LEAPS-MPS: Quantum Field Theories and Elliptic Cohomology

LEAPS-MPS：量子场论和椭圆上同调

• 批准号: 2316646
• 负责人: Laura Murray
• 金额: $ 13.05万
• 依托单位: Providence College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316646&HistoricalAwards=false
• 关键词: LEAPS; MPS; Quantum; Field; Theories

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.

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，经常揭示出深刻的统一结构，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，拓扑学询问形状的属性，这些属性在形状平滑变换时保持不变。在过去的世纪里，量子物理学的进步，特别是凝聚态物理学和超弦理论，依赖于拓扑学的创新。一个这样的创新是因式分解代数，一个（经典或量子）场论的可观测量的数学模型。这个项目使用因子分解代数来探索某种类型的量子场论和拓扑中称为椭圆上同调的对象之间的关系。这种关系的精确结构几十年来一直难以捉摸;这个猜想的解决方案将提供对椭圆上同调的几何解释和与弦理论相关的量子物理学基础问题的见解。该奖项还支持拓扑学和数学物理区域会议，杰出的系列讲座，以及PI机构本科生的研究机会，既增加了数学中代表性不足的少数民族的参与，又增强了PI机构的研究环境。更具体地说，这个项目探讨了等变因子分解代数，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分析光滑等变因子分解代数和超对称扭曲函域理论之间的关系，使用更高的操作工具。超对称扭曲场论已经被用来连接数学物理和上同调理论;在低维中，这些场论有已知的微分几何描述，使用德拉姆上同调和K理论。在下一个超对称维度中，扭曲场论与广义椭圆上同调理论，拓扑模形式（topological modular forms，简写为EMO）有着密切的联系。这一关系已经被调查了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子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。该项目的第二个组成部分是研究主丛的分类，其中对称性由光滑2-群给出。光滑2-群的主丛也与椭圆上同调的问题有关，因为弦群（光滑2-群作为范畴中心扩张的一个例子）给出了椭圆上同调的方向数据。该项目由LEAPS-MPS计划和刺激竞争研究的既定计划（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知识价值和更广泛的影响审查标准进行评估，被认为值得支持。