JHEP03(2018)192 - R gauge d Springer d,e,f March 30, 2018 January 5, 2018 : : February 23, 2018 : , , ) when the screenings n b b Received Published gl ( Accepted q,t Alexei Morozov, U c,d,e,f ) and . The correlation functions of these 1 n b b gl Published for SISSA by A ( https://doi.org/10.1007/JHEP03(2018)192 q,t to generic quantum toroidal algebras. A U k )-KZ equations. ) g b ( q, t q [email protected] d,g,h U , [email protected] Andrei Mironov, , a,b )-Knizhnik-Zamolodchikov (KZ) equation, which features the . 3 q, t Hiroaki Kanno, [email protected] and Yegor Zenkevich , a a 1712.08016 The Authors. Conformal and W Symmetry, Conformal Field Theory, Supersymmetric c

We describe the general strategy for lifting the Wess-Zumino-Witten model , [email protected] -KZ equations for quantum toroidal algebra and ) We also present an important application of the DIM formalism to the study of 6 Generalizing the construction of the intertwiner (refined topological vertex) of the [email protected] [email protected] National Research Nuclear University MEPhI, Moscow 115409, Russia Dipartimento di Fisica, Universit`adi Milano-Bicocca, Piazza della Scienza 3,INFN, I-20126 sezione Milano, di Italy Milano-Bicocca, I-20126E-mail: Milano, Italy Nagoya, 464-8602, Japan KMI, Nagoya University, Nagoya,Lebedev 464-8602, Physics Japan Institute, Moscow 119991,ITEP, Russia Moscow 117218, Russia Institute for Information Transmission Problems, Moscow 127994, Russia Graduate School of Mathematics, Nagoya University, q, t b c e g d a h f ArXiv ePrint: Open Access Article funded by SCOAP and periodicity properties of themodels gauge theories providing are solutions neatly to explained the by the ellipticKeywords: network ( matrix Gauge Theory, Topological Strings operators satisfy the ( matrix. The matching withspace the is Nekrasov worked function out for explicitly. the instanton counting on thetheories ALE described by the double elliptic integrable systems. We show that the modular nearly exhaustive presentation is givendo for both not exist and thushypergeometric all type the integrations/summations. correlators are purely algebraic, i.e. do notDing-Iohara-Miki include (DIM) additional algebra, wesentations obtain of the the intertwining quantum operators toroidal of algebra the of Fock type repre- Abstract: from the level of one-loop Kac-Moody Nekrasov partition functions on ALE spaces Hidetoshi Awata, Kazuma Suetake ( JHEP03(2018)192 ) 1 b b gl ( q,t U 30 8 7 3 34 37 2 13 11 gauge theories 25 d 49 15 22 53 26 19 16 51 35 ) ) and Fock representation – i – n n b b gl b b gl ( ( d , ) with general central charge d 31 q , 1 q U 5 b b gl U 45 ( )-KZ equation for unit central charge and its alge- 5 1 q,t 2 q, t -matrix U 37 47 5 )-KZ for general central charge 2 R 46 40 q, t -theoretic Nekrasov function for ALE space ) theory K )-KZ equations N q, t braic solutions 48 6.2.1 U(1)6.2.2 theory U( 1.1.2 Non-Abelian ( 1.1.3 Modular and periodic properties of1.2.1 6 DIM1.2.2 algebra ( 1.1.1 Abelian ( )-KZ equation for q, t 6.1 Adjoint mass6.2 shift Modular transformations 5.2 Relation to 5.3 Level one5.4 KZ equation for Nekrasov function as algebraic solutions to KZ equation network matrix model 4.4 Network matrix4.5 model and screening Abelianization operator of the DIM intertwiner 5.1 Shift operator and 3.2 Vertex operators and horizontal representation 4.1 Structure of4.2 the intertwining operator Vacuum component4.3 of the intertwiner Zero mode part and intertwining relations 3.1 Vertical representation and color selection rule 1.2 Tactics of computations 1.1 Strategic objectives A Combinatorics of the normalizationB factor Zero mode factor ofC the Recursion intertwiner relation for Nekrasov function 7 Discussion 6 Modular and periodic properties of double elliptic systems from 5 Level one KZ equation and Nekrasov function for ALE space 4 Construction of the intertwining operator 2 ( 3 Quantum toroidal algebra Contents 1 Introduction and outline JHEP03(2018)192 ]. 46 – )-KZ ] for a 40 31 q, t ] basing 85 to the low- 1 ] 7 – ) gauge theories 5 ], see also [ , we use solutions N ]. 30 6 – U( 68 – derived in [ 28 d 58 2 T , we describe the quantum ] and its various generaliza- 3 ] (WZW), which is the theory 70 ). In section , 39 – 69 , we propose a generalization of the 35 5.50 and therefore are once again at the 2 57 ]. This paper is a continuation of our ] and studied in [ 2 )–( ] 27 53 – 26 , 5.49 47 25 – 1 – that contain various technical details. 57 compactified on torus 55 E – ]. m 24 A – Mathematically the problem is that of the full-fledged 8 3 58 ]. 57 ] (CFT) are connected by the AGT relations [ – 4 ]. – 54 1 57 , we construct the operators that intertwine these representations. 4 ) with arbitrary central charge. In section 1 b b gl ( ] for these theories in the Ω-background. q,t 34 U – ], which we also refer to as DIM. In this paper, we focus on a small corner of , we derive the (level one) Knizhnik-Zamolodchikov equation for the correlation 32 84 5 are followed by appendices – 7 71 With the help of all the technical exercises, we would like to demonstrate a simple The plan of the paper is the following: in section )-KZ for For various AGT-related issues, seeIntegrability [ behind these theories was discoveredIt in goes [ in parallel with other important efforts in the same direction, see [ E.1 Quotients ofE.2 the Young diagram Decomposing characters 1 2 3 q, t idea: that the DIMtool, intertwiner which formalism can is find not its just use an in interesting gauge toy theories, but as an well as important review in and other [ related fields. to the elliptic KZwith equations adjoint to hypermultiplet obtain of modularon properties mass the of description the insection 6 terms of double elliptic integrable systems. Concluding remarks in ( toroidal algebra and its verticalthe and paper. horizontal (Fock) In representations that section In we section deal with in functions of these intertwining operators, eqs. ( representation theory of thetions Ding-Iohara-Miki [ algebra [ this very broad area andequations describe introduced two in generalizations [ and one application of the ( tions are straightforwardly handledNowadays by problem various is versions the of liftingalgebras of the (which this free-field corresponds model methods to torelations) [ the lifting and level from development of 4d of adequatelyscribe an to extended the efficient toroidal 6d resulting generalization on “network of the matrixstudy the Yang-Mills models” of free-field side [ this formalism of problem to the in de- AGT [ results of this ishigher-dimensional a generalizations new on interest the incentral Yang-Mills various personage side. extensions is and the In Wess-Zumino-Wittenwith deformations, conformal theory an needed field [ extended to theory, Kac-Moody match the including symmetry, of Liouville which and all Toda other theories, important are (if various not reductions. all) models, This model and its reduc- 1 Introduction and outline Conformal field theories [ energy supersymmetric Yang-Mills theoriescenter [ of attention in modern theoretical and mathematical physics. One of the immediate E From colored Young diagrams to quotients D Symmetry of Nekrasov function JHEP03(2018)192 - of R n 1). In ). The , ). The -matrix n ALE R -matrix in b b gl k, N ( R q,t ]. U 87 )-KZ equations, ) over the boxes z ( i q, t E ], where the )-KZ equation, we first 57 -component of intertwiner q, t λ ) is the “color selection” rules. -operator and the n ] and a review in [ T b b gl ] for their extensions. ( 103 – case is the appearance of the zero 68 q,t – -matrix and the Nekrasov partition 1 96 U . 66 b b gl R n Z )-KZ equations [ / is shifted according to the position of the 2 Its first generalization relaxes the condition z q, t C 4 ]. – 2 – 57 -shift of the argument of the intertwiner. The q 5 )-KZ equation is the much sought non-Abelian ver- ] can be used, and the KZ equations and [ 1). The intertwining relations for the intertwiners are 80 q ). In order to derive the ( q, t , n b b gl ( -difference of the operator product expansion (OPE) factor q case [ q,t ). )-KZ equation for unit central charge and its algebraic U 1 b b ) and (0 gl q, t 1.2 , where the argument ] and write down the ( λ ,N )-KZ for general central charge 56 , q, t 55 ] for the standard KZ and ) and then give some general description of the methods by which we are going 95 – )-KZ equation was introduced in [ , which is a resolution of the orbifold solutions 1.1 86 n q, t A ) can be expressed as the normal ordered product of the currents Once we obtain the intertwiners, we can introduce the In the remaining part of the introduction, we first describe these three objectives v See [ For solutions of various KZ equations, see original papers [ ( 4 5 λ agrees with the Nekrasov factor (thewe bi-fundamental contribution have to the a partition function), fundamentalfunction. relationship Basing on between this the relation, we can find explicit solutions to our ( a similar way to [ is featured as the connectionmatrix matrix can for be identified with of the intertwiners and is essentially diagonal. Since the OPE factor of the intertwiners modes are group algebra valued andcorrect their intertwining commutation relations. relation Another is new crucialSome aspect for combinatorial of obtaining arguments the for such rulesrelation are to required, the especially Nekrasov when we partitiontype establish function the for gauge theories on the ALE space same strategy asΦ in the of the Young diagram boxes. One ofmode the factor important in differences the with free the field realization of the horizontal Fock representation. The zero The second generalization of thesion, ( that forconstruct the the algebra intertwiners forcentral the charges, horizontal i.e. and (1 verticaldetermined Fock by representations the with unit coproduct structure of the quantum toroidal algebra the equations can also bescreening explicitly charges obtained. appear in The the solution answer. is algebraic, i.e.1.1.2 no integrals of Non-Abelian ( The ( on the central charge of theto “horizontal” be representation, the i.e. we Fock no space,vertical longer but representations require instead are this assume space still thatthis assumed it case, to has be the a Fock general modification spaces central of with charge the central ( charge KZ (0 equation is not hard to guess and the solution to (section to achieve them (section 1.1 Strategic objectives 1.1.1 Abelian ( JHEP03(2018)192 ) 1 . ], these . Since 1 2 S 109 – × n 107 -duality and the S ALE ) corresponding to the 1 b b gl ( )-brane diagram (figure q,t U p, q ]. We will explain this behavior 85 gauge theories d vertical lines. Notice that the picture of inter- of the Ω-background. Notice that we set the gauge theories with adjoint matter compactified – 3 – -dual. The ends of the lines should be identified t N d S the complex structure of the compactification torus , and the network of intertwiners, in figure and 1 q ]. Despite the recent important progress [ ) essentially factorize into products of noninteracting mixes n b b gl 106 can be also understood as the twisting parameters of the ( – q,q ⊥ 104 U Q , with respect to the Seiberg-Witten ( Q 2 π of the network diagram in the horizontal direction, and an extra sixth gauge theories and double elliptic systems under the d . These theories are, in a certain sense, the highest step in the hierarchy of 2 T ) intertwiners. The most complicated and interesting case of representations 1 b b gl ( n The “fugacities” The picture of DIM intertwiners corresponding to the double elliptic system is given Moreover, for the unrefined case with unit central charges, we actually demonstrate )-brane diagram associated with the gauge theory. The brane diagram for the case of compactification ,q n q p, q fibration giving the background forthem the and M-theory, equivariant which parameters hintspreferred at direction possible (determining duality the between vertical. coproduct structure However, and the thus final the answer intertwiners) for to the be character/partition function is independent of by the intersection of onetwiners horizontal is and rotated by usually given in theD5 literature. branes Of of course, Type this IIBwith does string each theory not other are change so the that answer the since picture NS5 is and essentially drawn on a two-dimensional torus. an elliptic fibration. In thetaking the algebraic trace language, of the the doublerepresentations. product compactification of The corresponds the network to intertwiners of over intertwiners both( is the modeled vertical after and the the Seiberg-Witten horizontal U(2) Type gauge IIB theory is shown in figure using the network matrix modelgauge of theories the Abelian in DIM question. algebra the Adding the adjoint matterdimension in the also gauge implies theory the“doubly corresponds compactified” to compactification network, of which the geometrically corresponds vertical to direction. a CY Thus, three-fold we with get the systems are still quite mysteriousbehavior and of require the more 6 explicitmodular description. transformations In of particular, thethe the compactification modular torus transformation is rule quitewith peculiar. the It complexified turns coupling out in that a very specific way [ on the torus gauge theories with eighttions supercharges, are for available. which Within the theelliptic Seiberg-Witten Seiberg-Witten integrable paradigm, and systems they Nekrasov with are solu- both describedthrough elliptic by coordinates functions the and [ double momenta entering the Hamiltonians with general central charges in non-Abelian DIM algebra is1.1.3 left for the Modular future. andTo periodic demonstrate properties the effectiveness of of 6 application DIM of formalism, network we matrix are models to going 6 to describe an important we consider only the setupare with still unit algebraic. central charges, all the solutions to the KZthat equations the intertwiners of U which turn out to be the Nekrasov functions for 5d gauge theories on JHEP03(2018)192 . ⊥ d ⊥ Q d elliptic Q count the ⊥ d ⊥ P d counts the states P U(2) gauge theory torus, but not for d βH 2 − and u , λ e 0) 2 , ⊥ z -duality transformation d ⊥ (1 S Q d Q 1) 1 , U(2) gauge theory with adjoint (1 , λ d 1 z compactification 0 Λ 2) 2 , 2 (1 , λ 2 P z z ⊥ Λ d ⊥ 1) 1 , P – 4 – d 1 (1 , λ P 1 m . The grading operators z P z 2 1 − is the mass of the adjoint field, Λ is the exponentiated com- a uP m ⊥ . The wavy (double wavy) lines are understood to be identified ⊥ d 2 Q T d Q )-brane web corresponding to the 6 2 0) is the exponentiated complex structure modulus of the compactification p, q , − 0 P (1 ⊥ -duality transformation corresponds to the modular transformation of S uQ compactified direction, where the trace is a lot harder to compute. However, is the Coulomb modulus, . Compactified network of intertwiners corresponding to the same 6 . Compactified ( . a 2 -duality. The T Using the intertwiner picture, we can also analyze periodicity properties of the partition vertical )-KZ equations. Since the result is expressed through the theta-functions, we can use S q, t as easily as the modular transformation. function. As an example considerunder moving the the trace). incoming The vertical linesThe lines necessarily around action have the to circle of pass (i.e. through grading the operators grading operators gives a shift in the positions of the incoming vertical it to effectively studystrategy works the only modular for properties modularthe transformation of on the the partitionthe function. Ofwe course, can this use the slicingdirection to invariance the of horizontal one. the partition This would function allow and us to safely analyze change the the preferred the preferred direction. Incan these be taken conventions, straightforwardly the giving a traceprevious combination over of investigations the theta-functions. we horizontal recall Moreover, from representation that our ( precisely this trace appeared as a solution to Figure 2 with adjoint hypermultiplet, asstates in of figure the Fock representationsin in a the quantum mechanical same partition way function. as the Boltzman factor with each other.branes: The parameters ofplexified the coupling, gauge and theory Λ torus are encoded in the distances between the Figure 1 hypermultiplet compactified on JHEP03(2018)192 ] 54 – (1.1) ] for a 50 Young 68 – N 66 ; we usually . t ) z , ( ) ) 2 and uz ,k ( q ) (0 2 k (e.g. a set of + is introduced in [ ⊗ F λ ,M ) k . In the horizontal direc- 1 ) are just the compositions ) k 1 ( b G z u/z vert ( ) 2 )Ψ( k 2 − z −→ F ,M ) 1 z k ( ( ) 2 ) carry the index ,k u ) ) and Ψ( ( ∗ λ (0 ) 2 u ( z −→ F ) ,M 2 ,Ψ – 5 – 1 ⊗ F ) λ k ,k )Ψ( ( ) u which is given in terms of operators acting in the (0 1 ( u F z ) ( F ] for various aspects of the DIM algebras and [ ) given by combinatorial formulas in some basis in an- ,M 84 1 ,M – , and two deformation parameters k 1 ( 2 k 69 ( k ] are: F F 57 ): and ): z is a quantum toroidal algebra (hence, two hats) with two central . We refer the reader to these papers for terminology and basic ( 1 z ) ) ( k 2 2 ) 2 ,k , k The intertwiners Ψ ,k 1 (0 k λ (0 ∗ λ ( ) Ψ Ψ g b b . The physical model, associated with it in just the same way as the ( n ] and used in [ q,t gl 88 U )-KZ equations network model = g q, t The basic ingredients of construction of the KZ equation for the network models Intertwiners. diagrams or a plaintion partition) space: which labels the element of the vertical representa- “Horizontal” representations infinite dimensional vector space “Vertical” representations other infinite dimensional space 6 For other related references, see [ • • • 6 -theory approach. In this paper, we concentratesetup, on we the can case supplement when the only items from algebraic the solutions list are above present. with some In more this concrete properties. K tion, they can beof easily operators. multiplied: Ψ( 1.2.2 ( They depend on the choice of the “vertical” coproduct ∆ WZW model is associatedand named with the Kac-Moodylogic. algebra U( generalizing [ 1.2.1 DIM algebra DIM algebra elements fixed to levels consider that it is actuallygauge theory. accompanied by a certain change in the complexified coupling1.2 of the Tactics of computations lines with respect to the outgoing vertical lines. Investigating this move in detail, we find JHEP03(2018)192 (1.6) (1.7) (1.9) (1.2) (1.3) (1.4) (1.5) (1.10) ) in the z ( i E -matrix. )-KZ equation for the R . . q, t ) ) . z z ) ( ( , , z µ λ ) ( 1 | ) (1.8) λ − T + pz z )Ψ ) ∅ T ( )) 2 h ) w w z q λ + z ( ( ( -matrix is to commute a pair of ( = ( , i λ µ Ψ λ − + ) λ R 1 -operators: − Ψ Ψ z T . − Ψ ) 1 ) T   |T 1 z )(Ψ − z )) ( − z z q q ∅ w w i z ( ( h pz ( )) i i   E ( λ + z 1 F F λ − ( ) ) − , λµ λµ identities. λ z z i − We need the commutation property as follows ( ( )) After we get the shift and commutation identi- R R T i i ∅ z | = Ψ – 6 – ( E E i ) = (Ψ 1 − = ) = ) = z − T RT T ( ) = ( i w w ) = ) = λ )) z ( ( + ∅ ( z z z | λ µ T ( ( ( λ i i + λ ) = ( − + − Ψ T )Ψ )Ψ z ) taken at certain discrete points specified by the Young T T z T 2 z z z ( ( ( q ( -operators. z∂ -operators. ( i µ λ p One way to obtain the i T T E Ψ T E ) over the boxes of the Young diagram. It is important that the -operators annihilate the vacuum: -operators. We can then move them to the ends of the string using We need to build up the shift operator, whose action on the inter- T T 1.10 to satisfy nontrivial -matrix. R  ), ( denote the creation and annihilation parts of the intertwiner. However, we T  1.9 -operators for the intertwiners are normal ordered products of the basic T Vacuum property of ties, we can act with theof shift a operator pair on of a the stringthe of commutation intertwiners and identities. get anensure insertion The that the last step in the derivation of the equation is to Diagonal intertwiners: The commutation can be done using the free boson formalism. also need Commutation of the To get these identities, we need first to find the non-Abelian This requirement by itself can be trivially satisfied, since we can simply write where Ψ Shift identity. twiner can be rewritten as the product of two Concretely for the non-Abelian DIM algebra, the intertwiner is built as a normal Namely, the short list of ingredients for the derivation of the ( 4. 3. 1. 2. -operators ( -operators in the non-Abelian case are still diagonal in the vertical Young diagram, so The T T that no extra sums over diagrams appear. product separately and then account for the normal ordering constants. We have where ordered product of elements diagram on the vertical leg. We can derive the shift identity for each operator non-Abelian DIM algebra is: JHEP03(2018)192 1) , (2.1) (2.3) -matrices. The R the usual solution contains an infinite ] contains an extra t q p 88 , = ) (2.2) ) 2 2 p   , z , z 1 2 1 1 2 1 z z can enter only as a shift of z z z z k ( ( k p p 2 )-KZ equation. This similarity 2 p )-KZ equation with the param- t q λ λ  1 1  λ q, t λ 2 q, t 2 ( λ G G )-KZ. Unfortunately, the intertwin- λ 1 1 λ   λ 2 q, t 1 1 2 G z G . z z pz k 0  z  1 elliptic ≥ ∂ 2 2 Y k k λ λ z . The vertical representations remain to be 1 1
 p λ λ t q 1 2 e e R z z -matrices featuring in the KZ equation are still R – 7 –  R ) with general central charge 1 , eq. (18)]. Obviously for f ) = ) = b b 2 2 gl 56 ( , z , z ) = 1 1 2 q,t z z ( ( , z U 2 2 1 λ λ z 1 1 ( λ λ 2 G G λ 1 1 2 z z λ ∂ ∂ G 1 2 z z p p with an arbitrary parameter k z ) can be found, e.g., in [ ∂ z k ( for the Fock space is recovered. The solution for general z p )-KZ equation for  λµ ). The shift operator determined by the first central charge of the horizontal 1 2 G z z encoding the central charges of the horizontal representation in the simplest case of -KZ equation for the conventional quantum affine algebra [ q, t  ,N q p 1 2 Having these two arguments, we can conjecture the ( Let us consider the combination of intertwiners similar to the Fock space case, but In this section, we will try to extend the central charge parameter to arbitrary values λ 1) Fock spaces. This implies that the 1 , λ G where product, which reminds us oflooks the solution mysterious to and the indeed might turn out to be only superficial. The solution up to a function independent of the Young diagrams is given by their arguments. eter two vertical incoming lines: with an arbitrarybecomes representation living on(0 the horizontal line.the The same (e.g. shift they operator are thus diagonal), and the new parameter and find the corresponding solutions toers the for Abelian general ( representationsas are that not for known, theseems so Fock to spaces. the give the solution However, only the cannotto way structure follow of be of introducing this found the route the as central and KZ charge easily investigate equation parameter the into is resulting it. very solutions We rigid for will and the try conformal blocks. conformal blocks of DIM areFock combinations spaces. of intertwiners These acting representations inor horizontal have (1 and definite vertical central chargesrepresentation of is therefore the fixed form and either reads (0 The parameter, which is missingis in the the central DIM charge caseconformal we of block. have the It considered “horizontal” so enters (highest the far. weight) shift This representation operator parameter running and in also the the shifts of the 2 ( JHEP03(2018)192 2 8 / 1 − . ) (3.2) (3.3) (3.4) (3.1) n ) and b ,j gl q/t , which d for other +1 , i = ( . q δ q 7             M + ], by introducing 1 . . . 0 0 ,j − 1 111 − ) for the equivariant , . . i 2 . . 0 1 0  δ . . ··· ( − 110 1) . . . , e − . . . 0 1 0 is the Kronecker delta − + 1) 1 . . . )  ··· . e j j j δ ) i,j . . 1 0 δ . . w 2). Explicitly the matrices ≡ ≡ ≡ . . ... − , ij i i i 1 a ) = ( . . ) coincides with that for the ( ( ( 1 0 = 2 − . . q . . q − (otherwise) n > q, t 1 ij − z, w 1 0 a , where − . . . ( 0 1 0 0 1 0 0 z w, w, w, d ,j , − 1 2 3 ij ij 1 q q q w, g = +1 − m             i (1) n d 3 − − − − δ = A z z z z − ],              , q ) = ( ,j ) and Fock representation 2 1 n q 82 − , i ) = b b ) has two deformation parameters z, w gl δ = – 8 – of type ( n w ( 81 ,M 2 d ij ij b b gl , = a A g ( q             q d ij , 1 1 ij U q ij . . . 0 0 , q m − − 1 U m m d − − . . 1 2 . . d 2 0 . . dq − ··· ) := − . . . with ) for a Lie algebra with a symmetrizable Cartan matrix. = . . . . z 0 g . . . 1 ··· 1 ( − q z, w q M t . . 1 0 ( . . U . . ij ... = − = 1), the structure function f ) := ( g 3 . . 1 0 1 2 . . d . . − − , and it is not known if the toroidal algebra for general affine algebra q, q z, w g b ( 1 0 1 2 = . . . 0 2 0 ij 1 − − g q ) can be regarded as the quantum affinization of the affine algebra n             ). or b b gl 2 = 1. We also use ( = / z, w 1 3 are given by ( ) . In this paper, we consider the generic case of ( =1 A q d ij n 2 , qt g q q M 1 U = ( q We introduce the structure function [ The quantum toroidal algebra d We use the Gothic letters for deformation parameters to keepIn ( general, we can define the quantum affinization based on the data of a quiver [ 7 8 and parameters for torus action, orand those for the Macdonald function. In the following, we identify Chevalley generators associated withfunction vertices and the corresponding Drinfeld currents with the structure with where we have defined quantum affinization of Thus It seems difficultaffine to Lie introduce algebras anallows analogue a of two the parameterparameter skew-symmetric is deformation. matrix trivial ( As we will see below, if the second deformation a skew-symmetric matrix modulo A For the future convenience,quantum we toroidal introduce algebras. here the basic definitions and notationare associated for with the the Cartan matrix 3 Quantum toroidal algebra JHEP03(2018)192 } n 0 z Z ∈ (3.5) (3.6) (3.7) \{ X n (3.12) (3.15) (3.16) (3.17) (3.18) (3.19) (3.20) Z ∈ ) =  z ) , r ( w Z δ ( − ∈ i )). z k K 2 ) / 1 . One is based on = 0 (3.10) z = 0 (3.11) w g − 2 . c  r  C , q , z 2 , z  , ( ) , / 2 2 ) k i, δ ! 1 2 z z r c/ − c/ ) (3.9) − 1 1 H ( c/ − z ∓ i z 2 1) C 1) C  ( ( z ) ) is defined by F ( ∓  q i,k r i z  i rc/ − i (1 ( (1 F  ⊗ E  K q q i, + i K Z q 3.12   ∈ ) ⊗ H ∆( ) = 0 (3.13) K k ) = 0 (3.14) X  , ) i  z w , , i z  1 z ( ( ) ) → ) + 1 ( =1 ∞ i −→ K 1 K i z z  r i X z z w + j ( ) 1 ) = )) ) r E 2 F C ) (3.8) ) c ) z z ) ( − − 1 1 i  z w K q w ( C ( ( ( i, ( w − ( − i ) i w C   i K ) ( j ( q j ( z +  i i H j δ z j K K c E F + c i ⊗  − E F − K K ,K q ) ) ) ) ) q 1 K ) q z ⊗ i,r ( 1 + − 1 w w, – 9 – ⊗ ,F ) ) q w, , ( ( ];  w, z z, w − i,j 2 k w, z z, w ( c ⊗ z z ) ( ( δ ( ( 2 q  2 ji j

