A universal counting of black hole microstates in AdS4

Journal of High Energy Physics, Feb 2018

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

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A universal counting of black hole microstates in AdS4

HJE in Francesco Azzurli 0 1 4 5 Nikolay Bobev 0 1 3 5 P. Marcos Crichigno 0 1 2 5 Vincent S. Min 0 1 3 5 Alberto Zaffaroni 0 1 4 5 0 Celestijnenlaan 200D , B-3001 Leuven , Belgium 1 I-20126 Milano , Italy 2 Institute for Theoretical Physics, University of Amsterdam 3 Instituut voor Theoretische Fysica, KU Leuven 4 Dipartimento di Fisica, Università di Milano-Bicocca 5 Science Park 904 , Postbus 94485, 1090 GL, Amsterdam , The Netherlands Many three-dimensional N = 2 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 AdS4 magnetically charged black holes in Mtheory and massive type IIA string theory. In this context we also discuss novel AdS2 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. AdS-CFT Correspondence; Black Holes in String Theory - A 1 Introduction A universal twist of 3d N = 2 SCFTs 2.1 M-theory and massive type IIA models 2 3 3.1 3.2 3.3 4.1 4.2 4.3 4.4 A simple 4d black hole On-shell action Uplift to M-theory Uplift to massive IIA 4 Uplift of universal solution to massive IIA C Details on the construction of the 11D solutions B.1 Determining the fluxes B.2 Proving supersymmetry C.1 BPS equations C.2 4-form equations C.3 Singularity analysis ing 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 state of affairs changed recently with the results in [3]. The authors of [3] employed a certain supersymmetric index, ZS1 g , defined in [4] to account for the entropy of a large class of supersymmetric asymptotically AdS4 black holes arising from M2-branes wrapped on Riemann surfaces. These results were later extended to dyonic black holes in [5]. See also [6–11] 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 N limit for a large class of N = 2 three-dimensional SCFTs.1 Namely, we find that the index is related to the three-sphere free energy, FS3 , of the three-dimensional theory through the simple relation log ZS1 g = (g 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 N = 2 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 g. 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 g such that the only non-zero flux is in the direction of the exact superconformal R-symmetry of the three-dimensional N = 2 SCFT. Such a simple universal relation in QFT should have an equally simple bulk realization for three-dimensional N = 2 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 N 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 elevendimensional 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 N = 2 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 N 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. 1The conditions on the N = 2 theory are discussed in detail in section 2. – 2 – . 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 AdS2 g geometry. This holographic construction was first employed for D3- and M5-branes in [15] and later generalized to M2-branes in [16].2 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 AdS2 near horizon region of the black hole. Employing this approach we find a large class of analytic explicit AdS2 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 AdS2 solutions which we find fall into the general classification of such backgrounds in M-theory discussed in [18, 19]. Other examples of AdS2 vacua in eleven-dimensional supergravity of the class discussed here can be found in [20–23]. We begin our story in the next section with a discussion of a universal partial topological twist that can be applied to three-dimensional N = 2 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 AdS2 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 N = 2 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 AdS4 black hole solution, with a near horizon AdS2 g geometry, in the four-dimensional maximal ISO(7) 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 AdS4 black hole solutions in massive type IIA supergravity were studied in [26, 27] and their microstates were counted using the topologically twisted index. 2For a review on branes wrapped on calibrated cycles and further references see [17]. – 3 – A universal twist of 3d N = 2 SCFTs on R g We consider N = 2 superconformal field theories compactified on a Riemann surface g of genus g 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 g for the background gauge field AR coupled to the R-symmetry current such that dAR = 2 (1 g). This condition ensures that the R-symmetry background field precisely cancels the spin connection on g and the resulting theory preserves two real supercharges. In general, since the R-symmetry can mix with global symmetries, the choice of AR is not unique and there is a family of different twists parametrized by the freedom to turn ing F -maximization [28, 29]. Following the terminology in [30, 31], we refer to a partial topological twist of the three-dimensional theory on g 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 Z, which is defined as the twisted supersymmetric partition function on g the salient features of the topologically twisted index. S1 [4, 8, 9]. Let us briefly summarize some of The index depends on a set of integer magnetic fluxes n = 21 R g F av 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 nI to each chiral field I in the theory with the constraint that, for each term Wa in the superpotential,4 Since the superpotential has R-charge 2, this condition ensures that R and supersymmetry is preserved. The Dirac quantization condition further restricts the g dAR = 2 (1 g) flux parameters nI to be integer. The index also depends on a set of complex fugacities y for the flavor symmetries. We can again assign a complex number yI to each chiral field I in the theory with the constraint that, for each term Wa in the superpotential, X I2Wa nI = 2(1 g) : Y I2Wa yI = 1 : (2.1) (2.2) It will be important also to consider the complexified chemical potentials by yI ei I . Notice that the chemical potentials are periodic, I I , defined I + 2 . 3By a flavor symmetry here we mean any global symmetry of the N = 2 SCFT which is not the however we suspect that the universal twist relation (2.9) below is valid more generally. 4In this paper we restrict to three-dimensional N = 2 SCFTs which admit a Lagrangian description, – 4 – Therefore (2.2) becomes Using this periodicity we can always choose 0 Re I 2 . With this at hand, (2.3) implies that P I2Wa I 6= 0 unless all I = 0. 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 N 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 N solution of the matrix model only when (2.3) (2.4) (2.5) (2.6) for each term Wa in the superpotential. Under this condition, the logarithm of the topologically twisted index is given by5 log Z( I ; nI ) = (1 g)i 2 V( I ) + X I nI 1 g I where the function V was called Bethe potential in [4] and is the Yang-Yang function of an associated integrable system [9, 33].6 Quite remarkably, in the large N limit, the Bethe potential V is related to the free energy on S3 of the three-dimensional N = 2 theory [6] through the simple identity 2i This relation might look puzzling at first sight and deserves some comments. Recall that the free energy FS3 is a function of a set of trial R-charges that parameterize a family of supersymmetric Lagrangians on S3 [28, 34]. The importance of this functional is that its extremization gives the exact R-charges of the theory [28]. In (2.6), V is a function of chemical potentials for the flavor symmetries while FS3 is a function of R-charges. However, although the I parameterize flavor symmetries in the three-dimensional theory on g S1, I = can be consistently identified with a set of R-charges the relation (2.4) ensures that for the theory on S3. There is a subtlety that arises when computing the topologically twisted index or the three-sphere free energy. In three-dimensional N = 2 SCFTs there are finite counterterms which affect the imaginary part of the complex function FS3 . These are given by ChernSimons 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 log Z is only defined modulo 2 and is effectively O(1) in the large N limit. The upshot 5The formula appears in [6] only for g = S2. The generalization to arbitrary g is straightforward and discussed in details for ABJM in [8]: the general rule is simply log Zg = (1 g) log Zg=0(nI =(1 g)). 