Symmetry TFTs for  Non-Invertible Defects

Justin Kaidi$^{1,2,3}$, Kantaro Ohmori$^4$, and Yunqin Zheng$^{3,5}$

$^1$   Department of Physics,
University of Washington, Seattle, WA, 98195, USA

$^2$   Simons Center for Geometry and Physics,
Stony Brook University, Stony Brook, NY 11794-3636, USA

$^3$   Kavli Institute for the Physics and Mathematics of the Universe,

University of Tokyo,  Kashiwa, Chiba 277-8583, Japan

$^4$   Department of Physics, The University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan

$^5$   Institute for Solid State Physics,

University of Tokyo,  Kashiwa, Chiba 277-8581, Japan

Given any symmetry acting on a $d$-dimensional quantum field theory, there is an associated $(d+1)$-dimensional topological field theory known as the Symmetry TFT (SymTFT). The SymTFT is useful for decoupling the universal quantities of quantum field theories, such as their generalized global symmetries and 't Hooft anomalies, from their dynamics. In this work, we explore the SymTFT for theories with Kramers-Wannier-like duality symmetry in both $(1+1)$d and $(3+1)$d quantum field theories.
After constructing the SymTFT, we use it to reproduce the non-invertible fusion rules of duality defects, and along the way we generalize the concept of duality defects to higher duality defects.  We also apply the SymTFT to the problem of distinguishing intrinsically versus non-intrinsically non-invertible duality defects in $(1+1)$d.

§ INTRODUCTION AND SUMMARY

Kramers-Wannier duality, originally identified in the $(1+1)$d Ising model, is the simplest example of a so-called “non-invertible symmetry." Non-invertible symmetries have been the subject of an extensive literature in $(1+1)$d (see e.g.  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]), but have only recently been generalized to spacetime dimensions greater than two  [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].
The constructions of non-invertible symmetries in higher dimensions that have appeared in the literature so far involve the following techniques,

* Gauging a discrete symmetry in a theory with particular `t Hooft anomaly [20];

* Gauging a non-anomalous symmetry in half of the spacetime [21, 23];

* Gauging a non-normal finite subgroup of the global symmetry [27, 47, 25, 31, 25];

* Gauging a higher form symmetry along a higher co-dimension submanifold [26];

* Gauging a diagonal symmetry between the quantum field theory and a lower dimensional topological theory on a defect [37].

Applications of non-invertible symmetries have ranged from constraints on the IR phases of supersymmetric and non-supersymmetric gauge theories [23, 21, 35] to the generation of new strongly-coupled theories via twisted compactification [28], and also include the obtaining of selection rules in real-world models such as QED and QCD [29, 30].

Despite their newfound utility, sometimes statements made using non-invertible symmetries in a theory $\cX$ can be recast as statements involving only invertible symmetries in a theory $\phi(\cX)$, where $\phi$ is some appropriate topological manipulation. The set of topological manipulations $\phi$ includes gauging of finite (non-anomalous) symmetries, as well as stacking with invertible phases. Non-invertible symmetries which cannot be recast as (potentially anomalous) invertible symmetries upon appropriate application of $\phi$ were dubbed “intrinsically" non-invertible in [28], whereas  non-invertible symmetries which can be recast as invertible symmetries were dubbed “non-intrinsically" non-invertible. The upshot is that the non-invertibility of a given symmetry, and indeed the full fusion (higher-)category, is not an invariant under topological manipulations.

With this in mind, it is interesting to search for an object which is invariant under topological operations $\phi$. Indeed, such an object is known to exist: given a $d$-dimensional theory with symmetry captured by a fusion $(d-1)$-category $\cC$, we may define a $(d+1)$-dimensional topological field theory SymTFT$(\cC)$ with the property that SymTFT$(\cC)=$SymTFT$(\phi(\cC))$ for any topological manipulation $\phi$. The topological theory SymTFT$(\cC)$ is known as the symmetry topological field theory [48, 49, 50, 51, 52, 53, 54]  (see also related developments in condensed matter physics [55, 56, 57, 58, 59, 60]), and will be the main focus of this paper.

§.§ Symmetry TFT and boundary conditions

The SymTFT of a theory $\cX$ in $d$-dimensions with global symmetry $\CC$ is a topological theory SymTFT($\cC)$ in $(d+1)$-dimensions which, when compactified on an interval with appropriate boundary conditions, gives back the theory $\cX$ of interest. One of the many nice properties of the SymTFT is that it decouples the dynamics of $\cX$ from the symmetries. This makes the action of the various topological manipulations $\phi$ more transparent: indeed, by choosing appropriate topological boundary conditions for the SymTFT before compactification to $d$-dimensions, we can obtain not only the original $\cX$, but any theory of the form $\phi(\CX)$.

[line width=2pt, top color=blue!30, bottom color=blue!5]
(0,0) to [out=90, in=-90]  (0,3)
to [out=0,in=180] (6,3)
to [out = -90, in =90] (6,0)
to [out=180, in =0]  (0,0);

[very thick] (-7,0) – (-7,3);
[below] at (-7,0) $Z_{\CX}[A]$;

[thick, snake it, <->] (-1.7,1.5) – (-5, 1.5);

[thick] (0,0) – (0,3);
[thick] (6,0) – (6,3);
at (3,1.5) SymTFT($\cC$);
[below] at (0,0) $\langle D(A)|$;
[below] at (6,0) $|\cX\rangle $;

[line width=2pt, top color=blue!30, bottom color=blue!5]
(0,0) to [out=90, in=-90]  (0,3)
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[very thick] (-7,0) – (-7,3);
[below] at (-7,0) $Z_{\CX/G}[A]$;

[thick, snake it, <->] (-1.7,1.5) – (-5, 1.5);

[thick] (0,0) – (0,3);
[thick] (6,0) – (6,3);
at (3,1.5) SymTFT($\cC$);
[below] at (0,0) $\langle N(A)|$;
[below] at (6,0) $|\cX\rangle $;

Schematic picture of the SymTFT. By imposing  Dirichlet (resp. Neumann) boundary conditions on the left and the non-topological boundary condition (<ref>) on the right, we may compactify to obtain $\cX$ (resp. $\cX/G$). In general, there exist other topological boundary conditions as well.

The basic idea is illustrated in Figure <ref>. The $(d+1)$-dimensional SymTFT is placed on an interval, where the right boundary is endowed with non-topological “enriched Neumann" boundary conditions capturing the dynamics of $\CX$, whereas the left boundary condition is topological. Both boundaries can be labeled by appropriate elements of the state space of SymTFT$(\cC)$.
The dynamical boundary of SymTFT$(\cC)$ is taken to be

| ⟩= ∑_a Z_[a] | a⟩ .
When $\CC$ is a finite group $G$, the $a$ above represents the collective set of flat connections of $G$, and $Z_{\cX}[a]$ denotes the partition function of $\cX$ coupled to gauge fields $a$ (on some fiducial $d$-manifold). When $\CC$ is a fusion (higher-)category rather than a group, the label $a$ is an appropriate collective label for the topological defects in the theory.

The topological boundary of SymTFT$(\cC)$ can take a number of forms, with common options being Dirichlet or Neumann boundary conditions. Dirichlet boundary conditions fix the fields $a$ to certain values $A$, whereas Neumann boundary conditions allow $a$ to fluctuate freely. In the state notation, these can be written as

Dirichlet:     |D(A)⟩= ∑_a δ(a-A) |a⟩ ,

Neumann:     |N(A)⟩= ∑_a exp(-i ∫a ∪A) |a⟩ .

The normalization factor in $\int a\cup A$ depends on the symmetry and is suppressed  here.  We will restore it in the main text.

Because the SymTFT is topological, the length of the interval is unimportant and can be shrank to zero size, upon which one obtains a $d$-dimensional theory whose partition function can be computed by taking the inner product between the states on the left and right boundaries. In the case of Dirichlet boundary conditions, one obtains
⟨D(A)| ⟩= ∑_a, a' Z_(a)  δ(a'-A) ⟨a' | a ⟩= ∑_a, a' Z_(a)  δ(a'-A) δ(a-a') = Z_(A)
and hence one reproduces the original $d$-dimensional theory $\cX$, coupled to background fields $A$.
On the other hand, by putting Neumann boundary conditions on the right, we obtain

\begin{equation}
\langle N(A)| \cX \rangle =\sum_{a, a'} Z_{\CX}(a)\,\exp\left(i \int a' \cup A\right) \langle a' | a \rangle = \sum_{a} Z_{\CX}(a) \,\exp\left(i \int a \cup A\right)  = Z_{\cX/G}(A)~.
\end{equation}

In other words, in this case we obtain a $d$-dimensional theory $\phi(\cX)$, where $\phi$ in this case represents gauging the discrete symmetry $G$ (which is indeed a topological operation).  Choosing mixed Dirichlet-Neumann boundary conditions will allow us to obtain $\cX$ with a subgroup of $G$ gauged.

More generally, one can dress either boundary condition in (<ref>) by a $d$-cocycle $\nu_d$.  The corresponding theories obtained by shrinking the slab are $\cX$ stacked with a counterterm, and $\cX/G$ with the gauging done with discrete torsion associated with the $d$-cocycle,

\begin{equation}
\begin{split}
\ket{D(A)_{\nu_d}}= \sum_{a} \delta(a-A)\exp\left(-i \int \nu_d(a)\right)\ket{a}  &\longleftrightarrow  \langle D(A)_{\nu_d}| \cX \rangle= Z_{\cX}(A) \exp\left(i \int \nu_d(A)\right)\\
\ket{N(A)_{\nu_d}}= \sum_{a} \exp\left(-i \int a\cup A -i \int \nu_d(a)\right)\ket{a}  &\longleftrightarrow  \langle N(A)_{\nu_d}| \cX \rangle= Z_{(\cX\times \nu_d)/G}(A) \\
\end{split}
\end{equation}

One therefore expects that any two theories $\CX$ and $\CX'$ that are related by a topological manipulation $\phi$, i.e. $\CX'=\phi(\CX)$, can be obtained by the slab construction with the same SymTFT but with different topological boundary conditions. In particular, this implies that the SymTFT in the bulk is  invariant under topological manipulations.

§.§ Invertible symmetry and Dijkgraaf-Witten theories

The SymTFT takes a particularly simple form when the symmetry in question is group-like. To see this, begin by considering a theory $\cX$ with background gauge fields $A$ and anomaly $\alpha$. The anomaly inflow paradigm [61], in modern language, says that such a theory may be realized on the boundary of an appropriate invertible phase Inv$^\alpha(A)$; see Figure <ref>(a).

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[below] at (1.5, -1) (a);

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[below] at (10, -1) (b);

at (1.5,1.5) Inv$^\alpha(A)$;
[below] at (3,0) $Z^\alpha_{\CX}[A]$;
at (9+3,1.8) $\mathrm{DW}(\alpha) =$;
[below] at (9+3,1.5) $ \sum_{a,b}  e^{i \int b \cup\delta a}\mathrm{Inv}^\alpha(a) $;
[below] at (7+3,0) $\langle D(A)| $;
[below] at (11+3,0) $\ket{\cX}$;
at (12.5-4,1.5) Inv$^\alpha(A)$;

A theory with anomaly $\alpha$ can be realized on the boundary of an invertible phase Inv$^\alpha(A)$. This is possible if and only if the Symmetry TFT is a (generalized) DW theory.

Given Figure <ref>(a), one can promote the background gauge field $A$ to a dynamical gauge field $a$ both on the right boundary and in a slab in the bulk. The bulk theory within this slab is then a nontrivial TQFT, namely a
(generalized) Dijkgraaf-Witten (DW)   theory[The qualifier “generalized" here refers to the fact that the relevant invertible phases are not restricted to group cohomology elements.]

DW(α) = ∑_a,b  exp(i ∫b ∪δa) Inv^α(a) ,
where both $a$ and $b$ are invariant $G$-valued cochains, and as commented above we suppress the normalization in the BF coupling for now.
On the right of the slab, one further imposes Dirichlet boundary conditions to pin the dynamical field $a$ to the background $A$.
The theory $\cX$ is recovered by taking the thin slab limit where the non-topological boundary condition $\ket{\cX}$ and the Dirichlet boundary condition $\ket{D(A)}$ collide.
We conclude that the SymTFT for an invertible symmetry with an anomaly is given by a (generalized) DW theory (<ref>), i.e. a discrete gauge theory, potentially with a non-trivial twist.

§.§ Symmetry TFT for duality defects

Having understood the SymTFT for theories with only group-like (i.e. invertible) symmetries, we may next proceed to consider theories with non-invertible symmetries. In $(1+1)$d, the simplest categories with non-invertible symmetries are the so-called Tambara-Yamagami categories $TY(G)$ with $G$ an Abelian group. The objects of this category consist of invertible lines $L_g$ for each $g \in G$, together with a single non-invertible line $\cN$ with fusion rules[To fully specify the category $TY(G)$, one must specify more than just the fusion rules; the additional data required is the Frobenius-Schur indicator $\varepsilon\in \ZZ_2$ together with a bicharacter $\chi \in H^2(G,U(1))$. In this paper we will mostly work with the case of trivial $\varepsilon$ and $\chi$.
L_g_1×L_g_2 = L_g_1 ·g_2 ,  L_g ×= ×L_g =  ,  ×= ∑_g ∈G L_g .
A higher-categorical analog of $TY(G)$ fusion rules exists in higher dimensions, as will be discussed for $G=\ZZ_N^{(1)}$ in $(3+1)$d in the main text. The goal of this paper is to construct the SymTFTs for $(1+1)$d theories with $TY(\ZZ_N)$ symmetry, as well as the analogs in $(3+1)$d.

Before beginning this analysis, we should note that the SymTFT is, at least implicitly, already known for any fusion (higher-)category. Indeed, given a theory with fusion (higher-)category $\cC$, the SymTFT is always given by Turaev-Viro theory on $\cC$, or equivalently as Reshetikhin-Turaev theory on the Drinfeld center $\cZ(\cC)$ (subject to some minor caveats[It is known that in $(2+1)$d, given a spherical fusion category $\CC$, one can define the Turaev-Viro theory of $\CC$ as an extended TQFT, which is equivalent to Reshetikhin-Turaev theory on the Drinfeld center $\cZ(\cC)$ [62]. But for general fusion categories in higher dimensions, the authors are unaware of a proof.
]). The latter definition makes it clear that the SymTFT of $\CC$ is the quantum double $\cZ(\cC)$ of $\cC$. This matches with the results obtained above for group-like symmetry $\cC = \mathrm{Vec}^\alpha(G)$ discussed above, since Turaev-Viro theory on $\mathrm{Vec}^\alpha(G)$ is known to be equivalent to DW theory with DW action $\alpha$.

For the case of $\cC = TY(\ZZ_N)$ in $(1+1)$d, various properties of the Drinfeld center $\cZ(TY(\ZZ_N))$ are known in detail in the mathematics literature [63, 64], as well as in the physics literature [65, 66]. As such, the results we present in $(1+1)$d are not new. However, the way in which we obtain these results will be new, and will carry the virtue of being more easily generalizable to the case of higher dimensions.

As will be described in the rest of this work, the SymTFTs for theories with duality symmetries can be obtained by starting with appropriate DW theories and gauging an electro-magnetic (EM) exchange symmetry. As part of our analysis, we will identify an explicit Lagrangian description of the EM-gauged theory, and will use it to obtain data about the spectrum of objects and (higher-)morphisms in the relevant categories $\mathcal{Z}(\cC)$.

§.§ Organization and Conventions

This paper is organized as follows. In the first half of the paper, we focus on the $(2+1)$d SymTFT for $(1+1)$d theories with $TY(\ZZ_N)$ symmetry. To prepare for this, we begin in Section <ref> by reviewing the construction of duality defects in $(1+1)$d via half-space gauging, as well as the derivation of their fusion rules. We pay special attention  to the normalizations appearing in the fusion rules, which we feel have not been carefully treated in the literature yet. After reviewing the fusion rules of $TY(\ZZ_N)$, we then construct the SymTFT in two steps. We begin in Section <ref> by constructing the SymTFT for a theory with non-anomalous $\ZZ_N^{(0)}$ symmetry. As discussed above, for theories with only group-like symmetries the SymTFT should be a DW theory, and in the current case we expect it to be $\ZZ_N$ gauge theory with trivial DW action. The spectrum of topological operators of this $\ZZ_N$ gauge theory, including the so-called “twist defects," are studied in detail. Interestingly, the twist defects admit an interpretation as higher duality interfaces, generalizing the constructions of non-invertible defects/interfaces listed in the beginning of this introduction.   To obtain the SymTFT for $TY(\ZZ_N)$, we next gauge the $\ZZ_2^{\mathrm{EM}}$ symmetry of the bulk $\ZZ_N$ gauge theory. This is done in Section <ref> in two different ways. In Section <ref>, the gauging is done at the level of the category by tracing the behavior of all of the simple objects of $\ZZ_N$ gauge theory under gauging of EM and carefully adding in the twisted sector objects. In Section <ref>, the gauging is done at the level of the Lagrangian by promoting cocycles to twisted cocycles.

The second half of the paper gives the analogous construction of the $(4+1)$d SymTFT for $(3+1)$d theories with duality defects. As before, we begin in Section <ref> by reviewing the construction of duality defects in $(3+1)$d via half-space gauging, as well as the derivation of their fusion rules. Once again, we pay special attention to the normalization appearing in these fusion rules. Having done so, we then construct the SymTFT for the duality defects in two steps. First in Section <ref> we construct the SymTFT for $\ZZ_N^{(1)}$ one-form symmetry in $(3+1)$d, which is again simply a $\ZZ_N^{(1)}$ gauge theory, and analyze its spectrum of topological operators as well as their fusions. Then in Section <ref> we gauge the $\ZZ_4^{\mathrm{EM}}$ EM duality of the $\ZZ_N^{(1)}$ gauge theory. This gauging is again done in two ways: in Section <ref> it is done by tracing the behavior of the various objects and morphisms of the ungauged theory under gauging, while in Section <ref> it is done by writing down an explicit Lagrangian in terms of twisted cocycles.

Finally, we close in Section <ref> by describing one application of the SymTFT, namely to the question of intrinsic vs non-intrinsic non-invertibility discussed in the beginning of this introduction. For example, we will see that when $N$ is not a perfect square, all duality defects in bosonic $(1+1)$d theories are intrinsically non-invertible.

