centertableaux,boxsize=.5em

1Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy, University of California, Los Angeles, CA 90095, USA

2 Department of Mathematics, King’s College London, Strand, London WC2R 2LS,
UK.

3Department of Physics, Condensed Matter Theory Center, and Joint Quantum
Institute, University of Maryland, College Park, Maryland 20742, USA

4Department of Physics, Harvard University, Cambridge, MA02138, USA

#  Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall
Conductance

Po-Shen Hsin1,2, Ryohei Kobayashi3, Carolyn Zhang4

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of results
  2. 2 Review of Invertible Symmetry Fractionalizations
    1. 2.1 Fractionalizations: modifying junctions of symmetry
    2. 2.2 Example: fractional quantum Hall systems with $U(1)$ symmetry
  3. 3 Fractionalization of Non-Invertible Symmetry: An Example
    1. 3.1 Coset Non-Invertible Symmetry From Outer Automorphism
    2. 3.2 Non-invertible symmetry as sandwich of invertible symmetry
    3. 3.3 Exotic quantum Hall conductance for continuous coset symmetry
      1. 3.3.1 $O(2)$ Chern-Simons theory
      2. 3.3.2 Gapless edge modes
    4. 3.4 More example of fractionalized coset symmetry: $S_{3}$ gauge theory
  4. 4 Lattice Model with Fractionalized Non-Invertible Symmetry
    1. 4.1 Cosine symmetry in $S_{3}$ quantum double model
    2. 4.2 Fractionalization of cosine symmetry in $S_{3}$ quantum double model
  5. 5 General Coset Symmetry Fractionalization and Bulk TQFT
    1. 5.1 General coset symmetry and its fractionalization in (2+1)D topological order
    2. 5.2 Non-invertible symmetry from invertible symmetry
    3. 5.3 Bulk TQFT for finite coset non-invertible symmetry
      1. 5.3.1 Anomalies of finite coset non-invertible symmetry
  6. 6 Application: Non-Invertible Symmetry in Spin Liquids
    1. 6.1 “Projective” symmetry action on Wilson lines
    2. 6.2 Deconfined critical points with non-invertible symmetry
  7. 7 Discussion and Outlook

##  1 Introduction

Symmetry fractionalization describes how zero-form symmetry can act
projectively in topologically ordered systems [1, 2]. Given an underlying
topological order and an invertible zero-form symmetry, one can enumerate all
of the mathematically consistent ways the symmetry can act projectively. In
particular, junctions of the symmetry defects can be modified by an Abelian
anyon.

Given the enormous body of work on fractionalization of invertible symmetries,
it is natural to consider the fractionalization of non-invertible symmetries
(see [3, 4, 5, 6, 7] for recent reviews of non-invertible symmetries).
Examples of non-invertible symmetries are discussed extensively in the
literature. They occur in various field theories such as discussed in [8, 9,
10, 11, 12, 13, 14, 15, 16] and lattice models such as discussed in [17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In this work we present a first
step toward extending symmetry fractionalization to non-invertible symmetries.

As in the case of invertible symmetries, the anomalies of non-invertible
symmetry constrain the low energy dynamics [30, 8, 31, 32, 33, 34]. Since the
anomalies depend on the fractionalization pattern [35, 36, 37, 33, 34], it is
important to understand how non-invertible symmetry fractionalizes in quantum
systems.

We will focus on a class of non-invertible symmetries, given by cosets $G/K$
for group $G$ and a non-normal subgroup $K$ [38, 6]. Such coset symmetries
arise naturally in spin liquids: when the microscopic model has symmetry $G$,
i.e. the “projective symmetry group”, a subgroup (the “invariant subgroup”)
can act trivially at low energy and it is gauged [39, 40, 41]. Such models
describe $K$ gauge theory of spin liquids, and they have coset global symmetry
$G/K$, which is non-invertible for non-normal $K$. Thus, understanding the
fractionalization of the coset symmetry is important for such quantum spin
liquids.

