Stringy origin of 4d black hole microstates

We derive a precise dictionary between micro-state geometries and open string condensates for a large class of excitations of four dimensional BPS black holes realised in terms of D3-branes intersecting on a six-torus. The complete multipole expansion of the supergravity solutions at weak coupling is extracted from string amplitudes involving one massless closed string and multiple open strings insertions on disks with mixed boundary conditions.


Introduction
The fuzzball proposal [1,2] relates black-hole micro-states to smooth and horizon-less geometries with the same asymptotics (the same mass and charges) as the putative black hole but differing from it in its interior. The (semi)classical black-hole geometry only results from a coarse-graining approximation of the micro-state solutions. For black holes corresponding to D-brane bound-states, the micro-states can be realised and counted in terms of open-string excitations of the D-branes constituents [3,4]. As a result, the fuzzball proposal suggests a one-to-one correspondence between open-string condensates and the geometries generated by them. The best understood example is the two-charge BPS system in type IIB theory for which the whole family of micro-state geometries is known [5,6,7]. The solutions can be written in several different U-duality related frames. In [8], the asymptotic form of the general micro-state geometries associated to a system of D1-and D5-branes has been derived from disk amplitudes computing the emission rate of closed string states from open string condensates at the D1-D5 intersections. The two-charge system represents a somewhat degenerate example of a black hole in that its classical geometry has a singular horizon of zero size. Smooth horizons in the supergravity limit require systems with 3-or 4-charges. Unfortunately, despite significant progress (see [9]- [18] and references therein for recent results and reviews), a similar analysis for 3-and 4-charge black holes is still missing since a family of micro-state solutions large enough to account for the black hole entropy is not yet known.
Aim of this paper is to partially fill in this gap, providing a direct link between micro-state geometries and open string condensates for a large class of 4-charge black hole microstates. We realise the four-dimensional BPS black holes in the 'democratic' frame of four mutually intersecting stacks of D3-branes on T 6 . We focus on a family of black hole solutions characterised by eight ('lovely' rather than 'hateful') harmonic functions in the type IIB theory. This family is dual to the extensively studied BPS solutions [19]- [24] describing D0-D2-D4-D6 brane bound-states in the type IIA theory or intersecting M2 and M5 branes in M-theory carrying non-trivial momentum and KK monopole charge. Amusingly, in the D3-brane frame, the eight harmonic functions are described entirely in terms of excitations of the metric and the fourform RR field. A subclass of this family of solutions involving two harmonic functions has been studied recently in [25] and shown to be regular and horizon-less.
The micro-state geometries will be derived from disk amplitudes computing the emission rate of massless closed string states from open string condensates binding the different stacks of D3 branes. The disk amplitude computes the leading correction to the supergravity solution in the limit of small string coupling g s . In principle the supergravity solution can be recovered by plugging the result into the type IIB equations of motion and solving recursively order by order in g s . Equivalently, one can expand the general micro-state solution to order g s and match it against the result of the string amplitude. Proceeding in this way, we identify three class of micro-states depending on whether the open string condensate involves either one D3-brane stack or the intersection of two or four different stacks. We dub the three classes as L, K and M solutions referring to the names of the harmonic functions underlying the solutions. We remark that disk contributions add and at this linear order both the metric and the four-form potential can be written as a sum of harmonic functions. In particular, we find that four of the harmonic functions (L) specifying the solution are associated to the four different disks with a single boundary, three harmonic functions (K) are associated to the three inequivalent 2-charge intersections and the last one (M) is sourced by a 4-charge open string condensate. The three types of micro-state geometries fall off at infinity as Q i /r, Q i Q j /r 2 and Q 1 Q 2 Q 3 Q 4 /r 3 respectively. Higher multipole modes are generated by insertions of untwisted (boundary-preserving) open strings on the boundary of the disk, much in the same way as for the D1-D5 system [12]. Remarkably, as we will see, the micro-state angular momentum is fully accounted for by the type K component of the solution.
Solutions of type M are particularly interesting since the number of disks with four different boundaries grows with the product of the four charges N = Q 1 Q 2 Q 3 Q 4 , the same quartic invariant entering in the black-hole entropy formula S = 2π √ N. It is tempting to speculate that after quantization the corresponding micro-states may account for the entropy of the putative black-hole much in the same way as the entropy of the 2-charge system with N = Q 1 Q 2 is reproduced by quantisation of the profile function describing D1-D5 intersections [5,26,27] or, equivalently, D3-D3' pair intersections, in the systems that we consider here 1 .
We remark that the class of 4-charge micro-state solutions generated by open string condensates that we find here is larger than the family of supergravity solutions explicitly known. This is at variant with the 2-charge systems, where the most general micro-state [7] is reproduced by a 2-charge intersection [8]. In the case of 4 charges we find new solutions looking pretty much like the K and M solutions, but tilted with respect to the background D3-brane geometry. It would be interesting to understand whether these could be accommodated in a larger class of micro-state geometries or should be excluded on the basis of supersymmetry or stability conditions (D-and F-flatness) for the open strings.
The plan of the paper is as follows. In Section 1 we introduce the 'eight-ful' harmonic family of supergravity solutions in the 'democratic' frame of D3-branes. We identify three sub-classes of solutions that we dub L solutions, K solutions and M solutions. In Section 2, we focus on solutions sourced by disks with a single type of boundary condition. We compute disk amplitudes involving the insertion of a closed string NS-NS or R-R vertex operator in the bulk and an arbitrary number of untwisted open string scalars. In Section 3, we compute disk amplitudes with two twisted open string insertions that intertwine between two different boundary conditions. In Section 4, we compute disk amplitudes with four different boundaries conditions. The computations are much more involved (they are equivalent to a 6point open string amplitude) but one can numerically integrate the final correlators and get a finite result both for NS-NS and R-R closed string insertions in the bulk. Remarkably, NS-NS and R-R correlators are proportional to the same integrals, so their numeric values are irrelevant for the identification of the dual supergravity solution. We finally draw our conclusions and indicate directions for further research. In the appendices we collect our conventions for string correlators and present the derivation of the D3-brane solution starting from its T-dual description in Type IIA theory.

