PPreprint typeset in JHEP style - PAPER VERSION
M-theoretic matrix models
Alba Grassi and Marcos Mari˜no
D´epartement de Physique Th´eorique et Section de Math´ematiques,Universit´e de Gen`eve, Gen`eve, CH-1211 Switzerland [email protected], [email protected]
Abstract:
Some matrix models admit, on top of the usual ’t Hooft expansion, an M-theory-likeexpansion, i.e. an expansion at large N but where the rest of the parameters are fixed, insteadof scaling with N . These models, which we call M-theoretic matrix models, appear in the local-ization of Chern–Simons–matter theories, and also in two-dimensional statistical physics. Gener-ically, their partition function receives non-perturbative corrections which are not captured bythe ’t Hooft expansion. In this paper, we discuss general aspects of these type of matrix integralsand we analyze in detail two different examples. The first one is the matrix model computingthe partition function of N = 4 supersymmetric Yang–Mills theory in three dimensions with oneadjoint hypermultiplet and N f fundamentals, which has a conjectured M-theory dual, and whichwe call the N f matrix model. The second one, which we call the polymer matrix model, computesform factors of the 2d Ising model and is related to the physics of 2d polymers. In both cases wedetermine their exact planar limit. In the N f matrix model, the planar free energy reproducesthe expected behavior of the M-theory dual. We also study their M-theory expansion by usingFermi gas techniques, and we find non-perturbative corrections to the ’t Hooft expansion. a r X i v : . [ h e p - t h ] N ov ontents
1. Introduction 12. General aspects of M-theoretic matrix models 33. The N f matrix model 8
4. The polymer matrix model 23
5. Conclusions 27A. The function G ( z ) A.1 General properties 29A.2 Limiting behavior 31
B. The O ( m ) model as a multi-trace matrix model 32 B.1 Multi-trace matrix models 32B.2 Examples 34B.2.1 Chern–Simons matrix model 34B.2.2 The N f and the polymer matrix models 35
1. Introduction
In the last years, a new window has opened to understand the properties of M-theory andstring theory on certain backgrounds: the combination of the AdS/CFT correspondence withsupersymmetric localization in gauge theories. This combination has provided conjectural, exactresults for quantities in M-theory, like for example Euclidean partition functions on certain AdSbackgrounds, and it has led to new checks of the AdS/CFT correspondence. In the case of AdS ,for example, the gauge theory computation of the Euclidean partition function on the three-sphere of Chern–Simons–matter theories reproduces at large N the gravity calculation, includingthe N / behavior of membrane theories predicted in [1]. This was first done in [2] in the case ofABJM theory [3], and it was extended to other models in many subsequent papers. One can even– 1 –est AdS/CFT beyond leading order and match logarithmic corrections to the partition function[4]. Perhaps the most interesting lesson for M-theory has been obtained in the study of non-perturbative corrections in ABJM theory. In this model, both worldsheet instanton correctionsand membrane instanton corrections to the partition function can be computed in detail, byusing the standard ’t Hooft expansion [2] and the so-called Fermi gas approach [5], respectively.Due to a hidden (and perhaps accidental) connection to topological string theory [6, 7, 8, 9], onecan obtain exact results for the full series of instanton corrections. A surprising aspect of thisexact result is that the total contribution of worldsheet instantons (i.e. of fundamental strings)has infinitely many poles at physical values of the string coupling constant. These divergencesare only cured if one adds membrane instantons and bound states of fundamental strings andmembranes [7, 10]. The cancellation of divergences in the partition function is known as theHMO mechanism. This mechanism shows, in a precise quantitative way, that a theory basedsolely on fundamental strings is radically incomplete, and a consistent theory is only obtainedwhen one considers M-theory together with its solitonic objects, like membranes.From a more formal point of view, the results obtained for ABJM theory show that the’t Hooft expansion does not capture the full physics of the model, since it only contains thecontribution of fundamental strings. On general grounds, it has been known for a long time thatthe ’t Hooft expansion is an asymptotic expansion, and in principle it has to be supplementedby non-perturbative contributions like large N instantons (see [11] for a review of these issues).The study of the ABJM matrix model has shown that these corrections are not just a luxuriouscommodity: they are needed for consistency.These conclusions are probably generic for a wide class of AdS /CFT duals, i.e. we expectthat the ’t Hooft expansion of the partition function of these models will miss an important partof the physics. Since the phenomena discovered in the study of the ABJM matrix model probefundamental aspects of string theory (and of the large N expansion), it is clearly important tostudy other examples where these aspects can be understood in detail, and where one can findexact results for the non-perturbative corrections.In this paper we take some steps towards an understanding of what we call “M-theoreticmatrix models,” i.e. matrix models which can be studied in both the ’t Hooft expansion andin an M-theory expansion in which N is large, but the coupling constants are kept fixed. Thematrix models appearing in the localization of Chern–Simons–matter theories are of this type,as required by their duality with M-theory backgrounds, and their M-theory expansion was firstconsidered in [12]. There are other contexts in which similar models appear. For example, manymatrix models considered in [13, 14], which describe ADE models and their affine extensions ona random lattice, are also M-theoretic matrix models. In all these models, the ’t Hooft expansionis likely to miss important ingredients, and it receives non-perturbative corrections which appearnaturally in the M-theory expansion.Unfortunately, the study of M-theoretic matrix models is difficult to pursue beyond ABJMtheory, since in many cases we don’t have a good control of their ’t Hooft expansion, let aloneof the non-perturbative corrections to it . As a matter of fact, even the planar limit of genericChern–Simons–matter matrix models is difficult to obtain explicitly. Examples exist where thislimit is more or less under control [19, 20, 21, 22], but the resulting expressions are often com-plicated and unilluminating.In this paper we analyze two matrix models whose planar limit can be determined exactlyand is relatively simple. They might be exactly solvable in both, the ’t Hooft expansion and ABJ theory [15] has been also extensively studied with similar techniques, see [2, 16, 17, 18]. – 2 –he M-theory expansion, and they represent interesting laboratories to start the exploration ofM-theoretic matrix models beyond ABJM theory. The first model, which we call the N f matrixmodel , calculates the partition function of a three-dimensional N = 4 gauge theory which consistsof an U ( N ) vector multiplet coupled to one hypermultiplet in the adjoint representation and N f hypermultiplets in the fundamental. This theory is dual to M-theory on AdS × S /N f , where thequotient by N f leads to an A N f − singularity [23, 24, 5]. When N f = 1, this theory is equivalentto the ABJM theory with k = 1, but for N f > polymer matrix model , is aparticular case of the models considered in [14], and it appears in the study of 2d polymers andin the calculation of correlation functions in the 2d Ising model. We solve exactly for its planarlimit, and we also study it from the point of view of the Fermi gas, where it displays againnon-perturbative effects which are not captured in the ’t Hooft expansion. As an interestingbonus, we give a Fermi gas derivation of the function determining the short-distance behaviorof the spin-spin correlation functions in the 2d Ising model. Both models, the N f matrix modeland the polymer matrix model, can be regarded as particular cases of the O (2) matrix model[26, 27], and we use the technology developed in [28, 29] to study their planar limit. They arealso closely related to the models with adjoint multiplets studied in [21], but they turn out to besimpler.This paper is organized as follows: in section 2 we give a general overview of M-theoreticmatrix models and their properties. In section 3 we study in detail the N f matrix model. Wesolve for its planar and genus one limit and we study it from the point of view of the Fermigas. In section 4 we study the polymer matrix model using a similar approach. In section 5we state some conclusions and prospects for future work. Appendix A contains some technicalingredients introduced in [28, 29] to solve the O ( m ) matrix model. Appendix B formulates thematrix models studied in this paper as Gaussian models perturbed by multi-trace potentials, andwe explain a method to compute the relevant quantities at small ’t Hooft coupling which can beused to check the exact solution.