For instance, in many condensed matter system the $U(1)$ electromagnetism is
usually treated as a classical probe field. When the system has charge
conjugation symmetry, we can consider the situation where the electromagnetism
can be extended to be Alice electromagnetism [42, 43, 44] where the
$\mathbb{Z}_{2}$ charge conjugation is gauged. While the fluctuations of the
$U(1)$ gauge field can be suppressed by the electric coupling, the
fluctuations of the flat $\mathbb{Z}_{2}$ gauge field are not suppressed,
leading naturally to a $\left(U(1)\rtimes\mathbb{Z}_{2}\right)/\mathbb{Z}_{2}$
electromagnetic coset non-invertible symmetry. This provides further
motivation for studying such class of non-invertible symmetry. We will show
that such a system with coset symmetry can also have a well-defined electric
Hall conductance, that cannot be explained by taking the symmetry to be
$U(1)$.

###  1.1 Summary of results

Here we summarize the main results of the paper. We consider the coset non-
invertible symmetry expressed as $G/K$, obtained by gauging a discrete
subgroup $K$ of the invertible zero-form symmetry $G$. The symmetry $G$ can be
either continuous or discrete. We show examples where the coset symmetry is
fractionalized in (2+1)D topological phases.

An example of such symmetry fractionalization is found in fractional quantum
Hall (FQH) systems with $\mathbb{Z}_{2}$ charge conjugation symmetry gauged.
By FQH system, we simply mean a topological order with a zero-form $U(1)$
symmetry that is fractionalized, leading to anyons carrying fractional $U(1)$
charge. For simplicity, we will consider bosonic FQH systems, but the
discussion can be extended to fermionic ones. In FQH states, the $U(1)$
fractionalization is understood by identifying an Abelian anyon $v$, called
the vison, with the vortex of $U(1)$ global symmetry. Fractional $U(1)$ charge
of each anyon in the system is determined by its braiding with $v$. In other
words, the junction of $U(1)$ symmetry defects is modified by a decoration of
an Abelian anyon $v$. When we gauge the charge conjugation of the (bosonic)
Abelian FQH state described by $U(1)_{2k}$ Chern-Simons theory, the resulting
theory is a non-Abelian topological order described by $O(2)_{2k}$ Chern-
Simons theory. The gauged FQH state now has a continuous non-invertible
symmetry $(U(1)\rtimes\mathbb{Z}_{2})/\mathbb{Z}_{2}$. This symmetry is
referred to as cosine symmetry, since its non-invertible fusion rule is
reminiscent of the product to sum formula of $\cos\theta$.

We point out that this cosine symmetry is fractionalized, where the anyons of
$O(2)_{2k}$ (for $k=1$ the anyons are Abelian) carries fractional charge under
the continuous non-invertible symmetry. This fractional charge of the anyon is
a certain superposition of the opposite fractional charges related by charge
conjugation.

Since the conductivity matrix is even under charge conjugation, one can still
define the electric Hall conductance in the gauged system, which is now
associated with the response under the continuous non-invertible symmetry.
While the fractional part of the $U(1)$ electric Hall conductance of the
standard (bosonic) FQH state is determined by the spin of the Abelian anyon as
$\sigma_{H}=2h_{v}$ mod 2, the vison after gauging the charge conjugation
typically behaves as a non-Abelian anyon. Hence, the Hall conductivity is no
longer associated with the spin of an Abelian anyon, but rather with the spin
of a non-Abelian anyon. Therefore, the value of Hall conductance for
continuous non-invertible symmetry cannot generally be computed in the same
way as that for an invertible $U(1)$ symmetry. We show that the nonzero Hall
conductance enforces the gapless edge state, once the boundary theory is
required to preserve the cosine symmetry.

Similar to Abelian FQH states, phases with fractionalization of non-invertible
coset symmetry can also be understood from modification of junctions of the
zero-form symmetry defects. To describe the junctions of the coset symmetry,
we first need to describe the individual defects. We show that the defects can
be obtained using a “sandwich” of topological operators. Specifically, the
coset symmetry defects are generally described by an invertible symmetry
defects sandwiched by the non-invertible topological interfaces. This
expression is useful for systematically describing the fractionalization of
the coset symmetry using the standard symmetry fractionalization theory of
invertible zero-form symmetries [1]. We establish an algebraic formalism of
the fractionalization for coset symmetry $G/K$ in (2+1)D bosonic TQFT, in
terms of the $G$-crossed braided fusion category interacting with the non-
invertible topological interfaces. When $G$ is a finite group, we also find
necessary conditions for the coset symmetry $G/K$ being free of ’t Hooft
anomalies.

