Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations

TL;DR

This paper develops a rigorous framework for irregular conformal blocks of the Virasoro algebra, enabling new expansions at irregular singular points and proposing conjectural formulas for Painlevé equations' tau functions.

Contribution

It provides precise definitions of irregular vertex operators and establishes their existence, advancing the understanding of irregular conformal blocks and their expansions.

Findings

Defined irregular vertex operators of two types.

Proved the unique existence of one type of irregular vertex operator.

Proposed conjectural series expansions for Painlevé tau functions.

Abstract

We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painlev\'e equations, using expansions of irregular conformal blocks at an irregular singular point.