The Geometrisation of $\mathbb N$-manifolds of degree 2

Abstract

This paper describes an equivalence of the canonical category of $\mathbb N$-manifolds of degree $2$ with a category of involutive double vector bundles. More precisely, we show how involutive double vector bundles are in duality with double vector bundles endowed with a linear metric. We describe then how special sections of the metric double vector bundle that is dual to a given involutive double vector bundle are the generators of a graded manifold of degree $2$ over the double base. We discuss how split Poisson $\mathbb N$-manifolds of degree $2$ are equivalent to \emph{self-dual representations up to homotopy} and so, following Gracia-Saz and Mehta, to linear splittings of a certain class of VB-algebroids. In other words, the equivalence of categories above induces an equivalence between so called \emph{Poisson involutive double vector bundles}, which are the dual objects to metric VB-algebroids, and Poisson $\mathbb N$-manifolds of degree $2$.