Decomposing Nekrasov decomposition

Journal of High Energy Physics, Feb 2016

AGT relations imply that the four-point conformal block admits a decomposition into a sum over pairs of Young diagrams of essentially rational Nekrasov functions — this is immediately seen when conformal block is represented in the form of a matrix model. However, the q-deformation of the same block has a deeper decomposition — into a sum over a quadruple of Young diagrams of a product of four topological vertices. We analyze the interplay between these two decompositions, their properties and their generalization to multi-point conformal blocks. In the latter case we explain how Dotsenko-Fateev all-with-all (star) pair “interaction” is reduced to the quiver model nearest-neighbor (chain) one. We give new identities for q-Selberg averages of pairs of generalized Macdonald polynomials. We also translate the slicing invariance of refined topological strings into the language of conformal blocks and interpret it as abelianization of generalized Macdonald polynomials.

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Decomposing Nekrasov decomposition

V Nekrasov decomposition A. Morozov 0 1 3 4 5 6 Y. Zenkevich 0 1 2 3 5 6 0 31 Kashirskoe highway, Moscow , 115409 Russia 1 19-1 Bolshoy Karetniy , Moscow, 127051 Russia 2 Institute for Nuclear Research of Russian Academy of Sciences 3 25 Bolshaya Cheremushkinskaya , Moscow, 117218 Russia 4 Institute for Information Transmission Problems 5 National Research Nuclear University MEPhI 6 6a Prospekt 60-letiya Oktyabrya , Moscow, 117312 Russia AGT relations imply that the four-point conformal block admits a decomposition into a sum over pairs of Young diagrams of essentially rational Nekrasov functions | this is immediately seen when conformal block is represented in the form of a matrix model. However, the q-deformation of the same block has a deeper decomposition | into a sum over a quadruple of Young diagrams of a product of four topological vertices. We analyze the interplay between these two decompositions, their properties and their generalization to multi-point conformal blocks. In the latter case we explain how Dotsenko-Fateev allwith-all (star) pair \interaction" is reduced to the quiver model nearest-neighbor (chain) one. We give new identities for q-Selberg averages of pairs of generalized Macdonald polynomials. aITEP - Decomposing HJEP02(16)98 1.1 1.2 1.3 2.1 2.2 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 1 Introduction 2 3.4 Identi cation with topological strings 4 Re nement q-Selberg measure Generalized bifundamental kernel Vertical slicing Generalized Kostka functions Seiberg-Witten theory and topological string pattern Re nement and slicing invariance Four point conformal block, no q-deformation Multi-point case. Star-chain duality Resolution of the star/chain problem. From chain to star. BifundaHorizontal slicing. DF representation and spectral dual Nekrasov function 5 Conclusions and discussion A Five-dimensional Nekrasov functions and AGT relations B Loop equations for matrix elements C Open topological string amplitude on resolved conifold D Useful identities E Dotsenko-Fateev integrals as contour and Jackson integrals { 1 { Introduction Conformal blocks [1{4] are among the most interesting and important quantities under study in modern theoretical physics. Perturbatively they are de ned as series of matrix elements in highest weight representations of Virasoro algebra, see [5, 6] for recent reviews. Non-perturbatively they are examples of matrix-model -functions [7{11], associated with peculiar conformal [12{14] (also known as Dotsenko-Fateev [15] or Penner [16{18]) matrix models, and exhibit non-trivial and almost unexplored behavior in various regions of moduli space [19]. Their modular transformations [20{32] are important for the study of knot polynomials (Wilson loop averages in Chern-Simons theory [33{35]), see [36{38] for a recent outline. AGT relations [39{41] connect conformal blocks to LMNS quantization [42{45] of the Seiberg-Witten theory [46{52] and express them in terms of Nekrasov functions [53{56]. Both the matrix model and Nekrasov function formalisms imply natural lifting of original conformal blocks to (q; t)-dependent quantities | looking from di erent perspectives this can be either a - or a q-deformation, associated with 5d generalization of Seiberg-Witten theory [57{59] and AGT relations [60, 61].1 It is at this level that the full duality pattern gets clear and manifest. Finally, as a quintessence of all this, conformal blocks are expressible through topological vertices [67, 68] | and this will be the story we concentrate on in the present paper. This relation involves not only the full-scale theory of Schur and Macdonald functions [73], but also conceptually important notions of star-chain duality and Selberg factorization. The ideniti cation between q-deformed CFT blocks and topological vertices has been used in [74] to prove the spectral duality [75{78] of the former. In the present paper we generalize this identi cation to the higher-point case. We also clarify the relation between preferred direction in re ned topological strings and the basis of states in conformal eld theory Hilbert space. Further generalizations to WN and elliptic cases would be given in [79]. 1.1 Conformal blocks and characters Conformal blocks are best described by the version of Dotsenko-Fateev (DF) conformal matrix model, introduced and investigated in [80{85] where BU(1) is an explicit function representing the contribution of an extra free boson. We nd it most convenient to use the number k of independent integration contours as 1For recent developments on the 6d generalizations of the AGT relations see [62{66]. { 2 { V 2( 1) of bifundamentals in the gauge theory description below will be k 2. The parameters of conformal block can be conveniently summarized in a diagram, such as one shown in External dimensions i = vi (vi + 1) are parameterized by the \momenta" vi, while internal dimensions 0+v1+:::+va 1+ N1+:::+ Na 1 0+v1+:::+va 1+ N1+:::+ Na 1+ are expressed through the numbers Na of screening integrations, i.e. conformal block is considered as analytical continuation of the integral in the number of integrations. It is important for this description that the integral is of Selberg type [86] and analytical continuation in Na is actually under control. The next important fact [87] is that the inter-screening coupling is reduced to a gure 1. e a= square of block to a bilinear combination of bi-character Selberg averages [86] over x and y, B4 = X A;B | A[x] B[x] A[y] B[y] {z ZAB } which are exactly calculable rational combinations of v-parameters, and are basically nothing but Nekrasov functions [53{56], labeled by arbitrary pairs A; B of Young diagrams. This line of reasoning reduces AGT relation [39{41] between conformal block and Nekrasov functions to Hubbard-Stratanovich resummation of Selberg integrals [80{85]. There are important details, making the story a little more technically involved, especially for 6= 1 (i.e. for the central charge c 6= 1) [74, 88{90], but in what follows we try =exp 8 < : X 1 X k k a;b xa yb =exp ( X pk[x]pk[y 1] ) k k =XJ A[x]JA[y] A k 9 = ; and { 3 { N fund U (N ) 1 bifund U (N ) 1 bifund U (N ) to separate concepts from technicalities, putting simpli ed general considerations before exact, but overloaded, formulas. After q-deformation (which in the Seiberg-Witten theory framework means going from 4d to 5d Yang-Mills theories [57{59]), the integral remains basically the same, only the integration is replaced by Jackson q-integration2 [91]: is not bi linear, but rather quadri linear: h A Bi = X SARSRB R =) B4 X Y1;Y2;Y3;Y4 SAR1 SR1BSBR2 SR2A (1.8) | and this is the decomposition which is related to topological vertex [67{72] and geometric engineering [92, 93]. The origin of two extra Young diagrams is simple: summation over them substitutes integration over x and y variables in the de nition of averages in (1.6) | this appears to be the right way to interpret the multiple Jackson integrals/sums in (1.7) (see appendix E for technical details). 