Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

TL;DR

This paper explores the connection between coset conformal field theories, the AGT correspondence, and Macdonald polynomials at roots of unity, providing new algebraic constructions and basis representations.

Contribution

It introduces a new construction of the algebra A(r,2) as a limit of the toroidal gl(1) algebra and expresses bases of representations via Macdonald polynomials at roots of unity.

Findings

Basis of A(r,2) algebra expressed through Macdonald polynomials at q,t → -1

Vertex operators have factorized matrix elements in this basis

Singular vectors of the N=1 Super Virasoro algebra realized via Macdonald polynomials

Abstract

We continue our study of the AGT correspondence between instanton counting on C^2/Z_p and Conformal field theories with the symmetry algebra A(r,p). In the cases r=1, p=2 and r=2, p=2 this algebra specialized to: A(1,2)=H+sl(2)_1 and A(2,2)=H+sl(2)_2+NSR. As the main tool we use a new construction of the algebra A(r,2) as the limit of the toroidal gl(1) algebra for q,t tend to -1. We claim that the basis of the representation of the algebra A(r,2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q,t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the $\mathcal{N}=1$ Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q,t tend to -1.