check on the proposed actions (4.21) of dualities on local operators on the
orbifold branch.

The relevant chiral algebra is given by the W-algebra $W(2,3,4)$ which
includes spin 3 and 4 generators,

$\displaystyle\cos({\sqrt{6}X_{L}})=\cos(3\theta+\phi)\quad{\rm and}\quad
j_{4}\,,$ (4.42)

in addition to the stress tensor. With respect to $W(2,3,4)$, the
$SU(2)_{k}/U(1)$ coset CFT has 10 primaries $\phi^{j}_{m}$ as listed in Table
7 with $j=0,{1\over 2},1,{3\over 2},2$ and $m+j\in{\mathbb{Z}}$ and subject to
the identification

$\displaystyle\phi^{j}_{m}=\phi^{j}_{m+4}=\phi^{2-j}_{2+m}\,.$ (4.43)

Correspondingly there are 10 TDLs in the $SU(2)_{4}/U(1)$ CFT realized by
Verlinde lines $\mathcal{L}_{(j,m)}$. In particular, from the fusion rules, we
find that the ${\mathbb{Z}}_{4}$ subgroup of $D_{8}$ is realized by the
Verlinde lines as $r^{i}=\mathcal{L}_{(0,i)}$ for $i=0,1,2,3$.

Coset $\phi^{j}_{m}$ | $U(1)_{6}/{\mathbb{Z}}_{2}^{C}$ | $h=\bar{h}$ |  ${\mathbb{Z}}_{4}$ charge
---|---|---|---
$\phi^{0}_{0}$ | 1 | 0 | 0
$\phi^{0}_{1}$ | $\cos{\sqrt{6}(X_{L}+X_{R})\over 2}=\cos(3\theta)$ | ${3\over 4}$ | 2
$\phi^{0}_{2}$ | $\partial X_{L}\bar{\partial}X_{R}=d\phi d\theta$ | ${1}$ | 0
$\phi^{0}_{3}$ | $\cos{\sqrt{6}(X_{L}-X_{R})\over 2}=\cos(\phi)$ | ${3\over 4}$ | 2
$\phi^{1\over 2}_{-{1\over 2}}$ | $\sigma_{1}$ | ${1\over 16}$ | $3$
$\phi^{1\over 2}_{1\over 2}$ | $\sigma_{2}$ | ${1\over 16}$ | $1$
$\phi^{1\over 2}_{3\over 2}$ | $\tau_{1}$ | ${9\over 16}$ | $3$
$\phi^{1\over 2}_{5\over 2}$ | $\tau_{2}$ | ${9\over 16}$ | $1$
$\phi^{1}_{-1}$ | $\cos{X_{L}+X_{R}\over\sqrt{6}}=\cos(\theta)$ | ${1\over 12}$ | 2
$\phi^{1}_{0}$ | $\cos{2(X_{L}+X_{R})\over\sqrt{6}}=\cos(2\theta)$ | ${1\over 3}$ | 0
Table 7: $SU(2)_{4}/U(1)$ coset primaries $\phi^{j}_{m}$ and their
descriptions in the $U(1)_{6}/{\mathbb{Z}}_{2}^{C}$ orbifold.

As we have described in Section 2.6, the $SU(2)_{k}/U(1)$ coset CFT is self-
dual under ${\mathbb{Z}}_{k}$ gauging, and there are two ${\mathbb{Z}}_{k}$ TY
symmetries corresponding to the fusion categories ${\rm
TY}({\mathbb{Z}}_{k},\chi_{\pm},1)$ with $\chi_{\pm}(a,b)\equiv\pm{2\pi
ab\over k}$. Here we explicitly verify this for the corresponding duality
defect $\mathcal{N}$ in the ${\mathbb{Z}}_{4}$ case (the analysis for the
other duality defect $s\mathcal{N}$ is similar). Unlike the ${\mathbb{Z}}_{4}$
symmetry TDLs, the duality defect $\mathcal{N}$ does not commute with the full
chiral algebra. Instead, it preserves the left-moving $W(2,3,4)$ while acting
on the right-moving generators by a ${\mathbb{Z}}_{2}$ automorphism that flips
the sign of the spin 3 generator. Furthermore, the ${\mathbb{Z}}_{4}$
invariant $W(2,3,4)$ primaries, namely the thermal operators
$\phi_{0}^{0},\phi^{1}_{0},\phi^{2}_{0}$, transform under the duality
as242424The overall factor of 2 is required by the fusion relation
$\mathcal{N}^{2}=I+r+r^{2}+r^{3}$.

$\displaystyle\mathcal{N}\cdot\phi^{j}_{0}=2(-1)^{j}\phi^{j}_{0}\,.$ (4.44)

The above two conditions suffice to completely determine how $\mathcal{N}$
acts on local operators (see Appendix B for details).

The $SU(2)_{4}/U(1)$ coset CFT has the diagonal modular invariant partition
function

$\displaystyle
Z(\tau,\bar{\tau})=|\eta|^{2}(|c^{0}_{0}|^{2}+2|c^{0}_{2}|^{2}+|c^{0}_{4}|^{2}+2|c^{1}_{1}|^{2}+2|c^{1}_{3}|^{2}+|c^{2}_{2}|^{2}+|c^{2}_{0}|^{2})\,.$
(4.45)

Here $\eta(\tau)c^{2j}_{2m}(\tau)$ are $W(2,3,4)$ characters (see Appendix
B.2). For the thermal operators $\phi_{0}^{j}$ with $j=0,1,2$, they are given
by

$\displaystyle\eta c^{0}_{0}(\tau)=$ $\displaystyle{1\over
2\eta(\tau)}\sum_{k\in{\mathbb{Z}}}((-1)^{k}q^{k^{2}}+q^{3k^{2}})\,,$ (4.46)
$\displaystyle\eta c^{2}_{0}(\tau)=$
$\displaystyle{1\over\eta(\tau)}\sum_{k\in{\mathbb{Z}}}q^{3(k+1/3)^{2}}\,,$
$\displaystyle\eta c^{4}_{0}(\tau)=$ $\displaystyle{1\over
2\eta(\tau)}\sum_{k\in{\mathbb{Z}}}(-(-1)^{k}q^{k^{2}}+q^{3k^{2}})\,.$

As reviewed above (also Appendix B), to write down the duality twisted
partition function, we just need the duality twisted characters for the
thermal operators,

$\displaystyle I_{1}(\tau)=$ $\displaystyle{1\over
2\eta(\tau)}\sum_{k\in{\mathbb{Z}}}(q^{k^{2}}+q^{3k^{2}})(-1)^{k}\,,$ (4.47)
$\displaystyle I_{2}(\tau)=$
$\displaystyle{1\over\eta(\tau)}\sum_{k\in{\mathbb{Z}}}q^{3(k+1/3)^{2}}(-1)^{k}\,,$
$\displaystyle I_{3}(\tau)=$ $\displaystyle{1\over
2\eta(\tau)}\sum_{k\in{\mathbb{Z}}}(-q^{k^{2}}+q^{3k^{2}})(-1)^{k}\,,$

where the extra sign comes from the odd parity of the spin 3 generator under
the duality.

The twisted partition function reads

$\displaystyle
Z_{1\mathcal{N}}(\tau,\bar{\tau})=2\bar{\eta}(\bar{c}_{0}^{0}I_{1}-\bar{c}^{2}_{0}I_{2}+\bar{c}^{4}_{0}I_{3})\,,$
(4.48)

and has the following explicit form

$\displaystyle
Z_{1\mathcal{N}}(\tau,\bar{\tau})={1\over|\eta|^{2}}\sum_{m,n\in{\mathbb{Z}}}\left((-1)^{m}q^{3m^{2}}\bar{q}^{3n^{2}}+(-1)^{m+n}q^{m^{2}}\bar{q}^{n^{2}}-2(-1)^{m}q^{3(m+1/3)^{2}}\bar{q}^{3(n+1/3)^{2}}\right)\,.$
(4.49)

Performing the modular $S$-transform using the Poisson resummation formula,

$\displaystyle Z_{\mathcal{N}1}(\tau,\bar{\tau})=$
$\displaystyle{1\over|\eta|^{2}}\sum_{m,n\in{\mathbb{Z}}}\left({1\over
6}q^{{1\over 12}(m+1/2)^{2}}\bar{q}^{{1\over 12}n^{2}}+{1\over 2}q^{{1\over
4}(m+1/2)^{2}}\bar{q}^{{1\over 4}(n+1/2)^{2}}-{1\over 3}q^{{1\over
12}(m+1/2)^{2}}\bar{q}^{{1\over 12}n^{2}}e^{(2(m+n)+1)\pi i\over 3}\right)\,,$
(4.50) $\displaystyle=$ $\displaystyle{1\over
2|\eta|^{2}}\sum_{m,n\in{\mathbb{Z}}}\left({1\over
3}(1-2\cos{(2(m+n)+1)\pi\over 3})q^{{1\over 12}(m+1/2)^{2}}\bar{q}^{{1\over
12}n^{2}}+q^{{1\over 4}(m+1/2)^{2}}\bar{q}^{{1\over 4}(n+1/2)^{2}}\right)\,,$
$\displaystyle=$ $\displaystyle{1\over
2|\eta|^{2}}\left(\sum_{m,n\in{\mathbb{Z}},m-n\in
2{\mathbb{Z}}}q^{{(n+3m+1)^{2}\over 16}}\bar{q}^{{(n-3m-2)^{2}\over
16}}+\sum_{m,n\in{\mathbb{Z}}}q^{{1\over 4}(m+1/2)^{2}}\bar{q}^{{1\over
4}(n+1/2)^{2}}\right)\,,$

which indeed decomposes into Virasoro characters with non-negative integer
degeneracies thanks to the overall factor of 2 in (4.48).252525This is also
the minimal choice. Since in the defect Hilbert space partition function
$Z_{\mathcal{N}1}$, certain operators appear with unit degeneracy, thus e.g.
${1\over 2}\mathcal{N}$ would not be a well-defined TDL. This determines the
defect Hilbert space $\mathcal{H}_{\mathcal{N}}$ completely. In particular,
the spectrum of spins satisfies

$\displaystyle e^{4\pi is}=\\{\pm e^{-{\pi i\over 4}},1\\}\,,$ (4.51)

which suggests, using the spin selections rules (4.25), that the duality
defect together with the ${\mathbb{Z}}_{4}$ symmetry TDLs furnishes the ${\rm
TY}({\mathbb{Z}}_{4},\chi_{+},1)$ category. By stacking the duality defect
$\mathcal{N}$ with the ${\mathbb{Z}}_{2}$ symmetry TDL $s$, we obtain ${\rm
TY}({\mathbb{Z}}_{4},\chi_{-},1)$ as explained in Section 4.2.

The marginal operator that moves away from the $SU(2)_{4}/U(1)$ CFT on the
orbifold branch is $\phi^{2}_{0}=\phi_{2}^{0}$ in Table 7, which commutes with
both ${\mathbb{Z}}_{4}$ TY symmetries. Consequently this verifies the presence
of these fusion category symmetries on the entire orbifold branch. In
particular, the twisted partition functions here (4.49) and (4.50) matches
with the general expressions (4.22) and (4.23) once we set $R=\sqrt{6}$ as
promised.

###  4.5 ${\rm Rep}(H_{8})$ and Ising2

At radius $R=2$ the $\mathbb{Z}_{2}$ orbifold is rational and equivalent to a
tensor product of two Ising CFTs, which we denote by ${\rm Ising}^{2}$. The
operators $\sigma_{1,2}$ of the orbifold (4.2) are identified with the spin
operators of the two Ising sectors, while $\sigma_{1}\sigma_{2}=\cos\theta$,
the sum of thermal operators $\epsilon_{1}+\epsilon_{2}=\cos\phi$, etc.

This rational point was discussed in detail in Section 4 of our previous work
[14]. We review it here for completeness. We will see that the self-duality of
the $\mathbb{Z}_{2}$ orbifold under $D_{4}^{B}$ gauging associated with the
bicharacter $\chi_{s}$ gives rise to a TDL with Frobenius-Schur indicator
$\epsilon=1$ (the TY category is therefore isomorphic to ${\rm Rep}(H_{8})$),
and verify our proposed action (4.35) on local operators.

The $D_{8}$ global symmetry is generated by the individual ${\mathbb{Z}}_{2}$
symmetries $\eta\sim sr^{3}$ and $\sigma\sim sr$ of the Ising factors, along
with the ${\mathbb{Z}}_{2}$ TDL $s$ that swaps them. Each Ising factor is
self-dual under gauging its $\mathbb{Z}_{2}$ symmetry, and realizes the
${\mathbb{Z}}_{2}$ TY category ${\rm TY}({\mathbb{Z}}_{2},+)$ with duality
defects $\mathcal{N}_{1,2}$. These TDLs satisfy the fusion rules

$\displaystyle\mathcal{N}_{1}^{2}=1+\eta\,,\quad\mathcal{N}_{2}^{2}=1+\sigma\,.$
(4.52)

The tensor product of these two duality defects thus gives a
$D_{4}^{B}=\langle sr,sr^{3}\rangle$ duality defect
$\mathcal{N}=\mathcal{N}_{1}\mathcal{N}_{2}$. The bicharacter of this defect
respects the product structure, and we recognize it as $\chi_{s}$ in (4.32).
Furthermore, since $\mathcal{N}_{1,2}$ both have $\epsilon=1$, so does
$\mathcal{N}_{1}\mathcal{N}_{2}$, so we recognize the TY category ${\rm
TY}(D_{4},\chi_{s},1)$, which is equivalent to ${\rm Rep}(H_{8})$.

Recall the twisted partition function in the Ising model,

$\displaystyle Z^{\rm Ising}_{1\mathcal{N}_{1}}(\tau,\bar{\tau})=$
$\displaystyle\sqrt{2}(|\chi_{0}|-|\chi_{1\over 2}|^{2})\,,$ (4.53)

where $\chi_{h}$ here denotes the Virasoro character at $c={1\over 2}$

$\displaystyle\chi_{0}=$ $\displaystyle{1\over
2}\left(\sqrt{\theta_{3}\over\eta}+\sqrt{\theta_{4}\over\eta}\right)\,,$
(4.54) $\displaystyle\chi_{1\over 2}=$ $\displaystyle{1\over
2}\left(\sqrt{\theta_{3}\over\eta}-\sqrt{\theta_{4}\over\eta}\right)\,,$
$\displaystyle\chi_{1\over 16}=$
$\displaystyle{1\over\sqrt{2}}\sqrt{\theta_{2}\over\eta}\,.$

Then in the Ising2 CFT, we have

$\displaystyle Z_{1\mathcal{N}}(\tau,\bar{\tau})=$ $\displaystyle
2(|\chi_{0}|^{2}-|\chi_{1\over 2}|^{2})^{2}\,.$ (4.55)

Using (4.54), this becomes

$\displaystyle
Z_{1\mathcal{N}}={1\over|\eta|^{2}}\left(\sum_{m,n\in{\mathbb{Z}}}(-1)^{m+n}q^{(m+n)^{2}/2}\bar{q}^{(m-n)^{2}/2}+\sum_{m,n\in{\mathbb{Z}}}(-1)^{m+n}q^{m^{2}}\bar{q}^{n^{2}}\right)\,,$
(4.56)

and the modular S-transform gives

$\displaystyle Z_{1\mathcal{N}}(\tau,\bar{\tau})=$ $\displaystyle{1\over
2|\eta|^{2}}\left(\sum_{m,n\in{\mathbb{Z}}}q^{{1\over 2}\left({n+m+1\over
2}\right)^{2}}\bar{q}^{{1\over 2}\left({n-m\over
2}\right)^{2}}+\sum_{m,n\in{\mathbb{Z}}}q^{{1\over
4}(m+1/2)^{2}}\bar{q}^{{1\over 4}(n+1/2)^{2}}\right)\,,$ (4.57)

which is agreement with our general proposal (4.36) at $R=2$. Moreover, we can
check the spin spectrum in $\mathcal{H}_{\mathcal{N}}$ satisfies

$\displaystyle e^{4\pi is}=\\{\pm i,1\\}$ (4.58)

as expected for the spin selection for ${\rm Rep}(H_{8})$ (4.40).

The tensor product of the thermal operators $\epsilon_{1,2}$ in the two Ising
models, $\epsilon_{1}\epsilon_{2}$ has weight $h=\bar{h}=1$ and is exactly
marginal. It is uncharged under $D_{4}^{B}$. Since $\epsilon_{1}\epsilon_{2}$
does not commute with the individual Ising ${\mathbb{Z}}_{2}$ dualities, the
duality defects $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ are no longer
topological when we move away from the Ising2 point. However this deformation
commutes with the duality defect $\mathcal{N}$,

$\displaystyle\mathcal{N}\cdot\epsilon_{1}\epsilon_{2}=\langle\mathcal{N}\rangle\epsilon_{1}\epsilon_{2}=2\epsilon_{1}\epsilon_{2}\,,$
(4.59)

thus the $D_{4}^{B}$ duality defect $\mathcal{N}$ and the associated category
${\rm Rep}(H_{8})$ will persist along the entire orbifold branch, in agreement
with our observation of self-duality under $D_{4}^{B}$ gauging.

###  4.6 ${\rm Rep}(D_{8})$ and the four-state Potts model

At $R=2\sqrt{2}$ the $\mathbb{Z}_{2}$ orbifold is again rational and is
equivalent to the four-state Potts model and a $D_{4}$ orbifold of the
$SU(2)_{1}$ WZW model,

$\displaystyle{U(1)_{8}\over\mathbb{Z}_{2}^{C}}\cong{SU(2)_{1}\over D_{4}}\,.$
(4.60)

The theory has $S_{4}$ global symmetry and an enhanced chiral algebra
$W(2,4,4)$ containing two spin 4 generators in addition to the stress tensor.
There are eleven chiral primaries $1,j_{a},\phi,\sigma_{a},\tau_{a}$ with
scaling dimension $h=\bar{h}=\\{0,1,{1\over 4},{1\over 16},{9\over 16}\\}$
respectively [74]. The index $a=1,2,3$ transforms in the 3-dimensional
irreducible representation of $S_{4}$.

This rational point gives us access to the other $D_{4}^{B}$ TY symmetry of
the $\mathbb{Z}_{2}$ orbifold, associated with the bicharacter $\chi_{a}$ in
(4.32), allowing us to determine the Frobenius-Schur indicator to be
$\epsilon=1$ and verify the action (4.34) on local operators. Indeed, the
Verlinde lines associated with the chiral primaries $1,j_{a},\phi$ have
$D_{4}$ TY fusion rules: $j_{a}$ correspond to the order 2 elements of
$D_{4}^{A}$, and $\phi$ corresponds to the duality defect.

The untwisted torus partition function of the four-state Potts model is

$\displaystyle Z(\tau,\bar{\tau})=|\chi_{0}|^{2}+3|\chi_{1}|^{2}+|\chi_{1\over
4}|^{2}+3|\chi_{1\over 16}|^{2}+3|\chi_{9\over 16}|^{2}\,,$ (4.61)

where $\chi_{h}$ denotes the $W(2,4,4)$ character for a weight $h$ primary.
The torus partition function twisted by the duality defect $\mathcal{N}$
wrapping the spatial circle is given by the $S$ matrix by the braiding
monodromy of $\phi$ (which is natural from the 2+1D TQFT perspective),

$\displaystyle Z_{1\mathcal{N}}(\tau,\bar{\tau})=$ $\displaystyle
2|\chi_{0}|^{2}+6|\chi_{1}|^{2}-2|\chi_{1\over 4}|^{2}\,.$ (4.62)

Its modular S-transform (see Table 9) is

$\displaystyle
Z_{\mathcal{N}1}(\tau,\bar{\tau})=(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over
4}+3\chi_{1\over 16}\bar{\chi}_{9\over 16})+c.c.)+3(|\chi_{1\over
16}|^{2}+|\chi_{9\over 16}|^{2})\,.$ (4.63)

The spin spectrum in $\mathcal{H}_{\mathcal{N}}$ satisfies

$\displaystyle e^{4\pi is}=\\{\pm 1\\}\,.$ (4.64)

Compared to the spin selection rules in (4.40), this implies $\mathcal{N}$ is
a duality defect with associated F-symbols determined by the bicharacter
$\chi_{a}$ and a FS indicator $\epsilon=\pm 1$. Below we will carry out a
refined version of the analysis around (4.40) to show that $\epsilon=+1$.

