Geometric Aspects of Quantum Field Theory and Topological String

在这个奖项下，首席研究员将继续他对复杂品种的量子几何的研究。此外，研究者将与共同负责人合作，对量子场论和拓扑弦产生的某些代数结构进行研究。本研究包含两个主要视角。第一个目标是在Calabi-Yau几何上发展b -扭曲拓扑弦量子几何的数学基础；完整的描述将包括开闭弦场论和朗道-金兹堡b模型。其次，PI和co-PI将分析二维sigma模型的微扰量化，并通过严格的可观测理论为更高的微分算子发展各种代数指标理论。量子场论和弦论的蓬勃发展为数学新领域的创造提供了巨大的推动力。提出的项目在数学严格的设置中探索与几何和拓扑相关的量子场论产生的例子。其目的是开发新的数学工具，并通过在数学的港湾中锚定物理直觉和论证来揭示量子现象的几何。所提议的项目的含义存在于一系列数学领域（微分几何、代数几何、拓扑）和理论物理中。

Under this award, the principal investigator will continue his research on the quantum geometry of complex varieties. Further, in collaboration with the co-PI, the investigator will undertake a study of certain algebraic structures arising from quantum field theory and topological strings. The research contains two major perspectives. The first aim is to develop the mathematical foundations of the quantum geometry of the B-twisted topological string on a Calabi-Yau geometry; a complete description will include open-closed string field theory and the Landau-Ginzburg B-model. Secondly, the PI and co-PI will analyze the perturbative quantization of two dimensional sigma models and develop various algebraic index theories for higher differential operators via a rigorous observable theory. The blossoming of quantum field theory and string theory has provided a great driving force in the creation of new areas in mathematics. The proposed project explores examples arising from quantum field theory relevant to geometry and topology in a mathematically rigorous setting. The aim is to develop new mathematical tools and reveal the geometry of quantum phenomenon by anchoring physical intuition and arguments in the harbor of mathematics. The implications of the proposed projects live both in an array of mathematical fields (differential geometry, algebraic geometry, topology) and theoretical physics.