Finalizing the proof of AGT relations with the help of the generalized Jack polynomials

TL;DR

This paper completes the proof of the AGT relations for GL(2) by introducing generalized Jack polynomials, which correctly reproduce Nekrasov functions for arbitrary beta, linking instanton moduli spaces and conformal blocks.

Contribution

It introduces generalized Jack polynomials depending on Young diagrams to resolve previous issues in AGT proofs for beta not equal to one.

Findings

Successful proof of AGT for GL(2) case.

Generalized Jack polynomials reproduce Nekrasov functions for arbitrary beta.

Connects instanton moduli spaces with conformal blocks.

Abstract

Original proofs of the AGT relations with the help of the Hubbard-Stratanovich duality of the modified Dotsenko-Fateev matrix model did not work for beta different from one, because Nekrasov functions were not properly reproduced by Selberg-Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instanton moduli spaces, and this also establishes connection to alternative ABBFLT approach to the AGT relations, studying the action of chiral algebras on the instanton moduli spaces. In this paper we describe a complete proof of AGT in the simple case of GL(2) (N=2) Yang-Mills theory, i.e. the 4-point spherical conformal block of the Virasoro algebra.