DD-modules on 1|11|1 supercurves

It is known that to every (1|1)(1|1)-dimensional supercurve XX there is associated a dual supercurve Xˆ, and a superdiagonal Δ⊂X×Xˆ. We establish that the categories of DD-modules on XX, Xˆ and ΔΔ are equivalent. This follows from a more general result about DD-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 XX is a super elliptic curve.