# Anyon condensation – what's the precise definition?

Say I have an anyonic system modelled as a Chern-Simons system with group $G$. If the centre of $G$ is non-trivial, one may also study the system described by $G/\Gamma$, where $\Gamma$ is a discrete subgroup of $Z(G)$.

In a recent conference, I've heard people referring to this process as anyon condensation (or gauging the symmetry $\Gamma$): the spectrum of $G/\Gamma$ can be obtained by identifying some anyons of the spectrum of $G$. What I understood from the conference is that the spectrum of $G/\Gamma$ is the set of equivalence classes of $\Gamma$-orbits of the lines of $G$ that are neutral under $\Gamma$. My understanding is probably incorrect to some level, because I got the impression that the $\Gamma$-orbits were sometimes broken up into smaller orbits, according to some prescription that was not clear to me. I didn't get the chance to ask the speaker at the time, and I cannot find any clear description online. Thus, my question: how is the spectrum of $G/\Gamma$ related to that of $G$? What is the precise definition of anyon condensation, and how can we implement this process in practice?

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As you are asking for Chern-Simons theory, my answer is perhaps too general and too abstract for your taste. But the mathematically precise treatment goes like this (without going into the details):

An anyon model can be described by a modular tensor category.

Given a modular tensor category $\mathcal{C}$ and a global symmetry $G$ (a finite group), gauging is the process of passing from $\mathcal{C}$ to $\mathcal{C}/G$, the equivariantization of $\mathcal{C}$ (https://arxiv.org/abs/1510.03475v3).

Anyon condensation is the inverse process of passing from $\mathcal{C}/G$ to $\mathcal{C}$. This defined as "taking the core" of the pair $\mathrm{Rep}(G) ⊂ \mathcal{C}/G$ (https://arxiv.org/abs/0906.0620), where $\mathrm{Rep}(G)$ (the symmetric category of $G$-representations) is a Tannakian subcategory of $\mathcal{C}/G$.

For more background, references and a physically oriented mathematical introduction to this formalism, see https://arxiv.org/abs/1410.4540v2 .