Geometric Structures in Field and String Theory

本研究关注的是使用数学和物理学的方法来研究场论和弦论中的几何结构。这两个领域的研究相互交织，产生了许多令人惊讶的联系和新的想法。一个丰富的见解来源是与物理量相关的几何结构变形的研究。这导致了对物理对偶性的理解，并揭示了丰富多样的新数学结构。在许多数学变形问题中，底层数学描述的某些特征的变化，称为壁交叉，是一个非常强大的特征。这些问题最好使用镜像对称来研究，它结合了数学各个领域的技术。这里提出的项目旨在获得与物理变形问题相关的新的数学思想，结构和工具，从而为物理理论以及不同数学结构之间的相互联系提供新的启发。超对称BPS（Bogomol'nyi-Prasad-Sommerfield）天体是一类适合于研究越壁现象的天体。这些在理解场论、对偶性和黑洞微观状态计数问题方面发挥了突出的作用。它们将在弦理论紧化的背景下进行研究，在那里它们与一组特殊的对象D-膜相关联，D-膜使用拓扑弦理论和镜像对称的工具进行研究。这使得人们可以系统地研究这些对理论模量的依赖性，从而得出强大的方程和重要的见解。理解BPS谱和它们的跳跃对于揭示它们在物理理论描述中的深层作用至关重要。通过将理论物理学和数学的不同领域（如微分和代数几何，表示论和数论）的思想联系起来，这些项目正在推动这些领域的知识边界。PI是弦理论的几何和数学方面的先驱和领导者。更广泛的影响：该项目位于物理学和数学前沿研究的交叉点，旨在加强两个社区之间的互动和沟通。一个不可分割的部分是培养博士后和研究生能够舒适地使用来自各个领域的方法和思想，并进行先进的跨学科研究。PI还积极通过各种讲座，演讲和出版物向公众介绍基础科学和研究。

This research is concerned with the study of geometric structures occurring in field and string theories using methods from mathematics as well as from physics. The intertwine of research in both fields has led to many surprising connections and new ideas. A rich source of insights is the study of the deformation of the geometrical structures associated to physical quantities. This has led to an understanding of physical dualities and has uncovered a rich variety of new mathematical structures. The change of certain characteristics of the underlying mathematical description known as wall crossing is a very powerful feature of many mathematical deformation problems. These questions are best studied using mirror symmetry which combines techniques from various fields of mathematics. The projects proposed here intend to derive new mathematical ideas, structures and tools associated to physical deformation problems shedding thus new light both on the physical theories as well as on the interconnections between different mathematical structures. A class of objects which is suitable for studying wall crossing phenomena are supersymmetric BPS (Bogomol'nyi-Prasad-Sommerfield) objects. These have played a prominent role in understanding field theories, dualities and black hole micro-state counting problems. They will be studied in the context of string theory compactifications where they are associated to a special set of objects, the D-branes, which are studied using tools from topological string theory and mirror symmetry. These allow one to systematically study the dependence of these on the moduli of the theory, leading to powerful equations and important insights. Understanding BPS spectra and their jumping is furthermore crucial to unravel their deep role in the description of physical theories. By connecting ideas from theoretical physics and very diverse areas of mathematics such as differential and algebraic geometry, representation theory and number theory, these projects are pushing the boundaries of knowledge in these fields. The PI is a pioneer and leader in the field of geometry and mathematical aspects of string theory. Broader Impacts: Lying at the intersection of cutting edge research in both physics and mathematics, the project seeks to enhance the interactions and communication between the two communities. An integral part is to train postdocs and graduate students to be comfortable using methods and ideas from various fields and to conduct advanced cross disciplinary research. The PI is also actively bringing fundamental science and research to the public through various talks, presentations and publications.