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 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.

arXiv comment41 pages, 5 figures, some figures in color, references added in v2, typos are corrected in v3

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• A slow review of the AGT correspondence
• Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries
• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds
• Quantum curves from refined topological recursion: the genus 0 case
• Topological recursion and uncoupled BPS structures II: Voros symbols and the $\tau$-function
• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• Resurgent Structure of the Topological String and the First Painlevé Equation
• Symplectic (Non-)Invariance of the Free Energy in Topological Recursion
• Refined topological recursion revisited -- properties and conjectures
• Deformation and quantisation condition of the $\mathcal{Q}$-top recursion
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants
• Painlevé Kernels and Surface Defects at Strong Coupling