Lie algebroids, non-associative structures and non-geometric fluxes

Abstract

In the first part of this article, the geometry of Lie algebroids as well as the Moyal-Weyl star product and some of its generalizations in open string theory are reviewed. A brief introduction to T-duality and non-geometric fluxes is given. Based on these foundations, more recent results are discussed in the second part of the article. On the world-sheet level, we will analyse closed string theory with flat background and constant H-flux. After an odd number of T-dualities, correlation functions allow to extract a three-product having a pattern similar to the Moyal-Weyl product. We then focus on the target space and the local appearance of the various fluxes. An algebra based on vector fields is proposed, whose structure functions are given by the fluxes. Jacobi-identities for vector fields allow for the computation of Bianchi-identities. Based on the latter, we give a proof for a special Courant algebroid structure on the generalized tangent bundle, where the fluxes are realized by the commutation relations of a basis of sections. As reviewed in the first part of this work, in the description of non-geometric Q- and R-fluxes, the B-field gets replaced by a bi-vector β, which is supposed to serve as the dual object to B under T-duality. A natural question is about the existence of a differential geometric framework allowing the construction of actions manifestly invariant under coordinate- and gauge transformations, which couple the β-field to gravity. It turns out that Lie algebroids are the right language to answer this question positively. We conclude by giving an outlook on future directions.