Various computations needed in the main text are relegated to appendices. In Appendix <ref> we derive the commutation relations between $k$-dimensional operators in $(2k+1)$d $\Z_N^{(k-1)}$ gauge theory, which are crucial for understanding the fusion rules of objects in the SymTFT. In Appendix <ref>, we explain how to measure the EM-charge of junctions in $\ZZ_N$ gauge theory, which is crucial for obtaining the fusion rules of the EM-gauged theory. In Appendix <ref> we provide details for the proof of gauge invariance of the $(2+1)$d Lagrangian describing the EM-gauged theory, which is written in terms of twisted cocycles. In Appendix <ref>, we review results from the math literature regarding the modular $S$ and $T$ matrices for the Drinfeld center of $TY(\ZZ_N)$. These results provide a useful check of the results obtained in the first half of this paper. In Appendix <ref> we give details on the form of the EM duality defects in the $(4+1)$d SymTFT before gauging, which depend crucially on whether $N$ is odd, $N=2$, or $N$ is even with $N>2$. Finally, in Appendix <ref> we provide a physical argument for the assignment of quantum defect lines after gauging $\Z_4^{\text{EM}}$ to operators whose constituents are invariant under the $\Z_2^{\text{EM}}$ normal subgroup of $\Z_4^{\text{EM}}$.

We close this introduction by listing some conventions that will be used throughout the paper:

* 1-form gauge fields are denoted as $a$ or $A$ (potentially with a hat or subscript) depending on whether they are dynamical or background fields. Likewise 2-form gauge fields are denoted by $b$ or $B$ depending on whether they are dynamical or background.

* Cup products are mostly suppressed, e.g. $aA := a\cup A$. We will explicitly write down the cup product only when it is necessary to distinguish it from the twisted cup product.

* We use $X_{d}$ to denote the $d$-dimensional spacetime, and $M_n$ to denote worldvolumes of $n$-dimensional topological defects/interfaces in spacetime.

* We denote the global fusion by $\times$ and the global direct sum as $+$. On the other hand, we denote the local fusion by $\otimes$ and the local direct sum as $\oplus$.  We will also use the notation $\mathcal{O}_c \subset\mathcal{O}_a\otimes \mathcal{O}_b$ to indicate that there exists a junction between incoming topological defects $\mathcal{O}_a$ and $\mathcal{O}_b$ and an outgoing topological defect $\mathcal{O}_c$.

§ DUALITY INTERFACES IN $(1+1)$D

Before analyzing the SymTFT for theories with duality defects, we begin with a detailed discussion of the duality defects themselves, together with their fusion rules. Special attention will be paid to normalizations, which are surprisingly subtle. We begin in $(1+1)d$ where things are somewhat simpler.

§.§ Duality interfaces from half-space gauging

Consider a non-spin QFT $\CX$ in $(1+1)$d with an anomaly free $\Z_N^{(0)}$ zero-form  global symmetry, defined on a closed two-dimensional spacetime $X_2$.  We denote the $\Z_N^{(0)}$  background gauge field as $A$, and the partition function as $Z_{\CX}[X_2,A]$. Gauging $\Z_N^{(0)}$ gives a new theory $\CX/\Z_N$,

\begin{eqnarray}\label{eq:2dgauging}
Z_{\CX/\Z_N}[X_2,A] = \frac{1}{|H^0(X_2, \Z_N)|} \sum_{a\in H^1(X_2, \Z_N)} Z_{\CX}[X_2,a]\, e^{ \frac{2\pi i}{N} \int_{X_2} a A}~,
\end{eqnarray}

where now $A\in H^1(X_2, \widehat \Z_N)$ is the background field of the quantum symmetry $\widehat\Z_N^{(0)}$ after gauging. The defect that generates this quantum symmetry is  the Wilson line of $a$,

\begin{eqnarray}
\eta(\gamma) = \exp\left(\frac{2\pi i}{N}\oint_{\gamma} a\right)~.
\end{eqnarray}

Let us comment on the normalization factor in (<ref>), which is introduced to subtract out gauge redundancies. It is straightforward to check that gauging $\Z^{(0)}_N$ twice maps the theory $\CX$ back to itself, up to an additional
Euler counterterm $\chi[X_2,\Z_N]^{-1}$, with

\begin{eqnarray}\label{eq:EulerCC2d}
\chi[X_2,\Z_N]=\frac{|H^0(X_2,\Z_N)||H^2(X_2,\Z_N)|}{|H^1(X_2,\Z_N)|}~.
\end{eqnarray}

The normalization of the partition function may be modified by multiplying by an arbitrary power of the Euler counterterm $\chi[X_2,\Z_N]^{\kappa}$. The case of $\kappa=\frac{1}{2}$ is of particular interest, since in this case the normalization in (<ref>) becomes $1/\sqrt{|H^1(X_2,\Z_N)|}$, and then gauging twice maps $\CX$ back to itself exactly, without any counterterm.[In obtaining this result, one uses the fact that $|H^n(X_2,\Z_N)|= |H^{2-n}(X_2,\Z_N)|$ for closed $X_2$. For the case of open manifolds $X_2^{\geq 0}$ with boundary $M_1|_0$, to be discussed below, absolute cohomology should be switched to relative cohomology, i.e. $|H^n(X_2^{\geq 0},\Z_N)|= |H^{2-n}(X_2^{\geq 0}, M_1|_{0}, \Z_N)|$.] We however will not include this factor, and will instead adopt the normalization in (<ref>).

[top color=gray!40, bottom color=gray!10] (0.03,-1)
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to[bend right] (-1,-2.5);

[red, thick] (-3.2,-1) arc(-90:90:0.5cm and 1cm);
[red, dotted, thick] (-3.2,-1) arc(270:90:0.5cm and 1cm);
[below] at (-3.2,-1) $M_1|_0$;

[below] at (-12.5,4) $X_2^{<0}$;
[below] at (6.5,4) $X_2^{\geq 0}$;

Decomposition of $X_2$ along a neck. The interface is located at $x=0$.

Instead of gauging $\Z_N^{(0)}$ over the entire $X_2$, one can gauge in half of the spacetime with Dirichlet boundary conditions. This defines a topological duality interface $\CN$ between $\CX$ and $\CX/\Z_N$ [21, 20].[The duality interface defined here by half-gauging should not be confused with the topological interface (also termed “duality wall") between two dual theories (e.g. mirror symmetry pairs) discussed in e.g. [67, 68, 69]. ] Let us decompose the spacetime manifold $X_2$ into left and right parts,

\begin{eqnarray}
X_2= X_2^{< 0} \cup X_2^{\geq 0}~,
\end{eqnarray}

where $\partial X_{2}^{\geq 0}= M_1$ is the interface. Locally around the interface, the topology of the manifold is $M_1\times \mathbb{R}$, and we can use a local coordinate $x$ to parameterize $\mathbb{R}$. The interface is located at $x=0$, as shown in Figure <ref>. The theory $\CX$ lives on $X_2^{<0}$, while the theory $\CX/\Z_N$ lives on $X_2^{\geq 0}$.

This is also shown in Figure <ref>, where the theory on the right side $X_2^{\geq 0}$ is defined to be

\begin{eqnarray}\label{eq:halfgauging}
Z_{\CX/\Z_N}[X_2^{\geq 0}, A]=\frac{1}{|H^0(X_2^{\geq 0},M_1|_0, \Z_N)|} \sum_{a\in H^1(X_2^{\geq 0}, M_1|_0, \Z_N)} Z_{\CX}[X_2^{\geq 0}, a]\, e^{ \frac{2\pi i}{N} \int_{X_2^{\geq 0} }a A }~.
\end{eqnarray}

The Dirichlet boundary condition implies that the dynamical gauge field $a$ is an element in relative cohomology $H^1(X_2^{\geq 0}, M_1|_0, \Z_N)$. Here, we use $M_1|_0$ to emphasize that $M_1$ is located at $x=0$.

[top color=red!20, bottom color=red!5]  (6,0)
to [out=0, in=180] (12,0)
to [out=90, in=270] (12,3)
to [out=180,in=0] (6,3)
to [out=270,in=90] (6,0);

[thick, ->] (0,0) – (12,0);
[thick, dashed] (6,3) – (6,-0.5);
[above] at (6,3) $\CN$;
[] at (3,2) $Z_{\CX}[X_2^{<0},A]$;
[] at (9,1.5)
$Z_{\CX/\Z_N}[X_2^{\geq 0}, A]$

[below] at (6,-0.5) $x=0$;
[right] at (12,0) $x$;

The duality defect from gauging $\Z_N$ over half of the spacetime $X_2^{\geq 0}$ with Dirichlet boundary conditions.

The orientation reversal of the duality defect $\overline{\CN}$ is defined by exchanging the theories on the two sides of Figure <ref>. Concretely, $\CX/\Z_N$ lives on the left and $\CX$ lives on the right. By redefining the theory $\widetilde{\CX}= \CX/\Z_N$, the defect  $\overline{\CN}$ can be rewritten as having $\widetilde{\CX}$ on the left and $\CX= \chi \cdot \widetilde{\CX}/\Z_N$ on the right.  In other words,

\begin{eqnarray}\label{eq:Ninvertible}
\overline{\CN} = \chi [X_2^{\geq 0}, \Z_N] \cdot \CN~.
\end{eqnarray}

Hence the duality interfaces in $(1+1)$d are orientation-reversal invariant, up to an Euler counterterm. This is also a consequence of the fact that gauging twice maps the theory to itself (up to a counterterm).

In the special case that the theory $\CX$ is self-dual under gauging, i.e. $\CX=\CX/\Z_N$, the duality interface discussed above which connects two different theories becomes a duality defect within a single theory. In this section we will not assume the self-duality condition; it will be the main focus of Section <ref>.

§.§ Fusion rules of duality interfaces

We now proceed to a discussion of the fusion rules of the duality interface $\CN$ defined in Section <ref>. In particular, we will find that $\CN$ is non-invertible. We warn the reader that the following analysis is somewhat technical, and those interested only in the final answer may skip to (<ref>).

We first discuss the fusion rule $\eta \times \CN$. Since $a$ has Dirichlet boundary conditions on $M_1$, $a|_{M_1|_0}=0$, the $\Z^{(0)}_N$ symmetry defect $\eta$ is trivial on $M_1|_0$. This justifies the fusion rule

\begin{eqnarray}\label{2deq:etaNN}
\eta \times \CN =\CN~.
\end{eqnarray}

It is more interesting to study the fusion rule between two duality interfaces $\CN\times \CN$. Let us place the two duality defects at $x=0$ and $x= \epsilon$, and let $\epsilon \to 0^+$.   The two duality defects divide the spacetime $X_2$ into three regions, $X_2=X_2^{<0} \cup X_2^{[0,\epsilon)} \cup X_2^{\geq \epsilon}$. Since $\epsilon$ is small, we may always take $X_2^{[0,\epsilon)}= M_1\times I_{[0,\epsilon)}$. The theories living on the three regions are as shown in Figure <ref>. Instead of defining $Z_{\CX/\Z_N}[X_2^{[0,\epsilon)}, A]$  and  $Z_{\CX/\Z_N/\Z_N}[X_2^{\geq \epsilon}, A]$  separately and discussing how to glue them together along $M_1|_{\epsilon}$, it suffices to discuss the theory on $X_2^{\geq 0}$ all together. The theory living on $X_2^{\geq 0}$ is given by

\begin{equation}\label{eq:NN}
{1 \over |H^0(X_2^{\geq 0}, M_1|_0, \Z_N)||H^0(X_2^{\geq \eps}, M_1|_\eps,\Z_N)|} \sum_{\substack{a \in H^1(X_2^{\geq 0}, M_1|_0, \Z_N) \\ \widetilde a \in H^1(X_2^{\geq \eps}, M_1|_\eps, \Z_N)}} Z_{\CX}[X_2^{\geq 0}, a]\, e^{{2 \pi i \over N}\int_{X_2^{[0,\eps)}} a A + {2 \pi i \over N}\int_{X_2^{\geq \eps}} (a-A)\widetilde a}
\end{equation}

and we will now evaluate (<ref>) in detail.

[top color=red!20, bottom color=red!5]  (4,0)
to [out=0, in=180] (8,0)
to [out=90, in=270] (8,3)
to [out=180,in=0] (4,3)
to [out=270,in=90] (4,0);

[top color=red!40, bottom color=red!10]  (8,0)
to [out=0, in=180] (12,0)
to [out=90, in=270] (12,3)
to [out=180,in=0] (8,3)
to [out=270,in=90] (8,0);

[thick, ->] (0,0) – (12,0);
[thick, dashed] (4,3) – (4,-0.5);
[thick, dashed] (8,3) – (8,-0.5);
[above] at (4,3) $\CN$;
[above] at (8,3) $\CN$;
[] at (2,2) $Z_{\CX}[X_2^{<0},A]$;
[] at (6,2)
$Z_{\CX/\Z_N}[X_2^{[0,\epsilon)}, A]$
[] at (10,2)
$Z_{\CX/\Z_N/\Z_N}[X_2^{\geq \epsilon}, A]$
[below] at (4,-0.5) $x=0$;
[below] at (8,-0.5) $x=\epsilon$;
[right] at (12,0) $x$;

Fusion of two duality interfaces. The partition function on $X^{\geq 0}$ is given by (<ref>).

Since $Z_{\CX}$ in the summand does not depend on $\widetilde{a}$, we would first like to sum over $\widetilde{a}$, which intuitively should enforce $a=A$ on $X_{2}^{\geq \eps}$. To perform this summation rigorously, one must convert the sum over cohomologies into a sum over cochains (for reasons that will be explained below). Note that $H^1= Z^1/B^1$, $|B^1|=|C^0|/|Z^0|$, and $|B^0|=1$;[We remind the reader that $C^n$ is the set of $n$-cochains, $Z^n\subset C^n$ is the set of $n$-cocycles, and $B^n\subset Z^n$ is the set of exact $n$-cocycles.] these equations hold for both absolute and relative cohomologies. The sum $\frac{1}{|H^0|}\sum_{a\in H^1}$ can thus be rewritten as $\frac{1}{|H^0||B^0|} \sum_{a\in Z^1}= \frac{1}{|C^0|} \sum_{a\in Z^1}$. Equation (<ref>) then becomes

\begin{equation}\label{eq:NN2d1}
{1 \over |C^0(X_2^{\geq 0}, M_1|_0, \Z_N)||C^0(X_2^{\geq \eps}, M_1|_\eps,\Z_N)|} \sum_{\substack{a \in Z^1(M_2^{\geq 0}, M_1|_0, \Z_N) \\ \widetilde a \in Z^1(X_2^{\geq \eps}, M_1|_\eps, \Z_N)}} Z_{\CX}[X_2^{\geq 0}, a]\, e^{{2 \pi i \over N}\int_{X_2^{[0,\eps)}} a A + {2 \pi i \over N}\int_{X_2^{\geq \eps}} (a-A)\widetilde a}~.
\end{equation}

The cocycle condition in the sum can further be relaxed by introducing additional dynamical scalars $\phi\in C^0(X_2^{\geq 0}, \Z_N)$ and $\widetilde{\phi}\in C^0(X_2^{\geq \eps}, \Z_N)$ with BF couplings acting as Lagrange multiplier fields. The relative condition can likewise be eliminated by including the couplings $\phi a$ and $\widetilde{\phi} \widetilde{a}$ on the subloci $M_1|_0$ and $M_1|_\eps$. Then all the variables in the sum are cochains,

\begin{equation}
\begin{split}
&{1 \over |C^0(X_2^{\geq 0}, M_1|_0, \Z_N)||C^0(X_2^{\geq \eps}, M_1|_\eps, \Z_N)|}{1 \over |C^0(X_2^{\geq 0}, \Z_N)| |C^0(X_2^{\geq \eps}, \Z_N)|}\times\\
&\sum_{\substack{a \in C^1(X_2^{\geq 0}, \Z_N), \widetilde a \in C^1(X_2^{\geq \eps}, \Z_N) \\ \phi \in C^0(X_2^{\geq 0}, \Z_N), \widetilde \phi \in C^0(X_2^{\geq \eps}, \Z_N) }} Z_{\CX}[X_2^{\geq 0},a] e^{{2 \pi i \over N}\int_{X_2^{[0,\eps)}} a A + {2 \pi i \over N}\int_{X_2^{\geq \eps}} (a-A)\widetilde a} e^{{2\pi i \over N}\int_{X_2^{>0}}\phi \delta a - {2\pi i \over N}\int_{M_1|_0} \phi a}e^{{2\pi i \over N}\int_{X_2^{>\eps}}\widetilde\phi \delta\widetilde a - {2\pi i \over N}\int_{M_1|_\eps} \widetilde\phi \widetilde a}
\end{split}
\end{equation}

Indeed, summing over $\phi$ in the bulk $X_2^{\geq 0}$ enforces $a$ to be a cocycle due to the coupling $\int_{X_2^{> 0}}\phi \delta a$, and summing over $\phi$ on the boundary $X_1|_0$ enforces $a=0$, which makes the cocycle relative to $M_1|_0$. The same comments apply to $\widetilde{a}$.

