Beyond Anyons

The theory of anyon systems, as modular functors topologically and unitary modular tensor categories algebraically, is mature. To go beyond anyons, our first step is the interplay of anyons with conventional group symmetry due to the paramount importance of group symmetry in physics. This led to the theory of symmetry-enriched topological order. Another direction is the boundary physics of topological phases, both gapless as in the fractional quantum Hall physics and gapped as in toric code. A more speculative and interesting direction is the study of Banados-Teitelboim-Zanelli black holes and quantum gravity in $3d$. The clearly defined physical and mathematical issues require a far-reaching generalization of anyons and seem to be within reach. In this short survey, I will first cover the extensions of anyon theory to symmetry defects and gapped boundaries. Then I will discuss a desired generalization of anyons to anyon-like objects---the Banados-Teitelboim-Zanelli black holes---in $3d$ quantum gravity.
Comment: To appear in the book "Topological Methods in Theoretical Physics"