The full space of BPS multicenter states with pure D-brane charges

We investigate the space of BPS states in type IIA string theory on a T$^6$ wrapped by one D6 brane and three D2 branes wrapping three disjoint 2-tori. This system of branes has 12 ground states. We show that these 12 states are all recovered as Coulomb branch BPS multicenter bound states, in which each center preserves 16 supercharges. Moreover, we show that these multicenter solutions can only exist with zero angular momentum, supporting the conjecture that all black hole microstates have zero angular momentum. For large charges, they might describe"near-horizon limit"of fuzzballs.


Introduction
One of the greatest successes of string theory has been to reproduce microscopically the Bekenstein-Hawking entropy of several classes of supersymmetric black holes [1]. This is supported by the idea that D-brane states describe the physics of black holes at low string coupling g s . Investigating these two possible viewpoints: the microscopic D-brane picture and the macroscopic supergravity picture, and using quantities which are protected under change of g s , such as the BPS index or the entropy, have brought a better understanding of the quantum structure of BPS black holes (see [2,3] for reviews).
Type II string theory on a six-dimensional Calabi-Yau manifold with D-branes wrapped on various cycles has been an extensively rich framework for state counting [4][5][6][7]. There are essentially three regimes in which one can work, depending on the value of g s and the number of D-branes N (see Fig.1): The Higgs branch is the one supporting the microscopic single D-brane picture of bound states and leads to exact results at infinitesimally small g s . By increasing g s , two scenarios are possible. The most admitted one is that the majority of the Higgs branch states are recovered as single center black hole solutions. The second possible scenario is that the D-brane charges gather in several centers forming a molecule-like BPS bound states described by quiver quantum mechanics. We denote this branch as the multicenter "Coulomb" branch 1 . One can still define two regimes inside this branch: the supergravity regime, valid as long as g s N ≫ 1 where BPS states are well-described as macroscopic multicenter configurations, and the quiver regime, in which the system is described by a quiver quantum mechanics 2 .
In the present paper, we are interested in BPS solutions in the multicenter "Coulomb" branch of type IIA compactified on T 6 wrapped by Q D6 D6 branes and three D2 branes wrapping three disjoint 2-tori for Q D6 = 1 or 2. In the Higgs branch, the number of states is 12 and 56 for Q D6 = 1 and Q D6 = 2 respectively [6,9,10]. At a generic point of moduli space, the angular momentum of the microscopic D-brane states was found to be strictly zero. Motivated by this result obtained in the regime where g s N ≪ 1, and by the fact that in the regime where g s N ≫ 1 (when the black hole description is trustable) all microstates of a single centered supersymmetric black hole carry zero angular momentum, it was conjectured that microstates of such black holes continue to carry zero angular momentum at a generic point of the moduli space 3 .
We test the conjecture in the multicenter "Coulomb" branch by building all the BPS 1 The Coulomb branch is not the same as the Coulomb branch obtained by moving the D2 and D6 branes away from each other. We are refering to the multicenter Coulomb branch where the separations between the centers cannot be modified freely.
2 See [8] for an exhaustive description of the different regimes. 3 Which includes g s .
Higgs branch Microscopic D-brane picture Multicenter "Coulomb" branch Quiver multi-particles Supergravity multi-centers Figure 1: The description of a D-brane system in different regimes of parameters where g s is the string coupling and N is the total number of branes.
multicenter bound states with the same D-brane charges and by investigating their possible moduli at infinity. Strictly speaking, we are in a multicenter "Coulomb" branch, far from the supergravity regime (g s N ≪ 1) and one should construct these configurations as solutions of quiver quantum mechanics. However, thanks to the work done in [8,[11][12][13][14][15], one knows that the conditions of existence for quiver BPS multi-particle solutions are exactly the same as the ones for BPS multicenter solutions in the supergravity picture even if the supergravity solution is not reliable. As long as the geometry is not considered, the picture of charge vectors located at some centers still holds true for g s N ≪ 1. Thus, for a counting problem, one can use the supergravity framework which is more commonly understood to build the BPS solutions and study their general properties.
A BPS multicenter solution is constructed by choosing a set of centers in a R 3 base space which carry magnetic and electric charges corresponding to branes wrapping cycles of the transverse space and by choosing a set of moduli at infinity. Requiring the solutions to be supersymmetric and free of closed timelike curves restricts significantly their number. We consider only 16-supercharge centers following the intuition that black hole microstates are built with maximally-supersymmetric objects [16]. For Q D6 = 1, we find exactly 12 BPS multicenter solutions. We show that they are not allowed to carry a non-zero angular momentum. This is in sharp contrast with the usual belief that it is easy to add momentum to multicenter solutions by slightly changing the moduli at infinity. From a supergravity point of view, it means that the possible asymptotics of the solutions are significantly restricted to the one preserving the zero angular momentum. In four dimensions, these solutions necessarily have AdS 2 ×S 2 asymptotics. Furthermore, each solution can be dualized to a six-dimensional D1-D5-P frame where it can have an AdS 3 ×S 3 asymptotics. However, the configurations we construct cannot give rise to asymptotically flat solutions in any frame.
The AdS 2 asymptotics of these geometries make them particularly interesting for the following reason. The near-horizon region of an extremal black hole develops an AdS 2 factor and this region decouples from the rest of the spacetime. This enables one to compute many physical black hole quantities, such as (quantum) entropy function [3,17,18], solely in this near horizon region. Thus, an asymptotically AdS 2 geometry, carrying appropriate charges, can be thought of as replacing the horizon by some structures in the "near horizon" region. It remains to be seen whether such "near horizon" structures can be thought of as "near horizon limit" of some full fledged fuzzball geometry, which lives in the same asymptotically flat ambient spacetime as the black hole. Despite this subtlety and the fact that we are really working in a regime where gravity can not be trusted, finding exactly e S BH number of such microstate geometries, does inspire curiosity about the possibility that they indeed correspond to "near-horizon limit" of some fuzzball geometries.
Moreover, it has been argued in [19] that, in string theory, the ground states of the CFT 1 dual to AdS 2 must break conformal invariance. Thus, their bulk duals which are asymptotically AdS 2 geometries must break conformal invariance by having a scale in the IR. This argument eliminates the possibility that the single-center asymptotically AdS 2 solution (which has an infinite throat and preserves conformal invariance in the IR) is dual to any ground states of a non-trivial CFT 1 . Our work confirms this expectation: From a microscopic counting the number of ground states of the CFT 1 corresponding to our D-brane system is 12. And indeed we have found exactly 12 asymptotically AdS 2 bulk multicenter solutions. Furthermore, all these solutions have a scale, given by the distances between the centers, and hence break conformal invariance in the IR, exactly as predicted by [19].
In section 2, we review the picture of BPS multicenter solutions as charge vectors Γ a localized at centers in a R 3 base space and we review the constraints on the bound states we are looking for. In section 3, we review the state counting obtained in the Higgs branch at g s N ≪ 1 [9,10]. In section 4, we explain the technical details of our own construction in the multicenter "Coulomb" branch. Finally, in section 5, we discuss the main features of the 12 solutions and we discuss the extension of our construction to D-brane bound states with Q D6 = 2.