We further find that fractionalized coset symmetry can be realized as an exact
symmetry of the microscopic lattice model of the non-Abelian topological
order. In particular, we find a fractionalized cosine symmetry in the $S_{3}$
quantum double model [17, 45], which is arguably the simplest exactly solvable
lattice model that hosts non-Abelian topological order. This cosine symmetry
of the $S_{3}$ quantum double model is understood as the resulting symmetry
from gauging the $\mathbb{Z}_{2}$ charge conjugation of
$U(1)\rtimes\mathbb{Z}_{2}$ symmetry in the $\mathbb{Z}_{3}$ toric code. In
the $S_{3}$ quantum double model, the fractionalized cosine symmetry acts on a
non-Abelian anyon to produce the cosine of the non-trivial $U(1)$ fractional
charge $q$, reflecting that the non-Abelian anyon carries a superposition of
the fractional charge $(q,-q)$.

The work is organized as follows. In section 2 we review the fractionalization
of invertible symmetry in terms of Abelian anyons. In section 3, we discuss a
class of non-invertible coset symmetry constructed from the outer
automorphism, such as $\mathbb{Z}_{2}$ charge conjugation of $U(1)$. In
section 4, we construct quantum double lattice model for $S_{3}$ enriched by
non-invertible coset symmetry. In section 5, we discuss general coset non-
invertible symmetry and a bulk TQFT that describes the symmetry, and use the
bulk TQFT to derive dynamical consequences from the coset symmetry. In section
6, we apply the discussion to provide construction of new quantum spin liquids
enriched with non-invertible coset symmetry, and give examples of new
deconfined quantum critical points with non-invertible symmetry. In section 7,
we discuss the results and mention several future directions.

##  2 Review of Invertible Symmetry Fractionalizations

Let us first briefly review the fractionalization of invertible symmetry in
(2+1)D. The subject is discussed extensively in the literature (e.g. [1, 46,
47, 48, 37]), and we will only summarize the properties relevant to our
discussion.

###  2.1 Fractionalizations: modifying junctions of symmetry

An invertible $q$-form symmetry is generated by codimension-$(q+1)$
topological defects in Euclidean spacetime111 Here, we will use defect and
operators interchangeably.  that each have an inverse. To fully specify the
$q$-form symmetry, we also need to specify the junctions of the domain walls.
A codimension $k>q+1$ junction where multiple defects meet can be modified
with the topological defects of $(k-1)$-form symmetry. The allowed
modification is constrained by the higher-group structure of the symmetries
[2]. In this work we will focus on $q=0$ and $k=2$. For zero-form symmetry $G$
and 1-form symmetry ${\cal A}$, the fractionalizations can be classified by
$H^{2}(G,{\cal A})$, which specifies the Abelian anyon at the codimension-two
junctions of $G$ defects in (2+1)D [1].222 In the presence of 2-group
symmetry, some fractionalization classes can be identified [36, 37].  If the
junction of three $G$ domain walls $g_{1},g_{2},g_{1}g_{2}$ are modified with
additional 1-form symmetry defect $\eta(g_{1},g_{2})\in{\cal A}$, then the
line operators that braid with the junction acquire additional projective
representation given by the braiding phase with $\eta(g_{1},g_{2})$. The
1-form symmetry operators inserted at the junction modify the correlation
functions of the zero-form symmetry. This can result in additional anomalies
depending on the fractionalization (see e.g. [35, 36, 37]).

In terms of symmetry operators on the Hilbert space, fractionalization means
that the fusion rules of the $q$-form symmetry generators $U_{q}(\Sigma)$
supported on a codimension-$q$ spatial region with boundary is modified
compared with the generators on region without a boundary. The spatial
boundary can be modified with another codimension-$(q+1)$ symmetry generator
$U_{q+1}(\partial\Sigma)$ for a $(q+1)$-form symmetry. Such boundary
modification affects the symmetry action on excitations that have braiding
with the $U_{q+1}(\partial\Sigma)$ symmetry.

