Holomorphic field realization of SH c and quantum geometry of quiver gauge theories

Journal of High Energy Physics, Apr 2016

In the context of 4D/2D dualities, SH c algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of \( \mathcal{N}=2 \) supersymmetricgaugetheories. Inthispaper,werewritetheSH c algebrainterms of three holomorphic fields D 0(z), D ±1(z) with which the algebra and its representations are simplified. The instanton partition functions for arbitrary \( \mathcal{N}=2 \) super Yang-Mills theories with A n and A n (1) type quiver diagrams are compactly expressed as a product of four building blocks, Gaiotto state, dilatation, flavor vertex operator and intertwiner which are written in terms of SH c and the orthogonal basis introduced by Alba, Fateev, Litvinov and Tarnopolskiy. These building blocks are characterized by new conditions which generalize the known ones on the Gaiotto state and the Carlsson-Okounkov vertex. Consistency conditions of the inner product give algebraic relations for the chiral ring generating functions defined by Nekrasov, Pestun and Shatashvili. In particular we show the polynomiality of the qq-characters which have been introduced as a deformation of the Yangian characters. These relations define a second quantization of the Seiberg-Witten geometry, and, accordingly, reduce to a Baxter TQ-equation in the Nekrasov-Shatashvili limit of the Omega-background.

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Holomorphic field realization of SH c and quantum geometry of quiver gauge theories

HJE eld realization of SHc and quantum Jean-Emile Bourgine 0 1 2 5 6 Yutaka Matsuo 0 1 2 3 6 Hong Zhang 0 1 2 4 6 Symmetry, String Duality 0 Sogang University 1 Hongo 7-3-1 , Bunkyo-ku, Tokyo , Japan 2 Via Irnerio 46 , 40126 Bologna , Italy 3 Department of Physics, The University of Tokyo 4 Department of Physics and Center for Quantum Spacetime , CQUeST 5 INFN Bologna, Universita di Bologna 6 35 Baekbeom-ro , Mapo-gu, Seoul 04107 , Korea In the context of 4D/2D dualities, SHc algebra, introduced by Schi mann and Vasserot, provides a systematic method to analyse the instanton partition functions of N = 2 supersymmetric gauge theories. In this paper, we rewrite the SHc algebra in terms of three holomorphic elds D0(z), D 1(z) with which the algebra and its representations are simpli ed. The instanton partition functions for arbitrary N = 2 super Yang-Mills theories with An and A(n1) type quiver diagrams are compactly expressed as a product which are written in terms of SHc and the orthogonal basis introduced by Alba, Fateev, Litvinov and Tarnopolskiy. These building blocks are characterized by new conditions which generalize the known ones on the Gaiotto state and the Carlsson-Okounkov vertex. Consistency conditions of the inner product give algebraic relations for the chiral ring generating functions de ned by Nekrasov, Pestun and Shatashvili. In particular we show the polynomiality of the qq-characters which have been introduced as a deformation of the Yangian characters. These relations de ne a second quantization of the Seiberg-Witten geometry, and, accordingly, reduce to a Baxter TQ-equation in the Nekrasov-Shatashvili limit of the Omega-background. of four building blocks; Gaiotto state; dilatation - Holomorphic 1 Introduction 2 Reformulation of SHc algebra SHc algebra in terms of holomorphic elds Rank N representations Adjoint action of the vertex operators Ward identities of SHc and qq-character Nekrasov instanton partition function Action of SHc operators on instanton partition functions A1 quiver qq-characters of higher representations for the A1 quiver Generalization to the AQ-type quiver Quantum Seiberg-Witten geometry Summary and concluding remarks 3 Instanton partition function and SHc algebra C Some calculations of commutation relations in the rank N representation 29 D Detailed analysis of the qq-characters D.1 Matter case D.2 Second qq-character of the A1 quiver D.2.1 Double action of SHc operators D (z) D.2.2 Derivation of the second qq-character 1 Introduction SHc is an algebra introduced by Shi mann and Vasserot in [1] (see also [2]) to describe the equivariant cohomology of the instanton moduli space of N = 2 gauge theories in four dimensions. It has been de ned as a spherical (symmetric) version of the degenerate double a ne Hecke algebra (DAHA) which has been developed by Cherednick for many years [3].1 While DAHA encodes the algebraic (recursive) properties of Macdonald polynomials [4] 1In fact, SHc is a short notation introduced in [1] for central extension of the Spherical degenerate double a ne Hecke algebra. It may be better referred to as Schi mann-Vasserot algebra, but we will use SHc in the text. { 1 { with two deformation parameters, degenerate DAHA is obtained by taking a limit q; t ! 1 such that one parameter remains, with t = q . In this limit, Macdonald polynomials degenerate into Jack polynomials. This algebra precisely describes the algebraic structure behind Nekrasov instanton partition functions [5] with the Omega background R21 R22 with the identi cation = 1= 2 and has been used to prove the 4D/2D correspondence which generalizes Alday, Gaiotto and Tachikawa's proposal [6] (AGT conjecture) for various types of quiver gauge theories | namely pure super Yang-Mills theory [1], the gauge theories with fundamental [7] and bifundamental hypermultiplets [8] (see also the recent preprint [9]). For pure super YangMills, the theory is characterized by a Gaiotto state [10], a coherent state a liated to the has also revealed itself particularly useful in the study of vortex dynamics [12]. In such developments, the important role devoted to SHc as a \universal" symmetry came principally from the fact that it contains all WN algebras for arbitrary N , together with an additional U(1) factor. The parameter is identi ed with a deformation parameter p the combination Q = p that de nes the central charge c = (N 1. In this sense, SHc should be regarded as a one parameter 1)(1+ Q2N (N +1)) of WN representations through deformation of the W1+1 algebra. The latter is known to have realizations in terms of N free fermions acting on a space of N -tuple Young diagrams. These representations, that we call here rank N representations, are identical to those de ned by Fateev and Lukyanov in [13]. The correspondence between the two algebras has also been con rmed in more general cases where the Hilbert space contains singular vectors. The most typical example is the minimal models of WN . There, it has been demonstrated explicitly in [14] that SHc reproduces the proper descriptions of the Hilbert space constrained by the so-called N-Burge conditions [15, 16]. This universality is essential when we have to treat a system that contains gauge groups of di erent rank, as it is the case for quiver theories. On the other hand, the action of SHc on instanton partition functions of quiver N = 2 gauge theories is very di erent from the representation of WN algebras. It is better understood after the introduction of an orthonormal basis constructed by Alba, Fateev, Litvinov and Tarnopolsky (AFLT basis) to prove the 4D/2D duality [ 17, 18 ]. AFLT basis should be regarded as a generalization of Jack polynomials [19{22], it is the proper basis to describe the action of degenerate DAHA. Instead of the description in terms of chiral primary elds with di erent spins, it is de ned more abstractly through generators Dm;n with two indices m; n. The rst index m 2 Z is identi ed with the index of the Virasoro generators Lm while the second one n 2 Z 0 corresponds to the spin n + 1 of the generator. Since it is a nonlinear symmetry with a reasonably complicated structure, we have to be careful how to organize the generators. One of the authors [23] has recently found that holomorphic expansion in terms of the second index, D 1(z); D0(z), gives a compact description of SHc { 2 { through the study of the Nekrasov-Shatashvili [24] limit of AGT conjecture. This turns out to be very useful and is a main tool of this paper. The focus of the paper is to provide an SHc description of Nekrasov partition functions for general AQ and A(Q1)-type quiver gauge theories. To do so, the action of SHc operators on Gaiotto states, together with the adjoint action on the intertwiner operator describing bifundamental elds, is worked out. These actions are conveniently expressed in terms of the vertex operators Y(z) associated with the current D0(z). They extend the work on the covariance of the partition function presented in [8] by giving us the possibility to consider quiver theories with gauge groups of di erent ranks. As a consequence of our results, several useful identities can be established among correlators of the gauge theories. In particular, we were able to recover the expression of the qq-characters recently introduced by Nekrasov, Pestun and Shatashvili (NPS) [25]. For the simplest A1 case with fundamental multiplets (4.10), (z) = i denotes an average weighted by the instanton partition function, which is de ned in (4.4). The operator Y(z) is interpreted as an operator version of the chiral ring generating function. These characters, presented as further deformation of the characters of Yangian algebras [26], encode in a compact form a recursion relation among the instanton partition functions [27]. Here we show that SHc provides a proper symmetry behind the qq-character formulae, as was already predicted by NPS, and that the polynomiality property naturally follows from our description. The qq-characters de ne a double deformation of the Seiberg-Witten geometry in a form of second quantization. In the above example, the Seiberg-Witten curve is expressed as (5.1), y + q m(z) y N = Y(z `=1 a`): Seiberg-Witten theory is well-known to provide an e ective description of the infrared sector of N = 2 gauge theories on R4 [28, 29]. The e ective Lagrangian is written in terms of an holomorphic function, the prepotential, obtained from the knowledge of an algebraic curve and a di erential form. This formulation is identi ed with the construction of nite gap solutions for classical integrable hierarchies, the algebraic curve corresponding to the spectral curve of the system [30].2 When the gauge theory is considered in the NekrasovShatashvili (NS) limit 2 ! 