The mathematics of generalised dualities in M-theory

作为爱因斯坦广义相对论量子分辨率的主要候选理论，m理论仍有许多未解之谜，但其最具争议的特征是u对偶性。对偶性是表面上截然不同的系统之间的深层关系和等价物。我们的主要动机是通过揭示对偶的数学结构来阐明它们的丰富性。Goals1。利用几何和/或变形量子化的数学工具，对弦理论中广义t二象性的量子方面有更精确的理解。为描述m理论的广义对偶性的新型例外德林费尔代数（EDAs）提供数学基础。利用泊松-李对偶与量子群之间的关系，给出弦对偶的量子视角4。发展EDAs中出现的广义Yang-Baxter方程与可积模型理论之间的联系。这项工作适合EPSRC数学科学主题内的许多研究领域。特别是与数学物理相关的进展（通过对弦理论对偶性背后的数学结构的潜在应用以及与可积模型的关系），几何和拓扑学（通过与希钦广义几何及其扩展，包括非交换几何的广义并行的发展），代数（通过理解与经典例外德林费尔代数相关的量子代数和与杨-巴克斯特方程相关的代数结构）

Whilst many mysteries remain about M-theory, the leading candidate for the quantum resolution of Einstein's general relativity, its most provocative features are U-dualities. Dualities are deep relationships and equivalences between seemingly distinct systems. Our central motivation is to illuminate the richness of dualities by exposing their mathematical structures. Goals1. Develop a refined understanding of quantum aspects of generalised T-dualities in string theory employing the mathematical tools of geometric and/or deformation quantisation2. Provide a mathematical underpinning of novel Exceptional Drinfel'd Algebras (EDAs) postulated to described generalised dualities of M-theory.3. Exploit the relation between Poisson-Lie dualities and Quantum Groups to give a quantum perspective on string dualities 4. Develop a linkage between the generalised Yang-Baxter equations that arise in EDAs and theory of integrable modelsThis work fits in a number of research areas within the Mathematical Science theme of EPSRC. In particular progress will be relevant to Mathematical Physics (through the potential applications to the mathematical structures sitting behind the dualities of string theory and through the relationships to integrable models), Geometry and Topology (through the development of generalised parallelisations with Hitchin's generalised geometry and its extensions, including non-commutative geometry), Algebra (through understanding the quantum algebras related to the classical exceptional Drinfel'd algebra and algebraic structures related to the Yang-Baxter equation)