Berezinian expansion and super exterior powers

Abstract

In supergeometry, the classical identification of top-degree differential forms with volume forms suitable for integration breaks down. To address this issue, several generalized notions of differential forms have been introduced, including integral forms and pseudo-differential forms of Bernstein-Leites and $r|s$-forms of Voronov-Zorich. The Baranov-Schwarz transformation maps pseudo-differential forms to $r|s$-forms of type II, and, on a supermanifold of dimension $n|m$, integral $r$-forms are isomorphic to $r|m$-forms. However, an explicit construction of $r|s$-forms for arbitrary $s$ has remained elusive. First, we observe that this problem can be related with expansions of $\mathrm{Ber}(E+zA)$ for a linear operator $A$ on a superspace $V$. It was already known that the coefficients of the expansions near zero and near infinity give the supertraces of the representations ${\Lambda }^{r|s}(A)$ in the two extreme cases $s=0$ and $s=m$, respectively. In this paper we show that the intermediate expansions of $\mathrm{Ber}(E+zA)$, taken in the annular regions between consecutive poles, encode the supertraces of representations on explicitly constructed vector spaces that model ${\Lambda }^{r|s}(V)$ for $0<s<m$. We introduce a formal analogue of the Baranov-Schwarz transformation and show that it relates these spaces to Voronov-Zorich $r|s$-forms. As a separate remark, we establish that $1|1$-forms at a point of a supermanifold of dimension $n|m$ can be realized as closed differential forms on the super projective space ${\mathbb{P}}^{m-1|n}$.