A direct proof of AGT conjecture at beta = 1

TL;DR

This paper provides a direct proof of the AGT conjecture at beta=1, establishing the equivalence between Dotsenko-Fateev beta-ensembles and Nekrasov functions for SU(2) with four fundamentals.

Contribution

It offers a novel, straightforward proof of the AGT conjecture at beta=1 using Selberg matrix models and character correlators, simplifying previous approaches.

Findings

Proved AGT conjecture at beta=1 for SU(2) with four fundamentals.

Connected Nekrasov functions to correlators of Schur polynomials in matrix models.

Extended the understanding of beta=1 case, highlighting difficulties for beta ≠ 1.

Abstract

The AGT conjecture claims an equivalence of conformal blocks in 2d CFT and sums of Nekrasov functions (instantonic sums in 4d SUSY gauge theory). The conformal blocks can be presented as Dotsenko-Fateev beta-ensembles, hence, the AGT conjecture implies the equality between Dotsenko-Fateev beta-ensembles and the Nekrasov functions. In this paper, we prove it in a particular case of beta=1 (which corresponds to c = 1 at the conformal side and to epsilon_1 + epsilon_2 = 0 at the gauge theory side) in a very direct way. The central role is played by representation of the Nekrasov functions through correlators of characters (Schur polynomials) in the Selberg matrix models. We mostly concentrate on the case of SU(2) with 4 fundamentals, the extension to other cases being straightforward. The most obscure part is extending to an arbitrary beta: for beta \neq 1, the Selberg integrals that we use do not reproduce single Nekrasov functions, but only sums of them.