Cohomology of Courant algebroids with split base

TL;DR

This paper investigates the cohomology of Courant algebroids, proving equivalences in transitive cases, introducing a spectral sequence for general cases, and explicitly describing the transgression homomorphism for split base algebroids.

Contribution

It establishes the equivalence of standard and naive cohomology for transitive Courant algebroids and introduces a spectral sequence and a transgression homomorphism for split base cases.

Findings

Standard cohomology equals naive cohomology for transitive Courant algebroids.

A spectral sequence converging to the standard cohomology is constructed.

Explicit formula for the transgression homomorphism T_3 in generalized exact Courant algebroids.

Abstract

We study the (standard) cohomology $H^\bullet_{st}(E)$ of a Courant algebroid $E$. We prove that if $E$ is transitive, the standard cohomology coincides with the naive cohomology $H_{naive}^\bullet(E)$ as conjectured by Stienon and Xu. For a general Courant algebroid we define a spectral sequence converging to its standard cohomology. If $E$ is with split base, we prove that there exists a natural transgression homomorphism $T_3$ (with image in $H^3_{naive}(E)$) which, together with the naive cohomology, gives all $H^\bullet_{st}(E)$. For generalized exact Courant algebroids, we give an explicit formula for $T_3$ depending only on the \v{S}evera characteristic clas of $E$.