does not lead to fusions of the form (2) or (9).

###  3.1 $G_{k}$ CS theory

Let us now consider the finite-level deformation of the fusion rules discussed
in the previous section. These are the fusion rules of Wilson lines in $G_{k}$
CS theory. We first consider $SU(2)_{k}$ as it is rather illustrative. We will
then generalize to $SU(N)_{k}$ and comment on more general $G_{k}$.

In the case of $SU(2)_{k}$, (105) becomes [44, 45]

$j_{1}\otimes j_{2}=\sum_{j=|j_{1}-j_{2}|}^{{\rm
min}(j_{1}+j_{2},k-j_{1}-j_{2})}j\leavevmode\nobreak\ .$ (110)

In addition to truncating the spectrum to the spins
$\left\\{0,1/2,1,\cdots,k/2\right\\}$, the above deformation abelianizes the
spin $k/2$ representation (since $k/2\otimes k/2=0$). However, these changes
do not alter the conclusion from the previous section: we cannot write
$j_{1}\otimes j_{2}=j_{3}$ for $j_{3}$ non-abelian irreducible $j_{1,2,3}$.
Indeed, consider

$j_{1}\otimes j_{1}=\sum_{j=0}^{{\rm
min}(2j_{1},k-2j_{1})}j\leavevmode\nobreak\ ,\ \ \ j_{2}\otimes
j_{2}=\sum_{j=0}^{{\rm min}(2j_{2},k-2j_{2})}j\leavevmode\nobreak\ ,\ \ \
j_{1,2}\neq 0,\ {k\over 2}\leavevmode\nobreak\ .$ (111)

The conditions $j_{1,2}\neq 0,\ {k\over 2}$ are to ensure that the
representation is non-abelian. In particular, we again see that the adjoint
representation appears in (111).

While the fusion rules discussed in [44, 45] apply to more general groups,
they are rather difficult to implement. Instead, using proposals suggested in
[46, 47] and finally proven in [48], the authors of [49] show that for
$\alpha$ an irreducible representation of $G_{k}$ (with $G$ a compact simple
Lie group), we have

${}^{(k)}N_{\alpha\bar{\alpha}}^{\rm
adj}=\left|\left\\{\hat{\lambda}^{(\alpha)}_{j}\neq
0\right\\}\right|-1\leavevmode\nobreak\ ,$ (112)

where $\hat{\lambda}^{\alpha}_{j}$ are the associated affine Dynkin labels.

In particular, for $SU(N)_{k}$, if $|\alpha|>1$, then
${}^{(k)}N_{\alpha\bar{\alpha}}^{\rm adj}\geq 1$.393939Here we define
$|\alpha|$ to be the quantum dimension. Indeed, the abelian representations,
$\gamma_{i}$, satisfy a $\mathbb{Z}_{N}$ fusion algebra and are characterized
by $\hat{\lambda}^{(\gamma_{i})}_{j}=k\delta_{ij}$, where
$i\in\left\\{0,1,...,N-1\right\\}$. On the other hand, all non-abelian
representations have at least two non-zero Dynkin labels. As a result, we
learn that

$\alpha\otimes\beta\neq\gamma\leavevmode\nobreak\ ,\ \ \
\forall\alpha,\beta,\gamma\in{\rm Irr}(SU(N)_{k})\leavevmode\nobreak\ ,\ \ \
|\alpha|,|\beta|>1\leavevmode\nobreak\ .$ (113)

Therefore, we see that we have the following fusions for non-abelian Wilson
lines in $SU(N)_{k}$ CS TQFT

$\mathcal{W}_{\alpha}\times\mathcal{W}_{\beta}=\mathcal{W}_{\gamma}+\cdots\leavevmode\nobreak\
,\ \ \ |\alpha|,|\beta|>1\leavevmode\nobreak\ ,$ (114)

where the ellipses necessarily include additional Wilson lines. This statement
is more generally true in any $G_{k}$ CS theory (with $G$ a compact and simple
Lie group) for which the lines in question correspond to affine
representations with at least two non-zero Dynkin labels.

Note that for certain $G_{k}$, non-abelian representations can have a single
non-vanishing Dynkin label. For example, consider the $(E_{7})_{2}$ CS
theory.404040We thank a referee for pointing out this example. It has Wilson
lines $\mathcal{W}_{\tau}$ and $\mathcal{W}_{\sigma}$ with quantum dimensions
$\frac{1+\sqrt{5}}{2}$ and $\sqrt{2}$, repectively, and they fuse to give a
unique outcome. The existence of this fusion rule follows from the fact that
$(E_{7})_{2}$ is not a prime TQFT. In fact, it resolves into the product of
prime theories $\text{Fib}\boxtimes\text{Ising}^{\prime}$, where Fib is the
Fibonacci anyon theory and $\text{Ising}^{\prime}$ is a TQFT with the same
fusion rules as the the Ising model.

We can apply the above arguments to learn about global properties of $G_{k}$
CS theory. For example, we can ask if $G_{k}$ CS theory is prime or not. The
answer is no in general. Indeed, consider the case $G=SU(2)$. For
$k\in\mathbb{N}_{\rm even}$, $SU(2)_{k}$ is prime. However, for
$k\in\mathbb{N}_{\rm odd}$, the abelian anyon generating the $\mathbb{Z}_{2}$
one-form symmetry forms a modular subcategory. By Müger’s theorem [10] (see
also [14] for a discussion at the level of RCFT), it then decouples and the
theory resolves into a product of two prime theories

$SU(2)_{k}\simeq\begin{cases}SU(2)_{1}\boxtimes SU(2)_{k}^{\rm
int}\leavevmode\nobreak\ ,&\text{if}\ k=1\ {(\rm mod4)}\\\
\overline{SU(2)_{1}}\boxtimes SU(2)_{k}^{\rm int}\leavevmode\nobreak\
,&\text{if}\ k=3\ {(\rm mod4)}\leavevmode\nobreak\ .\end{cases}$ (115)

where $SU(2)_{k}^{\rm int}$ is a TQFT built out of the integer spin
$SU(2)_{k}$ representations. Here $\overline{SU(2)_{1}}$ is the TQFT conjugate
to $SU(2)_{1}$ (these TQFTs are sometimes called the anti-semion and semion
theories in the condensed matter literature).

While $G_{k}$ CS theory is not prime in general, our arguments above readily
prove the following:

Claim 14: Non-abelian Wilson lines in $SU(N)_{k}$ CS theory must all lie in
the same prime TQFT factor. For more general $G_{k}$ CS theory (with $G$
compact and simple), all Wilson lines corresponding to affine representations
with at least two non-zero Dynkin labels must be part of the same prime TQFT
factor.

Proof: Suppose this were not the case. Then, we would find fusion rules of the
form (114) with no Wilson lines in the ellipses. $\square$

Clearly, to produce fusion rules of the form (2) for non-abelian Wilson lines
in the same prime TQFT, we will need to go beyond $SU(N)_{k}$ CS theory. One
way to proceed is to consider coset theories and use some intuition from
section 2. Indeed, since cosets can have fixed points (which we will describe
below), it is natural to think they can lead to fusion rules of the form (2).

###  3.2 Virasoro minimal models and some cosets without fixed points

We begin with a discussion of the Virasoro minimal models, as these are simple
examples of theories that are related to cosets. While these cosets do not
have fixed points, they turn out to produce factorized TQFTs that are
nonetheless illustrative. In the next section, we will focus on cosets that
have fixed points, and we will see how to engineer fusion rules of the form
(2).

One way to construct the Virasoro minimal models is to take a three-
dimensional spacetime $\mathbb{R}\times\Sigma$ and place $SU(2)_{k-1}\times
SU(2)_{1}$ CS theory on $I\times\Sigma$, where $I$ is an interval in
$\mathbb{R}$. We can place $SU(2)_{k}$ CS theory outside this region. At the
two $1+1$ dimensional interfaces between the CS theories (which form two
copies of $\Sigma$, call them $\Sigma_{1,2}$), we obtain the left and right
movers of the RCFT. Here the chiral (anti-chiral) primaries lie where
endpoints of Wilson lines from the $SU(2)_{k}$ and $SU(2)_{k-1}\times
SU(2)_{1}$ theories meet on $\Sigma_{1}$ ($\Sigma_{2}$).

Another way to think about the Wilson lines related to the Virasoro minimal
models is to start with $SU(2)_{k-1}\times SU(2)_{1}$ CS theory and change
variables to make an $SU(2)_{k}$ subsector manifest [50]. Integrating this
sector out leaves an effective coset TQFT.

