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

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 $(n+1/4)^2$.