6What we call V( I ) here is the extremal value with respect to the eigenvalues ui of the Bethe functional V[ I ; ui], defined in [3, 6, 7]. – 5 – of this discussion is that the physically unambiguous quantity in the large N limit are the real parts of the topologically twisted index and the free energy on S3. 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, nI , one has to extremize the function Z( I ; nI ) with respect to the fugacities I [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 nI proportional to the exact UV R-charges I and these, as we already mentioned, can be found by extremizing FS3 [28]. From the identification (2.6) it is clear that the Bethe potential V is also extremized at the values I . Given the normalizations (2.1) and (2.4), we find that the universal twist is determined by It is easy to see that, for this choice of fluxes, log Z in (2.5) is also extremized at I : I = 0 ; log Z( I ; nI ) = (g 1)FS3 I ; 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 N = 2 SCFT and the free energy on S3 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 AdS4 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 g > 1. 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 N = 1 SCFTs on R 2 g . 8 A notable difference is that for four-dimensional SCFTs the universal relation can be established at finite N . It would be most interesting to study subleading, i.e. non-planar, corrections to the universal relation in (2.9).9 7We note in passing that if one imposes 1nIg = I before extremizing FS3 then the second term in (2.5) vanishes. Thus, after using (2.6) we find that the extremization of FS3 is the same as the extremization of the topologically twisted index. 8Similar relations can be established for SCFTs with a continuous R-symmetry in various dimensions and with different amount of supersymmetry [31]. operator insertions were discussed in [39]. 9It is interesting to note that similar relations between FS3 and partition functions on S1 g with line – 6 – (2.7) I = I : (2.8) (2.9) The derivation of (2.9) is based on the large N 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 AdS4 SE7, where SE7 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 AdS5 SE5 dual, and then the twisted index of such theories scales as N 3=2 for N adding Chern-Simons terms and flavoring with fundamentals. In such theories, the sum of all Chern-Simons levels is zero, P a ka = 0. Holography predicts that the S3 free energy and ka. The large N behavior of the S3 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 antifundamentals. It turns out that, under the same conditions, the topologically twisted index scales like N 3=2 and the identities (2.5) and (2.6) are valid [6]. This particular class of quivers include all the vector-like examples in [41–43] and many of the flavored theories in [44, 45]. In particular, the latter includes the dual of AdS4 Q1;1;1=Zk and AdS4 N 0;1;0=Zk. The conditions are also satisfied for the N = 3 necklace and Db - and Eb-type quivers [46–50], as well as the quiver for the non-toric manifold V 5;2 discussed in [51]. The evaluation of the index in the large N 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 [41–43], on the other hand, it is not known how to properly take the large N limit in the matrix model to obtain the correct scaling predicted by holography. This applies both for the topologically twisted index and for the S3 partition function. This class of quivers includes interesting models, like the quiver for M 1;1;1 proposed in [43] and further studied in [52]. Consider now superconformal theories dual to warped AdS4 Y6 flux vacua of massive type IIA. A well-known example is the N = 2 U(N ) gauge theory with three adjoint multiplets and a Chern-Simons coupling k described in [53]. It corresponds to an internal manifold Y6 with the topology of S6. This has been generalised in [54] to the case where Y6 is an S2 fibration over a general Kähler-Einstein manifold KE4. The dual field theory is obtained by considering the four-dimensional theory dual to AdS5 SE5, where SE5 is the five-dimensional Sasaki-Einstein with local base KE4, reducing it to three dimensions and adding a Chern-Simons term with level k for all gauge groups.10 The large N limit at fixed k of the S3 free energy has been computed in [53] and [54] and shown to scale as N 5=3, as predicted by holography. The large N limit of the topologically twisted index has been 10The original example in [53] has KE4 = CP2, SE5 = S5, and the superconformal theory is obtained by reducing N = 4 SYM to three-dimensions and adding a Chern-Simons coupling with level k. The solution in [54] is obtained by replacing CP2 with more general KE4 manifolds. – 7 – 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 N = 2 SCFTs with rational R-charges. Indeed, since nI and g 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 N = 3 theories necessarily have rational R-charges this applies to the N = 3 necklace quivers [46, 47] as well as the Db - and Eb-type quivers [48–50], and the N = 3 quiver for N 0;1;0 [44, 55, 56]. In addition, one can check that the N = 2 quivers for Q1;1;1 [45] and V 5;2 [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.11 3 A simple 4d black hole Here we provide the holographic description of the universal twisted compactification of 3d N = 2 SCFTs discussed in the previous section. As we shall show, this corresponds to the supersymmetric magnetically charged AdS4 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 N = 2 gauged supergravity [58], with eight supercharges and bosonic content the graviton and an SO(2) gauge field, dual to the stress energy tensor and R-symmetry current, respectively. The bosonic action reads12 I = 1 with G(N4) the 4d Newton constant. We have chosen the cosmological constant such that the AdS4 vacuum of the theory has RAdS4 = 12 and LAdS4 = 1. The magnetically charged black hole solution of [12, 13] preserves two supercharges and is given by13 Z 2 ds42 = F = 1 2 11The R-charges for M-theory vacua can be computed using volume minimization [57] and for massive type IIA by a-maximization [36]. The result is generically irrational. However, we can easily find special classes of SE7 or SE5 where the result is rational. 12Here we follow the conventions in [3] and truncate the N = 8 supergravity to the minimal one by setting La = 1 ( ij = 0), Aa = A, and set the coupling constant g = 1=p2 which in turn amounts to setting the radius of the AdS4 vacuum to one. 13A generalization to include rotation while maintaining supersymmetry was also found in these references. – 8 – (3.1) (3.2) where ds2H2 = x12 dx21 + dx22 is the local constant-curvature metric on a Riemann surface 2 g of genus g > 1,14 normalized such that RH2 = 2. Using the Gauss-Bonnet theorem the volume of the Riemann surface is then vol( g) = 4 (g 1). Dirac quantization of the flux requires 21 R g F 2 Z, which holds for any genus g. 