The end result is that the TQFT we are interested in is414141This is the TQFT
analog of the classic result [51] for the corresponding affine algebras:
${\rm
Vir}_{p}\simeq{\widehat{\mathfrak{su}}(2)_{p-2}\times\widehat{\mathfrak{su}}(2)_{1}\over\widehat{\mathfrak{su}}{(2)_{p-1}}}\leavevmode\nobreak\
.$ (116)

$\mathcal{T}_{p}={SU(2)_{p-2}\boxtimes SU(2)_{1}\over
SU(2)_{p-1}}\leavevmode\nobreak\ ,\ \ \ p\geq 3\leavevmode\nobreak\ .$ (117)

Here, the natural number $p\geq 3$ labels the corresponding Virasoro minimal
model (so, for example, $p=3$ for the Ising model).424242In writing (117), we
have used the Deligne product to emphasize the fact that the
$SU(2)_{p-2}\times SU(2)_{1}$ CS theory is a product TQFT. We may construct
the MTC data underlying the RCFT and the coset TQFT by taking products (e.g.,
see [52])

$F_{\mathcal{T}_{p}}=F_{SU(2)_{p-2}}\cdot
F_{SU(2)_{1}}\cdot\bar{F}_{SU(2)_{p-1}}\leavevmode\nobreak\ ,\ \ \
R_{\mathcal{T}_{p}}=R_{SU(2)_{p-2}}\cdot
R_{SU(2)_{1}}\cdot\bar{R}_{SU(2)_{p-1}}\leavevmode\nobreak\ .$ (118)

In order to make (118) precise, we need to explain how the states in
$\mathcal{T}_{p}$ are related to those in the individual $SU(2)_{k}$ theories
that make up the coset. Let us denote the $SU(2)_{p-2}$, $SU(2)_{1}$, and
$SU(2)_{p-1}$ weights as $\lambda$, $\mu$, and $\nu$. Then, to build the coset
we should identify Wilson lines as follows

$\mathcal{W}_{\\{\lambda,\mu,\nu\\}}:=\mathcal{W}_{\lambda}\times\mathcal{W}_{\mu}\times\mathcal{W}_{\nu}\simeq\left(\mathcal{W}_{p-2}\times\mathcal{W}_{\lambda}\right)\times\left(\mathcal{W}_{1}\times\mathcal{W}_{\mu}\right)\times\left(\mathcal{W}_{p-1}\times\mathcal{W}_{\nu}\right)\leavevmode\nobreak\
,$ (119)

where $\mathcal{W}_{p-2}$, $\mathcal{W}_{1}$, and $\mathcal{W}_{p-1}$ are
abelian Wilson lines transforming in the weight $p-2$ (spin $(p-2)/2$), weight
$1$ (spin $1/2$), and weight $p-1$ (spin $(p-1)/2$) representations of the
different TQFT factors.434343At the level of the corresponding affine
algebras, this is the statement that [45]
$\left\\{\hat{\lambda},\hat{\mu},\hat{\nu}\right\\}\simeq\left\\{a\hat{\lambda},a\hat{\mu},a\hat{\nu}\right\\}\leavevmode\nobreak\
,$ (120) where the hat denotes affine weights, and $a$ is the generator of the
(diagonal) $\mathcal{O}(\widehat{\mathfrak{su}}(2))$ outer automorphism.
Moreover, in order to be a valid Wilson line in $\mathcal{T}_{p}$, we should
demand that our Wilson lines satisfy

$\mathcal{W}_{\left\\{\lambda,\mu,\nu\right\\}}\in\mathcal{T}_{p}\
\Leftrightarrow\ \lambda+\mu-\nu\in Q\ \Leftrightarrow\ \lambda+\mu+\nu=0\
({\rm mod}\ 2)\leavevmode\nobreak\ ,$ (121)

where $Q$ is the $SU(2)$ root lattice. This relation guarantees that all lines
that remain have trivial braiding with
$\mathcal{W}_{\left\\{p-2,1,p-1\right\\}}$ (which is a boson that is in turn
identified with the vacuum). It is in terms of these degrees of freedom that
(118) should be understood.

Before proceeding, let us stop and note that the fusion in (119) has no fixed
points. Indeed, this statement readily follows from the fact that $SU(2)_{1}$
is an abelian TQFT, and abelian theories cannot have fixed points since their
fusion rules are those of a finite abelian group (in this case
$\mathbb{Z}_{2}$).

Given this groundwork, we claim that $\mathcal{T}_{p}$ factorizes as follows

$\mathcal{T}_{p}\simeq\begin{cases}\left(SU(2)_{p-2}\boxtimes
SU(2)_{1}\right)^{\rm int}\boxtimes SU(2)_{p-1}^{\rm int}\leavevmode\nobreak\
,&\text{if}\ p=0\ {(\rm mod\ 2)}\\\ SU(2)_{p-2}^{\rm int}\boxtimes
SU(2)_{p-1}^{\rm conj}\leavevmode\nobreak\ ,&\text{if}\ p=1\ {(\rm mod\
2)}\leavevmode\nobreak\ .\end{cases}$ (122)

The various TQFTs appearing in (122) are

$\displaystyle\left(SU(2)_{p-2}\boxtimes SU(2)_{1}\right)^{\rm int}$
$\displaystyle:=$ $\displaystyle{\rm
gen}\left(\left\\{\mathcal{W}_{\left\\{\lambda,\mu\right\\}}\in
SU(2)_{p-2}\boxtimes SU(2)_{1}|\ \lambda+\mu=0\ {\rm(mod\
2)}\right\\}\right)\leavevmode\nobreak\ ,$ (123) $\displaystyle
SU(2)_{p-1}^{\rm int}$ $\displaystyle:=$ $\displaystyle{\rm
gen}\left(\left\\{\mathcal{W}_{\nu}\in SU(2)_{p-1}|\ \nu=0\ {\rm(mod\
2)}\right\\}\right)\leavevmode\nobreak\ ,$ (124) $\displaystyle
SU(2)_{p-1}^{\rm conj}$ $\displaystyle:=$ $\displaystyle{\rm
gen}\left(\left\\{\mathcal{W}_{\left\\{\lambda,\mu,\nu\right\\}}|\
\lambda+\mu+\nu=0\ {\rm(mod\ 2)\leavevmode\nobreak\ ,\
\mathcal{W}_{\lambda},\mathcal{W}_{\mu}\ {\rm
abelian}}\right\\}\right)\leavevmode\nobreak\ ,$ (125) $\displaystyle
SU(2)_{p-2}^{\rm int}$ $\displaystyle:=$ $\displaystyle{\rm
gen}\left(\left\\{\mathcal{W}_{\lambda}\in SU(2)_{p-2}|\ \lambda=0\ {\rm(mod\
2)}\right\\}\right)\leavevmode\nobreak\ ,$ (126)

where “${\rm gen(\cdots)}$" means that the TQFT is generated by the Wilson
lines enclosed. Notice that in the case that $p$ is even, $p-1$ is odd and
$SU(2)_{p-1}^{\rm int}$ is precisely the decoupled TQFT factor required by
Müger’s theorem in (115) containing integer spins (even Dynkin labels).
Similar logic applies to $SU(2)_{p-2}^{\rm int}$ in the case that $p$ is odd.
The TQFT $SU(2)_{p-1}^{\rm conj}$ has the same fusion rules as $SU(2)_{p-1}$,
but it is a different TQFT. Finally, for the case that $p=3$ (i.e., the Ising
model), we see that $\mathcal{T}_{3}$ does not factorize.444444Note also that
Ising shares the same fusion rules as $SU(2)_{2}$, though they are not the
same TQFTs. For example, the $\sigma$ fields have different twists.

Our strategy to prove the factorization in (122) is to construct the various
factors and then argue that they are well-defined TQFTs by Müger’s theorem
[10]. Although we will not pursue it in this paper, this same approach leads
to interesting generalizations for cosets built out of groups other than
$SU(2)$.

To that end, let us first take the case of $p\geq 3$ odd. Using the result in
(118), we have that the modular $S$ matrix also takes a product form

$S_{\left\\{\lambda,\mu,\nu\right\\}\left\\{\lambda^{\prime},\mu^{\prime},\nu^{\prime}\right\\}}=S^{(p-2)}_{\lambda\lambda^{\prime}}\cdot
S^{(1)}_{\mu\mu^{\prime}}\cdot S^{(p-1)}_{\nu\nu^{\prime}}\leavevmode\nobreak\
,\ \ \
\theta_{\left\\{\lambda,\mu,\nu\right\\}\left\\{\lambda^{\prime},\mu^{\prime},\nu^{\prime}\right\\}}=\theta^{(p-2)}_{\lambda\lambda^{\prime}}\cdot\theta^{(1)}_{\mu\mu^{\prime}}\cdot\bar{\theta}^{(p-1)}_{\nu\nu^{\prime}}\leavevmode\nobreak\
,$ (127)

where the superscripts on the righthand sides of the above equations refer to
the corresponding factors in the coset (117). From the $S$ matrix, Verlinde’s
formula yields (see also the discussion in [45])

$N_{\left\\{\lambda,\mu,\nu\right\\}\left\\{\lambda^{\prime},\mu^{\prime},\nu^{\prime}\right\\}}^{\left\\{\lambda^{\prime\prime},\mu^{\prime\prime},\nu^{\prime\prime}\right\\}}=N_{\lambda\lambda^{\prime}}^{(p-2)\lambda^{\prime\prime}}\cdot
N_{\mu\mu^{\prime}}^{(1)\mu^{\prime\prime}}\cdot
N_{\nu\nu^{\prime}}^{(p-1)\nu^{\prime\prime}}\leavevmode\nobreak\ ,$ (128)

where, again, the superscripts on the righthand side denote the different
coset factors in (117). The factor $SU(2)_{p-2}^{\rm int}$ in the second line
of (122) is clearly closed under fusion. So too is $SU(2)_{p-1}^{\rm conj}$.
To have factorization of the TQFT, we need only show that all Wilson lines can
be written in this way and, by Müger’s theorem, that one of these factors is
modular. The second part is trivial: we have already seen that
$SU(2)_{p-2}^{\rm int}$ is modular in the discussion surrounding (115). We can
confirm this statement by looking at the modular $S$-matrix for $SU(2)_{p-2}$

$S^{(p-2)}_{\lambda\lambda^{\prime}}=\sqrt{2\over
p}\sin\left({(\lambda+1)(\lambda^{\prime}+1)\pi\over
p}\right)\leavevmode\nobreak\ .$ (129)

and taking the submatrix involving the integer spins (even weights).