Supergravity solution
We consider four-dimensional BPS black hole solutions of type IIB supergravity consisting of D3-branes intersecting on T 6 . The solutions are characterised by a nontrivial metric and a five-form flux with all the other fields of type IIB supergravity vanishing. In this sector, the type IIB equations of motion take the extremely simple and elegant form 2 A family of solutions can be found by acting with three T-dualities on the more familiar class [22,24] describing systems of D0-D2-D4-D6 branes in type IIA theory (see Appendix for details). The solutions can be written in terms of eight harmonic functions (a = 1, . . . 8, I = 1, 2, 3) on (flat) R 3 associated to the four electric and four magnetic charges in the type IIA description of the system. These functions are conveniently combined into Here ǫ IJK characterise the triple intersections between two cycles on T 6 . In terms of these functions the type IIB supergravity solution can be written as and For K I = M = 0, the supergravity solution (2.5) describes a system of four stacks of intersecting D3-branes aligned according to the table below. The four harmonic functions V, L I describe the distribution of the D3-branes in the non compact spatial directions x ∈ R 3 . Solutions with non-zero K I and M will be associated to condensates involving twisted open strings. These excitations modify the geometry of the putative black hole in its interior while preserving the same asymptotic fall off at infinity. Consequently, K I and M should decay faster than |x| −1 at infinity. We thus focus on solutions involving harmonic functions with the following asymptotics with c i some constants, α D3 = 4 π g s (α ′ ) 2 /V T 3 the inverse D3-brane tension and V T 3 the volume of the 3-torus wrapped by the stack of D3-branes, that for simplicity we will take to be equal for all the stacks. We anticipate that the explicit string computations show that the coefficients c i 's entering in (2.8) satisfy the relation or equivalently the function µ defined in (2.4) starts at order 1/r 3 . Since at linear order in α D3 , the harmonic functions entering in the solution simply add, we can consider the contribution of each harmonic function separately. More precisely, we define three basic class of solutions: L solutions are associated to the harmonic functions L I , V with K I = M = 0. K solutions are generated by turning on a single K I harmonic function for a given I and taking M = −K I for the given I. Finally M solutions are associated to the orthogonal combination M + I K I that is generated at order 1/r 3 . In other words the dipole modes in (2.8) vanish: c M i = c K I i = 0. The family (2.5) can be seen as the "superposition" of these three basic classes of solutions. We remark that disks emissions simply add but massless closed string states interact with each other. This entails non-linearity, so the resulting solution will not be the naive "superposition" of its constituents as it is obvious from (2.5). In the rest of this section, we display the supergravity solutions for the three basic classes of micro-state geometries.