0 of the Omega-background, the associated integrable systems are quantized, with the remaining parameter 1 playing the role of the Planck constant [24]. The algebraic curve becomes the Baxter TQ-equation of the quantum system [31, 32] (see also [27, 33] for the extension to quivers), it is equivalent to a Schrodinger equation under a quantum change of variables [34], in a form of ODE/IM correspondence [35].3 In this 2These nite gap solutions can also be obtained from Hitchin systems. 3In this classical version of AGT correspondence, the Shrodinger equation is obtained as the semiclasframework, the two complex variables of the algebraic curve become non-commutative, thus de ning a rst quantization of the Seiberg-Witten curve [38{40]. In the full Omegabackground, the qq-character is an operator acting in a Hilbert space of quantum states. In the NS limit, the expectation value of this operator in the Gaiotto state (which plays the role of a coherent state) becomes the T-polynomial of the TQ-equation, while its de ning relation in terms of vertex operators reproduces Baxter's relation. In this sense, the qqcharacter formula presents a second quantization of the integrable system in which the TQ-relation emerges in the classical 2 ! 0 limit. We organize the paper as follows. In section 2, we introduce the holomorphic eld description of SHc algebra and the rank N representation. We also provide useful expressions for the adjoint actions of the vertex operators. In section 3, after a brief review of the general construction of the instanton partition functions, we introduce the building blocks (Gaiotto states, avor vertex operator, intertwiner) with which the partition functions are written as a product. We show that the Gaiotto state satis es stronger constraints which are compactly expressed in terms of SHc elds. A generalized intertwiner which connects di erent rank gauge groups is also de ned. It satis es similar conditions as the Gaiotto states and indeed it reduces to the Gaiotto state when the gauge group of one side is trivial. The avor vertex operator is used to include the fundamental hypermultiplets in the gauge theories. These results are used in section 4 to build an in nite number of constraints among the correlation functions of the vertex operator Y. The new characterizations of the Gaiotto states and the intertwiner play an essential role to give a closed and compact expression for these constaints | written in the form of qq-characters. Finally in section 5 we present their interpretation as quantum Seiberg-Witten geometry along the line of [23, 27, 41]. The concluding section proposes some perspectives for future research, and several technical details are gathered in the appendix. 2 2.1 Reformulation of SHc algebra SHc algebra in terms of holomorphic elds Z The SHc algebra is de ned on a set of operators Dm;n with the double grading (m; n) 2 Z 0 [1]. The rst index is called the degree and the second index the order. The algebraic relations involving D 1;n and D0;n are written as [D0;n; D 1;m] = D 1;n+m 1 ; n 1 ; [D 1;n; D1;m] = En+m n; m 0 ; [D0;n; D0;m] = 0 ; n; m 0 : (2.1) (2.2) (2.3) where Ek denotes a linear combination of powers of the generators D0;n that will be given shortly. Additional relations can be found in [2], but they will not be used here. The algebra is spanned by the operators of degree 0 and 1 upon the recursive use of the following commutation relations, 1 m D (m+1);0 = [D 1;1; D m;0]; D m;n = [D0;n+1; D m;0]; (2.4) { 4 { 1 n=0 1 n=0 for n product of the WN algebra and a U(1) current. The Heisenberg generators (U(1) currents) are related to Dm;0, the Virasoro generators to Dm;0; Dm;1, and operators Dm;n with n > 1 to the currents of spin n + 1 [1, 7]. It is useful to assemble the generators in the form of holomorphic elds [23], D 1(z) = X z n 1D 1;n; D0(z) = X z n 1D0;n+1; E(z) = 1 + + 1 X z n 1En; n=0 (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) holomorphic elds, [D0(z); D 1(w)] = to introduce where + = 1+ 2. We use here the Omega-background equivariant deformation parameters 1; 2 [5, 42] instead of the SHc deformation parameter = 1= 2 in order to simplify the comparison with gauge theories.4 It is noted that these Laurent series are vanishing at z = 1, in which they are di erent from usual holomorphic elds in CFT. We rewrite the de ning properties of the generators D 1;n and D0;n in terms of the D 1(w) This de nition of the eld (z) resembles the mode expansion of an holomorphic free bosonic eld in CFT and the series D0(z) can be interpreted as the associated current. As we noted, however, the usual U(1) current in CFT is expanded as a sum over the degree as J ( ) = P n2Z (coe :)D n;0 In addition, elds at di erent points are commuting, [ (z); (w)] = 0, as a consequence of (2.3). In this sense, the interpretation of the complex variable z is di erent from the holomorphic coordinate of a Riemann surface but it should rather be seen as the spectral parameter of an integrable model. We will come back to this description later. The following dressed combination of vertex operators will play a central role in our reformulation of the SHc algebra and the correspondence with gauge theories, n 1 while (2.7) is expanded with respect to the order. Y(z) = ec(z)e (z 1)e (z 2)e (z)e (z +): The function c(z) encodes the dependence in the in nite number of the central charges cn (n 0) of the algebra. It expands as The generating series E(z) can now be expressed using the newly de ned vertex operator, c(z) = c0 log(z) 1 X n=1 cn : nzn E(z) = Y(z + +)Y(z) 1: 4The correspondence between the convention of [8] and this paper is summarized in appendix A. { 5 { Among possible representations of SHc, the best studied one is the rank N representation where the Hilbert space is spanned by a basis labeled by N -tuple Young diagrams Y~ = (Y1; ; YN ). The representation is characterized by N complex numbers a` (` = 1; ; N ) that de ne the central charges cn through the relation N `=1 ec(z) = Y(z a`): To emphasize the dependence of the representation in the parameters a` through the central charges, they will be included in the notation of the vector basis j~a; Y~ i. These vectors form an orthonormal basis of the representation space, be identi ed with the generalized Jack polynomials introduced in [21] and studied in [22]. The vacuum state is obtained by taking the N -tuple of empty Young diagrams denoted ~;, it satis es D0;nj~a;~;i = D 1;nj~a;~;i = 0; or D0(z)j~a;~;i = D 1(z)j~a;~;i = 0: (2.13) The Hilbert space spanned by j~a; Y~ i will be denoted V~a. When several Hilbert spaces are considered, an extra label ~a will be inserted on the notation of the operators D~ra(z) to specify in which space V~a they act. The rank N representations of SHc are equivalent to the representations of WN U(1) [1, 14]. The action of SHc generators of degrees 1 on the state j~a; Y~ i involves the N -tuple Young diagram with a box added/removed. As such, they can be seen as an analog of creation/annihilation operators while the total number of boxes in Y~ represents the number of particles (later identi ed with the instanton charge). Following [23], N -tuple Young diagrams Y~ with a box x added/removed will be denoted Y~ x (respectively). We further introduce the sets A(Y~ ) and R(Y~ ) containing all the boxes that can be added to/removed from the Young diagrams composing Y~ . In gure 1, we illustrate the locations of the boxes in the sets A(Y ) and R(Y ) with the example of a single Young diagram Y . The boxes x 2 Y~ are characterized by a triplet of indices (`; i; j) where ` = 1 N and (i; j) 2 Y` gives the position of the box in the `th Young diagram. To each box x is associated a complex number x depending on the central charges using the map x = (`; i; j) 2 Y ~ ! x = a` + (i 1) 1 + (j 1) 2 2 C: With these de nitions, the action