the gapped boundary $b_{l}=0$ and hence these lines form a decoupled MTC

$\mathcal{N}_{21}=\mathcal{A}^{N,-t_{21}^{\perp}}\;.$ (4.54)

We have obtained the fusion rule

$D_{\mathcal{L}}[\mathcal{T}]\times
D_{\mathcal{L}}[\mathcal{T}]=\mathcal{A}^{N,-t_{21}^{\perp}}\;D_{\mathcal{L}}[\mathcal{T}_{21}]\;.$
(4.55)

Notice that this fusion rule is the same (with the same $\mathcal{N}_{21}$) on
all gapped boundaries $\rho(\mathcal{L})$ belonging to the same orbit under
$V[\mathcal{T}]$. This follows from footnote 23.

##### Generic case.

In order to treat the general case it is convenient to parametrize the lines
$(\xi_{1},\xi_{2})=(v,\eta-v)$ in terms of two vectors $v,\eta$, so that the
charge of a line under $\mathcal{B}$ is $\eta$, and redefine the numbers
$a_{1}=p+q$, $a_{2}=q$. The equations (4.50) become

$\displaystyle\bigl{(}\mathcal{T}_{1}+\mathcal{T}_{2}\bigr{)}v-\bigl{(}\mathcal{T}_{2}+\tfrac{\epsilon}{2}\bigr{)}\eta$
$\displaystyle=p\,l_{\perp}$ (4.56)
$\displaystyle\mathcal{T}_{2}\,\eta-\bigl{(}\mathcal{T}_{2}-\tfrac{\epsilon}{2}\bigr{)}v$
$\displaystyle=q\,l_{\perp}\;.$

Defining
$\Gamma=\bigl{(}\mathcal{T}_{2}-\frac{\epsilon}{2}\bigr{)}\bigl{(}\mathcal{T}_{1}+\mathcal{T}_{2}\bigr{)}^{-1}$,
the solutions are

$v=q\;\Gamma^{\mathsf{T}}\mathcal{T}_{21}^{-1}l_{\perp}+p\,\Bigl{[}\bigl{(}\mathcal{T}_{1}+\mathcal{T}_{2}\bigr{)}^{-1}+\Gamma^{\mathsf{T}}\mathcal{T}_{21}^{-1}\Gamma\Bigr{]}l_{\perp}\;,\qquad\eta=q\;\mathcal{T}_{21}^{-1}l_{\perp}+p\;\mathcal{T}_{21}^{-1}\Gamma
l_{\perp}$ (4.57)

and can be labelled by $q,p\in\mathbb{Z}_{N}$. Substituting in (4.33), the
spins of the lines are

$\theta\bigl{[}W_{(q,p)}\bigr{]}=\exp\biggl{(}-\frac{\pi
i}{N}\;(q,p)\mathcal{R}_{21}\Bigl{(}\\!\\!\begin{array}[]{c}q\\\\[-5.0pt]
p\end{array}\\!\\!\Bigr{)}\biggr{)}\qquad\text{with}\qquad\mathcal{R}_{21}=\begin{pmatrix}t_{21}^{\perp}&c_{\mathrm{o}}\\\
c_{\mathrm{o}}&c_{\mathrm{d}}\end{pmatrix}\;,$ (4.58)

where

$c_{\mathrm{o}}=l_{\perp}^{\mathsf{T}}\,\mathcal{T}_{21}^{-1}\Gamma\,l_{\perp}\;,\qquad
c_{\mathrm{d}}=l_{\perp}^{\mathsf{T}}\,\Bigl{[}\bigl{(}\mathcal{T}_{1}+\mathcal{T}_{2}\bigr{)}^{-1}+\Gamma^{\mathsf{T}}\mathcal{T}_{21}^{-1}\Gamma\Bigr{]}\,l_{\perp}=l_{\perp}^{\mathsf{T}}\,\bigl{(}\mathcal{T}_{1}+\tfrac{\epsilon}{2}\mathcal{T}_{2}^{-1}\tfrac{\epsilon}{2}\bigr{)}^{-1}\,l_{\perp}\;.$
(4.59)

The subset of lines $(q,0)$ have charges $\eta$ proportional to
$\mathcal{T}_{21}^{-1}l_{\perp}$ and thus couple to $b_{\perp}$. Their spins
show that they form the MTC $\mathcal{A}^{N,-t_{21}^{\perp}}(b_{\perp})$. On
the other hand, the subset of lines $(q,p)$ with
$q=-(t_{21}^{\perp})^{-1}c_{\mathrm{o}}\,p$ braid trivially with the former
subset and constitute the theory $\mathcal{N}_{21}$. Their charges $\eta$ are
such that $l_{\perp}^{\mathsf{T}}\eta=0$, therefore they are decoupled from
$\mathcal{B}$ on the gapped boundary. Their spins show that

$\mathcal{N}_{21}=\mathcal{A}^{N,-n_{21}}\qquad\text{with}\qquad
n_{21}=c_{\mathrm{d}}-(t_{21}^{\perp})^{-1}c_{\mathrm{o}}^{2}=(t_{21}^{\perp})^{-1}\det\mathcal{R}_{21}\;.$
(4.60)

One should recall that, in the absence of a coupling to $\mathcal{B}$, the
theories $\mathcal{A}^{N,-p}$ and $\mathcal{A}^{N,-pr^{2}}$ are equivalent for
any invertible $r\in\mathbb{Z}_{N}$, and thus for $N$ odd prime the only
physical information in $n_{21}\neq 0$ is whether it is a quadratic residue or
not. This is detected by the Legendre symbol $n_{21}^{(N-1)/2}\text{ mod
}N\in\\{1,-1\\}$.343434In the higher-rank case the situation is similar. For
$N$ odd prime, one can always bring a symmetric matrix $\mathcal{T}$ with
values in $\mathbb{Z}_{N}$ to a diagonal form
$U^{\mathsf{T}}\mathcal{T}U=\operatorname{diag}(t_{1},\dots,t_{r})$ using an
invertible matrix $U$ (see, e.g., [75]). The TQFT is then characterized by the
number of $+1$ and $-1$ Legendre symbols of the $t_{i}$’s.

##### Case $\boldsymbol{\mathcal{T}_{2}=-\mathcal{T}_{1}\equiv\mathcal{T}}$.

This is the case leading to condensation. The equations for lines in
$\mathcal{A}^{N,-\mathcal{R}_{21}}$ are just
$\mathcal{T}\xi_{1}+\frac{\epsilon}{2}\xi_{2}=-a_{1}l_{\perp}$ and
$\mathcal{T}\xi_{2}+\frac{\epsilon}{2}\xi_{1}=a_{2}l_{\perp}$. The general
solution is

$\xi_{1}=\biggl{[}a\,\Bigl{(}\mathcal{T}+\frac{\epsilon}{2}\Bigr{)}^{-1}-c\,\Bigl{(}\mathcal{T}-\frac{\epsilon}{2}\Bigr{)}^{-1}\biggr{]}\,l_{\perp}\;,\qquad\xi_{2}=\biggl{[}a\,\Bigl{(}\mathcal{T}+\frac{\epsilon}{2}\Bigr{)}^{-1}+c\,\Bigl{(}\mathcal{T}-\frac{\epsilon}{2}\Bigr{)}^{-1}\biggr{]}\,l_{\perp}$
(4.61)

where we redefined $a_{1}=c-a$ and $a_{2}=c+a$. For these lines:

$\theta\bigl{[}W_{(a,c)}\bigr{]}=\exp\biggl{(}-\frac{2\pi
i}{N}\,ac\;2l_{\perp}^{\mathsf{T}}\Bigl{(}\mathcal{T}+\frac{\epsilon}{2}\Bigr{)}^{-1}l_{\perp}\biggr{)}\;.$
(4.62)

Lines with either $a$ or $c=0$ have vanishing spin, which indicates that we
are dealing with a DW type theory. The lines with $a=0$ (electric) do not
couple to $\mathcal{B}$ since they have $\xi_{1}+\xi_{2}=0$. Redefining
$a\to\bigl{[}2l_{\perp}^{\mathsf{T}}\bigl{(}\mathcal{T}+\frac{\epsilon}{2}\bigr{)}^{-1}l_{\perp}\bigr{]}^{-1}$
gives the canonical braiding $B_{ac}=e^{\frac{2\pi i}{N}ac}$. Thus

$\mathcal{A}^{N,-\mathcal{R}_{21}}(b_{\perp})=(\mathbb{Z}_{N})_{0}(b_{\perp})=\mathcal{C}^{\mathbb{Z}_{N}}\;.$
(4.63)

We conclude that:

$D_{\mathcal{L}}[\mathcal{T}]\times\overline{D}_{\mathcal{L}}[\mathcal{T}]=\mathcal{C}^{\mathbb{Z}_{N}}\;.$
(4.64)

The condensate $\mathcal{C}^{\mathbb{Z}_{N}}$ is for the 1-form symmetry
$\mathcal{S}=(\mathbb{Z}_{N}\times\mathbb{Z}_{N})/\mathcal{L}\cong\mathbb{Z}_{N}$
that exists on the gapped boundary.

##### Examples.

We can now apply our formalism to the known cases of duality and triality
defects. We consider a generic boundary $\rho(\mathcal{L})$, but assume that
it is not invariant under any symmetry defect appearing below (apart from $C$,
which leaves every boundary invariant). For the application to self-duality
defects, we must compute the fusion $D_{\mathcal{L}}[S]\times
D_{\mathcal{L}}[S]$. This is a special case, since the r.h.s. involves charge
conjugation. The explicit computation is done in Appendix C (see also the
comments in Section 4.3.2). The complete fusion gives a coefficient which is a
product of DW theories, these all admit a universal boundary condition which
allows us to set them to one. This corresponds to the Dirichlet boundary of
the DW theory. After this we find:

$\displaystyle D_{\mathcal{L}}[S]\times D_{\mathcal{L}}[S]$
$\displaystyle=(\mathbb{Z}_{N})(\tilde{b}_{\perp},\phi_{\perp})\;D_{\mathcal{L}}^{\text{triv}}[0]$
(4.65) $\displaystyle D_{\mathcal{L}}[S]\times\overline{D}_{\mathcal{L}}[S]$
$\displaystyle=\mathcal{C}^{\mathbb{Z}_{N}}\;.$

For triality defects we compute:353535To get to the result we use the property
$\mathcal{A}^{N,pr^{2}}=\mathcal{A}^{N,p}$ for $\gcd(r,N)=1$.

$\displaystyle D_{\mathcal{L}}[ST]\times D_{\mathcal{L}}[ST]$
$\displaystyle=\mathcal{A}^{N,-p_{ST}}\;D_{\mathcal{L}}\bigl{[}(ST)^{2}\bigr{]}$
(4.66) $\displaystyle D_{\mathcal{L}}[CST]\times D_{\mathcal{L}}[CST]$
$\displaystyle=\mathcal{A}^{N,-p_{ST}}\;D_{\mathcal{L}}\bigl{[}(ST)^{2}\bigr{]}$
$\displaystyle D_{\mathcal{L}}\bigl{[}(ST)^{2}\bigr{]}\times
D_{\mathcal{L}}\bigl{[}(ST)^{2}\bigr{]}$
$\displaystyle=\mathcal{A}^{N,\;p_{ST}}\;D_{\mathcal{L}}[CST]$

where

$p_{ST}=\begin{cases}1&\text{if }\mathcal{L}=\mathcal{L}(e)\;,\\\
r^{2}+r+1&\text{if }\mathcal{L}=\mathcal{L}(m)_{r}\;.\end{cases}$ (4.67)

These fusions agree with those computed in [27] on the electric
boundary.363636One uses that $\mathcal{A}^{N,1}=U(1)_{N}$ [50]. Notice that
the conventions of [27] defined in their eqns. (6.7)–(6.9) differ from ours,
and their defects are the orientation reversal of ours, leading to a sign
change in the level. Notice that $p_{ST}\neq 0\text{ mod }N$ as long as the
boundary Lagrangian subgroup $\mathcal{L}$ is not invariant under $ST$.

##### Other defects.

Another interesting case is when the 4d defects $V_{M}=V[\mathcal{A},\xi]$ are
obtained by condensing a $\mathbb{Z}_{N}$ subgroup of
$\mathbb{Z}_{N}\times\mathbb{Z}_{N}$, corresponding to the elements $M\in
SL(2,\mathbb{Z}_{N})$ that are conjugate to $T^{k}$ for some $k$. For
simplicity let us consider the case $M=T^{k}$ with $k=1,\dots,N-1$. The
twisted sectors $D_{T^{k}\\!,\,\mathcal{L}}$ are described by the minimal
theories $\mathcal{A}^{N,k^{-1}}(b)$ for $\mathbb{Z}_{N}$ coupled to the bulk
field $b$. For the lines in these theories there is no extra contribution to
the braiding when we stack the theories, and thus they fuse in the standard
way:

$D_{T^{k}\\!,\,\mathcal{L}}\times
D_{T^{k^{\prime}}\\!,\,\mathcal{L}}=\mathcal{A}^{N,\,k^{-1}+k^{\prime-1}}\;D_{T^{k+k^{\prime}}\\!,\,\mathcal{L}}$
(4.68)

as long as $k+k^{\prime}\neq 0\text{ mod }N$. This formula is in agreement
with the fusion law of $N$-ality defects found in [21, 27]. Notice that these
twist sectors are not unique since they can be fused to 3d condensates for the
magnetic symmetry. However, on the magnetic boundaries $\mathcal{L}(m)$ (on
which the twisted sector $D_{T^{k}}$ hosts a minimal theory) we can take the
condensates to be generated by the magnetic symmetry
$l(m)\in\mathcal{L}(m)$.373737We can always arrive at this choice since any
two magnetic lattices differ by electric ones, which can be absorbed by
$D_{T^{k}}$. On the magnetic boundary these condensates however become all
decoupled DW theories since $l(m)^{\mathsf{T}}\mathcal{B}\big{|}_{X}=0$.

####  4.3.1 Twist defects and boundary-changing operators

$\rho(\mathcal{L})$$\rho(\mathcal{L})$$D_{M,\mathcal{L}}$$V_{M}$$\rho(\mathcal{L})$$\rho(M\mathcal{L})$$\mathcal{U}_{M}$$V_{M}$$D_{M,\mathcal{L}}^{\dagger}$$\mathcal{U}_{M}$$V_{M}$$=$$\varphi_{M}$

$\rho(\mathcal{L})$$\rho(M\mathcal{L})$$\rho(\mathcal{L})$$R[M\tau]$$R[\tau]$$\mathcal{U}_{M}^{\dagger}$$\varphi_{M}$$V_{M}$$\Rightarrow$$\Leftarrow$$A_{\rho}[M\tau]$$A_{\rho_{M}}[\tau]$$A_{\rho}[\tau]$$\Rightarrow$$A_{\rho}[\tau]$$A_{\rho}[M\tau]$$\rho(\mathcal{L})$$\rho(\mathcal{L})$$V_{M}$$R[\tau]$$R[M\tau]$$D_{M,\,\mathcal{L}}$

Figure 8: The way in which the symmetry TFT implements the construction of
[20, 28]. Above: definition of various morphisms. Below: construction of the
duality interface $D_{M,\,\mathcal{L}}$.

Consider starting with a twist defect $D_{M}$ (attached to a 4d symmetry
defect $V_{M}$) in the bulk and moving it on top of a gapped boundary
$\rho(\mathcal{L})$. We are here interested in the case that
$\rho(\mathcal{L})$ is not invariant under $M$. As discussed before (4.45),
the defect $D_{M}$ on the boundary decomposes into $D_{M,\,\mathcal{L}}$ and a
decoupled TQFT. We conclude that $D_{M,\,\mathcal{L}}$ is an interface between
two copies of $\rho(\mathcal{L})$, or using categorical terms, it defines a
morphism $D_{M,\,\mathcal{L}}:M\times\rho(\mathcal{L})\to\rho(\mathcal{L})$.
This is depicted in Figure 8 left. On the other hand, if we bring the symmetry
defect $V_{M}$ on top of the boundary we obtain an action
$V_{M}\times\rho(\mathcal{L})=\rho(M\mathcal{L})$. Thus, we can construct an
interface between $\rho(\mathcal{L})$ and $\rho(M\mathcal{L})$ by fusing the
boundary with $V_{M}$ only on a half-space and then letting $V_{M}$ escape in
the bulk, as in Figure 8 center. This defines a morphism
$\mathcal{U}_{M}:M\times\rho(\mathcal{L})\to\rho(M\mathcal{L})$. Since both
interfaces sit at the end of a symmetry defect $V_{M}$, it is possible to
define a local boundary-changing operator as the morphism
$\varphi_{M}=\mathcal{U}_{M}\circ
D_{M,\,\mathcal{L}}^{\dagger}:\rho(\mathcal{L})\to\rho(M\mathcal{L})$, as in
Figure 8 right.

Recall that, in the ungauged theory, one can expect to define only duality
_interfaces_. The interface is a composite object given by a discrete gauging
operation composed with an invertible duality transformation. On the TQFT side
this is described by acting with $\mathcal{U}_{M}^{\dagger}$ to map the
boundary to $\rho(M\mathcal{L})$ and then using $\varphi_{M}$ to go back to
$\rho(\mathcal{L})$. After compactifying the slab of symmetry TFT this gives
an interface : $A_{\rho}[\tau]\to A_{\rho_{M}}[\tau]\to A_{\rho}[M\tau]$
between absolute theories. Shrinking the middle part of the drawing gives the
duality interface. On the other hand the fusion
$\varphi_{M}\times\mathcal{U}_{M}^{\dagger}=D_{M,\,\mathcal{L}}$ holds, since
$D_{M,\,\mathcal{L}}$ is unique as a twist defect on the gapped boundary. We
can thus identify the defect $D_{M,\,\mathcal{L}}$ on the bottom-left of
Figure 8 with the duality interface in the absolute theory $A_{\rho}[\tau]$.

####  4.3.2 Boundaries with a stabilizer

Let us also discuss the properties of a twist defect
$D_{\mathcal{L}}[\mathcal{T}]$ on a gapped boundary $\rho(\mathcal{L})$ that
is invariant under the corresponding symmetry defect $V[\mathcal{T}]$. This
means that $M$ is in the stabilizer $H$ of $\mathcal{L}$ in
$SL(2,\mathbb{Z}_{N})$. We can gather information on the degrees of freedom
living on $D_{\mathcal{L}}[\mathcal{T}]$ by computing the anomaly inflow. The
invertible TQFT living on $V[\mathcal{T}]$ is (4.11). On the gapped boundary
we parametrize383838Here the normalization is different than before (4.46),
because $t_{\perp}=0$ when $\mathcal{L}$ is invariant under $M$.
$\mathcal{B}=\tilde{b}_{\perp}l_{\perp}$ and obtain

$I_{\mathcal{T}}\big{|}_{\rho(\mathcal{L})}=\frac{N}{2\pi}\int\biggl{[}\tilde{\gamma}_{\perp}\,d\tilde{b}_{\perp}-\frac{1}{2}\,t_{\perp}\,\tilde{b}_{\perp}\tilde{b}_{\perp}\biggr{]}$
(4.69)

(where $\tilde{\gamma}_{\perp}=l_{\perp}^{\mathsf{T}}\widetilde{\Gamma}$).
Since now $t_{\perp}=0$ (see footnote 30), the anomaly is trivialized.

What happens to the lines in the twisted sector can be understood using the
minimal theory description. We start with the twist defect $D[\mathcal{T}]$ in
the bulk and push it onto an invariant boundary $\mathcal{L}$. Normally we
would now separate the degrees of freedom which decouple on the boundary,
which form a $\mathcal{A}^{N,-t_{l}}(b_{l})$ factor. This is generated by
lines $L_{s}\equiv W_{n=s\;l}$. If the boundary is invariant then $t_{l}=0$
and this procedure is ill defined as the $L_{s}$ all have vanishing spin. They
thus form a Lagrangian subalgebra. This means that
$\mathcal{A}^{N,-\mathcal{T}}$ should rather be thought of as a DW theory
coupled to $\tilde{b}_{\perp}$. Since the lines with trivial spin are also
uncharged under $\tilde{b}_{\perp}$ this can be thought of as a condensate:

$\mathcal{A}^{N,\mathcal{T}}(\mathcal{B})\big{|}_{X}=\mathcal{C}^{\mathbb{Z}_{N}}\;.$
(4.70)

To be more precise we can choose a generator $u$ of $\mathcal{S}$. Since by
definition $u^{\mathsf{T}}l_{\perp}\neq 0$ lines $\tilde{L}_{r}\equiv
W_{n=ru}$ are charged under $\tilde{b}_{\perp}$. These lines have spin:

$\theta\left[\tilde{L}_{r}\right]=\exp\left(-\frac{\pi
i}{N}\;r^{2}\;u^{\mathsf{T}}\mathcal{T}u\right)$ (4.71)

and braid with the electric lines $L_{s}$:

$B_{r,s}=\exp\left(-\frac{2\pi
i}{N}\;rs\;u^{\mathsf{T}}\mathcal{T}l\right)\neq 1\,,$ (4.72)

since on the invariant boundary $\mathcal{T}\mathcal{L}=\mathcal{L}_{\perp}$.
Properly redefining $L_{s}$ we can make this braiding into canonical one. As
we have already commented there is no canonical choice for $u$, since we are
free to shift it by vectors in $\mathcal{L}$. The shift $u\to u+l$ does not
affect the braiding with $L_{r}$ but it does affect the spin of
$\tilde{L}_{s}$:

$\theta\left[\tilde{L}_{r}\right]\to\theta\left[\tilde{L}_{r}\right]\exp\left(-\frac{2\pi
i}{N}\;r^{2}\;u^{\mathsf{T}}\mathcal{T}l\right)$ (4.73)

For $N$ odd and on spin manifolds we can use this to set $\theta[L_{r}]$ to
one.

Since the defect $V[\mathcal{T}]$ has trivial anomaly on $\rho(\mathcal{L})$,
it can end there without adding new degrees of freedom. Therefore the twist
defect $D_{\mathcal{L}}[\mathcal{T}]$ is trivial (invertible) on an invariant
boundary:

$D[\mathcal{T}]\big{|}_{X}=\mathcal{C}^{\mathbb{Z}_{N}}\;D_{\mathcal{L}}^{\text{triv}}[\mathcal{T}]\,,$
(4.74)

where the superscript is useful to remember this fact.

