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}\pgfsys<EMAIL_ADDRESS>(121)

Here, $[k,0,0]$ represents the Dijkgraaf-Witten invariant in
$H^{3}(D_{2k},U(1))=\mathbb{Z}_{2k}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
that deforms the gauge theory Dijkgraaf and Witten (1990); Dijkgraaf  _et al._
(1991); Altschuler and Coste (1992); Bais  _et al._ (1993); de Wild Propitius
(1995) (see appendix C for the cohomology classification). The
$\mathbb{Z}_{2k}$ component comes from the cohomology classification
$H^{3}(\mathbb{Z}_{2k},U(1))=\mathbb{Z}_{2k}$ of the normal subgroup
$\mathbb{Z}_{2k}$ of $D_{2k}$. Each of the two $\mathbb{Z}_{2}$’s corresponds
to the cohomology classification $H^{3}(\mathbb{Z}_{2},U(1))=\mathbb{Z}_{2}$
for one of the two conjugacy classes $[\mu]$ and $[\mu m]$ of twofold
conjugation symmetries. The trivial components in $[k,0,0]$ signifies that the
$\mathbb{Z}_{2}$ fluxes – which are pair combinations of the $\sigma_{\pm}$ or
$\tau_{\pm}$ from $U(1)_{l}/\mathbb{Z}_{2}$ and $SU(n)_{1}/\mathbb{Z}_{2}$ –
are bosonic or fermionic (see table 2 and 3).

Next, we consider $n\equiv 4$ modulo 8. From (70), we saw that
$SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{l}\times SU(n)_{1}$ is equivalent, as an
UMTC, to $SO(8)_{1}\boxtimes_{\psi\bar{\psi}}D^{[k]}(\mathbb{Z}_{2k})$, where
$k=n/2$ and the relative tensor product $\boxtimes$ involves the anyon
condensation of a fermion pair $\psi\bar{\psi}$ (see section III.3.2). The
$\mathbb{Z}_{2}$ conjugation action applies simultaneously on $SO(8)_{1}$ and
$D^{[k]}(\mathbb{Z}_{2k})$. After gauging, the former becomes
$SO(8)_{1}/\mathbb{Z}_{2}=SO(8)_{1}\times D^{[1]}(\mathbb{Z}_{2})$ (see (56))
and the latter becomes $D^{[k,0,0]}(D_{2k})$. The simultaneous
$\mathbb{Z}_{2}$ action requires the $\mathbb{Z}_{2}$ gauge charges $\zeta$
from $D^{[1]}(\mathbb{Z}_{2})$ and $S$ from $D^{[k,0,0]}(D_{2k})$ to pair
“condense” in the vacuum sector in the anyon condensation picture. This
identifies

$\displaystyle D^{[1]}(\mathbb{Z}_{2})\boxtimes_{\zeta
S}D^{[k,0,0]}(D_{2k})=D^{[k,1,1]}(D_{2k})$ (122)

and modifies the Dijkgraaf-Witten invariant into $[k,1,1]$ (see appendix C) so
that the $\mathbb{Z}_{2}$ fluxes – which are pair combinations of the
$\sigma_{\pm}$ or $\tau_{\pm}$ from $U(1)_{l}/\mathbb{Z}_{2}$ and
$SU(n)_{1}/\mathbb{Z}_{2}$ – are all semionic with spins $h=\pm 1/4$ modulo 1
(see table 2 and 3). Combining with the remaining $SO(8)_{1}$, the
$SO(2n)_{1}/D_{k}$ twist liquid is identical, as an UMTC, to the relative
tensor product

$\displaystyle\frac{SO(2n)_{1}}{D_{k}}$
$\displaystyle=\frac{SO(2n)_{1}/\mathbb{Z}_{k}}{\mathbb{Z}_{2}}=\frac{SO(8)_{1}\boxtimes_{\psi\bar{\psi}}D^{[k]}(\mathbb{Z}_{2k})}{\mathbb{Z}_{2}}$
$\displaystyle=\frac{SO(8)_{1}}{\mathbb{Z}_{2}}\boxtimes_{\psi\bar{\psi},\zeta
S}\frac{D^{[k]}(\mathbb{Z}_{2k})}{\mathbb{Z}_{2}}$ (123)
$\displaystyle=\left(SO(8)_{1}\times
D^{[1]}(\mathbb{Z}_{2})\right)\boxtimes_{\psi\bar{\psi},\zeta
S}D^{[k,0,0]}(D_{2k})$
$\displaystyle=SO(8)_{1}\boxtimes_{\psi\bar{\psi}}D^{[k,1,1]}(D_{2k}),\quad\mbox{if
$n\equiv 4$ mod 8},$

where the vector fermion $\psi$ in $SO(8)_{1}$ is pair condensed with the
Abelian fermionic flux $\bar{\psi}=m^{n/2}$ in $D^{[k,1,1]}(D_{2k})$ (c.f. the
discussion above (70)). The anyon content is almost identical to
$D^{[k,1,1]}(D_{2k})$. The anyons are represented by a dyon $\chi$ in
$D^{[k,1,1]}(D_{2k})$ if $\chi$ and $m^{n/2}$ braid trivially, or the spinor-
dyon composite $s_{+}\chi$ if $\chi$ and $m^{n/2}$ have $\pi$ monodromy, where
$s_{+}$ is the even spinor in $SO(8)_{1}$. Therefore, the relative tensor
product with $SO(8)_{1}$ only modifies the topological spins of anyons in
$D^{[k,1,1]}(D_{2k})$, but leaves the fusion rules and modular $S$ matrix of
$D^{[k,1,1]}(D_{2k})$ unaltered.

Now, we examine the cases when $n\equiv 2$ modulo 4. From (70), we saw that
$SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{l}\times SU(n)_{1}$ is equivalent, as an
UMTC, to the decoupled product $SO(2n)_{1}\times D^{[0]}(\mathbb{Z}_{k})$,
where $k=n/2$ is odd. From (56),
$SO(2n)_{1}/\mathbb{Z}_{2}=SO(2n)_{1}\boxtimes_{\psi\bar{\psi}}D^{[2]}(\mathbb{Z}_{4})$,
where the vector fermion $\psi$ in $SO(2n)_{1}$ is pair condensed with the
fermionic gauge flux $\bar{\psi}=\bar{s}^{2}$ in the relative tensor product
$\boxtimes_{\psi\bar{\psi}}$ (see (55)). The $\mathbb{Z}_{2}$ gauging of
$D^{[0]}(\mathbb{Z}_{k})$ produces the un-deformed discrete gauge theory
$D^{[0]}(D_{k})$ with the trivial Dijkgraaf-Witten invariant $[0]$ in
$H^{3}(D_{k},U(1))=\mathbb{Z}_{2k}$. Since $\mathbb{Z}_{2}$ conjugation acts
simultaneously on $SO(2n)_{1}$ and $D^{[0]}(\mathbb{Z}_{k})$, the
$\mathbb{Z}_{2}$ charge pair $\zeta S$ from $D^{[2]}(\mathbb{Z}_{4})$ and
$D^{[0]}(D_{k})$ is local and belong in the vacuum sector in the anyon
condensation picture. This combines them into the discrete gauge theory

$\displaystyle D^{[2k]}(Q_{4k})=D^{[2]}(\mathbb{Z}_{4})\boxtimes_{\zeta
S}D^{[0]}(D_{k})$ (124)

of the dicyclic group

$\displaystyle Q_{4k}$
$\displaystyle=\left\\{\tilde{\mu}^{A}\tilde{m}^{a}:A=0,1,a=-k+1,\ldots,k\right\\}$
(125)
$\displaystyle=\left\langle\tilde{m},\tilde{\mu}|\tilde{m}^{2k}=\tilde{\mu}^{4}=1,\tilde{m}^{k}=\tilde{\mu}^{2},\tilde{\mu}^{-1}\tilde{m}\tilde{\mu}=\tilde{m}^{-1}\right\rangle.$

(Alternatively, the dicyclic group (125) is also the double cover
$\tilde{D}_{k}=Q_{4k}$ of the dihedral group $D_{k}$, where $\tilde{m}$ and
$\tilde{\mu}$ are represented by perpendicular rotations $e^{i(2\pi/k)S_{z}}$
and $e^{i\pi S_{x}}$, for $S_{x,z}=\sigma_{x,z}/2$, in a half-integral spin
representation, and $\tilde{m}^{k}=\tilde{\mu}^{2}=-1$ are $2\pi$ rotations.
Therefore, $Q_{4k}$ is a subgroup of the unit quaternions
$SU(2)=\\{a_{0}+a_{1}{\bf i}+a_{2}{\bf j}+a_{3}{\bf k}:|{\bf a}|=1\\}$, where
${\bf i}$, ${\bf j}$, ${\bf k}$ can be represented by $i\sigma_{x}$,
$i\sigma_{y}$, $i\sigma_{z}$.) Since we only encounter the dicyclic group with
odd degree $k$, (125) has the equivalent presentation

$\displaystyle Q_{4k}$ $\displaystyle=\mathbb{Z}_{4}\ltimes\mathbb{Z}_{k}$
(126)
$\displaystyle=\left\\{\tilde{\mu}^{A}m^{a}:\begin{array}[]{*{20}l}A=-1,0,1,2,\\\
a=-(k-1)/2,\ldots,(k-1)/2\end{array}\right\\}$ (129)
$\displaystyle=\left\langle
m,\tilde{\mu}|m^{k}=\tilde{\mu}^{4}=\tilde{\mu}m\tilde{\mu}^{-1}m=1\right\rangle.$

Here, both the $\mathbb{Z}_{4}$ flux $[\bar{s}]$ in $D^{[2]}(\mathbb{Z}_{4})$
and the $\mathbb{Z}_{2}$ flux $[\mu]$ in $D^{[0]}(D_{k})$ individually carries
the $\pi$ monodromy braiding phase with the $\mathbb{Z}_{2}$ charge pair
$\zeta S$, and therefore are separately confined. When appear simultaneously,
they combines into the fourfold gauge flux $[\tilde{\mu}]=[\bar{s}][\mu]$. We
associate $[\tilde{m}]=[\bar{s}^{2}][m]$ to the combination of fermion
$\bar{\psi}=\bar{s}^{2}$ in $D^{[2]}(\mathbb{Z}_{4})$ and the $\mathbb{Z}_{k}$
flux $[m]$ in $D^{[0]}(D_{k})$. Since $k$ is odd and $\bar{s}^{4}=1$,
$\tilde{m}^{k}=\bar{s}^{2}$, which can be identified as $\tilde{\mu}^{2}$
because $\mu^{2}=1$ as a group element in $D_{k}$. The discrete gauge theory
$D^{[2k]}(Q_{4k})$ is deformed by the Dijkgraaf-Witten cohomology invariant
$[2k]$ in $H^{3}(Q_{4k},U(1))=\mathbb{Z}_{4k}$ (see appendix C). The
deformation requires the spins of the fourfold fluxes to be $\pm 1/8$ and $\pm
3/8$ modulo 1. Combining with the remaining $SO(2n)_{1}$, the
$SO(2n)_{1}/D_{k}$ twist liquid is identical, as an UMTC, to the relative
tensor product

$\displaystyle\frac{SO(2n)_{1}}{D_{k}}$
$\displaystyle=\frac{SO(2n)_{1}/\mathbb{Z}_{k}}{\mathbb{Z}_{2}}=\frac{SO(2n)_{1}\times
D^{[0]}(\mathbb{Z}_{k})}{\mathbb{Z}_{2}}$
$\displaystyle=\frac{SO(2n)_{1}}{\mathbb{Z}_{2}}\boxtimes_{\zeta
S}\frac{D^{[0]}(\mathbb{Z}_{k})}{\mathbb{Z}_{2}}$ (130)
$\displaystyle=SO(2n)_{1}\boxtimes_{\psi\bar{\psi}}D^{[2]}(\mathbb{Z}_{4})\boxtimes_{\zeta
S}D^{[0]}(D_{k})$
$\displaystyle=SO(2n)_{1}\boxtimes_{\psi\bar{\psi}}D^{[2k]}(Q_{4k}),\quad\mbox{if
$n\equiv 2$ mod 4},$

where the vector fermion $\psi$ in $SO(2n)_{1}$ is pair “condensed” with the
Abelian fermionic flux $\bar{\psi}=\tilde{\mu}^{2}$ in $D^{[2k]}(Q_{4k})$. The
“condensate” $\psi\bar{\psi}$ forces the semionic spinor $s_{\pm}$ in
$SO(2n)_{1}$ to appear simultaneously with one of the $\mathbb{Z}_{2}$ fluxes
in $D^{[2k]}(Q_{4k})$. The $\mathbb{Z}_{2}$ flux-spinor composites have spins
$n/16$ or $(n+8)/16$ modulo 1. This matches with the spins of the pair
combination of the $\mathbb{Z}_{2}$ twist fields, $\sigma_{\pm}$ or
$\tau_{\pm}$, in $SU(n)_{1}/\mathbb{Z}_{2}$ and $U(1)_{l}/\mathbb{Z}_{2}$ (see
table 2 and 3).

Lastly, we present the $SO(2n)_{1}/D_{k}$ topological order when $n=k$ is odd.
Eq.(70) identifies $SO(2n)_{1}/\mathbb{Z}_{k}=SO(2n)_{1}\times
D^{[0]}(\mathbb{Z}_{k})$. Like the previous cases, the $\mathbb{Z}_{2}$
conjugation symmetry acts simultaneously on $SO(2n)_{1}$ and
$D^{[0]}(\mathbb{Z}_{k})$. The $\mathbb{Z}_{2}$ gauging of latter produces
$D^{[0]}(D_{k})$. From (64), the $\mathbb{Z}_{2}$ gauging of the former gives
$SO(2n)_{1}\boxtimes_{\psi\bar{\psi}}Z(\mathrm{Ising})$ for $n\equiv\pm 1$
modulo 8, or $SO(2n)_{1}\boxtimes_{\psi\bar{\psi}}Z(SU(2)_{2})$ for
$n\equiv\pm 3$ modulo 8. (See below (64) for the details of fermion pair
condensation.) The quantum doubles $Z(\mathrm{Ising})$ and $Z(SU(2)_{2})$ are
not discrete gauge theories because they support Ising anyons with non-
integral quantum dimension $\sqrt{2}$. This stems from the outer automorphism
nature of the $\mathbb{Z}_{2}$ conjugation, which switches the even and odd
spinors, $s_{+}\leftrightarrow s_{-}$, in $SO(2n)_{1}$, when $n$ is odd. The
$SO(2n)_{1}/D_{k}$ topological order therefore carries both Ising and discrete
gauge theory components

$\displaystyle\frac{SO(2n)_{1}}{D_{k}}$
$\displaystyle=SO(2n)_{1}\boxtimes_{\psi\bar{\psi}}\mathcal{Z}_{n}(D_{k})$
(131)

where the quantum double

$\displaystyle\begin{split}\mathcal{Z}_{n}(D_{k})&=D^{[0]}(\mathbb{Z}_{2k})\sslash\mathbb{Z}_{2}\\\
&=\mathcal{Z}_{n}(\mathbb{Z}_{2})\boxtimes_{\zeta S}D^{[0]}(D_{k})\end{split}$
(132)

is the non-chiral UMTC

$\displaystyle\begin{split}&Z(\mathrm{Ising})\boxtimes_{\zeta
S}D^{[0]}(D_{k}),\quad\mbox{if $n\equiv\pm 1$ mod 8},\\\
&Z(SU(2)_{2})\boxtimes_{\zeta S}D^{[0]}(D_{k}),\quad\mbox{if $n\equiv\pm 3$
mod 8}.\end{split}$ (133)

Like previous cases, the relative tensor product $\boxtimes_{\zeta S}$ anyon
“condenses” the $\mathbb{Z}_{2}$ charge pair $\zeta S$, where $\zeta$ is the
fermion pair in
$Z(\mathrm{Ising})=\mathrm{Ising}\times\overline{\mathrm{Ising}}$ or
$Z(SU(2)_{2})=SU(2)_{2}\times\overline{SU(2)_{2}}$ and $S=([1],A_{1})$ is the
$\mathbb{Z}_{2}$ charge in $D^{[0]}(D_{k})$. Consequently, any Ising anyon
(such as $\sigma$ and $\bar{\sigma}$) with quantum dimension $\sqrt{2}$ in
$Z(\mathrm{Ising})$ or $Z(SU(2)_{2})$ must appear simultaneously with a
$\mathbb{Z}_{2}$ flux $([\mu],\zeta^{\lambda})$, for $\lambda=0,1$, in
$D^{[0]}(D_{k})$. (See table 9 for the list of dyons in the $D^{[0]}(D_{k})$
discrete gauge theory.) These “Ising-fluxon” composites

$\displaystyle\begin{split}&\begin{array}[]{*{20}l}\Gamma^{\lambda}=\sigma\boxtimes([\mu],\zeta^{\lambda}),\\\
\overline{\Gamma}^{\lambda}=\bar{\sigma}\boxtimes([\mu],\zeta^{\lambda}),\end{array}\;\mbox{if
$n\equiv 1,3$ mod 16}\\\
&\begin{array}[]{*{20}l}\Gamma^{\lambda}=\bar{\sigma}\boxtimes([\mu],\zeta^{1-\lambda}),\\\
\overline{\Gamma}^{\lambda}=\sigma\boxtimes([\mu],\zeta^{1-\lambda}),\end{array}\;\mbox{if
$n\equiv 5,7$ mod 16}\\\
&\begin{array}[]{*{20}l}\Gamma^{\lambda}=\bar{\sigma}\boxtimes([\mu],\zeta^{\lambda}),\\\
\overline{\Gamma}^{\lambda}=\sigma\boxtimes([\mu],\zeta^{\lambda}),\end{array}\;\mbox{if
$n\equiv-1,-3$ mod 16}\\\
&\begin{array}[]{*{20}l}\Gamma^{\lambda}=\sigma\boxtimes([\mu],\zeta^{1-\lambda}),\\\
\overline{\Gamma}^{\lambda}=\bar{\sigma}\boxtimes([\mu],\zeta^{1-\lambda}),\end{array}\;\mbox{if
$n\equiv-5,-7$ mod 16}\end{split}$ (134)

in $\mathcal{Z}_{n}(D_{k})$ have quantum dimension $\sqrt{2}n$ and spins
$h_{\Gamma^{\lambda}}=k/16+\lambda/2$,
$h_{\overline{\Gamma}^{\lambda}}=-k/16+\lambda/2$ modulo 1. $\Gamma^{0}$ and
$\Gamma^{1}=S\times\Gamma^{0}$ have trivial statistics with the condensate
$\psi\bar{\psi}=\psi_{SO(2n)_{1}}\bar{\psi}_{\mathcal{Z}_{n}(D_{k})}$ in (131)
and are deconfined. Here, similar to (64),
$\bar{\psi}_{\mathcal{Z}_{n}(D_{k})}$ is the fermion in
$\overline{\mathrm{Ising}}$ when $n\equiv 1$ mod 8, in $\overline{SU(2)_{2}}$
when $n\equiv 3$ mod 8, in $\mathrm{Ising}$ when $n\equiv-1$ mod 8, or in
$SU(2)_{2}$ when $n\equiv 3$ mod 8. On the other hand, $\overline{\Gamma}^{0}$
and $\overline{\Gamma}^{1}=S\times\overline{\Gamma}^{0}$ have semionic mutual
statistics with $\psi\bar{\psi}$. Therefore, they must appear simultaneously
with a spinor $s_{\pm}$ in $SO(2n)_{1}$ in (131) to respect locality of
$\psi\bar{\psi}$. This modifies the spins of the Ising-fluxons to
$h_{\Gamma^{\lambda}}=h_{s_{\pm}\overline{\Gamma}^{\lambda}}=k/16+\lambda/2$
(mod 1) in $SO(2n)_{1}/D_{k}$. The quantum dimensions and spins of these
Ising-fluxons match the pair combination of the $\mathbb{Z}_{2}$ twist fields,
$\sigma$ and $\tau$, in $U(1)_{l}/\mathbb{Z}_{2}$ and
$SU(n)_{1}/\mathbb{Z}_{2}$ (see table 1 and 3). They can be identified
according to

$\displaystyle\begin{split}\Gamma^{0}&=\sigma_{SU(n)_{1}/\mathbb{Z}_{2}}\sigma^{+}_{U(1)_{l}/\mathbb{Z}_{2}}\\\
\Gamma^{1}&=\tau_{SU(n)_{1}/\mathbb{Z}_{2}}\sigma^{+}_{U(1)_{l}/\mathbb{Z}_{2}}\\\
s_{+}\overline{\Gamma}^{0}&=\sigma_{SU(n)_{1}/\mathbb{Z}_{2}}\sigma^{-}_{U(1)_{l}/\mathbb{Z}_{2}}\\\
s_{+}\overline{\Gamma}^{1}&=\tau_{SU(n)_{1}/\mathbb{Z}_{2}}\sigma^{-}_{U(1)_{l}/\mathbb{Z}_{2}}\end{split}$
(135)

(up to the $\mathbb{Z}_{2}$ charge pair condensate
$S_{SU(n)_{1}/\mathbb{Z}_{2}}S_{U(1)_{l}/\mathbb{Z}_{2}}$ and the fermion pair
condensate $\psi_{SO(2n)_{1}}\bar{\psi}_{\mathcal{Z}_{n}(D_{k})}$).