2. General aspects of M-theoretic matrix models
In [30, 31, 32], explicit expressions in terms of matrix integrals were found for the partition func-tions on the three-sphere of various Chern–Simons–matter theories with
N ≥ U ( N ) Chern–Simons theory, with levels k and − k , respectively. The partition function depends on N and k and it is given by the matrixintegral, Z ABJM ( N, k )= 1 N ! (cid:90) d N µ (2 π ) N d N ν (2 π ) N (cid:81) i 12 cosh (cid:16) µ i − ν j (cid:17) = (cid:88) σ ∈ S N ( − (cid:15) ( σ ) (cid:89) i 12 cosh (cid:16) µ i − ν σ ( i ) (cid:17) . (2.13)In this equation, S N is the permutation group of N elements, and (cid:15) ( σ ) is the signature of thepermutation σ . After some manipulations, one obtains [35, 5] Z ABJM ( N, k ) = 1 N ! (cid:88) σ ∈ S N ( − (cid:15) ( σ ) (cid:90) d N x (2 πk ) N (cid:81) i (cid:0) x i (cid:1) (cid:16) x i − x σ ( i ) k (cid:17) , (2.14)which can be immediately identified as the partition function of a one-dimensional ideal Fermigas with density matrix ρ ABJM ( x , x ) = 12 πk (cid:0) x (cid:1) / (cid:0) x (cid:1) / 12 cosh (cid:0) x − x k (cid:1) . (2.15)Notice that, by using the Cauchy identity with µ i = ν i , we can rewrite (2.14) as Z ABJM ( N, k ) = 1 N ! (cid:90) N (cid:89) i =1 d x i πk 12 cosh x i (cid:89) i 1. It is easy to see from (2.15) that k plays here the rˆole of Planck’s constant, therefore thisasymptotic behavior can be obtained by adapting WKB techniques [5], and one finds [5, 36, 9] E n ≈ kπ n, n (cid:29) . (2.18)It is easy to see that this behavior leads immediately to the N / behavior of the free energypredicted by classical supergravity [1].A similar analysis can be made for the necklace quivers considered above. The asymptoticbehavior of the energy levels is of the form E n ≈ n as in ABJM theory, but the precise coefficient(which was calculated in [5]) depends on the details of the quiver. The behavior of the energylevels at large n leads to the following behavior at large chemical potential, J ( µ ) ≈ C µ , µ (cid:29) , (2.19)where C depends on the quiver. For example, for ABJM theory one finds J ( µ ) ≈ µ π k , µ (cid:29) . (2.20)The ’t Hooft expansion and the M-theory expansion have been analyzed thoroughly only inthe case of ABJM theory. The results of this analysis can be summarized as follows:1. The free energy contains, at fixed k , a series of perturbative corrections in 1 /N , and on topof that a series of exponentially small corrections at large N , with an N -dependence of theform exp( −√ N ). The perturbative corrections can be obtained either from the ’t Hooftexpansion [2, 37, 38] or from the M-theory expansion [5]. They correspond conjecturallyto perturbative quantum gravity corrections in M-theory [37], and the first, logarithmiccorrection, has been tested against a one-loop calculation in supergravity in [4].2. There are two types of non-perturbative corrections. Worldsheet instanton corrections areobtained naturally in the ’t Hooft expansion, since they depend on N through the ’t Hooftparameter. There are however exponentially small corrections at large N which are non-perturbative in the string coupling constant, and are due to membrane instantons andbound states of membranes and fundamental strings. The pure membrane contributioncan be in principle calculated in the Fermi gas approach [5, 39, 9], while bound statesremain difficult to compute in both approaches [10, 9]. The analytic calculations have beenin addition tested against detailed numerical calculations [40, 36, 7]. The combinationof all these approaches has led to a precise conjectural answer for the full series of non-perturbative corrections, which turn out to be determined by topological string theory andits refinement on a particular local Calabi–Yau manifold [8].The analysis of the ABJM matrix model shows that the ’t Hooft expansion is fundamentallyincomplete , since important non-perturbative effects can not be obtained in this framework. Infact, as noticed in [7, 10], the ’t Hooft expansion leads to unphysical singularities in the freeenergy which need to be cured by the contribution of membranes and bound states (this is theHMO cancellation mechanism). From this point of view, the ’t Hooft expansion is an inconsistenttruncation of the theory. One of the advantages of the Fermi gas approach is that it gives analternatively framework to analyze the large N limit of these matrix models which captures someof these non-perturbative effects, and makes it possible to go beyond the ’t Hooft expansion.– 6 –lthough these results have been obtained for the ABJM model, we expect that similarfeatures will appear in the the matrix models describing Chern–Simons–matter theories, i.e.we expect that the ’t Hooft expansion of these models will miss important non-perturbativeinformation. One reason to believe this is that these models admit an M-theory expansion wherethe large N non-perturbative effects which are invisible in the ’t Hooft expansion are no longersuppressed.In the case of Chern–Simons–matter models, the M-theory expansion is directly related tothe existence of an M-theory dual. However, there are other matrix models which admit in anatural way an analogue of the M-theory expansion, in the sense that one can consider theirbehavior as N becomes large but the rest of the parameters are fixed (instead of scaling with N ,as in the ’t Hooft limit). We will call these models “M-theoretic matrix models.” For example,the matrix models discussed in [14, 13], as well as the matrix model of [41], are of this type. Theexamples considered in this paper are in fact particular examples of the (cid:98) A matrix model of [14]: Z (cid:98) A ( N, g s ) = 1 N ! (cid:90) N (cid:89) i =1 d z i π e − gs V ( z i ) (cid:89) i 3. The N f matrix model The theory we are going to consider is a supersymmetric U ( N ), N = 4 Yang–Mills theory in threedimensions, coupled to a single adjoint hypermultiplet and to N f fundamental hypermultiplets.When N f = 1, this theory is related by mirror symmetry to N = 8 super Yang–Mills theory,therefore to ABJM theory with k = 1 [35]. From the point of view of M-theory, this gauge theoryis supposed to describe N M2 branes probing the space [23, 24], C × (cid:0) C / Z N f (cid:1) , (3.1)where Z N f acts on C as e π i /N f · ( a, b ) = (cid:16) e π i /N f a, e − π i /N f b (cid:17) . (3.2)The corresponding quotient is an A N f − singularity, which can be resolved to give a multi-Taub-NUT space, as expected from the engineering of the theory in terms of D6 branes. The large N dual description of this theory is in terms of M-theory on AdS × S / Z N f , where the action of Z N f is the one inherited by the action on C × C .The standard rules for localization of Chern–Simons–matter theories [30, 31, 32] imply thatthe partition function on the three-sphere S is given by the matrix integral Z ( N, N f ) = 1 N ! (cid:90) N (cid:89) i =1 d x i π (cid:0) x i (cid:1) N f (cid:89) i 2. The reason is that there is a formalism to solve the O ( m ) model for generic m [29] which incorporates in an efficient way the elliptic geometry of the planar solution. We willnow solve the planar limit of the N f matrix model. In the approach of [28, 29], in order to solve the model (3.7), one introduces a planar resolventin the standard way, ω ( p ) = lim N →∞ N (cid:28) Tr 1 p − M (cid:29) , (3.8)where p is an exponentiated variable which lives in the z -plane. In terms of the density ofeigenvalues ρ ( p ), this reads ω ( p ) = (cid:90) d z ρ ( z ) p − z . (3.9)We will assume that our solution has one single cut in the z -plane, located at [ a, b ]. One importantingredient of the solution is of course to find the relationship between the endpoints of the cut andthe ’t Hooft parameter. The solution of the planar limit of the model is encoded in an auxiliaryfunction G ( z ), which was defined and used in [28] to solve the O ( n ) model. It was determinedexplicitly in terms of theta functions in [29]. Since this function will play an important rˆole inthe solution of the model, we list its most important properties in Appendix A. It depends on aparameter ν , which is in turn related to m through the equation m = 2 cos( πν ) . (3.10)Notice that the limit m → ν → 0. According to the results of [29], the endpoints of the cut are determined by thetwo equations M = 12 cos π (1 − ν )2 (cid:73) C d z π i V (cid:48) ( z ) G (1 − ν ) ( z ) = 0 ,M − = 12 cos πν (cid:73) C d z π i zV (cid:48) ( z ) G ( ν ) ( z ) = 12 (2 − m ) λ. (3.11) On the r.h.s of the second equation there is an overall factor of 1/2 w.r.t. the conventions in [29]. – 9 –here λ = g s N is the ’t Hooft parameter of the model, and C is a contour encircling the cut [ a, b ].The indices ν , 1 − ν indicate that the function G should be evaluated for these values of theparameter. These equations generalize the standard conditions determining the endpoints of thecut for the Hermitian one-matrix model. Once the endpoints of the cut have been determined,one should calculate the planar free energy. Our convention for the genus expansion of the freeenergy is F ( N, N f ) = (cid:88) g ≥ g g − s F g ( λ ) . (3.12)A useful result in [29] expresses the third derivative of the planar free energy w.r.t. the ’t Hooftparameter, in terms of the endpoints of the cut a, b :d F d λ = (cid:16) − m (cid:17) b − a (cid:18) e − a a d a d λ − e − b b d b d λ (cid:19) . (3.13)In this equation, e is a function of a , b and ν defined in (A.10). This expression has a well-definedlimit for m → 2, which corresponds to taking ν → 0. In this limit, the prefactor goes to zero,but e diverges, as shown in (A.26). Therefore, only the terms proportional to e (3.13) survive,and one obtains a finite result,d F d λ = − π b − a ) k ( K (cid:48) ( k )) (cid:18) d a d λ − a b d b d λ (cid:19) , (3.14)where K (cid:48) ( k ) = K ( k (cid:48) ) is the elliptic integral of the first kind, with k + ( k (cid:48) ) = 1, and k = a/b ,as in equation (A.3) of the Appendix A. It is possible to integrate this once w.r.t. λ to obtain,d F d λ = − π K ( k ) K (cid:48) ( k ) + constant = − π i τ + constant , (3.15)where τ is given in (A.5).In the matrix model corresponding to the N f model, one has, V (cid:48) ( z ) = 12 z z − z + 1 , (3.16)and we can calculate M and M − by residue calculus. One obtains,12 cos π (1 − ν )2 (cid:16) G (1 − ν ) ( − − G (1 − ν ) (0) (cid:17) = 0 , 12 cos πν (cid:16) − G ( ν ) ( − 1) + cos (cid:16) πν (cid:17)(cid:17) = 12 (2 − m ) λ. (3.17)These two equations have a non-trivial limit as ν → 0, which leads to the solution of the model.We will however analyze a slightly different set of equations which were obtained by Suyamain a closely related context. In [21], the planar limit of supersymmetric Chern–Simons with n adjoint multiplets was analyzed in detail, by using as well the correspondence with the O ( m )matrix model. However, the definitions of the resolvent and the map to the O ( m ) model wereslightly different from the ones explained above. To see how this goes, let us first extend ouroriginal matrix integral (3.3) to the case in which there are n adjoint multiplets, Z ( N, N f , n ) = 1 N ! (cid:90) N (cid:89) i =1 d x i π (cid:0) x i (cid:1) N f (cid:89) i 1, and one finds from [21] λ = − n − (cid:20) i2 e sin πν G ( − 1) + 12 (cid:21) , (3.29)where e is again given in (A.10) and G ( − 1) is the function G ( z ), evaluated at z = − 1. It is notobvious that the above expression has a smooth limit when n → 1, but this is the case, and inthis limit (3.29) is equivalent to the limit m → G ( z ) around the point ν = 1. After somecalculations, which are sketched in the Appendix, one finds the surprisingly simple equation λ = − 18 + (1 + k ) π K (cid:48) ( k ) , (3.30)where the elliptic modulus is k = a (3.31)This of course is a particular case of (A.3) when b = 1 /a .The above equation determines the ’t Hooft parameter as a function of the endpoint of thecut a . It is immediate to verify that, for a = 1, λ = 0, as it should. The free energy now followsfrom (3.14) and (3.30). One finds,d F d λ = 4 π (1 + a ) ( K (cid:48) ( k )) E (cid:48) ( k ) − a K (cid:48) ( k ) . (3.32)It is instructive to compare these results with a direct calculation of the endpoint of thecut and the free energy around λ = 0, by treating (3.3) as a multi-trace matrix model. Thisperturbative method is explained in Appendix B, and leads to the following expansions: A λ ) = 4 λ − λ λ − λ 105 + O ( λ ) ,F ( λ ) = λ (cid:18) log λ − (cid:19) − log(2) λ − λ λ − λ O ( λ ) , (3.33)where A = − log a is the endpoint of the cut in the x -plane, see (3.25). The weak-couplingexpansion, together with (3.32), determines completely F ( λ ). The integration constant in (3.15)can be fixed by the second equation in (3.33), and it turns out to be zero, therefore,d F d λ = − π i τ = − π K ( a ) K ( √ − a ) . (3.34) The next-to-leading term in the ’t Hooft expansion (3.12) is the genus one free energy F ( λ ). Ageneral expression for this free energy in the O (2) matrix model has been found in [28] . In ourcase, this reads: F ( λ ) = − 124 log( M J ) − 16 log(1 /a − a ) − 14 log( K ( k a ) K ( k b )) , (3.35) The expression written down in [28] seems to have some misprints: the term a / 48 should be log( a ) / 48, andfor general a, b , one should have k a = (1 − a /b ) / . – 12 –here k b = − /a + 1 , k a = − a + 1 , (3.36)and M and J are moments given by contour integrals, which can be computed explicitly interms of elliptic integrals of the first, second and third kinds, M = (cid:73) [ a,b ] d z π i z − z + 1) 1( z − a ) / ( z − a − ) / = 12 a ( − a ) (1 + a ) π (cid:40) − (cid:0) a (cid:1) E (cid:20) − a a (cid:21) +2 a (cid:18)(cid:0) a + 2 a + 3 (cid:1) K (cid:20) − a a (cid:21) − a Π (cid:20) ( − a ) a , − a a (cid:21)(cid:19) (cid:41) ,J = (cid:73) [ a,b ] d z π i z − z + 1) 1( z − a ) / ( z − a − ) / = − 12 ( − a ) (1 + a ) π a (cid:40) (cid:0) a (cid:1) E (cid:20) − a a (cid:21) − (cid:18)(cid:0) a + 3 a (cid:1) K (cid:20) − a a (cid:21) − a Π (cid:20) ( − a ) a , − a a (cid:21)(cid:19) (cid:41) . (3.37)The weak coupling expansion of (3.35) is given by F ( λ ) = − log( λ )12 − 16 log (cid:18) π (cid:19) + 3 λ − λ 24 + 25 λ − λ 48 + O ( λ ) . (3.38)This expansion matches with a direct perturbative computation in the matrix model. The function v ( z ), which contains all the information about planar correlators, can be alsoobtained from the results of [21]. It has the form, v ( z ) = 1 n − f ( z ) + nf ( − z )) + ω ( z ) , (3.39)where f ( z ) = − z − z + 1 , (3.40) ω ( z ) = − i (cid:16) e i πν/ ω + ( z ) − e − i πν/ ω + ( − z ) (cid:17) , (3.41)and ω + ( z ) = 12( n − e cn( u )dn( u ) z − G ( − (cid:0) z − G + ( z − ) + zG + ( z ) (cid:1) . (3.42)The variable u is related to z through (A.2). The above expressions are obtained for generic n .We can now take the limit n → 1. All the apparent divergences cancel, and we find the explicit– 13 –xpression v ( z ) = 12 π z ( − z ) (cid:110) π z (cid:0) − z + 2 λ (cid:0) z (cid:1)(cid:1) + a (cid:0) K (cid:48) ( k ) (cid:1) + z (cid:0)(cid:0) z + z (cid:1) A ( u )( π + A ( u )) − a cn( u )dn( u )( π + 2 A ( u )) K (cid:48) ( k ) − (cid:0) a + a (cid:1) z (cid:0) K (cid:48) ( k ) (cid:1) (cid:17)(cid:111) , (3.43)where A ( u ) = π K ( k ) u + K (cid:48) ( k ) (cid:18) ϑ (cid:48) ϑ (cid:19) (cid:18) u K ( k ) (cid:19) . (3.44)Notice that this function is not algebraic in z , in contrast to the resolvent of ABJM theory [49, 2].By using the expansion (3.21), we can extract the exact value of the Wilson loop vev,2 λ (cid:104) W (cid:105) = − 12 + 14 aK ( k ) + K (cid:48) ( k )2 aπ + aK (cid:48) ( k )2 π − E ( k ) K (cid:48) ( k )2 aK ( k ) π . (3.45)This can be expanded near λ = 0, and one finds, (cid:104) W (cid:105) = 1 + 2 λ + 4 λ + O ( λ ) , (3.46)which agrees with an explicit perturbative computation.Finally, we derive an explicit expression for the density of eigenvalues. From the standarddiscontinuity equation, we have ρ ( z ) = − π i v ( z )2 λz (cid:12)(cid:12)(cid:12)(cid:12) z +i0 z − i0 = 12 π i v ( w )2 λa sn( w ) (cid:12)(cid:12)(cid:12)(cid:12) K ( k ) − i wK ( k )+i w , (3.47)where z = a sn( K ( k ) + i w ) . (3.48)We find the explicit expression ρ ( z ( w )) = i4 a cd(i w ) ( − a cd(i w ) ) K ( k ) π λ × (cid:18) (cid:0) − a (cid:1) K ( k )nd(i w )sd(i w ) K (cid:48) ( k )+ (cid:0) cd(i w ) + a cd (i w ) (cid:1) (cid:0) iπw + 2 (cid:0) − a (cid:1) K ( k )nd(i w )sc(i w ) K (cid:48) ( k ) + 2 K ( k ) Z (i w ) K (cid:48) ( k ) (cid:1) − a cd(i w ) K ( k )sn(i w ) K (cid:48) ( k ) − a cd (i w ) K ( k )sn(i w ) K (cid:48) ( k ) (cid:19) . (3.49)It can be checked that ρ ( a ) = ρ ( z (0)) = 0 , ρ (1 /a ) = ρ ( z ( K (cid:48) ( k ))) = 0 , (3.50)as it should. The explicit form of ρ is shown in Fig. 1 for a = 1 / (cid:45) (cid:45) Figure 1: The density of eigenvalue (3.49) for a = 1 / x plane (left) and in the z = e x plane(right) . We will now explore the strong coupling limit of the planar and genus one solution found above.Since all the quantities depend on λ through a –the endpoint of the cut in the z plane– the firstthing to do is to find the relation between a and λ for λ large. It is convenient to define the newvariable ˆ λ = λ + 18 . (3.51)This shift is reminiscent of the shift in − / 24 which appears in the exact planar solution ofABJM theory [5, 3], and it might be explained along the same lines, i.e. it might correspond toa correction to the D-brane charge [50, 51]. If so, it would give a very interesting check of theproposed dual geometry C × (cid:0) C / Z N f (cid:1) .The exact relation (3.30) indicates that large ˆ λ corresponds to a → 0, and the leading orderbehavior is easily found to be a ≈ e − π √ λ , ˆ λ (cid:29) . (3.52)This means that A , the endpoint of the cut in the x -plane, grows like √ λ for large ’t Hooftparameter. This is similar to the behavior in ABJM theory [5, 2]. It is possible to invert (3.30)at large λ , to all orders, and find an expansion of the form a = 2 e − π √ λ ∞ (cid:88) k =1 k (cid:88) (cid:96) =0 a k,(cid:96) e − kπ √ λ ˆ λ (cid:96)/ . (3.53)For the first few terms, we find a = 2 e − π √ λ (cid:26) π (cid:112) λ e − π √ λ + (cid:16) π ˆ λ − − π (cid:112) λ (cid:17) e − π √ λ + · · · (cid:27) . (3.54)We can use the above results, together with (3.34) and (3.35), to obtain the expansion of theplanar and genus one free energy at strong ’t Hooft coupling. We find, for λ (cid:29) F ( λ ) = − π √ 23 ˆ λ / + c + F WS0 ( λ ) ,F ( λ ) = π (cid:112) λ − log(2ˆ λ )4 − 13 2 log(2) − π )12 + F WS1 ( λ ) , (3.55)– 15 – .0 0.1 0.2 0.3 0.4 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 2: Comparison of the exact result for d F / d λ given in (3.34), and plotted in a continuous blueline, with the strong and weak coupling behavior. The red dashed line represents the strong couplingbehavior (3.55), while the black dashed line represents the Gaussian weak coupling behavior (3.33). where c is a constant of integration which is determined by matching carefully the weak-couplingexpansion (3.33) to the above asymptotic expansion. One finds numerically c ≈ . . More-over, F WS g ( λ ) = ∞ (cid:88) k =1 k (cid:88) (cid:96) =0 f ( g ) k,(cid:96) e − kπ √ λ ˆ λ (cid:96)/ (3.56)is the contribution of non-perturbative corrections. We will refer to it as the worldsheet instantoncontribution. These corrections should be due, as in ABJM theory, to worldsheet instantons inthe type IIA superstring dual. For the first few terms, we find F WS0 ( λ ) = − e − π √ λ π (cid:16) π (cid:112) λ (cid:17) − e − π √ λ π (cid:16) π (cid:112) λ + 64 π ˆ λ (cid:17) + · · · ,F WS1 ( λ ) = (cid:18) − π (cid:112) λ (cid:19) e − π √ λ + (cid:18) π (cid:112) λ − (cid:19) e − π √ λ + · · · . (3.57)Some comments are in order concerning these expressions. First of all, the leading term in (3.55)has the same form as in ABJM theory, and N f plays the rˆole of k . This is in agreement withthe analysis in the strict large N limit, with N f fixed, performed in [53], and more recently in[25]. However, the structure of the subleading exponential terms is different: in ABJM theory,the powers of ˆ λ appearing in (3.56) are negative. It would be interesting to understand this interms of the expansion around