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 As will soon become clear the large N 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 N 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:15 SBH = Area 4G(N4) = (g 1) 2G(N4) is the induced metric on the boundary, defined by the radial cutoff = r, and K is the trace of the extrinsic curvature of this boundary metric. Taking r ! 1 leads to a divergence in IEinst+Max, which is cancelled by Ict+bdry, 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 IEucl for the extremal solution (3.2). Indeed, it is easy to see that this integral is naively not well defined, as the integrand of IEucl for this solution vanishes, while the integration over Euclidean time R01 d leads to infinity as a consequence of the solution being extremal and thus T = 0. To obtain the correct finite result we thus consider a non-extremal deformation of the solution, compute IEucl for the non-extremal solution and take the extremal limit at the end. 14As discussed in [13], there is no supersymmetric static black hole solution with g = 0; 1. 15See for example section 4.4 of [59] as a reference for the counterterms. With respect to their normalizations we have F there = F here=2. – 9 – There are two non-extremal deformations. One amounts to allowing for a generic magnetic charge Q under the graviphoton, and a second to adding a mass . This nonextremal generalization was discussed in [12, 13] and the solution reads The extremal solution is recovered for Q ! 1=2 and ! 0. The horizon radius 0 is obtained by solving the quartic equation V ( 0) = 0. The temperature T of the black hole can be obtained by requiring the Euclidean metric to be free of conical singularities, which gives T = jV 0( 0)j = 4 2 1 0 In the extremal limit 0 ! 1=p2 and T ! 0. Evaluating the on-shell action (3.4) for general values of Q and we find16 IEucl = (g 1) 2G(N4) 0 Q 2 = 1=T is the Euclidean time periodicity. Taking the extremal limit of this expression gives the finite answer Iextr = (g 1) 2G(N4) + O (Q 1=2)1=2 + O 1=2 : Comparing this to the entropy (3.3) of the extremal black hole, we thus have On the other hand, the holographic dictionary relates the gravitational on-shell action to the partition function of the dual CFT as I = log Z, leading to the satisfying expression 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 N . 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 16Based on general thermodynamics arguments one would expect the non-supersymmetric on-shell action to obey the relation I = (M ST Q), where M , Q, and , are the mass, charge and chemical potential of the black hole. See [60] for a recent discussion on this relation for asymptotically AdS4 black holes. Iextr = SBH : SBH = log Z : orders in N . S3 boundary is given by [61]17 which using (3.3) leads to the relation Finally, let us recall that the renormalized on-shell action for Euclidean AdS4 with an 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 AdS4 black hole (3.2) describes the RG flow from 3d N = 2 SCFTs with a universal topological twist on g 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.18 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 AdS4 to AdS2 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 [62]. The metric and four-form read19 (3.12) (3.13) (3.14) ds121 = L 2 ds42 + ds72 ; dimensional Kähler-Einstein space with RiKjE = 8giKjE and Kähler form J , with d where ds7jA=0 is a seven-dimensional Sasaki-Einstein with RiSjE7 = 6giSjE7 , dsKE is a six= 2J , F = dA = volH2 , ds24 is given in (3.2), and 4 is with respect to this four-dimensional metric. This solution of eleven-dimensional supergravity can be interpreted as the backreaction of N 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 S7 and thus the six-dimensional Kähler-Einstein space is CP3. 17With slight abuse of notation we define the CFT free energy as F = log Z = I. 18This can be checked, for instance, by setting na = 2 19Compared to equation (2.3) in [62] we have introduced an overall scale L. ; ~ = 0 ; La = 1 in the BPS equations (A.27) in [3]. The four-dimensional Newton constant is given by 1 Using (3.16) and (3.17) we compute the properly normalized horizon area, obtaining the This is consistent with (3.13) and the well-known expression for the holographic free energy 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 G4 reads where N is an integer determining the number of M2-branes and `11 is the Planck length in eleven dimensions. This translates into the quantization of the AdS4 length scale in terms of the Planck length,20 N = 1 energy on S3 is given by FS3 = p 3 2 k1=2N 3=2 [63] and thus It is instructive to unpack this equation for a couple of examples of well-known threedimensional SCFTs. For the special case of the ABJM theory, i.e., Y7 = S7=Zk, the free 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] p 3 log ZABJM = 2 k1=2N 3=2p I ; nI are, respectively, the chemical potentials and fluxes associated to the four chiral bi-fundamental fields of ABJM, subject to the constraints P4 20Upon a reduction to 10d type IIA supergravity the 11d Planck length is related to the string length and coupling constant via the equation `11 = lsgs1=3. PI nI = 2(1 whose dual SCFT was discussed in [45]. 4p k1=2N 3=2 [40]. Then, from (3.13) we have The exact R-symmetry corresponds to I = g)=2. We see that g must be odd. We obtain log ZABJM = SBAHBJM, As another example, we can consider the Sasaki-Einstein manifold Y7 = Q1;1;1=Zk The free energy on S3 is given by FS3 = SBH Q1;1;1 = (g 4 Ai is 2=3 and that of Bi is 1=3 [40]. Quantization of fluxes then requires g integer multiple of 3. The topologically twisted index was computed in [7] and it is easy to , again in agreement with the general result in section 2.22 check that log ZQ1;1;1 = SBH Q1;1;1 with the general result (2.9). Combining the results in [40] and [7] one can check many other examples, all in agreement 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 N 5=3 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 [64–66], where the uplift of the SU( 3 )-invariant sector of 4d N = 8 supergravity with ISO(7) gauging is given. The bosonic content of the 4d SU( 3 )-invariant sector is the graviton ea , 6 scalars '; ; ; a; ; ~, 2 electric vectors A0; A1 and their magnetic duals A~0; A~1, 3 two-forms B0; B1; B2 and 2 three-forms C0; C1. One should keep in mind that these are not all independent as, e.g., the field strengths of C0; C1 can be dualized into functions of the scalar fields, see [66] for more details. Since we are interested in solutions that asymptote to the N = 2 supersymmetric AdS4 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 AdS4 vacuum solution, provided in (B.3). The electric potential A1 is identified (up to a normalization, which we fix by using the massive IIA equations of motion) with the gauge field A of the minimal four-dimensional theory in (3.1). With this at hand we can use the uplift formulae of [64–66], 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: F^2, H^3, 21See formula (28) in [5], where the notations are: uI = I and pI = nI and the case k = 1 is considered. For k > 1 there could be ambiguities related to other saddle points, but we still expect (3.21) to hold. 22The free energy on S3 is given by formula (6.15) in [40] and the twisted index for g = 0 by formula (5.47) in [7]. The generalization to general genus g > 0 is obtained by log Zg = (1 g) log Zg=0(ni=(1 g)) [8]. As discussed in [40], the exact R-charge of the fields Ai is Ai = = 2=3 and that of the fields Bi is Ai = = 1=3, From formula (5.47) in [7] we find log ZQ1;1;1 = (g while m=0. The fluxes for the universal twist are then nAi = 2(1 g)=3, nBi = (1 1) 34p3 k1=2N 3=2. g)=3 and t + ~t = 0. (see (B.2)), A1 = 31g dxx21 is the connection on the Riemann surface, and where ds24 is the black hole metric (3.2), ds2CP2 is the standard Einstein metric on CP2 We note that (3.23) can be obtained from the AdS4 vacuum solution of [53] by the simple replacement ds2AdS4 ! ds24 and d ! d + gA1. 