The Landau paradigm for symmetry breaking phase transitions has been immensely successful at describing the theory of phase transitions at finite temperature. Universal properties of the transition are described by the pair (G,H), where G is the symmetry group of the original phase which breaks down to the subgroup H across the phase boundary. Any subgroup H<G is theoretically possible as the symmetry group of the new phase, and which subgroup H occurs depends on the microscopic details of the Hamiltonian. However, for topological phase transitions, there is more to the story. We will explain how the mathematical theory of fusion categories can be used to show that for the topological phase transitions described by condensing anyons, symmetry cannot be broken down to arbitrary subgroups H<G. There is a topological obstruction to this problem which is controlled by the splitting of a certain short exact sequence. Based on joint work with Marcel Bischoff, Yuan-Ming Lu, and David Penneys.