L solutions
A representative of the L class of micro-state solutions is given by taking in (2.5). This results into The Type IIB supergravity solution reduces to (discarding d-exact terms) This solution is generated by distributing D3-branes of type 0 in table 1 along R 3 . Similar solutions are found by turning on L I associated to the other three types of D3-branes in the table 3 .

K solutions
Next we consider solutions with For this choice one finds and the resulting supergravity solution reduces to (discarding d-exact terms) Similar solutions are found by turning on K 1 or K 2 . We would like to stress that K solutions in general carry angular momentum, unlike L and M solutions.

M solutions
Finally we consider solutions with Now one finds The Type IIB supergravity solution reads (discarding d-exact terms)

Harmonic Expansions
The harmonic functions H a entering into the supergravity solutions can be conveniently expanded in multi-pole modes. The general expansion can be written as with P i 1 ...in (x) a totally symmetric and traceless rank n tensor providing a basis of harmonic functions on R 3 . The explicit form of P i 1 ...in (x) can be found from the multi-pole expansion For example for n = 0, 1, 2 one finds and so on. We notice that even if the explicit form of P i 1 ...in (x) can quickly become messy as n grows, the form of the Fourier transform is very simply. Indeed writing one finds that the Fourier transform of the harmonic functions P i 1 ...in (x) are simply polynomials of the momenta divided by k 2 Disk amplitudes will be conveniently written in this basis. We notice that even if the single harmonic components P i 1 ...in (x) are singular at the origin, in analogy with the 2-charge case [28,2], one may expect that for an appropriate choice of the coefficients c i 1 ...in the infinite sum (2.20) produce a fuzzy and smooth geometry. This was explicitly shown in [25] for a class of solutions parametrized by one complex harmonic function H = H 1 + iH 2 obtained from our general solution by taking

String amplitudes and 1-charge microstates
At weak coupling, the gravitational background generated by a D-brane state can be extracted from a disk amplitude involving the insertion of a closed string states in the bulk and open string states specifying the D-brane micro-state on the boundary. The open string state, or condensate, is specified by a boundary operator O given by a trace along the boundary of the disk of a product of constant open string fields. The profile of a supergravity field Φ in the micro-state geometry can be reconstructed from the string amplitude A Φ,O (k) after Fourier transform in the momentum k of the closed string state. Moreover, since we consider constant open string fields we can take all open string momenta to vanish. For simplicity, we take the only non vanishing components k i of the closed string momentum to be along the 3 non compact space directions, i.e. k 0 = k y I = kỹ I = 0. In order to keep the mass-shell condition k 2 = 0, we analytically continue k i to complex values, and require k 2 i ≈ 0. More precisely, the deviation δΦ(k) from flat space of a fieldΦ(k) is extracted from the string amplitude via the bulk-to-boundary formula with (−i/k 2 ) denoting the massless closed string propagator for the (transverse) physical modes. We notice that the limit of zero momenta for open string states is well defined only in the case the disk diagram cannot factorize into two or more disks via the exchange of open string states. Indeed, in presence of factorization channels the string amplitude has poles in the open string momenta and the underlying worldsheet integral diverges. We will always carefullly choose the open string polarisations in such a way that no factorization channels be allowed. As a result, one finds that, up to 1/k 2 , δ Φ(k) is a polynomial in k i and therefore can be expressed as a linear combination of harmonic functionsP i 1 ...in (k) in momentum space In this section we illustrate the stringy description of the micro-state geometry for the very familiar case of parallel (wrapped) D3-branes. The vacua of this system are parametrised by the expectation values of the (three) scalar fields φ i , along the non-compact directions. The general open string condensate involves an arbitrary number of insertions of the 'untwisted' field φ i on the boundary of the disk. On the other hand, the associated micro-state geometry is described by a D3-brane solution characterised by a smooth multi-pole harmonic function determined by the open string condensate [29]. The string computation will allow us to fix the relative normalisation between the NS and RR vertices that will be used in the analysis of 2-and 4-charge intersections in the next sections.

