On the structure of graded symplectic supermanifolds and Courant algebroids

TL;DR

This paper explores the relationship between graded symplectic supermanifolds and Courant algebroids, extending the BRST formalism to describe their structures and proving the acyclicity of higher de Rham complexes.

Contribution

It establishes a canonical correspondence between graded symplectic supermanifolds and Courant algebroids, extending the classical BRST formalism to arbitrary pseudo-Euclidean bundles.

Findings

Constructs a graded symplectic supermanifold for any pseudo-Euclidean bundle.

Shows that BRST charges correspond to Courant algebroid structures.

Proves the acyclicity of higher de Rham complexes.

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,\Omega)), with (\textrm{deg}(\Omega)=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 Fr\"{o}hlicher-Nijenhuis, and derive several easy but useful corollaries.