Contents

A Universal Counting of Black Hole Microstates in AdS






Francesco Azzurli, Nikolay Bobev, P. Marcos Crichigno, Vincent S. Min, and Alberto Zaffaroni



Dipartimento di Fisica, Università di Milano-Bicocca, I-20126 Milano, Italy

INFN, sezione di Milano-Bicocca, I-20126 Milano, Italy


Instituut voor Theoretische Fysica, KU Leuven

Celestijnenlaan 200D, B-3001 Leuven, Belgium
Institute for Theoretical Physics, University of Amsterdam,

Science Park 904, Postbus 94485, 1090 GL, Amsterdam, The Netherlands
francesco.azzurli@mib.infn.it, nikolay.bobev@kuleuven.be, p.m.crichigno@uva.nl,

vincent.min@kuleuven.be, alberto.zaffaroni@mib.infn.it



Many three-dimensional SCFTs admit a universal partial topological twist when placed on hyperbolic Riemann surfaces. We exploit this fact to derive a universal formula which relates the planar limit of the topologically twisted index of these SCFTs and their three-sphere partition function. We then utilize this to account for the entropy of a large class of supersymmetric asymptotically AdS magnetically charged black holes in M-theory and massive type IIA string theory. In this context we also discuss novel AdS solutions of eleven-dimensional supergravity which describe the near horizon region of large new families of supersymmetric black holes arising from M2-branes wrapping Riemann surfaces.

1 Introduction

String theory offers a unique vantage point into the physics of black holes. One the one hand it gives rise to supergravity theories in ten and eleven dimensions as its low energy limits and these theories admit a plethora of classical black hole (or more generally black brane) backgrounds. These black holes are sourced by the massive strings and branes, the microscopic degrees of freedom in the theory. On the other hand one can understand the low energy physics of these branes in terms of the quantum field theory living on their worldvolume. This dichotomy has been the source of many advances in our understanding of black holes, as well as being ultimately responsible for the concrete realization of the holographic principle in string theory. A famous example of this success story is the microscopic counting of the entropy for a class of asymptotically flat black holes in string theory [1] (see also [2] for earlier important work). However, this perspective on black holes has not been put to good use in the context of asymptotically AdS black holes. This state of affairs changed recently with the results in [3]. The authors of [3] employed a certain supersymmetric index, , defined in [4] to account for the entropy of a large class of supersymmetric asymptotically AdS black holes arising from M2-branes wrapped on Riemann surfaces. These results were later extended to dyonic black holes in [5]. See also [6, 7, 8, 9, 10, 11] for further recent developments.

Our goal in this work is to build upon this recent success in a number of ways. First, we show that there is a universal expression for the topologically twisted index in the large limit for a large class of three-dimensional SCFTs.111The conditions on the theory are discussed in detail in Section 2. Namely, we find that the index is related to the three-sphere free energy, , of the three-dimensional theory through the simple relation

(1.1)

The universality of this relation stems from a universal partial topological twist that is used in the definition of the twisted index. In general the index depends on a set of background fluxes for the global symmetries of the SCFT as well as a set of complex fugacities. A given choice of Riemann surface and background fluxes represents a particular topological twist of the three-dimensional theory on . It was argued in [3] that for a given such choice of twist one obtains the degeneracy of vacua after extremizing the index as a function of the complex fugacities, and this extremal value appears in (1.1). The universal topological twist, for which the relation in (1.1) is valid, is singled out by choosing the background fluxes on such that the only non-zero flux is in the direction of the exact superconformal R-symmetry of the three-dimensional SCFT.

Such a simple universal relation in QFT should have an equally simple bulk realization for three-dimensional SCFTs with a holographic dual. Indeed, we show that there is a simple black hole solution with a hyperbolic horizon in minimal four-dimensional gauged supergravity which provides the holographic realization of the relation (1.1). In fact, the entropy of this black hole, originally found in [12] and later studied further in [13], in the large limit is simply given by the twisted index and thus we arrive at a microscopic understanding of the entropy for this simple universal black hole. An important point in our story is the fact that this simple black hole solution can be embedded in eleven-dimensional and massive type IIA supergravity in infinitely many ways. This provides further evidence that the relation (1.1) holds (at least in the planar limit) for a large class of three-dimensional SCFTs.