− − z C ) 82 ji g ( ( ji ij ji ij z z c/ C q g K ⊗ g g g g ( ( + − i i − i i i,k c ,H exp q K q E F K . ( E The delta function in ( 1 c ) = ) + ) + ) + ) + ) = i,k  i )] = Z  ] and [ q i 9 w w w w w w ∈ w K ( ( ( ( ( ( ) = k X K ), which implies ∆( 81 ⊗ ( j j c )) = )) = )) = )) = ) are j j +  j j j q z z z z n F E F E ( ( ( ( ) 10 ) i i b b 3.15 K K gl ) = ) = −→ ,F z − +  ,F z i i = 1  ∆( ( F ) ) ). Note that there is the following change of the scaling of the ( z z E ( z ) ) d z i i 1 ( ( 2 , z z K K 2 i,k z 2 z i q ( ( F − (  ( i E C ∆( ∆( c/ ) E c/ E U z i ) −  i i  ∆( ∆( i ( K 1) between [ 1) E δ [ K K  K ∓ w, z ) z, w (1 i,r (1 ) ( ( q q 1 and ji ]; H ij ) = (index set of simple roots or vertices of the cyclic quiver),   e g e g z, w z ⊗ z, w 82 c (  Z i  i c c δ − q q K K q /n ( ( ) ) corresponding to a simple finite dimensional Lie algebra Z = ij ij g g 1 b ∈ g w, z z, w C i is a central element. It is convenient to employ the generating currents; ( ( c The generators of As is well known, there are two equivalent constructions of the (untwisted) affine Lie ji ij In this paper, we doRemember not the use redefinition the ( Serre relations. e g e g 9 10 where algebra The coproduct is defined by with appropriate Serre relations. and satisfies Heisenberg part which satisfy [ where and JHEP03(2018)192 . . i ). g b ` K − ) in . In i q g b A ) and ( (3.21) (3.23) (3.24) (3.22) , Q q z k ( q ]. Later U − i := κ 115 ,K – ) ) = ( z q and 112 ( , κ c + c i q q 1) with symmetrizable K ). We can also see ) j = 0). The other way, − g + 1) 1 j 1 − ≡ j 3.13 A (and a redefinition of the − z i ≡ 2 z ), we see the difference is d ( ≡ q ) with i i ), when ( ( − ). We can check that ( w − z , ( 1.2 1 ( ) j 1 k, ` , , z ψ , F ) and ( 1 ) ( ) z − + z i ) in the form of OPE relations of ( q ( . + i K 3.10 + ) i ) = 1 K ] section 3.11 K w − ) z ) . The one-dimensional central extension ( z 1 j w g z ( w ( − 2 111 ( ) for a Lie algebra E j ψ deformed Serre relations [ q (otherwise) j − g b
E ( q E 1 − = z q w 1 ) and ( ) imply ( 1 , 2 1 1 U ) has two central elements, − ) z − q – 10 – n
q − ) that of level ( ( z
i ) ( n z b b w gl ). Note that, in this realization of the quantum z 3.10 z w = E ψ ( + 2 b b i g gl 2 1 d ( q (
, q z − q K 1 1 q d ( z , 1 w ) q q ). We see the relation q z U U ψ − with the same argument. z w ). Similarly, the OPE relations of the same form with ) or U q q ( − 1 ψ ψ ψ z ( j q 1 − −  i E        ( ) of 3.10 ψ ). That is, Drinfeld and Jimbo originally defined q g b − i K g b ( ( q K ) = ) = q U U w w ) := ) or ( ( ) by the second deformation parameter z j j z ( ] that the same algebra is obtained by introducing the generating ( ) follow from similar relations for E E ψ  i ) ) z, w z z 116 ( ). In fact, if we compare these relations with those of the Drinfeld’s K ( ( ij 3.14 )). + + i i g is replaced by K K 3.14 0) with the corresponding one of affine type (det 3.15 ψ ) is obtained by combining the above two ways of “affinization”. Namely, if ) with g ] of the finite dimensional Lie algebra b b ( 1 w ) and ( A > q ( ) to ( − j U 3.8 3.11 ,F z, z ) being replaced by ) [ z The quantum toroidal algebra It is convenient to introduce the following rational function ), where ( C w z ( ( + i j i ⊗ K that ( F We will call a representation of imply the exchange relation ( In fact, as webasis labelled shall by see partitions below, are the described by matrix the elements function of the vertical representation in the to rewrite the commutationE relations ( (generalized) Cartan matrix (seeonly a for change example of [ Cartan part ( affine algebra, we usealgebra the Cartan matrix ofwe finite use the type. Cartan In matrixfrom of a ( affine sense, type the in quantumrealization the toroidal Drinfeld of realization, the we obtain quantum the affine relations algebra Analogously there arequantum two affine methods algebra) toterms obtain of the the quantum ChevalleyDrinfeld enveloping generators observed algebra with [ (or the functions the of the generatorsthe (the quantum Drinfeld affinization currents). This is called Drinfeld realization, or this approach, the affinetype Lie (det algebra is obtainedwhich by is replacing more the familiarg Cartan among matrix physicists, of employs finite theof loop the algebra loop (the algebra current with algebra) the additional grading operator gives the affine Lie algebra the Chevalley generators with the Serre relations determined by the Cartan matrix JHEP03(2018)192 | λ , and ). We 1 ) by a − z ( (1) n ) becomes  i A ≥ · · · K 3.9 is introduced 1 theory. When v  which simultane- j K ) can be enhanced . Originally Naka- λ n 11 theory of the quiver b b ) and their relations +1 gl ( n ) has two subalgebras Z d , n Z K q / /n b b gl 2 U ( Z d C ≥ · · · ≥ , ] the defining relations with q ∈ 1 U ). The horizontal subalgebra i λ 1) representation vertical and 121 n ( , – b sl  ( i = ( , so that the empty partition and ) representation is mapped to the q λ j 117 K U ) transformation. It is interesting to 1 type, the affine algebra is k, ` ∈ of the Fock space, Z , )  n } and ) λ A i, j λ 0 6= 0. As in the case of DIM algebra, there i, {| i ,F ), let 0 i, 1) representation, the Heisenberg part is com- – 11 – , E ). In the following computation, we assume that with ) is canonically identified with the dual basis ( λ λ | 2 | ≥ · · · . Hence, in refs. [ c/ r to the box ( 2   λ q n i, , ]. The relevant geometry is the instanton moduli space of ≥ ) , we will call the level (0 H and related by SL(2 1 z 1 ( gl λ 121  i . We take a basis – k/` λ = ( modulo ,K ) ). Since the eigenvalues are non-degenerate, the freedom is only in 117 k λ z , z ( i ( + type, which is a resolution of the orbifold  ` j ,F n 111 ) , K A − z ) are completely commuting. Hence, there exist simultaneous eigenstates ( . The vertical representation with the spectral parameter i z i ( symmetry of vertical and horizontal subalgebra of 110 λµ ) representation horizontal. It is known that E  i 2 δ = 0 (corresponding to the vertical representation), the factor in ( Z K ,N ) representation. In the case of DIM, there exists the Heisenberg subalgebra c ). ) = in this case. In the geometric construction of representation of the quantum Z ) is generated by “zero modes” µ `, k ) is most conveniently described by restricting the Drinfeld currents to the “finite , | r n n − λ is an orthonormal basis. Then  ]. For a partition b sl b sl i, ( ( In the vertical representation, we can label simultaneous eigenstates of When } We reserve the standard bra-ket notation for states in the horizontal representation to be introduced 81 ) H 11 hor ver λ = 0 have been provided from the very beginning. q q with ( in [ assign the color in the next subsection. partition (Young diagram) ously diagonalizes the normalization of each{| eigenvector cohomology of the moduli space.to A be representation obtained, of thewe if quantum consider we affinization the is replace instantons expected thewe on have cohomology the a ALE with representation space thevariety of of (the corresponding the instanton quantum moduli toroidal space). algebra by the In the vertical representationpletely or commuting level and (0 theysentation are is diagonalizable what by wequiver simultaneous will variety eigenstates. [ obtain by This thethe repre- geometric ALE space construction of basedjima on constructed the a (Nakajima) representation of the affine Kac-Moody algebra from the (equivariant) toroidal algebra, we can obtaingive the simultaneous vertical eigenstates representation, of thec fixed points of torus action 3.1 Vertical representation and color selection rule see that the to SL(2 trivial, and of algebra” part is an algebra automorphismexchanges two that central exchanges elements the andlevel hence horizontal ( the and level vertical ( labeled algebra. by rational It number also the level (1 which are isomorphic toU the quantum affinetake algebra the originalU form by Drinfeld-Jimbo. On the other hand, the vertical subalgebra In analogy with the case of JHEP03(2018)192 , 1 ). s 3 − q ) s , we j , , such λ 1 1 1 x 3.21 q (3.25) (3.26) (3.29) (3.27) (3.28) ) from 1 − − 2 λ q k k q ) means + + ) may be / → ∞ are easily ∈ j j x, y s = λ λ y = ( ) 3.29 i,k 1 − − s, λ − F . i,j i,j s 3 i x, y δ δ q v/z 1 ) ) − and s λ 1 v/z v/z ). The difference of i,k q j j E x x ) is defined by ( , 1 1 ) z := and right for 3.27 q q ( , v ( ( 1 s s ψ δ δ x − k. ) ) ) s s z/x , has the color + 1 s → ∞ v/z s /x /x − λ 1 s x j j q ( x x x ≡ 6= 1 ( ( . Note that the box ( ψ i 1 − 3 ψ ψ k such that we can remove ( − q . Note that the matrix elements of k k + i s ) is the set of boxes ( + 1 ( s λ + + λ i λ , , ψ s s ≡ ( , k ∈ ≡ ≡ 1 + ∞ +1 =1 A + ) have the common factors in their matrix Y +1 j − s ) ∞ =1 i . Then the first condition of ( s s Y Y +1 +1 j z s , = i i + ( λ ≡ s i s + + ∞ λ s s =1 x, y F Y ∈ +1 s λ λ i 1 – 12 – . With the notation ) ) ) + − . Similarly, the second condition means the box j j s k ) λ v k, λ i, j +1 ) /x /x s s ) and s s + λ z x x ( 1 1 s v/z i z/x s ). We note that, except for the range of the product over − − 2 1 1 6= E q q x q -th row has the color ≡ s ( ( ( ( , s i λ v/z ψ ψ ψ ψ ) is due to the semi-infinite product construction of the Fock j ∞ k k k k + , let us define the set of addable (or concave) and removable the direction changes from up to right (from right to up) at z , x without violating the Young diagram condition. Similarly, the + + + + ( s 1 , , , λ s s s s ). The matrix elements of the generators i to the box ( 1 q λ λ +1 ≡ ≡ ≡ ≡ F s < ( j − ∞ ∞ ∞ i i i i =1 =1 =1 . The addable corner Y Y Y Y j n δ s s s = + + + + ) in the λ s ≤ 3.22 s s s s s ) is the set of boxes ( → ∞ ) to λ λ λ λ λ ( y ) and s, λ + 1 is called spectral parameter of the vertical representation. The factor R ) = ) = ) = ) = z x, y j j ( to λ λ λ i v | | | modulo ) ) ) E +1 z z z ]. There are the restrictions on the product in the right hand side: k ( ( ( λ i 1 or stands for the color selection rule stating that the color of the box that is | − + 81 i i + ) E → ∞ 1 | − z j K K is the charge (coloring) of the empty partition, and j ( | | − j i x k λ λ − k F ( ( + | +1 ≤ + 1), which may be added to the diagram if i ) are related by ( j λ z λ λ s For a Young diagram s ( ( ( − from without violating the Young diagram condition. That is, if we follow the boundary of − i ≤ i,j s, λ that we can add ( removable corner λ λ addable (removable) corner. Since the direction is up for color that the last boxremoved ( from the diagram( if (or convex) corner of module in [ The meaning of these restrictions becomes clear, if one recalls that we have assigned the identified by expanding 1 elements. In fact,imposed these by factors the emerge as delta-functionthe ranges a to for consequence the of corresponding substituting factors in ( where The parameter δ added to orK removed from the Young diagram is the non-vanishing matrix elements of the vertical representation can be written as follows; the box on the diagonal have the color JHEP03(2018)192 ). To (3.40) (3.35) (3.37) (3.38) (3.30) (3.31) (3.32) (3.34) (3.36) (3.39) (3.33) 3.15 . , | 1 w − | ) > | v/z s z | x 1 . . , − 3 } } q ` . ` +1 ( , and we see that the vertical , . 1 i,j i,j ψ ! ≡ ≡ δ , ij . δ − ) r ] ): )
 r q j λ j [ ∓ ij ) deformed Cartan matrix by ( λ w z z rm ) | w z w d ( d q C k ) − − , − = ) r , . ( ` ] 12 λ d i i ij ` q ] − (  A − − ] ( ) r j cr i, ∈ m k [ ] K + r + ij 1 1 ( ` ][ V Y = 1, we will employ the following ver- − r ` H  ) A [ r k k ) = 1 d [ ra | | +1) 1 z ] c [ # ) s Q ) λ 0 ( − i,j ij ( =1 ∞ − λ λ δ s, r ) X a i,j ( ( s,λ
= R λ + δ ( (+) ( i r ) = A R w
∓ z ) ) κ # q δ r V k ) is w ( ` ) as follows; z

d ) is invariant under the redefinition ( ∈ ∈ d – 13 – n − , R ) := [ − ) ) ): v/z r b b ) gl d ] = # s 1 − q ( , λ q 3.27 ( x 3.35 d i, j i, j ( q j,s 1 , ( ( ( ij ( q z, w A ψ ( { { 1). ij C U ,H ) ) = , ij # ) := exp λ C λ s ( z = i,r | ) = ) ). We introduce the ( ( ` ] k ) := ) := ) 1 ( ` H r [ [ , we introduce the set of addable and removable corners of ij K λ λ  − R ) = ( ( ( k q z, w i C ∈ ) ) Y ( ) w λ V k k s ( ( ` ( ` − ij ) s A R q − s,λ ( ( ( j / V ) ) n z ) = − ( λ q | ) the value of the center is ) (+) − z i ( V n + 3.30 ` q K | λ ( ] = ( ) can be rewritten similarly. by n ` 3.28 ( 12 where The fundamental OPE by normal ordering is Note that the commutationintroduce relation a ( vertex operatortex representation operators: with where [ and define 3.2 Vertex operators andThe horizontal Heisenberg representation subalgebra part of In particular, we have Hence, from ( representation has, in fact, level (0 With these notations, we can rewrite ( When the vacuum chargecolor is see for any Young diagram JHEP03(2018)192 (3.49) (3.50) (3.51) (3.52) (3.41) (3.46) (3.48) (3.42) (3.43) (3.44) (3.45) (3.47) and the 13 N , we see that N . | , we have ) is fixed by the i,k w j , . δ z | | | twist of the group ( ∓ z i ≡ i w | > 2 | α F , i ) is given by | ∂ Z ], where the level was > z >  | i,k | | ,N q δ z w
Dynkin diagram. | | , , 123 1 , n → ) and 1 − z i,k A − , . u ( δ : i  122
1 N , (0) 1 z , , u E 0 − −  1 2 i  − i,
q ) defines a N ) N N z z − H K q 1 2 N 1 w q of the currents is the same as the z i,k i,k , ( z q N j δ δ ∓ i  satisfy ) = i 3.48 α q  − α − − .   − , 0 1 e 1 z e ( i z j N i, j ) w (+) +1  w z − +1 ij i q α i is introduced for later convenience. q V 0 i i (+) α  ∂ i i,k H ). First of all, when m V ) α α ( i, e K z δ i q +1 V 1 2 − ∂ ∂ z 1 − j H ij 0 V ∓ (

