Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions

Journal of High Energy Physics, May 2015

We study five dimensional AGT correspondence by means of the q-deformed beta-ensemble technique. We provide a special basis of states in the q-deformed CFT Hilbert space consisting of generalized Macdonald polynomials, derive the loop equations for the beta-ensemble and obtain the factorization formulas for the corresponding matrix elements. We prove the spectral duality for SU(2) Nekrasov functions and discuss its meaning for conformal blocks. We also clarify the relation between topological strings and q-Liouville vertex operators.

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Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions

Received: February Published for SISSA by Springer Open Access 0 1 2 3 c The Authors. 0 1 2 3 0 7a Prospekt 60-letiya Oktyabrya , Moscow , Russia 1 Institute for Nuclear Research of the Russian Academy of Sciences 2 25 Bolshaya Cheremushkinskaya street , Moscow , Russia 3 31 Kasirskoe chaussee , Moscow , Russia We study five dimensional AGT correspondence by means of the q-deformed beta-ensemble technique. We provide a special basis of states in the q-deformed CFT Hilbert space consisting of generalized Macdonald polynomials, derive the loop equations for the beta-ensemble and obtain the factorization formulas for the corresponding matrix elements. dimensions; Matrix Models; Duality in Gauge Field Theories; Conformal and W Symmetry - Generalized Macdonald polynomials, spectral duality for conformal blocks and 1 Introduction 2 Generalized Macdonald polynomials 3 q-deformed Dotsenko-Fateev integrals 4 Loop equations for q-deformed beta-ensemble 5 Spectral duality for conformal blocks 6 Comparison with topological strings 7 Conclusions A Macdonald polynomials and Ruijsenaars Hamiltonians B Five dimensional Nekrasov functions and AGT relations C Ruijsenaars Hamiltonians and loop equations D Useful identities Introduction conformal blocks of the Liouville theory. This connection has provided insight in two dimensional CFT and gauge theory as well as other related areas [512]. In this paper we version of the AGT duality has been checked for many cases in [13][15]. We will focus on the simplest possible case of (q-deformed) Virasoro four-point conformal block. We briefly introduce the approach that we will use to study the five dimensional AGT correspondence. As in the four dimensional case the instanton part of the Nekrasov parfact a finer structure consisting of several factorized terms, so that the whole sum can be written as a sum over pairs of partitions (A, B): zvect(A, B) where zfund,vect are certain polynomials in the gauge theory parameters which we write to the expansion of four-point conformal block in terms of a certain complete system of zvect(A, B) 18], so that zvect(A, B) polynomials giving the desired expansion [19, 20]. However, for general t, q the basis turns out to be a lot more elaborate: in particular the naive deformation of Schur polynomials to Macdonald polynomials is not enough. In this paper we show that the right basis is in fact given by the generalization of the Macdonald polynomials MAB depending on two partitions (i.e. generalized Jack polynomials) were introduced in [21] and also studied in [22]. To compute the matrix elements in eq. (1.3) we use the q-deformed version of the cuts in the DF integrands pulverize into a set of poles, so that the integrals can be taken by residues. The sum over residues is captured by the Jackson q-integral, which is in fact a sum of the form dqzf (z) = (1 q) X qkaf qka = k0 1 q 1 qaa (af (a)). The matrix elements in this framework are given by the q-deformed Selberg averages of the generalized Macdonald polynomials: which provide the recurrence relations for the averages