Black hole microstates in AdS$_4$ from supersymmetric localization

This paper addresses a long standing problem, the counting of the microstates of supersymmetric asymptotically AdS black holes in terms of a holographically dual field theory. We focus on a class of asymptotically AdS$_4$ static black holes preserving two real supercharges which are dual to a topologically twisted deformation of the ABJM theory. We evaluate in the large $N$ limit the topologically twisted index of the ABJM theory and we show that it correctly reproduces the entropy of the AdS$_4$ black holes. An extremization of the index with respect to a set of chemical potentials is required. We interpret it as the selection of the exact R-symmetry of the superconformal quantum mechanics describing the horizon of the black hole.


Introduction
One of the great successes of string theory is the microscopic explanation of the entropy of a class of asymptotically flat black holes. An immense literature, which we will not try to refer to here, followed the seminal paper [1]. No similar result exists for asymptotically AdS black holes. This is curious since holography suggests that the microstates of the black hole should correspond to states in a dual conformal field theory. The AdS/CFT correspondence should be the natural setting where to explain the black hole entropy in terms of a microscopical theory.
Various attempts have been made to derive the entropy of a class of rotating black holes in AdS 5 in terms of states of the dual N = 4 super-Yang-Mills (SYM) theory [2,3] but none was completely successful. 1 In this paper we consider the analogous problem for asymptotically AdS 4 black holes. In AdS 4 there exist spherically symmetric static BPS black holes, 2 preserving at least two real supercharges. The first numeric evidence for these solutions was found in [4], but their analytic construction was discovered in [5] and further studied by many authors [6][7][8][9][10][11][12][13][14]. They occur in nonminimal N = 2 gauged supergravity in four dimensions and they reduce asymptotically to AdS 4 with the addition of magnetic charges for the gauge fields in vector multiplets. This background is sometimes called magnetic AdS 4 . The full spacetime can be thought of as interpolating between the asymptotic AdS 4 vacuum and the near-horizon AdS 2 × S 2 geometry, leading to a natural holographic interpretation as an RG flow across dimensions. In particular, we have a flow between a CFT 3 and a CFT 1 , from a three-dimensional theory compactified on S 2 to a superconformal quantum mechanics (QM).
To be concrete, we focus on a class of supersymmetric black holes that are asymptotic to AdS 4 × S 7 . The dual field theory is a topologically twisted ABJM theory [15] depending on a choice of magnetic fluxes n a for the four Abelian gauge fields U (1) 4 ⊂ SO(8) arising from the reduction on S 7 . The theory, dimensionally reduced on S 2 , gives rise to a supersymmetric quantum mechanics. The holographic picture suggests that it becomes superconformal at low energies. It also suggests that the original UV R-symmetry of the three-dimensional theory mixes in a non-trivial way with the flavor symmetries along the flow, and that some extremization principle is at work to determine the exact linear combination. The setting is indeed very similar to the one in [16,17], where the dual to the topologically twisted N = 4 SYM compactified on a Riemann surface Σ was studied. The gravity solution interpolates between AdS 5 and AdS 3 × Σ. In [16,17], the central charge of the dimensionally reduced CFT 2 has been computed via cextremization and successfully compared with the gravity prediction.
Here we focus our attention on the entropy of the black hole. We expect that it can be obtained with a microscopic computation in the dual field theory and we show indeed that this is the case. To this purpose, we evaluate the topologically twisted index introduced in [18] for the ABJM theory. This is the partition function of the topologically twisted theory on S 2 × S 1 and can be computed via localization [18]. The result depends on a set of magnetic fluxes n a and chemical potentials ∆ a for the global symmetries of the theory. It can be interpreted as the Witten index Z(n a , ∆ a ) = Tr (−1) F e −βH e iJa∆a of the dimensionally-reduced quantum mechanics. The magnetic fluxes n a precisely correspond to the magnetic charges of the black hole. The chemical potentials ∆ a parametrize the mixing of the R-symmetry with the flavor symmetries. We propose that, in order to find the R-symmetry that sits in the superconformal algebra in the IR, we need to extremize Z(n a , ∆ a ) with respect to ∆ a .
The main result of this paper is the evaluation of the topologically twisted index Z(n a , ∆ a ) for ABJM in the large N limit. We extremize Z(n a , ∆ a ) at large N and we show that the extremum exactly reproduces the black hole entropy: Re log Z| crit (n a ) = S BH (n a ) .
The critical values of the ∆ a 's coincide with the values of the bulk scalar fields at the horizon of the black hole, which parametrize the bulk dual to the R-symmetry, in perfect agreement with supergravity expectations.
One of the technical challenges of this paper is the evaluation of the topologically twisted index in the large N limit. The index can be expressed as a contour integral, of a meromorphic form Z int of Cartan-valued complex variables x, summed over a lattice Γ h of magnetic gauge fluxes. The form Z int encodes the classical and one-loop contributions to the path-integral, around BPS configurations. We first perform the sum over the magnetic flux lattice. Then we solve, at large N , an auxiliary set of equations-which have been dubbed "Bethe Ansatz Equations" in a similar context in [19]-that give the positions of the poles of the meromorphic integrand. This part of the computation bears many similarities with the large N evaluation of the S 3 partition function for N = 2 three-dimensional theories in [20,21], although it is much more complicated. We finally evaluate the partition function Z using the residue theorem.
Our result opens many questions and directions of investigation. Let us mention two of them.
First, it is tempting to speculate that, under certain conditions, the exact R-symmetry in N = 2 superconformal QM can be found by extremizing the corresponding Witten index. This fact would add to the other extremization theorems valid in higher dimensions. We know that even and odd dimensions work differently. In two and four dimensions, the exact R-symmetry is found by extremizing central charges: a-maximization works in four dimensions [22,23] and c-extremization in two [16]. In odd dimensions, we have so far the example of three dimensions where the partition function on S 3 is extremized [21,24,25]. The natural candidate for an extremization in one dimension is the partition function on S 1 , which is exactly the Witten index.
Secondly, it would be very interesting to understand better the superconformal quantum mechanics corresponding to the horizon of the black hole. Our computation is done in the topologically twisted three-dimensional theory. The dimensionally reduced QM has infinitely many states corresponding to different gauge fluxes on S 2 . The topologically twisted index defined in [18] depends on fugacities y a = e i∆a−σa and it counts the supersymmetric ground states, but it necessarily involves a regularization when the fugacities are pure phases, as it is in our case. 3 It would be interesting to understand more precisely how these supersymmetric ground states flow to the microstates of the black hole. This implies understanding in details the structure of the IR superconformal quantum mechanics. We leave these very interesting questions for the future.
The paper is organized as follows. In Section 2 we write the topologically twisted index for the ABJM theory. We evaluate it in the large N limit as a function of the magnetic fluxes n a and the fugacities y a = e i∆a . In particular we show that it scales as N 3/2 . Our computation is valid for N 1, which corresponds to the M-theory limit of ABJM. In Section 3 we review and discuss the general features of the static supersymmetric AdS 4 black holes. We emphasize in particular the holographic interpretation. In Section 4 we compare the field theory and supergravity results. We show that the critical value of the index correctly reproduces the black hole entropy. We also show that the critical values of the chemical potentials ∆ a match with the horizon values of the scalar fields and we show how this corresponds to the identification of the exact R-symmetry of the problem. In Section 5 we give a preliminary discussion of some open issues, like the Witten index extremization and the correct interpretation of the superconformal quantum mechanics. Finally, in the Appendices we give a derivation of the near horizon black hole metric from the BPS equations of gauged supergravity, we discuss the simplest case of a superconformal quantum mechanics-the free chiral field-and we discuss in details the attractor mechanism for our class of black holes. 2 The topologically twisted index of ABJM at large N A general 3d N = 2 supersymmetric theory with an R-symmetry and integer R-charges, can be placed supersymmetrically on S 2 × S 1 (in fact on Σ g × S 1 ) by performing a partial topological twist on S 2 . If the theory has also a continuous flavor symmetry, then there is a discrete infinite family of such twists obtained by mixing the R-symmetry with Abelian subgroups of the flavor symmetry, and twisting by these alternative R-symmetries. One can also turn on background flat connections along S 1 , and real masses. Both can be thought of as a background for the bosonic fields (the connection along S 1 and the real scalar) in external vector multiplets coupled to the flavor symmetry; we collectively call them "complex flat connections". One can then compute the path-integral of the theory on S 2 × S 1 with such a background: this defines the so-called topologically twisted index of the theory [18]. We briefly review its definition here and then we apply it to the ABJM theory.
We take a metric on S 2 × S 1 and a background for the R-symmetry given by where t ∼ = t + 1 . We take vielbein e 1 = R dθ, e 2 = R sin θ dϕ, e 3 = β dt. We can write supersymmetric Yang-Mills and Chern-Simons Lagrangians for a vector multiplet V = (A µ , σ, λ, λ † , D) and for matter chiral multiplets Φ = (φ, ψ, F ) transforming in a representation R of the gauge group where q is the R-charge of the chiral multiplet [18]. In the previous expression, for example, ψ † σψ is a shorthand for ψ † A σ α (T α ) A B ψ B , where the indices A, B run over the representation R and α, β over the Lie algebra. The covariant derivatives in (2.2) and (2.3) contain the R-symmetry background (2.1). Supersymmetry is preserved by a constant spinor satisfying γ 3 = .
Whenever the theory has flavor symmetries J f , we can turn on supersymmetric backgrounds for the bosonic fields in the corresponding vector multiplet A Cartanvalued magnetic background for the flavor symmetry is supersymmetric provided that F f 12 = iD f . We can also turn on an arbitrary Cartan-valued vacuum expectation value for σ f and A f t . The theory is deformed by various terms that can be read from the matter Lagrangian (2.3) where we consider the vector multiplet V = (A µ , σ, λ, λ † , D) appearing there as running over the gauge as well as flavor symmetries. The flavor gauge background appears in the covariant derivatives of the matter fields and in explicit mass term deformations. The magnetic flux n for the flavor symmetry will add up to the magnetic flux for the R-symmetry, providing a family of topological twists. The constant potential A f t is a flat connection (or Wilson line) for the flavor symmetry and σ f is a real mass for the three-dimensional theory. The nonvanishing value for D f induces extra bosonic mass terms in the Lagrangian [18].
One can then compute the path-integral of the theory on S 2 × S 1 with such a background using localization techniques [18]. The path integral is a function of the flavor magnetic fluxes n and fugacities y = e i(A f t +iβσ f ) for the flavor symmetries and it defines the so-called topologically twisted index of the theory [18]. It is explicitly given by a contour integral of a meromorphic form Z(n, y) summed over all magnetic fluxes m in the co-root lattice Γ h of the gauge group and integrated over the zero-mode gauge variables x = e i(At+iβσ) , where A t runs over the maximal torus of the gauge group and σ over the corresponding Cartan subalgebra. More precisely, we introduce a variable u = A t + iβσ on the complexified Cartan subalgebra g C and, given a weight ρ, we use a notation where x ρ = e iρ(u) . The form Z int (x, y; m, n) receives contributions from the classical action and the one-loop determinants. The contribution of a chiral multiplet to the one-loop determinant is given by where R is the representation under the gauge group G, ρ are the corresponding weights, q is the R-charge of the field, and ρ f is the weight of the multiplet under the flavor symmetry group.
The contribution of a vector multiplet to the one-loop determinant is instead given by where α are the roots of G. The classical action contributes a factor where k is the Chern-Simons coupling of G (each Abelian and simple factor has its own coupling). A U (1) topological symmetry with holonomy ξ = e iz and flux t contributes Supersymmetry selects the contour of integration to be used in (2.5) and determines which poles of Z int (x, y; m, n) we have to take. The result can be formulated in terms of the Jeffrey-Kirwan residue [26], and we refer to [18] for the details.

