Applications of Tensor Categories in Operator Algebras

对称性在数学科学的方方面面都扮演着重要的角色，尤其是作为物理学中的统一原理。传统上，物理系统的对称性是由称为群的代数对象来描述的，这些群作用于系统的可观测对象。然而，在量子系统中，可观测用非对易算子代数(C*和von Neumann代数)来描述。在这种背景下，一种新的对称性出现了。自然产生的代数对象被称为张量范畴，并被证明在描述低维量子场论的对称性、物质的拓扑相和量子统计力学方面非常成功。这个项目的目标是应用张量范畴理论来理解非对易算子代数之间的关系，以及探索张量范畴在低维量子系统中的作用。第一个是利用张量范畴来分类和构造von Neumann代数的离散包含，这是在这一领域的最新进展的基础上，进一步推进了PI与David Penney和Shamindra K.Ghosh的工作。二是从辫子张量范畴的角度研究有限von Neumann代数的Alain Connes chi不变量。我们利用不可逆双模给出了这个不变量的一个推广，并给出了这个不变量的新的计算方法，它将允许我们区分以前不可区分的von Neumann代数类。最后，我们研究了规范辫状张量范畴的代数过程，它描述了在凝聚态系统的拓扑相中量子对称性与经典对称性之间的相互作用。从物理角度来看，这是一个重要的结构，但在数学上很难理解。这个项目的主要问题是在物理相关的情况下建立测量类别的存在，如排列对称，推广特里·甘农的PI结果。该奖项反映了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 play a fundamental role across the spectrum of mathematical sciences, especially as a unifying principle in physics. Classically symmetries of a physical system are described by algebraic objects known as groups, which act on the observables of the system. In quantum systems, however, the observables are described by noncommutative operator algebras (C* and von Neumann algebras). In this setting a new kind of symmetry emerges. The algebraic objects that naturally arise are called tensor categories, and have proved to be very successful at describing symmetries of low dimensional quantum field theories, topological phases of matter, and quantum statistical mechanics. The goal of this project is to apply the theory of tensor categories to understand the relationship between noncommutative operator algebras, as well as exploring the role of tensor categories in low dimensional quantum systems.This project focuses on three main problems. The first is to use tensor categories to classify and construct discrete inclusions of von Neumann algebras building on recent progress in this area, furthering the work of the PI with David Penneys and with Shamindra K. Ghosh. The second is the study of Alain Connes' chi invariant for finite von Neumann algebras from the point of view of braided tensor categories. We propose a generalization of this invariant using non-invertible bimodules, along with new methods of computation of this invariant that will allow us to distinguish previously indistinguishable classes of von Neumann algebras. Finally, we investigate the algebraic process of gauging braided tensor categories, which described the interaction between quantum and classical symmetry in topological phases of condensed matter systems. This is an important construction from the physical point of view, but mathematically difficult to understand. The primary problem for this project is to establish the existence of gauged categories in physically relevant situations such as permutation symmetry, generalizing the results of the PI with Terry Gannon.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.