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 SU( 3 )-invariant fields consistent with the duality transformations of [66] 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: F^2 = H^3 = F^4 = + 1 m 1 g2 m g 2=3 4 sin2( ) cos( ) g(cos(2 )+3)(cos(2 )+5) 3 sin( )(cos(2 ) 3) g(cos(2 )+5)2 ^d 3 cos( ) cos(2 )+5 H(12) 1 p 2 3 cos( ) 4 H(12) ; m g m g 2=3 1=3 8 sin3( ) g(cos(2 )+3)2 vol4 + p3g sin4( )(3 cos(2 )+7) g(cos(2 )+3)2 J ^J d ^J + 3 sin( )d ^ 4H(12) ; ! and F^4, respectively. The metric and dilaton read g(cos(2 )+3)(cos(2 )+5) 4(cos(2 )+5) ^d ^ 4H(12) cos(2 )+3 4 H(12) ^J ; p3 sin2( ) where J = 12 d is the Kähler form on CP2, H(12) and 4 is the hodge dual with respect to ds4. dA1, vol4 is the volume form of ds4, We have explicitly checked that (3.23) and (3.25) indeed satisfy the equations of motion of massive IIA supergravity.23 At large the solution is locally asymptotic to the N = 2 AdS4 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 U(N ) Chern-Simons theory at level k with three adjoints superfields i . 23These equations can be found in equation (A.5) of [64]. 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 CP2 is replaced by a general Kähler-Einstein base. As an example, we have explicitly checked that replacing 1 6 1 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.24 The four-form quantization condition in massive IIA reads25 N = 1 (2 `s)5 Z M6 e 21 ^ 1 6 4 F^(4) + B^(2) ^ dA^( 3 ) + m B^(2) ^ B^(2) ^ B^(2) ; (3.28) where N is an integer determining the number of D2-branes, and the potentials B^(2); A^( 3 ) can be found in [65]. We can evaluate this for the solution in (3.23) and (3.25) for a general base B. Using the explicit expression for the background fluxes presented in appendix B.1 Sasaki-Einstein Y5, i.e., in (3.23) (along with suitable replacements of J and in (3.25)) is also a solution of the equations of motion. More generally, replacing CP2 by a generic Kähler-Einstein base B leads to a large family of new massive IIA black holes, which are asymptotic to the AdS4 solutions described in [54]. As discussed there, the 3d field theory duals are obtained from a 4d parent field Y5 dual, with B corresponding to the Kähler-Eistein base of the (3.26) (3.27) (3.29) (3.30) ds2Y 5 = 2 + ds2B : where we integrated over 0 near-horizon AdS2 radii: Using (3.24) this translates into a quantization condition on the asymptotic AdS4 and the and used the identity 12 J ^ J ^ d = dvol(Y5). LAdS2 = LAdS4 = 1 2 `sn1=24 255=4837=24 N vol(Y5) 5=24 ; where we have used that asymptotically ds24 ! ds2AdS4 , while close to the horizon ds24 ! 14 (ds2AdS2 + 2ds2 g ) and defined n 2 `sm. Note that the quantization condition on LAdS4 24In the case of ds42 = ds2AdS4 this was carried out in [54] for a general base B. The quantization for the black hole solution here is very similar and we provide it here for completeness. 25See for example equation (4.11) in [65]. Let us specialize this to the two simple cases considered above, namely CP2 and P 1 P1, with corresponding Y5’s equal to S5 and Y 1;0, respectively. We recall that for a U(N )G gauge theory, where each factor has the same Chern-Simons level k, the parameter n is related to the Chern-Simons levels by n = Gk [67]. For Y5 = S5 we have vol(Y5) = 3 and n = k and (3.32) reads SBSH5 = (g 1) 21=331=65 1N 5=3k1=3 : The dual three-dimensional field theory is a U(N ) Chern-Simons theory with three adjoints superfields i and superpotential W = 1 ; 3] [40]. The index is explicitly computed in appendix A; it is given by (A.15) and (A.16), where g). The exact R-symmetry corresponds to 1 + 2 + 3 = 2 and n1 + n2 + n3 = i = 2 =3 and the universal twist to g)=3. Quantization of fluxes requires g 1 to be an integer multiple of 3. We 2(1 ni = 2(1 then see that the index is given by while the free energy on S3 is given by equation (A.9), log Z = (g 1) 21=331=65 1N 5=3k1=3(1 i=p3) ; FS3 = 21=331=65 1N 5=3k1=3(1 i=p3) : 1 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 by26 Putting all this together, we can compute the properly normalized horizon area and obtain the following general formula for the black hole entropy: HJEP02(18)54 As expected this leads to log Z = (g 1)FS3 , in agreement with our general argument in section 2. We also note that both log Z and FS3 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; SBH = Re log Z = (g 1)ReFS3 . Similarly, for Y 5 = Y 1;0 one has vol(Y5) = 16273 , n = 2k and (3.32) reads SBYH1;0 = (g 1) 313=620 1N 5=3k1=3 : 26For a solution of the form ds120 = e2 L2 ds42 + ds62 , the effective four-dimensional Newton constant is given by 1=G(N4) = (1=G(N10)) L6 R d6xpg6e8 , with 16 G(N10) = (2 )7`s8, following the conventions used in [53]. Note in particular that in [53] they set gs = 1. The three-dimensional field theory dual is a U(N ) U(N ) gauge theory with four bifundamentals I with the same quiver and superpotential as the 4d N = 1 Klebanov-Witten theory [68]. In this case the exact R-symmetry corresponds to I = g)=2, which requires g to be odd. The topologically twisted index, computed explicitly in appendix A, correctly reproduces the entropy in (3.36). 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 [69] and read27 ) 1 p e 21 ^H^ Plugging in the background (3.23) and (3.25) and setting these variations to zero we obtain a set of differential equations for , with the following solution (see appendix B.2 for details): = jgttj1=4e 32 67 where gtt the time-time component of the 10d metric in (3.23). The quantity R( 1; 2) is an -dependent “rotation operator,” defined as where the functions 1;2 = 1;2( ) are given in (B.29), and is a constant ten-dimensional spinor obeying the projection conditions 1 = 2 = 3 = 4 = 0 : Here we have defined the projectors 1 3 1 2 1 2 1 1 + Using (3.39) one may alternatively express (3.40) as projection conditions on : Bpq $ B^pq. 1 = 2 = b 3 = b 4 = 0 ; 27The conventions of this reference are related to ours by $ 12 ^, Fmnpq $ 21 F^mnpq, Gpqr $ H^pqr, (3.38) (3.39) (3.40) (3.41) where we defined the “rotated projectors,” b 3 R( 1; 2) 3 R 1 Similar rotated projectors have appeared in [70, 71], albeit in a somewhat different class of supergravity solutions with internal fluxes. Note that the 3 projector can be understood as a dielectric deformation of the standard D2-brane projector due to the internal fluxes. The four projection conditions in (3.40) determine that the black hole solution of interest preserves 2 out of the 32 supercharges of the massive IIA supergravity.28 Note this is half the number of supercharges of the AdS4 solution of [53], since the 2 projector is absent in The discussion thus far has been restricted to the so-called universal twist of 3d N = 2 SCFTs and its holographic dual description in terms of a magnetically charged black hole in AdS4. As emphasized, the universal twist is characterized by the fact that the background magnetic flux in the CFT is only along the direction of the 3d superconformal R-symmetry. Most 3d N = 2 SCFTs with holographic duals, however, admit continuous flavor symmetries and it is natural to study the behavior of these theories in the presence of magnetic fluxes for these global symmetries, as well as their holographic