The index of ABJM
The low-energy dynamics of N M2-branes on C 4 /Z k is described by the so-called ABJM theory [15]: it is a three-dimensional supersymmetric Chern-Simons-matter theory with gauge group U (N ) k ×U (N ) −k (the subscripts are the CS levels) and matter in bifundamental representations. Using standard N = 2 notation, the matter content is described by the quiver diagram where i, j = 1, 2 and arrows represent bifundamental chiral multiplets, and there is a quartic superpotential For k = 1, 2 the theory has N = 8 superconformal symmetry, while for k ≥ 3 it has N = 6 superconformal symmetry. In the N = 2 notation, an SU (2) A × SU (2) B × U (1) T × U (1) R global symmetry is made manifest: the first two factors act on A i and B j , respectively, as on doublets; U (1) T is the topological symmetry associated to the topological current J T = * Tr(F − F ) where F, F are the two field strengths; U (1) R is an R-symmetry. Working in components, though, one finds the full SO(6) R symmetry; for k = 1, 2 the R-symmetry is further enhanced to SO(8) quantum mechanically [15,27,28].
To relate the symmetries of the theory to the isometries of C 4 , let us consider 4 N = 1 and k = 1 which describes a single M2-brane moving on C 4 . The theory has gauge symmetry 4 For N = 1, the superpotential vanishes and the manifest global symmetry is enhanced to SU (2)A × SU (2)B × U (1)D × U (1)T × U (1)R of rank 5, where U (1)D gives charge 1 to all chiral multiplets. In fact, in this case the theory is four free chiral multiplets describing C 4 , or a NLSM on the orbifold C 4 /Z k , which have a rank-4 flavor symmetry and a U (1)R symmetry. In view of the N > 1 case, we will neglect U (1)D.
U (1) g × U (1)g, and denoting U (1) A/B the Cartans of SU (2) A/B , the standard charge assignment is The monopole T corresponds to the magnetic flux m = (1, −1) while T to m = (−1, 1). These monopoles get their gauge charges from the CS terms. The chosen U (1) R is the superconformal R-symmetry of an N = 2 superconformal subalgebra. The gauge invariants are A i T and B j T , which are the coordinates of C 4 (their R-charge 1 2 signals that they are free). It is convenient to introduce a new basis for the Cartan of global symmetries, where the flavor symmetries J 1,2,3 act on a copy of C ⊂ C 4 respectively, J 4 is an R-symmetry, and they all have integer charges: (2.12) In terms of charges: We will use these symmetries to put the theory on S 2 × S 1 with a topological twist. In particular we call −n 1,2,3 the fluxes and y 1,2,3 the fugacities associated to J 1,2,3 . To restore the symmetry, we can introduce n 4 and y 4 as well, defined by a n a = 2 , a y a = 1 . (2.14) In the main body of the paper, we will not introduce separate parameters t, ξ for the topological symmetry, essentially because J 1 + J 2 − J 3 = J g − J T , i.e. the topological background is already included up to a gauge background. For the ABJM theory, the topologically twisted index is computed using the rules discussed above and we find b=3,4 x j For the moment, we introduced all possible parameters including redundant ones. However, the index has a set of symmetries and invariances, some of which correspond to the aforementioned redundancies. First of all, the index is actually nonvanishing only if t +t = 0 (mod k). This can be seen by performing the integral over the diagonal U (1). By a change of variables x i = zwx i and which can be non-zero only if t +t = 0 (mod k). Secondly, the index has nice properties under shift of the arguments: where each line represents a different transformation and λ,λ, p,p (with λ,λ ∈ C * and p,p ∈ Z) are the parameters. In the first column we indicated the transformation to be performed on the dummy variables in the expression of Z which gives the transformations reported in the last four columns. The first two transformations can be used to set ξ =ξ = 1, and for k = ±1 the last two can be used to set t =t = 0. For larger values of k, the best we can do is to set t +t = 0 since it is a multiple of k. However, since we will be mainly interested in the case k = 1, we will simply take t =t = 0 from the start. 5 Thirdly, the index is invariant under discrete involutions, which we write for simplicity for t =t = 0 and ξ =ξ = 1: (2.17) In the first two columns we indicated the transformation to be performed on dummy variables in the expression of Z. Combining the transformations we see that the index is invariant under change of sign of k (corresponding to a parity transformation [15]) and under inversion of the fugacities. We will assume then, without loss of generality, that k > 0. We thus study the index b=3,4 x j . (2.18) 5 In fact, it is simple to check that in the large N limit the free energy only depends on t +t.
The Jeffrey-Kirwan residue selects a middle-dimensional contour in (C * ) 2N . The integrand has no residues in the "bulk", and the only residues are at the boundaries x i = 0, ∞,x j = 0, ∞ of the domain. According to the rules discussed in [18], we need to choose reference covectors η, η that, combined with the sign of the Chern-Simons coupling, tell us which residues we have to take. The final result is independent of η, η. We choose the covectors −η = η = (1, . . . , 1) in such a way that we pick all residues at the origin [18]. Then the range of the sums over m i and m j are bounded above and below, respectively. We can take m i ≤ M − 1 and m j ≥ 1 − M for some large integer M . Performing the summations we get where we defined the quantities x j and After the summation, the contributions from the residues at the origin have moved to the solutions to the "Bethe Ansatz Equations" (BAEs) We borrow this terminology from [19], where a similar structure was found. Notice that if we take |y a | = 1, then the equations are invariant under the exchange x i ↔x * i . Moreover, taking the product of the equations immediately leads to the constraint 6 As generically all poles are simple, to take the residues we simply insert a Jacobian and evaluate everything else at the pole, hence we see that the dependence on M disappears. The partition function takes the compact expression (2.24) 6 In particular we can always find "obvious" solutions imposing xi = x,xi =x for all i. From the constraint, x = ω x where ω is a kN -th root of unity. Then These solutions, however, do not contribute to the original integral because they are killed by the vector multiplet determinant.
The sum is over all solutions I to the BAEs, modulo permutations of the x i 's andx j 's. All instances of x i ,x j have to be evaluated on those solutions.
The matrix B appearing in the Jacobian is 2N × 2N with block form It is the product of the matrix of derivatives and the diagonal matrix diag(x l ,x l ). The two blocks on the diagonal are diagonal matrices, ∂e iB j /∂x l = 0 and ∂e i B j /∂x l = 0 for j = l, while the off-diagonal blocks are more complicated and contain all components. We can introduce the function , (2.26) which allows to write the BAEs in a compact form: Then we can introduce the objects The blocks of B, imposing 1 = e iB i = e i B j , are Notice that, because of the relation between the y a 's, D(0) = D(∞) = 1. Moreover the logarithmic derivative ∂ log D(z)/∂ log z vanishes both at z → 0 and z → ∞. This behavior is sometimes called "absence of long-range forces" in the large N matrix model. For k > 1, given a solution {x i ,x j } to the BAEs (2.21)-(2.22), we can obtain more solutions multiplying all x i ,x j by a common k-th root of unity ω k : Thus, all solutions are "k-fold degenerate". One can also check that (2.24) receives the same contribution from those k solutions: both det B and the rest of the expression inside the summation are invariant under (2.30). Therefore in (2.24) we could sum over the orbits of (2.30) and multiply the result by k.

The Bethe potential
It is convenient to change variables to u i ,ũ j , ∆ a , defined modulo 2π: The relation a y a = 1 becomes a ∆ a = 0 (mod 2π). Then the Bethe ansatz equations become where n i ,ñ j are integers that parametrize the angular ambiguities. In the following we will take ∆ a real. We recall the polylogarithms Li n (z) defined by for |z| < 1, and by analytic continuation outside the disk. The first two cases are Li 0 (z) = z/(1−z) and Li 1 (z) = − log(1 − z). For n ≥ 1, the functions have a branch point at z = 1 and we shall take the principal determination with a cut [1, +∞) along the real axis. For different values of n, the polylogarithms are related by The functions Li n (e iu ) are periodic under u → u + 2π and have branch cut discontinuities along the vertical line [0, −i∞) and its images, as represented in Figure 1. For us the following inversion formulae will be important: 7 Li 0 (e iu ) + Li 0 (e −iu ) = −1 for 0 < Re u < 2π. The formulae in the other regions are obtained by periodicity. Also notice that Li 0 (z) and Li 1 (z) diverge at z = 1, while Li n (z) for n ≥ 2 have no divergences on the z-plane. All the equations in (2.32) can be obtained as critical points of the function (2.36) for some choice of n i ,ñ j and up to constants that do not depend on u i ,ũ j . We call this function the Bethe potential.

