Courant Cohomology, Cartan Calculus, Connections, Curvature, Characteristic Classes
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Communications in
Mathematical Physics
Courant Cohomology, Cartan Calculus, Connections, Curvature, Characteristic Classes Miquel Cueca1 , Rajan Amit Mehta2 1 Mathematics Institute, Georg-August-University of Göttingen, Bunsenstrasse 3-5, 37073 Göttingen,
Germany. E-mail: [email protected]
2 Department of Mathematics and Statistics, Smith College, 44 College Lane, Northampton, MA 01063, USA.
E-mail: [email protected] Received: 29 November 2019 / Accepted: 2 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We give an explicit description, in terms of bracket, anchor, and pairing, of the standard cochain complex associated to a Courant algebroid. In this formulation, the differential satisfies a formula that is formally identical to the Cartan formula for the de Rham differential. This perspective allows us to develop the theory of Courant algebroid connections in a way that mirrors the classical theory of connections. Using a special class of connections, we construct secondary characteristic classes associated to any Courant algebroid. 1. Introduction Courant algebroids were introduced by Liu et al. [28], axiomatizing the properties of brackets studied by Courant and Weinstein [6,7] and Dorfman [10] in the context of Dirac constraints. More recently, Courant algebroids have also appeared in the context of generalized geometry [21], double field theory [9,22], and AKSZ sigma models [3,32]. Associated to any Courant algebroid is a cochain complex, known as the standard complex. The existence of the standard complex arises immediately from the correspondence, due to Ševera [36] and Roytenberg [31], between Courant algebroids and degree 2 symplectic dg-manifolds. In some special cases, such as exact Courant algebroids, the corresponding symplectic dg-manifold can be described explicitly. However, in general, the symplectic dg-manifold associated to a Courant algebroid E → M is defined implicitly as the minimal symplectic realization of E[1], and explicit formulas for the standard complex and its differential are only available in local coordinates. This difficulty was nicely described by Ginot and Grutzmann [17], who wrote that the standard cohomology of a Courant algebroid “is quite different from the usual cohomology theories …where the cohomology is defined using a differential given by a Cartan-type formula.” As a way of circumventing the above difficulties, Stiénon and Xu [38] defined the naïve complex of a Courant algebroid. They proved that, in degree 1, the naïve coho-
M. Cueca, R. A. Mehta
mology is isomorphic to the standard cohomology; this result was sufficient for their construction of the modular class. In the case of a transitive Courant algebroid, Ginot and Grutzmann [17] proved that the naïve cohomology is isomorphic to the standard cohomology. However, for general Courant algebroids, the two cohomologies are different. The first main result of this paper is to show that there is indeed a description of the standard complex for which the differential has a Ca
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