Strong Coupling Limit of A Family of Chern-Simons-matter Theories
aa r X i v : . [ h e p - t h ] A ug June 2017KEK-TH-1985
Strong Coupling LimitofA Family of Chern-Simons-matter Theories
Takao Suyama KEK Theory Center, High Energy Accelerator Research Organization (KEK),Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan
Abstract
We investigate the strong coupling limit of a family of Chern-Simons-matter theories in the planarlimit. The family consists of N = 3 theories with the gauge group U( N ) k × U( N ) k coupled to n bi-fundamental hypermultiplets. All observables which can be determined from the planar resolvent turnout to have finite limits in the large ’t Hooft coupling limit. Possible gravity duals are briefly discussed.We observe that Kac-Moody algebras govern the structure of the planar spectral curves of the theories. e-mail address: [email protected] Introduction
ABJM theory [1] provides a prototypical example of AdS /CFT correspondence. It is an N = 6 Chern-Simons-matter theory, and its gravity dual is M-theory on AdS × S / Z k . It is natural to expect thatthis correspondence could be generalized by replacing the internal manifold S / Z k in the gravity sidewith a less symmetric seven-dimensional manifold B . One should choose a manifold B with some goodproperties in order to be able to perform the actual analysis. At the same time, one would like to choose B from a wide range of possible manifolds so that the insight into AdS/CFT correspondence [2] can begained by examining various examples.One such criterion for choosing B is to require that the corresponding CFT possesses N = 3 super-symmetry. Indeed, N = 3 supersymmetry for boundary theory is powerful enough to control quantumcorrections, and also it is flexible enough to allow the construction of theories with various gauge groupsand matter representations [3]. A more detailed characterization of B is given as follows. It is known thatM-theory on the background of the form R , × M preserves N = 3 supersymmetry in three dimensionsif and only if M is hyper-K¨ahler. Suppose that M has a conical singularity, and the base of the cone is B . One may put M2-branes to probe the singularity without breaking any supersymmetry, as long as theorientation of the M2-branes is appropriately chosen [4]. The near-horizon geometry of the M2-branes isthen AdS × B which is expected to be the dual gravity background for the worldvolume theory of theM2-branes. Therefore, one should choose B to be a 3-Sasakian manifold (see e.g. [5]).Suppose that the worldvolume theory of the M2-branes in the background discussed above is an N = 3 Chern-Simons-matter theory. Then, the hyper-K¨ahler manifold M should be obtained as themoduli space of vacua of this theory. The moduli spaces were investigated in [6] for N = 3 Chern-Simons-matter theories whose matter representation is specified by a circular quiver diagram. In [7],this analysis was extended to theories corresponding to more general quiver diagrams, and it was shownthat the moduli space is hyper-K¨ahler. It turned out that the dimension of the moduli space of a giventheory is determined by the number of loops in the quiver diagram, whose explicit formula depends on theChern-Simons levels. According to the formula in [7], theories with A-type quiver diagrams have eight-dimensional moduli spaces, as was shown explicitly in [6], while those with DE-type quiver diagramswould have four-dimensional ones. Later, the analysis in [7] was generalized in [8], in order to treatnon-toric hyper-K¨ahler manifolds as well, which showed that certain theories with D-type quivers alsohave eight-dimensional moduli spaces.A check of the correspondence can be performed by calculating the free energy. It was shown in [9]that the free energy of ABJM theory behaves as F ∼ √ √ kN (1.1)in the large N limit, which exactly reproduces the corresponding gravity result. This analysis can beextended to N = 3 cases. From the gravity side, the leading large N behavior of the free energy isexpected to be of the form [10] F ∼ N s π B ) (1.2)where vol( B ) is the volume of B . The formula for vol( B ) for a toric B was obtained in [12]. Thisvolume formula was extended in [8] to non-toric ones. The large N behavior (1.2) of the free energywas reproduced from the corresponding N = 3 Chern-Simons-matter theories exactly by employing thetechnique developed in [10] based on the supersymmetric localization [13]. The technique was applied tovarious theories in [14–16]. Similar analyses can be performed for N = 2 Chern-Simons-matter theoriesas well [17–24]. A similar technique was used in [11] in the ’t Hooft limit. N = 3 duals which does not seem to have been investigated indetail, compared with the other examples. The M-theory background of this case consists of a 3-Sasakianmanifold known as N , , which is a coset SU(3) / U(1). The cone over N , , is T ∗ P which is a quitesimple eight-dimensional hyper-K¨ahler manifold.The dual CFT was proposed in [25] to be a three-dimensional gauge theory with Chern-Simons termswhere the gauge group is U( N ) × U( N ) coupled to three bi-fundamental hypermultiplets. The Chern-Simons levels were expected to be determined by a flux in the M-theory background, but their values werenot specified explicitly. This proposal was further elaborated in [26] by showing the correspondence ofBPS operators in the boundary theory with the Kaluza-Klein spectrum of the M-theory on AdS × N , , .According to [7], the Chern-Simons levels must satisfy k + k = 0 in order to obtain T ∗ P as the modulispace. Recently, another dual CFT was proposed in [27]. The proposed theory is a deformation of ABJMtheory by adding fundamental hypermultiplets. It was shown that quantum corrections to the classicalmoduli space is crucial for this duality. This proposal was supported by calculating the superconformalindex which matches with the corresponding index in M-theory [28].With these developments in mind, in this paper, we investigate N = 3 Chern-Simons-matter theorieswith the gauge group U( N ) k × U( N ) k coupled to n ≥ n = 2 includes ABJM theory. Another family with n = 1was recently discussed in [29]. In our previous paper [30], we analyzed the saddle point equations forthe localized partition function, and obtained a closed formula for the resolvent in terms of the thetafunctions. From the resolvent, the free energy and the vevs of BPS Wilson loops, which are of the samekind investigated in [31–33] for ABJM theory, can be calculated as functions of the ’t Hooft couplings.We investigate the behavior of these observables in the large ’t Hooft coupling limit. Interestingly, wefind that the behavior is quite different from the one observed in ABJM theory. It turns out thatthe free energy scales as N , just like the ordinary gauge theories, and the vevs of BPS Wilson loopsapproach constant values in the limit. In addition, other observables which can be determined from theplanar resolvent are found to exhibit similar behaviors. This difference from the case of ABJM theoryis rather surprising since the difference at the level of Lagrangians is just to increase the number of thehypermultiplets. Curiously, a difference can be seen also at the level of spectral curves.This paper is organized as follows. In section 2, we show that the above-mentioned results for theChern-Simons-matter theories are derived from the resolvent obtained in [30]. Possible gravity duals forthe theories are briefly reconsidered in section 3, taking into account the results obtained in the previoussection. In section 4, we observe that differences among the N = 3 Chern-Simons-matter theoriesinvestigated in [7] can be seen in their spectral curves which are governed by associated Kac-Moodyalgebras. Section 5 is devoted to discussion. Some details on our resolvent are reviewed in Appendix A.Appendix B contains a proof for a fact on the spectral curve. In this section, we investigate theories in a family CSM( n ) of N = 3 Chern-Simons-matter theories for apositive integer n by using the corresponding matrix models. Each theory in CSM( n ) has a gauge groupof the form U( N ) k × U( N ) k and n hypermultiplets in the bi-fundamental representation of the gaugegroup. The action of the theory is completely specified by these data due to N = 3 supersymmetry [3].ABJM theory [1] [34] and GT theory [35] are members of CSM(2). We are mainly interested in thetheories in CSM(3) since there exists a proposal for a gravity dual [25] [26].3 .1 Planar resolvent The partition function of a theory in CSM( n ) defined on S can be given in terms of a finite-dimensionalintegral [13] as Z = Z d N u d N w exp " ik π N X i =1 ( u i ) + ik π N X a =1 ( w a ) N i 2) (2.2)In this limit, the integral in (2.1) is dominated by the saddle point which satisfies the following equations: k πi u i = N X j = i coth u i − u j − n N X a tanh u i − w a , (2.3) k πi w a = N X b = a coth w a − w b − n N X i =1 tanh w a − u i . (2.4)Let { ¯ u i , ¯ w a } denote the solution of these equations. Because of the presence of i in the left-hand side,¯ u i and ¯ w a are complex numbers in general. For later convenience, the eigenvalues ¯ u i and ¯ w a are labeledsuch that Re(¯ u i ) ≤ Re(¯ u i +1 ) , Re( ¯ w a ) ≤ Re( ¯ w a +1 ) (2.5)holds. We assume that both ¯ u i and ¯ w a approach zero in the small ’t Hooft coupling limit, since in thislimit the left-hand sides dominate in the equations.Some physical observables can be calculated in the planar limit. The free energy is obtained from thesaddle point value of the integral (2.1). In addition, there are two BPS Wilson loops for U( N ) gaugefields and U( N ) gauge fields. Their vevs in the planar limit are given as h W i = 1 N N X i =1 e ¯ u i , h W i = 1 N N X a =1 e ¯ w a . (2.6)The information of the theory in the planar limit is encoded in a row-vector-valued resolvent [30] v ( z ) := ( v ( z ) , v ( z )) . (2.7)The components v α ( z ) are defined as v ( z ) := lim t N N X i =1 z + e ¯ u i z − e ¯ u i , v ( z ) := lim t N N X a =1 z − e ¯ w a z + e ¯ w a , (2.8)where lim indicates the planar limit (2.2). The function v ( z ) has a square-root branch cut on an interval I in C , and holomorphic elsewhere, including the point at infinity. The branch points a and b on I are given as a := e ¯ u , b := e ¯ u N . (2.9)4ue to the symmetry of the equations (2.3)(2.4), they satisfy a b = 1 . (2.10)The other function v ( z ) has similar properties. The branch points a and b are given as a := − e ¯ w , b := − e ¯ w N (2.11)which also satisfy a b = 1 . (2.12)Note that | a α | ≤ v ( z ) is determined by specifying two complex parameters, say a α . These two parametersare related to the values of the ’t Hooft couplings t α for given κ α via v (0) = − ( t , t ) . (2.13)It was found in [30] that it is convenient to investigate a derivative zv ′ ( z ) of the resolvent, instead of v ( z ) itself, since the former can be obtained explicitly. In the following, we assume n = 2.Let c := ( c , c ) be the row vector satisfying(2 κ , κ ) = ( c , c ) (cid:20) − n − n (cid:21) . (2.14)Note that the solution of this equation exists for n = 2. Let us introduce another row-vector-valuedfunction f ( z ) such that zv ′ ( z ) can be written as zv ′ ( z ) = c + 1 s ( z ) f ( z ) , s ( z ) := p ( z − a )( z − b )( z − a )( z − b ) . (2.15)In order to determine the explicit form of f ( z ), we introduce a new variable u defined as u ( z ) := ϕ ( z )2 ϕ ( b ) , ϕ ( z ) := Z za dξs ( ξ ) , (2.16)where the integration contour for ϕ ( b ) lies above the segment I . The saddle point equations (2.3)(2.4)imply that, as a function of u , f ( z ) can be written as f ( z ( u )) = (cid:16) ˜ f ( u ) , ˜ f ( − u ) (cid:17) S − , S := (cid:20) − e πiν − e − πiν (cid:21) (2.17)where ν is related to n by n = 2 cos πν . The scalar function ˜ f ( u ) is required to satisfy˜ f ( u + 1) = ˜ f ( u ) , ˜ f ( u + τ ) = e πiν ˜ f ( u ) (2.18)where τ := 2 u ( a ). Therefore, ˜ f ( u ) can be written in terms of the theta functions. The explicit form of˜ f ( u ) is given in Appendix A.Since what we have obtained is zv ′ ( z ), the relation (2.13) cannot be used to determine t α for a given a α . An alternative way to recover t α is to use the formula t α = − Z C α dz πi log zz zv ′ α ( z ) , (2.19)where C ( C ) is a contour encircling the