The BAEs at large N
Our goal is to evaluate the twisted index (2.24) at large N . In order to do so, we first seek the dominant solution to the BAEs (2.22) at large N , and then evaluate its contribution to (2.24).
A convenient way to solve the BAEs at large N is to first evaluate the functional V and then extremize it. Even with this strategy, it is hard to compute all possible large N limits in full generality. Following a similar idea in [20], we first study the BAEs numerically for some large values of N , and extract a plausible ansatz for the large N solution; then we extremize the Bethe potential with respect to that ansatz. For the sake of clarity, in the following we focus on the case k = 1. Although most of the computations straightforwardly generalize to k > 1, there are in fact some subtleties related to the identification of the full set of solutions, and we defer the study of those cases to future work. Moreover we are interested in fugacities |y a | = 1, i.e. we will not consider the addition of real masses here.
The numerical analysis can by done with two different methods. The first one involves finding numerical solutions to the system (2.21)-(2.22) by iterating the transformation where C is some large positive integer. If e iB i is not 1, then to a first approximation we can move it towards 1 by rescaling x k i (and neglecting the effect on the product). The second method involves introducing a time coordinate and setting up a dynamical system  constants and initial conditions. However we were lucky enough to find a couple of enlightening examples that we show in Figure 2 and 3.
In Figure 2 we plot the distribution of eigenvalues u i andũ j in the symmetric case y a = i (i.e. ∆ a = π/2) for N = 25 and N = 101 (and k = 1). The distribution has been obtained with the iteration method (2.37). We see that the imaginary parts of u i andũ j grow with N . An analysis for many different values of N reveals that the scaling is consistent with a behaviour N 1 2 . On the other hand, the real parts of u i andũ j stay bounded when N grows. The difference Re(ũ i − u i ) has minimum value − π 2 and maximum value π 2 . For comparison, we also plot the analytical result that we will derive later in this section.
In Figure 3 we plot the distribution of u i andũ j for the case ∆ 1 = 0.3, ∆ 2 = 0.4, ∆ 3 = 0.5 with a ∆ a = 2π (and k = 1). The distribution has been obtained with the dynamical system method (2.38). The integers n i andñ j have been chosen in such a way that the distribution is "continuous" on the u-plane, as explained below. The plots for N = 50 and N = 75 are again consistent with a N 1 2 scaling of the imaginary parts of the eigenvalues. The real parts are not scaling with N , and there are two tails of the distribution where they are constant. One can check that the two tails occur whenũ i − u i + ∆ 3 = 0 andũ i − u i − ∆ 1 = 0. These values correspond to logarithmic singularities in the equations (2.32) and will play an important role in the following. Notice that both in this case and the previous one, the large N solutions is invariant under the symmetry x i ↔x * i . Thus, we consider an ansatz where the imaginary parts of u i andũ i are equal 8 and scale as N α for some power α (having in mind α ∼ 1 2 ), while the real parts remain of order one: Given the permutation symmetry, we can parametrize the points by the variable t instead of the index i, by introducing the density while at large N we assume that it becomes a continuous distribution. Summations are replaced by integrals: The density distribution is normalized: In the large N limit, we seek configurations where ρ(t), v(t),ṽ(t) and therefore also δv(t) are continuous functions. Inspecting the BAEs in (2.32) we see that they are singular whenever δv(t) hits ∆ 1,2 or −∆ 3,4 (or their periodic images), therefore on continuous solutions δv does not cross those values. We recall that all angular variables are defined modulo 2π. We can fix part of the ambiguity in ∆ a by requiring that We can fix the remaining ambiguity of simultaneous shifts δv → δv + 2π, ∆ 1,2 → ∆ 1,2 + 2π, ∆ 3,4 → ∆ 3,4 − 2π by requiring that δv(t) takes the value 0 somewhere (led by the numerical analysis, we assume that for k = 1, δv(t) = 0 (mod 2π) is always solved somewhere). Thus, our choice for the angular determination simply corresponds to Given the symmetry of all functions and equations under the exchange of a = 1 ↔ 2 and of a = 3 ↔ 4, without loss of generality we can order One could have considered a more general ansatz where the imaginary parts are unrelated: ui = iN α ti + vi andũi = iN αt i +ṽi. In the large N limit this leads to two different density distributions ρ(t) = di/N dt and ρ(t) = di/N dt. One can take the large N limit of the BAEs in (2.32) directly, without passing through the Bethe potential, as we do in (2.66). This leads to two copies of (2.66), one containing ρ(t) and oneρ(t). It follows, for generic values of δv(t), that ρ(t) =ρ(t).
Later on we will have to distinguish the cases that a ∆ a = 2π, 4π or 6π (while the cases 0 and 8π correspond to y a = 1 and are singular). Combining (2.44) with a ∆ a = 2π one finds ∆ 1 < 2π − ∆ 4 and ∆ 2 − 2π < −∆ 3 , therefore the inequalities (2.43) can be put in the stronger form: For a ∆ a = 4π one finds that there are two possibilities: At this point we should provide an estimate for what the constants n i ,ñ j are on solutions. Let us assume that , thus the function takes extremely small values. For large negative imaginary part of u, instead, the function approaches ∆ a − 4π. The dependence on u is exponentially suppressed and we observe an "absence of long-range forces" as in [20]. 9 We conclude that the integers n i ,ñ j take the values Given the ansatz, the Heaviside theta function could be replaced by Θ(i > j) if the points are ordered by increasing imaginary part. Hence, in the large N limit we will use the function Notice that from here on we set k = 1. The leading contribution from the first term in V is easy to compute: To compute the second term in V, we break (2.53) 9 There is a difference with respect to [20]. In the latter, the matrix model has long-range forces which cancel out if all species of eigenvalues have the same density distribution ρ(t). In our case, the BAEs do not have long-range forces at all, and the condition ρ =ρ is imposed by the local interactions among the eigenvalues.
The first term in this last expression is of O(N ) and apparently subleading. However it should be kept-as we will see-because its derivative is not subleading on part of the solution when δv approaches ∆ 1,2 or −∆ 3,4 . Therefore we keep We decompose into "Fourier modes", Li 2 (e iu ) = ∞ k=1 e iku /k 2 . Then we consider the integral (2.55) where in the second equality we have Taylor-expanded the integrand around the lower bound. Performing the integral in t we see that the leading contribution is for j = 0, thus With the second term in (2.53), where the summation is over i > j, we should be more careful because, in order to achieve a localization of the integral to the boundary, we should first invert the integrand. Consider first the case that 0 < Re(ũ j − u i + ∆ 3,4 ) < 2π: the formula to use is Following the same steps as before, the summation i>j of the first term in the latter expression gives something similar to (2.57) but with − Li 3 e −i(δv(t)+∆ 3,4 ) in place of Li 3 . The two contributions can then be combined using (2.35), and result in a cubic polynomial expression.
which differs from the previous one by a sign. Again, the result of the summation i>j can be combined with that of i<j to give a cubic polynomial expression. The remaining terms from (2.58) and (2.59), throwing away the constants which do not affect the critical points, are This term is precisely canceled by the last term in (2.51).
To have a competition between the leading terms of order N 1+α and N 2−α , we need Including a Lagrange multiplier µ to enforce the normalization of ρ(t), the final result is the following large N expression (up to constants independent of ρ and δv): We remind that the last term can be neglected when computing the value of the functional V, because Li 2 does not have divergences-however it becomes important when computing the derivatives of V because Li 1 (e iu ) diverges when u → 0.
The special case y a = i. This case, corresponding to ∆ a = π 2 , produces a particularly simple numerical solution that we reported in Figure 2. The function V simplifies to Setting to zero the variations with respect to ρ(t) and δv(t), we get the equations On the support of ρ(t), the solution is ρ(t) = 2µ/π 3 and δv(t) = π 2 t/4µ. Calling [t − , t + ] the support of ρ, then t + −t − = π 3 /2µ from the normalization. Plugging back into V and extremizing with respect to µ and t − we obtain µ = π 2 /2 √ 2 and t ± = ±π/ √ 2. Finally Since δv(t ± ) = ±π/2, in the solution δv(t) barely reaches ∆ 1,2 or −∆ 3,4 at the boundaries of the support, and the last term in (2.60) of order N − 1 2 can be safely neglected, as we did. This solution, corresponding to the solid grey line in Figure 2, precisely reproduces the numerical simulation.
The general case. To obtain the large N solution to the BAEs in the general case, we again set to zero the variations of V in (2.60) with respect to ρ(t) and δv(t). The latter equation, though, can be obtained as the large N limit of the BAEs directly, and it is instructive to do so first. 10 Consider the first equation in (2.32): we manipulate it as we did with the functional V. In particular, we break the sum N j=1 into j>i + j<i + (j → i). We find: The first term comes from u i with correction O(1); the second term comes from the summation j =i with corrections O(N 1−2α ) and 2πn i with corrections O(1); the terms on the second line come from j = i.
In order to have a competition between the leading terms it must be α = 1 2 . As long as δv(t)+∆ 3,4 = 0 and δv(t)−∆ 1,2 = 0, the terms on the second line are O(1) and can be neglected. One finds the equation (2.66) However, when δv(t) approaches ∆ 1,2 or −∆ 3,4 the terms on the second line blow up (in particular, smooth solutions never cross those values) and may compete with those on the first line. In order to have a competition it must be for some value of a = 1, 2, 3, 4 and with Y a (t) > 0 of order one. Then the second line contributes −N 1 2 ε a Y a (t) + π 2 + O(e −N 1/2 ), which competes with the other leading terms. One finds the equation (2.68) The equations (2.66) and (2.68) correspond to ∂V/∂ δv(t) = 0. The variation of V with respect to ρ(t) is not affected by the terms suppressed by N −1/2 in (2.60), because Li 2 (e iu ) has no divergences. Thus we find that the large N limit of the BAEs is the system of equations: as well as 1 = dt ρ, ρ > 0 on its support and Y a > 0.
The solution for ∆ a = 2π. We then proceed to solve the equations. First we solve the system (2.69) for generic values of δv, which we call the "inner interval". It turns out that ρ(t) is a linear function, while δv(t) is the ratio of two linear functions and the sign of its derivative equals the sign of µ. This solution is reliable until one of the conditions −2π < δv − ∆ 1,2 < 0 or one of 0 < δv + ∆ 3,4 < 2π is saturated. This defines the "inner interval" [t < , t > ]: one saturation happens on one side and one on the other side. The inequalities (2.46) fix that in the inner interval δv(t) goes from −∆ 3 to ∆ 1 . Imposing that ρ > 0 at the extrema fixes µ > 0, therefore δv(t) is increasing. Outside the inner interval, in two regions that we call the "left and right tails" respectively, δv remains frozen at its limiting values −∆ 3 and ∆ 1 up to exponentially small corrections, and the equations determine ρ(t) and the correction Y 1,3 (t). The end of the tails is where ρ(t) = 0. Reassuringly, ρ(t) turns out to be increasing in the left tail and decreasing in the right tail. Summarizing The points t < and t > are also those where Y 3,1 = 0. Then we define t and t as the values where ρ = 0 and those bound the left and right tails. Schematically: Finally we fix µ by requiring that dt ρ(t) = 1. The solution is as follows. The transition points are at In the left tail we have In the inner interval we have and δv > 0. In the right tail we have Finally, the normalization fixes The solution satisfies dt ρ(t) δv(t) = 0. In Figure 4 we consider a case with generic ∆ a 's-the same case considered in Figure 3-and compare the numerical simulation of the large N solution to the BAEs, with the analytical result: we plot the density of eigenvalues ρ(t) and the function δv(t).
The solution in the other ranges. For ∆ a = 4π, it turns out that there are no consistent solutions to the large N BAEs. One can run an argument similar to the one we had before, concluding that it is not possible to construct a solution with an inner interval where δv(t) transits between two singular values, and two tails where δv is frozen while ρ(t) dies off to zero. This implies that, for such a range of parameters, the order of the index Z(∆ a ) is smaller than for the other ranges.
The solution for ∆ a = 6π is very similar to the one in (2.71)-(2.75). The function δv(t) is decreasing from 2π − ∆ 4 to ∆ 2 − 2π, as prescribed by (2.47), and µ < 0. The solution is obtained from (2.71)-(2.75) by performing the substitutions In fact, notice that ∆ a = 6π is equivalent to ∆ a = 2π, therefore there is a pairing between points in the two ranges of the parameter space, and a corresponding map between BAE solutions. It turns out that, when evaluated on paired solutions, the twisted index Z takes the same value. This can be understood by the following argument. The matrix model for Z in (2.18) is invariant-possibly up to a sign-under the three involutions in (2.17). These transformations can be combined to show invariance of Z under each of the three operations: The last one, in particular, corresponds to ∆ a ↔ 2π − ∆ a and allows to map every solution for ∆ a = 2π to a solution for ∆ a = 6π, which produces the same value of the index Z.