BPS multicenter solutions
We would like to construct states with four supercharges carrying D-brane charges for g s ≪ 1 and N ∼ 1 in type IIA string theory on a T 6 . In this regime of parameters, the states are described as multi-particle quiver bound states [8,14,15]. As explained in the introduction, we prefer to use supergravity tools. This does not compromise our analysis since we are interested in counting the number of states and in the charge profiles carried at each center. At this level, both the supergravity and quiver frames give the same results [11,13,15].

BPS multicenter solutions
We work in the context of type IIA string theory on a T 6 = T 2 × T 2 × T 2 at the low energy limit g s ≪ 1. We consider multicenter BPS solutions where each center can carry D0-D2-D4-D6 charges wrapping the tori. They are stationary but generically non-static BPS bound states with four unbroken supersymmetries. They are specified by charge vectors Γ a at each center and an asymptotic vector Γ ∞ incorporating the moduli at spatial infinity. We denote The solutions can also be expressed in terms of a set of 8 harmonic functions in R 3 (we refer the reader to the Appendix A for a description of the field content and the resolution of the BPS equations.) where I = 1, 2, 3 and r a is the three-dimensional distance from the a th center. They can be incorporated to the charge and asymptotic vectors as The ten-dimensional type IIA metric is given by where ds 2 I is the internal metric of the I th 2-torus, Z I are the warp factors giving rise to the three charges of the solution and I 4 is the quartic invariant (2.5) As it has been argued previously, the geometry given by (2.4) is not reliable in a regime where N ∼ 1. In such a regime, the solutions should be seen as low-charge multi-particles which does not significantly backreact (2.4). However, the conditions of validity derived from the metric in the supergravity regime (2.4) are still valid in the low-charge regime. Furthermore, from an holographic point of view, it is also interesting to have access to the geometrical features of the multicenter solutions we want to build when they are sent to the macroscopic regime 4 . To solve the BPS equations, the warp factors Z I and the 1-form ω are constructed as follows, with C IJK = |ǫ IJK |.
All BPS solutions need to be free of closed timelike curves and Dirac-Misner strings. The first condition requires the positivity of the quartic invariant I 4 (see Appendix A), while the second restricts the inter-center distances r ab [11] We have defined a symplectic product , The equations (2.8) are known as the bubble equations and impose strong constraints on the space of solutions. The positivity of the quartic invariant is also a constraint which is difficult to manipulate. For our analysis, we will apply different versions which have been used in previous works. First, a necessary condition for the positivity of the quartic invariant is to satisfy the three inequalities This condition is easier to derive and check than (2.7). Moreover, in practice, it happens to be a sufficient condition. Second, we will apply a conjecture which reduces the condition regarding closed timelike curves to simpler algrebraic conditions on the bubble equations (we refer the reader to the section 3 of [20] for details).
Moreover, a generic BPS multicenter solution is stationary but not static. Its angular momentum, J, is given by The value of the angular momentum is closely related to the moduli at infinity of the solution. Indeed, using the bubble equations one can show that Finally, being interested in specific D-brane charge configuration we express the quantized asymptotic D-brane charges according to the charges at each center [21]. They can be derived from the RR gauge field forms C (1) and C (3) and their dual gauge fields C (5) and C (7) (see Appendix A) (2.13)

