Deser, Andreas (2013): Lie algebroids, non-associative structures and non-geometric fluxes. Dissertation, LMU München: Fakultät für Physik
In the first part of this thesis, basic mathematical and physical concepts are introduced. The notion of a Lie algebroid is reviewed in detail and we explain the generalization of differential geometric structures when the tangent bundle is replaced by a Lie algebroid. In addition, Lie bi-algebroids and Courant algebroids are defined. This branch of mathematics finds its application in deformation quantization, which in string theory is the dynamics of open strings in the presence of a background B-field. We explain how the Moyal-Weyl star product arises for constant background fields and how this can be generalized to arbitrary backgrounds and non-associative products. Non-commutative or even non-associative spaces are expected to play a role also in closed string theory: Starting with a compactification on toroidal backgrounds with non-trivial H-flux, T-duality leads on the one hand to configurations with geometric f-flux, but on the other hand to spaces which are only locally geometric in case of Q-flux, or even non-commutative or non-associative in case of the R-flux. We describe the action of T-duality in detail and review the motivation and structure of non-geometric fluxes. It will turn out, that in the local description of non-geometric backgrounds, a bi-vector $\beta$ is more appropriate than the original B-field. Based on these foundations, we will describe our results in the second part. On the world-sheet level, we will analyse closed string theory with flat background and constant H-flux. The correct choice of left- and right-moving currents allows for a conformal field theory description of this background up to linear order in the H-flux. It is possible to define tachyon vertex operators and T-duality is implemented as a simple reflection of the right-moving sector. In analogy to the open string case, correlation functions allow to extract information on the algebra of observables on the target space. We observe a non-vanishing three-coordinate correlator and after the application of an odd number of T-dualities, we are able to extract a three-product which has a structure similar to the Moyal-Weyl product. We then focus on the target space and the local structure of the H-,f-,Q- and R-fluxes. An algebra based on vector fields is proposed, whose structure functions are given by the fluxes and Jacobi-identities 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 $TM \oplus T^*M$, where the fluxes are realized by the commutation relations of a basis of sections. As was 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 $\beta$, 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 $\beta$-field to gravity. It turns out that we have to use the language of Lie algebroids to extend differential geometry from the tangent bundle of the target space to its cotangent bundle, equipped with a twisted version of the Koszul-Schouten bracket, to answer this question positively. This construction enables us to formulate covariant derivatives, torsion, curvature and gauge symmetries and culminates in an Einstein-Hilbert action for the metric and $\beta$-field. We observe that this action is related to standard bosonic low energy string theory by a field redefinition, which was discovered by Seiberg and Witten and which we described in detail in the first part. Furthermore it turns out, that the whole construction can be extended to higher order corrections in $\alpha'$ and to the type IIA superstring. We conclude by giving an outlook on future directions. After clarifying the relation of Lie algebroids to non-geometry, we speculate about the application of Lie algebroid constructions to supersymmetry and the extension to the case of Filippov three-algebroids, which could play a role in M-theory.