The entropy at large N
We are interested in the large N limit of the twisted index, or partition function, (2.24) and more precisely of its logarithm-the entropy. With the dominant solution to the BAEs at large N in hand, we can compute the large N limit of the expression in (2.24) and plug the solution in.
After various manipulations, we can recast the twisted index in a particularly convenient form: (2.77) This time we have already reorganized the products i,j into the diagonal parts i and the off-diagonal parts, the latter written in terms of j>i solely. Notice that the first two factors are just phases that can be neglected, as we will be interested in log |Z|.
We start with the products j>i . The terms on the third line are treated as in Section 2.3. For a = 3, 4 using 0 < δv + ∆ 3,4 < 2π we find (2.78) Instead, for a = 1, 2 using −2π < δv − ∆ 1,2 < 0 we find (2.79) The contribution of the Vandermonde determinant is similar: Then we consider the products i . The term only contributes in the tail where δv ≈ −∆ 3,4 , and the term Therefore, provided that B −1 exists and its entries are bounded, the leading term is log det B. Following the discussion after (2.25), the matrix B evaluated on the solutions to the BAEs takes the form The function G(z) diverges at z = y 1,2 and z = y −1 3,4 which are phases, therefore the only terms that can diverge are the diagonal ones G ii . We see that we can choose B to have diagonal matrices in all four blocks. Reorganizing the indices, B can be rewritten as a block-diagonal matrix made of 2 × 2 blocks M i : B = diag(M i ) with when G ii diverges, and M i = 1 2 when G ii does not. We made a choice of the O(1) terms such that M i is invertible and the inverse has bounded entries. We then compute using the behavior δv = ε a ∆ a − e −N 1/2 Ya in the tails.
Putting everything together we find the following functional for the entropy at large N : up to corrections of order N log N . We took the real part to get rid of irrelevant phases in Z.
Finally we should take the solution to the BAEs, plug it in the functional (2.88) and compute the integral. From the solution for ∆ a = 2π, we obtain the following surprisingly simple expression for the entropy: Notice that this expression is symmetric under permutations of the indices a = 1, 2, 3, 4. Such a symmetry is expected for k = 1, because the index parametrizes the four complex factors in the C 4 fiber of the normal bundle to the M2-branes.

AdS 4 black holes in N = 2 supergravity
We now move to discuss a class of supersymmetric static asymptotically AdS 4 black holes, holographically dual to the ABJM theory twisted on S 2 that we have discussed so far. We first present the general features of this class of black holes, and then we depict their holographic interpretation, focusing on the asymptotic AdS 4 region and the AdS 2 × S 2 horizon. The BPS black-hole solutions in AdS 4 -similarly to many higher dimensional solutionsà la Maldacena-Nuñez [4,17,[29][30][31][32][33][34][35]]-preserve supersymmetry due to the topological twist on the internal space S 2 (or more generally on any Riemann surface Σ). The noteworthy feature in four dimensions is the existence of full analytic solutions for a completely general set of parameters, as first discovered in [5], elaborated upon in [6,7] and further generalized in various directions in [8,[11][12][13][14] and references therein. The complete spacetime can be thought of as interpolating between the asymptotic AdS 4 vacuum and the near-horizon AdS 2 × Σ geometry, leading to a natural holographic interpretation of those black holes as RG flows across dimensions.
Here we are specifically interested in solutions to the maximal D = 4 N = 8 gauged supergravity, which can in turn be embedded in eleven-dimensional supergravity with an M-theory interpretation as wrapped M2-branes. In particular we focus on black holes that are asymptotic to AdS 4 × S 7 . The topological twist on the internal two-dimensional space requires a background SO(2) gauge field turned on, and therefore without loss of generality we can restrict our attention to the N = 2 truncation of the maximal supergravity [36,37]. 11 We follow the standard 11 See also [38] for the embedding of these black holes in 11D.
conventions of [39] and consider the so-called magnetic STU model with electric FI gaugings that arises exactly as a truncation of N = 8 supergravity. It consists of three vector multiplets (in addition to the gravity multiplet) with the prepotential and can be seen from the 11D point of view as a Kaluza-Klein reduction on S 7 (the X Λ are the holomorphic sections of the underlying special Kähler manifold). In addition, the gravitino R-symmetry is electrically gauged as specified by the FI parameters which complete the N = 2 data necessary for the unique definition of the Lagrangian and BPS variations. Further details about the supergravity model can be found in [5,7] and in Appendix A, where for completeness we present an explicit derivation of the BPS equations and the nearhorizon geometry that eventually leads to the crucial entropy formula. Before presenting the black hole solution, a word on notation is in order. The N = 2 STU model has four gauge fields that correspond to the Cartan subalgebra of the SO(8) isometry of S 7 . The standard N = 2 supergravity symplectic index Λ = {0, 1, 2, 3} used above is actually somewhat unnatural from the point of view of maximal supergravity and the field theory side, where the four gauge fields appear symmetrically. Therefore, with an abuse of notation we will introduce the index a = {1, 2, 3, 4}, and identify the original Λ = {0, 1, 2, 3} with a = {4, 1, 2, 3} in this order. The index a is the same as that used in Section 2 and it allows to write all formulae in a manifestly permutation-invariant way. We do not distinguish between the upper and lower position of the index a.
The 4D black hole metric 12 is compactly written as where g and c are parameters while the Kähler potential is The real sections X a are constrained in the range 0 < X a < 1 and satisfy a X a = 1. They are given by in terms of parameters β a subject to the above constraint and further ones spelled below. The solution for the sections above defines also the background values for the physical scalar fields, which are typically chosen as The parameters β a also specify the constant c, which is related to the value r h of the radial coordinate at the horizon: We already set the unit of the AdS 4 curvature g = 1/ √ 2, i.e. the parameters β a have been rescaled in the appropriate units more suitable for holographic use. The black hole has a regular horizon only for a restricted region in the parameter space of β a that ensures that r h is real and the scalars X a are positive.
Another crucial element of the solution is given by the background fluxes that carry magnetic charges through the sphere: The four magnetic charges of the black hole n a are integer and fulfil the twisting relation a=1,2,3,4 n a = 2 , (3.9) which ensures that two out of the original eight supercharges are preserved by the black hole solution. Supersymmetry further relates the magnetic charges to the parameters β a that specify how the scalars run along the RG flow: Let us define the following quantities: 13 Π = 1 8 n 1 + n 2 − n 3 − n 4 n 1 − n 2 + n 3 − n 4 n 1 − n 2 − n 3 + n 4 ) a<b n a n b − 1 4 a n 2 a , Θ = F 2 2 − 4n 1 n 2 n 3 n 4 . (3.11) It is easy to check that We can then invert the relations in (3.10), up to a common sign: Here the sign equals the sign of −(β 1 + β 2 )(β 1 + β 3 )(β 2 + β 3 ), in other words the sign is correlated with that of With a little bit of algebra, we find One can also write the first relation as .
(3. 16) Although both signs in the formulae above are compatible with supersymmetry, it turns out (see Appendix A) that smooth solutions exist only if three of the n a are negative, and in that case one should take the upper sign. The black hole above preserves two supercharges, packaged in the corresponding Killing spinor solution, written in terms of a constant spinor ε 0 obeying the following relations: where the hatted indices are flat. Note that the Killing spinors are constant in time and on the sphere and therefore the group of rotations on the sphere commutes with the fermionic symmetries, leading to the corresponding symmetry algebra U (1|1) × SO (3). This is the general black hole solution we want to describe holographically, and in the following we analyze separately the asymptotic region that defines our UV theory, and the nearhorizon IR region related to a 1D superconformal quantum mechanics. Afterwards we discuss the definitions of the black hole entropy and the R-symmetry from the N = 2 supergravity point of view.

The asymptotic AdS 4 vacuum
It is easy to take the limit r → ∞ of the full black hole solution (3.3)-(3.8): one gets the metric ds 2 −r 2 dt 2 + dr 2 r 2 + r 2 dθ 2 + sin 2 θ dφ 2 , constant scalars z 1 = z 2 = z 3 = 1 and non-vanishing magnetic field strengths as in (3.8). This background was dubbed "magnetic AdS 4 " in [40]: not all Killing vectors of AdS 4 are preserved by the magnetic fluxes, the usual supersymmetry enhancement does not take place, and the corresponding symmetry group remains U (1|1) × SO(3) as explained in detail in the reference. As standard in cases of twisting, the isometries of the internal manifold S 2 commute with the supersymmetries and therefore the fermions effectively become scalars under rotation. Of course, as we further go in the UV the background asymptotes to standard AdS 4 and the field strengths (which in vielbein coordinates read F â θφ = −n a sin θ/ √ 2 r 2 ) go to zero, since magnetic AdS 4 is a non-normalizable deformation of AdS 4 .
The complementary boundary picture is also clear: the dual boundary theory is a relevant deformation of the maximally supersymmetric ABJM theory, semi-topologically twisted by the presence of the magnetic charges. The fluxes n a give a family of twisted ABJM theories whose Euclidean version is precisely the one discussed in Section 2. The holographic dictionary can be made precise, as discussed in details in [41]. The boundary values of the gauge fields and the scalar fields z i correspond to relevant deformations of the ABJM Lagrangian: the gauge fields introduce a magnetic background for the R-and global symmetries, while the scalars z i induce mass deformations for the boundary scalar fields. In the Euclidean version the latter precisely correspond to the terms induced in the matter Lagrangian (2.3) by a constant auxiliary D f . Finally, the bulk spinor (3.17) restricts to a constant boundary spinor, as appropriate for a topological twist.

The near-horizon geometry AdS 2 × S 2
Taking the opposite limit, r → r h , leads instead to 14 the AdS 2 × S 2 metric with e −K(r h ) = 8 X 1 (r h )X 2 (r h )X 3 (r h )X 4 (r h ), and the same magnetic charges n a as before. We defined the unit-radius spaces ds 2 AdS 2 = (−dt 2 +dz 2 )/z 2 and ds 2 S 2 = dθ 2 +sin 2 θ dφ 2 . All isometries of AdS 2 are preserved by the background gauge field. This in turn leads to the appearance of new fermionic symmetries, and the full symmetry group becomes SU (1, 1|1) × SO(3) as discussed in [41]. The Killing spinors in this case are full Killing spinors on AdS 2 and are obtained from the general ones by dropping the first relation in (3.18), still keeping them constant on the sphere. We can therefore talk about a genuine superconformal symmetry in the IR, leading to a dual superconformal quantum mechanics.
Making use of the relations (3.10)-(3.15), we can express the near-horizon metric in terms of the magnetic charges n a (see also [13,14] for similar expressions in the literature). Recalling that smooth solutions are obtained only with the upper sign in those expressions, we find the IR metric where the quantities Π, F 2 , Θ are defined in (3.11). The physical scalars are given by . (3.23) The sections at the horizon are then obtained from (3.24) 14 One performs the standard change of variables r = r h + and expands at leading order in .
Smooth solutions are found if exactly three of the n a are negative. More details are given in Appendix A. These expressions can be further related to the different quartic invariants of the symplectic group and can be justified by the implicit electromagnetic duality of 4D N = 2 supergravity, see [13,14] for more details. Electromagnetic duality will likely play a more important role for generalizing our results to solutions with electric charges on top of the magnetic ones we consider.

The entropy and R-symmetry
At leading order, the entropy of the black hole is given by the area of the horizon via the Bekenstein-Hawking formula where G 4D is the four-dimensional Newton constant and we reinstated g for dimensional reasons. We can also write the entropy in a more suggestive form using the symplectic sections X a to compare more directly with the field theory expression (2.89), Let us stress that this is only the leading contribution to the gravitational entropy, which should be supplemented by the higher-derivative corrections following the Wald formalism, and possibly by other quantum corrections. The leading answer for the entropy was confirmed by verifying the first law of thermodynamics in the canonical and grand-canonical ensembles for black holes in AdS 4 [42]. Here we will not consider any corrections to the above formula, in accordance to the fact that we focused only on the leading N 3/2 contribution to the index on the field theory side. As a last important remark about the supergravity solutions, let us note that the theory under consideration has four U (1) gauge fields, which can be thought of as the four Cartan generators of the original SO(8) R-symmetry in the maximal gauged supergravity in 4D. The U (1) R-symmetry of N = 2 supergravity is gauged by a particular combination of those four U (1)'s, called the graviphoton. As shown in [43] for general matter-coupled N = 2 supergravities, in asymptotically AdS spacetimes the graviphoton field strength F gp µν is given by where F Λ are the field strengths of the four gauge fields. This formula is correct only in the case of purely real (or purely imaginary, depending on conventions) sections X Λ , which is the case here. In the context of the AdS/CFT correspondence, such a formula allows us to extract the exact R-symmetry from supergravity and it tells us how it changes from the boundary, where X Λ = 1/4, to the horizon, where we find X Λ (r h ). The notion of R-symmetry defined in (3.27) exists everywhere in the bulk, however it gets a clear holographic meaning only in the UV and the IR, where there is a corresponding exact R-symmetry for the superconformal 3D QFT and quantum mechanics, respectively. In the next section we will compare the field theory parameters ∆ a with the sections X a (r h ) at the horizon.

The attractor mechanism
The notion of attractor mechanism in black hole solutions refers to the way the expectation values of the scalars are fixed at the horizon in terms of the black hole charges. This has been explored carefully in the literature and we elaborate on it in Appendix C, while here we present a shortened version for the black holes we consider.
Let us first notice that there is a simple quantity that exists at generic points in spacetime, which is properly defined in an electromagnetic invariant way in Appendix C for more general black holes. The sections F a ≡ ∂F/∂X a are derived from the prepotential (3.1): It is therefore easy to see that |R| at the black hole horizon gives the entropy (3.26), up to a numerical prefactor. Unlike the entropy, R is defined for all values of the sections X a at any point in spacetime, and for a static geometry it is a function of the radial coordinate r only. It is therefore a natural measure of the holographic RG flow between the asymptotic AdS 4 and the near-horizon AdS 2 × S 2 geometry. We observe that R matches functionally the index (2.89), if we assume a proportionality between X a and ∆ a (see Section 4).
The quantity R is interesting for the attractor mechanism since it provides a function that the scalars extremize at the horizon, under the constraint a X a = 1, and this determines the sections X a (r h ) and correspondingly the physical scalars z i (r h ) in terms of the charges n a . We refer to Appendix C for the derivation of the above formula in the general context of half-BPS attractors in N = 2 gauged supergravity.

Comparison of index and entropy
We can finally compare the field theory and gravity results. We show that the topologically twisted index |Z| in the large N limit is extremized at a value of ∆ a which is proportional to the value of the sections X Λ at the horizon, and that the value of log |Z| at the critical point precisely reproduces the entropy of the black hole. The topologically twisted index is a function of the magnetic fluxes n a and the chemical potentials ∆ a , while the black hole entropy only depends on n a . The physical interpretation of the ∆ a is the following. The path integral of the topologically twisted theory can be interpreted as the Witten index Z(n a , ∆ a ) = Tr (−1) F e −βH e i 3 a=1 Ja∆a (4.1) of the supersymmetric quantum mechanics obtained by reducing the theory on S 2 in the presence of the magnetic fluxes n a [18]. Here J a denote the currents associated with the global symmetries, as defined in Section 2, and the Hamiltonian depends explicitly on the fluxes n a . The N = 2 quantum mechanics has supersymmetry algebra u(1|1): where Q = Q † and R is the R-symmetry generator. The R-symmetry R is not unique, however. The generators J a of flavor symmetries, by definition, commute with H, Q, Q, R-therefore any other symmetry R = R + a c a J a is an equally good R-symmetry. In particular, the fermion number (−1) F is a discrete R-symmetry transformation, which often is part of the continuous family of R-symmetries. In ABJM, the fermion number can be written in terms of the 3D superconformal R-symmetry R 0 that assigns charge 1 2 to the chiral multiplets A i and B j : The topologically twisted index can then be written as 15 as a function of the trial R-symmetry Thus, the fugacities ∆ a parametrize the mixing of the R-symmetry with the flavor symmetries, i.e. the space of trial R-symmetries. Given the AdS 2 factor at the horizon, we expect that our quantum mechanics becomes superconformal at low energies. The IR superconformal algebra will single out a particular R-symmetry-the one sitting in the algebra-and a particular value for ∆ a . It is natural to ask how to find the exact IR superconformal R-symmetry. We can probe the mixing of the R-symmetry with the flavor symmetries using the dual supergravity solution. As already discussed, the graviphoton field strength F gp µν = e K/2 X Λ F Λ,µν in (3.27) depends on the radial coordinate through the sections X Λ and it is different at the boundary and at the horizon. Its expression suggests the identification The constraint a ∆ a = 2πn is compatible with a X a = 1 valid everywhere in the bulk. Let us assume to be in the range a ∆ a = 2π. At the boundary, where the solution asymptotes to AdS 4 × S 7 , the scalar fields X a are all equal and we find ∆ a = π/2. This reproduces the UV superconformal R-symmetry of ABJM. At the horizon, on the other hand, the values of the scalars depend on the charges n a and, using (3.24), we find in terms of the horizon values of the scalars in (3.23). We can argue that∆ a determine, through (4.5), the exact R-symmetry of the IR superconformal quantum mechanics.
Here comes the main result of our paper. First, with an explicit computation one can check that∆ a is a critical point of the function |Z|: In fact,∆ a is the only critical point of log |Z| in the range 0 < ∆ a < 2π (with a ∆ a = 2π). Setting to zero the derivatives of (2.89) with respect of ∆ 1,2,3 and expressing them in terms of we find Re log Z crit (n a ) = BH Entropy (n a ) . (4.10) Thus, we have reproduced the black hole entropy with a microscopic counting of ground states in a dual field theory, at the leading order N 3/2 . Let us notice that∆ a is a critical point of the function Re log Z, but it is not a maximum. The Hessian of Re log Z has one negative and two positive eigenvalues, therefore the critical point is a saddle point. In fact, we should have expected this from the general large N expression (2.89) of Re log Z: since, generically, at least one of the integers n a is negative (and in fact three of them should be negative to have regular black hole solutions), it follows that Re log Z diverges to positive infinity when the corresponding ∆ a goes to zero.

