aa r X i v : . [ h e p - t h ] S e p RIKEN-MP-21
Matrix model from N = 2 orbifold partition function Taro Kimura ∗ Department of Basic Science, University of Tokyo, Tokyo 153-8902, JapanandMathematical Physics Lab., RIKEN Nishina Center, Saitama 351-0198, Japan
Abstract
The orbifold generalization of the partition function, which would describe the gaugetheory on the ALE space, is investigated from the combinatorial perspective. It is shownthat the root of unity limit q → exp(2 πi/k ) of the q -deformed partition function plays acrucial role in the orbifold projection while the limit q → R . Then startingfrom the combinatorial representation of the partition function, a new type of multi-matrix model is derived by considering its asymptotic behavior. It is also shown thatSeiberg-Witten curve for the corresponding gauge theory arises from the spectral curveof this multi-matrix model. ∗ E-mail address: [email protected] ontents N = 2 partition functions 43 Orbifold partition function 7 N = 2 ∗ theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 N limit and saddle point equation . . . . . . . . . . . . . . . . . . . . . 175.2 Relation to Seiberg-Witten theory . . . . . . . . . . . . . . . . . . . . . . . . 225.2.1 Four dimensional theory . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.2 Five dimensional theory . . . . . . . . . . . . . . . . . . . . . . . . . . 25 q -deformed CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 The recent progress on the four dimensional N = 2 supersymmetric gauge theory reveals aremarkable relation to the two dimensional conformal field theory [1]. This relation providesthe explicit interpretation for the partition function of the four dimensional gauge theory[2, 3] as the conformal block of the two dimensional Liouville field theory, and is naturallyregarded as a consequence of the M-brane compactifications [4, 5], which reproduces theresults of the four dimensional gauge theory [6, 7]. It is originally considered in [1] forSU(2) theory, and extended to the higher rank gauge theory [8, 9], the non-conformal theory[10, 11, 12] and the cases with the surface and loop operators [13], etc.According to this connection, established results on the two dimensional side can bereconsidered from the viewpoint of the four dimensional theory, and vice versa. One ofthe useful applications is the matrix model description of the supersymmetric gauge theory114, 15, 16, 17, 18, 19, 20, 21]. This is based on the fact that the conformal block on the spherecan be also regarded as the matrix integral, which is called the Dotsenko-Fateev integralrepresentation [22, 23]. In this direction some extensions of the matrix model description areperformed by starting with the Liouville correlators on the higher genus Riemann surfaces[24, 25]. Furthermore another type of the matrix model is also investigated so far [26, 27, 28][29]. This matrix model is directly derived from the combinatorial representation of thepartition function by considering its asymptotic behavior while the Dotsenko-Fateev typematrix model is obtained from the Liouville correlator, so that the conformal symmetry ismanifest. This treatment is quite analogous to the matrix integral representation of thecombinatorial object. Although they are apparently different from the Dotsenko-Fateevtype, both types of the matrix model correctly reproduce the results of the four dimensionalgauge theory, e.g. Seiberg-Witten curve.The purpose of this paper is to extend the remarkable connection between the two andfour dimensional theory to the orbifold theory. The four dimensional orbifold manifold isgiven by C / Γ where Γ is a finite subgroup of SU(2), e.g. Γ = Z k , and its minimal resolutionof the singularity gives the ALE space [34]. The gauge theory on the ALE space is wellinvestigated in [35, 36] with respect to the instanton moduli space. Their results show themoduli spaces of the instanton on the ALE spaces are deeply connected to the affine Liealgebras, and the quiver varieties. This interesting fact provides a guideline to search fora counterpart of the AGT relation for the orbifold theory. Furthermore this theory hasbeen reconsidered in terms of D-branes [37] in which some aspects of 2d/4d connection arepartially studied, and the theory on the Taub-NUT space has been investigated in [38] indetail.To obtain the solutions of the four dimensional gauge theory on the ALE space, we haveto deal with the partition function as the case of R by implementing the ADHM constructionand the localization method for the ALE space. By performing such a procedure, the orbifoldgeneralization of the combinatorial partition function, which would describe the gauge theoryon the ALE space, has been investigated [39] (see also [40]). They are defined as the invariantsector of the Young diagram under the orbifold action Γ, and the process to extract onlythe Γ-invariant sector is called the orbifold projection in [39]. This projection seems quitereasonable, but it is difficult to perform this in a systematic manner. In this paper we willshow the orbifold projection is naturally performed when one appropriately parametrizesthe partition function. This parametrization is just interpreted as the root of unity limit q → exp (2 πi/k ) of the q -deformed partition function while we have to take q → For example, the longest increasing subsequences in random permutations, the non-equilibrium stochasticmodel, so-called TASEP [30], and so on (see also [31]). Their remarkable connection to the Tracy-Widomdistribution [32] can be understood from the viewpoint of the random matrix theory through the Robinson-Schensted-Knuth (RSK) correspondence (see e.g. [33]). R theory. The similar approach is found in [41] in the context of the spinCalogero-Sutherland model, where the combinatorial method plays an important role incharacterization of the wavefunction, derivation of the dynamical correlation functions andso on (see, for example, [42]).In this paper we also propose a new matrix model description for the orbifold theory.Starting with the combinatorial representation and applying the method developed in [26,27, 28] we derive a new kind of matrix models, and discuss the corresponding gauge theoryconsequences. As a result of orbifolding, we get k -matrix models for the ALE space. Thismodel is quite similar to the Chern-Simons matrix model [43] for the lens space S / Z k [44, 45, 46] (see also [47]), which is recently applied to the ABJM theory [48, 49] [50, 51].This is so reasonable because the ALE space goes to the lens space S / Z k at infinity. Wethen discuss the large N limit of the matrix model, and show Seiberg-Witten curve is arisingfrom the spectral curve of the matrix model.Along the recent interest in the 2d/4d relation, it is natural to search a two dimensionalcounterpart of the orbifold partition function. While the original relation relies on theinterpretation of the SU( n ) partition function as the conformal block of the conformal fieldtheory described by W n algebra, its generalization is also proposed [52, 53] [54] [55]: thegeneralized