In the course of our analysis, we also clarify the relation proposed in [3] between the topologically twisted index and the entropy of the black hole. According to the holographic dictionary the logarithm of the partition function of the CFT in the large limit should correspond to the on-shell action of the dual gravitational background. We show that, for the universal black hole, the on-shell action indeed coincides with (minus) the entropy. This computation is subtle and it should be done by considering a non-extremal deformation of the black hole and carefully taking the extremal limit.

The string and M-theory embedding of the universal black hole can be interpreted as arising from D2- or M2-branes wrapped on . As pointed out in [14] these wrapped branes constructions implement the partial topological twist for the QFT which describes the low-energy dynamics of the branes. For a large number of branes one has to work in the supergravity limit in which one typically finds a black hole solution with a near horizon AdS geometry. This holographic construction was first employed for D3- and M5-branes in [15] and later generalized to M2-branes in [16].222For a review on branes wrapped on calibrated cycles and further references see [17]. This prompts us to seek generalization of the simple black hole solutions in M-theory. Due to the complicated nature of the BPS supergravity equations in eleven-dimensional supergravity we focus only on finding explicit solutions for the AdS near horizon region of the black hole. Employing this approach we find a large class of analytic explicit AdS solutions of the eleven-dimensional supersymmetry equations with non-trivial internal fluxes. The full black hole solution for which these backgrounds are a near-horizon limit should correspond to turning on background magnetic fluxes for the global non-R symmetry in the dual CFT. The entropy of the black hole should then be captured by the twisted topological index with the given background global symmetry fluxes. While we do not establish fully this holographic picture we provide some evidence for its consistency. We should point out that the explicit AdS solutions which we find fall into the general classification of such backgrounds in M-theory discussed in [18, 19]. Other examples of AdS vacua in eleven-dimensional supergravity of the class discussed here can be found in [20, 21, 22, 23].

We begin our story in the next section with a discussion of a universal partial topological twist that can be applied to three-dimensional SCFTs and we argue how for a large class of these twisted theories one can calculate the topologically twisted index of [4] in the planar limit. We then proceed in Section 3 with a holographic description of the RG flow induced by the topological twist, which is realized by a black hole in four-dimensional minimal gauged supergravity. In addition, we show how to embed this black hole in various ways in M-theory and massive type IIA string theory. In Section 4 we present a large class of AdS vacua of eleven-dimensional supergravity which can be viewed as near horizon geometries of black holes constructed out of wrapped M2-branes. We conclude with some comments and a discussion on interesting avenues for future research in Section 5. In the three appendices we collect various technical details used in the main text. In Appendix A we provide some details on the calculation of the topologically twisted index for three-dimensional SCFTs with massive IIA holographic duals. Appendices B and C contain details on the construction of our massive IIA and eleven-dimensional supergravity solutions.

Note added: While we were preparing this manuscript the papers [24, 25] appeared. In them the authors find a supersymmetric AdS black hole solution, with a near horizon AdS geometry, in the four-dimensional maximal gauged supergravity. We believe that upon an uplift to massive IIA supergravity this black hole is the same as the universal black hole discussed in Section 3.3 below. Soon after this paper appeared on the arXiv other supersymmetric AdS black hole solutions in massive type IIA supergravity were studied in [26, 27] and their microstates were counted using the topologically twisted index.

2 A universal twist of 3d SCFTs

We consider superconformal field theories compactified on a Riemann surface of genus with a topological twist. This is implemented by turning on a non-trivial background for the R-symmetry of the theory. More precisely, there is a magnetic flux on for the background gauge field coupled to the R-symmetry current such that . This condition ensures that the R-symmetry background field precisely cancels the spin connection on and the resulting theory preserves two real supercharges. In general, since the R-symmetry can mix with global symmetries, the choice of is not unique and there is a family of different twists parametrized by the freedom to turn on magnetic fluxes along the Cartan subgroup of the continuous global flavor symmetry group.333By a flavor symmetry here we mean any global symmetry of the SCFT which is not the superconformal R-symmetry. Later on we will make a distinction between different types of non-R symmetries. On the other hand, in a superconformal field theory there is an exact R-symmetry that is singled out by the superconformal algebra and can be fixed uniquely by performing -maximization [28, 29]. Following the terminology in [30, 31], we refer to a partial topological twist of the three-dimensional theory on along the exact superconformal R-symmetry as a universal twist.

