Polarisation of Graded Bundles

TL;DR

This paper introduces a full linearisation functor transforming graded bundles into symmetric k-fold vector bundles, characterising their structure and applications to supergeometry and symplectic structures.

Contribution

It constructs and characterises the full linearisation functor for graded bundles, revealing new structures like symplectical double vector bundles and applications to supergeometry.

Findings

Full linearisation functor maps graded bundles to symmetric k-fold vector bundles.

Characterisation of the image as symmetric k-fold vector bundles with symmetric group actions.

Introduction of symplectical double vector bundles as skew-symmetric analogues of metric bundles.

Abstract

We construct the full linearisation functor which takes a graded bundle of degree $k$ (a particular kind of graded manifold) and produces a $k$-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory of $k$-fold vector bundles consisting of symmetric $k$-fold vector bundles equipped with a family of morphisms indexed by the symmetric group ${\mathbb S}_k$. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a metric double vector bundle. We also discuss the related case of fully linearising $N$-manifolds, and how one can use the full linearisation functor to "superise" a graded bundle.