Courant algebroids which are not exact

Question

Does somebody have some interesting examples of Courant algebroids which are not exact? By exact I mean one which is of the form $TM\oplus T^\star M$ with the standard bracket twisted by a closed 3-form $H$.

Thank you!

Answer 1

The answer depends on what you mean by ``interesting."

For example the paper "On the Geometric Structure of Hamiltonian Systems with Ports" by Jochen Merker, J Nonlinear Sci (2009) 19: 717–738 DOI 10.1007/s00332-009-9052-3, may be considered as dealing with interesting examples of Courant algebroids. The algebroids there are not of the form $TM\oplus T^*M \to M$.

Grützmann's thesis (arXiv:1004.1487 [math.DG]) maybe another good place to look.

Answer 2

To add another example which hasn't already been mentioned, suppose that a Lie algebra $\mathfrak{g}$ carries an $\operatorname{ad}-$invariant non-degenerate symmetric 2-form, $\langle\cdot,\cdot\rangle$, then $\mathfrak{g}$ is a Courant algebroid (this example already appears in Liu-Weinstein-Xu's original paper). When $\mathfrak{g}$ is semi-simple, then the killing form works.

Now suppose that $\mathfrak{g}$ acts on a manifold $M$ and all the stabilizers are coisotropic (with respect to the quadratic form on $\mathfrak{g}$), then $\mathbb{E}:=\mathfrak{g}\times M$ is also a Courant algebroid (where the anchor is given by the action map). These examples occur frequently in nature, see: http://arxiv.org/abs/0811.4554 and http://arxiv.org/abs/1110.1525.

Answer 3

A very interesting class of non-exact Courant algebroids are transitive Courant algebroids. A transitive Courant algebroid is a Courant algebroid whose anchor map is surjective. A beautiful way to construct this class of Courant algebroids is via generalized reduction. Given a principal $G$-bundle $P$ with vanishing first Pontryagin class, its generalized tangent bundle $TP\oplus T^{\ast}P$ carries a canonical structure of an exact Courant algebroid endowed with a lifted $G$-action. The procedure of generalized reduction allows to construct a Courant algebroid $E$ with underlying vector bundle

$E = TM\oplus ad\, P\oplus T^{\ast}M$

over $M=P/G$. $E$ is the transitive Courant algebroid obtained by generalized reduction. You can check Severa's letters to Weinstein for more information. This procedure was then further investigated by Bursztyn, Cavalcanti and Gualtieri. Just if your curious, transitive Courant algebroids recently found an elegant application in Heterotic String Theory: the equations of motion of Heterotic Supergravity can be written has the Ricci-flatness of a particular metric-compatible generalized connection on $E$. You can check:

http://arxiv.org/abs/1304.4294

for more details.