SU(N) integrals and tau functions

Abstract

We present a family of solvable multi-matrix models associated with an arbitrary embedded graph $\Gamma$ with a single vertex. The graph with $n$ edges is equipped with $2n$ corner matrices. The partition function of each member of the family depends on the set of eigenvalues of monodromies of corner matrices around the vertices of the dual graph $\Gamma^*$ and sets of parameters attached to each vertex of $\Gamma$. We select the cases where the partition function of a model is a tau function of KP, 2KP and BKP hiearachies. We compare integrals over ${U}(N)$ and over ${SU}(N)$ groups. In $U(N)$ case there is no restriction on the number of vertices of $\Gamma$. We also consider mixed ensembles of matrices from $GL(N),U(N)$ and $SU(N)$.