Romans supergravity from five-dimensional holograms

Journal of High Energy Physics, May 2018

Abstract We study five-dimensional superconformal field theories and their holographic dual, matter-coupled Romans supergravity. On the one hand, some recently derived formulae allow us to extract the central charges from deformations of the supersymmetric five-sphere partition function, whose large N expansion can be computed using matrix model techniques. On the other hand, the conformal and flavor central charges can be extracted from the six-dimensional supergravity action, by carefully analyzing its embedding into type I’ string theory. The results match on the two sides of the holographic duality. Our results also provide analytic evidence for the symmetry enhancement in five-dimensional superconformal field theories.

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Romans supergravity from five-dimensional holograms

HJE Romans supergravity from ve-dimensional holograms Chi-Ming Chang 0 1 4 Martin Fluder 0 1 2 Ying-Hsuan Lin 0 1 2 Yifan Wang 0 1 3 One Shields Avenue 0 1 Davis 0 1 U.S.A. 0 1 Supersymmetric Gauge Theory 0 Jadwin Hall Washington Road, Princeton, NJ 08544-0708 , U.S.A 1 1200 E California Blvd , Pasadena, CA 91125 , U.S.A 2 Walter Burke Institute for Theoretical Physics, California Institute of Technology , USA 3 Joseph Henry Laboratories, Department of Physics, Princeton University 4 Center for Quantum Mathematics and Physics (QMAP), University of California Davis , USA We study ve-dimensional superconformal eld theories and their holographic dual, matter-coupled Romans supergravity. On the one hand, some recently derived formulae allow us to extract the central charges from deformations of the supersymmetric vesphere partition function, whose large N expansion can be computed using matrix model techniques. On the other hand, the conformal and avor central charges can be extracted from the six-dimensional supergravity action, by carefully analyzing its embedding into type I' string theory. The results match on the two sides of the holographic duality. Our results also provide analytic evidence for the symmetry enhancement in ve-dimensional superconformal eld theories. AdS-CFT Correspondence; Conformal Field Theory; Supergravity Models 1 Introduction 2 Review of key ingredients Supersymmetric ve-sphere partition function Central charges from deformations of ve-sphere partition function 2.2.1 2.2.2 Conformal central charge from metric deformations Flavor central charge from mass deformations Chern-Simons-like counter-terms Seiberg theories 3 The large N limit of Seiberg theories Free energy and conformal central charge Flavor central charges 3.3 Finite N numerics 4 Central charges from supergravity 2.1 2.2 2.3 2.4 3.1 3.2 4.1 4.2 4.3 4.4 5.1 5.2 5.3 6.1 6.2 3.2.1 3.2.2 3.2.3 4.4.1 4.4.2 4.4.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 Fundamental hypermultiplet symmetry Instantonic symmetry Mesonic symmetry Central charges and couplings Massive IIA supergravity solutions Conformal central charge from supergravity Flavor central charges from supergravity Mesonic symmetry Fundamental hypermultiplet symmetry Instantonic symmetry 5 Orbifold theories and their large N central charges Orbifold theories Free energy and conformal central charge Flavor central charges Fundamental hypermultiplet symmetries Bifundamental hypermultiplet symmetries Antisymmetric hypermultiplet symmetries Mesonic and baryonic symmetries Instantonic symmetries { i { 6 7 Conclusions Central charges from the supergravity dual of orbifold theories Central charges by comparison to Seiberg theories Baryonic avor central charges Numerical evaluation of central charges 1 Introduction 41 42 HJEP05(218)39 Since the discovery of AdS/CFT, the holographic equivalence between quantum gravity in anti-de Sitter space and conformal eld theory has elucidated key aspects on both sides of the duality [1]. Typically, strong coupling properties on one side are reincarnated with a simple weak coupling description on the other side, where physical observables can be computed in perturbation theory. However, there are also holographic dualities that do not have genuine weak coupling corners in their parameter spaces. Such is the case for interacting superconformal eld theories in ve dimensions, which do not admit marginal deformations, and have no known weak coupling limit [2]. On the gravity side, the dilaton coupling of type I' string theory diverges towards the boundary of the internal space [3]. Therefore, quantitative predictions of these holographic dualities are more di cult to come about. First steps towards closing this gap have been taken in [4{15]. On the gravity side, it has been realized that in an appropriate low energy regime, the type I' string theory admits an e ective supergravity description, which upon reduction to six dimensions becomes Romans F (4) supergravity [16, 17] plus additional matter [18, 19].1 It is expected that this much simpler six-dimensional e ective description captures key aspects of ve-dimensional superconformal eld theories. On the eld theory side, following the seminal work of [20], new advances in supersymmetric localization have provided extremely powerful tools for extracting supersymmetric observables even in these strongly interacting theories. Careful analyses of the localized formulae for these observables in the large N limit have provided checks of these holographic correspondences beyond kinematics [7, 9, 11{15]. The present paper aims at a more thorough study of an enlarged set of observables in ve-dimensional superconformal eld theories, and their manifestations in the bulk supergravity. The grander goal is to gain a clearer understanding of the strong coupling phenomena on both sides of the holographic correspondence, and to elucidate the general relationships connecting the rich landscape [21{24] of ve-dimensional superconformal eld theories, such as possible F - or C-theorems concerning renormalization group ows [25{27] (see also [28] and further references therein). Many of these theories exhibit avor symmetry enhancement, where extra conserved currents carrying instanton charges emerge when reaching the ultraviolet xed point, and enhance the avor symmetry [21{23]. In the bulk, this enhancement is a non-perturbative phenomenon that involves D0-branes 1Romans F (4) (gauged) supergravity contains only the elds dual to the ve-dimensional stress tensor dilaton diverges [3]. Thus, a deep understanding of avor symmetry enhancement may shed light on the non-perturbative dynamics in string theory. The physical observables that we shall pursue are the supersymmetric (squashed) vesphere partition function, the conformal central charge, and the various avor central charges. These charges appear in the superconformal block decomposition of the BPS fourpoint functions, and serve as inputs to the bootstrap analyses of these superconformal eld theories [29]. The avor central charges also serve as indicators of symmetry enhancement, including the aforementioned one. They also signify special features of the particular holographically dual pair. For instance, we shall see that the avor central charges for the mesonic U(1)M symmetry and the SU(2)R R-symmetry have a simple relation, which is a priori obscure from the eld theory description, but clear in the dual supergravity [6]. In previous work [29], the present authors used conformal perturbation theory to establish a precise relation between the central charges and the squashed ve-sphere partition function with mass deformations, and computed the latter via supersymmetric localization. The resulting central charges for the rank-one Seiberg theories led to a bootstrap analysis that revealed information about the non-BPS spectra in these theories. We further found that the quantities we computed have surprisingly small instanton corrections, and the comparison of avor central charges provided strong numerical evidence for avor symmetry enhancement. The present paper concerns the large N regime, in the hope to understand the holographic duality for ve-dimensional superconformal eld theories, and to what extent Romans F (4) supergravity (plus additional matter) captures the hologram. In addition to Seiberg theories, we extend our analysis to a larger class of orbifold theories proposed by [6]. Let us rst review prior progress and results in this direction. By localization, the perturbative (squashed) ve-sphere partition function can be reduced to an N -dimensional Coulomb branch integral [30{35], which has been computed analytically in the large N limit using matrix model techniques [7, 11]. It has further been argued that in this limit, the instanton contributions are exponentially suppressed, and thus, the perturbative results are exact [7]. The matrix model of the squashed ve-sphere partition function was studied in [11], and the result was matched holographically with the properly renormalized on-shell action of Romans F (4) supergravity in the bulk. The progress we make in the present paper is as follows. 1. Borrowing the results on the matrix model for the large N squashed ve-sphere partition function from [11], we compute the conformal central charge of the Seiberg theories in the large N limit. 