Types of center
Since we are looking for BPS multicenter solutions with four supercharges, the choice of charge vectors (2.1) is restricted to the one which preserves supersymmetry. The maximally-supersymmetric centers are the two-charge supertubes and the Gibbons-Hawking (GH) centers. They preserve 16 supercharges and the two U(1) isometries of the three-dimensional base space. This is not an exhaustive list of possible string theory objects. One can also imagine objects as wiggly supertubes, four-dimensional superstratum and so on. They are less supersymmetric and they may break some of the U(1) isometries [22]. Following the Bena-Warner ansatz [16], it is more likely that a system of low or pure D-brane charges (2.13) is fully or largely made of maximallysupersymmetric BPS objects.

Two-charge supertube centers
A two-charge supertube located at the a th center carries a dipole charge k a , two electric charges Q (b) a with b = a and a momentum charge m a [23]. In other words, a twocharge supertube of species I has a D0 charge, a D4 charge Q I D4 and two D2 charges Q J D2 and Q K D2 where I, J and K are all different and between 1 and 3. Consequently, there are three possible species of two-charge supertubes depending on which of the three possible D4 charges they can carry. If there is only one species of supertube in the center configuration, the solution can be dualized to a smooth spacetime in six dimensions. Otherwise, multi-supertube configurations are not smooth because each supertube sources different vector fields and one cannot render the geometry smooth using a particular vector as Kaluza-Klein vector, see [24].
As an illustration, let us consider a two-charge supertube of species 1 located at the a th center with the charge vector Γ a (2.14) In the analysis performed in [25][26][27], where it was derived that the condition to preserve 16 supercharges and the quantization of the charges require to fix the following parameters In anticipation of what will follow, one can have supertube centers with some of its charges being zero. For instance, on can construct a center with Γ a = (0, 0, 0, 0 ; 0, 0, Q a , m a ), (2.16) by simply imposing the D4 charge and one of the D2 charge of the supertube to be zero. We can similarly have Γ a = (0, k a , 0, 0 ; 0, 0, Q a , 0).
These types of centers should not be denoted as supertube centers since they are 16supercharge simple D-brane centers. However, in our construction, we will abusively use the generic term "supertube center" even for these objects.
To be maximally supersymmetric, it has to satisfy the following additional regularity Thus, a GH center located at the a th center has a charge vector The BPS multicenter solutions we focus on are bound-states of a certain number of such centers. They are sensible configuration when there are no Dirac-Misner string between centers and no closed timelike curves. This is achieved imposing the bubble equations (2.8), which fix the positions of the centers, and the global bound (2.5). Provided those conditions are satisfied, we have a physical solution. However, experience shows that finding a set of appropriate parameters can involve a vast exploration.

