D1D5 microstate geometries from string amplitudes

We reproduce the asymptotic expansion of the D1D5 microstate geometries by computing the emission amplitudes of closed string states from disks with mixed D1D5 boundary conditions. Thus we provide a direct link between the supergravity and D-brane descriptions of the D1D5 microstates at non-zero string coupling. Microscopically, the profile functions characterizing the microstate solutions are encoded in the choice of a condensate for the twisted open string states connecting D1 and D5 branes.


Introduction
The gravitational description of black hole microstates remains one of the fundamental and most debated problems, more than ten years after the seminal works [1,2] . While these works have shown that, at weak coupling, black hole microstates can be described and counted in terms of D-branes, their description when the gravitational coupling becomes finite is still considered an unsettled question. A possible answer to this question is provided by the "fuzzball proposal" (for reviews see [3,4,5,6]): in its essence, it states that the 'naive' geometry of black holes, in which the horizon is locally empty space, is modified by quantum gravity corrections up to scales of the horizon size. The proposal is motivated by the explicit construction of classical geometries with the same asymptotics than the black hole (or black string) solutions but with no horizon. The best understood example is that of the 2-charge BPS system in type IIB string theory, for which the geometries dual to the whole family of microstates has been constructed in [7,8,9,10,11,12]. The 2-charge system represents a somewhat degenerate example of black hole, in that its classical geometry has a singular horizon of zero size, and one needs higher derivative corrections to smooth out the singularity and produce a finite size horizon. It is thus of crucial importance to extend the construction of the geometries dual to microstates to the case of the 3-charge BPS black hole, which is likely to share the properties of general (extremal) black holes. Though many advances have been made towards this goal [13,14,15,16,17,18,19], a complete construction of 3-charge microstates is still missing. Part of the reason why this task has proved to be so challenging is that one lacks a systematic procedure to construct the geometry generated by a particular D-brane configuration.
To help close this gap, in this paper we provide the relation between the stringy description of the black hole microstates in terms of D-branes and the corresponding geometries of the gravitational description. In principle, once a D-brane configuration is fully understood at zero string coupling (g s = 0), it is sufficient to switch on a non-zero coupling in order to derive the gravitational backreaction of the system. When g s ≪ 1 it is possible to use the conformal field theory description of D-branes to compute the corresponding gravitational profile perturbatively. In practice, one needs to compute disc amplitudes with the insertion of a closed string vertex: the boundary conditions on the disc should carry all the information of the D-brane configuration under analysis, while the closed string emitted represents the gravitational backreaction. For the half-BPS geometries the situation is quite simple and this mechanism has been checked explicitly in [20]. However this idea should work for any supergravity state that admits a description in terms of D-branes and, for instance, in [21] this approach was used to study a non-BPS configuration.
The situation we want to study is that of D-brane bound states where the single constituents of the system cannot be freely separated. The first non-trivial case is represented by the 2-charge systems in type IIB string theory. There are various dual descriptions of these configurations. For instance the geometries corresponding to each microstate of this system have been derived [22,23,7] by using the description in terms of a fundamental string with a wave carrying a left (or right) moving momentum. These solutions are specified by a profile function, characterizing the profile of the fundamental string. Then by using a chain of dualities these geometries can be re-interpreted as solutions describing D1D5 bound states. Alternatively, these solutions can be derived by solving the 1 4 -BPS killing spinor equations of supergravity and can be thought as "bubblings" of the naive D1D5 geometry [24].
In this paper we address the study of the microstate geometries from the string theory perspective. In particular we will show how the leading asymptotics of the microstate geometries are reproduced by string amplitudes describing the emissions of closed string states from D1D5 disks. In principle we could start from the description of a D-string with a wave [25], derive the gravitational backreaction by using the boundary state for these D-branes [26] and then rewrite the result in the D1D5 duality frame. However, since our aim is to provide a direct link between the D-brane construction and the corresponding geometries, we will focus directly on the analysis of the D1D5 bound states. From the conformal field theory point of view these bound states are described by open string amplitudes with mixed boundary conditions. Mixed disks dual to the ones considered here were studied in [27], where a direct link between the open string emission and the gauge theory instantons was found. In this paper we will study the closed string emission from disks that have half of their boundary along the D1-branes and the other half along the D5-branes. We see that these simple amplitudes are sufficient to reproduce the first corrections that distinguish the microstate geometries from the naive superposition of D1 and D5-branes. These corrections display some of the fundamental properties of the fuzzball geometries, such as the fact that the non-trivial states carry some angular momentum that breaks the rotational invariance in the non-compact directions, and the presence of a Kaluza-Klein monopole dipole charge, which is ultimately responsible for the regularity of the geometries in their core. In the string description they can be associated to condensates of twisted open string states stretched between the D1 and D5 branes. The open string condensate, as we will see, encodes the information about the profile function characterizing the microstate supergravity solution.