2. We further study the matrix model for the mass-deformed ve-sphere partition function. In particular, we nd that the round sphere free energy, the conformal central charge, and the mesonic and baryonic avor central charges scale to leading order as N 5=2. Similarly, the hypermultiplet and instantonic avor central charges are found to scale as N 3=2. The coe cients of the latter two exactly agree in the large N limit, providing analytic evidence for avor symmetry enhancement in the Seiberg theories. { 2 { conformal symmetry. at large N . 3. A subtlety in the above analysis is the potential presence of Chern-Simons-like counter-terms. We explicitly determine a scheme under which the one-point function of the instanton number current vanishes in the ultraviolet, as is required by 4. Except the instantonic avor central charge, all the other central charges in the eld theory exactly agree with the couplings in Romans F (4) supergravity (coupled to additional vectors) obtained by a careful reduction from type I' supergravity.2 This match provides further evidence of the suppression of instanton contributions 5. Finally, we generalize the above considerations to a larger class of orbifold theories [6]. We match not only the conformal, hypermultiplet, and mesonic avor central charges, but also a set of baryonic avor central charges (up to an overall constant), on the two sides of the holographic duality. The rest of this paper is organized as follows. In section 2, we review our previous work [29], and highlight the main ingredients that are relevant to the present analysis. In section 3, we employ supersymmetric localization and matrix model techniques to the Seiberg theories, to compute the free energy and central charges in the large N limit. The relevant properties of the triple sine function and some nite N numerics are provided in appendices A and B. In section 4, we look at the dual type I' string theory, examine its reduction to matter-coupled Romans F (4) supergravity with additional vectors, and thereby compute the free energy and central charges. In section 5, a similar analysis is performed for a more general class of orbifold theories. section 7 ends with concluding remarks and future directions. 2 Review of key ingredients This section presents a review of the key ingredients for computing the central charges in interacting ve-dimensional superconformal eld theories. First, we present the general de nition of the (squashed) supersymmetric ve-sphere partition function, the key localization results, and its relation to the conformal central charge and avor central charges. We then discuss the admissible counter-terms that can appear (on a ve-sphere background). Finally, we introduce the theories of primary interest | the Seiberg theories. For the derivation and in-depth discussions, we refer the reader to an earlier paper by the present authors [29]. 2.1 Supersymmetric ve-sphere partition function A large class of ve-dimensional superconformal eld theories has an infrared gauge theory phase. This infrared Lagrangian description allows localization computations, the strong coupling limit of which recovers quantities at the ultraviolet xed point. This section 2We do not calculate the instantonic avor central charge from supergravity in this paper due to the subtleties explained in section 4.4.3, the resolution of which is left to future work. { 3 { HJEP05(218)39 is devoted to a review of the localization results of the ve-sphere partition function for general gauge theories. The present discussion largely omits instanton contributions, which are generally crucial for the consistency (e.g., symmetry enhancement) of the partition function. Nonetheless, as we shall argue in section 3 (see also [7]), in the large N limit which could be compared to the corresponding (weakly coupled) supergravity dual, the instanton contributions are exponentially suppressed. Consequently, we may simply deal with the perturbative part. The perturbative part of the squashed ve-sphere partition function (i.e. without instantons) has been computed in [31, 32, 35]. More precisely, the localization formula was derived for squashed supersymmetric backgrounds that retain U(1) SU(3) SO(6) isom etry, and then conjectured for the most generic squashing with U(1) U(1) U(1) isometry. The metric of a generically squashed (unit) ve-sphere is given by 3 i=1 ds2 = X(dyi2 + yi2d i2) + 2 X aj yj2d j A ; j=1 e coordinates, and yi are constrained such that P3 j=1 yj2 = 1. The round sphere (in terms of polar coordinates) is given by setting aj = 0. Thus, the latter part of the metric can be viewed as a perturbation of order O(aj2) to the round ve-sphere metric. For a general ve-dimensional gauge theory with a simple rank-N gauge group Gg and Nf hypermultiplets in the real representation Rf Rf of Gg, f = 1; : : : ; Nf , the perturbative squashed ve-sphere partition function is given by3 Zpert = where the products are taken over all the roots of Gg, the avors f = 1; : : : ; Nf that run over the hypermultiplets on which the avor symmetry group Gf acts, and the weights f of the particular representations Rf Rf . The (Roman font) subscripts g and f denote gauge and avor, respectively. We introduced the classical ( at space) prepotential F( ), which is given by F( ) = Tr 2 + 1 2gY2M k 24 2 Tr 3 ; with gYM the classical gauge coupling, k the Chern-Simons coupling, and Tr ( ) is the Killing form de ned as (h_ is the dual Coxeter number) Finally, S3 is the triple sine function, and 1 2h_ Tr ( ) tr adj( ) : S30(0 j !~) = lim x!0 S3(x j !~) x : 3We use to collectively denote the Cartan generators. In the case of Gg = USp(2N ), in the de ning representation is given by the diagonal matrix f 1; : : : ; N ; 1; : : : ; N g. { 4 { We refer the reader to appendix A for a de nition of S3, and some of its properties relevant for the present paper. There is another type of deformations. For a theory of given avor symmetry group Gf , we can introduce mass parameters into the partition function by coupling the hypermultiplets to background vector multiplets. For an Hermitian mass matrix M 2 gf Lie(Gf ), the mass term is explicitly given by Z d5xpg ij qiM qj + 2itij qiMqj 2 2 M ; (2.6) HJEP05(218)39 where qi and qi are the scalars, and the fermion, in the hypermultiplet.4 These masses arise from turning on vacuum expectation values for the scalars in the background vector multiplets, akin to Coulomb branch masses from turning on scalars in dynamical vector multiplets. Hence, these generic mass terms appear in the partition function (2.2) in the same way as the Coulomb branch parameters. We defer the presentation of the explicit formula for Zpert for Seiberg theories with a particular choice of mass deformations to section 2.4. 2.2 Central charges from deformations of ve-sphere partition function In [29], the present authors derived formulae for ve-dimensional superconformal eld theories that relate the conformal central charge and avor central charges to deformations of the ve-sphere partition function. The proof proceeds by coupling the ve-sphere background to the appropriate background supergravity multiplets. Here, we explain the rationale and present the resulting formulae. 2.2.1 Conformal central charge from metric deformations In order to extract the conformal central charge CT from a partition function, we study the superconformal eld theory on a ve-sphere background perturbed by coupling the stress tensor multiplet to a background supergravity multiplet on a generically squashed vesphere with metric given in (2.1). In order to preserve the full superconformal symmetry, we couple the theory to the ve-dimensional N = 1 standard Weyl multiplet g ; D; V ij ; v ; i ; i ; (2.7) which consists of the dilaton D, the metric g , an SU(2)R symmetry gauge eld V ij , a two-form eld v , together with their fermionic partners | the gravitino dilatino i. We deform the ve-sphere background by writing g = gS5 + h . Upon doing i and the so, the (bosonic) linearized action is given by 4Although the physical mass matrix M in (2.6) is Hermitian, when performing localization, one has to analytically continue M to an anti-Hermitian matrix for convergence [30, 31]. { 5 { where T , Jij , B for the given metric. and are the (bosonic) components of the stress tensor multiplet.5 The linearized background elds of the standard Weyl multiplet for the generic squashed ve-sphere can be explicitly determined by solving the relevant supersymmetry conditions From the linearized perturbation (2.8), it follows that the ve-sphere free energy at second order in the squashing parameters aj can be written in terms of the two-point functions of the currents B , J ij and on the round sphere. These two-point functions are related to the conformal central charge upon stereographic projection from at space. By evaluating them explicitly, and expanding the background values for the bosonic elds in the standard Weyl multiplet to leading order in the squashing parameters, we nd that the free energy is in fact related to the conformal central charge CT as F a2 = i 2CT Flavor central charge from mass deformations The same overall logic applies to extracting the avor symmetry group Gf , by considering mass deformations of the ve-sphere free energy. Working in conformal perturbation theory, we couple the avor current multiplet in the ve-dimensional superconformal eld theory to a background vector multiplet avor central charge CGf for a given J W a; M a; a i; Y a ij ; which contains the vector eld W a, the scalar M a, the gaugino a i and the triplet of auxiliary