1.2 Seiberg-Witten theory and topological string pattern To better understand the origin of the multi-character decomposition let us investigate the structure on the gauge theory side of the AGT duality. Conformal blocks correspond to instanton partition functions of quiver gauge theories which are given by Nekrasov formulas. The comb-like (k + 2)-point conformal blocks on a sphere correspond to linear quiver theories, in which the gauge group is a product of (k 1) U(N ) factors and the matter content is encoded in the quiver diagram as, e.g. in gure 2. Here a circle is a gauge group, a box denotes a collection of matter hypermultiplets, a outgoing (resp. incoming) link connecting a circle with a box indicates that the corresponding hypermultiplets transform as a fundamental (resp. antifundamental) under the gauge group. The structure of the corresponding Nekrasov function is modelled after the quiver diagram above: ZNek=X ~ jY1j 1 jY~kjzfund(Y~1) k ~ Ya 1 zvec(Y~1) zbifund(Y~1;Y~2) zbifund(Y~k 1;Y~k) 1 zvec(Y~k) zfund(Y~k); (1.9) 2Jackson q-integral is de ned as a sum R0a f (x)dqx = (1 q) Pk1 0 qkaf (qka). { 4 { where the de nitions of the rational factors zfund;vect;::: are given in appendix A. The structure of each term in the decomposition is linear, in particular for a (k + 2)-point conformal block there are k 1 vector multiplet contributions and k 2 bifundamental matter hy permultiplets. Such quiver or chain decomposition of the conformal block is obtained by inserting a special basis of states j ; Y~ i labelled by a pair of Young diagrams in the intermediate channels of the block: is then given by the Selberg average of a collection of orthogonal polynomials as in eq. (1.6). For c = 1 the special basis which reproduces the corresponding factor in the Nekrasov function (1.9) is given by Schur polynomials. We will compute the most general matrix element using q-Selberg averages and show that it is indeed given by the Nekrasov expression. For 5d gauge theories compacti ed on a circle of radius R5 the structure of Nekrasov function remains basically the same. The only change is that all the monomial factors in the rational functions zfund;vec;::: are transformed into q-analogues roughly as x ! q where q = e 2R5 . However, quite remarkably in this case Nekrasov partition function 1, | or conformal block | turns out to have yet another interpretation. Gauge theory in x ve dimensions can be obtained by compacti cation of M-theory on a toric Calabi-Yau threefold. Partition function of the resulting theory is equal to the (re ned) topological string partition function, which can be computed by the topological vertex technique as follows. One rst draws the toric diagram of the CY threefold and assigns to each internal edge the complexi ed Kahler parameter Q of the corresponding two-cycle. One also assigns a Young diagram to each internal edge, and an empty diagram to each external edge. There are in general only trivalent vertices in the diagram, and to each of them one assigns a certain function CY1Y2Y3 (q) | the topological vertex [67, 68] | depending in a cyclically symmetric way on three Young diagrams Ya residing on the adjacent edges and also on the parameter q = e 1R5 : CABC (q) = A where by qC+ P(i;j)2A 2(j B C (1.10) HJEP02(16)98 (1.11) (A) = we mean the in nite set of variables f qC1 21 ; qC2 23 ; : : :g and i). The partition function is computed by summing up over all the Young diagrams with weights given by the product of all topological vertices and the \propagators" of the form ( Q)jY jfY (q)n where n is the framing factor depending on the relative orientation of the edges adjacent to the given edge. = q 2 (A) C (q ) X AT=D qC+ B=D qCT+ ; D { 5 { t QF;4 t q Qe4 t q q Q4 QB;3 t QF;3 t q QB;3Q3=Qe3 q Qe3 t t q q Q3 QB;2 t QF;2 t q QB;2Q2=Qe2 q Qe2 t t q q Q2 QB;1 t QF;1 t q QB;1Q1=Qe1 q Qe1 t q Q1 t q is drawn using the recipe of geometric engineering. It is the crossing of N horizontal and k vertical lines, which intersect as shown e.g. in gure 3. There is a natural decomposition of the toric diagram depicted on gure 3 which leads to the same quiver structure as in gure 2 and the Nekrasov expression (1.9). One should perform the sums over all Young diagrams except those residing on the horizontal edges marked with QB;i, which are related to positions of the vertex operators in the conformal block and the gauge theory couplings i . In this way one obtains a sum over a chain of pairs of Young diagrams of certain rational factors, which turn out to coincide with zvect;fund;::: for t = q (we introduce t = q ). The resulting expression has exactly the form of Nekrasov function (1.9). Moreover, each term in the Nekrasov decomposition can now be decomposed into an in nite sum of simpler building blocks ZARB, related to the four-point topological string amplitude on resolved conifold. In the language of CFT this leads to the decomposition h e 1; Y~1jV 2 for the matrix element in the l.h.s. Another natural decomposition of the toric diagram | cutting along the vertical edges marked with QF;i (related to Coulomb moduli of the gauge theory and intermediate dimensions in the conformal block) | corresponds to the spectral dual Nekrasov function. The gauge theory origin of this dual description is that in 5d instantons are BPS particles as are the gauge bosons. Spectral duality exchanges these two sets of BPS objects and therefore leads to a nontrivial identi cation between two gauge theories. We will show { 6 { that the spectral dual decomposition of the toric diagram has a natural interpretation in terms of DF integrals of q-CFT | it is the sum featuring in the discrete Jackson integrals, each vertical leg corresponding to a separate integration contour in (1.1). Therefore, the spectral dual decomposition over horizontal lines of the diagram corresponds to the DF integrals themselves, while the original Nekrasov decomposition is the sum over a complete set of intermediate basis states in the CFT: Bk+2 BU(1) X R1;:::;Rk ZR1;:::;Rk ZR1;:::;Rk : (1.13) HJEP02(16)98 Our goal in this paper is to explain the relation between eq. (1.7), eq. (1.9), and gure 3. We will learn that the identi cation between conformal block and Nekrasov function requires a nontrivial rewriting of the Vandermonde determinant (which is the product of all-with-all form) into the sum of Nekrasov form (which is of nearest-neighbour form). We rst clarify the relation of the toric diagram and the DF integral schematically in the simplest case of the four-point conformal block (k = 2). Extension to arbitrary k involves an a priori non-trivial star-chain identity, which is in fact the key to understanding DF description of conformal blocks and relies upon the basic properties of representation theory. Another crucial property is Selberg factorization | a mysterious conspiracy between the integrands and integration measure in DF theory, between what is averaged and how it is done. This property guarantees that the averages of certain polynomials over the q-Selberg measure factorize into products of linear factors depending on the parameters of the integral. The last mystery is that the elementary building block in the quadrilinear decomposition of conformal blocks, i.e. the topological vertex, is closely related to the modular kernel and therefore to certain knot polynomials. 