We observe that the quantum double $\mathcal{Z}_{n}(D_{k})$ in (132) is the
$\mathbb{Z}_{2}$ twist liquid of the discrete gauge theory
$D^{[0]}(\mathbb{Z}_{2k})$ after gauging an unconventional twofold mixed
symmetry (see (138)) that is neither the electric-magnetic symmetry nor the
conjugation symmetry. It is a known result Barkeshli and Wen (2012); Teo  _et
al._ (2015); Chen  _et al._ (2017) that the quantum doubles
$Z(\mathrm{Ising})$ and $Z(SU(2)_{2})$ are $\mathbb{Z}_{2}$ twist liquids when
the electric-magnetic (e-m) symmetry in $D^{[0]}(\mathbb{Z}_{2})$ is gauged.
The $D^{[0]}(\mathbb{Z}_{2})$ discrete gauge theory can be described by a two-
component Chern-Simons field theory $\mathcal{S}=\int_{2+1}K_{IJ}a^{I}\wedge
da^{J}/(4\pi)$, where $K=2\sigma_{x}$. It exhibits the $\mathbb{Z}_{2}$ anyon
relabeling symmetry Khan  _et al._ (2014); Teo (2016) $M=\sigma_{x}$, which
exchanges the $\mathbb{Z}_{2}$ charge and flux (represented by the unit
vectors ${\bf e}_{1}$ and ${\bf e}_{2}$) and leaves the $K$ matrix invariant,
$MKM^{T}=K$. Gauging the symmetry gives the quantum double

$\displaystyle\begin{split}\mathcal{Z}_{n}(\mathbb{Z}_{2})&=D^{[0]}(\mathbb{Z}_{2})\sslash\mathbb{Z}_{2}\\\
&=Z(\mathrm{Ising})\quad\mbox{or}\quad Z(SU(2)_{2})\end{split}$ (136)

depending on the Frobenius-Schur indicator Kitaev (2006)
$\varkappa_{\sigma}=d_{\sigma}\left[F^{\sigma\sigma\sigma}_{\sigma}\right]^{1}_{1}=(-1)^{(n^{2}-1)/8}$
of the $\mathbb{Z}_{2}$ twist field (see discussion above (64)). Here, the
double quotient “$\sslash$” emphasizes the non-trivial flux-charge switching
nature of the $\mathbb{Z}_{2}$ symmetry, and consequently, the non-gauge
theory topological order of $D^{[0]}(\mathbb{Z}_{2})\sslash\mathbb{Z}_{2}$.
[Had the $\mathbb{Z}_{2}$ symmetry preserved the anyon types, the gauging
would only extend the gauge group and $D^{[0]}(\mathbb{Z}_{2})/\mathbb{Z}_{2}$
would be a discrete gauge theory $D^{[\omega]}(\mathbb{Z}_{4})$ or
$D^{[\eta]}(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$.]

With the help of (136), equation (132) can be re-expressed as

$\displaystyle\Big{(}D^{[0]}(\mathbb{Z}_{2})\sslash\mathbb{Z}_{2}\Big{)}\boxtimes_{\zeta
S}D^{[0]}(D_{k})$
$\displaystyle=\Big{(}D^{[0]}(\mathbb{Z}_{2})\sslash\mathbb{Z}_{2}\Big{)}\boxtimes_{\zeta
S}\frac{D^{[0]}(\mathbb{Z}_{k})}{\mathbb{Z}_{2}}$
$\displaystyle=\Big{(}D^{[0]}(\mathbb{Z}_{2})\times
D^{[0]}(\mathbb{Z}_{k})\Big{)}\sslash\mathbb{Z}_{2}.$ (137)

Here, the product theory $D^{[0]}(\mathbb{Z}_{2})\times
D^{[0]}(\mathbb{Z}_{k})$ has a four-component Chern-Simons description
$\mathcal{S}=\int_{2+1}K_{IJ}a^{I}\wedge da^{J}/(4\pi)$, the $K$-matrix is the
direct sum $K=(2\sigma_{x})\oplus(k\sigma_{x})$. The $\mathbb{Z}_{2}$ symmetry
acts simultaneously on $D^{[0]}(\mathbb{Z}_{2})$ and $D^{[0]}(\mathbb{Z}_{k})$
and can be represented by the matrix $M=\sigma_{x}\oplus(-\mathbb{I})$, which
combines the e-m symmetry in $D^{[0]}(\mathbb{Z}_{2})$ and the conjugation
symmetry in $D^{[0]}(\mathbb{Z}_{k})$. Since $k$ is odd, the product
$D^{[0]}(\mathbb{Z}_{2})\times D^{[0]}(\mathbb{Z}_{k})$ has identical
topological order as the discrete gauge theory $D^{[0]}(\mathbb{Z}_{2k})$ with
gauge group $\mathbb{Z}_{2k}=\mathbb{Z}_{2}\times\mathbb{Z}_{k}$. Labeling the
$D^{[0]}(\mathbb{Z}_{2k})$ dyons according to their flux-charge content,
$m^{a}z^{b}$, where $a,b=0,1,\ldots,2k-1$ modulo $2k$, the twofold action of
the mixed symmetry $\mathbb{Z}_{2}:m^{a}z^{b}\to m^{a^{\prime}}z^{b^{\prime}}$
can be represented by the matrix transformation

$\displaystyle\begin{pmatrix}a\\\
b\end{pmatrix}\to\begin{pmatrix}a^{\prime}\\\
b^{\prime}\end{pmatrix}=U\begin{pmatrix}a\\\ b\end{pmatrix},\quad
U=\begin{pmatrix}k-1&k\\\ k&k-1\end{pmatrix}.$ (138)

The symmetry has order 2 because the flux-charge labels have $2k$ periodicity
$a\equiv a+2k$, $b\equiv b+2k$, and $U$ squares to the identity matrix
$\mathbb{I}$ modulo $2k$. The symmetry preserves the spin and braiding data of
$D^{[0]}(\mathbb{Z}_{2k})$ because $U^{T}K^{-1}U\equiv K^{-1}$ modulo
$2\mathbb{Z}$, where $K=2k\sigma_{x}$. Gauging the $\mathbb{Z}_{2}$ symmetry
in $D^{[0]}(\mathbb{Z}_{2k})$ gives the $\mathbb{Z}_{2}$ twist liquids
$\mathcal{Z}_{n}(D_{k})=D^{[0]}(\mathbb{Z}_{2k})\sslash\mathbb{Z}_{2}$ in
(132), which we referred to as the “Ising-fluxon” phase. We conjecture in
passing that this non-chiral Ising-fluxon topological order may be identical
to the Drinfeld center of a (non-commutative) fusion category (see discussion
in section V).

Overall, we summarize the $SO(2n)_{1}/D_{k}$ twist liquid phases for general
integer $n$ by the UMTC equivalence

$\displaystyle\frac{SO(2n)_{1}}{D_{k}}=\frac{U(1)_{l}\times
SU(n)_{1}}{\mathbb{Z}_{2}}$ (139)
$\displaystyle=\left\\{\begin{array}[]{*{20}l}D^{[k,0,0]}(D_{2k}),&\mbox{for
$n\equiv 0$ mod 8}\\\ SO(8)_{1}\boxtimes D^{[k,1,1]}(D_{2k}),&\mbox{for
$n\equiv 4$ mod 8}\\\ SO(2n)_{1}\boxtimes D^{[2k]}(Q_{4k}),&\mbox{for $n\equiv
2$ mod 4}\\\ SO(2n)_{1}\boxtimes\mathcal{Z}_{n}(D_{k}),&\mbox{for $n$
odd}\end{array}\right.$ (144)

where $k=n$ and $l=4n$ when $n$ is odd, and $k=n/2$ and $l=n$ when $n$ is
even. The $D_{2k}$ and $Q_{4k}$ groups are “non-symmorphic” $\mathbb{Z}_{2}$
central extensions of $D_{k}$ and associates to non-trivial cohomology
elements in $H^{2}(D_{k},\mathcal{A})$. The super-script $[\omega]$ in
$D^{[\omega]}(G)$ labels the Dijkgraaf-Witten invariant in the cohomology
group $H^{3}(G,U(1))$ that deforms the gauge theory. Similar to (70), the
topological orders in the $n\equiv 0$ or 4 modulo 8 cases in (139) agrees with
(3) and admit the decomposition $SO(2n)_{1}\boxtimes D^{\omega}(D_{2k})$, for
$\omega=[k,0,0]$ or $[k,1,1]$. This is because, as UMTCs,
$SO(2n)_{1}=D^{[0]}(\mathbb{Z}_{2})$ or $SO(8)_{1}$ when $n\equiv 0$ or 4
modulo 8 respectively, and $D^{[0]}(\mathbb{Z}_{2})\boxtimes
D^{[\omega]}(D_{2k})=D^{[\omega]}(D_{2k})$ after condensing the
$\mathbb{Z}_{2}$ charge pair.

##  V Conclusion and discussion

We constructed the 2+1D electronic coupled-wire models that describe the twist
liquid topological phases $SO(2n)_{1}/G$ in the orthogonal family, for
$G=\mathbb{Z}_{2}$, $\mathbb{Z}_{k}$ and $D_{k}$, as well as
$SU(n)_{1}/\mathbb{Z}_{2}$ and $U(1)_{l}/\mathbb{Z}_{2}$ in the unitary
family. The degree, rank and level are related by $k=n$ and $l=4n$ ($k=n/2$
and $l=n$) when $n$ is odd (even). We presented their bosonic topological
orders and described the fusion and braiding properties of anyon excitations.
A more detailed summary of results can be found in section I.1 in the
introduction. Here, we discuss the implications, unanswered questions,
speculations, and possible future directions.

##### Boson wires with emergent Majorana fermions

In section II, we presented two electronic origins of the boson wires, whose
low-energy degrees of freedom are described by $N$ counter-propagating pairs
of emergent (non-local) massless Majorana fermions that constitute the
$SO(N)_{1}$ WZW CFT in 1+1D. The superconducting version relies on many-body
pairing interactions that break the charge $U(1)$ conservation. The spin
liquid version depends on many-body umklapp backscattering processes that
break the translation symmetry. The former can be facilitated by proximity to
a normal $s$-wave bulk superconductor and the latter can arise when the
electron filling is commensurate with a 1D lattice. Both interactions are
relevant in the RG sense when strong enough electron density-density
interactions are present. Since the interactions in both cases introduce a
finite energy gap to all odd charge and fermion excitations, we anticipate
these boson wires to exhibit unique spin-charge separated transport
properties. This may include shot-noise Schottky (1918); Blanter and Büttiker
(2000) signatures of charge and spin tunneling, and the violation of the
Wiedemann–Franz Franz and Wiedemann (1853) relation between electric and
thermal conductance. The dissipation-fluctuation behaviors and the effect of
disorder in these interacting boson systems are uncertain. These exotic
electron-based quasi-1D gapless boson systems may be supported in strongly-
correlated multi-channeled materials such as superconducting carbon nanotubes
Iijima (1991); Morpurgo  _et al._ (1999); Kociak  _et al._ (2001); Tang  _et
al._ (2001); Jarillo-Herrero  _et al._ (2006); Hamo  _et al._ (2016) and
ballistic Rashba spin-orbit coupled nanowires such as InSb/InAs Björk  _et
al._ (2005); Nilsson  _et al._ (2009); Plissard  _et al._ (2012); Liang and
Gao (2012) or LaAlO3/SrTiO3 waveguides Cheng  _et al._ (2016); Annadi  _et
al._ (2018); Briggeman  _et al._ (2020). They may also appear along gapless
edges or interfaces inside 2D topological states Zhang and Kane (2014); Orth
_et al._ (2015) such as biased (twisted) bilayer graphene Ju  _et al._ (2015);
Li  _et al._ (2016); Lee  _et al._ (2017); Efimkin and MacDonald (2018); Wu
_et al._ (2019a); Winn  _et al._ (2019); Tsim  _et al._ (2020) or transition
metal dichalcogenides Wu  _et al._ (2018, 2019b), and high Chern number
quantum anomalous Hall insulators Thouless  _et al._ (1982); Haldane (1988);
Liu  _et al._ (2016) such as magnetically dopped topological insulating
Bi2(Se,Te)3 film Jiang  _et al._ (2012); Wang  _et al._ (2013); Zhao  _et al._
(2020), topological crystalline insulating SnTe films Fang  _et al._ (2014),
and MnBi2Te4 flakes Ge  _et al._ (2020).

##### Generalized particle-hole conjugations

Some fractional quantum Hall states in a partially-filled Landau level are
pairwise related by particle-hole (PH) conjugation Girvin (1984); Balram and
Jain (2017); Nguyen  _et al._ (2018); Pan  _et al._ (2020). For example, the
Laughlin FQH state $U(1)_{3}$ of particles at $\nu=1/3$ is conjugate to the
Laughlin state of holes $U(1)_{1}\times\overline{U(1)_{3}}$ at $\nu=2/3$.
Here, $U(1)_{1}$ is the filled Landau level at $\nu=1$, and
$\overline{\mathcal{C}}$ is the time-reversal conjugate of $\mathcal{C}$. In
particular, PH conjugation relates FQH states at filling $\nu=1/2$. For
example, the Moore-Read Pfaffian state Moore and Read (1991); Greiter  _et
al._ (1991) $U(1)_{8}\otimes_{f}\mathrm{Ising}$ is conjugate to the anti-
Pfaffian state Levin  _et al._ (2007); Lee  _et al._ (2007)
$U(1)_{1}\times\overline{U(1)_{8}\otimes_{f}\mathrm{Ising}}=U(1)_{8}\otimes_{f}\overline{\mathrm{Ising}}^{3}$.
The PH symmetric Pfaffian state Son (2015)
$U(1)_{8}\otimes_{f}\overline{\mathrm{Ising}}$ is its own PH conjugate. The
notion of PH conjugation can be generalized by replacing the Landau level by a
prototype topological phase $\mathcal{P}$. Any topological state $\mathcal{C}$
that “partially occupies” $\mathcal{P}$ has a PH conjugate
$\mathrm{PH}(\mathcal{C})=\mathcal{P}\boxtimes\overline{\mathcal{C}}$, where
the tensor product, in general, is relative to a certain set of condensed
bosonic pairs of anyons. Generalized PH conjugation has been explored in
paired parton FQH states Sirota  _et al._ (2019) at $\nu=1/6$ that half-fill
the parton Landau level $\mathcal{P}=U(3)_{1}=U(1)_{3}\times SU(3)_{1}$ at
$\nu=1/3$, and Fibonacci QH states Lopes  _et al._ (2019) at $\nu=8$ that
half-fill $\mathcal{P}=(E_{8})_{1}$ at $\nu=16$. In section III.3.2 and IV, we
made use of the conformal embedding $U(1)_{l}\times SU(n)_{1}\subseteq
SO(2n)_{1}$ in (65) and the coset decomposition $SO(n)_{2}\times
SO(n)_{1}^{2}/SO(n)_{2}=SO(n)_{1}^{2}$ in (76). They imply the generalized PH
conjugations based on the prototype topological phases
$\mathcal{P}=SO(2n)_{1}$ or $SO(n)_{1}^{2}$. For example, (i) $U(1)_{l}$ and
$SU(n)_{1}$ is a PH conjugate pair based on $\mathcal{P}=SO(2n)_{1}$ because
$U(1)_{l}=SO(2n)_{1}\boxtimes\overline{SU(n)_{1}}$ and
$SU(n)_{1}=SO(2n)_{1}\boxtimes\overline{U(1)_{l}}$. The tensor product
$\boxtimes$ may be the decoupled one $\times$ or a relative one that condenses
a fermion pair or a $\mathbb{Z}_{2}$ charge pair. The PH conjugation in
example (i) extends to the corresponding $\mathbb{Z}_{2}$ twist liquids. (ii)
$U(1)_{l}/\mathbb{Z}_{2}=SO(n)_{1}^{2}/SO(n)_{2}$ and
$SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ is a PH conjugate pair based on
$\mathcal{P}=SO(2n)_{1}/\mathbb{Z}_{2}=SO(n)_{1}^{2}$. We expect the
generalized notion of PH conjugation can be made precise in the coupled-wire
setting Mross  _et al._ (2017); Fuji and Furusaki (2019a) by a non-local
representation of the PH operator, which may be interpreted as the 2+1D analog
of the Kramers-Wannier duality in 1+1D.

##### 2+1D quantum critical phase transitions

The coupled-wire models described in section III and IV encompass a variety of
states. In some cases, by tuning the relative strengths of intra- and inter-
wire backscattering interactions, a gap-closing phase transition can be driven
from one topological phase to another. For example, (36), (41), (44) and (45)
describe exactly-solvable models for the $U(1)_{l}$, $SU(n)_{1}$, trivial and
$SO(2n)_{1}$ topological order respectively. These models are special points
in a moduli space of Hamiltonians and can go from one to another by tuning the
density-density interaction strengths $u_{\perp}^{\mathrm{intra}}$,
$u_{\perp}^{\mathrm{inter}}$, $\tilde{u}^{\mathrm{intra}}$,
$\tilde{u}^{\mathrm{inter}}$ and the sine-Gordon backscattering bare strength
$\Delta_{\perp}^{\mathrm{intra}}$, $\Delta_{\perp}^{\mathrm{inter}}$,
$\Delta^{\mathrm{intra}}$, $\Delta^{\mathrm{inter}}$. The relevance of the
$\Delta$’s in the RG sense is controlled by the $u$’s in the sliding Luttinger
liquid theory O’Hern  _et al._ (1999); Emery  _et al._ (2000); Vishwanath and
Carpentier (2001); Sondhi and Yang (2001); Mukhopadhyay  _et al._ (2001) in
2+1D. The four exactly-solvable models are connected in a phase diagram shown
in figure 4. The RG flow depends on sliding Luttinger liquid parameters. This
determines the stability of the special fixed points where the exactly-
solvable models sit, as well as the stability of other critical fixed points
that control the phase transition between the gapped topological phases.
Depending on the RG flow or the presence of marginal operators, we speculate
that the PH symmetric point at the center may host previously unexplored
quantum critical points. On the other hand, if the PH symmetric point can be
gapped without breaking the symmetry, it may support a further fractionalized
topological order that is not supported by any of the special fixed points
(c.f. the PH symmetric Pfaffian FQH state Son (2015) at $\nu=1/2$ between the
vacuum and the filled Landau level). Similar phase diagram is expected to
arise for the $\mathbb{Z}_{2}$ twist liquids (see the right diagram in figure
4), although the qualitative critical behavior may differ because of the
complexity of the non-Abelian model Hamiltonians.

Figure 4: Possible phase diagrams connecting the exactly-solvable models at
special fixed points.

