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

Abstract

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.

14 pages, 6 figures, typos corrected, missing figure and references added

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• A slow review of the AGT correspondence
• Refined geometric transition and $qq$-characters
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• Trinion Conformal Blocks from Topological strings
• On generalized Macdonald polynomials
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions