On odd Laplace operators

Abstract

We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold $M$. We 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. This includes earlier results for odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on $M$ is partitioned into orbits by a natural groupoid whose arrows correspond to the solutions of the quantum Batalin--Vilkovisky equations. We give a comparison with the situation for Riemannian and even Poisson manifolds. In particular, the square of odd Laplace operator happens to be a Poisson vector field defining an analog of Weinstein's ``modular class''.

LaTeX2e, 18p. Exposition reworked and slightly compressed; we added a table with a comparison of odd Poisson geometry with Riemannian and even Poisson cases. Latest update: minor editing

Cited by in corpus (32)

• Higher derived brackets and homotopy algebras
• Characteristic classes of gauge systems
• Higher Derived Brackets for Arbitrary Derivations
• Higher Poisson Brackets and Differential Forms
• Schwinger-Dyson equation for non-Lagrangian field theory
• On Generalized Gauge-Fixing in the Field-Antifield Formalism
• Odd Scalar Curvature in Anti-Poisson Geometry
• Graded Geometry, $Q$-Manifolds, and Microformal Geometry
• Odd Scalar Curvature in Field-Antifield Formalism
• On volumes of classical supermanifolds
• "Nonlinear pullbacks" of functions and $L_{\infty }$-morphisms for homotopy Poisson structures
• Semidensities, Second-Class Constraints and Conversion in Anti-Poisson Geometry
• Unimodular homotopy algebras and Chern-Simons theory
• A Note on Semidensities in Antisymplectic Geometry
• Riemannian structures on ${\mathbb{Z}}_2^n$-manifolds
• Microformal geometry and homotopy algebras
• Gauge Independence in a Higher-Order Lagrangian Formalism via Change of Variables in the Path Integral
• Minimal models of quantum homotopy Lie algebras via the BV-formalism
• Graded manifolds and Drinfeld doubles for Lie bialgebroids
• Odd Connections on Supermanifolds: Existence and relation with Affine Connections
• Geometry of differential operators of second order, the algebra of densities, and groupoids
• On a Batalin--Vilkovisky operator generating higher Koszul brackets on differential forms
• A Comparative Study of Laplacians and Schroedinger-Lichnerowicz-Weitzenboeck Identities in Riemannian and Antisymplectic Geometry
• Projective Connections and Schwarzian Derivatives for Supermanifolds, and Batalin-Vilkovisky Operators
• Modular classes of Q-manifolds: a review and some applications
• Geometry of differential operators, odd Laplacians, and homotopy algebras
• Felix Alexandrovich Berezin and his work
• The existence of a canonical lifting of even Poisson Structures to the Algebra of Densities
• The canonical BV Laplacian on half-densities
• Quantization of (-1)-Shifted Derived Poisson Manifolds
• Supersymmetric sigma models and harmonic superfunctions
• Functionals and the Quantum Master Equation