##### Coupled-wire construction of related twist liquid topological phases

In this paper, we focus on the $D_{k}$ symmetry of $SO(2n)_{1}$ and the
$\mathbb{Z}_{2}$ conjugation of $SU(n)_{1}$ and $U(1)_{l}$. While the latter
is always an outer automorphism that non-trivially switches anyon types, the
former only transposes anyon types when $n$ is odd. In other words, the
$\mathbb{Z}_{2}$ symmetry in $SO(2n)_{1}$ considered in this paper is the
mirror symmetry of the $SO(2n)_{1}$ Dynkin diagram only when $n$ is odd.
Mirror outer automorphisms for even rank $n$, and the $S_{3}$ triality
symmetry of $SO(8)$ was not investigated. While the topological orders of the
$SO(2n)_{1}/\mathbb{Z}_{2}$, $SO(8)_{1}/\mathbb{Z}_{3}$ and $SO(8)_{1}/S_{3}$
are known Teo  _et al._ (2015); Chen  _et al._ (2017); Barkeshli  _et al._
(2019), microscopic descriptions of these chiral twist liquids states are
missing. (There exist non-chiral exactly-solvable string-net models Levin and
Wen (2005); Teo  _et al._ (2015), whose topological orders contain the above
states as chiral sub-sectors.) Moreover, the $D_{k}$ symmetry is only a
particular point group symmetry in $2n$ dimensions. In three dimensions, the
cyclic, dihedral, tetrahedral, octahedral, and icosahedral groups,
$\mathbb{Z}_{n}$, $D_{n}$, $T$, $O$ and $I$, are the orientation preserving
point groups in $SO(3)$. Their corresponding orbifold phases
$SU(2)_{1}/\mathbb{Z}_{n}=U(1)_{2n^{2}}$,
$SU(2)_{1}/D_{n}=U(1)_{2n^{2}}/\mathbb{Z}_{2}$, $SU(2)_{1}/T$, $SU(2)_{1}/O$,
and $SU(2)_{1}/I$, were studied in the CFT context. Ginsparg (1988b);
Dijkgraaf  _et al._ (1989); Ginsparg (1988a) The topological orders, as well
as their microscopic descriptions, of higher-dimensional point group orbifold
phases $SO(N)_{l}/G$ is largely unknown. Coupled-wire models of these
topological phases may be constructed by extending the methods used in this
paper. For example, the manipulation of fermion pair locality could be
generalized to $G$-invariant local fields in the $SO(N)_{l}$ WZW CFT. Further
investigation may lead to exotic symmetries or dualities and quantum critical
phase transitions similar to those discussed above.

##### Quantum doubles and non-chiral twist liquids

In (64), we saw that the quantum double $\mathcal{Z}_{n}(\mathbb{Z}_{2})$
appeared as the non-chiral component of the $SO(2n)_{1}/\mathbb{Z}_{2}$ twist
liquid when $n$ is odd. $\mathcal{Z}_{n}(\mathbb{Z}_{2})$ is the Drinfeld
center Kassel (1995); Bakalov and Kirillov (2001); Levin and Wen (2005) of
either the Ising or the $SU(2)_{2}$ fusion category, which are generated by
the simple fusion objects $1,\psi,\sigma$ with the inequivalent Frobenius-
Schur indicators Kitaev (2006)
$\varkappa_{\sigma}=d_{\sigma}\left[F^{\sigma\sigma\sigma}_{\sigma}\right]^{1}_{1}=1$
for the former or $\varkappa_{\sigma}=-1$ for the latter. The quantum double
is not a discrete gauge theory because the quantum dimension of the Ising
twist field, $d_{\sigma}=\sqrt{2}$, is non-integral. The distinction from a
gauge theory is a consequence of the $\mathbb{Z}_{2}$ symmetry of $SO(2n)_{1}$
being outer automorphic, i.e. exchanging anyon types. Similarly, we saw in
(131) that the $SO(2n)_{1}/D_{k}$ twist liquid, when $n$ is odd, carries the
non-chiral UMTC component
$\mathcal{Z}_{n}(D_{k})=\mathcal{Z}_{n}(\mathbb{Z}_{2})\boxtimes
D^{[0]}(D_{k})$. $\mathcal{Z}_{n}(D_{k})$ is not a gauge theory because it
supports the Ising-fluxon with non-integral dimension $d=\sqrt{2}k$. In this
paper, we also encountered the outer automorphic $\mathbb{Z}_{2}$ conjugation
symmetry in $SU(n)_{1}$ and $U(1)_{l}$. The corresponding twist liquids can be
decomposed tautologically into $SU(n)_{1}/\mathbb{Z}_{2}=SU(n)_{1}\boxtimes
Z(\mathcal{F}_{n})$ and $U(1)_{l}/\mathbb{Z}_{2}=U(1)_{l}\boxtimes
Z(\mathcal{F}_{l})$, where

$\displaystyle\mathcal{F}_{n}=\overline{SU(n)_{1}}\times\frac{SU(n)_{1}}{\mathbb{Z}_{2}},\quad\mathcal{F}_{l}=\overline{U(1)_{l}}\times\frac{U(1)_{l}}{\mathbb{Z}_{2}}$
(145)

and $\boxtimes$ anyon condenses $\Psi^{a}\overline{\Psi^{a}}$ and
$e^{a}\overline{e^{a}}$ respectively. We conjecture that the non-chiral UMTC
$\mathcal{Z}_{n}(D_{k})$ and $Z(\mathcal{F}_{n})$ are Drinfeld centers of the
following fusion categories, and therefore admit exactly-solvable string-net
model Levin and Wen (2005) Hamiltonians.

First, the non-chiral UMTC $Z(\mathcal{F}_{n})$ in (145) may be the Drinfeld
center of the defect fusion category Teo  _et al._ (2015); Teo (2016) (or
$G$-crossed category Etingof  _et al._ (2010); Barkeshli  _et al._ (2019))

$\displaystyle\mathcal{F}_{n}=\left\\{\begin{array}[]{*{20}l}\mathbb{Z}_{n}\cup\\{\sigma\\},\quad\mbox{if
$n$ is odd}\\\ \mathbb{Z}_{n}\cup\\{\sigma_{+},\sigma_{-}\\},\quad\mbox{if $n$
is even}\end{array}\right.$ (148)

Its simple fusion objects are the quasiparticles $\Psi^{a}$ in $SU(n)_{1}$ (or
$e^{a}$ in $U(1)_{l}$) that obey the $\mathbb{Z}_{n}$ fusion algebra
$\Psi^{a}\times\Psi^{a^{\prime}}=\Psi^{[a+a^{\prime}]}$, together with the
$\mathbb{Z}_{2}$ twist defect $\sigma$ when $n$ is odd, or $\sigma_{\pm}$ when
$n$ is even. The twist defects obey the (commutative) fusion rules

$\displaystyle\sigma\times\Psi^{a}=\sigma,\quad\sigma_{s}\times\Psi_{a}=\sigma_{s(-1)^{a}}$
$\displaystyle\sigma\times\sigma=\sum_{a=0}^{n-1}\Psi^{a},\quad\mbox{if $n$ is
odd}$ $\displaystyle\sigma_{s}\times\sigma_{s}=\sum_{a\mbox{
}\mathrm{even}}\Psi_{a},\quad\mbox{if $n\equiv 0$ mod 4},$
$\displaystyle\sigma_{s}\times\sigma_{s}=\sum_{a\mbox{
}\mathrm{odd}}\Psi_{a},\quad\mbox{if $n\equiv 2$ mod 4},$ (149)

where $s=\pm$. When $n$ is odd, $\mathcal{F}_{n}$ is a Tambara-Yamagami
category Tambara and Yamagami (1998).

Second, for the non-chiral Ising-fluxon topological order
$\mathcal{Z}_{n}(D_{k})$, it may be equivalent to the Drinfeld center
$Z(\mathrm{Ising}\ltimes\mathbb{Z}_{k})$, where
$\mathrm{Ising}\ltimes\mathbb{Z}_{k}$ is the fusion category

$\displaystyle\mathrm{Ising}\ltimes\mathbb{Z}_{k}=\left\langle\mathbb{I}^{r},\psi^{r},\sigma^{r}:r=0,1,\ldots,k-1\right\rangle.$
(150)

The (non-commutative) fusion rules Teo  _et al._ (2014a); Khan  _et al._
(2014); Teo  _et al._ (2015) are

$\displaystyle\mathbb{I}^{r}\times\mathbb{I}^{r^{\prime}}=\psi^{r}\times\psi^{r^{\prime}}=\mathbb{I}^{[r+r^{\prime}]},\quad\mathbb{I}^{r}\times\psi^{r^{\prime}}=\psi^{[r+r^{\prime}]},$
$\displaystyle\sigma^{r}\times\mathbb{I}^{r^{\prime}}=\sigma^{r}\times\psi^{r^{\prime}}=\sigma^{[r+r^{\prime}]},$
$\displaystyle\mathbb{I}^{r}\times\sigma^{r^{\prime}}=\psi^{r}\times\sigma^{r^{\prime}}=\sigma^{[r^{\prime}-r]}$
$\displaystyle\sigma^{r}\times\sigma^{r^{\prime}}=\mathbb{I}^{[r^{\prime}-r]}+\psi^{[r^{\prime}-r]},$
(151)

where $[r]$ wraps any integer $r$ to the range between 0 and $k-1$ by adding
or subtracting $k$ if necessary. The Frobenius-Schur indicator of
$\mathrm{Ising}\ltimes\mathbb{Z}_{k}$ is
$\varkappa_{\sigma^{0}}=d_{\sigma^{0}}\left[F^{\sigma^{0}\sigma^{0}\sigma^{0}}_{\sigma^{0}}\right]_{1}^{1}=(-1)^{(k^{2}-1)/8}$.
The Ising-fluxon fields group into the super-selection sector
$\Gamma=\\{\sigma^{0},\ldots,\sigma^{k-1}\\}$ in the Drinfeld center
$Z(\mathrm{Ising}\ltimes\mathbb{Z}_{k})$ and has the combined quantum
dimension $d=\sqrt{2}k$.

## Acknowledgments

We thank Steven H. Simon and Henry Chan for valuable discussion. JCYT is
supported by the National Science Foundation under Grant Number DMR-1653535.
YH is in part supported by grant EP/S020527/1 from EPSRC.

##  Appendix A The $SO(N)_{1}$ current algebra and its primary fields

We review the chiral $SO(N)$ WZW KM algebra at level 1 and its primary fields.
Ginsparg (1988a); Di Francesco  _et al._ (1999); Blumenhagen (2009) In section
II, we presented the electronic origins of the $SO(N)_{1}$ boson wires, which
are the fundamental building block of the topological models in this article.
The low-energy degrees of freedom of each boson wire is effectively described
by $N$ counter-propagating pairs of massless Majorana fermions
$\psi^{\sigma}_{r}=(\psi^{\sigma}_{r})^{\dagger}$, where $\sigma=L/R=+/-$ and
$r=1,\ldots,N$. The Hamiltonian density was given in (15). The fermions obey
the correlations

$\displaystyle\left\langle\psi^{r}_{L}(z)\psi^{s}_{L}(w)\right\rangle=\frac{\delta^{rs}}{z-w},\quad\left\langle\psi^{r}_{R}(\bar{z})\psi^{s}_{R}(\bar{w})\right\rangle=\frac{\delta^{rs}}{\bar{z}-\bar{w}},$
(152)

where $z$ and $w$ are complex Euclidean space-time parameters
$e^{2\pi(v\tau+ix)/l}$, $\tau$ is the Wick rotated Euclidean time, and $l$ is
the length of the 1D wire in a closed circle. The WZW KM current operators are
the Hermitian normal-ordered antisymmetric pairs of fermions

$\displaystyle
J^{rs}_{\sigma}=-J^{sr}_{\sigma}=i\psi^{r}_{\sigma}\psi^{s}_{\sigma}.$ (153)

They span the affine $SO(N)_{1}$ Lie algebra.

In this appendix, we mainly focus on the left-moving holomorphic sector
$J=J_{L}$. The anti-holomorhpic right-moving sector follows a similar
algebraic structure by conjugation $z\to\bar{z}$. The chiral current operators
obey the operator product expansion (OPE)

$\displaystyle\begin{split}&J^{rs}(z)J^{pq}(w)\\\
&=\frac{\delta^{(rs)(pq)}}{(z-w)^{2}}+\sum_{u<v}\frac{if_{(rs)(pq)(uv)}}{z-w}J^{uv}_{L}(w)+\ldots,\end{split}$
(154)

where $r<s$ and $p<q$. The structure constant of the $SO(N)$ Lie algebra is
$f_{(rs)(pq)(uv)}=\delta_{ur}\delta_{vq}\delta_{sp}-\delta_{ur}\delta_{vp}\delta_{sq}+\delta_{us}\delta_{rq}\delta_{vp}-\delta_{us}\delta_{vq}\delta_{rp}$.
The electronic origin of the boson wire dictates that all fermion pairs,
including the current operators $J^{\alpha}=J^{rs}$, are local and integral.

For even $N=2n$, it is sometimes convenient to express the current operators
in the complexified form by pairing the Majorana fermions into Dirac ones,
such as $d^{a}=(\psi^{a}+i\psi^{n+a})/\sqrt{2}=e^{i\phi^{a}}$, where
$a=1,\ldots,n$. The bosonized variables $\phi_{a}$ obey the correlations

$\displaystyle\left\langle\phi^{a}(z)\phi^{b}(w)\right\rangle=-\delta^{ab}\log(z-w)+\frac{i\pi}{2}\mathrm{sgn}(a-b)$
(155)

where the singular term implies (152) after exponentiation, and the non-
singular antisymmetric term ensures the Dirac fermions mutually anti-commute,
$d^{a}d^{b}=-d^{b}d_{a}$ for $a\neq b$. The Cartan-Weyl current generators are
the normal-ordered densities

$\displaystyle
H^{a}=i\psi^{2a-1}\psi^{2a}=(d^{a})^{\dagger}d^{a}=-i\partial\phi^{a}.$ (156)

The raising and lowering roots are the normal-ordered vertex operators

$\displaystyle E^{\boldsymbol{\alpha}}=\pm
i\left(d^{a}\right)^{\alpha_{a}}\left(d^{b}\right)^{\alpha_{b}}=e^{i\alpha_{c}\phi^{c}},$
(157)

where $d^{+}=d$ and $d^{-}=d^{\dagger}$. Here, each of the roots are labeled
by a vector $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{n})$ with integral
entries and of length $|\boldsymbol{\alpha}|=\sqrt{2}$. This means there are
only two non-zero entries $\alpha_{a},\alpha_{b}=\pm 1$, for $1\leq a<b<n$.
The $n(2n-1)$ roots form the simply-laced $D_{n}=SO(2n)$ root system of rank
$n$.

For odd $N=2n+1$, the $SO(2n+1)_{1}$ current algebra extends from $SO(2n)_{1}$
by including the raising and lowering operators

$\displaystyle E^{\pm{\bf e}_{a}}=i\left(d^{a}\right)^{\pm}\psi^{2n+1}=ie^{\pm
i\phi^{a}}\psi^{2n+1}$ (158)

that associate with the short root vectors $\pm{\bf e}_{a}$, which is an unit
integral vector of length $|{\bf e}_{a}|=1$ with its $a^{\mathrm{th}}$ entry
being the only non-zero one. Combining the long roots from $D_{n}$ and these
short roots, they form the non-simply-laced $B_{n}=SO(2n+1)$ root system of
rank $n$.

Primary fields $[V]$ of the $SO(N)_{1}$ WZW KM algebra are super-selection
sectors spanned by fields $V_{\mu}(z)$, for $\mu=1,\ldots,M$, that “rotate”
irreducibly under the action of the $SO(N)_{1}$ currents

$\displaystyle
J^{\alpha}(z)V_{\mu}(w)=\frac{1}{z-w}\left(t^{\alpha}\right)_{\mu}^{\nu}V_{\nu}(w)+\ldots,$
(159)

where $t^{\alpha}$ is a $M$-dimensional irreducible matrix representation of
the $SO(N)$ Lie algebra generator $J^{\alpha}$. At level 1, these irreducible
representations are the trivial, vector or spinor representations of $SO(N)$.
The vector representation $[\psi]$ is fermionic and carries the conformal
scaling dimension $h=1/2$. The super-selection sector is spanned by the
Majorana fermions $\psi^{r}$, for $r=1,\ldots,N$, and obeys the OPE

$\displaystyle
J^{pq}(z)\psi^{r}(w)=\frac{i}{z-w}\left(\delta^{qr}\delta^{ps}-\delta^{pr}\delta^{qs}\right)\psi^{s}(w)+\ldots.$
(160)

The fusion rule $[\psi]\times[\psi]=1$ indicates that fermion pairs belong to
the trivial super-selection sector 1, which consists of integral combinations
of local bosons. It abbreviates the OPE

$\displaystyle\psi^{p}(z)\psi^{q}(w)=\frac{\delta^{pq}}{z-w}-i(1-\delta^{pq})J^{(pq)}(w)+\ldots.$
(161)

Spinor fields transform according to the spinor representations of $SO(N)$.
For even $N=2n$, they decomposes into the even and odd spinors, $[s_{+}]$ and
$[s_{-}]$. The super-selection sectors are spanned by products of “half-
fermions”

$\displaystyle
s_{\boldsymbol{\varepsilon}}=e^{i\varepsilon_{a}\phi^{a}/2},\quad\boldsymbol{\varepsilon}=(\varepsilon_{1},\ldots,\varepsilon_{n})$
(162)
$\displaystyle\begin{split}[s_{+}]&=\mathrm{span}\left\\{e^{i\varepsilon_{a}\phi^{a}/2}:\varepsilon_{a}=\pm
1,\prod_{a=1}^{n}\varepsilon_{a}=+1\right\\},\\\
[s_{-}]&=\mathrm{span}\left\\{e^{i\varepsilon_{a}\phi^{a}/2}:\varepsilon_{a}=\pm
1,\prod_{a=1}^{n}\varepsilon_{a}=-1\right\\}.\end{split}$

The $SO(2n)_{1}$ currents transform the vertex operators into each other
within the same super-selection sector according to the OPE

$\displaystyle H^{a}(z)s_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\frac{-\varepsilon_{a}/2}{z-w}s_{\boldsymbol{\varepsilon}}(w)+\ldots$
$\displaystyle E^{\boldsymbol{\alpha}}(z)s_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\frac{Z^{\boldsymbol{\alpha}}_{\boldsymbol{\varepsilon}}}{z-w}s_{\boldsymbol{\varepsilon}+2\boldsymbol{\alpha}}(w)+\ldots,\quad\mbox{if
$\boldsymbol{\alpha}\cdot\boldsymbol{\varepsilon}=-2$}$ $\displaystyle
E^{\boldsymbol{\alpha}}(z)s_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\mbox{non-singular},\quad\mbox{if
$\boldsymbol{\alpha}\cdot\boldsymbol{\varepsilon}=0,2$}$ (163)

where
$Z^{\boldsymbol{\alpha}}_{\boldsymbol{\varepsilon}}=\exp\left[-i\pi\alpha_{a}\varepsilon_{b}\mathrm{sgn}(a-b)/4\right]$,
and $H^{a}$ and $E^{\boldsymbol{\alpha}}$ are the Cartan-Weyl generators and
raising/lowering operators defined in (156) and (157). The rotated spinor
$s_{\boldsymbol{\varepsilon}^{\prime}}=s_{\boldsymbol{\varepsilon}+2\boldsymbol{\alpha}}$
has the same parity as the original,
$\prod_{a}\varepsilon_{a}=\prod_{a}\varepsilon^{\prime}_{a}$, and belongs to
the same super-selection sector, because the raising/lowering operator flips
two signs in
$\boldsymbol{\varepsilon}=(\varepsilon_{a},\ldots,\varepsilon_{n})$. The above
OPE can be derived from the correlations (155) and the Baker-Campbell-
Hausdorff formula $e^{A(z)}e^{B(w)}=e^{A(z)+B(w)+\langle A(z)B(w)\rangle}$.