The moduli at infinity
The moduli at large distance of a multicenter BPS solution are encoded by the asymptotic vector Γ ∞ . From a macroscopic point of view, the constant terms in the harmonic functions given by Γ ∞ fix the asymptotics of the solution, which can be directly seen from the behavior of the metric at large distance (2.4). For instance, it has been showed in [19], that having no constant terms in the eight harmonic functions (Γ ∞ = 0) corresponds to asymptotically AdS 2 multicenter solutions. An asymptotically flat R 4,1 multicenter solution can be obtained by having a constant term in each L I function and a constant term in M (Γ ∞ = (0, 0, 0, 0; 1, 1, 1, m ∞ ) where m ∞ is fixed to have a Γ ∞ , Γ a = 0 ). Last but not least, a multicenter solution is asymptotically AdS 3 in the dual six-dimensional D1-D5 frame when one L I has a constant term turning on (Γ ∞ = (0, 0, 0, 0; 1, 0, 0, m ∞ ) for instance) [28].
In a regime far from the supergravity regime N ∼ 1, the moduli at infinity Γ ∞ lose their geometrical meaning. However, one can still relate the microscopic multicenter solutions we are building to their macroscopic equivalents by sending Γ → ΛΓ with Λ ≫ 1 where the geometry is trustable. So we can still relate our states to a certain type of asymptotics and then determine their holographic meaning.
The different choices of moduli at infinity can affect drastically the existence of a multicenter solution because of the bubble equations (2.8) [29]. Moreover, they affect significantly the value of the angular momentum (2.12).
In this paper, we will first build multicenter solutions with zero angular momentum J following the results in [9,10]. According to (2.12), one can impose Γ ∞ , Γ a = 0 for all centers. This does not necessarily imply that Γ ∞ = 0 and that the solutions are asymptotically AdS 2 . However, taking Γ ∞ = 0 is always a possibility. Consequently, one can just consider that Γ ∞ , Γ a = 0 in all our computations and build the charge vectors. Then, nothing stops us to test what moduli at infinity are indeed compatible with those charges. In particular, one can try to impose some moduli at infinity which give an angular momentum to the solution. This will give a non-trivial test to the zero-angular momentum conjecture. What we will find will confirm the conjecture: no moduli at infinity which gives angular momentum to our states are compatible with the charges.
3 Counting Higgs branch states with pure D2 and D6 charges In this section, we give a brief account of the state counting performed in [9,10]. In these papers the authors considered a D-brane system with four unbroken supercharges in four dimensions, in a duality frame where all charges were Ramond-Ramond charges. This theory is obtained by compactification over a T 6 and has 32 supercharges. Broken supersymmetries give rise to 32 − 4 = 28 Goldstinos and 28 partner Goldstone bosons. The Goldstones and Goldstinos can be arranged in 7 supersymmetric multiplets, in four dimensional termionology. Due to these Goldstino multiplets, the Witten index vanishes. In order to get something non-vanishing yet protected, one needs to consider the 14 th helicity supertrace [30,31] This essentially removes the contribution from the Goldstino multiplets and computes Witten index in the rest of the theory.
Microscopically, this D-brane system comprises 3 stacks of D2 branes, along three disjoint 2-cycles of the T 6 , and 1 stack of D6 branes along T 6 . In order to compute the index, it suffices to have the knowledge of only low energy dynamics of this configuration. This is described by a quantum mechanics living on the intersection point of the branes. Fields in this theory correspond to massless strings stretched between various branes. The spectrum can be arranged in 4 dimensional supersymmetric multiplets, with the understanding that they are 0 + 1 dimensional fields. For the rest of the section, we use four dimensional terminologies, with the understanding that everything is dimensionally reduced to 0 + 1 dimension.
The Lagrangian, L, has the following schematic form where i denotes the brane index. The first piece, L N =2 , denotes the interaction of i th and j th brane, which preserves 8 supercharges. These two pieces are determined by supersymmetry alone. Altogether these terms preserve only 4 supercharges, since each of them preserve different supersymmetries. Supersymmetry allows for, but does not determine, other interactions preserving 4 supercharges, denoted as L N =1 . In [9,10], the authors considered few terms of L N =1 and argued that higher order terms, although certainly present, do not affect the index computation.
Computing B 14 , is same as computing the Witten index in the theory obtained by throwing away Goldstino multiplets, which are non-interacting. In [9], authors identified the Goldstones, which are bosonic counterparts of the Goldstinos, using physical reasoning. The problem now reduces to one of computing Witten index in supersymmetric quantum mechanics, which is known to be given by Euler number of the vacuum manifold. For small charges, this was computed in [9,10] and was found to be in agreement with existing computations of the same in other duality frames [6].
Typically microscopic computations of the index are performed at a special point of moduli space, and in such cases the index receives contributions both from bosonic and fermionic states. The computations of [9,10] however required turning on some moduli and therefore going to a more generic point of moduli space. Consequently, all the ground states of the D-brane quantum mechanics were found to be bosonic, in particular they carried zero angular momentum 5 . This feature also holds for the regime of moduli space, where the brane system is better described as a single-center 6 supersymmetric black hole. Such black holes are also known to carry zero angular momentum. In view of these facts, the authors of [10] conjectured that at a generic point of moduli space, all microstates of a single-center BPS black hole carry zero angular momentum.

Counting zero-momentum multicenter states with pure D2 and D6 charges
In this section, we investigate the same D-brane configuration in a different region of parameters. We focus only on the space of BPS multicenter bound states even if it is possible that some or even all the Higgs branch states shift to different kinds of solutions as single-center solutions once g s is turned on. To start with, we do not expect that counting of the states in the multicenter "Coulomb" branch gives the same number as the counting in the Higgs branch. Our procedure is to build the BPS multicenter solutions first and then count them. Thus, we will be able to analyze directly the features of the solutions.
We build zero-momentum multicenter solutions with the following set of asymptotic D-branes charges: With such low charges, the supergravity regime should not be valid and the geometry of the center depicted in the section 2 should not be trustable. However, one can use the supergravity solutions to count states. In [8,11,12,14,15], it has been shown that BPS multicenter solutions with low charges can still be depicted by charge vectors (2.1) satisfying the bubble equations (2.8) and a condition equivalent to the absence of closed timelike curves (2.7). Consequently, one can still use the supergravity framework detailed in section 2 to build all the multicenter solutions satisfying (4.1). Our approach consists in scanning analytically or numerically all the valid multicenter solutions formed by either GH or supertube centers starting with the family of three-center solutions then the family of four-center solutions and finally the five-center solutions 7 . The growing complexity of the analysis does not allow to scan solutions with more than five centers but we have a strong intuition that adding centers increases necessarily the global D-brane charges of the solution. Thus, if solutions exist, they should consist in few centers. The constraints which restrict the possible solutions are: • The global D-brane charges (4.1).
• The constraints at each center depending on the nature of the center (2.18) or (2.15).
We have found exactly 12 distinct three-center solutions satisfying all those constraints. This matches exactly the microscopic counting. Readers interested only in the main ideas of our analysis can skip the next subsections until section 4.4.

