Argyres-Douglas theories, Painleve II and quantum mechanics

We show in details that the all order genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the H-1 Argyres-Douglas theory coupled to the background. In the self-dual limit we use the Painleve/gauge correspondence and we show that, after summing over all instanton sectors, the two-cut cubic matrix model computes the tau function of Painleve II without taking any double scaling limit or adding any external fields. We decode such solution within the context of transseries. 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.