6D attractors and black hole microstates

We find a family of AdS$_2\times \mathcal{M}_4$ supersymmetric solutions of the six-dimensional $\mathrm{F}(4)$ gauged supergravity coupled to one vector multiplet that arises as a low energy description of massive type IIA supergravity on (warped) AdS$_6\times S^4$. $\mathcal{M}_4$ is either a K\"ahler-Einstein manifold or a product of two Riemann surfaces with a constant curvature metric. These solutions correspond to the near-horizon region of a family of static magnetically charged black holes. In the case where $\mathcal{M}_4$ is a product of Riemann surfaces, we successfully compare their entropy to a microscopic counting based on the recently computed topologically twisted index of the five-dimensional $\mathcal{N}=1$ $\mathrm{USp}(2 N)$ theory with $N_f$ fundamental flavors and an antisymmetric matter field. Furthermore, our results suggest that the near-horizon regions exhibit an attractor mechanism for the scalars in the matter coupled $\mathrm{F}(4)$ gauged supergravity, and we give a proposal for it.

The five-dimensional topologically twisted index, which is the partition function of a five-dimensional N = 1 theory on M 4 × S 1 with an abelian topological twist along M 4 , has been recently computed when M 4 is a toric Kähler manifold [32] or the product of two Riemann surfaces [32,33]. Therein, the N = 1 USp(2N ) gauge theory with N f hypermultiplets in the fundamental representation and one hypermultiplet in the antisymmetric representation of USp(2N ), arising on the worldvolume of D4-branes near D8-branes and orientifolds [34], has been analyzed in the large N limit. With some assumptions on the relevant saddle-point, the large N limit of the index for M 4 = Σ g 1 × Σ g 2 has been evaluated as a function of magnetic charges and chemical potentials for the Cartan subgroup of the SU(2) M global symmetry of the theory [32]. 1 This result provides a prediction for the entropy of a family of AdS 6 magnetically charged black holes in massive type IIA supergravity. This prediction has been successfully tested for the only existing black hole solution, the so-called universal one [35,36] with a particular value of the magnetic charge, using the results for the entropy given in [37]. It is the purpose of this paper to find new black hole solutions, explicitly depending on a set of magnetic charges, and show that their entropy is correctly accounted by the topologically twisted index.
To this end, we will consider a six-dimensional truncation of the supersymmetric warped AdS 6 × S 4 background of massive type IIA supergravity [38] dual to the USp(2N ) theory. This truncation is described by an F(4) gauged supergravity coupled to vector multiplets [39][40][41][42]. Furthermore, we will restrict ourselves to one vector multiplet corresponding to the Cartan subgroup of the SU(2) M global symmetry of the five-dimensional superconformal field theory (SCFT).
As a warm-up, and to test the consistency of the truncation, we consider the background AdS 4 × Σ g which corresponds to a twisted compactification of the five-dimensional theory on Σ g . We successfully compare the free energy of the solution with the field theory computation in [33] and the ten-dimensional gravity computation in [43].
We then find new black hole horizon geometries of the form AdS 2 × Σ g 1 × Σ g 2 . We turn on an abelian gauge field inside the SU(2) R-symmetry that performs the topological twist by cancelling the spin connection, and two magnetic fluxes p 1 and p 2 (one along each Riemann surface) for the U(1) gauge field in the additional vector multiplet. We thus have a two-parameter family of magnetically charged black holes. We compare the entropy with the value of the topologically twisted index Z(p 1 , p 2 , ∆) that also depends on a chemical potential for the U(1) ⊂ SU(2) M symmetry and we find that the statistical entropy S BH of the black holes as a function of the magnetic charges is obtained by evaluating Z(p 1 , p 2 , ∆) at its critical point∆: With a convenient democratic parameterization for the fluxes and chemical potentials, see (5.7) and (5.11), the explicit form of the index can be written as [32] I SCFT (s I , t I , ∆ I ) = 4 √ 2 15 This structure is reminiscent of an analogous result for AdS 4 black holes [7-10, 17, 18]. This analogy and the relation to other interesting field theory quantities like the S 5 free energy and the effective twisted superpotential of the partial compactification on one of the Riemann surfaces, were discussed in detail in [32].
