LEAPS-MPS: Quantum Field Theories and Elliptic Cohomology

数学和物理之间的对话有着悠久的历史，在这两个学科中产生了深刻的见解，经常揭示出深刻的、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；特别是，拓扑学要求形状在平滑变换时保持不变的属性。在过去的一个世纪里，量子物理学的进步，特别是凝聚态物理和超弦理论，依赖于拓扑学的创新。其中一个创新就是因式分解代数，这是（经典或量子）场论中可观测的数学模型。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中称为椭圆上同调的对象之间的推测关系。几十年来，这种关系的确切结构一直难以捉摸；这一猜想的解决将为椭圆上同调的几何解释和与弦理论相关的量子物理学的基本问题提供见解。该奖项还支持一个关于拓扑和数学物理的区域会议，一个杰出的系列讲座，以及为PI所在机构的本科生提供研究机会，既增加了代表性不足的少数民族在数学领域的参与，又改善了PI所在机构的研究环境。更具体地说，本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平滑等变分解代数和超对称扭曲泛函场理论之间的关系，使用更高操作数的工具。超对称扭场理论已被用作数学物理和上同调理论的桥梁；在低维中，这些场论有已知的微分几何描述，使用deRham上同调和k理论。在下一个超对称维度中，扭曲场理论与广义椭圆上同调理论拓扑模形式（TMF）推测相关。这种推测的关系已经被研究了30多年，但仍未得到解决。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场论的例子。项目的第二个组成部分涉及到主束的分类，其中对称是由光滑2群给出的。光滑2群的主束也与椭圆上同调有关，因为弦群（光滑2群作为范畴中心扩展的一个例子）给出了TMF的方向数据。该项目由leap - mps计划和促进竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，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。

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.