Differential Forms and Odd Symplectic Geometry

We remind 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. Ševera. We study relations 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 attract attention to the fact that the de Rham complex on M naturally admits an action of the supergroup of all canonical transformations of ΠT ∗M . The infinitesimal generators of this action turn out to be the classical ‘Lie derivatives of differential forms along multivector fields’. Odd symplectic geometry (more generally, odd Poisson geometry) or geometry of odd bracket is the mathematical basis of the Batalin– Vilkovisky method [1, 2, 3] in quantum field theory. Odd symplectic geometry possesses features connecting it with both classical (“even”) symplectic geometry and Riemannian geometry. In particular, on an odd symplectic manifold naturally arise odd Laplace operators, i.e., the second order differential operators whose principal symbol is the odd quadratic form corresponding to the odd bracket [4]. The key difference with the Riemannian case is that the definition of an odd Laplace operator, in general, requires an extra piece of data besides the “metric”, namely, a choice of a volume form (even for a Laplacian acting on functions). This is due to the fundamental fact that on an odd symplectic manifold there is no invariant volume element [4]. However, as it was discovered by one of the authors, there is one isolated case where an odd Laplacian is defined canonically by the symplectic structure without any extra data [6, 7, 8]. It is an operator acting on densities of weight 1/2 (half-densities or semidensities). This fact is not obvious, and there is no simple explanation. A known proof is based on an analysis of the canonical transformations of the odd bracket. In works [9, 11, 10] further phenomena related with odd Laplacians on odd Poisson manifolds were discovered, such as the existence of a natural ‘master’ groupoid acting on volume forms, so that its orbits correspond to Laplacians on half-densities. The symplectic case is distinguished by the existence of a distinguished orbit, which gives the “canonical” operator.