− a α 1  − i, i − 1
z z w w e d α z 3.12   z H and  ). z 1) ∂ (+) i 1 2 dependence of is defined by flipping the sign of the exponential ij ij z z i
1
i α  i a a 1 2 − − – 14 – 1 V α  w z q z α z z − w q q ): ): ∂ q q 3.15 : q N − . Since ). We have also introduced the spectral parameter e e i z )   . Note that (  ) w 1 = ( = = ( z z ) ) ) α ( ) − − z ) i − ( i ,N j j j ( , z · z − −  η z ) ) 1 1 ξ i ( ( ( i α α α ( ) ( i i i ) (  i α e e e Z i  i z V V V w i i 0 i ξ ϕ η  V ( ( α α i, i e z, w i ∂ e is due to ( ( η ξ H P z → → → : : z ) = ) = ) = ij  = 0. The i ) ) s z z z =  replacing ( ( ( i z z K z ( ( i i α = 1 2 i Q  i q i ξ ) = ) = η ∂ q 1 z ϕ E i F w (  − ( i ) i P appears, only when the color  i E F w ) ) ( u K ) z w by ( ( − i i ( j N E F V − ) q z ( = The zero mode parts i (+) i 14 V K i Q Let us check the commutation relation ( The shift of theThe argument choice of of vacuum state breaks the cyclic symmetry of the affine 0). Here we generalize it to level (1 for the horizontal representation. Note that the modification by the level = , 13 14 commutation relations. An additional factor of and the same relationalgebra for of the rootκ lattice spectral parameter vacuum. The vertex operator representation(1 was originally given inu [ Then the vertex operator (horizontal) representation with level (1 The inverse of the vertex operator and satisfies, for example, Let us introduce notations for the oscillator part of the vertical operator representation: JHEP03(2018)192 , k α ∂ , (4.1) (4.2)  ) into (3.57) (3.53) (3.54) (3.55) (3.56) q ): ` w α . ( e 3.50 k ): k` η a z ) 1 2 z  ) ). 1 2 q q a. x ( ∗ 1 w − ) and ( ( − ` = q , − 1 k ( ) q ϕ ` − a ( ) + ` α E = Φ . δ q 3.48 e 1 ϕ ) w ∗ ) exist only when the k ( : − − − w v α k | q ( ∂ )Φ ) η q +1 a − :  i x ) and − , w q q `,k z +1 K ( q ], we define the intertwiners δ ), we see ∆( +1 q Φ = Φ∆( δ
( `,k + 1  80 δ `,k z +1 w − ` δ
z = 1
3.39 w `,k N q z w − ) δ 1 ( z . Note that, in our notation, the w , ,K 3 1 q
∗  ) by the shift of the color indices: 1 q 1) , a − . z w z δ , uv − − 3 ( x − 1 k q 1 (0 v 1 − q 1 − + ` +1) − − 1 − ) K q = 1 ,N `,k )] = 0 1 − z ⊗ F `,k δ ) and Φ (1 ( w q δ −
w ) w 1 v ). Following [
+ ( i | `,k z w − z − `,k 1 δ z ,N w ( δ 1 x K
), we obtain )(1  2 q k (1
i u → F − 1 2 y z q w + z ( q w N, u  – 15 –  ) − i 1 − ,F f 2 x ) as follows: . We assume that the color of the vacuum (the − − 2 ) w ) ,N q 1 z ) between q n u q 1 1 z → F q ,K (1 ( u ) − b b − gl 1 i  − − z 1 1 ( − ( 1 δ d E − 3.23 +1) , xy [ k (1 and q 1 `,k ( ⊗ F 1 `,k + δ − ,N δ i U δ −
v − 1)
(1 , w )] = `,k z w `,k ) ,F δ z w (0 F δ v 1 w )
) = x 1 ( q
z 1 F z i x w − ( 3 z w ( ): − − 3 q k 1 v q f q ,F 1 ): | + ) ) − i − 1 be the Fock spaces for the vertical and horizontal representa- v 1 − z − | q 1 E ( , w − 1 ) i − x E 1 ,N )(1 xy → [ N, u ( + 1 x ) = (1 ) u δ q z w F Φ( N ( ( ( ) = − k  i ∗ 1, by taking the commutation relations of zero modes ( z η 1 2 Φ ) (1  q z and ,K ( ) j 1 2 z − ` − ), we recover the relation ( 1) ( ≡ , i ϕ q ) ( i (0 v ,F + ` w F 3.49 ) Before concluding the section, we give another example that shows the role of the ( ϕ z k ( η i Later we will seespectral that the parameters intertwiners satisfy Φ( alevel conservation and law the spectralspace. parameter In of the the following, we horizontal often representation suppress refer them for to simplicity. the They source are indicated Fock explicitly, of the quantum toroidal algebra and its dual Let tions with spectral parameters highest weight state)Note in that the the verticalE color representation of is the 0 vacuum for can simplicity be of made expressions. By combining with the commutation relationsee of ( the zero modes: 4 Construction of the intertwining operator and commutation of the zero modes. From the OPE relation ( When account, we can check and the property Using the formula JHEP03(2018)192 | λ )( , we (4.8) (4.9) (4.3) (4.4) (4.5) (4.6) (4.7) λ | (4.10) (4.11) (4.12) 1) , λ (0 v : is a map P F λ λ , , ) ) z j ( , i ) E j + 1 , ) 1 ) v λ z ( | − ( ) λ i z λ F ( | ) are taken in the level i )Φ ) ) z v λ z F . If we employ a different ) which is related to the | | ( q q 3 ) ( λ ( λ z i λ,µ − , q i ( )( δ E 1 | v − i K q ( ]. ( . The component Φ λ j K ) + Φ λ | ) = )( 80 z 15 C +1 λ µ v q | 3.1 ∗ λ ( . ( j ) λ ) and Φ + 1 i λ , z − ) ( ) + ( K ∗ λ v ) )+1 + i ⊗ | =1 ( v Φ λ j X j ( ) ( ⊗ • ) ) of the vertical representation . ` , j • ) ,K λ ) , λ ) 1 =1 ( +1 . ( ) | ) z j ` λ X ) v λ ∗ λ − z | z q ( ) v λ ( ) + ( ( Φ i ( ∗ λ z v )Φ + − i i ∗ λ q ( – 16 – )Φ λ ( λ ,F ∗ λ )Φ | K K λ = Φ( ) X ) )Φ z − | ) ) i z z ) for the vertical and the horizontal representations, q z )Φ v v λ ( ( ( ( z = K ( ( i i z Φ q q λ λ ( + − i i ∗ E ( i E + | i Φ K K F introduced in section j = )Φ )Φ F ) ) ) Q | λ λ 2 ) λ λ λ j | | 1) | | | , ) ) v 1 + 1 C ) ) ) ( z z (0 v z z z ( ( ∗ λ λ − ( ( ( ( , we find the following intertwining relations: F + − i i + − + i i i λ ∗ λ )Φ is the spectral parameter of the vertical representation so that ( K K z ) )+1 K K K | | =1 ( | | | λ λ v j i X =1 λ λ ( ( λ λ λ j ` ` X E = 1 and 1 ) = ( ) = ( ) = ) = ) = ( ) = ( ) = ) = ( C v v v v = 1 for Φ z z z z ( ( ( ( ( ( ( q 2 λ λ λ λ i i ( − i F ]. Namely, with the normalization factor E + i ,C ) is the basis of )Φ )Φ )Φ )Φ ) K q v z z z z v ) 80 K } ( ( ( ( ( q ) v i i ) ∗ λ ( ∗ λ = ( + v i λ F E ( ∗ λ − Φ i Φ 1 K ∗ λ {| Φ C K + 1) representation, while, at the right hand side, the representations are at level Φ ). The argument ) represents the data of the state on the vertical side. We have used that 1 = The component of the dual intertwiner is defined by Here we normalize the dual intertwiner in a way distinct from [ ,N ,N 15 λ, v Since in order to derive the intertwiningnormalization, relations the assuming intertwining ( relation will involve the normalization factor. At the left hand(1 side, the currents (1 ( operators. Since the definition of the coproduct implies the following intertwining relations for Φ introduce components of the intertwiner by where between horizontal representations, and our task is to express it in terms of the vertex whenever it is helpful. In terms of the basis 4.1 Structure of theIt intertwining turns operator out thatDIM case the [ components of the intertwiner have the same structure as in the JHEP03(2018)192 ). 4.13 ). We (4.13) (4.14) (4.19) (4.15) (4.16) (4.17) (4.18) 4.13 by ) in ( z ), it is more , λ ( i ): ): v ∈ E ( v ) 4.13 ( λ ∅ i, j Φ · of Φ = ( λ .   t ) q in each row with respect  . , v , the intertwining relation ) ) 1 1 i, +1 . . − , i,j − ` i j 3 ( : ) 4.3  − ¯ )+1 c q ≡ 6=  =1 j 1 1 s λ 1 Y s 0 j ( s E ( q , λ − λ ` − λ q ` ) j 1 − s q v − 3 − ( 1 . In the case of ( q 1 )  ) 1 ) = − ) + 1 , which defines the coloring of boxes. − i 3  a i,j λ, q n ( q  ( 0 ∈ ` λ ¯ i, j c ( 1 ) , which means we take the product in ≡ , a ( 1 ≡ λ ) ··· − Y j E s q λ ` j λ 1 i,j 1 i ← ( λ ( =1 q Y Y ` λ s − ( − − q ) n ≤ s ) + 0 j ← a Y , the normalization factor appears in the norm (1 i,j ≡ ≤ n  ( `,j 0 λ ) n 1 ¯ c ) modulo ( δ a ∈ – 17 – ≡  j, ` η λ λ )  ( Y    ∈ a λ  i, j ) = ( Y − h  =1 +1 λ λ ∞ i ( j i Y ` h i,j λ 1 λ a ) = ≤ q ← i Y ) = n q  3 ≤ ≤ ( ) = i 1 − ← λ ) =: 3 , q ). As we will show in section ) := Y ≤ :  v h 1 1 x ( , q q ) ( = 1, we obtain ) 1 ( i, j = ) = 1 as the normalization condition. Then, later we will see 3 ∅ ) of the vertical representation, we have ) below more about the ordering of the product in ( λ 3 ( q 3 λ ∅ λ j ( P t Φ λ , q | t , q ) is a multiple of λ , q a 1 λ 1 1 +1 q t 4.22 C q  q λ ( ( ( t ( λ λ λ , then we order the blocks of each row from the first (rightmost) to ∅ t C h t j ) = ) = 3 v ( is the content of the box ( , q λ 1 j Φ q ( − ) gives the following recursion relation for the prefactor ∅ is the transpose of the Young diagram. Note that if we do not have the restriction i z 0 ( C ` λ ≡ E ) i, j ( ¯ With the initial condition of the Macdonald function with and where that the hook length We define the arm-length, the leg-length and the hook length of the last (leftmost). See ( impose the intertwining relation fixes the normalization factor as This is the reason why wethe used “reversed” the order, notation namely complicated, since we have doubleto indices. the second We index first order c Since the zero modes are non-commutative, we have to fix the ordering of where the vacuum component is (formally) given by an infinite product: normalization of the basis JHEP03(2018)192 . ) ) 3 +1) , q ,N 1 ,N (4.22) (4.23) (4.20) (4.21) q . The ; (1 1 w , 1 e u, v 2 , ( in the normal . 1 → F λ e ˜ t ). From now ) 3 Q +1 , 0 , 1 ,N ) j, e , ) in the product. (1 v u − 1 4.21 i , ( v 2 F ) in the representa- ( ): λ H ) on the level (1 e ) v z v 2 v i,j ( 1 1 , v ( e 2 1 ∅ − − λ i e 3 − N Φ q i 3 1 · q − = ) 1 − ) 3 j 1 − v q . As we emphasized before, j 1 , 2) 1 , ( , q q v 1 2 ) − 1 ( (3 i among the spectral parameters 3 q − − z q i i,j 3 j ( ( 1 q λ − ¯ c . Recall that the component of the i uv 1 2 − f j α 1 E 0 − . The monomial factor | − uv e q j and takes values in the group algebra 1 λ ( λ | q − ) ) and ) is = u N 3 v i,j 1) ) = − ( ( w , q ¯ c v ) − λ 1 ( η ( v q 0 λ λ ( i,j − ≡ ∈ ∈ λ ( ) ) ) 0 f | Y Y i,j ) is nothing but the framing factor of the refined – 18 – i,j i,j λ ( | ( ( 1 ¯ c u : − ) is required for the existence of the intertwiner. The , e ) 3 3 ) = q, t )   , q ) is for convenience of computing of the intertwining ( 3 uv ) 3 , q 1 λ v 1 q − , q f , q ( ( q 1 1 ; λ 4.22 q = q i,j t 0, ( ( e λ λ i w ≡ u, v f λ 16 ( C ]. The dependence of the intertwiner Φ ) = ) ). ≤ λ 3 j ← t e ) imposes the relation Y ). For example, it means that ≤ z i, j , q )) and it turns out that it is natural to use the vertex operators -dependence explicitly. The condition on the vacuum component z 4.13 1 ( ( 125 1 ( v , 0 c of the horizontal Fock space counts the number of boxes with the q `   ) = ; F ), we employ the vertex operator ) v ) u +1) and the spectral parameter 3.45 ,F λ ( 124 ) ( v ) λ ` ( u, v z ( ≤ ( ,N ← ) can be regarded as a map between two Fock spaces Φ ∅ 4.13 i Y ` λ v 4.21 t e ≤ ( E 1 λ ) (see ( z ) are non-commutative, we have to fix the ordering of ) = Φ ( ) = v i v ( v denotes the number of boxes with color 0 in ( E ( ∅ 0 i,j λ | e z λ )Φ | For the convenience of forthcoming computations, let us separate the zero mode part In formula ( Since there is no ordering problem in the oscillator part, we use the usual notation z ( 16 0 on, we write onlyF the of the horizontal and the vertical Fock spaces. product as compared with ( since Our choice of therelation ordering with in ( spectral parameter same color as the vacuum and only appears in the second factor of ( is the generalized framing factorimpose arising the from restriction the ¯ commutation oftopological zero vertex modes. [ If wecan do not be arranged simply in the powers of The factor in ( and now depends on the horizontalof spectral the parameter root lattice. The group algebra part of Φ where referring to the target Fock space. of the intertwiner as follows tion with level (1 intertwiner Φ We will see that thelevel relation and the spectral parameterpart of of the horizontal representation affect only the zero mode JHEP03(2018)192 target (4.29) (4.30) (4.31) (4.35) (4.32) (4.33) (4.34) (4.25) (4.26) (4.27) (4.28) , +1 0 ) : (4.24) j, v − , ( i , ) ∗ ∅ ) H z v ( − . Φ ): ) refers to the ( ` ) 1 · z ∗ λ , v E ( − . z 1 i ) q   ):  − , v N . 4.24 F ) i 3 j v ( ) 1 v  q ( 3 q ) +1 ∅ 1 , 1 ` q ∗ ∅ j  . = 1, we obtain − , q Φ − )+1 ( i ≡ 3 6= Φ − j 0 1 =1 : 1 ) by s λ λ ) in ( Y q s ∗ ∅ s q ( · q ` `, z  λ 1 v 1 t ` ( ( δ ( ) − 3 1 , − i ) − − λ λ, ) s q j v j 1 ) − 0 f ∈ E i z 3 1 . q − q ) v 0 ≡ i ( q | ) ( − 1 Y ` j z/v ) α i )+1 1 λ 3 z i,j + i , ( ) satisfies the following intertwining − ≡ ) ( q −  q − i 3 s i,j −| ( λ 1 u ψ j ( 1 e 1 K ( q ( λ =1 ( λ ¯ u c q Y ) ` − 1 s − ) and the horizontal spectral parameter − ` a 1 − ( 0 j 1 ∅ v | F q − s ) q q ( λ j i 1 K ) + 0 | ( ) = ) λ ∅ . q ,N i,j ) v − `,j v ≡ ( ( N λ v ( λ ≤ δ ) ¯ c ). Then we have the recursion relation ( Φ ) ) z ( i,j 1 j ← ∈ 0   Y ( F v ∅ ( Y (1)  ≤ i,j ¯ , c i,j  `, ( λ 1 ) δ ξ 0 − ¯ Φ c +1 Φ h – 19 – 4.23 z j − 0 λ ξ 0 ≡ |   ( )( λ ) ) λ 1 `, ∈ ` `, λ ) 3 ∈ ) δ q  δ λ =1 F ( Y )  Y q ( −| ) ∞ , q ) i,j λ ` Y ) , f v/z ( 1 i,j v h − ≤ q 2 ← q : ( i Y  q z/v (   v/z − ∅ ( ( ≤ ) ) ) ∗ λ ( 1 t 3 1 v δ ψ ψ ) = ) =: ) : ( 3 − 3 v , q q ) ( 1 i,j ) , q ) = ( 3 , q ) = q 3 ∗ ∅ f ) = ) = Φ ) = ) = 1 − 3 1 3 ; i q v v v v , q Φ q , q ( λ ( ( ( ( , q 1 ( , q 1 1 0 = ( ≤ ∅ ∅ ∅ ∅ λ q 0 1 λ q q j ← u, v ( q ( Y ( C ( j C 0 λ ≤ ; ∗ λ )Φ )Φ )Φ )Φ ∗ λ ∗ λ ), the vertex operator 1 t ∗ λ t z z z z t C e +1 v t ( ( ( q   ∗ λ ( ` ` ( u, v t + ) λ ` ). That is, it has the level (1 = ( F E λ − ` v ∗ λ ( ) = K ( ` t e v K ∗ λ ≤ ( ← i Y ∗ λ ≤ Φ 1 )). Let us decompose the monomial factor as before, ) = 3.46 v ( ∗ λ 17 Similarly, the dual intertwiner is given by replacing z The rule of ordering is the same as in the case of (see ( 17 4.2 Vacuum componentLet of us the intertwiner first checkrelations: that the vacuum component Φ By solving the recursion relation with the initial condition and the same generalized framing factor ( with As in the case ofFock space Φ of Φ u with The normalization of the dual intertwiners is JHEP03(2018)192 ) 1 q 18 z − 1 2 ) z v − 2 (4.40) (4.38) (4.39) (4.36) (4.37) q q ( 6= 0, all ( + ` ` ψ ). This is ϕ v · , we have ( ` q ∅ , , , . ≡ ): , since the level i v ): ): . q ) = 1, unless the ): ( z v z  ( ( ∅ ∅ ( 0 | ). ∅ w 0 ) η z j 1 )Φ ξ ) z ( q ) z )Φ ( v . Hence, if we take the (  v 0  ( z 1  ` ( 0 . When ( , ∅ ξ − ` K 0 ∅ . K )  : η | +1) z  j : :Φ ( 1 2 ) and the spectral parameter :Φ ∅ + 1). Taking this factor of z ). +1 v/z − 1 2 1 or − 0 i 1 , +1 +1 q q ) and η +1 0 ,N ) representation, when the color 0 0 − − , , ,N 0   i z 3 , 0 0 H  4.35 ` ( 0 q + ` − ` ) commute with the vacuum com- − H H 0 2 ,N w H z z z z q ϕ ϕ 1 ( − 0 ( 0 0 ) ) ,F ` − ) implies that non-trivial OPE factors z α α α ) j ψ 1 w w 0 F ) to (1 e e z − q · ( ( α ( ), which we chose 0. Hence, if e 1 1 ) = 0 and ( .  q 0 − j j η 3.56 − − ∅ ,N j  ∅ e | E − − |  (tri) (tri) i i  − v uv. ) ) implies i z e e  η η z v q z v z ) and η − ( ) to 1) has color 2 z v ( z z ` 0 q  q − ( − i, = ` ) = ) = E F – 20 – w 1 1 | ) − − E w w 2 4.39  w  v ( ( − , we have 1 1 ( 1 j 1 ` w j j ∅   q − = 0, we have 1 − − (tri) (tri) i i 1  ≡ u w − ` ) commute with the vacuum component Φ e e η η − ). We can also check ( 1 z 1) N N u   ( +1 combined with ( − − − z z ) = Φ − j  ` N N 2 1 2 1 ( 1 v   i 4.34 q ( q −   K − q q z z i ∅  q q q z z η ) cancel:    z ` −   )Φ 1 2 + ` ϕ z q ( ϕ ) and = 0. When ) = ) = ) := ( 0 z ) = ) = z v ` − ` w ( F ( ( z v ( + 1. Recall that, according to our choice of the color of the vacuum, ` 0 ( ϕ ( ∅ = 1, the non-trivial commutation relation comes only from the vertex 0 ` j ∅ E ,F F ) − ∅ )Φ ) ). And when (tri) i ) ≡ v t )Φ z z e η v ( ( ( z ( ` 0 ∅ ( ) and = v/z ∅ 0 z ), unless E E Φ 1 ∅ 1 2 F Φ v − 1) has color 0. From our definition of level (1 ( i 3 C ) arises only when the first box ( − , q ∅ v q 1 ) is the same as the vacuum, there is a change of the power of ( ( is the same as that of the vacuum state z − 1 ∅ + ` ( q ` ϕ ( + ` By the same reasoning, we see that Since ψ Note that we are looking at OPE with the inverse of K 1 18 − Hence, the condition On the other hand, the substitution of ( of of the horizontal representation changesinto from account, (1 we can confirm ( ponent Φ with Φ q product over rows, thesesurvives factors when cancel 1 inthe general. box (1 But a non-trivial factor Because of this “triplet” cancellation, for each row a non-trivial OPE factor of operator part. A crucialfact. point Let for the check of the intertwining relations isThen the the shift following of the powerwith of color the currents consistent with the fact that theof dependence the on horizontal the representation level appear (1 only in The color selection rule tells us that ((1) JHEP03(2018)192 (4.51) (4.45) (4.46) (4.50) (4.42) (4.44) (4.47) (4.48) (4.49) (4.41) (4.43) ); . 3.36 ): , . ) ) v j j ( ) can be also . ∅ v ≥ ≤ ( +1 i i )Φ ∅ 0 ( ( , z 0 ( , 0 H . i i η v ): − − : 0 j j ): α w d d ( ] ] +1 w ) i j ue 0 ( , , − ) : . 0 ( N − j − − , 1 H  ( − e V j j i z − v v ) ! V 0 1 2  z r ) + + α ( q v +1 . z ∓ e 3 2 ] n n (  z N  r r q r (+) q [ i z v   , (+) + [ + [ − V (+) i , i, k i,j  i e n : ) V e j,r δ V δ Λ − r (+) − : 0 j k i d H N i,j ) = s, 1 j,r e ] + V , − δ =1 3 r ∞ − i,j + − r n j ij [ r X H − δ r b q d d   , q ] ] δ (   uv 1 ∓ v 1 i v j ij q 3 2 − z z 1 2 =0 w w

