Extended Geometry and String Theory

We will explore global formulations of doubled and extended geometry in the language of para-Hermitian geometry and its generalizations, in analogy to related developments in generalized geometry. In particular, we will develop the notion of heterotic para-Hermitian manifold and its relation to heterotic double field theory and heterotic Courant algebroids, as well as generalizations to extended geometries underlying exceptional field theories in compactifications of M-theory and their relations to the higher algebroids of exceptional generalized geometry. In all generalizations we will work out the corresponding notions of DFT algebroid known from investigations of doubled geometry, as well as the properties of related sigma-models and homotopy algebras which control their automorphisms.