The family of three-center solutions
One needs at least one GH center in the configuration to have a non-zero D6 charges. We divide our analysis in three subfamilies: -Solutions with one GH center and two supertube centers.
-Solutions with two GH centers and one supertube.
-Solutions with three GH centers.
For the first subfamily, an analytic approach is possible. All the details of this analysis are given in Appendix B.1. The main result is that there exist 12 inequivalent solutions in this subfamily. As for the second and the third families, the number of parameters makes the analytic approach impossible. However, we have performed an efficient numerical analysis, fully detailed in Appendix B.2. We have scanned a significant part of the parameter space of the solutions by varying each GH charges q a , k 1 a , k 2 a , k 3 a and supertube charges k a , Q (b) a from -500 to 500. We didn't find any solution satisfying all the constraints in this domain of values.
In [32], the authors have tackled a similar issue by analyzing the parameter space of three-GH center solutions whose the total D6 charge is three and the three D2 charges are one. Interestingly, they have found that the total number of such multicenter solutions is also 12. This is half a coincidence with our computation. Even if the three-center solutions they study have a larger Q D6 8 , the form of their specific solutions are governed by the same type of permutations giving rise to the 12 states we found here.

The family of four-center solutions
A four-center solution has more degrees of freedom than the previous solutions and the constraints are more complicated to deal with. This makes any analytic investigation very hard to perform. However, we have done a numerical analysis of the following subfamilies: -Solutions with one GH center and three supertube centers.
-Solutions with two GH centers and two supertube.
-Solutions with three GH centers and one supertube.
-Solutions with four GH centers.
For each subfamily, we have analyzed a significant part of the parameter space by varying all the parameters from -10 to 10. The details are given in the Appendix C.1. Our final result is that there are no valid solutions with four centers.

The family of five-center solutions
For five-center solutions, even a scan of the parameter space is complicated. This is principally due to the number of parameters available and the complexity of the constraints for configurations with five centers. However, we have been able to pick randomly a huge number of solutions with the right global D-brane charges and check if they can satisfy the bubble equations and the absence of closed timelike curves at the same time. We did not find any. This gives good intuition that no solution with five centers exists as well.

Summary
We have analyzed analytically and numerically a huge number of BPS multicenter solutions to find only 12 solutions satisfying all the constraints. This exactly matches the exact counting of the 12 Higgs branch states [9]. They are all recovered as "Coulomb" branch multicenter bound states. They belong to the family of three-center solutions with one GH center and two supertubes of different species. The 12 solutions are given in full detail in Table 1. Moreover, as explained in section 2.2, for most of the solutions found, the two-charge-supertube centers are actually simple fluxed D-brane centers 9 . For instance, the six first solutions in Table 1 have a GH center and two D4-brane centers with an induced D2 charge. The six other solutions have one GH center, one two-charge-supertube center and one simple D2-brane center with an induced D0 charge.  We do not have indisputable arguments that having more centers will not give rise to other valid solutions but only good intuition. Usually, adding centers increases the global D-brane charges. Another difficulty in adding centers follows from the fact that these centers must carry negative D-brane charges, in order to keep the total D brane charges intact. However usually centers with negative D-brane charges are tricky, when it comes to the Z I V ≥ 0, i.e. absence of closed timelike curves. For these reasons, we can consider our analysis exhaustive even if we have analyzed configuration with few centers.

