BPS State Counting in N=8 Supersymmetric String Theory for Pure D-brane Configurations

Exact results for the BPS index are known for a class of BPS dyons in type II string theory compactified on a six dimensional torus. In this paper we set up the problem of counting the same BPS states in a duality frame in which the states carry only Ramond-Ramond charges. We explicitly count the number of states carrying the lowest possible charges and find agreement with the result obtained in other duality frames. Furthermore, we find that after factoring out the supermultiplet structure, each of these states carry zero angular momentum. This is in agreement with the prediction obtained from a representation of these states as supersymmetric black holes.


Introduction
Understanding the microscopic origin of Bekenstein-Hawking entropy is one of the important problems in any theory of quantum gravity, and in particular in string theory. In recent years there has been considerable progress towards this direction, including precision counting of microscopic states in certain string theories with 16 or more unbroken supersymmetries [1][2][3][4][5][6][7]. One of these theories is type IIA or IIB compactified on a six dimensional torus. In this theory, for certain configurations carrying a combination of Kaluza-Klein (KK) monopole charge, momentum along one of the circles of the torus and D-brane wrapping charges along some of the cycles of the torus, one can carry out the exact counting for the number of microscopic BPS states [7]. On the other hand, for large charges this system can be described by a supersymmetric black hole with a finite area event horizon. Thus, by comparing the logarithm of the number of microstates with the Bekenstein-Hawking entropy of the corresponding black hole, one can verify the equality of the macroscopic and microscopic entropy of the black hole.
Although the counting of microscopic states was carried out for a specific system of KK monopoles and D-branes carrying momentum along a compact circle, using duality symmetry we can map it to other systems. In particular it is possible to map this configuration to a system that contains only D-brane charges. Duality symmetry predicts that the BPS index of this system computed from microscopic counting should give the same result as the original system to which it is dual. Nevertheless, it is of some interest to count the number of microscopic states of the new system directly. At the least, this will provide us with another non-trivial test of duality symmetry which, although has been tested in many ways, has not been proven. Another motivation for this is that by learning how to count states of pure D-brane systems in type II string theory on T 6 we may eventually gain some insight into similar counting for Dbranes wrapped on various cycles of Calabi-Yau manifolds. Indeed, for type II compactification on Calabi-Yau manifolds, all charges are associated with D-branes wrapped on various cycles of Calabi-Yau manifolds as there are no non-contractible circles and hence no momentum, KK monopole charges or winding numbers of fundamental strings and NS 5-branes. Earlier attempts to count states of pure D-brane systems describing a black hole can be found in [8][9][10].
Another reason for studying representations of black holes as pure D-brane systems is as follows. One knows on general grounds that supersymmetric black holes in 3+1 dimensions describe an ensemble of states each of which carries strictly zero angular momentum [11,12] after factoring out the fermion zero mode contribution whose quantization generates the supermultiplet structure. This leads to many non-trivial conjectures about the sign of the index of supersymmetric black holes which have been verified explicitly [13,14]. However, in microscopic counting of the same system, one often finds BPS states carrying non-zero angular momentum. This does not represent a contradiction between microscopic and macroscopic results, since only the index, and not the detailed information about angular momentum, is protected as we go from the weak coupling regime where microscopic calculation is valid, to the strong coupling regime where the black hole computation is valid. Nevertheless, one could ask if there is a duality frame in which the detailed information about the angular momentum in the microscopic and macroscopic descriptions matches. Since in the macroscopic description all black holes carry zero angular momentum, in the microscopic description this will demand that all states are singlets under the SU(2) rotation group. Recent analysis of some microstates of N = 2 supersymmetric black holes revealed that when we describe them as D-branes wrapped on certain internal cycles of Calabi-Yau manifolds we indeed get exactly zero angular momentum for the microstates of single centered black holes [15][16][17]. Assuming this to be a general phenomenon led to the conjectured Coulomb branch formula for computing the spectrum of quiver quantum mechanics and of general systems of multicentered black holes [18][19][20]. Now in N = 2 supersymmetric string theories, the above analysis is made complicated due to the fact that the index receives a contribution from both single and multi-centered black holes. Since the latter do not necessarily carry zero angular momentum, we need to