A method of describing particular fractionalization class is breaking the
1-form symmetry ${\cal A}$ with additional microscopic degrees of freedom to
screen the 1-form symmetry generator, then the 1-form symmetry generator is no
longer topological and we do not have the freedom to modify the junction.
Rather, when the 1-form symmetry generator is stuck at the junction the entire
configuration is topological. Since there is no 1-form symmetry topological
operators that can modify the junction, there is no longer freedom of changing
the fractionalization, and the microscopic model without the 1-form symmetry
corresponds to a particular fractionalization class. For instance, in gauge
theories we can break the center 1-form symmetry by introducing heavy matter
fields in the fundamental representation, and we can break the magnetic 1-form
symmetry (for instance, consider continuous gauge group in (3+1)D) by
introducing dynamical magnetic monopoles, which replace the non-simply
connected gauge group such as $U(1)$ with a simply connected gauge group such
as $SU(2)$ of the same rank. In TQFT, the fractionalization class is an
additional data in TQFT enriched with symmetry.

###  2.2 Example: fractional quantum Hall systems with $U(1)$ symmetry

As an illustrative example, we can describe the fractionalization of $U(1)$
symmetry in FQH systems. For simplicity, we will only consider bosonic FQH
systems. These theories can be described as bosonic topological orders with
$G=U(1)$ zero-form symmetry.

The $U(1)$ symmetry fractionalization is described by an Abelian anyon $a$
inserted at the junction of three $U(1)$ transformations
$\theta_{1},\theta_{2},[\theta_{1}+\theta_{2}]_{2\pi}$ where
$\theta_{i}\sim\theta_{i}+2\pi$ are angles, and $[\theta]_{2\pi}$ is the
restriction of $\theta$ to $[0,2\pi)$. The Abelian anyon can be written as
$v^{\eta(\theta_{1},\theta_{2})}$, where $v$ is an Abelian anyon called a
vison, with
$\eta(\theta_{1},\theta_{2})=\frac{[\theta_{1}]_{2\pi}+[\theta_{2}]_{2\pi}-[\theta_{1}+\theta_{2}]_{2\pi}}{2\pi}$.
333 In fermionic systems, fractional quantum Hall states obey the spin/charge
relation where the $\mathbb{Z}_{2}$ subgroup of the $U(1)$ electromagnetism is
identified with the fermion parity symmetry. This implies that there is an odd
number $k$ such that $v^{k}=f$ is the transparent fermion. For instance, in
the fermionic Laughlin state $U(1)_{3}$, the simplest choice for $v$ has spin
$1/6$ and $k=3$.  This has the following consequences:

  * •

Particles can carry fractional $U(1)$ charge given by the mutual statistics
with $v$. The particles that braid with the $v$ with phase $e^{i\phi}$ carry
fractional charge given by $\phi/2\pi$ mod 1: the transformations by
$\theta_{1},\theta_{2}$ do not compose into that of
$[\theta_{1}+\theta_{2}]_{2\pi}$, but only up to a phase
$e^{i\frac{\phi}{2\pi}\left([\theta_{1}]_{2\pi}+[\theta_{2}]_{2\pi}-[\theta_{1}+\theta_{2}]_{2\pi}\right)}$.

  * •

The self-statistics of the Abelian anyon at the junction results in nontrivial
correlation function of the $U(1)$ symmetry defects. The fractional quantum
Hall conductance $\sigma_{H}$ for the $U(1)$ symmetry is given by the spin of
the Abelian anyon as $\sigma_{H}=h_{v}$ mod 2 (see e.g. [2]).

