Non-stationary difference equation for q-Virasoro conformal blocks

TL;DR

This work derives and analyzes a non-stationary difference equation for the $q,t$-Virasoro conformal block with one degenerate and four generic Verma modules, extending the familiar stationary and NS-limit cases to a genuine time-dependent setting. Using the ${\rm Vir}_{q,t}$ free-boson realization, screening charges, and the Dotsenko–Fateev integral, the author constructs a degenerate Higgsed block ${\psi}(\Lambda,x)$ and proves, for a specific degeneracy pattern, that it satisfies a non-stationary equation of the form $\Psi(t\Lambda,x) = {\cal A}_1 {\widehat{\gamma}} {\cal A}_2 {\widehat{\gamma}} {\cal A}_3 {\widehat{\gamma}} \Psi(\Lambda,x/(tqQ))$ with explicit ${\cal A}_i$ and a ${\widehat{\gamma}}$-action. In a Toda (quasi-classical) limit, this reduces to a simpler Hamiltonian evolution, and a commutativity result with the Toda Hamiltonian is shown. A notable feature is the exact, factorized form of the non-stationary evolution operator, expressible as infinite products, hinting at deep connections to ${\rm DAHA}$-type automorphisms and potential generalizations to higher rank and elliptic settings. The work also relates the degenerate conformal block to the surface defect wavefunction in 5d gauge theory and to Macdonald polynomials via a concrete two-variable reduction, providing a bridge between conformal blocks, gauge theories, and integrable systems. These results point toward a broader framework where non-stationary deformations of quantum integrable systems are governed by explicit operator-factorizations, with implications for refined topological strings and Ward identities in $q,t$-deformed matrix models.

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.