Courant Algebroids

TL;DR

This paper explores properties of Courant algebroids, linking them to gerbes, Dirac structures, and Lie algebroids, and proposes a new construction related to the WZNW-Poisson sigma model.

Contribution

It introduces the conducting bundle construction for Courant algebroids, connects them to gerbes and Dirac structures, and proposes a Lie algebroid construction on loop spaces with applications to sigma models.

Findings

The conducting bundle construction attaches Courant algebroids to gerbes.

WZNW-Poisson condition is equivalent to a Dirac structure in a Courant algebroid.

The proposed Lie algebroid construction on loop spaces applies to the WZNW-Poisson sigma model.

Abstract

This paper is devoted to studying some properties of the Courant algebroids: we explain the so-called "conducting bundle construction" and use it to attach the Courant algebroid to Dixmier-Douady gerbe (following ideas of P. Severa). We remark that WZNW-Poisson condition of Klimcik and Strobl (math.SG/0104189) is the same as Dirac structure in some particular Courant algebroid. We propose the construction of the Lie algebroid on the loop space starting from the Lie algebroid on the manifold and conjecture that this construction applied to the Dirac structure above should give the Lie algebroid of symmetries in the WZNW-Poisson $\sigma$-model, we show that it is indeed true in the particular case of Poisson $\sigma$-model.