Deser, Andreas (2013): Lie algebroids, nonassociative structures and nongeometric fluxes. Dissertation, LMU München: Faculty of Physics 

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Abstract
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 bialgebroids 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 Bfield. We explain how the MoyalWeyl star product arises for constant background fields and how this can be generalized to arbitrary backgrounds and nonassociative products. Noncommutative or even nonassociative spaces are expected to play a role also in closed string theory: Starting with a compactification on toroidal backgrounds with nontrivial Hflux, Tduality leads on the one hand to configurations with geometric fflux, but on the other hand to spaces which are only locally geometric in case of Qflux, or even noncommutative or nonassociative in case of the Rflux. We describe the action of Tduality in detail and review the motivation and structure of nongeometric fluxes. It will turn out, that in the local description of nongeometric backgrounds, a bivector $\beta$ is more appropriate than the original Bfield. Based on these foundations, we will describe our results in the second part. On the worldsheet level, we will analyse closed string theory with flat background and constant Hflux. The correct choice of left and rightmoving currents allows for a conformal field theory description of this background up to linear order in the Hflux. It is possible to define tachyon vertex operators and Tduality is implemented as a simple reflection of the rightmoving 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 nonvanishing threecoordinate correlator and after the application of an odd number of Tdualities, we are able to extract a threeproduct which has a structure similar to the MoyalWeyl product. We then focus on the target space and the local structure of the H,f,Q and Rfluxes. An algebra based on vector fields is proposed, whose structure functions are given by the fluxes and Jacobiidentities allow for the computation of Bianchiidentities. 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 nongeometric Q and Rfluxes, the Bfield gets replaced by a bivector $\beta$, which is supposed to serve as the dual object to B under Tduality. 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 KoszulSchouten bracket, to answer this question positively. This construction enables us to formulate covariant derivatives, torsion, curvature and gauge symmetries and culminates in an EinsteinHilbert 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 nongeometry, we speculate about the application of Lie algebroid constructions to supersymmetry and the extension to the case of Filippov threealgebroids, which could play a role in Mtheory.
Item Type:  Theses (Dissertation, LMU Munich) 

Keywords:  string theory, flux compactification, Lie algebroid, differential geometry 
Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 530 Physics 
Faculties:  Faculty of Physics 
Language:  English 
Date of oral examination:  30. July 2013 
1. Referee:  Blumenhagen, Ralph 
MD5 Checksum of the PDFfile:  b494d922724d23dc86e5fa1300d6fe84 
Signature of the printed copy:  0001/UMC 21473 
ID Code:  16048 
Deposited On:  04. Sep 2013 08:30 
Last Modified:  24. Oct 2020 00:38 