Performing a $T^{2}$ transformation on $Z_{\mathcal{N}1}$, we find

$\displaystyle Z_{\mathcal{N}1}(\tau+2,\bar{\tau}+2)={\epsilon\over
2}\left(Z_{\mathcal{N}1}+Z_{\mathcal{N}\eta}+Z_{\mathcal{N}\sigma}+Z_{\mathcal{N}\eta\sigma}\right)\,,$
(4.65)

which constrains the action of the $D_{4}^{B}$ TDLs on the defect Hilbert
space $\mathcal{H}_{\mathcal{N}}$. Moreover, from partial fusion and crossing,
one can derive that,

$\displaystyle\widehat{\eta}_{-}^{2}=\widehat{\sigma}_{-}^{2}=\widehat{\eta\sigma}_{-}^{2}=1\,,\quad\widehat{\eta}_{-}\widehat{\sigma}_{-}=\widehat{\sigma}_{-}\widehat{\eta}_{-}=-\widehat{\eta\sigma}_{-}$
(4.66)

as linear operators acting on $\mathcal{H}_{\mathcal{N}}$. Since
$\mathcal{H}_{\mathcal{N}}$ has a single defect chiral primary of weight
$(h,\bar{h})=(0,{1\over 4})$, from (4.65), we conclude that
$\widehat{\eta}_{-},\widehat{\sigma}_{-},\widehat{\eta\sigma}_{-}$ acting on
this operator has to be $-.-,-$ if $\epsilon=1$, and $+,+,-$ or its
permutations if $\epsilon=-1$.

We obtain additional constraints on
$\widehat{\eta}_{-},\widehat{\sigma}_{-},\widehat{\eta\sigma}_{-}$ from the
modular bootstrap equation (2.5) for the pair of TDLs $\eta$ and
$\mathcal{N}$, with two solutions

$\displaystyle Z_{\mathcal{N}\eta}=$
$\displaystyle{\gamma}\left[(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over
4}-\chi_{1\over 16}\bar{\chi}_{9\over 16})+c.c.)-(|\chi_{1\over
16}|^{2}+|\chi_{9\over 16}|^{2})\right]\,,$ (4.67) $\displaystyle
Z_{\eta\mathcal{N}}=$ $\displaystyle
2{\gamma}(\chi_{0}\bar{\chi}_{1}+\chi_{1}\bar{\chi}_{0}+2|\chi_{1}|^{2}-|\chi_{1\over
4}|^{2})\,,$

parametrized by ${\gamma}=\pm 1$. From the discussion around (4.66), for
$\epsilon=1$ we must have

$\displaystyle Z_{\mathcal{N}\eta}=$
$\displaystyle-\left[(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over 4}-\chi_{1\over
16}\bar{\chi}_{9\over 16})+c.c.)-(|\chi_{1\over 16}|^{2}+|\chi_{9\over
16}|^{2})\right]\,,$ (4.68) $\displaystyle Z_{\mathcal{N}\sigma}=$
$\displaystyle-\left[(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over 4}-\chi_{1\over
16}\bar{\chi}_{9\over 16})+c.c.)-(|\chi_{1\over 16}|^{2}+|\chi_{9\over
16}|^{2})\right]\,,$ $\displaystyle Z_{\mathcal{N}\eta\sigma}=$
$\displaystyle-\left[(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over 4}-\chi_{1\over
16}\bar{\chi}_{9\over 16})+c.c.)-(|\chi_{1\over 16}|^{2}+|\chi_{9\over
16}|^{2})\right]\,,$

while for $\epsilon=-1$ we end up with

$\displaystyle Z_{\mathcal{N}\eta}=$
$\displaystyle\left[(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over 4}-\chi_{1\over
16}\bar{\chi}_{9\over 16})+c.c.)-(|\chi_{1\over 16}|^{2}+|\chi_{9\over
16}|^{2})\right]\,,$ (4.69) $\displaystyle Z_{\mathcal{N}\sigma}=$
$\displaystyle\left[(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over 4}-\chi_{1\over
16}\bar{\chi}_{9\over 16})+c.c.)-(|\chi_{1\over 16}|^{2}+|\chi_{9\over
16}|^{2})\right]\,,$ $\displaystyle Z_{\mathcal{N}\eta\sigma}=$
$\displaystyle-\left[(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over 4}-\chi_{1\over
16}\bar{\chi}_{9\over 16})+c.c.)-(|\chi_{1\over 16}|^{2}+|\chi_{9\over
16}|^{2})\right]\,,$

or any permutations of the overall signs. However looking at the piece
$\chi_{1\over 16}\bar{\chi}_{9\over 16}$, we see $\epsilon=-1$ cannot be
compatible with (4.65). Thus we conclude $\epsilon=1$ for the $D_{4}$ TY
symmetry in the four-state Potts model.

We can also compute more general twisted partition functions that involve a
network of TDLs (see Figure 12) which provide additional consistency checks on
our identification of the TY symmetries.

Figure 12: Torus partition function $Z^{X}_{\mathcal{N}\eta}$ decorated by a
TDL network formed by duality TDL $\mathcal{N}$ and symmetry TDL $\eta$
(similarly when $\eta$ is replaced by other symmetry TDLs in $D_{4}^{B}$.

By solving the modular bootstrap equations (2.5), we find

$\displaystyle
Z^{X}_{\mathcal{N}\eta}=Z^{X}_{\mathcal{N}\sigma}=Z^{X}_{\mathcal{N}\eta\sigma}=\left[i(\chi_{0}+3\chi_{1})\bar{\chi}_{1\over
4}-\chi_{1\over 16}\bar{\chi}_{9\over 16})+c.c.)+(|\chi_{1\over
16}|^{2}+|\chi_{9\over 16}|^{2})\right]\,,$ (4.70)

which satisfy the following relations

$\displaystyle Z^{X}_{\mathcal{N}\eta}(\tau+1)=$ $\displaystyle{1\over
2}(Z_{\mathcal{N}}+Z_{\mathcal{N}\eta}-Z_{\mathcal{N}\sigma}-Z_{\mathcal{N}\eta\sigma})\,,$
(4.71) $\displaystyle Z^{X}_{\mathcal{N}\sigma}(\tau+1)=$
$\displaystyle{1\over
2}(Z_{\mathcal{N}}-Z_{\mathcal{N}\eta}+Z_{\mathcal{N}\sigma}-Z_{\mathcal{N}\eta\sigma})\,,$
$\displaystyle Z^{X}_{\mathcal{N}\eta\sigma}(\tau+1)=$ $\displaystyle{1\over
2}(Z_{\mathcal{N}}-Z_{\mathcal{N}\eta}-Z_{\mathcal{N}\sigma}+Z_{\mathcal{N}\eta\sigma})\,,$

that come from partial fusion and crossing.

Finally, we note that the four-state Potts model has three independent
exactly-marginal operators $j_{a}$ which transform as the three-dimensional
irreducible representation of $S_{4}$. Consequently, deformation from either
of the three moves the CFT along the orbifold branch. Now the duality defect
$\mathcal{N}$ commutes with all three marginal operators

$\displaystyle\mathcal{N}\cdot j_{a}\bar{j}_{a}=\langle\mathcal{N}\rangle
j_{a}\bar{j}_{a}\,.$ (4.72)

Thus the ${\rm Rep}(D_{8})$ category of TDLs is preserved along the entire
orbifold branch.

In particular we can compare to our general proposal (4.36), at $R=2\sqrt{2}$,

$\displaystyle Z_{1\mathcal{N}_{D_{8}}}=$
$\displaystyle{1\over|\eta|^{2}}\left(\sum_{m,n\in{\mathbb{Z}}}(-1)^{m}q^{{1\over
4}\left({2n}+{m}\right)^{2}}\bar{q}^{{1\over
4}\left({2n}-{m}\right)^{2}}+\sum_{m,n\in{\mathbb{Z}}}(-1)^{m+n}q^{m^{2}}\bar{q}^{n^{2}}\right)\,,$
(4.73)

using the explicit forms for the $W(2,4,4)$ characters,

$\displaystyle\chi_{0}(\tau)=$ $\displaystyle{1\over
2\eta}\sum_{m}(q^{4m^{2}}+(-1)^{m}q^{m^{2}})\,,$ (4.74)
$\displaystyle\chi_{1}(\tau)=$ $\displaystyle{1\over
2\eta}\sum_{m}(q^{4m^{2}}-(-1)^{m}q^{m^{2}})\,,$ $\displaystyle\chi_{1\over
4}(\tau)=$ $\displaystyle{1\over\eta}\sum_{m}q^{(1+4m)^{2}\over 4}\,,$

we find that it agrees with $Z_{1\mathcal{N}}$ in (4.62) as desired.

###  4.7 Continuum of Topological Defects

As we have mentioned in Section 2.5, and using the general method therein, we
can construct a continuum of TDLs for every $R$ from the continuous
$U(1)^{\theta}\times U(1)^{\phi}$ symmetry of the circle branch. In
particular, if $\mathcal{L}_{\alpha}^{\theta}$ is the TDL of the compact boson
corresponding to the symmetry $\theta\mapsto\theta+\alpha$, then
$\mathcal{L}_{\alpha}^{\theta}+\mathcal{L}_{-\alpha}^{\theta}$ is $C$
invariant and will define a noninvertible TDL $\mathcal{N}^{\theta}_{\alpha}$
of the $\mathbb{Z}_{2}^{C}$ orbifold with quantum dimension 2. It acts on
local operators (for generic ${\alpha}$) by

$\displaystyle\widehat{\mathcal{N}}^{\theta}_{\alpha}:\begin{cases}V^{+}_{n,w}\to
2\cos(n\alpha)V^{+}_{n,w}\,,\\\ j_{n^{2}}\bar{j}_{m^{2}}\to
2j_{n^{2}}\bar{j}_{m^{2}}\,,\\\ \text{other primaries}\to\text{non-local
operators}\,,\end{cases}$ (4.75)

and there is likewise a continuum family for the $\phi$ shift symmetry
$\mathcal{L}_{\phi}^{\alpha}$. See [18] for a recent discussion of these
lines.

The Noether currents associated with these families are $d\theta$ and $d\phi$
respectively, which live at the end of the Wilson line for the $C$ gauge
field. The TDL $\mathcal{N}_{D_{8}}$ of (4.34) appears in this family at
$\alpha={\pi\over 2}$. In fact the subalgebra of lines with $\alpha={2\pi
m\over k}$, $m=1,\ldots,\lfloor{k-1\over 2}\rfloor$ generates a ${\rm
Rep}(D_{2k})$ symmetry.262626These non-invertible TDLs are in one-to-one
correspondence with the $\lfloor{k-1\over 2}\rfloor$ two-dimensional
irreducible representations of $D_{2k}$. This can be seen by writing the
$\mathbb{Z}_{2}^{C}$ orbifold at radius $R$ as a $D_{2k}$ orbifold of the
compact boson on the circle branch at radius $R\over k$ by gauging
$\mathbb{Z}_{k}^{\phi}\rtimes\mathbb{Z}_{2}^{C}=D_{2k}$. By multiplying with
the $D_{8}$ generator $s$, we see that the ${\rm Rep}(H_{8})$ symmetry
similarly extends to an infinite family, as well as their T-dual TY
symmetries, which arise from $\mathcal{L}_{\phi}^{\alpha}$.

At the special radii $R=n\sqrt{2}$ this continuum is enhanced to an “$SO(4)$”
family, obtained from the $SO(4)$ invertible TDLs of $SU(2)_{1}$ upon gauging
the $D_{2n}$ subgroup $\mathbb{Z}_{n}^{\phi}\rtimes\mathbb{Z}_{2}^{C}$.
Besides $d\theta$ and $d\phi$ which live at the end of the Wilson line for the
sign representation, there are four Noether currents $V_{\pm n,\pm 1/n}$,
which live at the end of a Wilson line for two 2-dimensional representations
of $D_{2n}$. If $m$ is coprime to $n$, by first gauging
$\mathbb{Z}_{m}^{\theta}$ and then
$\mathbb{Z}_{n}^{\phi}\rtimes\mathbb{Z}_{2}^{C}$, we obtain a related family
at radius $R=\frac{n}{m}\sqrt{2}$, with Noether currents $V_{\pm n/m,\pm
m/n}$.

###  4.8 The Kosterlitz-Thouless point and triality

If we consider the $\mathbb{Z}_{2}^{C}$ orbifold of the $SU(2)_{1}$ point, we
land at the Kosterlitz-Thouless (KT) point, where the orbifold and circle
branches meet. This can either be described in terms of the $C$-gauged fields
$\theta$, $\phi$ and $C$-twist operators, or in terms of the (gauge invariant)
circle branch fields $\tilde{\theta}$, $\tilde{\phi}$ (see Appendix A.3).

Along with the continuous families of TDLs associated with the non-local
currents $d\theta$ and $d\phi$, which persist along the entire orbifold
branch, at the KT point we also have the four currents $V_{\pm 1,\pm 1}$
mentioned in the previous subsection (take $n=1$). Two of these are local:
$V_{1,\pm 1}+V_{-1,\mp 1}$, which we can identify with the currents
$d\tilde{\theta}$, $d\tilde{\phi}$. These generate the familiar
$U(1)_{\tilde{\theta}}\times U(1)_{\tilde{\phi}}$ symmetry of the KT point,
expressed as a circle branch theory. However, the other two currents $V_{1,\pm
1}-V_{-1,\mp 1}$ are non-local, and correspond to a two-parameter enhancement
of the TDLs at this special point. Note this is the same four-parameter family
we constructed in Section 3.5 by considering the KT point as an orbifold of
$SU(2)_{1}$ by $\mathbb{Z}_{2}^{\phi}$, since $\mathbb{Z}_{2}^{\phi}$ and
$\mathbb{Z}_{2}^{C}$ are conjugate in the full $SO(4)$ symmetry.

Some of the new TDLs at the KT point also have interpretations as self-
dualities under gauging. In part I [14], we noted that by gauging $D_{4}$
using the bicharacter $\chi_{a}$, $\beta=0$, and $\alpha$ the generator of
$H^{2}(D_{4},U(1))=\mathbb{Z}_{2}$ in (2.15) we obtain an order three
permutation of the $D_{4}$-symmetric gapped phases.

It turns out this triality is a category symmetry of the KT point. We can see
this as follows. We identify $D_{4}^{B}$ with
$\mathbb{Z}_{2}^{\tilde{\theta}}\times\mathbb{Z}_{2}^{\tilde{C}}$, where
$\tilde{C}$ is the charge conjugation symmetry of the circle branch variables
$\tilde{\theta}$, $\tilde{\phi}$. The theory is self-dual under gauging this
symmetry: gauging $\mathbb{Z}_{2}^{\tilde{\theta}}$ halves the radius, putting
us at the $SU(2)_{1}$ point, then gauging $\mathbb{Z}_{2}^{\tilde{C}}$ puts us
back at the KT point. This self-duality defines a $D_{4}^{B}$ TY symmetry. A
novelty for the KT point however is the chiral anomaly, one symptom of which
is that partition functions which might detect the $D_{4}^{B}$ discrete
torsion vanish identically (see Section 4.3). We can attribute this to a cubic
anomaly in
$\mathbb{Z}_{2}^{\tilde{\theta}}\times\mathbb{Z}_{2}^{\tilde{\phi}}\times\mathbb{Z}_{2}^{\tilde{C}}$,
such that $\mathbb{Z}_{2}^{\tilde{\phi}}$ “toggles” the discrete torsion,
mapping $\mathbb{Z}_{2}^{\tilde{C}}$-twisted operators to ones of opposite
$\mathbb{Z}_{2}^{\tilde{\theta}}$ charge, etc (see Appendix A.2). We expect
that the triality is therefore realized as a fusion product of a $D_{4}^{B}$
TY TDL with the $\mathbb{Z}_{2}^{\tilde{\theta}}$ TDL.

To explore this in detail, and to see its three-fold nature, we can examine
these symmetries as they arise from the $SO(4)$ symmetry of the $SU(2)_{1}$
point upon gauging $\mathbb{Z}_{2}^{C}$. In terms of the symmetries and the
fields $\theta,\phi$ before gauging, we have already identified the ${\rm
Rep}(D_{8})$ duality symmetry with a $\pi\over 2$ $U(1)^{\theta}$ rotation
$R^{\theta}({\pi\over 2})$ (see Section 4.7). Meanwhile
$\mathbb{Z}_{2}^{\tilde{\phi}}$ is generated by a conjugate ${\pi\over 2}$
rotation $R^{\phi^{\prime}}({\pi\over 2})$ (this can be considered a $\pi\over
2$ rotation of an $SO(4)$-conjugate field $\phi^{\prime}$). This squares to
$C$ and becomes a $\mathbb{Z}_{2}$ symmetry after gauging. As $SO(4)$ matrices
in the basis (3.7), these are two $2\times 2$ blocks overlapping in one
position, and we find their composition is a $2\pi\over 3$ rotation in the
combined $3\times 3$ block,

$\displaystyle R^{\theta}\left({\pi\over 2}\right)=$
$\displaystyle\begin{bmatrix}0&&1&&0&&0\\\ -1&&0&&0&&0\\\ 0&&0&&1&&0\\\
0&&0&&0&&1\end{bmatrix}\,,$ (4.76) $\displaystyle
R^{\phi^{\prime}}\left({\pi\over 2}\right)=$
$\displaystyle\begin{bmatrix}0&&0&&1&&0\\\ 0&&1&&0&&0\\\ -1&&0&&0&&0\\\
0&&0&&0&&1\end{bmatrix}\,,$ $\displaystyle Q=R^{\theta}\left({\pi\over
2}\right)R^{\phi^{\prime}}\left({\pi\over 2}\right)=$
$\displaystyle\begin{bmatrix}0&&1&&0&&0\\\ 0&&0&&-1&&0\\\ -1&&0&&0&&0\\\
0&&0&&0&&1\end{bmatrix}\,,$ $\displaystyle
C=R^{\phi^{\prime}}\left(\pi\right)=$
$\displaystyle\begin{bmatrix}-1&&0&&0&&0\\\ 0&&1&&0&&0\\\ 0&&0&&-1&&0\\\
0&&0&&0&&1\end{bmatrix}\,.$

See Appendix A.3 for more details. We can thus as well consider the triality
defect as coming from the invertible symmetry $Q$ of the $SU(2)_{1}$ point
upon gauging $\mathbb{Z}_{2}^{C}$, resulting in a triality TDL
$\mathcal{L}_{Q}$ of quantum dimension 2.

The subgroup of the $SO(4)$ symmetry at the $SU(2)_{1}$ point generated by $C$
and $Q$ is the alternating group $A_{4}$. We want to study what fusion
category results from $A_{4}$ after gauging $\mathbb{Z}_{2}^{C}$. This will be
a fusion subcategory of the symmetries of the KT point containing the triality
TDL we are interested in. Mathematically, $\mathbb{Z}_{2}^{C}$ defines a
module category $\mathcal{M}$ for the grouplike category $\mathcal{A}={\rm
Vec}_{A_{4}}$, such that the simple objects of $\mathcal{M}$ are labelled by
cosets in $A_{4}/\mathbb{Z}_{2}^{C}$ [76, 14]. The fusion category symmetry we
obtain after gauging is the “dual” category
$\mathcal{A}^{*}_{\mathcal{M}}={\rm End}_{\mathcal{A}}(\mathcal{M})$. Since
$\mathbb{Z}_{2}^{C}$ is not normal in $A_{4}$, $\mathcal{A}^{*}_{\mathcal{M}}$
is not grouplike (see Thm 3.4 in [77]). We find there are six simple TDLs:
four grouplike, corresponding to $\tilde{C}=sr$ (from $R^{\theta}(\pi)$) and
the magnetic symmetry $r^{2}$, which together generate the $D_{4}^{B}\subset
D_{8}$ symmetry of the KT point, as well as two triality defects of quantum
dimension 2, associated with $Q$ and $Q^{2}=Q^{-1}$. The fusion rules are

$\displaystyle\mathcal{L}_{Q}\mathcal{L}_{g}=\mathcal{L}_{g}\mathcal{L}_{Q}=\mathcal{L}_{Q}\,,\quad\mathcal{L}_{Q^{2}}\mathcal{L}_{g}=\mathcal{L}_{g}\mathcal{L}_{Q^{2}}=\mathcal{L}_{Q^{2}}\,,$
(4.77)

for any $g\in D_{4}^{B}$. We also have

$\displaystyle\mathcal{L}_{Q}\mathcal{L}_{Q^{2}}=\sum_{g\in
D_{4}^{B}}\mathcal{L}_{g}\,,\quad\mathcal{L}_{Q}^{2}=2\mathcal{L}_{Q^{2}}\,.$
(4.78)

By analyzing the action of $\mathcal{L}_{Q}$ on $D_{4}$-symmetric gapped
phases (see Section 3.2.3 of [14]), we find that the fusion category is
anomalous—there is no way to deform the KT point to a trivial gapped phase
while preserving the triality. This also means that the category above is not
${\rm Rep}(H)$ for any Hopf algebra (or group) $H$.

