$\tau$-Functions for a two-point version of the $KP$-hierarchy

In this paper a multipoint version of the linearization of the $KP$-hierarchy is described. Solutions of this system in the form of wave functions are constructed starting from a suitable Grassmann manifold and a group of commuting flows corresponding to the configuration of $n$ points in $\mathbb{C}$. The failure of equivariance at lifting these flows to the determinant bundle over this Grassmann manifold is measured by the determinants of certain Fredholm operators, the so-called $\tau$-functions. In the case of two points an explicit relation between these $\tau$-functions and the wave functions is derived.