elds Y a ij , all in the adjoint representation of Gf . The linearized perturbation of the bosonic action is then given by5 where Liaj , J a , and N a are the bosonic components of the avor current multiplet. On the ve-sphere, the vector multiplet has to satisfy supersymmetry conditions, which can be solved by W = i = 0 and Y ij = M tij , for tij = 2i 3 and some M 2 gf Lie(Gf ). Thus, the leading order piece in the mass deformation of the ve-sphere free energy can be expressed in terms of the two-point functions of Liaj and N a on the round sphere. Those two-point functions are related to the corresponding avor central charge CJGf in at space by stereographic projection, and we end up with F M2 = 3 2CGf J 256 abM aM b : It remains to translate the mass parameters M a appearing in conformal perturbation theory to the mass matrix appearing in the (infrared) gauge theory Lagrangian. We refer 5Notice that in the \rigid" limit, all fermionic elds of the background supergravity multiplets are set to zero. { 6 { the reader to [29] for a in-depth treatment of this. Here, we shall simply state the result, F M where RKf is the representation of the hypermultiplets under Kf | the manifest avor symmetry group of the infrared Lagrangian, IRKf is the Dynkin index associated with the represenation RKf , and M is the mass matrix in the action deformation (2.6). The global symmetry can be enhanced at the ultraviolet xed point, which is in fact a common phenomenon in ve dimensions. In that case, if G is simple, one can obtain CGf from CJKf , by use of the embedding index Ik,!g, i.e., J a physically relevant quantity, one must remove such divergences by introducing local di eomorphism-invariant counter-terms, which would generically break conformal invariance. For the case of ve-dimensional N = 1 superconformal eld theories, such counterterms would break the superconformal group to su(4j1), but renders the nite part of the ve-sphere partition function unambiguous and physical. More explicitly, the counter-terms are supersymmetric completions of (mixed) ChernSimons terms in Poincare supergravity, and have been classi ed in [29] by the present authors. The three possible counter-terms involving a single scalar M in the background vector multiplet (2.10) are given by (2.13) (2.14) (2.15) (2.16) (1) LT T J 3 i 1T T J (2) LT T J 3 i 2T T J (3) LT T J 3 i 3T T J hpgM R2 + i ; hpgM hpgM C R R C the theory restricts the constants jT T J to be real, and upon localization each of those terms will contribute a real piece at linear order in the (real) parameters mf , where mf are the demotion of Mf to constants, interpreted as mass parameters for the avor symmetry.6 If more than one background vector multiplet are present, then there exists another counterterm given by LJJJ 3 i aJbJcJ hpgM aM bM cR + i : As before, the constants aJbJcJ are real, and we conclude that such a counter-term contributes a real piece to the localized ve-sphere partition function, at cubic order in the mass parameters mf for the avor symmetry. As we shall see in section 3.2.2, one has to subtract such terms o the large N free energy to end up with a physically consistent result. 6Recall that localization requires M to be imaginary and here M is proportional to imf by a real number. { 7 { In the next few sections, we focus on a particular class of ve-dimensional superconformal eld theories, which were rst introduced by Seiberg [21].7 They can be constructed in type IIA string theory (or rather type I' string theory) by a D4-D8/O8-brane setup. We begin with two orientifold O8-planes located at x9 = 0 and x coupling, the D0-branes become massless, and together with the massless string degrees of freedom localized on the D4-branes yield the aforementioned Seiberg theories. The avor symmetries for the Seiberg theories of arbitrary rank are given by ENf +1 SU(2)M ; (2.17) where the rst factor is enhanced from the SO(2Nf ) symmetry of the Nf D8-branes and the U(1)I instanton particle (D0-brane) symmetry [3, 21]. The latter factor is the mesonic symmetry arising from rotations in the fx5; x6; x7; x8g directions. A renormalization group ow connects a Seiberg theory of avor symmetry ENf +1 in the ultraviolet to an infrared USp(2N ) gauge theory. The D4-D4 strings give rise to a vector multiplet and a hypermultiplet in the antisymmetric representation of USp(2N ). The D4D8 strings give rise to Nf 7 fundamental hypermultiplets. Starting from the ultraviolet, the infrared gauge theory can be reached by turning on a deformation that becomes the ve-dimensional Yang-Mills term towards the end of the ow. Although an in nite number of irrelevant terms arise along this ow, they are believe to be Q-exact, and thus do not contribute to the localized path integral [7, 32, 36]. Therefore, the supersymmetric partition function of the ultraviolet superconformal Seiberg theory is fully captured by the infrared USp(2N ) gauge theory. The supersymmetric ve-sphere partition function has been computed by localization using the infrared gauge theory description. The result is a sum over the contributions from in nitely many saddle points. We presently review the result of the perturbative saddle point, where all the hyper- and vector multiplets have trivial vacuum expectation value [31, 32, 35]. For simplicity, we choose the mass matrix (2.6) for the hypermultiplet SO(2Nf ) avor symmetry to be and for the mesonic SU(2)M symmetry acting on antisymmetric hypermultiplet to be M(SO(2Nf )) = mf (i 2 ve-sphere partition function for the infrared USp(2N ) gauge theories with Nf fundamental and one antisymmetric hypermultiplets is 7In section 5, we consider a more general class of theories labeled by n 2 Z 1, of which n = 1 corresponds to the Seiberg theories. { 8 { given by8 where jWj = 2N N ! is the order of the Weyl group of USp(2N ). Note that since the USp(2N ) gauge group has no cubic Casimir, the (gauge) Chern-Simons terms are absent, and the prepotential only consists of the classical Yang-Mills piece. The dependence of the supersymmetric ve-sphere partition function on the squashing and mass parameters encodes the central charges of the superconformal eld theory. The embedding indices, which by (2.14) relate the avor central charges of the infrared global symmetry group to the ultraviolet enhanced global symmetry group, in the case of Seiberg theories, are Iso(2Nf ),!eNf +1 = 1; Iu(1)I,!eNf +1 = 8 4 Nf : We refer to [29] for the explicit computations. 3 The large N limit of Seiberg theories Motivated by holography, we proceed to the computation of the deformed ve-sphere partition function for the Seiberg theories in the large N limit. As we shall see, the supersymmetric partition function, which is hard to evaluate at nite N , simpli es in this limit. To describe the partition function at the superconformal xed point, we are required to send gYM ! 1 under which all saddles in the instanton expansion contribute equally (without manifest suppression by a small parameter), and a perturbative analysis seems to break down. Nonetheless, it has been argued that the instanton contributions are exponentially suppressed in the large N limit [7]. To understand this, we write the full partition function schematically as (2.20) (2.21) (3.1) (3.2) (3.3) S3(x j !~)S3( x j !~). 1 X n=0 Znqn ; q = exp { 9 { where Z numbers, and n are functions of the fugacities that are only explicitly known for small instanton with gYM being the infrared gauge coupling. When moving out on the Coulomb branch, the e ective Yang-Mills coupling receives a one-loop correction where by we collectively denote the Coulomb branch parameters of the theory. Since the instanton particles are BPS, their masses are determined by the central charge (of the supersymmetry algebra), which is a linear combination of the bare gauge coupling and the Coulomb branch parameters. Accordingly, the e ective parameter for the instanton expansion is governed by not the bare gauge coupling but an e ective coupling, As we shall see below, in the large N limit, the Coulomb branch parameters asymptotically as Consequently, the contributions with nontrivial instanton numbers are exponentially suppressed in the large N limit. The upshot is that in the large N limit, the perturbative part of (squashed) vesphere partition function is exact. We can therefore explicitly determine the conformal central charge and the avor central charges for the ENf +1 SU(2)M avor symmetry, via the perturbative localization formula (2.20), as well as the prescription of section 2.2. The enhancement of the avor symmetry from the manifest infrared SO(2Nf ) U(1)I to the ultraviolet ENf +1 implies that the values of the exceptional avor central charge computed from the SO(2Nf ) and U(1)I embeddings should agree. Whereas numerical evidence for the agreement was found in [29] for the rank-one Seiberg theories, here we shall nd an exact agreement in the large N limit. In section 4, the results of the present section will be compared with the central charges extracted directly from the dual supergravity description. 3.1 Free energy and conformal central charge Let us rst recall the large N computation of the undeformed free energy in [7, 11, 12]. We start by rewriting the perturbative partition function (2.20) into the form Z where we have chosen to represent the exponent by the symbol F ( ) because when later evaluated at the large N saddle point, it becomes the large N free energy F . In the following, F ( ) will be referred to as the \localized action". For the Seiberg theories,9 N i=1 N + X i6=j 1 2 where GV and GH are the logarithms of the triple sine functions, i.e., GV (z j !