We also obtain AdS 2 ×M 4 horizon geometries where M 4 is a four-dimensional Kähler-Einstein manifold depending on a magnetic flux along M 4 . We find a simple and intriguing expression for the entropy suggesting that the computation in [32] could be generalized to this case too. We leave this for future work.
In gravity, the field theory chemical potential ∆ can be associated with the horizon value of the vector multiplet scalar field φ 3 . With a convenient parameterization, we find that the functional I SCFT (p 1 , p 2 ,∆) coincides with the area of the horizon divided by 4G N , where G N is the six-dimensional Newton's constant, as a function of φ 3 . This is the attractor mechanism in six-dimensional gauged supergravity: after expressing all the fields in the gravity multiplet in terms of vector multiplet scalars using the BPS equations, the remaining BPS equations are equivalent to the extremization of the area of the horizon as a functional of vector multiplet scalars, and the critical value of this functional is the entropy. We see that the I-extremization principle is equivalent to the attractor mechanism in sixdimensional gauged supergravity, 2 thus generalizing what was found for AdS 4 black holes in [7-9, 17, 18].
More explicitly, we find that a central role is played by the quantity (3.4) where X I (φ 3 ) (I = 1, 2), defined in (2.12), are the gravity counterpart of ∆ I in (1.3). This six-dimensional quantity is reminiscent and can be thought of as the analogue of the prepotential F sugra (X I ) in four-dimensional N = 2 gauged supergravity. Indeed, we will find that attractor equations for AdS 4 vacua correspond to extremizing and attractor equations for black holes correspond to extremizing for M 4 being a Kähler-Einstein manifold, and to extremizing for M 4 = Σ g 1 × Σ g 2 . Here s I and t I are the magnetic charges -see (4.6), (5.7) and (6.6). This is similar to the attractor mechanism in four-dimensional N = 2 gauged supergravity [1,2]. We thus expect that, in more general F(4) gauged supergravites coupled to vector multiplets, the attractor equations for AdS solutions supported by magnetic fluxes are given by extremizing expressions of the form (1.4)-(1.7) with a suitable function I AdS 6 (X I ), homogeneous of degree three. The structure of this paper is as follows. In section 2 we discuss general aspects of the F(4) gauged supergravity coupled to vector multiplets. In section 3 we discuss the AdS 6 vacuum and an interesting partially off-shell version of its free energy that we relate to its field theory counterpart. In section 4 we consider the background AdS 4 × Σ g with a topological twist on Σ g and we successfully compare the free energy of the solution with the field theory computation in [33] and the ten-dimensional gravity computation in [43]. In section 5 we obtain a two-parameter family of black hole horizons AdS 2 × Σ g 1 × Σ g 2 and successfully reproduce their entropy using the topologically twisted index. In section 6 we find a one-parameter family of black hole horizons AdS 2 × M 4 where M 4 is a fourdimensional Kähler-Einstein manifold. Our conventions and some useful formulae are collected in appendix A.
Note added: While we were writing this work, we became aware of [46] which has some overlaps with the results presented here.

Matter coupled F(4) gauged supergravity
We consider a six-dimensional truncation of the supersymmetric warped AdS 6 × S 4 background of massive type IIA supergravity [38] described by an F(4) gauged supergravity coupled to vector multiplets. The minimal F(4) gauged supergravity was written in [40] and coupled to matter in [41,42]. F(4) is the relevant superalgebra for five-dimensional superconformal field theories and its bosonic subalgebra is SO(5, 2) × SU(2) R .
