Supermanifold Forms and Integration. A Dual Theory

TL;DR

This paper develops a dual theory of forms on supermanifolds using Lagrangian systems, introducing infinite complexes and discovering new classes of forms with negative degrees, advancing the mathematical understanding of supergeometry.

Contribution

It introduces a novel dual framework for supermanifold forms, including infinite complexes and negative-degree forms, expanding the mathematical structure of supergeometry.

Findings

Established isomorphisms for forms on supermanifolds with added parameters and equations.

Defined an exterior differential using variational derivatives, with key properties.

Discovered new negative-degree forms on supermanifolds, previously unknown.

Abstract

We investigate forms on supermanifolds defined as Lagrangians of ``copaths'' (that is, systems of equations, which may or may not specify submanifolds). For this, we consider direct products $M^{n|m}\times\Bbb R^{r|s}$ and study isomorphisms corresponding to simultaneously advancing the number of additional parameters $r|s$ and the number of equations. We define an exteriour differential in terms of variational derivatives w.r.t. a copath and establish its main properties. In the resulting stable picture we obtain infinite complexes $\D:\Om{r}{s}\to\Om{r+1}{s}$ for $M^{n|m}$, where $0 \le s \le m$ and $r$ can be any integer. For $r\ge 0$ a canonical isomorphism with forms constructed as Lagrangians of $r|s$-paths is established. We discover the ``lacking half'' of forms on supermanifolds: $r|s$-forms with $r<0$, previously unknown except for $s=m$. (They have been partly replaced earlier by an augmentation of the ``non-negative'' part of the complexes.) All these results are new. The study of these questions is in progress now.