We are now ready to perform the sum over $\widetilde{a}$. By rewriting the final factor of the summand as
e^2πi N∫_X_2^>ϕδa - 2πi N∫_M_1|_ ϕa = e^- 2 πi N ∫_X_2^≥ δϕa ,
we see that summing over $\widetilde{a}$ produces a factor of $|C^1(X_2^{\geq \eps}, \Z_N)|$ and enforces that $a - A - \delta \widetilde \phi = 0$ on $X_2^{\geq \eps}$,
1 |C^0(X_2^≥0, M_1|_0, _N)||C^0(X_2^≥, M_1|_, _N)||C^1(X_2^≥, _N)| |C^0(X_2^≥0, _N)| |C^0(X_2^≥, _N)|×

∑_a ∈C^1(X_2^≥0, _N)
ϕ∈C^0(X_2^≥0, _N),    ϕ∈C^0(X_2^≥, _N)  Z_[X_2^≥0, a] e^2 πi N∫_X_2^[0,) a Ae^2πi N∫_X_2^>0ϕδa - 2πi N∫_M_1|_0 ϕa δ(a-A-δϕ)|_X_2^≥ .
Next we integrate out $\phi$, which produces a factor of $| C^0(X_2^{\geq 0},\Z_N)|$ and enforces $a \in Z^1(X_2^{\geq 0}, M_1|_0, \ZZ_N)$,

\begin{equation}
\begin{split}
&{1 \over |C^0(X_2^{\geq 0}, M_1|_0,\Z_N)||C^0(X_2^{\geq \eps}, M_1|_\eps,\Z_N)|}{|C^1(X_2^{\geq \eps},\Z_N)| \over |C^0(X_2^{\geq \eps},\Z_N)|} \\&\hspace{3cm}\sum_{\substack{a \in Z^1(X_2^{\geq 0}, M_1|_0,\Z_N)\\ \widetilde \phi \in C^0(X_2^{\geq \eps},\Z_N) }} Z_{\CX}[X_2^{\geq 0}, a] e^{{2 \pi i \over N}\int_{X_2^{[0,\eps)}} a A} \delta(a-A-\delta\widetilde \phi)|_{X_2^{\geq \eps}}~.
\end{split}
\end{equation}

The summand is now manifestly independent of $\widetilde \phi$, so we may set $\widetilde \phi$ to zero in the delta function and replace the sum with a factor of $|C^0(X_2^{\geq \eps},\Z_N)|$.  The remaining delta function then fixes $a = A$ in $X_2^{\geq \eps}$, which makes $a$ an element of $a \in Z^1(X_2^{[0,\eps]}, M_1|_0 \cup M_1|_\eps, \Z_N)$,

|C^1(X_2^≥,_N)| |C^0(X_2^≥0, M_1|_0,_N)||C^0(X_2^≥, M_1|_,_N)| ∑_a ∈Z^1(X_2^[0,], M_1|_0 ∪M_1|_,_N)  Z_[X_2^≥0, a+A|_M_2^≥] e^2 πi N∫_X_2^[0,] a A .
where $A|_{M_2^{\eps}}$ is equal to $A$ if we're on $M_2^{\geq \eps}$, and vanishes elsewhere. It is useful reorganize the normalization by factorizing out the Euler counter term,

\begin{eqnarray}\label{eq:NN2d3}
\begin{split}
{|C^0(X_2^{\geq \eps},\Z_N)| |C^2(X_2^{\geq \eps},\Z_N)| \over |C^0(X_2^{\geq 0}, M_1|_0,\Z_N)||C^0(X_2^{\geq \eps}, M_1|_\eps,\Z_N)|} \cdot {|C^1(X_2^{\geq \eps},\Z_N)| \over |C^0(X_2^{\geq \eps},\Z_N)| |C^2(X_2^{\geq \eps},\Z_N)|}~.
\end{split}
\end{eqnarray}

The second factor above is the inverse of the Euler counterterm $\chi[M_2^{\geq \eps}, \Z_N]^{-1}$ on $X_2^{\geq \eps}$, which can be seen by direct evaluation

\begin{eqnarray}
\chi[X_2^{\geq \eps}, \Z_N]= {|H^0(X_2^{\geq \eps},\Z_N)| |H^2(X_2^{\geq \eps},\Z_N)| \over |H^1(X_2^{\geq \eps},\Z_N)|} = {|C^0(X_2^{\geq \eps},\Z_N)| |Z^2(X_2^{\geq \eps},\Z_N)| \over |C^1(X_2^{\geq \eps},\Z_N)|}
\end{eqnarray}

and by further noting that $Z^2(X_2^{\geq \eps},\Z_N)= C^2(X_2^{\geq \eps},\Z_N)$ since all top forms are closed. Moreover, the first factor in (<ref>) can be simplified by using $|C^2(X_2^{\geq \eps},\Z_N)|= |C^0(X_2^{\geq \eps}, M_1|_\eps,\Z_N)|$ since the elements in $C^0(X_2^{\geq \eps}, M_1|_\eps,\Z_N)$ and the elements in $C^2(X_2^{\geq \eps},\Z_N)$ are Fourier partners. We finally use the fact that

\begin{eqnarray}\label{eq:NN2d4}
|C^n( X_2^{\geq 0}, M_1|_0, \Z_N)| = |C^n( X_2^{[0,\epsilon]}, M_1|_0 \cup  M_1|_\epsilon, \Z_N)| |C^n( X_2^{\geq \epsilon}, \Z_N)| ~,
\end{eqnarray}

which is simply a decomposition of cochains on $X_2^{\geq 0}$ into the sum of cochains on $X_2^{[0,\eps]}$ with fixed boundary condition at $M_1|_{\eps}$ and cochains on $X_2^{\geq \eps}$ with free boundary conditions. Note crucially that such decomposition can not be achieved at the level of cocycles or cohomologies, because the flatness condition is violated at $M_1|_{\eps}$. The ability to use this decomposition is the main reason for our reformulation of the sum over cohomologies in (<ref>) as a sum over cochains.

Using (<ref>), we may now simplify the first term in (<ref>) to $1/|C^0( X_2^{[0,\epsilon]}, M_1|_0 \cup  M_1|_\epsilon, \Z_N)|$. Substituting the simplified normalization in (<ref>) into (<ref>), we get

\begin{equation}
\label{eq:N218}
\chi[X_2^{\geq \eps}, \Z_N]^{-1} {1 \over |C^0( X_2^{[0,\epsilon]}, M_1|_0 \cup  M_1|_\epsilon, \Z_N)|} \sum_{\substack{a \in Z^1(X_2^{[0,\eps]}, M_1|_0 \cup M_1|_\eps,\Z_N) }} Z_{\CX}[X_2^{\geq 0}, a+A|_{X_2^{\geq \eps}}] e^{{2 \pi i \over N}\int_{X_2^{[0,\eps]}} a A}~.
\end{equation}

This formula admits a simple physical interpretation. First, in the region $M_2^{\geq \eps}$ we have two overlapping gaugings, which “annihilate" to produce a factor of $\chi[M_2^{\geq \eps}, \ZZ_N]^{-1}$. This is a consequence of the fact, mentioned before, that gauging twice  takes the theory $\CX$ back to itself up to an Euler counterterm.[As mentioned around (<ref>), this piece could be removed by modifying the definition of gauging by an appropriate Euler counterterm.] As for the rest of (<ref>), we notice that after the two gaugings “annihlate" to give the Euler counterterm, we are left with a gauging in the strip $X_2^{[0,\eps]}$, with Dirichlet boundary conditions on both boundaries $M_1|_{0} \cup M_1|_{\eps}$. This gives the remaining portions of the formula.

By taking the limit $\eps\to 0$, it now follows that the $\CN\times \CN$ fusion rule is

\begin{equation}\label{eq:NNfusionfinal}
\begin{split}
\CN\times \CN &= \chi[X_2^{\geq \eps}, \Z_N]^{-1} \frac{1}{|C^0( X_2^{[0,\epsilon]}, M_1|_0 \cup  M_1|_\epsilon, \Z_N)|}  \sum_{\substack{a \in Z^1(X_2^{[0,\eps]}, M_1|_0 \cup M_1|_\eps,\Z_N) }}  \eta(\mathrm{LD}(a))\\
&= \chi[X_2^{\geq \eps}, \Z_N]^{-1} \frac{1}{|H^0( X_2^{[0,\epsilon]}, M_1|_0 \cup  M_1|_\epsilon, \Z_N)|}  \sum_{\substack{a \in H^1(X_2^{[0,\eps]}, M_1|_0 \cup M_1|_\eps,\Z_N) }}  \eta(\mathrm{LD}(a))\\
&= \chi[X_2^{\geq \eps}, \Z_N]^{-1}  \sum_{\substack{\gamma \in H_1(M_1,\Z_N) }}  \eta(\gamma)
\end{split}
\end{equation}

where LD stands for the Lefschetz dual, and we have used $|H^0( X_2^{[0,\epsilon]}, M_1|_0 \cup  M_1|_\epsilon, \Z_N)|= 1$. Using (<ref>), we likewise find

\begin{eqnarray}
\CN\times \overline{\CN} = \sum_{\substack{\gamma \in H_1(M_1,\Z_N) }}  \eta(\gamma)~.
\end{eqnarray}

This fusion rule more straightforwardly corresponds to gauging in a slab, and indeed this “gauging in a slab" is precisely how the fusion rule of duality defects was originally derived in [21, 20]. The method used here was more roundabout, but gives important insight into the correct way to keep track of normalization factors.

In summary, the fusion rules for duality interfaces in $(1+1)$d are as follows,

×= ∑_γ∈H_1(M_1, _N) η(γ) ,η×=,

= χ[X_2^≥0,_N]   ,  η^N=1  .
These are, not coincidentally, the fusion rules of the $\Z_N$ Tambara-Yamagami fusion categories.

§ $(2+1)$D SYMMETRY TFT FOR $\Z_N^{(0)}$ SYMMETRY

In Section <ref>, we defined the duality interface $\CN$ and studied its fusion rules. In this section, we study the properties of the duality interface from the SymTFT point of view. In particular, we find that the duality interface in a $(1+1)$d QFT can be realized as a twist defect in a $(2+1)$d $\Z_N$ TQFT, from which the fusion rules (<ref>) can be reproduced. We also study the F-symbols of the duality interfaces using the SymTFT. Throughout this section, we will only assume that the $(1+1)$d QFT has a $\Z_N^{(0)}$ zero-form global symmetry; in particular, we do not assume invariance under gauging $\Z_N^{(0)}$.

§.§ $\Z_N$ gauge theory as the Symmetry TFT

[very thick, blue] (-7,0) – (-7,1.5);
[very thick, blue] (-7,1.5) – (-7,3);
[below] at (-7,0) $Z_{\CX}[A]$;
[above] at (-7,3) $x=0$;

[thick, snake it, <->] (-1.7,1.5) – (-5, 1.5);
[above] at (-3.3,1.6) Expand / Shrink;

[line width=2pt, top color=blue!30, bottom color=blue!5]
(0,0) to [out=90, in=-90]  (0,3)
to [out=0,in=180] (6,3)
to [out = -90, in =90] (6,0)
to [out=180, in =0]  (0,0);

[thick] (0,0) – (0,3);
[thick] (6,0) – (6,3);
[] at (3,1.5) $\frac{2\pi}{N} \widehat{a} \delta a$;
[below] at (0,0) $\langle D(A)| $;
[below] at (6,0) $|\cX\rangle $;
[above] at (0,3) $x=0$;
[above] at (6,3) $x=\varepsilon$;

A $(1+1)$d QFT $\cX$ with $\Z_N$ symmetry can be expanded into a $(2+1)$d slab. The bulk is the $(2+1)$d $\Z_N$ SymTFT, the right boundary encodes the dynamical information of the $(1+1)$d QFT, and the left boundary is a topological Dirichlet boundary condition for the bulk field $a$.

Suppose $\CX$ is a $(1+1)$d QFT with an anomaly-free $\Z_N^{(0)}$ global symmetry, whose partition function is $Z_{\CX}[X_2,A]$. As discussed in the introduction, we can expand the theory into a $(2+1)$d slab, as shown in Figure <ref>. The SymTFT living in the bulk of the slab is a $(2+1)$d $\Z_N$ gauge theory, whose action is

\begin{eqnarray}\label{eq:3dBFtheory}
S= \frac{2\pi}{N}\int_{X_3} \widehat{a} \delta a~.
\end{eqnarray}

Both $a$ and $\widehat{a}$ are dynamical $\Z_N^{(0)}$-valued 1-cochains. We will take the bulk to be a product $X_3= X_2\times I_{[0,\varepsilon]}$, and will use the coordinate $x$ to parameterize the interval $I_{[0,\varepsilon]}$. The two boundaries are $X_2|_{\varepsilon}$ and $X_2|_0$.

The boundary conditions on the left ($x=0$) and right ($x=\varepsilon$) boundaries of the slab are specified by appropriate boundary states. On the right boundary, the state is

\begin{eqnarray}
\ket{\CX} = \sum_{a\in H^1(X_2|_{\varepsilon}, \Z_N)} Z_{\CX}[X_2|_{\varepsilon}, a] \ket{a}~.
\end{eqnarray}

It corresponds to a non-topological enriched Neumann boundary condition that encodes all of the dynamical information of the $(1+1)$d QFT $\CX$.
On the left boundary, the state is

\begin{eqnarray}
\bra{D(A)} = \sum_{a\in H^1(X_2|_0, \Z_N)} \bra{a} \delta(a-A) ~,
\end{eqnarray}

which corresponds to a topological Dirichlet boundary condition.
Note that the background field dependence only enters in the left, topological boundary.
By shrinking the slab, i.e. taking $\varepsilon\to 0$, the partition function of the $(1+1)$d theory is reproduced by the inner product of the two boundary states,

\begin{eqnarray}
\braket{D(A)| \CX} = \sum_{a,a'\in H^1(X_2|_0, \Z_N)} Z_{\CX}[X_{2}|_{0}, a] \delta(a'-A)  \braket{a'|a}= Z_{\CX}[X_2|_0, A] ~,
\end{eqnarray}

where we have used $\braket{a'|a}=\delta(a'-a)$.

Gauging the $\Z_N^{(0)}$ symmetry of $\CX$ in $(1+1)$d corresponds to changing the topological boundary condition on the left from Dirichlet to Neumann. To see this, we define the Neumann boundary condition as

\begin{eqnarray}
\bra{N(A)} = \frac{1}{|H^0(X_2|_0, \Z_N)|}\sum_{a\in H^1(X_2|_0, \Z_N)} \bra{a} e^{\frac{2\pi i}{N}\int_{X_2|_0} aA }
\end{eqnarray}

and check explicitly that

\begin{equation}
\begin{split}
\braket{N(A)|\CX} =\frac{1}{|H^0(X_2|_0, \Z_N)|} \sum_{a,a'\in H^1(X_2|_0, \Z_N)}  Z_{\CX}[X_2|_0, a] e^{\frac{2\pi i}{N}\int_{X_2|_0} a'A } \braket{a'|a} = Z_{\CX/\Z_N}[X_2, A]~.
\end{split}
\end{equation}

More generally, when the $\Z_N^{(0)}$ symmetry has a nontrivial 't Hooft anomaly, the SymTFT will be a $\Z_N$ gauge theory with a non-trivial twist, i.e. a Dijkgraaf Witten theory. In this case, one cannot gauge $\Z_N^{(0)}$ in $(1+1)$d because of the anomaly. Correspondingly, there is no Neumann topological boundary condition.

§.§ Lines and surfaces in $\Z_N$ gauge theory

We now consider the line and surface operators in $\Z_N$ gauge theory, focusing for the moment on the ones without topological boundaries. The operators with boundaries will be discussed in subsequent subsections.

§.§.§ Line operators

The $\Z_N$ gauge theory (<ref>) has $N^2$ genuine topological line operators,

L_(e,m)(γ) =  exp(2πi/N ∮_γe a) exp( 2πi/N ∮_γm a) ,  (e, m) ∈_N ×_N ,
where $L_{(1,0)}$ and $L_{(0,1)}$ together generate a $\Z_N^{(1)}\times \Z_N^{(1)}$ 1-form symmetry. They satisfy the following fusion and braiding rules:

* Fusion rule:  The lines $L_{(e,m)}(\gamma)$ are invertible, and have straightforward fusion rules

L_(e,m)(γ) ×L_(e',m')(γ) = L_(e+e',m+m')(γ) .

* Commutation relation:
As derived in Appendix <ref>, the correlation function between $L_{(e,m)}$ on a closed loop $\gamma$ and $L_{(e',m')}$ on another closed loop $\gamma'$ is

\begin{eqnarray}\label{eq:linking}
\langle L_{(e,m)}(\gamma) L_{(e',m')}(\gamma')\dots\rangle = \exp\left( -\frac{2\pi i}{N}(em'+ me')\link(\gamma,\gamma')\right)\braket{\dots}~,
\end{eqnarray}

where $\link(\gamma,\gamma')$ is the Hopf linking number between $\gamma$ and $\gamma'$. The phase on the right-hand side gives the braiding phase between the two anyons whose worldlines are $L_{(e,m)}$ and $L_{(e',m')}$. Here, $\gamma$ and $\gamma'$ do not intersect each other and we also assume that the operators represented by “$\dots$" in (<ref>) do not link with $\gamma$ and $\gamma'$. Equation
(<ref>) can be equivalently rewritten as an equal time commutation relation  by pushing $\gamma$ and $\gamma'$ to a 2d plane [70],

L_(e,m)(γ) L_(e',m')(γ') = exp(-2 πi N (e m' +  m e') ⟨γ, γ'⟩) L_(e',m')(γ')  L_(e,m)(γ)  ,
where  $\langle \gamma, \gamma'\rangle$ represents the intersection number between $\gamma$ and $\gamma'$. Equation (<ref>) makes sense even when $\gamma$ and $\gamma'$ are line intervals.  Note that the order of lines in the correlation function matters: the operators are “time-ordered." The correlation function $L_{(e,m)}(\gamma) L_{(e',m')}(\gamma')$ on the left hand side of (<ref>)  means that the line segment $\gamma'$ is at time $t$, while the line segment $\gamma$ is at time $t+\epsilon$ with $\epsilon$  an infinitesimal  positive real number. Hence $L_{(e,m)}$ crosses $L_{(e',m')}$ from above. Likewise, the correlation function $L_{(e',m')}(\gamma')  L_{(e,m)}(\gamma) $ on the right-hand side of (<ref>)  means that the line  $L_{(e,m)}$ crosses $L_{(e',m')}$ from below. See Figure  <ref> for a pictorial explanation.

[dashed] (0,0) circle (1);
[thick,blue, ->] (-0.7,-0.7)  to (0.7,0.7);
[thick,dgreen, <-] (-0.7,0.7)  to (-0.1,0.1);
[thick,dgreen] (0.1,-0.1)  to (0.7,-0.7);
[below] at (-1,-0.9) $L_{(e,m)}(\gamma)$;
[below] at (1,-0.9) $L_{(e',m')}(\gamma')$;

[dashed] (0,0) circle (1);
[thick,rotate=45,blue, <-] (0,0.3) [partial ellipse=-80:260:0.4cm and 0.15cm];

[thick,dgreen,<-] (-0.7,0.7)  to (-0.4,0.4);
[thick,dgreen] (-0.2,0.2)  to (0.7,-0.7);
[thick,blue ] (-0.7,-0.7)  to (-0.1,-0.1);
[thick,blue, ->] (0.1,0.1)  to (0.7,0.7);
[below] at (-1,-0.9) $L_{(e,m)}(\gamma)$;
[below] at (1,-0.9) $L_{(e',m')}(\gamma')$;

$=$ $e^{-2\pi i (em'+me') /N}$
[dashed] (0,0) circle (1);
[thick,blue ] (-0.7,-0.7)  to (-0.1,-0.1);
[thick,blue,->] (0.1,0.1)  to (0.7,0.7);
[thick,dgreen,<-] (-0.7,0.7)  to (0.7,-0.7);
[below] at (-1,-0.9) $L_{(e,m)}(\gamma)$;
[below] at (1,-0.9) $L_{(e',m')}(\gamma')$;

Braiding between lines in the 3d bulk gives crossing between lines on a 2d plane.

* Quantum torus algebra: The correlation function (<ref>)  implies a quantum torus algebra on the plane. To see this, consider two lines of the same type but supported on two seperate segments, $L_{(e,m)}(\gamma)$ and $L_{(e,m)}(\gamma')$. From the definition in (<ref>), it is obvious that $L_{(1,0)}(\gamma+\gamma')= L_{(1,0)}(\gamma)L_{(1,0)}(\gamma')$ and similarly for $L_{(0,1)}$. However, this is not true for general $L_{(e,m)}$ due to the non-commutativity (<ref>).
By using the definition (<ref>) to expand $L_{(e,m)}= \left(L_{(1,0)}\right)^e \left(L_{(0,1)}\right)^m$, and applying the commutation relations mentioned above,  the quantum torus algebra is

\begin{eqnarray}\label{eq:quanttoral}
L_{(e,m)}(\gamma) L_{(e,m)}(\gamma') = \exp\left( -\frac{2\pi i}{N}  em \braket{\gamma,\gamma'}\right)  L_{(e,m)}(\gamma+\gamma')~.
\end{eqnarray}

This will be useful below when discussing the symmetry defect of the $\Z_2^{\text{EM}}$ electro-magnetic exchange symmetry.

