A direct proof of AGT conjecture at β = 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.