## 1

### 1.1

#### 1.1.1

Pavel Putrov 1 and Rajath Radhakrishnan

11institutetext: International Centre for Theoretical Physics,
Strada Costiera 11, Trieste 34151, Italy

#### Non-anomalous non-invertible symmetries in 1+1D from gapped boundaries of
SymTFTs

Pavel Putrov 1 and Rajath Radhakrishnan

###### Abstract

We study the anomalies of non-invertible symmetries in 1+1D QFTs using gapped
boundaries of its SymTFT. We establish the explicit relation between
Lagrangian algebras which determine gapped boundaries of the SymTFT, and
algebras which determine non-anomalous/gaugeable topological line operators in
the 1+1D QFT. If the Lagrangian algebras in the SymTFT are known, this
provides a method to compute algebras in all fusion categories that share the
same SymTFT. We find necessary conditions that a line operator in the SymTFT
must satisfy for the corresponding line operator in the 1+1D QFT to be non-
anomalous. We use this constraint to show that a non-invertible symmetry
admits a 1+1D trivially gapped phase if and only if the SymTFT admits a
magnetic Lagrangian algebra. We define a process of transporting non-anomalous
line operators between fusion categories which share the same SymTFT and apply
this method to the three Haagerup fusion categories.

##  2 Introduction

A quantum field theory contains a rich collection of extended operators of
various dimensions. Among them, topological operators are a particularly rich,
and yet relatively simple set of operators which can be often analysed
explicitly. These operators play a very important role in constraining the
dynamics of a QFT as they implement generalized symmetries Gaiotto:2014kfa .
The topological nature of these operators puts strong constraints on them to
the extent that a “topological bootstrap” approach can be used to classify
them. This reveals that such operators are captured by an algebraic structure
called a higher-fusion category Kapustin:2010ta ; Bhardwaj:2022yxj . In a
Topological Quantum Field Theory, which is a QFT in which all operators are
topological, the topological operators have to obey additional constraints
like modularity. This has led to the (partial) classification of TQFTs in
various dimensions Lan:2018vjb ; Lan:2018bui ; Johnson-Freyd:2020usu ;
Kong:2020jne ; Johnson-Freyd:2020twl ; Johnson-Freyd:2021tbq ; Ng:2023wsc .

While the topological operators that can appear in a $d$ dimensional QFT are
not as constrained as those in a TQFT, it is remarkable that their properties
can be captured by a $d+1$ dimensional TQFT called the SymTFT Freed:2012bs ;
Gaiotto:2020iye ; Kong:2020cie ; Apruzzi:2021nmk ; Freed:2022qnc ;
Chatterjee:2022kxb ; Kaidi:2022cpf ; Kaidi:2023maf . The SymTFT allows us to
study topological properties of a QFT with symmetry $\mathcal{C}$ and other
QFTs obtained from gauging “sub-symmetries” of $\mathcal{C}$ in a unified
setting.

Given a 1+1D QFT with a finite invertible symmetry $G$ and anomaly given by
$\omega$ in $H^{3}(G,U(1))$, a natural question to consider is whether $G$
contains non-anomalous subgroups. A subgroup $H\subset G$ is non-anomalous iff
the anomaly $\omega$ trivializes on the subgroup. More precisely, $H$ is non-
anomalous iff $\omega|_{H}$ is trivial in $H^{3}(H,U(1))$. An alternate
approach to this problem is to use the SymTFT of the symmetry $G$ with anomaly
$\omega$. This is the twisted Dijkgraaf-Witten theory DW$(G,\omega)$
determined by the gauge group $G$ and 3-cocycle $\omega$ Dijkgraaf:1989pz ;
Gaiotto:2020iye . Non-anomalous subgroups of $G$ are in correspondence with
gapped boundaries of this SymTFT. In 1+1D, gauging symmetries can produce dual
QFTs. For example, gauging non-anomalous subgroups $K,H\subseteq G$ such that
$K$ and $H$ are conjugate to each other are physically equivalent. This
defines an equivalence relation on the gaugeable subgroups of $G$. It is then
natural to ask:

Given a gapped boundary of the 2+1D SymTFT DW$(G,\omega)$ how do we determine
the equivalence class of physically equivalent gaugings in 1+1D?

More generally, if a 1+1D QFT has symmetries described by a fusion category
$\mathcal{C}$, then the appropriate generalization of “non-anomalous subgroup”
is a line operator $A$ (generically non-simple) which admits the structure of
an algebra Frohlich:2009gb ; Bhardwaj:2017xup . This constraint is an
algebraic generalization of the anomalies for invertible symmetries. For a
1+1D QFT on a spacetime 2-manfiold $\Sigma$ gauging a line operator $A$
involves constructing a mesh of $A$ lines on $\Sigma$. Since this mesh
involves trivalent junctions, we have to make a choice of a point operator at
a trivalent junction of $A$ lines (see Fig. 1).

$A$$A$$A$$\mu$ Figure 1: Gauging $A$ involves choosing a point operator $\mu$
at a trivalent junction of $A$ lines.

We want to choose $\mu$ such that the resulting theory is independent of the
choice of the mesh. This is guaranteed if the conditions in Fig. 2 are
satisfied.

$A$$A$$A$$A$$A$$A$$A$$A$$A$$A$$A$$\longleftrightarrow$$\longleftrightarrow$
Figure 2: Fig: Conditions on $\mu$ for the gauging to be consistent.

If there is at least one $\mu$ which satisfies these constraints, we can
regard $A$ as a non-anomalous line operator as it can be consistently gauged.
If there are multiple $\mu$ which satisfy these constraints, then the choice
of $\mu$ can be thought of as a choice of “generalized discrete torsion” on
$A$. In this case the object $A$ admits an algebra structure provided by
$\mu:A\times A\rightarrow A$. Once again, gauging two algebras $A_{1}$ and
$A_{2}$ might be physically equivalent. In fact, in some cases gauging $A$
might be a self-duality of the QFT. For example, in a 1+1D QFT with symmetries
described by the Ising fusion category, gauging $1+\psi$, where $\psi$ is the
non-trivial order two line is a self-duality Frohlich:2006ch ; Chang:2018iay .
While gauging $1+\psi$ produces the same 1+1 QFT, it is not a trivial
operation. Indeed, gauging $1+\psi$ in the Ising CFT is the famous Krammers-
Wannier duality which non-trivially exchanges local operators with twisted-
sector operators PhysRev.60.252 ; Frohlich:2006ch ; Chang:2018iay . Therefore,
it is important to keep track of physically equivalent gaugings. In this
general setting, it is natural to ask:

Given a gapped boundary $\mathcal{B}$ of the SymTFT,
$\mathcal{Z}(\mathcal{C})$, of a fusion category $\mathcal{C}$ how do we
determine the corresponding equivalence class of physically equivalent
gaugings in $\mathcal{C}$?

In this paper, we answer this question by studying the bulk-to-boundary map
which tells us how to map line operators in $\mathcal{Z}(\mathcal{C})$ to line
operators in $\mathcal{C}$, $F:\mathcal{Z}(\mathcal{C})\to\mathcal{C}$.
Physically, $F(x)$ for some line operator $x\in\mathcal{C}$ is the result of
perpendicular fusion of $x$ on the gapped boundary $\mathcal{B}_{\mathcal{C}}$
of $\mathcal{Z}(\mathcal{C})$ corresponding to the fusion category
$\mathcal{C}$ (see Fig. 3). We also study a boundary-to-bulk map
$K:\mathcal{C}\to\mathcal{Z}(\mathcal{C})$ which can be used to determine the
gapped boundary of $\mathcal{Z}(\mathcal{C})$ corresponding to a non-anomalous
line operator in $\mathcal{C}$.

$\mathcal{B}_{\mathcal{C}}$$x$$F(x)$$\mathcal{Z}(\mathcal{C})$ Figure 3: The
bulk-to-boundary map $F$ determines the outcome of fusing a bulk line operator
on the boundary $\mathcal{B}_{\mathcal{C}}$.

A gapped boundary of $\mathcal{Z}(\mathcal{C})$ is almost completely
determined by the line operators which can end on it. In fact, the lines which
can end on a gapped boundary form a Lagrangian algebra $L$ Kapustin:2010hk ;
kitaev2012models ; kong2014anyon . Conversely, the Lagrangian algebra $L$
completely determines the gapped boundary. Ending the full Lagrangian algebra
object $L$ on the gapped boundary $\mathcal{B}_{\mathcal{C}}$, produces
certain line operator $F(L)\in\mathcal{C}$. We show that

  * •

$F(L)$ is a non-anomalous line operator in $\mathcal{C}$. That is, it can be
gauged. (Theorem 5.1)

  * •

The equivalence class of non-anomalous line operators whose gauging is
physically equivalent is contained in $F(L)$. (Theorem 5.1)

  * •

The multiplication on an algebra $A$ in $\mathcal{C}$ can be determined from
the multiplication on the corresponding Lagrangian algebra $L$ through an
explicit formula. (Equation (32))

This has various applications:

  * •

If all Lagrangian algebras of a SymTFT are known, then the bulk-to-boundary
map $F$ determines all algebras in the fusion category describing the line
operators on the chosen boundary. This is an alternative to using NIM-reps to
classify algebra objects in a fusion category grossman2012quantum ;
grossman2016brauer ; Komargodski:2020mxz ; Diatlyk:2023fwf .

  * •

The map between Lagrangian algebras in the SymTFT $\mathcal{Z}(\mathcal{C})$
and non-anomalous line operators in $\mathcal{C}$ can be used to determine
necessary conditions for a line operator in $\mathcal{C}$ to be non-anomalous.
We use this condition to show that $\mathcal{C}$-symmetric trivially gapped
phases exist if and only if $\mathcal{Z}(\mathcal{C})$ contains a magnetic
Lagrangian algebra with respect to $\mathcal{B}_{\mathcal{C}}$. This relation
was first established in Zhang:2023wlu  through interval compactification of
the SymTFT.

  * •

We show that in many cases, the explicit structure of $F(L)$ also allows us to
classify line operators in $\mathcal{Z}(\mathcal{C})$ admitting the structure
of a Lagrangian algebra in terms of algebras in $\mathcal{C}$. We use this to
define the notion of transporting non-anomalous line operators from one fusion
category to another which share the same SymTFT.

  * •

When $\mathcal{C}$ is a modular tensor category, both a Lagrangian algebra $L$
in the SymTFT $\mathcal{Z}(\mathcal{C})$ and $F(L)$ have a purely bulk
interpretation. In this case, $L$ determines the action of a topological
surface operator, say $S_{L}$, on the line operators of $\mathcal{C}$. Then
$F(L)$ is the equivalence class of algebras which can be higher-gauged to
construct the surface $S_{L}$.

The plan of the paper is as follows. In section 3 we will briefly review non-
invertible symmetries in 1+1D and their gauging. In section 4 we will
introduce 2+1D SymTFTs $\mathcal{Z}(\mathcal{C})$ and describe their gapped
boundaries in terms of Lagrangian algebras. We will review
$\mathcal{C}$-symmetric TQFTs and explain the relation between 1D gapped
boundaries of a $\mathcal{C}$-symmetric TQFT and gapped interfaces between
gapped boundaries of $\mathcal{Z}(\mathcal{C})$. In section 5 we will explain
the explicit map between non-anomalous line operators in $\mathcal{C}$ and
gapped-boundaries $\mathcal{Z}(\mathcal{C})$. We begin by proving Theorem 5.1.
Then we provide an explicit formula relating the multiplication on a
Lagrangian algebra of $\mathcal{Z}(\mathcal{C})$ and the generalized discrete
torsion of the corresponding non-anomalous line operator in $\mathcal{C}$. We
then study a boundary-to-bulk map $K$ and give a physical picture of how it
can be used to determine the Lagrangian algebra corresponding to an algebra in
the boundary. Finally, we will discuss the special case of invertible
symmetries in detail and prove Theorem 5.2. In section 6 we discuss various
examples involving invertible and non-invertible symmetries. We end this
section with an example illustrating how Theorem 5.1 can be used to find
Lagrangian algebra objects in $\mathcal{Z}(\mathcal{C})$ using non-anomalous
line operators in $\mathcal{C}$. Finally, in section 7 we discuss various
applications of our results. We introduce the notion of “transporting algebra
objects” between fusion categories and illustrate it using the Haagerup fusion
categories. We conclude with some interesting open problems.

##  3 Review: Non-invertible symmetries in 1+1D and their gauging

In this section, we will briefly review non-invertible symmetries of 1+1D QFTs
and their gauging. We refer the reader to the excellent exposition of this
topic in the references (Huang:2021zvu, , Section 5), Choi:2023xjw  and
Diatlyk:2023fwf  for more details.

Symmetries of a 1+1D QFT $\mathcal{Q}$ are implemented by topological line
operators111We assume that $\mathcal{Q}$ has a unique vacuum. In this case,
there are no non-trivial topological local operators.. The structure of a
finite set of line operators is described by a fusion category
$\mathcal{C}$.222See the reviews Schafer-Nameki:2023jdn ; Brennan:2023mmt ;
Shao:2023gho . The simple objects in the category $a,b,c,\dots$ label simple
line operators of the QFT.333A line operator is simple if it cannot be
decomposed into a sum of other line operators. Equivalently, the only point
operator that a simple line operator hosts on it is the identity point
operator and its complex multiples. Their fusion rules are captured by the
non-negative integers $N_{ab}^{c}$ as follows.

$a\times b=\sum_{c}N_{ab}^{c}~{}c~{}.$ (1)

The fusion of four line operators obeys the Pentagon equations whose solutions
are given in terms of the $F$ matrices.

A line operator $A$ in $\mathcal{C}$ is non-anomalous if it can be gauged to
produce a new QFT $\mathcal{Q}/A$. For this gauging to be consistent, the line
operator $A$ must admit the structure of a symmetric separable Frobenius
algebra Frohlich:2009gb ; Bhardwaj:2017xup . To explain this structure,
suppose $A$ admits the decomposition

$A=\sum_{a}N_{A}^{a}~{}a~{},$ (2)

into simple line operators $a\in\mathcal{C}$, where $N_{A}^{a}$ are non-
negative integers. The object $A$ admits the structure of an associative
algebra if there exist complex numbers

$\mu_{(a,\alpha),(b,\beta)}^{(c,\gamma),\delta}~{},$ (3)

where $\alpha,\beta,\gamma,\delta$ denote indices running from
$1,\dots,N_{A}^{a}~{}$;$~{}1,\dots,N_{A}^{b}~{}$;$~{}1,\dots,N_{A}^{c}$ and
$1,\dots,N_{ab}^{c}$, respectively. These complex numbers should satisfy the
following constraint involving the $F$-symbol:

$\mu_{ab}^{c}\mu_{de}^{f}=\mu_{ag}^{f}\mu_{be}^{g}F_{abe}^{f}~{}.$ (4)

In writing the above constraint, we have assumed that both the fusion
coefficients and the algebra $A$ are multiplicity-free. If there is
multiplicity, extra indices need to be added. We will assume that
$N_{A}^{\mathbf{1}}=1$, where $\mathbf{1}$ is the trivial line operator. In
this case, $A$ is called a haploid algebra. A haploid algebra admits a unique
symmetric separable algebra structure (see (Choi:2023xjw, , Footnote 20) and
references therein). Therefore, we will not discuss these extra conditions in
this review.

Note that a line operator $A$ might admit distinct algebra structures. In
other words, there maybe more than one inequivalent solution to the
constraints (4). We will call this a choice of generalized discrete torsion
for gauging $A$. This is a generalization of the fact that when a non-
anomalous line operator $A$ is of the form

$A=\sum_{g\in G}g~{},$ (5)

for some group $G$, then the distinct multiplications on $A$ are classified by
the discrete torsion $H^{2}(G,U(1))$.

Two distinct algebras $A_{1}$ and $A_{2}$ might correspond to physically
equivalent gauging procedures leading to dual QFTs $\mathcal{Q}/A_{1}$ and
$\mathcal{Q}/A_{2}$. This is captured by Morita equivalence of algebras.444See
Diatlyk:2023fwf  for more details. We will denote a Morita equivalence class
with a representative algebra $A$ as $[A]$. Note that every Morita equivalent
class has a haploid representative. Therefore, the condition
$N_{A}^{\mathbf{1}}=1$ can be imposed without loss of generality. In the
following sections, we will use the terms “non-anomalous line operator” and
“algebra object” interchangeably.

##  4 SymTFTs and $\mathcal{C}$-symmetric TQFTs

###  4.1 SymTFTs

Let $\mathcal{C}$ be a finite subcategory of symmetries of a 1+1D QFT
$\mathcal{Q}$. The SymTFT of $\mathcal{C}$ is the Turaev-Viro-Barrett-Westbury
2+1D TQFT described by the Drinfeld centre $\mathcal{Z}(\mathcal{C})$ of
$\mathcal{C}$ Lan:2018vjb ; Lan:2018bui ; Johnson-Freyd:2020usu ; Kong:2020jne
; Johnson-Freyd:2020twl ; Johnson-Freyd:2021tbq ; Ng:2023wsc ; Turaev:1992hq ;
Barrett:1993ab . The simple line operators in $\mathcal{Z}(\mathcal{C})$ can
be written in the form

$(a,e_{a})~{},$ (6)

where $a$ is a (generically non-simple) line operator in $\mathcal{C}$ and
$e_{a}$ are isomorphisms called half-braidings

$e_{a}(b):a\times b\xrightarrow{\sim}b\times
a~{},\;\forall~{}b\in\mathcal{C}~{},$ (7)

satisfying several consistency conditions (see, for example,
muger2003subfactors ; etingof2016tensor ). The SymTFT allows us to separate
the data of the symmetry $\mathcal{C}$ from the 1+1D QFT $\mathcal{Q}$ on
which it is acting (see Fig. 4).

$\mathcal{B}_{\mathcal{Q}}$$\mathcal{B}_{\mathcal{C}}$$\mathcal{Z}(\mathcal{C})$
Figure 4: The SymTFT $\mathcal{Z}(\mathcal{C})$ with two boundary conditions.
The boundary $\mathcal{B}_{\mathcal{C}}$ is gapped while the
$\mathcal{B}_{\mathcal{Q}}$ is generically gappless. The category of line
operators on $\mathcal{B}_{\mathcal{C}}$ is $\mathcal{C}$.

Given the full data of the fusion category $\mathcal{C}$, the line operators
and modular data of $\mathcal{Z}(\mathcal{C})$ can be computed using the
string-net model Levin:2004mi ; Lin:2020bak . We refer the readers to the
review in (Cordova:2023bja, , Appendix A) for more details.

####  4.1.1 Lagrangian algebras and gapped boundaries

A gapped boundary of the SymTFT $\mathcal{Z}(\mathcal{C})$ is completely
determined by a Lagrangian algebra $L$ Kapustin:2010hk ; kitaev2012models ;
kong2014anyon . $L$ is a line operator in $\mathcal{Z}(\mathcal{C})$ and can
be written as

$L=\sum_{x}N_{L}^{x}~{}x~{},$ (8)

where $x$ are simple line operators and $N_{L}^{x}$ are non-negative integers.
If $N_{L}^{x}\neq 0$, then $x$ can end on the gapped boundary
$\mathcal{B}_{L}$ (see Fig. 5).

$\mathcal{B}_{L}$$\mathcal{Z}(\mathcal{C})$$x\in L$$\psi$ Figure 5: A simple
line operator $x\in L$ can end on the gapped boundary $\mathcal{B}_{L}$.