carefully subtract the contribution from multi-centered black holes before we can verify that D-brane microstates representing single centered black holes carry zero angular momentum. This can be done [18,21], and was used in the analysis of [18][19][20]. However, in type II string theory on T 6 , which has N = 8 supersymmetry, the multi-centered black holes do not contribute to the index, and hence we expect that only single centered black holes will survive at a generic point in the moduli space of the theory [22]. Generalization of the observations in N = 2 supersymmetric string theories made above would then suggest that representing a supersymmetric black hole in type II on T 6 as a system carrying only RR charges associated with various D-brane sources, we may get a system whose microstates would have strictly zero angular momentum after factoring out the goldstino fermion modes whose quantization generates the supermultiplet structure. Now, after factoring out these fermionic zero modes and the bosonic zero modes associated with various translational symmetries, the BPS states of the D-brane system correspond to the cohomology of the moduli space of classical solutions of the world-line theory of the system, with the space-time rotation group acting as the Lefshetz SU(2) action on the cohomology [15,23]. This shows that in order to get only zero angular momentum states, all states must come from the middle cohomology. Since any compact manifold has a non-trivial 0-form and a top form, the only way that a manifold can have only middle cohomology is if it becomes zero dimensional, i.e. a collection of points. 1 Verification of this conjecture is another motivation for our analysis.
In this paper we shall analyze a pure D-brane system in type II theory compactified on T 6 that is dual to the system for which the microscopic result is known, and test the result by direct computation of the microscopic index of the D-brane system. We introduce the system in §2, and derive its world-line theory for the lowest possible values of the charges in §3. In §4 we explicitly count the index of supersymmetric states of this system. This is shown to reduce to counting the number of independent solutions of a set of polynomial equations -a problem that can be easily solved. We find that the solution contains a set of isolated points provided we work at a generic point in the moduli space of the theory parametrized by constant background values of the metric and 2-form fields along the internal torus. Hence, at least in this example, the microstates carry strictly zero angular momentum in agreement with the macroscopic results. In §5 we briefly discuss possible generalization of our analysis to cases where we replace each D-brane of the system described in §2 by a stack of parallel D-branes. We conclude with a discussion of our results in §6. In appendix A we derive the relation between some of the parameters of the D-brane world-volume theory and the background values of the metric and 2-form field along T 6 . In appendix B we describe the chain of dualities that relate the system under consideration to the system analyzed in [7]. Finally, in appendix C we give explicit solutions to the polynomial equations which appear in the analysis of §4.

The system
Let us consider for definiteness a type IIA string theory on T 6 labelled by the coordinates x 4 , . . . , x 9 and in this theory we take a system containing N 1 D2-branes wrapped along 4-5 directions, N 2 D2-branes wrapped along 6-7 directions, N 3 D2-branes wrapped along 8-9 directions, N 4 D6-branes wrapped along 4-5-6-7-8-9 directions and N 5 D4-branes along 6-7-8-9 directions. By a series of duality transformations described in appendix B, this configuration is related to a system of N 1 KK monopoles associated with the 4-direction, −N 2 units of momentum along the 5-direction, N 3 D1-branes along the 5-direction, N 4 D5-branes along 5-6-7-8-9 directions and −N 5 units of momentum along the 4-direction. The microscopic index of this system was computed explicitly in [7] for N 1 = 1. By a further series of U-duality transformations reviewed e.g. in [24], this system can be mapped to a system in type IIA string theory on T 6 with only NS-NS sector charges, containing −N 2 units of momentum along the 5-direction, N 1 fundamental strings wound along the 5-direction, N 4 KK monopoles associated with the 4-direction, −N 3 NS 5-branes wrapped along 5-6-7-8-9 directions and N 5 NS 5-branes along 4-6-7-8-9 directions. In the notation of [24], the electric charge vector Q and magnetic charge vector P for this state in the latter description are represented as The T-duality invariant inner product matrix between charges was given by 0 I 2 I 2 0 . With this we get We also define We shall consider configurations for which In this case, following [25,26] one can show that there is a further series of duality transformations that map this system to one with N 1 = 1 [27] for which the microscopic index is known from the analysis of [7]. Expressed in terms of the more general set of variables (N 1 , · · · , N 5 ), the result for the BPS index for this system, which in this case corresponds to the 14-th helicity supertrace B 14 , takes the form [28] (2.6) ϑ 1 (z|τ ) and η(τ ) are respectively the odd Jacobi theta function and the Dedekind eta function.