b +1 j X n : − q − q – 21 – ] = N  i j  − − = [ [ = q ) j,s ) ] 1 1 ( − u, v r − − [ ij   ( ( ( i,r k k b ,H e e Λ V V (1) ) = i,r ) := exp ) = t e ) = ) = z z d ( ( [Λ , w w ) = ) 0 ) is ( ( q ) =: v ) ) (  v v E ( ( ( i − − ( ij ) , . Similarly, the vacuum component of the dual intertwiners ( ( e ∗ ∅ j j ∅ b V v ∅ e ( V V Φ `,k  ) ) Φ δ ∅ 2 n z z ( ( ) = 1 Φ − 3 3 ) holds with q  (+) (+) is a linear combination of , q z v i i − 1 e V V  q 4.32 2 n i,r ( 3 ψ q (1) − ) C   v 2 n ( ) are the components of the inverse of the deformed Cartan matrix ( − 1 ∅ d q , is the color of the vacuum. We can check the intertwining relation for the vacuum q )Φ − ( k z ij 2 ( n 1 b It may be useful to mention that the intertwining relation for Φ 0 q  E where component with general can be expressed as Then another formula for Φ The fundamental OPE relation is where and More explicitly, Λ with the commutation relation reproduced by introducing the dual vertex operator This means that ( gives JHEP03(2018)192 ) ) ) , ). v z ( N q 0 ∅ ( 3.23 4.32 `, (4.54) (4.52) (4.53) (4.56) (4.57) δ − ` − ` K , α ) ∂ z q ( `  , ), using ( E z . ) and ) ) z 1 2 ). i, z ) with the color ( v z +1) ( z i − ( ( q + ` ≡ + N q ` λ (   + ( `  ) 0 i, λ K Φ − K ` v `, K  + ` δ 1 ) + 1 ) still holds even after ) K ϕ v − − ` λ 1 i 3 ( ` ) | q α − λ ) v + ∂ ) 1 ( 3.23 z ) with ) a non-trivial OPE factor Φ ) with q i ( − i λ v j v 1 λ 1 −  1 ` ( Φ q ), we obtain v/z − z ( − λ s ) K i 3 ) 2 1 1 | x q − 1 − λ i,j 1 3.23 for (
( q − 3 ¯ − v/z c j q 1  s − ). Since ( E ( q x ) z i + ` ( v/z 1 1 ψ ( ) λ s ) also holds for λ ) when they satisfy the color selection ` − | ϕ − 3 i x ) ≤ i,j  q E +1 j 1 ← ( z ( , Y   ` ¯ c ) (4.55) v ( ≤ − 3 i 3.23 i, λ  z ≡ ψ )+1 1 q E + ` =1 s x λ v Y s ( λ λ 1   2 v/z ` K +1 ). To obtain the delta functions in the inter- + 1. From ( q − ) ∈ ψ | s , ` − ) s i 3 1 λ ` x λ ≡ )+1 ( q q ) ( ( =1 s ` +1 i,j λ – 22 – Y 1 v/z s ,  ` (  ( λ 1) exactly cancels the contribution from Φ ≤ ( ← or ψ ` − i Y ≡ )+1 − δ ` j 1 ), since ( i, Q =1 s v/z s ` ≤ λ ) Y , q s ≡ z s ( λ ) ) = 1 ) ` s q λ  x z − ( λ =1 ( ( ( s ) Y ` s v/z ` − − ` +1) ψ 2 v/z s × i,j ). Note that the first term at right hand side of ( E s q ` ( ). This implies i, ψ N K ¯ ) c ( ( , x ≡ ) 0 v ( s ψ E λ ) = ≡ ( `, ( λ =1 i 4.32 ψ v Y δ ` λ s 4.34 ` λ − ( ` − s 1) and the last box ( , ≤ λ ` ≡ ) )Φ + j 0 ← s α λ Y i, Φ i λ `, ( ∂ λ ≤ =1 ), we have | Y ` δ s 1 λ q ) − z ) ) has the color s i ( z +1)    ` ( representation, for each row (fixed index ) z N − E ( ` λ v/z 1 2 i, λ ( 0 2 for ) = uv ` − K `, q v | − δ ≤ q ← ( ( i Y λ −  λ ψ ) by ( ≤ `  1 z α + ` v − ( )Φ = ∂ + 1) i z ϕ ) z q + ` x v (  (  K + ` ,N z λ 1 2 ψ K )Φ − z q ( Let us move to the intertwining relation with  ` + ` E ϕ where we also usedvanishes relation due ( to the coefficient twining relation, we make use of the following formal series identity for a rational function Similar computation is valid for replacing where we also used relation ( and discussed above. Thus theselection remaining rule factor that comes ( from the last box ( and, hence, We can use the sameWhen idea we to compute compute the the OPE OPEfor relation relation of the of (1 Φ arises from the first boxrule. ( The factor from the first box ( 4.3 Zero mode part and intertwining relations JHEP03(2018)192 ) 1 z − ( `  . η j being ) (4.64) (4.59) (4.60) (4.61) (4.58) s x ): v 3 x ( z z dz q ( λ ) `  z E ( ) ψ . γ v ), we have to ) i. 1 t ( = 0. The sum z +1 ( : − λ , ]); ` ( | z z ≡ ` ≡  )+1 = w =1 s 72 i z | :Φ λ E s Y s ( λ ) j 1 ` x − v ≡ + 1 > 2 x and − s ( q | ` j  1 λ z  j s  | ∞ j s + x x = Res − + 1 )Φ ψ i x x i i , ) λ = − λ t | ):  ( ) 1 ≡ ≡ +1 z , , γ w z `  ψ j ) ( ( j j )+1 ) ≡ q j `, 6= =1 t − s at , λ ` Y , s ( η for s ) ( j ≡ λ ) ` s + 1 is satisfied in each row. The λ 1 1 ): K 6= +1 ( − =1 z λ , | Y ` ` x ` ∓ s − j s ( ): ): w z/z − λ i )+1 z ≡ ( 6= ) with ( s   ( =1  s η λ w w Y ) : (4.63) s j ≡ s δ z j ( ) λ ( ( j z v : z η ` ) ( s 1 x i i j j 3 s t − ( ) x − 1 x ( γ
` η η q − x 2 s λ 1 x z q ) in ) ) ) γ q z ( q w z E  v i z z − ( − ) of − ( 3 s ( ( − η  t i i j q v λ i 1 γ x 1 ) : ( X 1 x 3 η η t ψ (1 q ( q λ  ` : : q )Φ  or z  , ≡ z ) 1 1 − j  − w s – 23 – ( :Φ ` λ ) = s x − − 1 1 2 ` ( λ v =1 1 z 1 Y 1 x ` i,

s j ) (4.62)  z ( − q − E ≡ − z z s z x 1 w w q
−  ≡ ( i )  3 1 −  ` γ z 2 w λ q q j ` z, q ψ δ q E x ` ` 1 2 s ` − ) 1 − , , − − − + ≡ ≡ − x , q ) ) ≡ ) v − ) 2 s s i i λ λ j 1 1 1 (  λ q z ( ( λ λ =1 =1 (1 ( λ λ =1 Y Y ` ` λ ( s s ` X j 1 − − ij − − s s + s        j − ) implies 1 )Φ ) ) γ q q q λ  = |  for ) = 1 ) − − ) = and with simple poles at most (Lemma 3.3 of [ v 4.58 ( ( z λ − j | ( w    ) z x ( − ∞  ` s v v z 1 j j , j j ( x q v -th row is η K x 2 z z x x i i z | − ` q  ) 1 1 x q q λ z δ  = 0 K ( ( |   − i ψ ) out of the difference z λ − +1 δ δ , η , ` v ( 1 ) ( ≡ v )+1  +1 j +1 +1 − =1 j , ( , , λ ` ` ` j X ) denote the Taylor expansions of ) j ( λ λ )+1 ≡ ` ≡ ≡ )+1 )+1 z +1 6= λ =1 − =1 =1 s j j λ ( | Y s λ λ λ s j X X ( j j ( ( λ λ λ ) )Φ `  ` ` z z − − − + γ ( s j j ( ` + ` = = E ) regular at K | z ( λ ( Hence, the product over the rows gives The “triplet” cancellation alsoappears holds when in the this selection case.contribution rule of Thus, the a non-trivial OPE of Φ parameters into account. We need the following OPE relation for To get Φ compute the normal ordered product and take contributions of zero modes and spectral functions. The formula ( where at the right handthe side residues. runs One over can all prove the poles identity by the partial fraction decomposition of rational γ JHEP03(2018)192 ) ) 20 z , ( ) = i λ . , we ) v 4.18 C E ( v (4.66) (4.67) (4.68) j ( A j +1 +1 λ λ z Φ . j 19 λ  z λ j . +1 s λ C x ,` ` 1 x s C +1 q λ to get ,` j ≡ − s  s 1 i λ λ δ + 1, that is, when we +1 t − ψ − +1 ]. In appendix λ  ` s j ` t  δ j s , ≡ 80 j s 1 j,λ x x   s ≡ ): x x e − λ 1 =1 j s v Y j s j w 1 j − − x x 3 . ( , a necessary technical result − s λ z ) x q i + 1 − 1 v q η 1 − ` ( q B , ) ]. Taking this into account, we − ` − j  1 z  j 1 ( ≡ +1 ≡ ,  ` ` δ +1 j 120  +1 i ξ i λ )+1 , ` ≡ 6= 1 =1 1 : s λ +1 Y s ) ≡ s Φ ( )+1 +1 j λ − =1 s − ` λ 1 j j Y λ j Y s ( λ = − ( Y ` −  ) = s s ` ): λ ), we obtain the recursion relation ( ): − ): s +1 j s z t  s 1 λ w ( w z x x w t − ` ,` ( w ( q ( 4.4  +1 i s i ) involves the commutation relation of q i E ,` s λ  η − η ` u η s ) ) j − ) − x ( ) , v λ ≡ 1 ) s v z 3 x z j λ z s 1 δ ( λ − – 24 – ( ( q (  z ( λ s x =1 ` λ  ` Y ` `  δ ). s 1 ξ − q ξ ) (4.65) ξ ) appears when 1 v j s q s − :  ( : z )Φ : − x x ) , (  j s 1 -th row. Then we move +1

λ 1 ` 1 
`,j ` | j x x δ v/z j δ  − w z w ) ( +1 − z z λ 1 j E ≡ 2 )+1 w ) and Φ ) d q d ` j  z z s q ) q λ x Y z , ( (  ≡ λ = v 1 1 ( ` s s − − − ( j s +1 − − ` − q ` − − λ =1 j s λ ( z, w Y x x j 1 1 1 s F 1 λ 1 1 − ( K δ . Taking the level dependence of the zero modes part into ). in the | s q    `i )Φ  q λ s ` s          1 λ 1 B ( ψ j `,j | x − =1 − δ =1 − ) 3 x 4.19 Y j s = Y j s  −  q z ) by using Lemma 4 in appendix +1 , ( ` ) = ) v 1 = +1 ( +1 ≡ v − − ` w − , ` j =1 s ( ( Y j s j λ 1 λ 1 i λ ≡ +) )+1 K | =1 j q j η λ − λ  X ( j λ ( λ +1 ) q s λ )Φ ` z 1 λ ( C z − for the definition of ) z − ( j − × ( C ` v  − q B ` ( ξ , ` = ) E +1 ≡ v ) +1 , ` s +1 ) which also appeared in section 7.2.1 of [ ( j λ j ( λ λ ≡ )+1 Y ` ). But it can be performed similarly based on =1 j = λ − j,λ X j s ( λ s e w )Φ ` 4.66 ) − ( z j j × v ( ( ` F The OPE computation of Now we employ the following combinatorial identity for the normalization factor ) = To obtain trivial cancellations with this factor, we have chosen the product order of See appendix E − j 20 19 ( λ which gives the formula ( and By comparing with the intertwining relation ( arrive at See a relatedprove computation ( in the DIM case, Lemma 6.4 in [ may add a boxz with color is worked out inaccount, we appendix finally obtain Note that the delta-function JHEP03(2018)192 ] 139 – ) and (4.70) (4.71) (4.72) (4.69) ). z ( 132 , λ 4.72 49 – . 47 1.  − z u 3 j s q λ x q operator satisfies the , → − j = 2 quiver gauge theo-  1 T λ v ,  z j , N q ): x v +1 , `  ( , that is, when we may remove )+1 λ ` ≡ )-KZ equation, since it realizes δ =1 ]. The s ) with λ ) Y s ( . λ z ` ≡ )Φ ) 55 q, t − q z z j s ( ( ( 4.66 λ ` 1 + ` λ F − ϕ − Φ  j = ⊗ u )) : z ) by ( s ) z q z  x ]. The correlation functions of the model z ( ( operator [ ( v ` z − j ∗ λ − . However, we have changed the normalization q F γ x T 53 Φ λ , 1 – 25 –  −  λ 52 ). On the other hand, the gluing along the vertical δ ) ` X a z , ≡ ) ( s ): λ ( λ + =1 appears when v Y ` s ∆)( ) = γ j −  s z ◦ x ( v ( z j j q S x ∆) λ ) = ) = (  − ⊗ z z -th row. Using j δ ( ( j η ` γ and consequently there appears no weight factor in ( ) ), which we call F ]; z )((1 ) ∗ λ w ( 2 v ` ( 54 z ( ξ ∗ µ λ : Φ ) symmetry. There are two fundamental ways of gluing intertwiners )Φ( n 1 from the − z b b ` gl ) ( v d , ( ] for an original and generic issue of the conformal matrix models, and [ q λ U ). The gluing along the horizontal line is simply the successive action of op- . We can deduce that 131 | ) = Φ( )Φ – 2 3 z whose measure is determined by a trivalent planar diagram (5 brane-web) rep- w z | as a weight in the summation over ( ` 126 2 21 > F relation and plays an important role in deriving ( − )Φ( | -shift operator (see the next section). Note that such a product of the intertwiners 1 || Finally, the dual intertwining relations can be demonstrated in the same way. z q z | λ See [ 21 M Φ( In the DIM case, there|| should be the inverse ofof the the square norm dual of intertwiner the Φ Macdonald function for AGT-related conformal matrix models. line means taking the tensorintermediate Young product diagrams in on the the horizontalnetwork vertical matrix direction line. model with This summation [ gives over the the screening operator of the the dual intertwiner Φ RT T the along the horizontal line givesa again an intertwining operator which satisfies, for example, or five dimensional liftries. of the Using Nekrasov the partition intertwinersmodel function constructed with for in this(see section, figure we can defineerators a on network the matrix horizontal Fock space. A particular example is the product of Φ Network matrix model ismodel) a matrix model ofresenting the a Dotsenko-Fateev type toric (conformal Calabi-Yauare matrix threefold computed [ as theof (vacuum) the expectation intertwiners glued values or together. the They traces reproduce of refined appropriate topological products string amplitudes we can check the intertwining relation with 4.4 Network matrix model and screening operator This time the delta-function a box with color for JHEP03(2018)192 ) ), c ( w λ ( 2 ) for  1 u ` (4.73) X u . K

  = ) ) ) (4.74)

z z z ( ( ( X λ, z

λ j λ Φ ? Φ +1 ⊗ λ ⊗

) )  Φ z λ ∗ λ ). For z

( ( ⊗ ⊗ Φ Φ ∗ λ w j )

( 1  of the Young diagram. z Φ λ (  − |

` λ 1 ∗ λ ∗ λ Σ z λ u | K 2 P )Φ )Φ − zu j λ . | 1 − ) = ) N z − ) and w ( gl λ q S | w DIM algebra breaks down into a -operator (horizontal gluing) or the ( ) ( ` ` T . We will see that the intertwiner w E λ . | q ,F ( j ) ` ) so that it explicitly depends on the w E +1 | ( ` λ λ )] = 0 ( ( 4.20 E z ) ), we have ( gives the λ =1 ( S )+1 w non-Abelian j ` X =1 ∗ ( , λ j X ( ` ) – 26 – DIM algebra. ` λ E X X ). λ 22 = − w X ( [∆( `   u X F 1) ⊗ Abelian ) w of the vertical diagram ). For q z λ λ, z c ( ( ): Φ p λ ∗ − ` n ? K b b gl zu

