. Therefore we have only half of the total momentum available for distributing freely in the locally Type II theory, nII = n/2. Once we have identi?ed the correct number of degrees of freedom in the locally Type II theory we count as in [3] and obtain S = 2π
7

QII QII QII nII = 2π 2 6 5

Q1 Q5 QKK n ,

(3.3)

We also do not worry about (2,8) or (6,8) strings because, for large charges, there are not as

many ?avors of these as there are for (2,6) strings.

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which is the same formula as in the N = 4, 8 cases [3]. Again, this formula agrees with the classical result of equation (3.1) .

4. More General N=4 Examples In the previous two N = 2 examples the counting is similar to the maximally supersymmetric N = 8 cases. This is partly because the black holes we chose to analyze were present in N = 8 as well as N = 2 theories. In this section we analyze some new features that appear only when there is less than the maximal supersymmetry. In the N = 8 case all gauge charges are part of the supergravity multiplet, while in the N = 4 case we can have extra gauge multiplets. In this latter, more general case, the entropy formula in four dimensions can be written in terms of an O(6, 22) vector of magnetic charge P and an O(6, 22) vector of electric charges Q. We consider Type I/heterotic theory on T 6 , in which case the electric charges are carried by the Type I D-1-brane or heterotic fundamental 1 1 string. De?ne Q = ( 2 pR , 1 pL , √2 q) where pR,L are the right and left-moving momenta 2 of a heterotic string on T 6 and q are the 16 U(1) charges of a generic compacti?cation. In terms of D-branes these charges are carried by (1,9) strings. The black holes we considered n in the previous section had p5 = ( R ± Q1 R) (with other components of pR,L set to zero) R,L and q = 0; now let us consider q di?erent from zero. The magnetic charges are still carried by the D-5-brane and the Kaluza-Klein monopole. As in the preceding section, we go to the Type I ′ theory with Q1 D-2-branes, Q5 D-6-branes and QKK solitonic 5-branes8 . Now the open strings that carry momentum n will also have to carry some charge. The charge is carried by (2,8) strings, the D-8-branes appeared when we did the T-duality transformation to the Type I ′ theory. These (2,8) strings are left-moving fermions on the intersection onebrane. The (6,8) strings can also carry some charge but they are massive when the (2,6) strings are excited [16]. As in the case of rotating black holes [2] we conclude9 that the e?ective momentum that is left to distribute in (2,6) strings, after we have put enough (2,8) strings to account for the charge, is nef f = n ? q 2 /2Q1 , where the factor of Q1 arises as in [2] from the di?erent ?avors of (2,8) strings among which the charge is distributed. The entropy formula becomes S = 2π Q1 nef f QKK Q5 = 2π 1 Q1 n ? q2 QKK Q5 = 2π 2 Q2 P2 ? (Q · P)2 , (4.1)

1 since Q2 = Q1 n ? 2 q2 and in this case Q · P = 0. This is the classical formula [5]. Here q2 is an even integer, each left-moving (2,8) fermion carries one unit of charge and the total current carrying fermion number is restricted to even values [20]. It is also of interest to consider a black hole that is extremal (in the sense that the mass is such that the solution is on the threshold of developing a naked singularity) but 8

The subindex indicates what the object was in the original Type I theory, hopefully this will Similar observations have been made by C. Vafa (private communication).

not cause confusion.
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not BPS, as for example a black hole with Q2 < 0. It can be seen from the general black hole solutions in [5] that the classical entropy formula is just Sext = 2π |Q2 |P2 . (4.2)