In this paper we shall analyze the simplest of these configurations with For this, (2.6) predicts We shall verify this by direct counting of microstates of the D-brane system.

The low energy dynamics of the D-brane system
The combined system of four D-branes that we have introduced in §2 with the choice of N i 's given in (2.7) preserves 4 out of the 32 supersymmetries. This is equivalent to N = 1 supersymmetry in 3+1 dimensions. Since we are dealing with a quantum mechanical system, we can effectively regard this as an N = 1 supersymmetric theory in 3+1 dimensions, dimensionally reduced to 0+1 dimensions. Thus we can can use the N = 1 superfield formalism, but ignore all spatial derivatives and integration over spatial directions while writing the action. We shall follow the normalization conventions of [29] in constructing this action.
Since the four D-branes we have are related to each other by T-duality, each of them individually has the same low energy theory given by the dimensional reduction of N = 4 supersymmetric U(1) gauge theory from 3+1 to 0+1 dimensions. We begin with one of the four different D-branes. In the language of N = 1 supersymmetry in 3+1 dimensions, each D-brane has one U(1) vector superfield V and three chiral superfields Φ 1 , Φ 2 , Φ 3 . A vector multiplet, after dimensional reduction to 0+1 dimensions, has three scalars corresponding to three spatial components of the gauge field and a gauge field A 0 . We can use the gauge A 0 = 0 and interpret the three scalars as the coordinates giving the location of the D-brane along the three non-compact directions. We shall denote these three scalars by X 1 , X 2 , X 3 . The three chiral multiples {Φ i } contains three complex scalars {Φ i }. 2 These complex scalars give the coordinates or Wilson lines along x 4 + ix 5 , x 6 + ix 7 and x 8 + ix 9 directions respectively. For example, for the D6-brane all three complex scalars correspond to Wilson lines, while for the D2-brane wrapped along the 4-5 directions, Φ 1 corresponds to a Wilson line along x 4 + ix 5 but Φ 2 and Φ 3 correspond to positions of the brane along x 6 +ix 7 and x 8 +ix 9 respectively. Finally, we shall use a superscript (k) to label the four different D-branes, with k = 1, 2, 3 corresponding to D2-branes wrapped along the 4-5, 6-7 and 8-9 directions and k = 4 corresponding to the D6-brane along 4-5-6-7-8-9 directions. Besides these fields, for every pair of D-branes labelled by (k, ℓ) we have two chiral superfields Z (kℓ) and Z (ℓk) arising from open strings stretched between the two D-branes. They carry respectively 1 and −1 units of charge under the vector superfield V (k) and −1 and 1 units of charge under the vector superfield V (ℓ) .
We shall now write down the action involving these fields. To begin with we shall assume that the six circles of T 6 are orthonormal to each other, with each circle having radius √ α ′ and that there is no background 2-form field along T 6 . From now on, we shall set α ′ = 1. In this case the action takes the form where S kinetic denotes the kinetic terms for the vector superfields V (k) and the gauge neutral i . These have the standard form and will not be written down explic-2 Following usual notation, we shall use the same symbol to denote a superfield and its scalar component.