##  3 Fractionalization of Non-Invertible Symmetry: An Example

###  3.1 Coset Non-Invertible Symmetry From Outer Automorphism

Let us describe a mechanism for coset non-invertible symmetry. We start with a
system with an invertible symmetry $G$, which has automorphism group
$\text{Aut}(G)$. $G$ can be either continuous or discrete. We gauge a discrete
symmetry $K$ that acts on $G$ by $\rho:K\rightarrow\text{Aut}(G)$. If $\rho$
maps elements in $K$ to nontrivial elements of $\mathrm{Aut}(G)$, then the
symmetry $G$ becomes non-invertible. To see this, we note that an operator
$U_{g}$ for $g\in G$ is not gauge invariant under discrete gauging if $g$ is
permuted by $\rho$. Instead, the well-defined symmetry generator is

$\tilde{U}_{[g]}=\bigoplus_{k\in K}U_{\rho_{k}(g)}~{},$ (3.1)

where $[g]$ is the orbit under the action of $K$. The non-invertible fusion
rule of $\tilde{U}$ follows from the invertible fusion rules of
$\\{U_{\rho_{k}(g)}\\}$ up to condensation defects.444 As we will show below,
when the fusion of $U_{\rho(g)}$ produces the trivial element in $G$, the
fusion outcome is replaced by condensation defect for gauging $K$ symmetry on
the domain wall, i.e. condensing the Wilson lines of $K$ on the wall.

We can describe the non-invertible symmetry as the coset

$\text{Non-invertible coset symmetry}\quad\tilde{G}={G\rtimes_{\rho}K\over
K}~{},$ (3.2)

where the discrete group $K$ acts on $G$ by $\rho$. Since $K$ is not a normal
subgroup for nontrivial action $\rho$, the coset is not a group, but rather a
non-invertible symmetry. In addition, when $K$ is not normal, the left and
right cosets differ. Here, we will always refer to the right coset.

##### Example: cosine symmetry from gauging charge conjugation

Consider $G=U(1)$, and $K=\mathbb{Z}_{2}$ that acts on $G$ by charge
conjugation. Gauging $K$ results in the non-invertible symmetry

$\tilde{U}_{[\theta]}=U_{\theta}\oplus U_{-\theta}~{},$ (3.3)

where the $U(1)$ transform is $e^{i\theta}$, and $\theta\rightarrow-\theta$
under charge conjugation. It satisfies the fusion rule (here we take
$\theta\neq\pm\theta^{\prime}$)

$\tilde{U}_{[\theta]}\times\tilde{U}_{[\theta^{\prime}]}=\tilde{U}_{[\theta+\theta^{\prime}]}+\tilde{U}_{[\theta-\theta^{\prime}]}~{}.$
(3.4)

The above fusion rule is reminiscent of the formula
$2\cos\theta\cos\theta^{\prime}=\cos(\theta+\theta^{\prime})+\cos(\theta-\theta^{\prime})$,
so this continuous non-invertible symmetry is referred to as cosine symmetry.

The cosine symmetry can be described by the coset $O(2)/\mathbb{Z}_{2}$, where
$\mathbb{Z}_{2}$ is the charge conjugation. Since the $\mathbb{Z}_{2}$ is not
a normal subgroup, the coset is not a group, but rather a non-invertible
symmetry.

##### Example: non-invertible SWAP symmetry

Consider two copies of a system with $K$ symmetry, in total the theory has
$G=(K\times K)\rtimes\mathbb{Z}_{2}$ symmetry, where the $\mathbb{Z}_{2}$
exchanges the two copies and thus swaps the two $K$ groups. Let us then gauge
one of the $K$ non-normal subgroup symmetry: this results in the non-
invertible coset symmetry

${G\over K^{(1)}}={(K^{(1)}\times K^{(2)})\rtimes\mathbb{Z}_{2}\over
K^{(1)}}~{},$ (3.5)

where the $K$ symmetry in $j$th layer is written as $K^{(j)}$. In particular,
the $\mathbb{Z}_{2}$ SWAP symmetry becomes non-invertible, with the fusion
rule

$\displaystyle\widetilde{\text{SWAP}}\times\widetilde{\text{SWAP}}=C_{\mathrm{Rep}(K^{(1)})}\left(\sum_{k^{(2)}\in
K^{(2)}}U_{k^{(2)}}\right),$ (3.6)

where $C_{\mathrm{Rep}(K^{(1)})}$ is the condensation defect of the $K^{(1)}$
Wilson lines $\mathrm{Rep}(K^{(1)})$ in the first layer [49], and
$U_{k^{(2)}}$ is the generator of global $K^{(2)}$ symmetry in the second
layer.

