Decomposing Nekrasov Decomposition

TL;DR

This paper explores the complex decompositions of conformal blocks in AGT relations, analyzing their properties, generalizations, and connections to topological strings, Macdonald polynomials, and quiver models.

Contribution

It introduces new identities for q-Selberg averages, clarifies the relationship between different decompositions, and extends the analysis to multi-point conformal blocks.

Findings

Derived new identities for q-Selberg averages of Macdonald polynomials

Analyzed the interplay between Nekrasov and topological vertex decompositions

Connected slicing invariance of topological strings to conformal block properties

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.