# Discrete Painleve system for the partition function of $N_f =2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit

## TL;DR

This paper explores a discrete Painleve system linked to the partition function of a specific supersymmetric gauge theory, revealing new connections through a matrix model extension and its double scaling limit to study Argyres-Douglas theories.

## Contribution

It introduces a novel discrete Painleve system derived from a matrix model extension with a logarithmic potential, advancing the understanding of supersymmetric gauge theories and their critical behavior.

## Key findings

- Identifies a Painleve system associated with the $N_f=2$ $SU(2)$ gauge theory.
- Develops a method to analyze the double scaling limit for critical behavior.
- Extends orthogonal polynomial techniques to complex matrix models.

## Abstract

We continue to study the matrix model of the $N_f =2$ $SU(2)$ case that represents the irregular conformal block. What provides us with the Painlev\'{e} system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross-Witten-Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres-Douglas type theory from IR. We elaborate upon the method of orthogonal polynomial and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential.