Super Curves, their Jacobians, and super KP equations

TL;DR

This paper explores the geometry and cohomology of algebraic super curves, introduces a new contour integral for holomorphic differentials, and applies these findings to super KP hierarchy solutions, including super theta functions.

Contribution

It introduces a new contour integral for holomorphic differentials on super curves and studies their Jacobians, linking these to super KP hierarchy solutions and super theta functions.

Findings

The Jacobian of a generic SKP curve is a smooth supermanifold.

The odd part of the period matrix controls the cohomology of the dual curve.

The tau function can be expressed as a meromorphic section of a line bundle on the Jacobian.

Abstract

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.