The case with three equal fluxes
To give a concrete example, we consider the simple case where n 1 = n 2 = n 3 ≡ n , n 4 = 2 − 3n .
which lead to smooth supergravity solutions with regular horizon for n < 0. Consider the field theory expression in (2.89). For our particular choice of fluxes, we expect the critical point to lie along the submanifold ∆ 1 = ∆ 2 = ∆ 3 ≡ ∆, ∆ 4 = 2π − 3∆, with 0 ≤ ∆ ≤ 2π 3 . We can therefore restrict Z to such a submanifold: In the range 0 ≤ ∆ ≤ 2π 3 and for n < 0, which is the region in the flux parameter space where a black hole with regular horizon exists, the function has a critical point at that is also a positive maximum. 17 At the maximum the function takes the value which precisely matches the entropy of the black hole (3.25). Let us stress that, while restricted to the symmetric locus ∆ 1 = ∆ 2 = ∆ 3 ≡ ∆ the index has a maximum, in the full parameter space spanned by the three independent parameters ∆ 1 , ∆ 2 and ∆ 3 the critical point is a saddle point.

Discussion and Conclusions
In this paper we have computed the large N limit of the topologically twisted index of the 3D ABJM theory, which counts (with phases) the ground states of the theory compactified on S 2 with R-and flavor magnetic fluxes. We have argued that this is relevant for understanding the physics of magnetically charged BPS black holes in AdS 4 , arising in 4D maximal N = 8 gauged supergravity. Each black hole can be given a holographic interpretation as the RG flow from the 3D ABJM theory twisted by the corresponding magnetic fluxes to a 1D superconformal quantum mechanics, whose ground states are counted by the index. Indeed, the leading N 3/2 contribution to the index precisely reproduces the leading Bekenstein-Hawking entropy of the black hole.
The matching proceeds in two steps. First, the index Z(n a , ∆ a ) is a function of fugacities e i∆a as well as of magnetic fluxes n a for the flavor symmetries, and one has to extremize Z with respect to the ∆ a 's. Comparing with supergravity, we observe that this procedure selects the exact superconformal R-symmetry in the IR su(1, 1|1) superconformal algebra. Second, we observe that the index at the critical point, Z n a ,∆ a (n a ) , precisely reproduces the black hole entropy S BH (n a ).
A possible interpretation could be the following. We are evaluating a partition function with chemical potentials ∆ a for the flavor symmetries. The vanishing of the derivative with respect to ∆ a is equivalent to the vanishing of the electric charge of the system, which must be zero since the black hole is electrically neutral. It is then conceivable that we get the entropy by extremization. However this argument is not completely satisfactory. The partition function we are computing is supersymmetric and treats bosons and fermions with different sign. Moreover the argument makes no use of the exact superconformal R-symmetry, whose role in the game is strongly suggested by the supergravity analysis.
It would be more interesting to have a clear mapping of the states counted by the topologically twisted index of the 3D ABJM theory to the black hole microstates. Although we do not yet have a clear understanding of this point, let us make some general observations.

A naive argument
Let us first give a superficial argument that originally motivated our investigation. Suppose that the quantum mechanics describing the modes on S 2 is gapped with a finite number of ground states. Then the index reduces to Z(∆ a ) = Tr H=0 (−1) F e i Ja∆a = Tr H=0 (−1) R(∆a) , where the Hamiltonian H is a function of n a . In the last expression we have written the index as a function on the space of R-symmetries of the theory (assuming that all IR R-symmetries are visible in the UV, i.e. there are no accidental ones). Then further suppose that, at low energies, the system develops 1D N = 2 superconformal symmetry and the ground states are invariant under sl(2, R) conformal transformations: these assumptions follow from the fact that the supergravity solution develops an AdS 2 factor at the horizon. Then the su(1, 1|1) algebra implies that the ground states have R c = 0, where R c ∈ su(1, 1|1) is the superconformal R-symmetry. In other words, we conclude that in the space of all possible R-symmetries, there is one that assigns (−1) Rc = 1 to all ground states. But then, since (5.1) is a finite sum of phases, it is clear that it is maximized when all phases are 1. Since, as stressed in [18], the overall phase of the index defined through the path-integral is ambiguous because of fermionic Fock space quantizations, we conclude that |Z| is maximized: Thus, an argument of this kind "would prove" two statements: (1) that the index function Z(∆ a ) = Tr H=0 (−1) R(∆a) has a maximum at the point∆ a where the trial R-symmetry equals the IR superconformal R-symmetry, R(∆ a ) = R c ; (2) that the index evaluated at the maximum, Z(∆ a ) , computes the number of ground states (as opposed to a weighted sum). Unfortunately, this argument is too superficial and it does not apply to the black holes. First of all, if at low energies we just have a finite number of zero-energy ground states separated from the rest by a gap, then the low-energy theory is just H = 0: a bunch of states with no dynamics. An example is a collection of |n| 1D free Fermi multiplets (which can be obtained from a 3D free chiral multiplet on S 2 , with negative magnetic flux n): the index is Z chiral (n, ∆) = y 1/2 1 − y n with y = e i∆ which, for n < 0, is maximized at y = −1 with |Z chiral (n, π)| = 2 |n| (correct number of states in the fermionic Fock space). On such theories su(1, 1|1) simply does not act, and therefore it is hard to understand how this trivial superconformal quantum mechanics can be dual to AdS 2 (although compare with [45]). A non-trivial superconformal quantum mechanics with states with H > 0 necessarily has a continuous spectrum that spans R + , just because the spectrum must be invariant under dilations. Then the states are necessarily non-normalizable, and computing an index (for instance of L 2normalizable states as in [46]) is in general very difficult. In such cases, our index-which is an equivariant index as opposed to an L 2 index-is defined by first deforming the Hamiltonian with real masses σ a (that make the spectrum discrete), and then performing analytic continuation to σ a = 0 exploiting holomorphy in ∆ a + iβσ a . In this setup the argument above does not apply.
Indeed, the ABJM index in (2.89) diverges when some ∆ a vanish. 18 This excludes the possibility of a finite Hilbert space of normalizable ground states gapped from the rest, and so the superficial argument does not apply. In fact, the index has a saddle-not a maximum-at the point that corresponds to the superconformal R-symmetry and that reproduces the BH entropy.
The I-extremization principle We would like to propose that the I-extremization principle, stating that 1. the index is extremized at the superconformal R-symmetry, and 2. the value of the index at the extremum is the regularized number of ground states, has a general validity in N = 2 superconformal quantum mechanics, under certain assumptions suitable for the black holes. Obviously, it would be desirable to precisely understand what assumptions are necessary, and to have a rigorous proof.
A better understanding of all these issues necessarily involves a better understanding of the superconformal quantum mechanics with su(1, 1|1) symmetry. Here we just notice that a simple example of superconformal quantum mechanics with continuous spectrum is provided by a free chiral multiplet (this can be obtained from a 3D free chiral multiplet on S 2 with n > 0). We study this example in some details in Appendix B. It turns out that the index diverges at ∆ = 0, it has a minimum at the superconformal R-symmetry and its value gives the zeta-regularized number of states: 1 2 . In this case, extremization can be proven from time-reversal invariance and integrality of the R-charge spectrum.