We will focus on simple gapped boundaries on which the only non-trivial point
operator is the identity operator and its complex multiples.555A general
gapped boundary can be written as a sum of simple gapped boundaries.
Therefore, the trivial line operator in the bulk should end on the gapped
boundary in a unique way. In other words, we require that $L$ has exactly one
copy of the trivial line, $N_{L}^{\mathbf{1}}=1$, where $\mathbf{1}$ is the
trivial line operator. The line operator $L$ admits the structure of a
commutative and associative algebra. This algebra structure is determined by
the following complex numbers

$m_{(x,i),(y,j)}^{(z,k),l}$ (9)

where $i,j,k,l$ denote indices running from
$1,\dots,N_{L}^{x}~{}$;$~{}1,\dots,N_{L}^{y}~{}$;$~{}1,\dots,N_{L}^{z}$ and
$1,\dots,N_{xy}^{z}$, respectively. When the SymTFT $\mathcal{Z}(\mathcal{C})$
has fusion coefficients valued in $N_{xy}^{z}\in\\{0,1\\}$, we will denote the
multiplication coefficients in the algebra $L$ as $m_{(x,i),(y,j)}^{(z,k)}$.
Moreover, if $N_{L}^{x}\in\\{0,1\\}$, we will denote it as $m_{xy}^{z}$. Since
$L$ is an associative algebra. The complex numbers $m_{xy}^{z}$ must satisfy
the constraint (4). Also, for $L$ to be a commutative algebra, these complex
numbers must satisfy the following constraint involving the $R$-symbol
(braiding):

$m_{xy}^{z}=m_{yx}^{z}R_{xy}^{z}~{}.$ (10)

In writing this equation, we have assumed that $\mathcal{C}$ and $A$ are
multiplicity-free. Moreover, for $L$ to be a Lagrangian algebra, the non-
negative integers $N_{L}^{a}$ must satisfy

$N_{L}^{x}~{}N_{L}^{y}\leq\sum_{z}N_{xy}^{z}~{}N_{L}^{z},~{}~{}\text{and}~{}~{}\text{dim}(L)=\sum_{x}N_{L}^{x}d_{x}=\sqrt{\text{dim}(\mathcal{Z}(\mathcal{C}))}=\text{dim}(\mathcal{C})~{}.$
(11)

A line operator $L$ with multiplication specified by the set of complex
numbers $m_{(x,i),(y,j)}^{(z,k),l}$ is a Lagrangian algebra if and only if the
constraints (4), (10) and (11) are satisfied (Cong:2017ffh, , Corollary
3.8).666Strictly speaking, this result requires unitarity. We will assume that
the fusion category is unitary throughout this paper. Note that the conditions
(11) depend only on the fusion rules of $\mathcal{Z}(\mathcal{C})$ and a
choice of the line operator $L$. However, they are not a set of sufficient
conditions for $L$ to be Lagrangian davydov2016unphysical . The constraints
(11) can be used to enumerate a set of candidate Lagrangian algebras for which
we can try to solve (4), (11). However, this requires the knowledge of the
full $F$ and $R$ matrices of the SymTFT $\mathcal{Z}(\mathcal{C})$. In
sections 6.4 and 7, as a consequence of our main result, we will consider an
alternate method to determine line operators in the bulk SymTFT which are
guaranteed to admit a Lagrangian algebra structure using non-anomalous line
operators on a gapped boundary.

###  4.2 $\mathcal{C}$-symmetric TQFTs

In this section, we will briefly review the classification of $1+1$D
$\mathcal{C}$-symmetric TQFTs in terms of module categories over
$\mathcal{C}$. The topological local operators in a 1+1D TQFT form an algebra,
$E$, where the multiplication on $E$ captures the O.P.E of the local
operators. Moreover, the 2-point function of the local operators defines a
commutative Frobenius algebra structure on $E$ Moore:2006dw . Arbitrary
correlation functions of local operators on any closed 2-manifold can be
determined using $E$. The topological line operators in the TQFT are labeled
by bimodules of the Frobenius algebra, BiMod$(E)$ (for example, see
(Carqueville:2016nqk, , Section 2.4.1)).777BiMod$(E)$ is a multifusion
category etingof2005fusion  which captures the structure of both line
operators and local operators of the TQFT

A $\mathcal{C}$-symmetric TQFT is determined by a functor

$\mathcal{C}\to\text{BiMod}(E)~{}.$ (12)

which assigns line operators in $\mathcal{C}$ to line operators in BiMod$(E)$
in a consistent way. In other words, we must be able to identify line
operators in $\text{BiMod}(E)$ which form the chosen fusion category
$\mathcal{C}$. Let $\mathcal{M}$ be the category of boundary conditions of a
1+1D TQFT with simple objects
$\\{\operatorname{\text{m}}_{1},\dots\operatorname{\text{m}}_{n}\\}\in\mathcal{M}$
labeling the distinct boundary conditions. Then there is an action of the
fusion category $\mathcal{C}$ on $\mathcal{M}$ given by

$c\times\operatorname{\text{m}}_{i}=\sum_{j}N_{ci}^{j}~{}\operatorname{\text{m}}_{j}~{}.$
(13)

where $N_{ci}^{j}$ are non-negative integers. If $N_{ci}^{j}\neq 0$, we get
Fig. 6 where $\alpha$ labels a basis for the local operators at the junction
of $c,\operatorname{\text{m}}_{i}$ and $\operatorname{\text{m}}_{j}$.

$c$$\operatorname{\text{m}}_{i}$$\operatorname{\text{m}}_{j}$$\alpha$ Figure
6: Action of $\mathcal{C}$ on 1D gapped boundaries.

It is known that the full defining data of a 1+1D $\mathcal{C}$-symmetric TQFT
can be obtained from how $\mathcal{C}$ acts on the set of boundary conditions
Moore:2006dw ; Thorngren:2019iar ; Komargodski:2020mxz ; Huang:2021zvu . In
fact, $\mathcal{M}$ is a module category over $\mathcal{C}$. All physical data
in the $\mathcal{C}$-symmetric TQFT can be computed using the data of this
module category.

$\mathcal{C}\text{-module category }\leftrightarrow\text{1+1D
}\mathcal{C}\text{-symmetric TQFT}~{}.$ (14)

A $\mathcal{C}$-module category is in turn completely determined by an algebra
$A$ in $\mathcal{C}$ ostrik2003module  (see also Choi:2023xjw ;
Diatlyk:2023fwf ).

$\text{algebra $A$ in $\mathcal{C}$
}\longleftrightarrow\mathcal{C}\text{-module category
}\mathcal{M}\longleftrightarrow 1+1\text{D }\mathcal{C}\text{-symmetric
TQFT}~{}.$ (15)

In other words, every gaugeable line operator $A$ in $\mathcal{C}$ determines
a $\mathcal{C}$-module category which in turn determines a 1+1D
$\mathcal{C}$-symmetric TQFT. Given an algebra $A$, the module category
$\mathcal{M}$ is the category of modules of $A$ in $\mathcal{C}$. Note that
the module category only depends on the Morita equivalence class $[A]$ of the
algebra $A$. Conversely, given a module category $\mathcal{M}$, $[A]$ can be
determined as follows. Choose some $\operatorname{\text{m}}\in\mathcal{M}$,
then a line operator $a$ can form a junction on the gapped boundary
$\operatorname{\text{m}}$ if and only if $a\times\operatorname{\text{m}}$
contains $\operatorname{\text{m}}$. Let us define the non-simple line operator

$A_{\operatorname{\text{m}}}:=\sum_{a\in\mathcal{C}}N_{\operatorname{\text{m}}}^{a}~{}a~{},$
(16)

where $N_{\operatorname{\text{m}}}^{a}$ is the dimension of Hilbert space of
operators at the junction of $a$ with the gapped boundary
$\operatorname{\text{m}}$. The object $A_{\operatorname{\text{m}}}$ is called
the Internal Hom, and it admits the structure of a haploid symmetric separable
Frobenius algebra ostrik2003module . In other words,
$A_{\operatorname{\text{m}}}$ is non-anomalous and can be gauged. If we choose
a different boundary condition in $\mathcal{M}$, we get a Morita equivalent
haploid symmetric separable Frobenius algebra.

####  4.2.1 $\mathcal{C}$-symmetric TQFTs from SymTFTs

In this section, we will review the construction of 1+1D
$\mathcal{C}$-symmetric TQFTs from the SymTFT $\mathcal{Z}(\mathcal{C})$. Most
of this section is based on the papers Zhang:2023wlu ; Bhardwaj:2023idu ;
Bhardwaj:2023fca ; Bhardwaj:2023bbf , except for the last part where we
explain how to obtain the gapped boundaries of the 1+1D TQFT from the SymTFT.

Consider the sandwich picture in Fig. 7. Interval compactification of this
picture results in a 1+1D TQFT $\mathcal{T}_{L}$ with
$\mathcal{C}$-symmetry.888Similar procedures can be used to obtain TQFTs in
higher-dimensions Cordova:2023bja ; Antinucci:2023ezl . For example, interval
compactification of the Crane-Yetter TQFT with two fixed gapped boundaries
results in a 2+1D TQFT. This played a crucial role in proving that all Spin
TQFTs admit a modular extension johnson2024minimal . See also Nardoni:2024sos
. When $\mathcal{B}_{L}=\mathcal{B}_{\mathcal{C}}$, we get the regular
$\mathcal{C}$-symmetric TQFT defined in Huang:2021zvu  in which the symmetry
$\mathcal{C}$ is completely spontaneously broken. For other choices of
$\mathcal{B}_{L}$ we may get TQFTs where $\mathcal{C}$ is partially or fully
preserved.

$\mathcal{B}_{L}$$\mathcal{B}_{\mathcal{C}}$$\mathcal{Z}(\mathcal{C})$ Figure
7: SymTFT with two gapped boundaries $\mathcal{B}_{L}$ and
$\mathcal{B}_{\mathcal{C}}$ can be compactified to get a
$\mathcal{C}$-symmetric TQFT.

Various crucial properties of the TQFT can be deduced from this picture
Zhang:2023wlu ; Bhardwaj:2023idu ; Bhardwaj:2023fca . For example, the number
of local operators in the resulting TQFT can be counted by looking at the line
operators in $\mathcal{Z}(\mathcal{C})$ which can end on both
$\mathcal{B}_{L}$ and $\mathcal{B}_{\mathcal{C}}$. More precisely, the number
of local operators in the 1+1D TQFT is given by

$\sum_{x\in\mathcal{Z}(\mathcal{C})}N_{\mathcal{C}}^{x}~{}N^{x}_{L}~{}.$ (17)

where $N^{x}_{\mathcal{C}}$ counts the number of copies of $x$ in the
canonical Lagrangian algebra of $\mathcal{Z}(\mathcal{C})$ corresponding to
the gapped boundary $\mathcal{B}_{\mathcal{C}}$. Moreover, further details of
$\mathcal{T}_{L}$ including how the symmetry $\mathcal{C}$ is spontaneously
broken, the action of the symmetry on the vacua and order parameters can be
determined using this picture Bhardwaj:2023idu ; Bhardwaj:2023fca .

The gapped boundaries of $\mathcal{T}_{L}$ can also be determined from the
sandwich picture. To that end, we consider the configuration in Fig. 8 where
$r$ is a simple gapped interface between the gapped boundaries
$\mathcal{B}_{L}$ and $\mathcal{B}_{\mathcal{C}}$. Compactifying this diagram
gives $\mathcal{T}_{L}$ where the gapped interface $r$ becomes a 1D gapped
boundary $\operatorname{\text{m}}_{r}$ of $\mathcal{T}_{L}$.

$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{L}$$r$$\mathcal{T}_{L}$$\operatorname{\text{m}}_{r}$$\mathcal{Z}(\mathcal{C})$
Figure 8: A 1D gapped interface between the 2D gapped boundaries
$\mathcal{B}_{L}$ and $\mathcal{B}_{\mathcal{C}}$ turns into a 1D gapped
boundary of the 1+1D $\mathcal{C}$-symmetric TQFT $\mathcal{T}_{L}$.

In section 4.2, we described how the line operators in $\mathcal{C}$ act on
the 1D gapped boundaries of $\mathcal{C}$. This can be determined from the
SymTFT as follows. Consider a bulk line operator $x$ in the Lagrangian algebra
$L$. The line $x$ can end trivially on the gapped boundary $\mathcal{B}_{L}$.
Generically, $x$ cannot end on the gapped boundary $\mathcal{B}_{\mathcal{C}}$
but forms a junction with a line operator $a\in\mathcal{C}$. The action of $a$
on the gapped boundary $\operatorname{\text{m}}_{r}$ of the 1+1D TQFT
$\mathcal{T}_{L}$ is determined by the action of $a$ on the corresponding
gapped interface $r$ in the SymTFT as in Fig. 9.

$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{L}$$r$$\mathcal{T}_{L}$$\operatorname{\text{m}}_{r}$$\mathcal{Z}(\mathcal{C})$$x$$s$$\operatorname{\text{m}}_{s}$$a$$a$$\psi$$\chi$$\mathcal{O}_{(x,\psi,\chi)}$$\alpha$$\alpha$
Figure 9: The action of $a\in\mathcal{C}$ on the 1D gapped boundaries of
$\mathcal{T}_{L}$ can be determined by the action of $a\in\mathcal{C}$ on
gapped interfaces in the SymTFT. Upon collapsing the wedge on the left, the
line operator $x$ becomes a twisted-sector local operator
$\mathcal{O}_{(x,\psi,\chi)}.$

##  5 Gaugeable line operators in 1+1D and SymTFTs

Consider the symmetries of a 1+1D QFT described by a fusion category
$\mathcal{C}$. Given a line operator $A$ in the category $\mathcal{C}$, how do
we determine if it is anomalous? One approach is to determine whether $A$
admits the structure of an algebra. This involves solving the constraints (4)
which requires the $F$ matrices of the fusion category $\mathcal{C}$.

In this section, we will consider an alternate approach by studying gapped
boundaries of the SymTFT $\mathcal{Z}(\mathcal{C})$. An important defining
property of the SymTFT is that physically distinct gaugings in $\mathcal{C}$
are in one-to-one correspondence with gapped boundaries of
$\mathcal{Z}(\mathcal{C})$ (davydov2013witt, , Proposition 4.8).

$\text{$[A]\in\mathcal{C}$}\longleftrightarrow
L\in\mathcal{Z}(\mathcal{C})~{}.$ (18)

In the rest of this section, we will study this relation explicitly.

###  5.1 Non-anomalous line operators from SymTFT

In this section, we will explore the one-to-one correspondence between gapped
boundaries of $\mathcal{Z}(\mathcal{C})$ and non-anomalous line operators in
$\mathcal{C}$ from a physical perspective. We will describe various properties
of this correspondence which can be used to answer the following question:
Given a Lagrangian algebra $L$ of $\mathcal{Z}(\mathcal{C})$, which Morita
equivalence class of gaugeable algebra $[A]$ in $\mathcal{C}$ does it
correspond to? To answer this question, let us consider the perpendicular
fusion of a line operator $l$ on a gapped boundary $\mathcal{B}$

$F_{\mathcal{B}}(x)=\sum_{a}N_{F_{\mathcal{B}}(x)}^{a}~{}a~{},$ (19)

where the line operators $a$ live on the gapped boundary $\mathcal{B}$ and
$N_{l}^{a}$ are non-negative integers. In general, $F_{\mathcal{B}}(x)$ is
part of the data of a bulk-to-boundary map which relates the line operators
and fusion spaces of the bulk TQFT $\mathcal{Z}(\mathcal{C})$ to those of the
category of line operators on the gapped boundary $\mathcal{B}$. Consider the
canonical gapped boundary $\mathcal{B}_{\mathcal{C}}$ of
$\mathcal{Z}(\mathcal{C})$ on which the line operators form the fusion
category $\mathcal{C}$. In this case, we will denote
$F_{\mathcal{B}_{\mathcal{C}}}$ as $F$ to simplify notation.999This notation
is not to be confused with the associator of $\mathcal{C}$ given by the $F$
matrices. The action of $F$ on the simple line operators of
$\mathcal{Z}(\mathcal{C})$ has a simple description. Recall that simple line
operators in $\mathcal{Z}(\mathcal{C})$ can be written as

$(a,e_{a})~{},~{}a\in\mathcal{C}$ (20)

where $e_{a}$ are half-braidings. In this notation, $F$ is given by101010This
is called a forgetful functor from $\mathcal{Z}(\mathcal{C})$ to $\mathcal{C}$
since it forgets about the half-braidings $e_{a}$. More generally, for any
gapped boundary $\mathcal{B}$, $F_{\mathcal{B}}$ is a tensor functor
(Cong:2017hcl, , Section V. B).

$F((a,e_{a}))=a~{}.$ (21)

Since $F$ determines the relation between bulk and boundary line operators, it
is perhaps unsurprising that it can be used to determine the relation between
Lagrangian algebras in $\mathcal{Z}(\mathcal{C})$ and non-anomalous line
operators in $\mathcal{C}$. Indeed, consider a Lagrangian algebra $L$ in
$\mathcal{Z}(\mathcal{C})$. Consider the perpendicular fusion of $L$ with the
gapped boundary $\mathcal{B}_{\mathcal{C}}$, given by

$F(L)=\sum_{a}N_{F(L)}^{a}~{}a~{},~{}~{}a\in\mathcal{C}~{}.$ (22)

$\mathcal{B}_{L}$$\mathcal{B}_{\mathcal{C}}$$L$$\mathcal{Z}(\mathcal{C})$$F_{\mathcal{B}_{\mathcal{C}}}(L)$$F_{\mathcal{B}_{L}}(L)$$\psi$$\chi$
Figure 10: Fusion of line operators in $\mathcal{Z}(\mathcal{C})$ on gapped
boundaries.

We will show that $F(L)$ is a sum of algebra objects in $\mathcal{C}$ which
correspond to physically equivalent gaugings. In other words, $F(L)$ is a sum
of Morita-equivalent algebras. To see this recall that $\mathcal{T}_{L}$ is
the 1+1D TQFT obtained from the interval compactification of the SymTFT
$\mathcal{Z}(\mathcal{C})$ with gapped boundaries $\mathcal{B}_{\mathcal{C}}$
and $\mathcal{B}_{L}$. Consider the $\mathcal{C}$-module category
$\mathcal{M}_{L}$ describing the 1D gapped boundaries of $\mathcal{T}_{L}$.
Recall the definition of the algebra object $A_{\operatorname{\text{m}}}$ in
section 4.2. We will prove the following theorem.

######  Theorem 5.1.

Let $L$ be a Lagrangian algebra in the SymTFT $\mathcal{Z}(\mathcal{C})$. As
an object in $\mathcal{C}$

$F(L)=\bigoplus_{\operatorname{\text{m}}\in\mathcal{M}_{L}}A_{\operatorname{\text{m}}}~{}.$
(23)

Proof: Consider some simple line operator $a$ such that $N_{F(L)}^{a}\neq 0$.
Then, the configuration in Fig. 11, is non-trivial.

$\mathcal{B}_{\mathcal{C}}$$\mathcal{Z}(\mathcal{C})$$L$$\psi$$a$$\mathcal{B}_{L}$
Figure 11: Junctions between $L$ and $a$ on the gapped boundary
$\mathcal{B}_{\mathcal{C}}$ correspond to twisted sector states in the Hilbert
space $\mathcal{H}_{a}$ of the TQFT $\mathcal{T}_{L}$.

Compactifying this diagram gives $N_{F(L)}^{a}$ operators at the end of the
line operator $a$ in the TQFT $\mathcal{T}_{L}$.111111One could wonder whether
the choice of the junction of $L$ with the gapped boundary $\mathcal{B}_{L}$
affects this counting. Note that this is already taken into account in the
argument since the number of possible point junctions between a line operator
$x\in L$ and the gapped boundary $\mathcal{B}_{L}$ is precisely $N_{L}^{x}$.
The Lagrangian algebra object $L$ contains the term $N_{L}^{x}~{}x$, and on
acting with $F$ we get $N_{L}^{x}F(x)$. Now, using the state-operator
correspondence we find that dim$(\mathcal{H}_{a})=N_{F(L)}^{a}$, where
$\mathcal{H}_{a}$ is the twisted Hilbert space for the line operator
$a$.121212We have assumed that interval compactification of the configuration
in Fig. 11 for different $\psi$ produces all twisted sector local operators at
the end of the line operator $a$ in $\mathcal{T}_{L}$. Consider the
isomorphism Huang:2021zvu

$\mathcal{H}_{a}\cong\bigoplus_{\operatorname{\text{m}}\in\mathcal{M}_{L}}\text{Hom}_{\mathcal{M}_{L}}(\operatorname{\text{m}},a\times\operatorname{\text{m}})~{}.$
(24)

Note that
$\text{Hom}_{\mathcal{M}_{L}}(\operatorname{\text{m}},a\times\operatorname{\text{m}})$
is non-empty if and only if $a\in A_{\operatorname{\text{m}}}$. Indeed,

$|\text{Hom}_{\mathcal{M}_{L}}(\operatorname{\text{m}},a\times\operatorname{\text{m}})|=N_{\operatorname{\text{m}}}^{a}$
(25)

where, $N_{\operatorname{\text{m}}}^{a}$ is the number of copies of $a$ in
$A_{\operatorname{\text{m}}}$ (see definition (16)). Therefore,

$\sum_{\operatorname{\text{m}}\in\mathcal{M}_{L}}N_{\operatorname{\text{m}}}^{a}=\sum_{\operatorname{\text{m}}\in\mathcal{M}_{L}}|\text{Hom}_{\mathcal{M}_{L}}(\operatorname{\text{m}},a\times\operatorname{\text{m}})|=\text{dim}(\mathcal{H}_{a})=N_{F(L)}^{a}~{}.$
(26)

Therefore, we find that there are $N_{F(L)}^{a}$ copies of the line operator
$a$ in
$\bigoplus_{\operatorname{\text{m}}\in\mathcal{M}_{L}}A_{\operatorname{\text{m}}}$.
$\hfill\square$

When $\mathcal{C}$ is a modular tensor category, Theorem 5.1 follows from
Schellekens:2001xv  (see also (Kong:2007yv, , Proposition 4.3)). Recall from
the discussion in section 4.2 that each $A_{\operatorname{\text{m}}}$ admits
the structure of a haploid symmetric separable Frobenius algebra for each
$\operatorname{\text{m}}$. Therefore, their direct sums also admit the
structure of a symmetric separable Frobenius algebra (for example, see
(Fuchs:2002cm, , Section 3.5)). Hence, $F(L)$ is a non-anomalous line operator
in $\mathcal{C}$. It is clear from (23) that, in general, the number of copies
of the identity line $\mathbf{1}$ in $F(L)$ is equal to the number of distinct
boundary conditions of the TQFT $\mathcal{T}_{L}$ since
$A_{\operatorname{\text{m}}}$ has exactly one identity operator for every
$\operatorname{\text{m}}\in\mathcal{M}_{L}$. Therefore, in general, the
algebra $F(L)$ is not haploid. In order to identify the haploid subalgebras in
$F(L)$, we have to consider the action of the simple line operators in $F(L)$
on gapped interfaces between gapped boundaries $\mathcal{B}_{\mathcal{C}}$ and
$\mathcal{B}_{L}$ as in Fig. 9. Choose a simple gapped interface $r$ between
the gapped boundaries $\mathcal{B}_{\mathcal{C}}$ and
$\mathcal{B}_{L}$.131313A gapped interface is simple if it does not host any
non-trivial point operators on it. Define $A_{r}$ to be the sub-object/line
operator of $F(L)$

$A_{r}:=\sum_{a~{}\in~{}F(L)}N_{r}^{a}~{}a~{},$ (27)

where $N_{r}^{a}$ is the dimension of Hilbert space of operators at the
junction of $a$ with the gapped interface $r$ (see Fig. 12). Compactifying
this diagram, we clearly have that

$A_{r}=A_{\operatorname{\text{m}}_{r}}~{},$ (28)

where $\operatorname{\text{m}}_{r}\in\mathcal{M}_{L}$ is the 1D gapped
boundary of the 1+1D TQFT $\mathcal{T}_{L}$ corresponding to the gapped
interface $r$.

