(p, q)-webs of DIM representations, 5d \( \mathcal{N}=1 \) instanton partition functions and qq-characters

Journal of High Energy Physics, Nov 2017

Instanton partition functions of \( \mathcal{N}=1 \) 5d Super Yang-Mills reduced on S 1 can be engineered in type IIB string theory from the (p, q)-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels (1, n) ⊗ (0, m) → (1, n + m), thereby generalizing to higher rank m the original construction. It allows us to use a folded version of the usual (p, q)-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qq-characters of linear quivers based on the horizontal action of DIM elements. While fundamental qq-characters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl reflection acting on tensor products of DIM generators.

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(p, q)-webs of DIM representations, 5d \( \mathcal{N}=1 \) instanton partition functions and qq-characters

HJE (p; q)-webs of DIM representations, 5d instanton partition functions and qq-characters J.-E. Bourgine 0 1 2 4 M. Fukuda 0 1 3 K. Harada 0 1 3 Y. Matsuo 0 1 3 R.-D. Zhu 0 1 3 Bunkyo-ku 0 1 Tokyo 0 1 Japan 0 1 0 Via Irnerio 46 , 40126 Bologna , Italy 1 85 Hoegiro , Dongdaemun-gu, Seoul , South Korea 2 Sezione INFN di Bologna, Dipartimento di Fisica e Astronomia, Universita di Bologna 3 Department of Physics, The University of Tokyo 4 Korea Institute for Advanced Study (KIAS), Quantum Universe Center , QUC Instanton partition functions of N = 1 5d Super Yang-Mills reduced on S1 can be engineered in type IIB string theory from the (p; q)-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identi ed with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We de ne a new intertwiner acting on the representation spaces of levels (1; n) (0; m) ! (1; n + m), thereby generalizing to higher rank m the original construction. It allows us to use a folded version of the usual (p; q)-web diagram, bringing great simpli cations to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qq-characters of linear quivers based on the horizontal action of DIM elements. While fundamental qq-characters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl re ection acting on tensor products of DIM generators. Quantum Groups; Supersymmetric Gauge Theory; Topological Strings; D- - 4 Quantum Weyl re ection and qq-characters Horizontal intertwiner and qq-character for the A1 quiver 4.1.3 Higher qq-characters 4.2 Quantum Weyl re ection De nition of the horizontal intertwiner Horizontal intertwiner as screening current and fundamental qq- character Horizontal intertwiner for the A2 quiver Fundamental qq-character for the A2 quiver Quantum Weyl re ection and the second qq-character Generalization to the Ar quivers Inclusion of fundamental/antifundamental matter elds 1 Introduction 2 DIM algebra and representations 3 Generalized AFS intertwiners DIM algebra Quantum torus and DIM Vertical (0; m) representation Horizontal (1; n) representations Reminder on 5d N = 1 instanton partition functions Discrete symmetries of DIM algebra De nition of the generalized intertwiners AFS lemmas Gaiotto state Vertical intertwiner 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 4.1 4.1.1 4.1.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 5 Perspectives A Di erent expressions for the vertical representation B Useful formulas for the horizontal representation B.1 q-bosons vertex operators B.2 Commutation relations in horizontal representations C Derivation of the AFS lemmas D Connection with quiver W-algebras E Derivation of the qq-characters for the A3 quiver { 1 { Introduction Duality has been one of the most fundamental issues in string/gauge theories, and it has been studied from many di erent viewpoints and in various contexts. One of the standard approaches to the problem is to assign a brane con guration to the gauge dynamics, and interpret the duality in graphical ways. Such an approach has been taken in N = 2 superYang-Mills in 4 dimensions (and N = 1 in 5d). The corresponding graphical object, the Seiberg-Witten curve, is expressed through various con gurations of D- and NS-branes. For example, supersymmetric gauge theories with N = 1 supercharges in ve dimensions can be engineered in type IIB string theory using the methods developed in [1]. Linear quiver gauge theories with U(m) gauge groups are obtained from webs of (p; q)-branes that are bound states of p D5 branes and q NS5 branes [2, 3]. A useful algebraic tool to analyze brane con gurations is the topological vertex [4]. It has been introduced to reproduce the topological string amplitude on toric Calabi-Yau manifolds. In fact, the toric diagram of Calabi-Yau threefold can be identi ed with the (p; q)-branes web diagrams of type IIB string theory [5]. From this identi cation, it is possible to build the instanton partition function of the gauge theory using the machinery of topological string theory. In order to recover the Nekrasov partition function [6] in a general Omega-background, it is necessary to re ne the de nition of the topological vertex to include the gravi-photon background [7]. The importance of this representation of gauge theories in the context of the BPS/CFT correspondence was rst realized in [8, 9] where the connection with the decomposition of conformal blocks was also investigated. In [10, 11], Nekrasov partition functions have been studied from a di erent perspective, namely through the representation of underlying quantum algebras: the spherical double a ne Hecke algebra with central charges (SHc) [12] for the 4d gauge theory, and its quantum deformation, the Ding-Iohara-Miki (DIM) algebra [13{15] for the 5d gauge theory. The representations of these algebras coincide with those of (quantum) WN -algebra [12, 16{18] while the basis of the representation coincides with the set of xed points which represent the equivariant cohomology of the instanton moduli space. This observation was essential in the proof of the AGT conjecture elaborated in [12]. In addition, the presence of these algebras re ects the integrable nature of the BPS sector of the gauge theory. It led to the construction of R and T matrices satisfying the standard RT T relation of quantum integrable systems [19{22]. The main focus of our previous works [10, 11] was the derivation of the qq-characters from SHc/DIM algebras. These particular correlators of the gauge theory were rst introduced in [23], and further studied in [24{28]. They have the essential property to be polynomials, thus de ning a resolvent for the matrix model representing the localized gauge partition function. These quantities generalize the q-characters of quantum groups de ned in [29, 30], and naturally associated to the T -operators of integrable systems. In [10, 11], we have shown that the representation theoretical properties of the Gaiotto state and the intertwiner associated with bifundamental matter are directly translated into the regularity property of qq-characters. { 2 { A di erent construction of qq-characters has been presented by Kimura and Pestun (KP) in [31] (see also [32]). While both constructions are based on quantum W-algebras, the action of these algebras is seemingly di erent. In our approach, a copy of the DIM algebra is associated to each node of the quiver diagram, so that a quantum Wm algebra is attached to each gauge group U(m). On the other hand, the quiver W-algebras constructed in [31] is based on a Lie algebra whose Dynkin diagram coincides with the gauge theory quiver. In a sense, the two approaches are S-dual to each-other: in our approach the rank is the number of D-branes while it is the number of NS5-branes in KP's work. On the algebraic level, the S-duality is believed to be realized by Miki's automorphism [ 14 ] that exchanges the labels (l1; l2) of DIM representations, here identi ed with the (p; q) charge It was realized in [33, 34] that the two di erent pictures can be better understood using the re ned topological vertex. Indeed, in [35], Awata, Feigin and Shiraishi (AFS) have reconstructed this object using the generators of the DIM algebra where it plays the role of an intertwiner between vertical (0; m) (associated to m D-branes) and horizontal (1; n) (associated to a NS5-brane bound to n D-branes) representations [35]. Hence, like a string junction, it interpolates between the representations associated to di erent brane charges. In this way, di erent representations of DIM algebra can be combined to form a representation web [33, 34] that can be identi ed with the (p; q)-web diagram engineering the gauge theory.2 This presentation clari es the two approaches for the construction of qq-characters: KP employed DIM generators in horizontal representations [36] while we used similar generators but in vertical representations [16, 33]. The purpose of this paper is to propose a uni ed method to build qq-characters and prove their regularity property. As suggested in [33], the method is based on insertions of DIM operators in the horizontal representations. However, it also makes use of the commutation of vertical actions which was instrumental in our previous derivation [11]. The link is made by a set of lemmas that intertwines horizontal and vertical actions on AFS intertwiners. The AFS lemmas can be regarded as an uplift in the horizontal representation space of the relation characterizing Gaiotto state and vertical intertwiners obtained in [11]. With this new method, we derive all the (higher) qq-character of linear quiver theories with U(m) gauge groups, thus largely extending our previous results that were restricted to fundamental qq-characters. In order to achieve this general treatment of quivers, several new insights were necessary. First, we generalized the AFS intertwiner to higher level m of vertical representations, allowing us to treat gauge groups of arbitrary rank. Most of the previous considerations, including results on integrability, were restricted to gauge groups of rank one or two. Then, in order to build higher qq-characters, a Weyl re ection acting on (tensor products of) DIM generators has been introduced. Called quantum Weyl 1Due to a di erent choice of conventions for DIM representations, the usual representation of (p; q)branes webs is rotated by 90 degrees, with the NS charge q in the horizontal direction and the Ramond 2In [33, 34] this structure was called a network matrix model. However, since we do not use the matrix model presentation of partition functions, we prefer not to employ this terminology here. { 3 { transformation, it keeps the qq-character invariant. In practice, it is used to build operators commuting with a T -operator, the vacuum expectation value (vev) of which reproduces the instanton partition function. This commutation property is directly related to the regularity property of the qq-character, thus providing another link with the manifestation of integrability in supersymmetric gauge theories. This paper is organized as follows. The second section provides the main properties of DIM algebra and its vertical and horizontal representations. We put some emphasis on the various duality properties, including the SL(2,Z) automorphisms. This section also includes a brief reminder on N = 1 5d gauge theories. The third section starts from the de nition of the AFS intertwiners and proposes a generalization obtained from horizontal composition. The generalized intertwiners simplify the computation of amplitudes associated to the brane-web. In sections 3.3 and 3.4, these intertwiners are used to reconstruct the Gaiotto state and the vertical intertwiner built in [11]. In the fourth section, the horizontal intertwiner is de ned by taking vertical contractions of generalized AFS vertices. It is shown that it commutes with the co-product of DIM generators. The quantum Weyl transformation is de ned in the section 4.2 as an operation on the tensor product of generators. It leads to a systematic method to construct qq-character which is the main result of the paper. Finally, the details of computations can be found in the appendix, along with several useful identities. 2 2.1 DIM algebra and representations DIM algebra The Ding-Iohara-Miki algebra E [ 13, 14 ] can be presented in terms of the Drinfeld currents k are usually associated to points of a Z 2-lattice representing the elements of the algebra (see gure 1). We assign Z 2-degree for generators as deg(xn ) = ( 1; n), deg( n ) = (0; n). The notations and conventions used here are mostly borrowed from [21], up to minor di erences in the normalization of operators. The q-commutation relations satis ed by the currents read +(z) (w) = g(^w=z) g(^ 1w=z) (w) +(z) +(z)x (w) = g(^ 1=2w=z) 1x (w) +(z); (z)x (w) = g(^ 1=2z=w) 1x (w) (z) x (z)x (w) = g(z=w) 1x (w)x (z) [x+(z); x (w)] = (1 (1 q1)(1 q2) q1q2) (^ 1z=w) +(^1=2w) (^z=w) (^ 1=2w) ; (2.2) with ^ a central element. This algebra has two independent parameters encoded in the q with = 1; 2; 3 under the relation q1q2q3 = 1. It is sometimes more convenient to use instead the parameters q = q2 and t = q1 1, in particular in the context of representations { 4 { group SL(2,Z) acts a 90 degrees rotation. over Macdonald polynomials. We will also introduce the notation These parameters appear through the functions 3 = q1=2 = t1=2q 1=2. g(z) = Y of two operators, they are sometimes called scattering factors. In this paper, representations of level (l1; l2) 2 Z Z with a weight u will be denoted (l1;l2) or sometimes simply (l1; l2)u. The levels are de ned through the representations of u the central element ^ and the zero modes 0 , (l1;l2)(^) = q3l1=2; u (l1;l2)( 0 ) u (l1;l2)( 0+) u = q3l2 : Here, we will focus on the so-called vertical representations (0; m) and horizontal representations (1; n). The intertwiner de ned in the next section relates three di erent representations, it will be portrayed as a three-legged vertex. Products and tensor products of such operators can be described using diagrams resembling the (p; q)-web diagrams of brane con gurations in type IIB theory, they will be called representation webs. Note however, that here diagrams are rotated by 90 , such that vertical lines (0; m) are associated to m D5 branes and the horizontal line (1; 0) to a NS5 brane. Note also that we will take no care of the precise slope of horizontal lines: vertical lines in the diagrams will always refer to a vertical representation (0; m), while horizontal and inclined lines can represent any of the horizontal representations (1; n). To avoid confusion, the representation space associated to each line will be explicitly written on every gure. { 5 { (2.3) (2.4) The DIM algebra is a Hopf algebra with the following coproduct: tion the simplest representation with level (0; 0) which may be identi ed with the symmetry of a quantum mechanical system. We consider the noncommutative torus generated by the two operators U; V satisfying V U = q1U V: where q1 is not a root of unity. The enveloping algebra of U; V is generated by the elements wrs = U rV s identi ed as the degree (r; s) generators de ning the algebra [wr1s1 ; wr2s2 ] = (q1s1r2 q1s2r1 )wr1+s1;r2+s2 : This algebra has a SL(2; Z) duality realized as the rede nition of the basis, U 0 = In particular, the S-transformation is realized as S : (U; V ) ! (V; U 1). AU aV b, V 0 = BU cV d (a; b; c; d 2 Z) which satis es V 0U 0 = q1U 0V 0 as long as ad In this simple set-up, the DIM algebra with (l1; l2) = (0; 0) is realized as bc = 1.3 (0;0)(x+(z)) = u (0;0)( u (1 (V =z) 1)(1 (q1 1V t=z) 1) (0;0)(x (z)) = u X xn z n = n = exp X1 1 r=1 r (1 1 1 1 q 1 (V =z)U 1 ; q2 r)(1 q3 r)V r z r { 6 { n2Z with (z) = P zn. In this representation, the expression (2.5) of the coproduct simpli es as (u0;0(^) = 1. In the vector representation, the generators U; V are represented on a basis labeled by a single integer, (0;0)(U )ju; ii = ju; i + 1i; u (0;0)(V )ju; ii = uq1iju; ii ; u where u is the weight of the representation. 3A di erent but similar duality structure is realized by writing q1 =: e2 i 1 and de ning a SL(2,Z) modular transformation for 1: 10 = ac 11++db . Then, there exists two generators U~ ; V~ they satisfy the quantum torus algebra with q10 = e2 i 10 and that commute with the original generators [U nV m; U~ rV~ s] = This duality is known as the Morita equivalence, it plays a fundamental role in non-commutative geometry [37, 38], and is also relevant in more recent works such as [39]. (2.5) (2.6) (2.7) ! (2.8) (2.9) HJEP1(207)34 It is of some interest to compare the DIM algebra with the loop algebra g^ of a Lie algebra g. The generators of g^ (without the central extension) are de ned in terms of the generator ta of the Lie algebra as Jna = taU n where U is a formal variable. DIM algebra is a natural generalization of this setting in which two formal variables are introduced. It sometimes referred to as a two-loop symmetry. The algebra (2.7) depending on a single deformation parameter q1 can be extended by two central charges, c1; c2 [ 14 ], In this formulation, the SL(2,Z) symmetry is manifest, and the S-transformation is realized c1. The introduction of a second quantization representation of the two central charges, (ul1;l2)(c1) = q3l2 ; u parameter q2 leads to the DIM algebra [ 14 ]. One may identify the levels (l1; l2) with the The vertical representation (0; 1) has been formulated in [15, 16], it is equivalent to the rank m representation studied in [11] with m = 1 and up to a normalization. Here we employ conventions similar to the ones used in [11], but with di erent states normalization (the change of the convention is summarized in appendix A). The (0; m) representations depend on a m-vector of weights ~v = (v1; vm) and act on a space spanned by states in one-to-one correspondence with m-tuple Young diagrams ~ = ( (1); ; (m)), (0;m)(x+(z))j~v; ~ ii = ~v m 1z (m 1) (0;m)(x (z))j~v; ~ ii = ~v 2m+1zm 1 ~v (z))j~v; ~ ii = m ~ (z) j~v; ~ ii: X (z= x) zR=esx zY~ (z) j~v; ~ + xii; (z= x) zR=esx z 1Y~ (zq3 1)j~v; ~ xii; (2.11) where A(~ ) and R(~ ) denote respectively the set of boxes that can be added to or removed from the set of the Young diagrams 1 ; ; m. These expressions involve the functions ~ (z) and Y~ (z) that depend on a m-tuple Young diagram. Their expression can be factorized in terms of individual Young diagram contributions, m Y l=1 Y~ (z) = Y l (z); (z) = Y (zq3 1) ; Y (z) ~ (z) = Y m l=1 Y (z) = 1 (l) (z); v z x2 Y S( x=z) = Q Q x2A( ) 1 x2R( ) 1 z 1 x (zq3) 1 x Here, each box x 2 position of the box in (l). The associated box coordinate reads x = vlq1i 1qj 1 ~ is de ned by three integer labels (l; i; j) such that (i; j) indicates the As in [11], it will be important to add a set of diagonal operators Y (z) such that Y (z)j~v; ~ ii = Y~ (z) j~v; ~ ii : { 7 { (2.12) ; (2.13) (2.14) and ^(v0;m)(^) = 1. Strictly speaking, this is not a representation of the DIM algebra because some of the q-commutation relations are no longer satis ed. Instead, it should be seen as a representation on the dual states hh~v; ~ j that are orthogonal to the basis j~v; ~ ii,4 hh~v; ~ j~v; ~ 0ii = ~ ;~ 0 a~ 1; such that we have the property hh~v; ~ j ^~v (0;m)(e) j~v; ~ 0ii = hh~v; ~ j (0;m)(e)j~v; ~ 0ii ~v for any element e of the DIM algebra. The norm of the states involves the coe cients a~ which will play an important role in the construction of instanton partition functions. They are de ned in terms of the vector multiplet contribution to the instanton partition function Zvect.