itly. The superpotential W has two different components. The first component describes the coupling of the superfields Φ (k) to Z (kℓ) and takes the form where ε kℓm is the totally antisymmetric symbol with ε 123 = 1. The second component describes the cubic self-coupling between the Z (kℓ) 's and takes the form where C is a constant whose value can be computed in principle by analyzing the coupling between open strings stretched between different branes, but we shall not need it for our analysis. The sum over k, ℓ, m runs over all distinct values of k, ℓ and m which are not related by cyclic permutations of (k, ℓ, m). There could also be gauge invariant quartic and higher order terms involving the Z (kℓ) 's, but as we shall see, these can be ignored in our analysis. So far we have assumed that background metric along T 6 is diagonal and that there are no background 2-form fields. We shall now study the effect to switching on small background values of the off-diagonal components of the metric and 2-form fields. As reviewed in appendix A, this has two effects. First it introduces Fayet-Iliopoulos (FI) term with coefficient c (k) for each of the four vector superfields, satisfying Second, it generates a linear term in the superpotential of the form (3.5) Explicit expressions for c (k) and c (kℓ) for 1 ≤ k < ℓ ≤ 4 in terms of the off-diagonal components of the metric and 2-form fields have also been given in appendix A.
Let us now write down the potential involving the scalar fields derived from the above action. This consists of three pieces. The first comes from the usual quartic coupling between the gauge field components X (k) i and the charged scalars Z (kℓ) and takes the form where 'bar' denotes complex conjugation. The second component of the potential is the D-term contribution. This takes the form The third component is the F-term potential which takes the form For finding a supersymmetric configuration we have to look for configurations with vanishing potential. Since the potential is a sum of squares, this requires setting each of these terms to zero. In §4 we shall look for solutions to these conditions.
Note that the potential has the following flat directions where ξ m and ζ k are arbitrary complex numbers and a i are arbitrary real numbers. The a i 's represent overall translation of the system along the non-compact directions. The symmetries generated by ξ m and ζ k imply that the potential has six complex flat directions. 3 This corresponds to six exactly massless chiral multiplets. Since each chiral multiplet contains a Weyl fermion in 3+1 dimensions which has four real components, we have altogether 6 × 4 = 24 real fermion zero modes after dimensional reduction to 0+1 dimensions. The vector superfield 4 k=1 V (k) also decouples from the action, reflecting the symmetry parametrized by the a i 's. The Majorana fermion belonging to this multiplet gives 4 more fermion zero modes. Thus altogether we have 24 + 4 = 28 fermion zero modes. These are the Goldstino modes associated with supersymmetry breaking; since a 1/8 BPS black hole in N = 8 supersymmetric string theory preserves 4 out of 32 supersymmetries, we expect 32 − 4 = 28 broken supersymmetries. Quantization of these 28 fermion zero modes gives the 2 14 fold degenerate supermultiplet which is the right degeneracy for a 1/8 BPS state in a theory with 32 supersymmetries.

