Current algebra, generalised geometry and integrable models in string theory

One consequence of the extended nature of the string is that more general background geometries than Riemannian manifolds are possible in string theory. In particular, when gluing charts not only diffeomorphisms (or other gauge transformations of the background) but also (string) duality transformations are allowed. These geometries are called non-geometric spaces. Their mathematical formulation is based on Hitchin's and Gualtieri's generalised (or O(d,d)-)geometry. In this thesis it is shown that, despite previous results in the literature, the Poisson structure - to be more precise: the current algebra - of a string is not O(d,d)-invariant. Its correct treatment requires the so-called para-Hermitian geometry. Building on that, a Hamiltonian formulation of the classical world-sheet theory in a generic, geometric or non-geometric, background is proposed. The essence of this formulation is that the generalised fluxes, characterising such a background, describe a deformation of the current algebra. This formulation extends to backgrounds for which there is no Lagrangian description of the world-sheet theory - namely magnetically charged backgrounds and those that violate the section condition of generalised geometry, at the cost of violating the Jacobi identity of the current algebra. Two applications of this formulation are discussed. On the one hand, one can read off the non-commutative and non-associative interpretation directly from the deformed current algebra. On the other hand, one can derive two generalisations of non-abelian T-duality that go beyond the standard factorised Poisson-Lie T-duality. There is a non-abelian T-duality group, analogous to O(d,d) for abelian T-duality. Moreover, there are generalisations for Poisson-Lie T-duality for models with generic constant generalised fluxes. A generalisation of these results to M-branes in M-theory seems possible. For the membrane in M-theory compactified on a four-dimensional space, it is shown that the current algebra is not U-duality invariant. Exactly as for the string, a Lie bracket appears that is connected to para-Hermitian exceptional generalised geometry. In contrast to the string, even manifest covariance under the U-duality group, here SL(5), is only possible when introducing additional objects, the membrane charges. With the help of the typical double dimensional reduction from M-theory to type IIa superstring theory, one can relate the membrane and string currents. Another central topic of this thesis is integrability in context of string theory. In particularly symmetric backgrounds, like Minkowski spacetime or certain Anti-de Sitter compactifications, string theory is exactly solvable (integrable). Deformations of these backgrounds, that preserve integrability of the world-sheet theory, have been studied extensively in the last years. It turned out that many of these deformations can be described in terms of generalised geometry. In this thesis it is shown that a big class of these deformations, the homogeneous Yang-Baxter deformations, are nothing else than the beta-shifts of the non-abelian T-duality group mentioned above.