Features of the twelve solutions
The 12 solutions found are all BPS three-center solutions formed by one GH center and two other 16-supercharge centers. Although we looked for all possible center configurations, it so happens that the regular solutions we found, all have their centers lying on a line and hence are all axisymmetric. The fact that the index is reproduced by counting configurations with collinear centers was also observed in [33] and, given the very complicated algebra that required our physical solutions to be collinear, we do not believe this is a coincidence.
The center configuration has also a rescaling symmetry r ab → λ r ab . This is a consequence of having zero angular momentum which implies that the bubble equations have no right-hand terms (2.8) and so a scaling degree of freedom. When the solutions are asymptotically AdS 2 , this scaling symmetry arises from the conformal invariance of the dual theory (the only scale in the system is the one introduced by the breaking of the conformal symmetry in the IR). When the solutions are asymptotically AdS 3 , then the solutions with different values of the inter-center distances are physically inequivalent (the solutions have two scales, one given by the distances between the centers and one given by the constant in the harmonic function 10 ).
We can now investigate what kind of moduli at infinity determined by Γ ∞ are compatible with our multicenter solutions. The fact that microstates of black holes have necessarily zero angular momentum at any point of the moduli space has been conjectured in [10,34,35]. A single-center black hole solution clearly has this property. This is because when defining the near horizon AdS 2 path integral, that is needed to compute quantum entropy function, one needs to fix the charges. And angular momentum appears as a charge in this path integral. Nevertheless, one can also argue that a singlecenter black hole does not correspond to any pure state of the CFT 1 dual to AdS 2 [19], and the zero-angular momentum assymptotically AdS 2 solutions dual to pure states of the CFT 1 will have a non-trivial angular momentum when embedded in asymptotically AdS 3 geometries. This is what happens in the solutions constructed in [19]. We would like to understand whether our 12 multicenter solutions can also develop a non trivial angular momentum when embedded in an asymptotically AdS 3 space, or whether they are incompatible with J = 0.
• There are moduli at infinity which preserve J = 0. As it has been discussed in section 2.3, this implies that Γ ∞ , Γ a = 0 for all the centers. Due to the numerous zeroes in the charge vectors, there are many possible Γ ∞ . However, one needs to carefully check that these Γ ∞ do not induce closed timelike curves. We give a non-exhaustive list of possible moduli at infinity compatible with our solutions. Because all the 12 solutions have similar properties, it suffices to list the possibilities corresponding to the first solution in the Table 1. We list only the interesting moduli at infinity and their holographic meaning: -Γ ∞ = 0. As depicted in section 2.3, this corresponds to asymptotically AdS 2 microstate geometries. The 12 bound states are thus holographically dual to the 12 ground states of a CFT 1 . In [19], it has been argued that the "e S " ground states of a non-topological CFT 1 must break conformal invariance. So, all their bulk duals must have a scale, as it is actually the case for multicenter or superstratum solutions 11 . A single-center asymptotically AdS 2 solution does not have any scale and thus should not be dual to any ground state of a non-trivial CFT 1 . Our results corroborate this conjecture.
-Γ ∞ = (0, 0, 0, 0; 0, 0, 1, 0). The meaning of this choice can be seen only from a six-dimensional point of view in the dual D1-D5-P frame. In this frame, the solution turns out to be an asymptotically AdS 3 state.
Both kinds of moduli at infinity impose J = 0 and do not have any impact on the center configuration.
• There are also moduli at infinity which do not preserve the symplectic products Γ ∞ , Γ a and therefore give tise to a finite angular momentum, J, without affecting the D-brane charges. This type of moduli requires to solve once again the bubble equations (2.8). Because the inter-center distances can be as small as possible, one might expect that a change of Γ ∞ , Γ a can be absorbed by a very small change of distances in b Γa,Γ b r ab +δr ab . This is unfortunately not guaranteed especially for axisymmetric multicenter configuration 12 . Our multicenter solutions illustrate this feature. One can prove that any moduli at infinity which has Γ ∞ , Γ a = 0 and which does not induce closed timelike curves is incompatible with our solutions. Thus all our solutions are incompatible with having a non-zero angular momentum.
This corroborates in a different regime of g S the conjecture that every microstate of a single centered supersymmetric black hole must have zero angular momentum at any generic point of moduli space [10,34,35]. However, from an holographic point of view, the solutions can be either asymptotically AdS 3 13 or asymptotically AdS 2 solutions.
For the particular choice of moduli at infinity Γ ∞ = (0, 0, 0, 0; 0, 0, 1, 0), only 6 of the 12 solutions survive. Those are the solutions with K 3 = 0, i.e. the solutions labelled by 1, 2, 7, 8, 9 and 10 in the Table 1. The six others have unsolvable bubble equations. At this particular point on the moduli space, one can wonder where the other 6 solutions are to make the total number of solutions to be 12. Two scenarios are possible. The first one is that at this point of the moduli space, there are only 6 multicenter solutions and the 6 others are of a different nature. The second one is that 6 brand new multicenter solutions appear which reversely cannot be sent to the point of moduli space where Γ ∞ = 0. This interesting issue will be treated in future works.

The solutions with Q D6 = 2
As it has been done in [10], one can extend our construction to compute the number of states with global D-brane charges: The index is known to be 56 for these charges [6,10]. We have been able to find only 18 BPS bound states formed by three centers similar to the ones described in section 4 and 12 BPS unbound states formed by four axisymmetric centers. The special aspect of those 12 unbound states is that one of the centers does not interact with the other centers and can be placed anywhere on the axis of the center configuration. We did not consider these solutions since they are similar to the "real Coulomb" branch where the centers can move freely. Since we did not count the unbound Coulmb branch configuration with non-interacting D6 center and three D2 centers. We are not counting these unbound states either.
Our present construction technique does not allow easily to go beyond the multicenter solutions with GH centers or supertube centers and to find the 38 missing states. This will require more work in future projects. However, one can already have an idea of where these states may come from: • Adding extra gauge fields and preserving the U(1)×U(1) isometry of the base space. Following [36,37], one can add for example a fourth massless abelian gauge field to the configuration, which corresponds to changing the fluxes on the T 6 . The solutions will be slightly more complex but they will remain U(1)×U(1) invariant and one may hope, along the line of [33], that these will contribute to the index.
• Constructing configurations which break the U(1)×U(1) isometry. Such objects may include wiggly supertubes [38] or superstratum configurations [22,39]. In four dimensions, these solutions give rise to KK modes along the two U(1) fibers of the base space and do not correspond to supergravity solutions.