of any symmetric polynomials. One of our main results is the remarkable factorized formula for the averages of generalized Macdonald polynomials (4.12). It can be written schematically as (zfund(A))(zfund(B)) [zvect(A, B)]1/2 and evidently leads to the AGT conjecture for four-point blocks (1.3). Though we were not able to obtain a rigorous proof of this formula, we have checked it for several lower polynomials. This is the only missing step in the proof of the five dimensional AGT conjecture, however the conceptual picture is already apparent. Let us clarify the relation between our study and the alternative approach to the AGT duality proposed in [24, 25]. In these works the sum over residues in the DF integrals without any basis decomposition was shown to be the sum over pairs of partitions: qR+ t qR t , qR+ t, qR t , R+,R miraculously into the Nekrasov partition function However, this is not the Nekrasov function featuring in the AGT correspondence but a spectral dual [2628] thereof. One can also notice that the expansion in eq. (1.7) is not in We therefore have two different expansions of the q-deformed conformal block connected by the spectral duality. The original expansion in terms of a special basis corresponds to the AGT dual Nekrasov function and the spectral dual expansion in the intermediate momentum corresponds to the manifest sum over poles in the DF integrals: zvect(R+, R ) ZNek( = q2|a, mf, q, t) Generally the spectral dual Nekrasov functions describe gauge theories with differNevertheless, the parameters of the theories are reshuffled by the duality, e.g. the dual with the gauge theories. One can see that the AGT relation is a combination of the explicit DF expansion [24, 25] and the spectral duality. ZNek( = q2|a, mf, q, t) QF = P A,B Q|BA|+|B| (zfund(A))2(zfund(B))2 = ZNek zvect(A,B) R+,R QF corresponds to the vev of the scalar, Q1,2 represent the masses of the hypermultiplets and Generally the spectral dual conformal block B different conformal algebra (q-WM instead of q-Virasoro) compared to the original one B. Again, the case we consider here is extremely simple: the number of points and the algebra remain the same for the four-point conformal block of q-Virasoro. The dimensions of the We will mention a tentative application of this duality for conformal blocks in section 7. Why do these new duality features become visible only in the five dimensional version of the AGT correspondence and not in the original one? It turns out the in five dimensions gauge theory partition function has fine structure, which can be effectively analysed from the gauge theory.1 The geometry of this manifold is encoded in its toric diagram. For our case of SU(2) gauge theory with four fundamental hypermultiplets the diagram is shown in figure 1. The edges of the diagram correspond to the two-cycles in the threefold while Kahler parameters Qi of these cycles correspond to the gauge theory parameters. The topological string partition function can be computed using the topological vertex There are essentially two ways to carry out the sum over partitions in figure 1. One can cut the diagram vertically, compute the sum over R explicitly and leave the sum over A, B in the final answer. This sum over pairs of partitions is nothing but the sum in the Nekrasov function ZNek. Moreover, each half of the diagram corresponds to the matrix element in the conformal block expansion (1.2): However there is one more way to perform the summation: one can cut the diagram horizontally and do the sums over A and B first. In this way one gets the spectral dual corresponds to the sum in the DF integral representation (1.7): Q2 B = qR+ t qR t , qR+ t, qR t 1/2 . (1.12) The spectral duality in this picture is natural and corresponds to taking the mirror image In section 2 we introduce the