(~v; ~ ) as follows, The notation [ ] refers to an expansion in powers of z 1 of the quantity inside the brackets. They will be used to de ne the qq-character. The action on the bra states will be referred to as the dual vertical representation. In this representation, the roles of x+ and x are exchanged: hh~v; ~ j ^~v (0;m)(x+(z)) = hh~v; ~ j ^~v (0;m)(x (z)) = 1 X (z= x) zR=esx zY~ (z) hh~v; ~ + xj; hh~v; ~ j ^~v (0;m)( The vector contribution Zvect.(~v; ~ ) will be de ned in the section 2.5 below, it is expressed in terms of the Nekrasov factor (2.33) as a result of localization. As shown in [11, 40], the Nekrasov factor obeys a set of discrete Ward identities. Consequently, the coe cients a~ also obey similar identities. They can be written in terms of the function Y~ (z) as a~ +x = a~ x = a~ a~ (1 1 q1q2 q1)(1 1 q1q2 q2) (1 q1)(1 q2) m x m Res z= x Y~ (z)Y~ (zq3 1) ; 1 z= x m m 2 Res Y~ (z)Y~ (zq3 1): x 4Due to the change of states normalization performed in appendix A, and since the original states were orthonormal, the coe cient a~ is expected to be proportional to N (~ ) 2. The additional factors are chosen to simplify the formulation of the AFS lemmas below. m l=1 { 8 { Horizontal representations [36] of level (1; n) act as a vertex operator algebra in the Fock space of q-bosonic modes with the commutation relations5 The representations involve the positive/negative modes of the vertex operator and can be de ned in terms of the following operators: (z) = V (z)V+(z); (z) = V ( z) 1V+(z= ) 1; ' (z) = V ( 1=2z)V ( 3=2z) 1: Explicitly, we have Useful commutation relations involving these operators are presented in appendix B. The horizontal representation (1; n)u reads (1;n)(x+(z)) = u n u z n (z); (1;n)( +(z)) = u n'+(z); (1;n)(x (z)) = u 1 u nzn (z); (1;n)( u and (u1;n)(^) = . Similarly, it is possible to de ne the representation ( 1; n)u using the same vertex algebra, ( 1;n)(x+(z)) = u 1 nzn (z 1); u ( 1;n)( +(z)) = u n' (z 1); ( 1;n)(x (z)) = u u n z n (z 1); ( 1;n)( u 5Here we use parameters q; t instead of q to follow the convention of [35]: q = q2; t = q1 1. The oscillator modes can be represented on symmetric polynomials as follows: Macdonald(a k) = pk; Macdonald(ak) = k where pk denotes the power-sum symmetric polynomials. { 9 { (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.20) By de nition, the vacuum state j;i(1;n)u is annihilated by the positive modes ak, and '+(z)j;i(1;n)u = j;i(1;n)u . The dual vacuum state (1;n)uh;j is annihilated by negative modes, and (1;n)uh;j' (z) = (1;n)uh;j. The normal ordering, denoted : :, corresponds to write all the positive modes on the right, and all the negative modes on the left. Correlators of operators Oi(zi) acting in the Fock space are de ned as the vacuum expectation values hO1(z1) ON (zN )i = (1;n)uh;jO1(z1) ON (zN )j;i(1;n)u ; (2.27) with the radial ordering jz1j > jz2j > > jzN j. 2.5 Reminder on 5d N = 1 instanton partition functions The quiver Super Yang-Mills gauge theories with N = 1 in 5d reduced on S1 are characterized by a simply laced Dynkin diagram . Each node i 2 is associated to a vector multiplet with gauge group U(mi), and an exponentiated gauge coupling qi. To each edge < ij >2 corresponds a bifundamental matter multiplet of mass ij that transforms under the gauge group U(mi) U(mj ). In addition, a Chern-Simons term of level i coupled to the gauge group U(mi) can be introduced at each node i. Thus, each node bears two integer labels (mi; i) with mi > 0 that will later be related to the levels (l1; l2) of DIM representations. Extra matter elds in the fundamental/antifundamental representation of will be denoted i(;fj) with j = 1 fi, and i;j (af) with j = 1 f~i respectively. the gauge group U(mi) can also be attached to each node, and the corresponding masses The expression of the instanton contribution to the (K-theoretic) partition function re ects the particle content of the theory. It is written as a sum over mi-tuple Young diagrams ~ i, and each term is factorized into vector, Chern-Simons, (anti)fundamental and bifundamental contributions: Zinst.[ ] = X Y qj~ ijZvect.(~vi; ~ i)ZCS( i; ~ i)Zfund.(~ i(f); ~ i)Za.f.(~ i(af); ~ i) i f~ ig i2 Y <ij>2 Zbfd.(~vi~ i; ~vj ; ~ j j ij ); This operation will be very useful in order to express the qq-characters of the gauge theory. where the (exponentiated) Coulomb branch vevs ~vi are related to the vacuum expectation value of the scalar eld in the gauge multiplets. From this expression, it is possible to de ne a normalized trace of functions depending on the realization of the set of (tuple) Young diagrams f~ ig as follows, DF [f~ ig] E gauge = 1 Zinst.[ ] XF [f~ ig] Yqj~ ijZvect.(~vi; ~ i)ZCS( i; ~ i)Zfund.(~ i(f); ~ i) i f~ ig Za.f.(~ i(af); ~ i) i2 Y <ij>2 Zbfd.(~vi~ i; ~vj ; ~ j j ij ): (2.28) (2.29) HJEP1(207)34 The bifundamental contribution with U(m) U(m0) gauge group can be decomposed as a product of Nekrasov factors,6 Zbfd.(~v; ~ ; ~v0; ~ 0j ) = Y Y N (vl; (l); vl00 ; (l0)0): Various expressions of the Nekrasov factors have been written, the one given here has been obtained by solving the discrete Ward identities derived in [11, 40], N (v1; 1; v2; 2) = Y The vector multiplet contribution is expressed in terms of the Nekrasov factors as follows: Y Finally, the Chern-Simons and fundamental/antifundamental contributions are expressed in terms of a simple product over all boxes in the Young diagrams, ZCS( ; ~ ) = Y ( x) ; x2~ Zfund.(~ i(f); ~ i) = Za.f.(~ i(af); ~ i) = 1 Y fi Y The instanton partition functions de ned in (2.28) are invariant under the rescaling ~vi ! i~vi, qi ! i i qi, ~ (f) i ! i~ i(f), ~ (af) i ! i~ i(af) and ij ! ( i= j ) ij . This invariance can be used to set the bifundamental masses to a speci c value which simpli es the algebraic formulation developed here. Thus, from now on, all bifundamental masses will be set to ij = 1 . These N = 1 supersymmetric gauge theories can be engineered in type IIB string theory [1]. Linear quiver gauge theories with U(m) gauge groups are obtained from webs of (p; q)-branes that are bound states of p D5 branes and q NS5 branes [2, 3]. The branes occupy the dimensions 01234 corresponding to the space-time of the 5d gauge theory, plus an extra one-dimensional object (line) in the 56-planes. In order to preserve supersymmetry, the lines have the slope x6= x5 = p=q, so that the world-volume of D5-branes with charge (1; 0) occupy the dimensions 012346, i.e. they are vertical segments in the 56-plane. On the other hand, NS5 branes of charge (0; 1) are extended in the 012345 directions, and correspond to an horizontal line in the 56-plane. A representation of DIM algebra has been 6The Nekrasov factors enjoy the property N (v2; 2; v1q3 1; 1) = ( v1) j 2j( q3v2)j 1j Y x N (v1; 1; v2; 2): (2.30) associated to each brane of the (p; q)-web diagram [33, 34]. Representations of level (l1; l2) correspond to (l2; l1)-branes so that horizontal (1; 0) representations are associated to NS5 branes and vertical (0; 1)-representations to D5 branes. The topological vertex play the role of creation/annihilation operators for the (p; q)-branes, they will be identi ed with the generalized AFS intertwiners in the next section. In [ 14 ], Miki has introduced an automorphism of the DIM algebra that he denoted . Since it can be identi ed with the action of S-duality on the (p; q)-branes, it will be denoted by S here. This automorphism leaves the DIM algebra invariant, but map degree (r; l) generator into degree (l; r) and representations of di erent levels: ( 1; n)u, so that (u1;n)( 2 e) = in [ 14 ]), the square of the automorphism takes a rather simple form: and ^ $ ^ 1, or in terms of generating series, +(z) $ n+ $ n, xn $ x n (1=z) and x (z) $ x (1=z). 