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

TL;DR

This work addresses how the AGT correspondence can be realized inside conformal field theory by constructing an orthogonal basis in the Vir⊗H module that diagonalizes an infinite set of commuting integrals of motion. Focusing on the c=1 case, it shows that basis vectors factorize into products of two Jack polynomials, corresponding to the sum of two noninteracting Calogero-model integrals, and verifies this structure through explicit level-1 and level-2 checks against Nekrasov’s partition functions. For general c one must use two different Feigin–Fuks bosonizations, and the full basis with nonempty Y1,Y2 remains open. The results illuminate the integrable structure behind the AGT representation and provide concrete, checkable links between CFT blocks and gauge-theory Nekrasov functions, with potential implications for broader conformal-block computations.

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.