L solutions at weak coupling
We start by considering the supergravity solution generated by a single stack of D3branes, let us say of type 0. The gravity solution is given by (2.12). At linear order in α D3 one finds with δL = L − 1 and A both or order α D3 . In particular one can take corresponding to N 0 D3 branes on R 3 .

NS-NS amplitude
The relevant vertex operators are 4 with E = h + b the polarization tensor containing the fluctuation h of the metric and b of the B-field and c N S a normalisation constant. The momentum of the closed string state will be labeled by k. Henceforth we denote by X(z) the holomorphic part of the closed string field X(z,z) = X(z) + RX(z) with R the reflection matrix implementing the boundary conditions on the disk. More precisely, R is a diagonal matrix with plus one (+1) and minus one (-1) along the Neumann and Dirichelet directions, respectively. In particular for a disk of type 0, we have plus along t, y I and minus along x i ,ỹ I . Consequently kR = −k. We choose length units whereby α ′ = 2. Moreover we exploit SL ( The resulting string amplitude can be written as The basic contributions to the correlators are The asymptotic deviation from the flat metric can be extracted from After Fourier transform one finds with the identification We conclude that the function ξ(φ) codifies the complete multipole expansion of L(x) in terms of vacuum expectation values for untwisted scalar fields In particular, the harmonic function for a single D3-brane at position a, is realized by taking ξ(φ) ∼ e i a φ (3.15) that reproduces the multipole expansion (2.21) of L(x) according to (3.13). Similar results can be found by considering disks ending on D3-branes associated to the harmonic functions L I . We conclude by observing that D-and F-flatness conditions may restrict the choices for ξ i 1 ...in . Indeed for pointlike branes, the flatness conditions require [φ i , φ j ] = 0, so the matrices φ i can be simultaneously diagonalized and written in terms of N 0 positions in R 3 [29].

R-R amplitude
Next we consider the R-R string amplitude where F are the R-R field strengths and R is the reflection matrix in the spinorial representation.
The basic contributions to the correlators are given by the first two lines of (3.8) and Altogether one finds where we wrote F = ik i Γ i C. Here and below we will always restrict ourselves to the 'electric' components C tM 1 M 2 M 3 since the remaining 'magnetic' components are determined by the self-duality of F 5 . The R-R fields at the disk level is then determined from the corresponding variation of the string amplitudes and using (3.13) one finds δC ty 1 y 2 y 3 = −δL(x) (3.24) in agreement with (3.3) for an arbitrary harmonic function L(x) specified by ξ(k) via (3.13).

String amplitudes and 2-charge microstates
Next we consider K solutions. At leading order in α D3 , K solutions (2.15) reduce to and K a harmonic function starting at order |x| −2 . For example one can take K to be

NS-NS amplitude
In this section we compute the disk amplitude generating the metric of the K solutions. K solutions will be associated to string fermionic bilinears localised at the intersections of two D3-branes. The relevant string amplitude has been computed in [8] for the case of D1-D5-branes. Here we review this computation, adapting it to the case of D3-branes, and include the effect of untwisted open string insertions in order to account for higher multi-pole modes of the harmonic function K. The relevant disk amplitude can be written as