Therefore, we need only check that all states in the coset (117) can be
expressed in this way. To that end, we can see that

$|SU(2)^{\rm int}_{p-2}|={p-1\over 2}\leavevmode\nobreak\ ,\ |SU(2)_{p-1}^{\rm
conj}|=p\leavevmode\nobreak\ ,$ (130)

where the norm denotes the number of simple elements within. Therefore, we see
that we have $|\mathcal{T}_{p}|=p(p-1)/2$, which is precisely the number of
states in the coset (117) (note that in these computations we have used (119)
and (121)) and the corresponding $A$-type Virasoro minimal model.

To make contact with the fusion rules in (4), we need to explain precisely how
coset lines map onto the Virasoro primaries. The results above allow us to
realize the, say, Virasoro left-movers as states on the boundary of the bulk
TQFT, $\mathcal{T}_{p}\simeq SU(2)_{p-2}^{\rm int}\boxtimes SU(2)_{p-1}^{\rm
conj}$ with $p$ odd. Now, we need to see how we can map boundary endpoints of
lines in this theory to Virasoro primaries, $\varphi_{(r,s)}$. To that end, by
comparing the $S$-matrix for $\mathcal{T}_{p}\simeq SU(2)_{p-2}^{\rm
int}\boxtimes SU(2)_{p-1}^{\rm conj}$ with the corresponding expressions for
those of the Virasoro minimal models, we have that the labels of the Virasoro
primary, $\varphi_{(r,s)}$ map as follows (see also [45])

$r=\lambda+1\leavevmode\nobreak\ ,\ \ \ s=\nu+1\leavevmode\nobreak\ .$ (131)

In particular, we see that the $\varphi_{(r,1)}$ primaries are endpoints of
lines in $SU(2)_{p-2}^{\rm int}$ while the $\varphi_{(1,s)}$ are endpoints of
lines in $SU(2)_{p-1}^{\rm conj}$. This reasoning explains the fact that non-
abelian Virasoro primaries of these types have unique fusion
outcomes454545Though, again, we stress that this factorization is not a
factorization of RCFT correlators.

$\varphi_{(r,1)}\times\varphi_{(1,s)}=\varphi_{(r,s)}\leavevmode\nobreak\ ,$
(132)

discussed in the introduction (at least for $p$ odd). As an example, we have
$\mathcal{T}_{3}\simeq{\rm Ising}$ (i.e., the TQFT is the Ising MTC), which
does not factorize. On the other hand, for $p=5$, we have

$\mathcal{T}_{5}=(G_{2})_{1}\boxtimes SU(2)_{4}^{\rm conj}\leavevmode\nobreak\
,$ (133)

where $(G_{2})_{1}$ is the so-called “Fibnonacci" TQFT, and $SU(2)_{4}^{\rm
conj}$ is a TQFT with the same fusion rules and $S$-matrix as $SU(2)_{4}$.

Let us now consider $p\geq 4$ even. The modular data and fusion rules still
take a product form as in (127) and (128). Now, however, we should examine the
first line in (122). Using (128), it is again easy to see that both
$SU(2)_{p-1}^{\rm int}$ and $(SU(2)_{p-2}\boxtimes SU(2)_{1})^{\rm int}$ are
separately closed under fusion. Moreover, just as before, we can use the
discussion around (115) and Müger’s theorem to conclude that $SU(2)_{p-1}^{\rm
int}$ is indeed a decoupled TQFT as claimed in (122).

We should again check that all states in (117) can be reproduced. To that end,
we have

$|SU(2)^{\rm int}_{p-1}|={p\over 2}\leavevmode\nobreak\ ,\
|(SU(2)_{p-2}\boxtimes SU(2)_{1})^{\rm int}|=p-1\leavevmode\nobreak\ .$ (134)

As a result, we have $|\mathcal{T}_{p}|=p(p-1)/2$, which is the correct number
of states in the coset (117) and the corresponding $A$-type Virasoro minimal
model.

Our mapping is again as in (131), but now $\varphi_{(r,1)}$ primaries are
endpoints of lines in $(SU(2)_{p-2}\boxtimes SU(2)_{1})^{\rm int}$, and
$\varphi_{1,s}$ are endpoints of lines in $SU(2)_{p-1}^{\rm int}$. This again
explains the fusion outcomes in (132) for the case of $p$ even as well. As an
example, note that

$\mathcal{T}_{4}={\rm Ising^{\prime}}\boxtimes(F_{4})_{1}\leavevmode\nobreak\
,$ (135)

where the first factor is a rank three TQFT with the same fusion rules as
Ising (and $SU(2)_{2}$), and the second factor is the time reversal of the
Fibonacci theory in (133).

As a result, we conclude that, although the TQFTs discussed in this section do
have non-abelian anyons fusing to give a unique outcome, this is due to the
fact that the corresponding TQFTs factorize.

###  3.3 Beyond Virasoro: cosets with fixed points

In section 2 we saw that fixed points of various kinds gave rise to fusion
rules of the form (9) (in particular, see theorem 1 of section 2.2.3). In the
context of cosets, we can also naturally engineer fixed points under the
action of fusion with abelian anyons generating identifications of fields. In
the case of Virasoro, this didn’t happen (see (119)). Indeed, this statement
followed from the fact that we had an abelian factor in the coset (117).

The simplest way to get around this obstacle and generate fixed points is to
consider instead

$\widehat{\mathcal{T}}_{p}={SU(2)_{p-2}\boxtimes SU(2)_{2}\over
SU(2)_{p}}\leavevmode\nobreak\ ,$ (136)

where $p\geq 3$ (we should take $p\geq 4$ to avoid the problem of abelian
factors). By further identifying some of these coset fields, we get theories
related to the $\mathcal{N}=1$ super-Virasoro minimal models [53, 51]. Note
that the case of $p=3$ corresponds to the $\mathcal{T}_{4}$ case discussed
previously (i.e., to the TQFT related to the tri-critical Ising model).

For the theories in (136), we find the following generalization of the
identification condition in (119)464646We also require that
$\lambda+\mu+\nu=0\ {\rm(mod\ 2)}$ so that the lines in the coset theory have
trivial braiding with the bosonic line
$\mathcal{W}_{\left\\{p-2,2,p\right\\}}$. This line is in turn identified with
the vacuum.

$\displaystyle\mathcal{W}_{\\{\lambda,\mu,\nu\\}}$ $\displaystyle:=$
$\displaystyle\mathcal{W}_{\lambda}\times\mathcal{W}_{\mu}\times\mathcal{W}_{\nu}\simeq\left(\mathcal{W}_{p-2}\times\mathcal{W}_{\lambda}\right)\times\left(\mathcal{W}_{2}\times\mathcal{W}_{\mu}\right)\times\left(\mathcal{W}_{p}\times\mathcal{W}_{\nu}\right)$
(137) $\displaystyle=$
$\displaystyle\mathcal{W}_{p-2-\lambda}\times\mathcal{W}_{2-\mu}\times\mathcal{W}_{p-\nu}\leavevmode\nobreak\
,$ (138)

In particular, if $\lambda=(p-2)/2$, $\mu=1$, and $\nu=p/2$, we can have a
fixed point474747Note that the fixed points discussed in section 2 are fixed
points under 1-form and 0-form symmetry action. In the coset examples studied
here, fixed points refer to field identification fixed points.. Of course, if
$p$ is odd, we don’t have a fixed point. In this case, we can again run logic
similar to that used in the Virasoro case to argue that the TQFT factorizes.