$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{L}$$r$$T$$\operatorname{\text{m}}_{r}$$\mathcal{Z}(\mathcal{C})$$L$$r$$\operatorname{\text{m}}_{r}$$a$$a$$\psi$$\chi$$\mathcal{O}_{(x,\psi,\chi)}$$\alpha$$\alpha$
Figure 12: $a$ is a line operator in $\mathcal{C}$ which can form a non-
trivial junction on the gapped interface $r$. On closing the wedge, we find
that $a$ can form a junction on the corresponding gapped boundary
$\operatorname{\text{m}}_{r}\in\mathcal{M}_{L}$.

Suppose $L_{1},\dots,L_{n}$ is the full set of Lagrangian algebras in the
SymTFT $\mathcal{Z}(\mathcal{C})$. Then, $F(L_{1}),\dots,F(L_{n})$ is the full
set of equivalence classes of algebras in $\mathcal{C}$. Therefore, Theorem
5.1 provides an alternative method to using NIM-reps to classify algebra
objects in a fusion category grossman2012quantum ; grossman2016brauer ;
Komargodski:2020mxz ; Diatlyk:2023fwf . In fact, as will explore in detail in
section 7.3, if the SymTFT $\mathcal{Z}(\mathcal{C})$ and its Lagrangian
algebras are known, then Theorem 5.1 can be used to find algebra objects in
all fusion categories which share the same SymTFT $\mathcal{Z}(\mathcal{C})$.

###  5.2 Generalized discrete torsion from SymTFT

In the previous section, we studied the structure of
$F_{\mathcal{B}_{\mathcal{C}}}(L)$ in terms of the algebras
$A_{\operatorname{\text{m}}}$, $\operatorname{\text{m}}\in\mathcal{M}_{L}$.
This discussion only used the fact that $L$ is a line operator in the SymTFT
$\mathcal{Z}(\mathcal{C})$ which admits the structure of a Lagrangian algebra.
In particular, so far, we did not use the explicit algebra structure on $L$.
In this section, we will discuss how to determine the explicit algebra
structure on $F_{\mathcal{B}_{\mathcal{C}}}(L)$ from that on $L$.

Recall the bulk-to-boundary map

$F_{\mathcal{B}}:\mathcal{Z}(\mathcal{C})\to\mathcal{B}~{},$ (29)

introduced in (19) where we will use $\mathcal{B}$ to denote both a gapped
boundary as well as the fusion category of line operators living on the gapped
boundary. In order to describe how the full data of an algebra in
$\mathcal{Z}(\mathcal{C})$ is mapped to $\mathcal{B}$ under $F_{\mathcal{B}}$,
we have to determine how the fusion spaces in $\mathcal{Z}(\mathcal{C})$ get
mapped to fusion spaces in $\mathcal{B}$. This is determined by an isomorphism

$\Phi_{x,y}:F_{\mathcal{B}}(x)\times F_{\mathcal{B}}(y)\to
F_{\mathcal{B}}(x\times y)~{}.$ (30)

By choosing a basis, we can write these isomorphisms explicitly in terms of
some complex numbers as given in Fig. 13.

$\mathcal{B}$$\mathcal{B}$$x$$x$$y$$y$$z$$a$$b$$c$$c$$\psi$$\chi$$\xi$$\alpha$$i$$=\
\sum\limits_{(a,\psi)(b,\chi),\alpha}\left[\Phi_{x,y}^{c}\right]{{}_{(a,\psi)(b,\chi),\alpha}^{z,\xi,i}}$
Figure 13: The complex numbers
$\left[\Phi_{x,y}^{c}\right]{{}_{(a,\psi)(b,\chi),\alpha}^{z,\xi,i}}$ relate
the fusion spaces of $\mathcal{Z}(\mathcal{C})$ and $\mathcal{B}$.

The choice of isomorphisms $\Phi_{x,y}$ must be compatible with the
associativity of fusion rules in both $\mathcal{Z}(\mathcal{C})$ and
$\mathcal{B}$. This implies that $\Phi_{x,y}$ must satisfy the constraints in
Fig. 14. The isomorphisms $\Phi$ make $F_{\mathcal{B}}$ a strong monoidal
functor (see, for example baez2004some ; Kong:2009inh ).

$\mathcal{B}$$\mathcal{B}$$x$$y$$y$$z$$u$$d$$\psi$$\chi$$\sigma$$i$$\sum\limits_{(a,\psi)(b,\chi),\alpha}\left[\Phi_{x,y}^{e}\right]{{}_{(a,\psi)(b,\chi),\alpha}^{v,\varphi,k}}$$x$$\mathcal{B}$$y$$z$$w$$d$$\sigma$$x$$w$$v$$j$$k$$l$$\sum\limits_{k,v,l}\left[F_{xyz}^{w}\right]_{k,v,l}^{i,u,j}$$z$$\xi$$b$$c$$a$$d$$e$$\alpha$$\beta$$\mathcal{B}$$y$$z$$c$$\varphi$$x$$v$$k$$\xi$$d$$\sum\limits_{(e,\varphi)(c,\xi),i}\left[\Phi_{v,z}^{d}\right]{{}_{(e,\varphi)(c,\xi),\beta}^{w,\sigma,l}}$$e$$\beta$$\mathcal{B}$$\psi$$\chi$$\xi$$b$$a$$d$$\gamma$$\delta$$x$$y$$z$$\sum\limits_{\alpha,e,\beta}\left[F_{abc}^{d}\right]_{\alpha,e,\beta}^{\gamma,f,\delta}$$f$$\sum\limits_{(b,\chi)(c,\xi),\gamma}\left[\Phi_{y,z}^{f}\right]{{}_{(b,\chi)(c,\xi),\gamma}^{u,\rho,i}}$$f$$\psi$$d$$\sum\limits_{(a,\psi)(f,\rho),\delta}\left[\Phi_{x,u}^{d}\right]{{}_{(a,\psi)(f,\rho),\delta}^{w,\sigma,j}}$$a$$\delta$$\mathcal{B}$$x$$y$$z$$i$$u$$\rho$$c$
Figure 14: Associativity of fusion rules in $\mathcal{Z}(\mathcal{C})$ and
$\mathcal{B}_{L}$ implies that the above diagram must commute.

Consider a Lagrangian algebra $L$ in $\mathcal{Z}(\mathcal{C})$. Let
$m:L\times L\to L$ be the multiplication on $L$. By choosing a basis, we can
write the multiplication $m$ explicitly as in Fig. 15.

$x$$y$$z$$L$$L$$L$$=\
\sum\limits_{l}m_{(x,i),(y,j),l}^{(z,k)}$$i$$j$$k$$x$$y$$z$$l$ Figure 15: The
multiplication on $L$ is determined by the complex numbers
$m_{(x,i),(y,j),l}^{(z,k)}$.

A multiplication on $F(L)$ is determined by the complex numbers as defined in
Fig. 16.

$\mathcal{B}$$a$$b$$c$$F(L)$$F(L)$$F(L)$$=\
\sum\limits_{\delta}\mu_{(a,\alpha),(b,\beta),\delta}^{(c,\gamma)}$$\alpha$$\beta$$\gamma$$\mathcal{B}$$a$$b$$c$$\delta$
Figure 16: A multiplication on $F(L)$ is determined by the complex numbers
$\mu_{(a,\alpha),(b,\beta),\delta}^{(c,\gamma)}$.

In fact, $F(L)$ is an associative algebra in $\mathcal{B}$ given by the
multiplication

$\mu:m\circ\Phi~{}.$ (31)

By choosing a basis, this can be explicitly depicted as in the Fig. 17.

$=\
\sum\limits_{l}m_{(x,i),(y,j),l}^{(z,k)}$$\mathcal{B}$$a$$b$$c$$\delta$$\mathcal{B}$$a$$b$$c$$x$$y$$z$$\psi$$\chi$$l$$\xi$$=\
\sum\limits_{l}m_{(x,i),(y,j),l}^{(z,k)}\sum\limits_{\delta}\left[\Phi_{x,y}^{c}\right]_{(a,\psi),(b,\chi),\delta}^{z,\xi,l}$$\mathcal{B}$$a$$b$$c$$x$$y$$z$$\psi$$\chi$$\xi$$i$$j$$k$$L$$L$$L$$=$$\mathcal{B}$$a$$b$$c$$F(L)$$F(L)$$F(L)$$\alpha$$\beta$$\gamma$

Figure 17: The multiplication on $F(L)$ in terms of $m$ and $\Phi$. In the
first equality we used
Hom$(F(L),a)=\bigoplus_{x\in\mathcal{Z}(\mathcal{C})}\text{Hom}(L,x)\times\text{Hom}(x,a)$,
so that $\alpha=(\psi,x,i),\;\beta=(x,y,j),\;\gamma=(\xi,z,k)$. In the second
equality we used Fig. 15 and in the third equality we used Fig. 13

.

Comparing Fig. 16 and Fig. 17, we get the equation

$\mu_{(a,(\psi,x,i)),(b,(\chi,y,j)),\delta}^{(c,(\xi,z,k))}=\sum_{l}m_{(x,i),(y,j),l}^{(z,k)}\left[\Phi_{x,y}^{c}\right]{{}_{(a,\psi)(b,\chi),\delta}^{z,\xi,l}}~{}.$
(32)

We are in particular interested in the perpendicular fusion of line operators
on the canonical gapped boundary $\mathcal{B}_{\mathcal{C}}$. In this case, we
have the forgetful functor $F\equiv F_{\mathcal{B}_{\mathcal{C}}}$

$F((a,e_{a}))=a.$ (33)

Therefore, we have

$\displaystyle F((a,e_{a}))\times F((b,e_{b}))=a\times b~{},$ (34)
$\displaystyle F((a,e_{a})\times(b,e_{b}))=F((a\times b,e_{a\times
b}))=a\times b~{}.$ (35)

and the isomorphisms $\Phi_{x,y}$ thus can be chosen to be trivial. In other
words, there exists a basis in which the complex numbers
$\left[\Phi_{x,y}^{c}\right]{{}_{(a,\psi)(b,\chi),\alpha}^{z,\xi,k}}$ are all
equal to $1$. This significantly simplifies the expression (32).

Remark: The discussion above implies that the bulk-to-boundary map $F$ can
also be used to transport non-Lagrangian algebras in
$\mathcal{Z}(\mathcal{C})$ to $\mathcal{C}$. If $E$ is a general algebra in
$\mathcal{Z}(\mathcal{C})$, then (32) specifies a multiplication on $F(E)$. In
general, $F(E)$ is not haploid.

###  5.3 Boundary-to-bulk map: From $A\in\mathcal{C}$ to
$L\in\mathcal{Z}(\mathcal{C})$

Given an algebra $A$ with multiplication $\mu$ in $\mathcal{C}$ the
corresponding Lagrangian algebra with object $L$ and multiplication $m$ in
$\mathcal{Z}(\mathcal{C})$ can be found through a two-step process. Let
$K:\mathcal{C}\to\mathcal{Z}(\mathcal{C})$ be the boundary-to-bulk map whose
action on the line operators is given by

$K(a):=\bigoplus_{(x,e_{x})\in\mathcal{Z}(\mathcal{C})}N_{a}^{x}(x,e_{x})$
(36)

where $N_{a}^{x}$ counts the number of point operators at the junction of $a$
and $x$ in $\mathcal{C}$ (muger2003subfactors, , Section 8.1).141414$K$ is
sometimes called the induction functor (etingof2016tensor, , Section 9.2).
$K(a)$ has the interpretation as the sum of line operators of the SymTFT
$\mathcal{Z}(\mathcal{C})$ which can form a junction with $a\in\mathcal{C}$ on
the gapped boundary $\mathcal{B}_{\mathcal{C}}$. Given an algebra $A$ in
$\mathcal{C}$, consider the line operator $K(A)$ in $\mathcal{C}$. In general,
$K(A)$ does not admit the structure of a commutative algebra. This is in
particular because $K(A)$ may contain non-bosonic simple line operators in
$\mathcal{Z}(\mathcal{C})$. However, since $A$ is an algebra in $\mathcal{C}$,
$K(A)$ does admit the structure of an algebra in $\mathcal{Z}(\mathcal{C})$.
Indeed, the map $K$ gives an algebra structure on $K(A)$. Similar to the case
of $F$, to determine how $K$ transports algebras from $\mathcal{C}$ to
$\mathcal{Z}(\mathcal{C})$, we have to determine how $K$ relates the fusion
spaces of $\mathcal{C}$ and $\mathcal{Z}(\mathcal{C})$. We need to specify
maps

$\Xi_{a,b}:K(a)\times K(b)\to K(a\times b)~{}.$ (37)

$\Xi_{a,b}$ can be determined as follows. The bulk-to-boundary map $F$ and
boundary-to-bulk map $K$ together satisfies (muger2003subfactors, , Section
8.1)

$\text{Hom}_{\mathcal{C}}(F((a,e_{a})),b)\simeq\text{Hom}_{\mathcal{Z}(\mathcal{C})}((a,e_{a}),K(b))~{}.$
(38)

Let $\Theta_{(a,e_{a}),b}$ be this isomorphism. Let
$\text{id}_{\mathcal{Z}(\mathcal{C})}$ be the identity map on
$\mathcal{Z}(\mathcal{C})$. Consider the natural transformation

$\delta:\text{id}_{\mathcal{Z}(\mathcal{C})}\to K\circ F~{},$ (39)

specified by

$\delta_{(a,e_{a})}:=\Theta_{(a,e_{a}),F((a,e_{a}))}(\text{id}_{F(a,e_{a})}):(a,e_{a})\to
K\circ F((a,e_{a}))~{}.$ (40)

$\delta_{(a,e_{a})}$ is non-trivial because the action of the composition
$K\circ F$ on $\mathcal{Z}(\mathcal{C})$ is not the same as the identity map
idZ(C). Similarly, we have

$\rho:F\circ K\to\text{id}_{\mathcal{C}}~{},$ (41)

specified by

$\rho_{a}:=\Theta_{K(a),a}^{-1}(\text{id}_{K(a)}):F\circ K(a)\to a~{}.$ (42)

$\rho_{(a,e_{a})}$ is non-trivial because the action of the composition
$F\circ K$ on $\mathcal{C}$ is not the same as the identity map idC. Using the
data introduced above, we can write

$\Xi_{a,b}:K(a)\times K(b)\xrightarrow{\delta}K\circ F(K(a)\times
K(b))\xrightarrow{\Phi^{-1}}K\circ(F\circ K(a)\times F\circ
K(a))\xrightarrow{K(\rho)}K(a\times b)~{}.$ (43)

where $\Phi$ is the map introduced in (30) which relates the fusion spaces of
$\mathcal{Z}(\mathcal{C})$ and $\mathcal{C}$ under $F$.151515In other words,
$K$ is the adjoint of the forgetful functor $F$, and therefore $K$ itself is a
lax monoidal functor (see, for example, (Kong:2009inh, , Lemma 2.7),
nlab:monoidal_adjunction ).161616Since $K$ is a two-sided adjoint of
$F_{\mathcal{B}_{\mathcal{C}}}$, it is in fact both a lax and colax monoidal
functor (see, for example, Kong:2009inh ). $K$ is generally not strong
monoidal (445787  (see also Kong:2009inh )). However, $K$ still maps Frobenius
algebras to Frobenius algebras. This latter statement can be shown using
(yadav2024frobenius, , Theorem 3.3). Therefore, $\Xi$ can be used to determine
the multiplication on $K(A)$ given the multiplication on $A$ precisely as in
(32). However, note that unlike $\Phi$, $\Xi$ is not trivial in general.
Therefore, finding the multiplication on $K(A)$ requires finding $\Xi$
explicitly.

As we discussed above, though the map $K$ specifies a multiplication on
$K(A)$, it is not the Lagrangian algebra that we are looking for since $K(A)$
is not even commutative in general. However, $K(A)$ can be higher-gauged to
obtain a surface operator $S_{K(A)}$ of the SymTFT Roumpedakis:2022aik .
Consider the gapped boundary $\mathcal{B}_{L_{A}}$ of the SymTFT obtained from
fusing the surface $S_{K(A)}$ on the canonical gapped boundary
$\mathcal{B}_{\mathcal{C}}$ as in Fig. 18.

$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{\mathcal{C}}$$S_{K(A)}$$\mathcal{B}_{L_{A}}$$\mathcal{B}_{\mathcal{C}}$$=$
Figure 18: Parallel fusion of the surface operator $S_{K(A)}$ on the canonical
gapped boundary $\mathcal{B}_{\mathcal{C}}$ produces a new gapped boundary
$\mathcal{B}_{L_{A}}$.

The Lagrangian algebra object $L_{A}$ corresponding to $A$ is given by the set
of line operators in $K(A)$ that can end on $\mathcal{B}_{L_{A}}$. From Fig.
19 it is clear the a line operator $x\in K(A)$ can end on
$\mathcal{B}_{L_{A}}$ if

$S_{K(A)}\cdot x=1+\dots$ (44)

$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{L_{A}}$$\mathcal{B}_{\mathcal{C}}$$=$$S_{K(A)}$$a\in
A$$x\in K(A)$$a$$x$ Figure 19: The line operator $x\in K(A)$ can end on the
gapped boundary $\mathcal{B}_{L_{A}}$ if the action of $S_{K(A)}$ on $x$
produces the trivial line operator.

In other words, the action of the surface operator $S_{K(A)}$ on $x$ produces
the trivial line. As discussed in (Buican:2023bzl, , Section 5.2.2), the set
of line operators which satisfy (44) is given by the left center $C_{L}(K(A))$
of $K(A)$. The left center, $C_{L}(A)$ of $A$, is the maximal sub-object of
$A$ such that

$m_{x,i}^{y}~{}R_{i,x}^{y}=m_{i,x}^{y}~{},$ (45)

where $m$ is the multiplication in $A$, and $R_{i,x}^{y}$ is the braiding of
the lines $x\in A$ and $i\in C_{L}(A)$ in $\mathcal{Z}(\mathcal{C})$.
Diagrammatically, this is Fig. 20.

$K(A)$$C_{L}(K(A))$$K(A)$$C_{L}(K(A))$$K(A)$$K(A)$$=$ Figure 20: Diagrammatic
definition of left center of $K(A)$.

Therefore, the Lagrangian algebra $L_{A}$ corresponding to the algebra $A$ is
given by the left center of $K(A)$171717This agrees with the abstract
description of full center of an algebra given in Frohlich:2003hm ;
Kong:2007yv ; davydov2010centre ; Davydov:2013lma ..

