2-parameter $τ$-function for the first Painlevé equation -Topological recursion and direct monodromy problem via exact WKB analysis-

We show that a 2-parameter family of $τ$-functions for the first Painlevé 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.

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