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It is by now well understood how to associate a braided $C^*$-tensor category, the data describing `topological order', to gapped ground states of two-dimensional quantum spin systems. In the physics literature it is usually assumed that topological order is described by a modular fusion category, given by a finite set of anyons, their fusion rules, and F- and R-symbols satisfying the pentagon and the heptagon equations. This structure is slightly more restrictive than a braided $C^*$-tensor category. In this talk I will describe how to extract the data of a modular fusion category for a very simple class of states that nevertheless includes many important examples. I will then discuss the concrete example of the double semion state, which is the simplest example with non-trivial F-symbols.

Based on joint work with Boris Kjær and Alvin Moon.