Multicritical points of unitary matrix model with logarithmic potential identified with Argyres-Douglas points

In our recent publications, the partition function of the Gross-Witten-Wadia unitary matrix model with the logarithmic term has been identified with the τ function of a certain Painlevé system, and the double scaling limit of the associated discrete Painlevé equation to the critical point provides us with the Painlevé II equation. This limit captures the critical behavior of the su(2), Nf = 2, 𝒩 = 2 supersymmetric gauge theory around its Argyres-Douglas 4D superconformal point. Here, we consider further extension of the model that contains the kth multicritical point and that is to be identified with Â2k,2k theory. In the k = 2 case, we derive a system of two ODEs for the scaling functions to the free energy, the time variable being the scaled total mass and make a consistency check on the spectral curve on this matrix model.