of the spanning subalgebra takes the simple form [8, 23] D+1(z)j~a; Y~ i = D0(z)j~a; Y~ i = X X x2A(Y~ ) z x2Y~ z 1 { 6 { which are equivalent to their component form (n x ; i We note that the second relation in (2.16) implies that the moments of x2Y~ are the eigenvalues of the commuting charges D0;n. In the (generalized) Calogero-Sutherland system, D0;n plays the role of in nite commuting charges and x is interpreted as the momentum E(z), e (z). However it is reversed for the operators D 1(z), of each particle. The interpretation of z as the spectral parameter is natural in this sense. The left action of SHc generators on bra h~a; Y~ j is identical for the diagonal operators D0(z), h~a; Y~ jD+1(z) = X 2 R(Y ) X The series E(z) is also diagonal on the states j~a; Y~ i, with eigenvalues given by the function (z)2 at z = x with x 2 A(Y~ ) or R(Y~ ): x(Y~ ) in the action (2.15) of D (z) correspond to the residues of this Eventually, the action of the vertex operator is expressed in terms of a product over the boxes of Y~ , e (z)j~a; Y~ i = QY~ (z)j~a; Y~ i; with QY~ (z) = Y (z x2Y~ x): The speci c combination of vertex operators entering in the de nition (2.8) of Y(z) leads to a remarkable simpli cation of its eigenvalues Y(z)j~a; Y~ i = Y(z N `=1 a`) Y (z x2Y~ (z x 1)(z x)(z x x +)2) j~a; Y~ i = Q Q x2A(Y~ )(z We note that there is a cancellation of factors between the numerators and the denominators in the middle term, and the resulting expression in the r.h.s. bears contributions only from the edges of the Young diagrams. Taking the ratio (2.10) de ning the operator E(z), we recover the expression (2.18) for the function (z)2: In appendix C, we provide an explicit computation of the commutation relations of D0(z), D 1(z) in the rank N representation. Finally we would like to mention the existence of an automorphism of representation. Under the shift of ~a, ai ! ~a0 = ~a + ~e where ~e = (1; 1; ; 1), the representation (2.16) implies that D~+a+1 ~e(z)j~a + ~e; Y~ i = D~a+1 ~e(z)j~a + ~e; Y~ i = X spectral ow in the context of W1+1-algebra [43]. 2.3 Adjoint action of the vertex operators The coe cients appearing here may be identi ed with the representation of D~a 1(z implies that there is an automorphism of the algebra by shifting the variable z: Dr ~a+ ~e(z) ) for r = 0; 1. This shift symmetry of the representations is referred as the In order to prepare for the computations necessary in the next sections, we would like to evaluate the commutation relations between the vertex operators e (z) and the elements spanning the SHc algebra. The generators of degree zero form a commutative subalgebra, as a consequence the eld (z) commute with the series D0(z). The evaluation of the adjoint action on D 1 is slightly more involved. We introduce a vertex operator depending on two nite sets of points zi and wj with i 2 I, j 2 J , U (fzig; fwj g) := exp (zi) We claim the following identities: U (fzig; fwj g) 1D1(u)U (fzig; fwj g) = Pu=1;zi2I U (fzig; fwj g) 1D 1(u)U (fzig; fwj g) = Pu=1;wj2J Q X i2I { 8 { X j2J ! (wj ) : " Q Q " Q with the projector Pu=1;zi2I acting on functions of the variable u as Here Pz+ picks up the positive powers of a Laurent series in z, namely for a function powers of f (z). The second term in (2.27) also plays the role to remove singularities at u = zi. One may use a contour integration to write these projections in a compact form, where the contour C is de ned by jwj = R with R < Mini(jzij). The formula (2.26) formally resembles an OPE in CFT, up to the existence of the projection operator which is necessary here since there is no singularity except for u = 0 on the left hand side. Before proving (2.26), it may be instructive to give some speci c examples which will be used later. The rst one is when one of the sets I; J is null: (2.28) (2.29) ; (2.30) (2.31) (2.32) e PiM=1 (zi)D (z0)e PiM=1 (zi) = X D (zi) Y e PiM=1 (zi)D (w)e PiM=1 (zi) = Pw D (w) Y(zi M i=0 " j(6=i) zj M i=1 1 z i ; # w) : The other one is the adjoint action of Y(z) which is de ned as a product of vertex operators with shifted arguments: 1 Y(z) D 1(w)Y(z) = S(w z)D 1(w) + 1 Y(z + +) Y(z + +)D1(w) = S(z w)D1(w) where S(z) denotes a scattering factor S(z) de ned as of the SHc algebra. We use the property Proof of the formula (2.26). To end up this section, we would like to give a short derivation of the identity (2.26). Rather than working with the commutator (2.6) directly, it is easier to evaluate the action on the states j~a; Y~ i which form a faithful representation S(z) = (z + 1)(z + 2) : z(z + +) QY~ x(z) QY~ (z) = (z x) 1; { 9 { which is a direct consequence of (2.20), and the action (2.15) of D (z) on the states j~a; Y~ i. It follows that U (fzig; fwj g) 1D1(u)U (fzig; fwj g)j~a; Y~ i = X The product in the r.h.s. can be rewritten as a sum over single poles in x, with an extra polynomial term, using the algebraic identity, (u Q Here an are the coe cients appearing in Laurent expansion of the l.h.s. in , they depend on the parameter zi, wj and u: P+ (u Q can be used to reform D1(z), while the polynomial part in x gives the transformations D1;n, and (2.33) becomes Q Since the states j~a; Y~ i generate a faithful representation, the equality of vectors can be lifted at the level of operators U (z; w) 1D1(u)U (z; w) = Q Q The expression for U (z; w) 1D 1(u)U (z; w) is similarly obtained and is written as (2.37) with the substitution of the variables zi $ wj . The right hand side of (2.37) can be simpli ed by analyzing the rst term: the second term cancels the poles (and the constant part) at u = 1 of the rst term, while the third term cancels the simple poles at u = zi. The existence of such terms is natural since the left hand side of (2.37) is not singular at these points. The procedure of removing the unwanted poles is performed by the projector at in nity, the following property should be employed, de ned in (2.27), and (2.37) produces (2.26). It is noted that to analyze the pole (2.33) (2.34) (2.35) (2.36) (2.37) P+ r( ) u = P+r( ) Pu+ ur(u) (2.38) for any meromorphic function r(z). It implies in particular Instanton partition function and SHc algebra Nekrasov instanton partition function Q Y~1; Y~Q i=1 Class S gauge theories with N = 2 supersymmetry are obtained by compacti cation of the six dimensional N = (2; 0) theory on a Riemann surface. They are classi ed by a quiver diagram where each node i is in correspondence with the simple group component SU(Ni) of the total gauge group G = iSU(Ni). Thus, to each node corresponds a gauge multiplet containing a vector, two fermions and a scalar eld in the adjoint representation. The arrows i ! j of the quiver represents bifundamental matter elds, i.e. a chiral multiplet containing a fermion and a scalar eld, with mass mij , and transforming in the fundamental representation of SU(Ni) SU(Nj ). In addition, a number N~i of fundamental (or antifundamental) matter elds can be attached to each node i. They consist in chiral multiplets of masses m(f) with f = 1 i N~i, encoded in the N~i-vector m~i (see gure 2). The instanton partition functions of class S theories have been evaluated using localization in the Omega-background [5]. The theory is considered on the Coulomb branch where the adjoint scalar elds take non-zero vacuum expectation values. These complex parameters will be denoted a(`i) with ` = 1 Ni, they form the Ni-vector ~ai attached to the node i. Localization provides a sum over nested integrals that can be computed by residues. The residues are in one-to-one correspondence with the boxes of the Ni-tuple Young diagrams for each node i of the quiver. The resulting formula is a sum over realizations of these diagrams weighted by the multiplets contributions [42, 44{47]: Zinst. = X ~ Y qjYijZvect.(~ai; Y~i)Zfund.(~ai; Y~i; m~i) i Zbfd.(~ai; Y~i; ~aj ; Y~j jmij ); (3.1) Y i!j2EQ where Q is the number of nodes in the quiver, EQ its set of links, and jY~ j denotes the total number of boxes in the N-tuple Young diagram Y~ . The instanton counting parameter qi corresponds to the exponentiated gauge coupling at the node i, suitably renormalized in asymptotically free theories . It is known that the contribution from each representation can be systematically derived from that for the bifundamental representation. Taking a bifundamental eld of mass m12 coupled to the two gauge groups SU(N1) and SU(N2), the contribution reads: Zbfd.(~a; Y~ ;~b; W~ jm12) = Y gY`;W`0 (a` b`0 m12) Y N1 N2 `=1 `0=1 Y (i;j)2 Y (i;j)2 j + 1)) : (3.2) (3.3) fund. m~1 ~ N1 SU(N1) vevs a~1 fund. m~2 ~ N2 SU(N2) vevs a~2 bf. m12 SU(NQ) vevs a~Q Here i is the height of ith column and 0i is the length of ith row of Young diagram (see gure 3). The other building blocks can be written from (3.2) as follows. Fundamental hypermultiplets transforming under the gauge group SU(N ) and the avor group SU(N~ ), with masses m1; mass m12 = 0, for the rst node N1 = N~ , ~a1 = m~ := (m(1); and for the second node N2 = N , ~a2 = ~a and Y~2 = Y~ arbitrary:5 ; mN~ : we take a vanishing bifundamental ; m(N~ )) and Y~1 = ~;, Zfund:( m~; ~a; Y~ ) = Zbfd.