Anyon condensation and tensor categories

Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category CC; and the anyons in the condensed phase are given by another modular tensor category DD. By a bootstrap analysis, we derive a relation between anyons in DD-phase and anyons in CC-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A   in DD-phase is necessary to be a connected commutative separable algebra in CC, and the category DD is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A  -modules in CC. A more general situation is also studied in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin–Wen types of lattice models and some chiral topological phases.