Supermanifold Forms and Integration. A Dual Theory
Abstract
Description
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.
20 pages, LaTeX2e, resubmitted after TeX changes
20 pages, LaTeX2e, resubmitted after TeX changes