The same phenomenon appears if we consider a fusion
$D_{\mathcal{L}}[\mathcal{T}_{1}]\times D_{\mathcal{L}}[\mathcal{T}_{2}]$ in
which $V[\mathcal{T}_{21}]$ leaves the boundary $\rho(\mathcal{L})$ invariant,
but neither $V[\mathcal{T}_{1}]$ nor $V[\mathcal{T}_{2}]$ do. We proceed as in
the usual case by separating out the lines coupling to $b_{l}$ from both terms
in the fusion. This is a well defined procedure since
$t_{1}^{l},\,t_{2}^{l}\neq 0$ (due to $\mathcal{L}$ not being invariant under
neither $\mathcal{T}_{1}$ nor $\mathcal{T}_{2}$). Based on the previous
remarks we expect $\mathcal{A}^{N,-\mathcal{R}_{2,1}}$ to also be a
condensate. It is clear that the theory contains a Lagrangian algebra
generated by $W_{(q,0)}$ in (4.58). In the generic discussion these lines were
coupled to $b_{\perp}$, however if the boundary is invariant they are
not.393939The charge under the gauge symmetry for $\tilde{b}_{\perp}$ is
$ql_{\perp}^{\mathsf{T}}\mathcal{T}_{2,1}^{-1}l_{\perp}$, which vanishes when
the boundary is invariant. These form the set of “electric” lines. The
magnetic lines $W_{(0,p)}$ instead couple to $\tilde{b}_{\perp}$, but have
nontrivial spin:

$\theta\left[W_{(0,p)}\right]=\exp\left(-\frac{\pi
i}{N}c_{\mathrm{d}}\;p^{2}\right)\,,$ (4.75)

As before, we can redefine the magnetic lines by summing a multiple of the
electric ones to set this to zero. Notice that the discussion here is also
consistent with the example of $\mathcal{T}_{2}=-\mathcal{T}_{1}$ discussed
before, when the final result is a condensate and the identity defect leaves
all boundaries invariant.

We are now in a position to write down the full result of the boundary fusion
for $D_{\mathcal{L}}[\mathcal{T}]$:

$\displaystyle D_{\mathcal{L}}[\mathcal{T}_{2}]\times
D_{\mathcal{L}}[\mathcal{T}_{1}]$
$\displaystyle=\mathcal{N}_{21}\,D_{\mathcal{L}}[\mathcal{T}_{21}]\;,\qquad$
$\displaystyle\text{if
}V[\mathcal{T}_{2,1}]|\rho(\mathcal{L})\rangle\neq|\rho(\mathcal{L})\rangle\;,$
(4.76) $\displaystyle D_{\mathcal{L}}[\mathcal{T}_{2}]\times
D_{\mathcal{L}}[\mathcal{T}_{1}]$
$\displaystyle=\mathcal{C}^{\mathbb{Z}_{N}}\;D^{\text{triv}}_{\mathcal{L}}[\mathcal{T}]\;,\qquad$
$\displaystyle\text{if
}V[\mathcal{T}_{2,1}]|\rho(\mathcal{L})\rangle=|\rho(\mathcal{L})\rangle\;,$
$\displaystyle
D_{\mathcal{L}}[\mathcal{T}]\times\overline{D_{\mathcal{L}}[\mathcal{T}]}$
$\displaystyle=\mathcal{C}^{\mathbb{Z}_{N}}\;.$

This will have a more natural interpretation in the gauged theory. In that
case we will see that anomaly cancellation forces the Gukov Witten operator
$\text{GW}[\mathcal{T}]$ to exist only as a bound state with the twist defect
$D[\mathcal{T}]$ for $V[\mathcal{T}]$. When the boundary $\mathcal{L}$ is
$\mathcal{T}$-invariant there is no anomaly to cancel and
$\text{GW}[\mathcal{T}]$ can exist as a genuine defect on the gapped boundary.
The fusion rule above tell us that, when two bound operator fuse onto an
invariant one, such fusion is always accompanied by the appearence of a
condensation defect. This is consistent with the fact that defects
$D_{\mathcal{L}}[\mathcal{T}]$ absorb surface defects
$e^{i\\!\int\\!b_{\perp}}$, which survive on the gapped boundary. In the
absence of the condensation defect the r.h.s. cannot absorb such lines and
fusion would be inconsistent.

##  5 The gauged theory

Finally, we discuss the effect of gauging a discrete subgroup $G\subset
SL(2,\mathbb{Z}_{N})$ in the bulk TQFT. In the application to $\mathcal{N}=4$
SYM, the only relevant groups (including the action of charge conjugation) are
$\mathbb{Z}_{4}$ and $\mathbb{Z}_{6}$ generated by $S$ and $ST$, respectively.
Notice that they are both Abelian. The construction we present below applies
to a generic Abelian $G$, while the non-Abelian case requires modifications
that might be important in discussing theories of class $S$ (we comment on
that in the conclusions).

We will first describe abstractly the spectrum of operators in the gauged
theory. We follow the rules for discrete gauging described for 3d MTCs in [44]
and recently extended to higher dimensions in [25]. Particular care will be
needed in describing the Gukov-Witten operators of $G$ gauge theory, as they
get dressed by the corresponding twist defects $D[\mathcal{T}]$. We will
present the construction of these operators, that we dub
$\mathfrak{D}[\mathcal{T}]$. Finally, we will study gapped boundaries
$|\rho^{*}\rangle$ in the gauged theory in terms of orbits of boundaries
$|\rho\rangle$ in the ungauged theory. This allows for a simple derivation of
the fusion rules. We will also comment on the differences arising when the
boundary has a nontrivial stabilizer.

In the following we will restrict to the study of twisted sectors
$D[\mathcal{T}]$ for which $M(\mathcal{T})$ is an element of $G$. Together
with the assumption that $G$ is Abelian, this ensures that different twisted
sectors do not mix among each other and that the genuine codimension-2
operators $\mathfrak{D}[\mathcal{T}]$ of the gauged theory are still labelled
by group elements.404040In the general case they are labelled by conjugacy
classes under the adjoint action of $G$.

###  5.1 Spectrum of bulk operators

The spectrum of topological operators in the gauged theory can be obtained, at
least at a formal level, by applying standard rules for gauging a discrete
0-form symmetry to the ungauged theory. These are nicely summarized in [25].
Let us start with the surface defects $U_{n}$ that implement the 2-form
symmetry. These operators are in general not gauge invariant, as $G$ acts on
them nontrivially. We can build gauge-invariant combinations by considering
orbits under $G$:

$U^{*}_{[n]}=\frac{1}{\bigl{\lvert}\text{Stab}(n)\bigr{\rvert}}\,\sum_{g\in
G}U(g\,n)\;,$ (5.1)

where $\text{Stab}(n)$ is the stabilizer group for $n$ as an element of
$\mathbb{Z}_{N}\times\mathbb{Z}_{N}$. When $n$ admits a nontrivial stabilizer,
the surface $U^{*}_{[n]}$ supports nontrivial line defects labelled by
representations of $\text{Stab}(n)$. In the cases considered here, that is $N$
prime and $G=\mathbb{Z}_{4}$ or $\mathbb{Z}_{6}$, the only surface with a
nontrivial stabilizer is the identity, while all others ones do not host any
line.

As an example, in the case of the $\mathbb{Z}_{4}$ subgroup of
$SL(2,\mathbb{Z}_{N})$ generated by $S$, a dyon $(e,m)$ is mapped to an orbit

$[e,m]=(e,m)+(m,-e)+(-e,-m)+(-m,e)\;.$ (5.2)

These objects are non-invertible and their fusion is

$[e,m]\times[e^{\prime},m^{\prime}]=[e+e^{\prime},m+m^{\prime}]+[e+m^{\prime},m-e^{\prime}]+[e-e^{\prime},m-m^{\prime}]+[e-m^{\prime},m+e^{\prime}]\;.$
(5.3)

$g$$h^{-1}gh$$D[\mathcal{T}]$$h$$h^{-1}$$\leadsto$gauging$\mathfrak{D}[\mathcal{T}]$
Figure 9: In the gauged theory, a twist defect $D[\mathcal{T}]$ (with
$M(\mathcal{T})\equiv g\in G$) is dressed by codimension-1 surfaces of $G$
labelled by $h$.

More interesting is the situation for codimension-2 operators. We have already
discussed that in the ungauged theory, genuine 3d operators are necessarily
condensation defects. After the discrete gauging the situation is different.
The twist defects $D[\mathcal{T}]$ for surfaces $V[\mathcal{T}]$ generating
$G$ become “liberated” — in the sense that they become genuine 3d operators —
since the surfaces $V[\mathcal{T}]$ are transparent in the gauged theory. One
could think of the liberated defects as arising from the“lassoed”
configuration shown in Figure 9 after summing over $G$. Since $G$ is Abelian,
each twist sector is left fixed by the action of the lassos and it gives rise
to a single genuine operator $\mathfrak{D}[\mathcal{T}]$.414141When instead
$G$ is non-Abelian, twist defects also combine into orbits and the situation
is more subtle. The action of a lasso $V_{M^{\prime}}$ reduces to a 0-form
symmetry action on $D[\mathcal{T}]$, which maps $W_{n}\mapsto
W_{M^{\prime}n}$. This is indeed a symmetry of the theory, since

$M^{\prime\mathsf{T}}\,\mathcal{T}\,M^{\prime}=\mathcal{T}$ (5.4)

and thus it preserves the braiding. Summing over such action means that the
0-form symmetry on the defect is gauged, so we would like to conclude that
$\mathfrak{D}[\mathcal{T}]$ is $D[\mathcal{T}]/G$.

This description is slightly imprecise, because $D[\mathcal{T}]$ lives at the
boundary of $V[\mathcal{T}]$. Indeed, the gauging process can be thought of as
coupling the original system to a discrete $G$ gauge theory. Its gauge field
$a\in H^{1}(M_{5},G)$ couples minimally to the 0-form symmetry defects
$V_{M\in G}$ of the original theory (more details in Section 5.2). In this
setup, inserting a twist defect $D[\mathcal{T}]$ is only consistent at
locations where $a$ is not closed: it must instead satisfy $\delta a=g$
schematically. Another way of saying this is that $a$ exhibits a nontrivial
holonomy $g$ around the 3-cycle $Y$ on which $D[\mathcal{T}]$ lies. This is
the description of Gukov-Witten defect operators in $G$ gauge theory, that we
indicate as $\text{GW}_{g}$. We infer that a more precise definition of the
new operators is:

$\mathfrak{D}[\mathcal{T}]=\text{GW}_{M(\mathcal{T})}\times
D[\mathcal{T}]/G\;.$ (5.5)

The appearance of this “bound state” has a simple explanation: In the original
theory, the defect $D[\mathcal{T}]$ was not gauge invariant due to anomaly
inflow from $V[\mathcal{T}]$. The GW operator is not gauge invariant either,
as it carries the anomaly of $V[\mathcal{T}]$. Their combination is a well
defined operator in the gauged theory. This is a close cousin of the mechanism
described in [21]. We also learn that $\mathfrak{D}[\mathcal{T}]$ is charged
under the dual $\widehat{G}$ 3-form symmetry.

The exception is the twist defect $D[\mathcal{T}=0]\equiv D_{C}$ for charge
conjugation. In this case there is no anomaly inflow and therefore the GW
operator for $C$ defines a genuine, group-like object in the gauged theory.
This suggests that we should interpret the contributions from $D_{C}$ arising
upon fusion as decoupled condensates after gauging.

