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; 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具有悠久的历史，这些历史是在这两个学科中产生见解，通常会揭示出深层，统一的结构，例如一般相对论和riemannian几何形状。拓扑是形状的研究；尤其是，拓扑询问形状的特性，这些属性保持不变，因为形状被平稳地转化。在过去的一个世纪中，量子物理学的进步，尤其是凝结的物理和超弦理论，取决于拓扑创新。一种这样的创新是分解代数，这是一种数学模型，用于观察（经典或量子）场理论。该项目使用分解代数来探索某种类型的量子场理论与拓扑中的对象之间的猜想关系，称为椭圆形共同体。数十年来，这种关系的精确结构一直难以捉摸。解决这一概念的解决方案将洞悉与字符串理论相关的量子物理学中的元素同胞的几何解释和基础问题。该奖项还支持一场关于拓扑和数学物理学的区域会议，杰出的讲座系列，以及在PI机构中为大学生提供的研究机会，这既增加了代表性少数少数群体在数学方面的参与，又增强了PI机构的研究环境。更具体地说，该项目探讨了等效分解代数，超对称扭曲的田间理论和椭圆形的共同体之间的关系，目的是深入了解对后者的几何理解。该项目的第一个组成部分涉及分析使用高级操作员的工具，分析平滑等效分解代数和超对称扭曲功能场理论之间的关系。超对称扭曲的场理论已被用来弥合数学物理学和协同理论。在低维度中，使用Derham共同体学和K理论对这些现场理论进行了已知的差异几何描述。在下一个超对称维度中，扭曲的场理论与广义椭圆形的共同体学理论，拓扑模块形式（TMF）相关。猜想的关系已经进行了三十多年的调查，但仍未解决。将这些超对称场理论描述为等效分解代数将指出该猜想，从而可以更多地直接使用物理学的现场理论示例。该项目的第二个组成部分涉及查看一类主束，其中对称是由平滑的2组给出的。平滑2组的主要捆绑包也与椭圆形的问题有关，因为字符串组（作为分类中央扩展的平滑2组的一个示例）提供了TMF的方向数据。该项目由LEAPS-MPS计划和启发竞争研究的既定计划共同资助。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，通过评估被认为是宝贵的支持。