d-orthogonal polynomials, Toda Lattice and Virasoro symmetries

Abstract: The subject of this paper is a connection between d-orthogonal polynomials and the Toda lattice hierarchy. In more details we consider some polynomial systems similar to Hermite polynomials, but satisfying $d+2$-term recurrence relation, $d >1$. Any such polynomial system defines a solution of the Toda lattice hierarchy. However we impose also the condition that the polynomials are also eigenfunctions of a differential operator, i.e. a bispectral problem. This leads to a solution of the Toda lattice hierarchy, enjoying a number of special properties. In particular the corresponding tau-functions $\tau_m$ satisfy the Virasoro constraints. The most spectacular feature of these tau-functions is that all of them are partition functions of matrix models. Some of them are well known matrix models - e.g. Kontsevich model, Kontsevich-Penner models, $r$-spin models, etc. A remarkable phenomenon is that the solution corresponding to $d=2$ contains two famous tau functions describing the intersection numbers on moduli spaces of compact Riemann surfaces and of open Riemann surfaces.