1.3 Re nement and slicing invariance The calculation we have just described yields the Nekrasov function of the 5d gauge theory with the particular choice of -deformation parameters, i.e. 1 = 2, or equivalently t = q, which corresponds to c = 1 in CFT. To obtain the partition function in a general -background, one has to use re ned topological vertex3 [69, 70]: A t B q C CABC (t; q) = = q jjBjj2+jjCjj2 t jjBTjj2+jjCTjj2 M C(q;t) t 2 2 X D q jDj+jA2j jBj t AT=D q C t B=D t CT q ; (1.14) 3There is a slight historical mismatch of notations between the re ned and unre ned vertices. Reducing the re ned vertex (1.14) back to the unre ned case to compare with eq. (1.11) one needs to transpose all the diagrams and add some simple factors CABC(q; q) = ( 1)jAj+jBj+jCjq (A)+ (2B)+ (C) CATBTCT (q). { 7 { legs in the diagram is marked with a double stroke and the other two bear t and q labels on them. This is to indicate the right order of the indices and arguments of the re ned vertex, which depends on two deformation parameters and is not cyclically symmetric as was the case for t = q. The calculations generally get more technically involved, though the strategy remains the same. The only essentially new feature in this case is the naive loss of rotation symmetry of the diagram: the vertical and horizontal lines are no longer equivalent. However, it turns out that the symmetry in fact survives even for general t and q, though the individual vertices and propagators are not symmetric. This statement came to be known as the slicing HJEP02(16)98 invariance hypothesis. For toric geometries, which we consider, slicing invariance is also equivalent to spectral duality [75{78] of the corresponding Nekrasov partition functions, since the two sides of the duality are related to the 2 rotation of the whole toric diagram including the choice of preferred direction. We look at di erent choices of \slicing" of the toric diagram and relate them to di erent choices of the basis in conformal eld theory. One slicing direction corresponds to the \naive" basis of Schur polynomials A, the other | to the basis of generalized Macdonald polynomials MAB. The rst set of polynomials does not have factorized q-Selberg averages and does not reproduce the Nekrasov factors, while our calculations indicate that the second one does. Schematically * We investigate the connection between the two sets of polynomials and introduce generalized Kostka functions KACBD transforming one basis into the other: MAB = X KACBD C D: CD (1.15) (1.16) These functions are e ectively performing the 2 rotation of preferred direction. In more algebraic terms they are related to the abelianization map [98] acting on the basis in Ktheory of instanton moduli space. The paper is partitioned into a set of sections with increasing level of detail and complexity. After reviewing the basic steps of the construction at the simpli ed level in section 2 we ll in the details and provide full- edged formulas for the unre ned case in section 3. We then treat the re ned case in section 4. We provide a summary and point out future directions in section 5. 2 Basic steps In this section we introduce our approach to Dotsenko-Fateev integral expansion without q-deformation. We consider rst the most simple example of four-point conformal block { 8 { and show how decompose the integrand in terms of Schur polynomials. Next we consider the multi-point block and observe that a nontrivial star-chain duality is required in this case. We demonstrate this duality explicitly using skew Schur functions. Four point conformal block, no q-deformation In the case of four-point conformal block there are two contours of integration: C1 stretching from 0 to 1 and C stretching from 0 to in eq. (1.1) are divided into two groups: xi and yi and the inter-screening pairings are 1 . Therefore, the variables in the integration decomposed into a product HJEP02(16)98 The vertex operator contributions also decompose into a product of two factors: (z)2 ! (x)2 (y)2 N1+N2 Y i=1 (1 N1+N2 Y i=1 (1 zi)v1 ! zi)v2 ! N1 Y(1 i=1 N2 Y(1 i=1 N1 N2 Y Y yi we can write the cross terms which we denote by cross(x; y) = Y Y 1 N1 N2 Employing the Cauchy completeness identity (1.4) we get the expansion of the cross contributions in terms of Jack polynomials: 1 (2 pnq n + v1q n + pnv2)A jY1j+jY2jJY1 (pn)JY2 pn JY1 q n JY2 (q n); (2.1) v1 v2 0 X n 1 n n = X Y1;Y2 where pn = PiN=11 xin, qn = PiN=21 yin. polynomials After this decomposition the DF integral becomes the double Selberg average of Jack jY1j+jY2j JY1 (pn)JY2 pn v1 v2 JY1 q n JY2 (q n) u1;v1;N1; { 9 { u2;v2;N2; (2.2) where the averages are taken with respect to the measure QiN=a1 (xiua (1 xi)va ). For general the averages (2.1) do not give the Nekrasov expansion (xjua; va; Na; ) = 2 (x) of the conformal block (a more re ned basis of generalized Jack functions JAB depending on a pair of diagrams is required [88{90]). However, for the special case = 1 when Jack polynomials turn into Schur functions the structure of Nekrasov sum is indeed reproduced [80{85]. Thus, from the four-point case without q-deformation we learn that decomposing the inter-screening pairings in the DF integral in terms of characters and then taking the Selberg averages produces Nekrasov representation of the conformal block. We now move to the multi-point case where the star-chain duality is required to obtain Nekrasov decompoIf one approaches the multipoint case in a naive way one arrives at what seems to be a paradox. The DF representation contains a product of all pairings between screening operators, i.e. an expression of the form (2.3) (2.4) (2.5) (2.6) Y a<b 1 xa !2 i x b j : Y zbifund(Ya; Ya+1) : a A=W = def X cBW B A B C = X cBC A A : B A where cABC are the Littlewood-Richardson coe cients, describing multiplication of repreHowever, the gauge theory corresponding to the multipoint comb-like conformal block is a linear quiver of the form depicted in gure 2, and its Nekrasov partition function contains only the nearest neighbour pairings: Thus the multilinear decomposition of the DF integral should also have the nearestneighbor structure. In the four-point case there are only two term in the product, so that all-with-all (star) type interaction is the same as nearest-neighbour (chain) one. But how can one decompose the multi-point product (2.3) into a sum of nearest-neighbour products, how can star become equivalent to a chain? 2.2.2 Skew characters The resolution of the paradox is technically based on the properties of skew characters, Moreover, this is straightforwardly generalized to A pn(x(1))+:::+pn(x(m)) A=W [y]= W pn(x(1))+:::+pn(x(m)) m 1 X CVAW CVBW A V;W B (pn) B;W A ) =X A =exp k k pk(z) pk(x)+pk(y) A[z] A pn(x)+pn(y) At x = y we can apply (2.6) to the l.h.s. to get a doubling rule e.g. [1] (2pn) = 2 [1] (pn), [2] (2pn) = 3 [2] (pn) + [11] (pn), [11] (2pn) = [2] (pn) + 3 [11] (pn), [3] (2pn) = 4 [3] (pn)+2 [21] (pn), [21] (2pn) = 2 [3] (pn)+6 [21] (pn)+2 [111] (pn), [111] (2pn) = 2 [21] (pn) + 4 [111] (pn); : : : which can be further promoted to tripling, quadrupling and higher multiplication formulas. 2.2.3 Resolution of the star/chain problem. From chain to star. Bifundamental kernel We claim that the chain of skew characters indeed reproduces the star-like structure of the DF integrand. The basic building block of the chain decomposition is the bifundamental kernel X C NAB[y] = A=C [y] B=C [y]; where Y [x] = Y [ pn(x 1)]. Two such kernels, averaged over the Selberg measure like hNAB[y]NCD[y]i, correspond to a single bifundamental eld in Nekrasov partition function of the gauge theory depending on two pairs of diagrams (A; B) and (C; D). Observe that4 N?A[x] = A[x], NA?[x] = A[x]. Qk) 1 Qi;j 1(1 Qkxixj 1). 4There is another curious identity, which would be useful for toric blocks: PA QjAjNAA[x] = Q k 1(1 We start with the case of ve-point conformal block. Using the identities from the previous section, we can rewrite the chain answer into the Dotsenko-Fateev (star) form: X N?Y1 [x]NY1Y2 [y]NY2?