W algebra corresponds to more complicated gauge theories in four dimensions.In such a theory an embedding ρ : SU(2) → SU( n ) plays an important role in characterizingthe generalized W algebra [56]. On the other hand, in the orbifold theory, an embedding ρ : Γ ⊂ SU(2) → SU( n ) is also found to characterize the decomposition into the irreduciblerepresentations of Γ. We will discuss this similarity from the viewpoint of string theory andpropose a new kind of connection between 2d/4d theories.The organization of this paper is as follows. In section 2 we provide the combinatorialrepresentation of the four dimensional gauge theory, which is originally proposed in [2, 3],as preliminaries for some extensions discussed in this paper. In section 3 we then study theorbifold generalization of the partition function with focusing on its combinatorial aspect.We will see the root of unity limit of the q -deformed partition function implements theorbifold projection. Section 4 is devoted to derivation of the matrix model from the orbifoldpartition function by considering its asymptotic behavior. In section 5 we study the multi-matrix model in detail by taking the large N limit. We then extract Seiberg-Witten curveof the corresponding gauge theory via the spectral curve of the matrix model. In section6 we try to search a 2d/4d relation for the orbifold theory. The two dimensional theory isdiscussed from the viewpoint of string theory, and a relation to the generalized W algebra isalso proposed. We also discuss topics related to the q -deformation of the partition function.We finally summarize our results in section 7. Because the combinatorial partition function includes infinite products of the q -parameter, we haveto take care of its convergence radius. Thus, to obtain the root of unity limit q → exp(2 πi/k ), we firstparametrize it as q → ωq , and take the limit of q → ote added After submitting this article, some related papers, considering extensions of the AGT relationto the ALE spaces, appear in the preprint server [57, 58, 59, 60, 61]. N = 2 partition functions Let us start with the partition functions for N = 2 theories, originally studied by [2, 3],as preliminaries of discussions along this paper: its deformation is proposed in section 3and we will also see in section 4 the matrix model can be obtained from the combinatorialrepresentation by considering its asymptotic behavior.First we introduce the instanton part of the four dimensional partition function for N = 2SU( n ) theory with N f (anti)fundamental matters, Z = X ~λ Λ (2 n − N f ) | ~λ | Z Z , (2.1) Z = Y ( l,i ) =( m,j ) Γ( λ ( l ) i − λ ( m ) j + β ( j − i ) + b lm + β )Γ( λ ( l ) i − λ ( m ) j + β ( j − i ) + b lm ) Γ( β ( j − i ) + b lk )Γ( β ( j − i ) + b lk + β ) , (2.2) Z = n Y l =1 N f Y f =1 ∞ Y i =1 Γ( λ ( l ) i + b l + M f + βi + 1)Γ( b l + M f + βi + 1) , (2.3) Z = n Y l =1 N f Y f =1 ∞ Y i =1 Γ( λ ( l ) i + b l − M f + β ( i + 1))Γ( b l − M f + β ( i + 1)) . (2.4)These combinatorial expressions are written in terms of n -tuple partitions, ~λ = ( λ (1) , · · · , λ ( n ) ),and their parameters are related to those of the gauge theory as β = − ǫ ǫ , b l = a l ǫ , a lm = a l − a m , b lm = b l − b m , M f = m f ǫ (2.5)where ǫ > > ǫ are Ω-background parameters, a l is the Coulomb moduli parametrizing theU(1) n − vacua, and m f denotes the mass of the fundamental matter. We can also introducebifundamental matter fields, but we do not focus on them in this paper. For N = 2 ∗ theory,which includes the adjoint matter, we have to consider another contribution, Z = Y ( l,i ) =( m,j ) Γ( λ ( l ) i − λ ( m ) j + β ( j − i ) + b lm + M )Γ( λ ( l ) i − λ ( m ) j + β ( j − i ) + b lm + M + β ) Γ( β ( j − i ) + b lk + M + β )Γ( β ( j − i ) + b lk + M ) (2.6)with the mass of the adjoint matter M = m ǫ . (2.7)Note one can see a simple relation Z vec = 1 Z adj ( m = 0) . (2.8)4his N = 2 ∗ theory is given by the simple deformation of N = 4 theory by applying themass to the adjoint matter. We will comment on the massless limit of this N = 2 ∗ theorylater.The five dimensional extensions of these partition functions are also proposed, which canbe regarded as the q -deformed analogue of the original four dimensional one, Z = X ~λ Λ (2 n − N f ) | ~λ | Z Z , (2.9) Z = Y ( l,i ) =( m,j ) ( Q lm q λ ( l ) i − λ ( m ) j t j − i ; q ) ∞ ( Q lm q λ ( l ) i − λ ( m ) j t j − i +1 ; q ) ∞ ( Q lm t j − i +1 ; q ) ∞ ( Q lm t j − i ; q ) ∞ , (2.10) Z = n Y l =1 N f Y f =1 ∞ Y i =1 ( Q l Q m f qt − i ; q ) ∞ ( Q l Q m f q λ ( l ) i +1 t − i ; q ) ∞ , (2.11) Z = n Y l =1 N f Y f =1 ∞ Y i =1 ( Q l Q − m f t − i +1 ; q ) ∞ ( Q l Q − m f q λ ( l ) i t − i +1 ; q ) ∞ , (2.12) Z = Y ( l,i ) =( m,j ) ( Q m Q lm q λ ( l ) i − λ ( m ) j t j − i +1 ; q ) ∞ ( Q m Q lm q λ ( l ) i − λ ( m ) j t j − i ; q ) ∞ ( Q m Q lm t j − i ; q ) ∞ ( Q m Q lm t j − i +1 ; q ) ∞ . (2.13)Here we define ( x ; q ) ∞ = ∞ Y p =0 (1 − xq p ) , (2.14)and deformed parameters are given by q = e ǫ , t = e − ǫ = q β , Q l = e a l = q b l , Q lm = e a l − a m = q b lm , (2.15) Q m f = e m f = q M f , Q m = e m = q M . (2.16)The radius R of the compactified dimension S is implicitly included in these parametersby rescaling the parameters, e.g. q = e Rǫ . Thus one can check the four dimensional resultis reproduced by the five dimensional partition function by taking the limit R →
0, orequivalently q → N ( l ) for the partitions [26, 27, 28]. By decomposingthe partition functions (2.10), (2.13) as Z vec = n Y l,m Z ( l,m )vec , (2.17) Z adj = n Y l,m Z ( l,m )adj , (2.18)5ach part can be rewritten as Z ( l,l )vec = ∞ Y i = j ( q λ ( l ) i − λ ( l ) j t j − i ; q ) ∞ ( q λ ( l ) i − λ ( l ) j t j − i +1 ; q ) ∞ ( t j − i +1 ; q ) ∞ ( t j − i ; q ) ∞ = 1 A ( l,l ) m =0 N ( l ) Y i = j ( q λ ( l ) i − λ ( l ) j t j − i ; q ) ∞ ( q λ ( l ) i − λ ( l ) j t j − i +1 ; q ) ∞ N ( l ) Y i =1 ( q λ ( l ) i t N ( l ) − i +1 ; q ) ∞ ( q − λ ( l ) i t − N ( l ) + i ; q ) ∞ , (2.19) Z ( l,m )vec = ∞ Y i,j ( Q lm q λ ( l ) i − λ ( m ) j t j − i ; q ) ∞ ( Q lm q λ ( l ) i − λ ( m ) j t j − i +1 ; q ) ∞ ( Q lm t j − i +1 ; q ) ∞ ( Q lm t j − i ; q ) ∞ = 1 A ( l,m ) m =0 N ( l ) Y i =1 N ( m ) Y j =1 ( Q lm q λ ( l ) i − λ ( m ) j t j − i ; q ) ∞ ( Q lm q λ ( l ) i − λ ( m ) j t j − i +1 ; q ) ∞ Q N ( l ) i =1 ( Q lm q λ ( l ) i t N ( m ) − i +1 ; q ) ∞ Q N ( m ) i =1 ( Q lm q − λ ( m ) i t − N ( l ) + i ; q ) ∞ , (2.20) Z ( l,l )adj = A ( l ) m N ( l ) Y i = j ( Q m q λ ( l ) i − λ ( l ) j t j − i +1 ; q ) ∞ ( Q m q λ ( l ) i − λ ( l ) j t j − i ; q ) ∞ N ( l ) Y i =1 ( Q m q − λ ( l ) i t − N ( l ) + i ; q ) ∞ ( Q m q λ ( l ) i t N ( l ) − i +1 ; q ) ∞ , (2.21) Z ( l,m )adj = A ( l,m ) m N ( l ) Y i =1 N ( m ) Y j =1 ( Q m Q lm q λ ( l ) i − λ ( m ) j t j − i +1 ; q ) ∞ ( Q m Q lm q λ ( l ) i − λ ( m ) j t j − i ; q ) ∞ Q N ( m ) i =1 ( Q m Q lm q − λ ( m ) i t − N ( l ) + i ; q ) ∞ Q N ( l ) i =1 ( Q m Q lm q λ ( l ) i t N ( m ) − i +1 ; q ) ∞ . (2.22)We can check finite constants turn out to be A ( l,m ) m = (cid:18) ( Q m t ; q ) ∞ ( Q m ; q ) ∞ (cid:19) N ( l ) ( l = m )1 ( N ( l ) = N ( m ) , l = m ) N ( m ) − N ( l ) Y i =1 ( Q m Q lm t i ; q ) − ∞ ( N ( l ) < N ( m ) ) N ( l ) − N ( m ) Y i =1 ( Q m Q lm t − i +1 ; q ) ∞ ( N ( l ) > N ( m ) ) . (2.23)Therefore they can be represented as Z vec = n Y l,m A ( l,m ) m =0 Y ( l,i ) =( m,j ) ( Q lm q λ ( l ) i − λ ( m ) j t j − i ; q ) ∞ ( Q lm q λ ( l ) i − λ ( m ) j t j − i +1 ; q ) ∞ n Y l,m N ( l ) Y i =1 ( Q lm q λ ( l ) i t N ( m ) − i +1 ; q ) ∞ ( Q ml q − λ ( l ) i t − N ( m ) − i ; q ) ∞ , (2.24) Z adj = n Y l,m A ( l,m ) m Y ( l,i ) =( m,j ) ( Q m Q lm q λ ( l ) i − λ ( m ) j t j − i +1 ; q ) ∞ ( Q m Q lm q λ ( l ) i − λ ( m ) j t j − i ; q ) ∞ n Y l,m N ( l ) Y i =1 ( Q m Q ml q − λ ( l ) i t − N ( m ) − i ; q ) ∞ ( Q m Q lm q λ ( l ) i t N ( m ) − i +1 ; q ) ∞ , (2.25) Z fund = n Y l =1 N f Y f =1 N ( l ) Y i =1 ( Q l Q m f qt − i ; q ) ∞ ( Q l Q m f q λ ( l ) i +1 t − i ; q ) ∞ , (2.26) Z antifund = n Y l =1 N f Y f =1 N ( l ) Y i =1 ( Q l Q − m f t − i +1 ; q ) ∞ ( Q l Q − m f q λ ( l ) i t − i +1 ; q ) ∞ . (2.27)6hese expressions are convenient to give the matrix model description [26, 27, 28]. We willinvestigate deformed versions of the partition functions, starting from these expressions, andderive the corresponding matrix models in section 4. We then consider the orbifold generalization of the partition function for N = 2 theory[39], which describes the gauge theory on the ALE spaces [34, 35, 36], obtained from theminimal resolution of orbifolds C / Γ where Γ is a finite subgroup of SU(2). This resolutionof the singularity is performed by replacing two-spheres, where their intersecting numbersare related to a Cartan matrix of the corresponding Lie algebras g . This connection betweensubgroups Γ of SU(2) and Lie algebras g is known as the McKay correspondence. Especiallythe Abelian groups Γ = Z k correspond to A k − Lie algebras, and we focus on them in thispaper.To study such an extended partition function, we will show it is useful to deal with the q -deformed partition function which corresponds to the five dimensional theory, and takethe root of unity limit of the deformation parameter. We remark the procedure used in thispaper to obtain the orbifold partition function is closely related to the method proposedin [41] to study the spin Calogero-Sutherland model (e.g. see [42]), where combinatorialarguments are quite important as well as the four or five dimensional gauge theory. To construct the orbifold partition function, let us first clarify the orbifold action for thecombinatorial partition function. Recalling the partition function is obtained from the Cherncharacter of the fixed point in the moduli space under the torus action, or the equivariant K -theory class, it is reasonable to observe an effect of the orbifolding on the torus action.Their expressions are closely related to the five dimensional partition function, so that wefirst study the five dimensional function rather than the four dimensional version to discussthe meaning of the orbifold action.By considering ADHM construction and the localization method for the ALE space[62, 39] as in the case of the usual R space, it is shown the orbifold partition functionshould be defined as a sector of the q -deformed function (with q → q −→ ωq, t −→ ωt, Q l −→ ω p l Q l , (3.1)where ω = exp(2 πi/k ) is the primitive k th root of unity. p l is an integer p l ∈ { , · · · , k − } ,and p lm = p l − p m . It denotes the 1 st Chern class of the instanton bundle [39], and also isinterpreted as the linking number of the five branes [38]. Actually the boundary of the ALE7
315 1 1 1 12 2 2 23 3 345 6891011 138 445 689
Figure 1: Γ-invariant sector for U(1) theory with λ = (8 , , , , , , , h ( i, j ) = λ i − j + ˇ λ j − i + 1. Shaded boxes are invariant underthe action of Γ = Z .space is a lens space S / Γ, and thus we can assign non-trivial flat connection at infinity inthis case. Therefore the instanton solution is classified by both the 1 st and 2 nd Chern classes.We remark the instanton number defined as k/ | Γ | coincides with the 2 nd Chern class only ifthe 1 st Chern class is vanishing. This constraint is quite non-trivial, but it does not concernour derivation of the matrix model discussed later.It is easy to see this action in terms of a Young diagram. Since the five dimensionalpartition function is written in the following form Z ∼ n Y l,m ∞ Y i,j − Q lm q λ ( l ) i − j t ˇ λ ( m ) j − i +1 , (3.2)where ˇ λ j stands for the transposed partition, we can see the Γ-invariant sector satisfies λ ( l ) i + ˇ λ ( m ) j − i − j + 1 + p lm ≡ k ) . (3.3)For U(1) case the left hand side coincides with the hook length of the box, h ( i, j ) = λ i − j +ˇ λ j − i + 1. Thus the corresponding four dimensional function can be given by taking intoaccount this condition, Z ALEvec ∼ n Y l,m Y Γ-inv . a lm + ǫ ( i − ˇ λ ( m ) j −
1) + ǫ ( λ ( l ) i − j ) . (3.4)The product in this expression is explicitly taken over the Γ-invariant sector (3.3). Fig. 1shows Γ-invariant sector for U(1) theory with Γ = Z .Although this definition seems natural for the orbifold version of the partition function,it is not useful anyway because we have to extract the Γ-invariant sector by hands. Thisprocedure is called the orbifold projection in [39]. On the other hand, when we consider8nother deformation of the partition function, which is given by replacing the parameterswith (3.1) as Z ∼ n Y l,m ∞ Y i,j − ω λ ( l ) i +ˇ λ ( m ) j − i − j +1+ p lm Q lm q λ ( l ) i − j t ˇ λ ( m ) j − i +1 , (3.5)and take the limit of q →
1, namely the root of unity limit q → exp(2 πi/k ) of the original q -partition function, we can see only the Γ-invariant sector contributes to the partitionfunction and the others are decoupled in this limit. It is because, unless the power of ω becomes λ ( l ) i + ˇ λ ( m ) j − i − j + 1 + p lm ≡ k ) in (3.5), they just give a factor independentof the shape of the Young diagram, or the partition. Thus, if we take into account theadjoint matter contribution to regularize the singular behavior at q →
1, the weight functionbehaves as(1 − ω λ ( l ) i +ˇ λ ( m ) j − i − j +1+ p lm Q lm Q m q λ ( l ) i − j t ˇ λ ( m ) j − i +1 )(1 − ω λ ( l ) i +ˇ λ ( m ) j − i − j +1+ p lm Q lm q λ ( l ) i − j t ˇ λ ( m ) j − i +1 ) −→ a lm + ǫ ( i − ˇ λ ( m ) j − ǫ ( λ ( l ) i − j )+ m a lm + ǫ ( i − ˇ λ ( m ) j − ǫ ( λ ( l ) i − j ) if λ ( l ) i + ˇ λ ( m ) j − i − j + 1 + p lm ≡ k )1 if λ ( l ) i + ˇ λ ( m ) j − i − j + 1 + p lm k ) . (3.6)This means the orbifold projection is automatically assigned by this parametrization. There-fore let us define the partition function modified with (3.1) as the ( q -deformed) orbifold par-tition function from now. We note that the pure Yang-Mills contribution can be extractedby taking the decoupling limit m → ∞ .We now check this reduction with a simple example, SU(2) gauge theory on C / Z withthe adjoint matter. If we set ǫ = − ǫ = ~ , a = − a = a and p = p , lower degreeparts of the instanton partition function are obtained by replacing q -parameters as q → − q , Q → Q [39], Z = 2 (1 + Q m q )(1 + Q m q − )(1 + q )(1 + q − ) (1 − Q m Q )(1 − Q m Q )(1 − Q )(1 − Q ) , (3.7) Z , = (1 + Q m Q q )(1 + Q m Q q − )(1 + Q q )(1 + Q q − ) (1 + Q m Q q − )(1 + Q m Q q )(1 + Q q − )(1 + Q q ) (cid:18) (1 + Q m q )(1 + Q m q − )(1 + q )(1 + q − ) (cid:19) , (3.8) Z = 2 (1 + Q m q )(1 + Q m q − )(1 + q )(1 + q − ) (1 − Q m q )(1 − Q m q − )(1 − q )(1 − q − ) × (1 − Q m Q )(1 − Q m Q )(1 − Q )(1 − Q ) (1 + Q m Q q )(1 + Q m Q q − )(1 + Q q )(1 + Q q − ) . (3.9)Thus, taking the limit q →