the conjectural dual worldsheet instantons in the dual type IIAtheory background. The model with partition function (3.3) can be also studied in the Fermi gas approach [5]. Thisis a very useful formulation since one can study both the ’t Hooft expansion and the M-theoryexpansion, and they lead to two different types of non-perturbative effects. Therefore, the Fermigas approach makes it possible to go beyond the 1 /N expansion of the matrix model. After the first version of this paper appeared, Hatsuda and Okuyama conjectured in [52] that c = (cid:0) log(2) − (cid:0) ζ (3) /π (cid:1)(cid:1) / – 16 –o obtain the Fermi gas picture for the matrix model with partition function (3.3), we usethe Cauchy identity (2.13) for µ i = ν i . We find that (3.3) can be written as Z ( N, N f ) = 1 N ! (cid:88) σ ∈ S N ( − (cid:15) ( σ ) (cid:90) N (cid:89) i =1 d x i π (cid:0) (cid:0) x i (cid:1)(cid:1) N f (cid:16) x i − x σ ( i ) (cid:17) . (3.58)The corresponding kernel is given by ρ N f ( x , x ) = 12 π (cid:0) x (cid:1) N f / (cid:0) x (cid:1) N f / 12 cosh (cid:0) x − x (cid:1) . (3.59)In the Fermi gas approach, the basic quantity is the grand potential of the theory, ratherthan the partition function. As noted in [41], the ’t Hooft expansion of the partition function(3.3) leads naturally to a “genus” expansion of the grand potential, which is of the form J ( µ, N f ) = ∞ (cid:88) g =0 N − gf J g (cid:18) µN f (cid:19) . (3.60)This expansion contains exactly the same information than the ’t Hooft expansion of the (canon-ical) partition function, and it is related to it by the usual thermodynamic transform. In partic-ular, the genus zero piece J is just given by the Legendre transform of the planar free energy:we first solve for λ , the ’t Hooft parameter, in terms of µ/N f , through the equation µN f = − d F d λ , (3.61)and J (cid:18) µN f (cid:19) = F ( λ ) − λ d F d λ . (3.62)Similarly the genus one grand potential J is related to the genus one free energy F through aone loop saddle point: J (cid:18) µN f (cid:19) = F (cid:18) µN f (cid:19) + 12 log (cid:18) N f ∂ µ J (cid:18) µN f (cid:19)(cid:19) − 12 log(2 π ) . (3.63)Equivalently, since λ is in one-to-one correspondence with the endpoint of the cut a , and allrelevant quantities are expressed in terms of a , we can express a in terms of µ/N f . a = 2e − µNf (cid:88) n ≥ e − nµNf n (cid:88) l =0 b n,l (cid:18) µN f (cid:19) l , (3.64)and then plug this in the r.h.s. of (3.62). One obtains, N f J (cid:18) µN f (cid:19) = 2 µ π N f − µN f N f ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m =1 a (0) (cid:96),m (cid:18) µN f (cid:19) m e − (cid:96)µ/N f ,J (cid:18) µN f (cid:19) = 12 µN f + ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m =1 a (1) (cid:96),m (cid:18) µN f (cid:19) m e − (cid:96)µ/N f , (3.65)– 17 –here J WS g (cid:18) µN f (cid:19) = ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m =0 a ( g ) (cid:96),m (cid:18) µN f (cid:19) m e − (cid:96)µ/N f . (3.66)For the very first orders we find, J WS0 (cid:18) µN f (cid:19) = − π (cid:18) 14 + µN f (cid:19) e − µ/N f + 1 π (cid:32) − 732 + 14 µN f − (cid:18) µN f (cid:19) (cid:33) e − µ/N f + · · · ,J WS1 (cid:18) µN f (cid:19) = (cid:18) − − µN f (cid:19) e − µNf + (cid:18) − µN f (cid:19) e − µNf + · · · . (3.67)The exponentially small corrections in µ in (3.67) are due to the worldsheet instanton contribu-tions to the planar and genus one free energy (3.57). The structure of these corrections is quitedifferent from what is obtained in ABJM theory. In this theory, the rˆole of N f is played by k ,and after factoring out an overall factor k appearing in genus 0, one finds a simpler structurefor the worldsheet instantons J WS0 (cid:16) µk (cid:17) = 14 π ∞ (cid:88) (cid:96) =1 N (cid:96) e − (cid:96)µ/k , (3.68)where the coefficients N (cid:96) are related to the genus zero Gromov–Witten invariants of the non-compact Calabi–Yau local P × P [7].The planar limit gives us information about the behavior of the theory when N f large and N/N f is fixed. In the M-theory regime of the theory, we should take N large and N f fixed. In thisregime, based on the results of [5], we should expect new non-perturbative effects which are notdue to worldsheet instantons, but rather to membrane instantons. In order to study the M-theoryregime, one has in principle to obtain information about the spectrum of the operator (3.59) forfinite N f . This is however a difficult problem. One can then try to study the grand potentialof the theory in some approximation scheme. In [5, 39], various techniques were developed tounderstand the small k regime of the ABJM model. Since k is essentially the Planck constant ofthe Fermi gas, this is a WKB approximation However, in the model with density matrix (3.59),the Planck constant is fixed and set to 2 π , so in principle we can not use the WKB method.However, it was shown in [5] that the perturbative part in µ of the grand potential only receivesquantum corrections up to next-to-leading order, and this was recently used in [25] to calculateit. They obtain: J p ( µ ) = 23 π N f µ + (cid:18) N f − N f (cid:19) µ. (3.69)From the point of view of the ’t Hooft expansion, this expression contains information about thegenus zero and the genus one pieces of the grand potential: N f J (0)p ( µ/N f ) = 23 π N f µ − N f µ, J (1)p ( µ/N f ) = 12 µN f . (3.70)This is in agreement with the result in (3.65). Notice that, at large N (equivalently, large µ ), theleading part of the grand potential (3.69) is the cubic part coming from the planar limit. Thismeans that the M-theory limit agrees with the strong coupling expansion of the planar limit, inaccord with the planar dominance conjecture of [45].– 18 –e are interested in calculating non-perturbative corrections to (3.69), i.e. corrections whichare exponentially small in µ . In the model (3.3), N f plays the rˆole of k , and one could try tostudy the regime N f → 0. To understand the physical nature of this limit, notice that, for largeenergies, the Hamiltonian corresponding to (3.59) is of the form H ≈ log (cid:16) p (cid:17) + N f log (cid:16) q (cid:17) , (3.71)and the limit N f → Z = 14 π (cid:90) ∞−∞ d q (cid:0) q (cid:1) N f = 14 π Γ ( N f / N f ) = 1 πN f − πN f 24 + ζ (3)4 π N f + · · · , (3.72)which diverges as O ( N − f ) when N f → 0. We then have to extract the leading term in 1 /N f . Todo this, we rescale q = x/N f as in [25]. In this way we have an explicit Planck constant in themodel, (cid:126) = 2 πN f , but we also introduce an explicit N f dependence in the Hamiltonian: H ≈ log (cid:16) p (cid:17) + N f log (cid:18) x N f (cid:19) . (3.73)This prevents us from applying the WKB method to this problem. We can still extract thoughthe leading contribution to J ( µ ) as N f → 0, because, in this limit, quantum corrections aresuppressed. The Hamiltonian becomes H ≈ log (cid:16) p (cid:17) + | x | , N f → . (3.74)The function (2.11) becomes, in the limit N f → Z (cid:96) ≈ (cid:90) d p d x π (cid:126) e − (cid:96) | x | / (cid:0) p (cid:1) (cid:96) = 1(2 π ) N f Γ ( (cid:96)/ (cid:96) ) 4 (cid:96) , (3.75)and by using (2.12) we find J ( z ) ≈ − π N f (cid:88) (cid:96) ≥ Γ ( (cid:96)/ (cid:96) ) ( − z ) (cid:96) (cid:96) , N f → . (3.76)This infinite sum in the r.h.s. of (3.76) can be expressed in terms of hypergeometric functions, N f J ( z ) ≈ zπ F (cid:18) , , 12 ; 32 , 32 ; z (cid:19) − z π F (cid:18) , , , 1; 32 , , z (cid:19) , N f → , (3.77)and the derivative w.r.t. z has a simpler expression, ∂J∂z ≈ π zN f arcsin (cid:16) z (cid:17) (cid:16) π − arcsin (cid:16) z (cid:17)(cid:17) , N f → . (3.78)Although the building blocks of the functions appearing in (3.77) and (3.78) have branch cuts inthe complex z plane along the positive real axis, and starting at z = 2, the branch cut at positive– 19 – disappears in the final answer. This is as it should be, and in accord with previous examplesin [5, 39]: for a quantum Fermi gas, there is no physical source of non-analyticity in the grandpotential at large fugacity. One can then make an expansion at µ large to obtain, N f J ( µ ) ≈ µ π + µ ζ (3) π + J np0 ( µ ) , N f → , (3.79)where J np0 ( µ ) = (cid:88) (cid:96) ≥ (cid:16) a (cid:96) µ e − (cid:96)µ + b (cid:96) e − (cid:96)µ (cid:17) = 1 π (2 µ + 1) e − µ + 18 π (12 µ − 1) e − µ + · · · . (3.80)The perturbative part in µ of (3.79) agrees with the leading part of (3.69), at leading order in N f . We find, in addition, exponentially small corrections in µ . Since µ ≈ π (cid:114) N f N ≈ πN f (cid:114) λ , (3.81)these corrections are non-perturbative from the point of view of the ’t Hooft expansion, whichis an expansion in 1 /N f at λ fixed. They are presumably due to membrane instantons in theM-theory dual. All the information about the partition function (3.3) and the corresponding grand potential isencoded in the spectrum of the density matrix (3.59). As in the case of ABJM theory [5], thisdensity matrix can be regarded as a positive Hilbert–Schmidt kernel and its spectrum, definedby ∞ (cid:90) −∞ ρ N f ( x , x ) φ n ( x )d x = e − E n φ n ( x ) , n ≥ , (3.82)is discrete. We have ordered it as, E < E < · · · . (3.83)When N f = 1, the spectrum of this operator is the same as the spectrum of (2.15) for k = 1,and one can apply the results obtained in [9]. Unfortunately, for general N f it doesn’t seem tobe possible to obtain analytic results for the eigenvalues E n , or an exact quantization conditionas in [9]. The leading, large n behavior of E n can be obtained by using the techniques of [5], andit can be read immediately from (3.69). Indeed, since we are dealing with an ideal Fermi gas,the grand potential can be computed from the quantum volume of phase space vol( E ) as J ( µ, N f ) = 12 πN f (cid:90) ∞ E vol( E )d E e E − µ + 1 + · · · , (3.84)where E is the ground state energy, and the · · · denote subleading corrections which appearwhen we pass from the discrete sum over eigenvalues to the integration over the volume of phasespace. The pertubative part of µ computed in (3.69) comes from the polynomial part of vol( E ),vol p ( E )2 πN f = 2 π N f E − N f − N f , (3.85)– 20 –hich follows from the general results of [5]. Using now the WKB quantization conditionvol( E n ) = 2 πN f (cid:18) n + 12 (cid:19) , n ≥ , (3.86)we find the leading behavior, E (0) n = π (cid:18) N f (cid:19) / (cid:18) n + 12 + N f N f (cid:19) / . (3.87)The non-perturbative corrections to J ( µ, N f ) correspond to non-perturbative corrections tovol( E ), as shown in detail in [9]. As in ABJM theory, we expect two types of non-perturbativecorrections, of the form e − (cid:96)E , e − (cid:96)E/N f , (cid:96) ≥ . (3.88)The first type is due to membrane-type corrections, i.e. to the exponentially small correctionsappearing in (3.80), which are invisible in the ’t Hooft expansion. The second type is due toworldsheet instantons, i.e. to the exponentially small terms appearing in for example (3.67).Although we do not have an exact asymptotic expansion for generic N f , as we have in ABJMtheory, we have results at small N f for membrane corrections, coming from (3.79), as well asresults at large N f for worldsheet instanton corrections, coming from (3.67).Let us first analyze the behavior at large N f . It is clear that, in this regime, the leadingexponentially small correction is due to the first worldsheet instanton correction. By using (3.84)as well as the results of [9], we find that it leads to an exponentially small correction to thequantum volume of phase space of the form,vol( E )2 πN f ≈ vol p ( E )2 πN f + 4 π E e − E/N f , N f (cid:29) . (3.89)Using the WKB quantization condition, we find a correction to the spectrum of the form E n ≈ E (0) n − N f e − E (0) n /N f , N f (cid:29) . (3.90)Let us now look at the behavior at N f → 0. In this case, the grand potential is given by (3.77).The leading non-perturbative correction in (3.80) leads to a correction to the energy levels of theform, E n ≈ E (0) n − E (0) n e − E (0) n , N f (cid:28) . (3.91)We will now check some of these analytic results against explicit, numerical calculations ofthe spectrum. By using the techniques of [7], one can easily show that this integral equation isequivalent to an eigenvalue equation for an infinite dimensional Hankel matrix M with entries M nm = 14 π N f (cid:90) ∞−∞ d q tanh n + m ( q/ N f +2 ( q/ 2) = 12 π N f Γ (cid:0) + m + n (cid:1) Γ (cid:16) N f + 1 (cid:17) Γ (cid:16) + m + n + N f (cid:17) , (3.92)when m + n is even, m, n ≥ 0, otherwise it vanishes. The energy eigenvalues are obtained bydiagonalizing M nm . To implement this numerically, one truncates the matrix M to an L × L – 21 – E (0) n E num n Table 1: The numerical eigenvalues E num n against the theoretical leading order result E (0) n , for N f = 2. matrix. The eigenvalues of the truncated matrix, E n,L , will converge to E n as L → ∞ . In orderto improve our numerical approximation we apply Richardson extrapolation to E n,L = E n + (cid:88) i ≥ E in L i , (3.93)as in [9]. Using this procedure, we have computed the first energy levels for various values of N f .We will compare these numerical results with the predictions coming from (3.67) and (3.69). Asa first check we can compare the numerical eigenvalues E num n to (3.87). The results are shown inTable 1 for N f = 2, where we show only the first digits. As expected, (3.87) becomes increasinglygood as n is large.Next we would like to test exponentially small corrections. The study of the spectrum forsmall values of N f is more difficult, since one needs very good numerical precision. We will thenfocus on the study of the large N f , where the dominant non-perturbative correction is (3.90).This gives a nice numerical verification of the analytical result for the planar limit. To this endwe consider the following sequence: c n = − N f log (cid:32) E n − E (0) n E n − − E (0) n − (cid:33) E (0) n − E (0) n − . (3.94)According to (3.90) we should have c n → , n → ∞ . (3.95)This is confirmed by the numerical data, as shown in Figure 3 for N f = 100 (we have verified itfor other values of N f as well).As a last check of the large N f behavior, we test the − N f coefficient of the exponential in(3.90). Let us consider the following sequence:log (cid:16)(cid:12)(cid:12)(cid:12) a ( N f ) n (cid:12)(cid:12)(cid:12)(cid:17) = log (cid:16)(cid:12)(cid:12)(cid:12) E n − E (0) n (cid:12)(cid:12)(cid:12)(cid:17) + 4 N f E (0) n . (3.96)According to (3.90) we should have a ( N f ) n → − N f , n → ∞ . (3.97)This is confirmed by the numerical data, as shown in Figure (3) for N f = 100. Our best numericalapproximations, after applying Richardson transformations and taking n sufficiently large, give a (100) n ≈ − . , a (50) n ≈ − . , a (40) n ≈ − . , n (cid:29) , (3.98)which compare well to the theoretical value. – 22 – Figure 3: Left: the sequence (3.94) for N f = 100 with its 3 th and 4 th Richardson transform. The straightline is the analytic prediction. Right: the sequence (3.96) with its 3 th and 4 th Richardson extrapolation,again for N f = 100. The straight line is the analytic prediction. 4. The polymer matrix model The polymer matrix model is defined by the partition function Z ( N, t ) = 1 N ! (cid:90) N (cid:89) i =1 d x i π e − t cosh x i (cid:89) i 1, makes it possible to compute the universal scaling functions of a two-dimensional, self-avoiding, non-contractible polymer on a cylinder. This is why we refer to (4.1)as the polymer matrix model. It was also shown in [56, 57], based on previous insights in [58, 59],that the dependence on the coupling constant t is encoded in an integrable hierarchy of the KdVtype, which specializes to the sinh-Gordon and the Painlev´e III equations.In all these applications, the large N limit of the integral (4.1) does not play a crucial rˆole,since the grand potential has to be found at finite values of the fugacity ( z = − z = 2).However, a generalization of the matrix integral (4.1) can be used to study the six-vertex model– 23 –n a random lattice [60]. The planar limit of the model corresponds, as usual, to a lattice ofspherical topology.As we will see, the polymer matrix model displays a behavior which is very different fromthe one in the ABJM and the N f matrix models. However, it shares a common feature withthem: it can be studied in two different regimes, namely the ’t Hooft expansion and what wehave called an M-theory expansion. The ’t Hooft expansion is the regime in which N → ∞ , t → ∞ , λ = Nt fixed . (4.5)Indeed, when regarded as an integral over eigenvalues, the polymer matrix model has a tanhinteraction between them and a potential V ( x ) = cosh( x ). Therefore, the coupling t plays therˆole of 1 /g s , and λ is then the natural ’t Hooft parameter. On the other hand, by using again theCauchy identity (2.13), we can interpret the partition function (4.1) as the canonical partitionfunction of an ideal Fermi gas with kernel ρ ( x , x ) = 12 π e − t cosh( x ) − t cosh( x ) (cid:0) x − x (cid:1) . (4.6)The Hamiltonian of this gas is, in the leading semiclassical approximation, H ( x, p ) ≈ log (cid:16) p (cid:17) + t cosh( x ) . (4.7)The M-theory expansion of (4.1) is the regime in which N → ∞ , t fixed , (4.8)and it corresponds to the thermodynamic limit of the quantum Fermi gas, for a fixed value of thecoupling in the potential. It is also possible to interpret the matrix integral as a classical gas witha one-body interaction given by t cosh( x ), and a two-body interaction given by tanh(( x − x (cid:48) ) / We will first study the ’t Hooft expansion of the polymer matrix model, and we will determinethe exact planar limit of the free energy by using similar techniques to those used for the N f matrix model. Indeed, the polymer matrix model is closely related to the N f matrix modelstudied in detail in the previous section, since the interaction between eigenvalues is the same,and only the potential differs. Therefore, after the change of variables z = e x , we can also regardit as an O (2) model with potential V ( z ) = 12 ( z + z − ) , (4.9)We can immediately obtain the equations for the endpoints of the cut, by adapting (3.11) to oursituation. We find,12 cos π (1 − ν )2 (cid:73) C d z π i V (cid:48) ( z ) G (1 − ν ) ( z ) = 0 ⇒ 12 cos π (1 − ν )2 (cid:16) G (1 − ν ) (cid:17) (cid:48) (0) = 112 cos πν (cid:73) C d z π i zV (cid:48) ( z ) G ( ν ) ( z ) = 12 (2 − m ) λ ⇒ − 12 cos πν G ( ν ) (0) + c = λ (2 − m ) , (4.10)– 24 –here λ is the ’t Hooft parameter defined in (4.5), and c is defined by the asymptotics12 cos πν G ( ν ) ( z ) = 1 z + c z + · · · (4.11)as z → ∞ .The first equation is satisfied if a = 1 /b , as in the previous model. The ’t Hooft parameteris obtained by studying the limit ν → λ ( a ) = − π + (cid:0) E ( k ) + (cid:0) − k (cid:1) K ( k ) (cid:1) K (cid:48) ( k )4 k / πK ( k ) , (4.12)where k = a . (4.13)The planar free energy can be computed by using (3.15), and the integration constant can befixed against the behavior near the Gaussian point λ → 0. One finds,d F d λ = − π i τ . (4.14)By expanding around λ = 0 we get a =1 − √ λ + 2 λ − λ / − λ + O ( λ / ) ,F ( λ ) = 12 λ (cid:18) log (cid:18) λ (cid:19) − (cid:19) − λ − λ λ − λ 64 + O ( λ ) . (4.15)This agrees with an explicit perturbative computation. We will now look at the ’t Hooft expansion the limit λ → ∞ . The exact relation (4.12) indicatesthat large λ correspond to a → 0. More precisely, one finds a ( λ ) ≈ πλ W (cid:18) πλ e (cid:19) , λ (cid:29) , (4.16)where W ( x ) is the principal branch of the Lambert function W ( x ), defined by W ( x )e W ( x ) = x. (4.17)By using (4.14) this leads to F ( λ ) ≈ π λ W (cid:0) πλ e (cid:1) (cid:26) (1 − (cid:20) πλ W (cid:18) πλ e (cid:19)(cid:21)(cid:27) . (4.18)By expanding the Lambert function at infinity one gets F ( λ ) ≈ − π λ λ ) (cid:18) O (cid:18) log (log( λ ))log( λ ) (cid:19)(cid:19) . (4.19)Notice that, in this case, the leading order behavior of the free energy is very different from theone appearing in the ABJM matrix model and in the N f matrix model. In addition, we don’thave exponentially small corrections in λ . Of course, in the polymer matrix model we have avery different type of potential, which grows exponentially and not