To make the discussion above more transparent, let us consider the example
$\mathcal{C}=\text{Vec}_{\mathds{Z}_{2}}$. Suppose the two line operators in
$\mathcal{C}$ are labeled by $e,g$, where $g$ is order two. We have two
algebra objects in this case

$A_{1}=e~{},~{}A_{2}=e+g~{}.$ (46)

The SymTFT $\mathcal{Z}(\mathcal{C})$ is the $\mathds{Z}_{2}$ Dijkgraaf-Witten
theory with simple line operators

$([e],\mathds{1})~{},~{}([e],\pi)~{},~{}([g],\mathds{1})~{},~{}([g],\pi)~{}.$
(47)

where $\mathds{1},\pi$ are the two irreducible representations of
$\mathds{Z}_{2}$. $([e],\mathds{1}),([e],\pi)$ and $([g],\mathds{1})$ are
bosons while $([g],\pi)$ is a fermion. The boundary-to-bulk map $K$ is given
by

$K(e)=([e],\mathds{1})+([e],\pi)~{},~{}K(g)=([g],\mathds{1})+([g],\pi)~{}.$
(48)

Therefore, we get

$K(A_{1})=([e],\mathds{1})+([e],\pi)~{}.$ (49)

This is already a commutative algebra because both $([e],\mathds{1})$ and
$([e],\pi)$ are bosons, and they are closed under fusion. Indeed, $K(A_{1})$
is precisely the Lagrangian algebra corresponding to $A_{1}$.

Now, consider

$K(A_{2})=([e],\mathds{1})+([e],\pi)+([g],\mathds{1})+([g],\pi)~{}.$ (50)

Clearly, $K(A_{2})$ is not a commutative algebra because of the fermion
$([g],\pi)$. However, it admits the structure of an algebra and we can define
a surface operator $S_{K(A_{2})}$ by higher-gauging $A_{2}$ on a 2-manifold.
The action of $S_{K(A_{2})}$ on the line operators is given by
(Roumpedakis:2022aik, , Section 6.3.1)

$\displaystyle
S_{K(A_{2})}\cdot([e],\mathds{1})=S_{K(A_{2})}\cdot([g],\mathds{1})=([e],\mathds{1})+([e],\pi)~{},$
(51) $\displaystyle
S_{K(A_{2})}\cdot([e],\pi)=S_{K(A_{2})}\cdot([g],\pi)=0~{}.$ (52)

The set of lines on which the surface operator acts to produce the trivial
line is

$([e],\mathds{1})\text{ and }([g],\mathds{1})~{}.$ (53)

Therefore, we get a commutative algebra $A=([e],\mathds{1})+([g],\mathds{1})$
which is precisely the Lagrangian algebra object corresponding to $A_{2}$.

###  5.4 Invertible symmetry

When the symmetry is invertible, the category $\mathcal{C}$ is equivalent to
Vec${}_{G}^{\omega}$ for some group $G$ and anomaly given by a 3-cocycle
$\omega\in H^{3}(G,U(1))$. When $\omega$ is non-trivial, the symmetry $G$
cannot be gauged. Given a subgroup $H\subseteq G$, we can ask whether $H$ is
non-anomalous. This is true if and only if

$\omega|_{H}=1\in H^{3}(H,U(1))~{}.$ (54)

Given a non-anomalous group $H$, we can gauge it with a choice of discrete
torsion $\sigma\in H^{2}(H,U(1))$. The algebra objects in this case will be
denoted as $A(H,\sigma)$. Clearly, if $H$ is non-anomalous, so is the subgroup
$gHg^{-1}$ for any $g\in G$. In fact, the algebras $A(H,\sigma)$ and
$A(gHg^{-1},g\sigma g^{-1})$ are Morita equivalent for any $g\in G$ where
$g\sigma g^{-1}(h,k):=\sigma(ghg^{-1},gkg^{-1})$. Therefore, the equivalence
classes $[A(H,\sigma)]$ are given by conjugacy classes of subgroups of $G$ on
which $\omega$ trivializes and corresponding 2-cocycles.

So far we discussed how to determine whether $H\subset G$ is anomalous purely
from a 1+1D perspective. Given $H\subseteq G$, how do we determine whether $H$
is anomalous from the SymTFT
$\mathcal{Z}(\text{Vec}_{G}^{\omega})\equiv\text{DW}(G,\omega)$? We learned
that for $H\subseteq G$ to be non-anomalous there must be a Lagrangian algebra
$L_{A(H,\sigma)}$ in $\mathcal{Z}(\text{Vec}_{G}^{\omega})$ corresponding to
it. To determine this Lagrangian algebra object in terms of the simple objects
of $\mathcal{Z}(\text{Vec}_{G}^{\omega})$, we can use the theory of characters
for objects in $\mathcal{Z}(\text{Vec}_{G}^{\omega})$ davydov2010modular ;
davydov2017lagrangian . Let us define

$\omega(g,h|k):=\omega(g,h,k)^{-1}\omega(g,hgh^{-1},h)\omega(ghk(gh)^{-1},g,h)^{-1}~{}.$
(55)

Let $([g],\pi_{g})$ be a simple object in
$\mathcal{Z}(\text{Vec}_{G}^{\omega})$, where $[g]$ is a conjugacy class in
$G$ and $\pi_{g}$ is a projective representation of the centralizer $C_{g}$ of
$g$ satisfying

$\pi_{g}(h)\pi_{g}(k)=\omega(h,k|g)\pi_{g}(hk)~{}~{}\forall h,k\in C_{g}~{}.$
(56)

Note that for $h,k\in C_{g}$, the phases $\omega(h,k|f)$ is a 2-cocycle in
$H^{2}(C_{g},U(1))$. The character of $([g],\pi_{g})$ is defined as

$\chi_{([g],\pi_{g})}(h,k):=\begin{cases}\chi_{\pi_{h}}(k)&\mbox{if
}h\in[g]~{},\\\ 0&\mbox{otherwise}~{},\end{cases}$ (57)

where $h,k$ is a pair of commuting elements in $G$ and $\chi_{\pi_{g}}(k)$ is
the character of the representation $\pi_{h}$ of the centralizer $C_{h}$ of
$h$ evaluated on $k\in C_{h}$. See Appendix B for the definition of the
character of a general line operator in the SymTFT and its properties.

The Lagrangian algebra object $L_{A(H,\sigma)}$ can be determined by first
computing the character

$\chi_{\mathcal{Z}(A(H,\sigma))}(k,l):=\sum_{y\in
Y}\frac{\omega(y^{-1}ly,y^{-1}|k)\sigma(y^{-1}ky,y^{-1}ly)}{\omega(y^{-1},l|k)\sigma(y^{-1}ly,y^{-1}ky)}~{},$
(58)

where

$\displaystyle Y$ $\displaystyle:=$ $\displaystyle\\{y\in G,y^{-1}ky\in
H,y^{-1}ly\in H\\}/H\subset G/H~{},$ (59)

and then decomposing it in terms of the characters $\chi_{([g],\pi_{g})}$.

So far, we discussed how to get the Lagrangian algebra object
$L_{A(H,\sigma)}$ corresponding to an algebra $A(H,\sigma)$ in
Vec${}_{G}^{\omega}$. Given a subgroup $H\subseteq G$, we would like to
determine if $H$ is non-anomalous using the SymTFT
$\mathcal{Z}(\text{Vec}_{G}^{\omega})$. To this end, let us consider the
following set

$C(H):=\\{[h],h\in H\\}~{}.$ (60)

where $[h]$ is the conjugacy class in $G$ with representative $h$. As a set,
$C(H)$ contains all elements in the subgroup $H$ as well as elements of any
subgroup conjugate to $H$.

######  Theorem 5.2.

$H\subseteq G$ is non-anomalous if and only if there exists a Lagrangian
algebra $L$ in $\mathcal{Z}(\text{Vec}_{G}^{\omega})$ such that

$([h],\pi_{h})\in L~{}~{}~{}\forall~{}[h]\in C(H)~{},$ (61)

for some representation $\pi_{h}$ of the centralizer of $h$.

Proof: See Appendix B.

The above theorem shows that from a given object $L$ of
$\mathcal{Z}(\text{Vec}_{G}^{\omega})$ that admits the structure of a
Lagrangian algebra we can identify a conjugacy class of subgroups $[H]$ such
that any element of $[H]$ is a non-anomalous subgroup of $G$.

####  5.4.1 Lagrangian subcategory

In the previous section, we proved a theorem which can be used to determine
whether a subgroup $H\subseteq G$ is non-anomalous. To completely specify the
equivalence class of algebras $[A(H,\sigma)]$, we also need to find the
2-cocycle $\sigma$. In general, this requires not just the decomposition of
$L$ into simple objects, but also the multiplication of the algebra $L$.
Indeed, the same object $L$ can admit distinct Lagrangian algebra structures
davydov2014bogomolov  which then correspond to two non-equivalent algebra
structures on some subgroup $H\subseteq G$. In general, the 2-cocycle $\sigma$
can be determined using (32). However, when the simple line operators which
form $L$ are closed under fusion, the 2-cocycle $\sigma$ can be easily
obtained. In this case, these simple line operators form a Lagrangian
subcategory of $\mathcal{Z}(\text{Vec}_{G}^{\omega})$. Suppose we have

$L=\sum_{([g],\pi_{g})\in\mathcal{Z}(\text{Vec}_{G}^{\omega})}N_{L}^{([g],\pi_{g})}~{}([g],\pi_{g})~{}.$
(62)

Since the non-trivial simple line operators in $L$ are closed under fusion,
the conjugacy classes $[g]$ such that $N_{L}^{([g],\pi_{g})}\neq 0$ must all
be closed under fusion. The Lagrangian algebra $L$ corresponds to an algebra
object

$A=\sum_{([g]\pi_{g})\in\mathcal{Z}(\text{Vec}_{G}^{\omega})}N_{L}^{([g],\pi_{g})}~{}A_{g}~{},$
(63)

where $A_{g}=\sum_{h\in[g]}h$. Since the non-trivial conjugacy classes of
simple objects in $L$ are closed under fusion, we find that $A$ is in fact a
sum over elements of a normal subgroup of $G$ given by

$N=\bigcup_{N^{([g],\pi_{g})}_{L}\neq 0}[g]~{},$ (64)

Moreover, since the fusion of line operators in
$\mathcal{Z}(\text{Vec}_{G}^{\omega})$ is commutative, $N$ is an abelian
normal subgroup. Therefore, the subset of gapped boundaries of the SymTFT
$\mathcal{Z}(\text{Vec}_{G}^{\omega})$ determined by Lagrangian subcategories
correspond to non-anomalous abelian normal subgroups of $G$. In fact, there is
a one-to-one correspondence between Lagrangian subcategories of
$\mathcal{Z}(\text{Vec}_{G}^{\omega})$ and algebra objects $A(N,\sigma)$ in
Vec${}_{G}^{\omega}$ where $N\subseteq G$ is an abelian normal subgroup and
$\sigma\in H^{2}(N,U(1))$ naidu2008lagrangian .

The 2-cocycle $\sigma\in H^{2}(N,U(1))$ corresponding to the Lagrangian
subcategory (62) can be determined using the relation

$\frac{\sigma(h_{1},h_{2})}{\sigma(h_{2},h_{1})}=\frac{\beta_{g}(l,h_{2})\beta_{g}(lh_{2},l^{-1})}{\beta_{g}(l,l^{-1})}~{}\frac{\chi_{\pi_{g}}(l^{-1}h_{2}l)}{\text{dim}(\pi_{g})}$
(65)

where

$\beta_{g}(h,k):=\omega(g,h,k)\omega(h,h^{-1}gh,k)^{-1}\omega(h,k,(hk)^{-1}ghk)~{},$
(66)

$h_{1},l$ and $g$ satisfies $h_{1}=lgl^{-1}$ for some $l\in G$ and $g\in N$.
(See (naidu2008lagrangian, , Theorem 4.12) for more details)

##  6 Examples

In this section, we will go through various explicit examples to understand
the general discussion in the previous sections. We will start with the
familiar case of invertible symmetries. This is a good starting point to
understand Theorems 5.1 and 5.2. Then we will consider the case of a modular
tensor category $\mathcal{C}$, before moving on to the case of fusion
categories.

###  6.1 Invertible symmetries

####  6.1.1 $G=\mathds{Z}_{4}$

When $G=\mathds{Z}_{4}$, there is no anomaly since
$H^{2}(\mathds{Z}_{4},U(1))\cong\mathds{Z}_{1}$. Therefore, we have the
category Vec${}_{\mathds{Z}_{4}}$ whose SymTFT is the $\mathds{Z}_{4}$
Dijkgraaf-Witten theory. It has three Lagrangian algebras given by

$\displaystyle
L_{1}=\\{(e,\mathds{1}),(e,\pi_{1}),(e,\pi_{2}),(e,\pi_{3})\\}~{},$ (67)
$\displaystyle
L_{2}=\\{(e,\mathds{1}),(g,\mathds{1}),(g^{2},\mathds{1}),(g^{3},\mathds{1})\\}~{},$
(68) $\displaystyle
L_{3}=\\{(e,\mathds{1}),(g^{2},\mathds{1}),(e,\pi_{2}),(g^{2},\pi_{2})\\}~{}.$
(69)

where $g$ is the generator of $\mathds{Z}_{4}$ and $g^{4}=e$, and $\pi_{j}$
are the representations of $\mathds{Z}_{4}$ given by $\pi_{j}(g)=e^{\frac{2\pi
ij}{4}}$. Using Theorem 5.2, we can read off the non-anomalous subgroups of
$\mathds{Z}_{4}$ from the simple objects in the Lagrangian algebras above. We
get

$L_{1}\to\mathds{Z}_{1},~{}L_{2}\to\mathds{Z}_{4},~{}L_{3}\to\mathds{Z}_{2}~{}.$
(70)

Alternatively, using Theorem 5.1, the algebra objects which can be gauged in
the $\mathds{Z}_{4}$ group can be obtained by fusing the Lagrangian algebra
objects $L_{i}$ in the bulk TQFT with the gapped boundary
$\mathcal{B}_{L_{1}}$ corresponding to $L_{1}$. Fusing $L_{1}$ with the gapped
boundary $\mathcal{B}_{L_{1}}$, we get

$e+e+e+e~{}.$ (71)

We see that the resulting algebra is not haploid because of multiple copies of
the identity line in it. Fusing $L_{2}$ with the gapped boundary
$\mathcal{B}_{L_{1}}$, we get

$e+g+g^{2}+g^{3}~{}.$ (72)

In this case, we get a haploid algebra. Fusing $L_{3}$ with the gapped
boundary $\mathcal{B}_{L_{1}}$, we get

$e+g^{2}+e+g^{2}~{}.$ (73)

In this case we get a non-haploid algebra. A haploid subalgebra is $e+g^{2}$.

####  6.1.2 $G=\mathds{Z}_{2}\times\mathds{Z}_{2}$

Consider the $\mathds{Z}_{2}\times\mathds{Z}_{2}$ symmetry without anomaly.
The SymTFT is given by the $\mathds{Z}_{2}\times\mathds{Z}_{2}=\langle
g_{1},g_{2}\rangle$ Dijkgraaf-Witten theory. The line operators in this SymTFT
can be labeled as

$\displaystyle([e],\mathds{1}),~{}([e],\omega_{1}),~{}([e],\omega_{2}),~{}([e],\omega_{1}\omega_{2})~{},$
(74)
$\displaystyle([g_{1}],\mathds{1}),~{}([g_{1}],\omega_{1}),~{}([g_{1}],\omega_{2}),~{}([g_{1}],\omega_{1}\omega_{2})~{},$
(75)
$\displaystyle([g_{2}],\mathds{1}),~{}([g_{2}],\omega_{1}),~{}([g_{2}],\omega_{2}),~{}([g_{2}],\omega_{1}\omega_{2})~{},$
(76)
$\displaystyle([g_{1}g_{2}],\mathds{1}),~{}([g_{1}g_{2}],\omega_{1}),~{}([g_{1}g_{2}],\omega_{2}),~{}([g_{1}g_{2}],\omega_{1}\omega_{2})~{},$
(77)

where $\mathds{1},\omega_{1},\omega_{2}$ and $\omega_{1}\omega_{2}$ label the
four irreducible representations of $\mathds{Z}_{2}\times\mathds{Z}_{2}$. This
SymTFT has six gapped boundaries determined by the Lagrangian algebra objects

$\displaystyle
L_{1}=([e],\mathds{1}),~{}([e],\omega_{1}),~{}([e],\omega_{2}),~{}([e],\omega_{1}\omega_{2})~{},$
(78) $\displaystyle
L_{2}=([e],\mathds{1})+([g_{1}],\mathds{1})+([g_{2}],\mathds{1})+([g_{1}g_{2}],\mathds{1})~{},$
(79) $\displaystyle
L_{3}=([e],\mathds{1})+([g_{1}],\omega_{2})+([g_{2}],\omega_{1})+([g_{1}g_{2}],\omega_{1}\omega_{2})~{},$
(80) $\displaystyle
L_{4}=([e],\mathds{1})+([e],\omega_{2})+([g_{1}],\mathds{1})+([g_{1}],\omega_{2})~{},$
(81) $\displaystyle
L_{5}=([e],\mathds{1})+([e],\omega_{1})([g_{2}],\mathds{1})+([g_{2}],\omega_{1})~{},$
(82) $\displaystyle
L_{6}=([e],\mathds{1})+([e],\omega_{1}\omega_{2})([g_{1}g_{2}],\mathds{1})+([g_{1}g_{2}],\omega_{1}\omega_{2})~{}.$
(83)

The canonical gapped boundary corresponding to the symmetry
$\mathds{Z}_{2}\times\mathds{Z}_{2}$ is $\mathcal{B}_{L_{1}}$. Fusing the
eight Lagrangian algebra objects on the gapped boundary $\mathcal{B}_{L_{1}}$,
we get

$\displaystyle F_{\mathcal{B}_{L_{1}}}(L_{1})=e+e+e+e~{},$ (84) $\displaystyle
F_{\mathcal{B}_{L_{1}}}(L_{2})=e+g_{1}+g_{2}+g_{1}g_{2}~{},$ (85)
$\displaystyle F_{\mathcal{B}_{L_{1}}}(L_{3})=e+g_{1}+g_{2}+g_{1}g_{2}~{},$
(86) $\displaystyle F_{\mathcal{B}_{L_{1}}}(L_{4})=e+e+g_{1}+g_{1}~{},$ (87)
$\displaystyle F_{\mathcal{B}_{L_{1}}}(L_{5})=e+e+g_{2}+g_{2}~{},$ (88)
$\displaystyle F_{\mathcal{B}_{L_{1}}}(L_{6})=e+e+g_{1}g_{2}+g_{1}g_{2}~{}.$
(89)

Clearly, both Lagrangian algebras $L_{2}$ and $L_{3}$ correspond to the same
algebra object

$A=e+g_{1}+g_{2}+g_{1}g_{2}~{}.$ (90)

Since distinct Lagrangian algebras must correspond to non-equivalent gaugings
in 1+1D, the algebra object $A$ has two distinct multiplications on it. These
multiplications can be determined using the multiplications of the Lagrangian
algebras $L_{2}$ and $L_{3}$.

Recall that in the multiplicity-free case, the multiplication in a Lagrangian
algebra $L$ is determined by the complex numbers

$m_{xy}^{z}~{},$ (91)

where $x,y,z$ are three simple line operators in $L$. These have to satisfy
the constraints (4) and (10). For the $\mathds{Z}_{2}\times\mathds{Z}_{2}$
discrete gauge theory, there is a gauge in which the $F$ symbols are all
trivial. The $R$ matrices can be written as

$R_{([g],\pi_{g}),([h],\pi_{h})}^{([gh],\pi_{g}\times\pi_{h})}=\chi_{\pi_{h}}(g)~{}.$
(92)

where $\chi_{\pi_{g}}$ is the character of the irreducible representation
$\pi_{g}$. Using this expression for the $R$ matrix and constraint (10), we
find that for the Lagrangian algebra $L_{2}$, we get

$m_{xy}^{z}=1~{}\forall~{}x,y,z\in L_{2}~{}.$ (93)

Therefore, using (32), we find that the multiplication on the algebra $A$
corresponding to $L_{2}$ is trivial.

Using the constraint (10), we find that for the Lagrangian algebra $L_{3}$, we
get

$m_{([g_{1}],\omega_{2}),([g_{2}],\omega_{1})}^{([g_{1}g_{2}],\omega_{1}\omega_{2})}=-1$
(94)

and all other $m_{xy}^{z}$ are trivial. Therefore, using (32), we find that
the multiplication on the algebra $A$ corresponding to $L_{3}$ is given by

$m_{g_{1},g_{2}}^{g_{1}g_{2}}=-1~{},$ (95)

and all the other $m_{xy}^{z}$ are trivial. Therefore, the non-trivial
multiplication on the algebra object $A$ corresponds to the Lagrangian algebra
$L_{3}$.