generalized Macdonald Hamiltonian, compute its eigenpolynomials. In section 4 we derive the loop equations for the corresponding q-deformed section 5 and compare our results to the topological string partition function in section 6. We present our conclusions in section 7. Generalized Macdonald polynomials Generalized Macdonald polynomials are symmetric polynomials in two sets of variables xi and xi labelled by pairs of partitions Y1, Y2 and depending on an extra parameter Ding-Iohara coproduct acting on the of the original Macdonald Hamiltonian [15]:2 H1gen = t 1 ald eigenvalues H1genMY(q1,tY)2 (Q|p, p) = X2 Qa1 X(qYia 1)ti MY(q1,tY)2 (Q|p, p). a=1 i1 Eigenvalues of H1gen are non-degenerate, so the eigenfunctions are uniquely determined. In practice one finds Macdonald polynomials by solving linear equations for the coefficients of the polynomial eigenfunctions. One writes a general symmetric polynomial as a linear Y 0 cY Y 0 pY 0 . Acting with H1gen involves shifts of pn: X(qn 1)zn n1 This gives a rational function so it is straightforward to obtain the residues. The properties of generalized Macdonald functions are analogous to those of generalized Jack polynomials, though with minor variations. 1. Orthogonality, conjugation and normalization. There is a convenient way to normalize generalized Macdonald polynomials: through Macdonald polynomials themselves as follows MY(q1,tY)2 (Q|p, p) = mY 1 (p)mY 2 (p)+ with respect to Macdonald scalar product hMA~ , MB~ i = A~B~ CA01 CA02 hf (pk), g(pk)i = f 1 q n 1 tn pk pk=0 2We rescale pn by (q/t)n/2 compared to [15]. CA = (i,j)A 1 qAij+1tAjTi , CA0 = (i,j)A 1 qAij tAjTi+1 . Sometimes it is more convenient to use a different normalization 1 qYiaj t(Y a)jTi+1 MfY~ (p) = GY 2Y 1 Q1 MfY~ (p) = GY 1Y 2 (Q) a=1,2 (i,j)Y a a=1,2 (i,j)Y a where GAB is given in appendix B. This normalization is tailored so that the norms of the polynomials are given by the vector part of the Nekrasov function: Notice also that in this normalization generalized Macdonald polynomials depend polynomially on the parameters Q, q and t. 1 qYia+j t(Y a)jT+i1 MfY (p) , (2.8) 2. The inversion relation. 1 q = (1)|Y~ | CY 1 CY 2 MY~(tT1,q1) 3. Cauchy completeness identity. Generalized Macdonald polynomials form a complete basis in the space of symmetric functions in two sets of variables: = exp zvect(Y~ , Q) k1 a=1 1 qk 4. Specialization identities. For q 1, t = q become generalized Jack polynomials M~(q,t)(q2a|pk) J ~()(a|pk). For t = q the Y Y Hamiltonian H1gen turns into a sum of screening factors. Therefore, the eigenfunctions consistent with the results of [19, 20], where Schur polynomials were found to be the right basis for the case of t = q. One has the following reduction to ordinary Macdonald polynomials: MY(q1,tY)2 (Q|0, pk) donald polynomials on the Weyl vector can also be found and are nicely expressed through the ordinary Macdonald polynomials: Q pk = 1 Qn 1 tn , pk = 1 tn 1 qtnn ! Q pk = 0, pk = 1 tn 1 Qn = MY(q1,t) = MY(q2,t) 1 Qn 1 tn pk = pk = 1 tn 1 Qn M[2],[] = M[1,1],[] = M[1],[1] = M[],[2] = M[],[1,1] = = C i=1 j=1 k=0 non-integer values. The first few generalized polynomials are M[1],[] = p1, M[],[1] = p1 q (Q 1) p1(q t) 1q v+1 2 2 (q + 1)(t 1)p21 + 2(qt 1) (q 1)(t + 1)p2 2(qt 1) p21(q t) (qtQ + qQ tQ 2t + Q) 2q (qQ 1) (Q t) (q 1)(t + 1)Qp2(q t) 2q (qQ 1) (Q t) (q + 1)(t 1)p21(q t) q2 + qtQ qt tQ 2q2 (Q 1) (qt 1) (q Q) (q 1)(t + 1)p2(q t) q2 qtQ + qt tQ 2q2 (Q 1) (qt 1) (q Q) (q + 1)(t 1)p1p1(q t) (qt 1) (q Q) p21(q t) q t2Q + tQ t 2q2 (Q 1) (tQ 1) p2(q t) q t2Q tQ + t1 2q2 (Q 1) (tQ 1) p1p1(q t) q (tQ 1) 2 2 N v+1 yj j=1 n=0 q yj i=1 l=0 (q + 1)(t 1)p21 + 2(qt 1) (q 1)(t + 1)p2 2(qt 1) v1 v1 N+ v1 B = dqN z Y i6=j k=0 N+Y+N v+1 1qkzi q-deformed Dotsenko-Fateev integrals In the Dotsenko-Fateev approach conformal blocks are expressed in terms of