( m~; ~;; ~a; Y~ j0): Y~2 = ~;, Antifundamental hypermultiplet: in a symmetric way, we take m12 = 0, for the rst node N1 = N , ~a1 = ~a and Y~1 = Y~ and for the second one N2 = N~ , ~a2 = m~ and Zaf:( m~; ~a; Y~ ) = Zbfd.(~a; Y~ ; m~; ~;j0) = Zfund:( + m~; ~a; Y~ ): Adjoint hypermultiplet: we take N1 = N2 = N , Y~1 = Y~2 = Y~ and ~a1 = ~a2 = ~a, Vector multiplet: inverse of the adjoint hypermultiplet with zero mass, Zadj:(~a; Y~ jm) := Zbfd.(~a; Y~ ; ~a; Y~ jm): Zvect.(~a; Y~ ) := Zbfd.(~a; Y~ ; ~a; Y~ j0) 1: (3.4) (3.5) (3.6) (3.7) HJEP04(216)7 We note that the fundamental matter can be seen as bifundamental matter where one of the two gauge groups is taken in the weak coupling limit, e ectively becoming a avor group. In this limit, the corresponding exponentiated gauge coupling q is sent to zero, ~ and due to the presence of the factor qjY j, only empty Young diagrams contribute in the 5We have chosen to shift the de nition of the fundamental masses by + in order to simplify formulas: as they are equivalent to fundamental contributions with shifted masses. summations. As a result, the contribution of a fundamental matter multiplet is derived from the bifundamental contribution by attaching to each set of N~i fundamental avors an N~i-tuple of empty Young diagrams. The fact that the various contributions to the partition function of the di erent multiplets are derived from the bifundamental contribution implies important consequences for their SHc realization that will be presented in the next section. 3.2 Action of SHc operators on instanton partition functions In this paper we focus on the linear quiver AQ and its a ne version A(Q1), they are characterized by the set of arrows EQ = fi ! i + 1; i = 1 Q 1g and EQ = fi ! i + 1; i = 1 Qg respectively, with the identi cation of indices modulo Q. One of the goal of this paper is to formulate the action of SHc operators on the (a ne) linear quiver instanton partition function. For this purpose, we need to rewrite the partition function in terms of elements of the representation theory of SHc: Gaiotto states, intertwiner and the vertex operator ( gure 4). Gaiotto state. To each node i of the quiver diagram is associated a vector space of representation Va~i spanned by the vectors j~ai; Y~ii where Y~i takes values in all the possible realization of Ni-tuple Young diagrams. The set of complex parameters ~ai is xed in each Va~i and de ne the central charges of the representation of SHc. The Gaiotto state has been introduced in [10] as a speci c Whittaker vector of the Virasoro algebra with respect to the maximal nilpotent subalgebra fLn; n > 0g. This algebra is spanned by the two elements L1 and L2, and the Gaiotto state is de ned up to a normalization by the conditions,6 L1jGi = jGi; L2jGi = 0; (3.8) where is a constant. This de nition has been generalized to the case with fundamental avors [48], and to higher rank [49, 50], and eventually implemented in the space of 6As a consequence of the Virasoro commutation relations, the second condition implies LnjGi = 0 for n > 2. X q This state is known to provide the instanton partition function of pure N = 2 SYM (A1 where the operator D = D0;1 counts the number of boxes in Y~ , Dj~a; Y~ i = jY~ jj~a; Y~ i. It is identi ed with L0 in Virasoro algebra up to the zero mode and qD may be regarded as the propagator in string theory. In the following, we refer to the operator of the form qD as the dilatation operator. It satis es, qDD 1(z) = D 1(z)qD 1 : In terms of SHc operators written in the form of holomorphic elds, the Gaiotto state has a new characterization. This is one of the main results of the paper: hG; ~aj jG; ~ai U ( m~) representation of SHc (which contains a Virasoro sub-algebra) [1], These formulae are a consequence of a more general result, presented in (3.22) and (3.23) below, and proven in appendix B. These new expressions contain more information than the previously known relations given in (3.16). They reveal themselves powerful enough to derive several useful relations, presented in [27], among instanton partition function for arbitrary (A-type) quiver diagrams. The asymptotic of the operators Y(z) at in nity is deduced from (2.21): Y(z) zN since jA(Y~ )j jR(Y~ )j = N for any N -tuple Y~ . Expanding the rst relation at in nite spectral parameter z allows to recover the characterization of the Gaiotto states in [7, 8], D 1;njG; ~ai = 0; D 1;N 1jG; ~ai = p jG; ~ai; D 1;N jG; ~ai = p 1 2, and the last property has been obtained using the formula (A.3) in [23]. These identities suggest to see the Gaiotto state as a (partial) coherent state in the physical sense of eigenstate of the annihilation operators D 1;n. Flavor vertex operator. Due to the presence of the empty Young diagram in the de nition (3.4), the fundamental matter contribution can be written in a simpler form, 0 N~ X f=1 1 where QY~ (z) denotes the eigenvalue of the vertex operator de ned in (2.20). This expression implies that the vertex operator can be used to insert fundamental multiplets in the quiver gauge theories. U ( m~)j~a; Y~ i = Zfund.( m~; ~a; Y~ )j~a; Y~ i: (3.18) This operator generates the modi ed Gaiotto states in the presence of fundamental multiplets, as studied in [7]. Since it plays the role to add the contribution of fundamental hypermultiplets with avor group SU(N~ ), it will sometimes be referred to as the avor vertex operator. The instanton partition function for this theory can be written Zinst = hG; ~ajqDU ( m~)jG; ~ai = X qjY jZvect.(~a; Y~ )Zfund.( m~; ~a; Y~ ): ~ (3.19) It is noted that the vertex operator U ( m~) commutes with the dilatation operator qD. Intertwiner. Up to now, only partition functions of N = 2 theories with a single gauge group have been reproduced. To address the case of bifundamental matter coupled with multiple gauge groups, the construction of a new operator V12(~a1; ~a2jm12) : V~a2 ! V~a1 is required. This operator intertwines two SHc representations speci ed by ~a1; ~a2, with a di erent rank N1 for V~a1 and N2 for V~a2 , V12(~a1; ~a2jm12) = Zbfd.(~a1; Y~1; ~a2; Y~2jm12) j~a1; Y~1ih~a2; Y~2j; (3.20) Zfund.( m~; ~a; Y~ ) = m(f)) = Y( 1)jY~ jQY~ (m(f)); (3.17) ~ N f=1 where a renormalized version of the bifundamental contribution has been used, Zbfd.(~a1; Y~1; ~a2; Y~2jm12) = Zvect.(~a1; Y~1)Zvect.(~a2; Y~2)Zbfd.(~a1; Y~1; ~a2; Y~2jm12): (3.21) Several algebraic properties of the intertwiner operator were studied from the viewpoint of SHc in [8], in relation with a recursion formula satis ed by Zbfd..7 The intertwiner satis es a set of identities which resemble the conditions (3.12){(3.15) for the Gaiotto states: D~a11(z)V12(~a1; ~a2jm12) V12(~a1; ~a2jm12)D~a21(z m12) D~+a11(z)V12(~a1; ~a2jm12) V12(~a1; ~a2jm12)D~+a21(z + + Here the notation Y(i) (i = 1; 2) represents the action of the vertex operor Y in the space under the action of SHc. The proof of the formulae is summarized in appendix B. V~ai . These formulas characterize the transformation of the bifundamental contribution In the 4D/2D correspondence, the intertwiner is described as a vertex operator of the form, V = V COV Toda where V CO is the Carlsson-Okounkov vertex [11] for the U(1) factor and V Toda is the vertex operator of Toda eld theory associated with the WN algebra. This construction, however, has some limitations. One issue is the technical di culty to de ne the transformation properties of V Toda for higher spin generators. We have to face a nonlinear expression in terms of W generators or Toda elds which is usually not manageable. A more serious issue is the impossibility to de ne an intertwiner between WN and WM Toda systems with N 6= M since there is no obvious correspondence between the generators in WN algebras with a di erent N (see, for example [51, 52], for attempts to explore such a setup). At the level of SHc, the correspondence between the generators for representations of a di erent rank becomes obvious and the transformation properties (3.22) and (3.23) are compact and tractable. Furthermore, it was con rmed in [7, 8] that these conditions contain the modi ed Ward identities for the U(1) current and the Virasoro operator for V = V COV Toda when N1 = N2. In this sense, our characterizations of the intertwiner is a natural generalization of the conventional vertex operators in Toda eld theories to study the 4D/2D correspondence. Gaiotto state from intertwiner. As brie y recalled in the previous subsection, the study of the instanton partition functions for miscellaneous eld content can be reduced to the analysis of the bifundamental hypermultiplet. This fact has an important consequence for the SHc realization that we explain here. We rst consider the special case N1 = N , N2 = 0 and m12 = 0 for the intertwiner. Here the rank 0 representation means a trivial representation which consists of one state | the vacuum j; i (empty slots means that we have no Fock space). Since Zbfd.(~a; Y~ ; ; j0) = Zvect.