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方式。它们为描述从高能到凝聚态物理的整个光谱的物理理论提供了一个范例。传统上，对称是由称为群的数学对象来描述的。然而，近几十年来，有趣的量子理论已经出现，它们的基本对称性是不可逆的，这需要我们将经典的对称性思想扩展到群之外。量子理论可以用算符代数的语言来描述，这些新的对称性可以通过作用于算符代数的被称为张量范畴的数学对象来实现。这个项目的目标是为算子代数的范畴对称性提供分类结果，重点是与量子自旋系统相关的情况。这些结果尤其有助于对物质的拓扑有序相进行严格的理解。这个项目将包括北卡罗来纳州立大学研究生和本科生的研究机会，重点是招收代表不足的群体的学生。这个项目包括两个主要部分。在第一部分中，主要研究人员将利用K-理论不变量对近似有限维C*-代数上的从属张量范畴的近似有限维作用进行分类。顺应性张量范畴同时推广了离散顺从群和紧致群的表示范畴，而近似有限维作用提供了一维晶格自旋系统整体对称性的范畴推广。主要研究者将在这个方向上将他们以前的结果从融合范畴的情况扩展到无限可服从的情形，并应用这些结果来获得拓扑有序自旋系统的新分类。在第二个部分中，主要研究者将把他们以前发展的von Neumann代数的范畴X不变量推广到von Neumann代数上的群作用的不变量。首席研究员将使用这个不变量来区分针对McDuff因子的团体诉讼。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.