Conclusion
In this paper, we have investigated the space of states of the following D-brane configuration , 0, 0, 0; 1, 1, 1, 0), with Q D6 = 1 or 2. (6.1) We have reviewed a computation in the Higgs branch where the states are depicted as microscopic single D-brane bound states [9,10]. The number of states is 12 and 56 for Q D6 = 1 and Q D6 = 2 respectively. These states has been argued to carry zero angular momentum.
We have tackled the same issues in the multicenter "Coulomb" branch. Multicenter BPS bound states are characterized by specifying charge vectors carried by center-like particles in the three-dimensional base space and a moduli at infinity [8]. All the multicenter BPS bound states can be retrieved from the microscopic D-brane bound states when g s → 0 but the opposite is not necessarily true. For Q D6 = 1, we have found exactly 12 BPS multicenter bound states. In this instance, all the microscopic D-brane states are recovered as BPS multicenter bound states and no single-center state should exist. This conveys the idea that BPS multicenter microstates which are types of fuzzballs in the macroscopic regime do not correspond to a peculiar part of the overall space of states of a certain D-brane system. Furthermore, we have shown that the 12 multicenter solutions carry necessarily zero angular momentum at this point of the moduli space giving greater weight to the zero angular momentum conjecture. From a supergravity point of view, this means that they are incompatible with having flat asymptotics. For Q D6 = 2, only 18 BPS multicenter bound states have been found. However, we expect more multicenter solutions to exist. Indeed, our construction essentially focuses on U(1)×U(1) invariant centers carrying 16-supercharges. One can expect for configurations with more than pure D-brane charges to exist. Such centers may be more exotic and less supersymmetric, such as wiggly supertubes. This investigation will lead to future projects. Nevertheless, the 18 solutions found also confirm the zero angular momentum conjecture.
in this basis). We denote The eight harmonic functions {V, K I , L I , M} have poles at each center labeled by a, with charges given by the charge vectors Γ a ≡ (q a , k 1 a , k 2 a , k 3 a ; l 1 a , l 2 a , l 3 a , m a ) with integer components and an asymptotic behavior given by the asymptotic vector Γ ∞ depending on the moduli at spacial infinity.
All the timelike supersymmetric field configurations with two U(1) isometries have a metric where Z I are the warp factors giving rise to the three charges of the solution and I 4 is the quartic invariant From the metric (A.2), the absence of closed timelike curves is straightforwardly guaranteed when the quartic invariant is positive I 4 > 0.
The dilaton Φ, the Neveu-Schwarz (NSNS) potential B (2) , the Ramond-Ramond (RR) potentials C (1) and C (3) are given by [21,40] where dT I is the volume form on the I th 2-torus and A is a KK one-form satisfying ⋆ (3) dA = dV . One can derive from those expressions the NSNS 3-forms H (3) , the RR field strengths F (2) and F (4) and their dual gauge fields (C (5) , F (6) ) and (C (7) , F (8) ) (see section 2.2 of [40] for more details). These field configurations become a solution when the following set of BPS equations, defined on the three-dimensional base space, is satisfied The equations (A.5) and (A. 7) give where the symplectic product has been defined in (2.9).
Finally, one can compute the overall D-brane charges of the solution defined as integrals of the RR field strengths over cycles of the form S 2 ∞ × T 2 I where S 2 ∞ is the asymptotic two sphere of the three-dimensional base space: We review our method to construct zero angular momentum BPS multicenter solutions with global D0 and D4 charges being 0 and D6 and D2 charges being 1 starting from solutions with two supertube centers of different species and one GH center.
• We start with the full parameter space of solutions. The GH center is the 0 th center with charges q, κ 1 , κ 2 and κ 3 . We consider also a two-charge supertube of species 1 located at the 1 st center with charges k 1 , Q 1 and Q 1 and a two-charge supertube of species 2 located at the 2 nd center with charges k 2 , Q The general form of the eight harmonic function is: First, we want to impose the values of the eight global D-brane charges. This reduces the number of free parameters to two. One can express everything in terms of k 1 and k 2 : 2 } = {1, We have used the notation Γ IJ = Γ I , Γ J . We notice that the solutions are invariant under rescaling of inter-center distances r IJ → λ r IJ . That is why, r 01 remains a free parameter all along the construction. Furthermore, the solution corresponds to a physical center configuration if and only if it satisfies the triangle inequality (r 01 + r 02 − r 12 ) (r 01 − r 02 + r 12 ) (−r 01 + r 02 + r 12 ) ≥ 0 which constrainsthe two-dimensional parameter space of k 1 and k 2 significantly.
• The solution must have a positive quartic invariant (2.5) to guarantee the absence of closed timelike curve. One can either use the conjecture in [20] or the condition (2.10). Small number of centers makes the second option to be the simplest. We expand Z I V around each center. We find that which further constrains the parameter space defined by k 1 and k 2 . We remind the reader that these conditions are not necessarily sufficient to be free of closed timelike curves. One needs to check once those conditions satisfied that the quartic invariant is indeed positive.
• Last but not least, one has to impose all the charges in the harmonic functions (B.1) to be integer.
(B.7) Requiring each charge of the harmonic functions to be integer restricts (B.7) to six possible values (k 1 , k 2 ) = {(0, 1), (0, −1), (1, 0), (1, −1), (−1, 0), (−1, 1)}. We can repeat exactly the same procedure with solutions of two supertubes of species 1 and 3 and solutions of two supertubes of species 2 and 3. By carefully counting the redundancies, we have a final count of 12 inequivalent solutions. Their charge vectors as well as their center configuration are given in detail in Table 1. Their main and common features are that the center configurations are axisymmetric with a U(1) symmetry and all centers carry D-brane charges of value -1,0 or 1. Moreover, as explained in section 2.2, for most of the solutions found, the two-charge-supertube centers are actually fluxed D-brane centers. The six first solutions in Table 1 have a GH center and two D4-brane centers with an induced D2 charge. The six other solutions have one GH center, one two-charge-supertube center and one simple D2-brane center with an induced D0 charge. We have carefully checked that the quartic invariant is strictly positive for all solutions found and that they are not related by gauge transformations. One can also wonder why we do not consider configurations with two supertube centers of the same species. This is straightforward to check that such configurations are strictly incompatible with the global D-brane charges we impose (4.1).