(; ) = 1, we nd after omitting the trivial bra vacuum, V12(~a; j0) = jG; ~ai: (3.24) 7In [8] it was assumed that the ranks of the two representations are the same. However, the computation performed there can be straightforwardly generalized to the case N1 6= N2. SU(N ) vevs ~a Taking the opposite case of a rank zero representation for the rst node produces the bra Gaiotto state in a similar way. In this sense, the recursion properties of the Gaiotto states (3.12){(3.15) are straightforward consequences of (3.22) and (3.23). We note that we can take D(2)(z) = 0 for the trivial representation and the action operator Y on V2 is replaced by 1. Inclusion of (anti-)fundamental hypermultiplet is also straightforward. From (3.4) and (3.5), after xing N1 = N , N2 = N~ , m12 = 0 and ~a2 = m~ + +, we obtain that the action of the intertwiner on the vacuum produces the Gaiotto state with a avor vertex operator inserted: V12(~a; m~ + +j0)j m~ + +; ~;i = U ( m~)jG; ~ai : Full partition function. We now have all the elements to write down the instanton partition function of any linear quiver as a product of operators, Zinst = ( hG; ~a1jq1DU ( m~1)V12(~a1; ~a2jm12)q2DU ( m~2)V23(~a2; ~a3jm23) TrV~a1 q1DU ( m~1)V12(~a1; ~a2jm12)q2DU ( m~2)V23(~a2; ~a3jm23) jG; ~aQi VQ1(~aQ; ~a1jmQ1) for A(Q1): for AQ ; ~ Y ~ Y (3.25) (3.26) (3.27) To each arrow i ! j of the quiver is associated the intertwiner Vij (~ai; ~aj jmij ), and to each node i an operator qiDU ( m~i). For the linear quivers, the resulting operator is sandwiched between the Gaiotto states attached to the rst and the last node. On the contrary, a trace is directly obtained for the a ne quiver from the intertwiners, it is de ned as TrV~a = Xh~a; Y~ j As an example, the partition function of N bifundamental elds of mass m reads = 2 theory represented in gure 5 with Zinst = TrV~a qDV11(~a; ~ajm) = X qjY jZvect.(~a; Y~ )Zbfd.(~a; Y~ ; ~a; Y~ jm) ~ = X qjY jZvect.(~a; Y~ )Zadj(~a; Y~ jm); ~ (3.28) thanks to the orthonormality property of the states j~a; Y~ i. Alternative expressions. Although it will not be used in this paper, we would like to provide, as a side remark, a new expression for the bifundamental contribution (3.2) involving the vertex operator Y(z). This expression is a consequence of the property (B.1) expressing the variation of Zbfd.(~a; Y~ ;~b; W~ jm) under the addition of a box in the Young diagrams Y~ . It turns out that the right hand side of (B.1) is independent of the actual content of boxes in the Young diagrams Y~ . As a result, this formula can be used to build recursively Zbfd.(~a; Y~ ;~b; W~ jm) from Zbfd.(~a; ~;;~b; W~ jm), adding boxes one by one. Since, Zbfd.(~a; ~;;~b; W~ jm) can be further identi ed with a fundamental contribution of mass ~a m, it is shown that Zbfd.(~a; Y~ ;~b; W~ jm) = h~b; W~ jU (~a m)j~b; W~ i ~b; W~ jY( x m + +)j~b; W~ E = h~a; Y~ jU (~b + m +)j~a; Y~ i Y D~a; Y~ jY( x + m)j~a; Y~ E Y D x2Y~ x2 W~ where the second equality has been obtained by exploiting the symmetry under the exchange of (~a; Y~ ) $ (~b; W~ ) and m $ + formulae for the vector contribution and the Gaiotto states can also be deduced, m. As a special case of this expression, new * X ~ Y Zvect.(~a; Y~ ) = ~a; Y~ U (~a +) 1 Y Y( x) 1 ~a; Y~ + ; Ward identities of SHc and qq-character In the previous section, we have seen that the instanton partition function for any AQ type quiver gauge theories can be written by combining Gaiotto states, dilatation operators, avor vertex operators and intertwiners as in (3.26). The behavior of these states/operators under the action of SHc generators has been characterized through the set of relations (3.12){(3.15), (3.11), (2.26) and (3.22){(3.23). As a side result, one obtains a series of consistency conditions by inserting D 1(z) in the correlator and evaluating the inner product in two di erent ways, (hG; ~a1jO1D 1(z)) O2jG; ~aQi = hG; ~a1jO1 (D 1(z)O2jG; ~aQi) ; (4.1) where Oi denotes a combination of avor vertex operator, dilatation operators and intertwiners. These conditions may be regarded as the Ward identities for the correlation functions of SHc. Since it is written as a generating function with parameter z, it gives an in nite number of constraints. In the following, we evaluate the explicit form of these identities. We observe that their structure takes the form of a double quantum deformation of the character formulae for AQ, the so-called qq-character proposed by Nekrasov, Pestun and Shatashvili [25, 41]. In the next section, we will discuss another interpretation of these formulae as a quantum deformation of the Seiberg-Witten curve. (3.29) (3.30) HJEP04(216)7 We start from the expression (3.10) of the instanton partition function for pure SYM with SU(N ) gauge group, and consider the insertion of the operator of D 1(z), hG; ~ajD 1(z)qDjG; ~ai; evaluated in two di erent ways as in (4.1). After the use of the identities (3.11), (3.12), (3.14), we arrive at, hG; ~ajPz qDjG; ~ai = 0 : q Y(z) The insertion of Y has the e ect of adding extra factors to the partition function. For HJEP04(216)7 example, from (2.21), hG; ~ajY(z + +)qDjG; ~ai = X qjY j ~ Zvect.(~a; Y~ ): We will use the following notation for the expectation value of the Gaiotto state: 1 Zinst Y~ h i = X qjY jZvect.(~a; Y~ )h~a; Y~ j ~ (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) This de nes an average of operators acting on states j~a; Y~ i that is normalized to h1i = 1. The relation (4.2) is rewritten in the form: P z This condition is the generating function of an in nite number of constraints on the instanton partition function. At the same time, this formula implies that (z) := hY(z + +)i + = Pz+(Y(z + +)) q Y(z) has no negative powers of z in the Laurent expansion at z = 1. We note that Y(z) behaves as Y(z) zN as z ! 1. It implies that (z) thus de ned is a polynomial in z of degree N . The expression y + 1=y is reminiscent of the character of sl(2) for the fundamental representation. The formula (4.6) is deformed by two parameters 1;2 and was referred as a fundamental qq-character in [25, 27, 41] for the quantum deformed Yangian Y (sl(2)). The inclusion of fundamental hypermultiplets with N~ avor is a straightforward generalization. The only necessary modi cation is to insert a avor vertex operator U ( m~) in front of jG; ~ai. The commutator with D 1(z) is obtained from (2.29): D 1(z)U ( m~) = U ( m~)Pz [D 1(z)m(z)] ; m(z) = ~ N Y(z f=1 m(f)) : Inserting this relation between two Gaiotto states (with an operator qD), and then evaluating the action of D 1(z) through (3.12) and (3.14) leads to P z where the average acquired an extra factor Zfund.( m~; ~a; Y~ ), 1 i = X qjY jZfund.( m~; ~a; Y~ )Zvect.(~a; Y~ )h~a; Y~ j ~ After including the fundamental hypermultiplets, the qq-character is modi ed to (z) = As a consequence of (4.8), Pz (z) = 0 and the qq-character is again a polynomial of degree N in z. A more detailed discussion of the qq-character in the presence of fundamental hypermultiplets is presented in appendix D. In the case N~ < N , the ratio m(z)=Y(z) has no polynomial part and the character (z) equals the average of the operator Pz+Y(z + +). As a result, explicit expressions for the qq-character can be obtained by expansion of Y(z + +) at in nity using the properties (2.8), (2.21): (4.9) (4.10) (4.11) (4.12) (4.13) + : (4.14) N `=1 N `=1 N `=1 Y(z + +) = Y(z + + a`) 1 d 1 2 dz D0(z) + higher terms in = Y(z + + a`) + 1 2zN 2 D + O(zN 3): In the average (4.4) the operator D with eigenvalue jY~ j can be replaced by a logarithmic q-derivative, (z) = Y(z + + a`) + 1 2zN 2q@q log Zinst + O(zN 3) : Specializing to N = 1 and to N = 2 with a1 = a2 = a, we deduce the following expressions U(1) : SU(2) : (z) = z + + (z) = (z + +) 2 a; 4.2 qq-characters of higher representations for the A1 quiver In a series of recent lectures, Nekrasov proposed a generalization of the qq-character for higher representations of Y (sl(2)) [25]. Higher qq-characters involve a set of complex parameters 1 ; ; r 2 C, and they are de ned as r(zj 1; r) = X ItJ=f1; ;rg qjJj Y S( i i2I j2J j ) * Y i2I Y(z + + + i ) Y m(z + j ) j2J Y(z + j ) Here S(z) is the scattering factor (2.31). It is claimed in [25] that the expectation value of these operators is again a polynomial in z. This proposal has been veri ed using our formalism in appendix D for the second character of pure SU(N ) SYM in the restricted The polynomiality is further obtained in the cases N = 1 and N = 2 by employing the explicit expression (4.11) of the operator Y(z + +). In both cases, it was found that 2(z1; z2) = Pz+1 Pz+2 hY(z1 + +)Y(z2 + +)i + Generalization to the AQ-type quiver For simplicity here we will only treat explicitly the case of the A2 quiver without fundamental matter elds. For any operator O we introduce the index space V~a in which the operator acts, and we associate the expectation value = 1; 2 labeling the O D ( )(z)E = 1 ~ ~ X qjY1jqjY2jZvect.(~a1; Y~1)Zvect.(~a2; Y~2)Zbfd.