We consider a non-trivial open string condensate
Here σ I denotes the Z 2 -twist field along the I th T 2 inside T 6 with conformal dimension 1/8 and S A = e ± 1 2 (iϕ 3 ±iϕ 4 ±iϕ 5 ) even number − ′ s (4.9) the spin field on the six-dimensional plane along which D3-branes are NN or DD.
To evaluate the correlator one can consider a specific component and then use SO(6) invariance to reconstruct its covariant form. For instance, if we take A = B = 1 2 (+ + +), the open string condensate contributes a net charge +2 along the first three complex directions, so that only the ψ:ψψ: term can contribute to the correlator. The relevant correlators are with w = z 13 z 24 z 14 z 23 . We notice that the scalar fields factorize from the rest, since they are always oriented along the non compact space directions, while twist fields are along T 6 .
The z i dependence boils down to the worldsheet integral where γ is a small contour around the origin. On the other hand the open string condensate can be written as with v M N P a self-dual tensor in six-dimensions and c O a normalisation constant. Combining (4.12) with (4.10) and taking As before, the factor ξ(k) comes from untwisted insertions and is associated to higher multi-pole modes of the underlying harmonic function, so it is enough to match the leading term, i.e. we set ξ(k) = 1. Assuming a v IJK of the form v y 3ỹ3 3 = −v 12t = 4π v (4.15) and using (4.14) one finds δg 2t = −4πv k 1 δg 1t = 4πv k 2 δg y 3ỹ3 = −4πv k 3 (4.16) After Fourier transform one finds in agreement with (4.1) for the choice (4.4) with v i = δ i3 v. We conclude that the harmonic function K 3 describes a particular component of the fermion bilinear condensates connecting two branes parallel along the (y 3 ,ỹ 3 )-plane. Similarly K 1,2 represents fermion bilinear condensates connecting branes parallel along the 1 and 2 planes (tori).
We conclude by noticing that other choices for the 2-charge condensate give rise to solutions that do not belong to the 'eight-ful' harmonic family (2.5). In particular, taking non-zero values of v ty 3ỹ3 , v ty 3 i or v tỹ 3 i give rise to solutions with non-trivial b ij , g y 3 t or bỹ 3 t components, respectively. These components can be matched against a more general 2-charge solution obtained from [7] after two T-dualities. We refer the reader to [8] for a detailed match in the T-dual D1-D5 description.

R-R amplitude
Next we consider the disk amplitude with the insertion of a R-R vertex operator The ghost and untwisted bosonic correlators are given again by the first two lines in (4.10) while the fermionic and twist-field correlators are Again one finds that the amplitude is proportional to I given in (4.11) and can be written in the form where we used (4.12), (4.13) and (3.23). Taking ξ(k) = 1, specializing to the condensate in (4.15) and focussing on the C tM 1 M 2 M 3 components one finds Varying with respect to the four-form potential one finds δC ty 1ỹ2 y 3 = δC tỹ 1 y 2 y 3 = k 3 4πv (4.24) reproducing the Fourier transform of (4.1) for the choice (4.4) with v i = δ i3 v.

String amplitudes and 4-charge microstates
Finally we consider M solutions. At leading order in α D3 , M solutions (2.18) reduce to δg M N dx M dx N = 2M dy 1d y 1 + dy 3d y 3 + . . .
with M a harmonic function starting at order |x| −3 . For example one can take M to be of the form