However, if $p$ is even, then we need to properly define the coset. In
particular, we should resolve the fixed point Wilson line as follows (see [54,
15] for the dual RCFT discussion)

$\mathcal{W}_{\left\\{(p-2)/2,1,p/2\right\\}}\to\mathcal{W}_{\left\\{(p-2)/2,1,p/2\right\\}}^{(1)}+\mathcal{W}_{\left\\{(p-2)/2,1,p/2\right\\}}^{(2)}\leavevmode\nobreak\
.$ (139)

Let us consider what turns out to be the simplest interesting case, $p=6$

$\widehat{\mathcal{T}}_{6}={SU(2)_{4}\boxtimes SU(2)_{2}\over
SU(2)_{6}}\leavevmode\nobreak\ .$ (140)

The fixed point resolution in (139) becomes
$\mathcal{W}_{\left\\{2,1,3\right\\}}\to\mathcal{W}_{\left\\{2,1,3\right\\}}^{(1)}+\mathcal{W}_{\left\\{2,1,3\right\\}}^{(2)}$.
As in the cases of one-form gauging with fixed points discussed in section 2,
it is natural that there should be a zero-form symmetry exchanging
$\mathcal{W}_{\left\\{2,1,3\right\\}}^{(1)}\leftrightarrow\mathcal{W}_{\left\\{2,1,3\right\\}}^{(2)}$.

As a first step to better understand the theory after resolving the fixed
point, note that $\widehat{\mathcal{T}}_{6}$ has the following number of lines

$|\widehat{\mathcal{T}}_{6}|=28\leavevmode\nobreak\ .$ (141)

Of these fields, twenty-six come from identifying full length-two orbits in
(137) while two come from resolving the fixed point. In what follows,
$\left\\{\lambda,\mu,\nu\right\\}$ will denote fields in full orbits, while
labels of the form $\left\\{2,1,3\right\\}^{(i)}$ (with $i=1,2$) will denote
the fixed point lines.

To understand the fusion rules and the question of primality after fixed point
resolution, we can compute the $S$ matrix using the algorithm discussed in
[54] (let us denote the result by $\tilde{S}$). It takes the form

$\displaystyle\tilde{S}_{\left\\{\lambda,\mu,\nu\right\\}\left\\{\lambda^{\prime},\mu^{\prime},\nu^{\prime}\right\\}}$
$\displaystyle=$ $\displaystyle
2S_{\left\\{\lambda,\mu,\nu\right\\}\left\\{\lambda^{\prime},\mu^{\prime},\nu^{\prime}\right\\}}\leavevmode\nobreak\
,\ \ \
\tilde{S}_{\left\\{2,1,3\right\\}^{(i)}\left\\{\lambda^{\prime},\mu^{\prime},\nu^{\prime}\right\\}}=S_{\left\\{2,1,3\right\\}\left\\{\lambda^{\prime},\mu^{\prime},\nu^{\prime}\right\\}}\leavevmode\nobreak\
,$ (142)
$\displaystyle\tilde{S}_{\left\\{2,1,3\right\\}^{(i)}\left\\{2,1,3\right\\}^{(j)}}$
$\displaystyle=$ $\displaystyle{1\over 2}\begin{pmatrix}1&-1&\\\
-1&1\end{pmatrix}\leavevmode\nobreak\ ,$ (143)

where

$S_{\left\\{\lambda,\mu,\nu\right\\}\left\\{\lambda^{\prime},\mu^{\prime},\nu^{\prime}\right\\}}=S^{(p-2)}_{\lambda\lambda^{\prime}}\cdot
S^{(2)}_{\mu\mu^{\prime}}\cdot S^{(p)}_{\nu\nu^{\prime}}\leavevmode\nobreak\
,$ (144)

is the naive generalization of (127) to the cosets at hand. Note that the
fusion rules we obtain from $\tilde{S}$ for fields not involving
$\left\\{2,1,3\right\\}^{(i)}$ are the naive ones we get from $S$ via the
restrictions and identifications described above.

The above discussion is sufficient to prove that $\widehat{\mathcal{T}}_{6}$
is prime. Indeed, we see from (142) that the fields that come from identifying
length-two orbits have the quantum dimensions they inherit from $S$. The fixed
point resolution fields, on the other hand, have half the quantum dimension of
the fixed point field. We therefore have the following four abelian anyons
generating a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ fusion algebra

$\mathcal{W}_{\left\\{0,0,0\right\\}}\simeq\mathcal{W}_{\left\\{4,2,6\right\\}}\leavevmode\nobreak\
,\
\mathcal{W}_{\left\\{4,0,0\right\\}}\simeq\mathcal{W}_{\left\\{0,2,6\right\\}}\leavevmode\nobreak\
,\
\mathcal{W}_{\left\\{0,2,0\right\\}}\simeq\mathcal{W}_{\left\\{4,0,6\right\\}}\leavevmode\nobreak\
,\
\mathcal{W}_{\left\\{0,0,6\right\\}}\simeq\mathcal{W}_{\left\\{4,2,0\right\\}}\leavevmode\nobreak\
.$ (145)

By (142), we see that the braiding amongst abelian anyons is not affected by
taking $S\to\tilde{S}$. As a result, we see that the four abelian anyons all
braid trivially. Therefore, they cannot form a decoupled TQFT.

Wilson lines | ${\rm Quantum\ dimensions}$
---|---
$\mathcal{W}_{\\{0,0,0\\}},\mathcal{W}_{\\{4,0,0\\}},\mathcal{W}_{\\{0,2,0\\}},\mathcal{W}_{\\{0,0,6\\}}$ | $1$
$\mathcal{W}_{\\{0,0,2\\}},\mathcal{W}_{\\{0,0,4\\}},\mathcal{W}_{\\{4,0,2\\}},\mathcal{W}_{\\{4,0,4\\}}$ | $\cot\left(\frac{\pi}{8}\right)$
$\mathcal{W}_{\\{1,0,1\\}},\mathcal{W}_{\\{1,0,5\\}},\mathcal{W}_{\\{3,0,1\\}},\mathcal{W}_{\\{3,0,5\\}}$ | $\sqrt{\frac{3}{2}}\csc\left(\frac{\pi}{8}\right)$
$\mathcal{W}_{\\{0,1,3\\}},\mathcal{W}_{\\{2,1,3\\}^{(1)}},\mathcal{W}_{\\{2,1,3\\}^{(2)}}$ | $\sqrt{2}\csc\left(\frac{\pi}{8}\right)$
$\mathcal{W}_{\\{1,0,3\\}},\mathcal{W}_{\\{3,0,3\\}}$ | $\sqrt{3}\csc\left(\frac{\pi}{8}\right)$
$\mathcal{W}_{\\{2,0,0\\}},\mathcal{W}_{\\{2,0,6\\}}$ | $2$
$\mathcal{W}_{\\{0,1,1\\}},\mathcal{W}_{\\{0,1,5\\}}$ | $\csc\left(\frac{\pi}{8}\right)$
$\mathcal{W}_{\\{1,1,0\\}},\mathcal{W}_{\\{1,1,6\\}}$ | $\sqrt{6}$
$\mathcal{W}_{\\{2,0,2\\}},\mathcal{W}_{\\{2,0,4\\}}$ | $2\cot\left(\frac{\pi}{8}\right)$
$\mathcal{W}_{\\{1,1,2\\}},\mathcal{W}_{\\{1,1,4\\}}$ | $\sqrt{6}\cot\left(\frac{\pi}{8}\right)$
$\mathcal{W}_{\\{2,1,1\\}}$ | $2\csc\left(\frac{\pi}{8}\right)$
Table 1: The twenty-eight Wilson lines and associated quantum dimensions in
the $\widehat{\mathcal{T}}_{6}$ TQFT.

Given this discussion, what could a putative factorized theory look like?
Since $\widehat{\mathcal{T}}_{6}$ has order $28=7\cdot 2^{2}$, we see that the
only way to have a non-trivial factorization is to have a factorization of the
form $\tilde{\mathcal{T}}_{14}\boxtimes\tilde{\mathcal{T}}_{2}$ into prime
TQFTs with rank fourteen and rank two, or
$\tilde{\mathcal{T}}_{7}\boxtimes\tilde{\mathcal{T}}_{4}$ with prime TQFTs of
rank seven and four, or
$\tilde{\mathcal{T}}_{7}\boxtimes\tilde{\mathcal{T}}_{2}\boxtimes\tilde{\mathcal{T}}_{2}^{\prime}$
with prime TQFTs of rank seven, two, and two.

Let us consider the first factorization first. Since the abelian anyons (and
any subset thereof) cannot form a separate TQFT factor (this factor would be
non-modular), the classification in [7] implies that we have either
$\tilde{\mathcal{T}}_{2}\simeq(G_{2})_{1}$ or
$\tilde{\mathcal{T}}_{2}\simeq(F_{4})_{1}$. In any case, the non-trivial anyon
in $\tilde{\mathcal{T}}_{2}$ has quantum dimension $d_{\tau}=(1+\sqrt{5})/2$.
It is easy to check that no such quantum dimension can be produced from
products of quantum dimensions in the different coset factors (and so
restrictions cannot produce them either). Moreover, one can check that the
resolved fixed point fields cannot have this quantum dimension either. This
same logic applies to the
$\tilde{\mathcal{T}}_{7}\boxtimes\tilde{\mathcal{T}}_{2}\boxtimes\tilde{\mathcal{T}}_{2}^{\prime}$
factorization as well.