(~a1; Y~1; ~a2; Y~2jm12) 1 2 h~a ; Y~ jO (z)j~a ; Y~ i: To derive the qq-character relations, we consider the commutation relation (3.22) between the SHc generating series D (z) and the intertwiner operator. We consider the operator insertion of the following type, hG; ~a1jD~a11(z)q1DV12(~a1; ~a2jm12)q2DjG; ~a2i; hG; ~a1jq1DV12(~a1; ~a2jm12)q2DD~+a21(z)jG; ~a2i; and then using the action of the SHc modes on Gaiotto states, it is possible to derive the following identity, obtained respectively from the former and latter expressions: cases N = 1 and N = 2. The second character can be rewritten using the shifted spectral variables z1 = z + 1, z2 = z + 2 , ; z12 + Y(z1) with z12 = z1 z2. This condition is equivalent to (4.14). From the insertion of two operators D1(z1)D 1(z2) within two Gaiotto states, it is possible to show that Pz1 Pz2 2(z1; z2) = 0: P z P z Y Y (2)(z) = (1)(z + +) + q1 Y (2)(z + +) + q2 Y (2)(z + + m12) Y(1)(z) (1)(z + m12) Y(2)(z) + q1q2 Y(2)(z 1 1 + q1q2 Y(1)(z + m12 * * Y Y (1)(z + +) + q1 Y (2)(z + +) + q2 Y (2)(z + + m12) Y(1)(z) (1)(z + m12) Y(2)(z) + q1q2 Y(2)(z + q1q2 Y(1)(z + m12 m12) +) 1 1 + + = 0; = 0: m12) +) + + ; : These identities imply that the two following qq-characters are polynomials in z: (4.15) (4.16) Generalization of these formulae to the AQ quiver with fundamental multiplets is straightforward. For example, the rst one is generalized to ; j = j 1 X mk;k+1; k=1 (4.22) with 0 = 1 = 0, Y0 = YQ+1 := 1 and the mass polynomials mi(z) = QfN~=i1(z associated to the fundamental multiplet of the node i, with avor group SU(N~i). We note m(fi)) that for a linear quiver with N = N1 N2 NQ, the rank of the avor group need to satisfy N~i 2Ni Ni+1 Ni 1 with N0 = NQ+1 = 0. In this set-up, the character (1)(z) is a polynomial of degree at most N . As already stated, in the weak coupling limit q2 ! 0 the second node of the quiver diagram acts as a set of N2 fundamental avors of mass a(`2) coupled to the rst node. This relation can also be observed at the level of characters. As can be seen from (4.18) in this limit only the empty N2-tuple Y~2 = ~; contribute to the sum, Zvect.(~a2; ;) = 1 and ~ Zbfd. ! Zfund.. We further notice that the operator Y as such, can be identify with (2)(z), it reproduces a mass polynomial with masses a` (2)(z + +) becomes polynomial and, N2 Y(z `=1 (2)(z) ! a(2) + +) =: m(z): ` In this weak coupling limit, the rst equation in (4.21) becomes the equation (4.10) for the massive qq-character, with an extra shift of the fundamental masses by m12. 5 Quantum Seiberg-Witten geometry In the limit 1; 2 ! 0, the Omega-background reduces to R4 and the infrared theory is characterized by a complex algebraic curve. This curve, together with a di erential form, determines the prepotential of the theory through the Seiberg-Witten relations. It is also associated to the spectral curve of a classical integrable system in the Bethe/gauge correspondence (see for instance [30] and references inside). For simplicity here, we focus our discussion on the case of a single node with gauge group SU(N ) and a number N~ of fundamental multiplets. In this case, the algebraic curve can be written in the form (2) +, (4.23) y + q m(z) y N = Y(z `=1 a`): (5.1) This expression should be compared with the de nition (4.6) of the qq-character. It is then appealing to interpret the qq-character as a double deformation of the Seiberg-Witten geometry, where the expectation value of the operator Y(z) reduces to the complex parameter y of the curve E(y; z) = 0, while the qq-character (z) reproduces the gauge polynomial in the r.h.s. of (5.1). This is indeed the case, as we will demonstrate shortly. The discussion becomes even more illuminating if we introduce the intermediate background R21 R2 obtained in the Nekrasov-Shatashvili limit 2 ! 0 of the Omega-background. This 1-deformation of the Euclidean background is known to be responsible for the quantization of the classical integrable system associated to the N = 2 gauge theory [42]. In this background, the Seiberg-Witten curve is replaced by a Baxter TQ-equation that has been derived in [31, 32] (the derivation was later extended to quivers in [27, 33]), T (z)Q(z) = Q(z + 1) + qm(z)Q(z 1); Q(z) = Y(z ur); r where T (z) and Q(z) denote respectively the Baxter T- and Q-polynomials. The TQequation can be recast in a form more similar to the original Seiberg-Witten curve (5.1) by the introduction of the ratio Y (z) = Q(z)=Q(z (5.2) (5.4) (5.5) (5.6) (5.3) In this form, it readily reproduces (5.1) in the limit 1 ! 0. In order to show that the qq-character de nes a sort of second quantization of the Seiberg-Witten geometry, we will take the NS limit and reproduce the TQ-relation (5.4), the operator Y(z) being reduced to the rational function Y (z), and the qq-character to the T-polynomial. To perform the NS limit, we will follow the procedure described in [23] (see also [27]) and rst re-derive the Bethe equations. In the NS limit, the sum over Young diagrams entering the expression (3.19) of the partition function is dominated by a Young diagram with in nitely many boxes.9 This critical Young diagram minimizes the summation and its pro le is obtained by solving the discrete saddle point equations: qjY~ +xjZvect.(~a; Y~ + x)Zfund.( m~; ~a; Y~ + x) qjY~ jZvect.(~a; Y~ )Zfund.( m~; ~a; Y~ ) = 1; 8x 2 A(Y~ ): Taking into account the variation of the vector and fundamental contributions, we nd q 1 2 m( x) Q Q Following [23], we now consider only Young diagrams with in nitely high columns, and such that a box can be added to (or removed from) each column. Up to 2-corrections, the images under x of a box x 2 A(Y~ ) and the box immediately below x0 2 R(Y~ ) are 8The TQ-equation can also be written in an operatorial form, y^ + qm(z)y^ 1 Q(z) = T (z)Q(z); y^ = e 1@z T (z) = Y (z + 1) + q 1):8 m(z) Y (z) : where y^ is a shift operator. Here the non-commutativity of the variables y^ and z becomes manifest and the previous relation de nes a quantum curve. This di erence equation is actually equivalent to a Schrodinger equation under a quantum change of variables [34]. This correspondence goes under the name of bispectral duality [38{40] and can be seen as a degenerate version of the AGT correspondence relating the gauge theory in the NS background with the semiclassical Liouville/Toda theory. 9This argument is similar to the one employed by Nekrasov and Okounkov in [42] to perform the limit 1; 2 ! 0. The main di erence here is that the critical Young diagram doesn't have a continuous pro le but is instead described by a step-function where the plateaux are given by the Bethe roots. equal, they de ne the set of Bethe roots ur = x 2 A(Y~ ), except for N extra boxes (one for each diagram) that lie on the top right of the diagrams, and for which x = ` with l = a` + n` 1 and n` the number of columns for the Young diagram Y` .It is emphasized that these extra boxes are necessary to ful ll the relation jA(Y~ )j = jR(Y~ )j + N between the cardinal of the two sets. The number of columns n` in each diagram will play the role of a cut-o sent to in nity at the end of the computation. Under this identi cation, and taking into account the factor 1 2 from the box y of coordinate y = x 2 just below x, we nd in the limit 2 ! 0: x for x 2 R(Y~ ).10 This is true for all boxes 1 = q m(ur) (ur) (ur + 1) s=1 ur M Y ur s6=r us us + 1 1 (z) = Y(z (5.7) HJEP04(216)7 and the number of Bethe roots is M = P` n`. These equations resemble the Bethe equations of an inhomogeneous sl(2) XXX spin chain with a twist parameter q.11 The TQequation associated to this system of Bethe roots reads T (z)Q(z) = (z) (z + 1)Q(z + 1) + qm(z)Q(z 1); (5.8) eigenvalues:12 Introducing the Q-polynomial as in (5.2), it is indeed possible to show that the r.h.s. is a polynomial of degree M + 2N , with M zeros at z = ur as a consequence of the Bethe equations (5.7). The TQ-equation (5.2) is reproduced by further sending the number of Bethe roots M to in nity, together with the cut-o s ` after a proper rescaling of the T and Q polynomials. More details on this limit will be provided in the work [53] to appear. The NS limit of the expectation value (4.9) of operators is also dominated by the single state j~a; Y~ i, and diagonal operators in the basis j~a; Y~ i can be identi ed with their hOi D~a; Y~ jOj~a; Y~ E due to the simpli cation Zinst Zvect.(~a; Y~ )Zfund.( m~; ~a; Y~ ). From the action (2.21) of to /removed from Y~ by Bethe roots, we nd the operator Y(z) on states j~a; Y~ i, replacing the coordinate x of boxes that can be added D~a; Y~ jY(z)j~a; Y~ E Q(z) (z) Q(z 1) ~a; Y~ j Y(z) j~a; Y~ (5.9) Q(z Q(z) (z) 1) : (5.10) for an appropriate choice of masses mf . point equations. Denoting NS(z) the limit of the qq-character (z), the identity (4.6) reproduces the TQequation (5.8) with the T-polynomial T (z) = NS(z) (z) (the presence of the extra cut-o factor (z) will be explained in [53]). Our observation can be easily generalized to apply to linear quiver gauge theories. The NS limit for the A2 quiver has been performed in [23]. Using the same procedure, the qq-character identity (4.21) reproduces the TQ-relation for an inhomogeneous sl(3) XXX spin chain characterized by two sets of Bethe roots (equation (7.15) of [27]). 10In a Young diagram , the image x of x = (i; i) 2 R( ) is given explicitly by x = a + (i 1) 1 + ( i 1) 2, it is nite in the limit 2 ! 