The Hilbert series of N=1 SO(N_c) and Sp(N_c) SQCD, Painlev\'e VI and Integrable Systems

We present a novel approach for computing the Hilbert series of 4d N=1 supersymmetric QCD with SO(N_c) and Sp(N_c) gauge groups. It is shown that such Hilbert series can be recast in terms of determinants of Hankel matrices. With the aid of results from random matrix theory, such Hankel determinants can be evaluated both exactly and asymptotically. Several new results on Hilbert series for general numbers of colours and flavours are thus obtained in this paper. We show that the Hilbert series give rise to families of rational solutions, with palindromic numerators, to the Painlev\'e VI equations. Due to the presence of such Painlev\'e equations, there exist integrable Hamiltonian systems that describe the moduli spaces of SO(N_c) and Sp(N_c) SQCD. To each system, we explicitly state the corresponding Hamiltonian and family of elliptic curves. It turns out that such elliptic curves take the same form as the Seiberg-Witten curves for 4d N=2 SU(2) gauge theory with 4 flavours.