Argyres-Douglas theories, Painlev\'e II and quantum mechanics

TL;DR

This paper demonstrates that the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the $H_1$ Argyres-Douglas theory, connects Painlevé II tau functions to matrix models, and relates these theories to quantum mechanical models.

Contribution

It establishes a detailed link between matrix models, Argyres-Douglas theories, Painlevé equations, and quantum mechanics, providing new insights into their interrelations.

Findings

Matrix model reproduces Argyres-Douglas perturbative expansion.

Sum over instantons yields Painlevé II tau function.

Connections between theories and quantum mechanical models are clarified.

Abstract

We show in details that the all-orders genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the $H_1$ Argyres-Douglas theory coupled to the $\Omega$ background. In the self-dual limit we use the Painlev\'e/gauge correspondence and we show that, after summing over all instanton sectors, the two-cut cubic matrix model computes the tau function of Painlev\'e II without taking any double scaling limit or adding any external fields. We decode such solution within the context of trans-series. Finally in the Nekrasov-Shatashvili limit we connect the $H_1$ and the $H_0$ Argyres-Douglas theories to the quantum mechanical models with cubic and double well potentials.