On Curvature and Torsion in Courant Algebroids

TL;DR

This paper develops a graded geometric framework for understanding curvature and torsion in Courant algebroids, introducing new definitions and computing associated Ricci and scalar curvatures within this context.

Contribution

It provides a natural graded geometric definition of Courant algebroid curvature and torsion, extending classical notions and incorporating auxiliary affine connections.

Findings

Defined graded geometric curvature and torsion for Courant algebroids

Introduced K-curvature and K-torsion tensors based on an auxiliary connection

Computed Ricci and scalar curvature for these tensors

Abstract

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are conventional tensors on the body. Finally, we compute their Ricci and scalar curvature.