Cech and de Rham Cohomology of Integral Forms

TL;DR

This paper investigates the cohomology of integral forms on supermanifolds using sheaf theory, revealing new structures and differences from traditional superform cohomology, with implications for integration theory.

Contribution

It introduces a new sheaf of integral forms with delta distributions, expanding the cohomology framework for supermanifolds and highlighting novel relations and negative degree forms.

Findings

The sheaf of integral forms contains forms of negative degree.

The cohomology of integral forms differs from standard superform cohomology.

New relations in the sheaf affect the cohomological properties.

Abstract

We present a study on the integral forms and their Cech/de Rham cohomology. We analyze the problem from a general perspective of sheaf theory and we explore examples in superprojective manifolds. Integral forms are fundamental in the theory of integration in supermanifolds. One can define the integral forms introducing a new sheaf containing, among other objects, the new basic forms delta(dtheta) where the symbol delta has the usual formal properties of Dirac's delta distribution and acts on functions and forms as a Dirac measure. They satisfy in addition some new relations on the sheaf. It turns out that the enlarged sheaf of integral and "ordinary" superforms contains also forms of "negative degree" and, moreover, due to the additional relations introduced, its cohomology is, in a non trivial way, different from the usual superform cohomology.