The paper derives the resummed Nekrasov-Shatashvili free energy from blow-up equations and uses it to compute the band-gap structure, connection formulas, and stability chart of the Lamé equation.
Painleve equations from Nakajima-Yoshioka blowup relations
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abstract
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. As an application, we prove formula for the tau function of $q$-Painlev\'e $A_7^{(1)'}$ equation.
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Blowing-up the edge: connection formulae and stability chart of the Lam\'e equation
The paper derives the resummed Nekrasov-Shatashvili free energy from blow-up equations and uses it to compute the band-gap structure, connection formulas, and stability chart of the Lamé equation.