§.§.§ $\Z_2^{\text{EM}}$ symmetry and condensation defects

The $\Z_N$ gauge theory (<ref>) has an electromagnetic exchange symmetry $\Z_2^{\text{EM}}$ which interchanges the two gauge fields

a →a  ,  a →a .
Indeed, this manifestly leaves the action (<ref>) (on a closed manifold)  invariant. This symmetry is a zero-form symmetry, and there is a corresponding codimension-one surface defect $D_{\mathrm{EM}}$ in the bulk implementing the symmetry transformation. It was proven in [71] and rediscovered in [26] that the symmetry defect of any zero-form symmetry in $(2+1)$d TQFT with a single vacuum can be realized as a condensation defect, including the defect generating $\Z_2^{\text{EM}}$. In particular, it was pointed out in [26] that the defect for $\Z_2^{\text{EM}}$ in a $\Z_N$ gauge theory can be realized by condensing $L_{(1,N-1)}$ on a surface. Below, we provide further motivation for why $D_{\mathrm{EM}}$ should be a condensation defect, and then provide a rigorous definition. We then perform some consistency checks.

[top color=red!40, bottom color=red!10,rotate=90]  (0,-0.7) – (2,-0.7) – (2.6,-0.2) – (0.6,-0.2)– (0,-0.7);
[thick,rotate=90] (0,-0.7) – (2,-0.7);
[thick,rotate=90] (0,-0.7) – (0.6,-0.2);
[thick,rotate=90]  (0.6,-0.2)–(2.6,-0.2);
[thick,rotate=90]  (2.6,-0.2)– (2,-0.7);
[thick,blue,postaction=on each segment=mid arrow=blue] (-0.9,1.4)–++(1.3,0) coordinate (blue_r);
[thick,dgreen,dotted] (blue_r)–++(0.28,0) coordinate(green_dot);
[thick,dgreen,postaction=on each segment=mid arrow=dgreen] (green_dot) ++ (0.04,0)–++(1.3,0) coordinate(green_r);
[right] at (-2.1,1.4) $L_{(e,m)}$;
[right] at (2,1.4) $L_{(m,e)}$;
[below] at (0.4, -0) $D_{\text{EM}}$;

The piercing action of the $\Z_2^{\text{EM}}$ symmetry defect $D_{\text{EM}}$ on the line $L_{(e,m)}$.

[top color=red!40, bottom color=red!10,draw,thick]  (0,0) – (2,0) – (2,-2) – (0,-2)–cycle;
[anchor=north] at (1,-2) $D_{\text{EM}}$;

=   [baseline=-30]
[top color=red!40, bottom color=red!10,draw,thick]  (0,0) – (2,0) – (2,-2) – (0,-2)–cycle;
[thick,red,fill=white] (1,-1) circle(.3);
[anchor=north] at (1,-2) $D_{\text{EM}}$;

One can punch a hole in the $\mathbb{Z}_2^{\text{EM}}$ symmetry defect.

[top color=red!40, bottom color=red!10,rotate=90]  (0,-0.7) – (2,-0.7) – (2.6,-0.2) – (0.6,-0.2)– (0,-0.7);
[thick,rotate=90] (0,-0.7) – (2,-0.7);
[thick,rotate=90] (0,-0.7) – (0.6,-0.2);
[thick,rotate=90]  (0.6,-0.2)–(2.6,-0.2);
[thick,rotate=90]  (2.6,-0.2)– (2,-0.7);
[thick,blue,postaction=on each segment=mid arrow=blue] (-0.9,1.6)–++(1.3,0) coordinate (blue_r);
[thick,dgreen,dotted] (blue_r)–++(0.28,0) coordinate(green_dot);
[thick,dgreen,postaction=on each segment=mid arrow=dgreen] (green_dot) ++ (0.04,0)–++(0.6,0) coordinate(green_r) ..  controls ++(0.3,0) and ++ (0.3,0) .. ++(0,-.8) coordinate(green_d) ;
[thick,red,fill=white,thick] (green_d) ++(-.87,.25) arc(90:450:.1 and .25);
[thick,dgreen,postaction=on each segment=mid arrow=dgreen] (green_d) – ++(-.6,0) coordinate(green_d2);
[dotted,dgreen,thick] (green_d2) ++(-.04,0) – ++(-.14,0) coordinate(green_d3);
[dgreen,thick,postaction=on each segment=mid arrow=dgreen] (green_d3) – ++(-1.44,0);
[right] at (-2,1.4) $L_{(e,m)}$;
[right] at (1.7,1.4) $L_{(m,e)}$;
[below] at (0.4, -0) $D_{\text{EM}}$;

=  [baseline=35]
[top color=red!40, bottom color=red!10,rotate=90]  (0,-0.7) – (2,-0.7) – (2.6,-0.2) – (0.6,-0.2)– (0,-0.7);
[thick,rotate=90] (0,-0.7) – (2,-0.7);
[thick,rotate=90] (0,-0.7) – (0.6,-0.2);
[thick,rotate=90]  (0.6,-0.2)–(2.6,-0.2);
[thick,rotate=90]  (2.6,-0.2)– (2,-0.7);
[thick,blue,postaction=on each segment=mid arrow=blue] (-0.9,1.6)–(0.4,1.4) coordinate(r);
[thick,dgreen,postaction=on each segment=mid arrow=dgreen] (r)–(-0.9,1.2);
[right] at (-2.2,1.7) $L_{(e,m)}$;
[right] at (-2.2,1.2) $L_{(m,e)}$;
[below] at (0.4, -0) $D_{\text{EM}}$;

=  [baseline=35]
[top color=red!40, bottom color=red!10,rotate=90]  (0,-0.7) – (2,-0.7) – (2.6,-0.2) – (0.6,-0.2)– (0,-0.7);
[thick,rotate=90] (0,-0.7) – (2,-0.7);
[thick,rotate=90] (0,-0.7) – (0.6,-0.2);
[thick,rotate=90]  (0.6,-0.2)–(2.6,-0.2);
[thick,rotate=90]  (2.6,-0.2)– (2,-0.7);
[thick,violet,postaction=on each segment=mid arrow=violet] (-0.9,1.4)–(0.4,1.4);
[right] at (-2.2,1.5) $L_{(1,-1)}^{\otimes(e-m)}$;
[below] at (0.4, -0) $D_{\text{EM}}$;
The $\Z_2^{\text{EM}}$ symmetry defect $D_{\text{EM}}$ can absorb the line $L_{(1,-1)}$.

We first motivate the condensation construction of the $\Z_2^{\text{EM}}$ symmetry defect $D_{\text{EM}}$.
Note that in a $(2+1)$d TQFT with a single vacuum, there are no non-trivial local operators. This means that $D_\text{EM}$ acts only on line operators via the piercing action depicted in Figure <ref>, resulting from (<ref>).
Furthermore, it means that $D_\text{EM}$ admits topological boundary conditions that can be used to punch a hole in the defect (see Figure <ref>), since there are no local operators which could detect such a hole in the first place.

When a line $L_{(e,m)}$ enters $D_{\text{EM}}$ perpendicularly from the left, the line $L_{(m,e)}$ should leave $D_{\text{EM}}$ perpendicularly from the right. We may then consider a configuration in which the line $L_{(m,e)}$ is folded back through a hole to the left side of $D_{\text{EM}}$, as shown in Figure <ref>.
By reversing the orientation of the folded line, which flips the sign of both electric and magnetic charges, we arrive at a configuration where a line $L_{(e-m,m-e)}=L_{(1,-1)}^{\otimes(e-m)}$ enters from the left and is absorbed into the surface defect. In other words, $D_{\text{EM}}$ can absorb $L_{(1,-1)}$. This implies that $D_{\text{EM}}$ should be a condensate of $L_{(1,-1)}$.

With the above motivation, we now give a precise definition of the $\Z_2^{\text{EM}}$ condensation defect supported on a surface $M_2$, following [26],

D_EM(M_2) = 1|H^0(M_2, _N)| ∑_γ∈H_1(M_2, _N) L_(1,-1)(γ) .
Note that condensing $L_{(1,-1)}$ on $M_2$ amounts to gauging the $\Z_N^{(1)}$ one-form symmetry only on the surface $M_2$, which amounts to gauging a $\Z_N^{(0)}$ zero-form symmetry from the point of view of the surface. The normalization in (<ref>) comes precisely from the gauge redundancy of gauging the $\Z_N^{(0)}$ zero-form symmetry on $M_2$. However, as noted in Section <ref>, such a gauging is always subjected to an Euler counterterm ambiguity. For example, the convention adopted in [26] is to further multiply (<ref>) by $\chi[M_2, \Z_N]^{1/2}$ such that $D_{\text{EM}}^2=1$. It turns out that the standard convention as in (<ref>) is more convenient when discussing operators with boundaries, so we work with that convention.

With this definition, we may verify that the fusion of $L_{(e,m)}$ with $D_{\mathrm{EM}}$ from the left is equivalent to fusion of $L_{(m,e)}$ with $D_{\mathrm{EM}}$ from the right, as desired. To see this, we consider $L_{(e,m)}(M_1)$ and $D_{\mathrm{EM}}(M_2)$, where $M_1$ is located to the left of $M_2$ and parallel to it.  We begin by noting that

\begin{equation}
\begin{split}
L_{(e,m)}(M_1)\times D_{\mathrm{EM}}(M_2)&= \frac{1}{|H^0(M_2, \ZZ_N)|}\sum_{\gamma \in H_1(M_2, \Z_N)} L_{(e,m)}(M_1) L_{(1,-1)}(\gamma)\\
&=\frac{1}{|H^0(M_2, \ZZ_N)|} \sum_{\gamma \in H_1(M_2, \Z_2)} e^{{2 \pi i \over N}(e-m) \langle M_1, \gamma \rangle} L_{(1,-1)}(\gamma) L_{(e,m)}(M_1) \\
\end{split}
\end{equation}

where we have used (<ref>). We may then use the fusion rules (<ref>) to split $L_{(e,m)}(M_1) = L_{(e-m,m-e)}(M_1) \times L_{(m,e)}(M_1) = L_{(1,N-1)}(M_1)^{ (e-m)} \times L_{(m,e)}(M_1)$.

Using the quantum torus algebra (<ref>), we obtain

\begin{equation}
\begin{split}
L_{(e,m)}(M_1)\times D_{\mathrm{EM}}(M_2)&= \frac{1}{|H^0(M_2, \ZZ_N)|}\sum_{\gamma \in H_1(M_2, \Z_N)} L_{(1,-1)}(\gamma+(e-m)M_1) \times  L_{(m,e)}(M_1)
\\
&=D_{\mathrm{EM}}(M_2) \times L_{(m,e)}(M_1)~,
\end{split}
\end{equation}

giving the desired result.

We may also verify the invertibility of $D_{\text{EM}}$ by computing $D_{\mathrm{EM}}(M_2)\times D_{\mathrm{EM}}(M_2)$,

\begin{eqnarray}
\begin{split}
D_{\text{EM}}(M_2)\times D_{\text{EM}}(M_2)&= \frac{1}{|H^0(M_2, \ZZ_N)|^2}\sum_{\gamma, \gamma'\in H_1(M_2, \Z_N)}L_{(1,N-1)}(\gamma) L_{(1,N-1)}(\gamma')\\
&=\frac{1}{|H^0(M_2, \ZZ_N)|^2}\sum_{\gamma, \gamma'\in H_1(M_2, \Z_N)} e^{\frac{2\pi i}{N}\braket{\gamma,\gamma'}}L_{(1,N-1)}(\gamma+\gamma')\\
&= \frac{|H^1(M_2, \ZZ_N)|}{|H^0(M_2, \ZZ_N)|^2} = \chi[M_2, \Z_N]^{-1}
\end{split}
\end{eqnarray}

where we used the quantum torus algebra (<ref>) in the second equality. The result is an Euler counterterm, which confirms that $D_{\text{EM}}$  is an invertible condensation defect.

As discussed in [26], there are actually more condensation defects in $\Z_N$ gauge theory than just $D_{\mathrm{EM}}$. These are obtained by condensing different line operators with or without discrete torsion. Most of these condensation defects are non-invertible. In the present work, we will only study the particular invertible condensation defect $D_{\text{EM}}$ implementing the $\Z_2^{\text{EM}}$ symmetry.

§.§ Twist defects as higher duality interfaces

So far we have only discussed topological defects without boundaries. We now consider those with boundaries. By gauge invariance, the simple line operator $L_{(e,m)}$ cannot be given a boundary (unless it ends at a non-trivial junction involving other lines, as discussed below). On the other hand, there is a topological boundary condition for the condensation operator $D_{\text{EM}}$, which can be used construct a new topological defect while maintaining gauge invariance. This is known as a twist defect [66, 65]. As we will see below, such twist defects can be interpreted as higher duality interfaces.

§.§.§ Twist defects

A twist defect can be defined by condensing $L_{(1,-1)}$ on a surface $M_2$ with a nontrivial boundary $M_1=\partial M_2$ and imposing Dirichlet boundary conditions on $M_1$.
We denote this “minimal" twist defect by $\Sigma_{(0)}$. Concretely,

\begin{eqnarray}\label{eq:twistdefect}
\Sigma_{(0)}(M_1, M_2)= \frac{1}{{|H^0(M_2, M_1, \Z_N)|}} \sum_{\gamma\in H_1(M_2, \Z_N)} L_{(1,-1)}(\gamma)~,
\end{eqnarray}

where $M_2$ has a boundary. Equation (<ref>) is almost identical to the definition of the condensation defect (<ref>), with the only difference being that absolute cohomology $H^n(M_2,\Z_N)$ is replaced by relative cohomology $H^n(M_2,M_1,\Z_N)$ due to the Dirichlet boundary conditions.[Note that via Lefschetz duality $H^n(M_2,M_1,\Z_N) \simeq H_{2-n}(M_2,\Z_N)$, so the homology in the summand is still absolute. ] Moreover, thanks to the Dirichlet boundary conditions, the twist defect is topological. The twist defect $\Sigma_{(0)}(M_1,M_2)$ can be understood as a non-genuine topological line operator on $M_1$ with a condensation surface on $M_2$ attached it.

Whereas fusing $L_{(1,-1)}$ with $D_{\mathrm{EM}}$ gives a new quantum operator (i.e. a line supported on a surface), because of the Dirichlet boundary conditions  the non-genuine line operator $\Sigma^{(0)}$ can absorb $L_{(1,-1)}$, and more generally any line of the form $L_{(n,-n)}$. On the other hand, it cannot absorb lines $L_{(e,m)}$ for $e + m \neq 0 \,\,\, \mathrm{mod}\,\, N$. We may thus obtain new twist operators by beginning with $\Sigma_{(0)}$ and fusing with such $L_{(e,m)}$. Due to the Dirichlet boundary conditions, the resulting operator does not depend on $e$ and $m$ individually, but only on the combination $e+m\mod N$, and hence we have $N$ distinct twist defects, labeled by $e+m=0,1,...,N-1$, each being writable in $N$ equivalent ways. Picking one representative, we can define

\begin{eqnarray}\label{eq:defoftwdefects}
\Sigma_{(e)}(M_1,M_2):= L_{(n+e,-n)}(M_1)\times \Sigma_{(0)}(M_1,M_2), \hspace{1cm} \forall \,\,n=0, 1, \dots, N-1~.
\end{eqnarray}

It is useful to note that $\Sigma_{(0)}$ is $\Z_2^{\text{EM}}$ invariant. As a consequence, $\Sigma_{(e)}$ is also $\Z_2^{\text{EM}}$ invariant, since

\begin{eqnarray}
\begin{split}
\Z_2^{\text{EM}}:\,\, \Sigma_{(e)}(M_1, M_2) &\to L_{(-n,n+e)}(M_1)\times \Sigma_{(0)}(M_1,M_2)\\
&= L_{(n+e,-n)}(M_1)\times L_{(-2n-e, 2n+e)}(M_1)\times \Sigma_{(0)}(M_1,M_2)\\
&= L_{(n+e,-n)}(M_1)\times \Sigma_{(0)}(M_1,M_2) = \Sigma_{(e)}(M_1, M_2)~.
\end{split}
\end{eqnarray}

§.§.§ Higher duality interfaces

Before discussing the fusion of twist defects, let us compare their construction with that of duality interfaces [21, 20], which was reviewed in Section <ref>. Duality interfaces are constructed by gauging a global symmetry in half of the entire spacetime and imposing  Dirichlet boundary conditions for the gauge fields. Likewise, the twist defect is defined by  gauging a global symmetry (a one-form symmetry in the present context) along half of a codimension-$q$ spacetime submanifold ($q=1$ in the present context), with Dirichlet boundary conditions on the codimension-$(q+1)$ boundary. Hence the twist defects can naturally be interpreted as higher duality interfaces—duality interfaces associated with higher gauging.

In Section <ref>, we will gauge the $\Z_2^{\text{EM}}$ symmetry of the $(2+1)$d gauge theory. After gauging, the condensation defect (<ref>) becomes transparent, and the twist defects become genuine line operators on $M_1$. Accordingly, the higher duality interface becomes a higher duality defect since both sides of $M_1$ in $M_2$ support trivial operators after gauging.

\[{\begin{tikzpicture}[scale=0.8,baseline=-20]
\shade[top color=red!40, bottom color=red!10]  (0,0) -- (0,-3) -- (3,-2) -- (3,1)-- (0,0);
\draw[ultra thick,red] (0,0) -- (0,-3);

\shade[top color=red!70, bottom color=red!10]   (0.5,0.5/3) --(0.5,0.5/3-3) -- (3,-2)-- (3,1)-- (0.5,0.5/3);

\draw[ultra thick,red] (0.5,0.5/3) -- (0.5,0.5/3-3);

\draw[thick] (0,-3) -- (3,-2);
\draw[thick] (0,0) --(3,1);

\node[above] at (0,0) {$0$};

\node[above] at (0.5, 0.5/3) {$\eps$};

\node[left] at (0,-3) {$\Sigma_{(0)}$};
\node[left] at (1.65,-3.2) {$\Sigma_{(0)}$};
\end{tikzpicture}}
\hspace{0.7in}\Longrightarrow\qquad
\draw[ultra thick,red] (0,0) -- (0,-3);
\node[below] at (0,-3) {\,\,$\sum_{\gamma\in H_1(M_1,\Z_N)} L_{(1,-1)}(\gamma)$};
\end{tikzpicture}}
\]

Fusion of two twist defects $\Sigma_{(0)}$.

