Givental J-functions, Quantum integrable systems, AGT relation with surface operator
NNIKHEF-2014-028
Givental J -functions, Quantum integrable systems,AGT relation with surface operator Satoshi Nawata
NIKHEF theory group,Science Park 105, 1098 XG Amsterdam, The Netherlands
E-mail: [email protected]
Abstract:
We study 4d N = 2 gauge theories with a co-dimension two full surface oper-ator, which exhibit a fascinating interplay of supersymmetric gauge theories, equivariantGromov-Witten theory and geometric representation theory. For pure Yang-Mills and N = 2 ∗ theory, we describe a full surface operator as the 4d gauge theory coupled to a 2d N = (2 ,
2) gauge theory. By supersymmetric localizations, we present the exact partitionfunctions of both 4d and 2d theories which satisfy integrable equations. In addition, theform of the structure constants with a semi-degenerate field in SL( N, R ) WZNW model ispredicted from one-loop determinants of 4d gauge theories with a full surface operator viathe AGT relation. Keywords:
AGT relation, Surface operator, WZNW model a r X i v : . [ h e p - t h ] A p r ontents J -function of complete flag variety 112.1.3 One-loop determinant 142.2 N = 2 ∗ theory 162.2.1 Instanton partition function 162.2.2 J -function of cotangent bundle of complete flag variety 182.2.3 Twisted chiral ring 212.2.4 One-loop determinant 23 SL( N, R ) WZNW model from gauge theory 244 Discussions 28A Instanton partition function with surface operator 30B One-loop determinants 34C J -function of cotangent bundle of partial flag variety 39 In [1], a large family of 4d N = 2 superconformal field theories (SCFT), known as class S theories, has been constructed by compactifying the 6d N = (2 ,
0) theory on a Riemannsurface. This construction as well as advances in exact results of supersymmetric partitionfunctions has led to the celebrated AGT relation [2, 3], which amounts to the statementthat the partition function of a 4d N = 2 SCFT on S b [4, 5] can be identified with acorrelation function of 2d Toda CFT on the corresponding Riemann surface.The AGT relation becomes particularly enriched when we insert a half-BPS non-localoperator called a surface operator [6] supported on S ⊂ S b . One can characterize a surfaceoperator by specifying the boundary condition of the gauge field on S ⊂ S b which breaksthe gauge group to the Levi subgroup L ⊂ G . In this paper, we consider the SU( N ) gaugegroup so that the Levi subgroup L = S[U( N ) × · · · × U( N M )] is specified by a partition N = N + · · · + N M which we denote [ N , · · · , N M ]. Especially, the surface operator of[1 , N − , · · · , N = (2 ,
2) gaugetheory coupled to the 4d N = 2 gauge theory and identify the corresponding degenerateoperator in Toda CFT labelled by a Young diagram.On the other hand, the intersection of M5-branes spanning S ⊂ S b and wrappinga Riemann surface also gives rise to a surface operator in the 4d gauge theory, which wedenote a co-dimension two surface operator. The effect of wrapping the defect on theRiemann surface results in the change of the symmetry in 2d CFT. For a surface operatorof type (cid:126)N = [ N , · · · , N M ], it was conjectured [13, 14] that the 2d symmetry is the W-algebra W ( (cid:98) sl ( N ) , (cid:126)N ) obtained by the quantum Drinfeld-Sokolov reduction [15–17] for theembedding ρ (cid:126)N : sl (2) → sl ( N ) corresponding to the partition (cid:126)N . For the 4d gauge theoryside, the moduli space of instanton with the boundary condition of the gauge field on thesurface is called affine Laumon space . It was shown in [18, 19] that the affine Laumonspace is equivalent to instanton moduli space on an orbifold C × ( C / Z M ) so that it admitsquiver representations, called chain-saw quivers . Using the quiver representations, onecan demonstrate localization computations of the Nekrasov partition functions [18, 20].It was checked in [14, 20–22] [23, § (cid:126)N is equal to the norm of the Gaiotto-Whittaker state in the Verma module of the W-algebra W ( (cid:98) sl ( N ) , (cid:126)N ). In particular, for afull surface operator [1 , . . . , (cid:98) sl ( N ). In this paper, we shall provide the contour integral expressions of theNekrasov partitions functions for the pure Yang-Mills and the N = 2 ∗ theory with a surfaceoperator by using the supersymmetric non-linear sigma model with the chain-saw quiveras a target.The Nekrasov partition functions in the presence of a surface operator encode both4d and 2d non-perturbative dynamics. Hence, when we turn off the 4d instanton effect,the Nekrasov partition functions reduce to 2d vortex partition functions which containsthe non-perturbative dynamics on the support of the surface operator. In fact, when theinstanton number is zero, the chain-saw quivers demote to hand-saw quivers so that thegenerating function of equivariant cohomology of the hand-saw quivers becomes the vortexpartition function. On the other hand, a surface operator can also be described as a couplingof the 4d gauge theory with a 2d theory on the surface. In particular, the description on asurface operator in the pure Yang-Mills is given by a coupling of the N = (2 ,
2) non-linearsigma model with a flag manifold G/ L . In addition, for the N = 2 ∗ theory, the N = (2 , ∗ non-linear sigma model with the cotangent bundle T ∗ ( G/ L ) of the flag manifold depicts– 2 –he dynamics on the support of the surface operator. Since their ultra-violet descriptionsas N = (2 ,
2) gauged linear sigma models are known, one can also compute the vortexpartition functions by means of Higgs branch localizations [27, 28]. Therefore, we will seethe correspondence of vortex partition functions computed by the two methods.In this paper, we will also demonstrate explicit calculations for one-loop determinantswhen a full surface operator is inserted. The N = 2 partition functions on S b requireboth the Nekrasov partition functions and one-loop determinants over the instanton con-figurations [4, 5]. Since the Nekrasov partition functions in the presence of a full surfaceoperator can be computed by the orbifold method, it is plausible to expect that the one-loop determinants with a full surface operator is equivalent to those on the orbifold space C × ( C / Z N ). In fact, we show that the one-loop determinants calculated by using the indextheory on C × ( C / Z N ) correctly encode both 4d and 2d perturbative contributions.In the AGT relation, the Nekrasov partition functions correspond to the conformalblocks while the one-loop determinants is equivalent to the product of the three-pointfunctions of 2d CFT. When N = 2, the one-loop determinants of 4d gauge theories witha full surface operator computed by the orbifold procedure reproduce the structure con-stant of SL(2 , R ) WZNW model determined in [29–31]. Furthermore, using the one-loopdeterminants of 4d gauge theories with a full surface operator, we predict the form of thetwo-point and three-point functions of SL( N, R ) WZNW model.Let us also mention the algebro-geometric aspect of the AGT relation with a surfaceoperator. The fundamental idea of algebraic topology is to extract algebraic objects whichencode the information of a given space. Homology, cohomology groups and fundamentalgroups can be seen as typical examples for this idea. This idea has resulted in a greatsuccess in mathematics of the 20th century. From the late 80s, inspired by the idea com-ing from quantum field theory and string theory, “quantizations” of these invariants inalgebraic topology have been introduced, which opened up to the dawn of new geometryand quantum topology. In particular, one of significant steps to uncover deeper structuresbehind “quantization” has been made by Givental [32–34]. Since Givental’s theory playsan essential role in this paper, let us briefly review it by using a projective space P N − asan example.It is well-known that the cohomology ring of P N − is isomorphic to H ∗ ( P N − ) ∼ = C [ x ] / ( x N ) . (1.1)The cohomology ring relation x N = 0 can be resolved by using equivariant cohomology.To see that explicitly, let us define the S -equivariant action on P N − by λ [ z : · · · : z N − ] = [ λ r z : · · · : λ r N − z N − ] , (1.2)for λ ∈ S . Then, the S -equivariant cohomology ring of P N − is given by H ∗ S ( P N − ) ∼ = C [ x, (cid:126) ] / ( N − (cid:89) i =0 ( x − r i (cid:126) )) , (1.3)where (cid:126) represents the hyperplane class of the base manifold of the universal S -bundle S ∞ +1 = ES → BS = P ∞ , (1.4)– 3 –o that H ∗ ( BS ) = H ∗ ( P ∞ ) = C [ (cid:126) ]. Here the hyperplane class (cid:126) plays a similar roleto the Planck constant so that it resolves the cohomology ring relation. Moreover, thecohomology ring is quantized based on Gromov-Witten theory. The quantum cohomologyis ordinary cohomology with a quantum product defined by T i ◦ T j = (cid:88) k,(cid:96) C ijk ( t ) η k(cid:96) T (cid:96) , (1.5)for a basis T i of the cohomology group. Here the structure constants C ijk ( t ) := ∂ F ∂T i ∂T j ∂T k is the third derivative of the genus-zero prepotential depending on the complexified K¨ahlerparameter t and η ij := (cid:82) T i ∪ T j is the metric on the cohomology group. In fact, theWDVV equation is equivalent to the associativity of the quantum product, and thereforethe quantum product can be thought of as quantum deformation of the cup product ofcohomology. Writing q = e t , the quantum cohomology ring of P N − is isomorphic to QH ∗ ( P N − ) ∼ = C [ x, q ] / ( x N − q ) . (1.6)One of the most intriguing aspects of quantum cohomology is its relation with differentialequations. Actually, Givental’s profound insight perceived the relation in the equivari-ant Floer homology of the loop space. Roughly speaking, the Floer homology is the ∞ -dimensional homology theory of infinite-dimensional manifolds. In this example, it is suit-able to consider the universal covering (cid:94) L P N − of the loop space L P N − := Map( S , P N − )of the projective space. For this space, one can obtain the explicit expression of the S -equivariant Floer homology HF ∗ S ( (cid:94) L P N − ) = (cid:77) m ∈ Z N − (cid:77) k =0 C [ (cid:126) ] · ( x − m (cid:126) ) k · (cid:89) j 2, we provide a microscopic description of afull surface operator and give explicit formulae of the partition functions of 4d and 2dgauge theory for the pure Yang-Mills and the N = 2 ∗ theory. Most of the results inthis section have already been proven in literature of mathematics [19, 35–43]. Whatis mathematically new is that we conjecture the explicit expression of the J -function ofthe cotangent bundle of the complete flag variety by using the supersymmetric partitionfunction on S . In addition, we show the evidence that the one-loop determinants canbe computed by the orbifold method. In § 3, we predict the form of the two-point andthree-point function of