Relations with the literature and future directions
Let us briefly comment about the connection between our results and several other streams of ideas in the literature. The 3D topologically twisted index considered in this paper becomes an equivariant Witten index for the dimensionally reduced quantum mechanics. We should notice that there exist another chiral index in N = 2 superconformal quantum mechanics-the superconformal index-which makes use of L 0 that has discrete spectrum [47][48][49], as reviewed in Appendix B.3. The relation between the equivariant and the superconformal indices is not obvious and deserves investigation.
It would be interesting to better understand the relation of our procedure with other extremization mechanisms that appear in the physics of black holes. As we showed in Section 3 and Appendix C, the entropy can be obtained by extremizing with respect to the value of the scalar fields at the horizon. This has a natural interpretation in terms of an attractor mechanism [50], which plays an important role in asymptotically flat black holes. We also recognize many 18 Some divergence had to be expected. The BPS black holes are the near-horizon geometry of N M2-branes wrapping the S 2 in the Calabi-Yau geometry 4 a=1 La(−na) P 1 , which is the total space of four line bundles over P 1 with first Chern classes −na. When some na < 0, there are non-trivial holomorphic sections and the M2-branes can be well separated, giving rise to flat directions. This, however, only explains O(N ) divergences, not O(N 3/2 ). similarities with Sen's entropy function formalism [51], of which we might provide a supersymmetric version. In this context one could investigate the relation between the twisted index before extremization and Sen's entropy function.
If the I-extremization principle turned out to be correct, it should be added to the list of well-established theorems in other dimensions: a-maximization in 4D [22,23], F -maximization in 3D [21,24,25] and c-extremization in 2D [16,17].
To provide tests of the proposed I-extremization principle, one could study more general black holes in the same supergravity model, but with both magnetic and electric charges: we are currently investigating this direction. Other obvious generalizations are to look at the twisted index for CS level k > 1, and on higher-genus Riemann surfaces. In fact, as discussed in Appendix A, there are analogous families of BPS black holes with toroidal and higher-genus horizons. It would also be interesting to generalize our computations to other less symmetric theories, from the 11D point of view. For instance, starting with the geometries AdS 4 × SE 7 and their field theory duals (possibly considering toric Sasaki-Einstein cones as in [52][53][54][55][56][57]) and placing them on a Riemann surface, one can obtain 1 4 -BPS black holes in broad families of 4D N = 2 gauged supergravities.
A very important question is whether the index provides the exact number of black hole microstates, beyond the leading contribution in N . It is known that in some examples (e.g. [58]) the black hole represents only part of the conformally-invariant states, while other ones are represented by graviton waves or other modes. It would be interesting to compute 1/N corrections, both in supergravity and in the large N expansion of the index, to clarify the issue.
On a different note, let us also emphasize that the integral expression for the topologically twisted index found in [18], as the one for the elliptic genus in [59,60], provides a novel type of large N "matrix models": the integrands are standard, but they are integrated along non-trivial contours. These models probably have a rich mathematical structure deserving its own attention.
the general case with AdS 2 × Σ g horizon, where Σ g is a Riemann surface of arbitrary genus g.
A.1 4D N = 2 gauged supergravity from N = 8 We use the Lagrangian and BPS equations given in [31], which conveniently summarizes the results in [36,37]. Note that this is not the standard N = 2 gauged supergravity notation, but rather the natural notation imposed from the reduction of 11D supergravity on S 7 . For the bosonic fields we use the normalization and index structure from the main text, and make explicit comments about the relation with the conventions in [31] when needed.
The S 7 reduction of 11D supergravity gives the 4D N = 8 SO(8) gauged supergravity. Using the reduction ansatz of [36] one finds a consistent reduction to U (1) 4 gauged supergravity: Here a = 1, . . . , 4, the L a satisfy L 1 L 2 L 3 L 4 = 1 and parametrize the scalars, A a are 1-forms with field strengths F a = dA a , ∆ = a L a µ 2 a is the warp factor, a µ 2 a = 1 and 0 ≤ ϕ a < 2π parametrize S 7 , U (1) 4 ⊂ SO (8) is parametrized by ϕ a , * is the Hodge operator on ds 2 4 and 4 is its volume form. 19 The reduction gives a 4D theory with bosonic action where V = −4g 2 cosh φ 12 + cosh φ 13 + cosh φ 14 . In fact (A.2) is the bosonic action of 4D N = 2 U (1) 4 gauged supergravity with the three axions set to zero [36]. We stress that (A.2) is not a consistent reduction without the three axions [36]. They are sourced by F ∧ F , so it is consistent to set them to zero only if F ∧ F = 0. We can still consider either electric or magnetic charges. The fermionic fields of the N = 8 SO(8) gauged supergravity are the gravitini ψ I µ and the spin-1 2 fields χ [IJK] , where I, J, K are SO(8) indices. We can decompose I in the pair (a, i) with a = 1, . . . , 4 and i = 1, 2. The gravitini variations are (see (2.15) in [37]) where ε ij is the antisymmetric tensor, ai are the Killing spinors and The spin-1 2 fermions χ [IJK] are totally antisymmetric. It turns out [37] that δχ [IJK] = 0 unless at least two indices have the same a (then different i because of antisymmmetry), but they cannot all three have the same a because of antisymmetry. One can then write δχ ai bj ck = δχ ack δ ab ε ij + δχ bai δ bc ε jk + δχ cbj δ ca ε ki (A.9) which is automatically antisymmetric in the pairs (a, i) etc., where such that the UV metric is the unit-radius AdS 4 as in the main text; the coupling constant g can be reinstated at the end by sending L a → √ 2 gL a .

A.2 Wrapped M2-branes
The black-hole solutions can be thought of as the near-horizon geometry of a large number of M2-branes wrapping a Riemann surface Σ g . To construct them, we consider the metric ansatz where f 1,2 are functions of r and h is a function of x, y. We choose vielbein et = e f 1 dt, er = e f 1 dr, ex = e h+f 2 dx, eŷ = e h+f 2 dy. We fix for T 2 1 y 2 for H 2 (A.14) so that ds 2 Σ = e 2h (dx 2 + dy 2 ) is a constant curvature metric on the Riemann surface with and κ = 1 for S 2 , κ = 0 for T 2 , and κ = −1 for H 2 . The range of coordinates are (x, y) ∈ R 2 for S 2 , (x, y) ∈ [0, 1) 2 for T 2 , and (x, y) ∈ R × R >0 for H 2 . In the H 2 case the upper half-plane has to be quotiented by a suitable Fuchsian group to get a compact Riemann surface Σ g>1 . The ranges are chosen in such a way that where we defined the positive number η. The case of genus g > 1 follows from the Gauss-Bonnet theorem 1 2 R Σ dvol Σ = 4π(1 − g). The field strengths are taken as On curved Riemann surfaces, we can choose a gauge connection proportional to the spin connection: definingω µ = 1 2 ω ab µ ε ab on Σ g , we have The parameters n a will be quantized later. Notice that the ansatz considered here contains, for κ = 1, the supergravity solution presented in Section 3, however the radial coordinate used here is not the same as the one used in (3.3), as it is obvious by comparing with (A.13).
We choose the following projectors on spinors: x,y ai , γxŷ ai = −ε ij aj , ai = 0 for a = 2, 3, 4 . (A.19) The first two conditions generically select the Poincaré supercharges (versus possible conformal supercharges on AdS); the second is a symplectic reduction for M2-branes on Σ g ; the fourth oneto be compared with Ω in (A.8)-means that we only keep the diagonal supercharge coupled to all fluxes with charge +1 (additional supercharges arise if some fluxes are zero and so other rows of Ω vanish), as in [31].
Let us start with the gravitino variation. If a = 1 then δψ ai µ = 0 automatically. We then define i = 1i , and get From µ =t we get From µ =r, and using ∂r = e −f 1 ∂ r , we get Combining the two we get which gives two equations. We have an analogous equation for µ =ŷ. Combining the two we find a n a = 2κ , (A. 25) and an equation for f 2 . Now let us look at the gaugino variation δχ abi . Given our ansatz for bi , it follows that we obtain non-trivial equations only for b = 1 and therefore for a = 1. We get These are three equations for a = 2, 3, 4.
The final full set of BPS equations is: To understand the quantization condition, consider the case of M2-branes on T * Σ g i.e. take n 2,3,4 = 0 and n 1 = 2κ. In this case we know that on T * S 2 C 2 /Z 2 there are two (negative) units of flux, and on T * Σ g>1 there are 2(g − 1) units of flux. We conclude that the quantization condition is n a ∈ 2 η Z .
(A. 28) In the case of S 2 considered in the main text, the n a are integers. On a higher genus Riemann surface, a more refined quantization is possible.
We could solve the BPS equations in (A.27), which are a system of coupled ODEs, to find the complete black hole solutions discussed in the main text and their generalization with Σ g horizon.
Instead, we will here analyze only the near-horizon geometry AdS 2 × Σ g , for which the equations become algebraic. This will be enough to study the region in parameter space where smooth solutions with regular horizon exist.
We set e 2f 1 (r) = e 2f /r 2 and all other functions constant. We get the algebraic system: together with n a = 2κ. We have substituted L 1 L 2 L 3 L 4 = 1. First notice that it must be L a > 0 for all a. Then consider a linear combination of (A.30)-(A.33) with coefficients equal to the last row of Ω (A.8): it gives The combination (A.29) + (A. 30) gives We can define the positive non-vanishing variables which correspond to the physical scalars used in the main text. In fact, they are simply given by z 1,2,3 = L 1,2,3 /L 4 . 21 The relations above are inverted by The La are proportional to the Xa.
Solving the first or the second equation for z 2 , we get Each of the two expressions is valid if its numerator and denominator are both non-vanishing. Unless n 2 = 0 or z 1 = z 3 = 1, at least one of the two expressions is valid; we can then substitute in (A.38) or (A.39), respectively, obtaining If we substitute the first expression of z 2 (A.41) in (A.40) we obtain a complicated equation: .
Hence, we find two solutions for the scalars: Let us also define Π = 1 8 n 1 + n 2 − n 3 − n 4 n 1 − n 2 + n 3 − n 4 n 1 − n 2 − n 3 + n 4 (A. 48) as in the main text. The squares of the metric functions take the simple expressions As we show in Section A.3.2, in the special case that the n a 's are equal in pairs and κ = −1, a one-parameter family of solutions emerges.
To write down the metric functions directly, we first need to understand the positivity conditions on the fluxes n a , such that a smooth regular horizon can exist. Such conditions are that z 1,2,3 > 0, Θ ≥ 0 and Υ > 0 where With a little bit of algebra one can prove the following equalities: The first one shows that, under the assumption that n 4 = 0 and Π = 0 (those special cases are analyzed in Section A.3.2), z 1,2,3 > 0 guarantees that Θ = 0 and the square bracket is nonvanishing. The second one then guarantees that F 2 ± √ Θ > 0, and the third one shows that Π and Υ have the same sign. Summarizing: Under those conditions, the metric functions are which give the radii of AdS 2 and Σ g , respectively.

