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

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We find new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix model averages, which are themselves consequences of Ding-Iohara-Miki constraints. Factorized expressions for generalized Macdonald polynomials are identified with refined topological string amplitudes containing a toric brane on an intermediate preferred leg, surface operators in gauge theory and certain degenerate CFT vertex operators.

Section: Jhep09(2017)070mentioning

confidence: 99%

Refined toric branes, surface operators and factorization of generalized Macdonald polynomials

Zenkevich

2017

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“…2 Notice a slight change in the notation compared to [86]. We are now writing the generalized Macdonald polynomials as functions of the variable u 1 u 2 , which we call Q, whereas in [86] we denoted u 2 u 1 as Q.…”

Section: Jhep10(2016)047mentioning

confidence: 99%

“…Since the generalized Macdonald polynomials are actually known explicitly in many cases [65,66,[84][85][86], one can just use (1.8) to evaluate the first blocks of the R-matrix, and then promote these examples to the general formula. This is a much simpler way to get explicit expressions as compared with deducing them from the universal R-matrix [85,87], as was suggested in [88,89], and this will be the approach we adopt here.…”

Section: Jhep10(2016)047mentioning

confidence: 99%

“…Moreover, one can usually obtain two such descriptions related by the action of the spectral duality: either as a (k + 2)-point W N -block or as an (N + 2)-point W k -block, the corresponding toric diagrams being related to each other by π 2 rotation. The existence of two coinciding conformal blocks of different kinds is related to the AGT duality as shown in [86].…”

Section: Refined Topological Strings and Rt T Relationsmentioning

confidence: 99%

“…. We conform with the previous notation and denote the specially normalized polynomials by M AB as in [86], eq. (19).…”

Section: Jhep10(2016)047mentioning

confidence: 99%

See 2 more Smart Citations

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

Awata

Kanno

Миронов

et al. 2016

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R-matrix is explicitly constructed for simplest representations of the DingIohara-Miki algebra. Calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the R-matrix structure and the structure of DIM algebra intertwiners, i.e. of refined topological vertices and show that the R-matrix is diagonalized by the action of the spectral duality belonging to the SL(2, Z) group of DIM algebra automorphisms. We also construct the T -operators satisfying the RT T relations with the R-matrix from refined amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q-deformation of the reflection matrices of the Liouville/Toda theories.

Decomposing Nekrasov decomposition

Морозов

Zenkevich

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 functionsthis 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. 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.