Metric-Bourbaki algebroids: Cartan calculus for M-theory

Abstract

String and M theories seem to require generalizations of usual notions of differential geometry on smooth manifolds. Such generalizations usually involve extending the tangent bundle to larger vector bundles equipped with various algebroid structures such as Courant algebroids, higher Courant algebroids, metric algebroids, or G-algebroids. The most general geometric scheme is not well understood yet, and a unifying framework for such algebroid structures is needed. Our aim in this paper is to propose such a general framework. Our strategy is to follow the hierarchy of defining axioms for a Courant algebroid: almost-Courant - metric - pre-Courant - Courant. In particular, we focus on the symmetric part of the bracket and the metric invariance property, and try to make sense of them in a manner as general as possible. These ideas lead us to define new algebroid structures which we dub Bourbaki and metric-Bourbaki algebroids, together with their almost- and pre-versions. For a special case of metric-Bourbaki algebroids that we call exact, we construct a collection of maps which generalize the Cartan calculus of exterior derivative, Lie derivative and interior product. This is done by a kind of reverse-mathematical analysis of the Ševera classification of exact Courant algebroids. By abstracting crucial properties of this collection of maps, we define the notion of Bourbaki calculus. Conversely, given an arbitrary Bourbaki calculus, we construct a metric-Bourbaki algebroid by building up a standard bracket that is analogous to the Dorfman bracket. Moreover, we prove that any exact metric-Bourbaki algebroid satisfying some further conditions has to have a bracket that is the twisted version of the standard bracket; a partly analogous result to Ševera classification. We prove that many physically and mathematically motivated algebroids from the literature are examples of these new algebroids, and when possible we construct a Bourbaki calculus on them. In particular, we show that the Cartan calculus can be seen as the Bourbaki calculus corresponding to an exact higher Courant algebroid. We also point out examples of Bourbaki calculi including the generalization of the Cartan calculus on vector bundle valued forms. One straightforward generalization of our constructions might be done by replacing the tangent bundle with an arbitrary Lie algebroid A. This step allows us to define an extension of our results, A-version, and extend our main results for them while proving many other algebroids from the literature fit into this framework.