Fractionalization and anomalies in two-dimensional topological phases
Topological phases of matter in two dimensions support fractionalized excitations. They exhibit a number of fascinating phenomena, such as anyonic exchange and braiding statistics and fractional quantum numbers, or symmetry fractionalization. A famous example is the fractionally charged quasiholes in quantum Hall systems while the electrons making up the system have charge one. In this talk I will discuss recent advances in the theory of symmetry fractionalization and global anomalies for 2D topological phases. I will describe how to systematically classify symmetry fractionalization in a topologically ordered phase, and how certain seemingly consistent patterns of symmetry fractionalization are in fact “anomalous”. This kind of anomaly is deeply connected to symmetry-protected topological phases in three dimensions via bulk-boundary correspondence. With these ideas, I will discuss a new perspective on the classic Lieb-Schultz-Mattis theorem and its higher-dimensional version by Oshikawa and Hastings, and possible refinements and generalizations.