The degeneracy of ground states of the compactified theory after this partial topological twist can be extracted from the topologically twisted index , which is defined as the twisted supersymmetric partition function on [4, 8, 9]. Let us briefly summarize some of the salient features of the topologically twisted index.

The index depends on a set of integer magnetic fluxes for the Cartan generators of the flavor symmetry group, parameterizing the inequivalent twists. A convenient parameterization for the fluxes is the following. We can assign a magnetic flux to each chiral field in the theory with the constraint that, for each term in the superpotential,444In this paper we restrict to three-dimensional SCFTs which admit a Lagrangian description, however we suspect that the universal twist relation (2.9) below is valid more generally.

(2.1)

Since the superpotential has R-charge , this condition ensures that and supersymmetry is preserved. The Dirac quantization condition further restricts the flux parameters to be integer.

The index also depends on a set of complex fugacities for the flavor symmetries. We can again assign a complex number to each chiral field in the theory with the constraint that, for each term in the superpotential,

(2.2)

It will be important also to consider the complexified chemical potentials , defined by . Notice that the chemical potentials are periodic, . Therefore (2.2) becomes

(2.3)

Using this periodicity we can always choose . With this at hand, (2.3) implies that unless all . This will be important in the discussion below.

The topologically twisted index can be evaluated by localization and reduced to a matrix model [4]. The large limit of the matrix model has been analyzed in [3] for the ABJM theory [32] and generalized to other classes of quiver gauge theories with M-theory or massive type IIA duals in [6, 7]. The results of this analysis is surprisingly simple. One finds a consistent large solution of the matrix model only when

(2.4)

for each term in the superpotential. Under this condition, the logarithm of the topologically twisted index is given by555The formula appears in [6] only for . The generalization to arbitrary is straightforward and discussed in details for ABJM in [8]: the general rule is simply .

(2.5)

where the function was called Bethe potential in [4] and is the Yang-Yang function of an associated integrable system [33, 9].666What we call here is the extremal value with respect to the eigenvalues of the Bethe functional , defined in [3, 6, 7]. Quite remarkably, in the large limit, the Bethe potential is related to the free energy on of the three-dimensional theory [6] through the simple identity

(2.6)

This relation might look puzzling at first sight and deserves some comments. Recall that the free energy is a function of a set of trial R-charges that parameterize a family of supersymmetric Lagrangians on [28, 34]. The importance of this functional is that its extremization gives the exact R-charges of the theory [28]. In (2.6), is a function of chemical potentials for the flavor symmetries while is a function of R-charges. However, although the parameterize flavor symmetries in the three-dimensional theory on , the relation (2.4) ensures that can be consistently identified with a set of R-charges for the theory on .

There is a subtlety that arises when computing the topologically twisted index or the three-sphere free energy. In three-dimensional SCFTs there are finite counterterms which affect the imaginary part of the complex function . These are given by Chern-Simons terms with purely imaginary coefficients for the background gauge fields that couple to conserved currents, see [29, 35] for a detailed discussion. Moreover, the imaginary part of is only defined modulo and is effectively in the large limit. The upshot of this discussion is that the physically unambiguous quantity in the large limit are the real parts of the topologically twisted index and the free energy on .

There is an additional important point in the story. To obtain the degeneracy of vacua of the compactified theory for a given choice of the flux parameters, , one has to extremize the function with respect to the fugacities [3]. This prescription is analogous to the extremization principles that exist in four [36], three [28], and two-dimensional [37, 38] SCFTs with an Abelian R-symmetry.

