Backlund transformation of Painleve III($D_8$) tau function

TL;DR

This paper derives explicit formulas for the Painleve III(D8) tau function using Virasoro conformal blocks, revealing connections between bilinear forms, superalgebra representations, and algebraic solutions.

Contribution

It introduces a novel representation-theoretic approach to express Painleve III(D8) tau functions via Virasoro algebra embeddings and conformal blocks.

Findings

Explicit formulas for tau functions in terms of Virasoro conformal blocks.

Identification of Toda-like and Okamoto-like bilinear forms with superalgebra sectors.

Construction of algebraic solutions from special Virasoro representations.

Abstract

We study explicit formula (suggested by Gamayun, Iorgov, Lisovyy) for Painlev\'e III($D_8$) $\tau$ function in terms of Virasoro conformal blocks with central charge $1$. The Painlev\'e equation has two types of bilinear forms, we call them Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of a direct sum of two Virasoro algebra in a certain superalgebra. These two types of bilinear forms correspond to Neveu-Schwarz sector and Ramond sector of this algebra. We also obtain $\tau$ functions of algebraic solutions of Painlev\'e III($D_8$) from the special representations of the Virasoro algebra of highest weight $(n+1/4)^2$.