Bootstrapping bilinear relations of discrete Painlev\'e systems from 5d gauge theories

Abstract

We study tau-functions of elliptic and $ q $-Painlev\'e equations through their correspondence with 5d $ \mathcal{N}=1 $ supersymmetric field theories. Defining tau-functions associated with 5d field theories in terms of partition functions on the $ \Omega $-background, we propose a bootstrap method that determines their bilinear relations from the effective prepotential and the perturbative contributions, together with the global symmetry of the theory. Applying the method to 5d $ \mathrm{SU}(2) $ gauge theories coupled to $ N_f \leq 8 $ fundamental hypermultiplets, we obtain the bilinear relations of the elliptic and $ q $-Painlev\'e equations. This method can be applied to general 5d supersymmetric field theories beyond $ \mathrm{SU}(2) $ gauge theories.