Degeneration limits of Virasoro vertex operators and Painlev\'e tau functions

TL;DR

This paper develops a method to analyze degeneration limits of Virasoro vertex operators, leading to new insights into Painlevé tau functions through irregular conformal blocks.

Contribution

It introduces a novel approach to degeneration limits of Virasoro vertex operators and applies it to prove conjectural expansions of Painlevé tau functions.

Findings

Constructed degeneration limits of Virasoro vertex operators.

Derived a vertex operator between Verma modules as a degeneration.

Proved conjectural expansions of Painlevé tau functions.

Abstract

We construct degeneration limits of vertex operators for the Virasoro algebra. Our method relies on the rearranged expansion of compositions of vertex operators together with their integral representations. Using this framework, we obtain a vertex operator between Verma modules of rank $r+1$ as a degeneration of a composition of two vertex operators between Verma modules of rank $r$ ($r\in\mathbb{Z}_{\geq 0}$). Furthermore, we apply these degeneration limits to prove the conjectural expansions of the $\tau$ functions of the fifth and fourth Painlev\'e equations in terms of irregular conformal blocks [H. Nagoya, J. Math. Phys. 56, 123505 (2015)].