( ) for generating currents

d − , q )

shifts z )] = ( U ( ) and Φ 4.73

z λ z µ, w ( in ( Φ ? S and λ ∗ µ · , , though, of course, the DIM commutation relations are nontrivial. The non-Abelian DIM ) ) Φ X C c )) . Thus the intertwiner for the ( w c ( ) w λ = p ( ∗ µ . Gluing two intertwiners Φ and Φ 1 ` gl E z, w u ( = Φ λ z w µ An important property of the screening operator is a commutativity with ∆( [∆( We abuse the terminology and call Abelian the DIM algebra associated with the double loops on the T 22 in this terminology is the deformation of the double loop algebra on factorizes into a product ofand commuting shift operators, each dependingproduct on of its intertwiners own for quotient the Abelian A similar computation is valid for 4.5 Abelianization ofWe the would DIM like intertwiner toquotients reexpress the intertwiner ( The right hand side vanishes inductively in the number of boxes enough to check ( the commutativity isrelations easily for checked Φ by using the definition of ∆ and the intertwining any element This relation gives constraints (Schwinger-Dyson equations)the for network the matrix correlation model. functions of Since the coproduct ∆ is a homomorphism of the algebra, it is Figure 3 screening operator (vertical gluing). JHEP03(2018)192 , ) . 1 , c,r 0 − : 3 ˜ s, q H 4.20 +    (4.80) (4.78) (4.75) (4.77) (4.79) (4.76) r = ! # δ d ) 1 , r df q ), and we : rN 1 rδ rNp ) q 3.1 . − 1 #) r q ↔ − ) − rN − r 1 r − − 1 d, ( 1 q ˜ H − − ↔ − rN rd r r − − 1 1 − ( 1 . q q c, )(1 ) − 1 − H r ). We therefore conclude N − rN 1 =0 1 r − d X − q 1 N v c, q − ) − ) ! ij E.19 H − N 2 r m ) ) − rN 1 1 v − 1 q q N q . )(1 ! ( rN = (1 ) − ) ij − d N c 1 − 1 c,r 0 ( ( a q q N λ r 1 s, 1 H + )(1 ) − − ch r r )+ Nq N 1 − 1

(1 q (1 rδ 2 q ) ) rN 1 ( d ce q − − = − cd rN 1 d − rm ) q ij (1 L − 1
rN N 1 (1 . Let us first write down the expression ( Np − q ) – 27 – ( c − 1 d = − − ). We introduce modified Cartan generators 1 =0 r 1 ( c ce q c q (1 X ) ( q r N a ( c r 1 ) ) − 1 − r 1 − ) satisfy very simple commutation relations: : r 1 = c, cq q − E.16 1 1 q N ( ( ( L − H r 1 d,r cd ) T L 4.77 q ef ˜ L ) + H )) N L r 1 1 + 1 ) q ) expressing the colored character in terms of characters of r − =0 − 1 ( bosonic generators. d ) X − q 1 N ) + c ( q ( λ 2 1 ( − ) E.15 rN 1 − ch cd q

) using the colored characters: L rN 1 1 L − ) is from eq. ( ( v q − − 1 q ( =0 = 1 with arbitrary

( c X − λ ) N =0 1 − ) (1 q 1 e X cd N " 1 − are the Cartan and adjacency matrices introduced in eq. ( c ) =0 − 1 ) ( 1 c L (1 X r 1 − 1 N ) ij − c independent ,q =0 1 " , q c u,v 1 X − N ≥ m u, v 1 ( r 1 q r X ( N q N ( λ ( 1 ( ˜ λ t λ r 1 ˜ ( t λ ≥ ] = q C r X and non-Abelian DIM algebra. The unrefined limit corresponds to setting C are 1 f,s    − ij ˜ =0 H : exp ) = i,r c X a N , v ) = ˜ H ( × v We notice a further simplification which occurs when we rewrite the vacuum part of To minimize technical steps in the derivation in this section, we limit ourselves to the λ d,r ( : exp λ ˜ Φ H [ × Φ that the intertwiner in terms of the new Cartan generators: This property is easy to verify from the explicit expression ( where have used the crucial property of which are given by the following linear combinations of the original ones: The modified Cartan generators ( where the matrix Now we use thethe formula quotients: ( unrefined or, equivalently, to for the intertwiner Φ JHEP03(2018)192 . : , :    (4.87) (4.83) (4.84) (4.86) (4.82) (4.85) (4.81) # satisfy )    r r  ) ˜ H r ↔ − and the shift 0 r ↔ − ( ) d, d ˜ r ( H − DIM intertwiner! ( λ r  − d − r p d, − − ˜ H ˜ v H r f r p Abelian  z c v = 0). Then the zero modes f diagram  d ]: . − N 0 . The generators 0 ) d − 0 r , r 1 d, d, d b 54 q s, d ˜ 1 ˜ ˜ H H H ˜ ¯ + α − − Np ) 1 =0 r ) 1 N f 1 q 1 − )(1 rδ  1 =0 P N, r ) N, N d 0 quotient r + − 1  . d, d q ) ) as in [ − 1 0 P ˜ z q − H dp s, ( commute, the intertwiner is a product rN 1 ) mod ) mod = q + d (1 0 λ d − , r d − d − − N c c +1)+ ˜ ¯ and ( rδ α ( )+ d )(1 δ )(1 r δ 1 1 1 p d r 1 ( q − ˜ ¯ q α ) mod d ( − rN =0 − ] = j – 28 – N d λ s − 1 − − q i Np c,d ( c,d ch 1 2 ) into the intertwiner, we obtain P , a δ − δ H r ( ( − ) |− 1 (1 a 1 ) [ v − d i − =0 4.80 =0 (  ] = (1 d for different X d N − X λ N )+ s | j r 1 1 ) is correct. We can introduce a more convenient set of ˜ q N = 1 H = ( rN 1 r,d , − 1 ≥ q 0 c ˜ N r r q X ( H 4.82 c, ¯ α ) ˜ ) and ( H  d    [ ( H d λ ) mod ˜ ¯ α j d -dependence of the intertwiner and also indicated that it acts in 4.77 − ch p i 1 ( e q − ¯ α ) : exp 1  z − − , for which we have =0 ( 1 e Y r 1 d λ N ) − r q λ =0 c 1 ˜ d X ∈ − H N ) ) − = 1 1 " operators each depending on its own Y ,q i,j ) = ˜ r 1 u,v 1 ( r = ( q 1 ( λ ˜ r ) denotes the scalar prefactor, which we omit in what follows, and we have H ˜ t λ ≥ a z r X C ( z, ( λ    ) c 1 ) = q ( λ v ( Ψ : exp commuting λ , the latter entering only in the shift of the spectral parameter. Moreover, upon closer d Φ × which are independent (we assume that also factorize into a product of independent factors: Let us also introduce modified zero modes Notice that here the normalizationfor of the the generators Abelian is nonstandard, intertwineroperators though ( the expression explicitly written the the horizontal representation withthe the commutation bosonic relations generators where ˜ It is remarkable that,of since p examination each of the commutingLet operators us is denote nothing but the the Abelian DIM intertwiner by Ψ Plugging the identities ( JHEP03(2018)192 c  p N d,r ˜ H (4.88) . The powers shifts v, 4 quotients d − d . Np 1 d q integer  ) ) d N 1 ( q ( λ Ψ 0 -sheeted covering of the . Thus, the vertical legs d, ˜ v H -tuple of Young diagrams N  2 N d − p 2 p (2) − 3 1 horizontal Fock spaces and v λ picture for it, see figure on the vertical legs are the ) drawn in terms of the intertwiners f vq 3 ) p ) have the equivariant parameter N c d b b gl ( 1 f (1) ( λ − commute for different − N d λ ) has been omitted, since each factor 1 , d 4.88 p b 3 1 =1 1 d,r q − vq 4.88 ˜ =0 H U N f (0) P 0 λ p + 3 1 d vq dp . Notice also that the shifts are – 29 – d − d +1)+ d . Np p ( 1 q network matrix model d q ). The Young diagrams ∼ Np by 1 1 2 b b gl v ( |− ) d ) , v d ( ( λ =1 λ | q Φ N U − , and the spectral parameters of the vertical legs depend on the 1 . q λ  enter only as shifts of the spectral parameter E d d ˜ ¯ α p d p e 1 − =0 Y d shifts N . are also independent. , which are completely decoupled from each other. The vertical quotient diagrams 1 ) . The intertwiner of Fock representations of , i.e. one can view the corresponding Ω-background as a q ∼ d d,r ( ) N 1 ˜ of original one with parameter on which the intertwinercenter acts of do mass not position coincide, but are shifted with respect to their is already normal ordered, and the bosons q H λ v ( The The normal ordering in the product in eq. ( The Abelian intertwiners in the r.h.s. of eq. ( Each operator Ψ acts on its own horizontal Fock space with the bosonic operators Overall, since we have expressed the non-Abelian intertwiner as a product of the Eventually, we get the key result ) 1 4. 2. 3. 1. q ( λ Φ Abelian ones, we cannon-Abelian now intertwiner draw acts a invertical Fock the spaces. tensor The product latter have of the basis labelled by the where we have omitted the scalar prefactors. Several remarks are in order: Figure 4 of Fock representations of of the Young diagram obtained from thecollected quotient in construction. appendix More details on the quotients of Young diagrams are JHEP03(2018)192 , and (5.3) (5.1) (5.2) v ): w , ( ) ∗ ∅ horizontal ,N Φ · (1 zu/w N ) w )-KZ equation, 1 − i 3 q, t −→ F q it is nontrivial that 1 − j 1 +1) q . is for later convenience. ( ,N 1 ) zu ): (1 − − i,j q z ( ( ¯ c A priori λ ξ )-branes. If the branes in the λ ∈ )Φ ). −→ F ) p, q w N ) Y ( ( i,j ( b b gl ∗ µ ,N : ( N d (1 u 23 , ∼ q ) F of ):Φ U w ). ( ): 1 ∗ λ z ). The insertion of | z/w b b gl Φ 1 ( stack 5.2 − q,t q , – 30 – N, u ( U ( ): λ λµ z ( e G )Φ ∅ ] as a bilinear composition of the intertwiners w Φ | · ) = ) 56 z zu , of the horizontal Fock space. We also introduce a function z ( 1 − λ u 55 − , i 3 q )Φ DIM algebra 1 w + 1 − ( j 1 ∗ µ q N )-branes, each of them being represented by a Fock space in the ( ) label the states in the incoming and the outgoing vertical Fock ( Φ ) ∗ µ p, q i,j ( -operator [ Abelian ¯ w, µ c T η λ ) := Φ ∈ ) Y i,j ) plays a role similar to the two point function (the propagator) in the com- , while those on the horizontal legs are encoded in the momenta (zero modes) ( z, w ) and ( c | : p ∼ z/w z, λ ) 1 N, u z ) by the normal ordering of the oscillator part − ( ( q z λ λ One can consider a triple junction of a From the physical point of view, the phenomenon we observe in this computation is ( ( µ . The spectral parameters on the vertical legs are obtained from the original one We keep the ordering of the zero mode part in ( ) Φ and the spectral parameter T d 23 λµ λµ ( e e As we will seeG below, in the constructionputation of of algebraic correlation solutions functions tostructure based the of on ( the the intertwiners Wick theorem for the free fields. From the where ( space, respectively. In some ofN the computations below, itG is necessary to change the level 5 Level one KZ equationLet us and define Nekrasov the function for ALE space case: the branes pass throughto when each they other were and far form apart.protected the Perhaps quantities. junctions one just can in interpret this the effect way as they conservation used of certain stack are far apart thenwhen on we each move of them the there branescandidate is closer for still together, the an Abelian enhanced thethe algebra symmetry symmetry triple acting. algebra will junction However, is be ofjunctions. enhanced. stacks Our of The computation branes natural shows factorizes that at into least a in product the of unrefined non-interacting limit triple this is, indeed, the that of symmetry enhancement.of The three Abelian Type intertwiner corresponds IIBalgebraic to picture. ( a The triple DIM junction of algebra the plays brane. the role Since ofrepresented there by the the is “worldvolume gauge only symmetry” one brane, the symmetry is essentially Abelian, hence, the shifts of the corresponding bosonic fields.triple The topological Fock vertices spaces so areFock intertwined that, spaces. pairwise as by a the usual result, one gets a tensor product of λ JHEP03(2018)192 (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) (5.11) (5.10) (5.14) (5.15) (5.12) (5.13)  , w 1 1 2 − q   z . 1 2 (+) k −   e q V , ,  ) )  u 1 2 , 1 nr ,N ,N − u − (+) (1 (1 − ): 3  q q i , . . q z V w ( ): , 3 2 ! 1 λ − ): z ) q −  ( z , the diagonal components of r 3 2 ( ∗ λ 2   n 1 )Φ nr q −→ F −→ F 3 ∗ λ ) z  q − 3 w ) ) r 1 2 , )Φ − q ( − q ( z . ) )Φ q k µ ,N ,N ] w r z   e r  − ( V r w q ( [ ij r (1 (1 u u )(1 ( ) µ z 2 b :Φ 2 nr n r 1 3 ∗ µ F F − 0 ∅∅ q q ( 1 − ]. In the following, we renormalize the 1 i s, ) = ] e q − G V :Φ + n − ) ):Φ w 55 / [ r   ( − ) 1 δ 2 n ): ): (1 z − ∗ ∅ ( 2 − ) = z ) − 1 nr r z/w z/w 1 1 Φ z q q 2 1 q ( λµ z, z − − i  q z, e =1 − ] = ∞ – 31 – G | | q q z/w ] r X ] ( ( ( 2 n j,s 1 relation [ r

, [ nr q λµ λµ λµ Λ [ , ξ ) = 1  N, u N, u , e e e r z G G G 1  ( ( − ( ) gives the amplitude of the conifold and is the origin exp z λ λ i,r  λ λ z 1 2 z =1 λµ T T ∞ RT T ( ) = [Λ q r X ) = ) = ) = 3 2 → d G z z z  q , )   ∅∅ ( ( ( -shift in the vertex operators, we can see that when the ratio -matrix q  z λ ∗ λ ∗ λ q e ( ( ) := ) := G (+) R ); z z i ii | | )Φ )Φ )Φ (+) z b V ∅∅ k ( w w w e V e  G ( ( ( λµ z µ µ ∗ µ ) = exp N, u N, u  1 2 e ( ( G z Φ Φ Φ z ( − ) simplifies to + − λ λ 1 2 q z T T q (  ∅∅  ) e ) G ∅∅ − ( − e i ) = 1. ( G by k V z e V ( 1, λµ ∅∅ e ) = G → ) = z G z ( n ( i -operator have no positive or negative modes. Namely, if we define η ∅ T Φ of the incoming andthe the outgoing spectral parameters is so that 5.1 Shift operatorFrom and the combinations of the In topological string theory, of the “anomalous” factorfunction in the When Since the diagonal component of inverse of the deformed Cartan matrix is we find with the commutation relation Recall that the vacuum components are given by all the OPE relationsof (the the two point single function functions) of the intertwiners are expressed in terms with the free field realization JHEP03(2018)192 ). w and ( ∗ µ λ (5.22) (5.21) (5.16) (5.17) (5.18) (5.19) (5.20) z , )). Since
. z ) 2 5.20 ) and Φ ) so that we z − | z z . q ( ( | , u λ , , and the depen- u ! u 1 1 ∅∅ ) shift the horizontal − − e , + 1 G 2 |
nr q 1 , z − ∅ 3 ), we have to use the − N 1 h q ) ( − ∗ λ − λ ) ∓ + 1 q λ − c | Φ ) which follow from those 2 T 5.20 ]. It should be noticed that nr 0 z 2 N | nr | . 3 − 2 λ 57 ) q ) = . q N, u z ) + −| z λ -shift of the intertwining oper- | )( | 1 q and ( − 2 T N, u 2  + λ − nr ( ) 1 ) and (  2 λ case [ q nr T z  2 − 1 − λ | N, u N, u T ) = ) q 1 q q ( ( T ( z z ( . | , u |  gl − 5.19 λ λ − u λλ N T T q | 0 2 e | , nr + 1 G ) and the negative sign for ( 1 λ N, u ∅ q 0 | ( h ( N ∓| λ + 1 | λ ( r 1 q 5.19 λ Φ λ ∗ N f on both sides of these relations. We can see +1)
Φ ( =1 ∞ 0 λ – 32 – , z r X ) = N ∗ λ 1 i q ( 2 C z − Φ |  λ ∓ − ∅ |  q q ) and their inverses as follows: C

| , u z ∗ λ + λ z 1 t c ( λ uz −  t ) changes the level and the spectral parameter of the , λ + 1 q = − ) | T w , i z u = ( | N = exp z q ( ∗ µ ∅ | 1  λ  + 1 ) λ c −  z T N, u | ! ( N , while keeping the level
N, ) 1 λ z 2   2 Φ − λ − (the positive sign for ( q N, u 0 − − (1) ) and Φ | λ q ( T is simply q 0 λ ) to be discussed below, the prefactor can be simplified to ( z T | ) | | + ( λ 2 λ ∅∅ q