The bosonic part of the six-dimensional gravity multiplet consists of the metric g µν , four vectors A α , α = 0, 1, 2, 3, a two-form B µν and the dilaton σ. It is useful to split α = (0, r) where r = 1, 2, 3 is an index in the adjoint representation of SU(2) R . The fermionic components are a gravitino ψ A µ and a spin one-half fermion χ A , A = 1, 2, transforming in the fundamental representation of SU(2) R .
The vector multiplet in six-dimensions contains a gauge field A µ , four scalars φ α and a spin one-half fermion λ A . With n V vector multiplets, the 4n V scalar fields parameterize the coset space .
It is convenient to encode the scalar fields into a coset representative L Λ Σ ∈ SO(4, n V ), where indices are split as follows Λ = (α, I) with I = 1, . . . n V . A subgroup SU(2) R × G of dimension 3 + n V of SO(4, n V ) can be gauged.
The bosonic Lagrangian reads [42] 3) with f Λ ΠΓ the structure constants of the gauge group SU(2) R × G. The supersymmetry variations of the fermions are given by (2.4) where we suppressed the quadratic terms in fermions, σ rA B are the Pauli matrices, and we have defined In all the above formulae the indices Λ, Π, Γ, . . . are raised and lowered with the SO(4, n V ) invariant metric η ΛΣ = diag{1, 1, 1, 1, −1, . . . , −1} and the indices A, B, . . . with the SU(2) R tensor AB . We refer to the appendix for conventions, for the explicit form of the potential V , and the fermion mass matrices S AB , N AB , M I AB appearing in (2.4). The five-dimensional superconformal field theory dual to the warped background AdS 6 × S 4 has a gauge group USp(2N ), N f hypermultiplets in the fundamental representation and one hypermultiplet in the antisymmetric representation. The theory has an [34]. 4 The global SU(2) M acts on the antisymmetric field, SO(2N f ) on the fundamentals and U(1) I is the conserved instanton current. We just consider a supergravity containing one vector multiplet, n V = 1, corresponding to the U(1) subgroup of the global SU(2) M . We will consistently set to zero all gauge fields except A r=3 µ in SU(2) R and A I=1 µ that are needed for the twisting and to provide magnetic charges for the black holes. We will also require the scalar fields in the vector multiplet to be neutral under A r=3 µ and this restricts the nonzero components to φ 0 and φ 3 . For purely magnetic black holes we can find solutions with φ 0 = 0 and we further restrict to this case. 5 A convenient parameterization of the scalar coset is given by [47][48][49] The kinetic terms for the vectors can then be written as and the quantities in the fermionic variations read (2.8) The other fields that are turned on are the metric, the dilaton σ and the two-form B µν . It is consistent to set H µνλ = 0 but B µν is not in general zero and its value can be found by solving its equations of motion [37]. We believe that after all this simplification the theory is a consistent truncation of massive type IIA supergravity on the warped background AdS 6 × S 4 . We give evidence for this in section 4 where we match the ten-dimensional result found in [43].
We finish the discussion of the matter coupled theory with an argument about the definition of the R-symmetry for all asymptotically AdS 6 solutions in the theory. Let us first recall that the detailed match between supergravity and field theory for asymptotically AdS 4 black holes was facilitated by the gravitational answer for the R-symmetry along the holographic renormalization group (RG) flow [7], telling us explicitly how the Rsymmetry mixing is parametrized by the values of the scalar fields. In four-dimensional gauged supergravity the R-symmetry was carefully derived via the Dirac bracket of the supercharges Q obtained from the Noether procedure [50,51]. Following rigorously all these steps in six dimensions is out of our scope here; however, we can still provide some solid arguments and derive the expected gravitational R-symmetry as an explicit scalar dependent combination of the two U(1)'s mixing along the flow, F r=3 µν and F I=1 µν = F Λ=4 µν . This proposal is strongly backed up by the agreement with the field theory results we provide in the following sections.