B.2 Numerical analysis of solutions with one supertube and two GH centers and solutions with three GH centers
We review our numerical method which shows that there exists no valid solutions satisfying (4.1) with one supertube and two GH centers or with three GH centers. The number of parameters of such solutions makes an analytic approach difficult. The steps of our numerical analysis were the following: • First, we start with the most general solutions. The solutions with one supertube and two GH centers form a family of 11 parameters whereas the solutions with three GH centers form a family of 12 parameters.
• We fix 8 parameters by imposing the right global D-brane charges (4.1).
• We run the other free parameters from -500 to 500. Each value corresponds to one particular solution. For each one, firstly we check if the solution has integer charges, secondly if the the solutions of the bubble equations can give physical center configurations (B.5), and thirdly if the quartic invariant is positive. Checking the positivity of the quartic invariant is the hardest part. We have principally used the conjecture postulated in [20]. This conjecture drastically simplifies the loop computations. It allows to check the positivity of the quartic invariant all over the R 3 base space by just checking an algebraic condition on a matrix derived from the bubble equations. This conjecture should work for multicenter solutions with GH centers only. However, one can mathematically consider supertube center as a limit of a GH center. For instance, one can obtain (2.14) from (2.19) by taking the limit ǫ → 0 with q a = −ǫ k 1 , k 1 a = k 1 , k 2 a = ǫ Q (3) a , k 3 a = ǫ Q (2) a . (B.8) Thus, we can extend the conjecture to our solutions.
We did not find any solutions satisfying all the conditions in the huge range of parameters we have scanned. Furthermore, from the previous section we have a good intuition that if a solution exists the charges should be small. Consequently, one can say that our numerical analysis suggests that there is no solution of three GH centers or one supertube and two GH centers satisfying (4.1).

C Analysis of four-center and five-center solutions C.1 Analysis of four-center solutions
We perform a similar analysis as in section B.2. The main goal is to scan a significant part of the parameter space looking for BPS four-center solutions satisfying (4.1).
• As before, we start with the most general solutions. The solutions with three supertube and one GH centers form a family of 13 parameters, the solutions with two supertube and two GH centers form a family of 14 parameters, the solutions with one supertube and three GH centers form a family of 15 parameters and the solutions with four GH centers form a family of 16 parameters, .
• We fix 8 parameters by imposing the global D-brane charges (4.1).
• We run the remaining parameters from -5 to 5 (the range of values is smaller than in section B.2 due to the higher number of free parameters). For each value, we check if the solution is a valid BPS multicenter solution: -First, we check if all the harmonic-function charges are integer.
-Second, we check if the solution of the bubble equation can give rise to a physical center configuration. Because we have four centers and not three, this step is more complex than the one in the previous section. Indeed, we have to check four triangle inequalities as (B.5) for each face of the tetrahedron formed by the four centers plus an angle inequality at one vertex of the tetrahedron.
-Third, we check the absence of closed timelike curves as in the previous section.
No solution have been found in the range of values. One can realistically extend this result to all four-center BPS solutions.

C.2 Analysis of five-center solutions
The number of parameters and the complexity of the constraints for five-center configurations make the numerical scan of the parameter space impossible. However, we have randomly generated some solutions and checked if they are valid and physical. The main idea is to fix as many parameters as possible using the equations (the global D-brane charges, the bubble equations) and pick random values for the other parameters and check if they satisfy all the inequations (the absence of closed timelike curves, the triangle inequalities etc...). We have generated a significant number (∼ 10 3 ) of five-center solutions focusing on solutions with low charges at the centers , we find no valid solutions. This tends to argue that no five-center solutions with pure D6 and D2 charges exist.