Applications of Tensor Categories in Operator Algebras

对称性在整个数学科学中扮演着重要的角色，特别是作为物理学中的统一原则。物理系统的经典对称性由称为群的代数对象描述，它们作用于系统的可观测量。然而，在量子系统中，观测量是由非交换算子代数（C* 和冯诺依曼代数）描述的。在这种情况下，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，并已被证明在描述低维量子场论、物质的拓扑相和量子统计力学的对称性方面非常成功。本项目的目标是应用张量范畴理论来理解非交换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。本项目主要关注三个问题。第一个是使用张量范畴来分类和构造冯诺依曼代数的离散包含，这是基于这一领域的最新进展，进一步推进了PI与大卫彭尼和沙明德拉K.高希第二部分是从辫张量范畴的角度研究有限von Neumann代数的Alain Connes chi不变量。我们提出了一个推广的不变量使用不可逆的双模，沿着与新的方法计算这个不变量，这将使我们能够区分以前无法区分类的冯诺依曼代数。最后，我们研究了规范辫状张量范畴的代数过程，辫状张量范畴描述了凝聚态系统拓扑相中量子对称性与经典对称性之间的相互作用。从物理学的角度来看，这是一个重要的结构，但在数学上很难理解。该项目的主要问题是建立在物理相关的情况下，如置换对称性，推广PI与特里甘农的结果存在计量类别。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.