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Jeschek, Claus (2005): Background geometries in string and M-theory. Dissertation, LMU München: Fakultät für Physik



In this thesis we consider background geometries resulting from string theory compactifications. In particular, we investigate supersymmetric vacuum spaces of supergravity theories and topological twisted sigma models by means of classical and generalised G-structures. In the first part we compactify 11d supergravity on seven-dimensional manifolds due to phenomenological reasons. A certain amount of supersymmetry forces the internal background to admit a classical SU(3)- or G2-structure. Especially, in the case that the four-dimensional space is maximally symmetric and four form fluxes are present we calculate the relation to the intrinsic torsion. The second and main part is two-fold. Firstly, we realise that generalised geometries on six-dimensional manifolds are a natural framework to study T-duality and mirror symmetry, in particular if the B-field is non-vanishing. An explicit mirror map is given and we apply this idea to the generalised formulation of a topological twisted sigma model. Implications of mirror symmetry are studied, e.g. observables and topological A- and B-branes. Secondly, we show that seven-dimensional NS-NS backgrounds in type II supergravity theories can be described by generalised G2-geometries. A compactification on six manifolds leads to a new structure. We call this geometry a generalised SU(3)-structure. We study the relation between generalised SU(3)- and G2-structures on six- and seven-manifolds and generalise the Hitchin-flow equations. Finally, we further develop the generalised SU(3)- and G2-structures via a constrained variational principle to incorporate also the remaining physical R-R fields.