Non-stationary difference equation for \(q\)-Virasoro conformal blocks. (English) Zbl 07930035
Summary: 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 \(t \neq 1\) 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.
MSC:
| 16T05 | Hopf algebras and their applications |
| 17B68 | Virasoro and related algebras |
| 39A06 | Linear difference equations |
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