1, we have Z → (cid:18) − m a (cid:19) , (3.10) Z , → , (3.11) Z → (cid:18) − m ~ (cid:19) (cid:18) − m a (cid:19) . (3.12)9igure 2: Parametrization of q corresponding to N = 2 theories on C and C / Z with thelimit of | q | → | q | is related to the radius of the compactified dimension S .These correctly reproduce the result derived in [39].Let us consider the meaning of this parametrization without taking the four dimensionallimit | q | →
1. A candidate of the corresponding manifold is the Taub-NUT manifold becauseit can be obtained by compactifying the singularity on S , and reproduces the ALE space bytaking a certain limit. It is just a speculation, thus we have to come back to this identificationproblem in a future work.We then comment on ambiguity of this parametrization. In this paper we apply theprimitive root of unity to define the partition function as (3.5), but other roots of unity,represented as ω r = exp(2 rπi/k ), are also valid for the orbifold projection while k and r areco-prime. This arbitrariness reflects Z k symmetry of the system.Fig. 2 shows how to parametrize q to obtain the partition functions for the ALE spaces. | q | corresponds to the radius of the compactified dimension S . When we consider the fourdimensional limit, we have to approach | q | → q . In [39] a similar parametrizationis actually proposed, which is interpreted as an analytic continuated one, but they do nottake into account regularizing the infinite product appearing in the partition function. Thusthe parametrization used in this paper should be more suitable. As discussed before the partition function for SU( n ) theory on R is represented with n -tuple partition. For this case, we can reproduce the one-matrix model by blending n -tuplepartition to a single one [26, 27, 28]. Next we consider its natural generalization to theorbifold theory. In this case, on the other hand, it is convenient for the discussion below todivide a n -tuple partition into a kn -tuple one [40], n k ( λ ( l,r ) i − i + N ( l,r ) + p ( l,r ) ) + r (cid:12)(cid:12)(cid:12) i = 1 , · · · , N ( l,r ) o = n λ ( l ) i − i + N ( l ) + p l (cid:12)(cid:12)(cid:12) i = 1 , · · · , N ( l ) o k = 3. (3.13)where k − X r =0 p ( l,r ) = p l , k − X r =0 N ( l,r ) = N ( l ) . (3.14)This corresponds to the decomposition into irreducible representations of Γ.Let us practice with an example for U(1) theory as shown in Fig. 3. Starting with apartition λ = (8 , , , , , , , , , , , , , , ,
1) via themapping λ i → λ i + N − i + p with N = 8 and p = 0. This is interpreted as the mapping of aYoung diagram to a Maya diagram, or equivalently bosonic to fermionic variables. Classifyingentries by modulo 3, we have three configurations (15 , → (5 , , , → (3 , ,
0) and(11 , , → (3 , , N ( r ) and p ( r ) .We define the explicit relation between elements from both sides of (3.13) as k ( λ ( l,r ) i − i + N ( l,r ) + p ( l,r ) ) + r ≡ λ ( l ) j − j + N ( l ) + p l with j = c ( l,r ) i , (3.15)Here c ( l,r ) i means the mapping from the index of the divided kn -partition to that of theoriginal n -partition. Introducing another set of variables h ( l,r ) i ≡ k ( λ ( l,r ) i − i + N ( l,r ) + p ( l,r ) ) , h ( l ) i ≡ λ ( l ) i − i + N ( l ) + p l , (3.16) ℓ ( l,r ) i ≡ h ( l,r ) i + b l − p l + r, (3.17)and setting N ( l ) = N and N ( l,r ) = N for all l = 1 , · · · , n and ( l, r ) = (1 , , · · · , ( n, k −
1) forsimplicity, the partition function (2.17) is, up to constants, rewritten as, Z vec = Y ( l,i ) =( m,j ) ( ω h ( l ) i − h ( j ) j q h ( l ) i − h ( j ) j +( β − j − i )+ b lm − p lm ; ˜ q ) ∞ ( ω h ( l ) i − h ( j ) j +1 q h ( l ) i − h ( j ) j +( β − j − i )+ b lm − p lm + β ; ˜ q ) ∞ n Y l,m N Y i =1 ( ω h ( l ) i − p l +1 q h ( l ) i +( β − N − i )+ b lm − p l + β ; ˜ q ) ∞ ( ω − h ( l ) i + p l q − h ( l ) i − ( β − N − i ) − b lm + p l ; ˜ q ) ∞ = Y ( l,r,i ) =( m,s,j ) ( ω r − s q ℓ ( l,r ) i − ℓ ( m,s ) j +( β − c ( m,s ) j − c ( l,r ) i ) ; ˜ q ) ∞ ( ω r − s +1 q ℓ ( l,r ) i − ℓ ( m,s ) j +( β − c ( m,s ) j − c ( l,r ) i )+ β ; ˜ q ) ∞ × n Y l,m k − Y r =0 N Y i =1 ( ω r − p l +1 q ℓ ( l,r ) i − b m +( β − N − c ( l,r ) i )+ β ; ˜ q ) ∞ ( ω − r + p l q − ( ℓ ( l,r ) i − b m +( β − N − c ( l,r ) i )) ; ˜ q ) ∞ . (3.18)The former expression is in terms of the original n -tuple representation, and the latter iswritten with a kn -tuple partition. Blending the kn -tuple to k -tuple partitions as ℓ ( r ) i =1 , ··· , P nl =1 N ( l,r ) = (cid:16) ℓ ( n,r )1 , · · · , ℓ ( n,r ) N ( n,r ) , · · · , ℓ (1 ,r )1 , · · · , ℓ (1 ,r ) N (1 ,r ) (cid:17) , (3.19)we finally obtain an expression in terms of k -tuple partition, Z vec = Y ( r,i ) =( s,j ) ( ω r − s q ℓ ( r ) i − ℓ ( s ) j +( β − c ( s ) j − c ( r ) i ) ; ˜ q ) ∞ ( ω r − s +1 q ℓ ( r ) i − ℓ ( s ) j +( β − c ( s ) j − c ( r ) i )+ β ; ˜ q ) ∞ × n Y l =1 k − Y r =0 nN Y i =1 ( ω r − p l +1 q ℓ ( r ) i − b l +( β − N − c ( r ) i )+ β ; ˜ q ) ∞ ( ω − r + p l q − ( ℓ ( r ) i − b l +( β − N − c ( r ) i )) ; ˜ q ) ∞ . (3.20)Again c ( r ) i stands for the mapping from the index of the k -tuple partition to that of n -tupleone. We will discuss its matrix model description from the expression of (3.20) in section 4.We can also get explicit representations for matter parts in a similar way, Z adj = Y ( r,i ) =( s,j ) ( ω r − s +1 q ℓ ( r ) i − ℓ ( s ) j + M +( β − c ( s ) j − c ( r ) i )+ β ; ˜ q ) ∞ ( ω r − s q ℓ ( r ) i − ℓ ( s ) j + M +( β − c ( s ) j − c ( r ) i ) ; ˜ q ) ∞ × n Y l =1 k − Y r =0 nN Y i =1 ( ω − r + p l q − ( ℓ ( r ) i − b l + M +( β − N − c ( r ) i )) ; ˜ q ) ∞ ( ω r − p l +1 q ℓ ( r ) i − b l + M +( β − N − c ( r ) i )+ β ; ˜ q ) ∞ , (3.21) Z fund = n Y l =1 N f Y f =1 k − Y r =0 nN Y i =1 ( ω − i + p l +1 q b l + M f − βc ( r ) i +1 ; ˜ q ) ∞ ( ω r − N +1 q ℓ ( r ) i + M f − ( β − c ( r ) i − N +1 ; ˜ q ) ∞ , (3.22) Z antifund = n Y l =1 N f Y f =1 k − Y r =0 nN Y i =1 ( ω − i + p l +1 q b l − M f − βc ( r ) i + β ; ˜ q ) ∞ ( ω r − N +1 q ℓ ( r ) i − M f − ( β − c ( r ) i − N + β ; ˜ q ) ∞ . (3.23) We then derive matrix models from the orbifold partition functions. According to the resultthat the orbifold partition function can be written with a k -tuple partition, one can see the k -matrix model is naturally arising from the combinatorial expression. The multi-matrixmodel, derived in this section, is Z = Z D ~Xe − ǫ P k − r =0 P Ni =1 V ( x ( r ) i ) (4.1)12 ~X = k − Y r =0 N Y i =1 dx ( r ) i π ∆ ( x ) , (4.2) V ( x ) = V vec ( x ) + V (anti)fund ( x ) . (4.3)We remark this matrix model corresponds to only the contribution from the instanton part,thus we have to introduce the perturbative piece when we consider the whole prepotentialof the gauge theory. The deformed version of the