D-modules on 1|1 Supercurves

TL;DR

This paper proves an equivalence of categories of D-modules on 1|1 supercurves and their duals, revealing deep structural relationships and preserving key properties like tensor products and vector bundles.

Contribution

It establishes a categorical equivalence of D-modules on supercurves and their duals, extending to cases with purely odd submersions and analyzing line bundles with connection.

Findings

Categories of D-modules on X, X, and  are equivalent

Equivalences preserve tensor products and vector bundles

Examples include superelliptic curves

Abstract

It is known that to every 1|1 dimensional supercurve X there is associated a dual supercurve \hat{X}, and a superdiagonal \Delta in their product. We establish that the categories of D-modules on X, \hat{X}, and \Delta are equivalent. This follows from a more general result about D-modules and purely odd submersions. The equivalences preserve tensor products, and take vector bundles to vector bundles. Line bundles with connection are studied, and examples are given where X is a superelliptic curve.