Duality for graded manifolds

TL;DR

This paper explores duality in graded manifolds, establishing natural dual objects and characterizations for graded bundles and supergeometric counterparts, advancing the understanding of their algebraic and geometric structures.

Contribution

It introduces a duality framework for graded manifolds, defining dual objects as graded polynomial and Weil (co)algebra bundles, and characterizes double vector bundles and degree 2 graded bundles.

Findings

Defined dual objects as graded polynomial (co)algebra bundles.

Provided characterizations of double vector bundles.

Simplified the proof of N-manifolds of degree 2 characterization.

Abstract

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and free graded Weil (co)algebra bundles. Our results are then applied to obtain elegant characterizations of double vector bundles and graded bundles of degree 2. All these results have their supergeometric counterparts. For instance, we give a simple proof of a nice characterisation of $N$-manifolds of degree 2, announced in the literature.