Selberg Integral and SU(N) AGT Conjecture

TL;DR

This paper explores the connection between 2d Toda correlation functions and the SU(N) AGT conjecture by expressing Toda integrals as Selberg integrals involving Jack polynomials, proposing a new conjectural formula.

Contribution

It introduces a conjectural formula for Selberg averages of Jack polynomials that supports the SU(N) AGT correspondence and advances understanding of Toda correlation functions.

Findings

Derived a Selberg integral representation for Toda correlation functions.

Proposed a conjectural formula consistent with the SU(N) AGT conjecture.

Connected Toda integrals with Jack polynomial averages.

Abstract

An intriguing coincidence between the partition function of super Yang-Mills theory and correlation functions of 2d Toda system has been heavily studied recently. While the partition function of gauge theory was explored by Nekrasov, the correlation functions of Toda equation have not been completely understood. In this paper, we study the latter in the form of Dotsenko-Fateev integral and reduce it in the form of Selberg integral of several Jack polynomials. We conjecture a formula for such Selberg average which satisfies some consistency conditions and show that it reproduces the SU(N) version of AGT conjecture.