Geometric Structures in Field and String Theory

这项研究涉及使用数学和物理学的方法来研究发生在场论和弦理论中的几何结构。这两个领域的研究交织在一起，产生了许多令人惊讶的联系和新想法。一个丰富的洞察力来源是研究与物理量相关的几何结构的变形。这导致了对物理二元性的理解，并揭示了丰富多样的新的数学结构。被称为墙交叉的底层数学描述的某些特征的改变是许多数学变形问题的一个非常强大的特征。这些问题最好用镜像对称性来研究，它结合了不同数学领域的技术。这里提出的项目旨在得出与物理变形问题有关的新的数学思想、结构和工具，从而使人们对物理理论以及不同数学结构之间的相互联系有了新的了解。一类适合研究穿墙现象的物体是超对称BPS(Bogomol‘nyi-Prasad-Sommerfield)物体。这些理论在理解场论、对偶性和黑洞微态计数问题上起到了突出的作用。它们将在弦理论紧凑化的背景下进行研究，其中它们与一组特殊的对象-D-膜相关联，这些对象是使用拓扑弦理论和镜像对称的工具来研究的。这使得人们能够系统地研究这些方程对理论模数的依赖，从而产生强大的方程和重要的见解。此外，了解BPS光谱及其跳跃对于解开它们在描述物理理论中的深层次作用至关重要。通过将理论物理和非常不同的数学领域(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。圆周率是弦理论几何和数学领域的先驱和领导者。更广泛的影响：该项目位于物理学和数学前沿研究的交汇点，旨在加强两个社区之间的互动和沟通。一个不可或缺的部分是培养博士后和研究生熟练地使用不同领域的方法和想法，并进行高级跨学科研究。国际和平研究所还通过各种讲座、演讲和出版物，积极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.