The mathematics of generalised dualities in M-theory

虽然关于M理论仍然存在许多谜团，但它是爱因斯坦广义相对论量子分辨率的主要候选者，它最具挑衅性的特征是U对偶性。对偶是看似不同的系统之间的深层关系和等价。我们的主要动机是通过揭示对偶的数学结构来阐明对偶的丰富性。目标1.发展弦理论中广义T对偶的量子方面的精细理解，采用几何和/或变形量子化的数学工具2。提供一个数学基础的新的例外Drinfel'd代数（EDA）假设描述广义对偶的M-理论。利用Poisson-Lie对偶和量子群之间的关系，给出弦对偶的量子观点4。发展广义杨巴克斯特方程之间的联系，出现在EDAs和理论的可积modelsThis工作适合在一些研究领域内的数学科学主题的EPSRC。特别是进展将有关数学物理（通过潜在的应用程序的数学结构坐在后面的二元性弦理论，并通过可积模型的关系），几何和拓扑（通过发展广义平行与希钦的广义几何及其扩展，包括非交换几何），代数（通过理解与经典例外Drinfel'd代数相关的量子代数和与Yang-Baxter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)