A.3.1 Analysis of positivity
We want to precisely identify the region in the parameter space n a a n a = 2κ where the near-horizon solutions exist. First, let us impose the positivity constraints on the parameter space {n a }, with no restriction on a n a and assuming n 4 , Π = 0 (the special cases n 4 = 0 or Π = 0 are analyzed in Section A.3.2): where z ± 1,2,3 are the two solutions for the scalars in (A.46). It turns out that both domains are linear, in the sense that they are bounded by hyperplanes.
The domain D − is easy to write: This domain is unbounded. Actual solutions to the BPS equations follow from imposing the further constraint a n a = 2κ. On S 2 and T 2 clearly there are no solutions. On H 2 we can rewrite the region as D − (H 2 ) = n 1 +n 2 +1 n 1 +n 3 +1 n 2 +n 3 +1 < 0, n 1,2,3 < 0, n 1 +n 2 +n 3 > −2 (A. 56) in terms of n 1,2,3 . This domain in bounded.

A.3.2 The special cases
First, starting from the beginning, it is easy to see that if two, three or all four of the n a 's are zero, then there are no regular solutions. These are precisely the cases with enhanced supersymmetry. The case n a = κ = 0 corresponds to M2-branes on T 2 preserving 1D N = 16 supersymmetry. The case n 1 = 2κ = 0 and n 2,3,4 = 0 or permutations thereof corresponds to M2-branes on the (local) hyperkähler space T * Σ g , preserving 1D N = 8 supersymmetry. The case n 3,4 = 0 or permutations thereof corresponds to M2-branes on a local Calabi-Yau threefold, preserving 1D N = 4 supersymmetry.
For g > 0 the solutions are still written as in (3.3), with e −K(X) = 8 √ X 1 X 2 X 3 X 4 and however the constant c related to the horizon radius r h is and the relation between the parameters β a and the fluxes n a is implying a n a = 2κ. The latter can also be written as The inverse formula is where Π is the same as in (3.11). The expressions in (3.15) remain valid. From the radial profile of the scalars X a in (A.68) it is clear that whenever the near horizon solution is regular-in particular X a (r h ) > 0 and the horizon radius r h is positive-the full black hole solution is regular. Therefore the analysis of positivity we did in Section A.3.1 gives the region in parameter space where smooth black hole solutions with regular horizon exist. In particular, for the case of H 2 (i.e. κ = −1), when the parameters lie inside D − (H 2 ) one finds two black hole solutions with different entropy.
B I-extremization: the example of a free chiral multiplet In this appendix we examine in details the N = 2 quantum mechanics of a free chiral multiplet. Although seemingly trivial, the model contains some useful information. In particular, the index is extremized in correspondence with the exact R-symmetry of the model.

B.1 The massive case
Consider an N = 2 quantum mechanics with u(1|1) supersymmetry algebra is independent of β and it receives contributions only from "chiral" supersymmetric ground states that satisfy H = σJ. As a result, it is a holomorphic function of the complex fugacity y = e i(∆+iβσ) and it can be written as Such an index for N = 2 quantum mechanics has been considered in [61] and evaluated by localization therein. It is also related to the topologically twisted index of a three-dimensional theory by dimensional reduction on S 2 [18]. In the three-dimensional language, H is the Hamiltonian of the dimensionally reduced theory (and it depends on the magnetic fluxes), while ∆ and σ are expectation values for the background vector multiplet associated with the flavor symmetry J: ∆ is a flat connection on S 1 and σ is a real mass [18]. We consider a model with a complex scalar z = x 1 + ix 2 and a complex fermion ψ satisfying The Hamiltonian and the flavor symmetry J are Since y = e i∆−βσ , for σ > 0 the series is in fact convergent. For σ < 0, the ground state is |0 ⊗ |↓ : it is fermionic and has H = |σ|/2, J = − 1 2 . All excited states are obtained by acting with ψ, a † z , a † z . They all shift H → H + |σ|, but ψ and a † z shift J → J + 1 while a † z shifts J → J − 1. Therefore all states satisfy H ≥ |σJ|, while the normalized "chiral" states satisfying H = σJ are (a † z ) n √ n! |0, ↓ and are annihilated by Q and Q. The supersymmetric index is as before. This series is convergent for σ < 0 as it should. From the example, it appears that the index is only defined in the massive theory. The states counted by the index do not have a well-defined limit as σ → 0 (this is manifest in the Schrödinger representation), and the two series would not converge for σ = 0. Thus the index for zero real mass is defined as the limit of the index with σ = 0. However, for the free chiral case we can still make sense of the index in the massless σ = 0 case if we use generalized states | x (or | p ) in the Schrödinger representation. Let us compute The fermionic trace is easily computed to be y 1/2 − y −1/2 . To compute the bosonic trace, we notice that the operator e i∆J bos rotates the x-plane and p-plane by an angle ∆. We thus find d 2 p R −∆ p |e −βH | p = d 2 p e −β p 2 /2 δ (2) (R −∆ p − p) = 1 det(R −∆ − 1) .
Notice that the expression (B.13) is not a single-valued function of y due to an anomaly for the flavor symmetry. In three dimensions this is due to a parity anomaly and it can be cured by adding a Chern-Simons term for the background flavor field [18]. In quantum mechanics we should add a Wilson line [61].

B.2 The massless case
The case of interest for this paper is the massless case. By setting y = e i∆ we have (B.14) This is extremized at ∆ = (2k + 1)π with integer k, which corresponds to y = −1. The value of the index is |I(∆ = π)| = 1/2 which is not an integer. We can understand this value as a zeta-function regularization which is consistent with the geometric series expansion of (B.13) for y = −1. As we have seen, unfortunately, the index in the massless case should be suitably regularized. Only if we define it as a limit of the massive case we can make sense of it as a sum over states of the discrete spectrum.
On the other hand, we also have good news. In the massless case there can be a superconformal algebra and the free chiral theory provides an example of that. In the massless limit σ → 0 we find where we defined the holomorphic momentum p = p 1 + ip 2 . The theory also gains conformal symmetry. We can define the operators All together these operators satisfy an N = 2 superconformal algebra [47-49, 62, 63]. We can write it in a compact way by defining the operators The R-symmetry operator R is uniquely singled out by the superconformal algebra. This is the exact R-symmetry of the superconformal quantum mechanics. We can relate it to the extremization of the index as follows. The critical point of I(∆) is at ∆ = π. At that point, using (B.19), we have I(∆ = π) = Tr (−1) F (−1) J e −βH = i Tr (−1) R e −βH . (B.23) In other words, the extremization precisely singles out the exact R-symmetry! We may ask if there is some symmetry at work in this simple example behind the selection of R by extremization. The index I is purely imaginary and its extremization ∂ ∆ |I| = 0 is equivalent to − i ∂ ∆ I = Tr (−1) F J e i∆J e −βH = 0 . Notice that, since H has a continuum spectrum, all the previous traces must be regularized, using the generalized eigenstates of the momentum or by taking a suitable limit of the massive theory.
where in the second equality we set ξ Λ = 0 since we do not consider magnetic gaugings. 23 Now let us focus on the BPS equations that hold at the black hole horizon, as derived in [6] (eqns. (3.9) and (3.5) respectively), where the derivatives are with respect to the complex scalars z j , D j = ∂ j + K j /2 and K j = ∂ j K. These are the BPS attractor equations for AdS 4 black holes that are written in a completely general symplectic-invariant way. In particular, there is still a scaling symmetry for the choice of symplectic sections X Λ since the number of physical scalars is one less. This scaling symmetry is a remnant of the conformal symmetry in off-shell supergravity and one can always make a gauge choice for it, if needed. Here we decide to make the particular gauge choice This choice was already implicitly made in the main text, and it was built in the "ansatz" for the solutions in [7]. One can further see that the choice (C.4) leads to the explicit appearance of the function e K in the warp factor, which follows from the extra BPS flow equation we are not considering here. 24 This choice is made at the level of the theory, and it holds everywhere in spacetime, not just at the horizon. Such a choice does not lead to any physical observable, as the metric, scalars and gauge fields are gauge invariant. However it does change their functional dependence on the sections, and choosing (C.4) we put the physical solution in a form that is most convenient for us. Another reason for choosing (C.4) is the simplification in the holographic dictionary. As we saw in the dual field theory, the chemical potentials ∆ a obey a similar relation and can be identified with X a up to a proportionality constant. A different gauge choice would have led to a different identification and a more cumbersome notation. In this sense what we derive below for R is not a gauge-invariant statement, but this does not change the underlying physical picture. One can always refer back to (C.3) for the scale-invariant equations.
With the gauge choice (C.4), the first attractor equation in (C.3) gives at the horizon: meaning that |R| is equal to the entropy up to a proportionality constant. This result is valid in two-derivative supergravity, and will generically change with higher derivative corrections. Keeping in mind that R is a function of the sections, the second equation in (C.3) gives This is valid in the gauge (C.4) that determines, say, X 0 in terms of the other sections. Therefore the derivative with respect to the physical scalars z j can be traded for a derivative with respect 23 One can always symplectically rotate a given gauged theory to this choice of gauging frame, so there is no loss of generality in this choice. 24 The additional BPS equation (2.33) of [6] fixes 2R 2 AdS 2 = e −K on the horizon in accordance with the solution we presented in the main text.
to the sections, if we impose (C.4). We finally find ∂R ∂X Λ horizon = 0 , (C.7) that the function R is extremized at the horizon. This fixes the values of the complex scalars, and it can be thought of as an attractor equation. Furthermore the value of R at the extremum is proportional to the black hole entropy. This is valid for all supersymmetric asymptotically AdS 4 black holes, with a general choice of electric and magnetic charges and complex sections X Λ under the constraint (C.4). Due to the exact match between the twisted index and the quantity R in the particular case considered in the main text, it is natural to expect that this continues to hold for all AdS 4 black holes with a field theory dual (note that (C.7) holds for other BPS horizons as well). It is then tempting to speculate about a more general correspondence between R and the Witten index of the dual 1D superconformal quantum mechanics also in cases without AdS 4 asymptotics.