§.§.§ Fusion rules of the twist defects

We now proceed to study the fusion rules of the twist defects. Interestingly, although the condensation defect $D_{\text{EM}}$ on a closed surface is invertible, the twist defect obeys non-invertible fusion rules.

The fusion between $\Sigma_{(e)}$ and the invertible defects $L_{(e',m')}$ follows directly from the definition (<ref>),

\begin{eqnarray}\label{sigma2fusion2}
\Sigma_{(e)}(M_1,M_2)\times L_{(e',m')}(M_1) = \Sigma_{(e+e'+m')}(M_1, M_2)~.
\end{eqnarray}

More nontrivial are the fusion rules between the twist defects themselves $\Sigma_{(e)}\times \Sigma_{(e')}$. It is simplest to first understand the fusion rule for $\Sigma_{(0)}\times \Sigma_{(0)}$, and then to fuse the outcome with an appropriate invertible line  $L_{(e+e',0)}$.
The fusion rule $\Sigma_{(0)}\times \Sigma_{(0)}$ can be obtained by similar calculations as in Section <ref>. Near the boundary of $M_2$, the manifold is locally $M_1\times \mathbb{R}_{+}$. As before, we parameterize $\mathbb{R}_+$ by $x$ and label the manifolds as $M_2^{\geq 0}$ and $M_1|_0$ in the bulk and on the boundary, respectively. We take the two twist defects to be $\Sigma_{(0)}(M_1|_0, M_2^{\geq 0})$ and $\Sigma_{(0)}(M_1|_\eps, M_2^{\geq \eps})$. They overlap on the same two-dimensional manifold $M_2^{\geq \eps}$. Fusing two operators amounts to taking $\eps\to 0$, as seen in the left panel of Figure <ref>.

To proceed, we directly compute the fusion rule

\begin{eqnarray}\label{eq:TTfusion2d}
\begin{split}
&\Sigma_{(0)}(M_1|_0,M_2^{\geq 0})\times \Sigma_{(0)}(M_1|_{\eps},M_2^{\geq \eps})\\&= \frac{1}{|H^0(M_2^{\geq 0}, M_1|_{0},\Z_N)| |H^0(M_2^{\geq \eps}, M_1|_{\eps},\Z_N)|} \sum_{\substack{\gamma\in H_1(M_2^{\geq 0}, \Z_N)\\
\gamma'\in H_1(M_2^{\geq \eps}, \Z_N)}} L_{(1,-1)}(\gamma) L_{(1,-1)}(\gamma')~.
\end{split}
\end{eqnarray}

Evaluating this sum is analogous to evaluating (<ref>). The idea is to first rewrite the sum over lines $\gamma, \gamma'$ in absolute homology as a sum over the Lefschetz duals $a_{\gamma}, a_{\gamma'}$ in relative cohomology. Then we further convert the sums over relative cohomology into  sums over cochains by introducing additional fields $\phi, \phi'$ and appropriate BF terms. Integrating  out $a_{\gamma'}$ on $M_2^{\geq \eps}$, the right-hand side of (<ref>) becomes

\begin{eqnarray}
\begin{split}
\chi[M_2^{\geq \eps}, \Z_N]^{-1} \frac{1}{|H^0(M_2^{[0,\eps]}, M_1|_0 \cup M_1|_\eps, \Z_N)|} \sum_{a_{\gamma}\in H^1(M_2^{[0,\eps]}, M_1|_0\cup M_1|_{\eps}, \Z_N)} L_{(1,-1)}(\gamma)
\end{split}
\end{eqnarray}

which is of precisely the same form as (<ref>). The first factor is the Euler counterterm and can be removed by appropriate field redefinition. The remaining factors have the physical interpretation of condensing $L_{(1,-1)}$ only on $M_2^{[0,\eps]}$, with Dirichlet boundary conditions on both $M_1|_0\cup M_1|_{\eps}$. Interpreting this in terms of higher gauging, it is the 1-gauging of the one-form symmetry generated by $L_{(1,-1)}$ on the slab $M_2^{[0,\eps]}$.  Taking the limit $\eps\to 0$, we find the fusion rule

\begin{eqnarray}\label{sigma2fusion0}
\Sigma_{(0)}(M_1,M_2)\times \Sigma_{(0)}(M_1,M_2) =\chi[M_2, \Z_N]^{-1} \sum_{\gamma\in H_1(M_1,\Z_N)} L_{(1,-1)}(\gamma)~,
\end{eqnarray}

where we have used $|H^0(M_2^{[0,\eps]}, M_1|_0 \cup M_1|_\eps, \Z_N)|=1$. One can likewise define the orientation reversal of $\Sigma_{(e)}$ by switching the two sides of  $M_1|_0$. As in (<ref>), we have

\begin{eqnarray}
\overline{\Sigma}_{(e)}(M_1, M_2) = \chi[M_2, \Z_N]\cdot  \Sigma_{(-e)}(M_1, M_2)~.
\end{eqnarray}

Thus (<ref>) becomes

\begin{eqnarray}
\Sigma_{(0)}(M_1,M_2)\times \overline{\Sigma}_{(0)}(M_1,M_2) =\sum_{\gamma\in H_1(M_1,\Z_N)} L_{(1,-1)}(\gamma)~.
\end{eqnarray}

The right hand side can be interpreted as 2-gauging of the one-form symmetry generated by $L_{(1,-1)}$ on a codimension two manifold $M_1$. When $M_1= S^1$, the right-hand side is simply the sum over $L_{(e,-e)}$ for all $e\in \Z_N$.

Given (<ref>), we obtain the $\Sigma_{(e)}\times \Sigma_{(e')}$ fusion rule by further fusing with $L_{(e,0)}(M_1)$ and $L_{(e',0)}(M_1)$,

\begin{eqnarray}\label{sigma2fusion1}
\begin{split}
\Sigma_{(e)}(M_1,M_2)\times \Sigma_{(e')}(M_1,M_2) &= \chi[M_2, \Z_N]^{-1} \sum_{\gamma\in H_1(M_1,\Z_N)} L_{(e+e',0)}(M_1) L_{(1,-1)}(\gamma)\\
\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont \footnotesize $M_1\to S^1$ }}}{=}} \hspace{0.8cm} \chi[M_2, \Z_N]^{-1} \sum_{n=0}^{N-1} L_{(n+e+e', -n)}(S^1)~.
\end{split}
\end{eqnarray}

In the special case of $N=2$, these fusion rules were discussed in [66] from a lattice Hamiltonian perspective. It immediately follows from (<ref>) that the twist defect $\Sigma_{(e)}$ is non-invertible, and has quantum dimension $\sqrt{N}$.

§.§ Symmetry defects/interfaces in $(1+1)$d from topological defects in $(2+1)$d

Having discussed the closed and open topological defects in $\Z_N$ gauge theory, we now insert these operators into the $(2+1)$d slab in Figure <ref> and examine their behavior upon shrinking the slab. Schematically, we find the following correspondences

\begin{eqnarray}
\begin{split}
\text{Twist defects } \Sigma_{(e)} &\longleftrightarrow \text{Duality interface } \CN\\
\text{Magnetic line } L_{(0,1)} & \longleftrightarrow \text{$\Z_N^{(0)}$ symmetry defect } \eta\\
\text{Electric line } L_{(1,0)} & \longleftrightarrow \text{$\Z_N^{(0)}$ order parameter } \CO
\end{split}
\end{eqnarray}

These correspondences have also been discussed in e.g.  [50, 53, 60, 54].

§.§.§ $\Z_N$ symmetry defects and order parameters from the bulk line operators

We first insert a line operator into the $(2+1)$d slab. One can either place the line operator parallel to the boundary $X_2|_{0}$, or orthogonal to the boundary. Because of the Dirichlet boundary condition of $a$, i.e. $a|_{X_1|_{0}}=0$, the electric line $L_{(1,0)}$ can either end on the boundary perpendicularly, or be completely absorbed into the boundary (in the sense that it becomes a trivial operator) if it is parallel to it. Thus the only way that it can survive upon shrinking the slab is to place it orthogonal to the boundary, with one end anchored on the Dirichlet boundary on the left, and the other end anchored on the non-topological boundary on the right. As a consequence of ending on the non-topological boundary, one generically obtains a non-topological point-like operator $\CO$ upon shrinking the slab. This point operator will be charged under the $\Z^{(0)}_N$ symmetry of the $(1+1)$d theory, and hence serves as the non-topological $\Z^{(0)}_N$ order parameter.

On the other hand, the magnetic line $L_{(0,1)}$ can neither be absorbed into the Dirichlet boundary nor terminate on it. Hence it survives as a line defect upon shrinking the slab. This is the topological line $\eta$ generating the $\Z_N^{(0)}$ symmetry of the $(1+1)$d QFT $\CX$. Indeed, because $L_{(0,1)}$ and $L_{(1,0)}$ have a nontrivial correlation function when linked, it follows that the correlation function between $\eta$ and $\CO$ is also nontrivial when $\eta$ links  with the $\CO$. This link measures the $\Z^{(0)}_N$ charge of the order parameter $\CO$.

\[{\begin{tikzpicture}[scale=0.8,baseline=-20]
\shade[top color=blue!40, bottom color=blue!10]  (0,0) -- (0,-3) -- (3,-2) -- (3,1)-- (0,0);
\draw[ thick] (0,0) -- (0,-3);
\draw[thick] (0,-3) -- (3,-2);
\draw[thick] (0,0) --(3,1);
\draw[thick] (3,-2) -- (3,1);

\shade[top color=red!40, bottom color=red!10]   (0.5,-0.1) --(0.5,-3.1) -- (3.5,-2.1)-- (3.5,0.9)-- (0.5,-0.1);
\draw[ thick] (0.5,-0.1) -- (0.5,-3.1);
\draw[thick] (0.5,-3.1) -- (3.5,-2.1);
\draw[thick] (0.5,-0.1)-- (3.5,0.9);
\draw[thick] (3.5,-2.1)-- (3.5,0.9);

\node[left] at (0,-3) {$\bra{D(A)}$};
\node[left] at (0.95,-3.6) {$D_{\text{EM}}$};
\end{tikzpicture}}
\hspace{0.7in}\Longrightarrow\qquad
\shade[top color=violet!40, bottom color=violet!10]  (0,0) -- (0,-3) -- (3,-2) -- (3,1)-- (0,0);
\draw[ thick] (0,0) -- (0,-3);
\draw[thick] (0,-3) -- (3,-2);
\draw[thick] (0,0) --(3,1);
\draw[thick] (3,-2) -- (3,1);
\node[left] at (0,-3) {$\bra{N(A)}$};
\end{tikzpicture}}
\]

Colliding $D_{\text{EM}}$ with Dirichlet boundary condition produces Neumann boundary condition.

§.§.§ Duality interfaces from the twist defects

We next insert the condensation defect $D_{\text{EM}}(M_2)$ into the slab. When the condensation defect $D_{\text{EM}}(M_2)$ collides with the Dirichlet boundary condition $\bra{D(A)}$ on $X_2$, it produces the Neumann boundary condition $\bra{N(A)}$; see Figure <ref>.  One can see this explicitly as follows,

\begin{equation}\label{eq:emexchangebc}
\begin{split}
\bra{D(A)}  &= \sum_{a\in H^1(X_2, \Z_N)} \bra{a} \delta(a-A)\,\,\,\,
\xrightarrow{D_{\text{EM}}} \,\,\,\, \sum_{\widehat{a}\in H^1(X_2, \Z_N)} \bra{\widehat{a}} \delta(\widehat{a}-A)\\& = \frac{1}{|H^0(X_2,\Z_N)|} \sum_{a\in H^1(X_2, \Z_N)} \bra{a} e^{ \frac{2\pi i}{N} \int a A}  = \bra{N(A)}~,
\end{split}
\end{equation}

where we have used the fact that the EM dual bases are related by a discrete Fourier transform, i.e.
⟨a| = 1/|H^0(X_2,_N)| ∑_a∈H^1(X_2, _N) ⟨a| e^i 2π/N  ∫ab  .

[line width=2pt, top color=blue!30, bottom color=blue!5]
(0,0) to [out=90, in=-90]  (0,3)
to [out=0,in=180] (6,3)
to [out = -90, in =90] (6,0)
to [out=180, in =0]  (0,0);

[very thick, violet] (-7,0) – (-7,1.5);
[very thick, blue] (-7,1.5) – (-7,3);
[above] at (-7,3) $Z_{\CX}[A]$;
[below] at (-7,0) $Z_{\CX/\ZZ_N}[A]$;
at (-7,1.5) [circle,fill,red, inner sep=1.5pt];
[left,red] at (-7,1.5) $\cN$;

[thick, snake it, <->] (-1.7,1.5) – (-5, 1.5);
[above] at (-3.3,1.6) Expand / Shrink;

[thick] (0,0) – (0,3);
[thick] (6,0) – (6,3);
[ultra thick,red] (3,0) – (3,1.5);
[above] at (3,1.7) $\frac{2\pi}{N} \widehat{a} \delta a$;
at (3,1.5) [circle,fill,red, inner sep=1.5pt];
[below] at (0,0) $\langle D(A)| $;
[below] at (6,0) $|\cX\rangle $;

[above] at (0,3) $x=0$;
[above] at (6,3) $x=\varepsilon$;

A $(1+1)$d QFT $\cX$ with $\Z_N$ symmetry and another $(1+1)$d QFT $\cX/\Z_N$ with quantum $\widehat{\Z}_N$ symmetry are separated by a topological interface $\cN$. This setup can be expanded into a $(2+1)$d slab, where the $(2+1)$d $\Z_N$ SymTFT has an insertion of a twist defect parallel to the Dirichlet boundary.

We may now consider inserting a twist defect $\Sigma_{(e)}(M_1, M_2)$ into the slab. We place it parallel to the Dirichlet boundary, as shown in Figure <ref>.[In [53], a similar configuration is considered. Their configuration can be obtained from our Figure <ref> by bringing the endpoint of $D_{\text{EM}}$ to the left boundary and letting $D_{\text{EM}}$ stretch across the bulk towards the right boundary.]
For convenience, we use $y$ to parameterize the horizontal direction in the slab, and $x$ to parameterize the vertical direction near the boundary of the twist defect, with the boundary $M_1$ located at $x=0$.  By colliding the twist defect with the left boundary of the slab, the lower half of the Dirichlet boundary conditions (i.e. in $X_2^{\geq 0}$) is transformed to Neumann boundary conditions. Thus upon shrinking the slab, from Section <ref> the $(1+1)$d QFT on  $X_2^{\geq 0}$ becomes $\CX/\Z_N$, as shown in the left panel of Figure <ref>. This implies that the twist defect $\Sigma_{(e)}$, when collided with the Dirichlet boundary, becomes a duality interface $\CN$.

Because there are $N$ twist defects in $(2+1)$d,  one may initially expect $N$ different duality interfaces in  $(1+1)$d. However, all $N$ types of twist defects actually reduce to a single type of duality interface upon colliding with the Dirichlet boundary condition. To see this, note that
the generic twist defect $\Sigma_{(e)}$ is related to the minimal twist defect $\Sigma_{(0)}$ by fusing an electric line $L_{(e,0)}$ on its boundary. Such electric line can be absorbed by the Dirichlet boundary condition of the SymTFT. As a consequence, when $\Sigma_{(e)}$ is brought to the Dirichlet boundary, it reduces to $\Sigma_{(0)}$. We thus have

\begin{eqnarray}
\CN(M_1) = \Sigma_{(e)}(M_1, M_2)|_{x\to 0}~, \hspace{1.2cm} \forall~e\in \Z_N~.
\end{eqnarray}

Furthermore, the fusion rules of the twist defects given in (<ref>) and (<ref>) descend to the fusion rules of the duality interfaces. Upon colliding with the Dirichlet boundary, $\Sigma_{(e)}$ and $L_{(e',m')}$  reduce to $\CN$ and  $\eta^{m'}$ respectively, and  (<ref>) thus simplifies to

\begin{eqnarray}
\CN \times \eta^{m'} = \CN~,
\end{eqnarray}

which reproduces (<ref>). Moreover, (<ref>) simplifies to

\begin{eqnarray}
\CN \times \CN = \chi[M_2, \Z_N]^{-1}\sum_{\gamma\in H_1(M_1, \Z_N)} \eta(\gamma)~,
\end{eqnarray}

which reproduces (<ref>).

§.§ F-symbols of twist defects and duality interfaces

We close this section by discussing the F-symbols for lines, both genuine and non-genuine, of the $\Z_N$ gauge theory, as well as the F-symbols of the duality interfaces. Additional details on the techniques used here are given in Appendix <ref>.

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,0) – (1,1);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (-1,-1) – (0,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,-1) – (0,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1,1);

[blue] (0,0) circle (2pt);
[blue] (1,1) circle (2pt);

[below] at (-1.2,-1) $L_{(e_1,m_1)}$;
[below] at (1.1,-1) $L_{(e_2,m_2)}$;
[below] at (3.6,-1) $L_{(e_3,m_3)}$;
[left] at (0.4,0.8) $L_{(e_1+e_2,m_1+m_2)}$;

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,1) – (2,2);
[blue, thick] (0,0) – (1,1);
[blue, thick, ->] (-1,-1) – (0,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (2,0) – (1,1);

[blue] (2,0) circle (2pt);
[blue] (1,1) circle (2pt);

[below] at (-1.2,-1) $L_{(e_1,m_1)}$;
[below] at (1.1,-1) $L_{(e_2,m_2)}$;
[below] at (3.6,-1) $L_{(e_3,m_3)}$;
[right] at (1.3,0.8) $L_{(e_2+e_3,m_2+m_3)}$;

[blue,thick,decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,-.5)  to (0,0);
[blue,thick] (0,0) arc [radius=.3, start angle=240, end angle=120];
[blue,thick, decoration=markings, mark=at position 0.7 with >, postaction=decorate] (0,0.52) to +(30:.5) ;
[blue,thick] (0,0) arc [radius=.3, start angle=-60, end angle=42];
[blue,thick, decoration=markings, mark=at position 0.6 with >, postaction=decorate] (0,0.52) ++(150:.1) to +(150:.4) ;

[blue] (0,0) circle (1.2pt);

[below] at (0.8,0.7) $L_{(e_1,m_1)}$;
[below] at (-0.9,0.7) $L_{(e_2,m_2)}$;

=  $e^{-{2 \pi i \over N} e_1 m_2}$
[blue,thick,decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,-.8) to (0,0);
[blue,thick, decoration=markings, mark=at position 0.7 with >, postaction=decorate] (0,0) to (-.6,.5);
[blue,thick, decoration=markings, mark=at position 0.7 with >, postaction=decorate] (0,0)–(.6,.5) ;
[blue] (0,0) circle (1.2pt);

[below] at (0.8,0.4) $L_{(e_1,m_1)}$;
[below] at (-0.9,0.4) $L_{(e_2,m_2)}$;

[blue,thick,decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,-.5)  to (0,0);
[blue,thick] (0,0) arc [radius=.3, start angle=240, end angle=120];

[blue,thick] (0,0) arc [radius=.3, start angle=-60, end angle=42];

[blue,thick, decoration=markings, mark=at position 0.8 with >, postaction=decorate] (-0.1,0.42) – (0.4,0.7) ;

[white, line width = 0.15cm] (0.1,0.42) – (-0.4,0.7) ;

[blue,thick, decoration=markings, mark=at position 0.8 with >, postaction=decorate] (0.1,0.42) – (-0.4,0.7) ;

[blue] (0,0) circle (1.2pt);

[below] at (0.8,0.7) $L_{(e_1,m_1)}$;
[below] at (-0.9,0.7) $L_{(e_2,m_2)}$;

=  $e^{{2 \pi i \over N} e_2 m_1}$
[blue,thick,decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,-.8) to (0,0);
[blue,thick, decoration=markings, mark=at position 0.7 with >, postaction=decorate] (0,0) to (-.6,.5);
[blue,thick, decoration=markings, mark=at position 0.7 with >, postaction=decorate] (0,0)–(.6,.5) ;
[blue] (0,0) circle (1.2pt);

[below] at (0.8,0.4) $L_{(e_1,m_1)}$;
[below] at (-0.9,0.4) $L_{(e_2,m_2)}$;

F-symbols and half-braidings for genuine lines $L_{(e,m)}$.