SL( N, R ) WZNW model by using the one-loop determinants of the4d gauge theory. § J -function of the cotangent bundle of a partialflag variety is presented in Appendix C. – 5 – Gauge theory with full surface operator The surface operator was first introduced as a half-BPS non-local operator supported ona surface in the N = 4 SCFT by Gukov and Witten [6]. One way to define a surfaceoperator is to specify a singular behavior of gauge fields on the surface. To describe moreprecisely, let ( z , z ) be complex coordinate and the surface operator is supported on theplane C = { ( z , z ) | z = 0 } . If ( r, θ ) is the polar coordinate of z -plane, the singularbehavior of gauge fields is prescribed as A µ dx µ ∼ diag( α , . . . , α N ) idθ , (2.1)on the place C . Thus, the parameters (cid:126)α = ( α , . . . , α N ) can be considered as the mon-odromies of the abelian gauge fields around the operator. If the singular date has thestructure (cid:126)α = ( α (1) , . . . , α (1) (cid:124) (cid:123)(cid:122) (cid:125) N times , α (2) , . . . , α (2) (cid:124) (cid:123)(cid:122) (cid:125) N times , . . . , α ( M ) , . . . , α ( M ) (cid:124) (cid:123)(cid:122) (cid:125) N M times ) , (2.2)where α ( I ) > α ( I +1) , the gauge group is broken to the commutant of (cid:126)α on the surface C : L = S[U( N ) × U( N ) × · · · × U( N M )] , (2.3)which is called the Levi subgroup. In fact, the subgroup L is the Levi part of a parabolicsubgroup P of the complexified Lie group G C . For instance, if (cid:126)α = ( α, · · · , α, (1 − N ) α ),the Levi group is L = SU( N − × U(1), which is called simple. When all α i are distinct,which is called a full surface operator, the Levi group is L = U(1) N and the correspondingparabolic group is the Borel subgroup B of SL( N, C ). In addition to the M continuousparameters α ( I ) , there are “electric parameters” or “2d theta angles” η I correspondingto U(1) M ∈ L . These parameters enter into the path integral through the phase factorexp( iη I m I ) where m I are magnetic fluxes on C m I = 12 π (cid:90) C F I ( I = 1 , · · · , M ) , (2.4)where (cid:80) I m I = 0.The other way to describe a surface operator is to couple a 4d N = 2 gauge theory toan N = (2 , 2) supersymmetric gauge theory on the surface C [6]. For the surface operatorin the pure Yang-Mills, the 2d theory flows at infrared to the N = (2 , 2) supersymmetricnon-linear sigma model (NLSM) with the partial flag variety G C / P as a target. In thisdescription, the combined parameters (cid:126)t = 2 πi ( (cid:126)η + i(cid:126)α ) are identified with the complexifiedK¨ahler parameters of the NLSM. Furthermore, for the surface operator in the N = 2 ∗ theory, the infrared description of the 2d theory is given by an N = (2 , 2) NLSM with thecotangent bundle T ∗ ( G C / P ) of the flag variety.The instanton configurations F = − ∗ F on R \C with the singularity (2.1) are calledramified instantons. The moduli space of the ramified instantons is characterized by theLevi subgroup with (cid:126)N = [ N , · · · , N M ], the instanton number k and the magnetic fluxes m I so that we denote it by M (cid:126)N,k,(cid:126) m . The corresponding objects in algebraic geometry is– 6 –ctually rank- N torsion-free sheaves on P × P with coordinates ( z , z ), with framinggiven at { z = ∞} ∪ { z = ∞} and with parabolic structure of type P given at { z = 0 } ,called affine Laumon space [18, 19]. The affine Laumon space can be also regarded asthe smooth resolution of the space of quasi-maps from P into affine flag variety [43].Furthermore, using the equivalence between a parabolic sheaf on P × P of type P and a Z M -equivariant sheaf on P × P , the quiver description of M (cid:126)N,k,(cid:126) m is given by the ADHMquiver on the orbifold space C × ( C / Z M ). The resulting quiver is called a chain-saw quivershown in Figure 1. In this prescription, it is convenient to combine the instanton number k and the magnetic fluxes m I as follows: k M = k, k I +1 = k I + m I +1 , (2.5)where the index I is taken modulo M . Thus, we also denote the moduli space of ramifiedinstantons by M (cid:126)N,(cid:126)k with (cid:126)k = [ k , · · · , k M ]. To describe the ADHM construction of M (cid:126)N,(cid:126)k ,let V I and W I ( I = 1 , · · · , M ) vector spaces of dimensiondim W I = N I , dim V I = k I , (2.6)and we denote A I ∈ Hom ( V I , V I ), B I ∈ Hom ( V I , V I +1 ), P I ∈ Hom ( W I , V I ) and Q I ∈ Hom ( V I , W I +1 ). Then, the ADHM equations are E ( I ) C := A I +1 B I − B I A I + P I +1 Q I = 0 . (2.7)where the index I is taken modulo M . The moduli space is given by M (cid:126)N,(cid:126)k = { ( A I , B I , P I , Q I ) |E ( I ) C = 0 , stability condition } / GL( k , C ) ⊗ · · · ⊗ GL( k M , C ) . (2.8)As in the case without a surface operator, the moduli space M (cid:126)N,(cid:126)k of ramified instantonsreceives the action of the Cartan torus U(1) × U(1) N of the spacetime and the gaugesymmetry. Due the the orbifold operation, one of the equivariant parameters of U(1) actson the spacetime coordinate fractionally as( z , z ) → ( e i(cid:15) z , e i(cid:15) /M z ) . (2.9)In addition, since there are non-contractible cycles in the asymptotic region of C × ( C / Z M ),the gauge field can have a non-trivial holonomy. The non-trivial holonomy shifts theequivariant parameters ( a , · · · , a N ) of U(1) N by a s,I → a s,I − I − M (cid:15) , ( s = 1 , · · · , N I ) . (2.10)Fixed points under the equivariant action can be labeld by (cid:126)N -tuple of Young diagrams.For more detail, we refer the reader to [20]. Subsequently, the character of the equivariantaction at the fixed points yields the Nekrasov insanton partition function [18, 20] Z inst [ (cid:126)N ] = (cid:88) (cid:126)k M (cid:89) I =1 z k I I Z (cid:126)N,(cid:126)k ( (cid:15) , (cid:15) , a, m ) , (2.11)– 7 –here z I are instanton counting fugacity and Z (cid:126)N,(cid:126)k ( (cid:15) , (cid:15) , a, m ) depends on the mattercontent of the theory. In the context of the AGT relation, it was conjectured [13] that the2d symmetry is the W-algebra W ( (cid:98) sl ( N ) , (cid:126)N ) obtained by the quantum Drinfeld-Sokolovreduction [15–17] for the embedding ρ (cid:126)N : sl (2) → sl ( N ) corresponding to the partition (cid:126)N .In particular, when a full surface operator is present, it wan first proven in [35, 36] thatthe equivariant cohomology of the ramified instanton moduli space M [1 N ] ,(cid:126)k receives theaction of the affine Lie algebra (cid:98) sl ( N ). The checks of the correspondence between instantonpartition functions of 4d SCFTs and (cid:98) sl ( N ) conformal blocks have been carried out in [24–26]. For general W -algebras, it has been checked in [14, 20, 21] that the ramified instantonpartition functions of the pure Yang-Mills match with the norm of the Gaiotto-Whittakerstates in the Verma module of the corresponding W -algebra. · · ·· · · V W V W · · ·· · · V M W M V W · · ·· · · B Q P B Q P B M − Q M − B M P M Q P A A A M A Figure 1 . Chain-saw quiver By making change of variables z I = e t I − t I +1 ( I = 1 , · · · , M − , M (cid:89) I =1 z I = q , (2.12)the instanton partition function (2.11) can be re-arranged with (2.5) as Z inst [ (cid:126)N ] = ∞ (cid:88) k =0 (cid:88) m ∈ Λ L q k e t · m Z (cid:126)N,k,(cid:126) m ( (cid:15) , (cid:15) , a, m ) , (2.13)If a theory is superconformal, the fugacity of the instanton number k can be expressedin terms of the complexified gauge coupling τ by q = e πiτ . For an asymptotically freetheory, it is replaced by the dynamical scale Λ with appropriate mass dimension. Thechemical potentials (cid:126)t for the magnetic fluxes (cid:126) m are indeed the 2d complexified K¨ahlerparameters (cid:126)t = 2 πi ( (cid:126)η + i(cid:126)α ). Hence, when the instanton number is zero k = 0, the partitionfunction encodes only 2d dynamics on the support of the surface operator. Moreover, the k = 0 specialization of the chan-saw quiver in Figure 1 reduces to the hand-saw quiver[19] in Figure 2, which is equivalent to the smooth resolution of the space of quasi-mapsfrom P into the flag variety, called Laumon space [44]. The finite W -algebra that canbe obtained by quantum Drinfeld-Sokolov reduction of Lie algebra acts on the equivariantcohomology of the Laumon space [44]. We shall show that the generating function of– 8 –he equivariant cohomology of the Laumon space is actually the vortex partition functionof the N = (2 , 2) NLSM with the partial flag variety specified by the partition (cid:126)N . Inparticular, the generating function of the equivariant cohomology of the Laumon space canbe identified with the Givental J -function of the flag variety [35]. V V W · · ·· · · V M − W M − V M − W M − W M B Q B Q P P M − B M − Q M − P M − Q M − A A A M − A M − Figure 2 . Hand-saw quiver In this section, we concentrate on the pure Yang-Mills and the N = 2 ∗ theory with afull surface operator. For these theories, the Nekrasov partition functions and the vortexpartition functions on the support of the surface operator obey differential equations. Sincethey can be interpreted as quantum connections of Givental J -functions, they are writtenas integrable Hamiltonians. The pure Yang-Mills is related to the Toda integrable system[35, 36, 39, 40] whereas the N = 2 ∗ theory is connected to the Calogero-Moser integrablesystem [37, 38, 43]. When a general surface operator is placed, we present the partitionfunctions in Appendix A and C.Since an N = 2 supersymmetric path integral on S b localizes on the (anti-)instantonconfigurations on the north (south) pole, in order to obtain full exact partition functionson S b , one-loop determinants over the (anti-)instanton configurations have to be computedin addition to instanton partition functions [4, 5]. When a surface operator is present, thecalculations of one-loop determinants have not been demonstrated although the literature[14, 18, 20, 21, 24–26] has evaluated instanton partition functions. As in the case ofinstanton partition functions, it is natural to expect that the one-loop determinants can beevaluated by the orbifold method. In this paper, we propose that one-loop determinantsin the existence of a full surface operator can be obtained by means of the Atiyah-Singerindex theorem for transversally elliptic operators on the orbifold space C × ( C / Z N ). Tosupport this statement, we shall show that the one-loop determinants computed by thismethod correctly contain both the 4d and 2d perturbative contributions. The pure SU( N ) Yang-Mills theory is obtained by wrapping N M5-branes on a two-punctured sphere. Although the instanton partition function of the pure Yang-Mills with asurface operator is expressed as a character of the equivariant action at the fixed points of– 9 –he chain-saw quiver shown in Figure 1 [20], here we yield the contour integral representa-tion of