2 transforms horizontal representations (1; n)u into the representations By examination of the commutation relations, it is possible to de ne another transformation T acting on the Drinfeld currents as T Vertical representations are invariant under the action of T , and horizontal representations of level n are mapped to horizontal representations of level n + 1. The operations S and T obey the properties S 4 = 1 and (ST )3 = 1, so that they generate a group of SL(2; Z) transformations. To some extent, this group can be identi ed with the modular group of type IIB string theory. In particular, the Miki automorphism S would correspond to the S-duality that rotates the (p; q)-web diagrams by 90 , exchanging NS5 and D5 branes. The second duality symmetry in DIM algebra is permutation of three parameters q1; q2; q3, which is manifest at the level of algebra. This S3 symmetry is sometimes referred to as a \triality" [18] in connection with higher spin gravity [41]. We note that the representations of DIM are constructed with the reference to q1; q2. In this sense, the exchange between q1 and q2 is manifest. In 2D CFT language, such permutation is realized where parametrizes the central charge c = (n 1)(1 Q2n(n + 1)) with (l1; l2) = (l1; l1 + l2): or, in terms of modes, T xk (z) = ^ 1 x k 1 ; k = ^ 1 k : Again, the DIM algebra is invariant, but representations of di erent levels are mapped to each-other, Q = p p 1. In terms of the vertical representation basis, it is realized by taking the transpose of each Young diagram, $ 0. The other transformations, such as q1 $ q3, are less obvious. When the parameters are suitably chosen, they are identi ed with the levelrank duality [41{43] where the correspondence between basis is also more involved [18]. While the SL(2,Z) transformation may be regarded as a M-theoretical target space duality since it interchanges D-brane and NS-brane, the S3 duality may be interpreted as a worldsheet symmetry since it acts on the Hilbert space of equivalent 2D conformal eld theories. From the viewpoint of super Yang-Mills, q1; q2 represent the graviphoton background in the Euclidean planes (01) and (23). In this sense, we sometime denotes the symmetry q1 $ q2 as (01)(23). On the other hand q1 $ q3 does not have an immediate woldvolume 5 is obtained by replacing the parameters q by their inverse, e ectively exchanging S(z) with S(q3z), and g(z) with g(z) 1 = g(z 1).7 The transformation of DIM generators resemble the action of S2, except that x+ and x are not exchanged: (z) $ (1=z), x (z) $ x (1=z), and the central parameter ^ remains invariant. Thus, representations of level (l1; l2) are mapped to representations of level ( l1; l2) and vertical representations are left invariant.The 5 re ection of vertical (0; m) representations follows from the transformation of the functions m l=1 Y~ (z) ! z m Y( vl) 1 Y~ (z 1); ~ (z) ! q3 ~ (z 1); m (2.39) (1;n)(e) = u ! q 1 (2.40) (2.41) provided that the weights transform as vl ! 1=vl so that x ! 1= x for x 2 ~ . On the other hand, (1; n)u representations are sent to ( 1; n)u representations so that 5 ( 1;n)( 5 e) where the transformation u 5 sends the background parameters q together with the modes ak ! tk jkjak and the weights u ! and (z) are exchanged but ' (z) remain invariant. 2nu 1. In this manner, (z) The action of the 5 symmetry on instanton partition functions is closely related to the re ection symmetry studied in [45], where it relates two dual TQ equations in the Nekrasov-Shatashvili limit. However, here the Coulomb branch vevs behave di erently, since vl ! 1=vl. Vector and Chern-Simons contributions transform as Zvect.(~v; ~ ) ! q3 mj~ jZvect.(~v; ~ ); ZCS( ; ~ ) ! ZCS( ; ~ ); and the A1 pure U(m) partition function is invariant provided that the sign of the ChernSimons level is ipped, and the extra q3-factor is absorbed in the transformation q ! q3mq. On the other hand, the bifundamental contribution transforms into itself, but with the two nodes exchanged: Zbfd.(~v1; ~ 1; ~v2; ~ 2j ) ! Zbfd.(~v2; ~ 2; ~v1; ~ 1jq3 1 0) 7Obviously, the two re ections (01)(23) and 5 commute. The composition (01)(23) 5 acts on the DIM parameters as the exchange q1 $ q2 1, or t $ q. This is a well-known symmetry in the context of Macdonald polynomials, see for instance [44]. where 0 is the re ection of ( 0 = when = 1). As a result, the 5 symmetry for the instanton partition function of linear Ar quiver consists in re ecting the order of the nodes 123 r ! r(r 1) 1. Hence, this S2-symmetry can be interpreted as the re ection symmetry of the (p; q)-web diagram with respect to the horizontal (x5) axis. In fact, the symmetry 5 also acts as a re ection along the horizontal axis in the graphical representation of the DIM modes xn and n (see gure 1). 3 De nition of the generalized intertwiners The AFS intertwiner operator has been introduced in [35], it generalizes the free fermion presentation [46] of the topological string vertex to the re ned case. It is built over bosonic elds that coincide with those introduced in the horizontal representation of DIM algebra, thus providing a direct link with the representation theory. The original intertwiner acts in the tensor product of the representation spaces (0; 1)v and (1; n)u, and takes values in the space (1; n + 1) uv. The vertical space (0; 1)v is spanned by states in one to one correspondence with Young diagrams . Hence, the intertwiner is a vector in this space with index , (n)[u; v] = (n)[u; v] ; (n)[u; v] : (1; n)u ! (1; n + 1) uv: Both horizontal spaces (1; n)u and (1; n + 1) uv can be identi ed with the same Fock space of q-bosonic modes, and the elements (n) are expressed in terms of the modes as follows: The vacuum component is de ned in terms of a new vertex operator (n)[u; v] = tn( ; u; v) : ;(v) Y x2 ( x) : : ;(v) = V~ (v)V~+(v); V~ (z) = exp X1 1 k=1 k 1 z k q k ! a k ; which is related to the previous operator V (z) de ned in (2.22) as V (z) = V~ (q1z)V~ (q2z)V~ (z) 1V~ (q3 1z) 1: In fact, the vacuum operator can be obtained as a (normal-ordered) product of ( x) factors associated to an in nite Young diagram since 1 Y i;j=1 V~ (v) = V (vq1i 1qj 1) 1 2 ) ;(v) =: Y 1 i;j=1 (vq1i 1qj 1) 1 : : 2 (3.5) Thus, this operator is associated to the perturbative part of the partition function, extending the arguments developed in [47] for the degenerate limit relevant to 4d gauge theories. Indeed, the prefactors obtained from the normal ordering of two vacuum intertwiner and involving the function G(z) (de ned in appendix B) should be interpreted as (3.1) (3.2) (3.3) (3.4) perturbative (one loop) contributions to the gauge theory partition function. However, to keep our arguments simple, we will simply neglect these factors and no longer refer to this interpretation here. The normalization factor tn( ; u; v) is the vev of the operator D (n)[u; v]E. It is chosen in order to recover the exact form of the AFS relations presented below. Its explicit expression depends on the form of the vertical representation, which is (n), i.e. the correlator slightly di erent than the original one employed by AFS, tn( ; u; v) = ( uv)j j Y( = x)n+1: x2 (3.6) The reason for this di erent choice of normalization is that Awata, Feigin and Shiraishi were using the action on Macdonald polynomials to investigate the connection with the re ned topological vertex [48]. On the other hand, here we have chosen to keep a certain symmetry in the way the boxes of Young diagrams enter the formulas. It also makes the connection with our previous results on qq-characters more explicit [11]. The AFS intertwiner can be generalized to vertical representations of higher level, (n;m)[u; ~v] : (0; m)~v (1; n)u ! (1; n + m)u0 ; with where the vector in the vertical space (0; m)~v has components labeled by the m-tuple ~ that reads (n;m)[u; ~v] = tn;m(~ ; u; ~v) : Y ~ ( x) :; tn;m(~ ; u; ~v) = (u0)j~ j Y( = x)n+1: x2~ (3.8) This operator can be constructed as a product of vertical level one intertwiners coupled in the horizontal channel, as represented in the gure 3. The contraction in the horizontal channel simply corresponds to a product of operators in the q-boson Fock space. However, it is only possible if the weights of the intermediate representation spaces coincide. The resulting product can be normal ordered, and commutations produce a bifundamental contribution, (n+1)[u2; v2] (n1)[u1; v1] = 2 G(v1= 2v2) N (v1; 1; v2; 2) : (n1)[u1; v1] (n+1)[u2; v2] :; 2 (3.9) with the requirement u2 = u1v1. The function G(z) is de ned in appendix, formula (B.4). It only depends on the ratio v1=v2 and thus can be easily discarded. The (vertical) level m intertwiner is obtained by repeating this operation m times, m = m Y Qlm=1 tn+l 1( l; ul; vl) (n;m)[u; ~v]; ~ (3.10) with for each intermediate horizontal space the weight The extra factors in (3.10) and in (3.15) below will be absorbed in the replacement of products of a (l) by a~ in the de nition of the gauge theory operators (see the next section). The AFS dual intertwiner can be generalized in the same way. It is de ned as the operator8 HJEP1(207)34 (n;m) [u; ~v] : (1; n + m)u0 ! (1; n)u (0; m)~v; with with vertical components (n;m) [u; ~v] = tn;m(~ ; u; ~v) : Y ~ m l=1 x2~ ( x) :; where: ; (v) = V~ ( v) 1V~+( 1v) 1; tn;m(~ ; u; ~v) = ( u) j~ j Y( x= )n: Again, it can be constructed from the original dual intertwiners of vertical level one as a product in the horizontal channel using the relation 2 (n ) [u2; v2] (n +1) [u1; v1] 1 = G(v1=v2)( v2)j 1j( q3v1) j 2j x2 1 x 1 : (n +1) [u1; v1] (n ) [u2; v2] : 2 1 Q x2 2 x N (v2; 2; v1; 1) Repeating the operation m times reproduces the dual intertwiner of level m, m Qlm=1 tn+m l( l; ul; vl) (n;m) [u; ~v]; ~ with the intermediate horizontal weights ul such that ul = vl+1ul+1 and um = u: 8Note that we have exchanged the role of u and v with respect to the original de nition so that ~v is always the weight in the vertical space. Q m (3.11) (3.13) (3.14) (3.15) (3.16) Representation web. In order to form a particular web of representations relevant to a given gauge theory, the AFS interwiners can be coupled in two di erent ways: either along the horizontal (1; n) or the vertical (0; m) channels. These two contraction channels will be discussed below. To anticipate, an horizontal contraction corresponds to the operator product of the q-Heisenberg algebra describing the horizontal representation. On the other hand, the vertical contraction will be associated a scalar product that generates the trace of a tensor product. These contractions are represented by joining the vertex drawn in the gure 2 along horizontal/vertical legs.9 Taking only AFS intertwiners of rank m = 1, we recover the (p; q)-web diagram giving the brane con guration engineering the gauge theory. From the NS5 brane perspective, the operators and are interpreted as creation/annihilation operators of D5 branes since they increase/decrease the horizontal level by one respectively. Branes charge conservation takes the form of representation levels conservation, with horizontal edges oriented from left to right, and vertical edges from bottom to top. The introduction of intertwiners with higher rank allows us to simplify the diagram, e ectively folding it m times in the horizontal direction. In this way, it describes the creation/annihilation of m coinciding D5 branes in one go. The resulting web diagram corresponds to the (p; q)-web diagram of the gauge theory where all gauge groups have been replaced by U(1) groups. Although the information contained in this diagram become less visible, calculations are much more e cient in this setting. Mass-deformed intertwiners. For simplicity, the parameters of the gauge theory have been rescaled in order to set all the bifundamental masses to 1. It is however possible to keep arbitrary masses upon the introduction of mass-deformed intertwiners. This 9We remind the reader that we call \horizontal" every segment that is not vertical. can be achieved by using the twisted operators (z) = V (z ; ; (v) = V~ (z 1) 1V~+(zq3 1 the interwiner products of the form 1 ) 1 to construct the dual intertwiner (n;m) [u; ~v]. Then, 1 2 reproduces the bifundamental contribution cou ;~ 1) 1V+(zq3 1 1) 1 and pled to the nodes 1 and 2 with an arbitrary bifundamental mass . Since such a deformation brings only little new insight to our discussion, in the following we will keep all the bifundamental masses set to 1 to lighten the notations. In [35], a series of commutation relations were derived, involving the intertwiners , and the DIM generators in the appropriate representations. These relations can be extended HJEP1(207)34 to the generalized intertwiners de ned here. They are expressed formally as u0 (1;n+m)(e) (n;m)[u; ~v] = (n;m) [u; ~v] (u10;n+m)(e) = for any element e 2 E of the algebra. A proof is brie y sketched in appendix C. To be a little more explicit, the lemmas can be expressed as an action in the q-boson Fock space of the horizontal representation of DIM generators on the vertical components (3.17) ~ of the (3.18) (3.19) intertwiner operators: ( 1=2z) ~(n;m)[u; ~v] = z m m m 1 ~ (z) ~ (z) m X ~ ( ~ ~ +x x ( ~ 1z) = = zm 1 2m+1 X (z= x) zR=esx z 1Y~ (zq3 1) ~(n;mx)[u; ~v] + ( 1=2z): In these relations, the representation in the horizontal space of the DIM operators has been omitted: for instance x (z) reads (u1;n)(x (z)) on the right of the intertwiner ~ , and u0 (1;n+m)(x (z)) on the left. (1;n)(x (z)) on the left and u written for the dual intertwiner, ~ (n;m) [u; ~v] +( 1=2z) ~ 1=2z) u0 m ~ = 1 X For the dual intertwiner, x (z) is understood to be (1;n+m)(x (z)) on the right. Symmetric relations can be m ~ (z) ~ (z) 1z) +( 1=2z) ~(n;m) [u; ~v] = 0 ( x+( 1=2z) ~(n;m) [u; ~v] = 0; (z= x) zR=esx z 1Y~ (zq3 1) ( 1=2z) ~(n;mx) [u; ~v]; m z ~v ( m+1 X ~ +x z z the insertion of an operator e 2 E. ~ (n;m) [u; ~v]x (z) m ~ (z)x (z) ~(n;m) [u; ~v] = In fact, it is possible to re-write the r.h.s. of the relations involving x in a slightly more condensed way, ~ (z) ~(n;m)[u; ~v]x+(z) = ~(v0;m)(x+(z)) x ( ~ 1z) m x+( ~v 1=2z)h ^(0;m)(x+(z)) ~ (n;m) [u; ~v] ; i ~ (z)x (z) ~(n;m) [u; ~v] = ^~v (0;m)(x (z)) ~ where the dot denotes the action in the vertical space. The AFS lemmas are represented on the gure 4, in a very simpli ed manner. The insertion of the symbol z denotes the action of DIM generators e 2 E , and two insertions the action of the coproduct. 3.3 Gaiotto state The q-deformed Gaitto state is a Whittaker state for the q-Virasoro (or q-W) algebra [49{53], it is a deformed version of the original Gaiotto state de ned for the Virasoro algebra [54{57]. The intertwiners and de ned in the previous section can be interpreted as an uplift of the Gaiotto state to the horizontal representation space. Indeed, the Gaiotto state can be recovered by taking the vacuum expectation value in the horizontal spaces, ~ ~ (n ;m) [u ; ~v]j;i(1;n +m)u 0 hhG; ~vj = (1;n+m)u0h;j (n;m)[u; ~v]j;i(1;n)u = = X a~ ~ D (n ;m) [u ; ~v]E j~v; ~ ii; ~ X a~ ~ D (n;m)[u; ~v]E hh~v; ~ j; ~ where we have used the de nitions (n ;m) [u ; ~v] = hh~v; ~ j ~ (n ;m) [u ; ~v]; ~ (n;m)[u; ~v] = (n;m)[u; ~v]j~v; ~ ii and introduced the closure relation in the vertical space 1 = X a~ j~v; ~ iihh~v; ~ j: ~ (3.21) (3.22) (3.23) (0; m2)~v2 (1; n2)u2 Horizontal intertwiner for the A2 quiver The A2 quiver gauge theory, with gauge group U(m2) U(m1) is described by an operator TU(m2) U(m1) involving two vertical contractions (for the vector multiplet contributions), and a single horizontal contraction (the bifundamental contribution): TU(m2) U(m1) =Xa~1a~2 ~ with the constraints n2 = n1 and u2 = u1 in order to match the representation levels and weights in the horizontal channel. The relevant con guration of DIM representations can be seen on gure 9. Actually, this operator can be obtained from the combination of the operators TU(m1) and TU(m2) associated to the two gauge groups, using a new product to represent the horizontal contraction, TU(m2) U(m1) = TU(m1) TU(m2). The new product corresponds to the concatenation of two chains of tensor products of respective length r and s, in order to form a chain of length r + s 1: (a1 ar) (b1 bs) = a1 ar 1 arb1 b2 bs: (4.20) As before, the instanton partition function of the gauge theory corresponds to the vev in the horizontal spaces, up to a factor of G-functions depending only on the Coulomb branch vevs, m1 m2 Zinst.[A2] = Y Y G(vl(02)= vl(1)) TU(m2) U(m1) = X qj~1jqj~2jZvect.(~v1; ~ 1)ZCS( 1; ~ 1)Zvect.(~v2; ~ 2)ZCS( 2; ~ 2)Zbfd.(~v2; ~ 2; ~v1; ~ 1j 1 2 ~1;~2 (4.19) (4.21) 1); with the identi cation q = m u =u and = n n for the gauge coupling and the Chern-Simons level associated to the two gauge groups = 1; 2. Fundamental qq-character for the A2 quiver A di erent qq-character is associated to each node of the quiver diagram, in correspondence with antisymmetric representations of the Lie algebra. In the case of the A2 quiver, the two relevant representations are the fundamental and the fully antisymmetric ones. We focus rst on the construction of the fundamental qq-character +(z), leaving the antisymmetric representation for the next section. Within our conventions, the fundamental representation is associated to the action of x+(z) on the rst node. The commutation property (4.6) of the vertical channel is valid at each node in the form TU(m2) U(m1) = 1 TU(m2) U(m1) = 1 1 ~v1 (0;m1)(e) 1 ~v2 (0;m2)(e) The operator leading to the fundamental qq-character can be obtained using the double coproduct = ( 1) = (1 ) : The action of this double co-product on the DIM generators reads (z)) = (1 ^ 1=2 ^ 1=2 z) 1 (^ 1 1 z) ^1=2 ^ z) ^1=2 1 z) +(1 ^ 1=2 z) 1 1 +(1 1 z) 1 ^ 1 z) ^1=2 z) 1 ^1=2 z) Combining the general commutation properties of the vertical and horizontal contractions, together with the AFS lemmas (3.17), it is possible to show that the horizontal action of the generators commutes with the T -operator: (1;n1) u 1 (1;n1+m1) u0 1 = TU(m2) U(m1) (1;n2+m2) u0 2 (1;n1+m1) u0 1 (e) TU(m2) U(m1) (1;n2+m2) u0 2 (1;n2) u2 (e): The proof is a tedious but straightforward calculation, using the same method as in the single node case. Introducing the operator X +(z) = (x+(z)) that commutes with TU(m2) U(m1), the fundamental qq-character can be written (omitting the horizontal representations): = u =(q3m u0 ). Again, this quantity is a polynomial in z because X + commutes with T . The power of z in the prefactor is xed by consideration of the asymptotic behavior. (4.22) (4.23) (4.24) (4.25) (4.26) 1 in the : gauge (4.27) (4.28) ; (4.29) Evaluating the correlators in the q-bosonic Fock spaces, we recover the expression given in [11] for the fundamental A2 qq-character with a bifundamental mass = vertical channel:12 * Quantum Weyl re ection and the second qq-character There are two ways to obtain the second qq-character. The simplest one is to consider the insertion of the operator X (z) = (x (z 1)) that also commutes with the T -operator. De ning gives after evaluation of each horizontal actions, (z) = * which is indeed the second qq-character of the A2 quiver found in [11]. In fact, because of the re ection symmetry obeyed by the A2 quiver, we have why we have obtained the correct qq-character. This property seems to be a consequence of the 5 symmetry of the representation web combined with the S2 rotation of the DIM (z) = +(z) which explains algebra. There is a more natural way to derive the second qq-character, which is to start from insertions of x+(z) in two of the three horizontal lines at the end of the diagram, with spectral parameters ne-tuned to obtain the commutation with the T -operator. Indeed, the rst term of the qq-character, that is proportional to Y~ 2 (z the insertion of the operator 2), can be obtained from x+(z) 1 (4.30) term [31]. second node to on the left of the representation web. The other terms entering the expression of the qq-characters are known to be obtained by acting with the Weyl re ection on the rst The Weyl re ection for the A2 quiver diagram sends the co-weight w2 attached to the w2 !2 w1 1 w2 ! w1 () Y~ 2 ! 