###  3.2 Non-invertible symmetry as sandwich of invertible symmetry

Let us consider the $(d+1)$D theory $\mathcal{T}$ with zero-form symmetry $G$,
and gauge the discrete non-normal subgroup $K$. As discussed in section 3.1,
the gauged theory $\mathcal{T}/K$ has a non-invertible symmetry $G/K$ with
symmetry defects $\tilde{U}_{[g]}$. The symmetry defect $\tilde{U}_{[g]}$ can
be expressed as the invertible symmetry defect $U_{g}$ sandwiched by non-
invertible domain walls. To determine the appropriate choice of non-invertible
domain walls, we must first determine the “minimal” nonnormal subgroup
$K/\tilde{K}$. Specifically, there may be a subgroup $\tilde{K}\subset K$
which is a normal subgroup of $G$. Such a $\tilde{K}$ is obtained by a group
of elements $k\in K$ satisfying $gkg^{-1}\in K$ for any $g\in G$, which can be
checked to be a normal subgroup of $K,G$. In this case, we first gauge the
maximal normal subgroup $\tilde{K}$ of $K$ to obtain an invertible symmetry
$G/\tilde{K}$ and then gauge the remaining part $K/\tilde{K}$.555We use this
convention so that if $K$ is normal, then the defect under the sandwich
construction is invertible. If we do not gauge $\tilde{K}$ first, then
generically our construction would produce the $G/K$ defect together with a
condensation defect, even if $K$ is normal. Then the symmetry defect for $G/K$
takes the form

$\displaystyle\tilde{U}_{[g]}=D_{\mathrm{Rep}(K/\tilde{K})}\times
U_{g}\times\overline{D}_{\mathrm{Rep}(K/\tilde{K})}.$ (3.7)

Here, $U_{g}$ is a $g\in G/\tilde{K}$ symmetry defect of the theory
$\mathcal{T}/\tilde{K}$ before gauging $K/\tilde{K}$.
$D_{\mathrm{Rep}(K/\tilde{K})}$ condenses the Wilson line
$\mathrm{Rep}(K/\tilde{K})$ of the $K/\tilde{K}$ gauge theory
$\mathcal{T}/K=(\mathcal{T}/\tilde{K})/(K/\tilde{K})$, and regarded as a half
gauging defect that interpolates $\mathcal{T}/K$ and $\mathcal{T}/\tilde{K}$
[12] (see Figure 1 for the case with trivial $\tilde{K}$). Without loss of
generality, we can assume that $K$ is minimal with
$\tilde{K}=\\{\mathrm{id}\\}$, since $G/\tilde{K}$ is a group. In the
following we will use condensation defects $D_{\mathrm{Rep}(K)}$. We will
shortly see that the above defect $\tilde{U}_{[g]}$ only depends on the orbit
$[g]$ of $g\in G$ under the action of $K$. Let us first see a few simple
examples for the defect $\tilde{U}_{[g]}$:

  * •

The defect $\tilde{U}_{[g]}$ obviously gives an invertible symmetry defect if
and only if $D_{\mathrm{Rep}(K)}$, $\overline{D}_{\mathrm{Rep}(K)}$ are
trivial.

  * •

When the invertible defect $U$ in the middle is trivial, the defect
$\tilde{U}_{[g]}$ is a condensation defect condensing the electric particles
$\mathrm{Rep}(K)$. This corresponds to 1-gauging the algebra object
$\mathrm{Rep}(K)$ at a codimension-1 defect [49].

