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

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.