Natural quotients on split supercotangent bundles and their canonical supersymplectic structures
This paper is devoted to the generalization for graded manifolds of the notion of canonical 1-form on the cotangent bundle and the symplectic structure naturally defined by its exterior differential. This is an interesting problem and has been solved in the paper under the assumption that the supercotangent bundle of the graded manifold \((M,\bigwedge E)\) considered is split. The authors then construct two quotient graded manifolds of the supercotangent bundle with underlying manifolds \(TM\) and \(E\) respectively. The sum of the pull-backs of the corresponding \(\mathbb Z_2\)-graded 1-forms on these supermanifolds is a form \(\Theta\) on the supercotangent bundle which is the graded analogous to the canonical 1-form of the ordinary cotangent bundle. It enjoys an additional very nice property, and this is that the radical of \(d\Theta\) is a non-integrable involutive distribution. However, the radicals of the homogeneous components of \(d\Theta\) are integrable and give rise to supermanifolds of leaves which are shown to be the quotients of the supercotangent bundle formerly constructed. Moreover, it is proven that these supermanifolds carry natural supersymplectic structures. The splitness of the supercotangent bundle is used under the form of existence of a connection on \(E\oplus TM\oplus E^\ast\) that allows explicit expressions of the odd and even generators of the derivations on the supercotangent bundle.
scientific article; zbMATH DE number 1441611