Braided tensor categories, higher Picard groups, and classification of topological phases of matter

This project concerns the study of braided tensor categories and their higher-dimensional analogs. These are abstract algebraic structures consisting of objects that can be fused using certain natural rules. Such categories are indispensable in the study of classical and quantum symmetries. They are also used as mathematical models for a potential hardware design of a quantum computer. Namely, they correspond to topological phases of matter relevant to quantum computation and are used to predict the existence of new types of such phases, their behavior, and their physical realization. This project is motivated by these connections and deals with algebraic aspects of the theory of (higher) tensor categories: their structure, classification, and arithmetic properties. The emphasis is on categories that are most widely used in applications. Such categories admit an additional symmetry constraint called braiding that is used to model the interaction of quantum particles. Student recruitment and training are essential components of the proposed research activity. This project will address fundamental questions concerning the structure and classification of braided tensor categories and 2-categories. Categorical techniques will be employed to interpret and solve algebraic problems. Previously developed tools such as categorical Witt and Picard groups will be generalized and combined into higher-categorical groups, providing a useful homotopy-theoretic machinery. The concrete research problems include the following: (1) classification of braided fusion 2-categories and description of their Witt invariants, (2) computation of groups of minimal extensions corresponding to symmetry-protected topological phases of matter, (3) study of Hecke algebras associated with fusion categories and their integral forms, (4) classification of fiber functors on non-degenerate braided fusion categories using a categorical analog of Belavin-Drinfeld triples, and (5) classification of non-semisimple pointed braided tensor categories. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.