Applications of Tensor Categories in Operator Algebras

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.