Multimatrix models: Integrability properties and topological content

We analyze multi–matrix chain models. They can be considered as multi–component Toda lattice hierarchies subject to suitable coupling conditions. The extension of such models to include extra discrete states requires a weak form of integrability. The discrete states of the q–matrix model are organized in representations of slq. We solve exactly the Gaussian–type models, of which we compute several all-genus correlators. Among the latter models one can classify also the discretized c = 1 string theory, which we revisit using Toda lattice hierarchy methods. Finally we analyze the topological field theory content of the 2q–matrix models: we define primary fields (which are ∞q), metrics and structure constants and prove that they satisfy the axioms of topological field theories. We outline a possible method to extract interesting topological field theories with a finite number of primaries.