Differential forms and odd symplectic geometry

Details Differential forms and odd symplectic geometry Differential forms and odd symplectic geometry

2006-06-22

arxiv.org/abs/math/0606560v4

Mathematics

Differential Geometry

LaTeX, 14 pages. Minor editing

Scientific paper

We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {\v{S}}evera. We study the relationship of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformation for an odd symplectic form is a polynomial in matrix entries and a complete square. This is a simple but fundamental fact, parallel to Liouville's theorem for an even symplectic structure. We draw attention to the fact that the de Rham complex on $M$ naturally admits an action of the supergroup of all canonical transformations of $\Pi T^*M$. The infinitesimal generators of this action turn out to be the classical `Lie derivatives of differential forms along multivector fields'.

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