The two spinors have opposite fermion parities and can be flipped from one to
the other by absorbing or emitting a fermion. This is captured by the fusion
rule $[s_{\pm}]\times[\psi]=[s_{\mp}]$, which corresponds to the OPE

$\displaystyle d^{a}(z)^{-\varepsilon_{a}}s_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\frac{Z^{a}_{\boldsymbol{\varepsilon}}}{(z-w)^{1/2}}s_{\boldsymbol{\varepsilon}-2\varepsilon_{a}{\bf
e}_{a}}(w)+\ldots$ $\displaystyle
d^{a}(z)^{\varepsilon_{a}}s_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\mbox{non-singular}$ (164)

where
$Z^{a}_{\boldsymbol{\varepsilon}}=\exp\left[i\pi\sum_{b=1}^{n}\varepsilon_{a}\varepsilon_{b}\mathrm{sgn}(a-b)/4\right]$,
and $d^{+}=d$, $d^{-}=d^{\dagger}$. The $1/(z-w)^{1/2}$ factor indicates the
$\pi$ monodromy phase and the mutual semionic statistics between the fermion
and the spinors. The new spinor
$s_{\boldsymbol{\varepsilon}^{\prime}}=s_{\boldsymbol{\varepsilon}+2\varepsilon_{a}{\bf
e}_{a}}$ has opposite parity as
$\prod_{b}\varepsilon_{b}^{\prime}=-\prod_{b}\varepsilon_{b}$ because the
fermion flip one sign in
$\boldsymbol{\varepsilon}=(\varepsilon_{a},\ldots,\varepsilon_{n})$. The
conjugate pair

$\displaystyle
s_{\boldsymbol{\varepsilon}}(z)s_{-\boldsymbol{\varepsilon}}(w)=\frac{1}{(z-w)^{n/4}}+\ldots$
(165)

dictates the conformal scaling dimension of each spinor to be
$h_{s}=n/8=N/16$. A spinor $s_{\boldsymbol{\varepsilon}}$ has the same parity
as its conjugate $s_{-\boldsymbol{\varepsilon}}$, and the spinor super-
selection sectors are therefore self-conjugate with the fusion rule
$[s_{\pm}]\times[s_{\pm}]=1$, only when $n$ is even. In this case, the anyon
group $\mathcal{A}_{SO(2n)_{1}}=\\{1,[\psi],[s_{+}],[s_{-}]\\}$ has a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ Abelian group structure under fusion
product. Otherwise, when $n$ is odd, the spinor fusion rule become
$[s_{\pm}]\times[s_{\mp}]=1$, and the anyon group is
$\mathcal{A}_{SO(2n)_{1}}=\mathbb{Z}_{4}$ because
$[s_{\pm}]\times[s_{\pm}]=[\psi]$ and $[s_{\pm}]^{4}=1$, i.e. the spinors are
of order 4. The fusion rules of $SO(2n)_{1}$ are summarized below (also in
(26)).

$\displaystyle[\psi]\times[\psi]=1,\quad[\psi]\times[s_{\pm}]=[s_{\mp}],$
$\displaystyle[s_{\pm}]\times[s_{\pm}]=\left\\{\begin{array}[]{*{20}l}1,&\mbox{for
even $n$}\\\ {}[\psi],&\mbox{for odd $n$}\end{array}\right..$ (168)

For odd $N=2n+1$, the field content of $SO(2n+1)_{1}$ may be decomposed
according to $SO(2n)_{1}\times\mathrm{Ising}$, where the Ising sector is
generater by the additional Majorana fermion $\psi^{2n+1}$. The Ising CFT has
primary fields $1,\psi,\sigma$. The non-Abelian Ising twist field has
conformal scaling dimension $1/16$ and follows the multi-channel fusion rule

$\displaystyle\sigma\times\sigma=1+\psi,\quad\psi\times\sigma=\sigma,$ (169)

where $\psi=\psi^{2n+1}$ when embedded in $SO(2n+1)_{1}$. The Ising twist
field is non-local and has semionic mutual statistics with $\psi^{2n+1}$.
Since the fermion pairs $\psi^{r}\psi^{2n+1}$ are local, the Ising twist field
must appear simultaneously with a $SO(2n)_{1}$ spinor
$s_{\boldsymbol{\varepsilon}}$. The $SO(2n+1)_{1}$ spinor super-selection
sector $[\sigma]$ is spanned by the normal-ordered products
$\sigma_{\boldsymbol{\varepsilon}}=\sigma s_{\boldsymbol{\varepsilon}}$

$\displaystyle[\sigma]=\mathrm{span}\left\\{\sigma_{\boldsymbol{\varepsilon}}=\sigma
e^{i\varepsilon_{a}\phi^{a}/2}:\varepsilon_{a}=\pm 1\right\\},$ (170)

where the parity $\prod_{a=1}^{n}\varepsilon_{a}$ is not fixed. The conformal
scaling dimension is $h_{\sigma}=1/16+n/8=N/16$. The actions of the Cartan-
Weyl generators $H^{a}$ and long roots $E^{\boldsymbol{\alpha}}$ only affect
the Abelian component $s_{\boldsymbol{\varepsilon}}$ and are dictated by
(163).

$\displaystyle H^{a}(z)\sigma_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\frac{-\varepsilon_{a}/2}{z-w}\sigma_{\boldsymbol{\varepsilon}}(w)+\ldots$
$\displaystyle E^{\boldsymbol{\alpha}}(z)\sigma_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\frac{Z^{\boldsymbol{\alpha}}_{\boldsymbol{\varepsilon}}}{z-w}\sigma_{\boldsymbol{\varepsilon}+2\boldsymbol{\alpha}}(w)+\ldots,\quad\mbox{if
$\boldsymbol{\alpha}\cdot\boldsymbol{\varepsilon}=-2$}$ $\displaystyle
E^{\boldsymbol{\alpha}}(z)\sigma_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\mbox{non-singular},\quad\mbox{if
$\boldsymbol{\alpha}\cdot\boldsymbol{\varepsilon}=0,2$}.$ (171)

Using the fermion action (164), the short roots (158) transform the spinors
according to

$\displaystyle E^{\pm a}(z)\sigma_{\boldsymbol{\varepsilon}}(w)$
$\displaystyle=\frac{Z^{a}_{\boldsymbol{\varepsilon}}}{z-w}\sigma_{\boldsymbol{\varepsilon}-2\varepsilon_{a}{\bf
e}_{a}}(w)+\ldots,\quad\mbox{if $\varepsilon_{a}=\mp$},$ $\displaystyle E^{\pm
a}(z)\sigma_{\boldsymbol{\varepsilon}}(w)$ $\displaystyle=\mbox{non-
singular},\quad\mbox{if otherwise}.$ (172)

The fusion rules of $SO(2n)_{1}$ are summarized below.

$\displaystyle[\psi]\times[\psi]=1,\quad[\psi]\times[\sigma]=[\sigma],\quad[\sigma]\times[\sigma]=1+[\psi].$
(173)

##  Appendix B The anyon content of $SO(n)_{2}$ and $U(1)_{l}/\mathbb{Z}_{2}$

We first review the chiral $SO(n)$ WZW KM algebra at level 2 and its primary
field content. Ginsparg (1988a); Di Francesco  _et al._ (1999); Blumenhagen
(2009) The corresponding topological phase was introduced in section IV.1. We
focus on the holomorphic left-moving sector. The current algebra is generated
by the diagonal sums ${\bf J}={\bf J}^{A}+{\bf J}^{B}$ in the double
$SO(n)_{1}^{A}\times SO(n)_{1}^{B}$. They obey the current OPE (82). The
$SO(n)_{2}$ currents can also be identified as self-conjugate combinations of
the $SU(n)_{1}$ currents $J_{ab}\sim\cos(\phi_{a}-\phi_{b})\sim
d_{a}d_{b}^{\dagger}+d_{b}d_{a}^{\dagger}$ (see (84)), where
$d_{a}=(\psi_{a}^{A}+i\psi_{a}^{B})/\sqrt{2}\sim e^{i\phi_{a}}$ are the $n$
Dirac fermions that rotate under $SU(n)$. The hermitian $SO(n)_{2}$ currents
are symmetric under the internal $\mathbb{Z}_{2}$ symmetry,
$\phi_{a}\to-\phi_{a}$ (see also (79)), and therefore
$SO(n)_{2}=SU(n)_{1}/\mathbb{Z}_{2}$.

The primary fields of $SO(n)_{2}=SU(n)_{1}/\mathbb{Z}_{2}$ together with their
quantum dimensions and spins are listed in table 1 and 2. They consist of (a)
Abelian sectors with quantum dimension $d=1$, (b) non-Abelian super-selection
sectors $\Phi^{i}$ with dimension $d=2$, for $1\leq i<n/2$, and (c)
$\mathbb{Z}_{2}$ twist fields $\sigma$ and $\tau$ when $n$ is odd, or
$\sigma^{\pm}$ and $\tau^{\pm}$ when $n$ is even. Abelian primary fields
includes (i) the vacuum sector containing local fields such as the $SO(n)_{2}$
currents, (ii) the $\mathbb{Z}_{2}$ charge sector $S$ containing fermion pair
combinations that are odd under $\mathbb{Z}_{2}$ (see (85)), and (iii) when
$n$ is even, the $\mathbb{Z}_{2}$-closed $SU(n)_{1}$ primary field sectors
$\Psi^{n/2}_{\pm}$ (see (93)). When $n$ is odd, none of the $SU(n)_{1}$
primary fields, except the vacuum, are closed under $\mathbb{Z}_{2}$, and
$\Psi^{n/2}_{\pm}$ are absent. The Abelian sectors obey the fusion rules

$\displaystyle\begin{split}&\Psi^{n/2}_{\pm}\times S=\Psi^{n/2}_{\mp},\quad
S\times S=1,\\\
&\Psi^{n/2}_{\pm}\times\Psi^{n/2}_{\pm}=\left\\{\begin{array}[]{*{20}l}1,&\mbox{if
$n\equiv 0$ mod 4}\\\ S,&\mbox{if $n\equiv 2$ mod
4}\end{array}\right..\end{split}$ (174)

The set of Abelian anyons form the fusion group $\mathbb{Z}_{2}=\\{1,S\\}$
when $n$ is odd,
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}=\\{1,S,\Psi^{n/2}_{\pm}\\}$ when $n\equiv
0$ modulo 4, or $\mathbb{Z}_{4}=\\{1,S,\Psi^{n/2}_{\pm}\\}$ when $n\equiv 2$
modulo 4.

The internal $\mathbb{Z}_{2}$ symmetry conjugates the $SU(n)_{1}$ primary
fields $\Psi^{i}\leftrightarrow\Psi^{n-i}$, and therefore each conjugate pair
forms the $d=2$ super-selection sector $\Phi^{i}=\Psi^{i}+\Psi^{n-i}$, for
$1\leq i<n/2$. (See (42) for the bosonized vertex representation of
$\Psi^{i}$.) They do not carry a $\mathbb{Z}_{2}$ parity because

$\displaystyle\Phi^{i}\times S=\Phi^{i}.$ (175)

They obey the fusion rules

$\displaystyle\Phi^{i}\times\Phi^{j}=\Phi^{i+j}+\Phi^{j-i},\quad\mbox{for
$1\leq i<j<n/2$ and $i+j\neq n/2$}$
$\displaystyle\Phi^{i}\times\Phi^{j}=\Psi^{n/2}_{+}+\Psi^{n/2}_{-}+\Phi^{j-i},\quad\mbox{for
$1\leq i<j<n/2$ and $i+j=n/2$}$
$\displaystyle\Phi^{i}\times\Phi^{i}=1+S+\Phi^{2i},\quad\mbox{for $i\neq
n/4$}$ (176)
$\displaystyle\Phi^{n/4}\times\Phi^{n/4}=1+S+\Psi^{n/2}_{+}+\Psi^{n/2}_{-}$
$\displaystyle\Psi^{n/2}_{\pm}\times\Phi^{i}=\Phi^{n/2-i}$

where we identify $\Phi^{i}=\Phi^{n-i}$ if $n/2<i<n$.

When $n$ is odd, there are two $\mathbb{Z}_{2}$ twist fields, $\sigma$ and
$\tau$, that differ from each other by a $\mathbb{Z}_{2}$ charge,
$\sigma=\tau\times S$ and $\tau=\sigma\times S$. They have quantum dimension
$d=\sqrt{n}$. When $n$ is even, there are four $\mathbb{Z}_{2}$ twist fields
$\sigma^{\pm}$ and $\tau^{\pm}$. They have quantum dimension $d=\sqrt{n/2}$.
The $\pm$ sign is flipped when absorbing a $SU(n)_{1}$ primary field
$\Psi^{i}$ with odd degree $i$. In addition to the fusion rules (94-100), when
$n$ is even, the $\mathbb{Z}_{2}$ fluxes also obey the following product rules
when combining with $\Psi^{n/2}_{\pm}$.

$\displaystyle\left\\{\begin{array}[]{*{20}l}\Psi^{n/2}_{\pm}\times\sigma^{\pm}=\sigma^{\pm},&\Psi^{n/2}_{\pm}\times\tau^{\pm}=\tau^{\pm},\\\
\Psi^{n/2}_{\pm}\times\sigma^{\mp}=\tau^{\mp},&\Psi^{n/2}_{\pm}\times\tau^{\mp}=\sigma^{\mp},\end{array}\right.\quad\mbox{if
$n\equiv 0$ mod 4},$ (179)
$\displaystyle\left\\{\begin{array}[]{*{20}l}\Psi^{n/2}_{\pm}\times\sigma^{\pm}=\tau^{\mp},&\Psi^{n/2}_{\pm}\times\tau^{\pm}=\sigma^{\mp},\\\
\Psi^{n/2}_{\pm}\times\sigma^{\mp}=\sigma^{\pm},&\Psi^{n/2}_{\pm}\times\tau^{\mp}=\tau^{\pm},\end{array}\right.\quad\mbox{if
$n\equiv 2$ mod 4}.$ (182)

The braiding phase between anyon $x$ and $y$ with a fixed admissible fusion
channel $z$ is $R^{xy}_{z}R^{yx}_{z}=e^{2\pi i(h_{z}-h_{x}-h_{y})}$, where the
anyon spins, $h_{x}$, are listed in table 1 and 2. (For $SO(n)_{2}$, there is
no fusion degeneracies. The fusion number $N_{xy}^{z}$ is either 0 or 1
depending on whether $z$ is an admissible fusion channel of $x\times y$.
Hence, the exchange “matrices” $R^{xy}_{z}$ are $U(1)$ scalars.) The modular
$S$-matrix is

$\displaystyle S_{xy}=\frac{1}{\mathcal{D}}\sum_{z}d_{z}N_{xy}^{z}e^{2\pi
i(h_{z}-h_{x}-h_{y})}$ (183)

where the total quantum dimension of $SO(n)_{2}$ is $\mathcal{D}=2\sqrt{n}$.
The modular $T$-matrix is $T_{xy}=e^{i\pi(1-n)/12}e^{2\pi ih_{x}}\delta_{xy}$
where the anyon spins $h_{x}$ are listed in table 1 and 2. Together the $S$
and $T$ matrices generate a representation of the modular group
$SL(2,\mathbb{Z})$. They obey

$\displaystyle S^{2}=(ST^{\dagger})^{3},\quad S^{4}=\mathbb{I},$ (184)

where $C=S^{2}$ is the conjugation matrix so that $C_{x\bar{x}}=1$ if $x$ and
$\bar{x}$ are conjugate pair, $x\times\bar{x}=1+\ldots$, and
$C_{xx^{\prime}}=0$ if $x^{\prime}\neq\bar{x}$. The fusion rules (94-100) and
(174-182) are related to the $S$-matrix by the Verlinde formula Verlinde
(1988)

$\displaystyle
N_{x_{1}x_{2}}^{y}=\sum_{z}\frac{S_{x_{1}z}S_{x_{2}z}S_{yz}^{\ast}}{S_{1z}},$
(185)

where the fusion number $N_{x_{1}x_{2}}^{y}$ describes $x_{1}\times
x_{2}=\sum_{y}N_{x_{1}x_{2}}^{y}y$. The modular $S$-matrices of
$SO(n)_{2}=SU(n)_{1}/\mathbb{Z}_{2}$ are listed in table 4 and 5.

| 1 | $S$ | $\Phi^{i}$ | $\sigma$ | $\tau$
---|---|---|---|---|---
1 | 1 | 1 | 2 | $\sqrt{n}$ | $\sqrt{n}$
$S$ | 1 | 1 | 2 | $-\sqrt{n}$ | $-\sqrt{n}$
$\Phi^{i^{\prime}}$ | 2 | 2 | $4c(ii^{\prime})$ | 0 | 0
$\sigma$ | $\sqrt{n}$ | $-\sqrt{n}$ | 0 | $\sqrt{n}$ | $-\sqrt{n}$
$\tau$ | $\sqrt{n}$ | $-\sqrt{n}$ | 0 | $-\sqrt{n}$ | $\sqrt{n}$
Table 4: The modular $S$-matrix $\mathcal{D}S_{xy}=2\sqrt{n}S_{xy}$ from (183) of $SO(n)_{2}=SU(n)_{1}/\mathbb{Z}_{2}$, for odd $n$. The rows and columns are arranged in the same order of the primary fields in table 1. $c(ii^{\prime})=\cos(2\pi ii^{\prime}/n)$, for $i,i^{\prime}=1,\ldots,(n-1)/2$. | 1 | $S$ | $\Psi^{n/2}_{s}$ | $\Phi^{i}$ | $\sigma^{s}$ | $\tau^{s}$
---|---|---|---|---|---|---
1 | 1 | 1 | 1 | 2 | $\sqrt{n/2}$ | $\sqrt{n/2}$
$S$ | 1 | 1 | 1 | 2 | $-\sqrt{n/2}$ | $-\sqrt{n/2}$
$\Psi^{n/2}_{s^{\prime}}$ | 1 | 1 | $(-1)^{n/2}$ | $2(-1)^{i}$ | $\sqrt{n/2}e^{i\pi n/4}ss^{\prime}$ | $\sqrt{n/2}e^{i\pi n/4}ss^{\prime}$
$\Phi^{i^{\prime}}$ | 2 | 2 | $2(-1)^{i^{\prime}}$ | $4c(ii^{\prime})$ | 0 | 0
$\sigma^{s^{\prime}}$ | $\sqrt{n/2}$ | $-\sqrt{n/2}$ | $\sqrt{n/2}e^{i\pi n/4}ss^{\prime}$ | 0 | $\Sigma^{ss^{\prime}}_{\sigma\sigma}$ | $\Sigma^{ss^{\prime}}_{\sigma\tau}$
$\tau^{s^{\prime}}$ | $\sqrt{n/2}$ | $-\sqrt{n/2}$ | $\sqrt{n/2}e^{i\pi n/4}ss^{\prime}$ | 0 | $\Sigma^{ss^{\prime}}_{\tau\sigma}$ | $\Sigma^{ss^{\prime}}_{\tau\tau}$
Table 5: The modular $S$-matrix $\mathcal{D}S_{xy}=2\sqrt{n}S_{xy}$ from (183)
of $SO(n)_{2}=SU(n)_{1}/\mathbb{Z}_{2}$, for even $n$. The rows and columns
are arranged in the same order of the primary fields in table 2.
$c(ii^{\prime})=\cos(2\pi ii^{\prime}/n)$, for $i,i^{\prime}=1,\ldots,n/2-1$
and $s,s^{\prime}=\pm$. The $\Sigma$ matrices are defined in (186). The
$S$-matrix of $U(1)_{n}/\mathbb{Z}_{2}$ is obtained by replacing
$e^{n/2}_{\pm}\leftrightarrow\Psi^{n/2}_{\pm}$ and
$\mathcal{E}^{i}\leftrightarrow\Phi^{i}$.