The fusion rules of the half gauging defects are given by (up to
normalization)

$\displaystyle\overline{D}_{\mathrm{Rep}(K)}\times{D}_{\mathrm{Rep}(K)}=\sum_{k\in
K}U_{k},$ (3.8)

$\displaystyle{D}_{\mathrm{Rep}(K)}\times\overline{D}_{\mathrm{Rep}(K)}=C_{\mathrm{Rep}(K)},$
(3.9)

where $U_{k}$ is the $K$ symmetry defect of $\mathcal{T}$ and
$C_{\mathrm{Rep}(K)}$ is a condensation defect obtained by 1-gauging
$\mathrm{Rep}(K)$. Also

$\displaystyle
U_{k}\times\overline{D}_{\mathrm{Rep}(K)}=\overline{D}_{\mathrm{Rep}(K)},\quad{D}_{\mathrm{Rep}(K)}\times
U_{k}={D}_{\mathrm{Rep}(K)},$ (3.10)

with $k\in K$, and

$\displaystyle\overline{D}_{\mathrm{Rep}(K)}\times
W_{\rho}=\overline{D}_{\mathrm{Rep}(K)},\quad
W_{\rho}\times{D}_{\mathrm{Rep}(K)}={D}_{\mathrm{Rep}(K)},$ (3.11)

where $W_{\rho}$ is an Wilson line carrying the irreducible representation
$\rho\in\mathrm{Rep}(K)$.

From the fusion rule between $U_{k}$ and $D_{\mathrm{Rep}(K)}$ one can see
that the definition of $\tilde{U}_{[g]}$ only depends on the orbit $[g]$,

$\displaystyle\tilde{U}_{[g]}=D_{\mathrm{Rep}(K)}\times
U_{g}\times\overline{D}_{\mathrm{Rep}(K)}=D_{\mathrm{Rep}(K)}\times
U_{k}U_{g}U_{k^{-1}}\times\overline{D}_{\mathrm{Rep}(K)}=\tilde{U}_{[kgk^{-1}]}.$
(3.12)

The fusion algebra of $\tilde{U}_{[g]}$ follows from the fusion rules above:

$\displaystyle\begin{split}\tilde{U}_{[g]}\times\tilde{U}_{[g^{\prime}]}&=D_{\mathrm{Rep}(K)}\times
U_{g}\times\left(\sum_{k\in K}U_{k}\right)\times
U_{g^{\prime}}\times\overline{D}_{\mathrm{Rep}(K)}\\\
&=D_{\mathrm{Rep}(K)}\times\left(\sum_{k\in
K}U_{gkg^{\prime}k^{-1}}U_{k}\right)\times\overline{D}_{\mathrm{Rep}(K)}\\\
&=D_{\mathrm{Rep}(K)}\times\left(\sum_{k\in
K}U_{gkg^{\prime}k^{-1}}\right)\times\overline{D}_{\mathrm{Rep}(K)}\\\
&=\sum_{k\in K}\tilde{U}_{[gkg^{\prime}k^{-1}]}\end{split}$ (3.13)

When $[g^{\prime}]=[g^{-1}]$, one of the fusion channels is $\tilde{U}_{[1]}$
which is the condensation defect

$\displaystyle\tilde{U}_{[1]}=C_{\mathrm{Rep}(K)}.$ (3.14)

Finally, we can compute the fusion rules of $\tilde{U}_{[g]}$ with the Wilson
line of $K$ gauge theory using (3.11):

$\displaystyle\tilde{U}_{[g]}\times
W_{\rho}=\tilde{U}_{[g]}=W_{\rho}\times\tilde{U}_{[g]}.$ (3.15)

##### Junction of coset symmetry defects

Two coset symmetry defects $\tilde{U}_{[g]}$ can fuse at the junction into a
third defect, which corresponds to one of the fusion outcomes

$\displaystyle\tilde{U}_{[g]}\times\tilde{U}_{[g^{\prime}]}=\sum_{k\in
K}\tilde{U}_{[gkg^{\prime}k^{-1}]}~{}.$ (3.16)

For example, the junction for the fusion channel
$\tilde{U}_{[g]}\times\tilde{U}_{[g^{\prime}]}\to\tilde{U}_{[gkg^{\prime}k^{-1}]}$
can be obtained by the junction of invertible $G$ symmetry defects
$g,kg^{\prime}k^{-1}$ and $gkg^{\prime}k^{-1}$ sandwiched by the non-
invertible defects $D_{\mathrm{Rep}(K)}$, as shown Figure 2 (a).