~) = log S3 (iz j !~) ; GH (z j !~) = log S3 iz + !tot 2 j !~ ; with !tot = !1 + !2 + !3. By Weyl re ections of the USp(2N ) gauge group, we can restrict the integration region to i 0, and compensate by a symmetry factor of 2N . We now proceed to performing the large N saddle point approximation of the integral (3.6). These saddle points can be studied numerically at large but nite N , and the results suggest that the Coulomb branch parameters scale (to leading order) as i in the asymptotic limit of large N and nite x. This particular scaling of i with N can N 1=2xi be argued analytically, as follows. First, assume that i = N xi ; with > 0 : Then, note that the functions GV ( z j !~) and GH (z j !~) have the following asymptotic expansions at jzj ! 1, GV ( z j !~) GH (z j !~) 3 !1!2!3 j j which follow from the asymptotic formulae (A.9) and (A.10) for triple sine functions. As reviewed in appendix A.1, the expansions (3.11) have no subleading power law correction, a fact that will be important later. The leading order term in the rst line of (3.8) scales as N 1+3 , whereas the leading order term in the second line of (3.8) scales as N 2+ . In order to get a nontrivial saddle point, both terms must contribute to the same order, and we thereby determine In the large N limit, we introduce a density (x) for the rescaled Coulomb branch parameters xi (we use the Weyl re ections of the USp(2N ) gauge group to restrict to xi 0), (x) Z N 1 X N i=1 dx (x) = 1 : (x xi) ; Z x? 0 (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) normalized such that In the continuum limit, the localized action (3.8) becomes !t2ot (x+y +jx yj) (8 We are thus left with the simple variational problem of nding the function (x) that extremizes F ( ). We add a Lagrange multiplier term for the constraint (3.13), and solve F = = 0 for (x) and x . The resulting saddle point con guration is Evaluating (3.14) at this saddle gives the well-known result for the free energy [11, 12] (x) = 2x x2 ? ; x2? = !t2ot 2(8 Nf ) : F = p The leading corrections of order N 1=2 come from both the continuum approximation and further terms in the series expansion in mf . Using 3.2 3.2.1 Flavor central charges Fundamental hypermultiplet symmetry To compute the avor central charge for SO(2Nf ), let us now deform the round-sphere partition function (with squashing parameters !i = 1), by giving masses to the Nf fundamental hypermultiplets. For simplicity, we choose the following mass matrix (see (2.6) and the discussion in section 2.4), M = mf (i 2 Nf identity matrix, and 2 is the second Pauli matrix. By adding this particular mass term to the perturbative partition function, the localized action (3.8) is modi ed by the replacement We determine the round-sphere free energy to be HJEP05(218)39 and extract the conformal central charge from the !i-dependence of the free energy by the relation (2.9) to be GH ( i j !~) ! GH ( i + mf j !~) : Since the addition of this mass term only changes the saddle point to subleading order in 1=N , we may simply evaluate the mass term on the saddle point con guration (3.16) to nd the modi cation to the free energy, F = = mf2Nf N 3=2 Z x? 0 p 8 Nf p 2 mf2Nf N 3=2 + O(N 1=2) : dx (x)x + O(N 1=2) tr vec(SO(2Nf ))(M2) = 2Nf mf2 ; in the relation (2.13), we determine the avor central charge for SO(2Nf ) to be J CSO(2Nf ) = 256p2 3 p 8 Nf In this case, the embedding index into the enhanced avor symmetry group at the ultraviolet xed point is simply one, and therefore, (3.24) (3.25) (3.26) (3.27) (3.28) coupling by piece reads Next, we compute the avor central charge for the instantonic U(1)I symmetry. The avor symmetry SO(2Nf ) U(1)I in the infrared is expected to enhance to ENf +1 at the ultraviolet superconformal xed point. This implies that the value of CENf +1 predicted by the CJ of J the U(1)I subgroup and that of the SO(2Nf ) subgroup should agree (with the embedding indices properly accounted for). For general N , this statement is di cult to prove, as it requires re-summing all instanton contributions.10 However, we expect to obtain a precise agreement in the large N limit, since such non-perturbative contributions are exponentially suppressed as discussed before. To compute the CJ for U(1)I, we simply keep track of the dependence of the perturbative partition function on the instanton particle mass mI, which is related to the Yang-Mills HJEP05(218)39 mI = and only appears in the classical piece. The large N localized action including the classical up to corrections that are non-perturbative in 1=N , since the asymptotic expansions (3.11) of log S3 have no subleading power law correction. If we naively proceed as in the previous section, by evaluating the additional instanton mass term on the saddle point (3.16) without mass deformations, then we immediately run into a problem, which is that the dependence on mI truncates at linear order. Thus, to nd the CJ for U(1)I, we must include corrections to the saddle that are subleading in 1=N . We assume that (x) has support on the interval [x1; x2], with xi 0. The saddle point equation with inclusion of the instanton mass mI reads 0 = + 9 4 (8 3 10In [29], the present authors obtained some numerical evidence for the agreement in the case of N = 1. where as before is the Lagrange multiplier for the normalization of (x). Taking the derivative twice with respect to x shows that (x) is a linear function. The saddle point equation can be straightforwardly solved to give where11 2x a (x) = + b ; xed point. relation (2.12), We can now compute the leading order large N free energy F at the saddle point, F = p 9 2 N 5=2 p The leading correction of order N 1=2 comes from the continuum approximation of the discrete Coulomb branch parameters i by (x). By regarding the discrete sum over i = 1; 2; : : : ; N as a (midpoint) Riemann sum for the integral over x, we estimate the error of the continuum approximation to be of order N 2 relative to the leading N 5=2. In comparison, the correction coming from the Gaussian integral around the saddle point is of order log N . There appears to be an immediate problem. The term of order mI would imply a non-vanishing sphere one-point function of the avor currents, which violates conformal invariance. However, as discovered in [29] and reviewed in section 2.3, there are ambiguities in the free energy due to counter-terms. In particular, upon localization on the ve-sphere, the mixed Chern-Simons counter-terms contribute real pieces at order mI and mI3 [29]. Thus, we may pick a regularization scheme to remove the terms with linear and cubic dependence on mI in (3.31), to properly preserve conformal symmetry at the ultraviolet We can now compute the avor central charge for the instantonic U(1)I current by the J The avor central charge of the enhanced ENf +1 avor symmetry is related to that of the instantonic U(1)I by the embedding index (2.21), and we conclude that J CENf +1 = 8 4 Nf CU(1)I = J 256p2 3 p 8 Nf 11To leading order at large N , we recover the previous saddle point (3.16). (3.29) (3.30) (3.31) (3.32) (3.33) This precisely agrees with the avor central charge for ENf +1 computed using the SO(2Nf ) mass deformation given in (3.24) and (3.25), and con rms the avor symmetry enhancement in Seiberg theories. Finally, we compute the avor central charge for the mesonic SU(2)M symmetry. We deform the ve-sphere partition function by a mass for the antisymmetric hypermultiplet, which transforms in the fundamental of SU(2)M. As before, we couple it to a background vector multiplet with the following choice of mass matrix, HJEP05(218)39 Then the localized action (3.8) is modi ed by the replacement M = imas 3 2 su(2)M : GH ( i 1 j j !~) ! 2 GH ( i j mas j !~) : Again, we shall use the asymptotics for the triple sine functions as given in (3.11). The leading order modi cation to the large N localized action (3.14), due to the addition of the mass for the antisymmetric hypermultiplet, then reads 2 N 5=2m2as Z x? 0 Z x? 0 denotes the Heaviside theta function, and the leading corrections of order N 3=2 come from further terms in the series expansion in mas. This modi es the saddle point con guration to (3.34) (3.35) (3.36) (3.37) (3.38) (3.39) (3.40) This large N formula for the mesonic avor central charge is in fact equal to that for the R-symmetry avor central charge. The latter is related to the conformal central charge by Consequently, the large N free energy is given by As before, we apply the relation (2.13) with (x) = p 2 2x x2 ? ; x2? = 2 (8 9 + 4m2as : Nf ) F = 9 + 4m2as 3=2 15p8 Nf J CSU(2)M = 256p2 5 p 8 Nf to compute the avor central charge of the mesonic SU(2)M avor symmetry, superconformal Ward identities, since the R-symmetry currents reside in the same superconformal multiplet as the stress tensor. The agreement of these two avor central charges at large N is expected from the dual supergravity perspective, where SU(2)M and SU(2)R combine to form the SO(4) isometry of the internal four-hemisphere (see section 4), and are exchanged under a frame rotation as described in section 4.4.1. However, this agreement is obscure from eld theoretic considerations, and fails at nite N . We have argued that the perturbative partition function is exact in the large N