Duality Invariant Supergravity, String Geometry and Global Properties of T-Duality in arbitrary Dimension and Signature

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T-duality is one of the symmetries that sets string theory apart from theories based on point particles. It states that a string theory compactified on a circle of radius R is equivalent to another string theory compactified on a circle of radius 1/R. This shows that string theory requires us to replace the Riemannian geometry underlying general relativity with a new type of geometry, for which various working proposals exist. While T-duality with respect to spatial dimensions is well understood, T-duality transformations with respect to time-like dimensions are also possible and lead to new exotic string theories, which can have non-standard kinetic terms for some of the fields, or a non-standard number of time-like dimensions. Understanding the status of these exotic string theories is crucial for understanding what string theory fundamentally is, and what type of geometry should replace Riemannian geometry.The embedding of solutions into string theory will be used to study their lifts to 10 and 11 dimensions. By identifying the fundamental constituents of solutions in terms of D-branes, Euclidean branes and other string solitons, the underlying microscopic degrees of freedom will be identified. This will allow to relate the thermodynamic partition functions derived from four-dimensional solutions with statistical partition functions.The methods used in this project combine those of supergravity and string theory with differential geometry. Geometrical formalisms which aim at making T-duality a manifestly geometrical symmetry, such as doubled and exceptional geometry will be adapted to study black holes and cosmological spacetimes globally. A particularly interesting question is how horizons get mapped to other types of interfaces, and whether some singularities are removed by stringy effects. The formalism of doubled and exceptional field theory can be used to build string-effective supergravity theories which are manifestly invariant under string dualities. Another objective of this project is to investigate whether the exceptional formulations of maximal supergravities in five and four dimensions can be consistently truncated to N=2 supergravities. This will give insight into whether these frameworks can be applied more broadly to non-maximally supersymmetric string compactifications. It would be interesting to study whether these formalisms can provide new insights into black hole and cosmological space-time geometries. Since type-II double field theory naturally includes type-II* supergravity, a natural starting point is to investigate the action of T-duality on non-extremal Killing horizons and on singularities. Another point to investigate is how doubled spacetimes describing black holes and cosmologies can be characterised in terms of para-Hermitian geometry.