Therefore, it only remains to consider
$\tilde{\mathcal{T}}_{7}\boxtimes\tilde{\mathcal{T}}_{4}$. The other factor,
$\tilde{\mathcal{T}}_{4}$, has four anyons. By [7], this theory is either
$(G_{2})_{2}$ or its time reversal. In either case, we cannot produce the
requisite $d_{\alpha}=2\ {\rm cos}(\pi/9)$ quantum dimension from our coset.
Therefore, we conclude that $\widehat{\mathcal{T}}_{6}$ is indeed a prime
TQFT.

Moreover, we find the following fusion rules of non-abelian Wilson lines with
unique outcome

$\displaystyle\mathcal{W}_{\left\\{2,0,0\right\\}}\times\mathcal{W}_{\left\\{0,0,2\right\\}}$
$\displaystyle=$
$\displaystyle\mathcal{W}_{\left\\{2,0,2\right\\}}\leavevmode\nobreak\ ,\
\mathcal{W}_{\left\\{2,0,0\right\\}}\times\mathcal{W}_{\left\\{0,0,4\right\\}}=\mathcal{W}_{\left\\{2,0,4\right\\}}\leavevmode\nobreak\
,$ (146)
$\displaystyle\mathcal{W}_{\left\\{1,1,0\right\\}}\times\mathcal{W}_{\left\\{0,0,2\right\\}}$
$\displaystyle=$
$\displaystyle\mathcal{W}_{\left\\{1,1,2\right\\}}\leavevmode\nobreak\ ,\
\mathcal{W}_{\left\\{1,1,0\right\\}}\times\mathcal{W}_{\left\\{0,0,4\right\\}}=\mathcal{W}_{\left\\{1,1,4\right\\}}\leavevmode\nobreak\
,$ (147)
$\displaystyle\mathcal{W}_{\left\\{0,1,1\right\\}}\times\mathcal{W}_{\left\\{2,0,0\right\\}}$
$\displaystyle=$
$\displaystyle\mathcal{W}_{\left\\{2,1,1\right\\}}\leavevmode\nobreak\ .$
(148)

We can obtain additional such fusion rules by taking a product with some of
the abelian lines in (145).

Just as in the case of discrete gauge theories with fusion rules of the above
type, our theory also has a non-trivial symmetry of the modular data. Indeed,
from (142), it is clear that the $\tilde{S}$-matrix has a $\mathbb{Z}_{2}$
symmetry under the interchange

$g\left(\mathcal{W}_{\left\\{2,1,3\right\\}^{(1)}}\right)=\mathcal{W}_{\left\\{2,1,3\right\\}^{(2)}}\leavevmode\nobreak\
,\ \ \ 1\neq g\in\mathbb{Z}_{2}\leavevmode\nobreak\ .$ (149)

Note that this symmetry is not charge conjugation since $\tilde{S}$ is
manifestly real. Moreover, since we don’t change the twists, this action lifts
to a symmetry of the modular data (additionally, it should lift to a symmetry
of the full TQFT).

If we wish to make contact with the $\mathcal{N}=1$ minimal model, then we
should note that the fermionic $\mathcal{W}_{\left\\{0,2,0\right\\}}$ line
corresponds to the supercurrent of the SCFT. We can then organize the Neveu-
Schwarz (NS) sector into supermultiplets under fusion with this operator.
Doing so (and paying careful attention to the fields in the resolution of the
fixed point), we find nine NS sector fields and nine Ramond sector fields as
required.

There are many ways to generalize the example we have given here. Indeed, when
there are fixed points in the coset construction we expect to often be able to
generate fusion rules of the form (2). A deeper understanding of these
theories and some more general methods to characterize whether the cosets are
prime (along the lines of the general criteria we have in the case of discrete
gauge theories) would be useful. In any case, we see that, as in the case of
discrete gauge theories, symmetry fixed points and zero-form (quasi)
symmetries are deeply connected with fusion rules of the form (2).

##  4 Summary and Conclusions

In this paper, we have seen that the existence of fusions of non-abelian
anyons having a unique outcome is intimately connected with the global
structure of the corresponding TQFT.

Let us summarize our results for continuous gauge groups (and continuous
groups more generally):

  * •

Building on the well-known fact that $SU(2)$ spin addition / fusion of two
non-abelian representations (i.e., higher-dimensional / spin non-singlet
representations) is reducible (i.e., has multiple outcomes with different
total spin), we argued that a similar result holds in all compact simple Lie
groups.

  * •

We argued that the result in the previous bullet point on classical groups can
be extended to a theorem constraining $SU(N)_{k}$ CS theory: fusions of non-
abelian Wilson lines in these theories do not have unique outcomes. More
generally, Wilson lines corresponding to affine representations with at least
two non-vanishing Dynkin labels in any $G_{k}$ CS theory (for $G$ a compact
simple Lie group) do not have unique outcomes. These results have implications
for the global structure of these theories (claim 14): the Wilson lines
discussed here must all lie in the same prime factor (although $G_{k}$ CS
theories are not prime in general).

  * •

We showed that one way to produce $a\times b=c$ fusions involving non-abelian
$a$ and $b$ is to consider cosets. In the case of TQFTs underlying Virasoro
minimal models we argued that (as in the $(E_{7})_{2}$ case) such rules arise
from factorizations of the TQFTs into multiple prime factors. On the other
hand, if we include cosets with fixed points, we can obtain prime theories
with such fusion rules.

Next, let us summarize our results for discrete gauge groups (and discrete
groups more generally):

  * •

We argued that Zisser’s construction of irreducible products of higher-
dimensional irreducible $A_{N}$ representations [25] can be lifted to fusions
of non-abelian Wilson lines with unique outcomes in $A_{N}$ discrete gauge
theory. From the perspective of the closely related $S_{N}$ group and
corresponding discrete gauge theory, the $A_{N}$ result requires certain
1-form symmetry fixed points (where we define “one-form symmetry" in the
$S_{N}$ group to correspond to the $\mathbb{Z}_{2}\subset{\rm Rep(S_{N})}$
generated by the sign representation). We then derived theorem 1 that
generalizes this relation between the $A_{N}$ and $S_{N}$ discrete gauge
theories to other TQFTs.

  * •

Going to the $S_{N}$ discrete gauge theory by gauging the $\mathbb{Z}_{2}$
$0$-form outer automorphism symmetry of the $A_{N}$ discrete gauge theory
resolves the $a\times b=c$ non-abelian fusion rule into fusion rules not of
this type. However, we saw that in the case of $O(5,3)$ discrete gauge theory
such resolutions do not always occur via automorphism gauging. On the other
hand, a symmetry fixed point again plays a role: in the resulting
$O(5,3)\rtimes\mathbb{Z}_{2}$ discrete gauge theory, there is a 0-form
symmetry fixed point. We then proved theorem 2, which explains why this
phenomenon occurs in more general theories. In fact, the
$O(5,3)\rtimes\mathbb{Z}_{2}$ discrete gauge theory relative of the $a\times
b=c$ fusion equations in the $O(5,3)$ TQFT described in (49) also has a 1-form
symmetry fixed point for the anyon appearing on the right hand side. In the
original $O(5,3)$ TQFT this latter anyon becomes a set of two anyons related
by the 0-form symmetry. Our theorem 2A generalizes this observation to other
TQFTs.

  * •

We showed that one can lift Gallagher’s theorem to a statement on the fusion
of non-abelian Wilson lines involving unfaithful representations with a unique
outcome in TQFT. Moreover, we elucidated the roles that subcategory structure
and symmetries play in this result for various specific TQFTs. We then proved
theorem 3 that generalizes these observations to a broader set of theories. We
also argued that this subcategory structure helps explain the large ratio of
group orders in (58).

  * •

To gain a sense of how magnetic fluxes behave in general discrete gauge
theories, we proved theorem 5. In particular, we showed that in discrete gauge
theories with a non-abelian gauge group, $G$, the magnetic fluxes do not form
a fusion subcategory. This result immediately places constraints on electric-
magnetic self-dualities / quantum symmetries that constrain our symmetry
searches later in section 2.

  * •

At a more constructive level, we also proved theorem 10. This result gives
infinitely many generalizations of the well-known electric-magnetic self-
duality of the $S_{3}$ discrete gauge theory.

  * •

In order to better understand which discrete gauge theories are prime, we
proved theorem 7. This result allowed us to more easily analyze which prime
discrete gauge theories have fusions of non-abelian anyons with unique
outcomes.

  * •

In order to get a handle on the structure of discrete gauge theories with
fusion rules of our desired type involving anyons carying non-trivial flux, we
proved theorem 11 and corollary 12. These results give the subcategory
structure that arises when such fusions occur. In turn, this structure gives
an explanation of these fusion rules. Theorem 13 then partially extends these
results to the case in which one of the non-abelian anyons involved is a
Wilson line.

  * •

The software GAP was used to analyze the fusion rules of hundreds of untwisted
discrete gauge theories. In all the cases we checked, we find that discrete
gauge theories with $a\times b=c$ type fusion rules have quasi-zero-form
symmetries. This suggests that symmetries of the modular data are a
characteristic feature of such fusion rules.

