Painleve equations from Nakajima-Yoshioka blowup relations

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.