Painleve equations from Nakajima-Yoshioka blowup relations

Understand

Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlev\'e 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 Painlev\'e 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 $q$-Painlev\'e equation.