Courant Algebroids

By P. Bressler, A. Chervov. 2002

173 citations recorded by the source corpus.

. 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 σ -model, we show that it is indeed true in the particular case of Poisson σ -model.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7c58f9f233430013310646fce1da8b21fc1e5206

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. R. Mehta
2. M. Gualtieri
3. Y. Kosmann-Schwarzbach
4. A. Gracia-Saz
5. C. A. Abad
6. M. Crainic
7. Paul Koerber

In Geometry and Topology

1. Y. Kosmann-Schwarzbach
2. R. Mehta
3. M. Gualtieri
4. Anna Fino
5. G. Cavalcanti
6. M. Crainic
7. D. Waldram
8. M. Graña

In Mathematical Physics

1. R. Mehta
2. M. Gualtieri
3. Y. Kosmann-Schwarzbach
4. D. Waldram
5. M. Graña
6. Anna Fino
7. G. Cavalcanti
8. M. Crainic

Citations in the corpus

• Generalized Complex Geometry (2004, methodology): “The properties of this bracket were used in [27] and [32] to define the concept of exact Courant algebroid, which consists of a vector bundle extension”
• Supergroupoids, double structures, and equivariant cohomology (2006, background): “More generally, if G is a Poisson groupoid, then the double complex of the Q-algebroid [−1]T ∗A may be identified with Roytenberg’s [62] “commuting Hamiltonians” description of the Lie bialgebroid (A,A∗).”
• Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, background): “It was shown by Roytenberg [43] that a Courant algebroid can be equivalently defined as a vector bundle E → M with a Loday bracket on ΓE, an anchor ρ : E → TM and a field of non-degenerate, symmetric bilinear forms ( | ) on the fibers of E, related by a set of four additional properties.”
• Generalized Kähler Geometry (2010, background): “duli space modeled on H2 L . ApairoftransverseDirac structures,as (L,L¯)above, enjoyafurthertransversality property, which we now describe. We first recall the Baer sum operation on Courant algebroids [20, 19, 25], which is a realization of the additive structure on H1(Ω2,cl). Definition 2.3. Given tw”