0 since i tends to in nity such that 2 i remains nite. 11In fact, in the superconformal case N~ = 2N , it exactly reproduces the inhomogeneous XXX spin chain 12This is true for well-behaved operators for which the insertion does not modify signi cantly the saddle In this paper, we developed a holomorphic eld representation of SHc algebra. It has the merit to express the commutation relations and the nite rank representations of the operators in a compact form. Instanton partitions for AQ and A(Q1)-type quiver gauge theories can be expressed concisely in terms of Gaiotto state, intertwiner operator, and a newly introduced avor vertex operator that insert the contribution of fundamental hypermultiplets. A new characterization of the Gaiotto state and intertwiner has been established using the adjoint action of SHc holomorphic elds. It provided the in nite set of constraint on the instanton partition function from the chiral ring generating function proposed in [27]. These constraints are summarized in simple algebraic relations which were referred as the qq-character. The qq-characters describe a quantum version of the Seiberg-Witten geometry [41] in the NS limit 2 ! 0. In this setup, the 2-deformation introduces a form of second quantization of the TQ-system in which the T-polynomial is replaced by an operator acting in the Hilbert space of the rank N representation. This should be compared with the recent results obtained in [54, 55], where the subleading corrections to the NS limit have been derived. These corrections are compatible (up to a quantum correction) with the second quantization of the NS action that is interpreted as the Yang-Yang functional of the underlying quantum integrable system. By comparing these two di erent approaches, a uni ed interpretation for the 2-deformation of quantum integrable systems should emerge. There are some obvious generalizations of the current study for future work. In [27], similar algebraic relations for the chiral generating functional were proposed for ADE type quiver gauge theories. In order to describe the bifurcation in the quiver diagram, we need to nd a SHc description of trivalent vertex which takes of the form: ~ jbii = X ~ W Zvect.(~b; W~ ) 1=2j~b; W~ i j~b; W~ i j~b; W~ i : Here we added extra labels ; to specify the Hilbert spaces. With the help of such operator, one may give the partition for the D4 quiver partition function, for example, as 3 i=1 i hG; ~aijqi D0(i;)1 V i i (~ai;~bjmi) qD0(1;1) j~bii 1 2 3 4 = X Y~1;Y~2;Y~3; W~ 3 i=1 ~ ~ q4jW jZvect.(~b; W~ ) Y qijYijZvect.(~ai; Y~i)Zbfd.(~ai; Y~i;~b; W~ jmi) : (6.1) (6.2) In order to derive the qq-character for such extended cases, we need to nd an analog of (3.22), (3.23) for the trivalent vertex. At this moment, however, this seems not so simple and we would like to leave it for future study. Another possible direction proposed in [27] is the 5D version of the current analysis. It corresponds to the algebra studied by many authors in [56{59] and has implications in 4D [60, 61]. Since the building blocks are already known (for example, [62, 63]), it would not be so di cult to perform a similar analysis in such set-up. In addition, our formulation of SHc seems particularly suited to the generalization to the six-dimensional -background R21 2 R 2 R23 in which the instanton partition function of N = 2 theories are expressed as a sum over plane partitions [64]. In a di erent perspective, it is important to clarify the relations with integrable models. On one hand, a proposal of NPS in [27] suggests a connection between quantum geometry and the representation of the Yangian associated to the quiver Dynkin diagram. On the other hand, Maulik and Okounkov have proposed in [65] the expression of the Yangian of gbl(1). In their formalism, the Dynkin diagram is regarded as the nite lattice of a spin system, where the spin degree of freedom is actually described by a free boson Fock space. In [66], a short summary of [65] and a possible supersymmetric generalization were presented. While the two approaches are very di erent, the coproduct de ned by the authors of [65] coincides with the one employed in [1]. Our analysis which relates the NPS qq-character [27] to the SHc algebra [1] could provide an interesting link between the two Yangians. Note added. After the rst version of this paper was submitted to arXiv, N. Nekrasov published a paper [67] where he studied Dyson-Schwinger equations for the instanton partition functions. He evaluated the e ect of adding point-like instantons, and express this e ect by an operator Y which is identical to ours. There seems to be a direct relation with our analysis and we hope to provide a more detailed comparison in the near future. Acknowledgments J.-E. Bourgine thanks I.N.F.N. for his post-doctoral fellowship within the grant GAST, which has also partially supported this project, together with 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-RNP092 (PESC)) and the MPNS-COST Action MP1210. YM would like to thank Satoshi Nakamura and R.-D. Zhu for their comments and discussions. He is partially supported by Grants-in-Aid for Scienti c Research (Kakenhi #25400246) from MEXT, Japan. HZ thanks Chaiho Rim for instructions and comments. He is supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (NRF2014R1A2A2A01004951). A Comments on the notations In this paper, in order to ease the comparison with the gauge theory, we use the omega background parameters 1; 2 instead of the CFT parameter = 1= 2 in [7, 8]. Since some results of these papers are used here, we summarize the correspondence between the notations in this appendix. As we use two parameters instead of one, we need some rescaling and shift of parameters to compare with the results there. Adding a tilde to the notations in [7, 8], the comparison goes as follows: D0;n+1 = ( 2)nD~ 0;n+1; a` = (x)jx2A(Y`) = 2a~` + +; 2(a~` +A~t(Y`)); D 1;n = ( 2)nD~ 1;n; En = ( 2)nE~n ; z = 2= ~; (x)jx2R(Y`) = 2(a~` +B~t(Y`)): cn = ( 2)nc~n; (A.2) (A.1) (A.3) We note that under the rescaling (A.1), the algebra (2.1){(2.3) remains the same. HJEP04(216)7 Proof of the recursion formulae for Gaiotto states and intertwiner Since the Gaiotto state can be derived from the intertwiner (3.25), it will be su cient to prove (3.22), (3.23). We need a few formulae to characterize the behavior of the instanton partition function building blocks (here bifundamental and the vector contributions) under the variations of the number of boxes in the N -tuple Y~ . Zbfd.(~a; Y~ + x;~b; W~ jm) Zbfd.(~a; Y~ ;~b; W~ jm) Zbfd.(~a; Y~ x;~b; W~ jm) Zbfd.(~a; Y~ ;~b; W~ jm) Zbfd.(~a; Y~ ;~b; W~ + xjm) Zbfd.(~a; Y~ ;~b; W~ jm) Zbfd.(~a; Y~ ;~b; W~ xjm) Zbfd.(~a; Y~ ;~b; W~ jm) Zvect.(~a; Y~ + x) Zvect.(~a; Y~ ) Zvect.(~a; Y~ Zvect.(~a; Y~ ) = Q = Q Q = = q X Y~ ; W~ Q m) y y + + m) m) y + m y + m +) 1 1 1 2 Q Q Q y6=x y6=x y)( x y)( x y)( x y)( x y +) +) (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) These formulae were used in [8] to prove the recursive properties of the quiver gauge theories. Essentially the same computation shows up here. We evaluate the action of D (z) on the intertwiner: X Y~ ; W~ V (~a;~bjm) = Zbfd.(~a; Y~ ;~b; W~ jm)j~a; Y~ ih~b; W~ j ; with Zbfd.(~a; Y~ ;~b; W~ jm) := Zvect.(~a; Y~ )Zvect.(~b; W~ )Zbfd.(~a; Y~ : ~b; W~ jm). We rst evaluate the action of D 1(z) on the intertwiner from the left. It is easily deduced from its action on states j~a; Y~ i, D~a 1(z)V (~a;~bjm) = Zbfd.(~a; Y~ ;~b; W~ jm) xih~b; W~ j: (B.8) X z x An alternative expression can be obtained after noticing that the inverse images of a state j~a; Y~ i under the mapping D 1(z) are the states j~a; Y~ + xi for x 2 A(Y~ ) because the action of the operator removes one box. Since the states j~a; Y~ i form a basis of the vector space V~a, it is possible to write D~a 1(z)V (~a;~bjm) = X X Y~ ; W~ x2A(Y~ ) x(Y~ + x) z x Zbfd.(~a; Y~ + x;~b; W~ jm)j~a; Y~ ih~b; W~ j: (B.9) It is convenient to rewrite this expression as follows, using the fact that D~a 1(z)V (~a;~bjm) =X X x(Y~ ) (2.19): x(Y~ ) Zbfd.(~a; Y~ +x;~b; W~ jm) Zbfd.(~a; Y~ ;~b; W~ jm) = p x(Y~ ) Zbfd.(~a; Y~ +x;~b; W~ jm) y6=x +) Q y) Q y m+ +) We evaluate the ratio for Zbfd. from (B.1) and (B.5) and put the explicit form of The action of D 1 from the right can be evaluated similarly, V (~a;~bjm)D~b 1(z0) = X X Y~ ; W~ x2R( W~ ) z0 x( W~ ) Zbfd.(~a; Y~ ;~b; W~ xjm) Zbfd.(~a; Y~ ;~b; W~ jm) From (B.4) and (B.6), the factor in the middle takes the form: x( W~ ) Zbfd.(~a; Y~ ;~b; W~ Zbfd.