segment I ( I ) counterclockwise.In addition to t α , the expansion of zv ′ ( z ) provides the vevs h W α i of BPS Wilson loops as zv ′ ( z ) = − t h W i , − t h W i ) z + O ( z ) . (2.20)Combining (2.19) and (2.20), the vevs h W α i can be given as functions of t α .5 .2 Small ’t Hooft coupling limit Various quantities can be calculated perturbatively when the ’t Hooft couplings t α are small. For example,the vevs h W α i are given as [30] h W i = 1 + t κ + 16 (cid:18) t κ (cid:19) (cid:18) − N (cid:19) − n t t κ + O ( t ) , (2.21) h W i = 1 + t κ + 16 (cid:18) t κ (cid:19) (cid:18) − N (cid:19) − n t t κ + O ( t ) , (2.22)which can be obtained by using the method developed in [13]. This result was reproduced in [30] from theresolvent v ( z ) reviewed above by using the formulas (2.19)(2.20) for cases when N = N and k = ± k are satisfied. This can be regarded as a non-trivial check of the validity of the resolvent v ( z ) obtainedin [30].The method in [13], however, does not give us any information on the configuration of the eigenvalues { ¯ u i , ¯ w a } , and therefore, the positions a α of the branch points of v ( z ). To obtain such information, it isbetter to solve the saddle point equations (2.3)(2.4) perturbatively, as in [11].In the following, we will restrict ourselves to the case N = N =: N for simplicity. This implies that t = t =: t holds. The analysis of this case should be the first step toward the understanding of thegeneral case.Recall that the saddle point equations in this case are k πi u i = N X j = i coth u i − u j − n N X a =1 tanh u i − w a , (2.23) k πi w a = N X b = a coth w a − w b − n N X i =1 tanh w a − u i . (2.24)Assume that t is small, and introduce rescaled variables x i and y a defined such that u i = √ t x i , w a = √ t y a (2.25)holds. Then, the saddle point equations can be expanded in t . The leading order terms give the followingsimple equations: κ x i = 2 N N X j = i x i − x j , κ y a = 2 N X b = a y a − y b . (2.26)Each of these equations are the same as the saddle point equations of the Gaussian matrix model. Theplanar solution of them is encoded in ω ( z ) := lim 1 N N X i =1 z − x i = κ (cid:20) z − r z − κ (cid:21) , (2.27) ω ( z ) := lim 1 N N X a =1 z − y a = κ (cid:20) z − r z − κ (cid:21) . (2.28)The perturbative corrections to this solution can be systematically calculated, as shown in [11] for ABJMtheory. 6he discontinuity of ω ( z ) and ω ( z ) encodes the distributions of { x i } and { y a } which then give thedistributions of { u i } and { w a } . The definitions (2.9)(2.11) of the parameters a and a imply that theyare given as a ∼ exp (cid:18) − r tκ (cid:19) , a ∼ − exp (cid:18) − r tκ (cid:19) (2.29)for small t .One may notice that the relation between a and a becomes quite simple when κ = κ holds. Inthis case, the parameters are related simply as a = − a =: a. (2.30)This relation is satisfied when the equalities ¯ u i = ¯ w i hold. Since this equality is compatible with the fullsaddle point equations (2.23)(2.24) with k = k , the relation (2.30) should hold beyond perturbation.Therefore, any observables which can be derived from the resolvent v ( z ) are functions of a . Without lossof generality, we can choose κ = κ = 1.Note that the explicit form of zv ′ ( z ) shown in Appendix A is written in terms of the parameters u , u ∞ , and τ . The definition of u ( z ) implies u ∞ = u − . (2.31)When the equality a = − a holds, u is given by τ as u = 14 τ. (2.32)Therefore, all the observables are also functions of τ . The relation between a and τ can be obtained fromthe inverse of u ( z ) given as z ( u ) = − ϑ ( u − u ) ϑ ( u + u ) ϑ ( u − u ∞ ) ϑ ( u + u ∞ ) . (2.33)Since z = a corresponds to u = 0 by definition, a is related to τ as a = − (cid:18) ϑ ( τ ) ϑ ( τ − ) (cid:19) . (2.34)In the following, we choose τ as the parameter specifying the resolvent.It is important to notice that not all values of τ are physically relevant. Since the ’t Hooft couplingsare defined as (2.2), their physical values are purely imaginary. Therefore, τ must be chosen such thatthe integrals (2.19) take purely imaginary values. The physical values of τ form a curve γ in the complex τ -plane.The asymptotic behavior of γ in the small ’t Hooft coupling limit is determined as follows. When a approaches 1, the definition of τ implies that Im( τ ) diverges to + ∞ . In this limit, the relation (2.34)becomes a ∼ − e πiτ . (2.35)On the other hand, (2.29) implies a ∼ − √ t (2.36)for small t . Therefore, in the small ’t Hooft coupling limit, the physical curve γ approaches the lineRe( τ ) = 12 . (2.37)7igure 1: The plot of the curve γ for n = 3 on which t is purely imaginary. The curve γ approaches theline Re( τ ) = in the limit t → 0. It terminates on the imaginary axis at which t diverges. Our interest is in the large ’t Hooft coupling limit. The limit can be obtained by following the curve γ in the direction opposite to the small ’t Hooft coupling limit discussed above. Recall that the curve γ isdefined as Re( t ) = 0. Since t is given in terms of the integral of a complicated function, it looks quitedifficult to determine the curve γ analytically. Instead, we evaluate the integral (2.19) numerically, andfind out where Re( t ) = 0 holds in the τ -plane.The plot of the curve γ for n = 3 is shown in Figure 1. The curve γ terminates on the imaginary axisat which the ’t Hooft coupling t diverges as t ∼ rτ − ν , r = 1 . i, (2.38)where ν = 0 . i for n = 3. This pole comes from the factor (1 + z ( u ν )) − in the function g ( u ) definedin Appendix A. The singularity at τ = ν was pointed out in [30].The values of the vevs h W i and h W i are the same since ¯ u i = ¯ w i holds. Their values in the large t limit are h W i = h W i = − ϕ ( b )(1 + e πiν ) t [ g ′ ( u ) G ( u ) + g ( u ) G ′ ( u )] → . . (2.39)Note that these are finite even in the large t limit, in contrast to ABJM theory and GT theory. Thefiniteness can be anticipated from the eigenvalue distribution. Recall that the large t limit correspondsto the limit τ → ν . The relation (2.34) implies that a = 1 . × − in the limit which is small but finite.Then, the value of b = a − is also