SU(N) integrals and tau functions
A. Yu. Orlov
TL;DR
This work develops a unified framework for solvable SU$(N)$ multi-matrix models tied to embedded graphs, where corner matrices and their monodromies encode the spectral data driving the partition functions. By leveraging dressing by group elements and Schur-function (and Mac) identities, the authors derive explicit integral relations for $G\in\{GL(N),U(N),SU(N)\}$ and mixed ensembles, expressing correlators as sums over partitions and, in favorable cases, as $\tau$-functions of KP, 2KP, or BKP hierarchies. The KP case yields determinant forms, while BKP leads to Pfaffian expressions built from generalized hypergeometric functions, with hypergeometric $\tau$-functions arising under Mac-type restrictions and vertex-geometry constraints. They also provide a fermionic (2KP) construction for $SU(N)$ integrals and highlight when SU$(N)$ integrals can be represented as $\tau$-functions, clarifying the role of vertex topology and monodromy spectra. Overall, the paper links graph-based multi-matrix ensembles to integrable hierarchies, extending solvable models to SU$(N)$ and mixed groups and enabling explicit tau-function representations through spectral data and Mac relations.
Abstract
We present a family of solvable multi-matrix models associated with an arbitrary embedded graph $Γ$ 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 $Γ^*$ and sets of parameters attached to each vertex of $Γ$. 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 $Γ$. We also consider mixed ensembles of matrices from $GL(N),U(N)$ and $SU(N)$.