On the monodromy of the deformed cubic oscillator

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.