[z] Y1;Y2 Y1 [x] Y1=W [y] Y2=W [y] Y2 [z]=Y i;j xi yj Y2;W X Inverting this short derivation, we see that it is an iteration of the two-step procedure, which starts from m = k 1 with Fk 1[Y ] = Y [z] and ends at m = 2 with x = y0. In obvious notation: XFmfYmg Ym [y0;y1;:::;ym 1]= Ym X Ym;Wm 1 FmfYmg Ym=Wm 1 character is combined with the next product X Zm 1 Zm 1 HJEP02(16)98 ! (2.15) (2.16) (2.17) Since, whatever are the sets u and w, X Z and we are ready for the next iteration. At k = 3 this can be pictorially represented as FmfYmg Ym=Wm 1 Y2=W1 6 r r y x r = z Y1=W1 : Dots here stand for characters, which have labels according to the points at which they are evaluated, and arrows point from to . The arrows are labeled by the Young diagram, over which the sum is taken. The two encircled characters are evaluated at the same point, and can be transformed into one using eq. (2.6), thus fusing the two dots. The resulting trivalent vertex represents the Littlewood-Richardson coe cients, which depend on three Young diagrams. Note that only dots at the same place which are both either starting or end-points of the arrows can be merged in this way. Likewise at k = 4: z = y3 r r r r y1 6 r y2 r rkx r r W1 Ar r y = = z z Y3=W2 Y3=W2 W1 r KA A Y2=W1A Y2=W2 r KA A Y2=W1 A Y2=W2 6 r r y2 6 r r y2 rnx r A A Ar r? y1 x r r Cr rHHHHjr C r r r C C C C C Cr r r y3 r HJEP02(16)98 rr y1 r z r Cr r y3 r r OC C C C r OC C C Cr r y2 rkrx CrC rHHHHjr C C C C C C C C rrmy1 r rn r x r W3 r Y3 W2 r Y2 The main secret behind this derivation is that the structure constants in (2.6) are always the same | do not depend on the number of \Miwa variables" yi in [y0; : : : ; ym] | which allows to merge entire collections of points and parallel arrows in the examples above. This conspiracy between characters and the structure constants adds to associativity of multiplication and together they provide the star-chain equivalence. 2.3 Factorization of Selberg averages The \chain" decomposition of DF integrals (2.15) is also tied with the structure of the Selberg averages. More concretely, the averages of the bifundamental kernels (2.13) are given by the factorized formulas: hNAC [x]NBD[ pn(x) v]iu;v;N; =1=( 1)jBj+jDj zb4idfund([A;B];[C;D];u=2+v=2+N;u=2; v=2) ; CA4dCB4dCC4dCD4dG4AdB(u+v+2N )G4CdD(u) (2.20) where CA4d = (ArmA(i; j) + LegA(i; j) + 1) ; G4AdB(x) = x + Ai j + BT j i + 1 x Bi + j 1 AjT + i : Y zb4idfund A~; B~ ; ~a;~b; m = G4AdiBj (ai b j m) and ~a = (a; a), ~b = (b; b). This factorization means that expansion of the DF integrand in terms of the bifundamental kernels NAB indeed reproduces the Nekrasov decomposition. In the next section we will compute the q-deformed averages and show how to decompose them even further to obtain topological vertices. 3 Complete formulas for t = q In this section we esh out the basic formulas introduced in the previous section and incorporate q-deformation into our framework. After q-deformation, we obtain the natural identi cation of DF integral decompositions with topological vertices. We start with especially symmetric example of the four-point conformal block of q-Virasoro algebra and then consider multi-point blocks. We calculate q-deformed Selberg averages of the skew characters and identify the elements of the multilinear decomposition with topological string amplitudes. 3.1 Four point conformal block (t = q) The origin of the quadrilinear expansion of the four point conformal block is straightforward to see: two diagrams come from the character decomposition and two more represent the two integration contours in the DF integral (which becomes a sum in the q-deformed case [74]). The corresponding toric diagram is depicted in gure 4 and the four diagrams The DF representation is given by the sum over DF poles labelled by two partitions are denoted by A, B, R+ and R . R [94{96]5 (see appendix E for details): B4 = X R xR ;i = qR ;i i+N +1: (xR+ ) cross(xR+ ; xR ) (xR+ ); (3.1) (3.2) 5One should notice that since there are only N variables x ;i the number of columns in the diagrams R is not greater than N . This constraint is exactly parallel to a similar constraint in the topological strings, where it appears for quantized values of the complexi ed Kahler moduli, corresponding to N , i.e. Q = qN . 1 0 0 AT qn(N +1=2)q n X i=N++1 N +v X i=N +1 9 = ; 1 1 A The contribution of the q-Selberg measure (xju; v; N; q; q) = can be evaluated explicitly and is written as follows: (x) QiN=1 xiu Qvk=10(1 qkxi) =qjRj(2N+u+v 1) (RN)(q ) (RN+v)(q )=qjRj(u 1) (RN)(q ) (RN+v)(q ); (xRju;v;N;q;q) (x?ju;v;N;q;q) cross(x;y)=YY 1 N+ N i=1j=1 8 =exp<X :n 1 =X A;B X X R+;R A;B Using the identities xi yj 2 N v+ 1 Y Y j=1 n=0 n n pn+ 1 q nv+ 1 q n q n 1 N+ v 1 Y Y yj i=1 l=0 ( q n)+pn q n 1 qlxi 1 qnv 1 qn qN+ N jAj+jBj A@q (N++1=2)npn+ qn(1=2 i) A B q (N++1=2)npn and (AN)(x) denotes Schur polynomial in N variables xi. The cross contribution reads one gets where (3.3) is the open topological string amplitude for the resolved conifold. Pictorially SAB(Q) is given by one corner of the toric diagram from gure 4. It is also equal to the Chern-Simons (or WZW) S-matrix. identity A( pn) = ( 1)jAj AT (pn). Collecting all the contributions one obtains where pn = PiN=+1 xin, qn = PN j=1 yjn and we have employed the Cauchy identity and the B4= (x?ju+;v+;N+) (x?ju ;v ;N ) qN+ N jAj+jBjqjR+j(u++v++2N+ 1)+jR j(u +v +2N 1) (RN++)(q ) (RN+++v+)(q ) (N )(q ) (RN +v )(q ) (BN+)(qR++ ) (ANT )(q R R ) (AN++v+)(qR++ ) (BNT +v )(q R (RN)(q ) = ( 1)jRjq NjRj (RTN)(q ); (AN)(q R ) = ( 1)jAj (ATN)(qRT+ ) SR+B(q N+ )SART (qN )SRT B(qN +v )SAR+ (q N+ v+ ); B4= (x?ju+;v+;N+) (x?ju ;v ;N ) X X qN+ N jAj+jBjqjR+j(u++v++2N+ 1)+jR j(u 1) R+;R A;B Q2 QF R R+T QF A B QB Q2+ gauge theory with four fundamentals. Using the AGT relations (A.5) one immediately obtains the identi cation between the Kahler parameters of the toric Calabi-Yau, CFT and the gauge theory parameters: QB = qN+ N = QF = q2N++u++v+ 1 = qu Q1+ = q N+ v+ = qa+m1+ ; Q 1 = q N = qa+m2 ; q 2a+m2+ m2 ; 1 = q 2a 1; Q2+ = q N+ = qa m2+ ; Q 2 = q N +v = qa m1 : Let us state once more the result for the four-point q-deformed conformal block for t = q. This block can be simultaneously decomposed in two ways: DF integral and the decomposition in terms of the complete basis of states. Using the simplest choice of basis states (Schur functions) one gets a symmetric quadri linear decomposition in terms of characters. Moreover, this decomposition is naturally identi ed with the corresponding topological string amplitude, computed using the topological vertex technique. We now move on to describe the multipoint case. 3.2 Multipoint (t = q) In section 2.2.3 we understood the star-chain transformation for ordinary DF integrals. The q-deformed case goes along the same lines. Also, as in the four-point case above, we obtain a natural interpretation of the objects featuring in the decomposition from the point of view of the topological strings. Using the star-chain relation we rewrite the Vandermonde measure in the multipoint DF integral as a sum over chains of skew Schur functions. We consider the corresponding expansion of the k-point DF integral in terms of k q-deformed Selberg averages of skew Schur functions (cf. eq. (2.15)): Bk+2BU(1) = X k Y fY~ag a=1 * jaY~aj NYa 1;1Ya;1 [x]NYa 1;2Ya;2 pn 1 1 q nva + q n ua;va;Na;q;q (3.8) ? ? Z Ya;1 Ya;2 Ya 1;2 Ya 1;1 Q1; Q2; QF ; q; q ! = Ya;1 Ya 1;2 Ya;2 Q1 HJEP02(16)98 equal to q-deformed Selberg average of two bifundamental kernels as in eq. (3.8). where h: : :iua;va;Na;q;q denotes the q-Selberg average with the corresponding parameters and Y~0 = Y~k = (?; ?). Each average depends on two pairs of Young diagrams Y~a 1, Y~a and corresponds to the element of the toric diagram depicted on gure 5. Notice that a certain U(1) factor appears in the left hand site. It can be nicely eliminated in the four-point case, though not for higher multipoints. In the next section we show that each average in eq. (3.8) indeed reproduces the bifundamental part of the Nekrasov function. The whole sum thus becomes the Nekrasov function for linear quiver gauge theory, of the form depicted in gure 2. 