Decomposing Nekrasov Decomposition

Abstract

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.

29 pages, 7 figures

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• A slow review of the AGT correspondence
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• Rectangular superpolynomials for the figure-eight knot
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• A non-torus link from topological vertex
• Asymmetric Dark Matter Models and the LHC Diphoton Excess
• q-Vertex Operator from 5D Nekrasov Function
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• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
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• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• Hopf superpolynomial from topological vertices
• Singular Vector of Ding-Iohara-Miki Algebra and Hall-Littlewood Limit of 5D AGT Conjecture