NS-NS amplitude
We consider now string amplitudes on a disk with boundary on all four types of D3-branes. In particular we consider the insertions of four fermions µ a starting on a D3-brane of type (a) and ending on a D3-branes of type (a + 1) with a = 0, 1, 2, 3 (mod 4), in a cyclic order. We notice that unlike the case of two boundaries now the condensate is complex. Indeed, even if each intersection preserves N = 2 SUSY (1/4 BPS), so that each fermion µ a comes together with its charge conjugateμ a , the overall configuration preserves only N = 1 SUSY (1/8 BPS) and fermions connecting all four type of D3 brane form two pairs of opposite chirality. The charge conjugate condensate can be defined by replacing each µ a with its charge conjugate fieldμ a and running along the boundary of the disk with the same cyclic order but in reversed sense. The real and imaginary parts of the string amplitude can be selected by turning on the real and imaginary parts of the condensate respectively. We consider the following NS-NS amplitude We use the notation for internal and spacetime spin fields. The condensate is now the tensor where the sum over all the Chan-Paton indices µ 1 i 1 is understood. As before we discard components proportional to ǫ αβ and ǫαβ to ensure the irreducibility of the string diagram. As a result O αβαβ transforms in the (3, 3) of the SU(2) L × SU(2) R Lorentz group and can be better viewed as a symmetric and traceless tensor v µν . We notice that even though the presence of the four different D3-branes breaks the Lorentz symmetry down to SO(4), the string amplitude is invariant under SO(4) × U(1) 3 since the U(1) 3 rotations of the three internal tori do not change the relative angles between branes. We notice also that space-time chirality determines the internal U(1) 3 charges of the twisted fermions. In particular µ α 1 andμα 3 have opposite charges with respect to the first U(1) while µ α 2 andμα 4 have opposite charges with respect to the third internal U (1). Therefore the open string condensate v µν carries no U(1) 3 charges.
We notice that (ER) [11] and (ER) [33] always involve a Neumann and a Dirichlet direction, so only the metric contributes to the antisymmetric part of the matrix. Varying with respect to the symmetric part h ij of the polarization tensor E = h + b one finds that reproduces (5.1) after Fourier transform. We conclude this section by noticing that there other choices for the 4-charge condensate besides (5.6) lead to solutions that go beyond the eight harmonic function family (2.5). On the one hand one can consider different orderings of the branes along the disk. These condensates lead again to solutions of type M but with internal coordinates that are permuted among one another. Notice that some of these solutions can look very different from each other since L I and V do not enter in a symmetric fashion. On the other hand, turning on the real part of the condensate (5.6) will produce solutions with a non-trivial B-field given in terms of the real worldsheet integrals I 2 and I 4 . Finally turning one can consider condensates of type 9 These condensates generate a new class of solutions with off-diagonal terms mixing either two different T 2 's inside T 6 or space-time and internal components. We leave it as an open question to establish whether or not an extended family of supergravity solution accounting for these brane excitations can be explicitly written.
to the 128 of SO (1,9), with C Σ the spin field of negative chirality. Again taking α = β = 1 2 (++),α =β = 1 2 (−+), it is easy to see that the net charge −2 of the condensate can be only compensated by the kψψ M term in the closed string vertex with S Λ and C Σ both carrying charge − 1 2 along the fifth direction. This leads to eight choices for S Λ . On the other hand, once S Λ is chosen, the charges of Ψ M Σ (k) = :kψψ M C Σ : are completely determined by charge conservation. For example, for the choice one finds again the first four lines in (5.7), while the last four lines are replaced by Combining the various correlators and taking one finds that the string amplitude is again given by the integral (5.11) involving I 1 . A similar analysis can be performed for other choices of spin-field components (labelled by Λ, Σ) leading to similar answers in terms of the four characteristic integrals I 1 , . . . I 4 on C. The results are summarised in table 2. Since I 2 , I 4 are real and I 1 , I 3 purely imaginary, a purely imaginary condensate selects the I 1 , I 3 andĪ 1 ,Ī 3 components. The resulting string amplitude can then be written as A RR µ 4 ,ξ(φ) = 1 4 tr 16 (F R P Γ i )2πv ij k j ξ(k) (5.23) with P = 1 2 (1 − Γ y 1ỹ1 y 3ỹ3 )Γ y 2ỹ2 (5.24) a projector on I 1 , I 3 andĪ 1 ,Ī 3 components with +i and −i eigenvalues respectively. These are precisely the eigenvalues of the matrix P justifying our claim. Using R = Γ ty 1 y 2 y 3 one finds Taking derivatives with respect to C 4 one finds agreement with (5.1) after Fourier transform.

