Fermionic construction of the partition function for multimatrix models and the multicomponent Toda lattice hierarchy

We use $p$-component fermions, $p=2,3,\dots$, to represent $(2p-2)N$-fold integrals as a fermionic vacuum expectation. This yields a fermionic representation for various $(2p-2)$-matrix models. We discuss links with the $p$-component Kadomtsev–Petviashvili hierarchy and also with the $p$-component Toda lattice hierarchy. We show that the set of all but two flows of the $p$-component Toda lattice hierarchy changes standard matrix models to new ones.