$\displaystyle\begin{split}\begin{pmatrix}\Sigma^{ss^{\prime}}_{\sigma\sigma}&\Sigma^{ss^{\prime}}_{\sigma\tau}\\\
\Sigma^{ss^{\prime}}_{\tau\sigma}&\Sigma^{ss^{\prime}}_{\tau\tau}\end{pmatrix}&=\sqrt{n}\left(\begin{smallmatrix}1&0&-1&0\\\
0&1&0&-1\\\ -1&0&1&0\\\ 0&-1&0&1\end{smallmatrix}\right),\quad\mbox{if
$n\equiv 0$ mod 8},\quad\sqrt{n}\left(\begin{smallmatrix}0&1&0&-1\\\
1&0&-1&0\\\ 0&-1&0&1\\\ -1&0&1&0\end{smallmatrix}\right),\quad\mbox{if
$n\equiv 4$ mod 8},\\\
&=\sqrt{\frac{n}{2}}e^{\frac{i\pi}{4}}\left(\begin{smallmatrix}1&-i&-1&i\\\
-i&1&i&-1\\\ -1&i&1&-i\\\ i&-1&-i&1\end{smallmatrix}\right),\quad\mbox{if
$n\equiv 2$ mod
8},\quad\sqrt{\frac{n}{2}}e^{-\frac{i\pi}{4}}\left(\begin{smallmatrix}1&i&-1&-i\\\
i&1&-i&-1\\\ -1&-i&1&i\\\ -i&-1&i&1\end{smallmatrix}\right),\quad\mbox{if
$n\equiv 6$ mod 8}.\end{split}$ (186)

Next, we review the modular data of the $U(1)_{l}/\mathbb{Z}_{2}$ orbifold
Ginsparg (1988b); Dijkgraaf  _et al._ (1988, 1989) and its primary field
(anyon) content, where the level $l$ is $n$ ($4n$) when $n$ is even (odd). The
corresponding topological phase was introduced in section IV.2. The orbifold
CFT is identical to the coset $[SO(n)_{1}]^{2}/SO(n)_{2}$. Ginsparg (1988a)
The topological phase can therefore be identified (as an UMTC) as the relative
tensor product in (113), which we repeat here

$\displaystyle U(1)_{n}/\mathbb{Z}_{2}=\left[SO(n)_{1}^{A}\times
SO(n)_{1}^{B}\right]\boxtimes\overline{SO(n)_{2}},$ (187)

for even $n$. Here, $\overline{SO(n)_{2}}$ is the time-reversal of $SO(n)_{2}$
with the same fusion rules,
$N_{\overline{x}\overline{y}}^{\overline{z}}=N_{xy}^{z}$, but conjugated
spins, $h_{\overline{x}}=-h_{x}$. The following set of anyon pairs

$\displaystyle\mathcal{B}=\left\\{1,\psi^{A}\psi^{B}\overline{S},s_{\pm}^{A}s_{\pm}^{B}\overline{\Psi^{n/2}_{\pm}}\right\\}$
(188)

are condensed in the relative tensor product $\boxtimes$. The Abelian bosons
in $\mathcal{B}$ have trivial mutual statistics and follows
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ ($\mathbb{Z}_{4}$) fusion group structure
when $n\equiv 0$ ($n\equiv 2$) modulo 4.

The $U(1)_{n}/\mathbb{Z}_{2}$ primary fields together with their quantum
dimensions and spins are listed in table 3 (by substituting $l=n$). They
consists of (a) four Abelian fields $1,S,e^{n/2}_{\pm}$, (b) the non-Abelian
super-selection sectors $\mathcal{E}^{i}=e^{i}+e^{-i}$, for
$i=1,\ldots,n/2-1$, and (c) the $\mathbb{Z}_{2}$ twist fields $\sigma^{\pm}$
and $\tau^{\pm}$. In the anyon condensation picture (187), all bosons in (188)
belong to the vacuum sector. The non-vacuum anyons in
$U(1)_{n}/\mathbb{Z}_{2}$ can be identified with anyons in
$\left[SO(n)_{1}^{A}\times SO(n)_{1}^{B}\right]\boxtimes\overline{SO(n)_{2}}$
according to

$\displaystyle\begin{split}&S\equiv\psi^{A}\psi^{B}\equiv\overline{S},\quad\sigma^{\pm}\equiv
s_{\pm}^{A}\overline{\sigma^{\pm}},\quad\tau^{\pm}\equiv
s_{\pm}^{A}\overline{\tau^{\pm}},\\\
&e^{n/2}_{\pm}\equiv\left\\{\begin{array}[]{*{20}l}s_{\pm}^{A}s_{\pm}^{B}\equiv\overline{\Psi^{n/2}_{\pm}},&\mbox{if
$n\equiv 0$ mod 4}\\\
s_{\pm}^{A}s_{\mp}^{B}\equiv\psi^{A}\overline{\Psi^{n/2}_{\pm}},&\mbox{if
$n\equiv 2$ mod 4}\end{array}\right.,\\\
&\mathcal{E}^{i}\equiv\left\\{\begin{array}[]{*{20}l}\overline{\Phi^{i}},&\mbox{for
even $i$}\\\ \psi^{A}\overline{\Phi^{i}},&\mbox{for odd
$i$}\end{array}\right..\end{split}$ (189)

Consequently, $U(1)_{n}/\mathbb{Z}_{2}$ have the identical fusion rules as
$SO(n)_{2}=SU(n)_{1}/\mathbb{Z}_{2}$ (for even $n$) by replacing
$e^{n/2}_{\pm}\leftrightarrow\Psi^{n/2}_{\pm}$ and
$\mathcal{E}^{i}\leftrightarrow\Phi^{i}$ in (98-100) and (174-182). The
modular $S$-matrix of $U(1)_{n}/\mathbb{Z}_{2}$ can be computed using (183).
It is identical to that of $SO(n)_{2}=SU(n)_{1}/\mathbb{Z}_{2}$ and is
summarized in table 5 under the same anyon label replacement. 222We observe
the following differences in the $S$-matrices of $U(1)_{n}/\mathbb{Z}_{2}$
between our results and ref. Dijkgraaf  _et al._ , 1989. While they are
identical when $n\equiv 0,2$ mod 8, the $S$-matrices are conjugated when
$n\equiv 6$ and the $S_{\sigma^{s}\sigma^{s^{\prime}}}$,
$S_{\tau^{s}\tau^{s^{\prime}}}$, $S_{\sigma^{s}\tau^{s^{\prime}}}$ entries are
unequal when $n\equiv 4$ mod 8. Together with the $T$-matrix
$T_{xy}=e^{-i\pi/12}e^{2\pi ih_{x}}\delta_{xy}$, they represent the modular
group $SL(2,\mathbb{Z})$, and obey (184) and (185).

##  Appendix C Discrete gauge theories with cyclic, dihedral or dicyclic gauge
groups

We review the topological orders of discrete gauge theories Bais  _et al._
(1992); de Wild Propitius (1995); de Wild Propitius and Bais (1995); Preskill
(2004) relevant to the twist liquid orbifold phases of $SO(2n)_{1}$. The
discrete gauge theories encountered in this article include

  1. 1.

$D^{[u]}(\mathbb{Z}_{k})$ of the cyclic gauge group $\mathbb{Z}_{k}$ with
arbitrary order $k$ and Dijkgraaf-Witten invariant $[u]=[0]$ or $[k/2]$ in
$H^{3}(\mathbb{Z}_{k},U(1))=\mathbb{Z}_{k}$,

  2. 2.

$D^{[u,v,w]}(D_{2k})$ of the dihedral gauge group $D_{2k}$ with even degree
$2k$ (order $4k$) and Dijkgraaf-Witten invariant $[u,v,w]=[k,0,0]$ or
$[k,1,1]$ in
$H^{3}(D_{2k},U(1))=\mathbb{Z}_{k}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$,
and

  3. 3.

$D^{[u]}(Q_{4k})$ of the dicyclic gauge group $Q_{4k}$ with odd degree $k$
(order $4k$) and Dijkgraaf-Witten invariant $[u]=[2k]$ in
$H^{3}(Q_{4k},U(1))=\mathbb{Z}_{4k}$.

They appear in the non-chiral components in the $SO(2n)_{1}/\mathbb{Z}_{2}$,
$SO(2n)_{1}/\mathbb{Z}_{k}$ and $SO(2n)_{1}/D_{k}$ twist liquids (see (56),
(70) and (139) respectively). In this appendix, while focusing on the above
gauge theories, we present a more comprehensive review that (i) includes
$D^{[u]}(D_{k})$ of the dihedral gauge group $D_{k}$ with odd degree, and (ii)
covers all possible Dijkgraaf-Witten deformations Dijkgraaf and Witten (1990);
Dijkgraaf  _et al._ (1991); Altschuler and Coste (1992); Bais  _et al._
(1993); de Wild Propitius (1995) $[u]$ and $[u,v,w]$.

The global dihedral $D_{k}=\mathbb{Z}_{2}\ltimes\mathbb{Z}_{k}$ symmetry of
$SO(2n)_{1}$ was defined in (50), where $k=n$ (or $n/2$) when $n$ is odd
(resp. even). The dihedral group contains $\mathbb{Z}_{k}$ as a normal
subgroup. The conjugation symmetry group is the quotient group
$\mathbb{Z}_{2}=D_{k}/\mathbb{Z}_{k}$, which can also be (non-canonically)
embedded as an abnormal subgroup in $D_{k}$. (See the split exact
decompositions (270) for $k$ odd or (295) for $k$ even.) The gauge groups in
(1), (2) and (3) are subgroups of the extension $\widehat{D_{k}}$ of the
dihedral symmetry group $D_{k}$ by the anyon fusion group
$\mathcal{A}=\\{1,\psi,s\pm\\}$ of the Abelian $SO(2n)_{1}$ topological order,
where $\mathcal{A}=\mathbb{Z}_{2}^{2}$ when $n$ is even or
$\mathcal{A}=\mathbb{Z}_{4}$ when $n$ is odd. We first classify the group
extensions.

In general, a global symmetry group $G$ is extended by the fusion group
$\mathcal{A}$ of Abelian anyons into a quantum symmetry group Wen (2002);
Etingof  _et al._ (2010); Teo  _et al._ (2015); Barkeshli  _et al._ (2019)
$\widehat{G}$

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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.87155pt}{-13.89304pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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where the above sequence of (forward) group homomorphisms is exact so that the
inclusion map $i$ is injective, the projection map $p$ is surjective, and the
image of $i$ is identical to the kernel of $p$, $\mathrm{Im}(i)=p^{-1}(1)$.
$\mathcal{A}$ is a normal subgroup of $\widehat{G}$ and the global symmetry
group is identical to the quotient $G=\widehat{G}/\mathcal{A}$. The (non-
canonical) function $j:G\to\widehat{G}$ assigns to each $g$ in $G$, an element
$j(g)=\widehat{g}$ in the quantum group $\widehat{G}$ that projects back to
the same element $p(j(g))=g$. The function $j$ is not necessarily a group
homomorphism and may not preserve the group product. The group extension
respects the pre-assigned $G$-action on $\mathcal{A}$ so that
$\widehat{g}a\widehat{g}^{-1}=g\cdot a$. Most symmetries in this article are
inner automorphisms and associate to trivial $G$-actions, $g\cdot a=a$, except
for the conjugation symmetry, which is an outer automorphism that switches the
$SO(2n)_{1}$ spinors, $\mathbb{Z}_{2}:s_{+}\leftrightarrow s_{-}$, when $n$ is
odd.

Group extensions (190) are classified by the group cohomology
$H^{2}(G,\mathcal{A})$. When the $G$-action on $\mathcal{A}$ is trivial, the
Abelian group $\mathcal{A}$ sits inside the center of $\widehat{G}$ and
commute with all group elements. In this case, $H^{2}(G,\mathcal{A})$
classifies central extensions. The trivial cohomology corresponds to the
direct product $\widehat{G}=G\times\mathcal{A}$. When the $G$-action is non-
trivial, the extension is not central and must be non-Abelian. The trivial
cohomology in $H^{2}(G,\mathcal{A})$ corresponds to the semi-direct product
$\widehat{G}=G\ltimes\mathcal{A}$, where the group product is
$(g_{1},a_{1})\cdot(g_{2},a_{2})=(g_{1}g_{2},(g_{2}\cdot a_{1})a_{2})$. In
these cases, the exact decomposition (190) is said to split, the extension is
“symmorphic” and $j$ can be chosen to be a group homomorphism that embeds
$G\to G\times\mathcal{A}$ (or $G\to G\ltimes\mathcal{A}$). Non-trivial
cohomology elements in $H^{2}(G,\mathcal{A})$ corresponds to “non-symmorphic”
extensions where $\widehat{G}$ does not decompose into a (semi)direct product
and the function $j$ in (190) fails to be a group homomorphism.

The function $j$ preserves the group product when
$h(g_{1},g_{2})=j(g_{1})j(g_{2})[j(g_{1}g_{2})]^{-1}=\widehat{g_{1}}\widehat{g_{2}}(\widehat{g_{1}g_{2}})^{-1}$
is the identity. Since $p(h(g_{1},g_{2}))=1$, the exactness of (190) requires
$h(g_{1},g_{2})$ to belong in $\mathcal{A}$. The function
$h:G^{2}\to\mathcal{A}$ obey the 2-cocycle condition

$\displaystyle dh(g_{1},g_{2},g_{3})=\frac{\left(g_{1}\cdot
h(g_{2},g_{3})\right)h(g_{1},g_{2}g_{3})}{h(g_{1}g_{2},g_{3})h(g_{1},g_{2})}=1$
(191)

because $\mathcal{A}$ is Abelian and the group product is associative,
$\widehat{g_{1}}(\widehat{g_{2}}\widehat{g_{3}})=(\widehat{g_{1}}\widehat{g_{2}})\widehat{g_{3}}$.
The function $j$ in (190) is not unique, and there are other functions
$j^{\prime}=\lambda j$ that respect the projection, $p(j^{\prime}(g))=g$.
Since $p(\lambda(g))=p(j^{\prime}(g)j(g)^{-1})=gg^{-1}=1$, exactness of (190)
requires $\lambda=j^{\prime}j^{-1}$ to have image in $\mathcal{A}$. This
modifies the $h$ function to $h^{\prime}=hd\lambda$, where the 2-coboundary is

$\displaystyle
d\lambda(g_{1},g_{2})=\frac{\left(g_{1}\cdot\lambda(g_{2})\right)\lambda(g_{1})}{\lambda(g_{1}g_{2})}.$
(192)

$h^{\prime}$ obeys the cocyle condition (191) because $d(d\lambda)=1$. The
equivalent classes $[h]=[hd\lambda]$ of 2-cocycles (modulo 2-coboundaries)
form the group cohomology Brown (1982)

$\displaystyle
H^{2}(G,\mathcal{A})=\frac{\mathrm{ker}\left(d:C^{2}(G,\mathcal{A})\to
C^{3}(G,\mathcal{A})\right)}{\mathrm{Im}\left(d:C^{1}(G,\mathcal{A})\to
C^{2}(G,\mathcal{A})\right)}$ (193)

where $C^{r}(G,\mathcal{A})$ is the Abelian group of $r$-cochains – functions
that map $G^{r}\to\mathcal{A}$.

Here, we summarize the extension $\widehat{D_{k}}$ of the global symmetry
group $D_{k}=\left\langle\mu,m|\mu^{2}=m^{k}=(\mu m)^{2}=1\right\rangle$ of
$SO(2n)_{1}$. These quantum symmetry groups can be found in section III.3.1,
III.3.2, IV.3, and the corresponding cohomology classifications can be found
in appendix D. (1) When $n=2k\equiv 0$ modulo 4, the symmetry is “non-
symmorphically” extended to $\widehat{D_{k}}=D_{2k}\times\mathbb{Z}_{2}$,
which corresponds to the cohomology class $[x,1,1]$ in
$H^{2}(D_{k},\mathcal{A})=\mathcal{A}^{3}$ where $x=s_{\pm}$ (or $x=\psi$)
when $n\equiv 0$ (resp. $n\equiv 4$) modulo 8. The degree $2k$ dihedral group
is

$\displaystyle
D_{2k}=\left\langle\mu,\hat{m},x\left|\begin{array}[]{*{20}l}\hat{m}^{k}=x,\\\
\mu^{2}=\hat{m}^{2k}=(\mu\hat{m})^{2}=1\end{array}\right.\right\rangle.$ (196)

It contains the subgroups $\mathbb{Z}_{2}=\langle\mu|\mu^{2}=1\rangle$, which
is the conjugation symmetry group, and
$\mathbb{Z}_{2k}=\langle\hat{m}|\hat{m}^{2k}=1\rangle$, which non-trivially
extends the $\mathbb{Z}_{k}$ symmetry group. (2) When $n=2k\equiv 2$ modulo 4,
the symmetry is “non-symmorphically” extended to
$\widehat{D_{k}}=Q_{4k}\times\mathbb{Z}_{2}$, which corresponds to the
cohomology class $[\psi]$ in $H^{2}(D_{k},\mathcal{A})=\mathcal{A}$. The
dicyclic group is

$\displaystyle Q_{4k}$
$\displaystyle=\left\langle\hat{\mu},m,\psi\left|\begin{array}[]{*{20}l}\hat{\mu}^{2}=(\hat{\mu}m)^{2}=\psi,\\\
\psi^{2}=m^{k}=1\end{array}\right.\right\rangle$ (199)
$\displaystyle=\left\langle\hat{\mu},\hat{m},\psi\left|\begin{array}[]{*{20}l}\hat{\mu}^{2}=\hat{m}^{k}=(\hat{\mu}\hat{m})^{2}=\psi,\\\
\psi^{2}=\hat{\mu}^{4}=\hat{m}^{2k}=1\end{array}\right.\right\rangle,$ (202)

where $\hat{m}=m\psi$. It contains the subgroups
$\mathbb{Z}_{4}=\langle\hat{\mu}|\hat{\mu}^{4}=1\rangle$, which non-trivially
extends the conjugation symmetry group $\mathbb{Z}_{2}$, and
$\mathbb{Z}_{k}=\langle m|m^{k}=1\rangle$. (3) When $n=k$ is odd, the symmetry
is “symmorphically” extended to
$\widehat{D_{k}}=D_{k}\ltimes\mathcal{A}_{\mathrm{odd}}$, where the twofold
conjugation symmetry acts non-trivially on
$\mathcal{A}_{\mathrm{odd}}=\mathbb{Z}_{4}$ as the involution
$s_{+}\leftrightarrow s_{-}$. The extension corresponds to the trivial element
in $H^{2}(D_{k},\mathcal{A}_{\mathrm{odd}})=\mathbb{Z}_{2}$.