duals. In this section we begin exploring this question for a class of 3d N = 2 SCFTs arising from M2-branes placed at the tip of a CY4 conical singularity in M-theory. It is well-known that upon backreaction of the branes this setup leads to a supersymmetric Freund-Rubin vacuum of M-theory of the form AdS4 X7 where X7 is a Sasaki-Einstein manifold which is the base of the CY4 cone. We have already mentioned a few of these spaces above, for example M 1;1;1, Q1;1;1, and V 5;2. Many other examples of such Sasaki-Einstein manifolds with explicit metrics are known in the literature (see for example [72] and references thereof). In addition to the omnipresent Reeb vector, which exists on all such spaces and is dual to the superconformal R-symmetry, these examples of Sasaki-Einstein manifolds typically have additional isometries, which are dual to mesonic flavor symmetries in the dual CFT. In addition, if the seven-dimensional manifold has non-trivial two-cycles (and thus by Poincaré duality five-cycles) the dual CFT has continuous baryonic symmetries. After placing the 3d SCFT on S1 g we can turn on background magnetic fluxes for these mesonic and baryonic flavor symmetries while still preserving supersymmetry.29 The holographic dual description of this setup should be given by a magnetically charged supersymmetric black hole which preserves (at least) two supercharges and is asymptotic to the original AdS4 X7 background. The near horizon limit of this black hole should be given by a warped product of the form AdS2 w M9 where 28As usual here we discuss only the Poincaré type supercharges. For a supersymmetric background with an AdS factor in the metric the number of supercharges is doubled due to the presence of superconformal supercharges. 29From the point of view of the M2-brane picture this corresponds to wrapping the M2-branes on g which then by the general result in [14] automatically implements the topological twist and fibers the CY4 over the Riemann surface. the manifold M9 is a fibration of the Sasaki-Einstein space X7 over g . Our goal in this section is to construct explicit examples of such warped AdS2 supersymmetric vacua of eleven-dimensional supergravity. A few comments on our strategy to attack this technically challenging problem are in order. A classification of warped AdS2 solutions of 11d supergravity was given in [18, 19]. The result of these papers is that the manifold M9 should be given as a U(1) bundle over an eight-dimensional Kähler manifold. The metric on this eight-dimensional space should obey a certain fourth-order non-linear PDE.30 Finding explicit expressions for such metrics is a challenging problem to tackle (see however [20, 74, 75] for some examples) so we employ a different strategy, which closely follows the one in [30] where many explicit AdS3 vacua of type IIB supergravity with (0; 2) supersymmetry were constructed. To be more concrete we propose an Ansatz for the metric and four-form flux, G4 of the 11d theory and impose the vanishing of the gravitino supersymmetry variation as well as the equations of motion and the Bianchi identity for G4. We show that all solutions within our Ansatz are determined by a single function satisfying a nonlinear ODE. This nonlinear equation admits quartic polynomial solutions which lead to a rich class of supergravity solutions. This class contains the near horizon AdS2 region of the universal solution (3.14) as a special case. After this brief introduction let us proceed with the construction of our solutions. The eleven-dimensional space is of the form AdS2 w M9 where M9 is a nine-dimensional manifold given by a seven-dimensional space M7 fibered over a Riemann surface g : g to be compact (i.e. with no punctures) and have a constant-curvature M7 ,! M9 ds2 g = e2h(x1;x2) dx12 + dx22 ; h(x1; x2) = < 12 log 2 8 > > > > : log 1+x21+x22 2 log x2 g = 0 g = 1 : g > 1 (4.1) (4.2) We will take metric,31 given by with The Ansatz for M7 will be modelled on a large class of seven-dimensional Sasaki-Einstein manifolds found and studied in [72, 77], denoted by Y p;q(B), where B, in our case, is a fourdimensional Kähler-Einstein manifold that can be either CP1 CP1 or CP2, upon which the full manifold is constructed. M7 is then spanned by the four coordinates on B and (y; ; ), the latter two being angles. In particular, is the angle associated to the Reeb vector of M7. Thus, we employ an eleven-dimensional metric Ansatz which has explicit AdS2, g 30It is curious to note that such Kähler manifolds have appeared recently in a seemingly unrelated context 31In principle, this assumption could be relaxed but the analysis is more involved. In addition the results of [76] suggest that all the interesting physics in the IR is captured by the constant-curvature metric. We have limited the discussion in this work to the large N limit of the topologically twisted index which, for theories with a holographic dual, is captured by the supergravity approximation of string or M-theory. It is of great interest to go beyond this limit, both in the field theory as well as in the gravitational analysis. On the field theory side it is natural to ask whether some remnant of the universal twist relation in (1.1) survives beyond the large N approximation. There might be reasons to be cautiously optimistic, given that similar universal relations were derived in [30, 31] for the conformal anomaly coefficients of even-dimensional SCFTs related by RG flows across dimensions. On the gravity side the problem is equally challenging. One has to find subleading (in N ) corrections to the entropy of the universal black hole presented in section 3. Perhaps the methods employed in [85] can be useful in finding these corrections.42 It is worth noting that a similar question was addressed successfully in [88] for the subleading corrections to the S3 partition function for three-dimensional N = 2 theories with an AdS4 dual in M-theory. Resolving this question for the class of asymptotically AdS4 black holes studied here is bound to teach us important lessons about holography and the quantum structure of black holes. Acknowledgments We would like to thank Francesco Benini, Davide Cassani, Friðrik Gautason, Adolfo Guarino, Seyed Morteza Hosseini, Dario Martelli, Noppadol Mekareeya, Krzysztof Pilch, James Sparks, and Phil Szepietowski for interesting discussions. FA is partially supported by INFN. The work of NB is supported in part by the starting grant BOF/STG/14/032 from KU Leuven and by an Odysseus grant G0F9516N from the FWO. The work of VSM is supported by a doctoral fellowship from the Fund for Scientific Research — Flanders (FWO) and in part by the ERC grant 616732-HoloQosmos. NB and VSM are also supported by the KU Leuven C1 grant ZKD1118 C16/16/005, by the Belgian Federal Science Policy Office through the Inter-University Attraction Pole P7/37, and by the COST Action MP1210 The String Theory Universe. PMC is supported by Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) via a Vidi grant. The work of PMC is part of the Delta ITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW). AZ is partially supported by the INFN and ERC-STG grant 637844-HBQFTNCER. NB and AZ would like to thank KIAS, Seoul for warm hospitality in the beginning stages of this project. A The large N index for massive IIA theories The large N limit of the topologically twisted index for three-dimensional N = 2 SCFTs with massive type IIA holographic duals has been derived in [6]. Here we present some more details on this construction.43 The Bethe potential V( I ) is obtained by extremizing an auxiliary functional V[ I ; (t); v(t)] with respect to the density (t) and the distribution of eigenvalues u(t) = N 1=3(i t + v(t)) of the matrix model. The functional V[ I ; (t); v(t)] 42Other related work on corrections to black hole entropy beyond the leading order includes [86, 87]. 43Some of these results arise from discussions with S. M. Hosseini and N. Mekareeya. HJEP02(18)54 for a generic Yang-Mills-Chern-Simons theory with bi-fundamentals in the limit N constructed as follows. There is a contribution dt (t) t v(t) + for each Chern-Simons coupling ka and a contribution dt 1 (t)2 iv0(t) ; where g+(u) u3 6 2 u2 + 32 u, for any bi-fundamental field with chemical potential I . These formulae are derived under the assumption that 0 I 2 . Since the Bethe equations involve multivalued functions, one must treat this with care. Setting all ka = k, (A.1) (A.2) (t)2 ! ; (A.3) (A.4) (A.5) (A.6) Gk 2 dt v(t)2)) + X g+( I ) 1 I where G is the number of gauge groups. It is easy to extremize this functional with respect X I2a for each term Wa in the superpotential. The distribution of eigenvalues and the Bethe (t) = 3 4Gkt2 6p3(PI g+( I )) V = i 313=6 20 1 i p 3 ; y(t) = X g+( I ) I t p ; 3 !2=3 (Gk)1=3N 5=3 : = 3pGk X g+( I ) I !2=3 ; We can compare the result with the free energy on S3 derived in [40, 54]. Although the set of rules for constructing V and FS3 seem different, the final result is the same. Indeed we find again relation (2.6) 2i I In order to check this relation, it is important to remember that, although I are parameterizing global symmetries, due to (A.4), the quantities I = behave for all calculational purposes as R-symmetry parameters. To this end it is convenient to introduce trial central charges for the parent 4d quiver using the standard formulae [89] where we take a trace over all fermions and we assign R-charge multiplet and R-charge 1 to the gauginos. In the large N limit, for theories with an AdS dual, we have c = a, and therefore Tr R = G + X 1 N 2 = 0 : (A.8) Using this it follows that, in the large N limit, X g+( I ) = X I 2i = I 6 3 " 3 " 6 X I I 2G 3 (1 V( I I I # + G = 1 1 5 I I 3 3 I 1 ! 1 3 # 6N 2 Tr R3 = (t)2 : I I 1 (t)2 1 iv0(t) (t)2 (t)2 ; (A.13) = 25=331=6 i p 3 Comparing this result with (3.30) in [54] we finally find The index at large N is obtained by combining (6.9) and (6.10) in [6] and reads44 log Z = N 5=3 We want now to prove that g) + X(nI 1 + g)g+0( I ) log Z = i(1 1 " 2 V( I ) X I I dt X I = iN 5=3 Z = iN 5=3 Z dt X I " dt g+0( I ) X g+0( I )+ 44The introduction of g is straightforward — see section 6 of [8]. Proof. First notice that we can consider all the I in (A.11) as independent variables and impose the constraints P I2a I = 2 the differential operator in (A.11) and the topological twist constraint P only after differentiation. This is due to the form of as it is easy to check by an explicit computation. To prove (A.11), we promote the explicit factors of appearing in g+ to a formal variable . Notice that the “on-shell” Bethe potential V, at large N , is a homogeneous function of g+ and therefore of I and , i.e. I2a nI = 2(1 g), I ; ) = 2 V( I ; ) : X nI I I dt X nI g+0( I ) 1 (t)2 iv0(t) : (A.14) Multiplying (A.13) by (1 g) and using (A.14) and (A.10) we see that indeed (A.11) holds. It is always possible to choose a parametrization for the I , subject to the constraint in (A.4), such that the trial central charge a is a homogeneous function of degree three. In this case, (A.11) simplifies to log Z = i X nI I = c^(N Gk)1=3 X nI I ; where we defined c^ in [40], with three adjoints i and superpotential W = i=p3). For example, for the Chern-Simons theory I I = 2 we find U(N ) theory with equal Chern-Simons couplings k based on the conifold quiver with superpotential W = A1B1A2B2 A1B2A2B1 and the constraint P4 The previous derivation is based on the assumption (A.4). In principle, it is possible that there exist other extrema of V in regions where P might contribute to the index. For example, for the quiver in [53], since 0 I2a I = 2 n with n 6= 1, and they I 2 we can have 1 + 2 + 3 = 0; 2 ; 4 ; 6 . It is easy to see that the cases 0 and 6 give singular solutions for (t) and 4 is related to 2 by the redefinition ^ i = 2 I . We have checked in many models that, for the case of the universal twist, where the fluxes are proportional to the exact R-charges, the other solutions are singular or related to P I2a I = 2 by some discrete symmetry. However, we have no general proof of this fact. In general, one must alway check whether there are other saddle points for different values of the sum P For those other saddle points, the index theorem (A.11) does not hold and one needs to I2a I . perform an explicit computation to check whether they contribute or not. 2 X 1 2 1 1 1 ; + gA1 and the standard Einstein metric on CP2 is given by ds24 from section 3 and ranges of the angles used is 0 where we write des4 for the 4d metric, as it will turn out to be related by a rescaling to 2 , 0 . Furthermore, ds2CP2 = d 2 + sin2 4 12 + 22 + cos2 32 ; 1 2 3 cos 3d 1 + sin 1 sin 3d 2 ; sin 3d 1 sin 1 cos 3d 2 ; d 3 + cos 1d 2 ; sin2 2 3 ; J = d ; 1 2 In this appendix, we provide the details of the uplift of the universal solution (3.2) to massive IIA supergravity using the formulae of [64–66]. The metric and dilaton read F^(2) = dA^(1) + mB^(2) ; H^( 3 ) = dB^(2) ; F^(4) = dA^( 3 ) + A^(1) ^ dB^(2) + mB^(2) ^ B^(2) : 1 2 where J is the Kähler form on CP2, normalized such that RCP2 J ^ J = 2 volCP2 = Since we are interested in solutions that are asymptotic to the AdS4 vacuum of the theory with N = 2 supersymmetry we set the scalars to (see equation (7) in [53] or table 3 in [66]): 1 3 e ' = p 2 3 m g 1 3 ; = ~ = 0 ; e = p 1 2 m g 1 3 : We discuss the scalar a below. For the solution of interest, the 4d metric and connection read 2 des4 = m1=3 2p3g7=3 ds42 ; A1 = 1 dx1 3g x2 : B.1 Determining the fluxes Here we show how to obtain the uplifted fluxes (3.25). The massive IIA field strengths F^2; H^3; F^4 are given in terms of their potentials by (see, e.g., equation (A.4) in [64]) (B.1) d ^H(12) + 2m a sin( ) cos( )d ^H(03) + sin( ) cos( ) m 2=3 3 sin3( ) cos( ) m 1=3 sin3( ) cos( ) m 2=3 2gm g 2g2 2 sin2( )+3 cos2( ) sin( ) cos( )d ^H( 3 )1 + sin4( ) m 1=3 g sin2( ) cos2( ) m 2=3 2g 2 sin2( )+3 cos2( ) J sin2( ) cos( ) m 2=3 g sin2( )+2 cos2( ) 2 sin2( )+3 cos2( ) 2 sin3( ) m 2=3 d ^J gm sin2( )+2 cos2( ) sin3( ) cos( ) m 2=3 g2m sin2( )+2 cos2( ) Da^d ^J ^d ^H(02) + sin3( ) cos( ) m 1=3 ^d ^H(12) 4g2 2 sin2( )+3 cos2( ) 4 sin2( )+5 cos2( ) ^d ^J ^d ^H~(2)0 sin( ) cos( ) ^d ^H~(2)1 3g 3g2 H(02) ^J sin2( )J ^H~(2)1 3g2 2g3 sin2( )+2 cos2( ) 2 sin2( )+3 cos2( ) J ^J 2 sin2( )+5 cos2( ) 2g3 sin2( )+2 cos2( ) 2 sin2( ) cos2( ) m 2=3 gm sin2( )+2 cos2( ) + sin4( ) m 1=3 g H(12) ^J 2g2 sin2( )+2 cos2( ) ^H(03) + sin2( ) ^H( 3 )2 +H(04) cos2( )+H(14) sin2( ) ; The uplifted potentials A^(1); B^(2); A^( 3 ) are given in terms of SU( 3 )-invariant fields in equation (2.6) of [65]. Plugging these into (B.5), together with the values for the scalar fields (B.3), gives F^2 = sin( )Da^d 3 sin( ) m 2=3 2 sin2( )+cos2( ) ^d 2g 2 sin2( )+3 cos2( ) 2 H(02) cos( ) sin( )d ^H~(2)0 g H(03) cos( ) where we replaced exterior derivatives of the invariant potentials A0, A1, A~0, A~1, B0, B1, B2, C0, C1 by their field strengths, defined as, H(02) = dA0 +mB0 ; H(2)0 = dA~0 +gB0 ; ~ H(03) = dB0 ; 1 2 H(04) = dC0 +H(02) ^B0 mB0 ^B0 ; H( 3 )1 = dB1 gA0 ^B0 +mA~0 ^B0 +2g(C1 C0) + 1 2 A ^dA~0 +A~0 ^dA0 0 A ^dA~1 1 1 3 H(12) = dA1 ; H(2)1 = dA~1 ~ H( 3 )2 = dB2 ; H(14) = dC1 3 1 A~1 ^dA1 : 2gB2 ; 1 3 H(12) ^B2 ; (B.6) (B.7) (B.8) (B.9) (B.10) (B.11) (B.12) (B.13) The gauge potential A1, and corresponding field strength H(12), are identified