####  6.1.3 $G=S_{3}$

Consider the group

$S_{3}=\\{r,s|r^{3}=s^{2}=e;srs=r^{-1}\\}~{}.$ (96)

The conjugacy classes in this group are

$[e],~{}~{}[r]=\\{r,r^{2}\\},~{}~{}[s]=\\{s,rs,r^{2}s\\}~{}.$ (97)

$[e]$ has centralizer isomorphic to $S_{3}$ with the trivial representation
$\mathds{1}_{e}$, non-trivial 1-dimensional representation $\pi_{1}$ and non-
trivial two dimensional irreducible representation $\pi_{2}$. $[r]$ has a
centralizer isomorphic to $\mathds{Z}_{3}$ with irreducible representations
$\mathds{1}_{r},\omega,\omega^{2}$. Finally, $[s]$ has a centralizer
isomorphic to $\mathds{Z}_{2}$ which has the trivial representation
$\mathds{1}_{s}$ and non-trivial representation $\gamma$. The full set of line
operators in the $S_{3}$ DW theory are given by

$([e],\mathds{1}_{e}),([e],\pi_{1}),([e],\pi_{2}),([r],\mathds{1}_{r}),([r],\omega),([r],\omega^{2}),([s],\mathds{1}_{s}),([s],\gamma)~{}.$
(98)

The 1+1D gapped and gapless phases in this case and their relation to SymTFTs
were studied in detail in Bhardwaj:2023idu ; Bhardwaj:2023fca ;
Bhardwaj:2023bbf ; Bhardwaj:2024qrf .

There are four Lagrangian algebras given by davydov2017lagrangian ;
Cong:2017ffh

$\displaystyle L_{1}:=([e],\mathds{1}_{e})+([e],\pi_{1})+2([e],\pi_{2})~{},$
(99) $\displaystyle
L_{2}:=([e],\mathds{1}_{e})+([e],\pi_{2})+([s],\mathds{1}_{s})~{},$ (100)
$\displaystyle
L_{3}:=([e],\mathds{1}_{e})+([e],\pi_{1})+2([r],\mathds{1}_{r})~{},$ (101)
$\displaystyle
L_{4}:=([e],\mathds{1}_{e})+([s],\mathds{1}_{s})+([r],\mathds{1}_{r})~{}.$
(102)

Let us consider the fusion of all line operators on the $\mathcal{B}_{L_{1}}$
gapped boundary given by the map

$F_{\mathcal{B}_{L_{1}}}:\mathcal{Z}(\text{Vec}_{S_{3}})\to\text{Vec}_{S_{3}}~{}.$
(103)

In the following, we will denote $F_{\mathcal{B}_{L_{1}}}$ simply as $F$. We
get (see, for example, (Cong:2017ffh, , Section 2.6))

$\displaystyle
F(([e],\mathds{1}_{e}))=F(([e],\pi_{1}))=e~{},~{}F(([e],\pi_{2}))=2e~{},$
(104) $\displaystyle
F(([r],\mathds{1}_{r}))=F(([r],\omega))=e~{},~{}F(([r],\omega^{2}))=r+r^{2}~{},$
(105) $\displaystyle F(([s],\mathds{1}_{s}))=F(([s],\gamma))=s+sr+sr^{2}~{}.$
(106)

Note that $F$ preserves quantum dimensions. Using this, we can find the fusion
of all the Lagrangian algebras on $\mathcal{B}_{L_{1}}$. Fusing $L_{1}$ on the
gapped boundary $\mathcal{B}_{L_{1}}$ results in the line operator

$e+e+4e~{}.$ (107)

Fusing $L_{2}$ on the gapped boundary $\mathcal{B}_{L_{1}}$ results in the
non-anomalous line operator

$e+2e+s+sr+sr^{2}~{}.$ (108)

This is a sum of haploid subalgebras

$A_{1}=e+s~{},~{}A_{2}=e+sr~{},~{}A_{3}=e+sr^{2}~{},$ (109)

which are all Morita equivalent. Fusing $L_{3}$ on the gapped boundary
$\mathcal{B}_{L_{1}}$ results in the non-anomalous line operator

$e+e+2r+2r^{2}~{}.$ (110)

This is two copies of the haploid algebra

$e+r+r^{2}~{}.$ (111)

Fusing $L_{4}$ on the gapped boundary $\mathcal{B}_{L_{1}}$ results in the
non-anomalous line operator

$e+s+sr+sr^{2}+r+r^{2}~{}.$ (112)

###  6.2 When $\mathcal{C}$ is modular

When $\mathcal{C}$ is a modular tensor category the SymTFT has a particularly
simple form muger2003subfactors .

$\mathcal{Z}(\mathcal{C})\cong\mathcal{C}\times\bar{\mathcal{C}}~{},$ (113)

where $\bar{\mathcal{C}}$ is the category with the opposite braiding.
Therefore, the simple line operators in $\mathcal{Z}(\mathcal{C})$ can be
labeled as

$(a,b),~{}a,b\in\mathcal{C}~{}.$ (114)

However, for using the bulk-to-boundary map
$F:\mathcal{Z}(\mathcal{C})\to\mathcal{C}$, it is convenient to relabel the
simple line operators in the form $(a,e_{a})$ for some object
$a\in\mathcal{C}$ and half-braiding $e_{a}$. This can be done using the
explicit isomorphism between $\mathcal{Z}(\mathcal{C})$ and
$\mathcal{C}\times\bar{\mathcal{C}}$. We have (muger2003subfactors, , Theorem
7.10)

$(a,b)\mapsto(a\times b,e_{a\times b})~{},$ (115)

where

$e_{a\times b}(c):=(R_{a,c}^{-1}\times\text{id}_{b})\circ(\text{id}_{a}\times
R_{c,b}^{-1})~{},$ (116)

with $R_{a,b}:a\times b\xrightarrow{\sim}b\times a$ being the braiding on
$\mathcal{C}$.

####  6.2.1 Fibonacci Category

Let $\mathcal{C}$ be the Fibonacci MTC with the non-trivial simple line
operator $\tau$. The SymTFT in this case is

$\mathcal{Z}(\mathcal{C})=\mathcal{C}\times\bar{\mathcal{C}}~{}.$ (117)

Therefore, the simple line operators in the SymTFT can be labeled as

$(\mathbf{1},\mathbf{1}),(\mathbf{1},\tau),(\tau,\mathbf{1}),(\tau,\tau)~{}.$
(118)

The line operators $(\mathbf{1},\mathbf{1})$ and $(\tau,\tau)$ are bosonic.
For identifying gaugeable algebras in $\mathcal{C}$, we have to describe the
simple objects in the notation $(a,e_{a})$ where $a\in\mathcal{C}$ and $e_{a}$
is the half-braiding. Using (115) we get181818The line operators in
$\mathcal{Z}(\mathcal{C})$ can also be directly obtained in the form
$(a,e_{a})$ using the string-net construction (Levin:2004mi, , Section VI. B).

$(\mathbf{1},e_{\mathbf{1}}),(\tau,e_{\tau}),(\tau,e^{\prime}_{\tau}),(\mathbf{1}+\tau,e_{\mathbf{1}+\tau})~{}.$
(119)

We have a unique Lagrangian algebra

$L=(\mathbf{1},e_{\mathbf{1}})+(\mathbf{1}+\tau,e_{\mathbf{1}+\tau})~{}.$
(120)

Fusing $L$ with the gapped boundary $\mathcal{B}_{L}$, we find the algebra
object

$\mathbf{1}+\mathbf{1}+\tau~{},$ (121)

which is non-haploid. We can decompose this algebra into haploid subalgebras

$A_{1}=\mathbf{1},A_{2}=\mathbf{1}+\tau~{}.$ (122)

The fact that both of these algebras correspond to the same gapped boundary
implies that they are Morita equivalent. Indeed, gauging $\mathbf{1}+\tau$ is
a self-duality of a 1+1D QFT with $\mathcal{C}$ symmetry Choi:2023xjw ;
Diatlyk:2023fwf .

####  6.2.2 Ising Category

Let $\mathcal{C}$ be the Ising MTC with non-trivial line operators
$\psi,\sigma$. The SymTFT in this case is

$\mathcal{Z}(\mathcal{C})=\mathcal{C}\times\bar{\mathcal{C}}~{}.$ (123)

Therefore, the simple line operators in the SymTFT can be labeled as

$\displaystyle(\mathbf{1},\mathbf{1}),(\mathbf{1},\psi),(\mathbf{1},\sigma)~{},$
(124) $\displaystyle(\psi,\mathbf{1}),(\psi,\psi),(\psi,\sigma)~{},$ (125)
$\displaystyle(\sigma,\mathbf{1}),(\sigma,\psi),(\sigma,\sigma)~{}.$ (126)

The line operators $(\mathbf{1},\mathbf{1}),(\psi,\psi)$ and $(\sigma,\sigma)$
are bosonic. Using (115) we can relabel the simple line operators as

$\displaystyle(\mathbf{1},e^{(1)}_{\mathbf{1}}),(\psi,e^{(1)}_{\psi}),(\sigma,e^{(1)}_{\sigma})$
(127)
$\displaystyle(\psi,e^{(2)}_{\psi}),(\mathbf{1},e^{(2)}_{\mathbf{1}}),(\sigma,e^{(2)}_{\sigma})$
(128)
$\displaystyle(\sigma,e^{(2)}_{\sigma}),(\sigma,e^{(3)}_{\sigma}),(\mathbf{1}+\psi,e_{\mathbf{1}+\psi})~{}.$
(129)

The only Lagrangian algebra is

$L=(\mathbf{1},e^{(1)}_{\mathbf{1}})+(\mathbf{1},e^{(2)}_{\mathbf{1}})+(\mathbf{1}+\psi,e_{\mathbf{1}+\psi})~{}.$
(130)

Fusing $L$ with the gapped boundary $\mathcal{B}_{L}$ we get the algebra
object

$\mathbf{1}+\mathbf{1}+\mathbf{1}+\psi~{}.$ (131)

This algebra is a sum of three haploid subalgebras

$A_{1}=A_{2}=\mathbf{1},~{}A_{3}=\mathbf{1}+\psi~{}.$ (132)

This is consistent with the fact that $A_{3}$ is a Morita trivial algebra
Fuchs:2004dz ; Choi:2023xjw ; Diatlyk:2023fwf .

###  6.3 Non-invertible symmetries

####  6.3.1 Rep$(S_{3})$

For Rep$(S_{3})$ the SymTFT is dual to $\mathcal{Z}(\text{Vec}_{S_{3}})$ for
which we have already considered all the Lagrangian algebras. Recall the
simple line operators

$([e],\mathds{1}_{e}),([e],\pi_{1}),([e],\pi_{2}),([r],\mathds{1}_{r}),([r],\omega),([r],\omega^{2}),([s],\mathds{1}_{s}),([s],\gamma)~{}.$
(133)

and gapped boundaries

$\displaystyle L_{1}:=([e],\mathds{1}_{e})+([e],\pi_{1})+2([e],\pi_{2})~{},$
(134) $\displaystyle
L_{2}:=([e],\mathds{1}_{e})+([e],\pi_{2})+([s],\mathds{1}_{s})~{},$ (135)
$\displaystyle
L_{3}:=([e],\mathds{1}_{e})+([e],\pi_{1})+2([r],\mathds{1}_{r})~{},$ (136)
$\displaystyle
L_{4}:=([e],\mathds{1}_{e})+([s],\mathds{1}_{s})+([r],\mathds{1}_{r})~{}.$
(137)

For identifying the algebra objects in Rep$(S_{3})$, we have to choose a
gapped boundary of $\mathcal{Z}(\text{Vec}_{S_{3}})$ such that the boundary
line operators form Rep$(S_{3})$. For example, we can choose the gapped
boundary given by

$L_{2}=([e],\mathds{1}_{e})+([e],\pi_{2})+([s],\mathds{1}_{s})~{}.$ (138)

Let $\mathds{1},\pi_{1},\pi_{2}$ be the three irreducible representations of
$S_{3}$. Fusing the simple line operators in $\mathcal{Z}(\text{Vec}_{S_{3}})$
on $\mathcal{B}_{L_{2}}$, we get (see, for example, (Cong:2017ffh, , Section
2.6))

$\displaystyle
F_{\mathcal{B}_{L_{2}}}(([e],\mathds{1}_{e}))=\mathds{1}~{},~{}F_{\mathcal{B}_{L_{2}}}(([e],\pi_{1}))=\pi_{1}~{},~{}F_{\mathcal{B}_{L_{2}}}(([e],\pi_{2}))=\mathds{1}+\pi_{1}~{},$
(139) $\displaystyle
F_{\mathcal{B}_{L_{2}}}(([r],\mathds{1}_{r}))=F_{\mathcal{B}_{L_{2}}}(([r],\omega))=F_{\mathcal{B}_{L_{2}}}(([r],\omega^{2}))=\pi_{2}~{},$
(140) $\displaystyle
F_{\mathcal{B}_{L_{2}}}(([s],\mathds{1}_{s}))=\mathds{1}+\pi_{2}~{},~{}F_{\mathcal{B}_{L_{2}}}(([s],\gamma))=\pi_{1}+\pi_{2}~{}.$
(141)

Fusing $L_{1}$ on the gapped boundary $\mathcal{B}_{L_{2}}$ results in the
non-anomalous line operator

$3\mathds{1}+3\pi_{1}~{}.$ (142)

It decomposes into three copies of the haploid algebra $\mathds{1}+\pi_{1}$.
Fusing $L_{2}$ on the gapped boundary $\mathcal{B}_{L_{2}}$ results in the
non-anomalous line operator

$2\mathds{1}+\mathds{1}+\pi_{1}+\pi_{2}~{}.$ (143)

It decomposes into the haploid algebras

$A_{1}=\mathds{1}~{},~{}A_{2}=\mathds{1}+\pi_{1}+\pi_{2}~{}.$ (144)

Note that this implies that gauging $A_{2}$ is a self-duality as $A_{2}$ is a
Morita trivial algebra. Fusing $L_{3}$ on the gapped boundary
$\mathcal{B}_{L_{2}}$ results in the non-anomalous line operator

$\mathds{1}+\pi_{1}+2\pi_{2}~{}.$ (145)

This is a haploid algebra corresponding to the unique fibre functor admitted
by Rep$(S_{3})$. Fusing $L_{4}$ on the gapped boundary $\mathcal{B}_{L_{2}}$
results in the non-anomalous line operator

$2\mathds{1}+2\pi_{2}~{}.$ (146)

It decomposes into two copies of the haploid algebra object
$\mathds{1}+\pi_{2}$. The algebra objects obtained above agree with the
classification of algebras in Rep$(S_{3})$ given in Perez-Lona:2023djo .

###  6.4 Lagrangian algebras in $\mathcal{Z}(\text{Vec}_{D_{8}})$ from
algebras in Vec${}_{D_{8}}$

Given the non-anomalous line operators in a fusion category $\mathcal{C}$, we
can use Theorem 5.1 to put strong constraints on consistent Lagrangian algebra
objects in its SymTFT. In this section, we will show that all Lagrangian
algebra objects in the SymTFT $\mathcal{Z}(\text{Vec}_{D_{8}})$, where $D_{8}$
is the dihedral group of 8 elements, can be completely fixed using Theorem
5.1.

Consider the invertible symmetry Vec${}_{D_{8}}$ where

$D_{8}:=\langle r,s~{}|~{}r^{4}=s^{2}=e,srs=r^{3}\rangle~{}.$ (147)

Since this $D_{8}$ is chosen to be non-anomalous, any subgroup of $D_{8}$ can
be gauged. The physically inequivalent gaugings correspond to conjugacy
classes of subgroups of $D_{8}$ and a choice of discrete torsion. The
conjugacy classes of subgroups are as follows. The normal subgroups are

$\\{e\\}~{},~{}\\{e,r^{2}\\}~{},~{}\\{e,r,r^{2},r^{3}\\}~{},~{}\\{e,r^{2},s,r^{2}s\\}~{},~{}\\{e,r^{2},rs,r^{3}s\\}~{},~{}D_{8}~{}.$
(148)

We will denote these groups as
$\mathds{Z}_{1},\mathds{Z}_{2}^{(r^{2})},\mathds{Z}_{4}^{(r)},\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(s)},\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(rs)},D_{8}$,
respectively. There are four non-normal subgroups which form the following two
conjugacy classes of subgroups

$\\{\\{e,s\\},\\{e,r^{2}s\\}\\}~{},~{}\\{\\{e,rs\\},\\{e,r^{3}s\\}\\}~{}.$
(149)

All groups in these two conjugacy classes are isomorphic to $\mathds{Z}_{2}$,
and we will denote them by the representatives $\mathds{Z}_{2}^{(s)}$ and
$\mathds{Z}_{2}^{(rs)}$, respectively. A physically inequivalent gaugings in
Vec${}_{D_{8}}$ can be labeled as $(G,\sigma)$ where $G$ is a representative
of a conjugacy class of subgroups in $D_{8}$ and $\sigma\in H^{2}(G,U(1))$ is
the discrete torsion. Using the explicit data above, we get the following 11
physically inequivalent gaugings in Vec${}_{D_{8}}$

$(\mathds{Z}_{1},1)~{},~{}(\mathds{Z}_{2}^{(r^{2})},1)~{},~{}(\mathds{Z}_{4}^{(r)},1)~{},~{}(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(s)},1)~{},~{}(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(s)},\alpha)~{},~{}$
(150)
$(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(rs)},1)~{},~{}(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(rs)},\beta)~{},~{}(D_{8},1)~{},~{}(D_{8},\gamma)~{},~{}(\mathds{Z}_{2}^{(s)},1)~{},~{}(\mathds{Z}_{2}^{(rs)},1)~{}.$
(151)

where $\alpha,\beta,\gamma$ denote the choice of non-trivial discrete torsion
in
$H^{2}(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(s)},U(1))\cong\mathds{Z}_{2},$
$H^{2}(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(rs)},U(1))\cong\mathds{Z}_{2}$
and $H^{2}(D_{8},U(1))\cong\mathds{Z}_{2}$, respectively. We will relate these
$11$ algebra objects to gapped boundaries of the SymTFT of Vec${}_{D_{8}}$.

The SymTFT $\mathcal{Z}(\text{Vec}_{D_{8}})$ is the Dijkgraaf-Witten theory
with gauge group $D_{8}$. The conjugacy classes of $D_{8}$ are

$[e]=\\{e\\}~{},~{}[r^{2}]=\\{r^{2}\\}~{},~{}[s]=\\{s,r^{2}s\\}~{},~{}[rs]=\\{rs,r^{3}s\\}~{},~{}[r]=\\{a^{2},a^{3}\\}~{}.$
(152)

The centralizers of the representative of these conjugacy classes are

$D_{8}~{},~{}D_{8}~{},~{}\\{e,r^{2},s,r^{2}s\\}\cong\mathds{Z}_{2}\times\mathds{Z}_{2}~{},~{}\\{e,r^{2},rs,r^{3}s\\}\cong\mathds{Z}_{2}\times\mathds{Z}_{2}~{},~{}\\{e,r,r^{2},r^{3}\\}\cong\mathds{Z}_{4}~{},$
(153)

respectively. The simple line operators in $\mathcal{Z}(\text{Vec}_{D_{8}})$,
their quantum dimensions and spins are given by

Line operator | $([e],\mathds{1})$ | $([e],\pi_{1})$ |  $([e],\pi_{2})$, | $([e],\pi_{3})$ | $([e],\pi_{4})$
---|---|---|---|---|---
$d_{x}$ | 1 | 1 | 1 | 1 | 2
$\theta_{x}$ | 1 | 1 | 1 | 1 | 1
Line operator | $([r^{2}],\mathds{1})$ | $([r^{2}],\pi_{1})$ |  $([r^{2}],\pi_{2})$, | $([r^{2}],\pi_{3})$ | $([r^{2}],\pi_{4})$
---|---|---|---|---|---
$d_{x}$ | 1 | 1 | 1 | 1 | 2
$\theta_{x}$ | 1 | 1 | 1 | 1 | -1
Line operator | $([s],\mathds{1}_{s})$ | $([s],\omega_{1})$ |  $([s],\omega_{2})$, | $([s],\omega_{1}\omega_{2})$
---|---|---|---|---
$d_{x}$ | 2 | 2 | 2 | 2
$\theta_{x}$ | 1 | -1 | 1 | -1
Line operator | $([rs],\mathds{1}_{rs})$ | $([rs],\tilde{\omega}_{1})$ |  $([rs],\tilde{\omega}_{2})$, | $([rs],\tilde{\omega}_{1}\tilde{\omega}_{2})$
---|---|---|---|---
$d_{x}$ | 2 | 2 | 2 | 2
$\theta_{x}$ | 1 | -1 | 1 | -1
Line operator | $([r],\mathds{1}_{r})$ | $([r],\tilde{\omega})$ |  $([r],\tilde{\omega}^{2})$, | $([r],\tilde{\omega}^{3})$
---|---|---|---|---
$d_{x}$ | 2 | 2 | 2 | 2
$\theta_{x}$ | 1 | i | -1 | i

where $\mathds{1}$, $\pi_{1}$, $\pi_{2}$, $\pi_{3}$ are the 1-dimensional
irreducible representations of $D_{8}$ and $\pi_{4}$ is the 2-dimensional one;
$\pi_{1}$, $\pi_{2}$ and $\pi_{3}$ are the irreducible representations with
kernels
$\mathds{Z}_{4}^{(r)},\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(s)}$ and
$\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(rs)}$, respectively;
$\mathds{1}_{s}$, $\omega_{1}$, $\omega_{2}$ and $\omega_{1}\omega_{2}$ are
the four irreducible representations of the
$\mathds{Z}_{2}\times\mathds{Z}_{2}$ centralizer of $s$; $\mathds{1}_{rs}$,
$\tilde{\omega}_{1}$, $\tilde{\omega}_{2}$ and
$\tilde{\omega}_{1}\tilde{\omega}_{2}$ are the four irreducible
representations of the $\mathds{Z}_{2}\times\mathds{Z}_{2}$ centralizer of
$rs$; $\mathds{1}_{r}$, $\omega$, $\omega^{2}$ and $\omega^{3}$ are the four
irreducible representations of the $\mathds{Z}_{4}$ centralizer of $r$. The
fusion rules of these line operators and their modular $S$ matrix are given in
(Iqbal:2023wvm, , Appendices A,B).