Conclusions and outlook
We The M class of micro-states is particularly interesting since the number of disks with four different boundaries grows with the product of the four charges N = Q 1 Q 2 Q 3 Q 4 . In a first approximation, the function M can then be viewed as an Ncenter harmonic function with moduli space spanned by the N positions on R 3 up to S N -permutations. One may then expect that he number of micro-states in this class grow as e 2π √ N/6 , the number of conjugacy classes of S N or equivalently the partitions of N, at large N. This suggests that the 4-charge micro-states identified in this work may account for a large number of states contributing to the (putative) black hole entropy. A quantitative study of this issue remains one of the most challenging and exciting questions left open by our work.
Although we have worked mostly with D3-branes that are either parallel (along some directions) or perpendicular (along other directions), our analysis can easily be adapted to branes intersecting at arbitrary angles and preserving the same amount of supersymmetry (1/8 of the original 32 super-charges) [35,36] or branes intersecting in orbifolds [37,38]. The open strings living at the intersections will be described in terms of twist-fields [39,40,41] that generalise the Z 2 twist fields that appeared in the present analysis. We have explicitly checked that one-, two-and four-open string insertions source for classical supergravity profiles similar to those obtained in this paper. We defer a detailed analysis to future work. For the time being we would like to stress that off-diagonal components of the internal metric can mimic the effect of tilting the branes thus changing their intersection angles. Furthermore, it is amusing to observe that the 5 harmonic functions that appear in the static solutions found in [35,36,37,38] can be identified with V, L I and M. This is consistent with the fact that setting K I = 0 produces micro-state geometries with no angular momentum.
Our solution generating technique have useful physical applications, both for understanding the physics of multi-center extremal solutions and for constructing new smooth micro-states geometries for under-rotating extremal black holes in four and higher dimensions. Indeed, in addition to reproducing all known supergravity solutions in the 'eightful' harmonic family for certain choices of the condensates, we have shown that other choices are possible that can generate new 4-charge micro-state geometries beyond the ones available in the literature.

B. The supergravity solution B.1 D0-D2-D4-D6 solution
A family of under-rotating multi-center black hole solutions corresponding to a general system of D0-D2-D4-D6 branes in type IIA theory wrapped on T 6 was constructed in [22,24]. It is convenient to combine these functions into In term of these functions the metric in the string frame and the various p-form potentials are written as [24] ds 2 10 = −e 2U (dt + w) 2 + e −2U and * 3 dA = dV * 3 dw I = −d(K I ) * 3 dv 0 = dM * 3 dv I = dL I * 3 dw = V dµ − µdV − V Z I dP I (B.5) Here A, w, w I , v 0 , v I , α 0 and α I are one-forms in the (flat) 3-dimensional x-space. The R-R field strengths are defined by F n = dC n−1 − H 3 ∧ C n−3 , (B.6) and satisfy the duality relations * 10 F 2 = F 8 * 10 F 4 = −F 6 (B.7) We stress that the inclusion of C 5 and C 7 is required in order to get a self-dual fiveform field F 5 = dC 4 after T-dualities. Only after the inclusion of these terms, the Buscher's rules map solutions of type IIA to solutions of IIB and viceversa.

B.2 D3 4 -solution
The D0-D2-D4-D6 system can be mapped to a system of solely D3-branes applying three T-dualities alongỹ 1 ,ỹ 2 andỹ 3 . T-duality mixes the supergravity fields according to the generalised Buscher rules [5]: [m 1 ...m n−2 |z g z|m n−1 ] g zz (B.8) where by z we denote the direction along with T-duality is performed and the metric is understood in the string frame. Applying this rules one finds that after T-dualities alongỹ 1 ,ỹ 2 andỹ 3 the dilaton and the B-field becomes trivial while the metric maps to ds 2 = −e 2U (dt + w) 2 + e −2U [m 1 ...m n−2 |z g z|m n−1 ] cancels because the original metric is always diagonal along the two-torus along one of whose two coordinates the following T-duality transformation is performed. As a result, the T-duality transformations can be written in the simple form C ′ (n) = C (n−1) ⊥ ∧ẽ I + C into a components containing or not the direction dỹ I along which the T-duality is performed. After T-dualities alongỹ 1 ,ỹ 2 andỹ 3 , one finds that all RR fields are mapped to the four-form C 4 that can be written in the form C 4 = α 0 ∧ẽ 1 ∧ẽ 2 ∧ẽ 3 + 1 2 ǫ IJK α I ∧ dy I ∧ẽ J ∧ẽ K + β 0 ∧ dy 1 ∧ dy 2 ∧ dy 3 + 1 2 ǫ IJK β I ∧ẽ I ∧ dy J ∧ dy K (B.13) One can check that the five form flux obtained from (B.13) is self-dual as required.