###  4.9 Symmetric RG flows from the orbifold branch

Let us discuss deformations of the $c=1$ CFT on the orbifold branch preserving
certain fusion category symmetries. We focus on the $D_{4}$ and
${\mathbb{Z}}_{4}$ TY symmetries. Firstly, the $D_{4}$ TY symmetries presented
on the orbifold branch are all non-anomalous as they admit fiber functors [69,
14]. On the contrary, the ${\mathbb{Z}}_{4}$ TY symmetries are all anomalous
(see also discussion in Section 3.4). Consequently, while the $D_{4}$ TY
symmetries do not lead to nontrivial RG constraints, the ${\mathbb{Z}}_{4}$ TY
symmetries require the symmetric RG flow to end either in a nontrivial CFT or
a spontaneous symmetry broken phase. Below we will see explicitly that for RG
flows out of the orbifold branch preserving ${\rm
TY}({\mathbb{Z}}_{4},\chi_{\pm},1)$, the IR phase is described by a TQFT with
degenerate ground states, fulfilling the second scenario predicted by the
anomalous fusion category symmetry.

As discussed in Section 4.4, the ${\mathbb{Z}}_{4}$ TY symmetries are most
transparent at the ${\mathbb{Z}}_{4}$ parafermion point on the orbifold
branch, which is described by the $SU(2)_{4}/U(1)$ coset CFT. Even though the
${\mathbb{Z}}_{4}$ parafermion point does not have relevant deformations that
preserve the ${\mathbb{Z}}_{4}$ duality TDLs, once we have moved to large
enough $R$ by the duality invariant marginal deformation, there are plenty of
relevant deformations that do commute with these TDLs, namely the momentum
operators

$\displaystyle V^{+}_{4n,0}:~{}h=\bar{h}={8n^{2}\over R^{2}}\,,$ (4.79)

cf. (4.21). Note that $R=2\sqrt{2}$ corresponds to the four-state Potts model
in which case $V_{4,0}$ is marginal and related to the marginal operator
$\partial X\bar{\partial}X$ by the enhanced $S_{4}$ symmetry.

For $R>2\sqrt{2}$, $V^{+}_{4,0}$ is a relevant deformation that preserves the
${\mathbb{Z}}_{4}$ TY symmetries, and consequently the IR phase must realize
the ${\mathbb{Z}}_{4}$ TY symmetries and cannot be trivially gapped! Indeed,
in terms of the circle variable $\theta\sim\theta+2\pi$, the $V^{+}_{4,0}$
deformation takes the form of $\pm\cos{4\theta}$. With $+$ sign, the IR theory
is gapped with two vacua located at $\theta=\\{\pi/4,3\pi/4\\}$, while with
$-$ sign, there are three vacua located at $\theta=\\{0,{\pi\over
2},\pi\\}$.272727Note that we have taken into account the ${\mathbb{Z}}_{2}$
identification of $\theta$ in the orbifold theory. In either case, the IR
theory completes into a TQFT with two or three degenerate ground
states.282828Applying the self-duality condition (2.15) directly to the
putative TQFT on $\Sigma=T^{2}$, it is a simple exercise to derive a number of
constraints on the TY symmetric TQFT using modular invariance. One of them
requires the vacuum degeneracy to be $3d_{1}+2d_{2}$ where
$d_{1},d_{2}\in{\mathbb{Z}}_{\geq 0}$ count the number of ground states with
${\mathbb{Z}}_{4}$ charge 1 and 2 respectively.

For $\mathbb{Z}_{k}$ parafermions of general $k$, the $SU(2)_{k}/U(1)$ coset
CFT contains TY symmetric relevant deformations for $k>5$, for example given
by the leading duality-invariant thermal operator $\phi^{2}_{0}$ of weight
$h=\bar{h}={6\over k+2}$. Once again the anomalous ${\mathbb{Z}}_{k}$ TY
symmetries demands a nontrivial symmetric IR theory that reproduces the same
anomaly. In this case, this is achieved by a direct sum of $c=1$ CFTs,

$\displaystyle SU(2)_{k}/U(1)\xrightarrow[\text{deformed
by}~{}\phi^{2}_{0}]{{\rm TY}\text{-symmetric RG}}{\rm KT}\oplus{\rm
KT}/{\mathbb{Z}}_{k}\,,$ (4.80)

which is suggested by the numerical analysis of the ${\mathbb{Z}}_{k}$ clock
model in [78, 79, 80] and integrability analysis in [81].

##  5 Fusion Categories of the Exceptional Orbifolds

In the previous section we have studied TDLs of the circle and
$\mathbb{Z}_{2}$ orbifold branches of the $c=1$ moduli space. There are also
three isolated points, collectively known as the exceptional orbifolds, which
arise from gauging one of three special symmetry subgroups of the $SU(2)_{1}$
CFT. Recall this theory has a global symmetry

$\displaystyle SO(4)={SU(2)_{L}\times SU(2)_{R}\over{\mathbb{Z}}_{2}}\,,$
(5.1)

of which the diagonal subgroup $SO(3)_{\rm diag}$ is anomaly-free. Finite
subgroups of $SO(3)$ have an ADE classification, where the A series are cyclic
rotation groups around a fixed axis, whose orbifolds define the series
$R=n\sqrt{2}$ on the circle branch; the D series are dihedral groups of
rotations around a fixed axis along with a perpendicular $\pi$ rotation, whose
orbifolds define the series $R=n\sqrt{2}$ on the $\mathbb{Z}_{2}$ orbifold
branch; and the exceptional E series, with $E_{6}$ the symmetry group $T$ of
the tetrahedron, $E_{7}$ the symmetry group $O$ of the octahedron/cube, and
$E_{8}$ the symmetry group $I$ of the icosahedron/dodecahedron, whose
orbifolds define the exceptional orbifolds. These three exceptional groups are
isomorphic to alternating and symmetric groups

$\displaystyle T=A_{4}\,,\quad O=S_{4}\,,\quad I=A_{5}\,.$ (5.2)

The cases $G=D_{4n},A_{4},S_{4},A_{5}$ all admit nontrivial discrete torsion
$H^{2}(G,U(1))={\mathbb{Z}}_{2}$. However because of the ’t Hooft anomaly for
$SO(4)$ symmetry of the $SU(2)_{1}$ CFT, the discrete torsion can be removed
by a chiral ${\mathbb{Z}}_{2}$ rotation, and thus it does not lead to distinct
orbifold theories.292929This phenomena is a $D=2$ analog of what happens in
$D=4$ nonabelian gauge theories with an Adler-Bell-Jackiw (ABJ) anomaly for
certain chiral (axial) $U(1)$ rotation of the charged fermions. The role of
the discrete torsion is played by the usual theta angle term ${i\theta\over
8\pi^{2}}\int\mathop{\rm tr}\nolimits F\wedge F$ in the action. The presence
of the ABJ anomaly implies that the theta angle $\theta$ can be freely
adjusted by a chiral (axial) rotation of the fermions.

As discussed in Section 2.3, the resulting orbifold CFT contain $G$ Wilson
lines which make up a fusion category symmetry described by ${\rm Rep}(G)$.
For the A series, the corresponding ${\rm Rep}(\mathbb{Z}_{n})$ symmetry is
invertible and part of the $U(1)\times U(1)$ symmetry of the circle branch.
For the D series, the symmetry ${\rm Rep}(D_{2n})$ contains non-invertible
lines and are part of a continuous family of TDLs on the $\mathbb{Z}_{2}$
orbifold branch we discussed in Section 4.7. Likewise the ${\rm
Rep}(A_{4}),{\rm Rep}(S_{4})$ and ${\rm Rep}(S_{5})$ fusion category
symmetries of the exceptional orbifolds are part of a much larger algebra of
TDLs.

Some of the additional TDLs come from Verlinde lines associated to the
enhanced chiral algebra ${\mathcal{V}}_{G}$ of the orbifolds.
${\mathcal{V}}_{G}$ can be derived from the $G$-invariant part of the
$SU(2)_{1}$ Kac-Moody current algebra and are given by W-algebras of the
following types,

$\displaystyle{\mathcal{V}}_{D_{2n}}={\mathcal{W}}(2,4,n^{2}),~{}{\mathcal{V}}_{A_{4}}={\mathcal{W}}(2,9,16),~{}{\mathcal{V}}_{S_{4}}={\mathcal{W}}(2,16),~{}{\mathcal{V}}_{A_{5}}={\mathcal{W}}(2,36)\,,$
(5.3)

where we have listed the spins of the strong generators303030All other higher
spin currents in the chiral algebra are generated by the normal ordered
products of the strong generators and their derivatives. and the spin $2$
generator is the stress tensor with central charge $c=1$. The characters for
the ${\mathcal{V}}_{G}$ representations and the corresponding $S$ matrices
were derived in [74, 82]. The exceptional orbifold is described by the
diagonal modular invariant of the ${\mathcal{V}}_{G}$ chiral algebra.
Consequently, as explained in Section 2.2, they host a family of Verlinde
lines that are in one-to-one correspondence with the chiral primaries in the
orbifold CFT, and generate a fusion category symmetry corresponding to ${\rm
Rep}({\mathcal{V}}_{G})$, extending the subcategory ${\rm Rep}(G)$ of
$G$-Wilson lines.

Beyond these TDLs we also have a whole continuum obtained from the $SO(4)$
TDLs of the $SU(2)_{1}$ theory by the method of Section 2.5. The corresponding
Noether currents lie at the end of $G$ Wilson lines associated to the three
dimensional representation of each $G\subset SO(3)$. It is plausible that all
the Verlinde lines appear in the fusion products of this continuum of lines.
We leave this interesting conjecture to future work.

We summarize the situation below, giving ${\rm Rep}(G)$ fusion rules in these
theories. The relevant $S$ matrices are listed in Appendix E and we attach a
mathematica notebook that tabulates the complete fusion rules for Verlinde
lines in ${\rm Rep}({\mathcal{V}}_{G})$.

###  5.1 $A_{4}$ orbifold

The orbifold ${SU(2)_{1}/A_{4}}$ has ${\rm Rep}(A_{4})$ symmetry generated by
Wilson lines associated to the four irreducible representations of $A_{4}$:
the trivial representation $1$ and two nontrivial one-dimensional
representations $R_{1},R_{2}$ as well as one three-dimensional representation
$V$. The fusion rules are

$\displaystyle
R_{1}R_{2}=1\,,~{}~{}R_{1}^{2}=R_{2}\,,~{}~{}R_{2}^{2}=R_{1}\,,~{}~{}R_{i}V=V\,,~{}~{}V^{2}=1+R_{1}+R_{2}+2V\,.$
(5.4)

Note that $\\{1,R_{1},R_{2}\\}$ generates a ${\mathbb{Z}}_{3}$ symmetry of the
orbifold theory. This corresponds to the magnetic symmetry of the theory
regarded as an ${\mathbb{Z}}_{3}$ orbifold of the $SU(2)_{1}/D_{4}$ CFT (four-
state Potts), with $A_{4}=D_{4}\rtimes\mathbb{Z}_{3}$.

As an RCFT with respect to the ${\mathcal{V}}_{A_{4}}$ chiral algebra (5.3),
the theory contains $21$ chiral primaries which lead to $21$ Verlinde lines
defined by the modular $S$-matrix in Table 11 as in (2.6). They include the 4
Wilson lines in ${\rm Rep}(A_{4})$ and generate the ${\rm
Rep}({\mathcal{V}}_{A_{4}})$ fusion category symmetry.

###  5.2 $S_{4}$ orbifold

The orbifold ${SU(2)_{1}/S_{4}}$ has ${\rm Rep}(S_{4})$ symmetry generated by
Wilson lines associated to the five irreducible representations of $S_{4}$.
They are the trivial representation, the sign representation $S$, a two-
dimensional representation $U$, and two three-dimensional representations
$V,W$. The fusion rules are

$\displaystyle
S^{2}=1\,,~{}~{}SU=U\,,~{}~{}SV=W\,,~{}~{}U^{2}=1+S+U\,,~{}~{}UV=V+W\,,~{}~{}V^{2}=1+U+V+W\,.$
(5.5)

Note that the ${SU(2)_{1}/S_{4}}$ CFT is also realized by an
${\mathbb{Z}}_{2}$ orbifold of the ${SU(2)_{1}/A_{4}}$ theory and $S$
generates the corresponding magnetic ${\mathbb{Z}}_{2}$ symmetry.

The theory contains 28 chiral primaries with respect to the
${\mathcal{V}}_{S_{4}}$ chiral algebra (5.3) which give rise to 28 Verlinde
lines defined by the $S$-matrix in Table 12. They generate the fusion category
symmetry ${\rm Rep}({\mathcal{V}}_{S_{4}})$ which contains ${\rm Rep}(S_{4})$.

###  5.3 $A_{5}$ orbifold

The ${\rm Rep}(A_{5})$ symmetry of the ${SU(2)_{1}/A_{5}}$ orbifold is
generated by Wilson lines associated to the five irreducible representations
of $A_{5}$. They are the trivial representation, two three-dimensional
representations $V,W$, one four-dimensional representation $T$, and one five-
dimensional representation $U$. The fusion rules are

$\displaystyle
V^{2}=W^{2}=1+V+U\,,~{}~{}VW=T+U\,,~{}~{}VT=W+T+U\,,~{}~{}WT=V+T+U\,,$ (5.6)
$\displaystyle VU=WU=V+W+U+T\,,~{}~{}T^{2}=1+V+W+T+U\,,~{}~{}TU=V+W+T+2U\,,$
$\displaystyle U^{2}=1+V+W+2T+2U\,.$

The theory contains 37 chiral primaries with respect to the
${\mathcal{V}}_{A_{5}}$ chiral algebra (5.3) which give rise to 37 Verlinde
lines defined by the S-matrix in Table 13. They generate the fusion category
symmetry ${\rm Rep}({\mathcal{V}}_{A_{5}})$ which contains ${\rm Rep}(S_{5})$.

##  6 Conclusion and discussion

In this paper, we developed general methods to study non-invertible symmetries
in QFTs. In particular, we applied these tools to investigate the fusion
category symmetries of $c=1$ CFTs, and have found a surprisingly rich
structure, richer than can be accounted for by just studying Verlinde lines at
the rational points. We have not fully classified these symmetries, but we
have shown that many are within the grasp of methods which generalize broadly
and which we believe demonstrate the ubiquity of fusion category symmetry. It
is clear that in higher dimensions there will be interesting fusion category
(higher) symmetries to explore. One can take any theory with a group-like
symmetry and gauge a non-normal finite subgroup, obtaining a theory with non-
invertible TDLs. It will be very interesting therefore to investigate the
consequences of these symmetries in settings beyond 1+1D CFT. This is just
beginning to be explored, for instance, see [83, 41].

In light of the continua of topological defects, it is interesting to consider
how one might extend the usual mathematical definition of a fusion category to
these sets of lines. Intuitively the objects are parametrized by topological
spaces, and all of the fusion data, such as the fusion products, duals, and
F-symbols should be a suitable generalization of continuous maps. Indeed
continuity is too strong as the F-symbols for symmetry lines of continuous
symmetry groups are already not continuous, but only Borel measurable [84].
This is related to the question “what kind of category is the category of
QFTs?” if we demand that TDLs form the morphisms between 1+1D QFTs, with the
sort of fusion category we want to study being the endomorphisms of a given
object (such as a $c=1$ CFT). The parametrized fusion and F-symbol data appear
to insist that this category is enriched over measurable spaces. We leave this
interesting question to future work.

##  Appendix A Global Symmetries Of Orbifolds

###  A.1 The $SU(2)_{1}$ Point

The $SU(2)_{1}$ point where the compact boson is self-T-dual is the key to
unlocking the symmetries of the $c=1$ CFTs. It is convenient to write its
global symmetry group as $(SU(2)\times SU(2))/\mathbb{Z}_{2}=SO(4)$. The
$SO(4)$ symmetry is generated by six conserved currents in the CFT which can
be taken to be the chiral and anti-chiral $SU(2)$ currents

$\displaystyle\\{\partial
X_{L},\cos\sqrt{2}X_{L},\sin\sqrt{2}X_{L}\\}~{}~{}{\rm
and}~{}~{}\\{\bar{\partial}X_{R},\cos\sqrt{2}X_{R},\sin\sqrt{2}X_{R}\\}\,.$
(A.1)

The usual anomalous rotations generated by the (anti)holomorphic currents
$\partial X_{L}$ and $\bar{\partial}X_{R}$ are represented by

$e^{i{{\alpha}\over\sqrt{2}}\oint dz\partial
X_{L}}=\begin{bmatrix}R_{\alpha/2}&&0\\\ 0&&R_{\alpha/2}\end{bmatrix}\quad{\rm
and}\quad e^{i{{\alpha}\over\sqrt{2}}\oint d\bar{z}\partial
X_{R}}=\begin{bmatrix}R_{\alpha/2}&&0\\\ 0&&R_{-\alpha/2}\end{bmatrix}\quad\in
SO(4)\,,$ (A.2)

respectively, where $R_{\alpha}$ is a $2\times 2$ rotation matrix parametrized
by ${\alpha}\in[0,2\pi]$. Note that a $2\pi$ rotation generated by either
$\partial X_{L}$ or $\bar{\partial}X_{R}$ is the central element $-1\in
SO(4)$.

The non-holomorphic currents $(\partial\theta,\bar{\partial}\theta)$ and
$(\partial\phi,\bar{\partial}\phi)$ generate rotations by

$e^{i{\alpha}\oint(dz\partial\theta-d\bar{z}\bar{\partial}\theta)}=\begin{bmatrix}R_{\alpha}&&0\\\
0&&1\end{bmatrix}\quad{\rm and}\quad
e^{i{\alpha}\oint(dz\partial\phi-d\bar{z}\bar{\partial}\phi)}=\begin{bmatrix}1&&0\\\
0&&R_{\alpha}\end{bmatrix}\quad\in SO(4)\,,$ (A.3)

matching the identification $X_{L}={\theta+\phi\over\sqrt{2}}$,
$X_{R}={\theta-\phi\over\sqrt{2}}$. We denote the first group $U(1)^{\theta}$
and the second $U(1)^{\phi}$. We take charge conjugation to act by

$C=\begin{bmatrix}-1&&0&&0&&0\\\ 0&&1&&0&&0\\\ 0&&0&&-1&&0\\\
0&&0&&0&&1\end{bmatrix}\,,$ (A.4)

which we see negates $\alpha$ in all the above formulas when we conjugate by
it. It combines with the $U(1)$’s above to form the symmetry
$(U(1)^{\theta}\times U(1)^{\phi})\rtimes\mathbb{Z}_{2}^{C}$ of the whole
circle branch.

There are various symmetries of the $SU(2)_{1}$ point that might be called
T-duality. An $SO(4)$ matrix exchanging $U(1)^{\phi}$ and $U(1)^{\theta}$
takes the form

$\begin{bmatrix}0&&A_{2\times 2}\\\ B_{2\times 2}&&0\end{bmatrix}\,.$ (A.5)

For instance, one which appears in [57] is a $\pi$-rotation generated by the
current $\cos X_{R}$:

$\begin{bmatrix}0&&0&&1&&0\\\ 0&&0&&0&&1\\\ -1&&0&&0&&0\\\
0&&-1&&0&&0\end{bmatrix}\,,$ (A.6)

which is anomalous and squares to the central element, while in (3.6) we use

$T=\begin{bmatrix}0&&0&&1&&0\\\ 0&&0&&0&&1\\\ 1&&0&&0&&0\\\
0&&1&&0&&0\end{bmatrix}\,.$ (A.7)

The decomposition of the Hilbert space into $SO(4)$ representations may be
found in [85],

$\bigoplus_{\begin{subarray}{c}s,\bar{s}\in\mathbb{Z}\\\ s+\bar{s}\in
2\mathbb{Z}\end{subarray}}\left(\frac{s}{2},\frac{\bar{s}}{2}\right)\otimes\left(\frac{s^{2}}{4},\frac{\bar{s}^{2}}{4}\right)\,,$
(A.8)

where the first part denotes the $SU(2)\times SU(2)$ representation, and the
second part the Virasoro representation, ie. $h={s^{2}\over 4}$,
$\bar{h}={\bar{s}^{2}\over 4}$. Recall an operator of momentum $n$ and winding
$w$ has $s=n+m$, $\bar{s}=n-m$.

###  A.2 Chiral Anomaly and $D_{8}$ symmetry

When we vary the radius away from the $SU(2)_{1}$ point, the $SO(4)$ symmetry
is broken down to the subgroup

$G_{\rm bos}=(U(1)^{\theta}\times U(1)^{\phi})\rtimes\mathbb{Z}_{2}^{C}\,,$
(A.9)

where $U(1)^{\theta}$ (resp. $U(1)^{\phi}$) is embedded as the top left (resp.
bottom right) $2\times 2$ block of $SO(4)$, and $C$ is charge conjugation
defined above.