the U( N ) instanton partition function by using the supersymmetric NLSM with thechain-saw quiver as a target. Since the detail is presented in Appendix A, we just give theexpression (A.6) for the U( N ) instanton partition function of the N = 2 pure Yang-Millstheory with a full surface operator Z pureinst [1 N ] = (cid:88) (cid:126)k (cid:16) N (cid:89) I =1 z k I I (cid:17) Z pure[1 N ] ,(cid:126)k , (2.14)where Z pure[1 N ] ,(cid:126)k = (cid:15) − (cid:80) NI =1 k I (cid:73) N (cid:89) I =1 k I (cid:89) s =1 dφ ( I ) s ( φ ( I ) s + a I − ( I − (cid:15) N )( φ ( I ) s + a I +1 + (cid:15) − I(cid:15) N ) N (cid:89) I =1 k I (cid:89) s =1 k I (cid:89) t (cid:54) = s φ ( I ) st φ ( I ) st + (cid:15) N (cid:89) I =1 k I (cid:89) s =1 k I +1 (cid:89) t =1 φ ( I ) s − φ ( I +1) t + (cid:15)φ ( I ) s − φ ( I +1) t + (cid:15) N . (2.15)The SU( N ) instanton partition function could be obtained by simply dropping the “U(1)factor” [2, 20]. Then, the SU( N ) instanton partition function is dual to the norm of acoherent state, called the Gaiotto-Whittaker state, of the Verma module of the affine Liealgebra (cid:98) sl ( N ) [20, 25]. Interestingly, the instanton partition function satisfies the periodicToda equation [45] (cid:34) (cid:15) N (cid:88) I =1 ( z I ∂ I − z I +1 ∂ I +1 ) + (cid:15) N (cid:88) I =1 u I z I ∂ I − N (cid:88) I =1 z I (cid:35) Z pureinst [1 N ] = 0 , (2.16)where we impose the periodic condition z N + I = z I on z and u I = a I +1 − a I , u I + N = u I + (cid:15) . (2.17)In fact, making the change of variables as in (2.12) z I = e t I − t I +1 ( I = 1 , · · · , N − , N (cid:89) I =1 z I = Λ , (2.18)where Λ can be interpreted as the dynamical scale of the pure Yang-Mills, one can bringthe equation into the more familiar form (cid:34) (cid:15) (cid:15) Λ ∂∂ Λ + (cid:15) ∆ h − (cid:16) Λ e t N − t + (cid:88) α ∈ Π e (cid:104) t,α (cid:105) (cid:17)(cid:35) ( e − (cid:104) a,t (cid:105) (cid:15) Z pureinst [1 N ]) = (cid:104) a, a (cid:105) ( e − (cid:104) a,t (cid:105) (cid:15) Z pureinst [1 N ]) , (2.19)where Π represents the set of simple roots of sl ( N ) so that (cid:80) α ∈ Π e (cid:104) t,α (cid:105) = (cid:80) N − I =1 e t I − t I +1 ,and the rest of notations is as follows:∆ h = N (cid:88) I =1 ∂ ∂t I , (cid:104) a, t (cid:105) = N (cid:88) I =1 a I t I , (cid:104) a, a (cid:105) = N (cid:88) I =1 a I . (2.20)This was first derived in the context of geometric representation theory [35, 36] and laterreproduced in the context of the AGT relation [45, 46].– 10 – N − · · · 21 4d2d Figure 3 . Quiver diagram of the 2d-4d coupled system for the pure Yang-Mills in the presence ofa full surface operator where we use the hybrid node as in [10] to denote a 4d gauge group whichgauges a 2d flavor symmetry. The Higgs branch of N = (2 , 2) GLSM is the complete flag variety. J -function of complete flag variety Since the surface operator is a half-BPS operator, it preserves four supercharges. Moreover,the surface operator can be also described as a 2d N = (2 , 2) supersymmetric gauge theorycoupled to the 4d N = 2 gauge theory. For a full surface operator in the 4d N = 2pure Yang-Mills, the 2d N = (2 , 2) supersymmetric gauge theory coupled to the 4d pureYang-Mills is described by the quiver diagram above (Figure 3). At UV, the matter contentconsists of bifundamentals ( , ) ⊕ . . . ⊕ ( N − , N − ) and N fundamentals N − . The 2dquiver gauge theory is coupled to the 4d pure Yang-Mills by gauging the flavor symmetryU( N ). Hence, the Coulomb branch parameters a i in the 4d theory become the twistedmasses of the fundamentals in the 2d theory. Since the Higgs branch of the 2d theoryis given by the complete flag variety Fl N = SL( N, C ) /B where B is the Borel subgroupof SL( N, C ), the 2d theory flows to the NLSM with the complete flag variety Fl N in theinfrared. It is worth mentioning that there is another description for the complete flagvariety as an increasing sequence of linear subspaces of C N ⊂ C ⊂ C ⊂ · · · ⊂ C N − ⊂ C N , (2.21)which indeed yields the quiver description. In fact, the description by the gauged linearsigma model (GLSM) presented in Figure 3 enables us to compute the exact partitionfunction of the N = (2 , 2) quiver gauge theory on S [27, 28]. From the S partitionfunctions, one can extract the Givental J -function of the Higgs branch of the GLSM [47],which plays an important role in this paper.Therefore, let us briefly recall the definition of the Givental J -function of a compactK¨ahler variety X . Let T = 1 , T , · · · , T m be the basis of the cohomology group H ∗ ( X, Z ),and T , · · · , T r be the basis of the second cohomology group H ( X, Z ). We define thematrix g ij = (cid:82) X T i ∪ T j , and its inverse matrix g ij = ( g ij ) − , which provide the dual basis T a = m (cid:88) b =1 g ab T b , (2.22)so that (cid:82) X T i ∪ T j = δ ij . We denote by M g,n ( X, β ) the moduli space of stable mapsfrom connected genus g curves with n -marked points to X representing the class β ∈ H ( X, Z ). Let L , · · · , L n be the corresponding tautological line bundles over M g,n ( X, β ).– 11 –or γ , · · · , γ n ∈ H ∗ ( X, Z ) and non-negative integers d i , the gravitational correlation func-tion is defined (cid:104) τ d γ , · · · , τ d n γ n (cid:105) g,β = (cid:90) [ M g,n ( X,β )] vir n (cid:89) i =1 c ( L i ) d i ∪ ev ∗ ( γ i ) . (2.23)The J -function of X is defined by using the psi class ψ = c ( L ) J ( X ) = e δ/ (cid:126) (cid:88) β ∈ H ( X, Z ) m (cid:88) a =1 q β (cid:28) T a (cid:126) − ψ , (cid:29) ,β T a , (2.24)where δ = (cid:80) ri =1 t i T i and q β = e (cid:82) β δ . Thus, it is regarded as a generating function foronce-punctured genus zero Gromov-Witten invariants with gravitational descendants.Now, let us compute the partition function of the 2d gauge theory on S . The Coulombbranch formula of the partition function is given by Z [Fl N ] = 11! · · · ( N − (cid:88) (cid:126)B ( I ) I =1 ··· N − (cid:90) N − (cid:89) I =1 I (cid:89) s =1 dτ ( I ) s πi e πξ ( I ) τ ( I ) s − iθ ( I ) B ( I ) s Z vector Z bifund Z fund ,Z vector = N − (cid:89) I =2 I (cid:89) s 01 for k = 0 (cid:81) ki =1 x − i for k < . (2.29)As shown in [47], the vortex partition function in the massless limit a s = 0 is identical withthe Givental J -function of the complete flag variety [41] J [Fl N ] = (cid:88) (cid:126)k ( I ) (cid:126) − (cid:80) N − I =1 | k ( I ) | N − (cid:89) I =1 z | k ( I ) | I N − (cid:89) I =2 I (cid:89) s (cid:54) = t (cid:126) − H ( I ) st ) k ( I ) s − k ( I ) t (2.30) N − (cid:89) I =1 I (cid:89) s =1 I +1 (cid:89) t =1 1(1+ (cid:126) − H ( I ) s − (cid:126) − H ( I +1) t ) k ( I ) s − k ( I +1) t N − (cid:89) s =1 N (cid:89) t =1 1(1+ (cid:126) − H ( N − s − (cid:126) − H ( N ) t ) k ( N − s . Here we identify H ( I ) s ( s = 1 , ..., I ) with Chern roots to the duals of the universal bundles S I : 0 ⊂ S ⊂ S ⊂ · · · ⊂ S N − ⊂ S N = C N ⊗ O Fl N . (2.31)and we add H ( N ) t ( t = 1 , · · · , N ) to the last Pochhammer of (cid:101) Z v by hand. These additionalclasses are necessary to become an eigenfunction of the Toda Hamiltonian as we will seebelow.Performing the residue integral in (2.27), one obtains the Higgs branch formula Z [Fl N ] = 11! · · · ( N − (cid:88) σ ∈ S N N − (cid:89) I =1 ( z I z I ) − (cid:126) − (cid:80) It =1 a σ ( t ) Z ( a σ ( i ) ) Z v ( a σ ( i ) ) Z av ( a σ ( i ) ) , – 13 – = N (cid:89) s 2) GLSMdescription of the 2d theory coupled to the pure Yang-Mills whereas the description of thesurface operator by the boundary condition of the gauge field has led to the right handside. Thus, the identity (2.33) proves that the two descriptions for the surface operator areequivalent [6].It is straightforward to see from (2.19) that the vortex partition function becomes aneigenfunction of the Toda Hamiltonian (cid:16) (cid:126) ∆ h − (cid:88) α ∈ Π e (cid:104) t,α (cid:105) (cid:17)(cid:104) e − (cid:104) a,t (cid:105) (cid:126) Z v [Fl N ] (cid:105) = (cid:104) a, a (cid:105) (cid:104) e − (cid:104) a,t (cid:105) (cid:126) Z v [Fl N ] (cid:105) , (2.34)where we substitute z I = e t I − t I +1 . In addition, it is well-known that the J -function of thecomplete flag variety becomes an eigenfunction of the Toda Hamiltonian [48]. To see that,one has to identify H ( I ) s = H ( I +1) s ( s = 1 , · · · , I ) as the same cohomology class. Then,the J -function becomes equivalent to the generating function Z v [Fl N ] of the equivariantcohomology of the Laumon space by setting H s = − a s . The localization technique enables us to demonstrate exact evaluations of supersymmetricpartition functions by taking only the quadratic fluctuations over BPS configurations intoaccount. In the case of N = 2 supersymmetric gauge theories on S b , the BPS configu-rations correspond to the instantons at the north and south pole of S b [4, 5]. Then, the– 14 –uadratic fluctuations over the instanton configurations can be evaluated by the means ofthe Atiyah-Singer index theory for transversally elliptic operators. The minimum explana-tion is provided in Appendix B.Since the field content of the N = 2 pure Yang-Mills consists only of the vector multi-plet, the quadratic fluctuations of the theory is captured just by the one-loop determinant(B.8) of the vector multiplet over the instanton configurations, which can be obtained bythe equivariant indices of the self-dual (B.6) and anti-self-dual complex Z pure1-loop = (cid:89) α ∈ ∆ [Γ ( (cid:104) a, α (cid:105)| (cid:15) , (cid:15) ) Γ ( (cid:104) a, α (cid:105) + (cid:15) + (cid:15) | (cid:15) , (cid:15) )] − , = (cid:89) α ∈ ∆ Υ ( (cid:104) a, α (cid:105)| (cid:15) , (cid:15) ) , (2.35)where ∆ represents the set of roots of sl ( N ). Note that Γ ( x | (cid:15) , (cid:15) ) is the Barnes doubleGamma function (B.20) and Υ( x | (cid:15) , (cid:15) ) is the Upsilon function (B.22).Since the instanton partition function in the presence of a surface operator have beencomputed by the orbifold operation, it is natural to expect that the one-loop computationcan be obtained by the index theorem on C × ( C / Z N ). As in the case of the instantonpartition function, the equivariant parameters are shifted by (cid:15) → (cid:15) N , a i → a i − i − N (cid:15) , (2.36)due to the orbifold operation. This re-parametrization alters the one-loop determinant (cid:89) α ∈ ∆ Γ ( (cid:104) a, α (cid:105)| (cid:15) , (cid:15) ) Γ ( (cid:104) a, α (cid:105) + (cid:15) + (cid:15) | (cid:15) , (cid:15) ) → N (cid:89) i,j =1 ,i (cid:54) = j Γ (cid:16) a i − a j + j − iN (cid:15) | (cid:15) , (cid:15) N (cid:17) Γ (cid:16) a i − a j + (cid:15) + j − iN (cid:15) | (cid:15) , (cid:15) N (cid:17) . (2.37)To get its Z N -invariant part, we average over the finite group Z N as in (B.17), leaving theone-loop determinant in the existence of the full surface operator Z pure1-loop [1 N ] = N (cid:89) i,j =1 ,i (cid:54) = j (cid:104) Γ ( a i − a j + (cid:108) j − iN (cid:109) (cid:15) | (cid:15) , (cid:15) )Γ ( a i − a j + (cid:15) + (cid:108) j − iN (cid:109) (cid:15) | (cid:15) , (cid:15) ) (cid:105) − = N (cid:89) i,j =1 ,i (cid:54) = j Υ (cid:16) a i − a j + (cid:108) j − iN (cid:109) (cid:15) | (cid:15) , (cid:15) (cid:17) , (2.38)where (cid:100) x (cid:101) denotes the smallest integer ≥ x .As we have seen in the previous sections, the instanton partition function containsboth 4d and 2d dynamics, and the 2d vortex partition function is left when the 4d non-perturbative effect is switched off. This should be true for the perturbative contribu-tions. Namely, if the 4d contribution Z pure1-loop is subtracted from the one-loop determinant Z pure1-loop [1 N ], only the 2d effect Z [Fl N ] should be evident [28, § Z pure1-loop [1 N ] to– 15 – is independent of (cid:15) , and we have Z pure1-loop [1 N ] Z pure1-loop ( a, (cid:15) = (cid:126) ) = (cid:89) α ∈ ∆ + (cid:126) (cid:104) a,α (cid:105) (cid:126) − γ (cid:18) (cid:104) a, α (cid:105) (cid:126) (cid:19) “ = ” Z [Fl N ]( a, (cid:126) ) (2.39)where “ = ” means the equality up to a constant. This supports the validity of the orbifoldmethod even in the one-loop computations. N = 2 ∗ theory2.2.1 Instanton partition function The N = 2 ∗ theory is the deformation of the N = 4 SCFT by adding the mass µ adj to thehypermultiplet in the adjoint representation. From the 6d perspective, the SU( N ) N = 2 ∗ theory is obtained by wrapping N M5-branes on a once-punctured torus. Because thestandard ADHM description of the N = 2 ∗ theory [49] can be generalized to the orbifoldspace C × ( C / Z N ), one can write the contour integral representation of the U( N ) instantonpartition function of the N = 2 ∗ theory with a full surface operator: Z N =2 ∗ inst [1 N ] = (cid:88) (cid:126)k (cid:16) N (cid:89) I =1 z k I I (cid:17) Z N =2 ∗ [1 N ] ,(cid:126)k , (2.40)where Z N =2 ∗ [1 N ] ,(cid:126)k = (cid:20) (cid:15) − µ adj (cid:15) µ adj (cid:21) (cid:80) NI =1 k I (cid:73) N (cid:89) I =1 k I (cid:89) s =1 dφ ( I ) s ( φ ( I ) s + a I − ( I − (cid:15) N + µ adj )( φ ( I ) s + a I +1 + (cid:15) − I(cid:15) N − µ adj )( φ ( I ) s + a I − ( I − (cid:15) N )( φ ( I ) s + a I +1 + (cid:15) − I(cid:15) N ) N (cid:89) I =1 k I (cid:89) s =1 k I (cid:89) t (cid:54) = s φ ( I ) st ( φ ( I ) st + (cid:15) − µ adj )( φ ( I ) st + µ adj )( φ ( I ) st + (cid:15) ) N (cid:89) I =1 k I (cid:89) s =1 k I +1 (cid:89) t =1 ( φ ( I ) s − φ ( I +1) t + (cid:15) )( φ ( I ) s − φ ( I +1) t + (cid:15) N − µ adj )( φ ( I ) s − φ ( I +1) t + (cid:15) N )( φ ( I ) s − φ ( I +1) t + (cid:15) − µ adj ) . (2.41)It was proven in [38] that, by multiplying an appropriate factor, the instanton partitionfunction Z N =2 ∗ inst [1 N ] becomes an eigenfunction of a non-stationary deformation of thetrigonometric Calogero-Moser Hamiltonian. To avoid repetition, we refer the reader to[38] for the explicit expression of the differential equation. Instead, let us mention theconnection to the Knizhnik-Zamolodchikov-Bernard (KZB) equation [50–52].In the AGT relation, the partition function of the N = 2 ∗ theory is dual to the one-point correlation function on a torus. When a full surface operator is present, the instantonpartition function of the N = 2 ∗ theory is the one-point (cid:98) sl ( N ) conformal blocks on a torus.More precisely, the corresponding conformal block is a semi-degenerate field V κω N − ( x ; q )on a torus with the K operator [24, 25, 38] F K ( x ; q ) := Tr V j K ( x ; q ) V κω N − ( x ; q ) , (2.42) The author would like to thank Hee-Cheol Kim for suggesting this approach. – 16 –here V j is the Verma module of the affine Lie algebra (cid:98) sl ( N ) with the highest weight j .Note that the semi-degenerate field V κω N − ( x ; q ) labelled by the momentum proportionalto the fundamental weight ω N − depends on the isospin variables x i ( i = 1 , · · · , N − q . We refer the reader to [24, 25] for the explicit expressionof the K operator. Writing the instanton partition function in terms of q = e πiτ and t i ( i = 1 , · · · , N − 1) via (2.12), it is conjectured that it matches with the (cid:98) sl ( N ) conformalblock up to the U(1) factor ∞ (cid:89) i =1 (1 − q i ) − µ adj( N(cid:15) (cid:15) − Nµ adj) (cid:15) (cid:15) +1 Z N =2 ∗ inst [1 N ] = F K ( x (cid:96) = e t − t (cid:96) +1 ; q ) (2.43)Here, the parameters are identified by a(cid:15) = j + ρ , µ adj (cid:15) = − κN , − (cid:15) (cid:15) = k + N , (2.44)where ρ is the Weyl vector and k is the level. We further conjecture that, for the once-punctured conformal block on a torus, the effect of the insertion of the K operator resultsin the prefactor so that the ordinary conformal block F ( x ; q ) := Tr V j V κω N − ( x ; q ) is pro-portional to F K ( x ; q ) F ( x (cid:96) = e t − t (cid:96) +1 ; q ) = f ( t, q ) κN F K ( x (cid:96) = e t − t (cid:96) +1 ; q ) . (2.45)When N = 2, the explicit expression of the prefactor is found by computer analysis, whichis f ( t, q ) = 1 − e t − t − qe t − t [24, (4.20)]. We expect that this relation holds for higherrank gauge groups. Then, taking into account this prefactor and the U(1) factor, we candefine the function Y ( t, q, a, µ adj , (cid:15) , (cid:15) ) := e − (cid:104) a,t (cid:105) (cid:15) f ( t, q ) − µ adj (cid:15) +1 ∞ (cid:89) i =1 (1 − q i ) − µ adj( N(cid:15) (cid:15) − Nµ adj) (cid:15) (cid:15) +1 Z N =2 ∗ inst [1 N ](2.46)so that it should satisfy the KZB equation (cid:15) (cid:15) q ∂∂q + (cid:15) ∆ h + 2 µ adj ( µ adj − (cid:15) ) (cid:88) α ∈ ∆ + (cid:16) π ℘ ( (cid:104) t, α (cid:105) ; τ ) + 112 (cid:17) Y = (cid:104) a, a (cid:105) Y . (2.47)Note that the Weierstrass elliptic function ℘ ( u ; τ ) can be expressed as [53, § π ℘ ( u ; τ ) = T ( u ; τ ) − E ( τ ) , (2.48)where E ( τ ) is the Eisenstein series E ( τ ) = 1 − ∞ (cid:88) n =1 nq n − q n , (2.49)and we define T ( u ; τ ) := − (cid:88) (cid:96) ∈ Z q (cid:96) e u (1 − q (cid:96) e u ) . (2.50)– 17 –ince the solution (2.46) of the KZB equation should reduce to the eigenfunction (2.59) ofthe trigonometric Calogero-Moser Hamiltonian at q = 0, the q = 0 specialization of theprefactor is f ( t, q = 0) = (cid:89) α ∈ ∆ + (1 − e (cid:104) t,α (cid:105) ) . (2.51)When N = 2, the prefactor is subject to this condition. Nevertheless, it is crucial to findthe explicit expression of the prefactor f ( t, q ) for a higher rank gauge group. Moreover, dueto the insertion of the K operator, it is not obvious that the instanton partition functionsof class S theories generally satisfy the KZ equations [50]. Therefore, it is valuable to gaina better understanding of the meaning of the K operator. J -function of cotangent bundle of complete flag variety N N − · · · 21 4d2d Figure 4 . Quiver diagram of the 2d-4d coupled system for the N = 2 ∗ theory in the presence ofa full surface operator. The Higgs branch of the N = (2 , ∗ GLSM is the cotangent bundle of thecomplete flag variety. Since the N = 2 ∗ theory is a mass deformation of the N = 4 SCFT, the dynamicson the support of surface operator is also described by a deformation of an N = (4 , • bifundamentals Q ( I ) ∈ ( I , I + 1 ), (cid:101) Q ( I ) ∈ ( I , I + 1 ) ( I ∈ , · · · , N − • one adjoint Φ ( I ) for each gauge group U( I ) ( I ∈ , · · · , N − • N fundamentals Q ( N − and N antifundamentals (cid:101) Q ( N − of U( N − N = (4 , 4) supersymmetric gauge theory with thesuperpotential W = N − (cid:88) I =1 Tr (cid:101) Q ( I ) Φ ( I ) Q ( I ) + N − (cid:88) I =1 Tr Q ( I ) Φ ( I +1) (cid:101) Q ( I ) , (2.52)by turning on the twisted mass m of (cid:101) Q ( I ) and Φ ( I ) ( I = 1 , · · · , N − R -charges of Φ ( I ) are two and those of Q ( I ) and (cid:101) Q ( I ) are zero. The infrared dynamics of this– 18 –heory is described by the hyper-K¨ahler NLSM with the contangent bundle T ∗ Fl N of thecomplete flag variety [6]. Let us first compute the exact partition function of this theorywithout turning on the twisted masses coming from the Coulomb branch parameters. TheCoulomb branch formula of the partition function is given by Z [ T ∗ Fl N ] = 11! · · · ( N − × (cid:88) (cid:126)B ( I ) I =1 ··· N − (cid:90) N − (cid:89) I =1 I (cid:89) s =1 dτ ( I ) s πi e πξ ( I ) τ ( I ) s − iθ ( I ) B ( I ) s Z vect Z adj Z bifund Z fund-anti ,Z vect = N − (cid:89) I =2 I (cid:89) s 12 ln( − ir Σ) − (cid:36) ( ir Σ) − 12 ln( ir Σ) (cid:111) , (2.64)where (cid:36) ( x ) = x (ln x − r → ∞ [27, 58]. Using this prescription, we can write Z [ T ∗ Fl N ] ∼ · · · ( N − (cid:90) N − (cid:89) I =1 I (cid:89) s =1 d ( r Σ ( I ) s )2 π (cid:12)(cid:12)(cid:12) Q (Σ) e − (cid:102) W eff (Σ) (cid:12)(cid:12)(cid:12) , (2.65)where the logarithmic terms in (2.64) give the measure Q (Σ) = N − (cid:89) I =2 I (cid:89) s,t =1 s (cid:54) = t ( − ir Σ ( I ) s + ir Σ ( I ) t )( − ir Σ ( I ) s + ir Σ ( I ) t + ir ˆ m ) N − (cid:89) I =1 I (cid:89) s =1 I +1 (cid:89) u =1 ( − ir Σ ( I ) s + ir Σ ( I +1) u ) − ( ir Σ ( I ) s − ir Σ ( I +1) u − ir ˆ m ) − N − (cid:89) s =1 ( − ir Σ ( N − s ) − ( ir Σ ( N − s − ir ˆ m ) − , (2.66)and (cid:102) W eff (Σ) is the effective twisted superpotential of the mirror LG model in the Coulombbranch (cid:102) W eff (Σ) = N − (cid:88) I =1 I (cid:88) s =1 ( − πξ ( I ) + iθ ( I ) )( ir Σ ( I ) s ) + N − (cid:88) I =2 I (cid:88) s (cid:54) = t (cid:36) ( − ir Σ ( I ) s + ir Σ ( I ) t + ir ˆ m )+ N − (cid:88) I =1 I (cid:88) s =1 I +1 (cid:88) u =1 (cid:104) (cid:36) ( − ir Σ ( I ) s + ir Σ ( I +1) u ) + (cid:36) ( ir Σ ( I ) s − ir Σ ( I +1) u − ir ˆ m ) (cid:105) . (2.67)where Σ ( N ) s = a s . Here we redefine the twisted mass by m = i ˆ m . Then, the twisted chiralring is given by the equation of supersymmetric vacua [59, 60]exp (cid:32) ∂ (cid:102) W eff ∂ ( ir Σ ( I ) s ) (cid:33) = 1 . (2.68)Plugging (2.67) into (2.68), we obtain the following set of equations: for I = 1, (cid:89) t =1 Σ (1) s − Σ (2) t Σ (1) s − Σ (2) t − ˆ m = e − πξ (1) + iθ (1) , (2.69)for 1 < I < N , I (cid:89) t (cid:54) = s Σ ( I ) s − Σ ( I ) t − ˆ m Σ ( I ) s − Σ ( I ) t + ˆ m I − (cid:89) t =1 Σ ( I ) s − Σ ( I − t + ˆ m Σ ( I ) s − Σ ( I − t I +1 (cid:89) t =1 Σ ( I ) s − Σ ( I +1) t Σ ( I ) s − Σ ( I +1) t − ˆ m = ± e − πξ ( I ) + iθ ( I ) . (2.70)– 22 –hese equations are called nested Bethe ansatz equations [61, 62] for sl ( N ) spin chain. Forthe cotangent bundle T ∗ Gr( r, N ) of a Grassmannian, the vacuum equation (2.68) providesthe Bethe ansatz equation of an inhomogeneous XXX spin chain [59, 60]. Motivated bythis physical insight, it was proven in [63] that the algebra of quantum multiplication inthe equivariant quantum cohomology QH ∗ T ( (cid:96) r T ∗ Gr( r, N )) is isomorphic to the maximalcommutative subalgebra B q , so-called Baxter subalgebra , of Yangain Y ( sl (2)). Since (2.70)is the Bethe ansatz equation for sl ( N ) spin chain, it is natural to expect that the algebraof