§.§.§ F-symbols of twist defects

In the current context, it is possible to work in a basis of the junction vector space such that the F-symbols for the genuine lines $L_{(e,m)}$ are trivial, and the half-braiding is as shown in Figure <ref>. Note in particular that this half-braiding correctly reproduces the full braiding given in Figure <ref>. Our goal is to now incorporate twist defects.

Before considering the F-symbols involving twist defects, we must first discuss the relevant trivalent junctions.
From the fusion rules in (<ref>), it is clear that each junction should involve two twist defects $\Sigma_{(e)}$ and $\Sigma_{(e')}$, together with one genuine line $L_{(e+e'+n,-n)}$ for arbitrary $n$. A crucial fact is that these junctions actually depend on the angle $\theta$ between $L_{(e+e'+n,-n)}$ and the surface $D_{\mathrm{EM}}$ on which $\Sigma_{(e)}$ and $\Sigma_{(e')}$ are anchored, the signature of an isotopy anomaly. We will consider two angles for our current analysis, namely $\theta = 0^+$ and $\pi$.  The junction with $\theta = \pi$ will be labelled with a triangle, while the junction with $\theta = 0^+$ will be denoted by a square. The notation $\theta=0^+$ is to emphasize that the $D_{\text{EM}}$ surface is behind the line anchored on the square junction. Both of these configurations are illustrated in Figure <ref>. They will be studied in more detail in Appendix <ref>.

[top color=red!40, bottom color=red!10]  (0,-1) – (-1,-1) – (-1,1) – (0,1)– (0,-1);

[dgreen,thick](0,-1) – (0,1);
[blue, thick,decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,0) to[out=0,in=90 ]  (1,-1);
[dgreen, above] at (0,0.9) $\Sigma_{(e')}$;
[dgreen, below] at (0,-1) $\Sigma_{(e)}$;
[blue, below] at (1.4,-1) $L_{(e+e'+n,-n)}$;

[isosceles triangle,scale=0.4,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (0,0);

[baseline=0,square/.style=regular polygon,regular polygon sides=4]

[top color=red!40, bottom color=red!10]  (0,-1) – (-1,-1) – (-1,1) – (0,1)– (0,-1);
[dgreen,thick](0,-1) – (0,1);
[gray!30, line width=3pt] (0,0) to[out=180,in=90 ]  (-1,-1);
[blue, thick,decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,0) to[out=180,in=90 ]  (-1,-1);
[dgreen, above] at (0,0.9) $\Sigma_{(e')}$;
[dgreen, below] at (0,-1) $\Sigma_{(e)}$;
[blue, below] at (-1.5,-1) $L_{(e+e'+n,-n)}$;

at (0,0) [square,draw,fill=violet!60,scale=0.5] ;

[red, thick, snake it, -] (0,0) – (-2,0);
[dgreen] (0,0) circle (3pt);
[dgreen,right] at (0.1,0) $\Sigma_{(e)},\Sigma_{(e')}$;
[red,left] at (-2,0) $D_{EM}$;

[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,0) – (-0.9,-1.2);

[thick] (0,0)+(-0.6,0cm) arc[start angle=180, end angle=240, radius=0.5cm];
[blue,left] at (-0.9,-1.2) $L_{(e+e'+n,-n)}$;
[left] at (-0.5,-0.35) $\theta$;

The triangle junction with $\theta = \pi$, together with the square junction with $\theta = 0^+$. The $\ZZ_2^{\mathrm{EM}}$ duality surface $D_{\mathrm{EM}}$, which is a mesh of  algebra objects $\CA=\bigoplus_{n=0}^{N-1} L_{(n,-n)}$ that can end on $\Sigma_{(e)}$, is drawn in red.  When there is no ambiguity, we will suppress the $D_{\text{EM}}$ surface in our figures, taking it to go out to the left.

Another important fact is that the junctions should not depend on where along $\Sigma_{(e)}$ the line $L_{(e+e'+n,-n)}$ is anchored. To phrase this more concretely, we begin by defining the algebra object $\cA= \bigoplus_{n=0}^{N-1} L_{(n,-n)}$.
The condensation defect $D_{\mathrm{EM}}$ can be resolved into a mesh of such algebra objects, and in terms of this mesh the invariance under change in the anchor point is equivalent to Figure <ref>. The $\mu_L$ and $\mu_L^\vee$ appearing in Figure <ref> are certain junction vectors, defined in Appendix <ref>, which will not be important to us here. What is important for us is that, given a solution to the consistency condition for the triangle junction, a solution for the square junction can be obtained by defining the junction as in Figure <ref>. This is discussed in more detail in Appendix <ref>. We will take this as the definition of the square junction from now on.

[dgreen,thick](0,-1) – (0,1);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,0) to[out=0,in=90 ]  (1,-1);
[red, thick] (0,-0.5) to[out=180,in=270 ]  (-0.5,0);
[red, thick] (-0.5,0) to[out=90,in=180 ]  (0,0.5);

[red, left] at (-0.5,0) $\mathcal{A}$;
[dgreen, below] at (0,-1) $\Sigma_{(e)}$;
[blue, below] at (1.5,-1) $L_{(e+e'+n,-n)}$;
[red, right] at (0,0.5) $\mu_L$;
[red, right] at (0,-0.5) $\mu_L^\vee$;

[red] (0,0.5) circle (1.5pt);
[red] (0,-0.5) circle (1.5pt);

[isosceles triangle,scale=0.4,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (0,0);

[dgreen,thick](0,-1) – (0,1);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,0) to[out=0,in=90 ]  (1,-1);
[red, thick] (0,0.5) to[out=180,in=270 ]  (-0.5,1);

[dgreen, below] at (0,-1) $\Sigma_{(e)}$;
[blue, below] at (1.5,-1) $L_{(e+e'+n,-n)}$;
[red, right] at (0,0.5) $\mu_L^\vee$;

[red] (0,0.5) circle (1.5pt);

[isosceles triangle,scale=0.4,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (0,0);

[baseline=0,square/.style=regular polygon,regular polygon sides=4]

[dgreen,thick](0,-1) – (0,1);
[red, thick] (0,-0.5) to[out=180,in=270 ]  (-0.5,0);
[red, thick] (-0.5,0) to[out=90,in=180 ]  (0,0.5);
[white, line width=5pt] (0,0) to[out=180,in=90 ]  (-1,-1);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,0) to[out=180,in=90 ]  (-1,-1);

[red, left] at (-0.5,0) $\mathcal{A}$;
[dgreen, below] at (0,-1) $\Sigma_{(e)}$;
[blue, below] at (-1.5,-1) $L_{(e+e'+n,-n)}$;
[red, right] at (0,0.5) $\mu_L$;
[red, right] at (0,-0.5) $\mu_L^\vee$;

[red] (0,0.5) circle (1.5pt);
[red] (0,-0.5) circle (1.5pt);

at (0,0) [square,draw,fill=violet!60,scale=0.5] ;

[baseline=0,square/.style=regular polygon,regular polygon sides=4]

[dgreen,thick](0,-1) – (0,1);
[red, thick] (0,0.5) to[out=180,in=270 ]  (-0.5,1);
[white, line width=5pt] (0,0) to[out=180,in=90 ]  (-1,-1);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,0) to[out=180,in=90 ]  (-1,-1);

[dgreen, below] at (0,-1) $\Sigma_{(e)}$;
[blue, below] at (-1.5,-1) $L_{(e+e'+n,-n)}$;
[red, right] at (0,0.5) $\mu_L^\vee$;

[red] (0,0.5) circle (1.5pt);

at (0,0) [square,draw,fill=violet!60,scale=0.5] ;

Consistency conditions for the triangle and square junctions.

[baseline=0,square/.style=regular polygon,regular polygon sides=4]

[dgreen,thick](0,-1) – (0,1);
[white, line width=5pt] (0,0) to[out=180,in=90 ]  (-1,-1);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,0) to[out=180,in=90 ]  (-1,-1);

at (0,0) [square,draw,fill=violet!60,scale=0.5] ;

[baseline=0,square/.style=regular polygon,regular polygon sides=4]

[dgreen,thick](0,-0.45) – (0,1);
[dgreen,thick](0,-0.55) – (0,-1);
[white, line width=5pt] (0,0) to[out=180,in=90 ]  (-1,-1);
[blue, thick] (0,0) to[out=0,in=0,distance=0.2 in]  (0,-0.5);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,-0.5) to[out=180,in=90,distance=0.2 in]  (-1,-1);

[isosceles triangle,scale=0.4,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (0,0);

Defining the square junction in terms of the triangle junction.

We now return to the F-symbols. We begin with the case of two external genuine lines and two external twist defect. From the form of the fusion rules in (<ref>), it is clear that the internal line is necessarily non-genuine, i.e. it is a twist defect. We may now calculate the F-symbols using the tools developed above. The computation is shown in Figure <ref>. Summarizing it in words, one begins by using the definition of the square junction in terms of the triangle junction, and then decomposes the boundary of $\Sigma_{(e)}$ into $\cA\otimes L_{(e,0)} = \bigoplus_{n=0}^{N-1} L_{(e+n,-n)}$. The factor of $\cA$ is included here so that the mesh of $\cA$ making up $D_{\mathrm{EM}}$ can end on $\Sigma_{(e)}$  from the left. One then uses the half-braiding and fusion rules of the genuine lines, given in Figure <ref>, to obtain the  right-hand side of the second line of  Figure <ref>. One finally does a half-braiding and reassembles $\Sigma_{(e)}$ to get the end result.

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]

[top color=red!30, bottom color=red!5]  (1,-1) – (0,0) – (2,2) – (-1,2)– (-1,-1)– (1,-1);

[dgreen, thick] (0,0) – (1,1);
[dgreen, thick] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (-1,-1) – (0,0);
[dgreen, thick] (1,-1) – (0,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1,1);

[isosceles triangle,scale=0.5,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (1,1);

at (0,0) [square,draw,fill=violet!60,scale=0.6] ;

[below] at (-1,-1) $L_{(e_1,m_1)}$;
[below] at (1,-1) $\Sigma_{(e)}$;
[below] at (3,-1) $L_{(e_2,m_2)}$;
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[dgreen, thick] (1,1) – (2,2);
[dgreen, thick] (0.5,0.5)– (1,1);

[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1,1);

[isosceles triangle,scale=0.5,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (1,1);

[dgreen, thick] (0.5,0.5) to[out=225,in=90 ]  (-0,-1);
[white, line width = 0.05 in] (0.22,0) to[out=-1,in=0,distance = 0.5cm ] (0,-0.5);
[white,  line width = 0.05 in] (0,-0.5) to[out=180,in=90,distance = 0.5cm ] (-1,-1);
[blue, thick] (0.15,0) to[out=-1,in=0,distance = 0.5cm ] (0,-0.5);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,-0.5) to[out=180,in=90,distance = 0.5cm ] (-1,-1);

[isosceles triangle,scale=0.5,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (0.15,0);

[below] at (-1,-1) $L_{(e_1,m_1)}\,\,\,$;
[below] at (0.4,-1) $\Sigma_{(e)}$;
[below] at (3,-1) $L_{(e_2,m_2)}$;

$=\,\, \sum_n$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]

[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1,1);
[blue, thick] (0.5,0.5)– (1,1);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,1) – (2,2);

[blue] (1,1) circle (2pt);
[blue] (0.15,0) circle (2pt);

[blue, thick, <-, decoration=markings, mark=at position 0.85 with <, postaction=decorate] (0.5,0.5) to[out=225,in=90 ]  (-0,-1);
[white, line width = 0.05 in] (0.22,0) to[out=-1,in=0,distance = 0.5cm ] (0,-0.5);
[white,  line width = 0.05 in] (0,-0.5) to[out=180,in=90,distance = 0.5cm ] (-1,-1);

[blue, thick] (0.15,0) to[out=-1,in=0,distance = 0.5cm ] (0,-0.5);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0,-0.5) to[out=180,in=90,distance = 0.5cm ] (-1,-1);

[below] at (-1.6,-1) $L_{(e_1,m_1)}$;
[below] at (0.7,-1) $L_{(e+n, -n)}$;
[below] at (3.4,-1) $L_{(e_2,m_2)}$;

$=$   $ \footnotesize{\sum_n e^{ {2\pi i \over N}e_1 n}}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[blue,thick] (0.5,0.5)– (1,1);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,1) – (2,2);

[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1,1);

[blue] (1,1) circle (2pt);
[blue] (0.15,0) circle (2pt);

[blue, thick, <-, decoration=markings, mark=at position 0.75 with <, postaction=decorate] (0.5,0.5) to[out=225,in=90 ]  (-0,-1);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (0.15,0) to[out=180,in=90,distance = 0.5cm ] (-1,-1);

[below] at (-1.6,-1) $L_{(e_1,m_1)}$;
[below] at (0.7,-1) $L_{(e+n, -n)}$;
[below] at (3.4,-1) $L_{(e_2,m_2)}$;

$=$    $\footnotesize{\sum_n e^{ {2\pi i \over N}e_1 n}}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[blue, thick, decoration=markings, mark=at position 0.25 with >, mark=at position 0.65 with >,mark=at position 0.85 with >, postaction=decorate] (0.4,-1) – (2,2);

[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1.15,0.35);

[blue] (1.15,0.35) circle (2pt);

[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (1.6,1.2) to[out=180,in=90,distance = 0.9cm ] (-1,-1);

[blue] (1.6,1.2) circle (2pt);

[below] at (-1.6,-1) $L_{(e_1,m_1)}$;
[below] at (0.7,-1) $L_{(e+n, -n)}$;
[below] at (3.4,-1) $L_{(e_2,m_2)}$;
[right] at (1.15,0.4) $L_{(e_2+e+n,m_2-n)}$;

$=$    $ \footnotesize{\sum_n e^{{2\pi i \over N}e_1 n} e^{{2\pi i \over N}e_1(m_2-n)}}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[blue, thick, decoration=markings, mark=at position 0.25 with >, mark=at position 0.6 with >, mark=at position 0.9 with >, postaction=decorate] (0.4,-1) – (2,2);

[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1.15,0.35);

[blue] (1.15,0.35) circle (2pt);

[white, thick,line width = 0.05 in] (1.8,1.2) to[out=0,in=0,distance = 0.6cm ] (1.2,0.8);
[white, thick,line width = 0.05 in] (1.2,0.8) to[out=180,in=90,distance = 0.6cm ] (-1,-1);

[blue, thick] (1.6,1.2) to[out=0,in=0,distance = 0.7cm ] (1.2,0.8);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (1.2,0.8) to[out=180,in=90,distance = 0.6cm ] (-1,-1);

[blue] (1.6,1.2) circle (2pt);

[below] at (-1.6,-1) $L_{(e_1,m_1)}$;
[below] at (0.7,-1) $L_{(e+n, -n)}$;
[below] at (3.4,-1) $L_{(e_2,m_2)}$;

$=$  $\footnotesize e^{{2\pi i\over N } e_1 m_2}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[top color=red!30, bottom color=red!5]  (1,-1) – (2,0) – (1,1) – (2,2)–(-1,2)– (-1,-1)– (1,-1);
[dgreen, thick] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (-1,-1) – (1,1);
[dgreen, thick] (1,1) – (2,0);
[dgreen, thick] (1,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (2,0);

[isosceles triangle,scale=0.5,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (2,0);

at (1,1) [square,draw,fill=violet!60,scale=0.6] ;

[below] at (-1,-1) $L_{(e_1,m_1)}$;
[below] at (1,-1) $\Sigma_{(e)}$;
[below] at (3,-1) $L_{(e_2,m_2)}$;

Computation of the F-symbol for two genuine lines and one twist defect. The surface $D_{\mathrm{EM}}$ is suppressed at intermediate steps.

We next consider the F-symbols involving four external twist defects. Note that because of the fusion rules (<ref>), the internal line is guaranteed to be genuine. Let us take the three incoming twist defects to be $\Sigma_{(e_1)},\Sigma_{(e_2)},$ and $\Sigma_{(e_3)}$, and place the mesh of $\CA$ as shown in Figure <ref>.  The coefficient relating the configuration with internal line $L_{(e,m)}$ to that with internal line $L_{(\widetilde e, \widetilde m)}$ is then given in Figure <ref>, and derived in Figure <ref>. The various charges appearing here satisfy $e_1+e_2=e+m$, $e_2+e_3=\widetilde{e}+\widetilde{m}$. For simplicity we neglect real number normalization factors, and focus only on the phases. In the special case of $N=2$, the F-symbols in Figure <ref> and <ref> are consistent with those found in [66].

