Background geometries in string and M-theory

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.