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

Understand

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.