finite. The following inequality |h W i| ≤ N N X i =1 e Re(¯ u i ) ≤ | b | (2.40)8mplies that h W i must be finite.In fact, it turns out that all the observables derived from the resolvent v ( z ) are finite in the large t limit. To show this, consider a general situation in which ρ ( x ; t ) is a density function with a parameter t whose support is an interval I . The expectation value of O ( x ) is defined as hOi t := Z I dx ρ ( x ; t ) O ( x ) . (2.41)Define a resolvent ω ( z ; t ) := t Z I dx ρ ( x ; t ) z + xz − x . (2.42)The expectation value (2.41) can be written in terms of ω ( z ; t ) as h O i t = 12 t Z C dz πi O ( z ) ω ( z ; t ) z , (2.43)where C is a contour in the z -plane encircling I counterclockwise. Suppose that a function F ( x ) satisfies F ′ ( x ) = 1 x O ( x ) . (2.44)Then, hOi t can be written as hOi t = − t Z C dz πi F ( z ) z · zω ′ ( z ; t ) . (2.45)This integral has a finite value in the limit t → ∞ if t − zω ′ ( z ; t ) has a finite limit at any points on thecontour C .In the case of the Chern-Simons-matter matrix models, the condition for the finiteness turns out tobe lim t → + ∞ i (cid:12)(cid:12)(cid:12) t − ˜ f ( u ) (cid:12)(cid:12)(cid:12) < ∞ . (cid:0) − ≤ u ≤ + (cid:1) (2.46)This can be shown to be the case numerically. Since this implies that the eigenvalue distribution of thismatrix model has a finite limit, all the quantities calculated by using this eigenvalue distribution mustbe finite. Recall that the planar free energy is equal to N times a quantity calculated by the eigenvaluedistribution. Therefore, it scales as N , as for usual gauge theories.If n is large, then Im( ν ) becomes large. Then, since the pole of t is located at the point τ = ν , the q -expansion of the theta functions can be used for the analysis of the large t limit. The behavior of t around τ = ν turns out to be t ∼ rτ − ν , r = 2 iπ ( e − πiν − . (2.47)This pole structure is qualitatively the same as (2.38) observed in the numerical result for n = 3.The vevs h W α i in the large t limit are h W i = h W i → 11 + e πiν . (2.48)This limiting value approaches 1 in the large n limit. This can be anticipated from the saddle pointequations (2.23)(2.24). As Im( ν ) becomes large, the attractive forces among the eigenvalues due to thesecond terms in the right-hand side of (2.23)(2.24) become strong. Then, the eigenvalue distributionshrinks to two points in the limit. 9 .4 Small τ limit We observed in the previous subsection that the behavior of the vevs h W α i for the theories in CSM( n ≥ a (= a = − a ) approaches zero. Small a limit corresponds toa small τ limit. Due to the relation (2.34), they are related as − log a ∼ πi τ . (2.49)It turns out that it is not straightforward to take the limit τ → 0. Since the factor (1 + z ( u ν )) − ,which is the origin of the pole of t at τ = ν , has in fact infinitely many poles on the τ -plane for n > τ = ν l + 1 + m l + 1 . ( l, m ∈ Z ) (2.50)Therefore, the small τ limit taken along the imaginary axis, for example, is ill-defined.One way to avoid these singularities is to take the following limit: τ = εe iθ , ε → θ . It turns out that if θ is in the range0 < cot θ < 12 Im( ν ) , (2.52)then all the poles can be avoided. At the same time, the expansion of the theta functions, after thetransformation τ → − τ , behaves well.The integral formula (2.19) gives the small τ behavior of t . The leading order term turns out to be t ∼ Cτ , C := 1sinh (cid:0) π Im( ν ) (cid:1) ∞ X n =1 ( − i ) n n + ν . (2.53)Note that the sum in the coefficient C for n = 3 is finite: ∞ X n =1 ( − i ) n n + ν = − . − . i. (2.54)Then, an upper bound of the vevs h W α i is obtained as |h W α i| ≤ | b | . (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) πi C t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (2.55)As in the cases for ABJM theory and GT theory, this upper bound is actually saturated. The formula(2.20) implies h W α i ∼ exp (cid:18) πi C t (cid:19) (2.56)for large t . This behavior is different from the Wilson loops in ABJM theory and GT theory.10 .5 The case k = − k The weak coupling behavior (2.29) indicates that there is another case k = − k for which the analysisis rather simple. Assuming that t takes a physical value, that is t ∈ i R , the parameters a and a arerelated as a = − a ∗ . (2.57)This is realized when the equalities u i = w ∗ i are satisfied. This equality is compatible with the full saddlepoint equations (2.23)(2.24), and therefore, the relation (2.57) should hold beyond perturbation. Indeed,such eigenvalue distributions were discussed in [11] [10].When the relation (2.57) holds, the parameters u , u ∞ , and τ in the resolvent v ( z ) are related amongthem. In addition to the relation (2.31), one can show that τ is given as τ = 4 i Im( u ) . (2.58)Therefore, all the observables derived from the resolvent v ( z ) are functions of u . Let u be written as u = u R + 14 τ, u R ∈ R , (2.59)the parameter a is given as a ∼ − e πiτ e πiu R (2.60)for large Im( τ ). The relation (2.29) for small t implies that the physical curve γ , now on the u -plane,approaches the line Re( u ) = 18 (2.61)in the small t limit.The large t limit can be analyzed in the same way as in the case k = k . The plot of the curve γ for n = 3 turns out to be quite similar to figure 1. The curve γ terminates on the imaginary axis of the u -plane. The intersection point is at u = 14 ν, (2.62)where t diverges. The vevs h W α i are finite even in the large t limit. In this section, we briefly revisit the discussion on possible gravity duals of theories in CSM( n ≥ D of the moduli space for a given Abelian Chern-Simons-matter theory is given by the matterrepresentation and the Chern-Simons levels k i [7]. Let l be the number of loops in the quiver diagramdictating the matter representation of the theory. The dimension D is given as D = ( l + 1) , ( P i k i = 0)4 l. ( P i k i = 0) (3.1)For example, ABJM theory corresponds to a quiver diagram with l = 1 and the two Chern-Simons levelsare k and − k . Then, the above formula implies D = 8. This coincides with the dimension of the modulispace C / Z k . For the theory in CSM( n ≥ l = n − n = 3 and k + k = 0 