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

T对偶性是使弦论有别于基于点粒子的理论的对称性之一。它指出，在半径为R的圆上紧化的弦理论等价于在半径为1/R的圆上紧化的另一个弦理论。这表明，弦理论要求我们用一种新的几何来取代广义相对论所依赖的黎曼几何，对于这种新的几何，存在着各种各样的工作方案。虽然关于空间维度的T对偶性已经被很好地理解了，但是关于类时维度的T对偶性变换也是可能的，并导致了新的奇异弦理论，它可以对某些场具有非标准的动力学项，或者具有非标准数量的类时维度。了解这些奇异弦理论的地位，对于理解弦理论的本质是什么，以及什么样的几何应该取代黎曼几何至关重要。将解嵌入弦理论，将被用来研究它们提升到10维和11维。通过用D-膜、欧几里德膜和其他弦孤子来识别解的基本成分，将识别潜在的微观自由度。这将使从四维解导出的热力学配分函数与统计配分函数联系起来。这个项目中使用的联合收割机方法将超引力和弦理论与微分几何结合起来。旨在使T-对偶成为明显的几何对称性的几何形式主义，如加倍几何和例外几何，将被用于研究黑洞和宇宙时空的全局性。一个特别有趣的问题是视界如何映射到其他类型的界面，以及是否有些奇点会被弦效应消除。双重场论和例外场论的形式可以用来建立弦有效的超引力理论，这些理论在弦对偶下是明显不变的。这个项目的另一个目标是调查是否在五维和四维的最大超引力的特殊配方可以一致地截断为N=2的超引力。这将提供洞察这些框架是否可以更广泛地应用于非最大超对称弦紧化。研究这些形式是否可以为黑洞和宇宙时空几何提供新的见解将会很有趣。由于第二类双场论自然包含第二类 * 超引力，一个自然的出发点是研究T-对偶对非极值Killing视界和奇点的作用。另一个需要研究的问题是，如何用仿厄米几何来描述描述黑洞和宇宙学的双重时空。

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.