LEAPS-MPS: Quantum Field Theories and Elliptic Cohomology

数学和物理之间的对话在这两个学科中都有产生洞察力的悠久历史，往往揭示出深刻的、统一的结构，例如广义相对论和黎曼几何。拓扑学是对形状的研究；具体地说，拓扑学是指在形状平滑变换时保持不变的形状的属性。在过去的一个世纪里，量子物理的进步，特别是凝聚态物理和超弦理论，依赖于拓扑学的创新。其中一项创新是因式分解代数，这是一种(经典或量子)场论的可观测数据的数学模型。这个项目使用因式分解代数来探索某种类型的量子场论和拓扑学中的对象之间的一种猜想关系，称为椭圆上同调。几十年来，这一关系的确切结构一直难以捉摸；解决这个猜想将有助于我们深入了解椭圆上同调的几何解释，以及与弦理论相关的量子物理学的基本问题。该奖项还支持一个地区性的拓扑学和数学物理会议，一个杰出的讲座系列，并为国际数学学院的本科生提供研究机会，既增加了少数族裔在数学方面的参与，也改善了国际数学学院的研究环境。更具体地说，这个项目探索等变因式分解代数、超对称扭场理论和椭圆上同调之间的关系，目的是让我们对后者有一个几何上的理解。该项目的第一部分包括利用高等算术的工具分析光滑等变因式分解代数和超对称扭曲函数场理论之间的关系。超对称扭场理论被用来架起数学物理和上同调理论之间的桥梁；在低维中，已知有使用德勒姆上同调和K-理论对这些场理论的微分几何描述。在下一个超对称维中，扭场理论猜想地与一种广义的椭圆上同调理论--拓扑模形式(TMF)相联系。这种猜想的关系已经被研究了30多年，但仍然没有得到解决。将这些超对称场论描述为等变因式分解代数，将重新构建猜想，允许更直接地使用物理学中的场论例子。该项目的第二部分涉及到主丛的分类，其中对称性由光滑的2-群给出。光滑2-群的主丛也与椭圆上同调中的问题有关，因为弦群(作为范畴中心扩张出现的光滑2-群的一个例子)给出了TMF的方位数据。该项目由LEAPS-MPS计划和既定的激励竞争性研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The dialog between mathematics and physics has a long history of yielding insights in both disciplines, often revealing deep, unifying structure, such as in general relativity and Riemannian geometry. Topology is the study of shapes; in particular, topology asks about properties of shapes that remain unchanged as the shape is smoothly transformed. Over the past century, advances in quantum physics, especially condensed matter physics and superstring theory, depended on innovations in topology. One such innovation is factorization algebras, a mathematical model for the observables of a (classical or quantum) field theory. This project uses factorization algebras to explore a conjectured relationship between a certain type of quantum field theory and an object in topology called elliptic cohomology. The precise structure of this relationship has remained elusive for decades; a resolution to this conjecture would provide insight into both a geometric interpretation of elliptic cohomology and foundational questions in quantum physics related to string theory. This award also supports a regional conference on topology and mathematical physics, a distinguished lecture series, and research opportunities for undergraduates at the PI’s institution, both increasing participation of underrepresented minorities in mathematics and enhancing the research environment of the PI’s institution. More specifically, this project explores the relationship between equivariant factorization algebras, supersymmetric twisted field theories, and elliptic cohomology, with the goal of giving insight into a geometric understanding of the latter. The first component of the project involves analyzing the relationship between smoothly equivariant factorization algebras and supersymmetric twisted functorial field theories, using tools of higher operads. Supersymmetric twisted field theories have been used to bridge mathematical physics and cohomology theories; in low dimensions there are known differential geometric descriptions of these field theories using deRham cohomology and K-theory. In the next supersymmetric dimension, twisted field theories are conjecturally related to a generalized elliptic cohomology theory, topological modular forms (TMF). The conjectured relationship has been investigated for over thirty years but remains unresolved. Having a description of these supersymmetric field theories as equivariant factorization algebras would reframe the conjecture, allowing more direct use of examples of field theories from physics. The second component of the project involves looking at a categorification of principal bundles, where the symmetries are given by a smooth 2-group. Principal bundles for smooth 2-groups are also related to questions in elliptic cohomology, since the string group (an example of a smooth 2-group arising as a categorical central extension) gives orientation data for TMF. This project is jointly funded by the LEAPS-MPS program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.