[seminar on Dec. 25, 2019] Higher Dimensional Topological Order, Higher Category and A Classification in 3+1D

  • Published: 2019-12-24

Title: Higher Dimensional Topological Order, Higher Category and A Classification in 3+1D

Speaker: Dr. Tian Lan (University of Waterloo)

Time and Venue: 2pm Dec. 25, 2019 at S401



Topological orders are gapped quantum liquid states without any symmetry. Most of their properties can be captured by investigating topological defects and excitation of various dimensions.  Topological defects in n dimensions naturally form a (weak) n-category. In particular, anomalous topological order (boundary theory) is described by fusion n-category and anomaly-free topological order (bulk) is described by non-degenerate braided fusion n-category. Holographic principle works for topological orders: boundary always has a unique bulk. Another important property in 3+1D or higher is that point-like excitations must have trivial statistics; they must carry representations of certain group. Such a "gauge group" is hidden in every higher dimensional topological order. In 3+1D, condensing point-like excitations leads to a canonical boundary which in turn determines the bulk topological order. By studying such boundary, a rather simple classification is obtained: 3+1D topological orders are classified by the above "gauge group" together with some cocycle twists. These ideas would also play an important role in dimensions higher than 3+1D and in the study of higher categories, topological quantum field theories and other related subjects.