Proliferation Transitions for Quantum Doubles of Groups

Proliferation transitions are phase transitions between a 2+1d topological order T and another T/A, obtained by condensing anyons in an algebra A T. We consider such transitions in quantum doubles of groups, T=D(G, ω), for finite groups G possibly with three-cocycle ω, for which condensable algebras are fully classified. Given a condensable algebra A, we represent T in terms of a Symmetry Topological Field Theory (SymTFT), that is tailored to the A proliferation transition. The SymTFT bulk is constructed from the Müger center of the subcategory generated by the algebra, i.e. the category of transparent lines, which can be bosons or fermions. Properties of the transition can then be inferred directly from the algebra classification data. For gauged Landau-Ginzburg (LG*) type transitions, we show that they do not only depend on the anyons in A through their field content, but the cubic interactions are fixed in terms of the multiplication structure of A. Applied to twin algebras, which have the same anyon content but different multiplication structures, we show that twins result in distinct proliferation transitions.