) is independent of the horizontal spectral parameter u | λ T N ) 2 ∅∅ 1 − z z e G | (1) 2 q − e 5.23 − N ). In the computation of the commutation relation, it is important that G ( (1) q − q , which is consistent with the fact that ) = w λλ q ,

u z | ( | λλ λλ e 1 N, u G (1+2 ∗ µ u ( u e λλ e  G , the shift parameter is the same as in the G q  + 1 q e q  2 0 λ G q | / 0 T λ 1 | N N, | N, ) 2 λ -shift of the vertical spectral parameter is accompanied by a shift of the horizontal ( | 2 2 − λ ∗ λ λ q q t/q − ) and Φ Φ Φ Φ In order to derive the KZ equation based on ( q z ( = = = ( λ We want to requirehave a them simple to algebra commuteΦ of up two copies to ofthe the the exchange anomalous Zamolodchikov of algebra factor satisfied Φ horizontal separately representation. by It also involves the exchange of the zero mode factors the parameter spectral parameter by commutation relations between theamong intertwiners the and intertwiners. Let us begin with the commutation relations of Φ It is instructivethe to factor count theq power of Due to relation ( where we have used ators is realized as the action of Actually dence on the level A crucial point in deriving the KZ equation is that the with these operators satisfy JHEP03(2018)192 ) × n 5.27 (5.29) (5.28) (5.30) (5.31) (5.23) (5.24) (5.25) (5.26) (5.27) ALE , , ) ) , and ( ) on | v 1 ) z ( ( z µ − , ∗ | µ ( z ) z λµ ( 2 ) + N ∅∅ u . | ∅∅ ( e /w } λ G λ | 1 e 0 G z ) implies w , ) , ( ≡ ) 1 ) denotes the number of ) = z z ) − νρ ) = ( t 0 ( ) 5.23 | ( z ∗ µ µ R ( λ ) ) | λ, µ ], 1 ( )Φ λ,µ )Φ u/v − ( ( h w w | H 1 56 , , w ( ( , Υ µ 1 ) − ∗ λ λ , z ) stands for the anomalous factor q z ) 55 ) ( ∈ ( 2 z ), and , )Φ z )Φ ). w λ . ( t − ν ( ) ( ) µλ { q 1 T ) ∗ µ  ( ) , λµ G z − 4.23 2 z/w z/w 5.23 case [ 1 ) ) z (0 )Φ ( λµ G ( ) of the (dual) intertwiner. We find that 3 3 ) = 1 1 − 1 z + # 1 1 , w G q ( − λµ − λµ , q , q 2 } ( − ) gl λ q 24 1 1 ]. Υ z 0 z ( R R 4.29 q q ( ( ) ) ( ( λ,µ ∅∅ )Φ µ ( ≡ ρ λ µ 56 ∅∅ e µλ , G ), ( T f ) agrees with the Nekrasov factor f H ) e – 33 – G z s R − z/w z/w 0 ( 55 z/w | ( ) ( ( q ( ) µ z , we prove ( | λµ ) = 4.22 ( . − µ,λ 2 ) = + G (0) (+) ( 0 D h z | ) = λµ D | ( ) is justified by the fact that we can derive the following case [ λ , w z | λ ) = 1. Note that the relation ( R 2 ( 1 1  (0) z -matrix defined by ∈ ( − gl λµ v u ) = Υ ) = Υ ) = Υ 5.27 Υ µ z ) is a generalization of the usual symmetry of the Nekrasov factor s R z z z ρ  R ( ( ( ( { T ) λ λ ∗ λ ) ), we can write down the commutation relations of the inter- ,  5.23 1 , ) ) = 1 )Φ )Φ )Φ ) (0 u , w z − w w w 5.23 ( ) = # 1 2 z ( ( ( )Υ . In appendix λ z -matrix ( 2 − ∗ µ µ ∗ µ z z ( λ q − ( ) R λ Φ Φ Φ ( ) ν λ, µ q v ( ( (  T , 25 ) ∗ µ ∅∅ H 2 (0 z ∅∅ e ) G e ) is the generalized framing factor ( /z G 3 1 z u/v , q ( 1 1 ) = q − λµ z ( q ( λ ( R ). Thus the formula ( f relation by a computation similar to the λµ (+) Using the relation ( 5.33 G Υ As we will see in the next subsection, See also the computations in appendix ( , which are the group algebra parts ( 24 25 1 ∗ µ up to the anomalous factor from the vacuum contribution. S incorporating the contribution of zero modes. The definition of the RT T which satisfies Υ When there are no constraintsreduces on to the the definition relative for hook the length, where where we have introduced the boxes with color 0 in twiners as follows; the above requirement is satisfied if and only if where z JHEP03(2018)192 (5.36) (5.35) (5.32) (5.33) (5.34) , .  -deformed 3  q ) ] t , q ( 1 λ ` q 3 ; q . To define the 141 1 2 , z w − C ) 1 t ( 140 − µ q on a ) of the intertwiners is  z − 1 ( ∅ +1 λ uq . n λµ ) N Z 1 − G ]. Since our current problem − 3 1 =  , q ), we obtain .  1 149 is given by [ ) on the ALE space. By a direct 0 u – z − 1 ( w ≡ . 1 ]. ) q λ 1 ) ) µ t S ( ( 3 3 ∈ ) + 1 − 5.33 0 λ t Y 144 ) s 143 s × , q , q λ,µ ( C ( 1 1 ) h λ n ∅ q q 3 ` ` ( (  − 0 3 1 λ λ , q ) and Φ q 1 − C C v ) ) ) + q ALE s ( s s ( ( ( ) is related to the normalization factor of the -deformation. It is interesting to see how the ( ∗ ∅ λ λ =  µ 3 q µ a 1 ` – 34 – C 3 z q a w , q − 3 ,q 1 1 1 q 1 ) q − q − s i ; 3 ) q ) = ( ) = q 3 u λ λ s | 1 ( a − 1 ( , q λ − 1 1 j ( 1 λµ uq q λ,µ q  . At the CFT side, the uplift might be related to N 1 h 0 − − ≡ (1; A 1 q  λλ )+1 − 0 s N λ ( ≡ 1 ) ∈ ∅ λ  s Y s ` ( ∈ λ -theoretic Nekrasov function for ALE space 0 Y s version of the Jack polynomials (Uglov polynomials) is obtained by )+ ∈ ≡ s µ,λ ) j n K ( h Y λ − i,j gl i ( a ) = = = 3  , q ], the z 1 w q 1 ; − u q 143 ( ,  λµ ∅ Now we argue that the renormalized two-point function 142 λ N -coset models. We can expect the same level-rank duality as in the undeformed case, G level-rank duality is realized in the setting of quantumnothing toroidal but algebras. the bifundamental mattercomputation contribution of ( the OPE factors between Φ See also Lemma 2 in appendix W since the character is invariant under the factor of the intertwiner is closelyas related a to generalization the norm ofThough of the an such uplift uplift an of of the upliftfactor Uglov the polynomials from is Jack that not polynomials of available the to at Uglov the polynomials the Macdonald given moment, polynomials. in [ we can guess the normalization In [ taking the roots of unityan limit important of the role Macdonald in polynomials.the the Nekrasov The four-dimensional Uglov partition polynomials (Yangian) function play should version on of be the AGT related ALE correspondence to space for [ a five-dimensional uplift of this story, we expect the normalization The following specialization of intertwiners: Then the building blockNekrasov (the function bifundamental for matter instanton contribution) counting on of the five-dimensional Difference of the Nekrasov functions forrule the for flat the space boxes and of theinvariant the ALE part space Young of is diagram. in the The theselection selection selection character rule, rule we under is introduce the a the consequence orbifold relative of hook action taking length of the 5.2 Relation to JHEP03(2018)192 1 in ): z (5.39) (5.38) (5.40) (5.41) (5.42) (5.43) (5.37) − ( j λ Φ .  . i w
. − 1 ) − j 3 z/w q ) = z, w j 1 , ]. Here the vertical ( ) µ 1 − λ q i, j µ q 87 (  T -matrix for the tensor . ) ∅ ` z, w 0 ) ` ( ξ R λµ z λ w -difference of the vacuum : ) µ 2 ( 2 G z T ∗ ν − −  (  ) 0 q ) with respect to the second q Φ ( z 1  u λµ  ( ( − e z/w λµ κ G z/w ) and use 1 1 λµ e G . z/w − z/w j )Φ − ) ) j j G /w j j 3 0 ) − − − − − /z i i 3 3 i i 0 λ,µ 3 3 w z , q i ) ( q q ( q q 2 z 1 1 1 z λ ( 1 1 H q − ( − − νµ ; − − q − j j κλ j j ( u q µ µ R µ µ ∅∅ ( ) R − − ) − − e ) i i ∅∅ G i i λµ n λ λ 1 1 λ λ 1 1 e ) = /w G q q ) satisfies exactly the same recursion relation. /z 0 q q N b b z 0 gl 1 1 3 1 1 ( – 35 – ( z w − − − − ( ( d , q , q q ) λµ q q 1 ) = q ) := − q u (+) ( R z − − − − ; U ( ) u 1 1 1 1 z ( )Υ )Υ Υ( λµ     ` ` Υ( G λµ /z , /w , ) 0 ≡ +1 ) ≡ 0 +1 i , ` λ i , N ` λ z w ( λ =1 ( λ =1 )+1 Y ≡ ` i )+1 ( ( Y ≡ ` i i =1 − λ i =1 − Y λ i -matrix by Y i ) := ( i λ i ( λ ` -operator and the intertwiners: z (0) (0) ` − R − ( i i T λµ = R ) =
) = Υ ) = Υ 0 0 z ) to that over the arm length i ( z w ( λ λ ( κ z/w 1, we obtain a recursion relation for ∗ ν Φ 1 ≤ )Φ −  − )Φ , we prove that j q j w 1 ≤ C µ j z, w − z, w ( , j 3 ( − +1 λ q µ λ µ µ j j T λµ µ T 1 q G ≡  In general, from the normal ordering ` ` ξ It is amusing thatproduct of a evaluation similar representations of decomposition the takes quantum affine place algebra for [ the It is convenient to introduceanomalous a factor universal function that is the and define a renormalized mutation relations of the Thus, we have 5.3 Level one KZFrom equation the for commutation relations between the intertwiners, we obtain the following com- In appendix diagram for each row (1 where, in the second equality, we convert the summation over the co-arm length JHEP03(2018)192 ) ) we . , | ) ) 5.19 5.16 v v ∅ (5.44) (5.45) (5.46) (5.47) (5.48) ( ( h ) ) . , ) ) n,m n,m z w and ( ( | | to the right. , G G i ) ). Using the ,u + ,u λ , ) 1 ∅ w )-KZ) equation acting on each i | | k given by ( T − , ) representation. +1 ) ∅ +1 x ) | 5.20 k  λ q, t /w m ) z c N ` N x∂ | N,u )-KZ equation; n ( /z ( z 2 ( z − j q µ + − ( µ λ z ( q, t q n ( Φ k T N,u , n λ k µ ) ( )Φ  ` λ ) and ( Φ z j z λ w + | λ | | λ T R u u w ··· R ) q wu 5.19 ) to the left and , n − =1 w 1 − | Y ` z − , Y k

, λ t q P N (1 1 − P z z ) p Q  . 1 ⊥ ,N 1) −   2) , q,t (1 ⊥ ) uQ d ⊥ 2 − Υ (0 zQ : n , Q ⊥ − , λ d ) z, λ (1 t 2 ) d ⊥ n Q z , which involves the scalar ∗ Q − − 1 u ,N d x − (1 N 1) Q )(1 − − 1 − n + 1) n Q , q x , λ ⊥ w, µ (1 1 ( − z ,N (1 uzQ (1 2 z, λ u − n P n ) α z w 1 ⊥ ,N d ⊥ ≥ u X n Q (1 d = – 41 –   Q 0) ) = , u (1 u ,N 1) ) = exp , (1 x ) | (0 α × u ( ,N z, λ  (1 q,t j i + 1) Υ uz z P z − z, λ

,N t q (1 ) = ) through the grading operators r i ⊥ , ~z d ⊥ w, µ z  ⊥ ( ) Q q,t i d q,t z w λ Q ,N u Υ 1 (1 − =1 Q, P, P N N ( Y + 1) − i,j Z Q = ⊥ where we have indicatedcorresponding all to the the slopes legs explicitly. and spectral parameters of the Fock spaces The crucial point is thatΨ the so grading that operators satisfy the intertwining relations with with function Υ Here we use the commutation relation for Ψ with Ψ ,N Move Ψ (1 uzQ b) ( − JHEP03(2018)192 2) 0) 2 2 , − 2 2 (6.17) (6.16) , 2 z u , λ (1 , λ 1 2 uQ z (1 2 0) using Q z P z z , ), whereas ⊥ ⊥ 1) 1) ). d ⊥ 1 − 1 ⊥ − P , , λ , P,P d ) automorphism 1 1 Q (1 which used to be , λ Q z P (1 Z 1 ⊥ , P z z 2 z 1 0) u z slope, 2 , 2 P Q, P 2 P (1 , λ 2 ⊥ Q z P z d ⊥ ) d ⊥ ). There are two key differ- ⊥ Q 1) Q d 6.2 − sitting on the lower ones. ⊥ Q , 1 P ( 1 ⊥ (1 , λ d d 2 ⊥ ) 1 2 2 Q 2) z P z z P 1 , λ − uQ 2 z d 2 z PQ , 1 ( z P u z 2 (1 P 2 Q z -element from the SL(2 sitting on the upper vertical legs are effec- ⊥ ⊥ 1) d ⊥ d ⊥ T ⊥ Q 1 − Q d d ⊥ , d , λ Q Q Q (1 – 42 – 1 1 d Q z z Q 2 2 z 2) 0) 1 uQ z , u − , (1 ). To compare the parameters of the theory with that of (1 N − × , ×   j j i i z z P z P z

t q t q ) = ) through the trace so that they emerge on the right: r i , ~z r z  ⊥ (  i ) = q,t λ q,t Υ , ~z Υ ⊥ -transformation, i.e. act with the =1 Q, P, P N =1 T ( Y N 0) and has become (1 i,j Y i,j Z , = 26 the initial setup, we need tothe transform the slope ofgroup the Fock of space DIM. back to (1 The fugacities on theafter vertical the legs cyclic are movement of different. the intertwiners They theyThe used became Fock to space ( over be which ( (1 the trace is taken has a different Therefore, the grading operators tively multiplied with the grading operators Recall that the corresponding (double) wavy lines are identified with each other. = Q, P, P Move Ψ ( The horizontal spectral parameter has also changed, but this is inessential since there is only one c) a) b) Z 26 ( ( ( horizontal line and the overall shift eliminates this difference. The expression looks almostences: the same as the initial one ( After these steps one arrives at the following picture: JHEP03(2018)192 + 1) u (6.20) (6.21) (6.18) (6.19) , m (1 ⊥ d ) : ) Q ⊥ ,m ). Q (1 ( u Z d , ) of the DIM algebra F Q ). Let us consider the ) Z i to all the intermediate , Z − , 1 2 m N ( − z, λ T λ + 1) Q SL(2 ∈ N ) ⊥ 1) − ∈ , m , i,j ( T × uQ (1 (0 T ) in the basis of Macdonald poly- ) P t ) 1) q, t u , m , 1 2 ( = λ (0 (1 on the grading operators is explicitly z − f j | F ( λ | T λ ) ∈ z ) in turn: ) + 1) T − u i,j ( ( , m 6.17 P q (1 | + 1) – 43 – λ | u 1 = − , m transform as a doublet of SL(2 1) T ]. (1 − ⊥ The action of 80 ) Q z, λ z, λ , u T , m ) = ( Q 1) 1) (1 , , q, t ( (0 (0 ⊥ d ⊥ λ f Q The action of the automorphism d ) and commute them with the intertwiners and grading operators Q m -transformation naturally transforms the slope vector (which is the − T ) 6.17 Q ⊥ , m (1 uQ on the elements of the network ( T T m − Notice how the fugacities given by These actions are,intertwiners of written course, down in consistent [ with the explicit expression for the DIM where is the framing factor. It acts diagonally onnomials: the vertical Fock space vector of central charges) of the horizontal Fock representation + 1) Q On the grading operators. On the legs. -transformation. Now we are ready to insert the identity operators 1 = ⊥ T a) , m b) ( ( uQ (1 legs of the network ( on the vertical and horizontaloperators, representations as is well easy as to the deduce.action central of In charges form particular, doublets the under grading SL(2 2. JHEP03(2018)192 N T 0) , (6.22) 0) 2 (1 , 2 2 z and , λ 2 1 ) (6.23) (1 2 uQ z 1) − Q z , N , ~z T 2 − (0 N 2 − T T P ) intact. In this 1) 1 2 , N T 1) , λ (1 the 1 , Q Q z 6.23 ⊥ (1 2 1) − , T . What remains is the 2 part of the gauge theory (0 2) 2 , i.e. of the gauge theory ∅ − , ⊥ 2 ⊥ T d (1 , λ ) 2 2 P = 2 Q, QP, P Q z P z T ( 2) in between Q 2 , in the argument of the partition 2 Z λ i P (1 2 1) 1) z classical  ⊥ − , , j 1 = Q T i (1 (0 ⊥ 1 ⊥ 2 z , λ d 1 2 2 P z P 1 − ) Q z ( λ P z

T T d T ⊥ t q ) 2 Q T 1) r , PQ ⊥ ⊥ ( d  P (1 2 ) ( 2 − q,t d Q ) T – 44 – ⊥ Υ 2 2 2 Q − ( T T =1  d T PQ N j Y 2 ( Q i,j 1) i T , 0) z 2 2 , P z = z 0) (0

1 , uQ z (1 2 t q (1 − functions arises from the T r  q,t × ⊥ q,t d ⊥  j Υ Q i ): d z =1 P z N Q

Y i,j t 6.21 q 2 r ), ( ) = ) = T  0) ) and a similar one, but with shifted parameters: , ~z , ~z , q,t ⊥ ⊥ 6.18 , ~z (1 Υ ⊥ =1 N Y × i,j Q, P, P Q, P, P Finally, we obtain the equality between the initial gauge theory partition function ( ( = Q, P, P Z Z ( partition function. Indeed,coupling the constant, thus prefactor we can islimit, safely independent send the it of gauge to theory zerothe and instantons still vertical do keep lines not eq. ( contribute,strip become or, of uncompactified in intertwiners so the compactified that language only of along intertwiners, the horizontal direction. The prefactor can Notice that we have omitted an inessentialfunction. overall shift The of product of Υ Z The slopes indicatedoperators. on the picture are those appearing using eqs. ( JHEP03(2018)192 ]. 57 1, we (6.25) (6.28) (6.29) (6.30) (6.31) (6.26) N > ) gives the 6.28 . For ) (6.24) τ   , ~z ) (6.27) ) expression valid for ) partition function, ⊥ n , ~z N Q ⊥ U( − exact d n Q, P, P ( − )(1 = 0 in eq. ( x n Z Q, P, P , ]). 2 n − ( ) ε  t . . Q τ gauge coupling i j , ) 163 Q − ~ + + ˆ x – | Qz P z 2 n

α periodic ε )(1 = and x ( t q 160 n Z 1 + q 1 − q,t r ε 1 t − ε ,  ) = q + (1 = Υ 27 , ~z n ) N q,t,Q m n ) 2 α Ξ – 45 – x is invariant with respect to the following shift of (2 | 1 P Qx | τ, ≥ α N N i,j N Y ( α X n ( + ˆ Q . + − i ⊥ ˆ a m τ τ,   ) = periodic q,t,Q is symmetric in q,t,Q Z Ξ , ~z 7→ 7→ 7→ 7→ Ξ  ⊥ i x ˆ τ τ . This second transformation can also be easily analysed in a m 1 2 1 τ Q Q, QP, P ) = exp

( α x | 1 2 Q, P, P 7→ − α ( − ( ) τ Q ( periodic -deformation, this transformation law coincides with what was found  ε Z q,t,Q periodic Ξ Z q,t,Q ): 6.10 ] for the classical integrable system. Here we provide an Notice that Ξ 85 27 prefactor. This prefactor gives risewill to discuss an only extra one shift of of thethe the two second modular being transformations of the theour 6 framework, but we leave this task for the future. The properties of the partitioncompactification function with torus respect can to also theexplicit modular be transformations expression deduced of for from the the theIt network trace is matrix of written model. intertwiners as over We a take the product the horizontal of Fock theta-functions, space which from are [ modular invariant up to a simple in [ arbitrary Ω-background. Intransformation particular, law for setting thesystem (as exact usual prepotential in of the Nekrasov-Shatashvili the limit quantum [ 6.2 double elliptic integrable Modular transformations In absence of the Now the shift ofnary ( the gauge theory parameters can be easily extracted from the dictio- The resulting partition function the parameters: and we have the following difference equation where be absorbed into a simple redefinition of the partition function JHEP03(2018)192 (6.35) (6.36) (6.38) (6.39) (6.40) (6.32) (6.34) (6.37) (6.33) ,  
n ˆ τ
) 1 πi P ˆ τ , 2 t q e πi

 2 p e T j 2 ]. | λ πim )  57 (1 . 2 − ) n ). The term in the i e i t # λλ Q 1 2 . Θ 2 + − − λλ  6.33 i  .  ) T j Θ µ ) i 2 ) x λ Q +(1 −  j m k n + ) πiz −
2 ( + 1) n − Q j T xq 1 e τ i Q ( ( ε λ |  PQ + 1) − − λ − t q | − i Q πiτ 1 ( , 2 θ πi T j , )(1 1 p , e 2  2 − , µ n ε λ 1) + θ e ˆ τ 1 ( ˆ πi ∈ τ  − 2  ˆ T j πiε τ ) 2 t 1 πim ) 2 ˆ − n τ x λ 2 ε λ Y πiε τ − − k − ( i,j − j 2 X e 2 ( e − 1 prefactor used to construct the periodic 1) + e P e 1 Q ε )(1  + 2 ∼ n = ) + + = is multiplied with the exponential of the − n = i − ]. There was also a typo in [ = j z − q  +1 ˜ ) ( j λ ˜ ˜ P ˜ i q t λ 1 q,t,Q Q ) + 57 Q − − τ ) qt