Vandermonde determinant appearing inthe matrix measure is given by∆ ( x ) = k − Y r =0 N Y i 0. We study the asymptotics of the orbifold partition functionsin the following. Let us start with the measure part of the multi-matrix model, coming from the combinatorialexpression of (3.20). In this derivation we assume β = kγ + 1 ≡ k ) , (4.7)in order to satisfy the condition ωq β = ( ωq ) β . (4.8)This parametrization is proposed in [41], thus we call it the Uglov condition . One can see itis quite important to obtain the matrix model description from the combinatorial expressionbecause if we denote ˜ q = ωq , this condition implies( q, t ) −→ ( ωq, ωt ) = (˜ q, ˜ q β ) . (4.9) The reason why we assign this condition (4.7) here is just a technical one. We will see, in the forthcomingpaper [63], that this condition is not essential for deriving the matrix model description. Y ( r,i ) =( s,j ) ( ω r − s q ℓ ( r ) i − ℓ ( s ) j +( β − c ( s ) j − c ( r ) i ) ; ˜ q ) ∞ ( ω r − s +1 q ℓ ( r ) i − ℓ ( s ) j +( β − c ( s ) j − c ( r ) i )+ β ; ˜ q ) ∞ = Y ( r,i ) =( s,j ) β − Y p =0 (cid:18) − ω r − s + p q ℓ ( r ) i − ℓ ( s ) j +( β − c ( s ) j − c ( r ) i )+ p (cid:19) . (4.10)Taking the limit ǫ → (cid:18) − ω r − s e x ( r ) i − x ( s ) j (cid:19) kγ Y p =1 (cid:18) − ω r − s + p e x ( r ) i − x ( s ) j (cid:19) = (cid:18) − ω r − s e x ( r ) i − x ( s ) j (cid:19) (cid:18) − e k ( x ( r ) i − x ( s ) j ) (cid:19) γ (4.11)where we use the identity k − Y r =0 (1 − ω r z ) = (1 − z k ) . (4.12)As a result we obtain the deformed Vandermonde determinant ∆ ( x ) presented in (4.4).When we take the four dimensional limit, a factor including d ( r,s ) does not contribute inthe leading order because it is expanded as2 sinh x ( r ) i − x ( s ) j + d ( r,s ) ! = 2 i sin (cid:18) r − sk π (cid:19) " − i x ( r ) i − x ( r ) j ! cot (cid:18) r − sk π (cid:19) + · · · . (4.13)Thus the matrix measure goes to∆ ( x ) −→ k − Y r =0 nN Y i 1, it is convenient to introduce the useful identity, called the inversion formula,Li ( z ) + Li (1 /z ) = − 12 (log z ) + π − iπ log z. (4.18)Thus we can extend its domain to the whole complex plane.Utilizing the results obtained above, we can evaluate the factor contributing to the matrixpotential. The vector multiplet part in (3.20) yields n Y l =1 k − Y r =0 nN Y i =1 ( ω r − p l +1 q ℓ ( r ) i − b l +( β − N − c ( r ) i )+ β ; ˜ q ) ∞ ( ω − r + p l q − ( ℓ ( r ) i − b l +( β − N − c ( r ) i )) ; ˜ q ) ∞ ≡ exp k − X r =0 nN X i =1 − ǫ V ( x ( r ) i ) , (4.19) V ( x ) = − k n X l =1 h Li ( e k ( x − a l ) ) − Li ( e − k ( x − a l ) ) i + O ( ǫ ) ≃ n x + 2 k n X l =1 Li ( e − k ( x − a l ) ) . (4.20)Here we neglect a redundant constant term in this potential, and if x − a l < 0, we have toredefine this by using the identity (4.18). The derivative of this potential is given by V ′ ( x ) = 2 k n X l =1 log (cid:20) (cid:18) k x − a l ) (cid:19)(cid:21) . (4.21)We can also derive the potential for the four dimensional theory. Taking the four dimensionallimit, we have V ( x ) = 2 k n X l =1 [( x − a l ) log( x − a l ) − ( x − a l )] , (4.22)and its derivative V ′ ( x ) = 2 k n X l =1 log( x − a l ) . (4.23)We will show this potential plays an important role in obtaining Seiberg-Witten curve as thespectral curve of the matrix model. 15e then investigate the matrix model potentials for the fundamental matter from theexpression (3.22) in a similar way, Z fund = k − Y r =0 N f Y f =1 nN Y i =1 g ( ω − r q − ( M f +1 − βi ) ; ˜ q ) g ( ω − r q − ( ℓ ( r ) i + M f +( β − i +1) ; ˜ q ) ≡ exp k − X r =0 nN X i =1 − ǫ V ( x ( r ) i ) , (4.24) V = 1 k N f X f =1 Li ( e k ( x + m f ) ) , (4.25) V = − k N f X f =1 [( x + m f ) log( x + m f ) − ( x + m f )] . (4.26)We again neglect a finite constant independent of x for them. For k = 1 these potentialfunctions coincide with the usual matrix model potentials for N = 2 theories [26, 28]. Wenow remark this matrix potential is independent of the deformation parameter β , and itsdependence only appears in the matrix measure. N = 2 ∗ theory Let us derive another matrix model by taking into account the contribution from the adjointmatter, which should describe N = 2 ∗ theory [27]. After almost the same procedure discussedabove, we now obtain the following matrix model: Z = Z D ~Xe − ǫ P k − r =0 P Ni =1 V ( x ( r ) i ) , (4.27) D ~X = k − Y r =0 N Y i =1 dx ( r ) i π ∆ ( x )∆ m ( x ) , (4.28) V ( x ) = V vec ( x ) + V adj ( x ) . (4.29)Again N stands for the matrix size here. In this case the matrix measure requires anothercontribution,∆ m ( x ) = k − Y r =0 N Y i = j x ( r ) i − x ( r ) j + m ! k − Y r = s N Y i,j x ( r ) i − x ( s ) j + d ( r,s ) + m ! × k − Y r =0 N Y i = j k ( x ( r ) i − x ( r ) j + m )2 ! γ k − Y r = s N Y i,j k ( x ( r ) i − x ( s ) j + m )2 ! γ . (4.30)The matrix potential for the adjoint matter is V adj ( x ) = − n x + m ) − k n X l =1 Li (cid:16) e − k ( x − a l + m ) (cid:17) . (4.31)16s a result, the total potential function becomes V vec ( x ) + V adj ( x ) = − n m x + 2 k n X l =1 h Li (cid:16) e − k ( x − a l ) (cid:17) − Li (cid:16) e − k ( x − a l + m ) (cid:17)i . (4.32)This result is consistent with [27] for k = 1.We now comment on a relation to N = 4 theory on the ALE spaces [40]. Since N = 4theory is obtained from N = 2 ∗ theory by taking the massless limit of the adjoint matter m → 0, the result of [40] should be related to the result discussed above. In the masslesslimit the matrix measure and the potential become trivial∆ ( x )∆ m ( x ) −→ , V ( x ) −→ . (4.33)This means, when we go back to the combinatorial representation, the combinatorial weight,represented in terms of the hook length, becomes trivial in the partition function. As a resultit remains only the counting parameter, which is related to the dynamical scale.For U(1) theory, therefore it simply corresponds to the following partition function onlywith implementing the orbifold projection, Z ∼ X λ Λ { Γ-invariant boxes } . (4.34)This is similar to the partition function discussed in [40], but not identical. We shouldinvestigate this type of partition function and clarify the relation to N = 4 theory in afuture work. To study a connection between the gauge theory and the matrix model, in this section weconsider the multi-matrix model in detail, which is defined as Z = Z k − Y r =0 N Y i =1 dx ( r ) i π ∆ ( x ) e − gs P k − r =0 P Ni =1 V ( x ( r ) i ) (5.1)where ∆ ( x ) is defined in (4.4). In this section ǫ is replaced with g s , and we focus on thecase of β = 1. For a while we consider a generic potential V ( x ). The method used in thissection is partially based on that developed for the lens space Chern-Simons matrix model[45, 47]. N limit and saddle point equation We are interested in the ’t Hooft limit of this matrix model, in which g s −→ , N −→ ∞ , (5.2)17ith fixing the ’t Hooft coupling T = g s N. (5.3)Actually, in this large N limit, the evaluation of the matrix integral reduces to the calculationof the critical points.If we define the prepotential for the matrix model as − g s F = − g s k − X r =0 N X i =1 V ( x ( r ) i )+2 k − X r =0 N X i 2Λ sinh k z − a l ) (cid:19) X →∞ −→ X kn/ . (5.54)By setting the boundary condition w = e − y/ X →∞ −→ Λ n X kn/ , (5.55)Seiberg-Witten differential for this theory defined on the curve (5.53) reads dS = 14 πi y ( z ) dz = 12 πi log( X ) dww . (5.56)We can see the result of [66] is obtained by setting k = 1. In this case although the curve ismodified as (5.53), the form of the differential itself is not changed. In the four dimensionallimit R → k sets of n cuts are decoupled because distance between them is 2 πi/kR asshown in Fig. 4. A generalization to the theory with matter fields is straightforward. The recent remarkable progress on the four dimensional N = 2 theory gives the interestingrelation to the two dimensional conformal field theory. Thus, according to the results dis-cussed above, it is natural to search a two dimensional theoretical counterpart of the orbifoldtheory. In this section we discuss some proposals for such an interesting 2d/4d connection. We now discuss a string theory realization of the orbifold theory, and try to find its twodimensional description. It is well known that string theory gives us a lot of interesting fieldtheoretical consequences, e.g. the holographic approach to a strongly correlated system, themethod to construct instantons and monopoles and so on. Actually the 2d/4d relation [1]can be naturally understood in terms of M-theory.Let us consider the string theory realization of N = 4 SYM theory on the Taub-NUTspace [37], which is expected to be related to our case. The Taub-NUT manifold T N k isgiven by a S compactification of the A k − singularity. It approaches to the cylinder R × S 25t infinity, and the ALE space is obtained by taking the limit of R → ∞ where R standsfor the radius of S . We remark although the partition function of the gauge theory on theTaub-NUT manifold is not identical to that of the ALE space, we expect this difference doesnot seriously affect the corresponding two dimensional theory because the SU( n ) theory onboth of these manifolds is specified by a flat connection on the boundary S / Z k as discussedlater.We start with type IIA string theory on T N k × S × R with wrapping n D4-branes on T N k × S . Lifting this to M-theory, we then obtain the compactification of T N k × T × R andwrapping n M5-branes on T N k × T . By replacing the two-manifold T with various Riemannsurfaces Σ, we obtain the corresponding N = 2 theories [5]. To study this configuration wego back to type IIA theory with another compactification R × T × R . In this case thereare n D4-branes and k D6-branes wrapping R × T and T × R respectively because thecircle fibration of the Taub-NUT space has singular points. One can see these D4- and D6-branes are intersecting on T , and thus the chiral fermion is arising from these intersectingconfiguration. This chiral fermion plays an important role in considering the level-rankduality of the system.To discuss the two dimensional theory on T we then deal with the boundary of thefour dimensional manifold. By considering the radial quantization near the boundary and awavefunction for the time evolution along S / Z k × R , we have a Hilbert space with a state | ρ i for each n -dimensional representation ρ : Z k ⊂ SU(2) −→ SU( n ) . (6.1)Integrating out the flux coming from k D6-branes, we obtain the Chern-Simons term for I CS = 2 πk Z T × R CS ( A ) . (6.2)This means the boundary condition for the D4-brane requires specifying a state of the SU( n )Chern-Simons theory at level k living on T , and thus we have a state for each integrablerepresentation of the c SU( n ) k WZW model. Note that the same procedure can be performedfor the SU( k ) theory on k D6-branes, and the diagonal U (1) part is decoupled. Thereforewe obtain the embedding b U(1) nk × c SU( n ) k × c SU( k ) n ⊂ b U( nk ) , thus it is easy to see thelevel-rank duality.We also remark it is conjectured in [2] that the gauge theory partition function is relatedto the τ -function by utilizing the chiral fermion representation. Consequently the generatingfunction for instantons on the ALE space of the ADE type would be related to the ADEWZW theories on the Seiberg-Witten curve. It is consistent with the intersecting braneconfiguration because we obtain the N = 2 theories when we consider the intersectingbranes on general curves Σ, which turn out to be Seiberg-Witten/Gaiotto curves [5].We then discuss the connection between the two and four dimensional theory more con-cretely. The original proposals of [1, 8] elucidate the explicit relation between the SU( n )26artition function and the two dimensional conformal field theory described by W n alge-bra. This relation is extended to the theory with the generalized W algebra [52, 53] [54][55], which is obtained from the quantum Drinfeld-Sokolov reduction applied to an affineLie algebra c SU( n ) [56]. It is characterized by a choice of an embedding ρ : SU(2) → SU( n ),and this embedding ρ can be labeled by a partition of n , or equivalently a Young dia-gram Y . For example, it reproduces well-known algebras, W ( c SU( n ) , [1 , · · · , c SU( n ), W ( c SU( n ) , [ N ]) = W n , etc. Such an embedding has been discussed in the orbifold case (6.1).Indeed we consider the embedding of the finite subgroup Z k of SU(2) for the orbifold theory,and the decomposition into the irreducible representations of Z k corresponds to that of thegauge group SU( n ) → SU( n ) × · · · × SU( n k − ) with n + · · · + n k − = n . This showsthat this embedding is also labeled by a partition of n . Therefore it is expected that thetwo dimensional conformal field theory, corresponding to the SU( n ) gauge theory on theALE space C / Z k , is related to the generalized W algebra W ( c SU( n ) , [ n , · · · , n k − ]). Thisconnection is still speculation, and should be investigated in detail. q -deformed CFT It is shown in [69] that q -deformed CFT, which is described by q -Virasoro [70] and q - W algebra [71], corresponds to the q -deformed five dimensional partition function of the gaugetheory, and its matrix model description is also proposed in [72] by clarifying a connectionwith q -Virasoro algebra. This matrix model, which we now call q -Virasoro matrix model, isapparently different from the trigonometric one [26, 27, 28], but it is worth considering theroot of unity limit of such a q -Virasoro related models. The root of unity limit of q -Virasoroalgebra is investigated in [73] while that of q - W algebra is not yet well known.In this subsection let us focus on the matrix measure part of the q -Virasoro matrix model[72]. It is just given by the two-parameter deformed Vandermonde determinant, which isclosely related to the Macdonald polynomial [74],∆ q,t ( x ) = Y i = j ( x i /x j ; q ) ∞ ( tx i /x j ; q ) ∞ . (6.3)We can obtain the usual Vandermonde determinant by taking the limit q → t = q β ,∆ q,t ( x ) −→ Y i = j (cid:18) − x i x j (cid:19) β ≃ Y i 1, it becomes∆ q,t ( x ) −→ Y i = j (cid:18) − x i x j (cid:19) − x ki x kj ! γ ≃ Y i In this paper we have performed some extensions of the partition function and its matrixmodel description for