Multi-matrix models without continuum limit

Abstract

We derive the discrete linear systems associated to multi-matrix models, the corresponding discrete hierarchies and the appropriate coupling conditions. We also obtain the W 1 + ∞ constraints on the partition function. We then apply to the multi-matrix models the technique, developed in previous papers, of extracting hierarchies of differential equations from lattice ones without passing through a continuum limit. In a q-matrix model we find 2 q coupled differential systems. The corresponding differential hierarchies are particular versions of the KP hierarchy. We show that the multi-matrix partition function is a τ-function of these hierarchies. We discuss a few examples in the dispersionless limit.