LEAPS-MPS: Quantum Field Theories and Elliptic Cohomology

数学和物理学之间的对话有着悠久的历史，在这两个学科中都产生了深刻的见解，常常揭示深刻的、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；特别是，拓扑要求形状在平滑变换时保持不变的属性。过去一个世纪，量子物理学，特别是凝聚态物理学和超弦理论的进步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）场论可观测量的数学模型。该项目使用因式分解代数来探索某种类型的量子场论与称为椭圆上同调的拓扑对象之间的猜想关系。几十年来，这种关系的精确结构一直难以捉摸。这一猜想的解决将有助于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。该奖项还支持拓扑和数学物理区域会议、杰出讲座系列以及 PI 机构本科生的研究机会，既增加了代表性不足的少数群体对数学的参与，又改善了 PI 机构的研究环境。更具体地说，该项目探讨了等变因式分解代数、超对称扭曲场论和椭圆上同调之间的关系，目的是深入了解后者的几何理解。该项目的第一个组成部分涉及使用高级运算工具来分析平滑等变因式分解代数与超对称扭曲函子场论之间的关系。超对称扭曲场论已被用来连接数学物理和上同调理论；在低维中，已知使用德拉姆上同调和 K 理论对这些场论进行微分几何描述。在下一个超对称维度中，扭曲场论推测与广义椭圆上同调理论、拓扑模形式（TMF）相关。这种推测的关系已经被调查了三十多年，但仍未解决。将这些超对称场论描述为等变分解代数将重新构建猜想，从而允许更直接地使用物理学中的场论示例。该项目的第二个组成部分涉及研究主丛的分类，其中对称性由平滑的 2 群给出。平滑 2 群的主丛也与椭圆上同调中的问题相关，因为弦群（作为分类中心扩展出现的平滑 2 群的示例）给出了 TMF 的方向数据。该项目由 LEAPS-MPS 计划和刺激竞争研究既定计划 (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价值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.