Non-stationary difference equation for q-Virasoro conformal blocks

TL;DR

This paper derives a non-stationary difference equation for q,t-deformed Virasoro conformal blocks, extending known stationary cases and connecting to integrable systems and Macdonald polynomials.

Contribution

It explicitly formulates the non-stationary difference equation for q,t-Virasoro blocks with specific degenerate modules and proves it in certain cases, advancing understanding of these special functions.

Findings

Derived explicit non-stationary difference equation for q,t-Virasoro blocks.

Proved the equation for cases with three degenerate modules.

Connected the results to Macdonald polynomials.

Abstract

Conformal blocks of q,t-deformed Virasoro and W-algebras are important special functions in representation theory with applications in geometry and physics. In the Nekrasov-Shatashvili limit t -> 1, whenever one of the representations is degenerate then conformal block satisfies a difference equation with respect to the coordinate associated with that degenerate representation. This is a stationary Schrodinger equation for an appropriate relativistic quantum integrable system. It is expected that generalization to generic t <> 1 is a non-stationary Schrodinger equation where t parametrizes shift in time. In this paper we make the non-stationary equation explicit for the q,t-Virasoro block with one degenerate and four generic Verma modules, and prove it when three modules out of five are degenerate, using occasional relation to Macdonald polynomials.