Supersymmetric solutions
We shall now look for configurations preserving supersymmetry, i.e. configurations which make the potential vanish. As noted below (3.8), this requires setting each term in V gauge , V D and V F to zero. Furthermore, due to the U(1) 4 gauge symmetry of the original theory, we need to classify solutions up to equivalence relations under these U(1) gauge symmetries: where θ (k) for 1 ≤ k ≤ 4 are the gauge transformation parameters. Note that the overall U(1) -obtained by setting all the θ (k) 's equal -acts trivially on the Z (kℓ) 's. Furthermore, since we have fixed A It follows from this that as long as the c (kℓ) are non-zero for every k, ℓ in the range 1 ≤ k < ℓ ≤ 4, none of the Z (kℓ) 's can vanish. Eq.(3.6) now gives up to the translation symmetry parametrized by the constants a i in eq.(3.9). Next we consider the ∂W/∂Z (kℓ) = 0 equations. This gives  i 's are associated with flat directions and hence remain undetermined. Second they give the following relations among the Z (kℓ) 's: (4.6) Finally let us turn to the D-term constraints. It is well known that the effect of the D-term constraints together with quotienting by the U(1) gauge groups is to convert the space spanned by the coordinates Z (kℓ) to a toric variety. This is parametrized by the coordinates Z (kℓ) modded out by the complexified U(1) gauge groups after removing appropriate submanifolds of complex codimension ≥ 1 from the space spanned by the Z (kℓ) 's. These submanifolds are obtained by setting one or more Z (kℓ) 's to zero, and depend on the FI parameters c (k) . However, since we have seen that the F-term constraints force all the Z (kℓ) 's to be non-zero, removal of these complex submanifolds has no effect on the final solutions. 4 Thus, we can proceed by parametrizing the variety by an appropriate set of gauge invariant polynomials and forget about the D-term constraints. Since to start with there are 4 × 3 = 12 independent Z (kℓ) 's, and we quotient by a U(1) 3 gauge group -the overall U(1) having trivial action on all the Z (kℓ) 's -we need 12 − 3 = 9 independent gauge invariant coordinates. We take them to be (4.7) We now note that (4. respectively. The solutions to the system (4.10) are given in Table 1 of appendix C. The important point to note from Table 1 is that there are 12 distinct solutions. This shows that there are 12 supersymmetric ground states, in perfect agreement with the prediction (2.8) from the dual description. Furthermore since the moduli space of solutions is zero dimensional, all the solutions carry zero angular momentum after factoring out the contribution of the goldstino fermion zero modes. This is in agreement with the prediction from the black hole side. It is clear from the form of the potential as well as the solutions given in Table 1 that under a uniform scaling of all the c (k) 's and c (kℓ) 's by a real parameter λ, the Z (kℓ) 's and Φ (k) m 's at the solution (except the ones associated with flat directions) scale as λ 1/2 . Thus by taking λ to be small we can ensure that each Z (kℓ) and Φ (k) m at the solution is small. In this case the contributions from the quartic and higher order terms in the superpotential are small compared to the cubic terms that we have included. This justifies our ignoring such terms for studyng these solutions. This also justifies our ignoring the fact that Φ and Z (kℓ) 's are of order unity measured in the string scale. In such cases we must take into account possible higher order terms in the superpotential, and must also include the effect of Φ i 's being periodic variables so that we have to include in our analysis also open string states which wind around the various circles on their way from one D-brane to another. In other words, full stringy dynamics is needed for examining the existence of these states. Our experience with BPS state counting tells us however that the BPS states arise only from low energy fluctuations on the branes and hence it seems unlikely that there will be new BPS states from the stringy configurations of the type described above.

Non-abelian generalization
In this section we shall generalize the analysis of §3 to the case where some of the stacks have more than one brane, i.e. the N i 's introduced in §2 are not all equal to 1. We shall focus on the scalar fields and their potential since this is what is needed for the counting of supersymmetric solutions.
We begin with a discussion of how the scalar degrees of freedom change in this case.
First of all, the complex scalars Φ (k) i and the real scalars X (k) i become N k × N k hermitian matrices transforming in the adjoint representation of U(N k ). On the other hand, the complex scalar Z (kℓ) becomes N k × N ℓ complex matrix transforming in the (N k ,N ℓ ) representation of Let us now describe the modification of the potential. The superpotential W 1 given in (3.2) is generalized to The generalization of (3.3) takes the form The generalization of (3.5) is There is also an additional superpotential Tr [X Finally, the D-term potential (3.7) is generalized to where I N k denotes N k × N k identity matrix. The FI parameters c (k) now satisfy The coefficients c (kℓ) and c (k) are to be chosen so that they reproduce the masses of the Z (kℓ) 's correctly. The equations take the form of (A.4) and (A.5) with identical right hand sides, but the left hand sides are different since the masses of Z (kℓ) 's expressed in terms of c (k) 's and c (kℓ) 's have additional dependence on the N k 's. The potential given above has a shift symmetry generalizing (3.9) generalizing (4.2). The ∂W/∂Z (kℓ) equations give generalizing (4.4). It seems reasonable to assume that up to the translation symmetry described in the last line of (5.8), all the X (k) i 's vanish at the zeroes of the potential since this makes all the terms in V gauge vanish. This will also make the classical bound state have zero size in the non-compact directions. Furthermore, the effect of D-term constraints is to take the quotient of the space of solutions to (5.9), (5.10) by complexified 4 k=1 U(N k ) gauge transformations. Let M be the space of gauge inequivalent solutions to (5.9), (5.10) after factoring out the zero mode directions associated with the shift symmetry (5.8). The number of supersymmetric states (or more precisely the index B 14 ) will be given by the Euler number of M. Thus, duality symmetry of string theory predicts that whereĉ(u) has been defined in (2.6). If M is zero dimensional, then χ(M) just counts the number of solutions as in the abelian case. In that case all the microstates would carry strictly zero angular momentum after factoring out the contribution due to the goldstino fermion modes.