It is reasonable to expect that, in analogy to the four-dimensional arguments in [50,51], the anti-commutator between two supercharges for asymptotically AdS 6 solutions is given by a surface integral 6 where A is the Killing spinor preserved by AdS 6 , and the super-covariant derivative D includes all terms on the right hand side of the gravitino variation in (2.4), i.e. δψ Aµ = D µ A . The above anti-commutator is the explicit field dependent realization of the abstract AdS 6 superalgebra, F(4), generating a combination of different asymptotic bosonic charges of the SO(5, 2)×SU(2) R generators. We are interested in the term in the gravitino variation in (2.4) proportional to T (AB) , which precisely enters in the definition of the conserved SU(2) R-charge. We are further breaking the R-symmetry down to U(1) so we only need to look at the part proportional to σ 3 AB , cf. (2.5). We are then led to the following formula for the conserved U(1) R-symmetry charge of a given solution, (2.10) Note that we are only interested to know the R-symmetry at a given radial slice of the spacetime (that when interpreted as a holographic RG flow becomes a measure of how the R-symmetry changes along the flow), not at the value of the asymptotic conserved charge. Therefore, we can extract a normalized version of the integrand that we hope to match with the R-symmetry mixing in field theory. Considering that L −1 33 = cosh(φ 3 ), L −1 34 = − sinh(φ 3 ) and that the democratic choice of U(1)'s corresponds to taking F 1,µν ≡ F 3,µν + F 4,µν , F 2,µν ≡ F 3,µν − F 4,µν we finally define where the mixing of the democratic U(1) symmetries F 1,2 is given by the scalar dependent quantities (2.12) 6 We are evaluating the Dirac bracket of two conserved asymptotic supercharges. Therefore the resulting surface integral is defined on the asymptotic AdS boundary ∂V and the standard notation is

The AdS 6 vacuum
The F(4) supergravity discussed in the previous section has an AdS 6 vacuum if we set g = 3m [40][41][42]. Indeed, considering a background with metric a nontrivial scalar profile for σ(r) and φ 3 (r), and setting all other fields to zero, the BPS equations (2.4) reduce to 7 where prime denotes the derivative of the function with respect to the radial coordinate r.
With g = 3m, the AdS 6 background corresponds to e −2f = r 2 and σ = φ 3 = 0. We have further set m = 1/2 so that the AdS 6 radius is normalized to one. A more suggestive way of solving the above equations is by taking the Ansatz e 2f (r) = e 2f 0 /r 2 , and σ, φ 3 independent of r. We can write the BPS equations in an alternative form by using the parameterization (2.12). The BPS equations for the fields in the gravity multiplet in terms of X 1,2 can be solved as 3) The on-shell supergravity action is given by We then see that the BPS equation for φ 3 , which implies φ 3 = 0, is equivalent to extremizing I AdS 6 with respect to X I . The function I AdS 6 (X I ) has a natural field theory interpretation. The S 5 free energy of the USp(2N ) theory reads [52] This can be generalized to the case where a mass parameter is turned on for U(1)

6)
7 One can derive these equations by taking the ultraviolet limit of the more general flow equations (4.5), (5.6), or (6.5). 8 What we denote as chemical potentials here are actually mass parameters for the antisymmetric matter field in the S 5 free energy. Comparing to [53] we have ∆ 1 = π 1 + 2i 3 m as , ∆ 2 = π 1 − 2i 3 m as .
where ∆ 1 + ∆ 2 = 2π and the extremal value is recovered for ∆ 1 = ∆ 2 = π. Upon using the standard AdS 6 /CFT 5 dictionary [52] and identifying X I ≡ ∆ I , we find that F S 5 (∆ I ) = I AdS 6 (X I ) . (3.8) Interestingly, as shown in [32], the same quantity is also related to the Seiberg-Witten prepotential of the five-dimensional theory on R 4 × S 1 which can be written as As discussed in the introduction, the function I AdS 6 (X I ) in six dimensions plays a role similar to the prepotential of four-dimensional N = 2 gauged supergravity. In the AdS 4 black hole story, the supergravity prepotential is similarly related both to the twisted superpotential and to the S 3 free energy of the dual field theory [7,10,17]. 9 4 The AdS 4 × Σ g solution The F(4) gauged supergravity has also an AdS 4 × Σ g solution corresponding to the twisted compactification of the five-dimensional SCFT on a Riemann surface Σ g of genus g. In the infrared the theory flows to a three-dimensional SCFT.