The anomaly is captured by the Euler class of the fundamental representation
$V_{4}$,

$e(V_{4})\in H^{4}(BSO(4),\mathbb{Z})\,.$ (A.10)

To restrict the above anomaly to $G_{\rm bos}$, we need to equivalently study
the restriction of the representation $V_{4}$ to $G_{\rm bos}$. It is seen to
split into a sum of the two $O(2)$ fundamentals of the two $2\times 2$ blocks
$V_{4}=V_{2}^{\theta}\oplus V_{2}^{\phi}$,

$e(V_{4})=e(V_{2}^{\theta})e(V_{2}^{\phi})\,.$ (A.11)

Let us consider gauging $C$ at a generic radius. The commutant of $C$ in
$G_{\rm bos}$ is generated by the $\pi$-rotations

$(-1)^{n}=\begin{bmatrix}-1&&0&&0&&0\\\ 0&&-1&&0&&0\\\ 0&&0&&1&&0\\\
0&&0&&0&&1\end{bmatrix}\in SO(4)\,,$ (A.12)

and

$(-1)^{w}=\begin{bmatrix}1&&0&&0&&0\\\ 0&&1&&0&&0\\\ 0&&0&&-1&&0\\\
0&&0&&0&&-1\end{bmatrix}\in SO(4)\,,$ (A.13)

as well as $C$ itself. Thus our fundamental $SO(4)$ bundle splits over the
classifying space of the corresponding $\mathbb{Z}_{2}$ gauge fields $A_{n}$,
$A_{w}$, and $C$, respectively as $A_{n}+C$, $A_{n}$, $A_{w}+C$, and $A_{w}$,
so the Euler class is

$\displaystyle e(V_{4})=$
$\displaystyle(A_{n}+C)A_{n}(A_{w}+C)A_{w}=A_{n}^{2}A_{w}^{2}+C^{2}A_{n}A_{w}+CA_{n}^{2}A_{w}+CA_{n}A_{w}^{2}$
(A.14) $\displaystyle=$
$\displaystyle\frac{1}{2}d(A_{n}A_{w}^{2}+CA_{n}A_{w})\,.$

The last equality allows us to identify the anomaly with the element

$\frac{1}{2}A_{n}A_{w}^{2}+\frac{1}{2}CA_{n}A_{w}\in
H^{3}(B\mathbb{Z}_{2}^{3},U(1))\,.$ (A.15)

The first piece is the familiar $\mathbb{Z}_{2}$ shadow of the chiral anomaly,
while the second piece causes symmetry transmutation when we gauge $C$ to go
to the orbifold. Indeed, in that case we find that the
$\mathbb{Z}_{2}^{n}\times\mathbb{Z}_{2}^{w}$ commutant becomes $D_{8}$ because
of the extension class $A_{n}A_{w}$ which appears in the cubic piece. This is
the symmetry at a generic orbifold point

$G_{\rm orb}=D_{8}\,.$ (A.16)

In terms of the presentation

$D_{8}=\langle r,s\ |\ r^{4}=s^{2}=1,~{}srs=r^{-1}\rangle\,,$ (A.17)

we can write the two lifts of $\mathbb{Z}_{2}^{n}$ and $\mathbb{Z}_{2}^{w}$ as
$s$ and $sr$. Meanwhile the magnetic symmetry is represented as $r^{2}$.

In fact, whenever a symmetry such as $C$ only appears in a linear term in the
anomaly polynomial, the result is a kind of symmetry transmutation [45, 39].
To be precise, we introduce the $\mathbb{Z}_{2}$ gauge field $M$ of the
magnetic symmetry. It satisfies

$dM=A_{n}A_{w}\,,$ (A.18)

which leads to the $D_{8}$ group as claimed. Furthermore, and key for us is
that this allows us to write the chiral anomaly as a counterterm:

$\frac{1}{2}A_{n}A_{w}^{2}=d\left(\frac{1}{2}MA_{w}\right)\,.$ (A.19)

There are two choices here, which are evidently exchanged by $T$ duality,
since we may also take

$\frac{1}{2}A_{n}A_{w}^{2}=d\left(\frac{1}{2}MA_{n}+\frac{1}{4}A_{n}A_{w}\right)\,.$
(A.20)

These also differ by the unique choice of $D_{8}$ discrete torsion, which may
be written as

$\frac{1}{2}MA_{n}+\frac{1}{2}MA_{w}+\frac{1}{4}A_{n}A_{w}\,.$ (A.21)

###  A.3 The KT Point

The T-duality on the circle branch exchanges $U(1)^{\theta}$ and $U(1)^{\phi}$
but commutes with $C$. It thus descends to a duality of the orbifold which
switches the lifts of the $\mathbb{Z}_{2}$ subgroups of each of these
$U(1)$’s, hence it acts by the outer automorphism of $D_{8}$ which exchanges
$s$ and $sr$. This duality fixes the Kosterlitz-Thouless (KT) point, which is
the ${\mathbb{Z}}_{2}^{C}$ orbifold of the $SU(2)_{1}$ point, and we expect at
this point the $D_{8}$ symmetry is enhanced. In fact the KT point is a
junction between the circle and $\mathbb{Z}_{2}$ orbifold branches and we will
find the symmetry is enhanced to $G_{\rm bos}=(U(1)^{\tilde{\theta}}\times
U(1)^{\tilde{\phi}})\rtimes\mathbb{Z}_{2}^{\tilde{C}}$, where $\tilde{\theta}$
and $\tilde{\phi}$ are the circle branch fields describing the KT point and
$\tilde{C}$ is the charge conjugation $\tilde{\theta}\mapsto-\tilde{\theta}$,
$\tilde{\phi}\mapsto-\tilde{\phi}$. We reserve the untilded fields
$\phi,\theta$ to describe the circle branch fields at the $SU(2)_{1}$ point.

The $D_{8}$ symmetry is embedded into $G_{\rm bos}$ with $sr$ acting as
$\tilde{C}$ and $r$ acting as a ${\pi\over 2}$ rotation in
$U(1)^{\tilde{\theta}}$. The T-duality of the orbifold branch is realized by a
$\pi\over 4$ rotation in $U(1)^{\tilde{\theta}}$ (which swaps $s$ and $sr$).
We derive this as follows.

At the $SU(2)_{1}$ point, we represent the $SO(4)$ group elements in the basis
(3.7). The normalizer of $\mathbb{Z}_{2}^{C}$ is enlarged to $(U(1)\times
U(1))\rtimes\mathbb{Z}_{2}$. In particular, we see it includes a full rotation
group $SO(2)^{a}$ in the $1-3$ coordinates and another $SO(2)^{b}$ in the
$2-4$ coordinates. The elements $(-1)^{n},(-1)^{w}$ we studied above act as
simultaneous reflections in these $2\times 2$ blocks, while their product, as
well as $C$, is in the center. It is convenient to exchange the 2nd and 3rd
coordinates so we have:

$SO(2)^{a}=\begin{bmatrix}\star&&\star&&0&&0\\\ \star&&\star&&0&&0\\\
0&&0&&1&&0\\\ 0&&0&&0&&1\end{bmatrix}\subset SO(4)\,,$ (A.22)
$SO(2)^{b}=\begin{bmatrix}1&&0&&0&&0\\\ 0&&1&&0&&0\\\ 0&&0&&\star&&\star\\\
0&&0&&\star&&\star\end{bmatrix}\subset SO(4)\,,$ (A.23)
$(-1)^{n}=\begin{bmatrix}-1&&0&&0&&0\\\ 0&&1&&0&&0\\\ 0&&0&&-1&&0\\\
0&&0&&0&&1\end{bmatrix}\in SO(4)\,,$ (A.24)
$(-1)^{w}=\begin{bmatrix}1&&0&&0&&0\\\ 0&&-1&&0&&0\\\ 0&&0&&1&&0\\\
0&&0&&0&&-1\end{bmatrix}\in SO(4)\,,$ (A.25) $C=\begin{bmatrix}-1&&0&&0&&0\\\
0&&-1&&0&&0\\\ 0&&0&&1&&0\\\ 0&&0&&0&&1\end{bmatrix}\in SO(4)\,,$ (A.26)
$T=\begin{bmatrix}0&&1&&0&&0\\\ 1&&0&&0&&0\\\ 0&&0&&0&&1\\\
0&&0&&1&&0\end{bmatrix}\in SO(4)\,.$ (A.27)

Note that $C\in SO(2)^{a}$. The full normalizer group is $(SO(2)^{a}\times
SO(2)^{b})\rtimes\mathbb{Z}_{2}^{n}$.

In fact, we can re-express the $SU(2)_{1}$ point in terms of fields
$\phi^{\prime},\theta^{\prime}$ related to $\phi,\theta$ by an $SO(4)$
rotation which produces the change of basis used above. In these variables,
$SO(2)^{a}$ and $SO(2)^{b}$ act as shifts of $\phi^{\prime}$ and
$\theta^{\prime}$, respectively. $C$ is a $\pi$ rotation in $SO(2)^{a}$, and
when we gauge it the effect is to double the radius, such that the new circle
branch variables at the KT point may be expressed as
$\tilde{\phi}=\phi^{\prime}/2$, $\tilde{\theta}=2\theta^{\prime}$ (compare
with Section 3.3.1). The resulting symmetry group is $G_{\rm
bos}=(U(1)^{\tilde{\theta}}\times
U(1)^{\tilde{\phi}})\rtimes\mathbb{Z}_{2}^{\tilde{C}}$ as claimed.

##  Appendix B Parafermions and self-dualities

In this appendix, we discuss in detail the symmetries in the
${\mathbb{Z}}_{k}$ parafermion theory of [56] and the relation to TDLs in the
$SU(2)_{k}/U(1)$ coset CFT upon bosonization.

###  B.1 The ${\mathbb{Z}}_{k}$ parafermions

As briefly reviewed in Section 2.6, the fundamental fields in the
${\mathbb{Z}}_{k}$ parafermion theory consist of the chiral parafermions
$\psi_{n}(z)$ and its antichiral partners $\bar{\psi}_{n}(\bar{z})$ which obey
fractional mutual statistics as in Figure 8. The parafermions $\psi_{n}$
generate a chiral algebra that extends the Virasoro algebra at central charge
$c$ (and similarly for the antichiral fields $\bar{\psi}_{n})$,

$\displaystyle\psi_{n}(z)\psi_{m}(0)=$ $\displaystyle{c_{n,m}\over
z^{\Delta_{n}+\Delta_{m}-\Delta_{m+n}}}\times\begin{cases}\psi_{m+n}(0)+\mathcal{O}(z)&{\rm
for}~{}n+m<k\,,\\\ \psi_{n-m}(0)+\mathcal{O}(z)&{\rm for}~{}m<n~{}{\rm
and}~{}m+n>k\,,\\\ 1+{2\Delta_{n}\over c}z^{2}T(z)+\mathcal{O}(z^{3})&{\rm
for}~{}m<n~{}{\rm and}~{}m+n=k\,.\end{cases}$ (B.1)

In particular $\psi_{n}$ are Virasoro primaries with respect to the stress
tensor $T(z)$ and their scaling dimensions $\Delta_{n}$ satisfy

$\displaystyle\Delta_{n}+\Delta_{m}-\Delta_{m+n}={2mn\over k}~{}{\rm
mod\,}{\mathbb{Z}}\,,$ (B.2)

as required for the mutual statistics of the parafermions. The parafermion
algebra is completely specified by $(c,\Delta_{k})$ and the OPE coefficients
$c_{m,n}$ in (B.1), which are subjected to constraints from associativity.
There are infinitely many solutions to these constraints [56]. Here we focus
on the simplest parafermion theories given by

$\displaystyle c={2(k-1)\over k+2}\,,\quad\Delta_{k}={n(k-n)\over k}\,,$ (B.3)

and the explicit form of $c_{n,m}$ can be found in [56] which is not important
for our discussion here. The bosonization of this parafermion theory
corresponds to the $SU(2)_{k}/U(1)$ coset CFT which we review in the
following.

###  B.2 The $SU(2)_{k}/U(1)$ coset CFT

The $SU(2)_{k}/U(1)$ coset CFT is constructed by taking a quotient of the
$SU(2)_{k}$ WZW CFT by the current subalgebra $U(1)_{2k}$. The resulting
theory has central charge

$\displaystyle c={2(k-1)\over k+2}\,,$ (B.4)

and follows an ADE classification [71] which is inherited from a choice of
modular invariant in the parent $SU(2)_{k}$ WZW CFT. Here we will focus on the
diagonal modular invariant for the $SU(2)_{k}/U(1)$ coset which is often
referred to as the $A_{k+1}$ type.313131The coset CFTs with non-diagonal
modular invariants can be obtained from gauging a symmetric Frobenius algebra
in the diagonal CFT at special $k$ [10]. For $SU(2)_{2k-4}/U(1)$ CFT of the
$D_{k}$ type, this amounts to gauging the ${\mathbb{Z}}_{2}$ center of the
symmetry (B.6) in the $A_{2k-3}$ type coset CFT, and the resulting theory has
$D_{2k-4}\times{\mathbb{Z}}_{2}^{M}$ global symmetry with a mixed anomaly
between the two factors [39].

Physically the coset construction amounts to gauging an anomaly-free $U(1)$
subgroup of the $SO(4)={(SU(2)_{L}\times SU(2)_{R})/{\mathbb{Z}}_{2}}$ global
symmetry in the WZW model. Such a $U(1)$ subgroup is unique up to conjugation
by elements in $SO(4)$ and commonly referred to as the $U(1)_{V}$ vector
subgroup, whose commutant in $SO(4)$ is a $U(1)_{A}$ axial subgroup.

In general, potential global symmetries of a gauge theory with continuous
gauge group $G$ arise from the quotient $N_{G}(\hat{G})/G$ of the normalizer
of $G$ in the symmetry $\hat{G}$ of the ungauged theory. Here the normalizer
of $U(1)_{V}$ in $SO(4)$ is

$\displaystyle N_{U(1)_{V}}(SO(4))=(U(1)_{V}\times
U(1)_{A})\rtimes{\mathbb{Z}}_{2}\,,$ (B.5)

where the ${\mathbb{Z}}_{2}$ acts on both $U(1)$ factors by complex
conjugation. However the ’t Hooft anomaly of the WZW model governed by the
anomaly four-form ${\cal I}_{4}=ke_{4}(SO(4))$ induces an Adler-Bell-Jackiw
(ABJ) anomaly for $N_{U(1)_{V}}(SO(4))/U(1)_{V}$ upon gauging $U(1)_{V}$.
Consequently, the global symmetry of the coset theory is323232For $k=2$ which
corresponds to the Ising CFT, the second ${\mathbb{Z}}_{2}$ factor in (B.6)
acts trivially and the faithful global symmetry is just ${\mathbb{Z}}_{k=2}$.

$\displaystyle
G_{SU(2)_{k}/U(1)}={\mathbb{Z}}_{k}\rtimes{\mathbb{Z}}_{2}=D_{2k}\,.$ (B.6)

Coset CFTs are in general RCFTs with respect to certain chiral algebras that
descend from the parent current algebras by taking the commutant of the
subalgebras that we gauge. Here the $SU(2)_{k}/U(1)$ coset CFT bosonizes the
${\mathbb{Z}}_{k}$ parafermions, and consequently the relevant chiral algebra
${\mathcal{V}}_{k}\equiv{\mathcal{V}}(SU(2)_{k}/U(1))$ is also an integer-spin
subalgebra of the chiral parafermion algebra generated by $\psi_{n}$ [86, 87,
88, 89, 90, 91],

$\displaystyle{\mathcal{V}}_{k}=\begin{cases}W(2)\cong{\rm Vir}_{c={1\over
2}}&k=2\,,\\\ W(2,3)&k=3\,,\\\ W(2,3,4)&k=4\,,\\\ W(2,3,4,5)&k\geq
5\,,\end{cases}$ (B.7)

where $W(2,3,\dots)$ denotes a W-algebra with strong generators of spins $2,3$
and so on.333333The chiral algebra ${\mathcal{V}}_{k}$ is invariant under the
${\mathbb{Z}}_{k}$ symmetry of the coset CFT. Under the ${\mathbb{Z}}_{2}$
conjugation in (B.6), each generator of ${\mathcal{V}}_{k}$ picks up a sign
$(-1)^{s}$ depending on its spin $s$, similarly for the antichiral algebra.
The primary operators in the coset CFT with respect to ${\mathcal{V}}_{k}$
(and its anti-chiral partner) are obtained from the $SU(2)_{k}$ current
primaries $\Phi^{j,\bar{j}}_{m,\bar{m}}$ by factoring out $U(1)_{2k}$ charged
states which can be parametrized by a compact boson $(X_{L},X_{R})$ at radius
$R=\sqrt{2k}$ [71],343434Here our convention is that $j,\bar{j}=0,{1\over
2},\dots,{k\over 2}$ label the $SU(2)$ spins and $m,\bar{m}\in{\mathbb{Z}}/2$
satisfy $-j\leq m\leq j,-\bar{j}\leq\bar{m}\leq\bar{j}$.

$\displaystyle\Phi^{j,\bar{j}}_{m,\bar{m}}=\phi^{j,\bar{j}}_{m,\bar{m}}(z)e^{{2i\over\sqrt{2k}}(mX_{L}(z)+\bar{m}X_{R}(\bar{z}))}\,.$
(B.8)

The coset primary is denoted by $\phi^{j,\bar{j}}_{m,\bar{m}}$ and has scaling
dimensions,

$\displaystyle(h,\bar{h})=\left({j(j+1)\over(k+2)}-{m^{2}\over
k},{\bar{j}(\bar{j}+1)\over(k+2)}-{\bar{m}^{2}\over k}\right),\quad{\rm
for~{}}|m|<j~{}{\rm and}~{}|\bar{m}|<\bar{j}\,.$ (B.9)

With respect to the global symmetry (B.6) of the coset CFT, the operator
$\phi^{j,\bar{j}}_{m,\bar{m}}$ carries ${\mathbb{Z}}_{k}$ charge $m+\bar{m}$
and transforms to $\phi^{j,\bar{j}}_{-m,-\bar{m}}$ under the
${\mathbb{Z}}_{2}$ conjugation.353535Note that $j-\bar{j}\in{\mathbb{Z}}$
irrespective of the choice of the ADE modular invariant [71] and thus
$m+\bar{m}\in{\mathbb{Z}}$. It is convenient to extend the range of the
$m,\bar{m}$ indices with the identifications

$\displaystyle\phi^{j,\bar{j}}_{m,\bar{m}}=\phi^{k/2-j,k/2-\bar{j}}_{k/2+m,k/2+\bar{m}}=\phi^{j,\bar{j}}_{m+k,\bar{m}}=\phi^{j,\bar{j}}_{m,\bar{m}+k}\,.$
(B.10)

Consequently we use the following convenient set of indices $(j,m)$ with
$m+j\in{\mathbb{Z}}$ to parameterize the coset primaries,

$\displaystyle{\rm Even}~{}k~{}\\{(j,m)\\}=$ $\displaystyle\\{0\leq
j\leq{k-2\over 4},-j\leq m\leq k-j-1\\}\cup\\{j={k\over 4},-j\leq m\leq{k\over
2}-j-1\\}\,,$ (B.11) $\displaystyle{\rm Odd}~{}k~{}\\{(j,m)\\}=$
$\displaystyle\\{0\leq j\leq{k-1\over 4},-j\leq m\leq k-j-1\\}\,,$

and similarly for the antichiral indices $(\bar{j},\bar{m})$. There are
${k(k+1)\over 2}$ independent primaries on each side. The chiral fusion rule
of these primary operators is

$\displaystyle\phi^{j_{1}}_{m_{1}}\otimes\phi^{j_{2}}_{m_{2}}=\bigoplus_{j=|j_{1}-j_{2}|}^{\min(j_{1}+j_{2},k-j_{1}-j_{2})}\phi^{j}_{m_{1}+m_{2}}$
(B.12)

and similarly for the antichiral side.

The character for the coset chiral primary $\phi^{j}_{m}$ is often denoted by
$\eta(\tau)c^{2j}_{2m}(\tau)$, which is obtained from factorizing the
$SU(2)_{k}$ characters $\chi_{j}^{k}(\tau)$ in terms of the $U(1)_{2k}$
characters $K_{n}^{k}(\tau)$,

$\displaystyle\chi_{j}^{k}(\tau,z)=\sum_{n=-k+1}^{k}\eta(\tau)c^{2j}_{n}(\tau)K^{k}_{n}(\tau)\,.$
(B.13)

The so-called string functions $c^{\ell}_{n}(\tau)$ for $\ell-n\in
2{\mathbb{Z}}$ satisfy the identities

$\displaystyle c^{\ell}_{n}=c^{\ell}_{n+2k}=c^{k-\ell}_{k+n}=c^{\ell}_{-n}$
(B.14)

where the first three equalities follow from (B.10) and the last from the
${\mathbb{Z}}_{2}$ symmetry in (B.6).