2 Y~ 1 Y~ 2 1 ! 1 Y~ 1 ; (4.31) where i denote the roots of the Lie algebra with the Dynkin diagram A2. To each intermediate expression has been associated a term of the qq-character. A similar transformation 12To simplify formulas, the labels corresponding to the two nodes have been exchanged here with respect to the conventions employed in [11]. re ection on x+(z) x+(z) 1 1 1 ! 2 ! can be de ned on the DIM generator x+(z) involved in tensorial expressions like (4.30). It will be called the quantum Weyl re ection. The quantum Weyl re ection with respect to the root i consists in replacing the insertion of x+ in the ith tensor space with an insertion of in the ith space and x+ in the (i + 1)th space, together with the appropriate shifts of the spectral parameters:13 x+(z) i 1 i ! (^(1i=)2z) i x+(^(i)z) (4.33) where ^(i) = (1 )i 1 ^( 1)r+1 i. The transformation is forbidden if two x+ were to collide in the same space. In a sense, the generators x+ obey a fermionic statistics in the tensor spaces. It is further assumed that the operators are ordered on the left of operators x+, although this fact does not modify the derivation of the qq-characters. Before applying the quantum Weyl re ection to obtain the operator relevant to the second node of the A2 quiver, let us review how this transformation works in the known cases. The A1 quiver is described by two horizontal spaces, it has a single weight w ! w +(z) has only two terms. The application of the quantum Weyl 1 leads to the coproduct (x+(z)). Turning to the A2 quiver, the quantum Weyl re ections of reproduces the three terms in the expression of the operator X +(z) constructed from the application of the squared coproduct . Thus, the quantum Weyl re ection de nes a generalization of the coproduct that implements the action of an operator into three copies of the initial space. over the three terms in order to de ne Now, we apply the Weyl re ection to the operator (4.30) of the A2 quiver, and sum x+(z) + + 1 z) 1 1 z) 1 z): 1 (1 (1 ^1=2 1 z) x+(1 1 z) ^1=2 1 z) (^1=2 (^1=2 1 1 1 z) 1 z) 1 z) 1 1 z) x+(^ ^ 1 z) (4.34) After a tedious but straightforward computation, it can be shown that the operator (x+(z)) commutes with T . The commutation of the operator de ned in (4.14) with TU(m) (seen here as a subdiagram) is essential for the various cancellations to occur. The corresponding qq-character is de ned as 13The quantum Weyl transformation can be de ned in a similar manner on the generator x (z), i.e. in such a way that it reproduces the coproduct in the case of the fundamental representation: 1 i x (z) i ! x (^(i+1)z) i +(^(1i=+21)z) : (4.35) (4.36) (4.32) The evaluation of the vev in the horizontal spaces reproduces the expression of (z) given in (4.29), showing that indeed (z). The results obtained for the A2 quiver are easily generalized to linear quivers with an arbitrary number of nodes r. The T -operator is constructed using r the horizontal channel, rendered by the product de ned in (4.20), 1 contractions in where the order of indices labeling gauge groups has been reversed for convenience. (4.37) HJEP1(207)34 Schematically, it reads U(1) = X Y a~ ~ 1; ~ r s=1 1 1 2 r ; (4.38) where we have omitted all the weights and level parameters. This expression implies the constraints ns = ns+1 and us = us+1 in order to match the representation spaces in horizontal channels. Then, up to a prefactor of G-functions, the vev reproduces the instanton partition function (2.28) for the quiver n ns for the Chern-Simons levels, and qs = = Ar under the identi cation s = s ms us=us for the gauge couplings. As before, the fundamental qq-character, attached to the rst node, can be obtained by multiple applications of the coproduct. The Ar fundamental coproduct is de ned recursively as = ( ( 1)r 1 ) ( ( 1)r 2 (4.39) and acts on the DIM generators as follows: r+1 X s=1 r+1 s=1 (z)) = (1( ^ 1=2)r z) ^(s) = (^ )s1( 1)r s, and the tensorial transpose de ned as (a1 The fundamental coproduct of DIM generators commutes with the T -operator, (e) TU(mr) U(1) = TU(mr) U(1) (e); where we have omitted to indicate the horizontal representations. As a result, the fundamental qq-character de ned as (4.41) is a polynomial. Explicit evaluation of the correlators for each horizontal space provides the formula i=1 2) + X s=1 s i=1 1 Y qiz i 2mi ms+Pjs=i+1 j zm1 ms Y qiz i 2mi mr+Pjr=i+1 j zm1 mr mr ! ! 1 Y~ r (z r 1) + gauge : ms Y~ s+1 (z s 2) Y~ s (z s 1) (4.43) (4.44) The qq-character associated to the sth node corresponds to the antisymmetric representation denoted by the Young diagram (s) with s boxes, all in the rst column. The corresponding operator X(+s)(z) is obtained by application of quantum Weyl re ections on Note that since the operator is evaluated in horizontal representations, the position of the central element ^ in the arguments of operators is somewhat arbitrary here. As an illustration, the A3 quiver is treated in details in the appendix E. This construction can also be applied to the generator x (z). Because of the re ection symmetry of the quiver diagram, the corresponding qq-characters are expected to obey the relation (s)(z) = (r+1 s)(z). In fact, it is possible to de ne qq-characters associated to arbitrary representations of the Lie algebra. To a representation labeled by a Young diagram is associated the + operator X +(~z) obtained by taking the product over the columns X( i)(zi) de ned previously. This construction works if the i of the operators rst column of the Young diagram contains at most r boxes. By construction, these operators commute with the T operator of the gauge theory, and the vev by a monomial of the variables zi. X +(~z)T is a polynomial up to multiplication 4.2.5 Inclusion of fundamental/antifundamental matter elds Matter elds are introduced by semi-in nite D5 branes that are vertical edges in the representation web. These can be inserted either in the bottom or top part of the diagram, leading to fundamental ( ) or antifundamental ( ) matter respectively. It is well-known in gauge theory that such matter elds can be obtained by introducing extra gauge groups, sending the corresponding gauge coupling q to zero. This constraints the Young diagrams ~ associated to this gauge group in the partition function expansion (2.28) to be empty, hence generating the contributions Zfund.( 1~ (f); ~ ) = Zbfd.(~v; ~ ; ~ (f); ~;j 1); Za.f.( ~ (af); ~ ) = Zbfd.(~ (af); ~;; ~v; ~ j 1): (4.45) In this spirit, we can regard the massive A1 quiver as the limit of the A3 quiver as q1; q3 ! 0. This procedure corresponds to send two NS5 branes at in nity. Taking the massive A1 qq-character +(z) obtained in [11], formula from appendix E for the A3 qq-character sending the gauge couplings q1; q3 ! 0 while q2 = q is held +(z) associated to the second node, and xed, we indeed recover the since zmY~ (z Y~ (z) gauge ; z 1( l(f)) 1) = pfund.(z 2); (af)z 1) = pa.f.(z); l ; (n;f~) [u; ~(af)] ~ and we have dropped the label 2 of the middle node. In our formalism, the gauge coupling q is obtained as a ratio of horizontal weights u=u . The limiting procedure q ! 0 corresponds to send either u to zero for some intertwiner or u to in nity for the dual intertwiner . In either case, the normalization coe cients, tn;m or tn;m, vanishes except when the Young diagrams ~ are empty. The case of the , antifundamental matter is the easiest one to consider. Indeed, it is observed from the AFS lemma that since R(~;) = ;, the vacuum intertwiner the T -operator is simpli ed as the extra ; can be decoupled,14 x+(z). As a result, an additional horizontal contraction with this operator, as represented on the gure 10, does not spoil the commutation with the operator X +(z). In this case, ; commutes with the action of TU(a(mf) ) = X a~ ~ (n ;m) [u ; ~v] ~ (n;m)[u; ~v] ~(n;f~) [u; ~ (af)]: ~ ; (4.48) 14Said it otherwise, the action of x+(z) on ; being proportional to u ! 0, the extra horizontal channel can be dropped. (4.46) (4.47) HJEP1(207)34 On the other hand, in the case of fundamental matter, it does not seem possible to fully decouple the extra horizontal channel, and we are forced to de ne the T -operator within three di erent Fock spaces, X a~ ~ ~ ; (n;f) [u; ~ (f)] ; (n;f)[u; ~ (f)] ~(n ;m) [u ; ~v] ~ ~ (4.49) in order to observe the commutation relation with the operator X is related to the non-commutation of ~ with x+(z), it can be solved by considering the +(z).15 This problem commutation with the operator X (z) instead. However, the problem persists if both fundamental and antifundamental matter are introduced, in which case the only solution is to consider a third horizontal channel with a trivial vertical contraction as in (4.49). The treatment of fundamental matter here is rather di erent from the usual brane description. In particular, we do not observe a limitation on the number of elds in this algebraic construction, which may be an e ect of the presence of Chern-Simons terms. It would be advisable to achieve a deeper understanding of the precise di erence between the two constructions. Since our understanding of fundamental matter is based on gauging the avor group, the generalization of these results to all linear quivers would require to construct arbitrary quiver theories, which is way beyond the scope of our paper. However, we hope to be able to address this issue in a near future. 