On Regular Courant Algebroids

TL;DR

This paper introduces a new characteristic class for regular Courant algebroids, generalizing known invariants like the Severa class and the Cartan 3-form, and provides a classification of these structures.

Contribution

It constructs an intrinsic degree-3 characteristic class for regular Courant algebroids and offers a complete classification linking this invariant to existing known classes.

Findings

The characteristic class reduces to the Severa class for exact Courant algebroids.

It coincides with the Cartan 3-form class for quadratic Lie algebras.

A full classification of regular Courant algebroids is provided.

Abstract

For any regular Courant algebroid, we construct a characteristic class a la Chern-Weil. This intrinsic invariant of the Courant algebroid is a degree-3 class in its naive cohomology. When the Courant algebroid is exact, it reduces to the Severa class (in H^3_{DR}(M)). On the other hand, when the Courant algebroid is a quadratic Lie algebra g, it coincides with the class of the Cartan 3-form (in H^3(g)). We also give a complete classification of regular Courant algebroids and discuss its relation to the characteristic class.