multiple contour integrals of the degenerate field insertion. What is specific to the q-deformed case is that these integrals can be taken by residues and reduce to multiple Jackson qintegrals (1.4). We use the q-integral formalism which turns out to be more convenient throughout this paper. The four-point conformal block is given by the following integral3 Y (1 qkxi) (q,t)(y) Y yi Y (1 qkyi) The completeness (2.11) of the generalized Macdonald polynomials can be employed in the last line of eq. (3.1) and gives the following expansion i=1 j=1 k=0 = exp n1 n 1 qn N v+1 yj j=1 n=0 N+ v1 q yj i=1 l=0 pn qn 1 tn 1 qnv+ qn + pn qn 1 tn 1 qnv = tn 1 qnv+ = X |A|+|B| CACB MA(Bq,t) q2a qn pn 1 tn , pn A,B CA0CB0 M A(qB,t) q2a qn, qn t n 1 qnv 1 tn new parameter a has appeared in the last lines of eq. (3.2). This is an extra parameter of of the Selberg average according to eqs. (B.3). To check the AGT correspondence (1.3) one should therefore check that 1 qnv+ hf (x)i = R dqNx (q,t)(x) QiN=1 xiu Qvk=01(qkxi 1)f (x) R dqNx (q,t)(x) QiN=1 xiu Qv k=01(qkxi 1) and the parameters of the Selberg sum are identified with the gauge theory parameters with the help of eqs. (B.3). In the next section we develop the loop equations for the q-deformed beta-ensemble (3.4) in order to check eq. (3.3). Loop equations for q-deformed beta-ensemble Let us write down the loop equations for the DF integral. They provide the recurrence the average of any symmetric function. One first observes that the q-integral of a total q-derivative is zero tn 1 qnv AGT zvect(A, a) dqz z (1 qzz )g(z) = 0 i=1 xi xi q Y xi txj Y v1 where f (x) denotes a symmetric polynomial in xi corresponding to the insertion of extra vertex operators. Eq. (4.2) in this case is valid as a power series in negative powers4 of z. The q-derivative of the q-Vandermonde is given by qxixi q,t(x) = q N1 Y (qxi xj )(txi xj ) j6=i (xi xj ) xi qt xj Using this expression one can rewrite eq. (4.2) as follows: * N tN1qu+1(qvxi 1) qxii f (x) X z qxi Y txi xj j6=i xi xj (xi q)f (x) Y xi txj q(z xi) = 0, (4.4) where h. . .i denotes the q-Selberg average. Let us cast this equation into a more convenient form. We first observe that the sum in eq. (4.4) can be written as a contour integral powers of xi one expands eq. (4.2) in positive powers of z and obtains the recurrence 4Positive powers give rise to total derivatives with nonvanishing boundary values. ( q)f (pn) YN txj #+ = 0, (4.5) X 1 tn qnznpn = 0. where the contour Cx encircles all the points xi. Deforming the contour we pick up the powers of z and will not affect the recurrence relations for pn, so this term can be safely dropped. Other residues give q 1 f (pn) exp X 1 t znpn q q z + tN1qu+1 qv1 z f (pn + (1 qn)zn) exp n1 f (pn) exp (1tn)pn+ n1 X 1 znqn(1 tn)pn t k 1 qkv !+ 1 tk q(Y 1)ijt(Y 1)jTi+1 q1j+u+vt2i2(tNi+1qj1 1)(tN+2iqjuv2 1) (qv1ti)(tNiqu+j 1)(tNi+2qjv1 1) (B.3) q(Y 2)ijt(Y 2)jTi+1 1 Let us demonstrate how the recurrence relations work for the simplest example. Conage of p1: hp1i = q tN 1 tN1qu+1 1 (t 1) (t2N2qu+v+2 1) = 0. GY 2Y 1 (t1qu+1) 1 k=1 (i,j)Y k als [19, 20]:5 hMY (pn)i = hMY (pn)i = the factorized expressions MY(q1,tY)2 qu1t qk, qk (i,j)Y 1 (i,j)Y 2 Qf2=1 Q2k=1 fYk (mf + ak) GY 2Y 1(q2a) (i,j)Y qti1 1 tNi+1qj1 1 qu+jtNi The following simple relation for the negative power sums can be immediately deduced hpY iu,v,N = q|Y |(1v)hpY iuv2+22N,v,N . One can also check that the averages of generalized Macdonald polynomials are given by and analogously 1qkv+ = (1)|Y 2|qP(i,j)Y 1 i+P(i,j)Y 2(2i+2aj+1) q(Y 2)ijt(Y 2)jTi+1 1 q(Y 1)ijt(Y 1)jTi+1 1 qu+v+jt2N++2i(tN+iqu++j 1)(tN+i+1qj1 1) (B.3) GY 1Y 2 (t2N++1qu+v+1) Combining eqs. (4.12), (4.13) we obtain the desired result (3.3), though with additional MA(1qA,t2) q2a qn pn 1 qnv+ 1 qnv (i,j)Y 1 (i,j)Y 2 k=1 (i,j)Y k This relation