Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories

Abstract

Using the saddle point method, we give an explicit form of the planar free energy and Wilson loops of unitary matrix models in the one-cut regime. The multi-critical unitary matrix models are shown to undergo third-order phase transitions at two points by studying the planar free energy. One of these ungapped/gapped phase transitions is multi-critical, while the other is not multi-critical.The spectral curve of the k-th multi-critical matrix model exhibits an A 4k−1 singularity at the multi-critical point. Perturbation around the multi-critical point and its double scaling limit are studied. In order to take the double scaling limit, the perturbed coupling constants should be finetuned such that all the zero points of the spectral curve approach to the A 4k−1 singular point. The fine-tuning is examined in the one-cut regime, and the scaling behavior of the perturbed couplings is determined. It is shown that the double scaling limit of the spectral curve is isomorphic to the Seiberg-Witten curve of the Argyres-Douglas theory of type (A 1 , A 4k−1 ).

Construction of irregular conformal/W block and flavor mass relations of $\mathcal{N}=2$ SUSY gauge theory from the $A_{n-1}$ quiver matrix model

“…In the series of developments on the irregular limit of the A 1 (one-matrix) case [35][36][37][38][39][40][41], we have been led to the procedure of converting the hermitean matrix model into the unitary matrix model in order to make such procedure well-defined. The upshot from the case of N f = 2 is the well-known GWW model [42][43][44][45] (see also [46][47][48][49] for recent discussion) augmented by the log potential.…”