πi − j − ( + i ( (1 T j Q e P 7→ 7→ 7→ | i 7→   2 λ − t µ − q 1 2 ( ε λ – 46 – t 2 1 1 ˆ (1 τ i ) 2 j +1 p − λ ε πi k − − λλ πim  πiε πiε  ( ( iτ i 2 2 2 2 2 2 n Θ Q λ e τ − e e ε ε λλ − − e 2 − Θ τ P xq = − = = − − | 2 n ) = (  1 1 2 = λ t − ) | Q τ τ P  Q τ q 1 qt πiz 1  + θ 2 2 ( 2 − − ≥ ⊥ λ e − Y + P k 1 1 m ∈ P ( 1 ) t q     m Y πi τ . The modular transformation of the theta function is given by i,j 2 1 n ( ) =  + − + r − 1 x " e ) we exclude an extra Ξ  ( λ ≥ θ X n ∈ Q ) = ) λ θ periodic   Q X X | 6.32 i,j Z ( x × ( ˆ τ πi λµ ) = exp Θ ,Q ) is the Jacobi theta function: ⊥ x ( 28 Q P,P θ ( We slightly change the notations as compared to [ Z 28 we get a prefactor frominstanton each expansion theta-function labelled in by thefollowing the expression: product diagram in eq. ( Making the modular transformation Notice that in eq. ( partition function the standard formula: and where As a warm-up, we considermodular the properties U(1) theory, are where still thecase. nontrivial. expressions We are In have: simpler, particular, though there the are no vacuum moduli in this 6.2.1 U(1) theory JHEP03(2018)192   n , thus  (6.45) (6.44) (6.41) (6.43) j i m ). The z P z − t q 0) theory 2 , ε 6.40 p  − ) double elliptic 1 n . ε = (2 ) (6.42) Q  − N Q → − )

d +(1 Q ) P m | ˆ τ, τ, m n , t n , which we assume add q ,
) a quantized (1 Q 2 a q ˆ − τk λλ PQ , ε  is the same as that of the . t 1 q πi Θ . The spectral duality and .  )(1 2 ]. Notice that our derivation ε a  ˆ λλ τ ( e n τ p a ) ˆ a τ Q a ˆ − 85 τ πi Θ

) t Z 2 −  | i j πi πi e z z Q λ n − 2

2 |

) ∼ e − . This implies the second modular and e | i j P j 1 ) z z τ a t )(1 ) (1 q P j  ( τ n πim − = a 2 n ) i 2 q j p ˆ − τ a ( − m a i  ) − λ πiτk + ˜ a |− a , and ˆ ) ) ( 2 λ i j m qt ) | (1 ( πi ( e ( ( πi τ 2 λ 2 λ 2 ) πi ε e − e i ˆ Θ τ − 2 1) vacuum moduli ( ( πim 1 The partition function becomes a product of e λ ) 2 + = j  − ( (1 e Θ )  1 | λ a 29 j 1 ) = 0 the theory becomes 6 n ε λ ( ˜ – 47 – N i z ˆ | τ ( λ − ) ≥ =1 ) N λ Y 2 k i + Y P ( ε m i,j 7→ Θ λ 2 n | + ∼ m ) a λ 1 Θ | ( ε qt 0) + (   πia m N i,j 2 Y m − N ⊥ ( e ) 1 − P P m  ˆ τ, τ, = , 1 n Nm ˆ πi τ , τ 2 N 2 2 a 1 − ˆ 1 τ z  e τ ≥ , ε ( t q | − X n 1 = λ ,  ε |   ( 2 ˆ τ πi    ε : Z 2 = 0. The modular transformation of ˜ , Q . Notice that for e exp ) λ

~ 1 without any punctures. τ πiτ a , if we perform a shift of the complexified coupling of the gauge theory ˆ τ ˜ X ε ˜ 2 Q λ a P ~ τ 2 N | i,j e ˜ ˜ t q Y X ) gauge theory can be understood along the same lines as the U(1) one. ×  T (1 =1 = N ∼ Z q N a ) = λλ ) theory  ⊥ also leads to a symmetry enhancement. P Θ P N 2 λλ ε ,Q,~z Θ − : ⊥ 1 ε m -functions: η − As we have mentioned earlier, the U(1) theory is spectral self-dual, which, in this case, In the classical case, this matches the transformation law obtained from the Seiberg- P,P In the Ω-background, the theory is invariant with respect to the reflection ( = 29 Z up to zero, mass m Now we have a nontrivial dependence on ( where we omit anmodular invariance overall are prefactor evident in independent this of limiting6.2.2 case. ˆ U( The case of U( The partition function is equal to transformation for compactified on two Witten theory techniques and modularis anomaly valid for equations the [ general Ω-background,integrable in system. particular, it holds for the means that it is invariant under the exchange of transformation of ˆ encoded in final answer for the prefactor looks quite simple: The partition function is invariant (up to an overall scalar factor) under the modular There happen to be many cancellations between different terms in the sum ( JHEP03(2018)192 . d  and Q

) 2 (6.47) (6.46) i j z z Q T

i j P z z t q ( ) j p 2 (  λ .  ) ) i j ) ( ( # ) (6.48) i λ ) these solutions λ 2 ) n i + Θ  ( )-KZ equation for ]. b b ) λ cancels completely. gl 2 i 2 )T ( Θ a a 85 d  q, t ( j  , + a q λ =1 U )T N ˆ τ, τ, m,~a − Y a +1) , i,j +1) ( ( j i | 2 . i 1 λ λ n ε | − , ε − − Z 1 ( / )T ε )T 1 b 2 ( b ( j ε 1)+ ( j πimN C λ Z λ 2 − ( ( 1 j 1 |− 1)+ ∼ ε ε λ ), but with horizontal level one. In + | ]. The first generalization is the case ) − n  b )+ j ( i )+ b b ˆ gl τ 57 ~a j j ( λ ˆ + τ , d ) − , − − πimN b ˆ q ) τ ( m ) ( i 2 a a 2 U ( i , λ e ( i ε | ) λ λ − λ 2 ( | ( − ( ) 2 ε 2 2 2 – 48 – ε τ ), where we postulate the KZ equation using the ε ε ε + 1 2 − + − − − 1 1 b b gl 1 τ ε ε τ ( τ ) derived in [ ˆ τ + 2 1 + − 2 2 − − + q,t b m b b 1 gl 1 ( 1 a U ( m m + + + ( − b q,t b b a a U a m a ˆ a τ πiN − − − 2 + N a a a e a a m a − = +    ) ) )  , τ b a b m ( ( ( ˜ ˆ 1 Q τ λ λ λ

