Painleve equations from Nakajima-Yoshioka blowup relations

M. Bershtein; A. Shchechkin

doi:10.1007/s11005-019-01198-4

paper

Painleve equations from Nakajima-Yoshioka blowup relations

arXiv:1811.04050 · doi:10.1007/s11005-019-01198-4

Abstract

Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal to the series of $c = 1$ Virasoro conformal blocks. We study similar series of $c = - 2$ conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima-Yoshioka blowup relations on Nekrasov partition functions. We also study series of $q$ -deformed $c = - 2$ conformal blocks and relate it to $q$ -Painlevé equation. As an application, we prove formula for the tau function of $q$ -Painlevé $A_{7}^{(1)^{'}}$ equation.

v1 36 pages; v2 38 pages, many small corrections (including the title), references added