Conclusion
In this paper we have set up the general equations whose solutions describe the BPS states of type II string theory compactified on T 6 carrying only RR charges. We have been able to solve the equations explicitly when the charges take the lowest possible values. The result is in perfect agreement with the counting of the same states in a U-dual description.
Admittedly this is only a small beginning of the much more ambitious project. Nevertheless even at this level our analysis provides a non-trivial test of duality symmetry, since the counting leading to the magic number 12 is very different from the one that was used to arrive at the formula (2.6). As far as test of black hole entropy is concerned, a black hole carrying charges given in (2.7) has large curvature at the horizon and hence the Bekenstein-Hawking entropy is not expected to agree with ln 12. Nevertheless explicit computation of Bekenstein-Hawking entropy, together with one loop logarithmic corrections [30,31], give a macroscopic entropy In recent years there has also been progress in computing the macroscopic entropy of these black holes by evaluating the supergravity path integral in the near horizon geometry of the black hole using localization techniques [32][33][34][35][36]. In this approach one regards the N = 8 supersymmetric theory in 3+1 dimensions as an N = 2 supersymmetric theory with vector, hyper, gravitino and Weyl multiplets and evaluates the path integral. Although the arguments are not complete due to the inability to extend the analysis to include hypermultiplets and gravitino multiplets in the language of N = 2 supergravity, if we ignore this problem then the result of localization gives the following result for S macro from the leading saddle point [34] S macro ≃ ln √ 2π ∆ −7/4 I 7/2 (π √ ∆) , by the 0+1 dimensional fields Z (14) and Z (41) . The quadratic terms involving these fields, computed from the action in §3, is given by After diagonalization we find the renormalized (mass) 2 of the fields to be ± 16|c (14) On the other hand, we can calculate the mass of the open string stretched between the D2brane along the 4-5 directions and the D6-brane along the 4-5-6-7-8-9 directions as follows.

B Duality transformation
In [7], the counting of BPS states was done for a system consisting of N 1 KK monopoles associated with the 4-direction, −N 2 units of momentum along the 5-direction, N 3 D1-branes along the 5-direction, N 4 D5-branes along 5-6-7-8-9 directions and −N 5 units of momentum along the 4-direction. 5 Our goal will be to show that via a series of duality transformations this can be related to the system introduced in §2. During this analysis we shall ignore all the signs (which can in principle be determined by following some specific sign convention, e.g. the one given in appendix A of [24]). This way we shall at most miss the signs of the charges carried by the final D -brane configurations. However, our analysis of the world-line theory of the D-brane system is independent of the signs of these charges as long as the signs are chosen to give a configuration that preserves 4 out of 32 supersymmetries. Consider the following series of duality transformations: • T-duality transformations along the 4 and 5 directions: This gives a configuration of N 1 NS-5-branes along 5-6-7-8-9 directions, N 2 fundamental strings along the 5-direction, N 3 D1-branes along the 4-direction, N 4 D5-branes along 4-6-7-8-9 directions and N 5 fundamental strings along the 4-direction.
For N 5 = 0, this reduces to the configuration described in §3.