The paper is organized as follows. In Section 2 we review the solutions for the D1D5 microstate geometries of IIB supergravity on R 4,1 × S 1 × T 4 and write explicitly the first order corrections to the naive D1D5 superposition. In Section 3 we introduce all CFT ingredients we will need to compute the mixed disc amplitude mentioned above: we write the vertex operators for both the closed and the open strings, and discuss the identification between the holomorphic and antiholomorphic string coordinates induced by the D-branes. In Section 4 we compute the emission of one closed string state from a D1D5 disk and show that they reproduce the long distance behavior of the microstate geometries reviewed in Section 2. Finally, in the Conclusions, we discuss some possible generalisations and applications of our results.

Review of D1D5 geometries
Let us focus on type IIB string theory on R 4,1 × S 1 × T 4 . If one wraps n 1 D1 branes on S 1 and n 5 D5 branes on S 1 × T 4 , one obtains a system that preserves 1/4 of the 32 supersymmetries of type IIB strings and has, in the large n 1 n 5 limit, exp ( √ 2π √ n 1 n 5 ) states. The supergravity description of all these states was found in [7,8,9,10,11,12], exploiting the fact that the D1D5 system is U-dual to a fundamental string on R 4,1 × S ′1 × T ′4 wrapped n 5 times on S ′1 and carrying n 1 units of (left moving) momentum. The states of the fundamental string are described, in the semi-classical limit, by a curve in the space transverse to the string, R 4 × T ′4 . We denote the parametric representation of this curve by f A (v), with A = 1, . . . , 8, v = t − y and y the coordinate on S ′1 of radius R ′ . The length of the multiply wound fundamental string is L = 2πn 5 R ′ , and the curve f A (v) is taken to have trivial winding along the T 4 directions, so that f A (v + L) = f A (v), for any A. It will be convenient to distinguish the R 4 directions, denoted by the indices i, j, · · · = 1, . . . , 4, from the T 4 directions, labeled by a, b, . . . = 5, . . . , 8. Moreover the U-duality connecting the D1D5 and the fundamental string descriptions of the system requires to pick a particular direction in the four dimensional torus. So the internal components f a (v) of the curve describing the fundamental string are not labeled by a vector index of the T 4 of the D1D5 description, but by an indexâ running over the three self-dual 2-forms plus a scalar. In this notation the string profile is represented, in the D1D5 duality frame, by the functions We will choose the origin of our coordinates such as The string frame metric (ds 2 ), dilaton (Φ), B-field (b) and p-form RR fields (C (p) ) of the generic D1D5 state are given by In our conventions the RR field strengths F (p+1) are defined as with H (3) = db. The 5-form field strength is taken to be self-dual 1 F (5) = * F (5) . The various functions appearing above are defined as follows where * 4 denotes the Hodge dual with respect to flat R 4 . Finally the three 2-forms ωâ form a basis for the self-dual forms ⋆ 4 ωâ = ωâ, where ⋆ 4 is again defined with respect to a flat metric, but now acts on the indices in the T 4 . The length L can be expressed in terms of D1D5 quantities as with R the radius of S 1 in the D1D5 frame. The D1 charge is given by and the charges Q 1 and Q 5 are quantized as with V 4 the volume of T 4 and n 1 , n 5 the numbers of D1, D5 branes.

Asymptotic expansion of the geometry
In the limit in which the backreaction of the D-branes on the geometry is small, the D1D5 system can be described by the perturbative dynamics of open strings stretched between the D-branes. This is the description we will focus on in the next section. This description should capture the large distance expansion of the geometries (2.2), which looks like a small perturbation around flat space. We focus on the terms that distinguish the microstate geometries (2.2) from the naive D1D5 geometry, which is the singular, spherically symmetric, geometry with f A (v) = 0. These terms first appear at order 1/r 3 and are encoded in the functions A i , Aâ, A and in their duals B i , Bâ ij , B ij . Their large distance expansion is given by where we have used L 0ḟ A = 0, and In this paper we consider D1D5 geometries that are invariant under the SO(4) acting along the ND directions (i.e. the T 4 coordinates). Hence we will focus on the solutions that have Aâ = Bâ ij = 0. From (2.8-2.11) one finds that the asymptotic form of such solutions in the large distance limit is for the NSNS fields and for the RR fields.