To be de?nite consider a black hole with zero momentum n = 0, but with some gauge 1 1 charge q, so that Q = ( 1 Q1 R, ? 2 Q1 R, √2 q), Q2 = ? 1 q2 . In this case, we have to have 2 2 enough (2,8) strings to carry the charge, but since the (2, 8) strings can move only in one direction [20] they will carry some net momentum along the direction of the original string. This implies that there must be an equal amount of open strings moving in the opposite direction. These will be (2,6) strings since they carry the most entropy. The black hole, and the D-brane system, are not BPS but they are extreme in the sense that they carry the minimum amount of mass consistent with the given charges. Note that there is no BPS bound for the charges q under which this black hole is charged. In fact there can be light particles charged under this gauge group near an enhanced symmetry point. A real-world electrically charged extremal black hole will be of this type, in the sense that the electron is nearly massless compared to the string scale. Such black holes are not stable and will decay quickly by emitting charged particles. 5. Entropy for more general N = 2 cases Now we construct a black hole with charges that can exist only in the N = 2 case and not in the N = 4 case. In order to do this we use the Gimon-Polchinski model [21] which is connected to the Type I on K3 considered above. This model contains 9-branes and 5-branes. The 5-branes are oriented along the directions (012345). We compactify the directions (45) and T-dualize along the direction 4. Now we have 2 orientifold hyperplanes; the 9-branes (5-branes) transform into 16 8-branes (4-branes) between the orientifold hyperplanes. The 4-branes are oriented along (01235) and one can choose a point in moduli space where there are not any coinciding branes. There are also orbifold ?xed points on the internal torus (6789). These branes are “background” branes in the sense that they are completely extended along the macroscopic four dimensional space and are part of the vacuum state. Now we include the same con?guration that we had before: Q5 solitonic 5-branes along (056789), Q6 6-branes along (0456789), Q2 2-branes along (045) and momentum n along the direction 5. If these are all the charges we have, the state counting for this case is the same as in the previous case, since all “background” branes give contributions that are subleading in the limit of large charges. The new feature, relative to the N=4 case, is that we can have extra charges associated to 4-branes. These charges will be carried by (4,6) strings which are left-moving fermions, they are related by T-duality to the (2,8) or (1,9) strings of the previous section. The (2,4) strings also carry charge, but become massive when a condensate of (2,6) strings form. The (4,8) strings are a purely subleading contribution since they involve only the background branes, and, in any case, we can sit at a point in moduli space where they are massive. If the black hole also carries charge p under the 4-brane U (1)s and charge q under the 8-branes U (1)s, then we are forced to have some left-moving (2,8) and (4,6) strings thus 6

reducing the available momentum that we can distribute among the highly entropic (2,6) modes by nef f = n ? q2 /2Q2 ? p2 /2Q6 . The formula for the entropy then becomes S = 2π 5.1. Classical Solution The Gimon-Polchinski model is U-dual to a heterotic theory on K3 with a instanton numbers (12,12) embedded in the two E8 factors [22,23]. The six-dimensional low-energy lagrangian for this heterotic theory has been considered in [24]. This is equivalent to the Type I action. The relevant terms in this action, in heterotic variables, are: S= (2π)3 α′2 (2π)3 + ′2 α √ 1 α′ ? α′ d6 x ?g e?φ R + (?φ)2 ? H 2 ? F 2 ? F 2 12 8 8 ′2 α′ ? ? α ω3 ∧ ω3 , ? B∧F ∧F ? ? 4 8 M6 1 1 Q2 Q6 n ? Q2 p2 ? Q6 q2 Q5 . 2 2 (5.1)

(5.2)

? where F (F ) denotes the ?eld strength of the gauge ?elds arising from the 9-branes (5branes), φ is the six-dimensional dilaton, and ω3 is the Chern-Simons form de?ned by dω3 = ?F ∧ F/2, and likewise for ω3 . The ?eld strength for the antisymmetric tensor ?eld ? ′ is de?ned in the usual way as H = dB + α ω3 . Note we have dropped the higher derivative 2 terms that appear in the action of [24] which will only be relevant for large curvatures. The equations of motion that follow from this action are invariant under a Z 2 duality Z transformation which acts as φ → ?φ gmn → e?φ gmn ? A→A H → e?φ ? H ? A→A. This symmetry is actually just a T-duality symmetry on the Type I side which inverts the size of the K3. Now let us compactify on a torus down to four dimensions and consider the classical black hole solution which carries the charges mentioned above. It follows from [7,8] that there exists a solution with constant scalar ?elds, provided the asymptotic values of these scalars are adjusted to special values. Because the entropy does not depend on the asymptotic values of the moduli [7,8] there is no loss of generality in restricting our considerations to this case. For this solution, it may be seen from the equations of motion that H = ?H, where the Hodge dual ? is de?ned with respect to string metric. Taking into account the ? ? fact that F ∧ F and F ∧ F vanish for the solution at hand, the equations of motion take the same form as the usual N = 4 Type I (or heterotic) equations [25]. The extremal BPS 7 (5.3)

black hole solutions of the N = 4 equations have been classi?ed in [5]. Using these results it may be shown that the ?eld φ satis?es eφ = and the entropy is S = 2π 1 Q2 Q6 n ? Q6 (q2 + s2 ) Q5 , 2 (5.5) Q6 , Q2 (5.4)

where all charges are as de?ned above, and s = γp with γ a constant to be determined. ? The di?erence between the F and F ?elds arises when one when considers the relationship between the integer-valued quantized charges and the physical charges Qi (de?ned by F i = Qi /r 2 , with F i the four-dimensional ?eld strengths, as de?ned in [25]). Since the ? gauge kinetic terms for the F ?elds do not have the usual e?φ factor in front, we ?nd the ?φ/2 relation s = e p. Substituting this into (5.4) and (5.5) we ?nd the Bekenstein-Hawking entropy of the black hole agrees with the microscopic counting (5.1).