with the graviphoton A, and field strength F , of section 3 up to a normalization, provided in (B.4). To fully determine the uplifted fluxes (B.6)–(B.8) we must determine the remaining SU( 3 )invariant forms H(02); H~(2)0; H~(2)1; H(03); H( 3 )1; H( 3 )2; H(04); H(14) in terms of the data of the minimal supergravity. A naive guess would be to set these to zero to obtain the minimal theory, but this is inconsistent with the duality relations of [66]. To determine the consistent result we note that setting Da da + gA0 mA~0 = 0 is consistent with the equations of motion of the Lagrangian (3.7) in [66]. Taking an exterior derivative of this equation implies g dA0 = m dA~0 and from the definitions (B.9) and (B.10), HJEP02(18)54 Then, using (B.14) and (B.15) in the duality relations (3.17), (3.18), (3.19) in [66], we obtain H~(2)0 = 1 2 g 1=3 m H(03) = H( 3 )1 = H( 3 )2 = 0 ; 3 4 H(12)) ; H~(2)1 = 3 2 m 1=3 H(12) + p 3 4 H(12) ; H(04) = H(14) = p3g3 vol4 m 1=3 : (B.14) (B.15) (B.16) We see from here that it would have been inconsistent to set the fields H(02), H~(2)0, H~(2)1, H(04), H(14) to zero since they are related to vol4 and H(12) by the duality relations. Finally, plugging (B.15) and (B.16) into the massive IIA field strengths (B.6) and (B.7) and (B.8) we obtain the result presented in (3.25). To minimize the possibility of mistakes introduced during this unwieldy uplifting procedure we have checked explicitly that the uplifted massive IIA black hole background is a solution of the ten-dimensional equations of motion. We now proceed to show that it preserves two supercharges. B.2 Proving supersymmetry Here we give the details establishing supersymmetry of the black hole solution (3.23), (3.25) in massive IIA supergravity. It is useful to define the following basis of vielbeins dt ; e2 = e L 1 2 1 d ; dx2 x2 ; 3 ; (B.17) dx1 x2 1 2 ; e1 = e L e3 = e L e5 = ! sin( ) 1 e6 = ! sin( ) 1 e7 = ! sin( ) 1 e8 = ! sin( ) 1 e9 = ! e 21 (' 2 )X 1=2d ; e10 = ! sin( )X 1=2 e4 = e L 1=2d ; 1=2 sin 1=2 sin 2 2 1=2 sin cos 1 ; 2 ; 2 2 1=2 ; where e 2 (cos(2 ) + 3)1=2(cos(2 ) + 5)1=8 ; 2 5=8g 25=12m1=12 ; 21=231=4g1=6 L 2 ! 1 3 e L !0 : In our conventions 11 = The general supersymmetry variations of massive IIA are written in (3.37). Using the basis (B.17) and plugging in the massive IIA forms (3.25) we obtain a series of differential equations for . Assuming that the spinor is independent of the coordinates ; 3 leads to a series of algebraic equations along these directions. It is useful to take linear combinations of (3.37), such as 1 1 2 2 = N0 2 2 3 2 3 3 4 4 = 1 + 2 2 + 2 1 ; (B.19) 21 . We continue with 2N0 3 3 2 where N 1 0 211=16 p4cos(2 ) + 3 1p6cos(2 ) + 5 2q4 gm25 . The above equation is solved by taking the spinor be proportional to q which is solved by the -dependence Using this we find / e 23 34 : 5 5 8 8 / 678(3 cos(2 )) + 2 5 cos(2 ) + 6 348 sin2( ) : This is solved by imposing to be in the kernel of the following two projectors 1 2 1 1 Using these projectors to replace 34 and 58 by 67 we can write 2 2 5 5 = N0 1 + 2 3 1267 9(22 sin(2 ) + sin(4 ) 96 cot( )) p2(16 cos(2 ) + cos(4 ) + 31) + 4 1210 sin( ) (cos(2 ) + 3)pcos(2 ) + 5 p 3 6710(cos(2 ) + 3) csc( ) 2pcos(2 ) + 5 ! : This equation can be interpreted as imposing a further projection condition (one can check that the combination on the right hand side of (B.24) indeed squares to itself as a projector should). Now we can use the results so far to replace the 34; 58; 9 gamma matrices. (B.18) (B.20) (B.21) (B.22) (B.23) (B.24) Doing so, we see that all variations vanish, except for 1;2;3;4;9. The following combination is useful 2(cos(2 )+5)+2 3 167 1 1 + 3 3 = N0 4p3 6710 sin( )pcos(2 )+5 +12 167 +2 3 2(cos(2 )+5) 6(cos(2 )+3) : The two equations (B.24) and (B.25) can be solved by imposing to be in the kernel of the following two projectors: 2 2 3 cos2( ) 2 ! cos(2 ) + 3 ! ! ; where 3 ; 4 are the standard projectors and R( 1; 2) is a rotation operator defined by 3 4 1 2 1 2 1 + 1 + 134 ; 67910 ; with the “angles” 1 = 1( ); 2 = 2( ) defined as the following functions of b 3 b 4 1 2 1 2 cos( 1=2) cos( 2=2) p 1 1 b 3 = b 4 = 1 2 1 2 (B.25) (B.26) (B.27) (B.28) ; (B.29) (B.30) The reason for the notation with the “hat” in these two projectors will be made clear below. Together with (B.23), we have now imposed four projectors, leaving 32 2 4 = 2 supercharges. We note the projectors (B.26) can be rewritten, inspired by the discussion in [70, 71], in a more insightful way as 1 + cos( 2) 134 + sin( 2) cos( 1) 2 + sin( 1) 9 = R( 1; 2) 3R 1 ( 1; 2) ; = R( 1; 2) 4R 1 s pcos(2 )+5 p 6 cos( ) 2pcos(2 )+5 cos( ) pcos(2 )+3 ; ; sin( 1=2) sin( 2=2) s p 6 cos( )+pcos(2 )+5 2pcos(2 )+5 1 p 2 s cos(2 )+5 : The spinor can thus be written as = R( 1; 2)~; (B.31) where ~ is a spinor in the kernel of the unrotated projectors 1 ; 2 ; 3 ; 4. The only equation remaining at this point is 9 = 0, which can be solved by taking ~ to be proportional to the function e =2 defined in (B.18). We thus find that our massive type IIA black hole solution is indeed supersymmetric, with the following explicit Killing spinor = r 1 2 e =2e 32 67 (B.32) where is a constant spinor in the kernel of the four projectors there are a total of 2 supercharges preserved. Note that the factor q 1 ; 2 ; 3 ; 4 and hence 21 e =2 is precisely proportional to jgttj1=4, where gtt is the time-time component of the ten-dimensional metric Details on the construction of the 11D solutions In this appendix we give the details of the derivation of the family of solutions of eleven dimensional supergravity described in section 4. Before starting, it is convenient to choose (C.1) (C.2) e2 = f2ehdx1 ; e3 = f2ehdx2 ; e0 = f1 dt ; z e4 = f3dy ; e6 = f5E1 ; e10 = f7D ; e1 = f1 dz ; z e5 = f4D ; e7 = f5E2 ; e8 = f5E3 ; e9 = f5E4 ; where (t; z) are the Poincaré coordinates for AdS2 and E1;2;3;4 is a vierbein on the Kähler manifold B such that We can thus rewrite the 4-form as JB = E1 ^ E2 + E3 ^ E4 : G4 = e0 ^ e1 ^ g1 e2 ^ e3 + g2 e4 ^ e5 + g3 e5 ^ e10 + g4 e4 ^ e10 + g5 e6 ^ e7 + e8 ^ e + e6 ^ e7 + e8 ^ e BPS equations Imposing the vanishing of the gravitino variations in eleven-dimensional supergravity gives the following Killing spinor equation: = r " + G4 288 8 " = 0 : (C.3) Motivated by the symmetry of the problem and the underlying M2-brane interpretation we impose the following projection conditions on ": 23" = 45" = 67" = 89" = 1 10" = i" : Equation (C.3) leads to the system of differential equations:45 1 f7 (c~ f8c) 4 f 2 2 1 f7f80 4 f3f4 1 f7 (a 4 f 2 5 f 0 2 f2f3 f 0 5 f5f3 3 f1 f 0 1 2f1f3 6 + g4 = 0 ; 21 cff224 = 0 ; f4b 2f52 = 0 ; 1 f 0 2 f7f3 7 + g4 = 0 ; 6 + g1 + g2 + 2g5 = 0 ; 2g1 + g2 + 2g5 = 0 ; 12 12 12 some constraints on the in and ln coefficients, and a set of PDEs for ": i1 = 2i2 ; l1 = i2 ; l2 = i2 ; i5 = l4 = 0 ; 4 2 f4 f3f4 f 0 2 2cff242 + f4b f 2 5 " = 0 ; i rB i i 3 4 f7 +2 f7 (a f8b) f7f80 + g1 +g2 +2g5 " = 0 ; f 2 5 f3f4 45Here it is convenient to anticipate one of results from the Bianchi identity for G4, namely g3; i4; l3 = 0. (C.4) (C.5) (C.6) (C.7) (C.8) (C.9) (C.10) (C.11) (C.12) (C.13) (C.14) (C.15) (C.16) (C.17) (C.18) (C.19) (C.20) (C.21) where riB denotes the covariant derivative on B and i = 1; 2; 3; 4 spans its four real Let us now solve for the dependence of " on the various coordinates. From (C.14) we deduce that it is constant in time. We also assume that @x1" = @x2" = 0 and that " does not depend on the coordinates of B, checking a posteriori the consistency of such assumption. The coefficients f1 and f7 are related by (C.5) and (C.8), which can be combined to obtain and thus with !1 an integration constant. f 0 f1 1 = f 0 f7 7 ; f7 = 2!1f1 ; 