Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

Understand

Gravitational anomalies can be realized on the boundary of topologically ordered states in one higher dimension and are described by topological orders in one higher dimension.

• In this paper, we try to develop a general theory for both topological order and gravitational anomaly in any dimensions.
• (1) We introduce the notion of BF category to describe the braiding and fusion properties of topological excitations that can be point-like, string-like, etc.
• A subset of BF categories -- closed BF categories -- classify topological orders in any dimensions, while generic BF categories classify (potentially) anomalous topological orders that can appear at a boundary of a gapped quantum liquid in one higher dimension.