6.

Conclusions

We have found agreement between the macroscopic Bekenstein-Hawking entropies for BPS black holes in N = 2 supergravities and the microscopic entropy in string theory. In the ?rst two examples the counting is very similar to the counting for the N = 4, 8 cases. The only real di?erence is that the various branes are on a less supersymmetric background. The physical mechanism that gives rise to the large degeneracy is basically the same as in the more symmetric cases. We explored more intrinsically N = 4, 2 cases by considering black holes which carry gauge charges that exist in these less supersymmetric theories. It would be interesting to present a general argument testing the full N = 2 spectrum of charged black holes. In particular, D-brane counting for Type II string theories compacti?ed on generic Calabi-Yau 3-folds that are not orbifolds of more symmetric cases is an unexplored problem. The results of [8,9] describe a universal geometric formula for the entropy which must be somehow reproduced by D-branes. In particular the simple relationship [9] of the entropy formula to the minima of the central charge should have a microscopic explanation. Acknowledgements We thank J. Park and J.H. Schwarz for helpful discussions. The research of D.L. is supported in part by DOE grant DE-FG03-92ER40701; that of J. M. by DOE grant DE-FG02-96ER40559; and that of D. K. and A. S. by DOE grant DOE-91ER40618.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] A. Strominger and C. Vafa, hep-th/9601029, Phys. Lett. B379 (1996) 99. J. Breckenridge, R. Myers, A. Peet and C. Vafa, hep-th/9602065. J. Maldacena and A. Strominger, hep-th/9603060, Phys. Rev. Lett. 77 (1996) 428. C. Johnson, R. Khuri, R. Myers, hep-th/9603061, Phys. Lett. B378 (1996) 78. M. Cvetic and D. Youm, hep-th/9512127; M. Cvetic and A. Tseytlin, hep-th/9512031, Phys. Rev. D53 (1996) 5619. R. Kallosh and B. Kol, hep-th/9602014, Phys. Rev. D53 (1996) 5344; R. Dijkgraaf, E. Verlinde and H. Verlinde, hep-th/9603126. S. Ferrara, R. Kallosh and A. Strominger, hep-th/9508072, Phys. Rev. D52 (1995) 5412. A. Strominger, hep-th/9602111, Phys. Lett. B383 (1996) 39. S. Ferrara and R. Kallosh, hep-th/9602136, Phys. Rev. D54 (1996) 1514; hepth/9603090, Phys. Rev. D54 (1996) 1525. S. Ferrara, J.A. Harvey, A. Strominger and C. Vafa, hep-th/9505162, Phys. Lett. B361 (1995) 59. C. Vafa and E. Witten, hep-th/9507050. J. Polchinski, hep-th/9510017, Phys. Rev. Lett. 75 (1995) 4274. A. Tseytlin, hep-th/9601177, Mod. Phys. Lett. A11 (1996) 689. C. Callan and J. Maldacena, hep-th/9602043, Nucl. Phys. B472 (1996) 591. J. Maldacena, hep-th/9607235. M. Douglas, hep-th/9604198. J. Maldacena and L. Susskind, hep-th/9604042, Nucl. Phys. B475(1996) 679. A. Strominger, hep-th/9512059, Phys. Lett. B383 (1996) 44. P. Townsend, hep-th/9512062, Phys. Lett. B373 (1996) 68. J. Polchinski and E. Witten, hep-th/9510169, Nucl. Phys. B460 (1996) 506. E. Gimon and J. Polchinski, hep-th/9601038, Phys. Rev. D54 (1996) 1667. N. Seiberg and E. Witten, hep-th/9603003, Nucl. Phys. B471 (1996) 121. M. Berkooz, R.G. Leigh, J. Polchinski, J.H. Schwarz, N. Seiberg and E. Witten, hepth/9605184. M.J. Du?, R. Minasian and E. Witten, hep-th/9601036, Nucl. Phys. 465 (1996) 413. J. Maharana and J.H. Schwarz, hep-th/9207016, Nucl. Phys. 390 (1993) 3.

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