linearly, and this leads to adifferent structure for the free energy. – 25 – .3 Grand potential and non-perturbative effects Let us now analyze the polymer matrix model (4.1) from the point of view of the Fermi gas. Aswe reviewed in section 3.6, the ’t Hooft expansion of the matrix model leads to a genus expansionof the grand potential, which now has the structure J ( µ, t ) = ∞ (cid:88) g =0 t − g J g (cid:16) µt (cid:17) . (4.20)The relation (3.61) in the strong coupling regime leads to µ ( a ) = πt a + O ( a ) . (4.21)Together with (3.62) this leads to J ( µ ) = 12 π (cid:16) µt (cid:17) (cid:18) − − log (cid:18) π (cid:19) − (cid:18) tµ (cid:19)(cid:19) + O (cid:18) t µ (cid:19) . (4.22)Let us now compare this with a computation in the M-theory limit. As in the other models,working at finite t is difficult, but since t plays the rˆole of N f in this model, we can analyze theregime in which t → 0. As we argued before in the case of the N f model, in this limit we canuse the semiclassical approximation. Since2 (cid:90) ∞ e − (cid:96)t cosh q d q = 2 K ( (cid:96)t ) ≈ − t ) , t → , (4.23)where K ( z ) is a modified Bessel function of the second kind, we can calculate Z (cid:96) as Z (cid:96) ≈ − π log( t ) Γ ( (cid:96)/ (cid:96) ) , t → . (4.24)By using (2.12), and summing the resulting infinite series, we conclude that J ( z ) ≈ log( t ) π (arcsin( z/ − π ) arcsin( z/ , t → . (4.25)This is a well-known result [58, 61], although the above derivation seems to be simpler than theexisting ones. In particular, the quantum Fermi gas approach to (4.1) seems to be more powerfulin obtaining this result than the classical gas approach of [61, 62], where one has to treat theinteraction term by a Mayer expansion. Notice that J (2) ≈ − log( t )4 , t → , (4.26)and Ξ(2 , t ) ≈ (cid:18) t (cid:19) , t → , (4.27)which is the expected behavior for the correlator of order/disorder operators in the 2d Isingmodel (see [54]). – 26 –n the other hand, the expression (4.25) behaves at large µ as J ( µ ) ≈ log (cid:18) t (cid:19) µ π + 14 + (cid:88) (cid:96) ≥ ( a (cid:96) µ + b (cid:96) ) e − (cid:96)µ . (4.28)Again, the exponentially small terms at large µ are invisible in the ’t Hooft expansion, andcorrespond to non-perturbative effects in the M-theory regime of large N , small t .The term (4.25) is just the first term in an expansion of J ( z ) at small t but all orders in z . The next terms in this expansion can be computed systematically by using the integrablestructure of the KdV type underlying the matrix integral (4.1). The next terms in the expansionhave been computed in [56]: J ( µ ) = σ ( σ + 2)4 log (cid:18) t (cid:19) + B ( σ ) + O (cid:0) t ± σ (cid:1) , (4.29)where σ = − π arcsin (cid:16) z (cid:17) , B ( σ ) = 14 (cid:90) σ d x (1 + x ) (cid:20) ψ (cid:18) x (cid:19) + ψ (cid:18) − − x (cid:19) − (cid:21) . (4.30)By doing a large µ expansion of (4.29) one finds J ( z ) = − µ π + log( π ) π − log (cid:0) t (cid:1) π + log (cid:16) µ (cid:17) π + O (cid:0) t ± σ (cid:1) + O ( µ log( µ )) . (4.31)The terms of order µ match the result (4.22) obtained in the ’t Hooft expansion. This is similarto the phenomenon observed in [5] in the matrix integrals appearing in Chern–Simons–mattertheories, namely, that the leading, perturbative terms in µ are the same in both, the ’t Hooftexpansion and the M-theory expansion. This is again in agreement with the planar dominanceconjecture of [45]. 5. Conclusions In this paper we have studied matrix models which have, on top of the usual ’t Hooft regime, anM-theoretic regime. These models arise naturally in the localization of Chern–Simons–mattertheories with M-theory duals, but also in other contexts, like for example the statistical modelsconsidered in [14, 13]. An important property of these models is that their ’t Hooft expansion isinsufficient, and has to be complemented by considering non-perturbative effects which appearnaturally in the M-theory regime.Our main example has been the matrix model which computes the partition function on thesphere of an N = 4, 3d U ( N ) gauge theory with one adjoint and N f fundamental hypermultiplets.This theory has a proposed M-theory dual and shares many properties with ABJM theory. Wehave solved exactly for its planar and genus one limit and started the study of its non-perturbativecorrections beyond the ’t Hooft expansion. A similar model, the polymer matrix model, arisesin the study of statistical systems in two dimensions, and we have performed a similar analysis.The results presented here are just a first step in a more ambitious program which aims at afull understanding of M-theoretic matrix models. In this program, the two matrix models whichwe have studied will probably play an important rˆole and might be completely solvable, along– 27 –he lines of the proposed solution of the ABJM matrix model. However, it is clear that thereare many technical obstacles to face in order to deepen our understanding of M-theoretic matrixmodels. These obstacles were overcome in the study of the ABJM matrix model by a series ofhappy coincidences (mostly, the connection to topological string theory), but cannot be avoidedin the more general class of models which we would like to study.Indeed, one serious drawback of these models is the difficulty to obtain in a realistic waythe full ’t Hooft expansion. It has been shown in [46] that the technique of topological recursioncan be in principle applied to O ( m ) models like the one studied in this paper, but in practice itis not easy to apply it (indeed, even for ABJM theory, the ’t Hooft expansion was obtained in[2] by applying the technique of direct integration first proposed in [63], and not the topologicalrecursion). It is therefore important to develop further techniques and ideas to obtain the ’tHooft expansion.To understand the M-theoretic regime, we also need to resum the ’t Hooft expansion. Itis likely that the road to follow here is the one open by the Fermi gas method. In order tofollow this approach, we should develop techniques to compute the semiclassical expansion of thespectrum of the Fermi gas Hamiltonian, with exponential precision. This means that we haveto generalize the WKB method to the integral equations appearing in this type of problems. Aspointed out in [5, 9], one can obtain in this way a resummed ’t Hooft expansion, together withmembrane-like effects, but then quantum-mechanical instanton corrections have to be included,and these are difficult to compute.Another important open problem is to understand the membrane-like corrections from thepoint of view of the ’t Hooft expansion. These are, morally speaking, large N instantons ofthe matrix model (see for example [11]), but it is not clear how to make contact between thispoint of view and the Fermi gas calculation of these effects. This will probably need a betterunderstanding of exponentially small corrections in matrix models.Coming back to the concrete models studied in this paper, there are clearly some moreprecise questions that can be addressed. First of all, one could try to determine further termsin the ’t Hooft expansion, in both the N f matrix model and the polymer matrix model. In thisrespect, it would be interesting to see if the direct integration technique of [63] works also for the O ( m ) model. The non-perturbative study of the polymer matrix model is probably very muchfacilitated by the connection to classical integrable hierarchies, although a detailed study remainsto be done. For the N f matrix model, the preliminary results presented in this paper can beextended and deepened in many ways. One could use the TBA approach of [56], combined withthe results in [40, 36], in order to compute the exact values of Z ( N, N f ) for fixed values of N f and high values of N . This would lead to a reasonable ansatz for the first terms in the large µ expansion of the grand potential J ( µ, N f ), as in [7], and might be the starting point for a fullnon-perturbative study of the model. It would be also interesting to see if subleading correctionsto the N f → N f . We hope to report on some on these issues inthe near future. Acknowledgements We would like to thank Stefano Cremonesi for calling our attention to the N f matrix model,and we thank him, as well as the participants and organizers of the workshop STAL2013, for– 28 –nteresting discussions. We would also like to thank M. Mezai, G. Mussardo and S. Pufu foruseful communications. M.M. would like to thank the Banff Center for hospitality during theconference “Modern developments in M-theory.” This work is supported by the Fonds NationalSuisse, subsidies 200020-141329 and 200020-137523. A. The function G ( z ) A.1 General properties The function G ( z ) was introduced in [28, 29] as a technical tool to solve the O ( m ) matrix modelfor general m . It satisfies the defining equation G ( z + i0) + G ( z − i0) + 2 cos ( πν ) G ( − z ) = 0 , (A.1)and it is holomorphic on the whole z -plane except for the interval [ a, b ], where it has a branchcut. We will map the z -plane to the u -plane through the equation z = a sn( u, k ) = a ϑ ϑ ϑ (cid:0) u K (cid:1) ϑ (cid:0) u K (cid:1) , (A.2)where k = ab , (A.3)and K is the elliptic integral of the first kind with argument k . Here, ϑ a ( v ) = ϑ a ( v, τ ) , a = 0 , , , τ = i K (cid:48) K . (A.5)Our conventions for elliptic functions and theta functions are as in [64]. In the following we usethe notation G ( u ) and G ( z ) interchangeably. The relationship (A.2) can be inverted as u = (cid:90) z/a d x (cid:112) (1 − x )(1 − k x ) . (A.6)The function G ( u ) is obtained from the function, G + ( u ) = e π i ν G ( u ) + e − π i ν G ( − u )2 sin( πν ) , (A.7)as G ( u ) = − i (cid:104) e π i ν G + ( u ) − e − π i ν G + ( − u ) (cid:105) . (A.8)An explicit expression for G + ( u ) was found in [29] in terms of theta functions. Let us define, H + ( u ) = ϑ (cid:16) u − i K (cid:48) K (cid:17) ϑ (cid:0) u − ε K (cid:1) ϑ (cid:0) u − K K (cid:1) ϑ (cid:16) u − ( K +i K (cid:48) )2 K (cid:17) e − π i(1 − ν ) u K = − i ϑ (cid:0) u K (cid:1) ϑ (cid:0) u − ε K (cid:1) ϑ (cid:0) u K (cid:1) ϑ (cid:0) u K (cid:1) e − π i(1 − ν ) u K . – 29 –n going from the first to the second line, we have used various properties of the theta functions.This solution differs from the one given in [29] by an overall sign, and follows the conventions in[21]. The argument ε is given by ε = i (1 − ν ) K (cid:48) , (A.9)and we will denote e = a sn( ε, k ) . (A.10)The function G + ( z ) is proportional to H + ( z ), and satisfies the normalization conditionlim z →∞ zG + ( z ) = i . (A.11)Notice that this is the normalization condition chosen in [28], and it is different from the onechosen in [29]. One finds, G + ( z ) = ϑ aϑ ϑ (cid:0) ε K (cid:1) H + ( z ) . (A.12)This can be written in a useful form for the limit ε = 0, as follows. We havesn( u, k ) = ϑ (0) ϑ (0) ϑ ( u/ (2 K )) ϑ ( u/ (2 K )) , cn( u, k ) = ϑ (0) ϑ (0) ϑ ( u/ (2 K )) ϑ ( u/ (2 K )) , dn( u, k ) = ϑ (0) ϑ (0) ϑ ( u/ (2 K )) ϑ ( u/ (2 K )) . (A.13)Therefore, (cid:18) ϑ ϑ ϑ ϑ (cid:19) (cid:16) u K (cid:17) = (cid:18) ϑ ϑ (cid:19) sn( u )cn( u )dn( u ) , (A.14)and we can write the limit ε → G + ( u ) as − i a (cid:18) ϑ ϑ (cid:19) (cid:18) ϑ ϑ ϑ ϑ (cid:19) (cid:16) u K (cid:17) = − i z (cid:112) ( z − a )( z − b ) , (A.15)where we used that cn( u, k ) = (cid:112) − z /a , dn( u, k ) = (cid:112) − k z /a , (A.16)as well as (cid:18) ϑ ϑ (cid:19) = k = ab . (A.17)Using this result, we find G + ( z ) = − i z (cid:112) ( z − a )( z − b ) ϑ ϑ (cid:0) ε K (cid:1) ϑ (cid:0) u − ε K (cid:1) ϑ (cid:0) u K (cid:1) e − π i(1 − ν ) u/ (2 K ) . (A.18)Finally, we note that the function G + ( z ) satisfies the product formula G + ( z ) G + ( − z ) = z − e ( z − a )( z − b ) . (A.19)– 30 – .2 Limiting behavior In this paper we need to study the limits ν → , G ( z ). Let us first study thelimit ν → 1. For the function G + ( u ), one finds, at first order, G + ( z ) = − i z (cid:112) ( z − a )( z − b ) (cid:26) − i(1 − ν ) (cid:18) πu K + K (cid:48) ϑ (cid:48) ϑ (cid:16) u K (cid:17)(cid:19) + O (cid:0) (1 − ν ) (cid:1)(cid:27) . (A.20)In order to obtain an explicit expression for the ’t Hooft parameter λ in (3.29), we need to dothe expansion up to (and including) third order in (1 − ν ) , and evaluate the result at z = − (cid:18) K + i K (cid:48) (cid:19) = 1 √ k , (A.21)we find that, when b = 1 /a (which is the case in our model), the point z = 1 corresponds to u = K + i K (cid:48) , (A.22)The point corresponding to z = − u − = − u , since sn is an odd function. Inevaluating the coefficients of the expansion of G ( z ) at z = ± 1, it is convenient to use the Jacobizeta function, which is defined as Z ( u ) = dd u log ϑ (cid:16) u K (cid:17) . (A.23)It satisfies the two identities, Z ( u + v ) = Z ( u ) + Z ( v ) − k sn u sn v sn( u + v ) ,Z ( u + i K (cid:48) ) = Z ( u ) − i π K + cs u dn u. (A.24)From (A.24) one deduces, (cid:18) ϑ (cid:48) ϑ (cid:19) (cid:18) 12 + τ (cid:19) = i( k − − i π K ( k ) , (cid:18) ϑ (cid:48)(cid:48) ϑ (cid:19) (cid:18) 12 + τ (cid:19) = 1 − k − E ( k ) K ( k ) + (cid:18)(cid:18) ϑ (cid:48) ϑ (cid:19) (cid:18) 12 + τ (cid:19)(cid:19) , (cid:18) ϑ (cid:48)(cid:48)(cid:48) ϑ (cid:19) (cid:18) 12 + τ (cid:19) = 2i k ( k − 1) + 3 (cid:18) ϑ (cid:48) ϑ (cid:19) (cid:18) 12 + τ (cid:19) (cid:18) ϑ (cid:48)(cid:48) ϑ (cid:19) (cid:18) 12 + τ (cid:19) − (cid:18)(cid:18) ϑ (cid:48) ϑ (cid:19) (cid:18) 12 + τ (cid:19)(cid:19) . (A.25)In the limit ν → 0, the quantity e defined in (A.10) diverges. Indeed, one has that a sn( − i νK (cid:48) + i K (cid:48) ) = ak sn(i νK (cid:48) ) ≈ a i kνK (cid:48) (A.26)where in the first step we have used an standard identity for the Jacobi sn function. To calculatethe limit of G ( z ) as ν → 0, we use that [28] G (1 − ν ) ( z ) = − (cid:16) e i νπ/ g + ( z ) G ( ν )+ ( z ) + g + ( − z ) G ( ν )+ ( − z )e − i νπ/ (cid:17) , (A.27)– 31 –here g + ( z ) = (cid:112) ( z − a )( z − b ) + ze (cid:112) ( e − a )( e − b ) z − e . (A.28)The indices ν , 1 − ν indicate that the function G should be evaluated for these values of theparameter. It follows thatlim ν → G ( z ) = − lim ν → (cid:16) e i νπ/ g + ( z ) G + ( z ) + g + ( − z ) G + ( − z )e − i νπ/ (cid:17) , (A.29)and we can then use the expansion (A.20) around ν = 1. B. The O ( m ) model as a multi-trace matrix model Since the planar solution to the O ( m ) matrix model is relatively complicated, it is useful to makean independent computation of various planar quantities. Of course one can do a perturbativecomputation, but it is better to have a more systematic approach which captures the planar limitdirectly. Such an approach is obtained if one regards the O ( m ) model as a multi-trace matrixmodel. B.1 Multi-trace matrix models Let us consider a matrix model for a Hermitian N × N matrix, M , Z = 1vol( U ( N )) (cid:90) d M e − V ( M ) /g s , (B.1)where the potential is of the form V ( M ) = 12 Tr M + t ∞ (cid:88) k =1 a k k Tr M k + g s (cid:88) k,l ≥ c k,l Tr M k Tr M l (B.2)and it includes double-trace operators. We have denoted t = g s N, (B.3)and our conventions are as in [65]. The standard method to study this type of potentials inthe planar limit is to use an analogue of the Hartree–Fock approximation [66]. In terms of thedensity of eigenvalues ρ ( z ), the planar free energy becomes g − s F [ ρ ] = t − t ρ − (cid:88) k ≥ a k k ρ k − (cid:88) k,l ≥ c k,l ρ k ρ l + (cid:90) d λ d µρ ( λ ) ρ ( µ ) log | λ − µ | , (B.4)where ρ k = (cid:90) d λ ρ ( λ ) λ k . (B.5)The saddle point equation for ρ is obtained by varying w.r.t. ρ :12 t x + ∞ (cid:88) k =1 a k k x k + 2 (cid:88) k,l c k,l ρ l x k = 2 (cid:90) d yρ ( y ) log | x − y | + ζ, (B.6)– 32 –here ζ is a Lagrange multiplier. This equation can be written as1 t V eff ( x ) = 2 (cid:90) d yρ ( y ) log | x − y | + ζ, (B.7)which is the standard equation appearing in Hermitian matrix model, but it involves the “effec-tive” potential V eff ( x ) = 12 x + ∞ (cid:88) k =1 t a k k x k + 2 t (cid:88) k,l c k,l ρ l x k , (B.8)which can be written as V eff ( x ) = 12 x + t (cid:88) k ≥ ˜ a k k x k , (B.9)where ˜ a k = a k + 2 k (cid:88) l ≥ c k,l ρ l . (B.10)Therefore, we can solve for the density of eigenvalues by using this potential and then imposeself-consistency.We will restrict ourselves to even potentials. In this case, a l = 0 for l odd, and c k,l = 0 if k + l = odd. (B.11)This implies that the endpoints of the cut A, B where the eigenvalues condense are symmetric A = − B . It follows that ρ l = 0 l = odd , (B.12)and we have to pick only even terms in the effective potential, i.e. ˜ a l = 0 if l is odd.We can now treat the effective, even potential with the standard techniques of orthogonalpolynomials [67]. The basic quantity is R ( ξ, t ), which can be obtained from the equation ξ = 1 t R + (cid:88) k ≥ ˜ a k (cid:18) k − k − (cid:19) R k . (B.13)The moments ρ l can be computed as ρ l = (2 l )! l ! (cid:90) d ξR l ( ξ ) . (B.14)In practice we will calculate ρ l as a power series in t : ρ l = t l ∞ (cid:88) n =0 r l,n t n , r l, = (2 l )! l ! ( l + 1) . (B.15)We then obtain the following consistency conditions, ∞ (cid:88) n =0 r l,n t n = (2 l )! l ! (cid:90) d ξ (cid:16) R ( ξ ) t (cid:17) l , (B.16)where ˜ a k = a k + 4 k (cid:88) l ≥ (cid:88) n ≥ c k, l r l,n t n + l . (B.17)– 33 –ince the ˜ a k are themselves functions of the coefficients r l,n , as in (B.17), we obtain a set ofequations which determine the r l,n as functions of a k . This leads to expressions for many of theplanar quantities as power series in t . If we denote the endpoints of the cut as ( − A, A ), we find A R (1) = t − t a + t (cid:0) a − a − c , (cid:1) + t (cid:0) − a + 9 a a − a + 12 c , − c , − c , (cid:1) + · · · (B.18)Similarly, the planar free energy can be computed by evaluating the functional F [ ρ ] on theequilibrium distribution. One easily obtains F ( t ) = − t (cid:90) d xV eff ( x ) ρ ( x ) − t ζ + t (cid:88) k,l c k,l ρ k ρ l . (B.19)Notice that the last term is a correction to the single-trace case. If we use the formalism oforthogonal polynomials, we can rewrite the first two terms by using the function R ( ξ ). Ourfinal expression is F ( t ) − F G0 ( t ) = t (cid:90) d ξ (1 − ξ ) log (cid:18) R ( ξ ) tξ (cid:19) + t (cid:88) k,l c k,l ρ k ρ l , (B.20)where F G0 ( t ) is the planar free energy of the Gaussian matrix model. Like before, this quantitycan be computed perturbatively in t in terms of the coefficients of the potential. One finds, F ( t ) − F G0 ( t ) = − t a + 14 t ( a − a − c , ) + O ( t ) . (B.21) B.2 ExamplesB.2.1 Chern–Simons matrix model The Chern–Simons matrix model describing Chern–Simons theory on S [47] is a particular caseof the above multi-trace matrix model [68]. In this case, the coefficients are given explicitly bythe following expressions a k = − B k (2 k )! , c k, l = − B k + l ) k + l )(2( k + l ))! (cid:18) k + l )2 k (cid:19) , (B.22)where B k are Bernoulli numbers. Since this model is exactly solvable, we can test the aboveexpressions in detail. For example, (B.18) gives in this case, A t + t t − t − t O ( t ) , (B.23)which are precisely the first few terms of the perturbative expansion of the exact result A = 2 cosh − (cid:16) e t/ (cid:17) . (B.24)The perturbative result (B.21) for the planar free energy gives F ( t ) − F G0 ( t ) = t 12 + t − t O ( t ) , (B.25)– 34 –hich is the expansion of the exact result F ( t ) − F G0 ( t ) = − Li (e − t ) . (B.26)Using the above formalism we can also calculate the correlation functions W n ( t ) = g s (cid:10) Tr e nM (cid:11) = g s (cid:88) k ≥ n k k ! Tr M k = g s (cid:88) l ≥ n l (2 l )! ρ l , (B.27)which correspond to Wilson loops. B.2.2 The N f and the polymer matrix models The matrix models (3.3), (4.1) can be written as multi-trace matrix models. In the case of the N f matrix model, we have a k = − (cid:0) − k − (cid:1) B k (2 k )! + 1 t (1 − δ , k ) 4 (cid:0) k − (cid:1) B k (2 k )! ,c k, l = − (cid:0) − k + l ) − (cid:1) B k + l ) (2( k + l ))(2( k + l ))! (cid:18) k + l )2 k (cid:19) , (B.28)where t = 4 NN f = 4 λ. (B.29)The relative factor of 4 as compared to (3.4) is due to the fact that, in the formalism for multi-trace matrix models developed above, the Gaussian potential has the canonical normalization x / 2, while in the expansion of the potential in (3.3) around x = 0 we have instead x / c k, l , but a k is now, a k = − (cid:0) − k − (cid:1) B k (2 k )! + 1 t (1 − δ , k ) 1(2 k − . (B.30)and t = λ , where λ is given in (4.5) (the parameter t appearing in (4.1) should not be confusedwith the ’t Hooft-like parameter t used in this Appendix).One should take into account that the coefficients a k depend now on 1 /t , but it can be easilyseen that at each order in t only a finite number of terms in the above expansions contribute.After taking these two facts into account, one obtains the results (3.33), (4.15), in agreementwith the exact solution. References [1] I. R. Klebanov and A. A. Tseytlin, “Entropy of near extremal black p-branes,” Nucl. Phys. B ,164 (1996) [hep-th/9604089].