The Hilbert series of image image and image SQCD, Painlevé VI and integrable systems Original Research Article

We present a novel approach for computing the Hilbert series of 4d image supersymmetric QCD with image and image 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 Painlevé VI equations. Due to the presence of such Painlevé equations, there exist integrable Hamiltonian systems that describe the moduli spaces of image and image 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 image image gauge theory with 4 flavours.