The simplest example of this type of configurations is provided by a circular profile in the plane 1, 2 while fâ, f and all remaining components of f i are zero. The Eqs. (2.6) and (2.5) relate the amplitude a of the profile to Q 1 and Q 5 In this case one can compute exactly all functions in (2.4), but for our purposes it is sufficient to look at the order captured by (2.8-2.10). By using (2.14) we find By using this result in (2.12) and (2.13) and the quantization rule (2.7), one can see that all terms of order 1/r 3 are proportional to n 1 n 5 . This clearly suggests that the microscopic origin of these contributions is related to string diagrams involving both the D1 and the D5-branes at the same time.  [27]. Here our setup is different in two respects: first the ND directions are compactified on a T 4 , second we make a complementary choice for the open string condensate by turning on the states in the Ramond sector. As we will see, such a condensate does not generate any a non-trivial gauge (open string) profile, but only a supergravity (closed string) backreaction, reproducing the leading asymptotics of the microstate solutions. We restrict ourselves to systems with only two charges, (i.e. we set to zero the quantized momentum along the Neumann direction S 1 , that would appear as an additional charge) and so we focus on the open string states at zero momentum.
We denote the 10D coordinates (xM , ψM ) withM = t, y, 1, ..8. It is convenient to parametrize the coordinates in terms of the light cone directions Z ± and of four complex variables Z I : We will collectively denote the NN and DD directions by Z I , I = +, −, 1,1, 2,2 and the mixed ND directions Z a , a = 3,3, 4,4. Indices I, J, . . . and a, b, . . . label the vector representations of the SO(1, 5) and SO(4) Lorentz groups acting on the NN/DD and ND/DN planes respectively. In our conventions, the 10D Majorana-Weyl spinors ΘÂ satisfy Γ (10) ΘÂ = −ΘÂ, where Γ (10) = Γ 0 (10) Γ y (10) Γ 1 (10) . . . Γ 8 (10) . These spinors decompose with respect to the SO(1, 5) × SO(4) as where upper and lower indices A, B, · · · = 1, . . . , 4 denote Weyl SO(1, 5) spinors of opposite chirality; similarly α,α = 1, 2 are Weyl spinor indices of opposite chirality for the SO(4) group acting along the ND T 4 directions. We decompose the 10D Gamma matrices as follows Γ a (10) = 1 (6) where we use simply Γ I for the 6D Gamma matrices and Instead of the 6D Gamma matrices, we will often use the chiral components such as We will consider string amplitudes describing a closed string emission from a mixed D1D5 disk with twisted open string vertex insertions. We refer the reader to [28] for a general discussion of closed string amplitudes on disks with mixed boundary conditions and for details on the conventions we follow here. Twisted open string vertices are associated to string states stretched between D-branes with different boundary conditions and involve bosonic twist fields. Since there is no simple conformal field theory description when these fields have a non-zero expectation value, we will work perturbatively, and consider the leading contribution coming from the insertion of a single pair of twisted open string vertices. The open string vertices flip the boundary conditions on the disk from D1 to D5 types and viceversa and therefore split the disk boundary into two portions with D1 and D5 boundary conditions respectively. We restrict ourselves to massless physical states describing the lowest excitations of open strings stretched between D1 and D5 branes (for more details see, for example, [27]). In the NS sector, these states are generated by the zero-modes of the T 4 fermions, ψ a , and hence form a spinor representation of SO(4). The associated vertex operators are with wα,wα denoting n 1 × n 5 and n 5 × n 1 Chan-Paton matrices respectively. In the R sector, it is the fermions along the I directions that have zero modes, and hence the physical states form a spinor representation of SO (1,5). They correspond to the vertex operators where the Chan-Paton matrices µ A andμ A have n 1 × n 5 and n 5 × n 1 components respectively. In (3.5,3.6) and below, we denote by ϕ the free boson appearing in the bosonized language of the worldsheet superghost (β, γ). ∆ is the bosonic twist operator with conformal dimension 1 4 , that acts along the four mixed ND directions and changes the boundary conditions from Neumann to Dirichlet and viceversa. S α and S A are the SO(4) and SO(1, 5) spin fields. After bosonization these spin fields are simply exponentials of free bosons. If one introduces the bosons h I = (h + , h 1 , h 2 ) and h a = (h 3 , h 4 ) associated to the SO(1, 5) and SO(4) fermions respectively, fermions and spin fields are given by 3 As we mentioned before, we focus on open string condensates involving only states from the Ramond sector. Notice that states in the Ramond sector will break the SO(4) symmetry of the DD directions R 4 , while they are invariant under the SO(4) acting on the compact T 4 torus. They will be then associated with those supergravity solutions in (2.2) with Aâ = 0. The most general condensate of Ramond open strings can be written as: where the parenthesis on the indices A, B are meant to remind that the first term is automatically antisymmetric, while the second one is symmetric. Thus the open string bispinor condensate is specified by a one-form v I and an self-dual three-form v IJK . The self-duality of v IJK follows fromμ A and µ B having definite 6D chirality and can be written We will focus on symmetric open string condensates satisfying v I = 0. In fact, turning on a vacuum expectation value for v I would generate a tadpole for the D1D1 and D5D5 untwisted fields 4 A (1) I . This is not the case of v IJK components that do not have any trilinear coupling with open string states and therefore one can turn on a condensate v IJK = 0 without generating a tadpole in the open string theory. We will consider D1D5 geometries generated by the mixed disks involving these non-trivial condensates , i.e.
It has to be remembered that the spinorsμ A and µ B carry n 5 ×n 1 and n 1 ×n 5 Chan-Paton indices, and hence the condensateμ A µ B has to be thought of as the vev for the sum which, for generic choices of the Chan-Paton factors, is of order n 1 n 5 . We thus see that the amplitudes we compute are of the same order in n 1 n 5 as the gravity terms (2.12) and (2.13). In the following we will assume that the second identity in (3.11) already contains the trace over Chan-Paton indices, and thus that v IJK is of order n 1 n 5 .

Closed string vertex operators
The vertex operators in the closed string sector are 5 where all hatted indices are ten dimensional and GMN contains the 10D metric, the NSNS 2-form, and the dilaton, while FÂB is a RR field strength which can be expanded on a basis of ten dimensional Gamma matrices and contains a 1, a 3 and a self-dual 5-form The identification matrix R depends on the D-brane boundary conditions: RMN is a diagonal matrix with −1 along the Dirichlet directions and +1 otherwise, while the identification matrices with spinor indices is, up to a sign, the product of the (ten dimensional) Gamma matrices along the Neumann directions. In our case, we have two possible choices for R, depending whether we use the boundary conditions of the D1 or the D5 branes = −Γ +−3344 (10) , (3.17) with Γ 5678 (10) the chirality operator along T 4 written in terms of the 10D Gamma matrices. The final result of each correlator should not depend on the particular choice made for R. Here we will always take R = R D 1 .
As we are interested in configurations that are translationally invariant both along the world sheet directions t and y and the T 4 directions, we take the closed string momentum vector along R 4 : k i L = k i R = k i /2 . The asymptotics of the D1D5 geometries generated by the mixed disks will be captured by the leading term in the expansion of the string amplitude for small k i . We are interested only on the leading term in the momentum expansion, and therefore we can set to zero the momentum in the exponential of closed string vertices. Finally

Microstate geometries from disk amplitudes
In this section we show how the small g s , long distance behavior of the microstate solutions (2.2) is reproduced by using string amplitudes and the microscopic description of D-branes. In particular, we compute the emission of one closed string state from various mixed disks (i.e. disks that have half of their boundary along the D1-branes and the other half along the D5-branes) and extract from these amplitudes the leading deviation of the geometries (2.2) from the naive D1D5 metric, which appears at order 1/r 3 in the large r limit. We will restrict ourselves to open string condensates describing D1D5 geometries with trivial profile along the compact T 4 directions, i.e. Aâ = Bâ ij = 0. The correlators we compute reproduce the terms in the geometries (2.2) of order 1/r 3 , which are captured by the functions A i , A, and their duals B i , B ij given in eqs.  where ω is the ratio 6 ω = z 12 z 34 z 13 z 24 , (4.3) with z ij = z i − z j . Notice that ω is a pure phase since z 2 and z 3 are complex conjugate of each other, while z 1 and z 4 are real.

