AGT basis in SCFT for c=3/2 and Uglov Polynomials

TL;DR

This paper explores the AGT correspondence in N=1 superconformal field theory at c=3/2, establishing a link between basis elements and Uglov polynomials, and constructing four-point functions using this basis.

Contribution

It extends the understanding of the AGT basis in superconformal theories by explicitly connecting basis elements with Uglov polynomials at c=3/2 and constructing correlation functions.

Findings

Confirmed the connection between basis elements and Uglov polynomials at c=3/2

Constructed four-point correlation functions using the AGT basis

Clarified the role of bosonizations and reflection operators in this context

Abstract

AGT allows one to compute conformal blocks of d = 2 CFT for a large class of chiral CFT algebras. This is related to the existence of a certain orthogonal basis in the module of the (extended) chiral algebra. The elements of the basis are eigenvectors of a certain integrable model, labeled in general by N-tuples of Young diagrams. In particular, it was found that in the Virasoro case these vectors are expressed in terms of Jack polynomials, labeled by 2-tuples of ordinary Young diagrams, and for the super-Virasoro case they are related to Uglov polynomials, labeled by two colored Young diagrams. In the case of a generic central charge this statement was checked in the case when one of the Young diagrams is empty. In this note we study the N=1 SCFT and construct 4 point correlation function using the basis. To this end we need to clarify the connection between basis elements and Uglov polynomials, we also need to use two bosonizations and their connection to the reflection operator. For the central charge $c=3/2$ we checked that there is a connection with the Uglov polynomials for the whole set of diagrams.