U-dualities in Type II string theories and M-theory

In this thesis the recently developed duality covariant approach to string and M-theory is investigated. In this formalism the U-duality symmetry of M-theory or T-duality symmetry of Type II string theory becomes manifest upon extending coordinates that describe a background. The effective potential of Double Field Theory is formulated only up to a boundary term and thus does not capture possible topological effects that may come from a boundary. By introducing a generalised normal we derive a manifestly duality covariant boundary term that reproduces the known Gibbons-Hawking action of General Relativity, if the section condition is imposed. It is shown that the full potential can be represented as a sum of the scalar potential of gauged supergravity and a topological term that is a full derivative. The latter is conjectured to capture non-trivial topological information of the corresponding background, such as monodromy around an exotic brane. Next we show that the Scherk-Schwarz reduction of M-theory extended geometry successfully reproduces known structures of maximal gauged supergravities. Local symmetries of the extended space defined by a generalised Lie derivatives reduce to gauge transformations and lead to the embedding tensor written in terms of twist matrices. The scalar potential of maximal gauged supergravity that follows from the effective potential is shown to be duality invariant with no need of section condition. Instead, this condition, that assures the closure of the algebra of generalised diffeomorphisms, takes the form of the quadratic constraints on the embedding tensor.