We study the geometry and cohomology of algebraic super curves, using a new contour integral for holomorphic differentials. For a class of super curves ("generic SKP curves") we define a period matrix. We show that the odd part of the period matrix controls the cohomology of the dual curve. The Jacobian of a generic SKP curve is a smooth supermanifold; it is principally polarized, hence projective, if the even part of the period matrix is symmetric. In general symmetry is not guaranteed by the Riemann bilinear equations for our contour integration, so it remains open whether Jacobians are always projective or carry theta functions.These results on generic SKP curves are applied to the study of algebro-geometric solutions of the super KP hierarchy. The tau function is shown to be, essentially, a meromorphic section of a line bundle with trivial Chern class on the Jacobian, rationally expressible in terms of super theta functions when these exist. Also we relate the tau function and the Baker function for this hierarchy, using a generalization of Cramer's rule to the supercase.

For every partition of a positive integer n in k parts and every point of an infinite Grassmannian we obtain a solution of the k component differential-difference KP hierarchy and a corresponding Baker function. A partition of n also determines a vertex operator construction of the fundamental representations of the infinite matrix algebra gl^ and hence a r function. We use these fundamental representations to study the Gauss decomposition in the infinite matrix group Gloo and to express the Baker function in terms of r-functions. The reduction to loop algebras is discussed.

Pairs of n × n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r × r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case r = 1, this reproduces well-known results of Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. This new class of pairs (L, Λ) of bispectral matrix differential operators is different than those previously studied in that L acts from the left, but Λ from the right on a common r × r eigenmatrix.

The connection between τ functions and zero curvature equations for the homogeneous construction of the basic module L(Λ0) over the simplest affine Kac–Moody algebra A(1)1 is studied.

We introduce hierarchies of difference equations (referred to as nT -systems) associated to the action of a (centrally extended, completed) infinite matrix group GL (n) ∞ on n-component fermionic Fock space. The solutions are given by matrix elements (τ -functions) for this action. We show that the τfunctions of type nT satisfy bilinear equations of length 3, 4, . . . , n + 1. The 2T -system is, after a change of variables, the usual 3 term T -system of type A.Restriction from GL (n)∞ to a subgroup isomorphic to the loop group LGLn, defines nQ-systems, studied earlier in [1] by the present authors for n = 2, 3.