Berezinian expansion and super exterior powers

Abstract

In the supergeometric setting, the classical identification between differential forms of top degree and volume elements for integration breaks down. To address this, generalized notions of differential forms were introduced: pseudo-differential forms and integral forms (Bernstein-Leites), and $r|s$-forms (Voronov-Zorich). The Baranov-Schwarz transformation transforms pseudo-differential forms into $r|s$-forms. Also, integral $r$-forms are isomorphic to $r|m$-forms for a supermanifold of dimension $n|m$, yet the explicit construction of $r|s$-forms for arbitrary $s$ remains elusive. In this paper, we show that $1|1$-forms at a point can be realized as closed differential forms on a super projective space $\mathbb{P}^{m-1|n}$. We address a related problem involving the expansion of $\mathop{\mathrm{Ber}}(E + z A)$ for a linear operator on an $n|m$-dimensional space $V$, which generates supertraces of the representations $\Lambda^{r|s}(A)$ for $s=0$ and $s=m$ as the coefficients of the expansions near zero and near infinity, respectively. We demonstrate that the intermediate expansions in the annular regions between consecutive poles encode supertraces of representations on certain vector spaces that will be candidates for $\Lambda^{r|s}(V)$ for $0 < s < m$.