limit. Away from large N , instanton corrections are no longer suppressed, and computing the exact partition function or central charges becomes a di cult task. However, in [29], the present authors observed that for Seiberg theories of rank one | which is the opposite of large N | the instanton contributions to the central charges as extracted from the supersymmetric ve-sphere partition function are also small. Combining these two facts, it is natural to expect that instanton contributions are in fact small for all N . Under this assumption, we can approximate the exact partition function by the perturbative formula, and numerically compute the round-sphere free energy and the central charges at nite values of N . For 1 N 3, the integral (2.20) can be computed straightforwardly by direct numerical evaluation; for 1 N 40, we can estimate the integral by a saddle point approximation. In the absence of a large parameter, the latter is a priori illegal. However, after explicitly computing the round sphere free energy and the various central charges, we nd that the saddle point approximation agrees with direct numerical integration even for N = 3 to within 1% (these numerical results are presented in detail in appendix B). Thus, we believe that this approximation can in fact be trusted for N These results suggest that in situations where only approximate values of these quantities are needed, such as for numerical bootstrap, the nite N saddle point approximation serves as an e cient method to perform the perturbative localization integral (2.20). In gure 1, we juxtapose the results of direct numerical integration, nite N saddle point approximation, and the large N formula, in the case of the Seiberg E8 theory. 4 Central charges from supergravity We now move towards a study of the central charges in the holographic duals of the Seiberg theories. As reviewed in section 2.4, the Seiberg theories can be constructed by a system of D4-branes probing an orientifold singularity in type I' string theory [21], the con guration of which is given in table 1. In summary, the con guration has Nf < 8 D8-branes coinciding with an O8, on top of which lie N D4-branes. The decoupling limit suggests a holographic correspondence between Seiberg theories and type I' string theory on a background of the form M6 w HS4, a warped product of a four-hemisphere with a asymptotically locally AdS6 space M6 [4{6]. In an appropriate low energy regime, the gravity side is well approximated by type I' supergravity [3]; in particular, the region between D8-branes is described by Romans massive type IIA supergravity [3, 37]. This latter regime is what we mainly consider in the following. 106 105 104 1000 ■ 5 × 104 ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 50 CJSU (2)M 106 105 104 1000 100 ■ ■ ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 5 ■ ■ by numerically computing the perturbative partition function (squares), and by the large N formula (dashed line). The physical quantities considered here are the round sphere free energy conformal central charge CT , the mesonic avor central charge CSU(2)M , and the exceptional avor J central charge CJE8 . The numerical computations are done by direct integration up to N = 3, and F0, the by saddle point approximation up to N = 40. 0 1 2 3 4 5 6 7 8 9 This particular holographic correspondence has thus far been tested in various ways. Initial checks include comparisons of the symmetries on both sides [4, 5], and more recent checks include comparisons of the entanglement entropy [7] and the free energy/Wilson lines in the case of M6 = AdS6 [9], as well as for more general asymptotically locally AdS6 spacetimes [11, 12, 15]. 4.1 Central charges and couplings The conformal central charge CT and the avor central charges CJ in a d-dimensional superconformal eld theory with a weakly coupled AdSd+1 dual are related to speci c coupling constants in the supergravity action. The leading order derivative expansion of the weakly coupled bulk e ective action is schematically given by Z Sd+1 = dd+1xp gd+1 " 1 2 2d+1 (Rd+1 2 d+1) 1 2e2 Tr F (d+1)F (d+1) # + ; (4.1) where Rd+1 is the (d + 1)-dimensional Ricci scalar, F (d+1) is the (d + 1)-dimensional eld strength for the gauge eld which is sourced by the avor symmetry current at the conformal boundary, Tr ( ) is the Killing form de ned in (2.4), and nally, is the (d + 1)-dimensional cosmological constant, with ` the radius of AdSd+1. The gravitational coupling d+1 and the gauge coupling e are related to the central charge CT and the avor central charge CJ by [38, 39] HJEP05(218)39 These relations allow us to extract CT and CJ from the supergravity duals of the Seiberg theories. The key is then to determine the precise values of the couplings d+1 and e that properly embed the supergravity into the aforementioned type I' string theory background. Massive IIA supergravity solutions Let us rst review the AdS6 HS4 solutions in massive IIA supergravity [5]. In the string frame, the bosonic part of the ten-dimensional massive IIA action reads S1II0A = 2 210 d10x p where g10 is the ten-dimensional metric, R10 the corresponding Ricci scalar, and 10 = 8 7=2`4 s is the ten-dimensional gravitational coupling expressed in terms of the string length `s. Furthermore, we denote by the ten-dimensional dilaton, F6 the six-form ux, and mIIA the Romans mass. The string frame metric for the AdS6 HS4 background is explicitly (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) given by hemisphere, ds120 = 1 sin1=3 hL2ds2A\dS6 + R 2 d 2 + cos2 ds2S3 i ; where L is the AdS6 radius, R is the HS4 radius, and the standard metric of the unit-radius AdS6, and by ds2S3 the unit S3-slices of the four2 (0; =2]. We denote by ds2A\dS6 1 h 4 ds2S3 = d 12 + sin2 1 d 22 + (d 3 cos 1 d 2)2i ; with 1 2 [0; ], 2 2 [0; 2 ), and 3 2 [0; 4 ). Note that the isometry group of S3 is SO(4) = SU(2)M SU(2)R, where the former SU(2)M factor corresponds in the dual vedimensional superconformal eld theory to the mesonic global symmetry, and the latter to the R-symmetry. SO 2Nf(1) U N (2) f U N (k+1) f X12 USp(2N ) SU(2N ) X23 Xk(k+1) SU(2N ) A eld theories with odd n = 2k + 1. Their charges are related by The meson operator is A basis for the baryon operators is 1 2 Ba = det Xa(a+1) ; a = 1; : : : ; k 1 ; plus an additional one given by the Pfa an of the antisymmetric hypermultiplet A, Bk = Pf(A) = 1 2N A 1 2 A 2N 1 2N ; where j are the USp(2N )-indices, raised and lowered by the corresponding invariant tensor. Note that the alternative combination Pf(A0) is not independent of the chosen basis fM; B1; : : : ; Bkg of gauge-invariant operators composed purely of hypermultiplets. Odd orbifold ( gure 4) This nal case is a hybrid between the two even cases: there is a USp(2N ) SU(2N )k infrared gauge group, together with fundamental, bifundamental, and antisymmetric hypermultiplets. The resulting overall infrared avor symmetry group is G(n=2k+1) = U(1)I(1) f U(1)(b1) SO(2Nf(1)) U(1)I(k+1) U(1)A U(Nf(k+1)) : Yk hU(1)I(a) a=2 U(1)(ba 1) U(Nf(a))i (5.10) (5.6) (5.7) (5.8) (5.9) As before, we group the U(1) symmetries into mesonic U(1)M and baryonic U(1)(Ba) factors, with a = 1; ; k. Their charges are related by A basis for the baryon operators is ! # A : Again, the Pfa an Pf(A) is not independent of the basis fM; B1; : : : ; Bkg of gauge-invariant operators composed purely of hypermultiplets. In all three cases, the total number of fundamental hypermultiplets equals the number f of D8-branes in the type I' string theory setup, i.e., Nf = P be a UV xed point, this number must fall into the range 0 N (a) must satisfy certain conditions. Free energy and conformal central charge a Nf(a). In order for there to Nf < 8, and the distribution We begin by writing down the localized action of the Coulomb branch integral expression (3.6) for the perturbative ve-sphere partition function. For ease of notation, it is convenient to introduce a set of auxiliary Coulomb branch parameters i(a), where a labels the gauge node, i = 1; : : : ; 2N , and then perform identi cations on speci c gauge group. Schematically, the Coulomb branch integration measure is i (a) suitable for the Z [d ] e F ( ) (constraints) ; up to some ambiguity in the Jacobian factor that only a ects the free energy at order N in the large N limit. Since we only need to be accurate to order N 3=2 to extract the central charges of interest, we shall be liberal in our de nition of this integration measure. The forms of the localized action F ( ) for the three classes of orbifold theories are as follows. Even orbifold with vector structure (5.11) (5.12) (5.13) (5.14) (a+1) # (5.15) Fv(s2k)( ) = 2N " GV 1 2 + GV 2N " +X i=1 1 2 k a=2 (ja)) ( i(k+1) j (k+1)) # # j ; 1 2 GV 2 i(1) + GV 2 i(k+1) +X N (a)GH f with the constraints (1) N+i = i(1), (Nk++i1) = (k+1), and Pi2=N1 i i (a) = 0 for 2 a Fn(v2sk)( ) = X GV (ja)) +GH i(1) + (j1) +GH i(k) + (jk) (a+1) # +X X N (a)GH f i=1 a=1 (a) i # ; with Pi2=N1 i(=a)1 = 0 for 1 (ja)) 1 X2N " 2 2N " + X i=1 1 2 1 2 + GH (k+1) GV 2 i(1) + X N (a)GH f k+1 a=1 k+1 a=2 # # ; with N+i = (1) and Pi2=N1 i (a) = 0. Assuming that the Coulomb branch integral is convergent for these orbifold theories, their large N limit can be computed by the saddle point approximation.21 In the large N limit, provided that i(a) scale as N 1=2, the leading terms in the above F ( ) scale as N 7=2, and are minimized by [7] (a) i (a+1) j (5.17) # # (5.16) (5.18) (5.19) (a) = i ; i i = N+i i ; 1 1 i i 2N ; N ; i = N 1=2xi ; where i become the sole remaining Coulomb branch parameters. As before, de ne and rewrite the localized action F ( ) as a functional F [ ] of the density (x). The equations (5.18) actually render the SU(2N ) and the USp(2N ) gauge factors indistinguishable. Now we can take the large N limit of the exponent, and nd that for all three cases, it is simply given by n !t2ot (x+y +jx yj) (8 Nf ) 3 x The large N localized action F [ ] receives two types of corrections. First, there are N 3=2 terms coming from the asymptotic expansions (3.11) of F ( ). Second, there are order N 1 corrections to the saddle point con guration (5.18) due to the N 5=2 terms in F ( ), and 21While it requires extra work to understand the convergence of the Coulomb branch integral for general quiver gauge theories (see [50] for a discussion), the existence of a holographic dual here provides us certain con dence. (x) = 2x2? x2? = 9n 2(8 Nf ) ; F = 15 !1!2!3 p 8 2 !t3ot n3=2N 5=2 Nf + O(N 3=2) : CT = p 8 Nf + O(N 3=2) : and the free energy Invoking the general relation between the squashed free energy and the conformal central charge (2.9), we nd that the conformal central charges of the family of orbifold theories are given by is given by 2N i=1 1 Nf ) 2 so(2Nf(a)) F ( ) = N (a) X h f GH i (a) + m(a) f GH (a) i i : The leading order correction to the free energy is given by the large N asymptotics (3.11) of F ( ) evaluated on the leading order saddle (5.18) as well as (5.21) (from extremizing the leading order N 7=2 piece of the exponent), F = p 2 N (a)(mf(a))2 f p 8 Nf tr vec(SO(2Nf ))(M2) = 2Nf(a)(mf(a))2; in the relation (2.13), we determine the corresponding avor symmetry central charges to be they also give rise to order N 7=2 (N 1)2 = N 3=2 corrections to F [ ].22 The saddle point approximation to the Coulomb branch integral can be performed in complete analogy to the case of Seiberg theories, leading to the saddle point con guration Flavor central charges Let us now turn to the computation of the avor central charges for the orbifold theories. It su ces to consider the round sphere, so we set !i = 1 in the following. Fundamental hypermultiplet symmetries We rst compute the avor central charges for generic SO(2Nf(a)) factors | associated with the gauge nodes a = 1 and a = k + 1 for even orbifold with vector structure, and a = 1 for odd orbifold | in the avor symmetry groups of the orbifold theories. The modi cation to the localized action F ( ) due to the introduction of a mass matrix J CSO(2Nf(a)) 256p2 3 p 8 Nf n1=2N 3=2 + O(N 1=2) : 22A correction to the saddle point con guration (5.18) results in a correction of order localized action F ( ), since by de nition, F ( )= = 0 at the saddle point. (5.21) (5.22) (5.23) (5.24) (5.25) (5.26) (5.27) (5.28) 2 to the The fact that the leading 1=N corrections to the avor central charges are of order N 1=2 needs some explanation. Firstly, one may worry that the order N 3=2 corrections to the undeformed free energy F in (5.22) would contaminate the avor central charges that are also of order N 3=2. However, these corrections do not depend on the mass parameters, and therefore do not a ect the avor central charges. Secondly, since the mass deformations given by (5.25) scale as m2N 3=2 (N 2 relative to the leading N 7=2), there are corrections to the saddle point con guration (5.18) that are of order m2N 2, in addition to the mindependent corrections of order N 1 discussed below (5.22). The corresponding correction to the free energy F in (5.22) that is quadratic in m is from the mixed term, of order N 7=2 (m2N 2 N 1 = m2N 1=2. We now consider the remaining gauge nodes that are associated with U(Nf(a)) avor factors, by introducing the mass matrix, into the Lagrangian. Proceeding as before, we nd that the leading order correction to the F = 2 N (a)(mf(a))2 f p 8 1 2 ; Nf tr J CU(Nf(a)) = 256p2 3 p 8 Nf in the relation (2.13), we determine the corresponding avor symmetry central charges to be As will be discussed further in section 6.1, Hanany-Witten moves [51] relate an orbifold theory considered here to one that has SO(2Nf ) avor acting on hypermultiplets charged under the rst gauge node, and the nite shifts in the ranks of the gauge nodes cannot be detected in the large N limit. This large N avor symmetry enhancement explains the coincidence of the avor central charges for all gauge nodes in the original orbifold theory. 5.3.2 Bifundamental hypermultiplet symmetries We now turn towards the avor central charges for the U(1)(ba) Each bifundamental hypermultiplet Xa(a+1) has unit charge under the U(1)(ba) metry associated to a mass parameter m(a) for Xa(a+1). The mass deformation of the b avor symmetry factors. avor symlocalized action is i F ( ) on the large N saddle (5.18), and taking the continuum limit, the localized action becomes Z x? 0 This term modi es the saddle to which leads to the large N free energy, (x) = 2 2x x2 ? ; x2? = 9n + 32(m(ba))2 2(8 Nf ) ; F = 9n + 32(m(ba))2 3=2 15p8 Nf J Antisymmetric hypermultiplet symmetries The antisymmetric hypermultiplet A in the odd orbifold has unit charge under the U(1)A avor symmetry associated with a mass parameter mA. The mass deformation of the F ( ) on the large N saddle (5.18), and taking the continuum limit, the By the relation (2.12), we are led to the following U(1)(ba) avor central charge (5.35) (5.36) (5.37) HJEP05(218)39 (5.40) (5.41) (5.42) localized action becomes This term modi es the saddle to which gives the large N free energy, (x) = p 2 Z x? 0 Nf Nf 2x x2 ? ; x2? = 2(8 9n + 16m2A ; Nf ) F = 15p8 9n + 16m2A 3=2 J CU(1)A = 1024p2 5 p 8 n1=2N 5=2 + O(N 3=2) : By the relation (2.12), we are led to the following U(1)(Aa) avor central charge An analogue analysis shows that the antisymmetric hypermultiplets A and A0 in the even orbifold without vector structure have the same avor central charge as in (5.42). CU(1)B = BB J CU(1)B = BB J CU(1)B = BB J 0 2 B1 B 1 02 B1 B B B1 0 3 0 B2 B 2 1 2 1 2 3 2 1 1 . . . 2 2 . . . 1 2 1 1 C C C A 2 1 1 2 0 0 1 0 C C C C C 1 k k 2 1 C C A 3 k k J J J As speci ed in (5.2), (5.6) and (5.11), the mesonic U(1)M and baryonic U(1)B symmetries are linear combinations of the U(1) avor symmetry factors that act on the bifundamental and antisymmetric hypermultiplets. The mesonic avor central charge is then given by J CU(1)M = n 256p2 5 p 8 Nf n3=2N 5=2 + O(N 3=2) : (5.43) The baryonic symmetries are not orthogonal. Their avor central charges are given by matrices. These (symmetric) matrices can be computed by taking derivatives with respect to the mass parameters of di erent baryonic U(1) factors in the overall avor symmetry. We write these matrices in the bases (5.2), (5.6), and (5.11). In even orbifolds with vector structure, the baryonic avor central charge matrix is HJEP05(218)39 (5.44) (5.45) (5.46) (5.47) (5.48) In even orbifolds without vector structure, In odd orbifolds, To compare with the results from gravity in section 6.2, it is convenient to perform a change of basis for the charge lattice from the Q(Ba) de ned in (5.2), (5.6), and (5.11). For even orbifold with vector structure, we de ne For even orbifold without vector structure, we de ne 1 2 1 0 1 1 . . . 1 0 for a = 1; ; k 2 ; ; k Q a = Q(Ba) Q k 1 = Q(Bk 1) : Q a = Q(Ba) Q(Ba+1) Q k 1 = Q(Bk 1) ; Q k = Q(Bk) : For odd orbifold without vector structure, we de ne ; k (5.49) In these bases, we nd CJU;e(1v)eBn;vs = BB CJU;e(1v)eBn;nvs = BBBB 0 CJU;o(1d)dB = BB 0 2 B B 1 B 0 0 2 B 1 B B 0 0 2 B B 1 B 0 1 1 1 2 0 0 2 0 0 0 2 0 0 0 1 0 1 0 1 0 0 . . . 0 0 0 0 . . . 0 0 0 0 0 . . . 0 0 1 0 1 0 0 1 0 0 0 2 1 0 0 2 0 1 0 0 2 1 2 0 0 2 0 1 0 1 0 C C C C C 3 k k k k J J CU(1)b ; J CU(1)b ; (5.50) The above change of basis, (5.44), (5.45), and (5.46), from Q(Ba) to Q a is integral and uni-modular, and therefore preserves the charge lattice. While such a change of basis is convenient for the later holographic comparison, more fundamentally, we are matching the charge lattice, which is basis-independent. 5.3.5 Instantonic symmetries In order to extract the avor central charge for the U(1)I instantonic factor associated to the a-th gauge node, we proceed as in section 3.2.2 for the Seiberg theories, by keeping the contribution of the classical piece. The large N localized action is a Nf ) 3 I + + X m(a) 2 N 2 x1 x1 Z x2 x1 dx (x)x3 + O(N 3=2) dx (x)x2 + O(N ) ; (5.51) for either the USp or SU gauge nodes. The saddle point solution to the relevant 1=N order is (x) = x1 = 0 ; x2 = 4(8 Nf )x 9n ; a mI(a) + and gives rise to the large N free energy, HJEP05(218)39 F = p 9 2 p 5 8 Nf X m(a) I n3=2N 5=2 + O(N 3=2) + X m(a) I Nf ) nN 2 + O(N ) !2 " p 4 2 (8 Nf )3=2 n1=2N 3=2 + O(N 1=2) + O((mI(a))3) : q 16(P p 2 N (Nf a mI(a))2 + 18 nN (8 Nf ) ; 8) . . . 9 # 1 1 1 As in the case of Seiberg theories, the linear m(a) piece would be inconsistent with conformal symmetry, and must be removed by a counter-term. Using the relation (2.12), we nd that the instanton symmetries are not orthogonal, and their avor central charge matrix is J whose dimensionality equals the number of gauge nodes. The reasoning for the leading 1=N corrections to the avor central charges being of order N 1=2 proceeds as before. Since the mass deformations in (5.51) scale as mN 2 (N 3=2 relative to the leading N 7=2), there are corrections to the saddle point con guration (5.18) that are of order mN 3=2, in addition to the m-independent corrections of order N 1 discussed below (5.22). The correction to the free energy F in (5.22) that is quadratic in m is then of order N 7=2 (mN 3=2)2 = m2N 1=2. At order N 3=2, the avor central charge matrix is rank-one, meaning that we only observe one independent combination of avor central charges. In order to access the other independent avor central charges that are of order N 1=2, we are required to carefully study the subleading contributions to the matrix models (5.15), (5.16), and (5.17), and in particular modify the leading order saddle (5.18). 