AGT conjecture and Integrable structure of Conformal field theory for c=1

TL;DR

This paper explores the AGT correspondence for c=1 in conformal field theory, revealing that basis vectors are products of Jack polynomials and linking integrals of motion to noninteracting Calogero models.

Contribution

It demonstrates that at c=1, basis vectors are products of two Jack polynomials and integrals of motion correspond to noninteracting Calogero models, providing explicit computational verification.

Findings

Basis vectors at c=1 are products of two Jack polynomials.

Integrals of motion decompose into sums of two Calogero model integrals.

Different Feigin-Fuks bosonizations are needed for c≠1.

Abstract

AGT correspondence gives an explicit expressions for the conformal blocks of $d=2$ conformal field theory. Recently an explanation of this representation inside the CFT framework was given through the assumption about the existence of the special orthogonal basis in the module of algebra $\mathcal{A}=Vir\otimes\mathcal{H}$. The basis vectors are the eigenvectors of the infinite set of commuting integrals of motion. It was also proven that some of these vectors take form of Jack polynomials. In this note we conjecture and verify by explicit computations that in the case of the Virasoro central charge $c=1$ all basis vectors are just the products of two Jack polynomials. Each of the commuting integrals of motion becomes the sum of two integrals of motion of two noninteracting Calogero models. We also show that in the case $c\neq1$ it is necessary to use two different Feigin-Fuks bosonizations of the Virasoro algebra for the construction of all basis vectors which take form of one Jack polynomial.