1 2 Then, adding (C.10), (C.11) and (C.12) and using (C.9) and (C.22), (C.21) leads to Then, from equations (C.18) and (C.19) we find Using the constant curvature metric (4.2), the combination on the r.h.s. of this equation is constant and proportional to , the normalized curvature of the Riemann surface g , Thus, we find the dependence of " on is given by with !2 a constant satisfying 8 >1 > = < 0 for g = 0 for g = 1 : > >: 1 for g > 1 c!2 + c~!1 = 2 : f 0 2f1 1 " = d p dy f1 " ; " / p : 1 z " = ei(!2 +!1 )r f1 z "0 ; (C.22) (C.23) (C.24) (C.25) (C.26) (C.27) (C.28) HJEP02(18)54 while (C.15) the one on z: with "0 a constant spinor. One can think of this algebraic constraint as the supergravity implementation of the topological twist condition in the dual 3d N = 2 SCFT. Equations (C.16) and (C.5) allow us to determine also the dependence on y: Combining this result with (C.24), (C.25) and (C.27) we see that can be rewritten by (C.29) as since other combinations, such as !B12 + !34 = AB (a!1 + b!2) ; B coordinates on B, for both CP1 metric, we have that Let us now turn to condition (C.20). Referring to (C.1) and (C.2) for the indices of the CP1 and CP2, equipped with the standard Fubini-Study where !B is the spin connection in B. Therefore equation (C.20): !B13 = !B24 = !B23 = !41 ; B 4 B 1 !jk (j+5)(k+5)" = iAB (a!1 + b!2) " !31 86 + !24 79 " = !B31 + !24 B B B 78" = 0 ; vanish identically. On the other hand, the fact that B is Einstein, implies that its Ricci 2-form is proportional to its Kähler form by a constant 2q: 1 2 1 2 (C.29) (C.30) HJEP02(18)54 (C.31) (C.32) (C.33) (C.34) d!Bi j + !Bi k ^ !Bk j JB ij = 2qJB : We notice now that when one computes the only non-vanishing contributions to (C.31) from the Riemann 2-form, namely the components (1; 2) and (3; 4), the quadratic part vanishes because of (C.29). For instance one finds !B 3 ^ !B32 + !B1 4 ^ !B42 = 0 : 1 b!2 + a!1 = q : Therefore, taking the differential in (C.30), we can replace the first member with the second of (C.31) and, by (4.5), arrive at the following constraint: From equations (C.9), (C.10), (C.11), (C.12) and (C.5), we can express the functions gi in terms of the metric funtions: and find the constraint f7 (c~ f8c) f 2 2 1 f1 g1 = g4 = g5 = g2 = f7f80 + f3f4 1 f1 3 1 ; f1f3 f7 (a + 2 f7 (a f7f80 = f3f4 1 f1 which then can be solved by using (C.23) to eliminate f7. This results in the relation, f22f54f12 f8b) f22f12f52 (C.35) We note that f3 is in fact redundant, as it can be set to an arbitrary function by redefinitions of the coordinate y. A convenient choice is such that f1(y)f3(y)f4(y) = K ; (C.36) HJEP02(18)54 with K an arbitrary constant; we consider this as an equation determining f4. Taking also (C.23) into account, we are left with f1;2;3;5;8 as five independent functions. It is convenient to trade these for F1;2;3;5;6, defined by: F1 f22f54; F2 f1f22; F3 f1f52; F5 K f12f22f54 ; F6 f 2 3 f8f13f22f54 : (C.37) In terms of this new basis, equations (C.6) and (C.7) simplify to and thus with S2 and S3 integration constants. Furthermore, the complicated equation (C.35) becomes F20 = cK; F30 = bK ; F2 = cK y + S2; F3 = bK y + S3 ; F 0 K F1 = 2!1 6 + c~F32 + 2aF2F3 : F6 = F 0 4!2F2F32 : 5 4!1 Finally, F6 can be found through (C.17), which, after using (C.5), (C.25), (C.24) and (C.36), reads Thus, at this point the solution is completely controlled by the single function F5, which is determined by the Bianchi identity for G4, as we discuss next. C.2 4-form equations The equation of motion for the 4-form, (C.38) (C.39) (C.40) (C.41) d 11 G4 = 0 ; implies the following relations: g4i2f12f7f3f54 + i2f7f12f54 0 +bi2f3f4f7f12f52 = 0 ; i2f7f12f52f22 0 ci2f3f4f7f12f52 +2bi2f3f4f22f7f12 +g4i2f12f7f3f22f52 = 0 ; i2f80 f7f12f52 +i2f3f4f7f12f52 (a bf8) (g2i2 g5i2) f12f3f4f54 = 0 ; 2i2f3f4f22f7f12 (a bf8)+i2f80 f7f12f52f22 i2f3f4f7f12f52 (c~ cf8) g2f22f54f7 0 cg1f3f4f52f62f7 2bg5f3f4f22f52f7 = 0 ; f8g2f22f54f7 0 +2ag5f3f4f52f22f7 g4f22f4f54 0 6f22f3f4f54i22 = 0 : defining we arrive at which is solved by for some constant I0. I i 2 f 2 1 It turns out that after imposing the BPS equations studied above, all these equations are satisfied automatically. The first two are actually equivalent to the conditions (C.51) and (C.52) implied by the Bianchi identity that we examine later, after inserting (C.5) and (C.23). Equations (C.44), (C.45), (C.46) are identically satisfied when we plug (C.33) and (C.13) in. The same happens to (C.47) and (C.48) using (C.36) and (C.38). dG4 = 0 ; provides a wealth of information: it implies, as mentioned before, that g3 = i4 = l3 = 0 and the four additional equations bf22f3f4i1 cf52f3f4l1 + f2 f5 i2 0 f2 f3f7i5 (a bf8) f5 f3f7l4 (c~ cf8) = 0 ; 2 2 2 2 bf12f3f4g2 f12f3f7g4 (a bf8)+ f1 f5 g5 0 = 0 ; 2 2 2 2 2 f1 f2 g1 0 c f1 f3f4g2 2 f1 f3f7g4 (c~ cf8) = 0 ; bf52f3f4l1 + f5 l2 0 f5 f3f7l4 (a bf8) = 0 : 4 2 While (C.50) is satisfied automatically, the value of i2, the last unknown coefficient appearing in G4 we are left with, is determined by (C.51) and (C.52). Using (C.38) and (C.42) (C.43) (C.44) (C.45) (C.46) (C.47) (C.48) (C.49) (C.50) (C.51) (C.52) (C.53) (C.54) HJEP02(18)54 Thus, the only remaining equation is (C.49), which is a fourth-order nonlinear ODE for F5. Remarkably, this equation can be integrated twice into the second-order ODE:46 1 KF2F32 F502 + 2F5 F 00 2K 4qF2F3 + F 2 5 3 + P3 = 0 ; (C.55) where (y) and P3(y) satisfy the simple differential equations: 00 = 24KF2 FI04 ; 3 2 P300 = 32Kq ( F3 + qF2) : Using (C.39) and for b 6= 0 one may write (C.55) as (4.11). the most general polynomial solution is at most quartic, i.e., Although we have not found the most general solution to (C.55), one can show that n yn ; 4 n=0 (C.56) where n are real constants. Plugging this into (C.55) leads to a set of algebraic equations for the coefficients n and the other parameters specifying the solution: 6 42 3 4bK3(b +4cq)+4b2cK6q(b +cq) = 0 ; +2K2 2bK3q b2 S2 +5bc S3 +4bcqS2 +2c2qS3 12 K b K3qS32S2 +32bK3q2S3S22 +16c K3qS33 + 64 cK3q2S32S2 = 0 ; 3 4 K 0b2 bK3(b +4cq) 6 4 2 K 1b2 4K2(b S3 +2bqS2 +2cqS3) 3 3 +b2S3(KS3(cp1 +16qS2( S3 +qS2)) 4 2( S3 +4qS2)) 3 4(b S3 +2bqS2 +2cqS3) +16bcK3qS3(7 S3 +8qS2) = 0 ; +b4K2p0S2 +2b3KS3(cKp0 +p1S2) 12c2KI02 = 0 ; 4 0b3( 2 KS3( S3 +4qS2))+S2 b3Kp0S32 4bS2I02 8cS3I02 2b3 = 0 ; 1 K 4 1b3S3( S3 +4qS2)+2b4Kp0S3S2 +b3S32(cKp0 +p1S2) 16bcS2I02 8c2S3I02 4 0b3 2K2(b S3 +2bqS2 +2cqS3) 3 3 = 0 : (C.57) 46The same technical simplification happens in the analysis of supersymmetric AdS3 solution of type IIB supergravity in [30]. All the solutions discussed in this paper correspond to different solutions to this set of algebraic constraints. We note that assuming b 6= 0 the sixth equation above becomes cKI02(bS2 S3b 3 after solving the others. When instead b = 0, one of the equations is not independent of the others. In both cases, the system leaves (at least) one free parameter. C.3 Singularity analysis The metric (4.7) could in principle have conical singularities when y = y~ such that F5(y~) = 0. In this section, we analyze what happens in these points and show that they are regular. Let us consider the following linear combination of the angles and : = W 0 ! W = w1 w2 w3 w4 W invertible : Under this transformation the metric retains its structure, i.e., upon some redefinitions of the functions Fn, we can rewrite it in the same form, with new parameters given by c c~ = W c0 ! b a = W b0 ! 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Francesco Azzurli, Nikolay Bobev, P. Marcos Crichigno, Vincent S. Min, Alberto Zaffaroni. A universal counting of black hole microstates in AdS4, Journal of High Energy Physics, 2018, 54, DOI: 10.1007/JHEP02(2018)054