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]

[top color=red!30, bottom color=red!5]  (1,-1) – (3,-1) – (1,1) – (2,2)–(-1,2)– (-1,-1)– (0,0)–(1,-1);

[dgreen, thick] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,0) – (1,1);
[dgreen, thick] (-1,-1) – (0,0);
[dgreen, thick] (1,-1) – (0,0);
[dgreen, thick] (3,-1) – (1,1);

[below] at (-1,-1) $\Sigma_{(e_1)}$;
[below] at (1.1,-1) $\Sigma_{(e_2)}$;
[below] at (3.2,-1) $\Sigma_{(e_3)}$;
[left] at (0.8,0.9) $L_{(e,m)}$;

$= $ $ \sum_{\widetilde{m}=0}^{N-1}e^{-{2 \pi i \over N}e \widetilde m}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]

[top color=red!30, bottom color=red!5]  (1,-1) – (2,0) – (3,-1) –(1,-1);

[top color=red!30, bottom color=red!5]  (-1,-1) – (2,2) – (-1,2) –(-1,-1);

[dgreen, thick] (1,1) – (2,2);
[dgreen, thick] (0,0) – (1,1);
[dgreen, thick] (-1,-1) – (0,0);
[dgreen, thick] (1,-1) – (2,0);
[dgreen, thick] (3,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (1,1) – (2,0);

[below] at (-1,-1) $\Sigma_{(e_1)}$;
[below] at (1.1,-1) $\Sigma_{(e_2)}$;
[below] at (3.2,-1) $\Sigma_{(e_3)}$;
[right] at (1.5,0.8) $L_{(\widetilde{e},\widetilde{m})}$;

F-symbol for three incoming twist defects. The diagrams are non-vanishing only when the intermediate legs $L_{(e,m)}$ and $L_{(\widetilde{e},\widetilde{m})}$ satisfy $e_1+e_2=e+m$, $e_2+e_3=\widetilde{e}+\widetilde{m}$. The $D_{\text{EM}}$ surface attached to the twist defects are assumed to be to the left of $\Sigma_{(e_1)}$ as well as between $\Sigma_{(e_2)}$ and $\Sigma_{(e_3)}$.

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[top color=red!30, bottom color=red!5]  (1,-1) – (3,-1) – (1,1) – (2,2)–(-1,2)– (-1,-1)– (0,0)–(1,-1);
[dgreen, thick] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,0) – (1,1);
[dgreen, thick] (-1,-1) – (0,0);
[dgreen, thick] (1,-1) – (0,0);
[dgreen, thick] (3,-1) – (1,1);

[below] at (-1,-1) $\Sigma_{(e_1)}$;
[below] at (1.1,-1) $\Sigma_{(e_2)}$;
[below] at (3.2,-1) $\Sigma_{(e_3)}$;
[left] at (0.8,0.9) $L_{(e,m)}$;

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[dgreen, thick] (1.2,1.2) – (2,2);
[dgreen, thick] (3,-1) – (1.2,0.8);

[dgreen, thick] (1.2,1.2) to[out=225,in=135,distance = 0.2cm ] (1.2,0.8);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (1.1,1) to[out=180,in=180,distance = 1cm ] (-0.45,-0.5) ;

[dgreen, thick]  (-1,-1) to[out=45,in=135,distance = 1.5cm ] (1,-1);
at (1.1,1)[square,draw,fill=violet!60,scale=0.5] ;
at (-0.45,-0.5)[square,draw,fill=violet!60,scale=0.5] ;

[below] at (-1,-1) $\Sigma_{(e_1)}$;
[below] at (1.1,-1) $\Sigma_{(e_2)}$;
[below] at (3.2,-1) $\Sigma_{(e_3)}$;
[left] at (0.25,0.9) $L_{(e,m)}$;

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[dgreen, thick] (1.2,1.2) – (2,2);
[dgreen, thick] (3,-1) – (1.2,0.8);

[dgreen, thick] (1.2,1.2) to[out=225,in=135,distance = 0.2cm ] (1.2,0.8);

[dgreen, thick]  (-1,-1) to[out=45,in=135,distance = 1.5cm ] (1,-1);

[white, line width = 0.05in] (1.1,1) to[out=0,in=45,distance = 0.5cm ] (1.5,0.5) ;
[white, line width = 0.05in] (1.5,0.5) to[out=225,in=45,distance = 0.3cm ] (0.6,-0.5) ;
[white,line width = 0.05in] (-0.45,-0.5) to[out=-45,in=225,distance = 0.3cm ] (0.6,-0.5) ;

[blue, thick] (1.1,1) to[out=0,in=45,distance = 0.5cm ] (1.5,0.5) ;
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (1.5,0.5) to[out=225,in=45,distance = 0.3cm ] (0.6,-0.5) ;
[blue, thick] (-0.45,-0.5) to[out=-45,in=225,distance = 0.3cm ] (0.6,-0.5) ;

[isosceles triangle,scale=0.4,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (1.1,1);
[isosceles triangle,scale=0.4,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (-0.45,-0.5);

[below] at (-1,-1) $\Sigma_{(e_1)}$;
[below] at (1.1,-1) $\Sigma_{(e_2)}$;
[below] at (3.2,-1) $\Sigma_{(e_3)}$;
[left] at (0.25,0.9) $L_{(e,m)}$;

= $\sum_{n_1,n_2,n_3} e^{{2\pi i\over N} e (n_2 + n_3)}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,0) – (1,1);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (-1,-1) – (0,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,-1) – (0,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1,1);

[blue] (0,0) circle (2pt);
[blue] (1,1) circle (2pt);

[below] at (-2,-1) $L_{(e_1+n_1,-n_1)}$;
[below] at (1.1,-1) $L_{(e_2+n_2,-n_2)}$;
[below] at (4.2,-1) $L_{(e_3+n_3,-n_3)}$;
[left] at (0.8,0.9) $L_{(e,m)}$;

$=$ $\sum_{n_1,n_2,n_3} e^{{2\pi i\over N}e (n_2 + n_3)}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,1) – (2,2);
[blue, thick] (0,0) – (1,1);
[blue, thick, ->] (-1,-1) – (0,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (1,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.25 with >, mark=at position 0.75 with >, postaction=decorate] (3,-1) – (1,1);

[blue] (2,0) circle (2pt);
[blue] (1,1) circle (2pt);

[below] at (-2,-1) $L_{(e_1+n_1,-n_1)}$;
[below] at (1.1,-1) $L_{(e_2+n_2,-n_2)}$;
[below] at (4.2,-1) $L_{(e_3+n_3,-n_3)}$;
[right] at (1.5,0.8) $L_{(e_2+e_3+n_2+n_3,-n_2-n_3)}$;

$=$  $\sum_{\widetilde{m}=0}^{N-1}e^{-{2\pi i\over N}e \widetilde{m}}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[dgreen, thick] (1,1) – (2,2);
[dgreen, thick] (0,0) – (1,1);
[dgreen, thick] (-1,-1) – (0,0);
[dgreen, thick] (1,-1) – (2,0);
[dgreen, thick] (3,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (1,1) – (2,0);

[below] at (-1,-1) $\Sigma_{(e_1)}$;
[below] at (1.1,-1) $\Sigma_{(e_2)}$;
[below] at (3.2,-1) $\Sigma_{(e_3)}$;
[right] at (1.5,0.8) $L_{(\widetilde{e},\widetilde{m})}$;

[isosceles triangle,scale=0.4,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (1,1);

[isosceles triangle,scale=0.4,
isosceles triangle apex angle=60,
minimum size =0.01cm] at (2,0);

$=$$\sum_{\widetilde{m}=0}^{N-1}e^{-{2\pi i\over N}e \widetilde{m}}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]
[top color=red!30, bottom color=red!5]  (1,-1) – (2,0) – (3,-1) –(1,-1);

[top color=red!30, bottom color=red!5]  (-1,-1) – (2,2) – (-1,2) –(-1,-1);
[dgreen, thick] (1,1) – (2,2);
[dgreen, thick] (0,0) – (1,1);
[dgreen, thick] (-1,-1) – (0,0);
[dgreen, thick] (1,-1) – (2,0);
[dgreen, thick] (3,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (1,1) – (2,0);

[below] at (-1,-1) $\Sigma_{(e_1)}$;
[below] at (1.1,-1) $\Sigma_{(e_2)}$;
[below] at (3.2,-1) $\Sigma_{(e_3)}$;
[right] at (1.5,0.8) $L_{(\widetilde{e},\widetilde{m})}$;

Computation of the F-symbol for three incoming twist defects. For the diagrams in the second and third line to be non-vanishing, we require $e_1+e_2=m+e, m=-n_1-n_2, e_2+e_3=\widetilde{e}+\widetilde{m}$ and $\widetilde{m}=-n_2-n_3$.
The surface $D_{\mathrm{EM}}$ is suppressed at intermediate steps.

§.§.§ F-symbols of duality interfaces

We finally obtain the F-symbols of duality interfaces in $(1+1)$d. As explained in Section <ref>, upon shrinking the slab the twist defects of $\Z_N$ gauge theory become the duality interface of a $(1+1)$d QFT. As a consequence, the F-symbols of the duality interface naturally follow from those of the twist defects.

We first consider the F-symbols of duality interfaces descending from Figure <ref>. Upon shrinking, $\Sigma_{(e)}$ reduces to $\CN$. Moreover, note that $L_{(e_2,m_2)}$ collides with the Dirichlet boundary condition upon shrinking, and hence only its magnetic components survive the $\Z_N$ symmetry generators, i.e. $L_{(e_2,m_2)}$ reduces to $\eta^{m_2}$. On the other hand, $L_{(e_1, m_1)}$ is on top of $D_{\text{EM}}$, and hence upon shrinking it collides with the Neumann boundary condition and its electric component survives, i.e. $L_{(e_1, m_1)}$ reduces to $\widehat\eta^{e_1}$, where $\widehat{\eta}$ is the generator of the quantum $\Z_N$ symmetry.  Hence the F-symbol involving one duality interface and two $\Z_N$ symmetry generators distributed on its two sides is as shown in Figure <ref>. Note that the duality interface is unoriented, i.e. $\CN=\overline{\CN}$ (up to an Euler counterterm, c.f. (<ref>)), and hence we do not place an arrow on it. In the next section we will gauge $\ZZ_2^{\mathrm{EM}}$ to obtain the SymTFT for the $TY(\Z_N)$ category, for which the phase appearing in Figure <ref> implies a nontrivial bi-character.

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]

[top color=red!30, bottom color=red!5]  (1,-1) – (0,0) – (2,2) – (-1,2)– (-1,-1)– (1,-1);

[red, very thick] (0,0) – (1,1);
[red, very thick] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (-1,-1) – (0,0);
[red, very thick] (1,-1) – (0,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (1,1);

[below] at (-1,-1) $\widehat\eta^{e_1}$;
[below] at (1,-1) $\CN$;
[below] at (3,-1) $\eta^{m_2}$;

$=$  $\footnotesize e^{{2\pi i\over N } e_1 m_2}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]

[top color=red!30, bottom color=red!5]  (1,-1) – (2,0) – (1,1) – (2,2)–(-1,2)– (-1,-1)– (1,-1);

[red, very thick] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (-1,-1) – (1,1);
[red, very thick] (1,1) – (2,0);
[red, very thick] (1,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (3,-1) – (2,0);

[below] at (-1,-1) $\widehat\eta^{e_1}$;
[below] at (1,-1) $\CN$;
[below] at (3,-1) $\eta^{m_2}$;

F-symbol involving two external duality interfaces and two external $\Z_N$ symmetry generators.

[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]

[top color=red!30, bottom color=red!5]  (1,-1) – (3,-1) – (1,1) – (2,2)–(-1,2)– (-1,-1)– (0,0)–(1,-1);
[dgreen, thick] (1,1) – (2,2);

[red, very thick] (1,1) – (2,2);
[blue, thick, decoration=markings, mark=at position 0.5 with >, postaction=decorate] (0,0) – (1,1);
[red, very thick] (-1,-1) – (0,0);
[red, very thick] (1,-1) – (0,0);
[red, very thick] (3,-1) – (1,1);

[below] at (-1,-1) $\CN$;
[below] at (1.1,-1) $\CN$;
[below] at (3.2,-1) $\CN$;
[left] at (0.8,0.9) $\widehat\eta^{e}$;

$=$ $\sum_{\widetilde{m}=0}^{N-1}e^{-{2 \pi i \over N}e \widetilde m}$
[baseline=0,square/.style=regular polygon,regular polygon sides=4,scale=0.6]

[top color=red!30, bottom color=red!5]  (1,-1) – (2,0) – (3,-1) –(1,-1);

[top color=red!30, bottom color=red!5]  (-1,-1) – (2,2) – (-1,2) –(-1,-1);

[red, very thick] (1,1) – (2,2);
[red, very thick] (0,0) – (1,1);
[red, very thick] (-1,-1) – (0,0);
[red, very thick] (1,-1) – (2,0);
[red, very thick] (3,-1) – (2,0);
[blue, thick, decoration=markings, mark=at position 0.5 with <, postaction=decorate] (1,1) – (2,0);

[below] at (-1,-1) $\CN$;
[below] at (1.1,-1) $\CN$;
[below] at (3.2,-1) $\CN$;
[right] at (1.5,0.8) $\eta^{\widetilde{m}}$;

F-symbol for four external duality interfaces.

We next consider the F-symbols involving three duality interfaces, which follows from Figure <ref>. The result is shown in Figure <ref>. Note that $e$ and $\widetilde{m}$ can be arbitrary integers. The nontrivial phase on the right hand side of Figure <ref> implies that, once we gauge $\ZZ_2^{\mathrm{EM}}$ to make $\CN$ into a duality defect, the defect $\CN$ is “self-anomalous," in the sense that it is an obstruction to the existence of a trivially gapped phase. This is consistent with the results in [21].

§ $(2+1)$D SYMMETRY TFT FOR DUALITY DEFECTS

In Section <ref> we discussed the SymTFT for a theory with $\Z_N^{(0)}$ symmetry, and derived the fusion rules and F-symbols for the duality interface implementing $\Z_N^{(0)}$ gauging. In the current section, we will demand more symmetries of the $(1+1)$d theory $\cX$: namely, we require that it not only be $\Z_N$ symmetric, but also invariant under gauging $\Z_N^{(0)}$, i.e. $\cX=\cX/\Z_N^{(0)}$, or equivalently $D_\text{EM}\ket{\cX}=\ket{\cX}$. This means that the duality interface $\cN$ defined in Section <ref> is not just a topological interface between two distinct theories, but a topological defect within the theory $\cX$ itself. In this case the full symmetry is the Tambara-Yamagami fusion category $TY(\Z_N)$ (for particular choice of the Frobenius-Schur index and bicharacter). Our goal now is to find the SymTFT for  $TY(\Z_N)$.

§.§ Symmetry TFT for $TY(\ZZ_N)$

From a mathematical point of view, the SymTFT for $TY(\Z_N)$
is the Turaev-Viro theory specified by the Tambara-Yamagami $TY(\ZZ_N)$ category, or alternatively the Reshetikhin-Turaev theory for the Drinfeld center
$\cZ(TY(\ZZ_N))$. As we have mentioned already in the introduction, the simple objects and fusion rules of $\cZ(TY(\ZZ_N))$ have appeared already in the math and physics literature, so the results to be obtained below are not, strictly speaking, new.

The presentation given here emphasizes the fact that the SymTFT of $TY(\Z_N)$ can be obtained from that of $\mathrm{Vec}(\ZZ_N)$ by gauging the $\Z_2^{\text{EM}}$   electro-magnetic symmetry of the latter.[Let us mention that in $(2+1)$d, one can twist the gauging by stacking with an SPT for the $\Z_N^{(0)}$ zero-form symmetry, which is classified by $H^3(B\Z_N, U(1))$. This could give rise to different variants of the Tambara-Yamagami category. In this work, we consider only the untwisted gauging. ]
This fact is simple to see physically: indeed, recall that in the case of $\mathrm{Vec}(\ZZ_N)$, the duality interface $\cN$ descended from the twist defects $ \Sigma_{(e)}$ of the SymTFT, which were the boundaries of the $\Z_2^{\text{EM}}$ duality surface $D_{\mathrm{EM}}$. This was illustrated in Figure <ref>. Upon gauging  the $\Z_2^{\text{EM}}$ symmetry of the SymTFT this duality surface becomes transparent, and hence the duality interface $\cN$ becomes a duality defect, as shown in Figure <ref>.

Note that the procedure of gauging a zero-form symmetry in a $(2+1)$d TFT has been fully studied in the condensed matter literature [65, 66], where the power of modular tensor categories was utilized. Since the ultimate goal of this paper will  be to generalize to higher dimensions, and since a full-fledged theory of higher modular tensor categories has not been fully developed, we will not adopt this approach here. Instead, we will give a more physical rederivation of the known results from both the math [63, 64] and physics [65, 66] literature, being content with the operator contents and their fusion rules (without worrying about braiding properties). Our rederivation will have the virtue of being generalizable to higher dimensions.[Our derivation is similar to the approach in [27], where only the fusion rules of  topological lines with integer quantum dimensions were discussed. Here we will extensively discuss the fusion rules involving  topological lines with non-integer quantum dimensions, i.e. those descending from twist defects. ] We will also present a second derivation of the SymTFT of $TY(\Z_N)$ based on discrete twisted-cocycles. This will be (at least partially) generalizable to higher dimensions as well.

[thick, snake it, <->] (-1.7,-3.5) – (-5, -3.5);
[above] at (-3.3,-3.4) Expand/Shrink;

[line width=2pt,	top color=violet!30, bottom color=violet!5]
(0,-5) to [out=90, in=-90]  (0,-2)
to [out=0,in=180] (6,-2)
to [out = -90, in =90] (6,-5)
to [out=180, in =0]  (0,-5);

[thick] (0,-5) – (0,-2);
[thick] (6,-5) – (6,-2);
at (3,-3) $\cZ(TY(\Z_N))$;
at (3,-3.5) [circle,fill,red, inner sep=1.5pt];
[ thick,red,dashed] (3,-3.5) – (3,-5);
[below] at (0,-5) $\langle{D'}|$;

[below] at (6,-5) $|\cX\rangle $;

[very thick] (-7,-5) – (-7,-2);
[below] at (-7,-5) $Z_{\CX}$;
at (-7,-3.5) [circle,fill,red, inner sep=1.5pt];
[left,red] at (-7,-3.5) $\cN$;

The SymTFT of $TY(\Z_N)$ is obtained by gauging the $\Z_2^{\text{EM}}$ symmetry of the $\Z_N$ gauge theory in Figure <ref>. Mathematically, it is given by RT theory on the Drinfeld center of $TY(\Z_N)$.  The twist defect represented by the red dot attached to a red line (which eas actually a line living on the boundary of a 2d surface) in Figure <ref> now becomes a genuine line operator (represented by a red dot) in the current figure. After shrinking, this gives a genuine line defect $\cN$ within the theory $\cX$.