are satisfied.This is one reason that we focused our attention mainly on the theories with n = 3 and k = k .In the following, we assume n = 3 and k + k = 0. The moduli space turns out to be T ∗ P [25].This is an eight-dimensional hyper-K¨ahler cone whose base manifold is N , , = SU(3) / U(1). Then, onepossibility would be that the theories in CSM(3) would be dual to (suitable deformations of) M-theoryon AdS × N , , [25] [26]. Note that there could be quantum corrections to the moduli space, as shownin [27] for a flavored ABJM theory.This proposed gravity dual is, however, rather different from the dual of ABJM theory, that is, M-theory on AdS × S / Z k . In the latter, the planar limit corresponds to a Type IIA limit since the limitinvolves k → ∞ in which a circle direction in S shrinks. On the other hand, in the former, the planarlimit seems to be still eleven-dimensional since N , , is independent of the Chern-Simons levels.To understand the origin of the difference, it would be instructive to recall how the moduli space ofvacua is determined for Abelian Chern-Simons-matter theories. The relevant part of the action is ik π Z A ∧ dA + ik π Z A ∧ dA − Z d x | ∂ µ Y I − i ( A − A ) µ Y I | + · · · . (3.2)If the Chern-Simons levels satisfy k + k = 0, then by choosing a suitable linear combinations A and˜ A of the gauge fields A and A , the action can be written as ik π Z A ∧ d ˜ A − Z d x | ∂ µ Y I − iA µ Y I | + · · · . (3.3)This action defines a theory of matters Y I coupled to a Z k gauge theory constructed in terms of A and ˜ A [39]. This residual Z k gauge symmetry makes the moduli space to be an orbifold. Since thesuperpotential vanishes for the case k = − k , the moduli space is C / Z k for ABJM theory.On the other hand, if k + k is nonzero, then there is no choice of A and ˜ A for which the actionbecomes of the form (3.3). An alternative choice gives ik k ( k + k )4 π Z A ∧ dA + ik k π Z ˜ A ∧ d ˜ A − Z d x | ∂ µ Y I − i ( k + k ) AY I | + · · · . (3.4)Here, the gauge field ˜ A simply decouples. Since the gauge group is originally U(1) × U(1), there are twosets of F-term conditions and D-term conditions. In the theory under consideration, these two sets ofconditions are identical to each other. Therefore, the above theory can be regarded as a U (1) theorycoupled to three hypermultiplets, and as shown in [25], the resulting moduli space is T ∗ P , withoutorbifolding.If the gravity duals for theories in CSM(3) in the planar limit are really eleven-dimensional, thenthe dictionary between the AdS gravity and CFT should be quite different from the one for ABJMtheory. Then, the behavior of observables in the large ’t Hooft coupling limit might be quite different.For example, it is known that the dictionary for a CFT whose gravity dual is a massive Type IIA theoryis quite different from those for ABJM theory [40]. Due to this, the Wilson loop behaves as [41] h W i ∼ exp (cid:16) ct (cid:17) (3.5)for a constant c . The free energy scales as N in the planar limit. The leading term of the free energyin the large ’t Hooft coupling limit is multiplied by a power of t which reproduces the scaling in theM-theory limit [40] F ∼ c ′ N . (3.6)It seems that more detailed investigations are necessary to understand the issue of gravity duals fortheories in CSM(3). 12 Spectral curves In this section, we show that a difference among theories in CSM( n ) can be found in the structure ofplanar spectral curves obtained from the resolvent zv ′ ( z ). This observation can be extended to moregeneral Chern-Simons-matter theories. The structure turns out to be governed by an associated Kac-Moody algebra specified by the matter representation of the theory. The reader may consult [42] forresults on Kac-Moody algebras mentioned below. The spectral curves of the theories in CSM(2) werediscussed in [43]. n ) It was shown in [30] that the resolvent v ( z ) satisfies the following vector equations:(2 κ , 0) = x v ′ ( x +1 ) − x v ′ ( x − ) M , (4.1)(0 , κ ) = x v ′ ( x +2 ) − x v ′ ( x − ) M , (4.2)where x α ∈ I α , and x + α ( x − α ) is a point slightly above (below) x α . The matrices M α are defined as M := (cid:20) − n (cid:21) , M := (cid:20) n − (cid:21) . (4.3)In many cases, the left-hand sides of (4.1)(4.2) can be eliminated. This can be done by introducing ω ( z )such that zv ′ ( z ) = c + ω ( z ) (4.4)is satisfied, where the constant row vector c satisfies(2 κ , κ ) = cA, A := (cid:20) − n − n (cid:21) . (4.5)The solution exists as long as n = 2. Even in the case n = 2, there exists a solution of this equation ifand only if κ + κ = 0 (4.6)is satisfied. In the following, we assume that the equation (4.5) has a solution. Then, ω ( z ) satisfies ω ( x + α ) = ω ( x − α ) M α . (4.7)It was shown in [30] that, for the case n = 2, the vector equations (4.7) imply a simple scalar equation ω ψ ( x + α ) = − ω ψ ( x − α ) , (4.8)where ω ψ ( z ) := ω ( z ) · ψ, ψ := (cid:20) − (cid:21) . (4.9)Therefore, ω ψ ( z ) defines a two-sheeted covering of C , that is, a torus.For the case n = 1, one can check that the following equations are satisfied: ω ψ ( x +1 ) = ω M ψ ( x − ) , ω ψ ( x +2 ) = ω ψ ( x − ) ,ω M M ψ ( x +1 ) = ω M M ψ ( x − ) , ω M ψ ( x +2 ) = ω M M ψ ( x − ) , (4.10)13here ψ := (cid:20) (cid:21) . (4.11)The above equations imply that the three scalar functions ω ψ ( z ) , ω M ψ ( z ) , ω M M ψ ( z ) define a three-sheeted covering of C . After a compactification, the resulting curve is topologically a sphere. Note thatthese scalar functions are given in terms of the components of the resolvent v ( z ) as ω ψ ( z ) = zv ′ ( z ) − c , (4.12) ω M ψ ( z ) = zv ′ ( z ) − zv ′ ( z ) + c − c , (4.13) ω M M ψ ( z ) = zv ′ ( z ) − c . (4.14)Similar spectral curves appeared in a different context [44] [45].For the remaining cases n ≥ 3, it can be shown that ω ψ ( z ) do not define a finite-sheeted covering of C for any choice of ψ . A proof is given in Appendix B. The best one can find is a finite covering with atwist, defined by the following equations: ω ψ n ( x +1 ) = − ω ψ ′ n ( x − ) , ω ψ n ( x +2 ) = − e πiν ω ψ ′ n ( x − ) , (4.15)where ψ n := (cid:20) − e πiν (cid:21) , ψ ′ n = (cid:20) − e − πiν (cid:21) . (4.16)These equations are reduced to (4.8) when ν = 0, or in other words, n = 2.In summary, we have found that the spectral curve defined in terms of ω ( z ) is a Riemann surface ofgenus n − n = 1 , 2, and otherwise, the curve is an infinite-covering of C .There is also a difference between the case n = 1 and n = 2. For the case n = 1, it can be shown thatboth v ′ ( z ) and v ′ ( z ) are algebraic functions, while for the case n = 2, only the combination v ′ ( z ) − v ′ ( z )is an algebraic function.It is curious to notice that the matrix A in (4.5) is a generalized Cartan matrix. The correspondingKac-Moody algebra is su(3) for n = 1, affine su(2) for n = 2 and an algebra of indefinite type for n ≥ The structure of the spectral curves for theories in CSM( n ) observed in the previous subsection can alsobe found in more general Chern-Simons-matter theories. Consider a Chern-Simons-matter theory withthe gauge group Q n g a =1 U( N a ) k a coupled to bi-fundamental hypermultiplets discussed in [7] [38]. Let n ab be the number of the hypermultiplets coupled to U( N a ) × U( N b ) factor. Note that n ab is a symmetricmatrix.Assume that the Chern-Simons levels k a are chosen such that2( κ , · · · , κ n g ) = cA, A bc := 2 δ bc − n bc (4.17)has a solution. Again, the matrix A can be regarded as the generalized Cartan matrix of a Kac-Moodyalgebra g A . By construction, A is always symmetric.As in the previous subsection, the planar analysis of the Chern-Simons-matter theory is reduced tosolving the following vector equation [38]: ω ( x + a ) = ω ( x − a ) M a , (4.18)where ω ( z ) := ( ω ( z ) , · · · , ω n g ( z )) (4.19)14s a row-vector-valued function, and the matrices M a are defined as( M a ) bc := δ bc − δ ac A bc , (4.20)where the repeated indices are not summed. It can be shown that ( M a ) = I is satisfied for any a .Let W be a group generated by these M a . To construct a covering of C , we choose a column vector ψ ∈ C n g and consider the W -orbit: W ψ := { wψ | w ∈ W } . (4.21)Define scalar functions ω ψ ( z ) := ω ( z ) · ψ for each ψ ∈ W ψ . Then, the equations (4.18) imply ω ψ ( x + a ) = ω M a ψ ( x − a ) , (4.22)which define a covering of C . If W ψ is a finite set, then the covering gives a Riemann surface of a finitegenus.The following observation will be helpful for choosing an appropriate vector ψ which defines a simplespectral curve. Consider the following vectors:( α a ) b := A ab . (4.23)The action of M a on α b is M a α b = α b − A ab α a . (4.24)This indicates that M a act on α b as if α b are fundamental roots of g A , and M a are the fundamentalreflections of g A . Then, W gives a representation of the Weyl group of g A .If the matrix A is non-degenerate, then α a indeed define fundamental roots in a root system Φ of g A ,and M a define a faithful representation of the Weyl group of g A on C n g . Choosing ψ as one of the roots α ∈ Φ, the W -orbit W ψ is a subset of Φ.Suppose that g A is of finite type, that is, g A is a finite-dimensional Lie algebra. Then, W ψ is a finiteset since Φ is a finite set, and the covering defined above gives a Riemann surface of a finite genus.In fact, there is a simpler choice of ψ . Recall that the Weyl group also acts on the set of weightsin an irreducible representation. If ψ is chosen to be the highest weight of the smallest representation,then the resulting Riemann surface is the simplest possible one. The covering for the case n = 1 given inthe previous subsection is of this kind.Next, suppose that g A is of affine type. In this case, the rank of A is n g − 1. This implies that α a arenot linearly independent, and therefore, α a cannot define fundamental roots in a root system Φ of g A . Itis known that Φ can be realized in C n g +1 .The root system Φ of g A can be described explicitly as follows. Let g A be a finite-dimensional Liealgebra associated to g A , and let Φ be a root system of g A . Then Φ is given asΦ = { α + rδ | α ∈ Φ , r ∈ Z } , (4.25)where δ is defined as δ := n g X a =1 m a ˜ α a (4.26)in terms of the fundamental roots ˜ α a ∈ Φ of g A and integers m a satisfying n g X b =1 A ab m b = 0 , ( m , m , · · · , m n g ) = 1 . (4.27)15t is known that the one-dimensional subspace L ⊂ C n g +1 spanned by δ is invariant under the actionof the Weyl group of g A . Then, the Weyl group also acts on the quotient space C n g +1 /L . There existsan isomorphism of vector spaces π : C n g +1 /L → C n g , π (˜ α a ) = α a . (4.28)This is well-defined since π ( δ ) b = n g X a =1 m a ( α a ) b = n g X a =1 m a A ab = 0 (4.29)is satisfied. By this isomorphism, the matrices M a can be identified with the fundamental reflections ofthe Weyl group of g A acting on C n g +1 /L . In this quotient space, Φ becomes equivalent to Φ which isa finite set. If ψ is chosen to be one of α a , then the W -orbit W ψ is again finite. The covering for thecase n = 2 given in the previous subsection is of this kind.If g A is of indefinite type, there does not seem to exist a choice of ψ which gives us a Riemann surfaceof a finite genus as a spectral curve. It would be interesting if it would be possible to find a “twisted”covering like the one given in the previous subsection for the cases n ≥ 3. The existence of such coveringswould open the possibility to determine the resolvent explicitly. We have discussed the large ’t Hooft coupling limit of theories in CSM( n ≥ n . The structure is governed by a Kac-Moody algebra associated to the matter representation of eachChern-Simons-matter theory.It is curious that the behavior of observables of theories in CSM(3) is quite different from the oneobserved in ABJM theory, although they seem to share many properties, like the dimension of the modulispace which would suggest the existence of a gravity dual. We briefly pointed out that the differencecould be consistent with the observation that the dual gravity background would behave differently inthe planar limit. To know more about the possible gravity duals of theories in CSM(3), we need to knowmore detailed properties of the CFT side. One issue to be clarified is the consistency of the claim thatthe