Quantum Curve and the First Painlev\'e Equation

TL;DR

This paper demonstrates that the topological recursion applied to the spectral curve of the first Painlevé equation produces a WKB solution for its isomonodromy problem, establishing a link between quantum curves and Painlevé equations.

Contribution

It shows that the isomonodromy system for the first Painlevé equation can be viewed as a quantum curve derived from topological recursion.

Findings

Topological recursion yields WKB solutions for P_I

The isomonodromy system is a quantum curve

Connects quantum curves with Painlevé equations

Abstract

We show that the topological recursion for the (semi-classical) spectral curve of the first Painlev\'e equation $P_{\rm I}$ gives a WKB solution for the isomonodromy problem for $P_{\rm I}$. In other words, the isomonodromy system is a quantum curve in the sense of [Dumitrescu O., Mulase M., Lett. Math. Phys. 104 (2014), 635-671, arXiv:1310.6022] and [Dumitrescu O., Mulase M., arXiv:1411.1023].