A line operator in $\mathcal{Z}(\text{Vec}_{D_{8}})$ which admits the
structure of a Lagrangian algebra must be a direct sum of the bosonic line
operators in the list above. In fact, using Theorem 5.2, we can identify all
Lagrangian algebra objects in the SymTFT. For example, consider the algebra
$(\mathds{Z}_{2}^{(rs)},1)$. We have two Morita equivalent classes of algebras

$A_{1}:=e+rs~{},~{}A_{2}=e+r^{3}s.$ (154)

We need to choose a Lagrangian algebra object $L$ in the SymTFT such that
$F(L)$ produces the above two algebras. Note that the Lagrangian algebra $L$
has quantum dimension $8$. Since
$F:\mathcal{Z}(\text{Vec}_{D_{8}})\to\text{Vec}_{D_{8}}$ preserves quantum
dimensions, we must have

$F(L)=2A_{1}+2A_{2}~{},~{}\text{or }F(L)=A_{1}+3A_{2}~{},~{}\text{or
}F(L)=3A_{1}+A_{2}~{}.$ (155)

Therefore, $L$ must be of the form

$\displaystyle L=$
$\displaystyle([e],\mathds{1})+N_{L}^{([e],\pi_{1})}([e],\pi_{1})+N_{L}^{([e],\pi_{2})}([e],\pi_{2})+N_{L}^{([e],\pi_{3})}([e],\pi_{3})+N_{L}^{([e],\pi_{4})}([e],\pi_{4})+$
(157) $\displaystyle
N_{L}^{([rs],\mathds{1}_{rs})}([rs],\mathds{1}_{rs})+N_{L}^{([rs],\tilde{\omega}_{2})}([rs],\tilde{\omega}_{2})~{}.$

where $N_{L}^{*}$ are non-negative integers which need to be determined. Since
the conjugacy class $[rs]=\\{rs,r^{3}s\\}$, $F(L)$ contains equal number of
$rs$ and $r^{3}s$ terms. This implies that $F(L)=A_{1}+3A_{2}$ and
$F(L)=3A_{1}+A_{2}$ are inconsistent. Therefore, we must have
$F(L)=2A_{1}+2A_{2}$.

Suppose $N_{L}^{([rs],\mathds{1}_{rs})}\neq 0$. We will now show that
$N_{L}^{([rs],\mathds{1}_{rs})}=1$. To see this, suppose
$N_{L}^{([rs],\mathds{1}_{rs})}>1$. We have the fusion rule (Iqbal:2023wvm, ,
Appendix B)

$([rs],\mathds{1}_{rs})\times([rs],\mathds{1}_{rs})=([e],\pi_{4})+([r^{2}],\pi_{4})~{}.$
(158)

Using the constraint (11) we find that

$N_{L}^{([e],\pi_{4})}\geq 4~{}.$ (159)

However, this contradicts the requirement that $F(L)=2A_{1}+2A_{2}$ contains
exactly $4$ trivial line operators. Therefore, we must have
$N_{L}^{([rs],\mathds{1}_{rs})}=1$. Since $F(L)$ must contain two copies of
$rs$, we also find that $N_{L}^{([rs],\tilde{\omega}_{2})}=1$. Now, consider
the fusion rule

$([rs],\mathds{1}_{rs})\times([rs],\tilde{\omega}_{2})=([e],\pi_{4})+([r^{2}],\pi_{4})~{}.$
(160)

Since $([r^{2}],\pi_{4})$ is a fermion, using (11) we find that
$N_{L}^{([e],\pi_{4})}=1$. Combining the results above, we have determined
that the Lagrangian algebra must be of the form

$\displaystyle L=$
$\displaystyle([e],\mathds{1})+N_{L}^{([e],\pi_{1})}([e],\pi_{1})+N_{L}^{([e],\pi_{2})}([e],\pi_{2})+N_{L}^{([e],\pi_{3})}([e],\pi_{3})+([e],\pi_{4})+$
(162) $\displaystyle([rs],\mathds{1}_{rs})+([rs],\tilde{\omega}_{2})~{}.$

$L$ must have quantum dimension $8$. Therefore, only one among
$N_{L}^{([e],\pi_{1})}$, $N_{L}^{([e],\pi_{2})}$, $N_{L}^{([e],\pi_{3})}$ can
be non-zero. Using the fusion rules, we know that $([e],\pi_{3})$ is the only
line operator such that

$([rs],\tilde{\omega}_{2})\times([e],\pi_{2})\text{ contains
}([rs],\mathds{1}_{rs})~{},$ (163)

and

$([rs],\mathds{1}_{rs})\times([e],\pi_{2})\text{ contains
}([rs],\tilde{\omega}_{2})~{},$ (164)

which implies that $N_{L}^{([e],\pi_{3})}=1$. Therefore, the Lagrangian
algebra object corresponding to the algebra $(\mathds{Z}_{2}^{(rs)},1)$ is

$\displaystyle
L=([e],\mathds{1})+([e],\pi_{3})+([e],\pi_{4})+([rs],\mathds{1}_{rs})+([rs],\tilde{\omega}_{2})~{}.$
(165)

Similar analysis can be used to determine all Lagrangian algebra objects in
the SymTFT $\mathcal{Z}(\text{Vec}_{D_{8}})$. They are summarized in the table
below.

$\displaystyle\begin{array}[]{|c|c|}\hline\cr\text{Algebra objects in
Vec}_{D_{8}}&\text{Lagrangian algebra objects in
}\mathcal{Z}(\text{Vec}_{D_{8}})\\\
\hline\cr(\mathds{Z}_{1},1)&([e],\mathds{1})+([e],\pi_{1})+([e],\pi_{2})+([e],\pi_{3})+2([e],\pi_{4})\\\
\hline\cr(\mathds{Z}_{2}^{(r^{2})},1)&\begin{tabular}[]{@{}c@{}}$([e],\mathds{1})+([e],\mathds{\pi_{1}})+([e],\pi_{2})+([e],\pi_{3})$\\\
\+
$([r^{2}],\mathds{1})+([r^{2}],\pi_{1})+([r^{2}],\pi_{2})+([r^{2}],\pi_{3})$\end{tabular}\\\
\hline\cr(\mathds{Z}_{4}^{(r)},1)&([e],\mathds{1})+([e],\pi_{1})+2([r],\mathds{1}_{r})+([r^{2}],\mathds{1})+([r^{2}],\pi_{1})\\\
\hline\cr(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(s)},1)&([e],\mathds{1})+([e],\pi_{2})+2([s],\mathds{1})+([r^{2}],\mathds{1})+([r^{2}],\pi_{2})\\\
\hline\cr(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(s)},\alpha)&([e],\mathds{1})+([e],\pi_{2})+2([s],\omega_{2})+([r^{2}],\pi_{1})+([r^{2}],\pi_{3})\\\
\hline\cr(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(rs)},1)&([e],\mathds{1})+([e],\pi_{3})+2([rs],\mathds{1}_{rs})+([r^{2}],\mathds{1})+([r^{2}],\pi_{3})\\\
\hline\cr(\mathds{Z}_{2}^{(r^{2})}\times\mathds{Z}_{2}^{(rs)},\beta)&([e],\mathds{1})+([e],\pi_{3})+2([rs],\tilde{\omega}_{2})+([r^{2}],\pi_{1})+([r^{2}],\pi_{2})\\\
\hline\cr(D_{8},1)&([e],\mathds{1})+([r^{2}],\mathds{1})+([s],\mathds{1}_{s})+([r],\mathds{1}_{r})+([rs],\mathds{1}_{rs})\\\
\hline\cr(D_{8},\gamma)&([e],\mathds{1})+([r^{2}],\pi_{1})+([s],\omega_{2})+([r],\mathds{1}_{r})+([rs],\tilde{\omega}_{2})\\\
\hline\cr(\mathds{Z}_{2}^{(s)},1)&([e],\mathds{1})+([e],\pi_{2})+([e],\pi_{4})+([s],\mathds{1}_{s})+([s],\omega_{2})\\\
\hline\cr(\mathds{Z}_{2}^{(rs)},1)&([e],\mathds{1})+([e],\pi_{3})+([e],\pi_{4})+([rs],\mathds{1}_{rs})+([rs],\tilde{\omega}_{2})\\\
\hline\cr\end{array}$ (180)

##  7 Applications

###  7.1 Higher-gauging data from the action of surface operators

Consider a 2+1D TQFT whose line operators are described by a modular tensor
category $\mathcal{C}$. Let $S$ be a topological surface operator whose action
on the line operators is given by

$S\cdot a=\sum_{b\in\mathcal{C}}n_{S\cdot a}^{b}~{}b~{}.$ (181)

$S$ implements a non-invertible symmetry if the sum above is non-trivial. All
surface operators in a 2+1D TQFT can be constructed through higher-gauging
Kapustin:2010if ; Fuchs:2012dt ; Roumpedakis:2022aik . Given (181), we can
ask: Which algebra $A\in\mathcal{C}$ should be higher-gauged to construct the
surface operator $S$? When the higher-gauging involves only invertible line
operators, this was answered in Buican:2023bzl  by providing an explicit map
between the action of the surface operator and the abelian group of lines $+$
discrete torsion data for higher-gauging. In this section, we give a general
answer to this question for arbitrary higher-gaugings.

If $n_{S\cdot a}^{b}\neq 0$, then $a$ and $b$ can form a non-trivial junction
on the surface operator (see Fig. 21).

$S$$a$$b$$\mathcal{C}$$\mathcal{C}$ Figure 21: Action of a topological surface
operator $S$ on line operators.

Upon folding Fig. 21 the surface operator becomes a gapped boundary of the
TQFT $\mathcal{C}\times\bar{\mathcal{C}}$ on which the line operator
$(a,\bar{b})$ can end (see Fig. 22).

$\mathcal{B}$$\mathcal{C}\times\overline{\mathcal{C}}$$(a,\overline{b})$$\Longrightarrow$$S$$a$$b$$\mathcal{C}$$\mathcal{C}$
Figure 22: Folding trick to produce a gapped boundary from a surface operator.

Therefore, the action of the surface operator (181) specifies a Lagrangian
algebra object

$L_{S}=\sum_{a,b\in\mathcal{C}}n_{S\dot{a}}^{b}~{}(a,\bar{b})~{}.$ (182)

In fact, the action of $S$ on the fusion spaces of the line operators also
determines the Lagrangian algebra structure on $L_{S}$ (see Buican:2023bzl
for more details). Consider the fusion of $L_{S}$ on the canonical gapped
boundary $\mathcal{B}_{\mathcal{C}}$ of $\mathcal{C}\times\bar{\mathcal{C}}$.
From Theorem 5.1, we know that $F(L_{S})$ is a sum of Morita equivalent
algebras in $\mathcal{C}$. This is precisely the algebra which can be higher-
gauged to construct the surface $S$.

Remark: For a modular tensor category $\mathcal{C}$, one can define the
functors $T:\mathcal{C}\times\bar{\mathcal{C}}\to\mathcal{C}$ and
$\mathcal{C}\to\mathcal{C}\times\bar{\mathcal{C}}$ as follows Kong:2007yv ;
Kong:2009inh

$\displaystyle T:(a,b)\to a\times b~{},$ (183) $\displaystyle
R:a\to\sum_{c\in\mathcal{C}}~{}(a\times\bar{c},c)~{}.$ (184)

These functors can also be used to transport algebras between $\mathcal{C}$
and $\mathcal{C}\times\bar{\mathcal{C}}$. In fact, the functor $T$ is the same
as the forgetful functor $F:\mathcal{Z}(\mathcal{C})\to\mathcal{C}$ under the
equivalence
$\mathcal{Z}(\mathcal{C})\cong\mathcal{C}\boxtimes\bar{\mathcal{C}}$.
Nevertheless, it maybe useful to use $F$ instead of $T$ since the isomorphisms
$\Phi_{x,y}$ studied in section 5.2 are trivial for $F$ while it is non-
trivial for $T$.

###  7.2 A necessary condition for trivial anomaly

Let $A$ be a non-simple line operator in $\mathcal{C}$. Suppose $A$ is non-
anomalous. Then we know that there must be a Lagrangian algebra $L$
corresponding to $A$. Using Theorem 5.1, we must have

$A\subseteq F(L)~{}.$ (185)

More explicitly, if

$L=\sum_{(a,e_{a})\in\mathcal{Z}(\mathcal{C})}N_{L}^{(a,e_{a})}~{}(a,e_{a})~{},$
(186)

then, we must have

$\sum_{(a,e_{a})\in\mathcal{Z}(\mathcal{C})}N_{L}^{(a,e_{a})}N_{a}^{b}\neq
0~{}\forall~{}b\in A~{}.$ (187)

where $N_{a}^{b}$ is non-zero if the line operators $a$ and $b$ can form a
topological junction. For an arbitrarily chosen $A$, we may not have bosonic
line operators $(a,e_{a})\in\mathcal{Z}(\mathcal{C})$ satisfying (187)
implying that $A$ must be anomalous. In the next section, we will use this
necessary condition to prove a statement about $\mathcal{C}$-symmetric
trivially gapped phases.

####  7.2.1 Trivially gapped phase $\iff$ magnetic Lagrangian algebra

In Zhang:2023wlu  the authors argue that a fusion category $\mathcal{C}$
admits a trivially gapped phase only if the SymTFT $\mathcal{Z}(\mathcal{C})$
admits a magnetic Lagrangian algebra $L_{m}$ defined by the property that the
only simple line operator $L_{m}$ which can also end on the canonical gapped
boundary $\mathcal{B}_{\mathcal{C}}$ is the trivial line operator. This is
clear from the sandwich picture which shows that the local operators of the
1+1D TQFT $\mathcal{T}_{L}$ correspond to the line operators which can end
both on $\mathcal{B}_{L_{m}}$ and $\mathcal{B}_{\mathcal{C}}$ (see (17)). In
this section, we will use Theorem 5.1 to show that this is a necessary and
sufficient condition for the existence of a $\mathcal{C}$-symmetric trivially
gapped phase.

Recall the largest algebra object that a fusion category $\mathcal{C}$ can
admit is of the form

$A=\sum_{a\in\mathcal{C}}d_{a}~{}a~{}.$ (188)

In fact, $A$ admits the structure of an algebra (is non-anomalous) if and only
if $\mathcal{C}$ admits a trivially gapped phase. To see this, suppose
$\mathcal{C}$ admits a $\mathcal{C}$-symmetric TQFT that is trivially gapped.
Then this TQFT only has the identity local operator. Correspondingly, there is
only one simple 1D gapped boundary, say $\operatorname{\text{m}}$. Therefore
the module category $\mathcal{M}$ has only one simple object,
$\operatorname{\text{m}}$. Consider the algebra

$A_{\operatorname{\text{m}}}:=\sum_{a\in\mathcal{C}}N_{\operatorname{\text{m}}}^{a}~{}a~{},$
(189)

where $N_{\operatorname{\text{m}}}^{a}$ is the dimension of Hilbert space of
operators at the junction of $a$ with the gapped boundary
$\operatorname{\text{m}}$. Since $\operatorname{\text{m}}$ is the only gapped
boundary, all line operators in $\mathcal{C}$ must be able to form a junction
on it. Moreover, $N_{\operatorname{\text{m}}}^{a}=d_{a}$ to preserve quantum
dimensions. Therefore, we have

$A_{\operatorname{\text{m}}}=\sum_{a\in\mathcal{C}}~{}d_{a}~{}a~{}.$ (190)

Using ostrik2003module  (see section 4.2), we know that
$A_{\operatorname{\text{m}}}$ is non-anomalous.

Conversely, suppose

$A=\sum_{a\in\mathcal{C}}d_{a}~{}a~{},$ (191)

is non-anomalous. Consider the module category $\mathcal{M}$ corresponding to
this algebra. Let $\operatorname{\text{m}}\in\mathcal{M}$ and
$A_{\operatorname{\text{m}}}$ be the line operator as defined above. We have
the relation (etingof2010fusion, , Proposition 2.2)

$\sum_{\operatorname{\text{m}}}d_{\operatorname{\text{m}}}^{2}=\text{dim}(\mathcal{C})~{},$
(192)

where

$d_{\operatorname{\text{m}}}^{2}:=d_{A_{\operatorname{\text{m}}}}~{}.$ (193)

Now, since $\mathcal{M}$ is the module category defined using the algebra $A$,
there is some $\operatorname{\text{m}}^{\prime}\in\mathcal{M}$ such that

$A_{\operatorname{\text{m}}^{\prime}}=A~{}.$ (194)

Then, we have
$d_{A_{\operatorname{\text{m}}^{\prime}}}=d_{A}=\text{dim}(\mathcal{C})$. To
be compatible with the equation (192), $\operatorname{\text{m}}^{\prime}$ must
be the only simple object in $\mathcal{M}$. In other words, the
$\mathcal{C}$-symmetric TQFT defined using $\mathcal{M}$ has only one simple
gapped boundary. Consequently, it is trivially gapped and the whole symmetry
$\mathcal{C}$ is preserved. It follows from this argument that $A$ is not
Morita equivalent to any other haploid algebra.

From Theorem 5.1, we know that

$F(L)=\sum_{\operatorname{\text{m}}\in\mathcal{M}_{L}}A_{\operatorname{\text{m}}}~{}.$
(195)

The number of haploid algebras in the sum above is equal to the number of
identity line operators in $F(L)$. Since $F$ preserves quantum dimensions and
$d_{L}=\text{dim}(\mathcal{C})$ by definition, the largest haploid algebra
object that we can get on the R.H.S of the above equation is precisely $A$.
Therefore, we find

$F(L)=A$ (196)

if and only if $F(L)$ contains a single identity operator. Since $F(L)$ is
obtained from the perpendicular fusion of $L$ on the canonical gapped boundary
$\mathcal{B}_{\mathcal{C}}$, this implies that $L$ does not contain any non-
trivial line operators which can end on $\mathcal{B}_{\mathcal{C}}$.
Therefore, $L$ is a magnetic Lagrangian algebra with respect to
$\mathcal{B}_{\mathcal{C}}$.

###  7.3 Transporting non-anomalous line operators between fusion categories

Consider two fusion categories $\mathcal{C}$ and $\mathcal{D}$ which share the
same SymTFT. In many cases, $\mathcal{D}$ might be a simpler fusion category
in which the non-anomalous line operators can be explicitly classified, while
it could be hard to do the same in $\mathcal{C}$. For example, $\mathcal{C}$
could be a fusion category without multiplicity in its fusion coefficients
while $\mathcal{D}$ is not multiplicity-free.

Let $[B_{1}],...[B_{n}]$ be the $n$ Morita equivalence classes of algebra
objects/non-anomalous line operators in $\mathcal{D}$ for some integer $n$.
Using the data of this algebra, we can determine all the Lagrangian algebras
$L_{1},\dots,L_{n}$ of the SymTFT $\mathcal{Z}(\mathcal{D})$. As we will see
in explicit examples below, in many cases, even without knowledge of the
multiplication in the algebras in $\mathcal{D}$, we can determine the objects
$L_{1},...,L_{n}$ which admit the structure of a Lagrangian algebra. Now,
since the fusion category $\mathcal{C}$ share the same SymTFT, there must be a
boundary condition $\mathcal{B}_{\mathcal{C}}$ of $\mathcal{Z}(\mathcal{D})$
on which the line operators form the fusion category $\mathcal{C}$. The non-
anomalous line operators in $\mathcal{C}$ can then be determined by fusing
$L_{1},\dots,L_{n}$ with the gapped boundary $\mathcal{B}_{\mathcal{C}}$ to
get
$F_{\mathcal{B}_{\mathcal{C}}}(L_{1}),F_{\mathcal{B}_{\mathcal{C}}}(L_{2}),\dots,F_{\mathcal{B}_{\mathcal{C}}}(L_{n})$.
In this way, the non-anomalous line operators in $\mathcal{D}$ can be
‘transported’ through the SymTFTs to non-anomalous line operators in
$\mathcal{C}$.

