Categorical Symmetries of Operator Algebras

对称性是我们对物理世界进行数学建模的基本组成部分。它们为表征从高能到凝聚态物理学的各种物理理论提供了一个范例。传统上，对称性是用称为群的数学对象来描述的。然而，近几十年来，出现了一些有趣的量子理论，它们的基本对称性是不可逆的，这就要求我们将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，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。这个项目的目标是提供算符代数的范畴对称性的分类结果，重点是与量子自旋系统相关的情况。这些结果将，特别是，有助于提供一个严格的理解拓扑有序相的物质。 该项目将为北卡罗来纳州州立大学的研究生和本科生提供研究机会，重点是招收代表性不足群体的学生。在第一，主要研究者将提供一个分类的近似有限维的行动，顺从张量类别的近似有限维C*-代数的K-理论不变量。顺从张量范畴同时推广离散顺从群和紧群的表示范畴，而近似有限维作用提供了一维格点自旋系统的全局对称性的范畴推广。主要研究者将他们以前的结果在这个方向上从融合类别的情况下，无限的顺从设置，并应用这些结果，以获得新的分类拓扑有序自旋系统。在第二部分中，主要研究者将他们以前开发的冯诺依曼代数的范畴chi不变量推广到冯诺依曼代数上的群作用不变量。该奖项反映了NSF的法定使命，并被认为值得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的支持。

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.