After this short introduction to the topologically twisted index we are ready to discuss the universal twist. This is obtained by choosing fluxes proportional to the exact UV R-charges and these, as we already mentioned, can be found by extremizing [28]. From the identification (2.6) it is clear that the Bethe potential is also extremized at the values . Given the normalizations (2.1) and (2.4), we find that the universal twist is determined by

(2.7)

It is easy to see that, for this choice of fluxes, in (2.5) is also extremized at :

(2.8)

since .777We note in passing that if one imposes before extremizing then the second term in (2.5) vanishes. Thus, after using (2.6) we find that the extremization of is the same as the extremization of the topologically twisted index. The value at the extremum reads

(2.9)

where we made use of (2.6) and (2.7).

Equation (2.9) amounts to a universal relation between the value of the index of the universal twist of a 3d SCFT and the free energy on of the same superconformal theory in the planar limit. For theories with a weakly coupled holographic dual this identity should translate into a universal relation between the entropy of some universal black hole solution and the AdS supersymmetric free energy. As we discuss in detail in section 3 this expectation indeed bears out for the case of hyperbolic Riemann surfaces, i.e. for .

It is worth pointing out that the universal relation in (2.9) is the three-dimensional analog of the universal relation among central charges established in [30] for twisted compactifications of four-dimensional SCFTs on .888Similar relations can be established for SCFTs with a continuous R-symmetry in various dimensions and with different amount of supersymmetry [31]. A notable difference is that for four-dimensional SCFTs the universal relation can be established at finite . It would be most interesting to study subleading, i.e. non-planar, corrections to the universal relation in (2.9).999It is interesting to note that similar relations between and partition functions on with line operator insertions were discussed in [39].

2.1 M-theory and massive type IIA models

The derivation of (2.9) is based on the large identities (2.5) and (2.6), which in turn can be established for a large class of Yang-Mills-Chern-Simons theories with fundamental and bi-fundamental chiral fields with M-theory or massive type IIA duals. We now discuss the class of theories for which these identities are valid.

Consider first superconformal theories dual to M-theory on AdS SE, where SE is a Sasaki-Einstein manifold. Many quivers describing such theories have been proposed in the literature. Most of them are obtained by dimensionally reducing a “parent” four-dimensional quiver gauge theory with bi-fundamentals and adjoints with an AdS SE dual, and then adding Chern-Simons terms and flavoring with fundamentals. In such theories, the sum of all Chern-Simons levels is zero, . Holography predicts that the free energy and the twisted index of such theories scales as for . The large behavior of the free energy has been computed in [40] and successfully compared with the holographic predictions only for a particular class of quivers. In particular, for the method in [40] to work, the bi-fundamental fields must transform in a real representation of the gauge group and the total number of fundamentals must be equal to the total number of anti-fundamentals. It turns out that, under the same conditions, the topologically twisted index scales like and the identities (2.5) and (2.6) are valid [6]. This particular class of quivers include all the vector-like examples in [41, 42, 43] and many of the flavored theories in [44, 45]. In particular, the latter includes the dual of AdS and AdS. The conditions are also satisfied for the necklace and - and -type quivers [46, 47, 48, 49, 50], as well as the quiver for the non-toric manifold discussed in [51]. The evaluation of the index in the large limit for most of these examples is given in [7], where the identities (2.5) and (2.6) are also verified by explicit computation. For the “chiral” theories discussed in [41, 42, 43], on the other hand, it is not known how to properly take the large limit in the matrix model to obtain the correct scaling predicted by holography. This applies both for the topologically twisted index and for the partition function. This class of quivers includes interesting models, like the quiver for proposed in [43] and further studied in [52].