In the following subsections, we will go through three examples in detail to
illustrate the idea described above. First we will consider the fusion
categories

$\mathcal{C}=\text{Rep}(D_{8}),~{}~{}~{}\mathcal{D}=\text{Vec}_{D_{8}}~{},$
(197)

where $D_{8}$ is the dihedral group with $8$ elements. We will use the results
from section 6.4 to determine all algebra objects in Rep$(D_{8})$ recovering
the classification of algebra objects of Rep$(D_{8})$ in Perez-Lona:2023djo ;
Diatlyk:2023fwf . In Appendix C, we use a similar method to classify the
algebra objects in $A_{4}$, where $A_{4}$ is the alternating group of order
$12$. This group is the smallest for which the category Rep$(A_{4})$ has
multiplicities in its fusion rules. We will show how the non-anomalous line
operators in Vec${}_{A_{4}}$ can be used to determine the non-anomalous line
operators in Rep$(A_{4})$. The final example that we consider is the three
Morita equivalent Haagerup fusion categories.

$\mathcal{C}=\mathscr{H}_{1},~{}~{}~{}~{}\mathcal{D}=\mathscr{H}_{2},~{}~{}~{}\mathcal{E}=\mathscr{H}_{3}~{}.$
(198)

####  7.3.1 Rep$(D_{8})$ and Vec${}_{D_{8}}$

Rep$(D_{8})$ is a non-invertible symmetry of various CFTs Choi:2023vgk ;
Diatlyk:2023fwf  and lattice models Seifnashri:2024dsd . In this section, we
will identify the non-anomalous line operators (algebra objects) in
Rep$(D_{8})$ using the Lagrangian algebras in $\mathcal{Z}(\text{Rep}(D_{8}))$
and Theorem 5.1.

Note that the SymTFT $\mathcal{Z}(\text{Vec}_{D_{8}})$ is the same as
$\mathcal{Z}(\text{Rep}(D_{8}))$ since the symmetries Vec${}_{D_{8}}$ and
Rep$(D_{8})$ are dual to each other under gauging. Therefore, the Lagrangian
algebras in $\mathcal{Z}(\text{Rep}(D_{8}))$ are the same as in
$\mathcal{Z}(\text{Vec}_{D_{8}})$ which were constructed explicitly in section
6.4. Recall that the bosonic line operators in
$\mathcal{Z}(\text{Vec}_{D_{8}})$ are

$\displaystyle([e],\mathds{1})~{},~{}([e],\pi_{1})~{},~{}([e],\pi_{2})~{},~{}([e],\pi_{3})~{},~{}([e],\pi_{4})~{},~{}$
(199)
$\displaystyle([r^{2}],\mathds{1})~{},~{}([r^{2}],\pi_{1})~{},~{}([r^{2}],\pi_{2})~{},~{}([r^{2}],\pi_{3})~{},~{}([s],\mathds{1}_{s})~{},~{}$
(200)
$\displaystyle([s],\omega_{2})~{},~{}([rs],\mathds{1}_{rs})~{},~{}([rs],\tilde{\omega}_{2})~{},~{}([r],\mathds{1}_{r})~{}.$
(201)

All Lagrangian algebras in $\mathcal{Z}(\text{Vec}_{D_{8}})$ are certain
linear combinations of these bosons. The Lagrangian algebra

$L_{\text{Rep}(D_{8})}=([e],\mathds{1})+([r^{2}],\mathds{1})+([s],\mathds{1}_{s})+([r],\mathds{1}_{r})+([rs],\mathds{1}_{rs})~{}.$
(202)

specifies a gapped boundary on which the line operators form the fusion
category Rep$(D_{8})$. Therefore, in order to determine the algebra objects in
Rep$(D_{8})$ we can use the map

$F_{\mathcal{B}_{\text{Rep}(D_{8})}}:\mathcal{Z}(\text{Rep}(D_{8}))\to\text{Rep}(D_{8})~{},$
(203)

and using Theorem 5.1. In order to determine
$F_{\mathcal{B}_{\text{Rep}(D_{8})}}$ it is useful to write the bosonic line
operators in the form $(a,e_{a})$ for some $a\in\text{Rep}(D_{8})$ and half-
braiding $e_{a}$. Then the map $F_{\mathcal{B}_{\text{Rep}(D_{8})}}$ is just
given by

$F_{\mathcal{B}_{\text{Rep}(D_{8})}}((a,e_{a}))=a~{}.$ (204)

Using the construction of $\mathcal{Z}(\text{Rep}(D_{8}))$ as the Drinfeld
center of the Tambara-Yamagami category Rep$(D_{8})$ we have gelaki2009centers
; Zhang:2023wlu

$\displaystyle([e],\mathds{1})\to(\mathds{1},e_{\mathds{1}})~{},~{}([e],\pi_{1})\to(\pi_{1},e_{\pi_{1}})~{},~{}([e],\pi_{2})\to(\pi_{2},e_{\pi_{2}})~{},~{}([e],\pi_{3})\to(\pi_{3},e_{\pi_{3}})~{},~{}$
(205)
$\displaystyle([e],\pi_{4})\to(\pi_{4},e^{(1)}_{\pi_{4}})~{},~{}([r^{2}],\mathds{1})\to(\mathds{1},e^{\prime}_{\mathds{1}})~{},~{}([r^{2}],\pi_{1})\to(\pi_{1},e^{\prime}_{\pi_{1}})~{},~{}$
(206)
$\displaystyle([r^{2}],\pi_{2})\to(\pi_{2},e^{\prime}_{\pi_{2}})~{},~{}([r^{2}],\pi_{3})\to(\pi_{3},e^{\prime}_{\pi_{3}})~{},~{}([s],\mathds{1}_{s})\to(\mathds{1}+\pi_{2},e_{\mathds{1}+\pi_{2}})~{},~{}$
$\displaystyle([s],\omega_{2})\to(\pi_{4},e^{(2)}_{\pi_{4}})~{},~{}([rs],\mathds{1}_{rs})\to(1+\pi_{3},e_{1+\pi_{3}})~{},~{}([rs],\tilde{\omega}_{2})\to(\pi_{4},e^{(3)}_{\pi_{4}})~{},~{}$
(207)
$\displaystyle([r],\mathds{1}_{r})\to(\mathds{1}+\pi_{1},e_{\mathds{1}+\pi_{1}})~{}.$
(208)

where the explicit form of the half-braidings is not specified as they will
not play a role in the following discussion. In general, the line operators in
$\mathcal{Z}(\text{Vec}_{D_{8}})$ and $\mathcal{Z}(\text{Rep}(D_{8}))$ are
related by

$([g],\pi_{g})\to(\text{Ind}_{C_{g}}^{D_{8}}(\pi_{g}),e)~{},$ (209)

for some half-braiding $e$ and $\text{Ind}_{C_{g}}^{D_{8}}(\pi_{g})$ is the
induction of the representation $\pi_{g}$ of $C_{g}$ to the full group
$D_{8}$. The Lagrangian algebra objects in $\mathcal{Z}(\text{Rep}(D_{8}))$
and the corresponding non-anomalous line operators in Rep$(D_{8})$ are given
in the following table.

$\displaystyle\begin{array}[]{|c|c|}\hline\cr\text{Lagrangian algebra objects
in }\mathcal{Z}(\text{Rep}(D_{8}))&\text{Non-anomalous lines in Rep}(D_{8})\\\
\hline\cr(\mathds{1},e_{\mathds{1}})+(\pi_{1},e_{\pi_{1}})+(\pi_{2},e_{\pi_{2}})+(\pi_{3},e_{\pi_{3}})+2(\pi_{4},e_{\pi_{4}})&\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}+2\pi_{4}\\\
\hline\cr\begin{tabular}[]{@{}c@{}}$(\mathds{1},e_{\mathds{1}})+(\pi_{1},e_{\pi_{1}})+(\pi_{2},e_{\pi_{2}})+(\pi_{3},e_{\pi_{3}})$\\\
\+
$(\mathds{1},e_{\mathds{1}})+(\pi_{1},e^{\prime}_{\pi_{1}})+(\pi_{2},e^{\prime}_{\pi_{2}})+(\pi_{3},e^{\prime}_{\pi_{3}})$\end{tabular}&2\mathds{1}+2\pi_{1}+2\pi_{2}+2\pi_{3}\\\
\hline\cr(\mathds{1},e_{1})+(\pi_{1},e_{\pi_{1}})+2(\mathds{1}+\pi_{1},e_{\mathds{1}+\pi_{1}})+(\mathds{1},e^{\prime}_{\mathds{1}})+(\pi_{1},e^{\prime}_{\pi_{1}})\par&4\mathds{1}+4\pi_{1}\\\
\hline\cr(\mathds{1},e_{1})+(\pi_{2},e_{\pi_{2}})+2(\mathds{1}+\pi_{2},e_{\mathds{1}+\pi_{2}})+(\mathds{1},e^{\prime}_{\mathds{1}})+(\pi_{2},e^{\prime}_{\pi_{2}})\par&4\mathds{1}+4\pi_{2}\\\
\hline\cr(\mathds{1},e_{1})+(\pi_{2},e_{\pi_{2}})+2(\pi_{4},e^{(2)}_{\pi_{4}})+(\pi_{1},e^{\prime}_{\pi_{1}})+(\pi_{3},e^{\prime}_{\pi_{3}})\par&\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}+2\pi_{4}\\\
\hline\cr(\mathds{1},e_{1})+(\pi_{3},e_{\pi_{3}})+2(\mathds{1}+\pi_{3},e_{\mathds{1}+\pi_{3}})+(\mathds{1},e^{\prime}_{\mathds{1}})+(\pi_{3},e_{\pi_{3}})\par&4\mathds{1}+4\pi_{3}\\\
\hline\cr(\mathds{1},e_{1})+(\pi_{3},e_{\pi_{3}})+2(\pi_{4},e^{(3)}_{\pi_{4}})+(\pi_{1},e^{\prime}_{\pi_{1}})+(\pi_{2},e^{\prime}_{\pi_{2}})\par&\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}+2\pi_{4}\\\
\hline\cr\begin{tabular}[]{@{}c@{}}$(\mathds{1},e_{\mathds{1}})+(\mathds{1},e^{\prime}_{\mathds{1}})+(\mathds{1}+\pi_{2},e_{\mathds{1}+\pi_{2}})+$\\\
$(\mathds{1}+\pi_{1},e_{\mathds{1}+\pi_{1}})+(\mathds{1}+\pi_{3},e_{\mathds{1}+\pi_{3}})$\end{tabular}\par&5\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}\\\
\hline\cr\begin{tabular}[]{@{}c@{}}$(\mathds{1},e_{\mathds{1}})+(\pi_{1},e^{\prime}_{\pi_{1}})+(\pi_{4},e^{(2)}_{\pi_{4}})+$\\\
$(\mathds{1}+\pi_{1},e_{\mathds{1}+\pi_{1}})+(\pi_{4},e^{(3)}_{\pi_{4}})$\end{tabular}\par&2\mathds{1}+2\pi_{1}+2\pi_{4}\\\
\hline\cr(\mathds{1},e_{\mathds{1}})+(\pi_{2},e_{\pi_{2}})+(\pi_{4},e_{\pi_{4}})+(\mathds{1}+\pi_{2},e_{\mathds{1}+\pi_{2}})+(\pi_{4},e^{(2)}_{\pi_{4}})\par&2\mathds{1}+2\pi_{2}+2\pi_{4}\\\
\hline\cr(\mathds{1},e_{\mathds{1}})+(\pi_{3},e_{\pi_{3}})+(\pi_{4},e_{\pi_{4}})+(\mathds{1}+\pi_{3},e_{\mathds{1}+\pi_{3}})+(\pi_{4},e^{(3)}_{\pi_{4}})\par&2\mathds{1}+2\pi_{3}+2\pi_{4}\\\
\hline\cr\end{array}$ (228)

We find that there are three different Lagrangian algebra objects in the
SymTFT which gives the algebra object

$\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}+2\pi_{4}~{}.$ (229)

Therefore, there are three distinct multiplications on this algebra object.
This agrees with the fact that Rep$(D_{8})$ admits three fibre functors. The
algebra object

$2\mathds{1}+2\pi_{1}+2\pi_{2}+2\pi_{3}~{},$ (230)

decomposes into two copies of the algebra object
$\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}$. This corresponds to gauging the
$\mathds{Z}_{2}\times\mathds{Z}_{2}$ invertible line operators in
Rep$(D_{8})$. The algebra object

$4\mathds{1}+4\pi_{1}~{},$ (231)

decomposes into four copies of the algebra object $\mathds{1}+\pi_{1}$. The
same is true for the algebra objects $4\mathds{1}+4\pi_{2}$ and
$4\mathds{1}+4\pi_{3}$. These again correspond to gauging invertible line
operators in Rep$(D_{8})$.

We also have the algebra object

$5\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}~{},$ (232)

which decomposes into $4A_{1}+A_{2}$ where
$A_{1}:=\mathds{1},A_{2}:=\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}$. This shows that
$\mathds{1}+\pi_{1}+\pi_{2}+\pi_{3}$ is a Morita trivial algebra which agrees
with the fusion rules of Rep$(D_{8})$. Finally, we have the algebra object

$2\mathds{1}+2\pi_{1}+2\pi_{4}$ (233)

which decomposes into two copies of the algebra
$\mathds{1}+\pi_{1}+\pi_{4}$.191919At the level of the object we can also
consider the decomposition of $2\mathds{1}+2\pi_{2}+2\pi_{4}$ into two copies
of $\mathds{1}+\pi_{1}$ and $\mathds{1}+\pi_{4}$. $\mathds{1}+\pi_{1}$ admits
a unique multiplication on it. Therefore, this decomposition is not consistent
with the fact that $\mathds{1}+\pi_{1}$ corresponds to the Lagrangian algebra
object
$(\mathds{1},e_{1})+(\pi_{1},e_{\pi_{1}})+2(\mathds{1}+\pi_{1},e_{\mathds{1}+\pi_{1}})+(\mathds{1},e^{\prime}_{\mathds{1}})+(\pi_{1},e^{\prime}_{\pi_{1}})$.
The same is true for the algebra objects $2\mathds{1}+2\pi_{2}+2\pi_{4}$ and
$2\mathds{1}+2\pi_{3}+2\pi_{4}$. The haploid algebra objects obtained above
from Lagrangian algebra objects of the bulk SymTFT agree with Perez-
Lona:2023djo ; Diatlyk:2023fwf .

####  7.3.2 Haagerup fusion categories

The Haagerup fusion categories $\mathscr{H}_{1}$, $\mathscr{H}_{2}$ and
$\mathscr{H}_{3}$ is a Morita equivalence class of three fusion categories
Izumi:2001mi ; grossman2012quantum . The fusion category $\mathscr{H}_{1}$ has
line operators $\mathbf{1},\nu,\eta,\mu$ with fusion rules given by

$\begin{array}[]{|c||c|c|c|c|}\hline\cr&\mathbf{1}&\nu&\eta&\mu\\\
\hline\cr\hline\cr\nu&\nu&\mathbf{1}+2\nu+2\eta+\mu&2\nu+\eta+\mu&\mu\\\
\hline\cr\eta&\eta&2\nu+\eta+\mu&\mathbf{1}+\nu+\eta+\mu&\nu+\eta\\\
\hline\cr\mu&\mu&\nu+\eta+mu&\nu+\eta&\mathbf{1}+\nu\\\ \hline\cr\end{array}$
(234)

These line operators have quantum dimensions given by

$1,d+1,d,d-1~{},$ (235)

respectively where $d=\frac{3+\sqrt{13}}{2}$. The fusion rules are commutative
and have multiplicity.

Both $\mathscr{H}_{2}$ and $\mathscr{H}_{3}$ have line operators
$\mathbf{1},\alpha,\alpha^{2},\rho,\alpha\rho,\alpha^{2}\rho$ and the same
fusion rules given by

$\begin{array}[]{|c||c|c|c|c|c|c|c|}\hline\cr&\mathbf{1}&\alpha&\alpha^{2}&\rho&\alpha\rho&\alpha^{2}\rho\\\
\hline\cr\hline\cr\mathbf{1}&\mathbf{1}&\alpha&\alpha^{2}&\rho&\alpha\rho&\alpha^{2}\rho\\\
\hline\cr\alpha&\alpha&\alpha^{2}&\mathbf{1}&\alpha\rho&\alpha^{2}\rho&\rho\\\
\hline\cr\alpha^{2}&\alpha^{2}&\mathbf{1}&\alpha&\alpha^{2}\rho&\rho&\alpha\rho\\\
\hline\cr\rho&\rho&\alpha^{2}\rho&\alpha\rho&\mathbf{1}+H&\alpha^{2}+H&\alpha+H\\\
\hline\cr\alpha\rho&\alpha\rho&\rho&\alpha^{2}\rho&\alpha+H&\mathbf{1}+H&\alpha^{2}+H\\\
\hline\cr\alpha^{2}\rho&\alpha^{2}\rho&\alpha\rho&\rho&\alpha^{2}+H&\alpha+H&\mathbf{1}+H\\\
\hline\cr\end{array}$ (236)

where $H=\rho+\alpha\rho+\alpha^{2}\rho$. The quantum dimensions of the lines
are

$1,1,1,d,d,d~{},$ (237)

respectively. Note that the fusion rules are non-commutative and without
multiplicity.

The fusion category $\mathscr{H}_{2}$ has three module categories
$\mathcal{M}_{1},\mathcal{M}_{2}$ and $\mathcal{M}_{3}$ and we have
grossman2012quantum  (see also (Huang:2021ytb, , Section 7), Appendix E)

$\displaystyle\bigoplus_{\operatorname{\text{m}}\in\mathcal{M}_{1}}A_{\operatorname{\text{m}}}=4\mathbf{1}+\alpha+\alpha^{2}+3H~{}.$
$\displaystyle\bigoplus_{\operatorname{\text{m}}\in\mathcal{M}_{2}}A_{\operatorname{\text{m}}}=6\mathbf{1}+3H~{}.$
(238)
$\displaystyle\bigoplus_{m\in\mathcal{M}_{3}}A_{\operatorname{\text{m}}}=2\mathbf{1}+2\alpha+2\alpha^{2}+3H~{}.$

These three algebras correspond to three Lagrangian algebra objects in the
SymTFT $\mathcal{Z}(\mathscr{H}_{2})$.

$\mathcal{Z}(\mathscr{H}_{2})$ has line operators Izumi:2001mi

$\mathbf{1},\pi_{1},\pi_{2},\sigma_{1},\sigma_{2},\sigma_{3},\mu_{1},\mu_{2},\dots\mu_{6}~{}.$
(239)

These have quantum dimensions

$1,3d+1,3d+2,3d+2,3d+2,3d+2,3d,3d,3d,3d,3d,3d~{},$ (240)

respectively. The fusion rules and modular data of these line operators can be
found, for example, in hong2008exotic . The only bosonic line operators are
$\mathbf{1},\pi_{1},\pi_{2},\sigma_{1}$. All Lagrangian algebras in this
SymTFT must be supported on a linear combination of these bosonic line
operators. In order to identity the Lagrangian algebra corresponding to the
three classes of algebras in $\mathscr{H}_{2}$, it is useful to write the
bosonic line operators as $(a,e_{a})$ for some $a\in\mathscr{H}_{2}$ and half-
braidings $e_{a}$. This is given explicitly in (Evans:2010yr, , Section 2.2)
(see also (Lin:2022dhv, , Section 4.4.3)) and is given by

$\mathbf{1}\to(\mathbf{1},e_{\mathbf{1}}),~{}\pi_{1}\to(\mathbf{1}+\rho,e_{\mathbf{1}+\rho}),~{}\pi_{2}\to(\alpha+\alpha^{2}+H,e_{\alpha+\alpha^{2}+H}),~{}\sigma_{1}\to(\mathbf{1}+\mathbf{1}+H,e_{\mathbf{1}+\mathbf{1}+H})$
(241)

where $H=\rho+\alpha\rho+\alpha^{2}\rho$. Consider the Lagrangian algebra

$L=N_{L}^{\mathbf{1}}\mathbf{1}+N_{L}^{\pi_{1}}\pi_{1}+N_{L}^{\pi_{2}}\pi_{2}+N_{L}^{\sigma_{1}}\sigma_{1}.$
(242)

Moving $L$ to the canonical gapped boundary $\mathcal{B}_{\mathscr{H}_{2}}$
corresponding to the fusion category $\mathscr{H}_{2}$, we get

$F_{\mathcal{B}_{\mathscr{H}_{2}}}(L)=N_{L}^{\mathbf{1}}\mathbf{1}+N_{L}^{\pi_{1}}(\mathbf{1}+\rho)+N_{L}^{\pi_{2}}(\alpha+\alpha^{2}+H)+N_{L}^{\sigma_{1}}(\mathbf{1}+\mathbf{1}+H)~{}.$
(243)

where $N_{L}^{\pi_{1}},N_{L}^{\pi_{2}}$ and $N_{L}^{\sigma_{1}}$ are non-
negative integers. Using Theorem 5.1 we know that
$F_{\mathcal{B}_{\mathscr{H}_{2}}}(L)$ must agree with one of the algebra
objects in (238). Using this, we get the three Lagrangian algebras

$\displaystyle\mathcal{M}_{1}\to
L_{1}:=\mathbf{1}+\pi_{1}+\pi_{2}+\sigma_{1}~{},$ (244)
$\displaystyle\mathcal{M}_{2}\to L_{2}:=\mathbf{1}+\pi_{1}+2\sigma_{1}~{},$
(245) $\displaystyle\mathcal{M}_{3}\to L_{3}:=\mathbf{1}+\pi_{1}+2\pi_{2}~{}.$
(246)

The canonical gapped boundary corresponding to $\mathscr{H}_{2}$ is given by
the Lagrangian algebra $L_{2}$. $L_{3}$ corresponds to the fusion category
$\mathscr{H}_{3}$ and $L_{1}$ corresponds to $\mathscr{H}_{1}$.

