4:00pm,  Monday,     August 24, 2015
4:00pm,  Wednesday,  August 26, 2015

In 2D topological phases with symmetry, the fractional excitations in the system can transform under symmetry in a fractional way, e.g. by carrying fractional symmetry charges. With different types of topological order and different symmetries, what symmetry fractionalization (SF) patterns are possible in general? This question becomes particularly interesting with recent experimental progress towards realizing spin liquids and we try to answer this question in this talk. First, we discuss a simple consistency condition that all SF patterns have to satisfy. Surprisingly, some seemingly consistent SF patterns are actually anomalous, i.e. they cannot be realized in purely 2D systems. To exclude these cases, we discuss two anomaly detection methods: the flux fusion method, which is physically intuitive but applies only to special cases, and the gauging obstruction method, which is mathematically complete and straight-forward to apply. We give specific examples of anomalous SF patterns to be detected by these methods and discuss the interesting possibility of realizing them on the surface of 3D systems.

After obtaining a pretty complete understanding in 2D, we move on to address the same question in 3D. A new feature of 3D topological phases is the existence of loop excitations and we discuss first how to properly describe symmetry fractionalization on loops. Using a dimension reduction procedure, we show that loop excitations exist as the boundary between two 2D symmetry enriched topological phases and it is the difference in these phases that characterize the symmetry action. Moreover, similar to the 2D case, we find that some seemingly possible symmetry fractionalization patterns are actually anomalous and cannot be realized in 3D. To detect such anomalies, we generalize the flux fusion method and apply it to 3D. To illustrate these ideas, we use the 3D Z2 gauge theory with Z2 symmetry as an example and completely list the corresponding SET phases. In particular, we find four non-anomalous SETs and one anomalous SET which we show to be realizable as the surface of a 4D system.