§.§.§ Lines defects in the SymTFT of $TY(\ZZ_N)$

As we have just claimed, the SymTFT for $TY(\ZZ_N)$ can be obtained from $\ZZ_N$ gauge theory by gauging the $\ZZ_2^{\mathrm{EM }}$ zero-form symmetry. This gauging may be split into two steps. First, we begin by keeping only the lines invariant under $\ZZ_2^{\mathrm{EM }}$. This leaves us with lines $L_{(e,e)}$ with equal electric and magnetic charge. Next, we note that upon gauging the $\ZZ_2^{\mathrm{EM }}$ zero-form symmetry, we obtain a quantum one-form symmetry $\widehat \ZZ_2^{(1)}$ generated by an operator $K$. Note that $K$ is labelled by a representation of $\ZZ_2^{\text{EM}}$.  If in the pre-gauged theory a simple line is invariant under $\ZZ_2^{\mathrm{EM }}$, then in the gauged theory it can be assigned a $\ZZ_2$ representation distinguished by whether or not it is stacked with a $K$-line. We therefore denote the invertible lines of the gauged theory by $\widehat L_{(e)}^q$ where the $q=\pm$ labels whether $K$ is stacked with it or not, i.e. $\widehat L_{(e)}^+ := L_{(e,e)}$ and $\widehat L_{(e)}^- :=K\, L_{(e,e)}$. There are a total of $2N$ invertible lines.

In additional to the invertible lines, we should also allow for the combinations $L_{(e,m)}\oplus L_{(m,e)}$, which are invariant under $\ZZ_2^{\mathrm{EM }}$ when viewed as a single object.[We will be mainly discussing things at a local level, with each line operator $L_{(e,m)}$ is supported on a small interval. The direct sum/product between operators in such a small patch will be denoted by $\oplus, \otimes$. Note however that since all the operators in the SymTFT of $TY(\Z_N)$ are lines, the local fusion coincides with the global fusion, and thus it is equally fine to change $\oplus,\otimes$ to $+, \times$. The distinction will only be important in higher dimensions.]  Such combinations $L_{(e,m)}\oplus L_{(m,e)}$ are not simple before gauging, but become simple after gauging, and will be denoted by $\widehat L_{[e,m]}$.[We use the word “simple" here in the sense of category theory, where an object $a$ is simple if $\mathrm{Hom}(a,a)$ is one-dimensional.] Noting that when $e=m$ the two factors in the direct sum are individually $\Z_2^{\text{EM}}$ invariant, and because by definition $\widehat L_{[e,m]} \equiv \widehat{L}_{[m,e]}$, we may assume without loss of generality that $0\leq e<m\leq N-1$.
Unlike for $\widehat L_{(e)}^\pm$, in the current case we cannot stack with a $K$ defect since the constituent lines $L_{(e,m)}$ and $L_{(m,e)}$ in the pre-gauged theory are not invariant under $\ZZ_2^{\mathrm{EM}}$.
Another way to say this is that $\widehat L_{[e,m]}$ can absorb $K$.

\[
\raisebox{-2em}{\begin{tikzpicture}
\draw[thick] (0,0) circle (20pt);
\draw[thick, red] (0,0) circle (18pt);
\node at (-0.9,-0.7) {$\widehat L_{[e,m]}$};
\node[red] at (-0.2,-0.2) {$K$};
\end{tikzpicture}}
\hspace{0.5 in} = \hspace{0.5 in}
\raisebox{-2em}{\begin{tikzpicture}
\draw[thick] (0,0) circle (20pt);
\node at (1.1,-0.7) {$\widehat L_{[e,m]}$};
\end{tikzpicture}}
\]

The line $\widehat L_{[e,m]}$ can absorb the line $K$ since whenever the $K$ loop (red) is non-trivial, the $\widehat L_{[e,m]}$ loop (black) vanishes.

A physical picture for why this absorption can occur is given in Figure <ref>. We consider a configuration of coincident loops of $\widehat L_{[e,m]}$ and $K$ and ask if this configuration can be distinguished from a loop of $\widehat L_{[e,m]}$ in isolation. To answer this, we fix an arbitrary EM gauge configuration and compute the vev of the loops. In order for the $K$ loop to give a non-trivial result, it must link with an odd number of units of EM flux. However, in the presence of an odd number of units of EM flux, the loop of $\widehat L_{[e,m]}$ gives vanishing contribution, since each consitutent of $\widehat L_{[e,m]}$ is not EM invariant and hence cannot form a closed loop around such flux. The configuration of $\widehat L_{[e,m]}$ stacked with $K$ is thus indistinguishable from the configuration with only $\widehat L_{[e,m]}$. We conclude that there are only ${1 \over 2} N(N-1)$ distinct non-invertible defects of quantum dimension 2, labeled by the symmetric pair $[e,m]\equiv [m,e]$ with $e\neq m$.

Finally, there are lines descending from the twist defects in the pre-gauged theory. As shown in Figure <ref>, after gauging $\Z_2^{\text{EM}}$ the surface attached to the twist defect becomes transparent, and hence the twist defect become a genuine line defect.
Since, as discussed in Section <ref>, before gauging $\Z_2^{\text{EM}}$ the twist defect $\Sigma_{(e)}$ is $\Z_2^{\text{EM}}$ invariant, the genuine lines after gauging can be stacked with a $K$ line.
We will denote the resulting genuine lines by $\widehat{\Sigma}_{(e)}^q$, where $q=\pm$ again indicates whether or not we have stacked with $K$. There are $2N$ lines of this type. Below, we will find that the fusion rules of these genuine lines are almost identical to the fusion rules of the twist defects in (<ref>), (<ref>), and (<ref>), from which we can tell that the quantum dimensions are $\sqrt{N}$. The only new ingredient in determining the fusion rules is determining how to assign factors of $q$.

To summarize the discussion so far, the theory with $\ZZ_2^{\mathrm{EM }}$ gauged has the following simple objects:

* $2N$ invertible lines $\widehat L_{(e)}^\pm$ ,

* $ {1\over 2} N(N-1)$ lines $\widehat L_{[e,m]}$ of quantum dimension 2 ,

* $2N$ lines $\widehat{\Sigma}_{(e)}^\pm$ of quantum dimension $\sqrt{N}$  .

We now obtain the fusion rules for these objects.

§.§.§ Fusion rules involving only $\widehat{L}_{(e)}^{\pm}$ and $\widehat{L}_{[e,m]}$

The fusion rule involving only the lines $\widehat{L}_{(e)}^{\pm}$ and $\widehat{L}_{[e,m]}$ largely follow from those in the pre-gauged theory. The only new ingredient is to determine the  distribution of $K$ lines (i.e. the value of $q$'s).  To determine this, we need to determine the $\Z_2^{\text{EM}}$ charge localized at junctions in the pre-gauged theory; if the junction is $\Z_2^{\text{EM}}$ even, then there should an even number of $K$ lines anchored on the junction after gauging, whereas if the junction is $\Z_2^{\text{EM}}$ odd there should be an odd number of $K$ lines anchored on the junction. The junction charges of the pre-gauged theory can be measured by wrapping $D_{\text{EM}}$ surfaces around them, as discussed in Appendix <ref>.  Using the tools developed there, one finds that e.g. the junction between three operators ${L}_{(e,e)}, {L}_{(e',e')}$ and ${L}_{(e+e', e+e')}$ is $\ZZ_2^\mathrm{EM}$ even, and hence the number of $K$ lines is conserved under fusion, i.e. $\widehat{L}_{(e)}^{q}\otimes \widehat{L}_{(e')}^{q'}= \widehat{L}_{(e+e')}^{qq'}$.

Another interesting case is the fusion between $\widehat{L}_{[e,m]}\otimes \widehat{L}_{[e',m']}$ when $e+e'=m+m', m+e'=e+m'$. Before gauging $\Z_2^{\text{EM}}$, the right hand side is a direct sum of two lines $L_{(e+e',e+e')}$, $L_{(m+e', m+e')}$. Each of them are $\Z_2^{\text{EM}}$ invariant separately, and hence we must determine the assignment of $q$ after gauging. One could again determine this via computation of the junction charge, but a nice trick to circumvent this computation is to observe that since $\widehat{L}_{[e,m]}\otimes \widehat{L}_{[e',m']}$ can absorb a $K$ line, the right-hand side of the fusion rules must also be able to absorb a $K$ line. The only possibility is then to sum over all possible $q$ values, i.e. $\widehat{L}_{[e,m]}\otimes \widehat{L}_{[e',m']}= \widehat{L}_{(e+e')}^{+}\oplus \widehat{L}_{(e+e')}^{-}\oplus \widehat{L}_{(m+e')}^{+}\oplus \widehat{L}_{(m+e')}^{-}$. See [27] for more systematic discussions. In summary, we are able to determine the fusion rules as follows,

\begin{equation}
\label{eq:3daftergaugingLfusion}
\begin{split}
\widehat{L}_{(e)}^{q}\otimes \widehat{L}_{(e')}^{q'}&= \widehat{L}_{(e+e')}^{qq'}~,\\
\widehat{L}_{(e)}^{q}\otimes \widehat{L}_{[e',m']}&= \widehat{L}_{[e+e',e+m']}~,\\
\widehat{L}_{[e,m]}\otimes \widehat{L}_{[e',m']}&=
\begin{cases}
\widehat{L}_{(e+e')}^{+}\oplus \widehat{L}_{(e+e')}^{-}\oplus \widehat{L}_{(m+e')}^{+}\oplus \widehat{L}_{(m+e')}^{-}&
\widehat{L}_{(e+e')}^{+}\oplus \widehat{L}_{(e+e')}^{-}\oplus \widehat{L}_{[m+e',m+e']}&
e+e'=m+m',m+e'\neq e+m'\\
\widehat{L}_{[e+e',m+m']} \oplus
\widehat{L}_{(m+e')}^{+}\oplus \widehat{L}_{(m+e')}^{-}&
e+e'\neq m+m',m+e'= e+m'\\
\widehat{L}_{[e+e',m+m']} \oplus
\widehat{L}_{[m+e',m+e']}&
e+e'\neq m+m',m+e'\neq  e+m'\\
\end{cases}
\end{split}
\end{equation}

Note that since all of the operators above are lines, the local fusion is identical to the global fusion.

§.§.§ Fusion rules involving $\widehat{\Sigma}_{(e)}^{q}$

We now describe the fusion rules involving the $\widehat{\Sigma}_{(e)}^{q}$ lines of quantum dimension $\sqrt{N}$.

Fusion rule $\widehat{L}_{(e)}^{q}\otimes \widehat{\Sigma}_{(e')}^{q'}$:
We start by considering the fusion between the invertible line $\widehat{L}_{(e)}^{q}$ and the non-invertible line $\widehat{\Sigma}_{(e')}^{q'}$. The cases of $N$ odd and even are qualitatively different, and we begin by analyzing the former.

For odd $N$, it is useful to observe that before gauging, none of the twist defects map to themselves under fusing with a non-trivial $\Z_2^{\text{EM}}$ invariant invertible line,

L_(e,e)⊗Σ_(e') =  Σ_(2e+e') ,
where $e' \neq e'+2e$ for $e\neq 0\mod N$.  This implies that after gauging $\Z_2^{\text{EM}}$ one can define the twist defect $\widehat{\Sigma}_{(e)}^{q}$ by fusing $\widehat{L}_{(\widetilde{e})}^{q}$ with the minimal twist defect $\widehat{\Sigma}_{(0)}^{+}$,

\begin{eqnarray}\label{eq:LSigmaaftergauging}
\widehat{\Sigma}_{(e)}^{q}= \widehat{L}_{(\widetilde{e})}^{q}\otimes \widehat{\Sigma}_{(0)}^{+}~, \hspace{1cm} e=0, ..., N-1, \hspace{1cm} q=\pm
\end{eqnarray}

where $\widetilde{e}= e/2$ for even $e$, and $\widetilde{e}= (e+N)/2$ for odd $e$. Note that since the operators are always subject to relabeling, such a definition is always allowed. The fusion rule then follows from (<ref>),

\begin{eqnarray}\label{eq:LSfusionodd}
\widehat{L}^q_{(e)} \otimes \widehat{\Sigma}^{q'}_{(e')}= \widehat{L}^{qq'}_{(e+\widetilde{e}')} \otimes \widehat{\Sigma}^{+}_{(0)}= \widehat{\Sigma}^{qq'}_{(2e+e')}~, \hspace{1cm} N\,\, \text{ odd}~.
\end{eqnarray}

We now proceed to the case of $N$ even. The main conceptual difference is that we can no longer obtain all  defects $\widehat \Sigma^q_{(e)}$  from $\widehat \Sigma^0_{(0)}$ by fusing with an appropriate choice of $\widehat L_{(e')}^q$. Before gauging $\Z_2^{\text{EM}}$, every twist defect maps to itself upon fusing with $L_{(N/2, N/2)}$. This means that after gauging one can define the twist defects as

\begin{eqnarray}
\widehat \Sigma_{(2e)}^{q}=\widehat L_{(e)}^q \otimes \widehat \Sigma_{(0)}^{+} ~, \hspace{0.5 in} \widehat\Sigma_{(2e+1)}^{q}= \widehat L_{(e)}^q \otimes \widehat \Sigma_{(1)}^{+} ~, \hspace{1cm} e=0, 1, ..., N/2-1~.
\end{eqnarray}

Once this definition is fixed though, we are no longer free to choose the sign that appears in the fusion of $\widehat L_{(N/2)}^q$ with $\widehat \Sigma_{(0)}^{+}$ and $\widehat \Sigma_{(1)}^{+}$. Indeed, because before gauging the twist defects are stabilized under this fusion, the $q', q''$ in the following fusion rules

\begin{eqnarray}\label{eq:q'q''}
\widehat \Sigma_{(0)}^{q'}=\widehat L_{(N/2)}^q \otimes \widehat \Sigma_{(0)}^{+} ~, \hspace{0.5 in} \widehat\Sigma_{(1)}^{q''}= \widehat L_{(N/2)}^q \otimes \widehat \Sigma_{(1)}^{+}
\end{eqnarray}

are unambiguously defined. In other words, it is not possible to relabel the twist defects to change $q' $ and $q''$ once $q$ is given. To determine $q'$ and $q''$,
one must measure the $\ZZ_2^{\mathrm{EM}}$ charge of the relevant junction in the theory before gauging EM. Relegating the details to Appendix <ref>, we find that the junction associated with the first fusion rule in (<ref>) is $\Z_2^{\text{EM}}$ even, and hence the number of $K$ lines is conserved. On the other hand, the junction associated with the second fusion rule in (<ref>) is $\Z_2^{\text{EM}}$ odd, and hence the number of $K$ lines jumps by one. In terms of the relation between $q,q'$ and $q''$, we have

\begin{eqnarray}
q'=q~, \hspace{1cm} q''=-q~.
\end{eqnarray}

As a consequence, fusing $\widehat{L}^q_{(e)}$ and $\widehat\Sigma^+_{(0,1)}$ for the remaining choices of $e$, i.e. $e=N/2+1, ..., N-1$, are given by

\begin{eqnarray}
\widehat \Sigma_{(2e)}^{q}=\widehat L_{(e)}^q \otimes \widehat \Sigma_{(0)}^{+} ~, \hspace{0.5 in} \widehat\Sigma_{(2e+1)}^{-q}= \widehat L_{(e)}^q \otimes \widehat \Sigma_{(1)}^{+} ~, \hspace{1cm} e=N/2+1, ..., N-1~.
\end{eqnarray}

The fusion between generic $\widehat{L}^q_{(e)}$ and $\widehat\Sigma_{(e')}^{q'}$ then follows straightforwardly.

To summarize the results for even $N$,
we have

\begin{equation}\label{eq:LSfusion}
\begin{split}
\widehat{L}_{(e)}^{q}\otimes \widehat{\Sigma}_{(e')}^{q'}=
\begin{cases}
\widehat{\Sigma}_{(2e+e')}^{qq'}~, &  \{e'\in 2\Z\} \cup \{ e'\in 2\Z+1,\,\, 0\leq [e+\frac{e'-1}{2}]_N\leq N/2-1\}\\
\widehat{\Sigma}_{(2e+e')}^{-qq'}~, &  \{ e'\in 2\Z+1,\,\, N/2\leq [e+\frac{e'-1}{2}]_N\leq N-1\}
\end{cases}
\end{split}
\end{equation}

where $[x]_N$ is the mod $N$ reduction of $x$.

Fusion rule $\widehat{L}_{[e,m]}\otimes \widehat{\Sigma}_{(e)}^{q}$:
Since $\widehat{L}_{[e,m]}$ before gauging $\Z_2^{\text{EM}}$ consists of two lines exchanged under $\Z_2^{\text{EM}}$, after gauging $\Z_2^{\text{EM}}$ the fusion rules $\widehat{L}_{[e,m]}\otimes \widehat{\Sigma}_{(e)}^{q}$ contains two fusion channels which differ by a $\Z_2^{\text{EM}}$ representation. This leads to the fusion rule

\begin{eqnarray}\label{eq:hLSfusion}
\begin{split}
\widehat{L}_{[e,m]}\otimes \widehat{\Sigma}_{(e')}^{q'} = \widehat{\Sigma}_{(e+m+e')}^{+} \oplus \widehat{\Sigma}_{(e+m+e')}^{-}~, \hspace{1cm} q'=\pm~.
\end{split}
\end{eqnarray}

Another way to argue for the sum of $\widehat{\Sigma}_{(e+m+e')}^{\pm}$ on the right-hand side is to notice that since $\widehat L_{[e,m]}$ can absorb a factor of $K$, the right-hand side must also be able to absorb a factor of $K$. This can only be achieved by summing over the defects with and without $K$ stacked.

Fusion rule $\widehat{\Sigma}_{(e)}^{q}\otimes \widehat{\Sigma}_{(e')}^{q'}$:
The fusion rule $\widehat{\Sigma}_{(e)}^{q}\otimes \widehat{\Sigma}_{(e')}^{q'}$ can be obtained from the fusion rules derived in (<ref>), (<ref>), and (<ref>). When $N$ is odd, the fusion rules are

\begin{eqnarray}\label{eq:SSfusionNodd}
\begin{split}
\widehat{\Sigma}_{(e)}^{q} \otimes \widehat{\Sigma}_{(e')}^{q'} &= \widehat{L}_{(\tilde{e})}^{q} \otimes \widehat{\Sigma}_{(0)}^{+} \otimes \widehat{L}_{(\tilde{e}')}^{q'} \otimes \widehat{\Sigma}_{(0)}^{+}\\
&= \widehat{L}_{(\tilde{e}+ \tilde{e}')}^{qq'} \otimes \left( \widehat{L}_{(0)}^{+} \oplus  \bigoplus_{n=1}^{(N-1)/2} \widehat{L}_{[n,N-n]}\right)\\
&= \widehat{L}_{(\tilde{e}+\tilde{e}')}^{qq'} \oplus \bigoplus_{n=1}^{(N-1)/2} \widehat{L}_{[\tilde{e}+\tilde{e}'+n,\tilde{e}+\tilde{e}'-n]}~,
\end{split}
\end{eqnarray}