the four dimensional N = 2 gauge theory to the orbifold theory.We have shown that the orbifold projection, extracting the Γ-invariant sector of the Youngdiagram, is automatically performed by taking the root of unity limit q → exp (2 πi/k ) ofthe q -deformed partition function while we have to take q → R . For such an orbifold partition function of SU( n ) theory with Γ = Z k , it isconvenient to divide n -tuple to kn -tuple partitions. As a result, we have obtained the multi-matrix model by considering the asymptotic behavior of the combinatorial representation.This matrix model at the large N limit has been analysed in detail, which is equivalentto studying the limit shape of the Young diagram, and then we have seen Seiberg-Wittencurve is obtained as the spectral curve of the matrix model. We have also discussed thecorresponding two dimensional theory of the four dimensional orbifold theory. Focusing onthe embedding ρ : Z k ⊂ SU(2) → SU( n ), which characterizes the decomposition of SU( n )gauge group to the irreducible representations of Z k , we have suggested the generalized W algebra appears in the two dimensional theory.We now comment on some possibilities of extension beyond this study. We hope ourstudy becomes a step for understanding of M-theory itself. Actually the emergence of c SU( n ) k C / Z k isanalogous to the ABJM theory because the level of Chern-Simons theory in the ABJM the-ory is directly related to the background manifold C / Z k on which M2-branes are located.Indeed, in both cases, the level k is interpreted as the degree of singularity of the complemen-tary manifold of the corresponding world-volume theory. It is interesting to study a relationbetween M2- and M5-branes from this point of view. We are also interested in the level-rankduality of our model. Such a duality can be found in the two dimensional c SU( n ) k , or c SU( k ) n WZW theory, which is closely related to the four dimensional orbifold theory. It is expectedthat this duality plays an important role in understanding some aspects of M-theory.It is natural to consider some applications to related topics. One of them is the threedimensional duality, which is a recent hot topic on this subject [75, 76, 77, 78, 79, 80, 81].The q -parameter plays a similar role in such a theory, so that it is interesting to study thesingular limit of the q -parameter, i.e. the root of unity limit. Second is a relation to thequantized integrable models. It is shown in [82] that when we consider generic Ω-parameters, ǫ + ǫ = 0, correction to the prepotential can be interpreted as a quantization effect of thecorresponding integrable model. Searching an integrable model, which corresponds to theorbifold theory, would be also interesting. Acknowledgments The author would like to thank S. Hikami for reading the manuscript and useful comments.The author also would like to thank T. Azeyanagi, K. Hashimoto, Y. Kato, K. Maruyoshi,H. Shimada, T. Tai and M. Taki for valuable discussions and comments. This work issupported by Grant-in-Aid for JSPS Fellows. A Four dimensional matrix model In this appendix we summarize the results of the four dimensional limit of the matrix modelin order to fix our notations. Let us mainly consider the k = 1 theory, and shortly commenton simple generalizations to the cases of k > Z = Z N Y i =1 dx i π N Y i Liouville Correlation Functions fromFour-dimensional Gauge Theories , Lett. Math. Phys. (2010) 167–197,[ arXiv:0906.3219 ].[2] N. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting , Adv. Theor.Math. Phys. (2004) 831–864, [ hep-th/0206161 ].[3] N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions , hep-th/0306238 .[4] E. Witten, Solutions of four-dimensional field theories via M-theory , Nucl. Phys. B500 (1997) 3–42, [ hep-th/9703166 ].[5] D. Gaiotto, N = 2 dualities , arXiv:0904.2715 .[6] N. Seiberg and E. Witten, Monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory , Nucl. Phys. B426 (1994) 19–52, [ hep-th/9407087 ].[7] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD , Nucl. Phys. B431 (1994) 484–550, [ hep-th/9408099 ].[8] N. Wyllard, A N − conformal Toda field theory correlation functions from conformal N = 2 SU ( N ) quiver gauge theories , JHEP (2009) 002, [ arXiv:0907.2189 ].[9] A. Mironov and A. Morozov, On AGT relation in the case of U(3) , Nucl. Phys. B825 (2010) 1–37, [ arXiv:0908.2569 ]. 3110] D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks , arXiv:0908.0307 .[11] A. Marshakov, A. Mironov, and A. Morozov, On non-conformal limit of the AGTrelations , Phys. Lett. B682 (2009) 125–129, [ arXiv:0909.2052 ].[12] M. Taki, On AGT Conjecture for Pure Super Yang-Mills and W-algebra , JHEP (2011) 038, [ arXiv:0912.4789 ].[13] L. F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa, and H. Verlinde, Loop and surfaceoperators in N = 2 gauge theory and Liouville modular geometry , JHEP (2010)113, [ arXiv:0909.0945 ].[14] R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings, andN=2 Gauge Systems , arXiv:0909.2453 .[15] H. Itoyama, K. Maruyoshi, and T. Oota, The Quiver Matrix Model and 2d-4dConformal Connection , Prog. Theor. Phys. (2010) 957–987, [ arXiv:0911.4244 ].[16] T. Eguchi and K. Maruyoshi, Penner Type Matrix Model and Seiberg-Witten Theory , JHEP (2010) 022, [ arXiv:0911.4797 ].[17] R. Schiappa and N. Wyllard, An A r threesome: Matrix models, 2d CFTs and 4d N = 2 gauge theories , arXiv:0911.5337 .[18] H. Itoyama and T. Oota, Method of Generating q -Expansion Coefficients forConformal Block and N = 2 Nekrasov Function by β -Deformed Matrix Model , Nucl.Phys. B838 (2010) 298–330, [ arXiv:1003.2929 ].[19] T. Eguchi and K. Maruyoshi, Seiberg-Witten theory, matrix model and AGT relation , JHEP (2010) 081, [ arXiv:1006.0828 ].[20] H. Itoyama, T. Oota, and N. Yonezawa, Massive Scaling Limit of β -Deformed MatrixModel of Selberg Type , Phys. Rev. D82 (2010) 085031, [ arXiv:1008.1861 ].[21] H. Itoyama and N. Yonezawa, ǫ -Corrected Seiberg-Witten Prepotential Obtained FromHalf Genus Expansion in β -Deformed Matrix Model , Int. J. Mod. Phys. A26 (2011)3439–3467, [ arXiv:1104.2738 ].[22] V. S. Dotsenko and V. A. Fateev, Conformal algebra and multipoint correlationfunctions in 2D statistical models , Nucl. Phys. B240 (1984) 312–348.[23] V. S. Dotsenko and V. A. Fateev, Four-point correlation functions and the operatoralgebra in 2D conformal invariant theories with central charge c ≤ Nucl. Phys. B251 (1985) 691–734. 3224] K. Maruyoshi and F. Yagi, Seiberg-Witten curve via generalized matrix model , JHEP (2011) 042, [ arXiv:1009.5553 ].[25] G. Bonelli, K. Maruyoshi, A. Tanzini, and F. Yagi, Generalized matrix models andAGT correspondence at all genera , JHEP (2011) 055, [ arXiv:1011.5417 ].[26] A. Klemm and P. Su lkowski, Seiberg-Witten theory and matrix models , Nucl. Phys. B819 (2009) 400–430, [ arXiv:0810.4944 ].[27] P. Su lkowski, Matrix models for ∗ theories , Phys. Rev. D80 (2009) 086006,[ arXiv:0904.3064 ].[28] P. Su lkowski, Matrix models for β -ensembles from Nekrasov partition functions , JHEP (2010) 063, [ arXiv:0912.5476 ].[29] T.