Citation: Kong, Liang. "Anyon condensation and tensor categories." Nuclear Physics B 886 (2014): 436-482.
Web: https://arxiv.org/abs/1307.8244
Tags: Modular-tensor-categories, Defects/boundaries, Phase-transition, Foundational
This fantastic paper introduces the connection between algebras with special types of adjectives (which Kong calls "condensable algebras") and anyon condensation. This connection was discovered by Kitaev, and communicated to Kong privately.
A one-sentence summary of condensation, which the authors include as a lovely quote, is as follows: "Condensation is a process of selecting an energy-favorable subspace of the original Hilbert space". Condensation is some process whereby a homogenous collection of local projectors create a new optimal ground space. This ground space will certainly not be the ground space of the original phase. This means that it will be described as excited states in the original phase. That is, the ground state will be described by a large number of anyons! Zooming in at a tiny region in the condensed phase, we expect to see some finite collection of anyons, or some probabilistic combination of anyons. This means that the vacuum in the condensed phase will correspond to an object A in the modular category describing the original phase.
Natural physical considerations then imply that the object A which represents the vacuum in the original phase must canonically have the structure of an algebra, and that this algebra will have a large number of additional structures/properties. Such algebras are dubbed condensable algebras. It is argued that the condensed phase is described by the category of local A modules. The fact that the category of local A modules is a modular category is a quite deep result. It was originally proved in the context of subfactor theory, over three papers:
> Böckenhauer, Jens, David E. Evans, and Yasuyuki Kawahigashi. "On alpha-Induction, Chiral Generators and¶ Modular Invariants for Subfactors." Communications in Mathematical Physics 208 (1999): 429-487.
> Böckenhauer, Jens, David E. Evans, and Yasuyuki Kawahigashi. "Chiral structure of modular invariants for subfactors." Communications in Mathematical Physics 210 (2000): 733-784.
> Böckenhauer, Jens, David E. Evans, and Yasuyuki Kawahigashi. "Longo-Rehren subfactors arising from alpha-induction." Publications of the Research Institute for Mathematical Sciences 37.1 (2001): 1-35.
The connection between the subfactor language and the module theory language is found in
> Fröhlich, Jürg, et al. "Correspondences of ribbon categories." Advances in Mathematics 199.1 (2006): 192-329.
This paper gives a lovely way of thinking about a whole host of topics, and serves as a canonical example of phase transitions and gapped boundaries in the algebraic theory of topological order. The considerations also have mathematical consequences, such as the fact that each Witt equivalence class contains a unique representative which is completely anisotropic, which is one of the main results from
> Davydov, Alexei, et al. "The Witt group of non-degenerate braided fusion categories." Journal für die reine und angewandte Mathematik (Crelles Journal) 2013.677 (2013): 135-177.