The following table summarizes the properties of some objects in the gauged
theory:

Original object | Gauged object | Emergent lines | Grouplike?
---|---|---|---
$(0,0)$ | $[0,0]$ | $\text{Rep}(G)$ | YES
$(e,m)$ | $[e,m]=\oplus_{g\in G}\,g(e,m)$ | none | NO
$D[\mathcal{T}]$ | $\mathfrak{D}[\mathcal{T}]=\text{GW}_{M(\mathcal{T})}\times D[\mathcal{T}]/G$ | $\text{Rep}(G)$ | NO
$D_{C}$ | $\text{GW}_{C}$ | none | YES

###  5.2 Hybrid formulation of the gauged theory

In order to give a Lagrangian description of the gauging of the subgroup
$G\subset SL(2,\mathbb{Z}_{N})$ in the 5d Chern-Simons theory, we employ a
sort of hybrid formulation in which the Chern-Simons theory is described by
continuum gauge fields, while the gauge field for $G$ is described using
singular cochains (see, e.g., [76, 77] or the appendix in [78]).

First of all, on the spacetime manifold $M_{5}$ one chooses a simplicial
triangulation. This is made of vertices or 0-simplices $p_{i}$ with an
arbitrary ordering for the index $i$, edges or 1-simplices $p_{ij}$ (with
$i<j$) connecting the vertices $p_{i}$ and $p_{j}$, 2-simplices $p_{ijk}$
(with $i<j<k$) bounded by $p_{ij}$, $p_{jk}$ and $p_{ik}$, and so on. All
simplices are contractible, and $M_{5}$ is the union of all 5-simplices. A
gauge field $a$ for the discrete gauge group $G$ is a 1-cochain $a\in
C^{1}(M_{5},G)$ that assigns an element $a_{ij}\in G$ to each 1-simplex
$p_{ij}$ (with $i<j$), with the constraint that $da=\mathbbm{1}$. We use
multiplicative notation and define the differential as
$(da)_{ijk}=a_{jk}a_{ik}^{-1}a_{ij}$ (with $i<j<k$). We will only consider the
case that $G$ is Abelian. Gauge transformations then map
$a_{ij}\mapsto(d\lambda)_{ij}a_{ij}$ where
$d\lambda_{ij}=\lambda_{j}\lambda_{i}^{-1}$ and $\lambda\in C^{0}(M_{5},G)$ in
a 0-cochain. The gauging of $G$ is described by a sum over $a\in
H^{1}(M_{5},G)$ in cohomology.

Then we construct a covering of $M_{5}$ by closed patches that is dual to the
triangulation, as follows. Each patch $U_{i}$ is a 5d contractible manifold
with boundary that contains the 0-simplex $p_{i}$. Then each non-empty
intersection $U_{i_{1}\dots i_{k}}=U_{i_{1}}\cap\dots\cap U_{i_{k}}$ (with
$i_{1}<\ldots<i_{k}$ and $k=2,\ldots,6$) is a $(6-k)$-dimensional contractible
manifold with boundary that intersects the $(k-1)$-simplex $p_{i_{1}\dots
i_{k}}$ at one point. We give a graphical representation of this covering in
Figure 10.

$U_{jk}$$U_{ik}$$U_{ij}$$p_{i}$$U_{i}$$p_{j}$$U_{j}$$p_{k}$$U_{k}$$p_{ij}$$p_{jk}$$p_{ik}$$U_{ijk}$

$i$$j$$k$$\mathcal{B}_{i}$$\mathcal{B}_{j}$$\mathcal{B}_{k}$$a_{jk}$$a_{ik}$$a_{ij}$

Figure 10: Left: representation of the simplicial triangulation and the
covering by closed sets $U_{i}$, $U_{j}$, $U_{k}$ near a triple intersection.
Right: assignment of fields $\mathcal{B}_{i}$ and $a_{ij}$.

On every patch $U_{i}$ we define gauge fields $\mathcal{B}_{i}$ with values in
an Abelian group $\mathcal{A}$ (either continuous or discrete), and along the
intersections $U_{ij}$ we glue them using a group homomorphism
$\theta:G\to\operatorname{Aut}(\mathcal{A})$ and the gauge field
$a$:424242Besides, one could also have gauge transformations of
$\mathcal{B}_{i}$, but we keep them implicit here.

$\mathcal{B}_{i}=\theta(a_{ij})\,\mathcal{B}_{j}\qquad\qquad\text{across
}U_{ij}\;.$ (5.6)

The gauge field $\mathcal{B}$ is thus a piecewise-smooth field with
$\mathcal{B}\big{|}_{U_{i}}=\mathcal{B}_{i}$. Closeness of $a$ guarantees that
each $\mathcal{B}_{i}$ can be smooth and have a well-defined limit at triple
intersections $U_{ijk}$. In particular, we can always find a gauge in which
$a_{ij}=a_{jk}=a_{ik}=1$ around a given triple intersection $U_{ijk}$, and in
that gauge $\mathcal{B}$ can be smooth at the intersection.

The construction is quite general. In our case $\mathcal{B}_{i}$ are
continuous 2-form gauge fields valued in $\mathcal{A}=U(1)^{2}$, while
$SL(2,\mathbb{Z})$ has the natural action on $\mathcal{A}$ and $G\subset
SL(2,\mathbb{Z})$ is an Abelian subgroup. We should now understand how to
construct the action. Integrating
$S=\frac{N}{4\pi}\int\langle\mathcal{B},\,d\mathcal{B}\rangle$ with the
discontinuous gluing conditions (5.6) leads to singularities, in particular
the derivative $d\mathcal{B}$ has delta-function singularities along the
surfaces $U_{ij}$. To remedy, we introduce a covariant derivative $d_{a}$ that
removes those singularities:

$d_{a}\mathcal{B}=d\mathcal{B}-\sum_{U_{ij}}\delta^{(1)}(U_{ij})\bigl{(}\mathcal{B}_{j}-\mathcal{B}_{i}\bigr{)}=d\mathcal{B}-\sum_{U_{ij}}\delta^{(1)}(U_{ij})\,\sigma(a_{ij})\,\mathcal{B}_{j}\,\equiv\,\Bigl{(}d-\delta^{(1)}_{a}\sigma(a)\Bigr{)}\mathcal{B}\;,$
(5.7)

where $\delta^{(1)}(U_{ij})$ is a delta-1-form,
$\sigma(a)\equiv\mathbbm{1}-\theta(a)$, and in the last expression we used a
more compact notation. In this way, $d\mathcal{B}$ is a piecewise-smooth field
such that $d\mathcal{B}\big{|}_{\mathring{U}_{i}}=d\mathcal{B}_{i}$ with
discontinuities across $U_{ij}$ but no delta-function singularities. We can
then construct the action

$S=\frac{N}{4\pi}\int\langle\mathcal{B},d_{a}\mathcal{B}\rangle=\sum_{U_{i}}\frac{N}{4\pi}\int_{U_{i}}\langle\mathcal{B}_{i},d\mathcal{B}_{i}\rangle\;.$
(5.8)

The covariant derivative $d_{a}$ can be integrated by parts, and the action is
invariant under gauge transformations of $a$.

In order to discuss 1-form gauge transformations, we need to compute the
square $d_{a}^{2}$ of the covariant derivative. It turns out that, to do that,
we ought to be more careful and write
$d_{a}\mathcal{B}=d\mathcal{B}-\sum_{U_{ij}}\delta^{(1)}(U_{ij})\bigl{(}\mathcal{B}_{j}^{(ij)}-\mathcal{B}_{i}^{(ij)}\bigr{)}$
where the label $(ij)$ reminds us that we are taking the limit of
$\mathcal{B}_{i}$ or $\mathcal{B}_{j}$ towards $U_{ij}$. Then
$d_{a}d\mathcal{B}=-\sum_{U_{ij}}\delta^{(1)}(U_{ij})\bigl{(}d\mathcal{B}_{j}^{(ij)}-d\mathcal{B}_{i}^{(ij)}\bigr{)}$,
and finally

$\displaystyle d_{a}^{2}\,\mathcal{B}$
$\displaystyle=-\sum_{U_{ijk}}\delta^{(2)}(U_{ijk})\Bigl{[}\bigl{(}\mathcal{B}_{j}^{(ij)}-\mathcal{B}_{i}^{(ij)}\bigr{)}+\bigl{(}\mathcal{B}_{k}^{(jk)}-\mathcal{B}_{j}^{(jk)}\bigr{)}-\bigl{(}\mathcal{B}_{k}^{(ik)}-\mathcal{B}_{i}^{(ik)}\bigr{)}\Bigr{]}$
(5.9)
$\displaystyle\equiv\,-\sum_{U_{ijk}}\delta^{(2)}(U_{ijk})\,\sigma(da_{ijk})\,\mathcal{B}\;.$

In the first equality we used that
$d\bigl{(}\delta^{(1)}(U_{ij})\bigr{)}=\delta^{(2)}(\partial U_{ij})$ and that
the boundary of a double intersection is a collection of triple intersections
(with suitable signs due to orientations). In the second line we introduced a
compact notation. Indeed, if $a$ is closed ($da=\mathbbm{1}$) then each
$\mathcal{B}_{i}$ can be smooth and taking the limit towards $U_{ijk}$ in each
patch, the first line of (5.9) equals zero. If, instead, $a$ is not closed,
then the $G$ bundle can have non-trivial holonomies around the triple
intersections and the $\mathcal{B}_{i}$’s cannot be smooth there. Given
$(da)_{ijk}=g$, consider a gauge in which $a_{ij}=a_{jk}=1$, $a_{ik}=g^{-1}$
(see Figure 11 right). Then the contribution to (5.9) from $U_{ijk}$ becomes
$-\delta^{(2)}(U_{ijk})\,\sigma(da_{ijk})\,\mathcal{B}_{i}^{(ik)}$. We thus
write the compact formula

$d_{a}^{2}=-\delta^{(2)}_{a}\,\sigma(da)\;.$ (5.10)

In the presence of a background for $a$, 1-form gauge transformations of
$\mathcal{B}$ become

$\mathcal{B}\to\mathcal{B}+d_{a}\alpha\;,$ (5.11)

and the action (5.8) remains gauge invariant as long as $a$ is flat.

$a_{ik}=g^{-1}g_{ik}$$a_{jk}=g_{jk}$$a_{ij}=g_{ij}$$g$$i$$j$$k$$\mathcal{B}_{i}$$\mathcal{B}_{k}$$\mathcal{B}_{j}$
$g_{ij}$$g_{jk}$$g_{ik}$$g^{-1}g_{ik}$$g^{-1}$$g$
$1$$1$$g$$i^{\prime}$$j^{\prime}$$k^{\prime}$$g^{-1}$

Figure 11: Left: A triple intersection $U_{ijk}$ hosts a GW operator for $g\in
G$. We parametrized the gauge field $a$ in terms of $g_{ij}g_{jk}=g_{ik}$ and
$da_{ijk}=g$. Center: A refinement of the triangulation such that the GW
operator is pulled away from the junctions. Right: A zoom on the gauge field
configuration around the isolated GW operator.

The theory in which $G$ is gauged involves a sum over choices of $a$ on double
intersections $U_{ij}$ that satisfy the closeness condition $da=\mathbbm{1}$.
A single symmetry defect $U(\gamma)$ in the ungauged theory is mapped to a sum
over its $G$-orbit in the gauged theory. These are precisely the $[e,m]$
defect operators we introduced before.