The algebra objects/non-anomalous line operators in $\mathscr{H}_{3}$ can be
found by computing

$F_{\mathcal{B}_{L_{3}}}(L_{1}),~{}F_{\mathcal{B}_{L_{3}}}(L_{2})\text{ and
}F_{\mathcal{B}_{L_{3}}}(L_{3}).$ (247)

Using (Lin:2022dhv, , Section 4.4.3), we find

$F_{\mathcal{B}_{L_{3}}}(\pi_{1})=\mathbf{1}+H,~{}F_{\mathcal{B}_{L_{3}}}(\pi_{2})=\mathbf{1}+\mathbf{1}+H\text{
and }F_{\mathcal{B}_{L_{3}}}(\sigma_{1})=\alpha+\alpha^{2}+H~{}.$ (248)

Using this, we get

$\displaystyle
F_{\mathcal{B}_{L_{3}}}(L_{1})=\mathbf{1}+\mathbf{1}+H+\mathbf{1}+\mathbf{1}+H+\alpha+\alpha^{2}+H=4\mathbf{1}+\alpha+\alpha^{2}+3H~{}.$
$\displaystyle
F_{\mathcal{B}_{L_{3}}}(L_{2})=\mathbf{1}+\mathbf{1}+H+2(\alpha^{2}+\alpha^{2}+H)=2\mathbf{1}+2\alpha+2\alpha^{2}+3H~{},$
(249) $\displaystyle
F_{\mathcal{B}_{L_{3}}}(L_{3})=\mathbf{1}+\mathbf{1}+H+2(\mathbf{1}+\mathbf{1}+H)=6\mathbf{1}+3H~{},$

These three algebra objects are consistent with the three module categories in
$\mathscr{H}_{3}$ grossman2012quantum  (see Appendix E).

Finally, by computing

$F_{\mathcal{B}_{L_{1}}}(\pi_{1}),~{}F_{\mathcal{B}_{L_{1}}}(\pi_{2})\text{
and }F_{\mathcal{B}_{L_{1}}}(\sigma_{1})~{},$ (250)

we can find the non-anomalous line operators in $\mathscr{H}_{1}$. This
requires finding the bulk-to-boundary map for the boundary condition
$\mathcal{B}_{L_{1}}$. This can be determined by taking the quotient of
$\mathcal{Z}(\mathscr{H}_{2})$ by the Lagrangian algebra $L_{1}$ as described
in Cong:2017hcl . The details of this calculation are given in Appendix D. We
get,

$F_{\mathcal{B}_{L_{1}}}(\pi_{1})=\mathbf{1}+\nu+\eta+\mu,~{}F_{\mathcal{B}_{L_{1}}}(\pi_{2})=\mathbf{1}+2\eta+\nu\text{
and }F_{\mathcal{B}_{L_{1}}}(\sigma_{1})=\mathbf{1}+2\nu+\mu~{},$ (251)

Using this, we find

$\displaystyle
F_{\mathcal{B}_{L_{1}}}(L_{1})=\mathbf{1}+\mathbf{1}+\nu+\eta+\mu+\mathbf{1}+2\eta+\nu+\mathbf{1}+2\nu+\mu=4\mathbf{1}+4\nu+3\eta+2\mu~{}.$
$\displaystyle
F_{\mathcal{B}_{L_{1}}}(L_{2})=\mathbf{1}+\mathbf{1}+\nu+\eta+\mu+2(\mathbf{1}+2\nu+\mu)=4\mathbf{1}+5\nu+\eta+3\mu~{},$
(252) $\displaystyle
F_{\mathcal{B}_{L_{1}}}(L_{3})=\mathbf{1}+\mathbf{1}+\nu+\eta+\mu+2(\mathbf{1}+2\eta+\nu)=4\mathbf{1}+3\nu+5\eta+\mu~{}.$

These three algebra objects are consistent with the three module categories of
$\mathscr{H}_{1}$ grossman2012quantum  (see Appendix E).

##  8 Conclusion

In this paper, we studied the non-anomalous line operators in a 1+1D QFT with
fusion category symmetry $\mathcal{C}$ using gapped boundaries of the SymTFT
$\mathcal{Z}(\mathcal{C})$. Physically equivalent gaugings in 1+1D correspond
to the same gapped boundary of the SymTFT. Our analysis determines the
explicit map between them.

  * •

Given a Lagrangian algebra object $L$ in $\mathcal{Z}(\mathcal{C})$ the
corresponding physically equivalent gaugings in $\mathcal{C}$ can be found
from $F(L)$, where $F$ is the bulk-to-boundary map. (Theorem 5.1).

  * •

In fact, $F$ maps the algebra structure of $L$ to the line operator $F(L)$
through equation (32).

  * •

The explicit examples considered in this paper show that in some cases
Theorems 5.1 and 5.2 can be used to determine all non-anomalous line operators
in $\mathcal{C}$ even without the knowledge of the algebra structure on the
Lagrangian algebras in $\mathcal{Z}(\mathcal{C})$.

  * •

Conversely, knowledge of non-anomalous line operators in $\mathcal{C}$ puts
strong constraints on which line operators in the bulk SymTFT can form a
Lagrangian algebra. We used this to find a necessary condition for a line
operator $A\in\mathcal{C}$ to be non-anomalous. This was then used to prove
that $\mathcal{C}$-symmetric trivially gapped phases exist if and only if the
bulk SymTFT admits a magnetic Lagrangian algebra.

  * •

We discussed the notion of transporting non-anomalous line operators between
fusion categories which share the same SymTFT. This method can be used to
determine non-anomalous line operators in a fusion category from those in a
“simpler” fusion category. This provides an alternative to using NIM-reps to
determine algebra objects in a fusion category $\mathcal{C}$.

The following are some interesting future directions:

  * •

It will be interesting tp generalize the analysis in this paper to SymTFTs for
fermionic symmetries Wen:2024udn .

  * •

Generalization to higher dimensions. It will be interesting to apply the ideas
presented here to the setting in Putrov:2023jqi ; Antinucci:2023ezl ;
Cordova:2023bja ; Argurio:2024oym .

  * •

General relation between anomalies and gapped boundaries for SymTFTs of non-
finite and/or continuous symmetries kong2017boundary ; Putrov:2022pua ;
Brennan:2024fgj ; Antinucci:2024zjp ; Apruzzi:2024htg ; Bonetti:2024cjk .

  * •

For an invertible non-anomalous symmetry $G$, the discrete torsion takes
values in $H^{2}(G,U(1))$. It would be interesting to explore the structure of
generalized discrete torsions for a fixed non-anomalous line operator using
gapped boundaries of the SymTFT.

  * •

In Cordova:2024vsq , the authors show that massive 1+1D QFTs with
spontaneously broken non-invertible symmetries often have particle
degeneracies. A crucial part of finding the degeneracies involves determining
the kernel and cokernel of certain maps relating different 1D gapped
boundaries of the QFT. It will be interesting to use the map from gapped
interfaces between 1+1D gapped boundaries of $\mathcal{Z}(\mathcal{C})$ and 1D
gapped boundaries of 1+1D QFTs/TQFTs studied in this work to give a bulk
perspective on the results in Cordova:2024vsq .

###### Acknowledgements.

We thank Mahesh Balasubramanian, Matthew Buican, Arkya Chatterjee, Christian
Copetti, Clement Delcamp, Nicholas Holfester, Adrian Padellaro, Brandon
Rayhaun, Ingo Runkel, Sakura Schafer-Nameki, Yifan Wang for discussions and
comments related to this work. R.R. thanks Perimeter Institute for Theoretical
Physics for hospitality during the workshop ‘Higher Categorical Tools for
Quantum Phases of Matter’ during which part of this work was completed. P.P.
thanks the Galileo Galilei Institute for Theoretical Physics for the
hospitality and the INFN for partial support during the completion of this
work. We thank ULB Brussels and International Solvay Institutes for
hospitality during the workshop ‘Symmetries, Anomalies and Dynamics of Quantum
Field theory’ where some of the results in this paper were announced.

##  Appendix A Constraints on the action of line operators on gapped
interfaces

In Theorem 5.1, we learned that

$F(L)=\sum_{\operatorname{\text{m}}\in\mathcal{M}_{L}}A_{\operatorname{\text{m}}}~{}.$
(253)

From Fig. 12 and surrounding discussion we know that finding the haploid
algebra objects in $F(L)$ directly from the SymTFT requires determining how
the line operators on the gapped boundary $\mathcal{B}_{\mathcal{C}}$ act on
the gapped interfaces between $\mathcal{B}_{\mathcal{C}}$ and
$\mathcal{B}_{L}$. When $\mathcal{Z}(\mathcal{C})$ is a Dijkgraaf-Witten
theory, this can be determined using the results in Cong:2017hcl .

In this section, we will determine some general constraints on the action of
the line operators in $F(L)$ on gapped interfaces which may be used in
explicit computations to determine haploid sub-objects of $F(L)$. To this end,
let us consider the action of $x\in L$ on the gapped interface $r$. We get

$x\times r=\sum_{s}N_{xr}^{s}~{}s~{}.$ (254)

where $N_{xr}^{s}$ are non-negative integers and $s$ labels all the simple
gapped interfaces between $\mathcal{B}_{\mathcal{C}}$ and $\mathcal{B}_{L}$.
If $N_{xr}^{s}\neq 0$, then we get the configuration in Fig. 23.

$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{L}$$r$$\mathcal{Z}(\mathcal{C})$$s$$x$$\alpha$
Figure 23: $N_{xr}^{s}\neq 0$ implies a non-trivial junction between the lines
$x$ and the gapped interfaces $r$ and $s$ where $\alpha$ labels a basis of
point operators at the junction.

The condition $N_{xr}^{s}\neq 0$ also implies the non-trivial configurations
in Fig. 24.

$a$$b$$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{L}$$r$$\mathcal{Z}(\mathcal{C})$$s$$x$$\beta$$\mathcal{B}_{\mathcal{C}}$$\mathcal{B}_{L}$$r$$\mathcal{Z}(\mathcal{C})$$s$$x$$\gamma$$\psi$$\chi$
Figure 24: If $N_{xr}^{s}\neq 0$, then there must be some line operator $b\in
F_{\mathcal{B}_{L}}(L)$ such that the diagram on the left is non-trivial.
Similarly, there must be some line operator $a\in
F_{\mathcal{B}_{\mathcal{C}}}(L)$ such that the diagram on the right is non-
trivial.

Therefore, we have

$N_{xr}^{s}=\sum_{b\in\mathcal{C}_{L}}N_{x}^{b}N_{br}^{s}=\sum_{a\in\mathcal{C}}N_{x}^{a}N_{ar}^{s}~{}.$
(255)

where $\mathcal{C}_{L}$ is the category of line operators on the gapped
boundary $\mathcal{B}_{L}$. Since $x\in L$, the fusion of $x$ with the gapped
boundary $\mathcal{B}_{L}$ contains the identity line in $\mathcal{C}_{L}$. In
other words,

$F_{\mathcal{B}_{L}}(x)=1+\dots~{},$ (256)

Therefore, we have

$N_{xr}^{r}\neq 0~{}\forall~{}x\in L~{}.$ (257)

for any gapped interface $r$. Using (255), we find that

$\sum_{a\in\mathcal{C}}N_{x}^{a}N_{ar}^{r}\neq 0~{}.$ (258)

Therefore, for a fixed simple gapped interface $r$, there is some $a\in
F_{\mathcal{B}_{\mathcal{C}}}(x)$ which acts trivially on it.

##  Appendix B Proof of Theorem 5.2

In this section, we will review some properties of characters of simple line
operators in $\mathcal{Z}(\text{Vec}_{G}^{\omega})$ and give a proof of
Theorem 5.2.

Recall that Vec${}_{G}^{\omega}$ is the category of $G$-graded vector spaces
with associator given by the 3-cocycle $\omega$. An $\omega$-projective
$G$-action on a $G$-graded vector space $V=\bigoplus_{g\in G}V_{g}$ is a set
of linear maps

$f_{g}:V\to V~{},~{}g\in G~{},$ (259)

such that

$f_{g}(V_{h})=V_{ghg^{-1}}~{},\text{and
}(f_{g}f_{h})(v)=\omega(g,h|k)~{}f_{g}(f_{h}(v))~{},~{}\forall~{}v\in
V_{k}~{},$ (260)

where

$\omega(g,h|k):=\omega(g,h,k)^{-1}\omega(g,hgh^{-1},h)\omega(ghk(gh)^{-1},g,h)^{-1}~{}.$
(261)

It is known that any line operator in the SymTFT
$\mathcal{Z}(\text{Vec}_{G}^{\omega})$ can be labeled by a $G$-graded vector
space $V$ with $\alpha$-projective $G$-action (see (davydov2010centre, ,
Section 2.1, 3.1)). Its character is defined as

$\chi_{V}(g,h):=\text{Tr}_{V_{g}}(h)~{},$ (262)

where $g,h$ is a pair of commuting elements in $G$. For any two
$x,y\in\mathcal{Z}(\text{Vec}_{G}^{\omega})$, it can be shown that the
characters satisfy davydov2017lagrangian

$\displaystyle\chi_{x}(kgk^{-1},khk^{-1})=\frac{\omega(k,h|g)}{\omega(khk^{-1},k|g)}\chi_{x}(g,h)~{},$
(263) $\displaystyle\chi_{x\times
y}(g,h)=\chi_{x}(g,h)\times\chi_{y}(g,h)~{},$ (264)
$\displaystyle\chi_{x+y}(g,h)=\chi_{x}(g,h)+\chi_{y}(g,h)~{}.$ (265)

Consider the following inner product on the characters

$\langle\chi_{x},\chi_{y}\rangle=\frac{1}{|G|}\sum_{g,h\in
G,~{}gh=hg}\omega(h^{-1},h|g)\chi_{x}(g,h^{-1})\chi_{y}(g,h)~{}.$ (266)

We will use the following important property of this inner product
davydov2017lagrangian

$\langle\chi_{x},\chi_{y}\rangle=|\text{Hom}_{\mathcal{Z}(\text{Vec}_{G}^{\omega})}(x,y)|~{}.$
(267)

It will also be useful to recall some properties of characters of projective
representations.

$\displaystyle\overline{\chi_{\pi_{g}}(h)}=\omega(h^{-1},h|g)~{}\chi_{\pi_{g}}(h^{-1})~{},$
(268)
$\displaystyle\chi_{\pi_{g}}(khk^{-1})=\frac{\omega(k,h|g)}{\omega(khk^{-1},k|f)}~{}\chi_{\pi_{g}}(g)~{},$
(269)

where $\overline{\chi_{\pi_{g}}(h)}$ is the complex conjugate of
$\chi_{\pi_{g}}(h)$ (karpilovsky1985projective, , Proposition 2.2). Now, we
are ready to prove Theorem 5.2.

Theorem 4.2: $H\subseteq G$ is non-anomalous if and only if there exists a
Lagrangian algebra $L$ in $\mathcal{Z}(\text{Vec}_{G}^{\omega})$ such that

$([h],\pi_{h})\in L~{}~{}~{}\forall~{}[h]\in C(H)~{},$ (270)

for some representation $\pi_{h}$ of the centralizer of $h$.

Proof: Let $H$ is a non-anomalous group. Then $A(H,1)$ is an algebra object.
Using (58), we can compute the character of $\mathcal{Z}(A(H,1))$ to get

$\chi_{\mathcal{Z}(A(H,1))}(k,l):=\sum_{y\in
Y}\frac{\omega(y^{-1}ly,y^{-1}|k)}{\omega(y^{-1},l|k)}~{}.$ (271)

Let $([l],\pi_{l})$ be a simple object in
$\mathcal{Z}(\text{Vec}_{G}^{\omega})$ such that $[l]\in C(H)$. Consider the
inner product

$\langle\chi_{([l],\pi_{l})},\chi_{\mathcal{Z}(A(H,1))}\rangle=\frac{1}{|G|}\sum_{g,h\in
G,~{}gh=hg}\omega(h^{-1},h|g)\chi_{([l],\pi_{l})}(g,h^{-1})\chi_{\mathcal{Z}(A(H,1))}(g,h)~{}.$
(272)

Using (57), we get

$\displaystyle\langle\chi_{([l],\pi_{l})},\chi_{\mathcal{Z}(A(H,1))}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{|G|}\sum_{g\in[l],~{}gh=hg}\omega(h^{-1},h|g)\chi_{\pi_{g}}(h^{-1})\chi_{\mathcal{Z}(A(H,1))}(g,h)~{},$
$\displaystyle=$
$\displaystyle\frac{1}{|[l]|}\sum_{g\in[l]}\bigg{[}\frac{1}{|C_{g}|}\sum_{h\in
C_{g}}\omega(h^{-1},h|g)\chi_{\pi_{g}}(h^{-1})\chi_{\mathcal{Z}(A(H,1))}(g,h)\bigg{]}~{},$

where we used the fact that $|G|=|[l]||C_{g}|$ for any $l\in G$ and $g\in[l]$,
$|[l]|$ is the size of the conjugacy class $[l]$ and $|C_{l}|$ is the order of
the group $C_{l}$. Using (268), we can write the above expression as

$\frac{1}{|[l]|}\sum_{g\in[l]}\bigg{[}\frac{1}{|C_{g}|}\sum_{h\in
C_{g}}\overline{\chi_{\pi_{g}}(h)}\chi_{\mathcal{Z}(A(H,1))}(g,h)\bigg{]}~{}.$
(275)

Now, note that for a fixed $g$, $\chi_{\mathcal{Z}(A(H,1))}(g,h)$ is a
function on the group $C_{g}$. Moreover, if $k,h\in C_{g}$, we find that this
function satisfies

$\chi_{\mathcal{Z}(A(H,1))}(g,khk^{-1})=\chi_{\mathcal{Z}(A(H,1))}(kgk^{-1},khk^{-1})=\frac{\omega(k,h|g)}{\omega(khk^{-1},k|g)}\chi_{\mathcal{Z}(A(H,1))}(g,h)~{}.$
(276)

where in the first equality we used the fact that $k\in C_{g}$ commutes with
$g$ and in the second equality we used (263). Moreover, since the conjugacy
class $[l]\in C(H)$, $\chi_{\mathcal{Z}(A(H,1))}(g,h)$ is non-zero for some
$g$ and $h$. (For example, $\chi_{\mathcal{Z}(A(H,1))}(g,1)$ is non-zero.)
Therefore, for a fixed $g$, $\chi_{\mathcal{Z}(A(H,1))}(g,h)$ is a non-zero
projective character on the group $C_{g}$ and

$\frac{1}{|C_{g}|}\sum_{h\in
C_{g}}\overline{\chi_{\pi_{g}}(h)}~{}\chi_{\mathcal{Z}(A(H,1))}(g,h)~{},$
(277)

is the inner-product of the characters $\chi_{\pi_{g}}(h)$ and
$\chi_{\mathcal{Z}(A(H,1))}(g,h)$. Since the characters of irreducible
projective representations form a basis of projective class functions
(karpilovsky1985projective, , Theorem 3.1), we can always choose a $\pi_{l}$
in $([l],\pi_{l})$ such that the inner-product (277) is non-zero. Therefore,
there is always some $\pi_{l}$ for which
$\langle\chi_{([l],\pi_{l})},\chi_{\mathcal{Z}(A(H,1))}\rangle\neq 0$.

Conversely, consider some line operator $([l],\pi_{l})$ such that
$\langle\chi_{([l],\pi_{l})},\chi_{\mathcal{Z}(A(H,\sigma))}\rangle\neq 0$ is
non-zero. We have

$\displaystyle\langle\chi_{([l],\pi_{l})},\chi_{\mathcal{Z}(A(H,\sigma))}\rangle$
(278) $\displaystyle=$ $\displaystyle\frac{1}{|G|}\sum_{g,h\in
G,~{}gh=hg}\omega(h^{-1},h|g)\chi_{([l],\pi_{l})}(g,h^{-1})\chi_{\mathcal{Z}(A(H,\sigma))}(g,h)~{},$
(279) $\displaystyle=$