Consider now superconformal theories dual to warped AdS flux vacua of massive type IIA. A well-known example is the gauge theory with three adjoint multiplets and a Chern-Simons coupling described in [53]. It corresponds to an internal manifold with the topology of . This has been generalised in [54] to the case where is an fibration over a general Kähler-Einstein manifold KE. The dual field theory is obtained by considering the four-dimensional theory dual to AdS SE, where SE is the five-dimensional Sasaki-Einstein with local base KE, reducing it to three dimensions and adding a Chern-Simons term with level for all gauge groups.101010The original example in [53] has KE, SE, and the superconformal theory is obtained by reducing SYM to three-dimensions and adding a Chern-Simons coupling with level . The solution in [54] is obtained by replacing with more general KE manifolds. The large limit at fixed of the free energy has been computed in [53] and [54] and shown to scale as , as predicted by holography. The large limit of the topologically twisted index has been computed in [6]. The identities (2.5) and (2.6) also hold for massive type IIA. The explicit derivation was not reported in [6] and is given in Appendix A. Notice that for massive type IIA quivers there is no need to restrict to vector-like models.

The discussion above shows that there is a large number of three-dimensional superconformal theories with M-theory or massive type IIA duals for which relation (2.9) formally holds. However, it is important to notice that not all of them really admit a universal twist since we need to restrict ourselves to SCFTs with rational R-charges. Indeed, since and are integers, the relation in (2.7) implies that the exact R-charge of the fields must be rational. This slightly restricts the class of theories where we can perform the universal twist. However, we can still find infinitely many models for which the R-charges are rational and the universal twist exists. Since theories necessarily have rational R-charges this applies to the necklace quivers [46, 47] as well as the - and -type quivers [48, 49, 50], and the quiver for [44, 55, 56]. In addition, one can check that the quivers for [45] and [51] in M-theory have rational R-charges. The same holds for the theory in [53] and some of its generalizations in massive type IIA.111111The R-charges for M-theory vacua can be computed using volume minimization [57] and for massive type IIA by -maximization [36]. The result is generically irrational. However, we can easily find special classes of SE or SE where the result is rational.

3 A simple 4d black hole

Here we provide the holographic description of the universal twisted compactification of 3d SCFTs discussed in the previous section. As we shall show, this corresponds to the supersymmetric magnetically charged AdS black hole of [12, 13], thus providing the appropriate field theory interpretation of this solution. We also review the known uplift of this solution to M-theory and provide a new uplift to massive IIA supergravity.

The appropriate supergravity is 4d minimal gauged supergravity [58], with eight supercharges and bosonic content the graviton and an gauge field, dual to the stress energy tensor and R-symmetry current, respectively. The bosonic action reads121212Here we follow the conventions in [3] and truncate the supergravity to the minimal one by setting (), , and set the coupling constant which in turn amounts to setting the radius of the AdS vacuum to one.

(3.1)

with the 4d Newton constant. We have chosen the cosmological constant such that the AdS vacuum of the theory has and . The magnetically charged black hole solution of [12, 13] preserves two supercharges and is given by131313A generalization to include rotation while maintaining supersymmetry was also found in these references.

(3.2)

where is the local constant-curvature metric on a Riemann surface of genus ,141414As discussed in [13], there is no supersymmetric static black hole solution with . normalized such that . Using the Gauss-Bonnet theorem the volume of the Riemann surface is then . Dirac quantization of the flux requires , which holds for any genus . We note the solution has a fixed magnetic charge, set by supersymmetry. The entropy of this extremal black hole is given in terms of the horizon area by the standard Bekenstein-Hawking formula

(3.3)

As will soon become clear the large limit of the topologically twisted index reproduces exactly this entropy. However, there is a slight subtlety in this story. According to the standard holographic dictionary the logarithm of the partition function of the CFT (in the appropriate large limit) should correspond to a properly regularized on-shell action of the dual gravitational background, rather than the black hole entropy. In the next section we clarify this relation, showing that the black hole entropy in (3.3) is indeed closely related to the on-shell action and thus to the topologically twisted index.

3.1 On-shell action

We are interested in calculating the value of the Euclideanized action (3.1), evaluated on-shell for the solution (3.2). This action is divergent unless properly regularized by counterterms, following the standard holographic renormalization prescription, which we carry out explicitly next. As we show, this is intimately related to the entropy of the black hole. The regularized action we consider is given by:151515See for example Section 4.4 of [59] as a reference for the counterterms. With respect to their normalizations we have .

