The standard cohomology of regular Courant algebroids

TL;DR

This paper constructs a canonical dg manifold model for regular Courant algebroids, enabling the computation of their standard cohomology via spectral sequences, and applies this to specific classes of Courant algebroids.

Contribution

It introduces a minimal model for regular Courant algebroids that captures their cohomology, providing a new computational framework.

Findings

The minimal model encodes all cohomological information of regular Courant algebroids.

The standard cohomology can be computed using a Hodge-to-de Rham spectral sequence.

Applications include generalized exact Courant algebroids and those from regular Lie algebroids.

Abstract

For any regular Courant algebroid $E$ over a smooth manifold $M$ with characteristic distribution $F$ and ample Lie algebroid $A_E$, we prove that there exists a canonical homological vector field on the graded manifold $A_E[1] \oplus (TM/F)^\ast[2]$ such that the resulting dg manifold $\mathcal{M}_E$, which we call the minimal model of the Courant algebroid $E$, encodes all cohomological information of $E$. Indeed, the standard cohomology of $E$ can be identified with the cohomology of the function space on $\mathcal{M}_E$, which can be computed by a Hodge-to-de Rham type spectral sequence. We apply this result to generalized exact Courant algebroids and those arising from regular Lie algebroids.