• A
  • A
  • A
  • АБB
  • АБB
  • АБB
  • А
  • А
  • А
  • А
  • А
Обычная версия сайта

Статья

Bäcklund transformation of Painlevé III(D 8) τ function

Journal of Physics A: Mathematical and Theoretical. 2017. Vol. 50. No. 11. P. 115205.

We study the explicit formula (suggested by Gamayun, Iorgov and Lisovyy) for the Painlevé III(D 8τ function in terms of Virasoro conformal blocks with a central charge of 1. The Painlevé equation has two types of bilinear forms, which we call Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of the direct sum of two Virasoro algebras in a certain superalgebra. These two types of bilinear forms correspond to the Neveu–Schwarz sector and the Ramond sector of this algebra. We also obtain the τ functions of the algebraic solutions of the Painlevé III(D 8) from the special representations of the Virasoro algebra of the highest weight (n  +  1/4)2.