MULTIMATRIX MODELS - INTEGRABILITY PROPERTIES AND TOPOLOGICAL CONTENT

In this article, we analyze multimatrix chain models. They can be cons idered as multi-component Toda lattice hierarchies subject to suitable coupling conditions. The extension of such models to include extra di screte states requires a weak form of integrability. The discrete stat es of the q-matrix model are organized in representations of sl(q). We solve exactly the Gaussian type models, of which we compute several a ll-genus correlators. Among the latter models one can classify also th e discretized c = 1 string theory, which we revisit using Toda lattice hierarchy methods. Finally, we analyze the topological field theory c ontent of the 2q-matrix models: we define primary fields (which are co (q))1 metrics and structure constants. and prove that they satisfy the axioms of topological field theories. We outline a possible method fo r extracting interesting topological field theories with a finite numb er of primaries.