2-Parameter $$\tau $$-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis

Abstract

We show that a 2-parameter family of $\tau$-functions for the first Painlev\'e equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB theoretic computation of the Stokes multipliers of associated isomonodromy system assuming certain conjectures.