Splitting theorem for $\mathbb{Z}_2^n$-supermanifolds

TL;DR

This paper proves a splitting theorem for smooth $Z_2^n$-supermanifolds, showing they are diffeomorphic to a superization of a vector bundle, extending classical results to this generalized setting.

Contribution

It establishes a splitting theorem for $Z_2^n$-supermanifolds, demonstrating their structure as superizations of vector bundles, generalizing classical supermanifold results.

Findings

$Z_2^n$-supermanifolds follow a splitting theorem similar to classical supermanifolds.

The sign rule based on the scalar product of degrees leads to non-nilpotent even coordinates.

The theorem holds in the smooth category, extending classical results to $Z_2^n$-supergeometry.

Abstract

Smooth $\mathbb{Z}_2^n$-supermanifolds have been introduced and studied recently. The corresponding sign rule is given by the "scalar product" of the involved $\mathbb{Z}_2^n$-degrees. It exhibits interesting changes in comparison with the sign rule using the parity of the total degree. With the new rule, nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. The classical Batchelor-Gawcedzki theorem says that any smooth supermanifold is diffeomorphic to the "superization" $\Pi E$ of a vector bundle $E$. It is also known that this result fails in the complex analytic category. Hence, it is natural to ask whether an analogous statement goes through in the category of $\mathbb{Z}_2^n$-supermanifolds with its local model made of formal power series. We give a positive answer to this question.