On odd Laplace operators. (English) Zbl 1044.58042

Khudaverdian, Hovhannes M.; Voronov, Theodore

Lett. Math. Phys. 62, No. 2, 127-142 (2002).

The authors consider odd Laplace operators acting on densities of various weights on an odd Poisson (= Schouten) manifold M. They prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ’orbit space’ of volume forms. The space of volume forms on M is partitioned into orbits by the action of a natural groupoid whose arrows correspond to the solutions of the quantum Batalin-Vilkovisky equations. They compare their situation with that of Riemannian and even Poisson manifolds and show that the square of an odd Laplace operator is a Poisson vector field defining an analog of Weinstein’s ’modular class’.

Reviewer: Robert A. Wolak (Kraków)

MSC:

58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53D17	Poisson manifolds; Poisson groupoids and algebroids
58B20	Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58A50	Supermanifolds and graded manifolds
81T70	Quantization in field theory; cohomological methods

Keywords:

odd Laplace operator; Poisson manifold; Schouten manifold; volume form