(3.4)

where is the induced metric on the boundary, defined by the radial cutoff , and is the trace of the extrinsic curvature of this boundary metric. Taking leads to a divergence in , which is cancelled by , where we have collected the appropriate counterterms as well as boundary terms necessary for a well-defined variational principle. An important additional subtlety arises, however, in the explicit evaluation of for the extremal solution (3.2). Indeed, it is easy to see that this integral is naively not well defined, as the integrand of for this solution vanishes, while the integration over Euclidean time leads to infinity as a consequence of the solution being extremal and thus . To obtain the correct finite result we thus consider a non-extremal deformation of the solution, compute for the non-extremal solution and take the extremal limit at the end.

There are two non-extremal deformations. One amounts to allowing for a generic magnetic charge under the graviphoton, and a second to adding a mass . This non-extremal generalization was discussed in [12, 13] and the solution reads

(3.5)

where

(3.6)

The extremal solution is recovered for and . The horizon radius is obtained by solving the quartic equation . The temperature of the black hole can be obtained by requiring the Euclidean metric to be free of conical singularities, which gives

(3.7)

In the extremal limit and . Evaluating the on-shell action (3.4) for general values of and we find

(3.8)

where is the Euclidean time periodicity. Taking the extremal limit of this expression gives the finite answer

(3.9)

Comparing this to the entropy (3.3) of the extremal black hole, we thus have

(3.10)

On the other hand, the holographic dictionary relates the gravitational on-shell action to the partition function of the dual CFT as , leading to the satisfying expression

(3.11)

We have thus shown that the black hole entropy can be identified with the topologically twisted index of the dual CFT to leading order in . This relation was argued to hold more generally for a class of black holes in non-minimal gauged supergravity in [3]; it would be interesting to establish this explicitly by generalizing the computation above to this larger class of black holes. It would also be of interest to study these relations to subleading orders in .

Finally, let us recall that the renormalized on-shell action for Euclidean AdS with an boundary is given by [60]161616With slight abuse of notation we define the CFT free energy as .

(3.12)

which using (3.3) leads to the relation

(3.13)

With the identification (3.11) one recognizes this result as the holographic analog of the universal field theory relation (2.9). Thus, the result (3.13) provides strong evidence that the AdS black hole (3.2) describes the RG flow from 3d SCFTs with a universal topological twist on to superconformal quantum mechanical theories with two supercharges.

We emphasize that although the black hole solution (3.2) is derived as a supersymmetric solution of minimal gauged supergravity, it is also a solution to non-minimal gauged supergravity, with the additional vector and hyper multiplet bosonic fields set to zero.171717This can be checked, for instance, by setting in the BPS equations (A.27) in [3]. The fact that “freezing” the vector multiplets to zero is consistent corresponds to the property of universality of the topological twist (2.7). In other words, as the vector multiplet scalars in the gauged supergravity are not sourced along the flow from AdS to AdS realized by the black hole (3.2), there is no mixing between R-symmetry and flavor symmetry along the flow and the R-symmetry in the IR coincides with the one in the UV. This is consistent with the field theory discussion of Section 2.

3.2 Uplift to M-theory

Here we review the uplift of the universal solution in (3.2) to eleven-dimensional supergravity, which was carried out in [61]. The metric and four-form read181818Compared to Equation (2.3) in [61] we have introduced an overall scale .

(3.14)

where is a seven-dimensional Sasaki-Einstein with , is a six-dimensional Kähler-Einstein space with and Kähler form , with , , is given in (3.2), and is with respect to this four-dimensional metric. This solution of eleven-dimensional supergravity can be interpreted as the backreaction of M2-branes wrapping a supersymmetric cycle in a Calabi-Yau five-fold. This point of view was discussed in [16], following the ideas in [15], for the case where the Sasaki-Einstein manifold is and thus the six-dimensional Kähler-Einstein space is .

In order for this local supergravity solution to extend to a well defined M-theory background, some of its parameters should be properly quantized. The quantization condition on reads

(3.15)

where is an integer determining the number of M2-branes and is the Planck length in eleven dimensions. This translates into the quantization of the AdS length scale in terms of the Planck length,191919Upon a reduction to 10d type IIA supergravity the 11d Planck length is related to the string length and coupling constant via the equation .