gives the proof of the five dimensional AGT conjecture for four-point conlas (4.12), (4.13) for the q-Selberg averages. In appendix C we sketch some ideas which might lead to the proof of these identities. We have performed computerized checks of eqs. (4.12) and (4.13) on the first three levels. Spectral duality for conformal blocks and use combinatorial identities to cast them into the form of Nekrasov function. Let us examine the q-Selberg integral featuring in the DF representation of the q-deformed CFT (3.4) kNN zvect(A, a) . (4.14) of the conformal block (3.1) as a sum over pairs of partitions B R+,R BR+R , B Using the identities from appendix D one immediately obtains a compact expression not affect the final answer): BR+R = B (i,j)R+ qR+,ijtR+T,ji+1 1 qR+,ij+1tR+T,ji 1 (i,j)R 1 qj2tN++2i 1 qj+v1tN+i+1 qR,ijtRT,ji+1 qR,ij+1tRT,ji 1 1 qjv+1tiN+1 1 GR+R Qk= Qf2=1 fR+k ak + m,+ fRk ak + m, f f zvect(R+, R, ak) where f , GAB and zvect are given in appendix B, a+ = a = a 2a = 1 + (N+ N + 1), m1,+ = a + v + 1 + (N 1), m1, = a + 1 + N , m2, = a + v+ + 2 + N 2, Since we have proven both horizontal and vertical equalities in eq. (1.9), the diagonal line should also be true. We therefore obtain a (slightly indirect) proof of the spectral duality for the SU(2) Nekrasov partition functions. Eventually, this implies that all the dualities in the diagram (1.10) are valid. Comparison with topological strings Firstly we notice that the vertical half of the diagram from figure 1 indeed corresponds of Schur functions): q2i+m2+akj f=1 fYk (mf + ak) QF = q2a, Q1 = qm1a, Q2 = qm2a . This equality tells us that the vertical half of the toric diagram can be identified with the Of course, the computations are the same for the horizontal halves of the diagram, which correspond to the DF integrands (see eq. (1.12)): A R+ B R+ k= f=1 fR+k zvect R+, R, ak Conclusions the remarkable factorization formula for the averages of generalized Macdonald polynomials. We have proven the spectral duality for SU(2) Nekrasov functions and four-point vertex operators. It would be interesting to better understand other features of the CFT in the language of topological strings. In particular, the modular properties of conformal blocks should be encoded in the topological string partition function. The spectral duality in this case provides an alternative expansion, which can be used to find the nonperturbative results (cf. [41, 42]). Of course extensions of the above approach to the six dimensional gauge theories and to integrable systems, most interestingly to the double elliptic ones [43, 44], should also be Acknowledgments The author would like to thank A. Mironov, Al. Morozov, And. Morozov, S. Shakirov and hospitality of the International Institute of Physics, Natal, Brazil where part of this work and by D. Zimins Dynasty foundation stipend. Macdonald polynomials and Ruijsenaars Hamiltonians In this appendix we very briefly review some essential properties of the (trigonometric) Ruijsenaars Hamiltonians and list some useful expressions for them. Macdonald polynomials are the eigenfunctions of the set of commuting Ruijsenaars Hamiltonians Hk, which can be thought of as a maximal commutative subalgebra inside the Ding-Iohara algebra [15]: HkMY = ek 1)tNi MY where ek is the elementary symmetric polynomial of N variables and Hk = = t 2 k(k1) xia xj txia xj qPa xia ia . 1i1<...<ikN a=1 j6=i1...ik Note that for generic q and t the spectrum of H1 is non-degenerate and no higher Hk are needed to solve for the eigenfunctions. However, we give some expressions for higher Hk for the sake of completeness. Let us derive a compact expression for Ruijsenaars Hamiltonians acting on symmetric terms of power sums is given by The shift operators commute and we get Hk = t 2 k(k1) 1i1<...