(~a; Y~ ;~b; W~ jm) xjm) = p 1 2 Q Q y2R(W~ ) ( x y6=x y + +) Q Q y) (B.10) : m) (B.11) (B.12) y +m +) y + m) : (B.13) 1 To add (B.10) and (B.12), we use the following identity to simplify the formula (we put z0 = z m) X 1 Q y2R(Y~ )( x y6=x Q = P z X x2R( W~ ) 1 m Q Q y2A( W~ )( x Q y2R(W~ ) ( x +) Q y) y6=x Q y2A( W~ )( x y) y2R(Y~ )(z Q y2A(Y~ )(z y) +) Q Q y2A( W~ )(z y2R( W~ )(z m + +) m) y Q y + m +) y + m) m + +) ! y m) : (B.14) This formula is obtained by comparing the residue of the poles on both sides. Finally, we note that by using (2.21), we obtain P z Q y2R(Y~ )(z Q y2A(Y~ )(z +) Q y) Q y2A( W~ )(z y2R( W~ )(z y m + +) ! y m) j~a; Y~ ih~b; W~ j = P z Y 1 (1)(z) j~a; Y~ ih~b; W~ jY m + +) : (B.15) After combining everything, we prove (3.22). The proof of (3.23) is completely parallel and we omit it here. 13In this expression, and the analysis hereafter, the correct choice of sign is veri ed by comparing with the direct action of SHc generators on states with a small number of boxes. tation For completeness, we present here some explicit computations for the commutators of holomorphic generators. [D0(z); D1(w)]. From (2.15), D1(w)D0(z)j~a; Y~ i = D0(z)D1(w)j~a; Y~ i = X x2Y~ X X X y(Y~ ) x w y(Y~ ) j~a; Y~ + yi; j~a; Y~ + yi: y2A(Y~ ) x2Y~ +y The di erence is the inclusion of the box y in Y~ + y. So we obtain the rst relation in (2.6). [D0(z); D1(w)] j~a; Y~ i = X [D1(z); D 1(w)]. D1(z)D 1(w)j~a; Y~ i = w y(Y~ ) j~a; Y~ i: X X = X X x2R(Y~ ) y2A(Y~ x) x2R(Y~ ) y2A(Y~ ) + X y) j~a; Y~ i = X y2A(Y~ ) z w 1 y(Y~ ) z y(Y~ ) ! x(Y~ ) y(Y~ x z x(Y~ ) y(Y~ x z x) x) j~a; Y~ x + yi j~a; Y~ x + yi x(Y~ ) x(Y~ x z x) j~a; Y~ i; where we have assumed for Y~ a generic form such that A(Y~ explicit form of x(Y~ ), one may prove for y 2 A(Y~ ): x) = A(Y~ ) [ fxg. From the y(Y~ x)2 = y(Y~ )2 S( x S( y y) ; x) x)2 = x(Y~ )2 ; where S(z) is the scattering factor which appeared in (2.31). After combining them, D1(z)D 1(w)j~a; Y~ i becomes, X X x2R(Y~ ) y2A(Y~ ) x(Y~ ) y(Y~ ) S( x w x z y S( y x) j~a; Y~ x+yi+ X x2R(Y~ ) (w x)(z (C.1) (C.2) j~a; Y~ i (C.3) (C.4) (C.5) x) D 1(w)D1(z)j~a; Y~ i is evaluated similarly, assuming that R(Y~ + x) = R(Y~ ), x z S( x S( y x) x+yi+X (w x)(z In [D1(z); D 1(w)], the rst term cancels and the second term gives E(z) Ew( w)) j~a; Y~ i after +(z the use of the relations (2.18), (2.19) and (2.22). Degenerate situations should be evaluated case by case, but the general conclusion remains unchanged. x) (C.7) D D.1 Matter case Detailed analysis of the qq-characters There are di erent possibilities for the insertion position of the SHc operator D 1(z), the simplest option is to insert it on the left of the mass operator as in hG; ~ajqDD 1(z)U ( m~)jG; ~ai, and then use the braiding relation (4.7) to move it to the right. The two formulas (3.12) and (3.14) for the action of D 1(z) provides the identity (4.8). However, this is not the unique choice for the insertion of the D 1(z), a second possibility is to insert it on the right of the mass operator, deduced from (2.29): hG; ~ajqDU ( m~)D 1(z)jG; ~ai: A new braiding relation is needed in order to move the SHc operator to the left, which is U ( m~)D 1(z) = B 0 N~ N~ X D 1(m(f)) Y f=1 z m(f0) AC U ( m~): The action of D 1(z) on the Gaiotto state is then computed from (3.12) and (3.13), leading to the identity 1 m(z) Pz hY(z + +)i ~ N X f=1 z 1 ~ N Y f06=f 1 D m(f0) Y(m(f) + +) (m(f))E = q (D.1) (D.2) 1 Y(z) ; (D.3) where the operator (z) = Pz+Y(z + +) has been introduced to simplify the expression. As a result, the fundamental qq-character de ned in (4.6) obeys (z) = h (z)i + m(z) X ~ N f=1 1 ~ N Y f0=1 m(f) f06=f 1 D m(f0) Y(m(f) + +) (m(f))E : (D.4) In the last term, the apparent poles are cancelled by the zeros of m(z) and the r.h.s. is a polynomial. The compatibility with the identity (4.8) implies the following equality, m(z) X ~ N f=1 z 1 ~ N Y f0=1 m(f) f06=f 1 D m(f0) Y(m(f) + +) (m(f))E = Pz+q m(z) Y(z) : (D.5) D.2.1 Double action of SHc operators D (z) As a preliminary step it is necessary to derive the action of two operators D 1(z1) and D1(z2) with di erent arguments on a Gaiotto state jG; ~ai. The method is the same as in the case of a single operator, and upon using the property h~a; Y~ + xjY(z + +)j~a; Y~ + xi = S(z x)h~a; Y~ jY(z + +)j~a; Y~ i; (D.6) with S(z) the scattering factor de ned in (2.31), it is possible to write down HJEP04(216)7 D 1(z1)D1(z2)jG; ~ai 1 X q 1 2 Y~ Zvect.(~a; Y~ ) X x2A(Y~ )z1 1 x Q y2R(Y~ )( xy y2A(Y~ ) xy where the sign ambiguity has been xed by comparing the coe cient of the vacuum state with the direct action of the SHc generators. Inserting the pole decomposition of S(z2 z1 x) = x 1 2 +z12 z2 1 2 x +(z12 +) z2 1 x + + ; with the shortcut notation z21 = z2 z1, it is possible to perform the summation over 1 Xq 1 2 ~ Y Zvect.(~a; Y~ ) Pz2S(z21) Y(z2 + +) + Y(z1) 1 2 Y(z2 + +) +z12 Y(z2) 1 2 +(z12 +) j~a; Y~ i; where it has been used that the last two terms in the brackets have no polynomial part at in nity, and consequently in the last term the two factors Y(z2 + +) have cancelled each other. This result can be written in the compact form D 1(z1)D1(z2)jG; ~ai = 11 2 Pz2S(z21) Y(z2 + +) + Y(z1) 1 +z12 Y(z2 + +) Y(z2) 1 +(z12 The action of the commuted operators D1(z2)D 1(z1) is derived from the same method, D1(z2)D 1(z1)jG; ~ai = 1 Y(z1) 1 2 S(z21) Pz2Y(z2 + +) Pz1Y(z1 + +) + +z21 Pz1Y(z1) +(z21 + +) jG; ~ai: This expression is simpli ed employing the following identity, Pz2 [S(z21)Y(z2 + +)] = S(z21)Pz2Y(z2 + +)+ +1z221 Pz+1Y(z1 + +) 1 2 +(z21 + +) Pz+1Y(z1); Q 1 x)Y(z2 + +)j~a; Y~ i; (D.7) (D.8) (D.9) +) jG; ~ai: (D.10) (D.11) (D.12) obtained by decomposition of S(z21) as a sum over single poles and of Y(z2 + +) into positive and negative powers. As a result, we nd Pz2 S(z21) Y(z2 + +) + +z12 Y(z1 + +) Y(z1) 1 +(z12 +) jG; ~ai: (D.13) Taking the di erence between (D.10) and (D.13), the commutation relation (2.6) between D 1(z1) and D1(z2) is recovered, with the action of E(z ) on Gaiotto states given in (2.10) by the ratio of Y operators with shifted arguments. D.2.2 Derivation of the second qq-character The expression of the second qq-character follows from the consideration of the symmetrized action (D.10) of two SHc operators inside two Gaiotto states, hG~ ; ~ajqD [D 1(z1) D1(z2) + D 1(z2) D1(z1)] jG; ~ai: This quantity can be computed either using the right action (D.10) of two operators on the Gaiotto state, or from the right (3.12), (3.13) and left (3.14) actions of a single SHc operator. These two possible ways of calculation furnish the following identity: 2q 1 1 2 Pz1 Pz2 hY(z1 + +)Y(z2 + +)i 1 1 1 2 Y(z2 + +) Y(z2) Pz1 S(z12) Y(z1 + +) Y(z1) Y(z1 + +) Y(z2) Pz2 S(z21) 1 1 2 2 z12 2 2 + : The second line involves the commutator of D 1(z1) with D1(z2) evaluated in the Gaiotto states average. The same quantity can also be computed by direct right (3.12), (3.13) and left (3.14) actions on Gaiotto states, leading to a second identity: 1 +z12 Y(z2 + +) Y(z2) Y(z1 + +) Y(z1) q 1 1 2 1 2 Pz1 Pz2 hY(z1 + +)Y(z2 + +)i 1 Y(z1)Y(z2) : Replacing the average of the commutator in the rst identity, and introducing the positive part (z) = Pz+Y(z + +) produces 0 = q 1 h(Y(z1 + +) (z1))(Y(z2 + +) (z2))i + Pz1 S(z12) + Pz2 S(z21) Y(z2 + +) Y(z1) + q 1 Y(z1)Y(z2) 2 z12 + Y(z1 + +) Y(z2) This relation implies for the second qq-character (4.15), Pz1 Pz2 2(z1; z2) = 2 z12 + (D.14) (D.15) (D.16) (D.18) This is not enough to conclude on the polynomiality of the second qq-character, because of the presence of the cross-terms Pz1Pz+2 2(z1; z2) = (z2) Y(z1 + +) (z1) + q Y(z1) (z1 + +) (z21 + +)Y(z1) : On the other hand, we know the explicit expression of the second qq-character for small N , and can use it to deduce the expression of 2(z1; z2). First, it is noted that after the introduction of the orthogonal projector in (D.17), 2 can be rewritten as the average of 2(z1; z2) = h 2(z1; z2)i ; (z1) (z2) + Pz+1 (z1) Y(z2 + +) + q + Pz+2 (z2) Y(z1 + +) + q (D.19) (D.20) (D.21) (D.22) (D.23) (D.24) (D.25) z12 2 1 2 : 2 + q Y(z2) + 2 z12 S(z12) Y(z2) + 2 1 2 z12 2 + + 1 2 Y(z2) q 2 + 1 Zinst At rst sight, it is not clear whether this quantity is a polynomial and we had to check it case by case using the explicit expression of the polynomial operator (z) given in (4.13) for N = 1; 2. Case N = 1: in this case (z) is a scalar and can be taken out of the vacuum expectation values, i.e. h (z) i = (z) h i Pz+1S(z12) (z1) = (z1) and as a result . Since it is a polynomial of degree one, it satis es h 2(z1; z2)i = (z1) (z2) + (z1) (z2) + (z2) (z1) + q 2 where (z) is the fundamental qq-character given in (4.6). In this simple case, (z) = (z) which provides the nal result h 2(z1; z2)i = (z1) (z2) + q 2 Case N = 2: in this case, (z) is a polynomial of degree two that satis es Pz+1S(z12) (z1) = (z1) + 1 2. It follows that (z1) (z2) + (z1) Y(z2 + +) + + (z2) Y(z1 + +) + Y(z1) + 1 2 Y(z1) + q 2 z12 The explicit expression of (z) deduced from (4.11) allows to show that (z1) Y(z2 + +) + Y(z2) + 1 2 Y(z2) 1 Zinst Dz1 (Zinst (z2)) = Dz1Dz2Zinst: with the shifted derivative Using this expression we arrive at Dz = (z + +) 2 (z) = h (z)i = 2(z1; z2) = h 2(z1; z2)i = 1 Dz1 Dz2 + 2q z122 1 2 2 Zinst; h (z1) (z2)i = replacing z1 = z + 1 and z2 = z + 2, this quantity is obviously a polynomial in z. Open Access. 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. HJEP04(216)7 DzZinst : (D.26) (D.27) [1] O. Schi mann and E. 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Jean-Emile Bourgine, Yutaka Matsuo, Hong Zhang. Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, Journal of High Energy Physics, 2016, 167, DOI: 10.1007/JHEP04(2016)167