6 Central charges from the supergravity dual of orbifold theories Let us now study the massive IIA supergravity duals for the orbifold theories [6], and subsequently match the various avor central charges associated to the global symmetry group. M6 The holographic duals of the orbifold theories are the Zn orbifolds of the geometry w HS4 discussed in section 4.2. More precisely, for each S3-slice at constant of the hemisphere, we impose the identi cation 3 ! 3 + 4 =n on the coordinate 3 in (4.8). (5.52) (5.53) (5.54) This corresponds to replacing the three-sphere with the lens space L(n; 1) = S3=Zn. Notice that this action reduces the volume of the four-hemisphere by a factor of n, VolHS4=Zn = VolHS4 ; n and a ects the e ective six-dimensional couplings upon compacti cation. Due to the volume reduction, the F4 ux is multiplied by a factor of n to preserve the D4-brane charge quantization condition (4.11). Hence, the supergravity solution of the orbifold theory is given by the substitution N ! nN in the Seiberg theory background (4.7), (4.15), (4.10), and (4.14). Together these considerations immediately give us many of the central charges. Central charges by comparison to Seiberg theories Compared to the holographic duals of Seiberg theories, the free energy receives an extra factor of n 1 n5=2 = n3=2, where the n 1 comes from the reduction of the internal volume, and the n5=2 comes from the N ! nN shift due to the modi ed charge quantization. This precisely matches the large N conformal central charge (3.19) of the eld theory. By the same argument, the SO(2Nf ) and U(Nf ) avor central charges each receives a factor of n1=2, matching the eld theory results in (5.28) and (5.32), respectively. In the large N limit, we are not able to distinguish between the distinct theories that arise from di erent distributions of the Nf = X N (a) f a (6.1) (6.2) (6.3) fundamental hypermultiplets, for the following reason.23 On the gravity side, the orbifold solutions in [6] describe the con gurations where all the Nf D8-branes sit at the rst gauge node. However, starting with a di erent setup, in which the Nf D8-branes are distributed among di erent gauge nodes, and performing consecutive Hanany-Witten moves [51], one ends up with a eld theory description in which some of the ranks of the gauge nodes are shifted N ! N + `, where ` Nf . But this e ect is not expected to be visible in the leading order large N asymptotics. Thus, in the strict large N limit, we will only ever be able to probe the SO(2Nf ) avor symmetry associated to the nal gauge node.24 The mesonic U(1)M symmetry of the orbifold theories corresponds to the U(1)M factor in the isometry group of the orbifold hemisphere, SU(2)R U(1)M. Compared to the avor central charge of the U(1)M SU(2)M subgroup in the Seiberg theories, the orbifold theories receive the same extra n3=2 factor as discussed above for the conformal and hypermultiplet avor central charges. Taking into account the embedding index Iu(1)M,!su(2)M = 1 2 ; we nd accordance with the eld theory result (5.43). that do. charges at leading large N . 23Not all distributions give rise to ultraviolet superconformal xed points. Here, we restrict to the ones 24As a matter of fact, we cannot even distinguish between SO(2Nf ) and U(Nf ) by their avor central Note that the Kaluza-Klein modes have minimal U(1)M charge n=2, which follows from the periodicity 3 3 + 4n in the orbifold theory [6]. This holographic argument for the normalization of U(1)M is in agreement with the mesonic charge de ned in eld theory, given by (5.2), (5.6), and (5.11), of the meson operator, given in (5.3), (5.7), and (5.12). Let us brie y remark on the dual supergravity gauge elds for the U(1)I(a) instantonic symmetries. In [6], it was argued that all but one of them arise from the reduction of the R-R three-form C3 on the nontrivial two-cycles at the orbifold xed point of HS4=Zn. The remaining one is argued to arise by a reduction akin to the infrared U(1)I of the Seiberg theories (see section 4.4.3). We leave the matching of the instantonic avor central charge to future work, in light of the subtleties already present in the Seiberg theories, as discussed The baryonic gauge elds are obtained from the reduction of the R-R three-form C3 in ten-dimensional massive IIA on the internal two-cycles ea in HS4=Zn [6]. Thus, in order to match the large N baryonic avor central charge matrix with the coupling matrix of the gauge elds in the holographic dual, we are required to analyze the appropriate (baryonic) two-cycles, and their intersection forms. The orbifolded four-hemisphere, HS4=Zn, has an orbifold singularity at the north pole. Thus, the usual Kaluza-Klein reduction from IIA supergravity needs to be complemented by analyzing the twisted sectors of the string theory. We start by considering the covering space S4=Zn (before the orientifold projection), or more precisely a mirror pair of C2=Zn singularities, before the near horizon limit. We introduce two sets of vanishing anti-selfdual two-cycles i N;S, with i = 1; : : : ; n 1, for the two copies of C2=Zn, corresponding to the twisted sector ground states of the orbifold CFT. Their intersection forms are given by the An 1 Cartan matrices, (6.4) (6.5) (6.6) and north and south copies of C2=Zn as follows As we shall explain below, the twisted sector closed string states that survive the orientifold projection give rise to six-dimensional gauge elds dual to the baryonic symmetries. The orientifold action I, given by a composition of worldsheet parity and the re ection I : x9 x9 (before the near-horizon limit), maps the vanishing cycles in the N i $ S n i ; N;S ) = BB 0 2 B 0 C C C C C 2 (n 1) (n 1) for even n = 2k with i 6= k, and odd n = 2k + 1. For even n = 2k, the middle cycles k are allowed to transform in two di erent ways corresponding to with and without vector structure [52{55] vs : nvs : N N k $ k $ S k ; S k : (again, we emphasize that this is prior to taking the near-horizon limit)25 To make contact with the notation in reference [6], we identify the cycles i and ei as i < i > n 2 n 2 : : i = iN + iS ; i = N i S i ; ei = i N S ei = i S i ; iN : k = kN + k ; S ek = N S k : For even n = 2k, there is an additional middle cycle, which we de ne in terms of k Note that while there are D2-branes wrapping combinations of i and sitting on the O8plane, corresponding to (singular) instantons in the boundary theory, the baryonic cycles ei only exist away from the O8-plane. Then the six-dimensional baryonic gauge elds (coupled to the baryonic U(1)B currents in eld theory), are given by reducing the R-R three-form C3 on linear combinations of the two-cycles ei that are odd under the orientifold action I, that we call ea,26 Ba = C3 : Z ea Thus, the coupling matrix for the baryonic gauge elds are proportional to the intersection matrices ( ea ; eb ) which we compute for each case of the orbifolds below. Even orbifold with vector structure The odd two-cycles are ea = ea + en a = N N n a S S n a ; a = 1; ; k 1 : (6.11) (6.8) (6.7) N;S as (6.9) (6.10) The intersection matrix is ( ea ; eb ) = 4 0 2 B B B 0 1 2 0 0 0 1 0 0 0 C C C C C 2 (k 1) (k 1) : (6.12) 25In [6], upon Kaluza-Klein reduction of the R-R three-form C3 from massive IIA, the di erent cycles i and ei give rise to six-dimensional U(1) gauge elds sourced by instantonic and baryonic U(1) global conserved currents, respectively. 26Similarly, there are cycles a which are I-odd combinations of i. They give rise to six-dimensional gauge elds dual to instanton symmetries. However due to the mixing with other R-R and NS-NS elds from ten-dimensional supergravity as explained in section 4.4.3, the intersection matrix of a will not directly produce the CJ matrix for U(1)I. We leave the complete analysis in this case for future. The odd two-cycles are and Their intersection matrix is ( ea ; eb ) = 4 ( ea ; ek ) = 0 ; ( ek ; ek ) = 4 ; for a; b = 1; ; k Odd orbifold The odd cycles are ea = ea + en a = Their intersection matrix is ( ea ; eb ) = 4 0 0 2 1 0 1 C C C C 2 (k 1) (k 1) 1 C C C C k k N N n a S ; k ek = ek = N S k : 0 2 B 1 B B B B 0 1 2 0 0 N 0 2 B B B B 0 0 1 N n a 1 2 0 0 0 0 . . . 