5 We have proposed an algebraic method to derive qq-characters of linear quiver N = 1 gauge theories with U(m) gauge groups. It is based on the insertion of DIM generators in a tensored horizontal representation, symmetrized in order to de ne an operator commuting with the T -operator of the gauge theory. This method provides an e cient way to derive the explicit expression of the qq-characters as correlators in the gauge theory. There are several directions in which this work can be extended. The most natural one is the treatment of DE-type quivers. In the case of D-type quivers, the brane construction of Kapustin [60] involving an orientifold brane seems relevant. Progress along this direction will be reported elsewhere. A ne quivers could also be considered. There, the extra compact dimension seems to impose the consideration of a ring of tensor spaces in the horizontal representations in which an in nite number of quantum Weyl transformations can be applied. A much harder problem would consist in studying gauge theories with DE-type gauge groups, i.e. Sp(m) or SO(m) groups. The recent construction of Hayashi and Ohmori [61] could be helpful in this context. In [62], a deformation of the re ned topological vertex has been introduced, that corresponds to a further (q; t)-deformation of the horizontal representation. It would be interesting to further study the underlying algebraic structure. 15Taking the limit u1 ! 1 in the A3 operator X +(z), the dominant terms are those with a x+(z) generator inserted in the rst space. They are of order u1 and reproduce the two terms in the massive { 36 { HJEP1(207)34 We hope that the generalized intertwiners introduced here will also be useful in the description of the underlying integrability, leading to a generalization of the R-matrix construction [21, 22]. Finally, the action of a similar quantum algebra has been observed in the context of higher spins [63], and it would be interesting to investigate the role played by these fundamental objects that are interwiners and qq-characters. Acknowledgments J.-E.B. would like to thank A. Sciarappa and Joonho Kim for discussions. In the early stages of this project, he has been supported by an I.N.F.N. post-doctoral fellowship within the grant GAST, and the UniTo-SanPaolo research grant Nr TO-Call3-2012-0088 \Modern Applications of String Theory" (MAST), the ESF Network \Holographic methods for strongly coupled systems" (HoloGrav) (09-RNP-092 (PESC)) and the MPNS{COST Action MP1210. He also wishes to thank Tokyo University for their generous nancial support during his stay. YM is partially supported by Grants-in-Aid for Scienti c Research (Kakenhi #25400246) from MEXT, Japan. RZ is supported by JSPS fellowship and he is also grateful for the hospitality during his stay in KIAS. Part of the results of the paper were announced in the workshop \Progress in Quantum Field Theory and String Theory II" (March 27-31, 2017 Osaka City University). We would like to thank the participants of the workshop, especially H. Awata, H. Itoyama, H. Kanno, Y. Zenkevich with whom very useful discussion was made. A Di erent expressions for the vertical representation In [11], a di erent-looking vertical representation has been employed. At the level (0; 1), e(z)jv; i = z 1 f (z)jv; i = (z)jv; i = X x2R( ) X x2A( ) (z= x) x( )jv; + xi (z= x) x( )jv; x ; i (z)] jv; i; 1 = (1 q1)(1 q2)(1 q3); (A.2) with function residues x( )2 = tions as follows: 16Here the generators have been multiplied by a constant factor without altering the commutation relae(z) ! z 1p(1 q3)ve(z); f (z) ! zp(1 q3)vf (z); (z) ! (1 q3)v (z): (A.1) The de nition of the function (z) has also been modi ed in order to re ect this change of normalization. (z) de ned in (2.12), and the coe cients being the square root of the Resz= x (z). However, the normalization of the states can be modi ed by an arbitrary factor: letting jv; ii = N ( )jv; i, we have in general e(z)jv; ii = z 1 f (z)jv; ii = X N ( + x) jv; + xii; (z= x) x( ) N ( ) N ( x) jv; Note that the action of the Cartan is not modi ed since they are diagonal in this basis. Choosing the normalization factor as17 N ( ) = p 1 Zvect.(v; ) Y q3)v1=2 ! 1=2 1 x ; N ( ) N ( + x) x( ) Y (q3 1 x) ; using the fact that Res z= x2A( ) Res z= x2R( ) (z) = Y ( xq3 1) (z) = 1 1 Res z= x2A( ) Y (z) Res Y (zq3 1); Y ( x) z= x2R( ) e(z)jv; ii = f (z)jv; ii = X X x2R( ) 1 (z= x) zR=esx zY (z) jv; + xii; (z= x)Y x(q3 1 x)jv; xii: Y x(zq3 1) = Y (zq3 1)=S(q3 x=z), The second relation simpli es after a careful treatment of the limit z ! x in the expression f (z)jv; ii = 1q3 X (z= x) zR=esx z 1 Y (zq3 1)jv; xii: and the property x( ) = x), the new representation can be written Zvect.(v; + x) Zvect.(v; ) = (1 q3)2v 1 2x 1 Y ( xq3 1) zR=esx Y (z) 1 = (1 q3)2v 1 2x x( )2 Y ( xq3 1)2 : tion [11], Finally, we notice that the coe cient in front of the commutator [e; f ] is di erent from the one in (2.2) for [x+; x ]. In order to recover the same convention, we need to multiply f (z) ! (1 1q3 q3)2 f (z); (z) ! 1 1 (z): Under the identi cation of the renormalized f (z) with x (z), and e(z) with x+(z), we end up with the vertical representation (2.11). In addition, an extra cosmetic factor of has been added in front of x (z) and (z) in order to simplify some expressions. It is important to stress that our renormalized vertical representation here does not coincide with the one used in AFS's paper in which the normalization of the intertwiners and 17The recursive property is inherited from the discrete Ward identity obeyed by the vector contribuxii; ; (A.3) (A.5) (A.6) (A.7) (A.8) (A.9) 1 (A.4) B.1 q-bosons vertex operators The vertex operators , and ' satisfy the relations (z) (w) = S(w=z) 1 : (z) (w) :; (z) (w) = S( w=z) : (z) (w) :; '+( 1=2z) (w) = '+( 1=2z) (w) = S(z=w) S(w=z) : '+( 1=2z) (w) :; S(z=w) S(w=z) : '+( 1=2z) (w) :; ' ( 1=2z) (w) =: ' ( 1=2z) (w) :; ' ( 1=2z) (w) =: ' ( 1=2z) (w) :; (z) (w) = S(z=w) 1 : (z) (w) :; (w) (z) = S( z=w) : (z) (w) :; (w)'+( 1=2z) =: '+( 1=2z) (w) :; (w)'+( 1=2z) =: '+( 1=2z) (w) :; (w)' ( 1=2z) = (w)' ( 1=2z) = SS((zw==wz)) : ' ( 1=2z) (w) :; SS((zw==wz)) : ' ( 1=2z) (w) : : Explicitly, the vacuum intertwiners read ;(v) = exp ;(v) = exp ! qk vka k exp kvna k exp q k 1 q k ! v kak ; kv kak ; ! they obey the relations 1 1 1 1 1 (z) ;(w) = ;(w) (z) = (z) ;(w) = (1 ;(w) (z) = (1 1 w=z 1 : (z) ;(w) :; : (z) ;(w) :; w=z) : (z) ;(w) :; z=( w)) : (z) ;(w) : 1 2w=z w=z '+( 1=2z) ;(w) = : '+( 1=2z) ;(w) :; ;(w)'+( 1=2z) =: '+( 1=2z) ;(w) :; ' ( 1=2z) ;(w) =: ' ( 1=2z) ;(w) :; ;(w)' ( 1=2z) = 1 z=w : ' ( 1=2z) ;(w) :; ; (z) ;(w) = (1 (w) (z) = (1 (z) ;(w) = w=z) : (z) ;(w) :; z=( w)) : (z) ;(w) :; : (z) ;(w) :; 1 2w=z (B.1) (B.2) w=z 2w=z 1 z=w (w) (z) = : (z) ; (w) :; '+( 1=2z) ; (w) = : '+( 1=2z) ; (w) :; (w)'+( 1=2z) =: '+( 1=2z) ; (w) :; ' ( 1=2z) ; (w) =: ' ( 1=2z) ; (w) :; (w)' ( 1=2z) = : ' ( 1=2z) ; (w) : : Note also the properties with G(z) = exp and the fact that ;(z) ;(w) = G(w= 2z) : ;(z) ; (w) = G(w=( z)) 1 : ;(z) ;(w) :; ;(z) ; (w) :; (z) ; (w) = G(w=z) : (w) ;(z) = G(z=( w)) 1 : ; (z) ; (w) :; ;(z) ; ! ; ; = 1 Y i;j=1 k t k) 1 zq1i 1qj 1 ; 2 '+( 1=2z) =: (z) ( z) :; ' ( 1=2z) =: (z) ( 1z) : : B.2 Commutation relations in horizontal representations The simplest relations are the commutations between the operators and the intertwiners, they can be derived easily by combining the properties given previously and the formula (2.12):18 (n;m)[u; ~v] + (n;m)[u; ~v] 1=2z) ~(n;m)[u; ~v] = (n+m) ~ (z) : '+( 1=2z) ~(n;m)[u; ~v] : ( 1=2z) ~(n;m)[u; ~v] = n+m : ' ( 1=2z) ~(n;m)[u; ~v] : +( 1=2z) ~(n;m) [u; ~v] = ~ (z) 1 : '+( 1=2z) ~(n;m) [u; ~v] : 1=2z) ~(n;m) [u; ~v] = n : ' ( 1=2z) = n : '+( 1=2z) ~(n;m) [u; ~v] : 1=2z) ~(n;m)[u; ~v] : ( 1=2z) = ~ (z) 1 : ' ( 1=2z) ~(n;m)[u; ~v] : ~ (n;m) [u; ~v] +( 1=2z) = n m : '+( 1=2z) ~(n;m) [u; ~v] : ~ 1=2z) = n m ~ (z) : ' ( 1=2z) ~(n;m) [u; ~v] : In these expressions, the representation (1; n + m)u0 of the DIM generator is understood (but omitted) on the left of the operator ~ (n;m), while the representation on the right is (1; n)u. The two representations are exchanged for the dual operator: (1; n)u is on the left 18Operators are supposed to be radially ordered. (B.3) (B.4) (B.5) (B.6) (n;m) while (1; n + m)u0 is on the right. Similar expressions can be derived for x : ~ zn+mY~ (z) : (z) ~(n;m)[u; ~v] : x+(z) ~(n;m) [u; ~v] = u n z nY~ (z 1) : (z) ~(n;m) [u; ~v] : x (z) ~(n;m) [u; ~v] = n u nY~ (zq3 1) : (z) ~(n;m) [u; ~v] : (n;m)[u; ~v]x+(z) = (n;m) [u; ~v]x+(z) = u n (n;m) [u; ~v]x (z) = zn+m ~ (z) ~ (z) 1 u0 n+m zn+mY~ (z) n u n Y (zq3 1) : (z) ~(n;m) [u; ~v] : C Derivation of the AFS lemmas The proof of the relations involving (z) is a matter of writing the commutation relations (B.6). Hence the focus here is on the generators x (z). We rst examine the product of x+(z) and (n;m)[u; ~v], the proof is based on the following decomposition for the function 1 zY~ (z) X x2A(~ ) z 1 Res x z= x zY~ (z) : 1 As a consequence, we can write the right product of x+(z) on in (B.7) as x+(z) ~(n;m)[u; ~v] = u0 n+mz n m+1 = u0 n+mz n m+1 + u0 n+mz n m+1 X x2A(~ ) z X x2A(~ ) z X x2A(~ ) 1 1 Res Res x z= x zY~ (z) x z= x zY~ (z) 1 1 : (z) ~(n;m)[u; ~v] : : ( x) ~(n;m)[u; ~v] : x 1 Res z= x zY~ (z) : This expression is valid for jzj > j xj, however the second line of the last equality has no x and can be analytically continued to jzj < j xj. This is not true for the rst line, and the fraction should be expanded in positive powers of z. A similar expression can (B.7) HJEP1(207)34 (C.1) (C.2) be obtained for jzj < j xj by considering the left product of x+(z) on ~ (z) ~(n;m)[u; ~v]x+(z) + u0 n+mz n m+1 X x2A(~ ) x 1 Res z= x zY~ (z) : Taking the di erence of the two, the terms with no singularity cancel each-other. The remaining expression is a di erence of expansions in powers of z and z 1 that forms a delta function, x+(z) ~(n;m)[u; ~v] ~ (z) ~(n;m)[u; ~v]x+(z) = u0 n+mz n m X x2A(~ ) (z= x) zR=esx zY~ (z) 1 : ( x) ~(n;m)[u; ~v] : Then, since (n;m) is built as a product of operators ( x) for all x 2 , the vertex operator ~ : ( x) ~(n;m) : can be written as ~ +x (n;m). Taking into account the prefactor tn(~ ; u; v) tn(~ + x; u; v) n+1 u0 n+1 ; we recover the AFS lemma in the form (3.18). A similar argument can be employed to treat the action of x (z), with the poles located at the points z = 1 x2R(~ ), and the operator : (z) ~(n;m): simpli ed using the property (B.5) of the appendix B.19 However, in this case, a more elegant proof is also and (n) that can be found in [64] (formula (6.15)). By de nition, we have possible. It is based on the formula for the commutation relation between the modes xk k = I dz k 1 0 2i x (z) so that 19The following property is useful here, I jzj>j xj X I x2R(~ ) 2i dz k 1 x 1 2i x (z) ~(n;m)[u; ~v] dz zk+n+m 1 u0 n+m Y~ (z I z=0 jzj<j xj 2i 1) : (z) ~(n;m)[u; ~v] : ~ dz zk 1 (n;m)[u; ~v]x (z) z=Res1 f (z) = 1 Rz=es f (z 1): (C.3) (C.5) (C.7) (C.6) The second equality is the consequence of several cancellations between poles, such that only the poles of Y~ (z 1) will contribute. The expression for the product of operators is taken from (B.7). The contour integral can be reduced to the residue contributions of the integrand, which simpli es thanks to the properties (B.5) and (C.6) to give k+m 2 2m k+1 Res Y~ (zq3 1) ~(n;mx)[u; ~v] + z= x 1=2 x): (C.9) Summing over the index k with the spectral parameter at the power z k, we recover the AFS lemma (3.18). This short computation gives some insight on the interpretation of the AFS lemma: it is valid for each power of z in a formal expansion. D Connection with quiver W-algebras In [31, 32], Kimura and Pestun have introduced quantum W-algebras based on the Dynkin diagram of simple Lie algebras of ADE type. These algebras are constructed upon a set of q-bosonic modes s(ki) with k 2 Z and i 2 that obey the commutation relations [s(ki); s(i0k)0 ] = 1 1 q k k k;k0 cii0 ; [k] k > 0; where c[iki0] denotes the k-th Adams operation applied to the mass-deformed Cartan matrix. For instance, in the case of the A3 quiver with bifundamental masses ii0 = 1, this matrix reads 0 1 + q3k k 1 + q3k A 1 k C : Since this algebra is also acting on Nekrasov partition functions, it should be related to the DIM algebra considered in our paper. The aim of this appendix is to highlight this connection. It is based on the decomposition of the tensor product of two (1; 0) DIM representations into q-Heisenberg q-Virasoro algebras. This decomposition has been described by Mironov, Morozov and Zenkevich in [33], and this appendix is just a reformulation of their results in our notations. In order to simplify the discussion, we will neglect the role of zero modes, TU(1) = tr will also restrict ourselves to U(1) gauge groups at each node of the quiver diagram. It is an easy exercise to extend the argument to more general cases. We rst focus on the A1 quiver for which the T -operator is built as a vertical contraction of two intertwiners, . Since two horizontal spaces are involved, we need two copies of the q-bosonic modes in order to represent the horizontal action of the intertwiner and its dual. We denote these modes a(ki) with i = 1; 2. By de nition, modes with a di erent value of the label i commute, while modes with the same label obey the commutation relation (2.21): ;,. . . We [a(ki); a(i0k)0 ] = k 1 1 q t k k;k0 i;i0 ; k > 0: (D.1) (D.3) (z) ': S(z) 1S(q2z) :; S(z) =: exp ! X zks k : : k2Z Taking the product over the boxes x 2 , several cancellations occur, and the nal result is expressed in terms of a product over each column i of height i , : Y x2 ( x) ( x) : ' : Y S(vq1i 1q2 i ) : i where we have neglected the boundary terms S(vq1i 1) that can be taken care of using zero modes. In the r.h.s. , the product is taken over the elements of the set X de ned in [31], and we can formally identify the state jZT i representing the partition function with the action of TU(m) over the horizontal vacuum states: jZT i ' X : Y S(vq1i 1q2 i ) : j;i ' TU(1) (j;i j;i) : i The modes sk can be used to build the stress-energy tensor of the q-Virasoro algebra. The orthogonal combination kbk = a(1) + jkja(k2), which by de nition commutes with sk, k obeys the q-bosonic commutation relation [bk; b k0 ] = 1 1 k 1 t k k;k0 (1 + q3k) k > 0: (1; 0) (1; 0) = q-Heisenberg q-Virasoro: It leads to identify the modes ksk = jkja(k1) + a(k2). They indeed obey the commutation relation (D.1) with the deformed A1 Cartan matrix c[k] = 1 + q3k. As a result, the operator (D.4) can be expressed in terms of the screening operator de ned in [31], (D.5) (D.7) (D.8) (D.9) (D.10) (D.11) The operator T involves a trace over Young diagram realizations of a product over the box content of the diagram. Each factor contains the following operator evaluated at z = for some x 2 , (z) = exp X1 1 k=1 t k zk a(1k) k exp k=1 z k a(1) k k It is also interesting to rewrite the coproduct of x+(z) in terms of the modes bk; sk: : (x+(z)) : ' : exp X 1 k2Z 1 + q3j j k z k bk : Y (z 1)+ : Y (z ) 1 : where, following Kimura and Pestun, we have introduced the operator Y (z) =: exp X z k k2Z yk :; yk = t 1 1 + q3k sk: In this expression, a di erent set of modes a(i) is attached to each tensor space, with i = 1 r + 1. It leads to identify the modes as follows: ks(ki) = jkja(ki) + a(ki+1); kbk = jkjia(ki): Under this identi cation, the modes s(ki) reproduce the commutation relation (D.1) with the deformed Cartan matrix of the Ar Dynkin diagram. In addition, they all commute with the modes bk. Thus, for a general linear quiver, we have the formal decomposition (1; 0) (r+1) = q-Heisenberg Wr: E Derivation of the qq-characters for the A3 quiver The qq-characters associated to the three nodes of the A3 quiver are labeled by the Young diagrams , r 1 r r ( x) ( x)A X (D.12) ( x)A : (D.13) (D.14) Hence, up to a U(1) factor, we recover in (x+(z)) the operator T of Kimura and Pestun, identi ed with the fundamental current (stress-energy tensor) of q-Virasoro [59].20 For a general linear quiver diagram Ar, the modes s(ki) i = 1 r are associated to the nodes of the diagram. On the other hand, the T -operator is written as an (r + 1)th tensorial product X Y (1); ; (r) x2 (1) U(m1) = tr 12 r 1 ( x) 1 2 1 0 Y x2 (2) x2 (r) ( x) :1=2x:2 x:1 : 1=2 :3=2x:3 :1=2x:2 x:1 20Note that we have chosen to denote the Q-operator of Kimura and Pestun as T since the partition function is obtained as the vev of this operator. On the other hand, their T-operator has been denoted X to emphasize the fact that it comes from the generators x of the DIM algebra, and that the TQ-relation only holds if we forget about the di erence of representations. 1 + :3=2 1 : 1=2 :3=2 : 1=2x:1 : 3=2 :1=2x:2 : 3=2 :1=2x:2 : 3=2 :1=2 5=2 + x:3 :1=2 :5=2 1 1 + :1=2 :5=2 : 1=2x:1 : 1=2x:1 : 1=2x:1 x:+2 + :3=2x:3 :1=2 :1=2 x:1 1 + :1=2 :3=2 :3=2 : 1=2 :3=2 :1=2x:2 :3=2 x:2 (1; n3)u3 (n3;m3) [u3;~v3] (1; n3 + m3)u03 (0; m3)~v3 (1; n1 + m1)u01 where we have introduced the shortcut notations x:+k = x+( kz), :k = ( kz). Note that the argument of operators has been simpli ed taking advantage of the fact that they act in the horizontal representations where ^ becomes . After a long and tedious computation, it is possible to show that these expressions do commute with the operator TU(m3) U(m2) U(m1) represented on gure 11. In practice, we have used a short program in Python to perform the algebraic manipulations. TU(m3) U(m2) U(m1) 2 u01u1 2n1+2n1+2m1 zn1+n1+m1+m2 3 u01u1u02 2n1+2n1+2m1+2n2+2m2 zn1+n1+n2+m1+m2+m3 we nd the following expressions after evaluation in the four independent Fock spaces, * 1zm1Y~1(z 2) + q1z 1 Y~2(z 1 + q1q2 Y~1(z) 1 2+2m1 m2z 1+ 2+m1 m2 Y~3(z) Y~2(z ) 1 2+2 3+2m1+2m2 2m3 z 1+ 2+ 3+m1 m3+ E TU(m3) U(m2) U(m1) (E.2) gauge 2+m2 (z ) 2+ 3+m1+m2 m3 Y~ 1 (z 2+m2 (z ) 1+ 2+ 3+m2 m3 1 2m1 (z ) 1+2 2+ 3+m1 m3 + Y~ 2 (z 2) Y~ 2 (z) Y~ 3 (z ) 1 Y~ 2 (z) Y~ 1 (z )Y~ 3 (z ) + q1q22q3 1 2 3 2 3 3 2)+q2 1 1 z 1+ 2 Y~ 3 (z Y~ 1 (z ) * 3zm3 Y~ 3 (z 2) + q3 2 m2 z 3+m2 Y~ 2 (z gauge 1 Y~ 3 (z) + q2q3 1 2+2m1 z 2+ 3+m1 Y~ 1 (z) Y~ 2 (z ) + q1q2q3 2 1+ 2 z 1+ 2+ 3 + Y~ 1 (z 2) gauge : Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [hep-th/9711013] [INSPIRE]. 7 (2004) 831 [hep-th/0206161]. [hep-th/0701156] [INSPIRE]. [arXiv:1510.01896] [INSPIRE]. [arXiv:1603.00304] [INSPIRE]. 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J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo, R.-D. Zhu. (p, q)-webs of DIM representations, 5d \( \mathcal{N}=1 \) instanton partition functions and qq-characters, Journal of High Energy Physics, 2017, 34, DOI: 10.1007/JHEP11(2017)034