Next, we describe the topological orders of the discrete gauge theory
components $D^{[\omega]}(\mathbb{Z}_{k})$, $D^{[\omega]}(D_{k})$ and
$D^{[\omega]}(Q_{4k})$ of the $SO(2n)_{1}/\mathbb{Z}_{2}$,
$SO(2n)_{1}/\mathbb{Z}_{k}$ and $SO(2n)_{1}/D_{2}$ twist liquids in (56), (70)
and (139). In general, anyons in a discrete gauge theory $D(G)$ are dyons
$\chi=([g],\rho)$. $[g]=\\{hgh^{-1}:h\in G\\}$ is the conjugacy class of the
group element $g$, and $\rho:Z_{g}\to U(N)$ is a unitary irreducible
representation of the centralizer subgroup $Z_{g}=\\{h\in G:gh=hg\\}$, which
carries group elements that commute with $g$. The conjugacy class $[g]$ and
the representation $\rho$ specify the gauge flux and gauge charge of the dyon.
When the gauge group is the Abelian cyclic group
$G=\mathbb{Z}_{k}=\\{1,m,\ldots,m^{k-1}\\}$, a conjugacy class only contains a
single element $[m^{j}]=\\{m^{j}\\}$, and its centralizer group is the entire
group $Z_{m^{j}}=\mathbb{Z}_{k}$. The irreducible representations are the
homomorphism $z^{b}:\mathbb{Z}_{k}\to U(1)$ sending $m^{a}\to
z^{b}(m^{a})=e^{2\pi iab/k}$, where $a,b=0,1,\ldots,k-1$. Dyons in
$D(\mathbb{Z}_{k})$ are the flux-charge composites $\chi^{ab}=(m^{a},z^{b})$.
Table 6, 6 and 6 summarize the conjugacy classes and irreducible
representations of the dihdral group $D_{k}$ with odd and even degree $k$, and
the dicyclic group $Q_{4k}$ with odd degree $k$. Possible centralizer
subgroups are $\mathbb{Z}_{2}$, $D_{2}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$,
$\mathbb{Z}_{4}$, $\mathbb{Z}_{k}$, $\mathbb{Z}_{2k}$, and the entire group
$D_{k}$ or $Q_{4k}$. Irreducible representations of the Abelian centralizers
$Z$ form the groups of homomorphisms $\mathrm{Hom}(Z,U(1))$, which is
isomorphic to $Z$. The quantum dimension of a dyon $\chi=([g],\rho)$ is the
integer $d_{\chi}=|[g]|\dim(\rho)$. The total quantum dimension of the
discrete gauge theory is the order of the group

$\displaystyle|G|=\sqrt{\sum_{\chi}d^{2}_{\chi}}.$ (203)

The number of dyons (i.e. the ground state degeneracy on a torus) is
$\mathrm{GSD}(D(\mathbb{Z}_{k}))=k^{2}$, $\mathrm{GSD}(D(D_{k}))=(k^{2}+7)/2$
when $k$ is odd, $\mathrm{GSD}(D(D_{k}))=(k^{2}+28)/2$ when $k$ is even, and
$\mathrm{GSD}(D(\mathrm{Dick}_{k}))=2(k^{2}+7)$ when $k$ is odd. The dyons and
their quantum dimensions of $D(D_{k})$ and $D(Q_{4k})$ are listed in table 9,
10 and 11.

| $[1]$ | $[\mu]$ | $[m^{j}]$
---|---|---|---
$A_{0}$ | 1 | 1 | 1
$A_{1}$ | 1 | $-1$ | 1
$E_{l}$ | 2 | 0 | $2\cos(2\pi jl/k)$
Table 6: Character table of $D_{k}$ when $k$ is odd. $\rho=A_{0,1},E_{l}$ are irreducible representations. $j,l=1,\ldots,(k-1)/2$. Numerical entries are $\mathrm{Tr}(\rho(g))$. | $[1]$ | $[\mu]$ | $[\mu m]$ | $[m^{j}]$ | $[m^{k/2}]$
---|---|---|---|---|---
$A_{0}$ | 1 | 1 | 1 | 1 | 1
$A_{1}$ | 1 | $-1$ | $-1$ | 1 | 1
$B_{0}$ | 1 | 1 | $-1$ | $(-1)^{j}$ | $(-1)^{k/2}$
$B_{1}$ | 1 | $-1$ | 1 | $(-1)^{j}$ | $(-1)^{k/2}$
$E_{l}$ | 2 | 0 | 0 | $2\cos(2\pi jl/k)$ | $2(-1)^{l}$
Table 7: Character table of $D_{k}$ when $k$ is even $\rho=A_{0,1},B_{0,1},E_{l}$ are irreducible representations. $j,l=1,\ldots,k/2-1$. Numerical entries are $\mathrm{Tr}(\rho(g))$. | $[1]$ | $[\hat{\mu}]$ | $[\hat{\mu}^{2}]$ | $[\hat{\mu}^{3}]$ | $[\hat{m}^{j}]$
---|---|---|---|---|---
$A_{0}$ | 1 | 1 | 1 | 1 | 1
$A_{1}$ | 1 | $e^{i\pi/2}$ | $-1$ | $e^{-i\pi/2}$ | $(-1)^{j}$
$A_{2}$ | 1 | $-1$ | 1 | $-1$ | 1
$A_{3}$ | 1 | $e^{-i\pi/2}$ | $-1$ | $e^{i\pi/2}$ | $(-1)^{j}$
$E_{l}$ | 2 | 0 | $2(-1)^{l}$ | 0 | $2(-1)^{jl}\cos(2\pi jl/k)$
Table 8: Character table of $Q_{4k}$ when $k$ is odd. $\rho=A_{0,1,2,3},E_{l}$ are irreducible representations. $j,l=1,\ldots,k-1$. Numerical entries are $\mathrm{Tr}(\rho(g))$. dyons |  | $d_{\chi}$ | $\theta_{\chi}=e^{2\pi ih_{\chi}}$
---|---|---|---
$([1],A_{\nu})$ | $\nu=0,1$ | 1 | 1
$([1],E_{l})$ | $l=1,\ldots,(k-1)/2$ | 2 | 1
$([m^{j}],z^{b})$ |  $j=1,\ldots,(k-1)/2$, | 2 | $e^{2\pi ijb/k}\vartheta_{m^{j}}$
$b=0,\ldots,k-1$
$([\mu],\zeta^{\lambda})$ | $\lambda=0,1$ | $k$ | $(-1)^{\lambda}\vartheta_{\mu}$
Table 9: Dyons $\chi=([g],\rho)$ in $D^{[u]}(D_{k})$, their quantum dimensions $d_{\chi}$ and topological spins $\theta_{\chi}$, when $k$ is odd. $\zeta^{\lambda}(\mu^{A})=(-1)^{\lambda A}$ are irreducible representations of $Z_{\mu}=\mathbb{Z}_{2}={\mu^{A}:A=0,1}$. The deformation phases $\vartheta_{m^{j}}$ and $\vartheta_{\mu}$ depend on the Dijkgraaf-Witten invariant $[u]$ and are computed in (223). dyons |  | $d_{\chi}$ | $\theta_{\chi}=e^{2\pi ih_{\chi}}$
---|---|---|---
$([1],A_{\nu})$ | $\nu=0,1$ | 1 | 1
$([m^{k/2}],A_{\nu})$ | 1 | $\vartheta_{m^{k/2}}$
$([1],B_{\nu})$ | 1 | 1
$([m^{k/2}],B_{\nu})$ | 1 | $(-1)^{k/2}\vartheta_{m^{k/2}}$
$([1],E_{l})$ | $l=1,\ldots,k/2-1$ | 2 | 1
$([m^{k/2}],E_{l})$ | 2 | $(-1)^{l}\vartheta_{m^{k/2}}$
$([m^{j}],z^{b})$ |  $j=1,\ldots,k/2-1$, | 2 | $e^{2\pi ijb/k}\vartheta_{m^{j}}$
$b=0,\ldots,k-1$
$([\mu],\zeta^{\lambda_{1},\lambda_{2}})$ | $\lambda_{1},\lambda_{2}=0,1$ | $k/2$ | $(-1)^{\lambda_{1}}\vartheta_{\mu}$
$([\mu m],\zeta^{\lambda_{1},\lambda_{2}})$ | $k/2$ | $(-1)^{\lambda_{1}}\vartheta_{\mu m}$
Table 10: Dyons $\chi=([g],\rho)$ in $D^{[u,v,w]}(D_{k})$ their quantum dimensions $d_{\chi}$ and topological spins $\theta_{\chi}$, when $k$ is even. $\zeta^{\lambda_{1},\lambda_{2}}(\mu^{A_{1}}m^{kA_{2}/2})=(-1)^{\lambda_{1}A_{1}+\lambda_{2}A_{2}}$ are irreducible representations of $Z_{\mu}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}=\\{1,\mu,m^{k/2},\mu m^{k/2}\\}$. The deformation phases $\vartheta_{m^{j}}$, $\vartheta_{\mu}$ and $\vartheta_{\mu m}$ depend on the Dijkgraaf-Witten invariant $[u,v,w]$ and are computed in (226). dyons |  | $d_{\chi}$ | $\theta_{\chi}=e^{2\pi ih_{\chi}}$
---|---|---|---
$([1],A_{\nu})$ | $\nu=0,1,2,3$ | 1 | 1
$([\hat{\mu}^{2}],A_{\nu})$ | 1 | $(-1)^{\nu}\vartheta_{\hat{\mu}^{2}}$
$([1],E_{l})$ | $l=1,\ldots,k-1$ | 2 | 1
$([\hat{\mu}^{2}],E_{l})$ | 2 | $(-1)^{l}\vartheta_{\hat{\mu}^{2}}$
$([\hat{m}^{j}],z^{b})$ |  $j=1,\ldots,k-1$, | 2 | $e^{i\pi jb/k}\vartheta_{\hat{m}^{j}}$
$b=0,\ldots,2k-1$
$([\hat{\mu}],\zeta^{\lambda})$ | $\lambda=0,1,2,3$ | $k$ | $e^{i\pi\lambda/2}\vartheta_{\hat{\mu}}$
$([\hat{\mu}^{-1}],\zeta^{\lambda})$ | $k$ | $e^{-i\pi\lambda/2}\vartheta_{\hat{\mu}^{-1}}$
Table 11: Dyons $\chi=([g],\rho)$ in $D^{[u]}(Q_{4k})$ their quantum
dimensions $d_{\chi}$ and topological spins $\theta_{\chi}$, when $k$ is odd.
$z^{b}(\hat{m}^{j})=e^{i\pi bj/k}$ and
$\zeta^{\lambda}(\hat{\mu}^{A})=e^{i\pi\lambda A/2}$ are irreducible
representations of
$Z_{\hat{m}^{j}}=\mathbb{Z}_{2k}=\\{\hat{m}^{j}:j=0,1,\ldots,2k-1\\}$ and
$Z_{\hat{\mu}}=\mathbb{Z}_{4}=\\{\hat{\mu}^{A}:A=-1,0,1,2\\}$ respectively.
The deformation phases $\vartheta_{\hat{m}^{j}}$, $\vartheta_{\hat{\mu}}$ and
$\vartheta_{\hat{\mu}^{2}}$ depend on the Dijkgraaf-Witten invariant $[u]$ and
are computed in (228).

A finite group $G$ can be promoted to a fusion category
$\mathcal{F}^{[\omega]}(G)$. The simple objects of the tensor category are the
group elements. Fusion rules are group multiplications, $g_{1}\times
g_{2}=g_{1}g_{2}$. The associativity equivalence $(g_{1}\times g_{2})\times
g_{3}\equiv g_{1}\times(g_{2}\times g_{3})$ is described by the $U(1)$ basis
transformation

$\displaystyle\left|\vbox{\hbox{\includegraphics[width=36.135pt]{F1.pdf}}}\right\rangle=f^{g_{1}g_{2}g_{3}}\left|\vbox{\hbox{\includegraphics[width=36.135pt]{F2.pdf}}}\right\rangle.$
(204)

The collection of $f$-symbols are 3-cochains in $C^{3}(G,U(1))$ that obey the
cocycle condition (also known as the pentagon identity) Dijkgraaf and Witten
(1990); Dijkgraaf  _et al._ (1991); Altschuler and Coste (1992); Bais  _et
al._ (1993); de Wild Propitius (1995)

$\displaystyle
df^{g_{1}g_{2}g_{3}g_{4}}=\frac{f^{g_{1}g_{2}g_{3}}f^{g_{1}(g_{2}g_{3})g_{4}}f^{g_{2}g_{3}g_{4}}}{f^{(g_{1}g_{2})g_{3}g_{4}}f^{g_{1}g_{2}(g_{3}g_{4})}}=1.$
(205)

This is because the following pentagon diagram is commutative. The $f$-symbols
are gauge dependent and can be changed under a gauge transformation of the
quantum states, $f^{g_{1}g_{2}g_{3}}\to
f^{g_{1}g_{2}g_{3}}d\Lambda^{g_{1}g_{2}g_{3}}$, where

$\displaystyle
d\Lambda^{g_{1}g_{2}g_{3}}=\frac{\Lambda^{g_{1}g_{2}}\Lambda^{(g_{1}g_{2})g_{3}}}{\Lambda^{g_{1}(g_{2}g_{3})}\Lambda^{g_{2}g_{3}}}$
(207)

is the coboundary of $\Lambda^{g_{1}g_{2}}$, the gauge transformation of the
fusion vertex $|g_{1}\times g_{2}\rangle\to\Lambda^{g_{1}g_{2}}|g_{1}\times
g_{2}\rangle$. Equivalent classes of $f$-symbols $[\omega]=f\cdot
dC^{2}(G,U(1))$ are cohomology elements in

$\displaystyle
H^{3}(\mathbb{Z}_{n},U(1))=\frac{\mathrm{ker}\left(d:C^{3}(G,U(1))\to
C^{4}(G,U(1))\right)}{\mathrm{Im}\left(d:C^{2}(G,U(1))\to
C^{3}(G,U(1))\right)}.$ (208)

The cohomology class $[\omega]$ is also referred to as a Dijkgraaf-Witten
invariant. Dijkgraaf and Witten (1990); Dijkgraaf  _et al._ (1991); Altschuler
and Coste (1992); Bais  _et al._ (1993); de Wild Propitius (1995) The group
cohomology classifications and explicit cocycle representatives of cohomology
elements of the relevant gauge groups $G=\mathbb{Z}_{k}$, $D_{k}$, and
$Q_{4k}$ are presented in appendix D.

The discrete gauge theory $D^{[\omega]}(G)$ is the Drinfeld center Kassel
(1995); Bakalov and Kirillov (2001); Levin and Wen (2005) of the fusion
category $\mathcal{F}^{[\omega]}(G)$. The trivial cohomology class $[0]$
associates to the conventional (or un-deformed) gauge theory $D^{[0]}(G)$,
where all the $f$-symbols can be chosen to be trivial,
$f^{g_{1}g_{2}g_{3}}=1$. In this case, all pure gauge fluxes, whose charge
components (i.e. centralizer representations) are trivial, carry bosonic
exchange statistics. When a gauge theory $D^{[\omega]}(G)$ is deformed by a
non-trivial Dijkgraaf-Witten invariant $[\omega]$, the $f$-symbols are
necessarily non-trivial. They modify the exchange statistics of gauge fluxes.
The collection of $U(1)$ exchange phases $r^{gh}$, each relates

$\displaystyle|g\times h\rangle=r^{gh}|h\times\left(h^{-1}gh\right)\rangle,$
(209)

obeys the hexagon identity Dijkgraaf  _et al._ (1991); Altschuler and Coste
(1992); Bais  _et al._ (1993); de Wild Propitius (1995)

$\displaystyle\begin{split}&r^{g_{1}g_{2}}f^{g_{2}(g_{2}^{-1}g_{1}g_{2})g_{3}}r^{(g_{2}^{-1}g_{1}g_{2})g_{3}}\\\
&=f^{g_{1}g_{2}g_{3}}r^{g_{1}(g_{2}g_{3})}f^{g_{2}g_{3}(g_{3}^{-1}g_{2}^{-1}g_{1}g_{2}g_{3})}.\end{split}$
(210)

This is because the following hexagon diagram is commutative. The self-
exchange phase of a pure gauge flux $g$ is $\vartheta_{g}=r^{gg}$. In general,
the topological spin $\theta_{\chi}=e^{2\pi ih_{\chi}}$ of a dyon
$\chi=([g],\rho)$ is the $U(1)$ phase

$\displaystyle\theta_{\chi}=\left\langle\rho(g)\right\rangle\vartheta_{g},$
(212)

where $\rho:Z_{g}\to U(N_{\rho})$ is the irreducible representation of the
centralizer subgroup $Z_{g}$,
$\rho(g)=\langle\rho(g)\rangle\mathbb{I}_{N_{\rho}\times N_{\rho}}$, and
$\langle\rho(g)\rangle=\mathrm{Tr}\left(\rho(g)\right)/N_{\rho}$. The modular
$T$-matrix is the diagonal unitary matrix
$T_{\chi\chi^{\prime}}=\delta_{\chi\chi^{\prime}}\theta_{\chi}$. The modular
$S$-matrix is the symmetric unitary matrix Dijkgraaf  _et al._ (1991); Bais
_et al._ (1993); de Wild Propitius (1995)

$\displaystyle
S_{\chi\chi^{\prime}}=\frac{1}{|G|}\sum_{\begin{subarray}{c}g_{i}\in[g],g^{\prime}_{j}\in[g^{\prime}]\\\
g_{i}g^{\prime}_{j}=g^{\prime}_{j}g_{i}\end{subarray}}\mathrm{Tr}\left(\rho\left(x_{i}^{-1}g^{\prime}_{j}x_{i}\right)\right)\mathrm{Tr}\left(\rho^{\prime}\left({x^{\prime}}_{j}^{-1}g_{i}x^{\prime}_{j}\right)\right)r^{g_{i}g^{\prime}_{j}}r^{g^{\prime}_{j}g_{i}},$
(213)

where $\chi=([g],\rho)$ and $\chi^{\prime}=([g^{\prime}],\rho^{\prime})$ are
dyons, whose flux components are conjugacy classes
$[g]=\\{g_{i}=x_{i}gx_{i}^{-1}:i=1,\ldots,|[g]|\\}$ and
$[g^{\prime}]=\\{g^{\prime}_{j}=x^{\prime}_{j}g^{\prime}{x^{\prime}}_{j}^{-1}:j=1,\ldots,|[g^{\prime}]|\\}$.
Together the $S$ and $T$ matrices generate a representation of the modular
group $SL(2,\mathbb{Z})$. They obey

$\displaystyle S^{2}=(ST^{\dagger})^{3},\quad S^{4}=\mathbb{I},$ (214)

where $C=S^{2}$ is the conjugation matrix so that $C_{\chi\bar{\chi}}=1$ if
$\chi$ and $\bar{\chi}$ are conjugate pair, $\chi\times\bar{\chi}=1+\ldots$,
and $C_{\chi\chi^{\prime}}=0$ if $\chi^{\prime}\neq\bar{\chi}$. The fusion
rule

$\displaystyle\chi_{1}\times\chi_{2}=\sum_{\chi}N_{\chi_{1}\chi_{2}}^{\chi}\chi$
(215)

is determined by the non-negative integral fusion number
$N_{\chi_{1}\chi_{2}}^{\chi}$, which is related to the $S$-matrix by the
Verlinde formula Verlinde (1988)

$\displaystyle
N_{\chi_{1}\chi_{2}}^{\chi}=\sum_{\chi^{\prime}}\frac{S_{\chi_{1}\chi^{\prime}}S_{\chi_{2}\chi^{\prime}}S_{\chi\chi^{\prime}}^{\ast}}{d_{\chi^{\prime}}/|G|}.$
(216)

The $S$-matrix can in turn be computed by plugging in the fusion numbers and
topological spins into the ribbon identity Kitaev (2006)

$\displaystyle
S_{\chi_{1}\chi_{2}}=\frac{1}{|G|}\sum_{\chi}d_{\chi}N_{\chi_{1}\chi_{2}}^{\chi}\frac{\theta_{\chi}}{\theta_{\chi_{1}}\theta_{\chi_{2}}}.$
(217)

###  C.1 The $\mathbb{Z}_{k}$ gauge theory

The $f$-symbols of the quantum double $D^{[u]}(\mathbb{Z}_{k})$ that represent
the Dijkgraaf-Witten invariant $[u]$ in
$H^{3}(\mathbb{Z}_{k},U(1))=\mathbb{Z}_{k}$ can be chosen to be

$\displaystyle\begin{split}f^{a_{1}a_{2}a_{3}}&=\omega^{3}_{u}(a_{1},a_{2},a_{3})\\\
&=\exp\left(\frac{2\pi
iu}{k^{2}}a_{1}\left(a_{2}+a_{3}-[a_{2}+a_{3}]\right)\right)\end{split}$ (218)

from the cocycle representative $\omega^{3}_{u}$ in (265), where
$a_{i}\in(-k/2,k/2]\cap\mathbb{Z}$ label group elements in $\mathbb{Z}_{k}$
and the square bracket $[a_{2}+a_{3}]$ wraps the number back to the
appropriate range $(-k/2,k/2]$ by adding or subtracting $k$ to $a_{2}+a_{3}$
if necessary.