The above discussion leads to various natural questions:

  * •

In the discussion around (58) we explained the large hierarchy between the
size of simple and non-simple groups whose corresponding discrete gauge
theories have non-abelian Wilson lines satisfying (2) by using symmetries and
subcategory structure. It would be interesting to explore whether other
related hierarchies can be explained in a similar way.

  * •

We saw that in almost all the prime untwisted discrete gauge theories we
studied, if there was a fusion rule of the form (2), then the theory had non-
trivial zero-form symmetries. The only exceptions where discrete gauge
theories based on the $M_{23}$ and $M_{24}$ Mathieu groups discussed in
section 2.5. Here we argued that there were zero-form symmetries of the
modular data that did not lift to symmetries of the full theory. It would be
interesting to understand if gauge theories based on certain finite simple
sporadic groups are the only prime theories with fusion rules of the form (2)
that exhibit this phenomenon.

  * •

In section 3.1, we proved that the non-abelian lines of $SU(N)_{k}$ CS theory
don’t have fusion rules of the form (2). While $(E_{7})_{2}$ CS theory does
have such fusion rules, we do not know of an example of such a fusion in a
prime $G_{k}$ CS theory with $G$ a compact and simple Lie group. It would be
interesting to either find an example of such a fusion or prove a more general
theorem forbidding one. Given such fusions are common for discrete gauge
theories, it would be interesting to understand how these two statements
interact with each other.

  * •

As we saw in section 3.3, it would be useful to develop new tools to
understand primality in theories built on cosets. One promising direction is
to study the role of Galois actions in such theories.

## Acknowledgements

We thank D. Aasen, M. Barkeshli, S. Ramgoolam, and I. Runkel for useful
discussions and correspondence. M. B. is funded by the Royal Society grant,
“New Constraints and Phenomena in Quantum Field Theory." M. B. and R. R. are
funded by the Royal Society grant, “New Aspects of Conformal and Topological
Field Theories Across Dimensions." The STFC also partially supported our work
under the grant, “String Theory, Gauge Theory and Duality."

##  Appendix A Wilson line $a\times b=c$ in gauge theories with order forty-
eight discrete gauge group

Let us study groups of order $48$ for which the corresponding discrete gauge
theories have Wilson line $a\times b=c$ type fusions484848We won’t discuss the
direct product groups $S_{3}\times S_{3},D_{8}\times S_{3}$ and $Q_{8}\times
S_{3}$ which also have such fusions (the corresponding discrete gauge theories
factorize). Since we have already discussed the case of $BOG$ and $GL(2,3)$,
we won’t be discussing them here.

$(48,15)$ ($(\mathbb{Z}_{3}\times D_{8})\rtimes\mathbb{Z}_{2}$);

$\displaystyle\mathcal{W}_{2_{2}}\times\mathcal{W}_{2_{4}}$ $\displaystyle=$
$\displaystyle\mathcal{W}_{4}\leavevmode\nobreak\ ,\
\mathcal{W}_{2_{2}}\times\mathcal{W}_{2_{5}}=\mathcal{W}_{4}\leavevmode\nobreak\
,\
\mathcal{W}_{2_{3}}\times\mathcal{W}_{2_{4}}=\mathcal{W}_{4}\leavevmode\nobreak\
,\ \mathcal{W}_{2_{3}}\times\mathcal{W}_{2_{5}}=\mathcal{W}_{4}$ (150)
$\displaystyle\mathcal{W}_{2_{4}}\times\mathcal{W}_{2_{6}}$ $\displaystyle=$
$\displaystyle\mathcal{W}_{4}\leavevmode\nobreak\ ,\
\mathcal{W}_{2_{4}}\times\mathcal{W}_{2_{7}}=\mathcal{W}_{4}\leavevmode\nobreak\
,\
\mathcal{W}_{2_{5}}\times\mathcal{W}_{2_{6}}=\mathcal{W}_{4}\leavevmode\nobreak\
,\
\mathcal{W}_{2_{5}}\times\mathcal{W}_{2_{7}}=\mathcal{W}_{4}\leavevmode\nobreak\
.\ \ \ $ (151)

We have $\text{Out}((\mathbb{Z}_{3}\times
D_{8})\rtimes\mathbb{Z}_{2})=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. Let $r_{1}$
and $r_{2}$ be the generators of this group. They act on the Wilson lines
involved in the fusion above as follows

$\displaystyle
r_{1}:\mathcal{W}_{2_{2}}\leftrightarrow\mathcal{W}_{2_{2}};\leavevmode\nobreak\
\mathcal{W}_{2_{3}}\leftrightarrow\mathcal{W}_{2_{3}};\leavevmode\nobreak\
\mathcal{W}_{2_{4}}\leftrightarrow\mathcal{W}_{2_{4}};\leavevmode\nobreak\
\mathcal{W}_{2_{5}}\leftrightarrow\mathcal{W}_{2_{5}};\leavevmode\nobreak\
\mathcal{W}_{2_{6}}\leftrightarrow\mathcal{W}_{2_{7}};\leavevmode\nobreak\ $
(152) $\displaystyle
r_{1}:\mathcal{W}_{2_{2}}\leftrightarrow\mathcal{W}_{2_{2}};\leavevmode\nobreak\
\mathcal{W}_{2_{3}}\leftrightarrow\mathcal{W}_{2_{3}};\leavevmode\nobreak\
\mathcal{W}_{2_{4}}\leftrightarrow\mathcal{W}_{2_{5}};\leavevmode\nobreak\
\mathcal{W}_{2_{6}}\leftrightarrow\mathcal{W}_{2_{6}};\leavevmode\nobreak\
\mathcal{W}_{2_{7}}\leftrightarrow\mathcal{W}_{2_{7}};\leavevmode\nobreak\ $
(153)

Since this group has complex characters we also have a non-trivial quasi-zero-
form symmetry given by complex conjugation.
$\mathcal{Z}(\text{Vec}_{(\mathbb{Z}_{3}\times D_{8})\rtimes\mathbb{Z}_{2}})$
also has all other $a\times b=c$ type fusions (involving fluxes and dyons)
discussed in this paper.

$(48,16)$ ($(\mathbb{Z}_{3}:Q_{8})\rtimes\mathbb{Z}_{2}$); This has fusions
identical to (150). The only difference is that now $\mathcal{W}_{2_{4}}$ and
$\mathcal{W}_{2_{5}}$ are conjugates. The outer automorphism group and
symmetry action is identical to $\mathcal{Z}(\text{Vec}_{(\mathbb{Z}_{3}\times
D_{8})\rtimes\mathbb{Z}_{2}})$. Since this group has complex characters we
also have a non-trivial quasi-zero-form symmetry given by complex conjugation.
We additionally have all other $a\times b=c$ type fusions (involving fluxes
and dyons) discussed in this paper.

$(48,17)$ (($\mathbb{Z}_{3}\times Q_{8})\rtimes\mathbb{Z}_{2}$); This has
identical character table to $(48,16)$, so same fusion rules. The properties
are identical to the two cases above.

$(48,18)$ ($\mathbb{Z}_{3}\rtimes Q_{16}$); Identical characters to $(48,15)$,
so shares (150). The discussion is identical to the case above.

$(48,39)$ ($(\mathbb{Z}_{4}\times S_{3})\rtimes\mathbb{Z}_{2}$);

$\displaystyle\mathcal{W}_{2_{1}}\times\mathcal{W}_{2_{5}}$ $\displaystyle=$
$\displaystyle\mathcal{W}_{4}\leavevmode\nobreak\ ,\
\mathcal{W}_{2_{1}}\times\mathcal{W}_{2_{6}}=\mathcal{W}_{4}\leavevmode\nobreak\
,\
\mathcal{W}_{2_{2}}\times\mathcal{W}_{2_{5}}=\mathcal{W}_{4}\leavevmode\nobreak\
,\ \mathcal{W}_{2_{2}}\times\mathcal{W}_{2_{6}}=\mathcal{W}_{4}$ (154)
$\displaystyle\mathcal{W}_{2_{3}}\times\mathcal{W}_{2_{5}}$ $\displaystyle=$
$\displaystyle\mathcal{W}_{4}\leavevmode\nobreak\ ,\
\mathcal{W}_{2_{3}}\times\mathcal{W}_{2_{6}}=\mathcal{W}_{4}\leavevmode\nobreak\
,\
\mathcal{W}_{2_{4}}\times\mathcal{W}_{2_{5}}=\mathcal{W}_{4}\leavevmode\nobreak\
,\
\mathcal{W}_{2_{4}}\times\mathcal{W}_{2_{6}}=\mathcal{W}_{4}\leavevmode\nobreak\
.\ \ \ $ (155)

We have $\text{Out}((\mathbb{Z}_{4}\times
S_{3})\rtimes\mathbb{Z}_{2})=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. Let $r_{1}$
and $r_{2}$ be the generators of this group. They act on the Wilson lines
involved in the fusion above as follows

