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arXiv · 1608.02568

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

M. A. Bershtein·A. I. Shchechkin
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Abstract

We study explicit formula (suggested by Gamayun, Iorgov, Lisovyy) for Painlevé III($D_8$) $τ$ function in terms of Virasoro conformal blocks with central charge $1$. The Painlevé 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 $τ$ functions of algebraic solutions of Painlevé III($D_8$) from the special representations of the Virasoro algebra of highest weight $(n+1/4)^2$.

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BibTeXRIS

M. A. Bershtein, A. I. Shchechkin. 2017-01-03. Backlund transformation of Painleve III($D_8$) tau function. https://doi.org/10.1088/1751-8121%2Faa59c9

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