quantum multiplication on the equivariant quantum cohomology QH ∗ T ( (cid:96) (cid:126)d T ∗ Fl( (cid:126)d )) isisomorphic to the Baxter subalgebra of Y ( sl ( N )) [64]. (See Appendix C for the definition ofa partial flag variety Fl( (cid:126)d ).) In addition, similar Bethe ansatz equations have been obtainedin the system of multiple M2-branes ending on M5-branes [65]. It would be interesting toinvestigate whether there is a duality between the two systems. The quadratic fluctuations in the N = 2 ∗ theory receive the contributions from both thevector multiplet and the hypermultiplet in the adjoint representation. Particularly, theone-loop determinant (B.12) of the hypermultiplet in the adjoint representation can beread off from the index of the Dirac complex tensored with the adjoint bundle. Thus, theone-loop determinant of the N = 2 ∗ theory is expressed as Z N =2 ∗ = (cid:89) α ∈ ∆ Γ (cid:0) (cid:104) a, α (cid:105) + m adj + (cid:15) + (cid:15) | (cid:15) , (cid:15) (cid:1) Γ (cid:0) (cid:104) a, α (cid:105) − m adj + (cid:15) + (cid:15) | (cid:15) , (cid:15) (cid:1) Γ ( (cid:104) a, α (cid:105)| (cid:15) , (cid:15) ) Γ ( (cid:104) a, α (cid:105) + (cid:15) + (cid:15) | (cid:15) , (cid:15) ) . (2.71)Actually, the mass parameter of the hypermultiplet in the instanton partition function isgiven by µ adj = m adj + (cid:15) + (cid:15) , and then we can re-write the one-loop determinant with µ adj in terms of the Upsilon functions: Z N =2 ∗ = (cid:89) α ∈ ∆ Υ ( (cid:104) a, α (cid:105)| (cid:15) , (cid:15) )Υ ( (cid:104) a, α (cid:105) + µ adj | (cid:15) , (cid:15) ) . (2.72)With the insertion of a full surface operator, the one-loop determinant can be computedby the same way as in § Z N -invariant part (B.17), we get Z N =2 ∗ [1 N ] = N (cid:89) i,j =1 ,i (cid:54) = j Γ (cid:18) a i − a j + µ adj + (cid:24) j − iN (cid:25) (cid:15) | (cid:15) ,(cid:15) (cid:19) Γ (cid:18) a i − a j − µ adj + (cid:15) + (cid:24) j − iN (cid:25) (cid:15) | (cid:15) ,(cid:15) (cid:19) Γ (cid:18) a i − a j + (cid:24) j − iN (cid:25) (cid:15) | (cid:15) ,(cid:15) (cid:19) Γ (cid:18) a i − a j + (cid:15) + (cid:24) j − iN (cid:25) (cid:15) | (cid:15) ,(cid:15) (cid:19) = N (cid:89) i,j =1 ,i (cid:54) = j Υ (cid:16) a i − a j + (cid:108) j − iN (cid:109) (cid:15) | (cid:15) , (cid:15) (cid:17) Υ (cid:16) a i − a j + µ adj + (cid:108) j − iN (cid:109) (cid:15) | (cid:15) , (cid:15) (cid:17) . (2.73)This one loop determinant encodes both 4d and 2d quadratic fluctuations. Indeed, if wesubtract the 4d contribution Z N =2 ∗ from Z N =2 ∗ [1 N ], then we obtain the 1-loop determi-nant Z [ T ∗ Fl N ] of the 2d theory on the surface operator: Z N =2 ∗ [1 N ] Z N =2 ∗ ( a i , µ adj = m + (cid:126) , (cid:15) = (cid:126) ) = (cid:89) α ∈ ∆ + (cid:126) − m − (cid:126) γ (cid:18) (cid:104) a, α (cid:105) (cid:126) (cid:19) γ (cid:18) −(cid:104) a, α (cid:105) − m (cid:126) (cid:19) – 23 – = ” Z [ T ∗ Fl N ]( a, m, (cid:126) ) , (2.74)where “ = ” means the equality up to a constant. Here we use the same change of themass parameter as in (2.58). SL( N, R ) WZNW model from gauge theory In a CFT, two-point and three-point functions encode the information about the dynam-ics of the CFT although the conformal blocks are universal since they are determinedby algebras. In the AGT relation, the instanton partition functions coincide with theconformal blocks of the W N /Virasoro algebra, while the one-loop determinants of gaugetheory reproduce the product of the three-point functions (the structure constants) of theToda/Liouviile theory [66–68]. When a full surface operator is inserted, various checks havebeen carried out for the equivalence between SU( N ) ramified instanton partition functionsand (cid:98) sl ( N ) conformal blocks [20, 24–26]. The natural candidate for the corresponding partof the one-loop determinants with a full surface operator is the three-point function ofSL( N, R ) WZNW model. In § § , R ) WZNW model de-rived in [29–31]. Since the three-point function of SL( N, R ) WZNW model has not beendetermined yet, we predict the forms of two-point and three-point function with a semi-degenerate field of SL( N, R ) WZNW model by using the one-loop determinants with a fullsurface operator.Let us first review the correspondence between a one-loop determinant of a 4d gaugetheory without a surface operator and a product of three-point functions of Toda CFT.To this end, we consider the one-loop determinant of the SU( N ) SCFT with N F = 2 N . i.e. N fundamentals of mass m i and N anti-fundamentals of (cid:101) m i . The dual correlationfunction in Toda CFT is the four point function (cid:104) V β V κω N − V (cid:101) κω N − V (cid:101) β (cid:105) [3, § 3] with twosemi-degenerate fields. Making use of the one-loop determinants (B.14) with redefinitionsof the mass parameters µ i = − m i + (cid:15) + (cid:15) , (cid:101) µ i = − (cid:101) m i − (cid:15) + (cid:15) , (3.1)the one-loop determinant of the SU( N ) SCFT with N F = 2 N can be expressed by theproduct of the Upsilon function Z N F =2 N = (cid:81) α ∈ ∆ + Υ( (cid:104) a, α (cid:105)| (cid:15) , (cid:15) )Υ( −(cid:104) a, α (cid:105)| (cid:15) , (cid:15) ) (cid:81) i,j Υ( (cid:104) a, h i (cid:105) + µ j | (cid:15) , (cid:15) )Υ( −(cid:104) a, h i (cid:105) − (cid:101) µ j | (cid:15) , (cid:15) ) , (3.2)where h i ( i = 1 , · · · , N ) are the weights of the fundamental representation. On the otherhand, the corresponding part of the correlation function in Toda CFT can be determinedby the conformal symmetry and W N -symmetry [68]. For example, the reflection amplitude– 24 –n the two point function of primary fields is expressed as (cid:104) V β ( z ) V β ∗ ( z ) (cid:105) = R − ( β ) | z | β ) , R ( β ) = ( πµγ ( b )) (cid:104) Q − β,α (cid:105) b (cid:89) α ∈ ∆ + Γ(1+ b (cid:104) β − Q,α (cid:105) )Γ( b − (cid:104) β − Q,α (cid:105) )Γ(1 − b (cid:104) β − Q,α (cid:105) )Γ( − b − (cid:104) β − Q,α (cid:105) ) , (3.3)where z = z − z , the conformal dimension is given by ∆( β ) = (cid:104) Q − β, β (cid:105) / 2, and theconjugated vector parameter β ∗ is defined in terms of simple roots ( α , · · · , α N − ) ∈ Π of sl ( N ) ( β, α k ) = ( β ∗ , α N − k ) . (3.4)In addition, since the conformal symmetry fixes the form of the three-point functions ofprimaries (cid:104) V β ( z ) V β ( z ) V β ( z ) (cid:105) = C ( β , β , β ) | z | | z | | z | , where ∆ kij = ∆( β i ) + ∆( β j ) − ∆( β k ), it amounts to specifying the structure coefficient C ( β , β , β ). Although the general structure coefficient in Toda CFT is not known yet,the structure coefficient of the three-point function (cid:104) V β V κ ω N − V β (cid:105) of Toda CFT with asemi-degenerate field V κ ω N − [68] is given by C ( β , κ ω N − , β ) (3.5)= (cid:104) πµγ ( b ) b − b (cid:105) (cid:104) Q − (cid:80) βi,ρ (cid:105) b (Υ b ( b )) N − Υ b ( κ ) (cid:81) α ∈ ∆ + Υ b (cid:0) (cid:104) Q − β , α (cid:105) (cid:1) Υ b (cid:0) (cid:104) Q − β , α (cid:105) (cid:1)(cid:81) ij Υ b (cid:0) κ N + (cid:104) β − Q, h i (cid:105) + (cid:104) β − Q, h j (cid:105) (cid:1) , where we use the short-hand notation of the Upsilon function for Toda CFTΥ b ( x ) = Υ( x | b, b − ) . (3.6)Using the shift relation (B.25) of the Upsilon function, one can convince oneself that the re-flection amplitude (3.3) can be obtained from the structure coefficient (3.5) in the followingway: R − ( β ) = C ( β, , β ∗ ) . (3.7)Therefore, the relevant part in the correlation function of Toda CFT can be expressed as C ( β , κ ω N − , β ) R ( β ) C ( β ∗ , (cid:101) κ ω N − , (cid:101) β ) (3.8)= A (cid:81) α ∈ ∆ + Υ b (cid:0) (cid:104) Q − β, α (cid:105) (cid:1) Υ b (cid:0) (cid:104) β − Q, α (cid:105) (cid:1)(cid:81) ij Υ b (cid:0) κ N + (cid:104) β − Q, h i (cid:105) + (cid:104) β − Q, h j (cid:105) (cid:1) Υ b (cid:0) (cid:101) κ N − (cid:104) β − Q, h i (cid:105) + (cid:104) (cid:101) β − Q, h j (cid:105) (cid:1) . where we confine the unnecessary part to the coefficient A . Then, it is easy to see thecorrespondence between (3.2) and (3.8) upon the identification of the parameters a = β − Q , µ i = κ N + (cid:104) β − Q, h i (cid:105) , (cid:101) µ i = − (cid:101) κ N − (cid:104) (cid:101) β − Q, h j (cid:105) . (3.9)The natural candidate of the 2d CFT dual to N = 2 class S theories with a full surfaceoperator is SL( N, R ) WZNW model. So far, the two-point and three-point function of– 25 –L(2 , R ) WZNW model are known [29–31]. The primary field V j ( x ; z ) of SL(2 , R ) WZNWmodel specified by a highest weight j of the affine Lie algebra (cid:98) sl (2) depends on the isospincoordinate x and the worldsheet coordinate z . Then, the two-point function takes the form (cid:104) V j ( x ; z ) V j ( x ; z ) (cid:105) = B ( j ) | x | j | z | j ) (3.10)where ∆( j ) = j ( j +1)2( k +2) is the conformal dimension of the primary field and the reflectionamplitude B ( j ) is given by B ( j ) = − k + 2 π ν j γ (cid:16) j +1 k +2 (cid:17) , ν = π Γ (cid:16) k +2 (cid:17) Γ (cid:16) − k +2 (cid:17) . (3.11)In addition, the conformal invariance and the affine symmetry determine the three-pointfunction (cid:104) V j ( x ; z ) V j ( x ; z ) V j ( x ; z ) (cid:105) = D ( j , j , j ) | x | j | x | j | x | j | z | | z | | z | (3.12)where the structure coefficient D ( j , j , j ) is given by D ( j , j , j )= − ν j + j + j +12 (cid:101) Υ k +2 (1) (cid:101) Υ k +2 ( − j − (cid:101) Υ k +2 ( − j − (cid:101) Υ k +2 ( − j − π γ (cid:16) k +1 k +2 (cid:17) (cid:101) Υ k +2 ( − j − j − j − (cid:101) Υ k +2 ( j − j − j ) (cid:101) Υ k +2 ( j − j − j ) (cid:101) Υ k +2 ( j − j − j ) . (3.13) Here we use the short-hand notation of the Upsilon function for SL( N, R ) WZNW model (cid:101) Υ k + N ( x ) = Υ( x | , − k − N ) . (3.14)As in (3.7), the reflection amplitude can be obtained from the structure coefficient via B ( j ) = D ( j, , j ) . (3.15)Yet, the two-point and three-point function in SL( N, R ) WZNW model are not avail-able even with a semi-degenerate field. It was pointed out in [25] that, although theprimary field V j ( x, z ) of SL( N, R ) WZNW model is dependent of N ( N − / N − x i ( i = 1 , · · · , N − V j ( (cid:126)x, z ) labelled by a highest weight j of (cid:98) sl ( N ) has its conformal dimension∆( j ) = (cid:104) j,j +2 ρ (cid:105) k + N ) . Besides, the conformal invariance and the affine symmetry constrain theform of the three-point function with a semi-degenerate field [25, (4.18)] (cid:104) V j ( x (1) ; z ) V j = κω N − ( x (2) ; z ) V j ( x (3) ; z ) (cid:105) = D ( j , κω N − , j ) | z | | z | | z | N − (cid:89) i =1 | x (12) i | (cid:104) j ,h i (cid:105) | x (13) i | (cid:104) j ,h i (cid:105) | x (23) i | (cid:104) j ,h i (cid:105) , (3.16)– 26 –here we define (cid:104) j mk(cid:96) , h i (cid:105) = (cid:104) j k + j (cid:96) − j m , h i (cid:105) . In the following, let us predict the formof the two-point and three-point function in SL( N, R ) WZNW model by making use ofthe one-loop determinant of the SU( N ) SCFT with N F = 2 N in the existence of a fullsurface operator. Since we have derived the one-loop determinant (2.38) of the vectormultiplet, we need to determine the one-loop determinant of the hypermultiplet in the(anti-)fundamental representation. As the Coulomb branch parameters are shifted by theholonomy (2.36), we shift the mass parameters due to the orbifold method µ i → µ i + N − iN (cid:15) , (cid:101) µ i → (cid:101) µ i + i − N (cid:15) . (3.17)As a result, the one-loop determinant of the hypermultiplet in the fundamental represen-tation is modified as N (cid:89) i,j =1 Γ ( a i + µ j | (cid:15) , (cid:15) ) Γ ( − a i − µ j + (cid:15) + (cid:15) | (cid:15) , (cid:15) ) → N (cid:89) i,j =1 Γ (cid:16) a i + µ j + N − i − j +1 N (cid:15) | (cid:15) , (cid:15) N (cid:17) Γ (cid:16) − a i − µ j + (cid:15) + i + j − NN (cid:15) | (cid:15) , (cid:15) N (cid:17) . (3.18)Averaging over the finite group Z N as in (B.17), in the presence of a full surface opera-tor, the one-loop determinant of hypermultiplet in the fundamental representation can bewritten as Z hm,fund1-loop [1 N ] = Υ (cid:16) a i + µ j + (cid:108) N − i − j +1 N (cid:109) (cid:15) | (cid:15) , (cid:15) (cid:17) . (3.19)After performing the same manipulation for the anti-fundamental representation, the one-loop determinant of the the SU( N ) SCFT with N F = 2 N in the presence of a full surfaceoperator can be written as Z N F =2 N [1 N ] (3.20)= (cid:81) α ∈ ∆ + Υ( (cid:104) a, α (cid:105) + (cid:15) | (cid:15) , (cid:15) )Υ( −(cid:104) a, α (cid:105)| (cid:15) , (cid:15) ) (cid:81) p,q Υ (cid:16) (cid:104) a, h p (cid:105) + µ q + (cid:108) N − p − q +1 N (cid:109) (cid:15) | (cid:15) , (cid:15) (cid:17) Υ( −(cid:104) a, h p (cid:105) − (cid:101) µ q + (cid:6) p − qN (cid:7) (cid:15) | (cid:15) , (cid:15) ) . When the correspondence between the instanton partition function of the the SU( N )SCFT with N F = 2 N and the (cid:98) sl ( N ) conformal block part of the four point function (cid:104) V j V κω N − V (cid:101) κω N − V (cid:101) j (cid:105) was checked in [25], the parameters between the 4d gauge theoryand SL( N, R ) WZNW model are identified with a(cid:15) = j + ρ , − (cid:15) (cid:15) = k + N , µ i (cid:15) = − κ N + (cid:104) j + ρ, h i (cid:105) , (cid:101) µ i (cid:15) = (cid:101) κ N − (cid:104) (cid:101) j + ρ, h i (cid:105) . (3.21)Using this identification, one can easily deduce the form of three-point function D ( j , κ ω N − , j ) (3.22)= A (cid:16) (cid:101) Υ k + N (1) (cid:17) N − (cid:101) Υ k + N ( − κ − (cid:81) α ∈ ∆ + (cid:101) Υ k + N ( −(cid:104) j + ρ, α (cid:105) ) (cid:101) Υ k + N ( −(cid:104) j + ρ, α (cid:105) ) (cid:81) Np,q =1 (cid:101) Υ k + N (cid:16) − κ N + (cid:104) j + ρ, h q (cid:105) + (cid:104) j + ρ, h p (cid:105) − (cid:108) N − p − q +1 N (cid:109) ( k + N ) (cid:17) . Here we scale the momenta j and κ by two and there are trivial sign differences from [25] due to thenotation change. – 27 –ubsequently, the form of reflection coefficient can be obtained from the three-point func-tion B ( j ) = D ( j, , j ∗ ) = A (cid:81) α ∈ ∆ + γ (cid:16) (cid:104) j + ρ,α (cid:105) k + N (cid:17) . (3.23)In fact, the relevant part of the four-point correlation function of SL( N, R ) WZNW modelcan be written as D ( j , κ ω N − , j ) D ( j ∗ , (cid:101) κ ω N − , (cid:101) j ) B ( j ) (3.24)= A (cid:89) α ∈ ∆ + (cid:101) Υ k + N (cid:0) (cid:104) j + ρ, α (cid:105) − ( k + N ) (cid:1) (cid:101) Υ k + N (cid:0) − (cid:104) j + ρ, α (cid:105) (cid:1)(cid:89) p,q (cid:34) (cid:101) Υ k + N (cid:0) − κ N + (cid:104) j + ρ, h p (cid:105) + (cid:104) j + ρ, h q (cid:105) − (cid:108) N − p − q +1 N (cid:109) ( k + N ) (cid:1)(cid:101) Υ k + N (cid:0) − (cid:101) κ N − (cid:104) j + ρ, h p (cid:105) + (cid:104) (cid:101) j + ρ, h q (cid:105) − (cid:6) p − qN (cid:7) ( k + N ) (cid:1)(cid:35) − , which is equivalent to (3.20) upon the identification (3.21) of the parameters. Furthermore,the corresponding part of the one-point correlation function on a torus is equal to D ( j, κ ω N − , j ∗ ) B ( j ) = A (cid:89) α ∈ ∆ + (cid:101) Υ k + N (cid:0) (cid:104) j + ρ, α (cid:105) − ( k + N ) (cid:1) (cid:101) Υ k + N (cid:0) − (cid:104) j + ρ, α (cid:105) (cid:1)(cid:101) Υ k + N (cid:0) − κ N + (cid:104) j + ρ, α (cid:105) − ( k + N ) (cid:1) (cid:101) Υ k + N (cid:0) − κ N − (cid:104) j + ρ, α (cid:105) (cid:1) . (3.25)By the identification (2.44) of the parameters, this corresponds to the one-loop determinant(2.73) of the N = 2 ∗ theory. When N = 2, it is easy to see that (3.22) and (3.23) reduce to(3.13) and (3.11), respectively. This confirms that, when a full surface operator is inserted,a one-loop determinant of an SU(2) N = 2 gauge theory coincides with a product of thethree-point functions of SL(2 , R ) WZNW model. Nonetheless, the one-loop determinant ofthe 4d gauge theory cannot determine the coefficients A and A so that it is importantto obtain these coefficients by studying SL( N, R ) WZNW model directly. The study of the AGT relation with a surface operator that we have implemented raisesseveral questions. An obvious direction for future work is to study the correlation functionsof SL( N, R ) WZNW model. Although the gauge theory side has been investigated to someextent, SL( N, R ) WZNW model has not been explored at all. In particular, the immediateproblem left in this paper is to determine the coefficients A in (3.22) and A in (3.23) ofSL( N, R ) WZNW model as well as the prefactor f ( t, q ) in (2.45). It is desirable to obtaina better comprehension of the effect of the K operator.In this paper, we study only the pure Yang-Mills and the N = 2 ∗ theory with asurface operator. The extensive study is needed to provide more complete microscopicdescriptions of co-dimension two surface operators in terms of an N = (2 , 2) GLSM coupledto a 4d N = 2 theory as done for co-dimension four surface operators [10]. Since the AGT– 28 –elation tells us that an instanton partition function with a full surface operator obeys aKZ equation, the quantum connection for the Higgs branch of the 2d theory on the supportof the surface operator can be obtained by a certain limit of the KZ equation. For example,in the case of the SU( N ) SCFT with N F = 2 N , the J -function of the Higgs branch of the2d theory should become an eigenfunction of the Painlev´e VI Hamiltonian [45].It is intriguing to study K-theoretic J -functions [40] in terms of N = 2 gauge theorieson S × S . K-theoretic vortex partition functions (a.k.a. holomorphic blocks) [69–71]should compute K-theoretic J -functions of the Higgs branches of 3d N = 2 gauge theo-ries. It is well-known that the K-theoretic J -function of the complete flag variety becomesan eigenfunction of the q -difference Toda operator [40]. Recently, it is shown that theK-theoretic J -function of the cotangent bundle of the complete flag variety is actually aneigenfunction of a certain Macdonald difference operator [72]. Hence, it is important toextend these results to the infinite-dimensional version, namely, to find q -difference opera-tors of the 5d instanton partition functions with a full surface operator, which should belinked to q -KZ equations [73, § N = 2 gauge theory with Chern-Simons term is related to equivariantquantum K-theory of the tautological bundle of a Grassmannian. Further study is requiredto examine this relationship in order to clarify it.Another important problem concerns the relation between co-dimension two and foursurface operators. The Liouville correlation functions with appropriate number of degener-ate field insertions correspond to SL(2 , R ) WZNW correlation functions [75, 76], which canbe thought of the correspondence between co-dimension two and four surface operators inSU(2) gauge theories. Nevertheless, the relation in higher rank gauge theories is not under-stood at all. Since the W -algebras are complicated, it would be more amenable to examinethe relation by using the microscopic description of surface operators by a coupling of the4d theories to 2d gauge theories. Acknowledgement The author is indebted to Jaume Gomis for suggesting this project in 2012. Since then,he has benefited through discussion with various people at various occasions. He wouldlike to thank F. Benini, G. Bonelli, H-Y. Chen, B. Dubrovin, D. Gaiotto, V. Ginzburg, D.Honda, K. Hosomichi, Bumsig K., Hee-Cheol. K., K. Maruyoshi, Sunjay L., V. Pestun,S. Shadrin, R. Suzuki, Y. Tachikawa, A. Tanzini, J. Teschner, P. Vasko, Y. Yamada, forvaluable discussions and correspondences. The preliminary versions of the results werepresented in “ N = 2 JAZZ workshop 2012” at McGill University, Algebra and Geometryseminar at University of Amsterdam and the workshop “Quantum Curves and QuantumKnot Invariants” at Banff so that the author deeply appreciates J. Seo, G. van der Geerand M. Mulase, respectively, for the kind invitations and their warm hospitality. He wassupported by an STSM Grant from the COST action MP1210 for the stay at SISSA sothat he is grateful to both the COST action and SISSA for the support and hospitality.During the stay at SISSA, the result in Appendix A has been obtained with A. Sciarappaand J. Yagi so that he would like to express his special thanks to them. The work of S.N. is– 29 –artially supported by the ERC Advanced Grant no. 246974, “Supersymmetry: a windowto non-perturbative physics” . A Instanton partition function with surface operator In this appendix, we provide contour integral expressions for the Nekrasov instanton par-tition function of the chain-saw quiver by making use of the S partition functions as donein [77]. The result in this appendix has been obtained with Antonio Sciarappa and JunyaYagi.A D-brane engineering of the N = 2 U( N ) pure Yang-Mills is provided by a stackof fractional N D3-branes at the singular point of the orbifold geometry C / Z . Thenon-perturbative instanton contributions are indeed encoded by D(-1)-branes [78]. In par-ticular, the open string sectors of the D(-1)-D3 system provides the ADHM description ofthe instanton moduli space where the ADHM constraints are provided by the D-term andF-term equations. Hence, the Nekrasov partition function can indeed be computed fromthe D(-1)-branes point of view as a supersymmetric matrix integral [49, 79].A more sophisticated description of the construction has been given by resolving theorbifold geometry C / Z to T ∗ S . More specifically, the N = 2 U( N ) pure Yang-Millsis now engineered by N space-time filling D5-branes wrapped on S ⊂ T ∗ S in Type IIBbackground C × T ∗ S × C . Now the instanton contributions are encoded by D1-braneswrapped on S ⊂ T ∗ S . From the D1-branes perspective, the D1-D5 system is describedby an N = (2 , 2) GLSM on S which flows to the NLSM with the instanton moduli space.In fact, the exact partition function of this GLSM computed in [77] captures the S -finitesize corrections to the Nekrasov partition function. Furthermore, it was shown that thesecorrections encode the equivariant quantum cohomology of the instanton moduli space interms of Givental J -functions. The ordinary instanton partition function can be obtainedby taking the zero radius limit of S .Although the instanton partition function can be obtained by the D(-1)-D3 system,the D1-D5 system contains richer information. Hence, we shall compute the Nekrasovpartition function of the affine Laumon space by using the GLSM description. We considerType IIB background on C × ( C / Z M ) × T ∗ S × R with the D1-branes wrapping S andspacetime filling D5-branes wrapped on S . To illustrate the GSLM description of theD1-D5 system, let us briefly recall the chain-saw quiver. The chan-saw quiver M (cid:126)N,(cid:126)k islabelled by (cid:126)N = [ N , N , . . . , N M ] and (cid:126)k = [ k , · · · , k M ] where the vector spaces V and W are decomposed according to the representation under the Z M action, W = M (cid:77) I =1 W I , V = M (cid:77) I =1 V I , (A.1)with dim W I = N I , dim V I = k I . (A.2)In the language of branes, W I and V I are the Chan-Paton spaces of D5- and D1-braneswhich give rise to U( k I ) gauge symmetry and U( N I ) flavor symmetry in the GLSM.