(3.16)

The four-dimensional Newton constant is given by

(3.17)

3.2.1 Entropy

Using (3.16) and (3.17) we compute the properly normalized horizon area, obtaining the black hole entropy:

(3.18)

This is consistent with (3.13) and the well-known expression for the holographic free energy on [47, 62]

(3.19)

It is instructive to unpack this equation for a couple of examples of well-known three-dimensional SCFTs. For the special case of the ABJM theory, i.e., , the free energy on is given by [62] and thus

(3.20)

This result is correctly reproduced by the topologically twisted index for the ABJM theory. This follows from the general discussion in Section 2, which is model independent. It is nonetheless instructive to see explicitly how this works. The topologically twisted index for ABJM is given by [3, 5]

(3.21)

where are, respectively, the chemical potentials and fluxes associated to the four chiral bi-fundamental fields of ABJM, subject to the constraints and .202020See formula (28) in [5], where the notations are: and and the case is considered. For there could be ambiguities related to other saddle points, but we still expect (3.21) to hold. The exact R-symmetry corresponds to and the universal twist to . We see that must be odd. We obtain , as expected.

As another example, we can consider the Sasaki-Einstein manifold whose dual SCFT was discussed in [45]. The free energy on is given by [40]. Then, from (3.13) we have

(3.22)

The dual field theory is a flavored version of ABJM and the exact R-charge of the fields is and that of is [40]. Quantization of fluxes then requires to be an integer multiple of . The topologically twisted index was computed in [7] and it is easy to check that , again in agreement with the general result in Section 2.212121The free energy on is given by formula (6.15) in [40] and the twisted index for by formula (5.47) in [7]. The generalization to general genus is obtained by [8]. As discussed in [40], the exact R-charge of the fields is and that of the fields is , while =0. The fluxes for the universal twist are then , and . From formula (5.47) in [7] we find . Combining the results in [40] and [7] one can check many other examples, all in agreement with the general result (2.9).

As discussed in Section 2, the universal twist is not only possible for theories with an M-theory dual, but also for field theories with massive IIA duals with an scaling of the free energy. We discuss this next.

3.3 Uplift to massive IIA

Here we discuss new black hole solutions in massive IIA supergravity, obtained by uplifts of the 4d solution (3.2). This can be done by using the formulae of [63, 64, 65], where the uplift of the SU-invariant sector of 4d supergravity with ISO gauging is given. The bosonic content of the 4d SU-invariant sector is the graviton , 6 scalars , 2 electric vectors and their magnetic duals , 3 two-forms and 2 three-forms . One should keep in mind that these are not all independent as, e.g., the field strengths of can be dualized into functions of the scalar fields, see [65] for more details.

Since we are interested in solutions that asymptote to the supersymmetric AdS vacuum of the theory described in [53], and we expect the scalars to be set to constant values, we set them to the values of the AdS vacuum solution, provided in (B.3). The electric potential is identified (up to a normalization, which we fix by using the massive IIA equations of motion) with the gauge field of the minimal four-dimensional theory in (3.1). With this at hand we can use the uplift formulae of [63, 64, 65], reproduced in (B.1), to obtain explicit expressions for the bosonic fields of the ten-dimensional massive IIA supergravity, i.e., the metric, the dilaton, , as well as the two-, three-, and four-form fluxes: , , and , respectively. The metric and dilaton read

(3.23)

where is the black hole metric (3.2), is the standard Einstein metric on (see (B.2)), is the connection on the Riemann surface, and

(3.24)

We note that (3.23) can be obtained from the AdS vacuum solution of [53] by the simple replacement and . The massive IIA form fields, however, are not so easily obtained and require more work. The basic point is that in order to determine these one must find a further truncation of the remaining -invariant fields consistent with the duality transformations of [65] and the equations of motion, which due to the connection on the Riemann surface is non-trivial. We discuss this in detail in Appendix B.1. Here we simply present the final answer:

(3.25)

where is the Kähler form on , , is the volume form of , and is the hodge dual with respect to . We have explicitly checked that (3.23) and (3.25) indeed satisfy the equations of motion of massive IIA supergravity.222222These equations can be found in Equation (A.5) of [63]. At large the solution asymptotes to the AdS solution of [53] and thus (3.23) and (3.25) describe the twisted compactification of the corresponding three-dimensional field theory dual, consisting of a Chern-Simons theory at level with three adjoints superfields .