When the Dijkgraaf-Witten invariant is trivial, $[u]=[0]$, the solutions to
the hexagon equation (210) are homomorphisms $r^{a_{1}\ast}:\mathbb{Z}_{k}\to
U(1)$ sending $a_{2}$ to $r^{a_{1}a_{2}}=\exp(2\pi ip_{a_{1}}a_{2}/k)$, where
$p_{a}$ are integers. These exchange phases can be canceled by attaching the
gauge charge $z^{-p_{a}}$ to the gauge flux $m^{a}$. Therefore, $p_{a}$ can be
chosen to be 0 without loss of generality. In this case, all pure fluxes
$m^{a}$ are bosonic because their self-exchange phases (212) are trivial,
$\theta_{m^{a}}=\vartheta_{a}=r^{aa}=1$. The topological spin of a general
dyon $\chi^{ab}=m^{a}z^{b}$ is $\theta_{m^{a}z^{b}}=e^{2\pi iab/k}$. The
modular $S$-matrix defined in (213) summarizes the $2\pi$ braiding
$kS_{\chi^{ab}\chi^{a^{\prime}b^{\prime}}}=e^{2\pi
i(ab^{\prime}+ba^{\prime})/k}$ between $\chi^{ab}$ and
$\chi_{a^{\prime}b^{\prime}}$. The $D^{[0]}(\mathbb{Z}_{k})$ gauge theory can
be described by a topological Chern-Simons field theory with the action
$\mathcal{S}=\int_{2+1}K_{IJ}\alpha^{I}\wedge d\alpha^{J}/(4\pi)$, where the
$K$-matrix is $K=2\sigma_{x}$. The dyon $\chi^{ab}=m^{a}z^{b}$ is associated
to the integral vector ${\bf v}=(a,b)^{T}$, and the spin and braiding
statistics are captured by $\theta_{\bf v}=e^{\pi i{\bf v}^{T}K^{-1}{\bf v}}$
and $nS_{{\bf u}{\bf v}}=e^{2\pi i{\bf u}^{T}K^{-1}{\bf v}}$.

For a general Dijkgraaf-Witten deformation $[u]$, the solution to the hexagon
equation (210) can be chosen to be de Wild Propitius (1995)

$\displaystyle r_{u}^{a_{1}a_{2}}=\exp\left(\frac{2\pi
iu}{k^{2}}a_{1}a_{2}\right).$ (219)

The discrete gauge theory depends only on $u$ modulo $k$ because
$r_{u+k}=r_{u}r_{k}$ differs from $r_{u}$ by $r_{k}^{a_{1}a_{2}}=e^{2\pi
ia_{1}a_{2}/k}$, which can be canceled attaching gauge fluxes. These exchange
phases $r^{a_{1}a_{2}}_{u}$ modify the spin and braiding statistics of the
dyons. Pure fluxes $m^{a}$ are no longer bosonic but carry non-trivial
exchange statistics $\vartheta_{a}=r^{aa}=e^{2\pi iua^{2}/k^{2}}$. The
topological spins (212) and the modular $S$-matrix (213) of dyons
$\chi^{ab}=m^{a}z^{b}$ are

$\displaystyle\begin{split}\theta_{\chi^{ab}}&=\vbox{\hbox{\includegraphics[width=36.135pt]{AQPexchange.pdf}}}=e^{2\pi
i\left(\frac{ab}{k}+\frac{ua^{2}}{k^{2}}\right)},\\\
kS_{\chi^{ab}\chi^{a^{\prime}b^{\prime}}}&=\vbox{\hbox{\includegraphics[width=43.36243pt]{AQPbraiding.pdf}}}=e^{2\pi
i\left(\frac{ab^{\prime}+ba^{\prime}}{k}+\frac{2uaa^{\prime}}{k^{2}}\right)}.\end{split}$
(220)

In section III.3.1 and III.3.2, we encounter the discrete gauge theories
$D^{[0]}(\mathbb{Z}_{2})$, $D^{[1]}(\mathbb{Z}_{2})$ and
$D^{[2]}(\mathbb{Z}_{4})$ in (56) as well as $D^{[0]}(\mathbb{Z}_{k})$ and
$D^{[k]}(\mathbb{Z}_{2k})$ in (70). In general, for any even order $n$, by
choosing $u=-n/2$ to represent the Dijkgraaf-Witten invariant $[n/2]$, the
discrete gauge theory $D^{[n/2]}(\mathbb{Z}_{n})$ has the non-chiral
$U(1)_{n}\times\overline{U(1)_{n}}$ topological order. The theory can be
described by the topological Chern-Simons field theory
$\mathcal{S}=\int_{2+1}K_{IJ}\alpha^{I}\wedge d\alpha^{J}/(4\pi)$ with the
$K$-matrix $K=n\sigma_{z}$. The dyon $\chi^{ab}=m^{a}z^{b}$ is represented by
the integral vector ${\bf v}=(b,a-b)$ so that the spin and braiding statistics
(220) agrees with $\theta_{\bf v}=e^{\pi i{\bf v}^{T}K^{-1}{\bf v}}$ and
$kS_{{\bf v}{\bf v}^{\prime}}=e^{2\pi i{\bf v}^{T}K^{-1}{\bf v}^{\prime}}$.

###  C.2 The $D_{k}$ gauge theory

The cohomology classification $H^{3}(D_{k},U(1))$ of $f$-symbols depends on
the parity of $k$ (see (258) in appendix C). We begin with dihedral groups
with odd degree $k$. The $f$-symbols of $D^{[u]}(D_{k})$ that represent the
Dijkgraaf-Witten invariant $[u]$ in $H^{3}(D_{k},U(1))=\mathbb{Z}_{2k}$ can be
identified with the cocycle representative $\xi^{3}_{u}$ in (284),

$\displaystyle f^{g_{1}g_{2}g_{3}}=\xi^{3}_{u}(g_{1},g_{2},g_{3}).$ (221)

The solution to the hexagon equation (210) can be chosen to be de Wild
Propitius (1995)

$\displaystyle r^{g_{1}g_{2}}_{u}=\exp\left(\frac{2\pi
iu}{k^{2}}\left\\{a_{2}\left[(-1)^{A_{2}}a_{1}+2A_{1}a_{2}\right]-A_{1}a_{2}^{2}\right\\}\right)\exp\left(\frac{i\pi
u}{2}A_{1}A_{2}\right),$ (222)

where $g_{i}=(A_{i},a_{i})=\mu^{A_{i}}m^{a_{i}}$ are group elements in
$D_{k}$, for $A_{i}=0,1$ and $a_{i}\in(-k/2,k/2)\cap\mathbb{Z}$. The square
bracket wraps $\left[(-1)^{A_{2}}a_{1}+2A_{1}a_{2}\right]$ back to the
appropriate range $(-k/2,k/2)$ by adding or subtracting $k$ to
$(-1)^{A_{2}}a_{1}+2A_{1}a_{2}$ if necessary. The exchange phases
$r_{u=2k}^{g_{1}g_{2}}$ and $(2k)^{\mathrm{th}}$ roots of unity and can be
absorbed by attaching gauge charges. Therefore $r_{u+2k}$ and $r_{u}$
correspond to the equivalent gauge theory $D^{[u]}(D_{k})$. The dyons and
their quantum dimensions and topological spins are listed in table 9 under the
deformation phases

$\displaystyle\vartheta_{m^{j}}=r^{m^{j}m^{j}}=e^{2\pi
iuj^{2}/k^{2}},\quad\vartheta_{\mu}=r^{\mu\mu}=i^{u}.$ (223)

The modular $S$-matrix can be computed with the help of the character table 6
and by substituting the exchange phases (222) in (213). It is presented in
table 12.

| $([1],A_{\nu})$ | $([1],E_{l})$ | $([m^{j}],z^{b})$ | $([\mu],\zeta^{\lambda})$
---|---|---|---|---
$([1],A_{\nu^{\prime}})$ | 1 | 2 | 2 | $k(-1)^{\nu^{\prime}}$
$([1],E_{l^{\prime}})$ | 2 | 4 | $4c(jl^{\prime})$ | 0
$([m^{j^{\prime}}],z^{b^{\prime}})$ | 2 | $4c(j^{\prime}l)$ | $4s^{u}_{jbj^{\prime}b^{\prime}}$ | 0
$([\mu],\zeta^{\lambda^{\prime}})$ | $k(-1)^{\nu}$ | 0 | 0 | $k(-1)^{\lambda+\lambda^{\prime}+u}$
Table 12: The $S$-matrix $2kS_{\chi\chi^{\prime}}$ from (213) of the
$D^{[u]}(D_{k})$ gauge theory, for odd $k$. We abbreviate $c(x)=\cos(2\pi
x/k)$ and
$s^{u}_{jbj^{\prime}b^{\prime}}=c(jb^{\prime}+j^{\prime}b+2ujj^{\prime}/k)$.
The row and column entries are arranged in the same order of the dyons in
table 9.

When the degree $k$ is even, dihedral gauge theories $D^{[u,v,w]}(D_{k})$ are
classified by the Dijkgraaf-Witten invariant $[u,v,w]$ in
$H^{3}(D_{k},U(1))=\mathbb{Z}_{k}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$.
Their $f$-symbols can be identified with the cocycle representative
$\xi^{3}_{u,v,w}$ in (298),

$\displaystyle f^{g_{1}g_{2}g_{3}}=\xi^{3}_{u,v,w}(g_{1},g_{2},g_{3}).$ (224)

The solution to the hexagon equation (210) can be chosen to be foo

$\displaystyle r^{g_{1}g_{2}}_{u,v,w}=\exp\left(\frac{2\pi
iu}{k^{2}}\left\\{a_{2}\left[(-1)^{A_{2}}a_{1}+2A_{1}a_{2}\right]_{k}-A_{1}a_{2}^{2}\right\\}\right)\exp\left(\frac{i\pi}{2}\left(vA_{1}A_{2}+(v+w)[a_{1}]_{2}A_{2}\right)\right),\quad\mbox{if
$(A_{1},a_{1})\neq(0,k/2)$},$ $\displaystyle
r^{g_{1}g_{2}}_{u,v,w}=\exp\left(-\frac{i\pi
u}{k}a_{2}\right)\exp\left(\frac{i\pi}{2}(v+w)\left[\frac{k}{2}\right]_{2}A_{2}\right),\quad\mbox{if
$(A_{1},a_{1})=(0,k/2)$}.$ (225)

Here, $A_{i}=0,1$ and $a_{i}\in(-k/2,k/2]\cap\mathbb{Z}$ label group elements
$g_{i}=(A_{i},a_{i})=\mu^{A_{i}}m^{a_{i}}$ in $D_{k}$. The square brackets
$[\ast]_{k}$ and $[\ast]_{2}$ wrap the numbers back to the appropriate ranges,
$(-k/2,k/2]$ for the former and $0,1$ for the latter, by respectively adding
or subtracting $k$ or 2 if necessary. Like the previous case, the Dijkgraaf-
Witten invariants $u,v,w$ have periodicities $k,2,2$ respectively because
$r_{k,0,0}$, $r_{0,2,0}$ and $r_{0,0,2}$ are $(2k)^{\mathrm{th}}$ roots of
unity and can be absorbed by attaching gauge charges. The dyons and their
quantum dimensions and topological spins are listed in table 10 under the
deformation phases $\vartheta_{g}=r^{gg}$

$\displaystyle\begin{split}&\vartheta_{m^{j}}=e^{2\pi
iuj^{2}/k^{2}},\quad\mbox{for $0<j<k/2$},\\\
&\vartheta_{m^{k/2}}=i^{-u},\quad\vartheta_{\mu}=i^{v},\quad\vartheta_{\mu
m}=i^{2v+w}.\end{split}$ (226)

The modular $S$-matrix can be computed with the help of the character table 7
and by substituting the exchange phases (225) in (213). It is presented in
table 13.

The dihedral gauge theories encountered in this article include
$D^{[n/2,0,0]}(D_{n})$ for $n\equiv 0$ modulo 8, and $D^{[n/2,1,1]}(D_{n})$
for $n\equiv 4$ modulo 8 (see (139)). The Dijkgraaf-Witten invariants
$[u,v,w]\in\mathbb{Z}_{k}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ are fixed
by the subgroup inclusion $i:\mathbb{Z}_{n}\hookrightarrow D_{n}$ and the
(non-canonical) injections $j_{0},j_{1}:\mathbb{Z}_{2}\to D_{n}$ into the two
inequivalent reflections in $D_{n}$ (see (295)). The inclusion/injections
induce the restriction homomorphisms in the group cohomology level
$i^{\ast}:H^{3}(D_{n},U(1))\to H^{3}(\mathbb{Z}_{n},U(1))=\mathbb{Z}_{n}$ and
$j_{0}^{\ast},j_{1}^{\ast}:H^{3}(D_{n},U(1))\to
H^{3}(\mathbb{Z}_{2},U(1))=\mathbb{Z}_{2}$ (see (296)), all being surjective
in this case and extract $i^{\ast}[u,v,w]=[u]$, $j_{0}^{\ast}[u,v,w]=[v]$ and
$j_{1}^{\ast}[u,v,w]=[w]$ (see (299)). The invariant $[u]=[n/2]$ is fixed by
the $D^{[n/2]}(\mathbb{Z}_{n/2})$ theory in (70) when $n\equiv 0$ modulo 4,
while $[v]=[w]=[0]$ (or $[1]$) is fixed by $D^{[0]}(\mathbb{Z}_{2})$ (or
$D^{[1]}(\mathbb{Z}_{2})$) in (56) when $n\equiv 0$ (resp. $n\equiv 4$) modulo
8. From (226), the two $\mathbb{Z}_{2}$ invariants dictate the topological
spins of the $\mathbb{Z}_{2}$ fluxes $([\mu],\zeta^{\lambda_{1},\lambda_{2}})$
and $([\mu m],\zeta^{\lambda_{1},\lambda_{2}})$ to be bosonic/fermionic (or
semionic) when $n\equiv 0$ (resp. $n\equiv 4$) modulo 8. Choosing the
Dijkgraaf-Witten invariants $(u,v,w)=(-n/2,0,0)$ or $(-n/2,1,1)$, the
topological spins of the $\mathbb{Z}_{k}$ fluxes $[m^{j}]$ are
$\vartheta_{m^{j}}=e^{-\pi ij^{2}/n}$, which match with that of the
$D^{[n/2]}(\mathbb{Z}_{n})$ gauge theory.

| $([1],A_{\nu})$ | $([m^{k/2}],A_{\nu})$ | $([1],B_{\nu})$ | $([m^{k/2}],B_{\nu})$ | $([1],E_{l})$ | $([m^{k/2}],E_{l})$ | $([m^{j}],z^{b})$ | $([\mu],\zeta^{\lambda_{1},\lambda_{2}})$ | $([\mu m],\zeta^{\lambda_{1},\lambda_{2}})$
---|---|---|---|---|---|---|---|---|---
$([1],A_{\nu^{\prime}})$ | 1 | 1 | 1 | 1 | 2 | 2 | 2 | $(-1)^{\nu^{\prime}}k/2$ | $(-1)^{\nu^{\prime}}k/2$
$([m^{k/2}],A_{\nu^{\prime}})$ | 1 | $(-1)^{u}$ | $(-1)^{k/2}$ | $(-1)^{u+k/2}$ | $2(-1)^{l}$ | $2(-1)^{u+l}$ | $2(-1)^{b}$ | $(-1)^{\nu^{\prime}+\lambda_{2}}s^{uvw}_{k/2,\mu}k/2$ | $(-1)^{\nu^{\prime}+\lambda_{2}}s^{uvw}_{k/2,\mu}k/2$
$([1],B_{\nu^{\prime}})$ | 1 | $(-1)^{k/2}$ | 1 | $(-1)^{k/2}$ | 2 | $2(-1)^{k/2}$ | $2(-1)^{j}$ | $(-1)^{\nu^{\prime}}k/2$ | $-(-1)^{\nu^{\prime}}k/2$
$([m^{k/2}],B_{\nu^{\prime}})$ | 1 | $(-1)^{u+k/2}$ | $(-1)^{k/2}$ | $(-1)^{u}$ | $2(-1)^{l}$ | $2(-1)^{u+l+k/2}$ | $2(-1)^{j+b}$ | $(-1)^{\nu^{\prime}+\lambda_{2}}s^{uvw}_{k/2,\mu}k/2$ | $-(-1)^{\nu^{\prime}+\lambda_{2}}s^{uvw}_{k/2,\mu}k/2$
$([1],E_{l^{\prime}})$ | 2 | $2(-1)^{l^{\prime}}$ | 2 | $2(-1)^{l^{\prime}}$ | 4 | $4(-1)^{l^{\prime}}$ | $4c(jl^{\prime})$ | 0 | 0
$([m^{k/2}],E_{l^{\prime}})$ | 2 | $2(-1)^{u+l^{\prime}}$ | $2(-1)^{k/2}$ | $2(-1)^{u+l^{\prime}+k/2}$ | $4(-1)^{l}$ | $4(-1)^{u+l+l^{\prime}}$ | $4(-1)^{b}c(jl^{\prime})$ | 0 | 0
$([m^{j^{\prime}}],z^{b^{\prime}})$ | 2 | $2(-1)^{b^{\prime}}$ | $2(-1)^{j^{\prime}}$ | $2(-1)^{j^{\prime}+b^{\prime}}$ | $4c(j^{\prime}l)$ | $4(-1)^{b^{\prime}}c(j^{\prime}l)$ | $4s^{u}_{jbj^{\prime}b^{\prime}}$ | 0 | 0
$([\mu],\zeta^{\lambda^{\prime}_{1},\lambda^{\prime}_{2}})$ | $(-1)^{\nu}k/2$ | $(-1)^{\nu+\lambda^{\prime}_{2}}s^{uvw}_{k/2,\mu}k/2$ | $(-1)^{\nu}k/2$ | $(-1)^{\nu+\lambda^{\prime}_{2}}s^{uvw}_{k/2,\mu}k/2$ | 0 | 0 | 0 | $\Sigma^{\lambda\lambda^{\prime}}_{00}$ | $\Sigma^{\lambda\lambda^{\prime}}_{01}$
$([\mu m],\zeta^{\lambda_{1},\lambda_{2}})$ | $(-1)^{\nu}k/2$ | $(-1)^{\nu+\lambda^{\prime}_{2}}s^{uvw}_{k/2,\mu}k/2$ | $-(-1)^{\nu}k/2$ | $-(-1)^{\nu+\lambda^{\prime}_{2}}s^{uvw}_{k/2,\mu}k/2$ | 0 | 0 | 0 | $\Sigma^{\lambda\lambda^{\prime}}_{10}$ | $\Sigma^{\lambda\lambda^{\prime}}_{11}$
Table 13: The $S$-matrix $2kS_{\chi\chi^{\prime}}$ from (213) of the
$D^{[u,v,w]}(D_{k})$ gauge theory, for even $k$. The row and column entries
are arranged in the same order of the dyons in table 10. We abbreviate
$c(x)=\cos(2\pi x/k)$,
$s^{u}_{jbj^{\prime}b^{\prime}}=c(jb^{\prime}+j^{\prime}b+2ujj^{\prime}/k)$
and $s^{uvw}_{k/2,\mu}=i^{-u+(v+w)[k/2]_{2}}$. Here, $[k/2]_{2}=0$ (or 1) when
$k/2$ is even (odd). If $k\equiv 0$ mod 4,
$\Sigma^{\lambda\lambda^{\prime}}_{00}=\frac{k}{2}(-1)^{\lambda_{1}+\lambda^{\prime}_{1}+v}\left[1+(-1)^{\lambda_{2}+\lambda^{\prime}_{2}}i^{-u}\right]$,
$\Sigma^{\lambda\lambda^{\prime}}_{11}=\frac{k}{2}(-1)^{\lambda_{1}+\lambda^{\prime}_{1}+w}\left[1+(-1)^{\lambda_{2}+\lambda^{\prime}_{2}}i^{-u}\right]$
and
$\Sigma^{\lambda\lambda^{\prime}}_{01}=\Sigma^{\lambda\lambda^{\prime}}_{10}=0$.
If $k\equiv 2$ mod 4,
$\Sigma^{\lambda\lambda^{\prime}}_{00}=\frac{k}{2}(-1)^{\lambda_{1}+\lambda^{\prime}_{1}+v}$,
$\Sigma^{\lambda\lambda^{\prime}}_{11}=\frac{k}{2}(-1)^{\lambda_{1}+\lambda^{\prime}_{1}+w}$
and
$\Sigma^{\lambda\lambda^{\prime}}_{01}=\Sigma^{\lambda\lambda^{\prime}}_{10}=\frac{k}{2}(-1)^{\lambda_{1}+\lambda_{2}+\lambda^{\prime}_{1}+\lambda^{\prime}_{2}}i^{-u+3v+w}$.

