258 research outputs found

    Transitive Courant algebroids

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    We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal Whitney sum E⊕CE\oplus C where E is a given Courant algebroid and C is a flat, pseudo- Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, i.e., a Courant algebroid with a surjective anchor, and describe a class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric connection. In particular, this class contains all the transitive Courant algebroids of minimal rank; for these, the flat term mentioned above is zero. The results extend to regular Courant algebroids, i.e., Courant algebroids with a constant rank anchor. The paper ends with miscellaneous remarks and an appendix on Dirac linear spaces.Comment: LaTex, 27 page

    AV-Courant algebroids and generalized CR structures

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    We construct a generalization of Courant algebroids which are classified by the third cohomology group H3(A,V)H^3(A,V), where AA is a Lie Algebroid, and VV is an AA-module. We see that both Courant algebroids and E1(M)\mathcal{E}^1(M) structures are examples of them. Finally we introduce generalized CR structures on a manifold, which are a generalization of generalized complex structures, and show that every CR structure and contact structure is an example of a generalized CR structure.Comment: 18 page

    Twisted Courant algebroids and coisotropic Cartan geometries

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    In this paper, we show that associated to any coisotropic Cartan geometry there is a twisted Courant algebroid. This includes in particular parabolic geometries. Using this twisted Courant structure, we give some new results about the Cartan curvature and the Weyl structure of a parabolic geometry. As more direct applications, we have Lie 2-algebra and 3D AKSZ sigma model with background associated to any coisotropic Cartan geometry

    On Regular Courant Algebroids

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    For any regular Courant algebroid, we construct a characteristic class a la Chern-Weil. This intrinsic invariant of the Courant algebroid is a degree-3 class in its naive cohomology. When the Courant algebroid is exact, it reduces to the Severa class (in H^3_{DR}(M)). On the other hand, when the Courant algebroid is a quadratic Lie algebra g, it coincides with the class of the Cartan 3-form (in H^3(g)). We also give a complete classification of regular Courant algebroids and discuss its relation to the characteristic class.Comment: Section 3.3 and references added; An error about classification is correcte
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