NSNS amplitude
Let us start from the emission of the NSNS state The transversality condition k I G IJ = 0 implies that the term of the closed vertex (3.13) proportional to ∂XM does not contribute to the correlator: only the term proportional to k L · ψψM contributes. Only one Lorentz invariant can be built out of k K , (GR) IJ and v IJK and so the form of the string amplitude is determined from Lorentz invariance up to a constant I where the last equality follows from the second relation in (3.11). It is important to stress that the CFT correlator is SO(1, 5) invariant, and only the explicit form of the matrix R breaks this invariance down to SO(1, 1) × SO(4). The constant I has to be computed from the explicit evaluation of the CFT correlator. To evaluate I, we specify to the term in (4.5) with K = 1, I = 2, J = + and ǫ A = ǫ B = (− − −). Notice that for this choice of open string polarizations, the indices A, B are automatically symmetric consistently with the desired form of theμµ-condensate (see Eq. (3.11)). Charge conservation implies that only the cubic fermionic term of the vertex (3.13), in which all the three fermionic fields carry SO(1, 5) indices, can contribute to this amplitude and therefore one can write Thus the relevant correlators are 14 .
The last equation can be written in the covariant form 7 14 .
with the integral running over the unitary circle. Substituting the value of I in (4.5), one finds that the string amplitude is As we will now show, this amplitude exactly reproduces the 1/r 3 contributions to the metric and the B-field in (2.2) associated with functions A i , B i , A and B. In order to identify the fields appearing in (4.10), we need to decompose the SO(1, 5) vector indices into SO(1, 1) × SO(4) indices (I = (t, y, i), where, as before, i = 1, ..4 label the R 4 Dirichlet directions and t, y the Neumann directions). Notice that only antisymmetric components (GR) [JK] contribute to the amplitude. Using the fact that the matrix R is +1 along t, y and −1 otherwise, one finds that the matrix (GR) JK is antisymmetric if and only if G is antisymmetric (B-field) and the (J, K) indices are of the same type (i.e. (t, y) or (i, j)), or G is symmetric (metric) and the (J, K) indices are of the different type (i.e. (t, i) or (y, i)). This implies that only the components g ti , g yi and b ty , b kl are emitted from the mixed disk. In addition we recall that the momentum k I of the closed string is non-zero only in the Dirichlet directions i. The amplitude (4.10) can then be written as The origin of the factors of √ 2 lies in the OPE's of the ψ I 's and the twist fields [31,27]: Note that that equation and on the other side which agrees with (4.8).
where in the second line we used the self-duality conditions of the 3-form v IJK given by From (4.11) we can read the profile of the induced metric and B field, for instance As in [20], the space-time configuration associated with a closed string emission amplitude is obtained by multiplying the derivative of the amplitude with respect to the closed string field by a free propagator and taking the Fourier transform. In general for a field a µ 1 ...µn we have with a µ 1 ...µn (k) given in terms of derivatives of A as in (4.13). In our case, the Fourier transform has the following form where we used Thus from (4.13) we get the following results for the large distance behavior of g ti and b ty at order 1/r 3 We can follow the same steps for the variation of A NS in (4.11) with the respect to g yi and b ij leading to Comparing the values of g ti and b ty derived above with the 1/r 3 terms of the gravity solution (2.12), one fixes the identification between the string condensate parameter v IJK and the parametersf ij andf i that characterize the gravity solution at this order: Using this identification, one verifies that the components g yi and b ij predicted by the string computation agree with the gravity result (2.12).

RR amplitude
Let us now consider the amplitude describing the emission of RR states 20) with W R given by (3.14). As discussed at the end of Section 3, the open string condensate under analysis contributes only to the emission of with where, in the last line, we have used Γ 5678 (10) = −Γ 3344 (10) = 1 (6) ⊗ (−γ N D ) and that −γ N D is 1 on the indicesα (see (3.4)). Lorentz invariance again fixes the form of A R up to a constant where the factor of two in evidence comes from the trace over the 4D spinor indices. Notice that both the D1 and the D5 reflections matrices (3.17) reduce in 6D to the expression R = Γ ty , since they differs by Γ 5678 (10) which is just the identity on the spinor components (α) entering in this computation. Due to the symmetry properties of the open string condensate, this result is non-vanishing only when the Gamma matrices in F and R reconstruct a 3-form (3.11). This is possible for all terms in (4.22), thus 1, 3 and 5-forms components of the RR field will all contribute to the string amplitude. We use A R [n] to indicate the contribution of the form of degree n. By using again the second relation in (3.11), we get i5678 .