0 0 0 1 1 0 S 0 HJEP05(218)39 (6.13) (6.14) (6.15) (6.17) The intersection matrices (6.17), (6.15), and (6.12) are in agreement with the baryonic avor central charge matrices (5.50), up to an overall normalization that can be xed. 7 In this paper, we study the central charges of a certain class of holographic ve-dimensional superconformal eld theories that have constructions in type I' string theory, from both the eld theory and supergravity perspective. On the eld theory side, we employ the formulae discovered by the present authors in previous work [29], that relate particular deformations of the ve-sphere partition function to the conformal and avor central charges. In order to compare to the corresponding supergravity quantities, we take the large N limit of the eld theory, where it is argued that the instanton contributions are exponentially suppressed. Thus, we obtain exact large N results purely based on the perturbative part of the partition function, and compute the large N conformal and avor central charges for our class of theories. The comparison of the large N avor central charges for the manifest infrared instantonic U(1)I and hypermultiplet SO(2Nf ) symmetries provides evidence for the avor symmetry enhancement to ENf +1 in the ultraviolet. We further support our large N results by juxtaposing the results from direct numerical integration, the saddle point method, and the analytic large N formulae. We then compare and explicitly match these large N eld theory results against their holographic duals. The central charges are related to certain couplings in the sixdimensional e ective gravity dual that arise from the reduction of massive IIA supergravity. Although the corresponding ten-dimensional supergravity backgrounds have a curvature singularity in the internal manifold, by explicitly reducing the relevant terms in the ten-dimensional action to six dimensions, we obtain nite values for the e ective sixdimensional couplings, that precisely match with the conformal and avor central charges in the eld theory. The matching of the instantonic avor central charges is left for future work. On the gravity side, as argued in section 4.4.3, in order to reproduce the correct e ective sixdimensional kinetic term from which the corresponding central charge can be extracted, we are required to take into account the reduction of all the elds in ten-dimensional supergravity. On the eld theory side, we found only one independent instantonic avor central charge to leading order in the large N limit, i.e., the avor central charge matrix is rank-one. To capture the remaining (independent) central charges, we need to carry the matrix model analysis for the orbifold theories to further subleading order in 1=N . Finally, we are con dent that a similar large N analysis should provide evidence and checks for some proposed large N dualities of ve-dimensional theories [48], as well as the recently discovered type IIB AdS6 solutions [ 56, 57 ].27 Acknowledgments We are grateful to Oren Bergman, Daniel L. Ja eris, Igor R. Klebanov, Silviu S. Pufu, and Diego Rodr guez-Gomez for helpful discussions and correspondence. CC, YL, and YW thank the Aspen Center for Physics, MF thanks the Simons Summer Workshop, and YL thanks National Taiwan University for hospitality during the course of this work. CC is supported in part by the U.S. Department of Energy grant DE-SC0009999. MF is supported by the David and Ellen Lee Postdoctoral Scholarship, YL is supported by the Sherman Fairchild Foundation, and both MF and YL by the U.S. Department of Energy, O ce of Science, O ce of High Energy Physics, under Award Number DE-SC0011632. YW is supported in part by the US NSF under Grant No. PHY-1620059 and by the Simons Foundation Grant No. 488653. This work was partially performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. A Triple sine function In this appendix, we introduce the triple sine function as well as its asymptotic limit, which is important when taking the large N limit of the squashed ve-sphere free energy. The multiple sine function is de ned as SN (z j !1; : : : ; !N ) = N (z j !1; : : : ; !N ) 1 N (!tot z j !1; : : : ; !N )( 1)N ; (A.1) 27While AdS6 solutions in massive IIA supergravity are rather rare [58], there is a variety of solutions in type IIB that contains a warped AdS6 factor. Some early works in this direction include [59, 60]. N is the multiplet gamma function de ned as Here, we have introduced yet another special function N (z j !1; : : : ; !N ) = exp [ N (z j !1; : : : ; !N )] : N (z j !1; : : : ; !N ) = N (s; z j !1; : : : ; !N ) ; where the multiple zeta-function is given as follows d ds s=0 1 X m1;:::;mN =0 N (s; z j !1; : : : ; !N ) = (z + m1!1 + mN !N ) s ; (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) where Re z > 0, Re s > N and !1; : : : ; !N > 0. The function N is a meromorphic function with simple poles at s = 1; : : : ; N . A.1 Asymptotics of the triple sine function In order to compute the large N limit of the (squashed) ve-sphere free energy, we require the asymptotic jzj ! 1 expansion of log S3 (z) and we shall quickly mention how to compute those. In [61], the author proved that there is an alternative de nition of N (z) N (z j !1; : : : ; !N ) = BN;N (z) log z + ( 1)N X N ! M X k=N+1 ( 1)k BN;k (0) zN k(k 1)! + RN;M (z) ; k=0 N N 1 BN;k(0)zN k N k X 1 where Re z > 0, and M was shown in [61] to behave as N is an arbitrary integer. RN;M (z) is some remainder, which RN;M (z) O z N M 1 ; in the asymptotic limit jzj ! 1 as long as jarg zj < . Similarly it is straightforward to see that the third term in (A.5) is of order O z 1 in the asymptotic limit jzj ! 1. Furthermore, we denoted by BN;N (z j !~) the generalized/multiple Bernoulli polynomials, which can be explicitly computed by expanding and solving tN ezt QbN=1 (e!bt 1) X1 t n BN;n (z) order-by-order. For the case of interest in the present paper, i.e. N = 3, we have B3;3 (z j !~) = z 3 !1!2!3 2!1!2!3 3!tot z2 + !t2ot + (!1!2 + !1!3 + !2!3) z 2!1!2!3 !tot (!1!2 + !1!3 + !2!3) : 4!1!2!3 Thus, it is now easy to compute (A.5) in the asymptotic limit and equivalently for the triple sine function. Explicitly one obtains log S3 (iz j !~) + log S3 ( iz j !~) 3 !1!2!3 jzj3 + jzj + O(jzj2 M ) ; log S3 iz + z 3 3 jzj + O(jzj2 M ) ; (A.10) where z 2 R. In other words, for arbitrary M , the contributions from the third term in (A.5) exactly cancel. Since the integer M can be chosen arbitrarily large, the expansions (A.9) and (A.10) have no subleading power law corrections jzj n, for n Numerical evaluation of central charges We present tabulated values for the round sphere free energy F0, the conformal central charge CT , the mesonic J avor central charge CSU(2)M , and the exceptional avor central charges CJGf for Gf = E1; E2; : : : ; E8, for Seiberg theories up to rank three. Each quantity is extracted from the perturbative partition function (2.20), computed via direct numerical integration and an a priori illegal saddle point approximation. An important conclusion we draw is that the nite N saddle point method actually produces good approximations for the integrals. Integral Saddle Integral Saddle Integral Saddle Error Gf F0 E1 E2 E3 E4 E5 E6 E7 E8 5:0967 6:1401 7:3949 8:9590 11:007 13:898 18:538 28:473 1 5:2612 6:2817 7:5109 9:0441 11:052 13:886 18:440 28:215 Error 1:6% 1:1% 0:78% 0:47% 0:20% 0:041% 0:27% 0:46% 22:190 25:425 29:335 34:233 40:391 49:543 64:642 96:712 Error 0:067% 0:052% 0:16% 0:25% 0:016% 0:13% 0:49% 0:53% 55:114 61:896 70:122 80:430 93:965 113:63 143:41 214:24 3 54:960 61:645 69:760 79:965 93:454 112:70 144:08 212:68 0:14% 0:20% 0:26% 0:29% 0:27% 0:41% 0:23% 0:36% 2 22:220 25:398 29:243 34:061 40:404 49:413 64:010 95:699 F0 in the rank-one to rank-three Seiberg theories, computed by numerical integration and by the saddle point approximation. Gf CT E1 E2 E3 E4 E5 E6 E7 E8 333:39 422:94 529:78 662:00 834:00 1075:1 1459:5 2274:4 J CSU(2)M Gf E1 E2 E3 E4 E5 E6 E7 E8 0:92% 0:75% 0:60% 0:43% 3673:0 4197:1 4829:4 5619:4 6653:6 8126:6 10504: 15651: 3 3741:8 4266:1 4898:5 5688:9 6727:3 8199:4 10580: 0:36% 15736: 0:27% 0:93% 0:82% 0:71% 0:61% 0:55% 0:45% 1 365:53 455:28 562:40 694:99 867:48 1109:2 1494:3 2309:8 66:277 70:839 76:440 83:556 93:064 106:81 129:60 180:73 4:6% 3:7% 3:0% 2:4% 2:0% 1:6% 1:2% 0:77% 1477:0 1737:1 2049:6 2438:4 2946:9 3663:9 4815:9 7287:9 2 66:218 70:733 76:268 83:300 92:708 106:34 129:06 180:35 2 1529:8 1790:2 2102:9 2492:2 3001:4 3719:6 4873:8 7351:0 avor central charge CSU(2)M in the rank-one J to rank-three Seiberg theories, computed by numerical integration and by the saddle point approximation. Seiberg theories, computed by numerical integration and by the saddle point approximation. 0:045% 0:074% 0:11% 0:15% 0:19% 0:22% 0:21% 0:11% 218:62 233:82 252:45 275:74 307:17 354:67 431:93 605:43 3 218:15 233:23 251:77 275:41 307:14 353:31 430:43 604:63 Error 0:11% 0:13% 0:13% 0:061% 0:0044% 0:19% 0:17% 0:067% 21:638 23:700 26:413 30:131 35:587 44:657 64:752 18:409 20:582 23:120 26:190 30:128 35:664 44:707 64:756 1 Saddle 22:275 24:400 27:158 30:895 36:324 45:256 64:766 1 17:966 19:956 22:342 25:307 29:192 34:722 43:783 63:825 J CSO(2Nf ) Gf CU(1)I J Iu(1)I,!gf Gf E2 E3 E4 E5 E6 E7 E8 E1 E2 E3 E4 E5 E6 E7 E8 Seiberg theories, obtained using the instanton particle mass, computed by numerical integration and by the saddle point approximation. 0:79% 0:78% 0:75% 0:67% 0:54% 0:35% 0:019% 88:687 96:877 107:22 121:22 142:94 177:82 255:52 3 Saddle 89:681 97:981 108:66 123:07 144:01 178:69 255:60 0:56% 0:57% 0:67% 0:76% 0:37% 0:24% 0:015% 0:28% 0:051% 0:11% 0:19% 0:21% 0:18% 0:14% 0:10% 79:666 87:144 96:095 107:25 121:96 143:05 177:89 255:42 3 80:481 87:726 96:490 107:51 122:14 143:22 178:07 255:64 Error 0:51% 0:33% 0:21% 0:12% 0:075% 0:058% 0:051% 0:043% Seiberg theories, obtained using the hypermultiplet masses, computed by numerical integration and by the saddle point approximation. 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Chi-Ming Chang, Martin Fluder, Ying-Hsuan Lin, Yifan Wang. Romans supergravity from five-dimensional holograms, Journal of High Energy Physics, 2018, 39, DOI: 10.1007/JHEP05(2018)039