Categorical Symmetries of Operator Algebras

算子代数的分类对称性

• 批准号: 2247202
• 负责人: Corey Jones
• 金额: $ 26.81万
• 依托单位: North Carolina State University
• 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: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247202&HistoricalAwards=false
• 关键词: Categorical; Symmetries; Operator; Algebras

Symmetries are a fundamental part of the way we mathematically model the physical world. They provide a paradigm for characterizing physical theories across the spectrum, from high energy to condensed matter physics. Traditionally, symmetries are described by mathematical objects called groups. However, in recent decades, interesting quantum theories have emerged whose fundamental symmetries are not reversible, requiring an extension of our classical ideas of symmetry beyond groups. Quantum theories can be described in the language of operator algebras, and these new kinds of symmetry can be realized by mathematical objects called tensor categories acting on operator algebras. The goal of this project is to provide classification results for categorical symmetry of operator algebras with an emphasis on situations relevant to quantum spin systems. These results will, in particular, help provide a rigorous understanding of topologically ordered phases of matter. This project will incorporate research opportunities for graduate and undergraduate students at North Carolina State University, with an emphasis on the recruitment of students from underrepresented groups.This project has two main components. In the first, the principal investigator will provide a classification of approximately finite dimensional actions of amenable tensor categories on approximately finite dimensional C*-algebras in terms of K-theoretic invariants. Amenable tensor categories simultaneously generalize discrete amenable groups and the representation categories of compact groups, while approximately finite dimensional actions provide categorical generalizations of global symmetries of 1D lattice spin systems. The principal investigator will extend their previous results in this direction from the case of fusion categories to the infinite amenable setting and apply these results to obtain new classifications for topologically ordered spin systems. In the second component, the principal investigator will generalize their previously developed categorical chi invariant for von Neumann algebras to an invariant for group actions on von Neumann algebras. The principal investigator will use this invariant to distinguish group actions on McDuff factors.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.

对称性是我们对物理世界进行数学建模的基本组成部分。它们提供了表征从高能物理到凝聚态物理等各个领域的物理理论的范例。传统上，对称性是通过称为群的数学对象来描述的。然而，近几十年来，出现了有趣的量子理论，其基本对称性是不可逆的，需要将我们经典的对称性思想扩展到群之外。量子理论可以用算子代数的语言来描述，而这些新的对称性可以通过作用于算子代数的称为张量范畴的数学对象来实现。该项目的目标是提供算子代数的分类对称性的分类结果，重点是与量子自旋系统相关的情况。这些结果尤其有助于提供对物质的拓扑有序相的严格理解。 该项目将为北卡罗来纳州立大学的研究生和本科生提供研究机会，重点是从代表性不足的群体中招募学生。该项目有两个主要组成部分。首先，主要研究者将根据 K 理论不变量提供近似有限维 C* 代数上合适张量类别的近似有限维作用的分类。顺应张量类别同时概括离散顺应群和紧群的表示类别，而近似有限维作用提供一维晶格自旋系统的全局对称性的分类概括。首席研究员将在这个方向上将他们之前的结果从融合类别的情况扩展到无限适合的设置，并应用这些结果来获得拓扑有序自旋系统的新分类。在第二部分中，主要研究者将把他们之前开发的冯诺依曼代数的分类卡不变量推广到冯诺依曼代数的群作用的不变量。主要研究者将利用这一不变量来区分 McDuff 因素的群体行为。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。