Anyon condensation and tensor categories

Instead of studying anyon condensation in concrete models, we take an abstract approach. Assume that a system of anyons, which form a modular tensor category D, is obtained via an anyon condensation from another system of anyons (i.e. another modular tensor category C). By a bootstrap analysis, we derive the relation between C and D from natural physical requirements. It turns out that the tensor unit of D can be identified with a connected commutative separable algebra A in C. The modular tensor category D consists of all deconfined particles and can be identified with the category of local $A$-modules in C. If this condensation occurs in a 2d region in the C-phase, then it also produces a 1d gapped domain wall between the C-phase and the D-phase. The confined and deconfined particles accumulate on the wall and form a fusion category that is precisely the category of right A-modules in C. We also consider condensations that are confined to a 1d line. We show how to determine the algebra A from physical macroscopic data. We provide examples of anyon condensation in the toric code model, Kitaev quantum double models and Levin-Wen types of lattice models and in some chiral topological phases. In the end, we briefly discuss Witt equivalence between 2d topological phases. We also attach to this paper an Erratum and Addendum to the original version of "Anyon condensation and tensor categories" published in [Nucl. Phys. B 886 (2014) 436-482].