<ikN a=1 j6=i1...ik xia xj a=1 txia xj Y ePn1(qn1)xina pn . One can see that this sum can be expressed as a sum over residues:6 Hk = k(k1) where all k integrals are taken over the same contour encircling the points xi. The origin of the t-deformed factorial lies in the useful symmetrization formula Y tx(i) x(j) = [k]t! . Using (A.6) once again for za variables we get Hk = k(k1) Yk dza Y za zb Y Yk tza xi ePka=1 Pn1(qn1)zan pn . N We expand the rational factors containing xi in terms of power sums Hk = k(k1) +kN I (t1)kk! Cx a=1 za a6=b za tzb H1 = h[1] , H2 = H3 = 2 h[1,1] 2 h[2] , 6 h[1,1,1] 2 h[2,1] + h[1k] = h[k] = h[2,1] = tkN+ k(k21) (t 1)k C (za zb)2 a=1 za a<b (tza zb)(za tzb) ePka=1((za)(tza))ePka=1(+(qza)+(za)), (tk 1) (t 1)(t2 1) C C dz e(z)(tkz)ePka=01(+(qtaz)+(taz)) 2 dz1 dz2 (z1 z2)(tz1 z2) z1 z2 (z1 tz2)(t2z1 z2) 7To be precise one should ensure the convergence of the expansions. We assume that za are radially ordered, i.e. |z1| < . . . < |zk|. understood as the commutative subalgebra in the Ding-Iohara vertex algebra. n1 zn pn and (z) = P n1 als degenerate into Shur polynomials which are independent of q. In this limit all the Hamiltonians can be explicitly expressed through the first one. For example: h[1k](q) = (h[1](q))k, h[k](q) = h[1](qk), h[2,1](q) = h[1](q2)h[1](q). The first integral also simplifies and becomes h[1](q) = : e(z) qN+zz 1 e(z) : q 1 that h[1](q) with different values of q commute: [h[1](q), h[1](q0)] = 0 . h[1](q)R = qN 12 CR(q) + qN 1 q 1 where CR(e~) = P tion of the Casimirs CR(n) = P i1 Ri i + 12 n (i + 12 )n . i1 e~(Rii+ 12 ) e~(i+ 12 ) = P n n0 ~n! CR(n) is the generating func Five dimensional Nekrasov functions and AGT relations multiplets is given by ZN5dek = X |A~| QiN=1 QfN=1 fA+i(mf+ + ai)fAi(mf + ai) zvect(A~, ~a) G(AqB,t)(x) = (i,j)A (i,j)B 1 qxqAijtBjTi+1 1 qxqAijtBjTi+1 (i,j)B (i,j)A 1 qxqBi+j1tAjT+i = 1 qxqBi+j1tAjT+i . fermionic construction of GL() Casimirs. The AGT relations for N = 2 are: u+ = m1+ m2+ 1 + , v+ = m1+ m2+ , v = m1 m2 , Ruijsenaars Hamiltonians and loop equations Let us rewrite the loop equations (4.4) in terms of the Ruijsenaars Hamiltonian (A.2). We first write down a useful identity: i=1 z xi j6=i xi xj Y txi xj = (1 t)z 1 which is proven by expanding the right hand side as a sum over poles in z. Using eq. (C.1), one can rewrite eq. (4.4) as follows tN1qu+1 n1 qn 1 z qtN qz(1 t) z q f (x) qz(1 t) j=1 pn f (x)+ z xj where H1 is the first Ruijsenaars Hamiltonian.9 The expansion of eq. (C.2) in negative of any symmetric polynomials in xi. In fact, only two of the infinite family of constrains are needed. Indeed, the Virasoro generators Ln with positive n can be obtained by commuting L1 and L2. The same holds for the q-deformed case. We may therefore consider only the tN1qu+vH1f (x) + * tN1qu+2 tN1qu+v+1 q 1 [H1, p1] f (x) + 1 tN 1 t f (x) q1p1f (x) = 0, (C.3) For f (x) a Macdonald polynomial the Hamiltonian H1 acts diagonally. However multiplication with pn produces a sum over multiple Macdonald polynomials. To see this one should use the Pieri idenity: where the sum is over diagrams W such that the skew diagram W \Y is a horizontal strip (i.e. there are no more then one box belonging to each column) of k boxes; CW \Y (resp. RW \Y ) is the set of columns (resp. rows) intersecting W \Y and completely in terms of the averages of Macdonald polynomials. We are sure that one can obtain a proof of the identities (4.9), (4.10), (4.12), (4.13) along these lines, though we were not able to find it. Useful identities bY (i, j) = (i,j)A Qqj1t1i (i,j)B Qqjti YN vY1 1 Qqk+Riti i=1 k=0 1 Qqkti k=0 i=1 j=1 (i,j)R 1 Qqktji where GRP is given by eq. (B.2). 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Yegor Zenkevich. Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions, Journal of High Energy Physics, 2015, 131, DOI: 10.1007/JHEP05(2015)131