We consider the following Ansatz for the metric and for the gauge fields U(1) × U(1) ⊂ SU(2) R × U(1): with ζ = ±1. There is a nontrivial profile for the scalars σ(r), φ 3 (r) and all other fields are set to zero. Here, Σ g is a Riemann surface with metric normalized as R µν = κg µν , with κ = 1 for S 2 , κ = 0 for T 2 , and κ = −1 for g > 1. 10 The U(1) ⊂ SU(2) R gauge field is chosen in order to cancel the spin connection while the magnetic flux p parameterizes a family of three-dimensional SCFTs. If we choose spinors satisfying For black holes in AdS 4 × S 7 the prepotential is proportional to the function √ X 1 X 2 X 3 X 4 with 4 I=1 X I = 2π and for massive type IIA black holes to (X 1 X 2 X 3 ) 2/3 with 3 I=1 X I = 2π. 10 With this normalization vol(Σ g ) = 2πη g with η g = 2|g − 1| for g = 1 and η g = 1 for g = 1.
where the frame indices 3, 4 refer to the Riemann surface, the U(1) ⊂ SU(2) R gauge field cancels the spin connection along Σ g . This is precisely the topological twist. Requiring in addition that wherer is a frame index along the radial direction, the BPS equations (2.4) reduce to 11 (4.5) We choose the parameterization of the scalar field φ 3 as in (2.12) and, in addition, we introduce a redundant but democratic parameterization for the flux with s 1 + s 2 = 2(1 − g). We look for AdS 4 × Σ g vacua where e f (r) = e f 0 /r and h(r), σ(r) and φ 3 (r) are constant. Using the BPS equations (4.5), the fields in the gravity multiplet can be solved in terms of the X I 's as where we set g = 3m and m = 1/2. The on-shell supergravity action can be written as It turns out that the BPS equation for φ 3 is equivalent to the extremization of I AdS 4 (X I ) with respect to X I . The previous expression can be more elegantly rewritten as As expected, using (3.7) and identifying X I ≡ ∆ I , we find that F S 3 ×Σg (∆ I ) = I AdS 4 (X I ) , (4.10) 11 Here we correct a numerical factor in the gaugino variation in [47].
where F S 3 ×Σg is the S 3 × Σ g free energy of the same theory, as a function of R-charges, computed in [33] 12 Here, ∆ 1 + ∆ 2 = 2π. Moreover, as noticed in [32], this expression is also related to the effective twisted superpotential W of the theory compactified on Σ g × S 1 : The extremization of F S 3 ×Σg (∆ I ) with respect to ∆ I determines the exact R-symmetry of the three-dimensional field theory that is obtained by twisted compactification on Σ g . The critical value of F S 3 ×Σg (∆ I ) is the free energy of the theory and coincides with the value derived directly in ten-dimensional massive type IIA supergravity in [43]. This is an evidence that the gauged supergravity provides a consistent truncation of the tendimensional theory.

The AdS
Now we search for black hole horizon solutions of the form AdS 2 × Σ g 1 × Σ g 2 . We consider the following Ansatz for the metric ds 2 = e 2f (r) dt 2 − dr 2 − e 2h 1 (r) ds 2 Σg 1 − e 2h 2 (r) ds 2 Σg 2 , (5.1) and the gauge fields with ζ = ±1 and the previous conventions for Riemann surfaces. The U(1) ⊂ SU(2) R gauge field is chosen in order to cancel the spin connection and p 1 and p 2 are magnetic charges, one for each Riemann surface. There is as usual a nontrivial profile for the scalars σ(r), φ 3 (r). This time the two-form B µν cannot be set to zero. Assuming H µνλ = 0, the equations of motion require that which is solved by Comparing to [33] we have s 1 = (1−g)(1+n M ), s 2 = (1−g)(1−n M ), ∆ 1 = π(1+ ν AS ), ∆ 2 = π(1− ν AS ).