3.3 Factorization of averages Let us check that the DF averages of the bifundamental kernel NAB[x] indeed factorizes. To do it we use the loop equations, which are given in appendix B. We obtain a formula for the average of four Schur polynomials: NAC[x]NBD pn(x) 1 q nv 1 q n ; (3.9) (3.10) (3.11) (3.12) and the de nition of G(AqB;t) and zb(qif;ut)nd are collected in appendix A. This indeed proves that the average of four Schur polynomials gives the bifundamental Nekrasov contribution. Identi cation with topological strings We would like to further decompose the q-Selberg average in eq. (3.9) to observe the structure of the corresponding topological string amplitude from gure 5. Notice that each average contains a product of two bifundamental kernels NAB[x]. This corresponds to the product of two four-point functions each having the form: ? RT Z C A Q2;q;q = ? A C Q2 RT =X( Q2)jW jCATW T?(q)CCW RT (q) W =Z ? ? ? ? Q2;q;q q C 2 A ( Q2)jCj( 1)jRj (RN)(q )N A(NC) q R ; (3.13) HJEP02(16)98 where Q2 = qN . Recall the expression for the q-deformed Selberg measure (3.3), which consists of two Schur polynomials. The product of two four-point amplitudes (3.13) therefore gives exactly the product of q-Selberg measure with two bifundamental kernels as in the average (3.8). More explicitly, gluing two amplitudes (3.13) one obtains the amplitude from gure 5, which is given by: R ? ? R R ? ? ? ? ? ? ? ? ? ? ? ? Z DC BA Q1;Q2;QF ;q;t RT =X( QF )jRjZ C R?T A Q2;q;q Z D ? B Q1;q;q =Z ? ? Q1;q;q Z ? ? Q2;q;q q C A+2 D B ( Q1)jCj( Q2)jDj X( QF )jRj (RN)(q ) (RTN v)(q )N A(NC)hq R iN D(TNBTv)hq RT i =Z ? ? Q1;q;q Z ? ? Q2;q;q q C A+2 D B ( Q2)jCjQj1Dj( 1)jBjq ( 21 +N)(jAj jCj+jBj jDj) X(QF q N v)jRj (RN)(q ) (RN+v)(q )N A(NC)hq R 21 N iN B(ND+v)h pn q R 12 N i =Z ? ? Q1;q;q Z ? ? Q2;q;q q C A+2 D B ( 1)jBj+jCjq(2N+ 12 )jCjq( 21 v)jDjq ( 21 +N)(jAj+jBj) Su;v;N;q;q N A(NC)[x 1]N B(ND) where Q1 = q N v, Q2 = qN , QF = qu+v+N 1 and Su;v;N;q;q is the q-Selberg integral without character insertions. Notice how in the last equality we have used the fact that the q-Selberg averages can be calculated as sums over partitions R with the measure given where given by6 jjM E(q;t)jj2 = CE (q; t) CE0 (q; t) = Y 1 (i;j)2E 1 qEi j+1tEjT i qEi j tEjT i+1 is the norm of Macdonald polynomials and generalized skew Macdonald polynomial is M A(qB;t=)EF (Qjpn; pn) = M E(q;t) n 1 1 q MF(q;t) n 1 1 q M A(qB;t)(Qjpn; p): One can immediately notice that for t = q HJEP02(16)98 Ne A(qB;q;)CD(u; v; N jx) = NAC [x]NBD pn(x) 1 1 qnv qn ; exactly reproducing the unre ned case (3.9). Generalized Macdonald polynomials are obtained from the kernel by forgetting about one of the two pairs of Young diagrams: Ne A(qB;t;)??(u;v;N jx)=MA(Bq;t) qu+1t 1 Ne ?(q?;t;)CD(u;v;N jx)=M C(qD;t) qu+v+1t2N 1 pn; pn 1 q nv 1 t n : t n q p n 1 tn 1 (t=q)n qnv;p n+ (t=q)n qnv 1 tn One can also get the product of two ordinary Macdonald polynomials by setting the two \cross-wise" diagrams to be empty: Ne ?(qB;t;)C?(u; v; N jx) = M B(q;t) p n + The most remarkable property of the generalized bifundamental kernel is that its qSelberg average is factorized into a product of simple monomials (B.10). More concretely, it is given by the bifundamental contribution to the Nekrasov function (hence the name of the kernel). Schematically DNe A(qB;t;)CD(u; v; N jx) E Averages of this kind can be obtained by using the loop equations for q-Selberg integral (or (q; t)-matrix model). The full form of the average (4.7) and technical details are summarized in appendix B. Thus, we prove that generalized bifundamental kernel is indeed the relevant object to be averaged to get the chain-like decomposition of the DF integral. Let us now try to nd similar objects in re ned topological strings. 6One can ask why we \subtract" ordinary Macdonald polynomials from the generalized ones to obtain the skew polynomials. In fact eq. (B.10) is independent of the concrete choice of \subtracted" polynomials as long as they constitute a complete system and the Cauchy identity holds. We will return to this issue in section 4.4. ; (4.4) (4.5) (4.6) (4.7) 4.3 We start from the re nement of the basic building block, i.e. the four-point amplitude (3.13) and set preferred direction to be vertical. The general re ned amplitude depending on four Young diagrams is given by eq. (C.1). For our purposes we need the following specializations: ? R Z t pn(t ) pn Q2 r q t t CT=E pn(q R t ) + pn Q2 r q t t r t Q 1 q 2 t pn(qRt ) q n tn pn ; (4.8) and R ? Z t q AT 1 1 t n qn R t q BT CT Q2 R ? t ? X E AT=E DT Q1 q ? ? ? t ? = Z ? Q1; q; t q jjRjj2 t jjRT2jj2 ( Q1)jDjM (t;q) 2 RT pn(q ) pn Q1 r t q q X F BT=F DT=F q n tn pn(q R 1 1 t n qn r q r t t q t t t ) pn Q 1 1 pn(qRt ) pn Q1 ; (4.9) To make contact with DF integrals we make the identi cation q qt Q1 = t1 N q v 1, q qt Q2 = tN . and (4.9) each give precisely one \half" of the q-Selberg measure (4.1). We immediately notice that two Macdonald polynomials in eqs. (4.8) Skew Schur functions in eqs. (4.8), (4.9) can be rewritten through the discrete q-Selberg \integration" variables xR;i = qRi+1tN i in the following way: Z CT AT tN+ 12 q 21 ;q;t =Z ? ? tN+ 12 q 21 ;q;t ( 1)jAj q 21 t N jAj qt2N 12 jCj q jjR2jj2 t jjRT2jj2 M R(q;t) pn(t ) pn tN t X E t jEj A=E 1 t 1 qn Z DT BT t 21 N q v 12 ;q;t =Z ? ? t 21 N q v 12 ;q;t ( 1)jBj q 1t 12 N jBj q v+ 32 t 1 jDj q jjRjj2 t jjRTjj2 M (t;q) pn(q ) pn t 2 2 RT 1 N q 1 v q X F t jF j q B=F n 1 t 1 qn pn(xR;i)+ t 1 t n t n q pn(xR;i) 1 t 1 t n C=E (p n(xR;i)); (4.10) HJEP02(16)98 D=E t n q 1 qnv 1 tn : (4.11) ? R ? ? R ? ? ? =X( QF )jRjZ CT AT tN+ 12 q 21 ;q;t Z DT BT t 21 N q v 12 ;q;t R ? Su;v;N;q;tZ ? ? tN+ 12 q 21 ;q;t Z ? ? t 21 N q v 12 ;q;t (x?ju;v;N;q;t) ? ? ( 1)jAj+jBj q 21 t N jAj qt2N 12 jCj q 1t 12 N jBj q v+ 32 t 1 jDj Let us glue two four-point functions (4.8) and (4.9) to obtain the q-Selberg average: Z DT t R t q q Q2 CT QF t q Q1 t q BT ? ? ? R where Su;v;N;q;t is the q-Selberg integral without insertions. We have used the identity (4.1) HJEP02(16)98 to obtain the q-Selberg measure evaluated at discrete points labeled by R from the product of two Macdonald functions, and made the identi cation QF = qu+v+ 12 tN 2 . The sum 3 over representations R corresponds to the q-Selberg average as shown in appendix E. Notice also that the two pairs of Schur functions under the average are skew, just as in the de nition of the bifundamental kernel (2.13), and should be summed over the \intermediate" representations E and F . At this point one observes that the expression under the average is not the generalized xi ! q i bifundamental kernel (4.3), which would give the bifundamental contribution as an average. Instead it is simply a product of two pairs of Schur polynomials. However, closer look n reveals that the arguments of the Schur polynomials exactly match (up to the factors 11 qtn ) the arguments of the generalized bifundamental kernel after the elementary transformation 1 vx 1 as can be seen e.g. from eq. (B.11). If we could expand the generalized Macdonald polynomials in terms of Schur polynomials with the same arguments, this would produce a transformation between the expression under the average in eq. (4.12) and the average of generalized bifundamental kernel (4.3). This average (but not the original one in eq. (4.12)) in turn is given by the bifundamental Nekrasov function, as shown in eq. (4.7) and in appendix B. Going further, we notice that the bifundamental Nekrasov contribution is given by the re ned topological amplitude with horizontal preferred direction, whereas the preferred direction is vertical in the diagrams (4.8), (4.9) and thus in the average (4.12). This leads us to the relation between horizontal and vertical slicing of the re ned amplitude. To get this relation we will need to introduce generalized Kostka functions, which are the expansion coe cients of the generalized Macdonald polynomials in terms of Schur polynomials. 