On the other hand, we can introduce Gukow-Witten operators in the gauged
theory [54]. These are codimension-2 disorder operators defined by a
nontrivial holonomy $g\in G$ for $a$ around a 3d submanifold
$\gamma^{\prime}$. In the hybrid formulation, such a GW operator displaced
along a collection of triple intersections $U_{ijk}$ is defined by a sum in
the path integral over cochains $a$ such that

$da_{ijk}=g\qquad\text{whenever $U_{ijk}\subset\gamma^{\prime}$}\;,$ (5.12)

as in Figure 11. More generally, a collection of GW operators is described by
an exact cochain $h\in C^{2}(M_{5},G)$, and it prescribes to sum over cochains
$a$ with $da=h$ in the gauged theory. As mentioned above, requiring the
$\mathcal{B}_{i}$’s to be smooth in their own patches in a neighborhood of a
triple intersection, forces them to be invariant under $g$ there.434343If
$\mathcal{B}_{i}$ is smooth at $U_{ijk}$, then it has a well-define limit
there. The limits in the three patches $U_{i}$, $U_{j}$, $U_{k}$ are related
by
$\mathcal{B}_{i}=\theta(a_{ij})\,\mathcal{B}_{j}=\theta(a_{ik})\,\mathcal{B}_{k}$
and $\mathcal{B}_{j}=\theta(a_{jk})\,\mathcal{B}_{k}$. Recalling that $G$ is
Abelian, this implies $\theta(da_{ijk})\,\mathcal{B}_{i}=\mathcal{B}_{i}$ and
similarly for $\mathcal{B}_{j}$ and $\mathcal{B}_{k}$. This is a boundary
condition naturally implemented on the GW operators, consistent with the fact
that $g$-twisted sectors absorb the surfaces of $\mathcal{B}$ not stabilized
by $g$.

Indeed, we can identify a double intersection $U_{ij}$ with gauge field
$a_{ij}$ as an alternative description for the 4d symmetry defect $V_{M}$ with
$M=\theta(a_{ij})^{\mathsf{T}}$. This is already apparent if we compare the
relation $\mathcal{B}_{i}=\theta(a_{ij})\,\mathcal{B}_{j}$ between the fields
on the two sides of the intersection and (3.53), but it also follows from the
action. Let us rewrite (5.8) as

$S=\frac{N}{4\pi}\int\langle\mathcal{B},d\mathcal{B}\rangle-\frac{N}{4\pi}\sum_{U_{ij}}\int_{U_{ij}}\langle\mathcal{B},\mathcal{B}_{j}-\mathcal{B}_{i}\rangle\;.$
(5.13)

The first term imposes that $\mathcal{B}$ is a
$\mathbb{Z}_{N}\times\mathbb{Z}_{N}$ gauge field. When $\mathcal{T}(M)$ is
invertible, we can identify the second term with the reduced defect action
(3.51). Recall from the discussion in Section 3.3 that the relation between
$M$ and the torsion matrix $\mathcal{T}$ follows from determining the field on
the defect
$\mathcal{B}(0)=\frac{1}{2}(\mathcal{B}_{\text{L}}+\mathcal{B}_{\text{R}})=-\mathcal{T}\Phi$,
where $\epsilon\,\Phi=\mathcal{B}_{\text{R}}-\mathcal{B}_{\text{L}}$, in terms
of the left/right fields $\mathcal{B}_{\text{L/R}}$. Substituting
$\mathcal{B}_{\text{R}}-\mathcal{B}_{\text{L}}=-\epsilon\,\mathcal{T}^{-1}\mathcal{B}(0)$
into (5.13), the second term becomes

$-\frac{N}{4\pi}\sum_{U_{ij}}\int_{U_{ij}}\mathcal{B}^{\mathsf{T}}\mathcal{T}^{-1}\mathcal{B}$
(5.14)

that reproduces (3.51).

To compute how a GW operator transforms under gauge transformations (5.11) we
simply evaluate the variation of the action (5.8) on a non-closed gauge
configuration as in (5.12):

$\delta_{\alpha}S=-\sum_{U_{ijk}}\frac{N}{4\pi}\int_{U_{ijk}}\bigl{\langle}2\mathcal{B}+d_{a}\alpha,\,\sigma(da_{ijk})\,\alpha\bigr{\rangle}\;.$
(5.15)

In the gauge of Figure 11 right, as above,
$\sigma(da_{ijk})\,\alpha=\alpha_{i}^{(ik)}-\alpha_{k}^{(ik)}=\epsilon\,\mathcal{T}^{-1}\alpha(0)$
in terms of the gauge transformation parameter on the defect. Substituting in
(5.15) and using that the boundary conditions fix $\mathcal{B}=0$ on the GW
operator, we obtain

$\delta_{\alpha}S=\sum_{U_{ijk}}\frac{N}{4\pi}\int_{U_{ijk}}\alpha^{\mathsf{T}}\mathcal{T}^{-1}d\alpha\;.$
(5.16)

As in the description of Section 4 in terms of symmetry defects $V_{M}$, also
in the hybrid formulation we find that pure GW operators are not gauge
invariant in this theory. We can construct gauge-invariant operators by
dressing the GW operators with the twisted sectors $D[\mathcal{T}]$, whose
variation (4.12) is opposite to (5.16).

In the case of the symmetry defect $V_{C}$, the field on the defect is simply
$\mathcal{B}(0)=0$ and thus its gauge transformation parameter $\alpha(0)$
vanishes as well. This means that the gauge variation (5.15) vanishes and the
GW operator for $C$ is a well-defined gauge-invariant (invertible) topological
operator in the gauged theory.

###  5.3 Gapped boundaries and non-invertible fusion rules

We consider now gapped boundaries in the gauged theory. We can use to our
advantage the study and classification we already did in the ungauged theory.
In order to construct a gauged boundary $|\rho^{*}\rangle$, we proceed in two
steps. First we take a boundary $|\rho\rangle$ in the ungauged theory and make
it invariant under the $G$ action:

$|\rho\rangle\;\;\to\;\;\frac{1}{\bigl{\lvert}\text{Stab}(\rho)\bigr{\rvert}}\,\sum_{g\in
G}|\rho_{g}\rangle\;.$ (5.17)

As long as $G$ is Abelian, we can associate a stabilizer $H\subset G$ in a
consistent way also to the gauged boundary $|\rho^{*}\rangle$, since
$\text{Stab}(\rho_{g})=\text{Stab}(\rho)$. This does not specify a boundary
condition completely, since it does not prescribe boundary conditions for
neither the $\text{Rep}(G)$ dual symmetry lines, nor the codimension-2 defects
$\mathfrak{D}[\mathcal{T}]$. They form a canonically-conjugated pair of
variables, since they braid nontrivially. Therefore, the second step is to
choose boundary conditions for them. We choose to impose Dirichlet boundary
conditions on $a$:444444This is the same choice made in the holographic setup
of [17].

$|\text{Dir}\rangle:\quad a=0\;.$ (5.18)

Then the operators $\mathfrak{D}[\mathcal{T}]$ still exist on the gapped
boundary as confined excitations.

These are not the only meaningful boundary conditions one could consider.
Indeed it would be interesting to understand the effect of Dirichlet boundary
conditions on the $\mathfrak{D}[\mathcal{T}]$’s, or of mixed ones. That they
might be useful to describe theories in which either charge conjugation $C$
(this has been studied, e.g., in [79, 80, 25]) or the full categorical
symmetry, are gauged. We hope to come back to these questions in the
future.454545In the same spirit, we could consider boundaries twisted by the
dual $\check{G}$ symmetry. This amounts to choosing a representation $\alpha$
of $G$ and define, for a boundary with a trivial stabilizer,
$|\rho_{\alpha}^{*}\rangle=\,\sum\nolimits_{g\in
G}\chi_{\alpha}(g)\;|\rho_{g}\rangle\times|\text{Dir}\rangle\;.$ (5.19) These
boundaries have vanishing overlap with the relative theory if we assume
absolute theories in the same orbit to have the same partition function. When
a stabilizer is present we can only twist by characters of
$G/\text{Stab}(\rho)$, while boundaries split into copies labelled by
representations of $\text{Stab}(\rho)$. We do not know how to interpret these
splitted boundaries from the point of view of the 4d QFT, thus we only
consider the ones labelled by the trivial representation.

With Dirichlet boundary conditions on $a$, we define:

$|\rho^{*}\rangle=\frac{1}{\bigl{\lvert}\text{Stab}(\rho)\bigr{\rvert}}\,\Biggl{(}\;\sum_{g\in
G}|\rho_{g}\rangle\Biggr{)}\times|\text{Dir}\rangle\;.$ (5.20)

The Dirichlet boundary condition on $a$ greatly simplifies the discussion. The
operators $\mathfrak{D}[\mathcal{T}]$, which away from the boundary host a
$\text{Rep}(G)$ worth of lines constructed with the gauge field $a$, on the
gapped boundary reduce to a direct product $\text{GW}_{M(\mathcal{T})}\times
D[\mathcal{T}]$. The Gukov-Witten operators still exist on $|\rho^{*}\rangle$
and have group-like fusion. We will now show that the fusion of the twist
operators $D_{\mathcal{L}}[\mathcal{T}]$ is the same on each gapped boundary
$|\rho_{g}\rangle$ in the $|\rho^{*}\rangle$ orbit. This allows to use the
results already derived for the boundary fusion.

We need to show that the various minimal theories we constructed in Section
4.3 in order to study the fusion of twist defects, are isomorphic for
boundaries in the same $G$-orbit. Let $M_{g}\in G\subset SL(2,\mathbb{Z}_{N})$
be a generator of $G$, and $M(\mathcal{T})$ be the element of $G$ associated
to the twist defect $D[\mathcal{T}]$ we want to study. Let
$\bigl{|}\rho(\mathcal{L})\bigr{\rangle}$ be a gapped boundary defined by the
Lagrangian subgroup $\mathcal{L}$ with generator $l$. The Lagrangian subgroup
of $|\rho_{g}\rangle$ is $M_{g}\mathcal{L}$, and since
$\mathcal{L}^{\mathsf{T}}\mathcal{L}_{\perp}=0$, we have

$\mathcal{L}_{\perp}^{g}=M_{g}^{-1\mathsf{T}}\,\mathcal{L}_{\perp}\;.$ (5.21)

The generators $l$ and $l_{\perp}$ transform in a similar way. Since $G$ is
Abelian and $\epsilon M_{g}=M_{g}^{-1\mathsf{T}}\epsilon$, then
$\mathcal{T}=M_{g}^{\mathsf{T}}\mathcal{T}M_{g}$ and so both $t_{l}$ and
$t_{\perp}$ are invariant along the orbit. Besides, $\Gamma M_{g}=M_{g}\Gamma$
and thus the theory $\mathcal{R}_{21}$ is invariant as well. Since all
relevant building blocks are isomorphic on boundaries that sit inside the same
$G$-orbit, we conclude that fusion only depends on the orbit
$|\rho^{*}\rangle$.

A new ingredient appears when fusion produces a defect
$\mathcal{D}[\mathcal{T}_{21}]$ such that $M(\mathcal{T}_{21})$ stabilizes
$|\rho\rangle$. As we discussed, in these cases the minimal theory is replaced
by a condensate. After gauging $G$, we are left with the GW operator
$\text{GW}_{M(\mathcal{T}_{21})}$.