[2] N. Drukker, M. Mari˜no, P. Putrov, “From weak to strong coupling in ABJM theory,” Commun.Math. Phys. , 511-563 (2011). [arXiv:1007.3837 [hep-th]].[3] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, “N=6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals,” JHEP , 091 (2008)[arXiv:0806.1218 [hep-th]]. – 35 – 4] S. Bhattacharyya, A. Grassi, M. Mari˜no and A. Sen, “A One-Loop Test of Quantum Supergravity,”Class. Quant. Grav. , 015012 (2014) [arXiv:1210.6057 [hep-th]].[5] M. Mari˜no and P. Putrov, “ABJM theory as a Fermi gas,” J. Stat. Mech. , P03001 (2012)[arXiv:1110.4066 [hep-th]].[6] M. Mari˜no and P. Putrov, “Exact Results in ABJM Theory from Topological Strings,” JHEP ,011 (2010) [arXiv:0912.3074 [hep-th]].[7] Y. Hatsuda, S. Moriyama and K. Okuyama, “Instanton Effects in ABJM Theory from Fermi GasApproach,” JHEP , 158 (2013) [arXiv:1211.1251 [hep-th]].[8] Y. Hatsuda, M. Mari˜no, S. Moriyama and K. Okuyama, “Non-perturbative effects and the refinedtopological string,” JHEP , 168 (2014) [arXiv:1306.1734 [hep-th]].[9] J. K¨all´en and M. Mari˜no, “Instanton effects and quantum spectral curves,” arXiv:1308.6485[hep-th].[10] Y. Hatsuda, S. Moriyama and K. Okuyama, “Instanton Bound States in ABJM Theory,” JHEP , 054 (2013) [arXiv:1301.5184 [hep-th]].[11] M. Mari˜no, “Lectures on non-perturbative effects in large N gauge theories, matrix models andstrings,” arXiv:1206.6272 [hep-th].[12] C. P. Herzog, I. R. Klebanov, S. S. Pufu and T. Tesileanu, “Multi-Matrix Models and Tri-SasakiEinstein Spaces,” Phys. Rev. D , 046001 (2011) [arXiv:1011.5487 [hep-th]].[13] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and S. Pakuliak, “Conformal matrix modelsas an alternative to conventional multimatrix models,” Nucl. Phys. B , 717 (1993)[hep-th/9208044].[14] I. K. Kostov, “Solvable statistical models on a random lattice,” Nucl. Phys. Proc. Suppl. , 13(1996) [hep-th/9509124].[15] O. Aharony, O. Bergman and D. L. Jafferis, “Fractional M2-branes,” JHEP , 043 (2008)[arXiv:0807.4924].[16] H. Awata, S. Hirano and M. Shigemori, “The Partition Function of ABJ Theory,” Prog. Theor.Exp. Phys. , 053B04 (2013) [arXiv:1212.2966].[17] M. Honda, “Direct derivation of ”mirror” ABJ partition function,” JHEP , 046 (2013)[arXiv:1310.3126 [hep-th]].[18] S. Matsumoto and S. Moriyama, “ABJ Fractional Brane from ABJM Wilson Loop,” JHEP ,079 (2014) [arXiv:1310.8051 [hep-th]].[19] T. Suyama, “On Large N Solution of Gaiotto-Tomasiello Theory,” JHEP , 101 (2010)[arXiv:1008.3950 [hep-th]].[20] R. C. Santamaria, M. Mari˜no, P. Putrov, “Unquenched flavor and tropical geometry in stronglycoupled Chern-Simons-matter theories,” JHEP (2011) 139 [arXiv:1011.6281 [hep-th]].[21] T. Suyama, “On Large N Solution of N = 3 Chern-Simons-adjoint Theories,” Nucl. Phys. B ,887 (2013) [arXiv:1208.2096 [hep-th]].[22] T. Suyama, “A Systematic Study on Matrix Models for Chern-Simons-matter Theories,” Nucl.Phys. B , 528 (2013) [arXiv:1304.7831 [hep-th]].[23] D. Bashkirov and A. Kapustin, “Supersymmetry enhancement by monopole operators,” JHEP , 015 (2011) [arXiv:1007.4861 [hep-th]].[24] F. Benini, C. Closset and S. Cremonesi, “Chiral flavors and M2-branes at toric CY4 singularities,”JHEP , 036 (2010) [arXiv:0911.4127 [hep-th]]. – 36 – 25] M. Mezei and S. S. Pufu, “Three-sphere free energy for classical gauge groups,” JHEP , 037(2014) [arXiv:1312.0920 [hep-th], arXiv:1312.0920].[26] I. K. Kostov, “O( n ) Vector Model on a Planar Random Lattice: Spectrum of AnomalousDimensions,” Mod. Phys. Lett. A , 217 (1989).[27] I. K. Kostov and M. Staudacher, “Multicritical phases of the O ( n ) model on a random lattice,”Nucl. Phys. B , 459 (1992) [hep-th/9203030].[28] B. Eynard and C. Kristjansen, “Exact solution of the O ( n ) model on a random lattice,” Nucl. Phys.B , 577 (1995) [hep-th/9506193].[29] B. Eynard and C. Kristjansen, “More on the exact solution of the O ( n ) model on a random latticeand an investigation of the case | n | > , 463 (1996) [hep-th/9512052].[30] A. Kapustin, B. Willett and I. Yaakov, “Exact Results for Wilson Loops in SuperconformalChern-Simons Theories with Matter,” JHEP , 089 (2010) [arXiv:0909.4559 [hep-th]].[31] N. Hama, K. Hosomichi and S. Lee, “Notes on SUSY Gauge Theories on Three-Sphere,” JHEP (2011) 127 [arXiv:1012.3512 [hep-th]].[32] D. L. Jafferis, “The Exact Superconformal R-Symmetry Extremizes Z,” JHEP , 159 (2012)[arXiv:1012.3210 [hep-th]].[33] D. L. Jafferis and A. Tomasiello, “A Simple class of N=3 gauge/gravity duals,” JHEP , 101(2008) [arXiv:0808.0864 [hep-th]].[34] Y. Imamura, K. Kimura, “On the moduli space of elliptic Maxwell-Chern-Simons theories,” Prog.Theor. Phys. , 509-523 (2008). [arXiv:0806.3727 [hep-th]].[35] A. Kapustin, B. Willett, I. Yaakov, “Nonperturbative Tests of Three-Dimensional Dualities,” JHEP , 013 (2010). [arXiv:1003.5694 [hep-th]].[36] Y. Hatsuda, S. Moriyama and K. Okuyama, “Exact Results on the ABJM Fermi Gas,” JHEP , 020 (2012) [arXiv:1207.4283 [hep-th]].[37] N. Drukker, M. Mari˜no and P. Putrov, “Nonperturbative aspects of ABJM theory,” JHEP ,141 (2011) [arXiv:1103.4844 [hep-th]].[38] H. Fuji, S. Hirano and S. Moriyama, “Summing Up All Genus Free Energy of ABJM MatrixModel,” JHEP , 001 (2011) [arXiv:1106.4631 [hep-th]].[39] F. Calvo and M. Mari˜no, “Membrane instantons from a semiclassical TBA,” JHEP , 006(2013) [arXiv:1212.5118 [hep-th]].[40] P. Putrov and M. Yamazaki, “Exact ABJM Partition Function from TBA,” Mod. Phys. Lett. A ,1250200 (2012) [arXiv:1207.5066 [hep-th]].[41] V. A. Kazakov, I. K. Kostov and N. A. Nekrasov, “D particles, matrix integrals and KP hierarchy,”Nucl. Phys. B , 413 (1999) [hep-th/9810035].[42] N. A. Nekrasov and S. L. Shatashvili, “Quantization of Integrable Systems and Four DimensionalGauge Theories,” arXiv:0908.4052 [hep-th].[43] C. Meneghelli and G. Yang, “Mayer-Cluster Expansion of Instanton Partition Functions andThermodynamic Bethe Ansatz,” JHEP , 112 (2014) [arXiv:1312.4537 [hep-th]].[44] J. -E. Bourgine, “Notes on Mayer Expansions and Matrix Models,” Nucl. Phys. B , 476 (2014)[arXiv:1310.3566 [hep-th]].[45] T. Azeyanagi, M. Fujita and M. Hanada, “From the planar limit to M-theory,” Phys. Rev. Lett. , no. 12, 121601 (2013) [arXiv:1210.3601 [hep-th]]. – 37 – 46] G. Borot and B. Eynard, “Enumeration of maps with self avoiding loops and the O ( n ) model onrandom lattices of all topologies,” J. Stat. Mech. , 01010 (2001) [hep-th/0910.5896].[47] M. Mari˜no, “Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants,”Commun. Math. Phys. , 25 (2004) [arXiv:hep-th/0207096].[48] M. Tierz, “Soft matrix models and Chern-Simons partition functions,” Mod. Phys. Lett. A , 1365(2004) [hep-th/0212128].[49] N. Halmagyi and V. Yasnov, “The Spectral curve of the lens space matrix model,” JHEP , 104(2009) [hep-th/0311117].[50] O. Bergman and S. Hirano, “Anomalous radius shift in AdS /CFT ,” JHEP , 016 (2009)[arXiv:0902.1743].[51] O. Aharony, A. Hashimoto, S. Hirano and P. Ouyang, “D-brane Charges in Gravitational Duals of2+1 Dimensional Gauge Theories and Duality Cascades,” JHEP , 072 (2010) [arXiv:0906.2390[hep-th]].[52] Y. Hatsuda and K. Okuyama, “Probing non-perturbative effects in M-theory,” JHEP , 158(2014) [arXiv:1407.3786 [hep-th]].[53] D. L. Jafferis, I. R. Klebanov, S. S. Pufu and B. R. Safdi, “Towards the F-Theorem: N=2 FieldTheories on the Three-Sphere,” JHEP , 102 (2011) [arXiv:1103.1181 [hep-th]].[54] G. Mussardo, Statistical Field Theory , Oxford University Press, Oxford, 2010.[55] P. Fendley and H. Saleur, “N=2 supersymmetry, Painlev´e III and exact scaling functions in 2-Dpolymers,” Nucl. Phys. B , 609 (1992) [hep-th/9204094].[56] A. B. Zamolodchikov, “Painlev´e III and 2-d polymers,” Nucl. Phys. B , 427 (1994)[hep-th/9409108].[57] C. A. Tracy, H. Widom, “Fredholm determinants and the mKdV/sinh-Gordon hierarchies,”Commun. Math. Phys. , 1-10 (1996).[58] B. M. McCoy, C. A. Tracy and T. T. Wu, “Painlev´e Functions of the Third Kind,” J. Math. Phys. , 1058 (1977).[59] S. Cecotti, P. Fendley, K. A. Intriligator and C. Vafa, “A new supersymmetric index,” Nucl. Phys.B , 405 (1992) [hep-th/9204102].[60] I. K. Kostov, “Exact solution of the six vertex model on a random lattice,” Nucl. Phys. B , 513(2000) [hep-th/9911023].[61] V. P. Yurov and A. B. Zamolodchikov, “Correlation functions of integrable 2-D models ofrelativistic field theory. Ising model,” Int. J. Mod. Phys. A , 3419 (1991).[62] J. L. Cardy and G. Mussardo, “Form-factors of Descendent Operators in Perturbed ConformalField Theories,” Nucl. Phys. B , 387 (1990).[63] M. x. Huang and A. Klemm, “Holomorphic anomaly in gauge theories and matrix models,” JHEP , 054 (2007) [hep-th/0605195].[64] N.I. Akhiezer, Elements of the theory of elliptic functions , Americal Mathematical Society,Providence, 1990.[65] M. Mari˜no, “Les Houches lectures on matrix models and topological strings,” hep-th/0410165.[66] S. R. Das, A. Dhar, A. M. Sengupta and S. R. Wadia, “New critical behavior in d = 0 large N matrix models,” Mod. Phys. Lett. A , 1041 (1990).[67] D. Bessis, C. Itzykson and J. B. Zuber, “Quantum Field Theory Techniques In GraphicalEnumeration,” Adv. Appl. Math. , 109 (1980).[68] M. Aganagic, A. Klemm, M. Mari˜no and C. Vafa, “Matrix model as a mirror of Chern-Simonstheory,” JHEP , 010 (2004) [hep-th/0211098]., 010 (2004) [hep-th/0211098].