4.4 Generalized Kostka functions The basis of generalized Macdonald polynomials can be reexpanded in terms of Schur M A(qB;t)(Qjpn; sn) = X KACBD(Q; q; t) C (pn) D(sn); C;D MA(Bq;t)(Qjpn; sn) = jjM A(q;t) 2 jj jjM B(q;t)jj2 X KACBD(Q; q; t) C 1 1 q n tn pn D 1 1 (4.13) q n tn sn ; (4.14) * q A=E n 1 qn C=E(p n(xR;i)) B=F D=E pn(xR;i) 1 t n t 1 t 1 qn pn(xR;i)+ t 1 t n t n 1 qnv + q 1 tn ; (4.12) C;D analogy with the ordinary Kostka polynomials,7 de ned as where the coe cients KCADB(Q; q; t) can naturally be called generalized Kostka functions by As an explicit example we give here generalized Kostka functions for the rst level: M A(q;t)(pn) = X KAB(q; t) B(pn): B KACBD(Q; q; t) jAj+jBj=jCj+jDj=1 = KACBD(Q; q; t) jAj+jBj=jCj+jDj=1 = 1 0 ! 11 Qqt 1 ; 1 0 1 qt ! (1 Q) : 1 (4.15) (4.16) (4.17) q B 1 1 t n qn rn : (4.18) 1 qn We would like to use generalized Kostka functions to transform the two pairs of Schur polynomials inside the average in eq. (4.12) into two generalized Macdonald polynomials and thus obtain the bifundamental kernel. The Schur functions in eq. (4.12) are skew and are summed over the \intermediate" representations E and F , so we use the following identity, which is the consequence of the Cauchy completeness theorem, to transform the sum over Schur polynomials into that over Macdonald ones: X E A=E (pn) B=E X E 0 X 1 1 n 1 n 1 M E(q;t) 1 1 t qn rn q 1 1 q A(pn) B 1 1 A(pn) M E(q;t) t n qn rn n 1 1 Now the sum over \intermediate" representations in the last line includes the combination of Macdonald functions depending on the conjugates of the power sums pyn = n which exactly reproduces the sum in the de nition of generalized bifundamental kernel (4.3). The Schur polynomials A, B remaining in eq. (4.18) are no longer skew and can be transformed into generalized Macdonald polynomials using the generalized Kostka functions, as in eqs. (4.13), (4.14). 7Our de nition of generalized Kostka functions can be modi ed slightly to turn them into polynomials. This is achieved by using a di erent normalization of generalized Macdonald polynomials, called MfA(qB;t) in eqs. (19), (20) in [74]. Eventually, we obtain the connection between re ned topological string amplitude with vertical slicing and bifundamental Nekrasov function: KWCD1W2 (qQF Q1) 1; q; t Z CT DT BATT Q1; Q2; QF ; q; t KY1AYB2 (qQF Q2) 1; q; t ? ? ! X A;B;C;D 2 Q17!t 21 N q v 12 3 Su;v;N;q;t Z Q3 t q QB;2 t q R2 Q2 t q QB;1 Q1 t q t q R1 ? t 21 N q v 12 ; q; t ? ? ? (x(?)ju; v; N; q; t) ( 1)jAj+jBj qv 23 t N jAj qvt2N 12 jCj qv 2t 12 N jBj q 12 t 1 jDj D NeAB;CD(u; v; N jq1 vxR1)E (q;t) zbifund [A; B]; [C; D]; u v 12 2 N+ ; u 21+ ; v2 CA0(q; t)CB0(q; t)CC0 (q; t)CD0(q; t)G(AqB;t)(q u v 1t1 2N )G(DqC;t)(qu+1t 1) Note that generalized Kostka functions in this formula depend on the \distance" (in the sense of Kahler parameters) between the pairs of horizontal external legs of the toric diagram. Let us also point out that our Kostka functions are q-deformation of the coe cients of the abelianization map acting on the instanton moduli space. 4.5 Horizontal slicing. DF representation and spectral dual Nekrasov function Let us glue three pieces (4.8) together horizontally to obtain the DF integrand for ve-point conformal block and its AGT dual | U(2)2 quiver gauge theory. The resulting amplitude is equal to \half" of the total DF measure DF(x(1); x(2); x(3)) evaluated at discrete points xa;i = qRa;i+1tNa i. We get: Z ? ? ? ? ? Q1;Q2;Q3;QB;1;QB;2;q;t (4.19) Z ? Y 1 2 N1 N2 YY m=0 i=1j=1 4 4 Y 2 2 N1 Y m=0 i=1 1 ? tNa+ 12 q 21 ;q;t q jjR2ajj2 t jjRaTjj2 M R(qa;t) pn(t ) pn tNat 2 1 1qm xR2;j N2 N3 Y Y xR1;i j=1k=1 1q m 1 2q m Y xR2;j i=1k=1 1 2qm xR3;k xR1;i N1 N3 YY 2q m xR2;j 1 1 3 5 : 3 5 (4.20) Three Macdonald polynomials in the second line can be thought of as a \half" of the three q-Selberg measures, corresponding to three integration contours in the DF representation. Moreover, the measure (4.20) can be evaluated explicitly and also gives the \half" of the spectral dual Nekrasov function with gauge group U(3), cf. (A.1) (the other half of the factors comes from the lower half of the diagram): HJEP02(16)98 Z ? ? ? ? ? Q1;Q2;Q3;QB;1;QB;2;q;t q jjR1jj22+jjR3jj2 tjjR1Tjj2+ jjR2Tjj2 Z?(Q1)Z?(Q2)Z?(Q3)Z?(QB;2)Z?(QB;2Q2Q3)Z?(QB;1) 2 CR01(q;t)CR02(q;t)CR03(q;t)Z? q qt QB;2Q2 Z? q qt QB;2Q3 Z? q qt Q1QB;2 (Q2B;1QB;2Q2)jR1j( QB;2)jR2jZ?(Q1Q2QB;1)Z?(QB;1QB;2Q2)Z?(Q1Q2Q3QB;1QB;2) Z? q qt Q2QB;1 Z? q qt Q1Q2QB;1QB;2 Z? q qt QB;1QB;2Q2Q3 G(Rq1;t?) G(Rq2;t?) r q r q t Q1 G(Rq1;t?) t Q2 G(Rq2;t?) r q r q t QB1;1 G(Rq1;t?) t QB1;2 G(Rq2;t?) r q t r q (QB;1QB;2Q2) 1 t Q1Q2QB;1 G(Rq3;t?) q qt Q3 G(Rq3;t?) q qt QB;2Q2Q3 G(Rq3;t?) q qt Q1Q2Q3QB;1QB;2 G(Rq3;tR)2(QB;1Q3)G(Rq2;tR)1(Q2QB;1)G(Rq3;tR)1(QB;1QB;2Q2Q3) (4.21) ? where Z?(Q) = Z ? Q; q; t . This is a manifestation of the spectral duality for Nekrasov functions [75{78]: while vertical slicing of the toric diagram gives Nekrasov function for U(2)3 quiver gauge theory, the horizontal slicing yields its spectral dual | gauge theory with a single U(3) gauge group. In the language of conformal blocks [74] this means that both the Jackson integral and the sum over complete basis of generalized Macdonal polynomials have the form of Nekrasov decompositions, which are spectral dual to each other. For re ned topological strings only one Nekrasov decomposition can be obtained for a given choice of preferred direction | the cut should dissect the preferred edges. If one cuts along a di erent direction the amplitudes do not reproduce the Nekrasov functions, as can Nekrasov/generalized Macdonald decomposition (1.10) Vertical slicing, horizontal preferred direction (4.21) Conformal block Bk, gure 1 q-Selberg measure (4.1) DF integral (1.1) Decomposition in Schur polynomials Rotation of preferred direction by 2 Topological string Closed string amplitude on toric CY Ztop, gure 3 Two four-point conifold amplitudes (3.13), (3.14) Horizontal slicing, vertical preferred direction (3.16), (4.20) Vertical slicing, vertical preferred direction (4.12) Generalized Kostka function (4.16), (4.17) be seen in eq. (4.12). However, there is still a way to see the dual decomposition: preferred direction can be changed with the help of generalized Kostka functions (4.16), (4.17). HJEP02(16)98 5 Conclusions and discussion We have investigated the connection between q-deformed conformal blocks and topological strings. This connection arises in the following way. Due to the AGT relation conformal blocks are equal to Nekrasov partition functions, which can be obtained by the geometric engineering technique, as compacti cations of type IIA strings (or, more generally, Mtheory) on toric CY threefold. String partition function on the threefold is equal to partition function of topological strings. We obtain an explicit dictionary between the objects in CFT and elements of the corresponding toric diagram, summarized in table 1. For the case of t = q we introduce the bifundamental kernel (2.13), compute its q-Selberg averages (2.20) and show that they reproduce Nekrasov partition function. We also study spectral duality of conformal blocks and generalize the statements of [74] to multipoint blocks. Most importantly, we study the ever-troublesome case of t 6= q, where we introduce generalized bifundamental kernel (4.3). We compute the average of the generalized kernel | it satis es the most general of all the so far encountered \factorization of averages" type identities (B.10) | and is again given by Nekrasov function. We interpret the change of preferred direction of re ned topological strings as a change of basis between generalized Macdonald and Schur polynomials, which is performed by generalized Kostka functions (4.13), (4.14). Of course the expansion we have considered is not limited to the case of U(2) gauge theories and Virasoro conformal blocks. U(N )/WN story goes along the same lines. In this