 ) ) i ∈ ∈ ∈ j ˜ − ˜ ˜ ) ) ) z z Q a X X X , (

˜ i λ j P i,j i,j i,j 2 ˜ ˜ z ˆ z τ ( ( ( ˜ ˜ t q ∈ ε horizontal level in ( ) X , ) + − − j q 1 i,j ( ). Thus, the networks of the “non-Abelian” intertwiners can be redrawn as ˆ  τ ( λ ε 1 ) gauge theory with massive adjoint hypermultiplet compactified on ) ) " i j b b ( ( gl  N λ λ ( ) N a,b n i Z Θ X ( U( ,q λ n ˆ τ arbitrary πi Θ d q In both of these cases, we still find only algebraic solutions. For We also consider an application of network matrix models and KZ equations to 6 U =1 N Y i,j are related to the Nekrasov functions on the ALEgauge spaces theories. We identify thethe compactified 6 network of the intertwiners corresponding to the “non-Abelian” quantum toroidalthis algebra setup, we findshow that the (at expressions least for infor the the unrefined intertwiners case) of theymore the factorize complicated Fock into networks representations of products and the of Abelian the ones. intertwiners We call this procedure Abelianization. We have presented two generalizationsthe and quantum one toroidal algebra application ofof the an ( analogy with the quantum affine case. The second generalization is the KZ equation for The transformation is consistent with the classical case discussed in7 [ Discussion The partition function is therefore invariant under the following modular transformation: Again there are many cancellations, inEventually, particular, we the have dependence a on simple transformation law: actor, which is the exponential of After the modular transformation, the theta-functions in the instanton series give a pref- JHEP03(2018)192 both (A.3) (A.1) (A.2) integral , ,` j +1 λ ,` − gauge theory. j j λ δ d −  j δ j k . . x ) ) x  -corrections to the known ) 1 )-KZ equations, when 1 j k ε  − − x 3 x ( ) ) by the following factor: q λ q, t  ` u ( − ( − 3 − λ λ ` 1 q 1 − 3  q  ) )+1 +1   ) )+1 +1 ( ( k λ λ k λ λ Y -th row, the change of the normalization ( = ( Y a a ` 1 1 = ` j k q q j ,` ) is slightly different and given by − − +1 j u ,` λ ( j (1 (1 − in the λ ∗ λ 0 0 j δ − – 49 – ` ≡ ≡ j λ λ ) )  δ ∈ ∈   j k ( ( Y Y    λ λ x x j k h h 1 x x − 1 2 q q ) = ) = 3 3 − − still remains to be understood. In this case, new 1 , q , q 1 1 1   q q 1 ( ( 1 0 λ λ − =1 − =1 Y j k Y j k C C = arbitrary k is given by λ +1 0 λ λ C C C When we add a box with color ) modular transformations and spectral duality of the 6 and Z λ , C Our work is supported in part by Grants-in-Aid for Scientific Research (# 17K05275) The most interesting and nontrivial generalization of ( In this subsection, we will prove technicalLemma 1. lemmas on these normalizationfactors factors. Note that in the product thereof is the a component restriction of on the the length dual of intertwiner the Φ hook. Our normalization A Combinatorics of theWe normalization have normalized factor the components of the intertwiner Φ of the FoundationRFBR for grants the 16-01-00291 Advancement (A.Mir.)17-51-50051-YaF, of 15-51-52031-NSC-a, and 16-51-53034-GFEN, Theoretical 16-51-45029-IND-a 16-02-01021 (A.M.’s Physics (A.Mor. and Y.Z.). “BASIS” and The (A.Mor.), Y.Z.), work of by by 637844-HBQFTNCER. Y.Z. joint was grants supported in part by INFN and by the ERC Starting Grant S. Yanagida. (H.A.), (# 15H05738)ration) (H.K.) “Topological and Field JSPS TheoriesQuantum Bilateral Toroidal and Joint Algebra” from String Projects MEXT, Theory: (JSPS-RFBR Japan. collabo- It from is Topological also Recursion partly supported to by the grant solutions should arise, whichplan generalize to study the these Nekrasov intriguing functions cases in elsewhere. aAcknowledgments nontrivial way. We We very much appreciate correspondence and discussions with M. Jimbo, S. Minabe, and transformations of theclassical compactification answer. torus. It wouldtwo We SL(2 be find interesting all to understand the origin andcentral interplay charges between are study the properties of the partition function under shifts of the adjoint mass and modular JHEP03(2018)192 ) 3 , q +1) 1 (A.5) (A.6) (A.7) (A.4) ) of a j q t ( λ , y j, λ t C x appears, if + 1)-th row. k . In terms of j • +1 , we may have . . . λ • n , let us introduce λ C λ +1 , that is, when we / ,` ∈ ,` j j j ) λ λ t in the ( − − j j , if the up-shifted hook ` > λ , y . δ δ + 1) has the same color k  1  ) and the tail (  1 +1 h . Then the head ( j ) + 1. On the other hand, k − j k +1 + 1 − λ j j -th row, a newly appearing n x x λ t x x t , y ( λ k x C . This is the case only when 1 1 x h ` q − ( λ − 2 x j, λ h , q ≤ ) x − 3 h q j 1 . − 1 + 1). Let us first consider the case , y in the  1 −  k , t ≤ h 1 y  ` ) x  − +1 − λ j k + 1) with color k k, λ ( h +1 Y ` − − y 1 = + 1 )+1 k j k 3 3 j q λ Y +1 = q q ( k j ) by identifying the heads and the tails of ` j and ( j 1 − 3 λ − ,` λ , λ . This time a new factor in 1 k − j +1 , q k k λ λ  ,` 1 . Since the tail ( ∈ and the tail is the red box with λ j 1 λ − q – 50 – − 0 j j λ + 1 q ( j The hook consists of boxes with ) δ ≡ t µ − λ . This takes place if λ 1 λ ) ≤ j × j q −  ∈ λ δ  C , y j ( Y 1 so that j k  t 1  λ − + 1) with color x x x  h j k − ( 1 3 k -th row. Since the head and the tail of such a new hook x < µ x + 2) or the tail ( , q j  j k 1) ) = − − +1 k, λ 3 → j 1 1 − × k, λ λ j , q   h a hook in the original diagram 1 . This hook gives a new factor to 1 1 in the q ` 0, the head and the tail have the same color. In the above hook , y • ( − − =1 =1 ` + 1), the new factor is h Y Y j j k k and λ ≡ x not k j C ) = t ··· ≤ k k, λ , y + 2) is the head and the hook length is a multiple of h 0 k λ ··· +1 x k 0 λ C ( . . . • • λ C has the head ( + 2), we can add a box ( h k, λ ) for k k j +1) is the tail and the hook length is a multiple of 1 has the color + 1, the minimum of allowed k + 1 + 1) and ( k, λ − j, µ λ j k +1 k, λ j a hook in the original diagram ). The color of the boxes is increasing along a hook from the head to the tail. If a t λ ≤ j, λ Now when we add a box ( Since we have to deal with the coloring and the length of the hooks in 1 as ( j , y = not h − j x ` The new factor is where we have made athinking shift in the opposite way, we see that the hooks that cease to contribute Next, when ( the tail ( the left-shifted hook is µ for is can add a box withare color ( Thus, we can evaluatethe each hooks. factor of hook in when ( diagram, the head isthese the notations, red the box normalization with factor is Note that the tail( belongs to the hook,hook but satisfies the head does not. The corner of the hook is convenient notations for this purpose.hook We by define the the condition head ( ( and JHEP03(2018)192 ) , ) z , ( A.4 i ) (B.1) (B.2) (B.3) (B.4) (B.5) (A.8) (A.9) ) and f z 3 (A.11) (A.10) ) ( i , q w f 1 ( ) j q ( w f ( 30 λ . These are j i,j ,j 1, or the head C e . m ) +1 i,j − − i λ ) | m δ , k d +1 ) ) ), we see that (  0 i d z − − 3 i, ≤ ( k ( ` − ) H j , q ( ,j z i,j E 1 − 1 ( | a , i,j q z j − ≤ k i − a ( i f  i ) ) 0 α δ λ k i,j − C + 1 a e = ∓ λ w/z w/z ) = ( q → z 3 i,j ( of the Fock space, which simulta- ) ,q i , ). In fact, if we define ) and 1 z m ) = ) = ( ) = ( e , , ) ) 3 } q + 1) for 1 ( λ z ) 3 3 | i ) ) w w (   k z ( ( , q k 3 λ j 1 k 1 , q , q j j 1 ,q f 1 1 − e q − f , λ {| and ∓ 1 j j ) q q ( ) 3  i q 1 q , f + 1 ( ( q ) z λ z ( − ,j 0 i λ λ ( 1 ). The selection rule is that we can remove ( k 3 λ λ − i i C +1 | | f q λ − ). The normalization factors C C 0 +1 1 k j ( k k λ i i, . λ ` λ ( λ ∓ | δ x 3 H q , f − − := q z ) + 1 j k ≤ – 51 – y − i 1 3 λ λ z 1 1 q α , f j λ ( q q ,q 1 ) ,j i ( e ) = 1 1 e z q − z 2 ≤ − − ) ( ) − ( q (0) ` i i i → 1 1 λ is given by the matrix elements of the vertical repre- w e R δ | f ) ( )   ]. ) # +1 → 3 λ j z z ( -th row. The corresponding factors are w − e ( ,q − k x 80 1 3 ( i j 1 j q q i,j  (0) ` i,j ) y f 1 q m A δ q λ ( | i i,j − # , e ) for e ) λ i + 1) and the tail ( q j m ). Since the eigenvalues are non-degenerate, the freedom is only , k 1 d α ) ). Then the first lemma is stated as below j z = 2 ∂   i d ( − q  k j, λ (  ) = − ` ) from the q j, λ i,j ( k 3.50 z i,j ` a K ( a i,j j → , − a e )–( + 1 ) ) Z  i ) by a shift i,j λ k | a /n ) 3.48  w/z w/z z A.3 Z q ( ` ∈ F | ) = ) = ( ) = ( ). Considering the difference between λ z ) are related to the relative normalization of w w ( ( 3 ( ( j i, j j j e +1) and the tail ( , q A.3 e f 1 k ) )  i By using Lemma 1, the formula is easily checked by direct computation. This is a In the vertical representation, we take a basis q This is motivated by the formula for the norms of the Macdonald functions. z z k ( ( ( 30 0 i λ i k, λ e e represented by and relations ( where First of all, we recall the zero mode algebra generalization of Lemma 6.1 in [ B Zero mode factorIn of this the appendix, intertwiner we prove the lemmas concerning the zero mode part of the intertwiner. the recursion relation forsentation ( as follows: Lemma 2. the change of theC norm of each eigenvector for the lattertain one. ( Byfollows taking from above ( four factors with theneously color diagonalizes selection rule, we ob- ( a box with the color for the former case and have either the head ( JHEP03(2018)192 1 − ) 1 − (B.6) (B.8) (B.9) s 3 (B.16) (B.12) (B.13) (B.14) (B.15) (B.10) (B.11) q v/u 1 q , − ) − s , λ 1 u ) , ( q , u `  ) (B.7) ` ) ( j, f ` = v k v ) f 1 1 − ) v s ) using the same ) ( − − v i v v ] i i 3 3 ( ( ( k q q ] [ λ , x ∗ ∗ ≡ 1 1 ] ] k z [ λ
k k − − 1 [ [ λ λ z z j j 1 1 z z − s q q ) ) 1 ( ( x , ` k 1 k ∓ ) + − + 3 .  u v/u v/u j j ] q ) and ( ( ( k ` 1 − − u ] ] [ λ,` i i . e ( k k − + 1 ˜ [ [ e f π λ,` λ,` ` q ) ` `, e π π  v + 1 − ( ≡ ≡ ` `, ) = ) = 1 v v ) = ) = ) = − k k ( ( +1 ) is ≡ ≡ v v v ] ] ` i,j ( ( ( k k λ + + ] , [ i,j [ k k ≡ e i,j ∗ ( ∗ ] ) is k ) s s k ] ). On the other hand, the factor which ` [ λ )+1 k k λ =1 [ v λ + + λ λ λ z + [ λ Y λ s ( ( ( s z ) ( ≤ z ` ` s ) λ ` u − − i,j  ` s u ( − e s ≤ ` ( s ≡ ) 1 ` 31 ≤ 1 . s v − ` , f ( − , e Z 1
– 52 – ,
≤ − 1 , f z   +1   )
/n − ) s ) v/u s v/u u i,j q Z v x v s ( 1 , f ( e when ) for 1 1 ( ` ] ) ] x − v/u s ∈ , k = 0. k e − +1 k 1 s 3 [ s u v/u s i,j [ i,j ` ) q  ` ( s q k 3 k − x 3 f e , v ≡ ` ) = 1) for 1 k q q 1 q i i − ( s, λ v/u k e )+1 v q ) 1 λ λ ∗ =1 ) − − ] ( λ + s, Y v s ` ( − q ≤ ≤ v s − k 1 ( ` [ λ j j )+1 ← ← ( q λ ≡ Y Y ] ] − − z λ ( , k ≤ ≤ k ) − ] when k ( 1 − [ [ λ 1 1 λ ` s i,j λ + k z z − ( =1 s e x ( q Y ` ) )     s λ ) 1 1 ` ) ) v −  − 3 λ λ ≡ ( s ( ( q ,  k ) ` ` v/u v/u ] 1 i,j ( ( λ + ≤ ≤ k ← ← `,k ( ] ] =1 − s [ e λ,` i i Y Y Y ` δ s k k ) λ q ˜ [ [ π λ,` λ,` ≤ ≤ − v 1 1 − π π  ) − ( s z q +1 `,k ) = ) = δ ) = ) = ) = − i,j and v v q v v v ( ( e ( ( ( ] ] ∗ k ] ∗ ] k ) = ] k [ λ k [ k λ [ k λ [ u λ z ] ) = ( [ λ ≡ z ( z z k z z ) [ ` λ,` ] denotes the vacuum color ` ( )  ` e u ˜ π ] k u ( k Now we shall check the commutation relation between k ` ( [ λ,` We often omit the symbol [ ` e π e 31 comes from the right-most box ( The non-trivial factors survivingwhen after we take the product over the rows are ( we only have tocomes consider from the the left-most left- box and ( right-most boxes for each row. The factor which and [ idea as for the oscillator parts. Thanks to the triplet cancellation in each row where satisfy Lemma 3. JHEP03(2018)192 ) . )  λ ( j s ` x x q (B.25) (B.26) (B.17) (B.18) (B.19) (B.20) (B.23) (B.24) (B.21) (B.22) −  we divide , . . . , λ j ) (C.2) +1 ) , ` t +1 1 ( j ≡ , − λ =1 s λ Y ` j ) 3 s , λ . q ) v − 1 1 v − s = ( , ( − − , , k . 1 ) ) 1 ) q  t − +) ( ( v j s v −} S j . For each µ j  ( , ↔ j j 3 ( x x λ a ) j s { q z q × v j,λ ( + x x − 1 ) +1 λ ( j f 3 k − 4.3 n v , ) −} ( q zq ), )  , j +) v j,λ ] j q v ∗ { j ( j e ( ( − +1 λ [ , λ z ) +1 − ) j } z z ` − ALE v  ) + . +1 j (1 ≡ ( )+1 j ` j,λ ( ↔ v j ) λ 0 s ) = { λ Y , e 1 z ≡ v ( λ [0] = 1 λ ≡ − z s , . . . , λ ` ( s ) 1 − ) (C.1) 1 j z − − µ ) t 1 λ   =1 , ( λ 3 q s Y ( j − − s ) ∈ − λ ( − z f t , Y 1 =  j v s  ) j , q ( λ ) ( λ,µ ) + − ` 1 1 1 z h 1 q k ↔ `,j j,λ    = ( − j [0] i,k ) − j δ ) ; + ) − j ` j s f ). x e } e 1 v ) z x v k x − x x 3 j 3 i ( ( 1 , we have ( + 1 ). The computation of other commutation j − 3 q 3 1  λ ) j ` q , q − j 1 q − 1 s { − +) λµ [0] i,k ≤ 1 q ( q q x − j 1 +1 λ ← k e  ( q ≡ µ ( ( λ Y j q v/u q N ` i ; − ≤ ,` ,` z ( ( λ ,` ( 1 1 λ 1 z – 53 – ] − 3 }  q ,` ,` ,` ( k q , ` ≤ −} −} + } } [ q λ,`   j j − ) j ← k ] ] ) = ≡ + + ) { { −} Y s π { ) λµ j j − j j j s +1 j ( ≤ z [ [ λ λ λ [0] ] λ λ { { { j ( 1 λ ( λ  λ j ( N ˜ Y [0] [0] π π λ λ ` [ π λ a ` 1 = − 0  π π    s π  s λµ ≤ , we have − `, i j zq ` δ ← G j Y ≤ = `,j ≤ − i δ − ≡ ← ) = ) = ) = ˜ ) = ) =  Y + ≤ ` +1  v v v j v v j 1 j k ( ( (1 1 ( ( x λ ) q ) ) 0 ) into two parts +1 1 − ) v j q − +) − ≡ +1 q ( λ − λ j j 1 ) j j ( ) λ ( ( ( λ λ ( s λ ) := ) := q − ` j j ( ∈ z z − z q v v  Y s j,λ ) j ) ( ( + ` ( λ ) e j,λ ) v − µ,λ v ) k z q f 1 ( h +) −  ) v j j − − ( ( ( v λ λ +1 q , . . . , λ ( z z j ( 1 ) = +) ) = j j 1 +) 3 λ ( j,λ λ j ) = ( − j ( z e j,λ λ 1 , q x f z 1 − j = ( j 3 q Under the condition q x ; λ 1 1 z − − 1 1 ( q q ( ( λµ ,` ,` } N We can write down the factors explicitly, for example, Consequently, we obtain the following results used in section + −} j j ] { { j [0] [ λ λ π π for the Nekrasov function (bifundamental contribution) on by obtaining a recursion relation for C Recursion relation forIn Nekrasov this function appendix, we give a proof of We can also check the same result, if we replace and under the condition Lemma 4. relations can be performed in the same way, hence,a we partition omit them here. so that Hence, gathering these factors, we get JHEP03(2018)192 ) j j j − +1 +1 and and i 3 q (C.3) (C.4) (C.5) λµ λµ j j 1 µ µ N N − j µ ) consists ≤ ≤ . − i i . λ 1 λ ` +1 5.33 i . ) zq ) , λ j ≡ i ) +1. Conversely, − i 1 . Hence, suppose ` + 1) λ − µ j, µ i, λ j j ) appears in ) appears in ≡ − − 1 j i i 3 , µ i = ( − = ( − q λ i ) 3 1 j ) = (1 1 i q x y 1 − 0 is imposed. Only the − − 1 i 3 + 1. Conversely, for the − − j ) satisfying i ( q . In any case, it is enough ) − µ ` + 1) to i, λ j λ s 1 j ≡ µ ) satisfying ( ( − j µ − . The corresponding relative i µ ` 0, where only the arm length λ i j ≡ x ` − ∈ λ 1 ( = ( = ( µ λ ` − 3 ≡ j, µ +1 ≤ − zq y x q i i +1 ) ≤ λ 1 i ) + 1 λ + 1). 1 i s ≤ ··· ( − ( s + 1) λ j λ ( i zq zq j . . . a t 1 ≤ ) + 1 y µ (1 + 1 − t x ` − − ( j j, µ ≤ zq +1 j, µ λ µ , ` ` − , )+1 ≡ + 1 ) + ··· i =1 λ Y for 1 i s i ( λ ( ` ) + ) = (1 ) = (1 . . . for 0 − ) satisfies the following recursion relation s y t λ ) µ i 1 ( 3 t a j ( 1 − µ 0, which is equivalent to ) λ , q ∈ ), − a ` s 3 – 54 – + 1) 1 ) ( , q ) q j ≡ µ + 1 1 + 1) i 0, which is equivalent to ` ; ) = − C.5 ) +1 ) satisfying − i j 3 s z ) = µ − i 3 ) s ( ( t q λ t ≡ q ( ( ( j, λ ( 1 ) ∈ µ s ` λµ j, µ i, λ − ( µ,λ a µ,λ j λ,µ λ + 1) h N − µ 1 ≤ h ) a = ( 1 ), h j i )+1 = ( − = ( i ) i s zq t if ), the old factor (1 zq ( + 1) z . However, since a pair of the Young diagrams is involved, the λ 1 y x ( ≤ C.4 i, λ µ j, µ − − ( ` λµ A zq C.4 +1 λµ i N = ( = ( − )+ x ), set /N for 1 x y s ) j, λ (1 ( ` z ( λ λ , ( 0, which is equivalent to C.5 ) ≡ a j ··· ) may change, when we add the box ( i λ ∈ = ( 1 is the color of the added box ( ( s λ =1 Y ` ≡ x i ( see the left hook in ( +1 − . . . t t y i µ − ) ) = ` λµ t j 32 s + 1) , ( = ( µ ··· N j j ,λ +1 The Nekrasov factor j = 0. − i +1 . . . s y -part. 0 -part. λ i, µ j +1 λ λ ) may change, and there is the new box ( µ λ,µ µ t ∈ ∈ 6= = ( h s t h = ( i µ ` hook is the left one in ( Q In this case, theleg length condition s λ Q This time the constraint is a to consider Then the new factor (1 if for the right hook inceases ( to contribute Then the new factor (1 if right hook in ( To obtain the recursion relation, we can proceed in the way similar to the proof of We set 1. 2. 32 of two parts, let us consider each part separately. where Lemmas 1 and 2 in appendix argument necessarily becomes more sophisticated. As the Nekrasov factor ( Lemma 5. JHEP03(2018)192 ). . 1 − , ) D.5 ) (D.6) (D.3) (D.4) (D.5) (D.1) (D.2) z ) t , ( u/v λ ) ` 3 +1 t . ( 0 q λ , 1 ) disappears j, ` 3 +1 ) j − − 0 q i v ) − j, 1 t ( i 3 H -part and those ( − ∗ − q µ i µ µ − ) 1 z t a ) ∈ H ) ( t − v ) − 1 µ j u 1 q a u ( µ Q 1 1 − λ − 1 − i 3 − i z − q q i 3 ) q λ 1 1 1 q − 1 − − is the generalized framing zq , j q ( 1 − ) 0 v/u q λ j 1 − − 1 ( ≡ q f ( ) − ( µ 0 t j ( q u/v ), ∈ − ≡ j ( t Y ( i ) µ − t α i λ,µ ( ∈ µλ (0) µλ h α − t Y ) = (1 5.33 e e ) 1 N λ,µ G ( t ( h ) ) − 1 λ ) . Note we can take these factors even 3 3 ) come from the ) = ` 3 ` j z S ) = ) = q , q , q 1 1 µ v u 1 1 1 ≡ − − × ( ( u/v q q − ) ) i ) ≤ 1 ( ( s n t i,j i,j ( u λ λ µ ( − ( µ ) µ f f ` λ s +1 a , we can take the same strategy which we used − ( i z − 0 3 | µ − 1 i 1 λ q ALE ∅ ` – 55 – µ ) ( | − s − j 3 zq ) ( + , f , e = q µ 0 v λ ) | − ( a 1 s µ     λ ( | ∗ -part. Combining these two contributions, we arrive µ q ≤ ) ) λ 1 z λ  v u i a 1 ) ∈ ( − ( v q u λ s u q 1 ( i,j  i,j − Q λ − f e ( q z i i 0 ) µ λ ). − ) = v ≡ ( ≤ ≤ ) ( λ u j j 0 ← ← s ∗ Y Y µ ( ( ∈ C.3 ≤ ≤ ≡ z Y s λ ) 1 1 λ s z µ,λ ( ∈ )     h Y s v due to the cancellation. ) ) ( µ,λ λ µ λ µ ∗ h ( ( µ f f ` ` z +1 0 | ) i ≤ ≤ ← ← = 0, which is equivalent to i i Y Y µ λ | come from the ≤ ≤ ) ) + , then the old factor (1 1 1 ≡ i . Thus we first derive a recursion relation for the left hand side of ( u/v 0 = | ) 1 λ i λ +1 t ) is the Nekrasov function on | u/v v/u i − C ( λ ) and z 1 1 z ) = ) = q λ ≤ ( ) is trivially satisfied, when ( v − − u λ,µ ( ( q q 6= λµ ∗ µ ( ( λµ λ h 4.23 ) = i z z D.5 N + 1 z N if when λ λµ µλ ( j Since a direct computation leads us to N N µ (0) µλ G Since ( in appendix it suffices to show that where are the group algebra parts of the (dual) intertwiner. Lemma 6. where factor ( D Symmetry of Nekrasov function In this appendix, we prove the following relation for the Nekrasov function: Thus we see that thefor changes for at the recursion relation ( JHEP03(2018)192 ) , ) λ,µ v ( ) ( H (D.8) (D.9) (D.7) λ,µ − +1 (D.13) (D.10) (D.11) (D.12) ( j q 0 H , . | ) j,µ ) q µ | f 0 v | ) ( + λ,µ µ ( 0 u µ , | , ( ) z −| λ ) H | ) λ 0 v | − ( q z u λ q λ,µ ( µ ) ( × . z λ −| , × ) ) } H z ) ) q ) v 0 1 q u v u/v ( v ( × ( − 1 ( ∗ µ 1 × ∗ 1 λ ) µ ≡ ∗ µ u/v − z ) z )) − v − z j j ) z ) ) v ( ) ) t  − − (  u µ u ( i i where we translated the 3 3 ) ) ( u µ z ( q q u/v 1 ( 1 ∗ λ z ) ∗ λ u/v ( 1 1 ∗ λ z ) − C λ,µ − ( u z z − − ) 1 ) u ( ) h j j (0) µλ 1 | ( v − 1 λ (0) µλ µ µ ) − λ q µ z G − ) − − z G 1 u/v ) i i 1 ) ) ∈ q u/ λ λ − 1 1 ( ( u − v ) v/u q q t 1 ( ) 1 v/u 2 q 1 1 { u/v ) = ) = ( ∗ λ ( 2 − − ( − − − u u z µ q 1 − q q v/u q q ( ( ). Hence, the desired recursion relation 1 ( z ( ( v/u q ) to the condition on the relative hook ( − ∗ ∗ λ λ ) v ( ( − − − + # q ( z z ) ( ( µλ u µλ µλ ( µλ ) ) ` } ( u µλ µλ D.7 , v v N ) satisfies the same recursion relation. This N ) 0 +1 λ ≡ N N +1 q µλ ( ( i , j ` λ N N λ µ z ( λ µ λ µ λ µ ( ∗ λ µ µ =1 f f 1 f f )+1 ∗ Y λ λ µ µ ≡ f λ f N ≡ ` f f i i =1 j,µ − f f f f λ ) in the way similar to the proof of Lemma 4 z D.6 Y − 0 0 i λ µ i , since all the commutation relations of the zero 0 0 ) ( ) as follows: | λ | f ) | ) | – 56 – f f ` 0 0 ∗ µ s µ z µ | | µ − v µ v | | ( i ( , z 0 µ µ ( | ∼ q D.7 + ) + −| −| ∗ 0 µ ( µ 0 0 0 ) | −| −| v 0 | (0) µλ | µ,λ z `, | | µ 0 0 ( λ v λ δ ) | | + λ λ z | h | G µ ( , z 0 | λ λ ) u  | )  z  ∗ µ −| −| λ v q λ ) v −| −| | v u ( (  v  u ( q q u /z    ∗ λ µ ∈ u v ∗ ( µ u/v ) v u z z   v v v q u u q u z λ v s ) ( z        +1 { × × j 1 j u ( = = = = − − 3 λ +1 q ) = ) = ) = z )) ∗ µ j u u u ) universally appears in the commutation relations among the ) z µ ( ( ( ) = # z ∗ ∗ − λ λ λ 1 ( u/v q z z z u/v ( ) ) ) q 1. We can check ( (0) µλ ( λ, µ v v v ( − ( ( ( (0) µλ G − (0) µλ ∗ µ µ µ  H G j z z z G ) 1 1 µ and the dual intertwiner Φ . Then by the same argument as in appendix − − ) = ) ) λ − u ) = ) = ( u/v B ( j 1 u u λ ), we see that ( ( z − u/v u/v ) ( λ λ 2 = q z z ( v − ) ) D.3 ( ` λµ q v v λµ ( ( ( N +1 The function ∗ N µ µ j λµ z z j,µ N f where Using these relations, we can see intertwiner Φ modes can be expressed in terms of in appendix color selection rule inlength, the we right see that hand thecompletes side right the of hand proof ( side of of Lemma ( 6. where follows from the relation From ( JHEP03(2018)192 , i d (E.3) (E.4) (E.5) (E.1) (E.2) shifts contains -tuple of . c , N   ) plus an equal ) . The general Y Y ( N ( , , ,... , and the electrons ) ) 3 2 } Y Y 1 2 − ( ( mod ,...,n is given by , ,...,n , , − 2 1 c , c = 3, then we get three   5 2 = 1 − = 1 1). Subsector 2 1 2 1 . ≡ determines the = N ∅   , . . . , n , . . . , n {− − 1 2 Y k = = = = N − N, i = 1 = 1 N, i ( ∗ ∗ ∗ d k (2) (0) (1) mod mod ,..., and the vector of integer-valued , i , d , d . , i c 2 1 c   1 2 Y = 0 ≡ 1 2 1 2 + ≡ By the boson-fermion correspondence, the c such that − 1 2 , d − − 1 2 T i i The lattice of fermionic momenta is divided k −   ∅ Y − ∗ i − i quotients d = = = i − d – 57 – color Y i

2], the holes are at (2) (0) (1)

= = , d d d i i ∗ called d 1 2 d = 0. The correspondence is described in steps: a natural number. Then 1 2 } = [3 c + + p 1) N  Y 1  − c c 2: − N =0 , ( − 2] as in the example above and = 0 : = 1 : = 2 : N c − 1 , 1 2 c c c , 1 2 holes at: P − − i ∗ i . = 0 = [3 d d } electrons at: ,...,Y c 3 2   Y , (0) 1 1 1 2 N N Y { {   let ) is the length of the diagonal of for the diagram = , satisfying Y = = subsectors labelled by the ( ∗ } ) ) : c c 1 n d ( ( Y − ∗ d N d N of ∗ i be a Young diagram and Example: subsectors with the electrons and holesformula with for momenta the momenta of electrons and holes in subsector Example: are at Division of theinto momenta lattice. where Young diagram determines thestate Maya consists diagram of a specifying Dirac thenumber sea of of fermionic electrons electrons state. and with holes, momenta d This with momenta given by the Frobenius coordinates Transformation into Maya diagram. Y , . . . , p 2. 1. 0 p Let Young diagrams { In this appendix, wethen first use introduce some the combinatorial notion identities toin of express terms the the of quotient characters their of of colored quotients. the Young Young diagrams diagram. We E.1 Quotients of the Young diagram E From colored Young diagrams to quotients JHEP03(2018)192 (E.8) (E.9) (E.6) (E.7) (E.11) (E.10) of this ). The 33 E.5 value negative . ) c , ( } } Y . ∅ 1 , . ,  1 ∅ c o , ∗ − , . We also have ) cp , c 1 [1] ( i , . 0 Y ) actually gives the character of the − { . d − { i 2 c n − = p = 0 = = 1 = j . ) # is reversed. q c 0 1 2 | E.10 2 c N ( p p p p c Y Y − p | ∈ Y + ) o | ) X Young diagrams ) i,j c , eq. ( c ( ( i ( ~ d (1) = Y ⇔ – 58 – n | Y ) = q N = 0 : = 1 : = 2 : = 3. The subsectors are listed in eq. ( The collection of electrons and holes from each sub- ( ch c c c  2] Y = # quotient 1 , N vanishes since the charge of the original vacuum state c − ch =0 p c c X N p = [3 = | Y 2] and Y , | : c and expanding in p ~ = [3 e = Y denotes the value of the momentum shift needed to eliminate the vacuum charge. q c p let Finally, the collection of quotient Young diagrams together with shifts is shifts read: Notice that the sumis of zero. all Eliminating thesponding vacuum charges fermionic in states each into sector, one canExample: transform the corre- sector determines a fermionic state.equal However, to though the the total total number numberseparately. of of electrons holes, Equivalently, is their the numbers Diracthe might not seas states match may in in have the each nonzerovacuum subsector subsectors vacuum charge charge. by have We different denote levels, the i.e. Shifting the vacuum charge. representation associated with the Young diagram One can easily write down the relation between the total number of boxes in the In other words, 3. 33 ∞ gl Of course, one can use the convention, where the sign of The (uncolored) character of the Young diagram is defined as follows: After substituting In the next subsection,Young we diagram work and out its a quotients. more general relation betweenE.2 the character of Decomposing the characters original Young diagram and its quotients: JHEP03(2018)192 ) ) c q ( ( ) Y c : Y ( (E.16) (E.17) (E.18) (E.19) (E.12) (E.13) (E.14) (E.15) L ,  d .  N Np q c − q − N Np q − − 1 q 1 − , − 1 N          1 , we used this result for the − q q . . . 0 0 1 N − . − − 4.5 i , . . . q 1 1 . . . 1 0 1 − . . . . j ··· . − d − q ) − + q 1 q N c,d c
0 ( N ¯ q δ − ) ··· N c q − c− ( Y q c q −
Y 0 − − N mod ∈ 1 ch − d ) N ) d 1 b q cd . In section ( X q c i,j − δ N 1 0 1 0 00 1 =0 Y ( ≡ ) − c N X N − j c ( q = ch − – 59 –          i Y  . One observes that the colored characters ch cd character ) = d N ) ch x q ) = q 1 − (  q ) = 1 d − ( c Y q − ) ( − q cd Np ) with very special coefficients forming a matrix ) 1 c ch ( c q q L ( Y d ( L ), which permits any use, distribution and reproduction in colored = ) Np ( + c ch q ( c 1 1 Y − c− − ) =0 c q c X N − N ( d L b N q q N − − CC-BY 4.0 − q 1 . This, indeed, can be done and we obtain the following expression: 1 1 c This article is distributed under the terms of the Creative Commons − =0 p d X N ) = turns out to have a particularly nice inverse: q ( L ) = Y q denotes the floor function of is the Kronecker symbol modulo ( ) ch c c cd Y ( x ¯ δ b ch One can also introduce the inverse matrix in orderinto to a transform product the of expression commuting for intertwiners. theOpen non-Abelian DIM Access. intertwiner Attribution License ( any medium, provided the original author(s) and source are credited. or, in the index notation, where The matrix where are linear combinations of ch The colored characters can also be expressed in terms of the quotient Young diagrams: Naturally and of the shifts We would like to rewrite the character as a manifest function of the quotient diagrams JHEP03(2018)192 ] = 2 ] ] ]. N ] ]. ]. B 825 ]. SPIRE ]. IN Helv. 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