The local operator content of the $SU(2)_{k}/U(1)$ CFT of the $A_{k+1}$ type
consists of primaries $\phi^{j,j}_{m,m}$ for $(j,m)$ given in (B.11) and their
${\mathcal{V}}_{k}$ descendants.363636In the main text, we denote the local
operators $\phi^{j,j}_{m,m}$ as $\phi^{j}_{m}$ for convenience. The torus
partition function reads

$\displaystyle
Z(\tau,\bar{\tau})=|\eta(\tau)|^{2}\sum_{(j,m)}|c^{2j}_{2m}(\tau)|^{2}\,.$
(B.15)

As explained in Section 2.2, the $SU(2)_{k}/U(1)$ CFT contains obvious TDLs in
the form of Verlinde lines $\mathcal{L}_{(j,m)}$ associated to each of the
coset primaries above labelled by $(j,m)$. These TDLs preserve the extended
${\mathcal{V}}_{k}$ algebra (and its antichiral version) and follow the same
fusion rules as in (B.12), therefore furnishing a ${\rm
Rep}({\mathcal{V}}_{k})$ fusion category symmetry in the coset CFT. In
particular ${\rm Rep}({\mathcal{V}}_{k})$ contains the ${\mathbb{Z}}_{k}$
symmetry TDLs which correspond to $\mathcal{L}_{(0,m)}$ with
$m=0,1,\dots,k-1$. The full category symmetry of the coset CFT is much richer
and come from TDLs that only commute with a subalgebra of ${\mathcal{V}}_{k}$
that contains the Virasoro subalgebra. This includes as independent
generators, the ${\mathbb{Z}}_{2}$ conjugation symmetry in (B.6) and also a
duality defect $\mathcal{N}$ which generalizes the Kramers-Wannier duality of
the Ising CFT (see Table 8).

While only the diagonal coset primaries appear as local operators in the
$A_{k+1}$ type $SU(2)_{k}/U(1)$ CFT, more general coset operators of the type
$\phi^{j,j}_{m,\bar{m}}$ reside in the sector twisted by an element
$a_{1}=m-\bar{m}\in{\mathbb{Z}}_{k}$. They contribute to the twisted partition
function with ${\mathbb{Z}}_{k}$ holonomies $a_{1},a_{2}$ around the spatial
and temporal cycles on $T^{2}$ respectively [71],

$\displaystyle Z(\tau,\bar{\tau};a_{1},a_{2})={1\over
2}|\eta(\tau)|^{2}\sum_{n=-k+1}^{k}\sum_{\ell=0}^{k}\omega^{(n+a_{1})a_{2}}c^{\ell}_{n+2a_{1}}(\tau)c^{\ell}_{n}(\bar{\tau})\,.$
(B.16)

The twisted partition function satisfies

$\displaystyle Z(\tau,\bar{\tau};a_{1},a_{2})={1\over
k}\sum_{b_{1},b_{2}\in{\mathbb{Z}}_{k}}Z(\tau,\bar{\tau};b_{1},b_{2})\omega^{b_{1}a_{2}-b_{2}a_{1}}={1\over
k}\sum_{b_{1},b_{2}\in{\mathbb{Z}}_{k}}Z(\tau,\bar{\tau};b_{1},b_{2})\bar{\omega}^{b_{1}a_{2}-b_{2}a_{1}}$
(B.17)

which confirms the two self-dualities of the theory under ${\mathbb{Z}}_{k}$
gauging with the bicharacter in (2.15) given by $\chi_{\pm}(a,b)\equiv\pm{2\pi
ab\over k}$ .

###  B.3 Operators and symmetries in the parafermion theory

Similar to the case of Majorana fermion, a general operator $\mathcal{O}$ in
the parafermion theory leads to a ${\mathbb{Z}}_{k}$ valued monodromy for the
parafermion field $\psi_{1}$,

$\displaystyle\psi_{1}(e^{2\pi
i}z)\mathcal{O}(0)=\omega^{s}\psi_{1}(z)\mathcal{O}(0)$ (B.18)

and corresponds to a state in the Hilbert space $\mathcal{H}^{PF}_{s}$ on
$S^{1}$ with the corresponding twisted periodicity for $\psi_{1}$. For $k=2$,
the relations to the usual fermionic Hilbert spaces are
$\mathcal{H}^{PF}_{0}=H^{F}_{NS}$ and
$\mathcal{H}^{PF}_{1}=H^{F}_{R}$.373737Because of the nontrivial mutual
statistics among $\psi_{n}$ (see Figure 8), the chiral parafermion $\psi_{1}$
is not local with respect to itself unless $k=2$ and thus $\psi_{1}$ (and
general $\psi_{n}$) are not contained in $\mathcal{H}_{0}^{PF}$.

Thanks to the bosonization/parafermionization map (2.31), we can describe the
parafermion operators using the coset primaries $\phi^{j,\bar{j}}_{m,\bar{m}}$
reviewed in the last section. We also impose an additional constraint
$j=\bar{j}$ which is consistent with the OPE and write for simplicity

$\displaystyle\phi^{j}_{m,\bar{m}}\equiv\phi^{j,j}_{m,\bar{m}}\,.$ (B.19)

In particular, the parafermion fields in Section B.1 are realized by

$\displaystyle\psi_{n}\equiv\phi^{0}_{n,0}\,,\quad\bar{\psi}_{n}\equiv\phi^{0}_{0,n}\,.$
(B.20)

The mutual statistics for two such operators (generalizing Figure 8) is

$\displaystyle\phi^{j}_{m,\bar{m}}(z,\bar{z})\phi^{j^{\prime}}_{m^{\prime},\bar{m}^{\prime}}(w,\bar{w})=\omega^{{\bar{m}\bar{m}^{\prime}-mm^{\prime}}}\phi^{j^{\prime}}_{m^{\prime},\bar{m}^{\prime}}(w,\bar{w})\phi^{j}_{m,\bar{m}}(z,\bar{z})\,.$
(B.21)

Consequently, the operators corresponding to the states in the twisted
parafermion Hilbert space $\mathcal{H}^{PF}_{s}$ are $\phi^{j}_{s/2,\bar{m}}$.
There are two distinguished operators in each $\mathcal{H}^{PF}_{s}$ which are
a pair of parafermion primaries

$\displaystyle\sigma_{s}=\phi^{s/2}_{s/2,s/2}\,,\quad\mu_{s}=\phi^{s/2}_{s/2,-s/2}\,,$
(B.22)

that generalize the order and disorder spin operators for the Majorana fermion
and its bosonization, the Ising CFT at $k=2$. All other operators arise from
consecutive actions of the modes of the basic parafermions
$\psi_{1},\bar{\psi}_{1}$ on these (generalized) spin operators (including the
identity operator at $s=0$) [56]. In particular, they include energy (thermal)
operators $\epsilon_{r}$ for $r=1,2,\dots,\left\lfloor{k\over
2}\right\rfloor$,

$\displaystyle\epsilon_{r}=\phi^{r}_{0,0}\,.$ (B.23)

| ${\mathbb{Z}}_{k}\times\bar{\mathbb{Z}}_{k}$ | ${\mathbb{Z}}_{2}^{C}$ | ${\mathbb{Z}}_{2}^{S}$
---|---|---|---
$\phi^{j}_{m,\bar{m}}$ | $({m+\bar{m}},m-\bar{m})$ | $\phi^{j}_{-m,-\bar{m}}$ | $(-1)^{m+j}\phi^{j}_{m,-\bar{m}}$
$\psi_{n}$ | $(n,n)$ | $\psi_{k-n}$ | $(-1)^{n}\psi_{n}$
$\bar{\psi}_{n}$ | $(n,-n)$ | $\bar{\psi}_{k-n}$ | $\bar{\psi}_{k-n}$
$\sigma_{n}$ | $(n,0)$ | $\sigma_{k-n}$ | $\mu_{n}$
$\mu_{n}$ | $(0,n)$ | $\mu_{k-n}$ | $\sigma_{n}$
$\epsilon_{r}$ | $(0,0)$ | $\epsilon_{r}$ | $(-1)^{r}\epsilon_{r}$
Table 8: Global symmetries of the ${\mathbb{Z}}_{k}$ parafermion theory.

The full global symmetry of the parafermion theory is,

$\displaystyle
G_{PF}=({\mathbb{Z}}_{k}\times\bar{\mathbb{Z}}_{k})\rtimes({\mathbb{Z}}_{2}^{C}\times{\mathbb{Z}}_{2}^{S})\,,$
(B.24)

where ${\mathbb{Z}}_{2}^{C}$ and ${\mathbb{Z}}_{2}^{S}$ act on the
${\mathbb{Z}}_{k}\times\bar{\mathbb{Z}}_{k}$ by conjugating and swapping the
two factors respectively. The operators of the parafermion theory transform
under (B.24) as summarized in Table 8. Here $\bar{\mathbb{Z}}_{k}$ is the
generalization of the $(-1)^{F}$ symmetry at $k=2$, while ${\mathbb{Z}}_{k}$
is identified with the ${\mathbb{Z}}_{k}^{B}$ symmetry of the bosonized
theory.383838For notational simplicity, ${\mathbb{Z}}_{k}^{B}$ and
${\mathbb{Z}}_{2}^{C}$ are denoted as ${\mathbb{Z}}_{k}$ and
${\mathbb{Z}}_{2}$ in the last section. The bosonization of the parafermion
theory amounts to summing over $\bar{\mathbb{Z}}_{k}$ twisted sectors and
projecting to the $\bar{\mathbb{Z}}_{k}$ invariant states. Consequently, the
bosonized theory only contain operators of the form $\phi^{j}_{m,m}$, which
constituents the diagonal modular invariant (of the $A_{k+1}$ type) for the
$SU(2)_{k}/U(1)$ coset CFT. The other more general parafermion operators carry
nonzero $\bar{\mathbb{Z}}_{k}$ charges reside in the ${\mathbb{Z}}_{k}^{B}$
twisted sectors, as reviewed in the last section.

The full global symmetry of the bosonic $SU(2)_{k}/U(1)$ coset CFT is given in
(B.6) and we see the ${\mathbb{Z}}_{2}$ conjugation is also inherited from the
${\mathbb{Z}}_{2}^{C}$ symmetry in the parafermion theory. The
${\mathbb{Z}}_{2}^{S}$ symmetry in (B.24) however does not lead to an ordinary
symmetry in the bosonized theory. For $k=2$, this is simply the left fermion
parity $(-1)^{F_{L}}$ which is known to generate the Kramers-Wannier duality
under bosonization. Here we will see that for general $k$, the
${\mathbb{Z}}_{2}^{S}$ symmetry of the ${\mathbb{Z}}_{k}$ parafermions
corresponds to a generalized Kramers-Wannier duality for $SU(2)_{k}/U(1)$
coset CFTs, namely self-duality under gauging the ${\mathbb{Z}}_{k}^{B}$
symmetry with the bicharacter $\chi_{+}(a,b)={2\pi ab\over k}$, as evident
from the first equality in (B.17) satisfied by the torus partition function in
general ${\mathbb{Z}}_{k}^{B}$ background.

The parafermionic partition function with the paraspin structure
$(s_{1},s_{2})$ on $T^{2}$ is obtained from (2.31) applied to (B.16) [54],

$\displaystyle Z_{PF}(\tau,\bar{\tau};s_{1},s_{2})={1\over
2}|\eta(\tau)|^{2}\sum_{a=0}^{k-1}\sum_{\ell=0}^{k}\omega^{-(a+s_{1})s_{2}}\left(c^{\ell}_{s_{1}}(\tau)c^{\ell}_{s_{1}+2a}(\bar{\tau})+c^{\ell}_{s_{1}+k}(\tau)c^{\ell}_{s_{1}+k+2a}(\bar{\tau})\right)\,,$
(B.25)

which counts the states in the twisted sector $\mathcal{H}_{s_{1}}^{PF}$ with
an insertion of the ${\mathbb{Z}}_{k}$ group element corresponding to $s_{2}$
along the spatial cycle of the $T^{2}$. The condition (2.35) for self-duality
in the parafermion theory, after projecting to sectors with definite
$\bar{\mathbb{Z}}_{k}$ charges, simply requires a symmetry between states of
charge $q$ and $s-q$ in the twisted sector $\mathcal{H}_{s}^{PF}$, which
follows from the ${\mathbb{Z}}_{2}^{S}$ symmetry in Table 8. Furthermore, the
${\mathbb{Z}}_{2}^{S}$ symmetry exchanges the order spin operators
$\sigma_{n}$ and disorder spin operators $\mu_{n}$, which is expected for the
${\mathbb{Z}}_{k}$ duality since the latter live in the ${\mathbb{Z}}_{k}^{B}$
twisted sector of the bosonic coset CFT. Finally the ${\mathbb{Z}}_{k}^{B}$
symmetric operators (i.e. the energy operators $\epsilon_{r}$) also carry
definite parity under the duality [92]. The duality defect is not a Verlinde
line in the $SU(2)_{k}/U(1)$ coset CFT since ${\mathbb{Z}}_{2}^{S}$ acts
nontrivially on the chiral algebra ${\mathcal{V}}_{k}$ (B.7). Under the
duality, all left-moving generators are invariant, whereas on the right-moving
side, the generators of spin $\ell$ transform by a sign $(-1)^{\ell}$. This
needs to be taken into account when writing down the duality twisted partition
functions of the coset CFT. A similar analysis as above applies to the
diagonal
${\mathbb{Z}}_{2}\subset{\mathbb{Z}}_{2}^{C}\times{\mathbb{Z}}_{2}^{S}$
symmetry which leads to the condition (2.37), corresponds to the second self-
duality in the coset CFT, and explains the second equality in (B.17). For this
latter self-duality, it is the left-moving chiral algebra generators that
transform with a sign $(-1)^{\ell}$.

###  B.4 The duality defects and ${\mathbb{Z}}_{k}$ Tambara-Yamagami
Symmetries

As explained in the last section, the ${\mathbb{Z}}_{2}^{S}$ symmetry
generator $S$ of the ${\mathbb{Z}}_{k}$ parafermions, upon bosonization, gives
rise to a ${\mathbb{Z}}_{k}$ duality defect $\mathcal{N}$. For $k>2$ the same
procedure also works for the other chiral symmetry $CS$ where $C$ generates
the ${\mathbb{Z}}_{2}^{C}$ in (B.24). Therefore we have in fact two such
duality defects $\mathcal{N}$ and $C\mathcal{N}$ in the $SU(2)_{k}/U(1)$ coset
CFT, which lead to the two equalities between the ${\mathbb{Z}}_{k}$ twisted
partition functions in (B.17) following the general discussion in Section 2.4.
Since $C$ acts as an automorphism on ${\mathbb{Z}}_{k}$, the two duality
defects have identical fusion rules and generate two ${\mathbb{Z}}_{k}$ TY
symmetries.

Such fusion categories are classified in [36]. For $k>2$, they are
parametrized by the bicharacter $e^{i\chi(a,b)}=\omega^{ab}$ where $\omega$ is
a primitive $k$-root of unity and the Frobenius-Schur indicator
$\epsilon=\pm$. Comparing the ${\mathbb{Z}}_{k}$ and $\bar{\mathbb{Z}}_{k}$
charges of the order and disorder spin operators given in Table 8 that are
related by the parafermionic symmetries $S$ and $CS$, we see that the
corresponding bicharacters for the duality defects $\mathcal{N}$ and
$C\mathcal{N}$ are $\chi_{+}$ and $\chi_{-}$ respectively (see Section 2.4).
Below we provide further evidence that the TY symmetries in the coset CFT
correspond to the two categories ${\rm TY}({\mathbb{Z}}_{k},\chi_{\pm},1)$
(see Table 4).

Firstly, the two duality TDLs $\mathcal{N}$ and $C\mathcal{N}$ must have the
identical FS indicator but conjugate bicharacters. The former follows since
$C$ has trivial F-symbols (no ’t Hooft anomaly) whereas the latter can be
derived from F-moves as in [14] (see Figure 13).393939We note that these
relations between the F-symbols of the underlying fusion categories do not
depend on specifics of the underlying CFT and it would be interesting to
understand more general relations of this type. Next, the spin spectrum in the
defect Hilbert space $\mathcal{H}_{\mathcal{N}}$ (or
$\mathcal{H}_{C\mathcal{N}}$) of the duality TDL is constrained by both $\chi$
and $\epsilon$. For $\chi=\chi_{\pm}$, the spin selection rules are given in
(3.45),

$\displaystyle e^{4\pi is}=\begin{cases}\epsilon e^{\pm{\pi i\over
4k}(2n+1)^{2}}e^{\mp\pi i/4}&k\in 2{\mathbb{Z}}\,,\\\ \epsilon e^{\pm{\pi
i\over k}n^{2}}e^{\mp\pi i/4}&k\in 2{\mathbb{Z}}+1\,.\end{cases}$ (B.26)

Therefore knowledge of the partition function for the defect Hilbert space
$\mathcal{H}_{\mathcal{N}}$ can be used to fix $\epsilon$. For $k=2,3$, by
inspecting the spin spectrum of $\mathcal{H}_{\mathcal{N}}$ in the Ising and
3-state Potts CFT, one finds that $\epsilon=1$ [12]. For $k=4$, we deduce that
$\epsilon=1$ from the bootstrap analysis in Section 3.3.4. We conjecture that
the duality defects for general $k$ in the $SU(2)_{k}/U(1)$ coset CFT have
$\epsilon=1$. This can be proven for example by working out the duality
twisted partition functions $Z_{1\mathcal{N}}$ and we leave that to the
interested readers.

$e^{i\chi_{\mathcal{N}}(g_{1},g_{2}^{-1})}$$e^{i\chi_{C\mathcal{N}}(g_{1},g_{2})}$
Figure 13: Two sequences of F-moves that go from the TDL configuration in the
leftmost figure to the top figure. The equivalence between them relates the
bicharacters associated to the two duality defects $\mathcal{N}$ and
$C\mathcal{N}$ by
$e^{i\chi_{C\mathcal{N}}(g_{1},g_{2})}=e^{i\chi_{\mathcal{N}}(g_{1},g_{2}^{-1})}=e^{-i\chi_{\mathcal{N}}(g_{1},g_{2})}$.

##  Appendix C $c=1$ CFT along the orbifold branch and $D_{8}$ discrete
gauging

At a generic point along the orbifold branch of the $c=1$ moduli space, the
CFT has $D_{8}$ global symmetry. Here we consider discrete gauging by
subgroups of $D_{8}$.

We adopt the standard representation $D_{8}=\langle
r,s|r^{4}=s^{2}=(sr)^{2}=1\rangle$, whose conjugacy classes and corresponding
stabilizers are

$\displaystyle\\{1\\}:$ $\displaystyle~{}D_{8}\,,$ (C.1)
$\displaystyle\\{r^{2}\\}:$ $\displaystyle~{}D_{8}\,,$
$\displaystyle\\{s,r^{2}s\\}:$
$\displaystyle~{}{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}=\\{1,r^{2},s,r^{2}s\\}\,,$
$\displaystyle\\{rs,r^{3}s\\}:$
$\displaystyle~{}{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}=\\{1,r^{2},rs,r^{3}s\\}\,,$
$\displaystyle\\{r,r^{3}\\}:$
$\displaystyle~{}{\mathbb{Z}}_{4}=\\{1,r,r^{2},r^{3}\\}\,.$

The distinct gaugings of the CFT will be labeled by subgroups $G\subset D_{8}$
up to conjugation, and possible discrete torsion classes from $H^{2}(G,U(1))$
which we will refer to as $\iota(G)\in{\mathbb{Z}}_{2}$. In particular, we
have 3 conjugacy classes of ${\mathbb{Z}}_{2}$ subgroups, 2 conjugacy classes
of ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ subgroups (which we will label as
$A$ and $B$ types) and 1 conjugacy class of ${\mathbb{Z}}_{4}$ subgroup. As we
will see, depending on these data, the discrete gauging may map one point on
the orbifold branch with partition function $Z_{\rm orb}(R_{1})$ to a point on
the unorbifolded branch $Z_{\rm circ}(R_{2})$ or another point on the orbifold
branch $Z_{\rm orb}(R_{2})$. Let us briefly summarize our findings below and
the details will be giving in the subsequent sections.

