On the structure of graded symplectic supermanifolds and Courant algebroids

Abstract

This paper is devoted to a study of geometric structures expressible in terms of graded symplectic supermanifolds. We extend the classical BRST formalism to arbitrary pseudo-Euclidean vector bundles (E\to M_{0}) by canonically associating to such a bundle a graded symplectic supermanifold ((M,Ω)), with (\textrm{deg}(Ω)=2). Conversely, every such manifold arises in this way. We describe the algebra of functions on (M) in terms of (E) and show that ``BRST charges'' on (M) correspond to Courant algebroid structures on (E), thereby constructing the standard complex for the latter as a generalization of the classical BRST complex. As an application of these ideas, we prove the acyclicity of ``higher de Rham complexes'', a generalization of a classic result of Fröhlicher-Nijenhuis, and derive several easy but useful corollaries.