Based on our field theory analysis we actually expect (3.23) and (3.25) to be part of a more general class of solutions, where the is replaced by a general Kähler-Einstein base. As an example, we have explicitly checked that replacing

(3.26)

in (3.23) (along with suitable replacements of and in (3.25)) is also a solution of the equations of motion.

More generally, replacing by a generic Kähler-Einstein base leads to a large family of new massive IIA black holes, which are asymptotic to the AdS solutions described in [54]. As discussed there, the 3d field theory duals are obtained from a 4d parent field theory with an AdS dual, with corresponding to the Kähler-Eistein base of the Sasaki-Einstein , i.e.,

(3.27)

Thus, the black hole solutions reported here describe the compactifications of these 3d SCFTs on a Riemann surface, twisted by the exact superconformal R-symmetry.

To ensure that the local supergravity solution given in (3.23) and (3.25) extends to a well defined string theory background, it should be properly quantized.232323In the case of this was carried out in [54] for a general base . The quantization for the black hole solution here is very similar and we provide it here for completeness. The four-form quantization condition in massive IIA reads242424See for example Equation (4.11) in [64].

(3.28)

where is an integer determining the number of D2-branes, and the potentials can be found in [64]. We can evaluate this for the solution in (3.23) and (3.25) for a general base . Using the explicit expression for the background fluxes presented in Appendix B.1 we find

(3.29)

where we integrated over and used the identity . Using (3.24) this translates into a quantization condition on the asymptotic AdS and the near-horizon AdS radii:

(3.30)

where we have used that asymptotically , while close to the horizon and defined . Note that the quantization condition on coincides with the one obtained in [54]. This is expected since our solution is asymptotic to the solutions discussed there. Similarly, the four-dimensional Newton constant is given by252525For a solution of the form , the effective four-dimensional Newton constant is given by , with , following the conventions used in [53]. Note in particular that in [53] they set .

(3.31)

3.3.1 Entropy

Putting all this together, we can compute the properly normalized horizon area and obtain the following general formula for the black hole entropy:

(3.32)

Let us specialize this to the two simple cases considered above, namely and , with corresponding ’s equal to and , respectively. We recall that for a gauge theory, where each factor has the same Chern-Simons level , the parameter is related to the Chern-Simons levels by [66].

For we have and and (3.32) reads

(3.33)

The dual three-dimensional field theory is a Chern-Simons theory with three adjoints superfields and superpotential [40]. The index is explicitly computed in Appendix A; it is given by (A.15) and (A.16), where and . The exact R-symmetry corresponds to and the universal twist to . Quantization of fluxes requires to be an integer multiple of . We then see that the index is given by

(3.34)

while the free energy on is given by equation (A.9),

(3.35)

As expected this leads to , in agreement with our general argument in Section 2. We also note that both and are complex but as discussed above Equation (2.7) we should focus on the real part. We then arrive at the following, by now familiar, relation between the black hole entropy and the SCFT partition functions; .

Similarly, for one has , and (3.32) reads

(3.36)

The three-dimensional field theory dual is a gauge theory with four bifundamentals with the same quiver and superpotential as the 4d Klebanov-Witten theory [67]. In this case the exact R-symmetry corresponds to and the universal twist to , which requires to be odd. The topologically twisted index, computed explicitly in Appendix A, correctly reproduces the entropy in (3.36).

3.3.2 Checking supersymmetry

As a final consistency check of the supergravity discussion of these new massive IIA black hole solutions here we show explicitly that the solution in (3.23) and (3.25) preserves two supercharges. The supersymmetry variations of the fermionic fields are given in [68] and read262626The conventions of this reference are related to ours by .