On the monodromy of the deformed cubic oscillator

TL;DR

This paper investigates the monodromy properties of the deformed cubic oscillator, linking it to Painlevé equations and solving complex Riemann-Hilbert problems relevant to Donaldson-Thomas theory.

Contribution

It introduces the first solutions to infinite-dimensional Riemann-Hilbert problems associated with the deformed cubic oscillator beyond the uncoupled case.

Findings

Solutions to infinite-dimensional Riemann-Hilbert problems are obtained.

The monodromy map's asymptotics are analyzed using WKB methods.

Connections between the oscillator's monodromy and Donaldson-Thomas theory are established.

Abstract

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlev{\'e} equation. We use the generalised monodromy map for this equation to give solutions to the infinite-dimensional Riemann-Hilbert problems arising from the Donaldson-Thomas theory of the A2 quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map.