Non-stationary difference equation for q-Virasoro conformal blocks

Abstract

Conformal blocks of q, t-deformed Virasoro and \({\mathcal {W}}\)-algebras are important special functions in representation theory with applications in geometry and physics. In the Nekrasov–Shatashvili limit \(t \rightarrow 1\), 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 \(t \ne 1\) 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.