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

## TL;DR

This paper constructs a 2-parameter family of tau-functions for the first Painlevé equation using topological recursion and elliptic curves, and computes Stokes multipliers via exact WKB analysis under certain conjectures.

## Contribution

It introduces a novel construction of tau-functions through Fourier transform of topological recursion and advances the understanding of monodromy via exact WKB methods.

## Key findings

- Construction of tau-functions from elliptic curve topological recursion
- Explicit computation of Stokes multipliers using WKB analysis
- Connection between tau-functions and monodromy data

## 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.