“Anyon condensation” (or better, “gauging one-form symmetry”. The reason is explained here) is a mathematical operation that relates different 3d topological quantum field theories (TQFTs). When the condensed anyons are Abelian bosons, the effect on topological lines (equivalently, anyon types) is often described as a “three-step procedure”:
- Remove all lines that braid nontrivially with the condensed lines.
- Arrange the remaining lines into orbits under fusion with the condensed lines. Lines in the same orbit are identified.
- If a (non-Abelian) line is invariant under fusion with nontrivial condensed lines, it “splits” into multiple descendant lines.
I recently encountered an example where naively applying the rules, especially rule 3, leads to incorrect results (there is no problem with rules 1 and 2). The upshot is that if the condensed bosons form a cyclic group under fusion, then the procedure can be safely applied. Otherwise the procedure does not always work.
The example can be described in several ways. Most compactly, it is the TQFT of the $\mathrm{Spin}(8)_{-2}$ Chern-Simons theory. To understand the source of the problem, it is more useful to think of it as the $\bZ_2\times \bZ_2$ orbifold of ${\rm SU}(2)_1$. Here “orbifold” means gauging a zero-form symmetry. ${\rm SU}(2)_1$ has a single nontrivial line, which is the semion line. The $\bZ_2\times \bZ_2$ symmetry fractionalizes, in such a way that the semion carries the two-dimensional projective representation. Gauging this symmetry leads to our example. This description is slightly ambiguous, as we have not specified which Dijkgraaf-Witten term is used, but that does not affect the discussion.
To understand what is going on, we do not need to know the full details of the gauged theory (or $\mathrm{Spin}(8)_{-2}$). All we need is the following subcategory of lines, which is isomorphic to ${\rm Rep}(Q_8)$ as a fusion category. Let us denote the lines in this subcategory by $1, b_1, b_2, b_3$ and $s$. $b_1, b_2$ and $b_3$ are Abelian bosons, which generate the $\bZ_2\times \bZ_2$ group that can be condensed. $s$ is a non-Abelian anyon with quantum dimension 2. The most important fusion rule is:
\[s\times s = 1+b_1+b_2+b_3.\]$s$ has quantum dimension $d_s=2$, and is invariant under fusion with any $b_i$. Since it has trivial braiding with $b_i$’s, this whole subcategory survives the condensation. Then applying rule 3 naively would mean that $s$ should split into 4 lines, each with quantum dimension $1/2$, which is clearly impossible.
The reason that the “splitting” rule fails becomes clearer if we reverse the gauging. In general, gauging an (invertible) one-form symmetry ${\cal A}^{(1)}$ in a TQFT $\mathcal{T}$ leads to a new theory $\mathcal{T}’$, which has a dual zero-form symmetry ${\cal A}^{(0)}$. Gauging ${\cal A}^{(0)}$ in ${\cal T}’$ (with appropriate Dijkgraaf-Witten terms) restores $\mathcal{T}$. This is completely general and applies to non-topological quantum field theories as well.
In our example, $\mathcal{T}’$ is the semion TQFT, or ${\rm SU}(2)_1$. It is enriched by the dual $\bZ_2^2$ zero-form symmetry, in such a way that the semion carries the two-dimensional projective representation of $\bZ_2^2$. This is an example of “symmetry fractionalization”. Hence after gauging the zero-form $\bZ_2^2$, the semion is promoted to a non-Abelian anyon with $d=2$, which is the anyon $s$. So the condensation/one-form gauging does not split the non-Abelian $s$. $s$ simply turns into an Abelian anyon.
On the other hand, a zero-form symmetry can act on a TQFT by permuting anyon types. Then after gauging, the orbit of anyon types under the permutation action is promoted to a single anyon with a larger quantum dimension (more precisely, the orbit is a direct sum). The reverse gauging/condensation splits this “orbit” anyon back into the members of the orbit. This is precisely rule 3.
This perspective also explains why rule 3 always works if the group of condensed bosons/the one-form symmetry is cyclic: in the dual theory, the dual zero-form symmetry does not have multi-dimensional irreducible projective representations. Thus the kind of mechanism that makes rule 3 fail in our example never occurs.
How does this example fit into the mathematical theory of anyon condensation? In general, given a 3d TQFT $\mathcal{T}$ (as a modular tensor category) and an etale (or “condensable”) algebra $A$ in $\mathcal{T}$, the objects of the new theory are local $A$-modules in $\mathcal{T}$. Equivalently, they are “ambichiral” $A-A$-bimodules in $\mathcal{T}$.
To be continued.