Geometric Aspects of Quantum Field Theory and Topological String

根据该奖项，首席研究员将继续研究复杂品种的量子几何。此外，在合作PI，研究人员将承担从量子场论和拓扑弦产生的某些代数结构的研究。研究包含两个主要视角。第一个目标是在卡-丘几何上发展B-扭曲拓扑弦的量子几何的数学基础;一个完整的描述将包括开-闭弦场论和朗道-金兹伯格B-模型。其次，PI和co-PI将分析二维sigma模型的微扰量子化，并通过严格的可观测理论发展各种高阶微分算子的代数指标理论。 量子场论和弦论的蓬勃发展为数学新领域的创建提供了巨大的推动力。拟议的项目探讨了量子场论产生的例子相关的几何和拓扑结构在数学上严格的设置。目的是开发新的数学工具，并通过将物理直觉和论证锚定在数学的港湾中来揭示量子现象的几何结构。所提出的项目的影响生活在一系列的数学领域（微分几何，代数几何，拓扑）和理论物理。

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.