Courant Algebroids

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 σ-model, we show that it is indeed true in the particular case of Poisson σ-model. 0 COURANT ALGEBROIDS 1 ContentsAt the end of this section we make a remark that WZNW-Poisson condition of Klimcik and Strobl [16] coincides with Dirac structure in the Courant algebroid twisted by the 3-form (this was discovered independently by P.Severa and A.Weinstein [27]).Section 5 is motivated by the papers [16,17] and we try to give more geometric understanding of some of their constructions. Consider the diagram:We will show that if A is an algebroid on Y then there pr * ev * A has a natural structure of algebroid, considering the T * M as an algebroid with the bracket constructed from Poisson bracket on M and considering its pullback pr * ev * T * M on the loop space LM we see that this algebroid is the same as algebroid considered in [17] which plays the role of symmetries in Poisson σ-model. Analogous algebroids were considered in [9,10]. We also hope that for a Courant algebroid on M one can naturally construct Lie algebroid on LM this construction should be agreed with the construction of line bundle on LM from gerbe on M and the transgression of characteristic class of gerbe to characteristic class of line bundle on LM.Remark on Notations. In this work will denote by O functions on our manifolds. Most of the considerations works for analytic or algebraic as well as for smooth functions. The basic situation in our paper that O are smooth functions. We are sorry, that it may cause some inconvenience for those readers, who get used to think about O only as about analytic or algebraic functions. The same words should be said about our notation T -it is tangent bundle, basic example is smooth tangent bundle for smooth manifold, but most considerations works well for the holomorphic tangent bundle as well.Remark on Further Developments. This paper was basically finished in summer 2001. Since that time there have been some further developments in the subject. One of the authors (P.B.) in [28] proposed the analogy between Courant algebroids and "vertex algebroids" i.e. algebroids that appeared in [22,23] and which gives rise to the chiral de Rham complex and also he gave a "coordinate free" construction and proved the uniqueness of the vertex algebroid. The paper [27] contains many important ideas related to the subject of present work.Acknowledgements. The first part of this work owes its existence to...

2-Plectic geometry, Courant algebroids, and categorified prequantization

“…In other words, a splitting of an exact Courant algebroid is an isotropic splitting of the sequence of vector bundles. Bressler and Chervov call splittings 'connections' [6]. If s is a splitting and B ∈ Ω 2 (M ) is a 2-form then one can construct a new splitting:…”

“…Furthermore, one can show that any two splittings on an exact Courant algebroid must differ by a 2-form on M in this way. Hence the space of splittings on an exact Courant algebroid is an affine space modeled on the vector space of 2-forms Ω 2 (M ) [6]. The failure of a splitting to preserve the bracket gives a suitable notion of curvature:…”

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“…A section λ of the associated algebroid A(G Σ ) (this algebroid has been defined, at least for Σ one-dimensional, in [4] intrinsically in terms of the Lie algebroid T * M , so it exists also for nonintegrable Poisson manifolds) is defined by giving a section…”

Extended Riemannian geometry III: global Double Field Theory with nilmanifolds

“…13 The Courant algebroid is to the abelian gerbe roughly what the Atiyah algebroid is to a principal circle bundle, see e.g. [31,32]. We start with a description of the Lie 2-algebra associated to C j,k , before we interpret it as the appropriate symmetry Lie 2-algebra.…”