$\displaystyle
r_{1}:\mathcal{W}_{2_{1}}\leftrightarrow\mathcal{W}_{2_{1}};\leavevmode\nobreak\
\mathcal{W}_{2_{2}}\leftrightarrow\mathcal{W}_{2_{2}};\leavevmode\nobreak\
\mathcal{W}_{2_{3}}\leftrightarrow\mathcal{W}_{2_{3}};\leavevmode\nobreak\
\mathcal{W}_{2_{4}}\leftrightarrow\mathcal{W}_{2_{4}};\leavevmode\nobreak\
\mathcal{W}_{2_{5}}\leftrightarrow\mathcal{W}_{2_{6}};\leavevmode\nobreak\ $
(156) $\displaystyle
r_{1}:\mathcal{W}_{2_{1}}\leftrightarrow\mathcal{W}_{2_{2}};\leavevmode\nobreak\
\mathcal{W}_{2_{3}}\leftrightarrow\mathcal{W}_{2_{3}};\leavevmode\nobreak\
\mathcal{W}_{2_{4}}\leftrightarrow\mathcal{W}_{2_{4}};\leavevmode\nobreak\
\mathcal{W}_{2_{5}}\leftrightarrow\mathcal{W}_{2_{5}};\leavevmode\nobreak\
\mathcal{W}_{2_{6}}\leftrightarrow\mathcal{W}_{2_{6}};\leavevmode\nobreak\ $
(157)

Since this group has complex characters we also have a non-trivial quasi-zero-
form symmetry given by complex conjugation.
$\mathcal{Z}(\text{Vec}_{(\mathbb{Z}_{4}\times S_{3})\rtimes\mathbb{Z}_{2}})$
also have all other $a\times b=c$ type fusions (involving fluxes and dyons)
discussed in this paper.

$(48,41)$; ($(\mathbb{Z}_{4}\times S_{3})\rtimes\mathbb{Z}_{2}$)

Fusion of Wilson lines giving unique output is same as $\eqref{4839W}$. We
have $\text{Out}((\mathbb{Z}_{4}\times S_{3})\rtimes\mathbb{Z}_{2})=D_{12}$.

Since this group has complex characters we also have a non-trivial quasi-zero-
form symmetry given by complex conjugation.
$\mathcal{Z}(\text{Vec}_{(\mathbb{Z}_{4}\times S_{3})\rtimes\mathbb{Z}_{2}})$
also have all other $a\times b=c$ type fusions (involving fluxes and dyons)
discussed in this paper.

##  Appendix B Genuine zero-form symmetries and quasi-zero-form symmetries in
$A_{9}$ discrete gauge theory

Recall from section 2.2.1 that $A_{9}$ is the simplest example of an $A_{N}$
(with $N=k^{2}\geq 9$) discrete gauge theory with fusion rules involving non-
abelian Wilson lines having unique outcome. Here our goal is to disentangle
the genuine zero form symmetries

${\rm Aut}^{\rm br}(\mathcal{Z}({\rm Vec}_{A_{9}}))\simeq
H^{2}(A_{9},U(1))\rtimes{\rm
Out}(A_{9})\simeq\mathbb{Z}_{2}\times\mathbb{Z}_{2}\leavevmode\nobreak\ ,$
(158)

from a charge conjugation quasi zero-form symmetry [38].

Let us first discuss the outer automorphisms. To that end, recall that $A_{9}$
has an outer automorphism corresponding to conjugation by odd elements of
$S_{9}\triangleright A_{9}$. Acting with the outer automorphism generated by
$(89)\in S_{9}$, we see that the following lines are exchanged

$\mathcal{L}_{([(123456789)],\pi_{p})}\leftrightarrow\mathcal{L}_{([(123456798)],\pi_{p})}\leavevmode\nobreak\
,\ \ \
\mathcal{L}_{([(12345)(678)],\pi_{n})}\leftrightarrow\mathcal{L}_{([(12345)(679)],\pi_{n})}\leavevmode\nobreak\
,$ (159)

where the relevant conjugacy classes are listed in table 2, and $0\leq p\leq
8$, $0\leq n\leq 14$ label representations of the corresponding
$\mathbb{Z}_{9}$ and $\mathbb{Z}_{14}$ centralizers (they are also listed in
table 2).

In fact, as described in the main text, the symmetry in (159) generates an
action on some of the Wilson lines involved in (35)

$\mathcal{W}_{[3^{3}]_{+}}\leftrightarrow\mathcal{W}_{[3^{3}]_{-}}\leavevmode\nobreak\
.$ (160)

This action can be read off from the character table of $A_{9}$ or,
equivalently, from the braiding

$\displaystyle{S_{\mathcal{W}_{[3^{3}]_{+}}\mathcal{L}_{([(12345)(678)],\pi_{n})}}\over
S_{\mathcal{W}_{1}\mathcal{L}_{([(12345)(678)],\pi_{n})}}}$ $\displaystyle=$
$\displaystyle\chi_{[3^{3}]_{+}}([(12345)(678)])^{*}=-{1\over
2}(1-i\sqrt{15})\leavevmode\nobreak\ ,$ (161)
$\displaystyle{S_{\mathcal{W}_{[3^{3}]_{-}}\mathcal{L}_{([(12345)(678)],\pi_{n})}}\over
S_{\mathcal{W}_{1}\mathcal{L}_{([(12345)(678)],\pi_{n})}}}$ $\displaystyle=$
$\displaystyle\chi_{[3^{3}]_{-}}([(12345)(678)])^{*}=-{1\over
2}(1+i\sqrt{15})\leavevmode\nobreak\ ,$ (162)
$\displaystyle{S_{\mathcal{W}_{[3^{3}]_{+}}\mathcal{L}_{([(12345)(679)],\pi_{n})}}\over
S_{\mathcal{W}_{1}\mathcal{L}_{([(12345)(679)],\pi_{n})}}}$ $\displaystyle=$
$\displaystyle\chi_{[3^{3}]_{+}}([(12345)(679)])^{*}=-{1\over
2}(1+i\sqrt{15})\leavevmode\nobreak\ ,$ (163)
$\displaystyle{S_{\mathcal{W}_{[3^{3}]_{-}}\mathcal{L}_{([(12345)(679)],\pi_{n})}}\over
S_{\mathcal{W}_{1}\mathcal{L}_{([(12345)(679)],\pi_{n})}}}$ $\displaystyle=$
$\displaystyle\chi_{[3^{3}]_{-}}([(12345)(679)])^{*}=-{1\over
2}(1-i\sqrt{15})\leavevmode\nobreak\ .$ (164)

Note that, since the $[(12345)(678)]$ and $[12345)(679)]$ conjugacy classes
are complex, we also have a non-trivial $\mathbb{Z}_{2}$ charge conjugation
that acts on the modular data and swaps
$\mathcal{W}_{[3^{3}]_{+}}\leftrightarrow\mathcal{W}_{[3^{3}]_{-}}$ and
$\mathcal{L}_{([(123456789)],\pi_{p})}\leftrightarrow\mathcal{L}_{([(123456798)],\pi_{p})}$.
Recall from the discussion in (75) that elements of
$H^{2}(A_{9},U(1))\simeq\mathbb{Z}_{2}$ act trivially on the Wilson lines.
Hence, we learn that charge conjugation cannot be a genuine symmetry of the
TQFT (this statement is also confirmed by the analysis in [38]).

However, this is not a contradiction with what we have written, because ${\rm
Out}(A_{9})$ also interchanges the real conjugacy classes $[(123456789)]$ and
$[(123456798)]$ along with the corresponding lines in (159). Since charge
conjugation leaves these degrees of freedom untouched, it is a distinct
operation.

Conjugacy class | ${\rm Length}$ | ${\rm Centralizer}$
---|---|---
$1$ | $1$ | $A_{9}$
$[(12)(34)]$ | $378$ | ${\rm SmallGroup}(480,951)$
$[(12)(34)(56)(78)]$ | $945$ | ${\rm SmallGroup}(192,1493)$
$[(123)]$ | $168$ | ${\rm SmallGroup}(1080,487)$
$[(123)(45)(67)]$ | $7560$ | ${\rm SmallGroup}(24,10)\leavevmode\nobreak\ ,\ (D_{8}\times\mathbb{Z}_{3})$
$[(123)(456)]$ | $3360$ | ${\rm SmallGroup}(54,13)$
$[(123)(456)(789)]$ | $2240$ | ${\rm SmallGroup}(81,7)\leavevmode\nobreak\ ,\ ((\mathbb{Z}_{3}\times\mathbb{Z}_{3}\times\mathbb{Z}_{3})\rtimes\mathbb{Z}_{3})$
$[(1234)(56)]$ | $7560$ | ${\rm SmallGroup}(24,5)\leavevmode\nobreak\ ,\ (S_{3}\times\mathbb{Z}_{4})$
$[(1234)(567)(89)]$ | $15120$ | ${\rm SmallGroup}(12,2)\leavevmode\nobreak\ ,\ (\mathbb{Z}_{12})$
$[(1234)(5678)]$ | $11340$ | ${\rm SmallGroup}(16,13)\leavevmode\nobreak\ ,\ ({\rm central\ product}\ D_{8},\ Z_{4})$
$[(12345)]$ | $3024$ | ${\rm SmallGroup}(60,9)$
$[(12345)(67)(89)]$ | $9072$ | ${\rm SmallGroup}(20,5)\leavevmode\nobreak\ ,\ (\mathbb{Z}_{10}\times\mathbb{Z}_{2})$
$[(12345)(678)]$ | $12096$ | ${\rm SmallGroup}(15,1)\leavevmode\nobreak\ ,\ (\mathbb{Z}_{15})$
$[(12345)(679)]$ | $12096$ | ${\rm SmallGroup}(15,1)\leavevmode\nobreak\ ,\ (\mathbb{Z}_{15})$
$[(123456)(78)]$ | $30240$ | ${\rm SmallGroup}(6,2)\leavevmode\nobreak\ ,\ (\mathbb{Z}_{6})$
$[(1234567)]$ | $25920$ | ${\rm SmallGroup}(7,1)\leavevmode\nobreak\ ,\ (\mathbb{Z}_{7})$
$[(123456789)]$ | $20160$ | ${\rm SmallGroup}(9,1)\leavevmode\nobreak\ ,\ (\mathbb{Z}_{9})$
$[(123456798)]$ | $20160$ | ${\rm SmallGroup}(9,1)\leavevmode\nobreak\ ,\ (\mathbb{Z}_{9})$
Table 2: The eighteen conjugacy classes of $A_{9}$, their order, and their
centralizers (recall that the centralizers of elements in the same conjugacy
class are isomorphic). The centralizer is labeled by its GAP ID (for
sufficiently small groups) as “${\rm SmallGroup}(a,b)$" along with a more
common name in certain cases.