| $([1],A_{\nu})$ | $([\mu^{2}],A_{\nu})$ | $([1],E_{l})$ | $([\mu^{2}],E_{l})$ | $([m^{j}],z^{b})$ | $([\mu],\zeta^{\lambda})$ | $([\mu^{3}],\zeta^{\lambda})$
---|---|---|---|---|---|---|---
$([1],A_{\nu^{\prime}})$ | 1 | $(-1)^{\nu^{\prime}}$ | 2 | $2(-1)^{\nu^{\prime}}$ | $2(-1)^{j\nu^{\prime}}$ | $ki^{\nu^{\prime}}$ | $ki^{-\nu^{\prime}}$
$([\mu^{2}],A_{\nu^{\prime}})$ | $(-1)^{\nu}$ | $(-1)^{u+\nu+\nu^{\prime}}$ | $2(-1)^{l}$ | $2(-1)^{u+l+\nu^{\prime}}$ | $2(-1)^{uj+j\nu^{\prime}+b}$ | $ki^{u+\nu^{\prime}+2\lambda}$ | $ki^{-u-\nu^{\prime}+2\lambda}$
$([1],E_{l^{\prime}})$ | 2 | $2(-1)^{l^{\prime}}$ | 4 | $4(-1)^{l^{\prime}}$ | $4(-1)^{jl^{\prime}}c(jl^{\prime})$ | 0 | 0
$([\mu^{2}],E_{l^{\prime}})$ | $2(-1)^{\nu}$ | $2(-1)^{u+l^{\prime}+\nu}$ | $4(-1)^{l}$ | $4(-1)^{u+l+l^{\prime}}$ | $4(-1)^{uj+jl^{\prime}+b}c(jl^{\prime})$ | 0 | 0
$([m^{j^{\prime}}],z^{b^{\prime}})$ | $2(-1)^{j^{\prime}\nu}$ | $2(-1)^{uj^{\prime}+j^{\prime}\nu+b^{\prime}}$ | $4(-1)^{j^{\prime}l}c(j^{\prime}l)$ | $4(-1)^{uj^{\prime}+j^{\prime}l+b^{\prime}}c(j^{\prime}l)$ | $4s^{u}_{jbj^{\prime}b^{\prime}}$ | 0 | 0
$([\mu],\zeta^{\lambda^{\prime}})$ | $ki^{\nu}$ | $ki^{u+\nu+2\lambda^{\prime}}$ | 0 | 0 | 0 | $ki^{\lambda+\lambda^{\prime}}e^{i\pi u/4}$ | $ki^{\lambda-\lambda^{\prime}}e^{-i\pi u/4}$
$([\mu^{3}],\zeta^{\lambda^{\prime}})$ | $ki^{-\nu}$ | $ki^{-u-\nu+2\lambda^{\prime}}$ | 0 | 0 | 0 | $ki^{-\lambda+\lambda^{\prime}}e^{-i\pi u/4}$ | $ki^{-\lambda-\lambda^{\prime}}e^{i\pi u/4}$
Table 14: The $S$-matrix $4kS_{\chi\chi^{\prime}}$ from (213) of the
$D^{[u]}(Q_{4k})$ gauge theory, for odd $k$. We abbreviate $c(x)=\cos(2\pi
x/k)$ and
$s^{u}_{jbj^{\prime}b^{\prime}}=c((jb^{\prime}+j^{\prime}b)/2+u(2/k+k/2)jj^{\prime})$.
The row and column entries are arranged in the same order of the dyons in
table 11.

###  C.3 The $Q_{4k}$ gauge theory

We only consider the dicycle group $Q_{4k}$ with odd degree $k$. Its group
elements are labeled by $g=(A,a)=\hat{\mu}^{A}m^{a}$, where $A=-1,0,1,2$ and
$a\in(-k/2,k/2)$. We also abbreviate $\hat{m}=\hat{\mu}^{2}m=(2,1)$. (See
(202) for the group presentations.) The Dijkgraaf-Witten invariants of the
$D^{[u]}(Q_{4k})$ gauge theory are cohomology elements $[u]$ in
$H^{3}(Q_{4k},U(1))=\mathbb{Z}_{4k}$. The $f$-symbols can be identified with
the cocycle representative
$f^{g_{1}g_{2}g_{3}}=\xi^{3}_{u}(g_{1},g_{2},g_{3})$ in (285). The solution to
the hexagon equation (210) can be chosen to be

$\displaystyle r^{g_{1}g_{2}}_{u}=\exp\left(\frac{2\pi
iu}{k^{2}}\left\\{a_{2}\left[(-1)^{A_{2}}a_{1}+2\left[A_{1}\right]_{2}a_{2}\right]-\left[A_{1}\right]_{2}a_{2}^{2}\right\\}\right)\exp\left(\frac{i\pi
u}{8}A_{1}A_{2}\right),$ (227)

where the square brackets, $[\ast]_{k}$ and $[\ast]_{2}$, wrap the numbers
back to the appropriate ranges in $(-k/2,k/2]$ and $0,1$ respectively.
Exchange phases $r^{g_{1}g_{2}}_{4k}$ can be absorbed by attaching gauge
charges to fluxes. Therefore, $u$ has period $4k$. The dyons and their quantum
dimensions and topological spins are listed in table 10 under the deformation
phases $\vartheta_{g}=r^{gg}$

$\displaystyle\vartheta_{\hat{m}^{j}}=\exp\left(2\pi
iu\left(\frac{j^{2}}{k^{2}}+\frac{[j]_{2}}{4}\right)\right),\quad\vartheta_{\hat{\mu}^{A}}=\exp\left(\frac{i\pi
uA^{2}}{8}\right),$ (228)

where $[j]_{2}=0,1$ is congruent to $j$ modulo 2. (Here, as a simplification,
we have replaced $e^{2\pi iu[j]_{k}^{2}/k^{2}}$ by $e^{2\pi iuj^{2}/k^{2}}$ in
$\vartheta_{\hat{m}^{j}}$ by attaching the $\mathbb{Z}_{k}$ charge
$\hat{m}^{j}\to\hat{m}^{j}z^{2}$ when $j>k/2$ to absorb the phase difference
$e^{4\pi ij/k}$.) The modular $S$-matrix can be computed with the help of the
character table 8 and by substituting the exchange phases (227) in (213). It
is presented in table 14.

The dicyclic gauge theory component in $SO(2n)_{1}/D_{k}$ is
$D^{[2k]}(Q_{4k})$, when $k=n/2$ is odd (see (139)). The Dijkgraaf-Witten
invariant $[u]=[2k]$ is fixed by the subgroup inclusion
$i:\mathbb{Z}_{k}\hookrightarrow Q_{4k}$ and the (non-canonical) injection
$j:\mathbb{Z}_{4}\to Q_{4k}$ (see (270)). They induce the restriction
homomorphisms $i^{\ast}:H^{3}(Q_{4k},U(1))\to
H^{3}(\mathbb{Z}_{k},U(1))=\mathbb{Z}_{k}$ and $j^{\ast}:H^{3}(Q_{4k},U(1))\to
H^{3}(\mathbb{Z}_{4},U(1))=\mathbb{Z}_{4}$ (see (272)), both being surjective
in this case and relate $i^{\ast}[u]_{4k}=[u]_{k}$ and
$j^{\ast}[u]_{4k}=[u]_{4}$. The Dijkgraaf-Witten invariants of
$D^{[0]}(\mathbb{Z}_{k})$ in (70) and $D^{[2]}(\mathbb{Z}_{4})$ in (56), when
$n=2k\equiv 2$ modulo 4, therefore restrict $u=2k$ modulo $4k$. The
topological spins of the fluxes in (228) become
$\vartheta_{\hat{m}^{j}}=(-1)^{j}$ and $\vartheta_{\hat{\mu}^{A}}=e^{i\pi
kA^{2}/4}$. In particular, this shows $\hat{m}^{k}=\hat{\mu}^{2}$ is fermionic
and can be identified with the fermion $\psi$ in $SO(2n)_{1}$.

##  Appendix D Useful group cohomology identities

We summarize some relevant tools in computing group cohomologies $H^{r}(G,M)$.
Brown (1982); Hatcher (2001); Adem and Milgram (2004); Adem (2006); Chen  _et
al._ (2013) We only consider the following finite groups $G$: (i) the cyclic
group $\mathbb{Z}_{k}$ of arbitrary order $k$, (ii) the dihedral group
$D_{k}=\mathbb{Z}_{2}\ltimes\mathbb{Z}_{k}$ of arbitrary degree $k$, and (iii)
the dicyclic group $Q_{4k}=\mathbb{Z}_{4}\ltimes\mathbb{Z}_{k}$ with odd
degrees $k$ (see (126)). We only examine the following Abelian coefficients
group $M$: (i) the ring of integers $\mathbb{Z}$, (ii) the (multiplicative)
group $U(1)$, and (iii) the anyon fusion group
$\mathcal{A}=\mathbb{Z}_{2}^{2}$ of $SO(2n)_{1}$ when $n$ is even. We always
assume $G$ acts trivially on $M$. (We omit the unnecessary odd $n$ cases when
the $\mathbb{Z}_{2}$ conjugation acts non-trivially on
$\mathcal{A}_{SO(2n)_{1}}=\mathbb{Z}_{4}$.) The group cohomology is defined to
be the quotient

$\displaystyle H^{r}(G,M)=\frac{\mathrm{ker}\left(d:C^{r}(G,M)\to
C^{r+1}(G,M)\right)}{\mathrm{Im}\left(d:C^{r-1}(G,M)\to C^{r}(G,M)\right)}.$
(229)

The set of $r$-cochains, $C^{r}(G,M)$, consists of functions $\omega:G^{r}\to
M$ that are not necessarily group homomorphisms. $C^{r}(G,M)$ is an Abelian
group under addition $\omega_{1}+\omega_{2}$ if $M=\mathbb{Z}$ or
multiplication $\omega_{1}\omega_{2}$ if $M=U(1)$ or $\mathcal{A}$. The
differential map is defined by

$\displaystyle d\omega(g_{1},\ldots,g_{r+1})$
$\displaystyle=\omega(g_{2},\ldots,g_{r+1})+\sum_{j=1}^{r}(-1)^{j}\omega(g_{1},\ldots,g_{j-1},g_{j}g_{j+1},\ldots,g_{r+1})+(-1)^{r+1}\omega(g_{1},\ldots,g_{r})$
$\displaystyle d\omega(g_{1},\ldots,g_{r+1})$
$\displaystyle=\omega(g_{2},\ldots,g_{r+1})\left[\prod_{j=1}^{r}\omega(g_{1},\ldots,g_{j-1},g_{j}g_{j+1},\ldots,g_{r+1})^{(-1)^{j}}\right]\omega(g_{1},\ldots,g_{r})^{(-1)^{r+1}}$
(230)

depending whether $M$ is additive or multiplicative. The differential map is
nilpotent and squares to zero, $d(d\omega)=0$. A $d$-exact cochain $\omega$,
satisfying $d\omega=0$, is called a cocycle. The derivative of a cochain,
$\omega=d\eta$, is called a coboundary. Cohomology elements in (229) are
equivalent classes (i.e. cosets) $[\omega]=\omega+dC^{r-1}(G,M)$ for additive
coefficients, or $[\omega]=\omega dC^{r-1}(G,M)$ for multiplicative
coefficients.

By construction, $H^{r}(G,M)=0$ for any negative order $r<0$, and
$H^{0}(G,M)=M$. The group cohomologies for $G=\mathbb{Z}_{k}$, $D_{k}$ and
$Q_{4k}$ with integral coefficient are known. For the cyclic groups and
positive orders $r>0$,

$\displaystyle
H^{r}(\mathbb{Z}_{k},\mathbb{Z})=\left\\{\begin{array}[]{*{20}l}\mathbb{Z}_{k},&\mbox{if
$r$ is even}\\\ 0,&\mbox{if $r$ is odd}\end{array}\right..$ (233)

For the dihedral groups and positive orders $r>0$, when $k$ is odd,

$\displaystyle
H^{r}(D_{k},\mathbb{Z})=\left\\{\begin{array}[]{*{20}l}\mathbb{Z}_{2k},&\mbox{if
$r\equiv 0$ mod 4}\\\ \mathbb{Z}_{2},&\mbox{if $r\equiv 2$ mod 4}\\\
0,&\mbox{if $r$ is odd}\end{array}\right.,$ (237)

and when $k$ is even, Handel (1993)

$\displaystyle
H^{r}(D_{k},\mathbb{Z})=\left\\{\begin{array}[]{*{20}l}\mathbb{Z}_{k}\times\mathbb{Z}_{2}^{r/2},&\mbox{if
$r\equiv 0$ mod 4}\\\ \mathbb{Z}_{2}^{(r-1)/2},&\mbox{if $r\equiv 1,3$ mod
4}\\\ \mathbb{Z}_{2}^{(r+2)/2},&\mbox{if $r\equiv 2$ mod
4}\end{array}\right..$ (241)

For the dicyclic groups and positive orders $r>0$, we only need the cases when
$k$ is odd,

$\displaystyle
H^{r}(Q_{4k},\mathbb{Z})=\left\\{\begin{array}[]{*{20}l}\mathbb{Z}_{4},&\mbox{if
$r\equiv 1$ mod 4}\\\ \mathbb{Z}_{4k},&\mbox{if $r\equiv 3$ mod 4}\\\
0,&\mbox{if $r$ is even}\end{array}\right..$ (245)

These results can be computed using technique such as the Lyndon-Hochschild-
Serre spectral sequence Lyndon (1948); Hochschild and Serre (1953). The
derivation is not needed and will be omitted.

The group cohomologies with the multiplicative coefficient $M=U(1)$ or
$\mathbb{Z}_{2}^{2}$ can be derived from the above integral coefficient ones
using the universal coefficient theorem (or the Künneth formula, by using
$G\cong G\times 1$ and $M\cong\mathbb{Z}\otimes M$) for cohomologies Hatcher
(2001); Chen  _et al._ (2013)

$\displaystyle H^{r}(G,M)=\left[H^{r}(G,\mathbb{Z})\otimes
M\right]\oplus\mathrm{Tor}_{1}^{\mathbb{Z}}(H^{r+1}(G,\mathbb{Z}),M),$ (246)

where $\otimes$ is the tensor product that treats Abelian groups as modules
over the ring $\mathbb{Z}$. The direct sum $\oplus$ is the Cartesian product
$\times$ and the notation is used here to avoid confusion with $\otimes$.
Tensor products between (additive) Abelian groups can be computed using

$\displaystyle(A_{1}\oplus A_{2})\otimes B=(A_{1}\otimes B)\oplus(A_{2}\otimes
B),$ $\displaystyle\mathbb{Z}\otimes B=B,\quad\mathbb{Z}_{k}\otimes B=B/kB.$
(247)

The tor functor $\mathrm{Tor}_{1}^{\mathbb{Z}}$ between (additive) Abelian
groups can be computed using the following properties

$\displaystyle\mathrm{Tor}_{1}^{\mathbb{Z}}(A_{1}\oplus
A_{2},B)=\mathrm{Tor}_{1}^{\mathbb{Z}}(A_{1},B)\oplus\mathrm{Tor}_{1}^{\mathbb{Z}}(A_{2},B),$
$\displaystyle\mathrm{Tor}_{1}^{\mathbb{Z}}(\mathbb{Z}_{k},B)=B[k]=\\{b\in
B:kb=1\\},$ (248)
$\displaystyle\mathrm{Tor}_{1}^{\mathbb{Z}}(\mathbb{Z},B)=0,$

where $B[k]$ is known as the $k$-torsion subgroup of $B$. In particular, we
are interested in (i) the $H^{2}(G,\mathcal{A})$ classification of quantum
symmetry group $\widehat{G}$ – the $\mathcal{A}=\mathbb{Z}_{2}^{2}$ central
extension (190) of $G=\mathbb{Z}_{k}$ and $D_{k}$ – and (ii) the Dijkgraaf-
Witten invariants in $H^{3}(G,U(1))$, for $G=\mathbb{Z}_{k}$, $D_{k}$ and
$Q_{4k}$. They are given by

$\displaystyle H^{2}(\mathbb{Z}_{k},\mathcal{A})$
$\displaystyle=\left\\{\begin{array}[]{*{20}l}0,&\mbox{for $k$ odd}\\\
\mathcal{A},&\mbox{for $k$ even}\end{array}\right.,$ (251) $\displaystyle
H^{2}(D_{k},\mathcal{A})$
$\displaystyle=\left\\{\begin{array}[]{*{20}l}\mathcal{A},&\mbox{for $k$
odd}\\\ \mathcal{A}^{3},&\mbox{for $k$ even}\end{array}\right.,$ (254)
$\displaystyle H^{3}(\mathbb{Z}_{k},U(1))$ $\displaystyle=\mathbb{Z}_{2k},$
$\displaystyle H^{3}(D_{k},U(1))$
$\displaystyle=\left\\{\begin{array}[]{*{20}l}\mathbb{Z}_{2k},&\mbox{for $k$
odd}\\\ \mathbb{Z}_{k}\times\mathbb{Z}_{2}^{2},&\mbox{for $k$
even}\end{array}\right.,$ (257) $\displaystyle H^{3}(Q_{4k},U(1))$
$\displaystyle=\mathbb{Z}_{4k},\quad\mbox{for $k$ odd}.$ (258)

Next, we present explicit cocycle representatives of the elements in the above
cohomologies. We begin with group cohomologies of the cyclic group
$\mathbb{Z}_{k}$. 1-cocycles in $C^{1}(\mathbb{Z}_{k},U(1))$ are group
homomorphisms $\omega^{1}_{u}:\mathbb{Z}_{k}\to U(1)$ that send an (additive)
group element $a$ in $\mathbb{Z}_{k}=\mathbb{Z}/(k\mathbb{Z})$ to

$\displaystyle\omega^{1}_{u}(a)=e^{2\pi iua/k},$ (259)

where $u=0,1,\ldots,k-1$ (mod $k$) represents a cohomology element in
$H^{1}(\mathbb{Z}_{k},U(1))=\mathrm{Hom}(\mathbb{Z}_{k},U(1))=\mathbb{Z}_{k}$.
In general, cohomologies with integral coefficients can be identified with
ones with $U(1)$ coefficients. The short exact sequence of coefficient groups
$0\to\mathbb{Z}\hookrightarrow\mathbb{R}\xrightarrow{\exp(2\pi i\ast)}U(1)\to
1$ induces the long exact sequence of cohomologies

$\displaystyle\ldots\to H^{r}(G,\mathbb{R})$ $\displaystyle\to H^{r}(G,U(1))$
(260) $\displaystyle\xrightarrow{\delta}H^{r+1}(G,\mathbb{Z})\to
H^{r+1}(G,\mathbb{R})\to\ldots.$

The differential map (also called connecting homomorphism) is defined by
$\delta\omega=d\log\omega/(2\pi i)$. The universal coefficient theorem (246)
implies $H^{r}(G,F)=H^{r}(G,\mathbb{Z})\otimes F$ for any characteristic zero
field $F$ such as $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$. In particular,
if $H^{r}(G,\mathbb{Z})$ is finite like the case when $r>0$ and
$G=\mathbb{Z}_{k}$, $D_{k}$ or $Q_{4k}$, then $H^{r}(G,F)=0$. Therefore,

$\leavevmode\hbox to150.08pt{\vbox
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