The geometry of the hermitian matrix model and lattices for the NLS and dNLS hierarchies

Abstract

The geometrical description of the Nonlinear Schrödinger-Toda system hierarchy in the Sato Grassmannian with the action of the translation group is applied to the Hermitian one-matrix model. A family of derivative Nonlinear Schrödinger system hierarchies with its lattices—associated with the Volterra chain—which are auto-Bäcklund transformations, is analyzed from a geometrical point of view. The Sato periodic flag manifold with the line bundles over it turns out to be the proper infinite-dimensional manifold in this case. The lattice appears as a square root of the action of the translation group; this can be understood as a reduction of the action of a translation group of a larger loop group. The reduction t 2 n+1 = 0 of the Hermitian one-matrix model, essential in the double scaling limit, is shown to be described in terms of the derivative Nonlinear Schrödinger-Volterra system hierarchy. The role of the heat hierarchy, self-similarity and auto-Bäcklund transformations is pointed out. A characterization in Sato's Grassmannian and periodic flag manifold of the Hermitian one-matrix model is given. In the latter case we are concerned with the t 2 n+1 = 0 reduction.