dual gravity background is AdS × N , , with [35] since the Chern-Simons levels are chosen suchthat k + k = 0 is satisfied.The observation that Kac-Moody algebras may play important role in Chern-Simons-matter matrixmodels seems to be quite interesting. The relation of Kac-Moody algebras and Chern-Simons-mattertheories would imply a classification of the latter in terms of the classification of the former. Thisalso suggests that detailed knowledge of Kac-Moody algebras of indefinite type would give us a crewto investigate various Chern-Simons-matter theories. For example, if one could find a representationof a Kac-Moody algebra of indefinite type which may give the twisted covering like (4.15) given insubsection 4.1, then the planar resolvent of the corresponding Chern-Simons-matter theory could beobtained explicitly in terms of the theta functions on a higher-genus Riemann surface.16 cknowledgements We would like to thank H. Itoyama, Y. Matsuo, T. Okazaki, T. Oota, R. Yoshioka for valuablediscussions. This work was supported in part by the Grant-in-Aid for Scientific research, No 16H06490and Fujukai Foundation. A Details on the planar resolvent In this appendix, we review some details on the planar resolvent obtained in [30].The derivative of the planar resolvent can be written as zv ′ ( z ) = c + 1 s ( z ) f ( z ) , (A.1)where s ( z ) is a square-root function defined in (2.15), and c is a constant vector satisfying (2.14). Ex-plicitly, c = − κ + 2 nκ n − , c = − nκ + 4 κ n − n = 2. The row-vector-valued function f ( z ) is given in terms of a scalar-valued function ˜ f ( u ) as in(2.17).One can check that a function G ( u ) defined as G ( u ) := ϑ ( u − u ν ) ϑ ( u − u ν + ) ϑ ( u − u ∞ ) ϑ ( u + u ∞ ) (A.3)satisfies the conditions (2.18), provided that the parameters are defined as u := u (0) , u ∞ := u ( ∞ ) , u ν := 12 ν + 14 . (A.4)In terms of G ( u ), the function ˜ f ( u ) is given as˜ f ( u ) = g ( u ) G ( u ) (A.5)where g ( u ) is an elliptic function. The function g ( u ) is then determined by requiring that the derivative zv ′ ( z ) has the right analytic structure. The explicit form of g ( u ) is g ( u ) = r (1 + z ( u )) + r ( g ( u ) + g ( u )) , (A.6)where z ( u ) is defined in (2.33), and g ( u ) := ϑ ( u − u ) ϑ ( u − u ν + ) ϑ ( u − u ∞ ) ϑ ( u − u ν ) , g ( u ) := − ϑ ( u + u ) ϑ ( u − u ν ) ϑ ( u + u ∞ ) ϑ ( u − u ν + ) . (A.7)The coefficients r , r are given as r = 1 g ( − u ) − g ( u ) (cid:20) ˜ c g ( − u ) G ( u ) − ˜ c g ( u ) G ( − u ) (cid:21) , (A.8) r = 1 g ( − u ) − g ( u ) (cid:20) − ˜ c G ( u ) + ˜ c G ( − u ) (cid:21) , (A.9)17here ˜ c := (˜ c , ˜ c ) is defined as ˜ c := cS . Note that the equality g ( − u ) − g ( u ) = g ( − u ) (1 + z ( u ν )) (A.10)holds. B Infinite number of sheets for n ≥ Let W be a group generated by M and M . As explained in subsection 4.1, one can construct a spectralcurve from the equations (4.7) in terms of the scalar functions ω ψ ( z ) , ψ ∈ W ψ (B.1)for a choice of ψ ∈ C .Suppose that W ψ is finite for a suitable ψ . Then, there exists an element w ∈ W such that wψ = ψ (B.2)holds. The element w has one of the following forms:( M M ) m , ( M M ) m , M ( M M ) m , M ( M M ) m (B.3)for a non-negative integer m . One may choose a basis of C such that M and M can be written as M = (cid:20) − − (cid:21) , M = (cid:20) − e − πiν − e πiν (cid:21) . (B.4)For n ≥ 3, the parameter ν is purely imaginary. We choose Im( ν ) > M M ) m = (cid:20) e πimν e − πimν (cid:21) . (B.5)Apparently, if w is of this form, then the condition (B.2) cannot be satisfied for any ψ . By the samereasoning, w is not of the form ( M M ) m .Now, suppose w = M ( M M ) m . The explicit form of this matrix is (cid:20) − e πi ( m +1) ν − e πi ( m +1) ν (cid:21) . (B.6)Then, ψ is determined to be ψ = (cid:20) − e πi ( m +1) ν (cid:21) (B.7)up to an overall factor. This ψ gives( M M ) l ψ = (cid:20) e πilν − e πi ( m +1 − l ) ν (cid:21) , (B.8) M ( M M ) l ψ = (cid:20) e πi ( m − l ) ν − e πi ( l +1) µ (cid:21) (B.9)for 0 ≤ l ≤ m . If W ψ is finite, then ( M M ) l ψ for a positive integer l must have one of the above forms.However, this is impossible since its form is( M M ) l ψ = (cid:20) e − πilν − e πi ( m +1+ l ) ν (cid:21) . (B.10)18herefore, w is not of the form M ( M M ) m . The same argument excludes the possibility for w to be ofthe form M ( M M ) m . It is concluded that an element w ∈ W satisfying (B.2) does not exist, implyingthat W ψ is infinite for any choice of ψ . References [1] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, “N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,” JHEP , 091 (2008) doi:10.1088/1126-6708/2008/10/091 [arXiv:0806.1218 [hep-th]].[2] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J.Theor. Phys. , 1113 (1999) [Adv. Theor. Math. Phys. , 231 (1998)] doi:10.1023/A:1026654312961[hep-th/9711200].[3] D. Gaiotto and X. Yin, “Notes on superconformal Chern-Simons-Matter theories,” JHEP , 056(2007) doi:10.1088/1126-6708/2007/08/056 [arXiv:0704.3740 [hep-th]].[4] J. P. Gauntlett, G. W. Gibbons, G. Papadopoulos and P. K. Townsend, “Hyper-Kahler manifoldsand multiply intersecting branes,” Nucl. Phys. B , 133 (1997) doi:10.1016/S0550-3213(97)00335-0[hep-th/9702202].[5] J. Sparks, “Sasaki-Einstein Manifolds,” Surveys Diff. Geom. , 265 (2011)doi:10.4310/SDG.2011.v16.n1.a6 [arXiv:1004.2461 [math.DG]].[6] Y. Imamura and K. Kimura, “On the moduli space of elliptic Maxwell-Chern-Simons theories,” Prog.Theor. Phys. , 509 (2008) doi:10.1143/PTP.120.509 [arXiv:0806.3727 [hep-th]].[7] D. L. Jafferis and A. Tomasiello, “A Simple class of N=3 gauge/gravity duals,” JHEP , 101(2008) doi:10.1088/1126-6708/2008/10/101 [arXiv:0808.0864 [hep-th]].[8] P. M. Crichigno and D. Jain, “Non-toric Cones and Chern-Simons Quivers,” arXiv:1702.05486 [hep-th].[9] N. Drukker, M. Marino and P. Putrov, “From weak to strong coupling in ABJM theory,” Commun.Math. Phys. , 511 (2011) doi:10.1007/s00220-011-1253-6 [arXiv:1007.3837 [hep-th]].[10] C. P. Herzog, I. R. Klebanov, S. S. Pufu and T. Tesileanu, “Multi-Matrix Models and Tri-Sasaki Ein-stein Spaces,” Phys. Rev. D , 046001 (2011) doi:10.1103/PhysRevD.83.046001 [arXiv:1011.5487[hep-th]].[11] T. Suyama, “On Large N Solution of ABJM Theory,” Nucl. Phys. B , 50 (2010)doi:10.1016/j.nuclphysb.2010.03.011 [arXiv:0912.1084 [hep-th]].[12] H. U. Yee, “AdS/CFT with Tri-Sasakian Manifolds,” Nucl. Phys. B , 232 (2007)doi:10.1016/j.nuclphysb.2007.03.031 [hep-th/0612002].[13] A. Kapustin, B. Willett and I. Yaakov, “Exact Results for Wilson Loops in SuperconformalChern-Simons Theories with Matter,” JHEP , 089 (2010) doi:10.1007/JHEP03(2010)089[arXiv:0909.4559 [hep-th]]. 1914] D. R. Gulotta, C. P. Herzog and S. S. Pufu, “From Necklace Quivers to the F-theorem, OperatorCounting, and T(U(N)),” JHEP , 077 (2011) doi:10.1007/JHEP12(2011)077 [arXiv:1105.2817[hep-th]].[15] D. R. Gulotta, C. P. Herzog and S. S. Pufu, “Operator Counting and Eigenvalue Distributionsfor 3D Supersymmetric Gauge Theories,” JHEP , 149 (2011) doi:10.1007/JHEP11(2011)149[arXiv:1106.5484 [hep-th]].[16] P. M. Crichigno, C. P. Herzog and D. Jain, “Free Energy of D n Quiver Chern-Simons Theories,”JHEP , 039 (2013) doi:10.1007/JHEP03(2013)039 [arXiv:1211.1388 [hep-th]].[17] D. Martelli and J. Sparks, “The large N limit of quiver matrix models and Sasaki-Einstein manifolds,”Phys. Rev. D , 046008 (2011) doi:10.1103/PhysRevD.84.046008 [arXiv:1102.5289 [hep-th]].[18] S. Cheon, H. Kim and N. Kim, “Calculating the partition function of N=2 Gauge theorieson S and AdS/CFT correspondence,” JHEP , 134 (2011) doi:10.1007/JHEP05(2011)134[arXiv:1102.5565 [hep-th]].[19] D. L. Jafferis, I. R. Klebanov, S. S. Pufu and B. R. Safdi, “Towards the F-Theorem: N=2 Field The-ories on the Three-Sphere,” JHEP , 102 (2011) doi:10.1007/JHEP06(2011)102 [arXiv:1103.1181[hep-th]].[20] D. Martelli, A. Passias and J. Sparks, “The gravity dual of supersymmetric gauge theories ona squashed three-sphere,” Nucl. Phys. B , 840 (2012) doi:10.1016/j.nuclphysb.2012.07.019[arXiv:1110.6400 [hep-th]].[21] A. Amariti, C. Klare and M. Siani, “The Large N Limit of Toric Chern-Simons Matter Theories andTheir Duals,” JHEP , 019 (2012) doi:10.1007/JHEP10(2012)019 [arXiv:1111.1723 [hep-th]].[22] D. Gang, C. Hwang, S. Kim and J. Park, “Tests of AdS /CFT correspondence for N = 2 chiral-liketheory,” JHEP , 079 (2012) doi:10.1007/JHEP02(2012)079 [arXiv:1111.4529 [hep-th]].[23] A. Amariti and S. Franco, “Free Energy vs Sasaki-Einstein Volume for Infinite Families of M2-BraneTheories,” JHEP , 034 (2012) doi:10.1007/JHEP09(2012)034 [arXiv:1204.6040 [hep-th]].[24] S. Lee and D. Yokoyama, “Geometric free energy of toric AdS /CFT models,” JHEP , 103(2015) doi:10.1007/JHEP03(2015)103 [arXiv:1412.8703 [hep-th]].[25] S. Gukov, C. Vafa and E. Witten, “CFT’s from Calabi-Yau four folds,” Nucl. Phys. B , 69 (2000)Erratum: [Nucl. Phys. B , 477 (2001)] doi:10.1016/S0550-3213(01)00289-9, 10.1016/S0550-3213(00)00373-4 [hep-th/9906070].[26] M. Billo, D. Fabbri, P. Fre, P. Merlatti and A. Zaffaroni, “Rings of short N=3 superfields inthree-dimensions and M theory on AdS(4) x N**(0,1,0),” Class. Quant. Grav. , 1269 (2001)doi:10.1088/0264-9381/18/7/310 [hep-th/0005219].[27] D. Gaiotto and D. L. Jafferis, “Notes on adding D6 branes wrapping RP**3 in AdS(4) x CP**3,”JHEP , 015 (2012) doi:10.1007/JHEP11(2012)015 [arXiv:0903.2175 [hep-th]].[28] S. Cheon, D. Gang, S. Kim and J. Park, “Refined test of AdS4/CFT3 correspondence for N=2,3theories,” JHEP , 027 (2011) doi:10.1007/JHEP05(2011)027 [arXiv:1102.4273 [hep-th]].[29] T. Nosaka and S. Yokoyama, “Complete factorization in minimal N=4 Chern-Simons-matter theory,”arXiv:1706.07234 [hep-th].[30] T. Suyama, “Notes on Planar Resolvents of Chern-Simons-matter Matrix Models,” JHEP , 049(2016) doi:10.1007/JHEP11(2016)049 [arXiv:1605.09110 [hep-th]].2031] N. Drukker, J. Plefka and D. Young, “Wilson loops in 3-dimensional N=6 supersymmetricChern-Simons Theory and their string theory duals,” JHEP , 019 (2008) doi:10.1088/1126-6708/2008/11/019 [arXiv:0809.2787 [hep-th]].[32] B. Chen and J. B. Wu, “Supersymmetric Wilson Loops in N=6 Super Chern-Simons-matter theory,”Nucl. Phys. B , 38 (2010) doi:10.1016/j.nuclphysb.2009.09.015 [arXiv:0809.2863 [hep-th]].[33] S. J. Rey, T. Suyama and S. Yamaguchi, “Wilson Loops in Superconformal Chern-Simons The-ory and Fundamental Strings in Anti-de Sitter Supergravity Dual,” JHEP , 127 (2009)doi:10.1088/1126-6708/2009/03/127 [arXiv:0809.3786 [hep-th]].[34] O. Aharony, O. Bergman and D. L. Jafferis, “Fractional M2-branes,” JHEP , 043 (2008)doi:10.1088/1126-6708/2008/11/043 [arXiv:0807.4924 [hep-th]].[35] D. Gaiotto and A. Tomasiello, “The gauge dual of Romans mass,” JHEP , 015 (2010)doi:10.1007/JHEP01(2010)015 [arXiv:0901.0969 [hep-th]].[36] M. Marino and P. Putrov, “Exact Results in ABJM Theory from Topological Strings,” JHEP ,011 (2010) doi:10.1007/JHEP06(2010)011 [arXiv:0912.3074 [hep-th]].[37] T. Suyama, “On Large N Solution of Gaiotto-Tomasiello Theory,” JHEP , 101 (2010)doi:10.1007/JHEP10(2010)101 [arXiv:1008.3950 [hep-th]].[38] T. Suyama, “A Systematic Study on Matrix Models for Chern-Simons-matter Theories,” Nucl. Phys.B , 528 (2013) doi:10.1016/j.nuclphysb.2013.06.008 [arXiv:1304.7831 [hep-th]].[39] A. Kapustin and N. Seiberg, “Coupling a QFT to a TQFT and Duality,” JHEP , 001 (2014)doi:10.1007/JHEP04(2014)001 [arXiv:1401.0740 [hep-th]].[40] O. Aharony, D. Jafferis, A. Tomasiello and A. Zaffaroni, “Massive type IIA string theory cannotbe strongly coupled,” JHEP , 047 (2010) doi:10.1007/JHEP11(2010)047 [arXiv:1007.2451 [hep-th]].[41] T. Suyama, “Eigenvalue Distributions in Matrix Models for Chern-Simons-matter Theories,” Nucl.Phys. B , 497 (2012) doi:10.1016/j.nuclphysb.2011.11.013 [arXiv:1106.3147 [hep-th]].[42] R.W. Carter, “Lie Algebras of Finite and Affine Type,” Cambridge University Press.[43] H. Itoyama, T. Oota, T. Suyama and R. Yoshioka, “Cubic constraints for the resolvents ofthe ABJM matrix model and its cousins,” Int. J. Mod. Phys. A , no. 11, 1750056 (2017)doi:10.1142/S0217751X17500567 [arXiv:1609.03681 [hep-th]].[44] R. Dijkgraaf and C. Vafa, “On geometry and matrix models,” Nucl. Phys. B , 21 (2002)doi:10.1016/S0550-3213(02)00764-2 [hep-th/0207106].[45] H. Itoyama, K. Maruyoshi and T. Oota, “The Quiver Matrix Model and 2d-4d Conformal Connec-tion,” Prog. Theor. Phys.123