Using all of the above, we finally obtain the categorical fusion rules in the
boundary theory specified by the gapped boundary $|\rho^{*}\rangle$:

$\displaystyle\mathfrak{D}_{\rho^{*}}[\mathcal{T}_{2}]\times\mathfrak{D}_{\rho^{*}}[\mathcal{T}_{1}]$
$\displaystyle=\mathcal{N}_{21}\;\mathfrak{D}_{\rho^{*}}[\mathcal{T}_{21}]\qquad$
$\displaystyle M_{21}$ $\displaystyle\notin\text{Stab}(\rho^{*})\;,$ (5.22)
$\displaystyle\mathfrak{D}_{\rho^{*}}[\mathcal{T}_{2}]\times\mathfrak{D}_{\rho^{*}}[\mathcal{T}_{1}]$
$\displaystyle=\mathcal{C}^{\mathbb{Z}_{N}}\;\text{GW}_{M(\mathcal{T}_{21})}\qquad$
$\displaystyle M_{21}$ $\displaystyle\in\text{Stab}(\rho^{*})\;,$
$\displaystyle\text{GW}_{M(\mathcal{T}_{2})}\times\text{GW}_{M(\mathcal{T}_{1})}$
$\displaystyle=\text{GW}_{M(\mathcal{T}_{21})}\qquad$ $\displaystyle
M_{1},M_{2}$ $\displaystyle\in\text{Stab}(\rho^{*})\;.$

In the second line, the condensate is for the 1-form symmetry $\mathcal{S}$ on
the gapped boundary, and the DW description couples to $\tilde{b}_{\perp}$.
For defects in which only one $\mathbb{Z}_{N}$ factor is gauged, on the other
hand, the fusions are as follows:464646These can be thought of as the case of
$T^{k}$ modulo conjugation.

$\displaystyle\mathfrak{D}_{T^{k}\\!,\,\rho^{*}}\times\mathfrak{D}_{T^{k^{\prime}}\\!,\,\rho^{*}}$
$\displaystyle=\mathcal{A}^{N,\,k^{-1}+k^{\prime-1}}\;\mathfrak{D}_{T^{k+k^{\prime}}\\!,\,\rho^{*}}\qquad$
$\displaystyle T$ $\displaystyle\notin\text{Stab}(\rho^{*})\;,$ (5.23)
$\displaystyle\text{GW}_{T^{k}}\times\text{GW}_{T^{k}}$
$\displaystyle=\text{GW}_{T^{k+k^{\prime}}}\qquad$ $\displaystyle T$
$\displaystyle\in\text{Stab}(\rho^{*})\;.$

The same can be said for conjugacy classes $g=\mathcal{H}^{-1}T\mathcal{H}$.

##  6 Conclusions and future directions

In this paper we have studied how non-invertible self-duality symmetries arise
in holography, through the presence in the gravitational bulk of emergent
discrete gauge fields at self-dual points on the moduli space. Although we
have focused on the specific example of the 4d $\mathcal{N}=4$ super-Yang-
Mills theory with gauge algebra $\mathfrak{su}(N)$, our methods are rather
general and should be applicable to a wide range of other theories, for
instance to $\mathcal{N}=2$ theories of class $S$.

The key role is played by a topological low-energy sector of type IIB string
theory on $S^{5}$: a 5d Chern-Simons-like topological field theory of 2-form
gauge fields — equivalent to a $\mathbb{Z}_{N}$ discrete 2-form gauge theory —
further orbifolded by a discrete Abelian symmetry $G$. It is essentially the
symmetry TFT for $SU(N)$ $\mathcal{N}=4$ SYM. This theory appears to be quite
interesting and rich in its own right, both before and after gauging $G$. We
have studied various aspects of the theory in this paper. Before orbifolding,
we have analyzed the 4d symmetry defects associated to an
$SL(2,\mathbb{Z}_{N})$ 0-form global symmetry of the theory, the associated
twisted sectors that live at the boundary of the defects, and their fusion. We
have also investigated topological (gapped) boundaries, and how the various
defects reduce when they are brought there. Then, we have studied the effect
of gauging a subgroup $G\subset SL(2,\mathbb{Z}_{N})$. In particular, the
“liberated” twist defects $\mathfrak{D}_{g,\,\rho^{*}}$ that live on gapped
boundaries turn out to be the self-duality defects of $\mathcal{N}=4$ SYM. We
derived their fusion rules using our formalism, confirming the results
previously obtained in field theory.

We conclude listing a few open questions for future research.

##### Extension to class $\boldsymbol{S}$.

Our formalism can naturally be extended to study self-duality non-invertible
symmetries in other theories, for instance on the conformal manifold of
$\mathcal{N}=2$ theories of class $S$ (see [32] for related investigations).
The symmetry TFT of the 6d $\mathcal{N}=(2,0)$ theory of ADE type
$\mathfrak{g}$ is a 7d Chern-Simons theory of 3-form gauge fields with level
matrix equal to the Cartan matrix $A_{ij}$ of $\mathfrak{g}$. After reducing
on a genus-$g$ Riemann surface $\Sigma_{g}$, one obtains a 5d CS theory with
$2g$-tuples of 2-form gauge fields $\mathcal{B}_{i}$ and braiding:

$B_{v,w}=\exp\biggl{(}\frac{2\pi i}{N}\sum_{i,j}A_{ij}\,\langle
v^{i},w^{j}\rangle\biggr{)}\;,$ (6.1)

where $\langle v,w\rangle$ is the homology intersection form on $\Sigma_{g}$.
The simplest example is given by theories of type $A_{N-1}$, which give rise
to $g$ copies of the CS theory we studied in this paper. Then the (projected)
mapping class group $Sp(2g,\mathbb{Z}_{N})$ of $\Sigma_{g}$ acts on the TQFT
as a 0-form symmetry.

##### The case of $\boldsymbol{N}$ not prime.

For the sake of simplicity, we have restricted our analyses to the case of $N$
prime throughout our paper. This technical assumption allowed us to exploit
the multiplicative group structure of $\mathbb{Z}_{N}^{*}$, simplifying many
formulas. When $N$ is not prime, the situation is technically more
complicated, both because the number of subgroups and global structures grows
with the number of prime factors in $N$, and because the fusion relations for
minimal theories become more involved.

##### General formulation of the symmetry TFT.

In the last part of our work, we have resorted to a hybrid formulation of the
gauged TQFT that uses both discrete and continuous gauge fields. It would be
pleasant to give a completely general description in terms of the correct
cohomology theory. A promising route could be to employ Deligne-Beilinson
twisted cocycles.

##### Anomalies for $\boldsymbol{N}$-ality symmetries.

In spite of the many recent developments, a clear understanding of ’t Hooft
anomalies for non-invertible symmetries in $d>2$ is still lackluster. The main
obstacle is to give a concrete implementation of the associativity conditions
for $n$-categories. The higher-dimensional TQFT approach might help to give an
alternative concrete route to such questions: instead of choosing Dirichlet
boundary conditions for the discrete 1-form gauge fields $a$, one might try to
define Neumann boundaries instead. On these, the non-invertible defects
$\mathfrak{D}[\mathcal{T}]$ are effectively gauged and they define an absolute
theory which is obtained from the ones we have studied here by gauging the
non-invertible symmetry. The failure to find such a boundary would signal an
’t Hooft anomaly.

## Acknowledgments

We thank Riccardo Argurio, Michele Del Zotto, Davide Gaiotto, Justin Kaidi,
Kantaro Ohmori, Sakura Schafer-Nameki, Luigi Tizzano, and Matthew Yu for
helpful discussions. We are especially grateful to Davide Gaiotto for pointing
out the possible role of emergent discrete gauge symmetries in the string
theory setup in relation to the non-invertible defects of $\mathcal{N}=4$,
providing inspiration for this work. We gratefully acknowledge support from
the Simons Center for Geometry and Physics, Stony Brook University, where some
of the research for this paper was performed. We thank the Perimeter Institute
for Theoretical Physics and the organizers of the workshop “Global Categorical
Symmetries” for hospitality. Research at Perimeter Institute is supported in
part by the Government of Canada through the Department of Innovation, Science
and Economic Development in Canada and by the Province of Ontario through the
Ministry of Colleges and Universities. The authors are partially supported by
the INFN “Iniziativa Specifica ST$\&$FI”. A.A., F.B., C.C. and G.R. are
supported by the ERC-COG grant NP-QFT No. 864583 “Non-perturbative dynamics of
quantum fields: from new deconfined phases of matter to quantum black holes”,
by the MIUR-SIR grant RBSI1471GJ, and by the MIUR-PRIN contract 2015 MP2CX4.

##  Appendix A Basic manipulations with symmetry TFT

We review here some basic facts about the symmetry TFT approach to global
variants of gauge theories. The idea is simple: in order to describe an
$n$-dimensional gauge theory $\mathcal{T}$, we introduce an auxiliary system,
comprised of a $n$-dimensional _relative_ theory $R$ together with an
$(n+1)$-dimensional non-invertible TQFT $Sym$. The relative theory contains
all the information about $\mathcal{T}$ which is insensitive to the global
structure, such as correlators of local operators, possibly charged under
flavor symmetries. Other properties of the theory, such as its 1-form
symmetry, depend on the choice of the global structure and thus both the
topological defects generating them and the charged objects are not part of
$R$. The geometric setup is as follows:

$Sym$$R$

From the point of view of the $(n+1)$-dimensional theory, the output of the
$n$-dimensional boundary manifold is not a complex number but a vector in a
finite-dimensional vector space, namely the Hilbert space of $Sym$. From the
point of view of $R$, this Hilbert space is the vector space of partition
functions. This does not define an absolute theory, as the bulk Hilbert space
of $Sym$ is in general not one-dimensional (i.e., $Sym$ in not invertible).
This can be fixed by the choice of a topological boundary $\rho$. In general
there will be multiple independent such $\rho$’s, each one specifying an
absolute theory.

Since $\rho$ is topological, we take $Sym$ in a slab with $R$ and $\rho$
boundary conditions on the two sides, and collapse the picture onto $X$, thus
obtaining a local (absolute) theory $A_{\rho}$:

$Sym$$R$$\rho$$=$$A_{\rho}$

Using standard arguments, one can view the choice of $\rho$ as the gauging of
a “maximal” non-anomalous generalized symmetry $\mathcal{A}_{\rho}$ inside
$Sym$. On the other hand, one can expand the state on the r.h.s. as
$|\mathcal{R}\rangle=\sum_{\gamma}\mathcal{Z}[\gamma]\,|\gamma\rangle$, with
$|\gamma\rangle$ an orthonormal basis for the TQFT Hilbert space. Computing
the overlap gives the partition function of the absolute theory:

$\mathcal{Z}[A_{\rho}]=\langle\rho|\mathcal{R}\rangle=\sum\nolimits_{\gamma}\;\langle\rho|\gamma\rangle\;\mathcal{Z}[\gamma]\;.$
(A.1)

In our case $\gamma\in H^{2}(X,\mathbb{Z}_{N}\times\mathbb{Z}_{N})$ is a
surface in the TQFT and

$\mathcal{Z}[A_{\rho}]=\sum\nolimits_{\gamma\in\mathcal{L}(\rho)}\mathcal{Z}[\gamma]\;.$
(A.2)

Using the symmetry TFT construction we can define various objects in the
absolute theory on the slab geometry:

$\mathcal{A}$A topological object $\mathcal{A}$ inside
$A_{\rho}$$\mathcal{O}$An $\mathcal{A}$-neutral operator $\mathcal{O}$ inside
$A_{\rho}$An $\mathcal{A}$-charged object $\mathcal{W}_{\mathcal{B}}$ inside
$A_{\rho}$$\mathcal{B}$$\mathcal{W}$

The fact that $\mathcal{O}$ is neutral while $\mathcal{W}_{\mathcal{B}}$ is
charged under $\mathcal{A}$ follows from sliding the $\mathcal{A}$ operator in
the pictures above before squashing the setup into the absolute theory.