setting q-Selberg integrals are replaced by the AN q-Selberg integrals, their measure being given by the product of several basic building blocks (C.1). Generalized Macdonald polynomials, bifundamental kernels and Kostka functions can also be found for the U(N ) case. It would be extremely interesting to understand the relation of the character/topological string decomposition of conformal blocks from the point of view of Seiberg-Witten integrable systems. One relation is provided by the quantum spectral curve for DF integrals [97], which in Nekrasov-Shatashvili limit reproduces the quantum spectral curve (Baxter TQ equation) of the relevant Seiberg-Witten system, the XXZ spin chain. Of course, a general method to obtain quantum spectral curves from the toric data is desirable. Let us also mention that in this way one can study the mirror symmetry between the B-model CY, encoded in the spectral curve and the A-side toric CY described by the topological vertex formalism. In the four dimensional limit generalized Kostka polynomials coincide with the coe cients of the abelianization map acting on the xed points in the cohomology of the instanton moduli space. Explicit combinatorial expressions for these coe cients were obtained in [98]. It would be interesting to understand these formulas from the point of view of re ned topological strings. The product of generalized Kostka matrices turns out to be an interesting algebraic object. We can reason in the following way. Let us rst expand generalized Macdonald polynomials in terms of products of Schur polynomials using the Kostka matrix. Then we exchange the two Schur polynomials and apply the reverse Kostka transformation. Thus we obtain another set of generalized Macdonald polynomials. However the two sets are clearly related. Recall that generalized Macdonald polynomials are eigenfunctions of the operator H1gen = (H1), which is given by the Ding-Iohara coproduct, acting on trigonometric Ruijsenaars Hamiltonian H1. The second set of generalized Macdonald polynomials is obtained by acting on the same Ruijsenaars Hamiltonian with the opposite coproduct As in any quasitriangular Hopf algebra, there is an R-matrix performing the transformation from one coproduct to the other. The two sets of generalized Macdonald polynomials are therefore also related by the same R-matrix. This is the K-theoretic version of the instanton R-matrix [99, 100] with spectral parameter being the parameter of generalized Macdonald polynomials. The implications of this observation and the relation between toric CY and integrable systems will be studied elsewhere. Acknowledgments We thank Andrey Smirnov for clarifying the concept of instanton R-matrix to us. We would also like to thank the anonymous referee for many insightful comments, which led to serious improvement in the presentation of the results. Our work is partly supported by grants NSh-1500.2014.2, 15-31-20484-Mol-a-ved and 15-31-20832-Mol-a-ved, by RFBR grants 13-02-00478, by joint grants 15-52-50034-YaF, 15-51-52031-NSC-a, by 14-01-92691Ind-a and by the Brazilian National Counsel of Scienti c and Technological Development. Y.Z. is supported by the \Dynasty" foundation stipend. A Five-dimensional Nekrasov functions and AGT relations The Nekrasov partition function for the U(N ) theory with Nf = 2N fundamental hypermultiplets is given by ~ A ZN5de;kU(N)=X jA~j zfund(A~; m~+;~a)zfund(A~; m~ ;~a) zvect(A~;~a) ~ A =X jA~j QiN=1QfN=1fA+i (qmf++ai )fAi (qmf +ai ) zvect(A~;~a) (A.1) where fA (qx) = Q(i;j)2A 1 q xt (i 1)q (j 1) , zvect(A~; ~a) = QiN;j=1 G(Aqi;At)j (qai aj ) and G(AqB;t)(qx) = Y 1 + ; v+ = n+ = m2+ ; qxqj 1t1 i = fA (q x); qxq1 jti 1 = fA+(qx); u = v = 1 + m1 2a ; m2 ; n = a + m2 ; (A.2) (A.3) (A.4) (A.5) b2, (A.6) a2 = a. Masses ma, vevs ai, radius R5 of the fth dimension and 1;2 all have dimensions of mass. In this paper we set the overall mass scale so that 1 = 2 = 1 and q = e R5. The t parameter in Macdonald polynomials is related to q by t = q More generally, one can consider quiver gauge theories with gauge groups U(N )k and bifundamental matter hypermultiplets as shown in gure 2. The corresponding Nekrasov ~ Ya ZN5de;kU(N)k =X j1Y~1j jY~kj Y YfY+1;i qmf++a1;i k N N f=1i=1 zbifund Y~k 1;Y~k;~ak 1;~ak;mbifund;k 1 zvec(Y~k;~ak) f=1i=1 where the bifundamental contribution is given by zbifund(Y~ ; W~ ; ~a;~b; m) = QiN=1 QjN=1 G(Yqi;Wt)j qai bj m . B Loop equations for matrix elements We would like to compute the q-Selberg average of a function f (pn) which is polynomial both in pn and p n. To do this we use an improved version of the loop equations obtained in [74]. Concretely, we use the fact that the q-integral of a total q-derivative vanishes: 1 1 zbifund Y~1;Y~2;~a1;~a2;mbifund;1 N N Y YfYk;i qmf +ak;i We will write a instead of ~a = (a; a) for N = 2. The AGT relations for N = 2 are: Z 1 0 { 32 { 1)g(x) = 0; (B.1) if g(1) = g(0) = 0. We write down the following judiciously chosen total derivative: Z N dqN x X 1 i=1 xi 2 1)xi 4 z xi q Y xi txj Y xi j6=i xi N xj k=1 v 1 a=0 x u Y(qaxk k 1) q;t(x)f (x)5 = 0: The di erence with [74] is that we assume f (pn) to be a Laurent polynomial, i.e. the function of pn for n both positive and negative. Writing down the action of the q-derivative in eq. (B.2) we get the following identity: * 1 1 q z +t2N 1qu+1 qv 1 1 z * X 1 t n qnz npnA n 1 Res =04 f (pn+(qn 1) n)exp@ 0 n 1 0 X 1 tn n n n X 1 t n npnA5 n 1 13 0 X 1 t n n 1 n n 13 p nA5 z npnA 1 1 tN f (pn) z f (pn) =0: p n, e.g.: p 1 + v 1 t More generally MA p n+ MB(pn) tn=qn qnv 1 tn The expression in the average in eq. (B.3) looks complicated and not too suitable for explicit calculations. However, expanding in powers of z and taking f (pn) to run over products of pn (with n both positive and negative) one gets the recurrence relations determining the averages of any symmetric function. Let us note that the expansion in positive and negative powers of z lead to the same recurrence relations as it should. In addition to the usual factorized formulas for the generalized Macdonald polynomials these equations give the averages of the products of two Macdonald polynomials, one in pn the other in p1 t N+1(1 tN )(1 qutN 1)(1 q1+utN )(1 q1+vtN 1 q2+u+vt2N 2) Y tiq 1(1 qArmA(i;j)tLegA(i;j)+1) 1 Y q1+j+utN 1(qArmB(i;j)tLegB(i;j)+1 1) 1 G(Aq?;t)(q u)G(Bq?;t)(qu+v+2t2N 2) ; (B.2) One can also transform the averages of positive power sums pn to negative ones p n and vice versa: which leads to hf (xi)iu;v;N;q;t = hf (q1 vxi 1)i u v 2+2 2 N;v;N;q;t (B.6) MA pn + q nv 1 t n q(jAj jBj)(1 v) MA p n + We remind the result from [74]: M A(qB;t) qu+v+1t2N 1 pn; pn MB(p n) tn=qn qnv 1 tn MB(pn) 1 q nv 1 t n = ( 1)jAjq (v+1)jAj (u+2v+3)jBj+P(i;j)2A j+2 P(i;j)2B jtjCj (2N+3)jBj P(i;j)2B i We also give an alternative average of generalized Macdonald polynomial (notice the difference in shifts of the power sums pn) M A(qB;t) q u 1t p n + 1 tn ; p n = ( 1)jAjq 2jBj+ujAjtjBj jAjtP(i;j)2B i+2 P(i;j)2A iq t n 1 (t=q)n 1 tn P(i;j)2A j u;v;N;q;t f A tN qu f A t1 N q v 1 f B tN+1q 1 f B t2 N q u v 2 CA0(q; t)CB0(q; t)G(BqA;t) (qu+1t 1) : (B.9) which gives all the averages above as special cases:8 Finally, we were able to nd the most general factorized formula for the average of two generalized Macdonald polynomials (or generalized bifundamental kernel NeA(qB;t;)CD(u; v; N jx)), : (B.7) u v 2+2 2 N;v;N;q;t hNeA(qB;t;)CD(u; v; N jx)iu;v;N;q;t * X E;F t jEj+jF j MA(Bq;=tE)F M C(qD;t=)EF qu+1t 1 qu+v+1t2N 1 pn; pn jjM E(q;t) 2 jj jjMF(q;t)jj2 1 t n q 8We have checked this formula up to the third level. = ( 1)jBj+jCjq 2jAj (v+1)jCj+ujBj (u+2v+3)jDjtjAj jBj+jCj (2N+3)jDj p n 1 tn 1 (t=q)n qnv; p n + 1 t n q nv + u;v;N;q;t P R Z A B Q; q; t = Q t q t R where CA0(q; t) = Q(i;j)2A(1 qAi jtAjT i+1). hNeA(qB;t;)CD(u; v; N jx)i u v 2+2 2 N;v;N;q;t = hNeA(qB;t;)CD(u; v; N jq1 vx 1)iu;v;N;q;t = q(v 1)(jAj+jBj jCj jDj) Re ned open string amplitude depends on four