$\displaystyle Z_{\rm orb}(R)\xrightarrow{{\mathbb{Z}}_{2}^{r^{2}}}Z_{\rm
circ}(R)\,,$ (C.2) $\displaystyle Z_{\rm
orb}(R)\xrightarrow{{\mathbb{Z}}_{2}^{s}}Z_{\rm orb}(2R)\,,\quad Z_{\rm
orb}(R)\xrightarrow{{\mathbb{Z}}_{2}^{sr}}Z_{\rm orb}(R/2)\,,$ $\displaystyle
Z_{\rm
orb}(R)\xrightarrow[\iota=0]{({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2})_{A}}Z_{\rm
circ}(2R)\,,\quad Z_{\rm
orb}(R)\xrightarrow[\iota=0]{({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2})_{B}}Z_{\rm
orb}(R)\,,$ $\displaystyle Z_{\rm
orb}(R)\xrightarrow[\iota=1]{({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2})_{A}}Z_{\rm
orb}(R)\,,\quad Z_{\rm
orb}(R)\xrightarrow[\iota=1]{({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2})_{B}}Z_{\rm
circ}(2R)\,,$ $\displaystyle Z_{\rm
orb}(R)\xrightarrow{{\mathbb{Z}}_{4}}Z_{\rm orb}(R)\,,\quad$ $\displaystyle
Z_{\rm orb}(R)\xrightarrow[\iota=0]{D_{8}}Z_{\rm orb}(2R)\,,\quad Z_{\rm
orb}(R)\xrightarrow[\iota=1]{D_{8}}Z_{\rm orb}(R/2)\,.$

Note that T-duality acts on this set of transformation rules by swapping $A$
and $B$, $s$ and $sr$, $\iota=0$ and $\iota=1$.

###  C.1 The $D_{8}$ twisted torus partition functions

We label the torus partition function in the $r^{a}$ twisted sector with a
$sr^{b}$ symmetry line inserted along the spatial cycle as $Z_{a\bar{b}}$.
Similarly we define the twisted partition functions
$Z_{ab},Z_{\bar{a}\bar{b}},Z_{\bar{a}b}$.

Let ${\mathcal{Z}}^{{\alpha}{\beta}}_{{\gamma}{\delta}}$ denote twisted
partition function of the (unorbifolded) $c=1$ compact boson at radius $R$:

$\displaystyle{\mathcal{Z}}^{{\alpha}{\beta}}_{{\gamma}{\delta}}(\tau,R,\omega)={1\over|\eta|^{2}}\sum_{n,m\in{\mathbb{Z}}}\omega^{(2n+{\gamma})(2m+{\delta})}q^{{1\over
2}\left({n+{\alpha}/2\over R}+{(m+{\beta}/2)R\over
2}\right)^{2}}\bar{q}^{{1\over 2}\left({n+{\alpha}/2\over
R}-{(m+{\beta}/2)R\over 2}\right)^{2}}\,,$ (C.3)

with $\omega=e^{\pi i\over 2}$. Under $T$ and $S$ transformations, they
transform as

$\displaystyle{\mathcal{Z}}^{{\alpha}{\beta}}_{{\gamma}{\delta}}(\tau+1,R,\omega)=$
$\displaystyle\omega^{-{\alpha}{\delta}-{\beta}{\gamma}}{\mathcal{Z}}^{{\alpha}+{\gamma},{\beta}+{\delta}}_{{\gamma}{\delta}}(\tau,R,\omega)\,,$
(C.4)
$\displaystyle{\mathcal{Z}}^{{\alpha}{\beta}}_{{\gamma}{\delta}}(-1/\tau,R,\omega)=$
$\displaystyle\omega^{({\alpha}-{\gamma})({\beta}-{\delta})+2{\alpha}{\beta}}{\mathcal{Z}}^{{\gamma}{\delta}}_{{\alpha}{\beta}}(\tau,R,\omega)\,.$

We also recall [75]

$\displaystyle
W\equiv{1\over\eta}\sum_{r\in{\mathbb{Z}}}(-1)^{r}q^{r^{2}}\,,\quad
W_{\pm}\equiv{1\over\eta}\sum_{r\in{\mathbb{Z}}}(\pm 1)^{r}q^{\left(r+{1\over
4}\right)^{2}}\,.$ (C.5)

which transform as

$\displaystyle T:\begin{pmatrix}\sqrt{1\over 2}W(\tau+1)\\\ W_{+}(\tau+1)\\\
W_{-}(\tau+1)\end{pmatrix}=\begin{pmatrix}e^{-2\pi i/24}&0&0\\\ 0&e^{\pi
i/24}&0\\\ 0&0&e^{\pi i/24}\end{pmatrix}\begin{pmatrix}\sqrt{1\over
2}W(\tau)\\\ W_{-}(\tau)\\\ W_{+}(\tau)\end{pmatrix}\,,$ (C.6) $\displaystyle
S:\begin{pmatrix}\sqrt{1\over 2}W(-1/\tau)\\\ W_{+}(-1/\tau)\\\
W_{-}(-1/\tau)\end{pmatrix}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}\sqrt{1\over 2}W(\tau)\\\ W_{+}(\tau)\\\
W_{-}(\tau)\end{pmatrix}\,.$

The explicit forms of these twisted torus partition functions of the
$\mathbb{Z}_{2}$ orbifold are

$\displaystyle Z_{0\bar{0}}=$ $\displaystyle{1\over
2}{\mathcal{Z}}_{00}^{10}+{1\over 2}|W|^{2}\,,~{}Z_{\bar{0}0}={1\over
2}{\mathcal{Z}}_{10}^{00}+|W_{+}|^{2}\,,~{}Z_{\bar{0}\bar{0}}={1\over
2}{\mathcal{Z}}_{10}^{10}+|W_{-}|^{2}\,,$ (C.7) $\displaystyle Z_{2\bar{0}}=$
$\displaystyle{1\over 2}{\mathcal{Z}}_{00}^{10}-{1\over
2}|W|^{2}\,,~{}Z_{\bar{0}2}={1\over
2}{\mathcal{Z}}_{10}^{00}-|W_{+}|^{2}\,,~{}Z_{\bar{0}\bar{2}}={1\over
2}{\mathcal{Z}}_{10}^{10}-|W_{-}|^{2}\,,$ $\displaystyle Z_{0\bar{1}}=$
$\displaystyle{1\over 2}{\mathcal{Z}}_{00}^{01}+{1\over
2}|W|^{2}\,,~{}Z_{\bar{1}0}={1\over
2}{\mathcal{Z}}_{01}^{00}+|W_{+}|^{2}\,,~{}Z_{2\bar{1}}=-{1\over
2}{\mathcal{Z}}_{00}^{01}+{1\over
2}|W|^{2}\,,~{}Z_{\bar{1}2}=Z_{\bar{3}2}=-{1\over
2}{\mathcal{Z}}_{01}^{00}+|W_{+}|^{2}\,,$ $\displaystyle Z_{\bar{1}\bar{1}}=$
$\displaystyle{1\over
2}{\mathcal{Z}}_{01}^{01}+|W_{-}|^{2}\,,~{}Z_{\bar{3}\bar{1}}=-{1\over
2}{\mathcal{Z}}_{01}^{01}+|W_{-}|^{2}\,,~{}$ $\displaystyle Z_{00}=$
$\displaystyle{1\over 2}{\mathcal{Z}}_{00}^{00}+{1\over
2}|W|^{2}+|W_{+}|^{2}+|W_{-}|^{2}\,,~{}Z_{20}={1\over
2}{\mathcal{Z}}_{00}^{00}-{1\over 2}|W|^{2}+|W_{+}|^{2}-|W_{-}|^{2}\,,~{}$
$\displaystyle Z_{02}=$ $\displaystyle{1\over
2}{\mathcal{Z}}_{00}^{00}+{1\over
2}|W|^{2}-|W_{+}|^{2}-|W_{-}|^{2}\,,~{}Z_{22}={1\over
2}{\mathcal{Z}}_{00}^{00}-{1\over 2}|W|^{2}-|W_{+}|^{2}+|W_{-}|^{2}\,,~{}$
$\displaystyle{Z_{10}}=$ $\displaystyle{Z_{30}={1\over
2}{\mathcal{Z}}_{11}^{00}+|W_{+}|^{2}}\,,~{}Z_{01}=Z_{03}={1\over
2}\omega^{-1}{\mathcal{Z}}_{00}^{11}+{1\over 2}|W|^{2}\,,~{}$ $\displaystyle
Z_{12}=$ $\displaystyle-{1\over
2}{\mathcal{Z}}_{11}^{00}+|W_{+}|^{2}\,,~{}{Z_{21}}={1\over
2}\omega{\mathcal{Z}}_{00}^{11}+{1\over 2}|W|^{2}\,,~{}$ $\displaystyle
Z_{13}=$ $\displaystyle-{1\over
2}{\mathcal{Z}}_{11}^{11}+|W_{-}|^{2}\,,~{}Z_{23}={1\over
2}\omega{\mathcal{Z}}_{00}^{11}+{1\over 2}|W|^{2}\,,~{}$
$\displaystyle{Z_{31}}=$ $\displaystyle{-{1\over
2}{\mathcal{Z}}_{11}^{11}+|W_{-}|^{2}}\,,~{}Z_{32}=-{1\over
2}{\mathcal{Z}}_{11}^{00}+|W_{+}|^{2}\,,~{}$ $\displaystyle{Z_{11}=}$
$\displaystyle{{1\over
2}{\mathcal{Z}}_{11}^{11}+|W_{-}|^{2}}\,,~{}Z_{33}={1\over
2}{\mathcal{Z}}_{11}^{11}+|W_{-}|^{2}\,.$

Note the following identities since $Z(g,h)=Z(cgc^{-1},chc^{-1})$,

$\displaystyle
Z_{10}=Z_{30}\,,~{}Z_{\bar{1}0}=Z_{\bar{3}0}\,,~{}Z_{\bar{0}0}=Z_{\bar{2}0}\,,~{}Z_{\bar{0}2}=Z_{\bar{2}2}\,,~{}Z_{\bar{\bar{1}}2}=Z_{\bar{3}2}\,,~{}$
(C.8) $\displaystyle
Z_{\bar{2}\bar{2}}=Z_{\bar{0}\bar{0}}\,,~{}Z_{\bar{0}\bar{2}}=Z_{\bar{2}\bar{0}}$

and that under S-transform

$\displaystyle Z_{11}(-1/\tau,-1/\bar{\tau})=Z_{31}(\tau,\bar{\tau})\,,$ (C.9)

since the TDLs are oriented.

###  C.2 ${\mathbb{Z}}_{2}$ gauging

The ${\mathbb{Z}}_{2}$ subgroups of $D_{8}$ fall into 3 conjugate classes
whose representatives are generated by $r^{2}$ (center ${\mathbb{Z}}_{2}$),
$s$ or $rs$.

The center ${\mathbb{Z}}_{2}$ orbifold returns the original partition function
between ${\mathbb{Z}}_{2}$ orbifold

$\displaystyle Z^{r^{2}}_{{\mathbb{Z}}_{2}}(R)={1\over
2}(Z_{00}+Z_{02}+Z_{20}+Z_{22})=Z^{00}_{00}(R)=Z_{\rm circ}(R)\,.$ (C.10)

The reflection ${\mathbb{Z}}_{2}$ orbifold preserves the orbifold branch but
increases or decreases $R$ by 2 respectively

$\displaystyle Z^{s}_{{\mathbb{Z}}_{2}}(R)=$ $\displaystyle{1\over
2}(Z_{00}+Z_{0\bar{0}}+Z_{\bar{0}0}+Z_{\bar{0}\bar{0}})$ (C.11)
$\displaystyle=$ $\displaystyle{1\over
4}({\mathcal{Z}}_{00}^{00}+Z^{10}_{00}+Z^{00}_{10}+Z^{10}_{10})+{1\over
2}|W|^{2}+|W_{+}|^{2}+|W_{-}|^{2}=Z(2R)\,,$ $\displaystyle
Z^{rs}_{{\mathbb{Z}}_{2}}(R)=$ $\displaystyle{1\over
2}(Z_{00}+Z_{0\bar{1}}+Z_{\bar{1}0}+Z_{\bar{1}\bar{1}})$ $\displaystyle=$
$\displaystyle{1\over
4}({\mathcal{Z}}_{00}^{00}+Z^{01}_{00}+Z^{00}_{01}+Z^{01}_{01})+{1\over
2}|W|^{2}+|W_{+}|^{2}+|W_{-}|^{2}=Z(R/2)\,.$

Here we have used

$\displaystyle{\mathcal{Z}}_{00}^{00}+{\mathcal{Z}}^{01}_{00}+{\mathcal{Z}}^{00}_{01}+{\mathcal{Z}}^{01}_{01}$
(C.12) $\displaystyle=$
$\displaystyle{1\over|\eta|^{2}}\sum_{n,m\in{\mathbb{Z}}}(1+(-)^{n})q^{{1\over
2}\left({n\over R}+{mR\over 2}\right)^{2}}\bar{q}^{{1\over 2}\left({n\over
R}-{mR\over 2}\right)^{2}}+(1+(-)^{n})q^{{1\over 2}\left({n\over
R}+{(m+1/2)R\over 2}\right)^{2}}\bar{q}^{{1\over 2}\left({n\over
R}-{(m+1/2)R\over 2}\right)^{2}}\,,$ $\displaystyle=$
$\displaystyle{2\over|\eta|^{2}}\sum_{n,m\in{\mathbb{Z}}}q^{{1\over
2}\left({2n\over R}+{mR\over 4}\right)^{2}}\bar{q}^{{1\over 2}\left({2n\over
R}-{mR\over 4}\right)^{2}}\,.$

###  C.3 ${\mathbb{Z}}_{4}$ gauging

The orbifold branch is point-wise invariant under ${\mathbb{Z}}_{4}$ gauging

$\displaystyle Z_{{\mathbb{Z}}_{4}}(R)={1\over
4}\sum_{a,b=0}^{3}Z_{ab}={1\over
4}(2Z^{00}_{00}+2|W|^{2}+4|W_{+}|^{2}+4|W_{+}|^{2})=Z_{00}(R)=Z(R)$ (C.13)

###  C.4 ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ gauging

There are two ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ subgroups of $D_{8}$ up
to conjugation. We will refer to them as

$\displaystyle({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2})^{A}=$
$\displaystyle\\{1,r^{2},s,r^{2}s\\}\,,$ (C.14)
$\displaystyle({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2})^{B}=$
$\displaystyle\\{1,r^{2},rs,r^{3}s\\}\,.$

Furthermore
$H^{2}({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2},U(1))={\mathbb{Z}}_{2}$, so we
may couple a nontrivial SPT when gauging.

#### Trivial SPT

In one case the orbifold branch is mapped to the unorbifolded branch with $R$
doubled

$\displaystyle Z^{A}_{{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}}=$
$\displaystyle{1\over
2}(Z_{0\bar{0}}+Z_{2\bar{0}}+Z_{\bar{0}0}+Z_{\bar{0}2}+Z_{\bar{0}\bar{0}}+Z_{\bar{0}\bar{2}})+{1\over
4}(Z_{00}+Z_{02}+Z_{20}+Z_{22})\,,$ (C.15) $\displaystyle=$
$\displaystyle{1\over
2}({\mathcal{Z}}_{00}^{00}+{\mathcal{Z}}_{10}^{00}+{\mathcal{Z}}_{00}^{10}+{\mathcal{Z}}_{10}^{10})\,,$
$\displaystyle=$ $\displaystyle Z^{00}_{00}(2R)=Z_{\rm circ}(2R)\,,$

while in the other case the orbifold branch is invariant

$\displaystyle Z^{B}_{{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}}(R)=$
$\displaystyle{1\over
2}(Z_{0\bar{1}}+Z_{2\bar{1}}+Z_{\bar{1}0}+Z_{\bar{1}2}+Z_{\bar{1}\bar{1}}+Z_{\bar{1}\bar{3}})+{1\over
4}(Z_{00}+Z_{02}+Z_{20}+Z_{22})\,,$ (C.16) $\displaystyle=$
$\displaystyle{1\over
2}({\mathcal{Z}}_{00}^{00}+{\mathcal{Z}}_{10}^{00}+{\mathcal{Z}}_{00}^{10}+{\mathcal{Z}}_{10}^{10})\,,$
$\displaystyle=$ $\displaystyle{1\over 2}Z^{00}_{00}+{1\over
2}|W|^{2}+|W_{+}|^{2}+|W_{-}|^{2}=Z(R)\,.$

#### Nontrivial SPT

In one case, the orbifold branch is point-wise invariant

$\displaystyle Z^{A\star}_{{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}}(R)=$
$\displaystyle{1\over
2}(Z_{0\bar{0}}-Z_{2\bar{0}}+Z_{\bar{0}0}-Z_{\bar{0}2}+Z_{\bar{0}\bar{0}}-Z_{\bar{0}\bar{2}})+{1\over
4}(Z_{00}+Z_{02}+Z_{20}+Z_{22})\,,$ (C.17) $\displaystyle=$
$\displaystyle{1\over
2}({\mathcal{Z}}_{00}^{00}+{\mathcal{Z}}_{10}^{00}+{\mathcal{Z}}_{00}^{10}+{\mathcal{Z}}_{10}^{10})-Z_{2\bar{0}}-Z_{\bar{0}2}-Z_{\bar{0}\bar{2}}\,,$
$\displaystyle=$ $\displaystyle
Z^{00}_{00}(2R)-Z_{2\bar{0}}-Z_{\bar{0}2}-Z_{\bar{0}\bar{2}}\,,$
$\displaystyle=$ $\displaystyle{1\over
2}({\mathcal{Z}}_{00}^{00}+|W|^{2}+|W_{+}|^{2}+|W_{-}|^{2})=Z(R)\,,$

and in the other case, the orbifold branch is mapped to the unorbifolded
branch with $R$ doubled

$\displaystyle Z^{B\star}_{{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}}(R)=$
$\displaystyle{1\over
2}(Z_{0\bar{1}}-Z_{2\bar{1}}+Z_{\bar{1}0}-Z_{\bar{1}2}+Z_{\bar{1}\bar{1}}-Z_{\bar{1}\bar{3}})+{1\over
4}(Z_{00}+Z_{02}+Z_{20}+Z_{22})\,,$ (C.18) $\displaystyle=$
$\displaystyle{1\over
2}({\mathcal{Z}}_{00}^{00}+{\mathcal{Z}}_{10}^{00}+{\mathcal{Z}}_{00}^{10}+{\mathcal{Z}}_{10}^{10})=Z^{00}_{00}(2R)=Z_{\rm
circ}(2R)\,.$

###  C.5 $D_{8}$ gauging

Recall that the nonabelian $G$ orbifold partition function involves sum over
the conjugacy classes of $G$ and in each $g$ twisted sector a insertion of
projector consisting of stablizers of $g$. Since
$H^{2}(D_{8},U(1))={\mathbb{Z}}_{2}$, we may consider coupling to a nontrivial
SPT (discrete torsion) when gauging.

#### Trivial SPT

$\displaystyle Z_{D_{8}}=$ $\displaystyle{1\over
8}(\sum_{a=0}^{4}Z_{0a}+\sum_{\bar{a}=0}^{4}Z_{0\bar{a}}+\sum_{a=0}^{4}Z_{2a}+\sum_{\bar{a}=0}^{4}Z_{2\bar{a}})+{1\over
4}(Z_{\bar{0}0}+Z_{\bar{0}2}+Z_{\bar{0}\bar{0}}+Z_{\bar{0}\bar{2}})$ (C.19)
$\displaystyle+$ $\displaystyle{1\over
4}(Z_{\bar{1}0}+Z_{\bar{1}2}+Z_{\bar{1}\bar{1}}+Z_{\bar{1}\bar{3}})+{1\over
4}\sum_{a=0}^{4}Z_{1a}\,,$ $\displaystyle=$ $\displaystyle{1\over
2}Z_{{\mathbb{Z}}_{4}}+{1\over
8}(\sum_{\bar{a}=0}^{4}Z_{0\bar{a}}+\sum_{\bar{a}=0}^{4}Z_{2\bar{a}})+{1\over
4}(Z_{\bar{0}0}+Z_{\bar{0}2}+Z_{\bar{0}\bar{0}}+Z_{\bar{0}\bar{2}})+{1\over
4}(Z_{\bar{1}0}+Z_{\bar{1}2}+Z_{\bar{1}\bar{1}}+Z_{\bar{1}\bar{3}})\,.$

We can rewrite the above as a combination of partition functions from gauging
subgroups of $D_{8}$,

$\displaystyle Z_{D_{8}}=$ $\displaystyle{1\over
2}Z_{{\mathbb{Z}}_{4}}+{1\over
2}Z^{A}_{{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}}+{1\over
2}Z^{B}_{{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}}-{1\over
2}Z^{r^{2}}_{{\mathbb{Z}}_{2}}\,.$ (C.20)

Hence

$\displaystyle Z_{D_{8}}(R)=Z(R)+{1\over
2}({\mathcal{Z}}_{00}^{00}({2R})-{\mathcal{Z}}_{00}^{00}({R}))=Z(2R)\,.$
(C.21)

Namely, the $D_{8}$ gauging preserves the orbifold branch but doubles $R$.