-S. Tai, Instanton Counting and Matrix Model , Prog. Theor. Phys. (2008)165–177, [ arXiv:0709.0432 ].[30] K. Johansson, Shape Fluctuations and Random Matrices , Commun. Math. Phys. (2000) 437–476, [ math/9903134 ].[31] T. Sasamoto, Fluctuations of the one-dimensional asymmetric exclusion process usingrandom matrix techniques , J. Stat. Mech. (2007) P07007, [ arXiv:0705.2942 ].[32] C. Tracy and H. Widom, Level-spacing distributions and the Airy kernel , Commun.Math. Phys. (1994) 151–174, [ hep-th/9211141 ].[33] R. P. Stanley, Enumerative Combinatorics: Volume 2 . Cambridge University Press,2001.[34] P. B. Kronheimer, The Construction of ALE Spaces as hyper-K¨ahler Quotients , J.Diff. Geom. (1989) 665–683.[35] H. Nakajima, Moduli spaces of anti-self-dual connections on ALE gravitationalinstantons , Invent. Math. (1990) 267–303.[36] P. B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitationalinstantons , Math. Ann. (1990) 263–307.[37] R. Dijkgraaf, L. Hollands, P. Sulkowski, and C. Vafa, Supersymmetric Gauge Theories,Intersecting Branes and Free Fermions , JHEP (2008) 106, [ arXiv:0709.4446 ].[38] E. Witten, Branes, Instantons, And Taub-NUT Spaces , JHEP (2009) 067,[ arXiv:0902.0948 ].[39] F. Fucito, J. F. Morales, and R. Poghossian, Multi instanton calculus on ALE spaces , Nucl. Phys. B703 (2004) 518–536, [ hep-th/0406243 ].3340] R. Dijkgraaf and P. Su lkowski, Instantons on ALE spaces and orbifold partitions , JHEP (2008) 013, [ arXiv:0712.1427 ].[41] D. Uglov, Yangian Gelfand-Zetlin bases, gl N -Jack polynomials and computation ofdynamical correlation functions in the spin Calogero-Sutherland model , Commun.Math. Phys. (1998) 663–696, [ hep-th/9702020 ].[42] Y. Kuramoto and Y. Kato, Dynamics of One-Dimensional Quantum Systems:Inverse-Square Interaction Models . Cambridge University Press, 2009.[43] M. Mari˜no, Chern-Simons theory, matrix integrals, and perturbative three-manifoldinvariants , Commun. Math. Phys. (2004) 25–49, [ hep-th/0207096 ].[44] M. Aganagic, A. Klemm, M. Mari˜no, and C. Vafa, Matrix model as a mirror ofChern-Simons theory , JHEP (2004) 010, [ hep-th/0211098 ].[45] N. Halmagyi and V. Yasnov, The spectral curve of the lens space matrix model , JHEP (2009) 104, [ hep-th/0311117 ].[46] N. Halmagyi, T. Okuda, and V. Yasnov, Large N duality, lens spaces and theChern-Simons matrix model , JHEP (2004) 014, [ hep-th/0312145 ].[47] M. Mari˜no, Lectures on localization and matrix models in supersymmetricChern-Simons-matter theories , arXiv:1104.0783 .[48] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, N = 6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals , JHEP (2008)091, [ arXiv:0806.1218 ].[49] O. Aharony, O. Bergman, and D. L. Jafferis, Fractional M2-branes , JHEP (2008)043, [ arXiv:0807.4924 ].[50] A. Kapustin, B. Willett, and I. Yaakov, Nonperturbative Tests of Three-DimensionalDualities , JHEP (2010) 013, [ arXiv:1003.5694 ].[51] N. Drukker, M. Mari˜no, and P. Putrov, From weak to strong coupling in ABJM theory , Commun. Math. Phys. (2011) 511–563, [ arXiv:1007.3837 ].[52] L. F. Alday and Y. Tachikawa, Affine SL (2) conformal blocks from 4d gauge theories , Lett. Math. Phys. (2010) 87–114, [ arXiv:1005.4469 ].[53] C. Kozcaz, S. Pasquetti, F. Passerini, and N. Wyllard, Affine sl( N ) conformal blocksfrom N = 2 SU( N ) gauge theories , JHEP (2011) 045, [ arXiv:1008.1412 ].[54] Y. Tachikawa, On W-algebras and the symmetries of defects of 6d N = (2 , theory , JHEP (2011) 043, [ arXiv:1102.0076 ].3455] H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and thechain-saw quiver , JHEP (2011) 119, [ arXiv:1105.0357 ].[56] J. de Boer and T. Tjin, The Relation between quantum W algebras and Lie algebras , Commun. Math. Phys. (1994) 317–332, [ hep-th/9302006 ].[57] V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quivergauge theories , JHEP (2011) 079, [ arXiv:1105.5800 ].[58] T. Nishioka and Y. Tachikawa, Para-Liouville/Toda central charges from M5-branes , Phy. Rev. D84 (2011) 046009, [ arXiv:1106.1172 ].[59] G. Bonelli, K. Maruyoshi, and A. Tanzini, Instantons on ALE spaces and SuperLiouville Conformal Field Theories , JHEP (2011) 056, [ arXiv:1106.2505 ].[60] A. Belavin, V. Belavin, and M. Bershtein, Instantons and 2d Superconformal fieldtheory , arXiv:1106.4001 .[61] G. Bonelli, K. Maruyoshi, and A. Tanzini, Gauge Theories on ALE Space and SuperLiouville Correlation Functions , arXiv:1107.4609 .[62] M. Bianchi, F. Fucito, G. Rossi, and M. Martellini, Explicit Construction of Yang-MillsInstantons on ALE Spaces , Nucl. Phys. B473 (1996) 367–404, [ hep-th/9601162 ].[63] T. Kimura, β -ensembles for toric orbifold partition function , arXiv:1109.0004 .[64] B. Eynard, All orders asymptotic expansion of large partitions , J. Stat. Mech. (2008) P07023, [ arXiv:0804.0381 ].[65] R. Dijkgraaf and C. Vafa, Matrix models, topological strings, and supersymmetricgauge theories , Nucl. Phys. B644 (2002) 3–20, [ hep-th/0206255 ].[66] N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems , Nucl.Phys. B531 (1998) 323–344, [ hep-th/9609219 ].[67] A. Marshakov, On Gauge Theories as Matrix Models , arXiv:1101.0676 .[68] A. Marshakov and N. Nekrasov, Extended Seiberg-Witten theory and integrablehierarchy , JHEP (2007) 104, [ hep-th/0612019 ].[69] H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the DeformedVirasoro Algebra , JHEP (2010) 125, [ arXiv:0910.4431 ].[70] J. Shiraishi, H. Kubo, H. Awata, and S. Odake, A Quantum deformation of theVirasoro algebra and the Macdonald symmetric functions , Lett. Math. Phys. (1996)33–51, [ q-alg/9507034 ]. 3571] H. Awata, H. Kubo, S. Odake, and J. Shiraishi, Quantum W(N) algebras andMacdonald polynomials , Commun. Math. Phys. (1996) 401–416, [ q-alg/9508011 ].[72] H. Awata and Y. Yamada, Five-dimensional AGT Relation and the Deformed β -ensemble , Prog. Theor. Phys. (2010) 227–262, [ arXiv:1004.5122 ].[73] P. Bouwknegt and K. Pilch, The Deformed Virasoro Algebra at Roots of Unity , Commun. Math. Phys. (1998) 249–288, [ q-alg/9710026 ].[74] I. G. Macdonald, Symmetric Functions and Hall Polynomials . Oxford UniversityPress, 2nd ed., 1997.[75] Y. Terashima and M. Yamazaki, SL (2 , R ) Chern-Simons, Liouville, and Gauge Theoryon Duality Walls , JHEP (2011) 135, [ arXiv:1103.5748 ].[76] F. A. H. Dolan, V. P. Spiridonov, and G. S. Vartanov, From 4d superconformal indicesto 3d partition functions , arXiv:1104.1787 .[77] D. Galakhov, A. Mironov, A. Morozov, and A. Smirnov, On 3d extensions of AGTrelation , arXiv:1104.2589 .[78] A. Gadde and W. Yan, Reducing the 4d Index to the S Partition Function , arXiv:1104.2592 .[79] A. Gadde, L. Rastelli, S. S. Razamat, and W. Yan, The 4d Superconformal Index fromq-deformed 2d Yang- Mills , Phy. Rev. Lett. (2011) 241602, [ arXiv:1104.3850 ].[80] Y. Imamura, Relation between the 4d superconformal index and the S partitionfunction , arXiv:1104.4482 .[81] T. Nishioka, Y. Tachikawa, and M. Yamazaki, 3d Partition Function as Overlap ofWavefunctions , JHEP (2011) 003, [ arXiv:1105.4390 ].[82] N. A. Nekrasov and S. L. Shatashvili, Quantization of Integrable Systems and FourDimensional Gauge Theories , arXiv:0908.4052arXiv:0908.4052