##  Appendix B Properties of minimal TQFTs

Three-dimensional TQFTs with discrete 1-form symmetry group $\mathbb{Z}_{N}$
(or, more generally, products of the form $\prod_{I}\mathbb{Z}_{N_{I}}$) and
fixed anomaly for said 1-form symmetry admit powerful classification results
in terms of “minimal” TQFTs $\mathcal{A}^{\\{N_{I}\\},\\{p_{IJ}\\}}$, as
pioneered in [50]. Here we review some important consequences of the
classification results for $\mathbb{Z}_{N}$ (theories $\mathcal{A}^{N,p}$) and
$\mathbb{Z}_{N}^{r}$ (theories $\mathcal{A}^{N,\mathcal{T}})$. In the main
text we only use $r=1$ and $r=2$.

The possible anomalies for a $\mathbb{Z}_{N}$ 1-form symmetry in 3d are
labelled by an integer $p$ defined modulo $2N$ (or modulo $N$ on spin
manifolds) and can be represented by the following 4d inflow
action[50]:474747The anomaly is generated by $-I_{\mathcal{T}}$ as customary.

$I_{\mathcal{T}}=\frac{2\pi p}{2N}\int_{X}\mathfrak{P}(B)\;,\qquad\qquad B\in
H^{2}(X,\mathbb{Z}_{N})\;.$ (B.1)

If we assume that there are no non-anomalous subgroups, that is $\gcd(N,p)=1$,
to such anomaly we associate a minimal TQFT $\mathcal{A}^{N,p}$. This theory
has $N$ line operators $W_{l}$, $l=0,\ldots,N-1$ that form a $\mathbb{Z}_{N}$
fusion algebra, with spins

$\theta_{l}=\exp\biggl{(}\frac{2\pi i\,p}{2N}\,l^{2}\biggr{)}\;.$ (B.2)

If the theory is bosonic then $\theta_{N}=1$ and $Np\in 2\mathbb{Z}$. In this
paper we deal with spin theories, in which case $W_{N}=\psi$ can be a
transparent fermion. In bosonic theories $p$ is identified with $p+2N$, while
in the spin case $p+N$ gives rise to the same spin theory. We stress that
$\mathcal{A}^{N,p}$ is a well defined 3d TQFT if and only if $\gcd(N,p)=1$, as
otherwise the theory has transparent bosonic lines, which give a non-unitary
$S$ matrix. Otherwise, the $S$-matrix is given by

$S_{l\,l^{\prime}}=\frac{1}{N^{1/2}}\,\exp\biggl{(}\frac{2\pi
i\,p}{N}\,l\,l^{\prime}\biggr{)}\;.$ (B.3)

An important result of [50] is that a 3d TQFT $T$ with a $\mathbb{Z}_{N}$
1-form symmetry with anomaly $p$ admits an expansion in the
$\mathcal{A}^{N,p}$ (hence the name “minimal”):

$T=\mathcal{A}^{N,p}\times T^{\prime}\;,\qquad\qquad
T^{\prime}=\frac{\mathcal{A}^{N,-p}\times T}{\mathbb{Z}_{N}}\;,$ (B.4)

which can be derived using the identity

$\bigl{(}\mathbb{Z}_{N}\bigr{)}_{-Np}=\mathcal{A}^{N,p}\times\mathcal{A}^{N,-p}\;.$
(B.5)

The product (or stacking) of two minimal theories
$\mathcal{A}^{N,p}\times\mathcal{A}^{N,q}$ is also simple to compute, as long
as $\gcd(p+q,N)=1$:

$\mathcal{A}^{N,p}\times\mathcal{A}^{N,q}=\mathcal{A}^{N,(p^{-1}+q^{-1})^{-1}}\times\mathcal{A}^{N,p+q}\;,$
(B.6)

where inverses are taken in $\mathbb{Z}_{N}$. Minimal theories have a large
degree of redundance, indeed let $r$ be coprime with $N$, then

$\mathcal{A}^{N,p}\sim\mathcal{A}^{N,r^{2}p}$ (B.7)

as MTCs. The transformation is equivalent to choosing a different generator
for $\mathbb{Z}_{N}$. Note that this implies that
$\mathcal{A}^{N,p}\sim\mathcal{A}^{N,p^{-1}}$. Using these conventions, we can
set the line $W_{l=1}$ as the generator of $\mathbb{Z}_{N}$. It then follows
from the $S$-matrix that the generator has charge $p$ under $\mathbb{Z}_{N}$.
Alternatively, we could use the line $W_{l=p^{-1}}$ as the fundamental line.
This line has unit charge under $\mathbb{Z}_{N}$. The change of variables
affects the inflow action, which is then labelled by $p^{-1}$ instead. In the
main text we choose to work under this choice of generator.

The construction can be generalized to multiple $\mathbb{Z}_{N}$ factors. For
simplicity we treat the case in which $N$ is prime, as in the main text. A
theory $\mathcal{A}^{N,\mathcal{T}}$ is then described by a symmetric $r\times
r$ matrix $\mathcal{T}$, whose lines have spins

$\theta_{l}=\exp\left(\frac{2\pi
i}{2N}\,l^{\mathsf{T}}\,\mathcal{T}\,l\right)=\exp\left(\frac{2\pi
i}{2N}\left[\sum_{i}\mathcal{T}_{ii}l_{i}^{2}+2\sum_{i>j}\mathcal{T}_{ij}l_{i}l_{j}\right]\right)\,.$
(B.8)

A bosonic theory requires $N\mathcal{T}_{ii}\in 2\mathbb{Z}$ and
$N\mathcal{T}_{ij}\in\mathbb{Z}$, while a spin theory can have
$\mathcal{T}_{ii}\in\mathbb{Z}$ and $\mathcal{T}_{ij}\in\mathbb{Z}$. The
condition of having a well defined $S$-matrix requires that $\mathcal{T}$ is
an invertible matrix over $\mathbb{Z}_{N}^{r}$, this is the natural
generalization of the $\gcd$ condition for $r=1$. To this TQFT we can
associate an anomaly theory as in the previous case:

$I_{\mathcal{T}}=\frac{2\pi}{2N}\mathfrak{P}^{\mathcal{T}}(B)\,,\ \
\mathfrak{P}^{\mathcal{T}}(B)=\sum_{i}^{r}\mathcal{T}_{ii}\mathfrak{P}(B_{i})+\sum_{i>j}^{r}2\mathcal{T}_{ij}B_{i}\cup
B_{j}\,,\ \ \ B\in H^{2}(X,\mathbb{Z}_{N}^{r})\,.$ (B.9)

As in the previous case there is a large degree of redundancy in these
theories. Let $\mathcal{N}$ be an invertible matrix over $\mathbb{Z}_{N}$,
then:

$\mathcal{A}^{N,\;\mathcal{N}^{\mathsf{T}}\mathcal{T}\mathcal{N}}=\mathcal{A}^{N,\;\mathcal{T}}\,,$
(B.10)

as $\mathcal{N}$ just implements a redefinition of the generators. Since
$\mathcal{T}$ is a nondegenerate symmetric quadratic form it can be
diagonalized with coefficients in $\mathbb{Z}_{N}$ by a suitable
$\mathcal{N}$: $\mathcal{T}=\operatorname{diag}(t_{i},\;...,\;t_{r})$ with
$\gcd(t_{i},N)=1$. The only relevant information about the theory (without
specifying the coupling to the two-form gauge field $B$) are thus the
quadratic residue classes for the $t_{i}$. As for the one-dimensional case, in
the main text we use a slightly different convention in which the fundamental
lines have charge one. To go back to the standard convention one has to
substitute $\mathcal{T}$ by $\mathcal{T}^{-1}$ in the formulas.

An important novelty with respect to the one-dimensional case is that
generalized minimal theories can have anomaly free subgroups. For spin
theories these are generated by vectors $l$ such that:

$l^{\mathsf{T}}\mathcal{T}l=0\mod N\,.$ (B.11)

Let us take $r=2$ and $N$ prime. For $r=2$ $\mathcal{A}^{N,\mathcal{T}}$
theories also contain twisted $\mathbb{Z}_{N}$ DW theories $\text{DW}(\alpha)$
but with a twist matrix which is a multiple of $N$. If $N$ is prime every $l$
will generate a Lagrangian subgroup, so a solution to (B.11) implies that the
theory is DW for a certain choice of torsion. This is important in the main
text, as it implements the correct fusion laws for twisted sectors on
invariant boundaries.

Notice that the factorization theorem for $r=1$ still applies. This means
that, given a $\mathbb{Z}_{N}$ subgroup with nontrivial anomaly $p$, we can
write:

$\mathcal{A}^{N,\mathcal{T}}=\mathcal{A}^{N,p}\times\mathcal{A}^{N,\mathcal{T}^{\prime}}\,,$
(B.12)

And $\mathcal{T}^{\prime}$ is an $r-1\times r-1$ matrix. We use this
decomposition property multiple times throughout our work.

##  Appendix C The case of charge conjugation

Here we expand on the case of charge conjugation $C$, which is the only 0-form
symmetry defect with vanishing torsion. Since the 4d defect theory (3.48) or
(4.1) with $\mathcal{T}=0$ is a non-invertible TQFT, the twisted sector of the
charge-conjugation defect does not host a well defined MTC of line operators
[50]. Consider the case of two defects $V[\mathcal{T}_{1}]$ and
$V[\mathcal{T}_{2}]$ whose fusion is $C$. If $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$ are such that $\mathcal{T}_{21}=0$ (they fuse onto $C$),
then

$\mathcal{T}_{2}=-\frac{\epsilon}{2}\,\mathcal{T}_{1}^{-1}\,\frac{\epsilon}{2}\;.$
(C.1)

Using our formalism, we indeed find that the braiding in
$\mathcal{A}^{N,-\mathcal{T}_{1}}\times_{\mathcal{B}}\mathcal{A}^{N,-\mathcal{T}_{2}}$
is degenerate, because the braiding matrix $\mathcal{K}_{21}$ in (4.33) has
$\det\mathcal{K}_{21}=0$.484848One uses that if
$K=\big{(}\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\big{)}$ and $A$ is
invertible, then $\det(K)=\det(A)\,\det(D-CA^{-1}B)$. Another simple way to
see this is to perform the redefinition
$n_{1}\to\mathcal{T}_{1}^{-1}\frac{\epsilon}{2}n_{1}$, then the matrix
becomes: $\mathcal{K}_{2,1}=\begin{pmatrix}\mathcal{T}_{2}&\mathcal{T}_{2}\\\
\mathcal{T}_{2}&\mathcal{T}_{2}\end{pmatrix}$, which indeed has half rank.

##### Bulk fusion.

Lines in the kernel of $K$ couple to $\mathcal{B}$, but have vanishing spin
and do not braid with anything else and thus do not form a well defined MTC.
They are the naive restriction of the lines $\Psi$ of the $C$-twisted sector
when we decouple the lines charged under $\Phi$.