With the spinor projections where the frame indices 1, 2 refer to the first Riemann surface and 3, 4 to the second, the U(1) ⊂ SU(2) R gauge field cancels the spin connection, and the BPS equations (2.4) reduce to We choose the parameterization (2.12) for the scalar field φ 3 and a democratic parameterization for the fluxes with s 1 +s 2 = 2(1−g 1 ) and t 1 +t 2 = 2(1−g 2 ). To have a black hole horizon AdS 2 ×Σ g 1 ×Σ g 2 we set e f (r) = e f 0 /r and h 1 (r), h 2 (r), σ(r) and φ 3 (r) constant. Using the BPS equations (5.6), the fields in the gravity multiplet can be solved in terms of X I as e σ = (X 1 X 2 ) 1/8 (s 1 X 2 + s 2 X 1 )(t 1 X 2 + t 2 X 1 ) + 2X 1 X 2 (s 2 t 1 + s 1 t 2 ) 3π(s 1 X 2 + s 2 X 1 )(t 1 X 2 + t 2 X 1 ) where we set g = 3m and m = 1/2. The Bekenstein-Hawking entropy can be written as as a function of X I . It is quite remarkable that the BPS equation for φ 3 is equivalent to the extremization of I AdS 2 (X I ) with respect to X I . This is the attractor mechanism in six-dimensional gauged supergravity: once the fields in the gravity multiplet are expressed in terms of the scalars in the vector multiplet, the entropy is obtained by extremizing the functional I AdS 2 (X I ).
We can now compare the entropy of the six-dimensional black holes with the prediction of the topologically twisted index computed in [32]. The index, at large N , is given by [32] I SCFT (∆ I ) = 4 √ 2 15 The index depends on a chemical potential ∆ for the U(1) subgroup of the SU(2) global symmetry. As in [32], we find it convenient to use a pair of redundant but democratic parameters with ∆ 1 + ∆ 2 = 2π. In the spirit of the microscopic counting for magnetically charged AdS black holes in four dimensions, we expect that the entropy is obtained by extremizing I SCFT (∆ I ) with respect to ∆ I . This was called I-extremization principle in [7,8]. Using (3.7) and identifying X I ≡ ∆ I , we find that and we see that the field theory I-extremization precisely corresponds to the attractor mechanism in supergravity.

The AdS 2 × M 4 solution
It is easy to find more general black hole horizons with abelian twists. We consider the following metric where M 4 is a Kähler-Einstein manifold with metric normalized as R µν = κg µν (κ = ±1, 0), and gauge fields F r=3 = ζ g κ (e 12 + e 34 )e −2h(r) , F I=1 = ζ g p (e 12 + e 34 )e −2h(r) , where e i , i = 1, 2, 3, 4, are vierbeins in the directions corresponding to the manifold M 4 . The reduced holonomy group on the manifold, U(2), splits into U(1) that we choose to correspond to the selfdual part of the spin connection, ω + , and SU(2) for the anti-selfdual part, ω − . As in the previous section, there is a nontrivial profile for the scalars σ(r), φ 3 (r) and the two-form With the spinor projections (6.5) We choose the parameterization (2.12) for the scalar field φ 3 and a democratic parameterization for the fluxes s 1 ≡ κ + p , with s 1 + s 2 = 2κ. To have an AdS 2 × M 4 horizon topology we set e f (r) = e f 0 /r and h(r), σ(r) and φ 3 (r) constant. Using the BPS equations (6.5), the fields in the gravity multiplet can be solved in terms of X I as e σ = (X 1 X 2 ) 1/8 (s 1 X 2 + s 2 X 1 ) 2 + 4X 1 X 2 s 1 s 2 3π(s 1 X 2 + s 2 X 1 ) 2