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

2-parameter $\tau$-function for the first Painlev\'e equation -Topological recursion and direct monodromy problem via exact WKB analysis-