#### Nontrivial SPT

The nontrivial SPT associated with the $D_{8}$ discrete torsion has nontrivial
torus partition functions

$\displaystyle Z^{\rm SPT}_{\bar{a}b}=i^{b}\,,~{}~{}Z^{\rm
SPT}_{a\bar{b}}=i^{-a}\,,~{}~{}Z^{\rm SPT}_{\bar{a}\bar{b}}=i^{b-a}\,.$ (C.22)

Consequently

$\displaystyle Z^{\star}_{D_{8}}=$ $\displaystyle{1\over
2}Z_{{\mathbb{Z}}_{4}}+{1\over
4}(Z_{0\bar{0}}-Z_{2\bar{0}}+Z_{0\bar{1}}-Z_{2\bar{1}})+{1\over
4}(Z_{\bar{0}0}-Z_{\bar{0}2}+Z_{\bar{0}\bar{0}}-Z_{\bar{0}\bar{2}})+{1\over
4}(Z_{\bar{1}0}-Z_{\bar{1}2}+Z_{\bar{1}\bar{1}}-Z_{\bar{1}\bar{3}})$ (C.23)
$\displaystyle=$ $\displaystyle Z_{D_{8}}-{1\over
2}(Z_{2\bar{0}}+Z_{2\bar{1}}+Z_{\bar{0}2}+Z_{\bar{1}2}+Z_{\bar{1}\bar{3}}+Z_{\bar{0}\bar{2}})\,,$
$\displaystyle=$ $\displaystyle{1\over
2}|W|^{2}+|W_{+}|^{2}+|W_{-}|^{2}+Z^{00}_{00}(R/2)=Z(R/2)\,.$

Thus the $D_{8}$ gauging with discrete torsion preserves the orbifold branch
but halves $R$.

##  Appendix D Some Lattice Defects

###  D.1 Ising Duality Defect

Let us use our recipe from Section 2.4 to define the Ising duality defect. We
start with the Hamiltonian

$H=-\sum_{j}Z_{j}Z_{j+1}+X_{j}\,,$ (D.1)

($X,Z$ etc are Pauli matrices) and apply the $R_{x}=\prod_{j}X_{j}$ gauging
procedure

$X_{j}\mapsto Z_{j}Z_{j+1}\,,$ (D.2)

and

$Z_{j}\mapsto\prod_{0\leq k\leq j}X_{k}\,,$ (D.3)

to all $j\geq 0$. The Hamiltonian is invariant away from the resulting defect
at $j=0$, but near it we have

$X_{-1}\mapsto X_{-1}\,,$ (D.4) $Z_{-1}Z_{0}\mapsto Z_{-1}X_{0}\,,$ (D.5)
$X_{0}\mapsto Z_{0}Z_{1}\,,$ (D.6) $Z_{0}Z_{1}\mapsto X_{1}\,,$ (D.7)

so we obtain

$H^{\prime}=-\left(\sum_{j\leq-1}Z_{j-1}Z_{j}+X_{j}\right)-Z_{-1}X_{0}-\left(\sum_{j\geq
1}Z_{j-1}Z_{j}+X_{j}\right)\,,$ (D.8)

in agreement with [38, 9].

We see that $H^{\prime}$ enjoys two anti-commuting $\mathbb{Z}_{2}$
symmetries,

$U_{L}=\left(\prod_{j\leq-1}X_{j}\right)Z_{0}\,,$ (D.9)

and

$U_{R}=\left(\prod_{j\geq 0}X_{j}\right)\,,$ (D.10)

which we recognize as the left and right $\mathbb{Z}_{2}$ action in the
duality twisted sector, with the anti-commutation reflecting the nontrivial TY
F-symbol.

###  D.2 Continuum of topological defects in Ising2

We begin with the XY chain

$H=-\sum_{j}X_{j}X_{j+1}+Y_{j}Y_{j+1}\,.$ (D.11)

This can be related to the Ising2 chain by gauging $R_{x}=\prod_{j}X_{j}$. In
terms of the operator algebra, this amounts to the transformation

$X_{j}\mapsto Z_{j}Z_{j+1}\,,$ (D.12)

and

$Z_{j}\mapsto\prod_{k\leq j}X_{k}\,,$ (D.13)

from which we obtain

$H^{\prime}=-\sum_{j}Z_{j}Z_{j+2}+Z_{j}X_{j+1}Z_{j+2}\,.$ (D.14)

This is related to a more familiar Hamiltonian $H^{\prime\prime}$ after
applying the cluster entangler transformation

$X_{j}\mapsto Z_{j-1}X_{j}Z_{j+1}\,,$ (D.15)
$H^{\prime\prime}=-\sum_{j}Z_{j}Z_{j+2}+X_{j}\,,$ (D.16)

for which the even and odd sites define two decoupled critical Ising chains.

To construct the symmetry defect for a rotation $U(\theta)=\prod_{j}e^{i\theta
Z_{j}/2}$, we apply this symmetry to sites $j>0$. The bond terms are unchanged
except for the 01 bond, which becomes

$-\cos\theta(X_{0}X_{1}+Y_{0}Y_{1})-\sin\theta(X_{0}Y_{1}-Y_{0}X_{1})\,.$
(D.17)

For $\theta\neq 0,\pi$, this defect is not invariant under the symmetry
$R_{x}$, indeed $R_{x}U(\theta)R_{x}=U(-\theta)$. As in Section 2.5, we want
to replace it with a sum of defects labelled by $\theta$ and $-\theta$. We do
this by introducing an extra qubit at $j=1/2$, and writing the modified bond

$-\cos\theta(X_{0}X_{1}+Y_{0}Y_{1})-\sin\theta(X_{0}Y_{1/2}Y_{1}-Y_{0}Y_{1/2}X_{1})\,,$
(D.18)

with the new symmetry $R_{x}=X_{1/2}\prod_{j}X_{j}$.

We gauge this symmetry as above and obtain

$-\cos\theta(Z_{0}Z_{1/2}Z_{1}Z_{2}+Z_{0}Y_{1/2}Z_{1}Y_{2})-\sin\theta(Z_{0}X_{1}Z_{2}+Z_{0}X_{1/2}Z_{2})\,.$
(D.19)

If we then apply the cluster entangler, it becomes

$-\cos\theta(Z_{0}Z_{1/2}Z_{1}Z_{2}+Y_{1/2}Z_{1}Y_{2}Z_{3})-\sin\theta(Z_{0}Z_{1/2}X_{1}+X_{1/2}Z_{1}Z_{2})\,.$
(D.20)

##  Appendix E Modular $S$ matrices for nonabelian orbifolds

The modular S-matrices for nonabelian orbifolds of the $SU(2)_{1}$ CFT were
derived for $G=D_{2n}$ in [74] and for $G=A_{4},S_{4},A_{5}$ in [82]. Here for
completeness we record their results after correcting a number of typos. As
explained in Section 2.2, they determine the Verlinde lines in these orbifold
theories.

The $SU(2)_{1}/D_{2n}$ CFT contains $n^{2}+7$ chiral primaries with respect to
the ${\mathcal{V}}_{D_{2n}}$ chiral algebra denoted as

$\displaystyle\\{u_{\pm},~{}\phi_{\pm},~{}\chi_{a},~{}\sigma_{r},~{}\tau_{r}\\}$
(E.1)

with $1\leq a\leq n^{2}-1$ and $r=1,2$. The vacuum representation corresponds
to $u_{+}$. The modular $S$ matrix takes slightly different forms for $n$ even
versus for $n$ odd and are given in Table 9 and Table 10 respectively. Here
$\sigma_{rs}=2{\delta}_{rs}-1$.

| $u_{\pm}$ | $\phi_{\pm}$ | $\chi_{b}$ | $\sigma_{s}$ | $\tau_{s}$
---|---|---|---|---|---
$u_{\pm}$ | 1 | 1 | 2 | $\pm n$ | $\pm n$
$\phi_{\pm}$ | 1 | 1 | $2(-1)^{b}$ | $\sigma_{rs}n$ | $\sigma_{rs}n$
$\chi_{a}$ | 2 | $2(-1)^{a}$ | $4\cos{\pi ab\over n}$ | 0 | 0
$\sigma_{r}$ | $\pm n$ | $\sigma_{rs}n$ | 0 | ${\delta}_{rs}n\sqrt{2}$ | $-{\delta}_{rs}n\sqrt{2}$
$\tau_{r}$ | $\pm n$ | $\sigma_{rs}n$ | 0 | $-{\delta}_{rs}n\sqrt{2}$ | ${\delta}_{rs}n\sqrt{2}$
Table 9: The modular $S$ matrix for ${\mathcal{V}}_{D_{2n}}$ multiplied by $2\sqrt{2n}$ for $n\in 2{\mathbb{Z}}$. | $u_{\pm}$ | $\phi_{j}$ | $\chi_{b}$ | $\sigma_{s}$ | $\tau_{s}$
---|---|---|---|---|---
$u_{\pm}$ | 1 | 1 | 2 | $\pm n$ | $\pm n$
$\phi_{\pm}$ | 1 | 1 | $2(-1)^{b}$ | $i\sigma_{rs}n$ | $i\sigma_{rs}n$
$\chi_{a}$ | 2 | $2(-1)^{a}$ | $4\cos{\pi ab\over n}$ | 0 | 0
$\sigma_{r}$ | $\pm n$ | $i\sigma_{rs}n$ | 0 | $e^{{\pi i\over 4}\sigma_{rs}}n$ | $-e^{{\pi i\over 4}\sigma_{rs}}n$
$\tau_{r}$ | $\pm n$ | $i\sigma_{rs}n$ | 0 | $-e^{{\pi i\over 4}\sigma_{rs}}n$ | $e^{{\pi i\over 4}\sigma_{rs}}n$
Table 10: The modular $S$ matrix for ${\mathcal{V}}_{D_{2n}}$ multiplied by
$2\sqrt{2n}$ for $n\in 2{\mathbb{Z}}+1$.

The $SU(2)_{1}/A_{4}$ CFT contains $21$ chiral primaries with respect to the
${\mathcal{V}}_{A_{4}}$ chiral algebra which are denoted as

$\displaystyle\\{u_{i}\,,~{}j\,,~{}\phi_{i}\,,~{}\sigma\,,~{}\tau\,,~{}\omega_{i}^{\pm}\,,~{}\theta_{i}^{\pm}\\}$
(E.2)

with $i=0,1,2$. The vacuum representation corresponds to $u_{0}$. The modular
$S$ matrix is given in Table 11. For convenience, we introduce the following
parameters given by various roots of unity,

$\displaystyle\omega=e^{2\pi i\over 3}\,,~{}{\alpha}=e^{\pi i\over
9}\,,~{}{\beta}=e^{\pi i\over 16}\,,~{}\zeta=e^{2\pi i\over 5}\,.$ (E.3)

| $u_{j}$ | $j$ | $\phi_{j}$ | $\sigma$ | $\tau$ | $\omega_{j}^{+}$ | $\omega_{j}^{-}$ | $\theta_{j}^{+}$ | $\theta_{j}^{-}$
---|---|---|---|---|---|---|---|---|---
$u_{j}$ | 1 | 3 | 2 | 6 | 6 | $4\omega^{i}$ | $4\bar{\omega}^{i}$ | $4\omega^{i}$ | $4\bar{\omega}^{i}$
$j$ | 3 | 9 | 6 | $-6$ | $-6$ | 0 | 0 | 0 | 0
$\phi_{j}$ | 2 | 6 | $-4$ | 0 | 0 | $-4\omega^{i}$ | $-4\bar{\omega}^{i}$ | $4\omega^{i}$ | $4\bar{\omega}^{i}$
$\sigma$ | 6 | $-6$ | 0 | $6\sqrt{2}$ | $-6\sqrt{2}$ | 0 | 0 | 0 | 0
$\tau$ | 6 | $-6$ | 0 | $-6\sqrt{2}$ | $6\sqrt{2}$ | 0 | 0 | 0 | 0
$\omega_{i}^{+}$ | $4\omega^{j}$ | 0 | $-4\omega^{j}$ | 0 | 0 | $4\bar{\alpha}^{4}\bar{\omega}^{i+j}$ | $4{\alpha}^{4}\omega^{i+j}$ | $4{\alpha}^{2}\omega^{2i+j}$ | $4\bar{\alpha}^{2}\bar{\omega}^{2i+j}$
$\omega_{i}^{-}$ | $4\bar{\omega}^{j}$ | 0 | $-4\bar{\omega}^{j}$ | 0 | 0 | $4{\alpha}^{4}\omega^{i+j}$ | $4\bar{\alpha}^{4}\bar{\omega}^{i+j}$ | $4\bar{\alpha}^{2}\bar{\omega}^{2i+j}$ | $4{\alpha}^{2}\omega^{2i+j}$
$\theta_{i}^{+}$ | $4\omega^{j}$ | 0 | $4\omega^{j}$ | 0 | 0 | $4{\alpha}^{2}\omega^{i+2j}$ | $4\bar{\alpha}^{2}\bar{\omega}^{i+2j}$ | $4\bar{\alpha}\omega^{i+j}$ | $4{\alpha}\bar{\omega}^{i+j}$
$\theta_{i}^{-}$ | $4\bar{\omega}^{j}$ | 0 | $4\bar{\omega}^{j}$ | 0 | 0 | $4\bar{\alpha}^{2}\bar{\omega}^{i+2j}$ | $4{\alpha}^{2}\omega^{i+2j}$ | $4{\alpha}\bar{\omega}^{i+j}$ | $4\bar{\alpha}\omega^{i+j}$
Table 11: The modular $S$ matrix for ${\mathcal{V}}_{A_{4}}$ multiplied by
$12\sqrt{2}$.

The $SU(2)_{1}/S_{4}$ CFT contains $28$ chiral primaries with respect to the
${\mathcal{V}}_{S_{4}}$ chiral algebra which are denoted as

$\displaystyle\\{u_{\pm},~{}u_{f}\,,~{}j_{\pm}\,,~{}\phi_{\pm}\,,~{}\phi_{f}\,,~{}\mu_{s}\,,~{}\sigma_{\pm}\,,~{}\tau_{\pm}\,,~{}\omega_{i}\,,~{}\theta_{i}\,,~{}{\alpha}_{k}\,,~{}{\beta}_{k}\\}$
(E.4)

with $s=1,2$, $i=0,1,2$ and $k=0,1,2,3$. The vacuum representation corresponds
to $u_{+}$. The modular $S$ matrix is given in Table 12 with the parameters
$\omega,{\alpha},{\beta}$ as in (E.3), and in addition

$\displaystyle c_{k}=(-1)^{k}12{\rm Re\,}{\beta}^{2}\,,~{}s_{k}=(-1)^{k}12{\rm
Im\,}{\beta}^{2}\,,~{}$ (E.5) $\displaystyle a_{ij}=16{\rm
Re\,}\bar{\alpha}^{4}\bar{\omega}^{i+j}\,,~{}b_{ij}=16{\rm
Re\,}{\alpha}^{2}\omega^{2i+j}\,,~{}d_{ij}=16{\rm
Re\,}\bar{\alpha}\omega^{i+j}\,,~{}$ $\displaystyle q_{kl}=12{\rm
Re\,}\bar{\beta}i^{k+l}\,,~{}r_{kl}=12{\rm
Re\,}\bar{\beta}^{3}i^{l-k}\,,~{}s_{kl}=12{\rm
Re\,}\bar{\beta}^{9}i^{-l-k}\,.$

| $u_{\pm}$ | $u_{f}$ | $j_{\pm}$ | $\phi_{\pm}$ | $\phi_{f}$ | $\mu_{s}$ | $\sigma_{\pm}$ | $\tau_{\pm}$ | $\omega_{j}$ | $\theta_{j}$ | ${\alpha}_{l}$ | ${\beta}_{l}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$u_{\pm}$ | 1 | 2 | 3 | 2 | 4 | $\pm 12$ | 6 | 6 | 8 | 8 | $\pm 6$ | $\pm 6$
$u_{f}$ | 2 | 4 | 6 | 4 | 8 | $0$ | 12 | 12 | $-8$ | $-8$ | 0 | 0
$j_{\pm}$ | 3 | 6 | 9 | 6 | 12 | $\pm 12$ | $-6$ | $-6$ | 0 | 0 | $\mp 6$ | $\mp 6$
$\phi_{\pm}$ | 2 | 4 | 6 | $-4$ | $-8$ | $0$ | 0 | 0 | $-8$ | 8 | $\pm 6\sqrt{2}$ | $\mp 6\sqrt{2}$
$\phi_{f}$ | 4 | 8 | 12 | $-8$ | $-16$ | $0$ | 0 | 0 | 8 | $-8$ | 0 | 0
$\mu_{r}$ | $\pm 12$ | 0 | $\pm 12$ | 0 | 0 | $12\sqrt{2}(-1)^{r+s}$ | 0 | 0 | 0 | 0 | 0 | 0
$\sigma_{\pm}$ | 6 | 12 | $-6$ | 0 | 0 | $0$ | $6\sqrt{2}$ | $-6\sqrt{2}$ | 0 | 0 | $\pm c_{l}$ | $\pm s_{l}$
$\tau_{\pm}$ | 6 | 12 | $-6$ | 0 | 0 | $0$ | $-6\sqrt{2}$ | $6\sqrt{2}$ | 0 | 0 | $\pm s_{l}$ | $\mp c_{l}$
$\omega_{i}$ | 8 | $-8$ | 0 | $-8$ | $8$ | 0 | 0 | 0 | $a_{ij}$ | $b_{ij}$ | 0 | 0
$\theta_{i}$ | 8 | $-8$ | 0 | $8$ | $-8$ | 0 | 0 | 0 | $b_{ji}$ | $d_{ij}$ | 0 | 0
${\alpha}_{k}$ | $\pm 6$ | 0 | $\mp 6$ | $\pm 6\sqrt{2}$ | $0$ | 0 | $\pm c_{k}$ | $\pm s_{k}$ | 0 | 0 | $q_{kl}$ | $r_{kl}$
${\beta}_{k}$ | $\pm 6$ | 0 | $\mp 6$ | $\mp 6\sqrt{2}$ | $0$ | 0 | $\pm s_{k}$ | $\mp c_{k}$ | 0 | 0 | $r_{lk}$ | $s_{kl}$

Table 12: The modular $S$ matrix for ${\mathcal{V}}_{S_{4}}$ multiplied by
$24\sqrt{2}$.

