Manchester Geometry Seminar 1999/2000

Odd Poisson manifolds are super analogues of Poisson manifolds with a Poisson bracket changing parity (bracket of even functions is odd, bracket of even function and odd function is even). We study various generating operators of a given odd Poisson bracket on a supermanifold. They arise as the operators that map a function to the divergence of the associated Hamiltonian derivation, where divergences of derivations can be defined either in terms of Berezin volumes or of graded connections. Examples include the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd), the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres) and the "odd Laplacian" (the delta-operator) of Batalin-Vilkovisky quantization. All necessary notions will be introduced in the course of the talk.