Note that in the $A_{9}$ discrete gauge theory we can also turn on a large
variety of twists

$\omega\in
H^{3}(A_{9},U(1))\simeq\mathbb{Z}_{2}\times\mathbb{Z}_{3}^{2}\times\mathbb{Z}_{4}\simeq\mathbb{Z}_{6}\times\mathbb{Z}_{12}\leavevmode\nobreak\
.$ (165)

Since the charge conjugation quasi-symmetry is a property of the Wilson line
fusion rules, it remains regardless of the twist.

##  Appendix C GAP code

The following GAP code defines the function checkdyon() which takes in a group
as an argument. It checks for $a\times b=c$ type fusions for non-abelian
anyons $a,b,c\in\mathcal{Z}(\text{Vec}_{G})$ and ouputs all such fusions.
Moreover, if such fusions exist, it outputs $\text{Out}(G)$ as well as
$H^{2}(G,U(1))$. Note that it requires the package HAP to function.

In order to define checkdyon() we need to first define the functions comconj()
and conjprof().

$\displaystyle>\text{conjcom:=function(a,b)}$ $\displaystyle>\text{local
com,i,j;}$ $\displaystyle>\text{com:=[];}$ $\displaystyle>\text{for i in
[1..Size(AsList(a))] do }$ $\displaystyle>\text{for j in [i..Size(AsList(b))]
do}$ $\displaystyle>\text{Append(com,
[AsList(a)[i]*AsList(b)[j]*Inverse(AsList(b)[j]*AsList(a)[i])]);}$
$\displaystyle>\text{od; od;}$ $\displaystyle>\text{return
DuplicateFreeList(com)=[AsList(a)[1]*Inverse(AsList(a)[1])]; end;}$

This function takes two conjugacy classes of a group $G$ as inputs and outputs
true if they commute element-wise and false otherwise. Now, let us define the
function conjprod()

$\displaystyle>\text{conjprod:=function(a,b,c)}$ $\displaystyle>\text{local
prod,i,j,k;}$ $\displaystyle>\text{prod:=[];}$ $\displaystyle>\text{for i in
[1..Size(AsList(a))] do}$ $\displaystyle>\text{for j in [i..Size(AsList(b))]
do}$ $\displaystyle>\text{for k in [1..Size(c)] do}$ $\displaystyle>\text{if
AsList(a)[i]*AsList(b)[j] in AsList(c[k]) then }$
$\displaystyle>\text{Append(prod, [k]); break; fi; od; od; od;}$
$\displaystyle>\text{if Size(DuplicateFreeList(prod))=1 then}$
$\displaystyle>\text{return DuplicateFreeList(prod)[1]; else return 0; fi;
end;}$

This function takes three arguments. The first two arguments $a,b$ are two
conjugacy classes of a group $G$ and the third argument $c$ is the set of all
conjugacy classes of $G$. The function outputs an integer $k>1$ if the product
of two input conjugacy is a single conjugacy class (which is at position $k$
in the list of conjugacy classes $c$). The function outputs $0$ otherwise.

Using these two functions, we finally define the checkdyon() function.

checkdyon:=function(G) $\displaystyle>\text{local
cn,i,j,k,a,l,cen1,cen2,cen3,cenint,irrcenint,irrcen1,irrcen2,irrcen3, }$
cen1res,cen2res,cen3res,x,y,z,w,a1,a2,A,I,F,R;
$\displaystyle>\text{cn:=ConjugacyClasses(G);}$ $\displaystyle>\text{a:=0;}$
$\displaystyle>\text{for i in [1..Size(cn)] do}$ $\displaystyle>\text{for j in
[i..Size(cn)] do}$ $\displaystyle>\text{if conjcom(cn[i],cn[j]) then}$
$\displaystyle>\text{k:=conjprod(cn[i],cn[j],cn); }$ $\displaystyle>\text{if
k$<>$0 then }$ $\displaystyle>\text{cen1:=Centralizer(G,AsList(cn[i])[1]);}$
$\displaystyle>\text{cen2:=Centralizer(G,AsList(cn[j])[1]);}$
$\displaystyle>\text{cen3:=Centralizer(G,AsList(cn[k])[1]);}$
$\displaystyle>\text{cenint:=Intersection(cen1,cen2,cen3);}$
$\displaystyle>\text{irrcen1:=Irr(cen1);}$
$\displaystyle>\text{irrcen2:=Irr(cen2);}$
$\displaystyle>\text{irrcen3:=Irr(cen3);}$
$\displaystyle>\text{cen1res:=RestrictedClassFunctions(irrcen1,cenint);}$
$\displaystyle>\text{cen2res:=RestrictedClassFunctions(irrcen2,cenint);}$
$\displaystyle>\text{cen3res:=RestrictedClassFunctions(irrcen3,cenint);}$
$\displaystyle>\text{irrcenint:=Irr(cenint);}$ $\displaystyle>\text{for x in
[1..Size(cen1res)] do}$ $\displaystyle>\text{for y in [1..Size(cen2res)] do}$
$\displaystyle>\text{if Size(AsList(cn[i]))*DegreeOfCharacter(cen1res[x])$>$1
and}$ Size(AsList(cn[j]))*DegreeOfCharacter(cen2res[y])$>$1 then
$\displaystyle>\text{for z in [1..Size(cen3res)] do}$
$\displaystyle>\text{a1:=[\leavevmode\nobreak\ ]; a2:=[\leavevmode\nobreak\
];}$ $\displaystyle>\text{for w in [1..Size(irrcenint)] do}$
$\displaystyle>\text{Append(a1,[ScalarProduct(irrcenint[w],cen1res[x]*cen2res[y])]);
}$ $\displaystyle>\text{Append(a2,[ScalarProduct(irrcenint[w],cen3res[z])]);}$
$\displaystyle>\text{od;}$ $\displaystyle>\text{if a1*a2=1 and }$
Size(AsList(cn[i]))*DegreeOfCharacter(cen1res[x])*
Size(AsList(cn[j]))*DegreeOfCharacter(cen2res[y])=
Size(AsList(cn[k]))*DegreeOfCharacter(cen3res[z]) then
$\displaystyle>\text{a:=1;}$ $\displaystyle>\text{Print(IdSmallGroup(G), `` ",
StructureDescription(G), ``\textbackslash n");}$
$\displaystyle>\text{Print(``Anyon a: ", cn[i], `` , ", irrcen1[x],
``\textbackslash n");}$ $\displaystyle>\text{Print(``Anyon b: ", cn[j], `` ,
", irrcen2[y], ``\textbackslash n");}$ $\displaystyle>\text{Print(``Anyon c:
", cn[k], `` , ", irrcen3[z], ``\textbackslash n","\textbackslash n");}$
$\displaystyle>\text{fi; od; fi; od;od; fi; fi; od; od; }$
$\displaystyle>\text{if a=1 then}$
$\displaystyle>\text{A:=AutomorphismGroup(G);}$
$\displaystyle>\text{I:=InnerAutomorphismsAutomorphismGroup(A);}$
$\displaystyle>\text{F:=FactorGroup(A,I);}$
$\displaystyle>\text{Print(``Out(G): ",StructureDescription(F),
``\textbackslash n");}$ $\displaystyle>\text{R:=ResolutionFiniteGroup(G,3);}$
$\displaystyle>\text{Print(``H2(G,U(1)):
",Homology(TensorWithIntegers(R),2),``\textbackslash n");}$
$\displaystyle>\text{Print(``\textbackslash n",``\textbackslash n"); fi;}$
$\displaystyle>\text{end;}$

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