– 30 –ence, in the chain-saw quiver (Figure 1), the linear maps A I ∈ Hom( V I , V I ) and B I ∈ Hom( V I , V I +1 ) are realized from D1-D1 open strings, P I ∈ Hom( W I , V I ) from D1-D5 openstrings and Q I ∈ Hom( V I , W I +1 ) from D5-D1 open strings. The superpotential of thismodel is given by W = (cid:80) I Tr V I { χ I ( A I +1 B I − B I A I + P I +1 Q I ) } that yields the ADHMequations (2.7). Here the indices I are taken to be modulo M . In addition, the equivariantparameters of the torus action U(1) × U(1) N become the twisted masses of the chiralfields. Since the chiral fields A I and B I are transformed as the coordinate z and z (2.9)respectively under the spacetime rotation U(1) , their twisted masses are given by − (cid:15) and − (cid:15) M . It follows from the fact that the superpotential W is trivial under the equivariantaction that the chiral fields has the twisted mass (cid:15) = (cid:15) + (cid:15) M . Because the weight of theequivariant action on W I is given by the Cartan torus U(1) N of SU( N ) with the holonomyshift (2.10), the chiral fields P I and Q I − possess the twisted mass M ( s ) P I := − a s,I + I(cid:15) M and M ( s ) Q I − := a s,I − I(cid:15) M − (cid:15) , respectively. All in all, the data about the GLSM is summarizedin Table 1. χ I A I B I P I Q I − D-brane sector D1/D1 D1/D1 D1/D1 D1/D5 D5/D1gauge ( k I , k I + ) Adj ( k I , k I + ) k I k I − flavor I N I twisted mass (cid:15) = (cid:15) + (cid:15) M − (cid:15) − (cid:15) M − a s,I + I(cid:15) M a s,I − I(cid:15) M − (cid:15)R -charge 2 0 0 0 0 Table 1 . Data of GLSM for chain-saw quiver With these data, it is straightforward to write the Coulomb branch representation ofthe S partition function of the GLSM Z [ (cid:126)N , (cid:126)k ; a, (cid:15) , (cid:15) ] = 1 k ! . . . k M ! (cid:88) (cid:126)B ( I ) ∈ Z kI I =1 ,...,M (cid:90) M (cid:89) I =1 k I (cid:89) s =1 d ( rσ ( I ) s )2 π e − πir ˆ ξ I σ ( I ) s − i (cid:98) θ I B ( I ) s M (cid:89) I =1 k I (cid:89) s 0, the partition function receives the contribution only from (cid:101) Z , leaving– 32 –he generating function of the equivariant cohomology of the chain-saw quiver M (cid:126)N,(cid:126)k Z pure (cid:126)N,(cid:126)k = M (cid:89) I =1 k I !(2 πi(cid:15) ) k I (cid:73) M (cid:89) I =1 k I (cid:89) s =1 dφ ( I ) s (cid:81) N I j =1 ( φ ( I ) s − M ( j ) P I ) (cid:81) N I +1 j =1 ( φ ( I ) s + M ( j ) Q I ) M (cid:89) I =1 k I (cid:89) s =1 k I (cid:89) t (cid:54) = s φ ( I ) st φ ( I ) st + (cid:15) M (cid:89) I =1 k I (cid:89) s =1 k I +1 (cid:89) t =1 φ ( I ) s − φ ( I +1) t + (cid:15)φ ( I ) s − φ ( I +1) t + (cid:15) M . (A.6)The poles of this contour integral are classified by the N -tuple of Young diagrams (cid:126)Y =( Y s,I ) ( I = 1 , · · · , M, s = 1 , · · · , N I ) where the boxes in the j -th column of Y s,I contributeto the instanton number k I + j − . We verify that the residues match with the result [20,Mathematica file] in various values of ( (cid:126)N , (cid:126)k ).Furthermore, since the N = 4 ADHM data is given [49, § N = 2 ∗ theory ina similar manner. For brevity, we just present the final result: Z N =2 ∗ (cid:126)N,(cid:126)k = M (cid:89) I =1 ( (cid:15) − µ adj ) k I k I !(2 πi(cid:15) µ adj ) k I (cid:73) M (cid:89) I =1 k I (cid:89) s =1 dφ ( I ) s (cid:81) N I j =1 ( φ ( I ) s − M ( j ) P I + µ adj ) (cid:81) N I +1 j =1 ( φ ( I ) s + M ( j ) Q I − µ adj ) (cid:81) N I j =1 ( φ ( I ) s − M ( j ) P I ) (cid:81) N I +1 j =1 ( φ ( I ) s + M ( j ) Q I ) M (cid:89) I =1 k I (cid:89) s =1 k I (cid:89) t (cid:54) = s φ ( I ) st ( φ ( I ) st + (cid:15) − µ adj )( φ ( I ) st + µ adj )( φ ( I ) st + (cid:15) ) M (cid:89) I =1 k I (cid:89) s =1 k I +1 (cid:89) t =1 ( φ ( I ) s − φ ( I +1) t + (cid:15) )( φ ( I ) s − φ ( I +1) t + (cid:15) M − µ adj )( φ ( I ) s − φ ( I +1) t + (cid:15) M )( φ ( I ) s − φ ( I +1) t + (cid:15) − µ adj ) . (A.7)Let us conclude this appendix by mentioning a relation between quantum cohomologyof the affine Laumon space and quantum integrable system. It was found in [58] thatthere is the relation between the gl ( N ) intermediate long wave integrable system and thequantum cohomology of the ADHM instanton moduli space. More precisely, the authors of[58] shows that the effective twisted superpotential in the Landau-Ginzburg mirror of theGLSM with the standard ADHM instanton moduli space coincides with the Yang-Yangpotential of the gl ( N ) intermediate long wave integrable system [81].Thus, to see quantum integrable structure behind the quantum cohomology of theaffine Laumon space, we can perform the same analysis done in § ( I ) s ≡ σ ( I ) s − i B ( I ) s r , (A.8)we can obtain the effective twisted superpotential of the Landau-Ginzburg mirror of thechain-saw quiver by taking the large radius limit of (A.3): Z [ (cid:126)N , (cid:126)k ; a, (cid:15) , (cid:15) ] ∼ k ! . . . k M ! (cid:90) M (cid:89) I =1 k I (cid:89) s =1 d Σ ( I ) s π (cid:12)(cid:12)(cid:12) Q (Σ) e − (cid:102) W eff (cid:12)(cid:12)(cid:12) , (A.9)– 33 –here the measure is written as Q = M (cid:89) I =1 k I (cid:89) s,t =1 s (cid:54) = t k I +1 (cid:89) u =1 N I (cid:89) j =1 N I +1 (cid:89) (cid:96) =1 (Σ ( I ) st )(Σ ( I ) s − Σ ( I +1) u + (cid:15) )(Σ ( I ) st − (cid:15) )(Σ ( I ) s − Σ ( I +1) u + (cid:15) M )(Σ ( I ) s + M ( j ) P I )(Σ ( I ) s − M ( (cid:96) ) Q I ) , (A.10)and the effective twisted superpotential is given by (cid:102) W eff = M (cid:88) I =1 k I (cid:88) s =1 (cid:34) − (2 πξ ( I ) − iθ ( I ) )( ir Σ ( I ) s ) (A.11)+ k I +1 (cid:88) u =1 − (cid:36) (cid:16) ir (Σ ( I ) s − Σ ( I +1) u + (cid:15) ) (cid:17) + (cid:36) (cid:16) ir (Σ ( I ) s − Σ ( I +1) u + (cid:15) M ) (cid:17) + k I (cid:88) t =1 (cid:36) (cid:16) − ir (Σ ( I ) st − (cid:15) ) (cid:17) + N I (cid:88) j =1 (cid:36) (cid:16) − ir (Σ ( I ) s + M ( j ) P I ) (cid:17) + N I +1 (cid:88) ( (cid:96) =1 (cid:36) (cid:16) ir (Σ ( I ) s − M (cid:96) ) Q I ) (cid:17) (cid:35) . It would be interesting to find the quantum integrable system whose Yang-Yang potentialcoincides with (A.11). For instance, in the case of N = 2 and [1 , 1] partition, the vacuumequation exp (cid:32) ∂ (cid:102) W eff ∂ ( ir Σ ( I ) s ) (cid:33) = 1 , (A.12)leads to the Bethe equation k (cid:89) t (cid:54) = s (Σ (1) s − Σ (1) t − (cid:15) )(Σ (1) s − Σ (1) t + (cid:15) ) k (cid:89) t =1 (Σ (1) s − Σ (2) t − (cid:15) )(Σ (1) s − Σ (2) t + (cid:15) )(Σ (1) s − Σ (2) t + (cid:15) )(Σ (1) s − Σ (2) t − (cid:15) ) = ± e − πξ (1) + iθ (1) (Σ (1) s − M Q )(Σ (1) s + M P ) k (cid:89) t (cid:54) = s (Σ (2) s − Σ (2) t − (cid:15) )(Σ (2) s − Σ (2) t + (cid:15) ) k (cid:89) t =1 (Σ (2) s − Σ (1) t − (cid:15) )(Σ (2) s − Σ (1) t + (cid:15) )(Σ (2) s − Σ (1) t + (cid:15) )(Σ (2) s − Σ (1) t − (cid:15) ) = ± e − πξ (2) + iθ (2) (Σ (2) s − M Q )(Σ (2) s + M P ) . (A.13) This can be interpreted as the spin version of the Bethe ansatz equation for the interme-diate long wave integrable system [58, 81]. B One-loop determinants In this appendix, we shall elaborate the computation of one-loop determinants on theorbifold space C × ( C / Z N ). We start with a brief review of the one-loop computationsusing the Atiyah-Singer equivariant index theorem. For more detail, we refer the reader to[4, 5, 82].The exact partition functions of N = 2 supersymmetric Yang-Mills theories on S b canbe evaluated by applying supersymmetric localization. The value of an infinite-dimensionalfunctional integral is invariant under the deformation S → S + t ˆ Q ˆ V of the action S bya ˆ Q -exact term where ˆ Q = Q + Q BRST is the combination of a supercharge and a BRSToperator and ˆ V = V + V ghost is the combination of V = (Ψ , Q Ψ) and the gauge fixing– 34 –erm V ghost . In the limit of t → ∞ , the term t ˆ Q ˆ V dominates in the infinite-dimensionalfunctional integral, which renders the one-loop approximation at the BPS configurationsˆ Q ˆ V = 0: Z = (cid:90) ˆ Q ˆ V =0 Z , Z = (cid:20) det K fermion det K boson (cid:21) , (B.1)where K boson and K fermion are the kinetic operators ofˆ Q ˆ V = ( X boson , K boson X boson ) + ( X fermion , K fermion X fermion ) . (B.2)In this one-loop determinant, there occurs the cancellation between the bosonic and thefermionic fluctuations when they are paired by the supercharge Q . Hence it receives thecontribution only from the kernel and cokernel spaces of the transversal elliptic operator D that is the quadratic operator in ˆ V so that Z = (cid:20) det Coker D R det Ker D R (cid:21) , (B.3)where ˆ Q = R is the generator of the product SO(4) × SU( N ) × G F of the spacetime,guage and flavor symmetry. Therefore, the one-loop determinants can be obtained by theproduct of weights for the group action R on the kernel and cokernel spaces of D . This isencoded in the R -equivariant indexind D = tr Ker D e R − tr Coker D e R , (B.4)which can then be calculated from the equivariant Atiyah-Singer index theorem [83]. Sincethe index ind D is expressed as the sum over weights, we can convert the index into thedeterminant via (cid:88) j c j e w j ( (cid:15) ,(cid:15) ,a,m f ) → (cid:89) j w j ( (cid:15) , (cid:15) , a, m f ) c j , (B.5)where ( (cid:15) , (cid:15) , a, m f ) denote the equivariant parameters for SO(4) × SU( N ) × G F .For N = 2 supersymmetric gauge theories S b , the critical points ˆ Q ˆ V = 0 consist ofself-dual connections F + = 0 at the north pole and anti-self-dual connections F − = 0 atthe south pole so that we consider the equivariant index around these configurations [4].Let us first compute the index for the vector multiplet. Near the north pole, the operator D vm for the vector mutiplet is actually the complex of vector bundles associated withlinearization of the self-dual equation F + = 0 on R D SD : Ω d → Ω d + → Ω . (B.6)where d + is the composition of the de Rham differential and self-dual projection operator.Then, tensoring the adjoint representation of the gauge group with this complex, the– 35 –(1) × U(1) N -equivariant index for the vector multiplet can be computed by the Atiyah-Singer index theorem [83] in a simple wayind( D vm )( (cid:15) , (cid:15) , a ) = (1 + e i(cid:15) + i(cid:15) )(1 − e i(cid:15) )(1 − e i(cid:15) ) (cid:88) w ∈ adj e i (cid:104) a,w (cid:105) . (B.7)where w is a weight of the adjoint representation of SU( N ). At the south pole, we expand(B.7) in terms of negative powers of e i(cid:15) and e i(cid:15) , which results in the sign change ( (cid:15) , (cid:15) ) → ( − (cid:15) , − (cid:15) ). This can be absorbed into the reflection of weights w → − w . Hence, it givesrise to the identical contribution to the one-loop determinant. Then, using (B.5), one canwrite the one-loop determinant of the vector multiplet Z vm1-loop = (cid:89) α ∈ ∆ [Γ ( (cid:104) a, α (cid:105)| (cid:15) , (cid:15) ) Γ ( (cid:104) a, α (cid:105) + (cid:15) + (cid:15) | (cid:15) , (cid:15) )] − , (B.8)where the Barnes double Gamma function Γ ( x | (cid:15) , (cid:15) ) can be considered as the regularizedinfinite product Γ ( x | (cid:15) , (cid:15) ) ∝ ∞ (cid:89) n,m =0 ( x + m(cid:15) + n(cid:15) ) − . (B.9)The precise definition of the Barnes double Gamma function Γ ( x | (cid:15) , (cid:15) ) is given in theend of this section.Next, we shall evaluate the hypermultiplet contribution to the one-loop determinant.The transversal elliptic operator D hm for a hypermultiplet is the Dirac operator D Dirac thatmaps the spinor bundle S + of positive-chirality to the spinor bundle S − of negative-chirality D Dirac : S + → S − . (B.10)An equivariant index for a hypermultiplet depends on the representation of the gaugegroup. For a hypermultiplet in the adjoint representation, the Dirac complex is tensoredwith the adjoint bundle on which the G F = SU(2) flavor symmetry acts on. Therefore theU(1) × SU( N ) × G F equivariant index is given byind D hmadj ( (cid:15) , (cid:15) , a, m adj ) = − e ( i(cid:15) + i(cid:15) ) (1 − e i(cid:15) )(1 − e i(cid:15) ) ( e im adj + e − im adj ) (cid:88) w ∈ adj e i (cid:104) a,w (cid:105) . (B.11)where m adj is the equivariant parameter of the SU(2) flavor symmetry. Since the contribu-tion from the south pole is the same as that from the north pole, the one-loop determinantof a hypermultiplet in the adjoint representation is given by Z hm, adj1-loop = (cid:89) α ∈ ∆ Γ (cid:0) (cid:104) a, α (cid:105) + m adj + (cid:15) + (cid:15) | (cid:15) , (cid:15) (cid:1) Γ (cid:0) (cid:104) a, α (cid:105) − m adj + (cid:15) + (cid:15) | (cid:15) , (cid:15) (cid:1) . (B.12)The equivariant index for a hypermultiplet in an arbitrary representation R of thegauge group is rather subtle since there occurs an enhancement of a flavor group in somerepresentations. We refer the reader to [82] in which the detail analysis is provided. In– 36 –onclusion, for a hypemultiplet in an arbitrary representation R , the U(1) × SU( N ) × G F -equivariant index can be expressed asind D hm R ( (cid:15) , (cid:15) , a, m f ) = − e ( i(cid:15) + i(cid:15) ) (1 − e i(cid:15) )(1 − e i(cid:15) ) N F (cid:88) f =1 (cid:88) w ∈ R (cid:16) e i (cid:104) a,w (cid:105)− im f + e − i (cid:104) a,w (cid:105) + im f (cid:17) . (B.13)where N F mass parameters m f with f = 1 , . . . N F parametrizes the Cartan subalgebra of G F . Therefore, the one-loop determinant of a hypermultiplet in a representation R can beexpressed as Z hm R = N F (cid:89) f =1 (cid:89) w ∈ R Γ (cid:0) (cid:104) a, w (cid:105) − m f + (cid:15) + (cid:15) | (cid:15) , (cid:15) (cid:1) Γ (cid:0) −(cid:104) a, w (cid:105) + m f + (cid:15) + (cid:15) | (cid:15) , (cid:15) (cid:1) . (B.14)Since the instanton partition functions with a full surface operator can be obtained byapplying the localization method to the instanton moduli space on the orbifold space C × ( C / Z N ), it is reasonable to expect that the one-loop determinant can be also computed bythe orbifold procedure. Due to the orbifold space C × ( C / Z N ), we need to take the fractionalequivariant parameter (cid:15) → (cid:15) N (2.9), and the coulomb (2.10) and mass parameters (3.17)with holonomy shift. Hence, the part of a one-loop determinant that takes the formΓ ( x | (cid:15) , (cid:15) ) on C is generally altered in the following way:Γ ( x ( a, m f , (cid:15) , (cid:15) ) | (cid:15) , (cid:15) ) → Γ (cid:16) ˜ x ( a, m f , (cid:15) ) + I(cid:15) N (cid:12)(cid:12)(cid:12) (cid:15) , (cid:15) N (cid:17) (B.15)Then, its Z N -invariant part becomes the one-loop determinant on C × ( C / Z N ). To takethe Z N -invariant part, it is easy to use the index. Writing t = e i(cid:15) /N , the index thatcorresponds to the right hand side of (B.15) is g ( t ) = e i ˜ x t I − t . (B.16)The Z N -invariant part can be taken by averaging over the Z N group1 N N − (cid:88) k =0 g ( ω k t ) = e i ˜ x t (cid:100) IN (cid:101) N − t N , (B.17)where ω = exp(2 πi/N ) is the N -th root of unity and (cid:100) x (cid:101) denotes the smallest integer ≥ x .Subsequently, the one-loop determinant on C × ( C / Z N ) can be written as Z [ C × ( C / Z N )] = Γ (cid:16) ˜ x ( a, m f , (cid:15) ) + (cid:6) I(cid:15) N (cid:7) (cid:12)(cid:12)(cid:12) (cid:15) , (cid:15) (cid:17) . (B.18)For concrete illustration, let us show simple examples in the case of C × ( C / Z ). Thefractional equivariant parameter (cid:15) and the holonomy shift generally ends up with theBarnes double gamma function Γ ( x | (cid:15) , (cid:15) ) whose pole structure is depicted in Figure 5.Roughly speaking, we need to take the even modes from them. For instance, the evenmodes can be read off by averaging over the Z groupΓ (cid:0) x + (cid:15) | (cid:15) , (cid:15) (cid:1) → Γ ( x + (cid:15) | (cid:15) , (cid:15) ) 12 (cid:20) t − t + ( − t ) − ( − t ) (cid:21) = t − t – 37 – (cid:0) x + (cid:15) | (cid:15) , (cid:15) (cid:1) → Γ ( x + (cid:15) | (cid:15) , (cid:15) ) 12 (cid:20) t − t + ( − t )1 − ( − t ) (cid:21) = t − t Γ (cid:0) x | (cid:15) , (cid:15) (cid:1) → Γ ( x | (cid:15) , (cid:15) ) 12 (cid:20) − t + 11 − ( − t ) (cid:21) = 11 − t Γ (cid:0) x − (cid:15) | (cid:15) , (cid:15) (cid:1) → Γ ( x | (cid:15) , (cid:15) ) 12 (cid:20) t − − t + ( − t − )1 − ( − t ) (cid:21) = 11 − t . (B.19) (cid:15) (cid:15) Figure 5 . The distribution of poles of Γ ( x | (cid:15) , (cid:15) ). Only poles with black color are Z -invariant. Let us conclude this section by providing the definitions of the special functions thatappear in this paper. The Barnes double Gamma function Γ ( x | (cid:15) , (cid:15) ) is defined byΓ ( x | (cid:15) , (cid:15) ) := exp (cid:20) dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ζ ( s ; x | (cid:15) , (cid:15) ) (cid:21) , (B.20)where the double zeta function is provided as ζ ( s ; x | (cid:15) , (cid:15) ) = (cid:88) m,n ( m(cid:15) + n(cid:15) + x ) − s = 1Γ( s ) (cid:90) ∞ dtt t s e − tx (1 − e − (cid:15) t )(1 − e − (cid:15) t ) . (B.21)In this paper, we also use the Upsilon function which is the product of the Barnes doubleGamma functions Υ( x | (cid:15) , (cid:15) ) := 1Γ ( x | (cid:15) , (cid:15) )Γ ( (cid:15) + (cid:15) − x | (cid:15) , (cid:15) ) , (B.22)and therefore it obeys Υ( x | (cid:15) , (cid:15) ) = Υ( (cid:15) + (cid:15) − x | (cid:15) , (cid:15) ) . (B.23)Besides, it admits the following line integral representationlog Υ( x | (cid:15) , (cid:15) ) = (cid:90) ∞ dtt (cid:34) ( (cid:15) + (cid:15) − x ) e − t − sinh ( (cid:15) + (cid:15) − x ) t sinh( (cid:15) t ) sinh( (cid:15) t ) (cid:35) . (B.24)The characteristic property of the Upsilon function is the shift relationΥ( x + (cid:15) | (cid:15) , (cid:15) ) = (cid:15) x/(cid:15) − γ ( x/(cid:15) )Υ( x | (cid:15) , (cid:15) )Υ( x + (cid:15) | (cid:15) , (cid:15) ) = (cid:15) x/(cid:15) − γ ( x/(cid:15) )Υ( x | (cid:15) , (cid:15) ) , (B.25)which plays an important role in this paper.– 38 – J -function of cotangent bundle of partial flag variety N d M − · · · d d Figure 6 . Quiver diagram for N = (2 , ∗ GLSM whose Higgs branch is the cotangent bundle T ∗ Fl( (cid:126)d ) of a partial flag variety. The partial flag variety Fl( (cid:126)d ) = Fl( d , · · · , d M − , d M = N ) is an increasing sequenceof linear subspaces of C N ⊂ C d ⊂ · · · ⊂ C d M − ⊂ C d M = C N . (C.1)Thus, the GLSM given in Figure 6 flows to NLSM with T ∗ Fl( (cid:126)d ). As in § J -function of T ∗ Fl( (cid:126)d ) from the S partition function of the GLSM: J [ T ∗ Fl( (cid:126)d )] = (cid:88) (cid:126)k ( I ) M − (cid:89) I =1 z | k ( I ) | I M − (cid:89) I =1 d I (cid:89) s (cid:54) = t (1+ (cid:126) − H ( I ) st + (cid:126) − m ) k ( I ) s − k ( I ) t ( (cid:126) − H ( I ) st ) k ( I ) s − k ( I ) t M − (cid:89) I =1 d I (cid:89) s =1 d I +1 (cid:89) t =1 ( (cid:126) − H ( I ) s − (cid:126) − H ( I +1) t − (cid:126) − m ) k ( I ) s − k ( I +1) t (1+ (cid:126) − H ( I ) s − (cid:126) − H ( I +1) t ) k ( I ) s − k ( I +1) t d M − (cid:89) s =1 N (cid:89) t =1 ( (cid:126) − H ( M − s − (cid:126) − H ( M ) t − (cid:126) − m ) k ( M − s (1+ (cid:126) − H ( M − s − (cid:126) − H ( M ) t ) k ( M − s . (C.2)Here we identify H ( I ) s ( s = 1 , ..., d I ) with Chern roots to the duals of the universal bundles S I : 0 ⊂ S ⊂ S ⊂ · · · ⊂ S M − ⊂ S M = C N ⊗ O Fl N . (C.3)Furthermore, the Higgs branch formula of the vortex partition function can be written as Z v [ T ∗ Fl( (cid:126)d )] = (cid:88) (cid:126)k ( I ) M − (cid:89) I =1 z | k ( I ) | I M − (cid:89) I =1 d I (cid:89) s (cid:54) = t (1 − (cid:126) − a st + (cid:126) − m ) k ( I ) s − k ( I ) t ( − (cid:126) − a st ) k ( I ) s − k ( I ) t (C.4) M − (cid:89) I =1 d I (cid:89) s =1 d I +1 (cid:89) t =1 ( − (cid:126) − a st − (cid:126) − m ) k ( I ) s − k ( I +1) t (1 − (cid:126) − a st ) k ( I ) s − k ( I +1) t d M − (cid:89) s =1 N (cid:89) t =1 ( − (cid:126) − a st − (cid:126) − m ) k ( M − s (1 − (cid:126) − a st ) k ( M − s . With the identification d I = (cid:80) IJ =1 N J , this can be regarded as k M = 0 specialization ofthe instanton partition function (A.7) Z v [ T ∗ Fl( (cid:126)d )]( z I , a i , m, (cid:126) ) – 39 – (cid:88) k , ··· ,k M − (cid:16) M − (cid:89) I =1 z k I I (cid:17) Z N =2 ∗ (cid:126)N,k , ··· ,k M − ,k M =0 ( a i , µ adj = m − (cid:126) , (cid:15) = (cid:126) ) . (C.5)Among partial flag varieties, the projective space P N − and the Grassmannian Gr( r, N )play a distinctive role since they are particularly simple. Hence, we write the J -functionsof their cotangent bundles explicitly. 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