Young diagrams A, B, R and P and the Kahler parameter Q of the conifold. HJEP02(16)98 CA0(q; t)CB0(q; t)CC0 (q; t)CD0(q; t)G(AqB;t)(qu+1t 1)G(DqC;t)(q u v 1t1 2N ) ; (B.10) Again, one can use the symmetry (B.6) to write eq. (B.10) in an alternative form: * X E;F t jEj+jF j jjM E(q;t) 2 jj jjMF(q;t)jj2 MA(Bq;=tE)F M C(qD;t=)EF q u v 1t1 2N q u 1t p n; p n t n pn 1 (q=t)n 1 t n ; pn + q nv 1 t n t n 1 1 tn qnv + u;v;N;q;t : (B.11) C Open topological string amplitude on resolved conifold In this appendix we write down the basic building block of the toric diagrams related to 5d quiver gauge theories. It is given by an open re ned topological string amplitude in the resolved conifold background depicted in gure 7. Using the IKV re ned topological vertex one gets the following answer for this amplitude: Z A B Q;q;t C ? Q;q;t q jjRjj2 2 jjPjj2 t jjPTjj2 2 jjRTjj2 q jAj 2jBj M R(q;t)(t )MP(tT;q)(q )G(RqP;t) r q X( Q)jCj AT=CT pn(t q R) pn B=C pn(q t P T ) pn r qt Qt q P r qt Qq t RT ? Q; q; t = Qi;j 1 1 Qqi 21 tj 21 is the closed re ned string amplitude One can perform a op transformation on the conifold geometry. We employ the following symmetry of the closed string amplitude: Z ? Q; q; t = Z ? Q 1; q; t : HJEP02(16)98 The opped open string amplitude is related to the original amplitude with Kahler (C.1) (C.2) (C.3) (C.4) (D.1) (D.2) (D.3) where Z on the conifold. parameter reversed: Z opped A B Q;q;t = R P where MY(t;q)(pn) = ( 1)jY jhY T (q; t)M (q;t) Y T hY T (q; t) = CY0 (q; t) = CY0 T (q; t) CY0 (t; q) Y 1 (i;j)2Y 1 Y (i;j)2Y 1 qYi j tYjT i+1 : 1 1 q n tn pn ; tYi j+1qYjT i tYi j qYjT i+1 ; Q 1pq 1t 1 112 q jjRjj2 jjPjj2+j2jATjj2 jjBTjj2 t jjRTjj2 jjPTjj22+jjAjj2 jjBjj2 q jAj 2jBj ( Q)jRj+jP j+jAj+jBjZ A B Q 1;q;t : P R The multipliers in eq. (C.3) combine with the change of framing in the adjacent edges induced by the op. The answer for any closed string amplitude, which includes the opped part is given simply by Z opped(Q; Qadjacent; Qi) = 1 Q 1pq 1t 1 12 Z(Q 1; QQadjacent; Qi); where in the right hand side the original Kahler parameter of the conifold is reversed and the Kahler parameters of the two-cycles adjacent to the opped conifold are shifted by Q. D Useful identities One has the following identity for the power sum symmetric functions pn(x) = P i 1 xin: pn(qY t ) = t n=2 1 1 n tn pn(t Y T q ) where i = 12 i and Y is a Young diagram. Macdonald polynomials satisfy the following \transposition" identities: Combining eqs. (D.1) and (D.3) we get the identity, which will be useful in re ned topological string computations: M (t;q) pn(q t ) pn(Qt ) = ( 1)j jh T(q; t)M (qT;t) pn(t T q ) pn(Qq ) : (D.4) The following identity involving Nekrasov functions is also useful: YN1 YN2 1 Qqj Wi 12 ti YjT 21 1 Qqj 21 ti 21 G(YqW;t)(q 21 t 21 Q) G(Yq?;t)(Qq 21 tN1 21 )G(?qW;t)(QqN2) ; (D.5) and in particular for N1;2 ! 1 (we assume jqj; jtj < 1): One can exchange the diagrams in Nekrasov function using the identity i;j 1 Qqj Wi 12 ti YjT 21 1 Qqj 21 ti 21 = G(q;t) Y W r q Q : = ( Q)jAj+jBjq jjBjj2 2 jjAjj2 t jjATjj22 jjBTjj G(AqB;t) r q Q 1 ; G(BqA;t) r q Q where jjRjj2 = Pi Ri2. G(Rq?;t) r q G(?qP;t) r q and also t When one of the diagrams is zero, there is a nice expression in terms of Macdonald Q = Y (1 Qqj 21 t 12 i)= Q = Y (1 Qq 21 jti 21 )= M R(q;t) 1 Qnq n2 t n2 t n2 t 2 n M R(q;t) t n21 t n2 MP(tT;q) 1 Qnt n2 qn n2 q n2 q 2 MP(tT;q) q n2 q 2 1 n M R(q;t) pn(t ) pn Qp q t t M R(q;t)(t ) MP(tT;q) pn(q ) pn Qq t q q MP(tT;q)(q ) (D.6) (D.7) ; (D.8) ; (D.9) (D.10) M R(q;t) t t CR0(q; t) Dotsenko-Fateev integrals as contour and Jackson integrals In this appendix we show that the q-deformed Dotsenko-Fateev integrals can be understood both as the contour integrals (as in [60, 61, 94{96]) or as Jackson integrals, i.e. discrete sums (as in [74, 91]). More concretely, we show that in both description the DF representation essentially reduces to the sum over (tuples of) Young diagrams. Let us rst consider the contour integral description for the q-deformed (M + 2)-point Virasoro conformal block on a sphere: BM+2 BU(1) = I C1;:::;CM dN1x dNM x (q;t)(x)Vu(x; 0)Vva(x; za); (E.1) where the contour Ca encircles the points xi = zaqmtn for n; m Vv(x; z) = Y YN 1 k 0 i=1 1 q k v z qk xzixi ; Vu(x; 0) = Y xiu; N i=1 0, and t = q , (q;t)(x) = Y Y N i6=j k 0 1 1 and N = PM of Young diagrams Ra, a = 1; : : : ; M , i.e. xa;i = xa;i(R~ ) = zaqRa;i tNa i, i = 1; : : : ; Na so a=1 Na. As shown in [94{96], the poles of the integrand are labeled by M -tuples that the integral is reduced to the sum over residues. One can compute the ratio of the residues corresponding to the given set of diagrams Ra and the set of empty diagrams tqqkkxxxxjiji ; (E.2) HJEP02(16)98 = llnnqt , u and va are positive integers, so that the i=1 ! N 1 Yxi (1 N) Y Y xi qkxj : i6=jk=0 =Resx=x(?~) (q;t)(x)Vu(x;0)Vva (x;za) X R1;:::;RM Resx=x(R~) (q;t)(x)Vu(x;0)Vva (x;za) Resx=x(?~) (q;t)(x)Vu(x;0)Vva (x;za) The rst term is an inessential normalization constant N , which can be calculated separately. One can evaluate the ration of the residues by simply evaluating the integrands at the poles: BM+2 BU(1) = N X R1;:::;RM (q;t)(x(R~ ))Vu(x(R~ ); 0)Vva (x(R~ ); za) : (q;t)(x(?~))Vu(x(?~); 0)Vva (x(?~); za) Since there are only Na variables xa;i(R~ ), the sum in eq. (E.4) is actually over the partitions Ra having at most Na columns. The ratio of the integrands turns out to be given by the Nekrasov formula for the SU(M ) gauge theory with 2M fundamental hypermultiplets depending on the M -tuple of partitions R~ [94{96]. The vacuum moduli and masses of the gauge theory are related to the momenta of the primary elds in the conformal block, which can be expressed in terms of u, va and Na. In particular, since Na are integers, the gauge theory is at a particular point in the vacuum moduli space, where ak + mi are integers. At this point the contributions of the fundamental hypermultiplets vanish for all partitions Ra having more than Na columns, which conforms with the DF result. Now we consider the Jackson integral version of the DF representation: (q;t)(x)Vu(x;0)Vva (x;za) The Jackson integral is de ned as follows We assume that the parameters integrand turns into a (Laurent) polynomial: Vv(x;z)=( z)vq v(v2+1) YNxi v Y v 1 i=1 1 qk+1 xi ; z BM+2 BU(1) = Z zM dNM x q (q;t)(x)Vu(x; 0)Vva (x; za); Z z1 dqN1 x 0 Z a 0 dqxf (x) = (1 q) X aqnf (aqn): n 0 In this case all the integrals are well-de ned geometric progressions, and evaluating them amounts to evaluating the integrand at discreet points xa;i(na;i) = zaqna;i with na;i 2 N: X na;i 0 (q;t)(x(~n))Vu(x(~n); 0)Vva (x(~n); za); One immediately notices that some terms in the sum vanish, e.g. if na;i = na;j , then (q;t)(x) = 0. More generally, for (q;t) not to vanish, the corresponding na;i and na;j should di er at least by . This gives rise to the following form of na;i for nonvanishing xi;a(Ra) = zaqRa;i+(Na i) ; (E.9) where Ra are Young diagrams (we have used the symmetry of the integrand and chosen a particular ordering of xa;i). The sum in eq. (E.8) is reduced to the sum over M -tuples of Young diagrams. Of course one can rewrite this sum using the same trick as in eq. (E.3): (q;t)(x(R~ ))Vu(x(R~ ); 0)Vva (x(R~ ); za) (q;t)(x(?~))Vu(x(?~); 0)Vva (x(?~); za) (q;t)(x(R~ ))Vu(x(R~ ); 0)Vva (x(R~ ); za) : (q;t)(x(?~))Vu(x(?~); 0)Vva (x(?~); za) (E.10) X R1;:::RM X R1;:::RM We arrive at the same sum over Young diagrams Ra (each with no more than Na columns) as in eq. (E.4), obtained from contour integrals. Thus, the two approaches to the DF integrals give exactly the same decomposition, leading to a sum over Young diagrams. In the main sections of the paper we relate this sum to the sum in the topological vertex formalism. Open Access. 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A. Morozov, Y. Zenkevich. Decomposing Nekrasov decomposition, Journal of High Energy Physics, 2016, 98, DOI: 10.1007/JHEP02(2016)098