The $SU(2)_{1}/A_{5}$ CFT contains $37$ chiral primaries with respect to the
${\mathcal{V}}_{A_{5}}$ chiral algebra which are denoted as

$\displaystyle\\{u_{m}\,,~{}\phi_{k}\,,~{}\sigma\,,~{}\tau\,,~{}\omega_{i}\,,~{}\theta_{i}\,,~{}\pi_{m}\,,~{}\rho_{m}\,,~{}\lambda_{m}\,,~{}\xi_{m}\\}$
(E.6)

with $m=0,1,2,3,4$, $i=0,1,2$ and $k=0,1,2,3$. The vacuum representation
corresponds to $u_{0}$. The modular $S$ matrix is given in Table 12 with
$g_{\pm}=6(1\pm\sqrt{5})$ and

$\displaystyle P_{mn}^{1}={\rm Re\,}e^{-{4\pi i\over
25}}\zeta^{2(n+m)}\,,~{}P_{mn}^{2}={\rm Re\,}e^{-{2\pi i\over
25}}\zeta^{2n+m}\,,~{}P_{mn}^{3}={\rm Re\,}e^{-{8\pi i\over
25}}\zeta^{2n-m}\,,~{}P_{mn}^{4}={\rm Re\,}e^{-{6\pi i\over
25}}\zeta^{2(n-m)}\,,~{}$ (E.7) $\displaystyle R_{mn}^{1}={\rm Re\,}e^{-{\pi
i\over 25}}\zeta^{n+m}\,,~{}R_{mn}^{2}={\rm Re\,}e^{-{4\pi i\over
25}}\zeta^{n-m}\,,~{}R_{mn}^{3}={\rm Re\,}e^{-{3\pi i\over
25}}\zeta^{n-2m}\,,~{}L_{mn}^{1}={\rm Re\,}e^{{16\pi i\over
25}}\zeta^{n+m}\,,~{}$ $\displaystyle L_{mn}^{2}={\rm Re\,}e^{{12\pi i\over
25}}\zeta^{2m+n}\,,~{}X_{mn}={\rm Re\,}e^{{9\pi i\over 25}}\zeta^{2(n+m)}\,.$

| $u_{0}$ | $u_{1}$ | $u_{2}$ | $u_{3}$ | $u_{4}$ | $\phi_{1}$ | $\phi_{2}$ | $\phi_{3}$ | $\phi_{4}$ | $\sigma$ | $\tau$ | $\omega_{j}$ | $\theta_{j}$ | $\pi_{n}$ | $\rho_{n}$ | $\lambda_{n}$ | $\xi_{n}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$u_{0}$ | 1 | 3 | 3 | 4 | 5 | 2 | 2 | 4 | 6 | 30 | 30 | 20 | 20 | 12 | 12 | 12 | 12
$u_{1}$ | 3 | 9 | 9 | 12 | 15 | 6 | 6 | 12 | 18 | $-30$ | $-30$ | 0 | 0 | $g_{-}$ | $g_{+}$ | $g_{+}$ | $g_{-}$
$u_{2}$ | 3 | 9 | 9 | 12 | 15 | 6 | 6 | 12 | 18 | $-30$ | $-30$ | 0 | 0 | $g_{+}$ | $g_{-}$ | $g_{-}$ | $g_{+}$
$u_{3}$ | 4 | 12 | 12 | 16 | 20 | 8 | 8 | 16 | 24 | 0 | 0 | $20$ | $20$ | $-12$ | $-12$ | $-12$ | $-12$
$u_{4}$ | 5 | 15 | 15 | 20 | 25 | 10 | 10 | 20 | 30 | 30 | 30 | $-20$ | $-20$ | 0 | 0 | 0 | 0
$\phi_{1}$ | 2 | 6 | 6 | 8 | 10 | $-4$ | $-4$ | $-8$ | $-12$ | 0 | 0 | $-20$ | $20$ | $-g_{+}$ | $g_{-}$ | $-g_{-}$ | $g_{+}$
$\phi_{2}$ | 2 | 6 | 6 | 8 | 10 | $-4$ | $-4$ | $-8$ | $-12$ | 0 | 0 | $-20$ | $20$ | $-g_{-}$ | $g_{+}$ | $-g_{+}$ | $g_{-}$
$\phi_{3}$ | 4 | 12 | 12 | 16 | 20 | $-8$ | $-8$ | $-16$ | $-24$ | 0 | 0 | 20 | $-20$ | $-12$ | $12$ | $-12$ | $12$
$\phi_{4}$ | 6 | 18 | 18 | 24 | 30 | $-12$ | $-12$ | $-24$ | $-36$ | 0 | 0 | 0 | $0$ | $12$ | $-12$ | $12$ | $-12$
$\sigma$ | 30 | $-30$ | $-30$ | 0 | 30 | 0 | 0 | 0 | 0 | $30\sqrt{2}$ | $-30\sqrt{2}$ | 0 | 0 | 0 | 0 | 0 | 0
$\tau$ | 30 | $-30$ | $-30$ | 0 | 30 | 0 | 0 | 0 | 0 | $-30\sqrt{2}$ | $30\sqrt{2}$ | 0 | 0 | 0 | 0 | 0 | 0
$\omega_{j}$ | 20 | $0$ | $0$ | 20 | $-20$ | $-20$ | $-20$ | $20$ | 0 | 0 | 0 | ${5\over 2}a_{ij}$ | ${5\over 2}b_{ij}$ | 0 | 0 | 0 | 0
$\theta_{j}$ | 20 | $0$ | $0$ | 20 | $-20$ | $20$ | $20$ | $-20$ | 0 | 0 | 0 | ${5\over 2}b_{ji}$ | ${5\over 2}d_{ij}$ | 0 | 0 | 0 | 0
$\pi_{m}$ | 12 | $g_{-}$ | $g_{+}$ | $-12$ | 0 | $-g_{+}$ | $-g_{-}$ | $-12$ | $12$ | 0 | 0 | 0 | 0 | $P_{mn}^{1}$ | $P_{mn}^{2}$ | $P_{mn}^{3}$ | $P_{mn}^{4}$
$\rho_{m}$ | 12 | $g_{+}$ | $g_{-}$ | $-12$ | 0 | $g_{-}$ | $g_{+}$ | $12$ | $-12$ | 0 | 0 | 0 | 0 | $P_{nm}^{2}$ | $R_{mn}^{1}$ | $R_{mn}^{2}$ | $R_{mn}^{3}$
$\lambda_{m}$ | 12 | $g_{+}$ | $g_{-}$ | $-12$ | 0 | $-g_{-}$ | $-g_{+}$ | $-12$ | $12$ | 0 | 0 | 0 | 0 | $P_{nm}^{3}$ | $R_{nm}^{2}$ | $L_{mn}^{1}$ | $L_{mn}^{2}$
$\xi_{m}$ | 12 | $g_{-}$ | $g_{+}$ | $-12$ | 0 | $g_{+}$ | $g_{-}$ | $12$ | $-12$ | 0 | 0 | 0 | 0 | $P_{nm}^{4}$ | $R_{nm}^{3}$ | $L_{nm}^{2}$ | $X_{mn}$

Table 13: The modular $S$ matrix for ${\mathcal{V}}_{A_{5}}$ multiplied by
$60\sqrt{2}$.

## References

  * [1] S. Gukov and A. Kapustin, _Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories_ , 1307.4793.
  * [2] A. Kapustin and R. Thorngren, _Higher symmetry and gapped phases of gauge theories_ , 1309.4721.
  * [3] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, _Generalized Global Symmetries_ , _JHEP_ 02 (2015) 172 [1412.5148].
  * [4] A. Kapustin and R. Thorngren, _Higher symmetry and gapped phases of gauge theories_ , _Progress in Mathematics_ (2017) 177–202.
  * [5] D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, _Theta, Time Reversal, and Temperature_ , _JHEP_ 05 (2017) 091 [1703.00501].
  * [6] A. Kapustin and M. Tikhonov, _Abelian duality, walls and boundary conditions in diverse dimensions_ , _JHEP_ 11 (2009) 006 [0904.0840].
  * [7] A. Kapustin and N. Saulina, _Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory_ , 1012.0911.
  * [8] J. Fuchs, C. Schweigert and A. Valentino, _Bicategories for boundary conditions and for surface defects in 3-d TFT_ , _Commun. Math. Phys._ 321 (2013) 543 [1203.4568].
  * [9] D. Aasen, R. S. K. Mong and P. Fendley, _Topological Defects on the Lattice I: The Ising model_ , _J. Phys. A_ 49 (2016) 354001 [1601.07185].
  * [10] L. Bhardwaj and Y. Tachikawa, _On finite symmetries and their gauging in two dimensions_ , _JHEP_ 03 (2018) 189 [1704.02330].
  * [11] R. Vanhove, M. Bal, D. J. Williamson, N. Bultinck, J. Haegeman and F. Verstraete, _Mapping topological to conformal field theories through strange correlators_ , _Phys. Rev. Lett._ 121 (2018) 177203 [1801.05959].
  * [12] C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, _Topological Defect Lines and Renormalization Group Flows in Two Dimensions_ , _JHEP_ 01 (2019) 026 [1802.04445].
  * [13] Y.-H. Lin and S.-H. Shao, _Duality Defect of the Monster CFT_ , 1911.00042.
  * [14] R. Thorngren and Y. Wang, _Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases_ , 1912.02817.
  * [15] T. Lichtman, R. Thorngren, N. H. Lindner, A. Stern and E. Berg, _Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States_ , 2003.04328.
  * [16] D. Aasen, P. Fendley and R. S. K. Mong, _Topological Defects on the Lattice: Dualities and Degeneracies_ , 2008.08598.
  * [17] Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, _Symmetries and Strings of Adjoint QCD 2_, 2008.07567.
  * [18] C.-M. Chang and Y.-H. Lin, _Lorentzian Dynamics and Factorization Beyond Rationality_ , 2012.01429.
  * [19] T.-C. Huang and Y.-H. Lin, _Topological Field Theory with Haagerup Symmetry_ , 2102.05664.
  * [20] W. Ji and X.-G. Wen, _Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions_ , 1912.13492.
  * [21] L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang and H. Zheng, _Algebraic higher symmetry and categorical symmetry – a holographic and entanglement view of symmetry_ , _Phys. Rev. Res._ 2 (2020) 043086 [2005.14178].
  * [22] A. Kapustin, _Topological Field Theory, Higher Categories, and Their Applications_ , in _International Congress of Mathematicians_ , 4, 2010, 1004.2307.
  * [23] T. Johnson-Freyd, _On the classification of topological orders_ , 2003.06663.
  * [24] P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, _Tensor Categories_ , Mathematical Surveys and Monographs. American Mathematical Society, 2015.
  * [25] E. P. Verlinde, _Fusion Rules and Modular Transformations in 2D Conformal Field Theory_ , _Nucl. Phys. B_ 300 (1988) 360.
  * [26] V. Petkova and J. Zuber, _Generalized twisted partition functions_ , _Phys. Lett. B_ 504 (2001) 157 [hep-th/0011021].
  * [27] P. H. Ginsparg, _Curiosities at c = 1_ , _Nucl. Phys. B_ 295 (1988) 153.
  * [28] G. W. Moore and N. Seiberg, _Classical and Quantum Conformal Field Theory_ , _Commun. Math. Phys._ 123 (1989) 177.
  * [29] G. W. Moore and N. Seiberg, _LECTURES ON RCFT_ , in _1989 Banff NATO ASI: Physics, Geometry and Topology_ , 9, 1989.
  * [30] J. Fuchs, I. Runkel and C. Schweigert, _TFT construction of RCFT correlators 1. Partition functions_ , _Nucl. Phys. B_ 646 (2002) 353 [hep-th/0204148].
  * [31] J. Fuchs, I. Runkel and C. Schweigert, _TFT construction of RCFT correlators. 2. Unoriented world sheets_ , _Nucl. Phys. B_ 678 (2004) 511 [hep-th/0306164].
  * [32] J. Fuchs, I. Runkel and C. Schweigert, _TFT construction of RCFT correlators. 3. Simple currents_ , _Nucl. Phys. B_ 694 (2004) 277 [hep-th/0403157].
  * [33] J. Fuchs, I. Runkel and C. Schweigert, _TFT construction of RCFT correlators IV: Structure constants and correlation functions_ , _Nucl. Phys. B_ 715 (2005) 539 [hep-th/0412290].
  * [34] J. Fjelstad, J. Fuchs, I. Runkel and C. Schweigert, _TFT construction of RCFT correlators. V. Proof of modular invariance and factorisation_ , _Theor. Appl. Categor._ 16 (2006) 342 [hep-th/0503194].
  * [35] C. Bachas and S. Monnier, _Defect loops in gauged Wess-Zumino-Witten models_ , _JHEP_ 02 (2010) 003 [0911.1562].
  * [36] D. Tambara and S. Yamagami, _Tensor categories with fusion rules of self-duality for finite abelian groups_ , _Journal of Algebra_ 209 (1998) 692 .
  * [37] A. Recknagel and V. Schomerus, _Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes_ , Cambridge Monographs on Mathematical Physics. Cambridge University Press, 11, 2013, 10.1017/CBO9780511806476.
  * [38] M. Oshikawa and I. Affleck, _Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line_ , _Nucl. Phys. B_ 495 (1997) 533 [cond-mat/9612187].
  * [39] Y. Tachikawa, _On gauging finite subgroups_ , _SciPost Phys._ 8 (2020) 015 [1712.09542].
  * [40] Y.-H. Lin and S.-H. Shao, _Anomalies and Bounds on Charged Operators_ , _Phys. Rev. D_ 100 (2019) 025013 [1904.04833].
  * [41] B. Heidenreich, J. Mcnamara, M. Montero, M. Reece, T. Rudelius and I. Valenzuela, _Non-Invertible Global Symmetries and Completeness of the Spectrum_ , 2104.07036.
  * [42] R. Thorngren, _Anomalies and Bosonization_ , _Commun. Math. Phys._ 378 (2020) 1775 [1810.04414].
  * [43] A. Karch, D. Tong and C. Turner, _A Web of 2d Dualities: ${\bf Z}_{2}$ Gauge Fields and Arf Invariants_, _SciPost Phys._ 7 (2019) 007 [1902.05550].
  * [44] W. Ji, S.-H. Shao and X.-G. Wen, _Topological Transition on the Conformal Manifold_ , _Phys. Rev. Res._ 2 (2020) 033317 [1909.01425].
  * [45] R. Thorngren and C. von Keyserlingk, _Higher SPT’s and a generalization of anomaly in-flow_ , 1511.02929.
  * [46] L. Fidkowski and A. Kitaev, _The effects of interactions on the topological classification of free fermion systems_ , _Phys. Rev. B_ 81 (2010) 134509 [0904.2197].
  * [47] S. Ryu and S.-C. Zhang, _Interacting topological phases and modular invariance_ , _Phys. Rev. B_ 85 (2012) 245132 [1202.4484].
  * [48] H. Yao and S. Ryu, _Interaction effect on topological classification of superconductors in two dimensions_ , _Phys. Rev. B_ 88 (2013) 064507 [1202.5805].
  * [49] X.-L. Qi, _A new class of (2 + 1)-dimensional topological superconductors with $\mathbb{Z}$ topological classification_, _New J. Phys._ 15 (2013) 065002 [1202.3983].
  * [50] A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, _Fermionic Symmetry Protected Topological Phases and Cobordisms_ , _JHEP_ 12 (2015) 052 [1406.7329].
  * [51] T. Numasawa and S. Yamaguchi, _Mixed global anomalies and boundary conformal field theories_ , _JHEP_ 11 (2018) 202 [1712.09361].
  * [52] I. Runkel and L. Szegedy, _Topological field theory on $r$-spin surfaces and the Arf invariant_, 1802.09978.
  * [53] D. Radicevic, _Spin Structures and Exact Dualities in Low Dimensions_ , 1809.07757.
  * [54] Y. Yao and A. Furusaki, _Parafermionization, bosonization, and critical parafermionic theories_ , 2012.07529.
  * [55] E. H. Fradkin and L. P. Kadanoff, _DISORDER VARIABLES AND PARAFERMIONS IN TWO-DIMENSIONAL STATISTICAL MECHANICS_ , _Nucl. Phys. B_ 170 (1980) 1.
  * [56] V. A. Fateev and A. B. Zamolodchikov, _Parafermionic Currents in the Two-Dimensional Conformal Quantum Field Theory and Selfdual Critical Points in Z(n) Invariant Statistical Systems_ , _Sov. Phys. JETP_ 62 (1985) 215.
  * [57] J. A. Harvey and G. W. Moore, _An Uplifting Discussion of T-Duality_ , _JHEP_ 05 (2018) 145 [1707.08888].
  * [58] N. Carqueville, I. Runkel and G. Schaumann, _Line and surface defects in Reshetikhin-Turaev TQFT_ , 1710.10214.
  * [59] R. Longo and K.-H. Rehren, _Nets of subfactors_ , _Rev. Math. Phys._ 7 (1995) 567 [hep-th/9411077].
  * [60] J. Bockenhauer and D. E. Evans, _Modular invariants, graphs and alpha induction for nets of subfactors. 1._ , _Commun. Math. Phys._ 197 (1998) 361 [hep-th/9801171].
  * [61] J. Bockenhauer and D. E. Evans, _Modular invariants, graphs and alpha induction for nets of subfactors. 2._ , _Commun. Math. Phys._ 200 (1999) 57 [hep-th/9805023].
  * [62] J. Bockenhauer and D. E. Evans, _Modular invariants, graphs and alpha induction for nets of subfactors. 3._ , _Commun. Math. Phys._ 205 (1999) 183 [hep-th/9812110].
  * [63] J. Bockenhauer, D. E. Evans and Y. Kawahigashi, _Chiral structure of modular invariants for subfactors_ , _Commun. Math. Phys._ 210 (2000) 733 [math/9907149].
  * [64] V. Ostrik, _Module categories, weak Hopf algebras and modular invariants_ , _arXiv Mathematics e-prints_ (2001) math/0111139 [math/0111139].
  * [65] J. Milnor and D. Husemöller, _Symmetric Bilinear Forms_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1973.
  * [66] C. Wall, _Quadratic forms on finite groups, and related topics_ , _Topology_ 2 (1963) 281 .
  * [67] L. Wang and Z. Wang, _In and around abelian anyon models_ , 2020.
  * [68] V. TURAEV, _Reciprocity for gauss sums on finite abelian groups_ , _Mathematical Proceedings of the Cambridge Philosophical Society_ 124 (1998) 205–214.
  * [69] D. Tambara, _Representations of tensor categories with fusion rules of self-duality for abelian groups_ , _Israel Journal of Mathematics_ 118 (2000) 29.
  * [70] R. Dijkgraaf, E. P. Verlinde and H. L. Verlinde, _C = 1 Conformal Field Theories on Riemann Surfaces_ , _Commun. Math. Phys._ 115 (1988) 649.
  * [71] D. Gepner and Z.-a. Qiu, _Modular Invariant Partition Functions for Parafermionic Field Theories_ , _Nucl. Phys. B_ 285 (1987) 423.
  * [72] V. A. Fateev and A. B. Zamolodchikov, _Integrable perturbations of Z(N) parafermion models and O(3) sigma model_ , _Phys. Lett. B_ 271 (1991) 91.
  * [73] J. Fuchs, M. R. Gaberdiel, I. Runkel and C. Schweigert, _Topological defects for the free boson CFT_ , _J. Phys. A_ 40 (2007) 11403 [0705.3129].
  * [74] R. Dijkgraaf, C. Vafa, E. P. Verlinde and H. L. Verlinde, _The Operator Algebra of Orbifold Models_ , _Commun. Math. Phys._ 123 (1989) 485.
  * [75] S.-K. Yang, _$Z$(4) X $Z$(4) Symmetry and Parafermion Operators in the Selfdual Critical Ashkin-teller Model_, _Nucl. Phys. B_ 285 (1987) 639.
  * [76] E. Meir and E. Musicantov, _Module categories over graded fusion categories_ , _Journal of Pure and Applied Algebra_ 216 (2012) 2449–2466.
  * [77] D. Naidu, _Categorical Morita equivalence for group-theoretical categories_ , _arXiv Mathematics e-prints_ (2006) math/0605530 [math/0605530].
  * [78] F. C. Alcaraz and R. Koberle, _Duality and the Phases of $Z$(n) Spin Systems_, _J. Phys. A_ 13 (1980) L153.
  * [79] F. C. Alcaraz and R. Koberle, _The Phases of Two-dimensional Spin and Four-dimensional Gauge Systems With $Z(N)$ Symmetry_, _J. Phys. A_ 14 (1981) 1169.
  * [80] F. C. Alcaraz, _The Critical Behavior of Selfdual $Z(N$) Spin Systems: Finite Size Scaling and Conformal Invariance_, _J. Phys. A_ 20 (1987) 2511.
  * [81] P. Dorey, R. Tateo and K. E. Thompson, _Massive and massless phases in selfdual Z(N) spin models: Some exact results from the thermodynamic Bethe ansatz_ , _Nucl. Phys. B_ 470 (1996) 317 [hep-th/9601123].
  * [82] A. Cappelli and G. D’Appollonio, _Boundary states of c = 1 and 3/2 rational conformal field theories_ , _JHEP_ 02 (2002) 039 [hep-th/0201173].
  * [83] M. Nguyen, Y. Tanizaki and M. Ünsal, _Non-invertible 1-form symmetry and Casimir scaling in 2d Yang-Mills theory_ , 2104.01824.
  * [84] D. Baker, _Differential characters and borel cohomology_ , _Topology_ 16 (1977) 441.
  * [85] M. R. Gaberdiel, A. Recknagel and G. M. T. Watts, _The Conformal boundary states for SU(2) at level 1_ , _Nucl. Phys. B_ 626 (2002) 344 [hep-th/0108102].
  * [86] A. B. Zamolodchikov, _Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory_ , _Theor. Math. Phys._ 65 (1985) 1205.
  * [87] R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel and R. Varnhagen, _W algebras with two and three generators_ , _Nucl. Phys. B_ 361 (1991) 255.
  * [88] K. Hornfeck, _W algebras with set of primary fields of dimensions (3, 4, 5) and (3, 4, 5, 6)_ , _Nucl. Phys. B_ 407 (1993) 237 [hep-th/9212104].
  * [89] J. de Boer, L. Feher and A. Honecker, _A Class of W algebras with infinitely generated classical limit_ , _Nucl. Phys. B_ 420 (1994) 409 [hep-th/9312049].
  * [90] R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck and R. Hubel, _Coset realization of unifying W algebras_ , _Int. J. Mod. Phys. A_ 10 (1995) 2367 [hep-th/9406203].
  * [91] C. Dong, C. H. Lam, Q. Wang and H. Yamada, _The Structure of parafermion vertex operator algebras_ , _Commun. Math. Phys._ 299 (2010) 783 [0904.2758].
  * [92] V. A. Fateev, _Integrable deformations in Z(N) symmetrical models of conformal quantum field theory_ , _Int. J. Mod. Phys. A_ 6 (1991) 2109.