Irregular conformal blocks and connection formulae for Painlev\'e V functions

TL;DR

This paper establishes a new representation of the Painlevé V tau function using Fredholm determinants and series, explores its relation to irregular conformal blocks, and derives connection formulas for its asymptotic behavior.

Contribution

It provides a novel Fredholm determinant and series representation of the Painlevé V tau function, linking it to irregular conformal blocks and deriving explicit connection formulas.

Findings

Fredholm determinant and series representation of $ au(t)$

Connection formulas between asymptotic expansions at different points

Irregular conformal blocks as confluent limits of regular blocks

Abstract

We prove a Fredholm determinant and short-distance series representation of the Painlev\'e V tau function $\tau(t)$ associated to generic monodromy data. Using a relation of $\tau(t)$ to two different types of irregular $c=1$ Virasoro conformal blocks and the confluence from Painlev\'e VI equation, connection formulas between the parameters of asymptotic expansions at $0$ and $i\infty$ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as $t\to 0,+\infty,i\infty$ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks.