The Geometrisation of $\mathbb N$-manifolds of degree 2
| Autor: | Lean, Madeleine Jotz |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.geomphys.2018.07.007 |
| Popis: | 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 $2$. Comment: This is the improved and completed first part of the paper arXiv:1504.00880 (N-manifolds of degree 2 and metric double vector bundles). The improved second and third parts will be uploaded here in due time. v2: inconsistencies fixed in Example 3.18, as well as some typos. v3: Section 3.4.2 had a faulty formula that is now fixed |
| Databáze: | arXiv |
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