To evaluate I, one can take a specific choice of the open and closed string polarizations; for instance, it is convenient to choose forμ A , µ B the following weights ǫ A = ǫ B = (−−−).
The relevant correlators are (4.25) Assembling the various correlators together, and using again the measure (4.2), one finds We can now show that the amplitude (4.24) contains the 1/r 3 contributions to the RR fields in the solution (2.2) that are characterized by the functions A i , B i , A and B. We first need to rewrite the result (4.24) in terms of the gauge potentials C (n−1) by using So we get We then extract from the amplitude the gauge field profile attach a free propagator to each profile and take the Fourier transform, as explained in (4.14). Hence (4.28) yields the following results for the large distance behavior of the RR fields Using the identifications (4.19), the values above exactly reproduce the supergravity result (2.13).

Conclusions
We have shown how the asymptotic expansion of the 2-charge fuzzball geometries (2.2) is reproduced by computing string amplitudes for the emission of a closed string state from a disk with mixed D1D5 boundary conditions. Each fuzzball geometry is completely determined by a curve f A (v) that captures how different the solution is from the naive D1D5 superposition. Microscopically the information about the curve f A (v) is encoded in a condensate for the open strings stretched between the two types of D-branes. In order to derive the exact dictionary between f A (v) and the string condensate one should compute the closed string emission from a disk in presence of a finite value for the open condensate. This is a challenging task since the open string states stretched between the D1 and the D5 branes contain twist fields. However we could explicitly check this dictionary by treating the open condensates perturbatively: in the large distance limit the 1/r 3 terms of the fuzzball solutions match the gravitational backreaction of the D-brane system when the open string condensates are included at first order. We believe that a similar pattern exists also at higher orders and that it is possible to construct the dictionary term by term in the perturbative expansion without changing the identifications established at lower orders. Our analysis is not complete: in this paper we have reproduced only a subset of the D1D5 geometries, those that are invariant under the SO(4) rotations of the compact space T 4 (in the notation of Section 2 these are the geometries with Aâ = 0). These geometries are dual to those Ramond ground states of the D1D5 CFT that are associated with the "universal sub-sector" of the cohomology of T 4 (i.e. the (0, 0), (2, 2), (2, 0), (0, 2) forms, and the Kähler (1, 1) form). Hence the geometries we consider do not use in any way the properties of the compact space T 4 and the results of our computation apply, with no modification, to the case in which the compactification manifold is K3.
We have left open the problem of identifying which open string condensates generate the geometries associated with the remaining even cohomology of T 4 . We expect these latter configurations to be related to the ones we have considered here by the action of the supersymmetries broken by the D1 and D5 branes. A similar approach was used in [32] to study spin potentials of 1 2 -BPS Dp-branes. Moreover, in the case of T 4 , one also has the microstates associated with the odd cohomology, which correspond, in the duality frame of the fundamental string, to fundamental string states with pairs of fermionic excitations [11]. One can show that these states are distinct form the ones we have considered here: indeed, all the geometries in the class of [11] display 1/r 3 corrections in the g tt and g yy metric components, which are absent for the states we identify here. There must exist open string condensates generating also those geometries, and it is an interesting problem to find them.
From a more general point of view, we think that our computation provides an important relation between the gravitational description of the D1D5 microstates and their microscopic description in terms of D-branes. Most of the studies of the fuzzball solutions focused on the near-"horizon" geometry and its relation with the dual CFT description. However the full microstate geometry is asymptotically flat and the large r limit represents a regime where both the gravitational and the D-brane descriptions are valid. Clearly, in the string amplitude computation, we are able to explore only perturbatively the Higgs branch, by inserting a finite number of twist vertex operators associated to the strings stretched between the D1 and the D5 branes. However, this is sufficient to capture some of the distinctive properties of the microstate geometries and provides a direct support to the idea that the D-brane configurations used to compute the entropy at g s = 0 evolve into fuzzballs when the string coupling is turned on.
An advantage of our approach is that it is completely systematic and allows, in principle, to investigate a large number of open problems. In our opinion, two of the most important open issues are the study of string corrections and the construction of the geometry dual to a general 3-charge microstate. With our approach one could derive the asymptotic expansion for such a general 3-charge geometry, and this could provide an important clue for constructing the full exact solution. Within our framework, one could also obtain the higher order string corrections to the microstate geometries without the need of guessing what term in the gravitational effective action are the relevant one in the different configuration. We hope to come back to these issues in a subsequent work.