To understand the fusion we must also take into account nonlocal lines. The
$D[\mathcal{T}_{1}]\times D[\mathcal{T}_{2}]$ system is described by a
braiding matrix:

$K^{-1}=\frac{1}{N}\begin{pmatrix}0&\mathbbm{1}&0&0\\\
\mathbbm{1}&\mathcal{T}_{1}&0&\frac{\epsilon}{2}\\\ 0&0&0&\mathbbm{1}\\\
0&-\frac{\epsilon}{2}&\mathbbm{1}&\mathcal{T}_{2}\end{pmatrix}\,,$ (C.2)

with a basis made up of
$\left(\Gamma_{1},\Psi_{1},\Gamma_{2},\Psi_{2}\right)$. Thus we label a line
by its charges $\left(n_{1},m_{1},n_{2},m_{2}\right)$ under the above
generators. The vectors $(\mathcal{T}_{1}m_{1},-m_{1},0,0)$ and
$(0,0,\mathcal{T}_{2}m_{2},-m_{2})$ are charged only under $\mathcal{B}$
transformations, while the vectors $(n_{1},0,0,0)$ and $(0,0,n_{2},0)$ are
charged under $\Phi_{1}$ and $\Phi_{2}$ respectively. In the variables $\Phi$,
$\Phi_{1}$ the lines charged only under $\Phi$ are $(n_{1},0,n_{1},0)$.

We want to decompose this system. First, lines of the form:

$\widetilde{L}=\begin{pmatrix}\mathcal{T}_{1}n\\\ -n\\\ -\mathcal{T}_{2}n\\\
n\end{pmatrix}\,,$ (C.3)

are neutral w.r.t. all gauge transformations of the 4d bulk and form an
$\mathcal{A}^{N,-(\mathcal{T}_{1}+\mathcal{T}_{2})}$ decoupled theory. Lines
which do not braid with them must satisfy the condition

$n_{1}-n_{2}+\frac{\epsilon}{2}(m_{1}+m_{2})=0.$ (C.4)

We choose the basis:

$L_{n_{1}}=\begin{pmatrix}n_{1}\\\ 0\\\ n_{1}\\\ 0\end{pmatrix}\,,\ \
L_{n_{2}}=\begin{pmatrix}\frac{\epsilon}{2}n_{2}\\\ n_{2}\\\
\frac{\epsilon}{2}n_{2}\\\ -n_{2}\end{pmatrix}\,,\ \
L_{n_{3}}=\begin{pmatrix}\frac{\epsilon}{2}(\Gamma-1)n_{3}\\\ \Gamma n_{3}\\\
\frac{\epsilon}{2}\Gamma n_{3}\\\ (1-\Gamma)n_{3}\end{pmatrix}$ (C.5)

with
$\Gamma=\left(\mathcal{T}_{1}+\mathcal{T}_{2}\right)^{-1}(\mathcal{T}_{2}+\frac{\epsilon}{2})$.
Notice that the relevant definitions can be read off from our Lagrangian
computations in Section 4.1. The line $L_{1}$ is charged only under $\Phi$,
$L_{n_{2}}$ under $\Phi$ and $\Phi_{1}$ while $L_{n_{3}}$ only under
$\mathcal{B}$ and $\Phi$. Between these lines, $L_{n_{2}}$ have nontrivial
spin :

$\theta_{n_{2}}=\exp\left(\frac{\pi
i}{N}n_{2}^{\mathsf{T}}(\mathcal{T}_{1}+\mathcal{T}_{2})n_{2}\right)\,,$ (C.6)

and does not braid with both $L_{1}$ and $L_{3}$. Therefore it forms a
decoupled $\mathcal{A}^{N,\mathcal{T}_{1}+\mathcal{T}_{2}}(\widetilde{\Phi})$
factor where $\widetilde{\Phi}=(T_{1}+T_{2})(\Phi_{1}+\Gamma\Phi)$. Since
$\theta_{n_{1}}=1$, $\theta_{n_{3}}=\exp\left(\frac{\pi
i}{N}n_{3}^{\mathsf{T}}\mathcal{T}_{2,1}n_{3}\right)$,
$B_{n_{1}n_{3}}=\exp\left(\frac{2\pi i}{N}n_{1}^{\mathsf{T}}n_{3}\right)$ they
form a $(\mathbb{Z}_{N\times N})_{N\mathcal{T}_{2,1}}(\mathcal{B},\Phi)$
theory, which is the twisted sector for $V[\mathcal{T}_{2,1}]$:

$D[\mathcal{T}_{1}]\times
D[\mathcal{T}_{2}]=\left[\mathcal{A}^{N,\mathcal{T}_{1}+\mathcal{T}_{2}}(\widetilde{\Phi})\times\mathcal{A}^{N,-\mathcal{T}_{1}-\mathcal{T}_{2}}\right]D[\mathcal{T}_{21}]\,.$
(C.7)

Notice that:

$\mathcal{A}^{N,\mathcal{T}_{1}+\mathcal{T}_{2}}\times\mathcal{A}^{N,-\mathcal{T}_{1}-\mathcal{T}_{2}}=(\mathbb{Z}_{N}\times\mathbb{Z}_{N})_{N(\mathcal{T}_{1}+\mathcal{T}_{2})}$
(C.8)

and also that the decoupled coefficient
$\mathcal{A}^{N,-\mathcal{T}_{1}-\mathcal{T}_{2}}$ is the same decoupled TQFT
as in the normal bulk fusion, which is the correct leftover coefficient once
$\Phi_{1}$ is integrated out. This formula generalizes smoothly to the case of
charge conjugation for which $\mathcal{T}_{2,1}=0$.

##### Boundary fusion

Now we can understand fusions on a gapped boundary $\mathcal{L}$. All gapped
boundaries are $C$-invariant, thus the twisted sector will host a genuine GW
operator in the gauged theory, plus a condensate coming from the fusion.

First we must discuss what happens to the full defect $D[\mathcal{T}]$ when it
approaches the gapped boundary. In the 4d zero form symmetry defect
$V[\mathcal{T}]$ we have a coupling $\mathcal{B}^{\mathsf{T}}\;\Phi$. As we
move to the boundary this becomes
$\mathcal{B}^{\mathsf{T}}\;\Phi\rightarrow\tilde{b}_{\perp}\;l_{\perp}^{\mathsf{T}}\Phi$.
We expand

$\displaystyle\Phi=l\phi_{l}+\frac{u\phi_{\perp}}{l_{\perp}^{\mathsf{T}}u}\;\Rightarrow\;l_{\perp}^{\mathsf{T}}\Phi=\Phi_{\perp}\,,$
(C.9)
$\displaystyle\mathcal{B}=l_{\perp}\tilde{b}_{\perp}+\frac{u_{\perp}b_{l}}{u_{\perp}^{\mathsf{T}}l}\;\Rightarrow\;l^{\mathsf{T}}\mathcal{B}=b_{l}\,,$

where $u$ is the generator of $\mathcal{S}$ defined in Section 3 and
$u_{\perp}=\langle\,,u\rangle$ is such that
$l_{\perp}^{\mathsf{T}}u=l^{\mathsf{T}}u_{\perp}\not=0$ 494949When
$\mathcal{T}\not=0$ we can choose $u=\mathcal{T}^{-1}l_{\perp}$ and we find
$b_{\perp}\equiv
l_{\perp}^{\mathsf{T}}\mathcal{T}^{-1}\mathcal{B}=t_{\perp}\tilde{b}_{\perp}$..
Labelling a line $L_{n,m}$ in the twisted sector by its charges $(n,\,m)$
under $\Gamma$ and $\Psi$ we find that the lines

$\begin{pmatrix}u_{\perp}\\\ 0\end{pmatrix}\,,\begin{pmatrix}-\mathcal{T}l\\\
l\end{pmatrix}\,$ (C.10)

are charged only under $\phi_{l}$ and $b_{l}$ respectively. They form a DW
theory with braiding matrix

$K^{-1}=\begin{pmatrix}0&l^{\mathsf{T}}u_{\perp}\\\
l^{\mathsf{T}}u_{\perp}&-t_{l}\end{pmatrix}\,.$ (C.11)

After a rescaling of the electric generator this becomes a
$(\mathbb{Z}_{N})_{Nt_{l}}(\phi_{l},b_{l})$ DW theory. Remaining lines need to
have trivial braiding with these generators. They have a basis given by

$\begin{pmatrix}0\\\ u\end{pmatrix}\,,\ \ \begin{pmatrix}l_{\perp}\\\
0\end{pmatrix}\,,$ (C.12)

with braiding matrix

$\tilde{K}^{-1}=\begin{pmatrix}0&l_{\perp}^{\mathsf{T}}u\\\
l_{\perp}^{\mathsf{T}}u&u^{\mathsf{T}}\mathcal{T}u\end{pmatrix}\,.$ (C.13)

To get a more familiar result notice that $u=\mathcal{T}^{-1}l_{\perp}$ is a
good choice for $u$ as long as the boundary is not invariant. In these
variables the lines

$\begin{pmatrix}l_{\perp}\\\ -\mathcal{T}^{-1}l_{\perp}\end{pmatrix}\,,$
(C.14)

have spin $\exp\left(-\frac{\pi i}{N}t_{\perp}\right)$ and couple only to
$b_{\perp}$ with unit charge. They thus correctly reproduce the sub-theory
$\mathcal{A}^{N,-t_{\perp}}(b_{\perp})$. We conclude that this procedure is
consistent with the one used in section 4.3 where the field $\Phi$ was
integrated out. However this procedure is more general and in particular it
can be extended to the case of charge conjugation. We have shown that in
general

$D[\mathcal{T}]=(\mathbb{Z}_{N})_{-Nt_{l}}(b_{l},\phi_{l})\times(\mathbb{Z}_{N})_{Nu^{\mathsf{T}}\mathcal{T}u}(\tilde{b}_{\perp},\phi_{\perp})\,.$
(C.15)

We want now to discuss the fusions of two twist defects on the gapped
boundary. A simple way to derive the boundary fusion is to start from the
formula (C.7), impose boundary conditions which set the decoupled DW theories
to one on the boundary (which is a consistent boundary condition) and divide

$(\mathbb{Z}_{N}\times\mathbb{Z}_{N})_{0}(\mathcal{B},\Phi)=(\mathbb{Z}_{N})_{0}(\phi_{l},b_{l})\times(\mathbb{Z}_{N})_{0}(\tilde{b}_{\perp},\phi_{\perp})\,.$
(C.16)

The first term is generated by lines:

$\begin{pmatrix}u_{\perp}\\\ 0\end{pmatrix}\,,\ \ \begin{pmatrix}0\\\
l\end{pmatrix}\,,$ (C.17)

While the second by lines:

$\begin{pmatrix}0\\\ u\end{pmatrix}\,,\ \ \begin{pmatrix}l_{\perp}\\\
0\end{pmatrix}\,.$ (C.18)

Notice that the braiding are non-degenerate owning to
$l_{\perp}^{\mathsf{T}}u=l^{\mathsf{T}}u_{\perp}\neq 0$. The first term is
also a decoupled DW theory which can be set to one on the boundary, while the
second term is a condensate for the $\mathbb{Z}_{N}$ surviving there. One
would then conclude

$D_{\mathcal{L}}[\mathcal{T}_{1}]\times
D_{\mathcal{L}}[\mathcal{T}_{2}]=(\mathbb{Z}_{N})_{0}(\tilde{b}_{\perp},\phi_{\perp})\,D_{\mathcal{L}}[0]\,,$
(C.19)

for a trivial $D_{\mathcal{L}}[0]$.

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