As mentioned earlier, the sandwich construction for the coset symmetry defect
has a redundancy $\tilde{U}_{[g]}=\tilde{U}_{[kgk^{-1}]}$. In other words, the
invertible defect $g\in G$ or $kgk^{-1}\in G$ leads to the same coset symmetry
defect. This leads to the redundancy in expressing the junction of the coset
symmetry defects: the junction of invertible defects $g,g^{\prime}$ into
$gg^{\prime}$ gives an identical junction of coset symmetry defects as the
junction of $kgk^{-1},kg^{\prime}k^{-1}$ into $kgg^{\prime}k^{-1}$ for $k\in
K$ (see Figure 2 (b)).

Figure 1: The continuous non-invertible symmetry defect of the gauged theory
$\mathcal{T}/K$ is understood as an invertible defect sandwiched by a pair of
half gauging defects. Figure 2: (a): The junction of the cosine symmetry
defects that corresponds to the fusion channel
$\tilde{U}_{[g]}\times\tilde{U}_{[g^{\prime}]}\to\tilde{U}_{[gkg^{\prime}k^{-1}]}$.
(b): Conjugating the invertible defects by $k\in K$ in the network of coset
symmetry defects leads to an another expression of the same defect network in
terms of a sandwich.

##### Example: cosine symmetry

For example, when $G=O(2)=U(1)\rtimes\mathbb{Z}_{2}$ and
$K=\mathbb{Z}_{2}=\\{1,c\\}$ that acts on $G$ by charge conjugation, the
fusion rule of the symmetry defects of $\mathcal{T}/K$ is given by

$\displaystyle\begin{split}\tilde{U}_{[\theta]}\times\tilde{U}_{[\theta^{\prime}]}&=D_{\mathrm{Rep}(K)}\times
U_{\theta}\times\left(1+U_{c}\right)\times
U_{\theta^{\prime}}\times\overline{D}_{\mathrm{Rep}(K)}\\\
&=D_{\mathrm{Rep}(K)}\times\left(U_{\theta+\theta^{\prime}}+U_{\theta-\theta^{\prime}}U_{c}\right)\times\overline{D}_{\mathrm{Rep}(K)}\end{split}$
(3.17)

We then have the fusion rule
$U_{c}\times\overline{D}_{\mathrm{Rep}(K)}=\overline{D}_{\mathrm{Rep}(K)}$, so

$\displaystyle\tilde{U}_{[\theta]}\times\tilde{U}_{[\theta^{\prime}]}=\tilde{U}_{[\theta+\theta^{\prime}]}+\tilde{U}_{[\theta-\theta^{\prime}]}$
(3.18)

This reproduces the fusion rule of the cosine symmetry (3.4), and is a
particular case of (3.13). When $[\theta]=[\theta^{\prime}]$ we get

$\displaystyle\tilde{U}_{[\theta]}\times\tilde{U}_{[\theta]}=\tilde{U}_{[2\theta]}+C_{\mathrm{Rep}(K)}.$
(3.19)

###  3.3 Exotic quantum Hall conductance for continuous coset symmetry

In the section above, we described how to construct defects of non-invertible
0-form symmetry. We now study the possible decorations of junctions of such
defects; these decorations specify fractionalization patterns of non-
invertible 0-form symmetry. We begin with an example. Let us start with an
ordinary fractional quantum Hall system with $U(1)$ symmetry in (2+1)D, and
then gauge the charge conjugation symmetry. This converts the $U(1)$ symmetry
into the non-invertible cosine symmetry, which can be expressed as the coset
$O(2)/\mathbb{Z}_{2}$ for non-normal charge conjugation $\mathbb{Z}_{2}$. The
non-invertible symmetry obeys the fusion rule (3.4). We note that the cosine
symmetry does not permute the anyons, just as the ordinary continuous $U(1)$
symmetry.

The current is odd under $\mathbb{Z}_{2}$ charge conjugation and therefore
becomes not gauge invariant. Similarly, the electric and magnetic fields are
also odd under $\mathbb{Z}_{2}$ charge conjugation. Thus the conductivity
matrix is gauge invariant after gauging the charge conjugation and well-
defined.

Another way to see that the Hall conductance remains well defined is by using
the current two point function. This quantity is invariant under charge
conjugation, and its contact term gives the Hall conductivity:

$\langle
j_{\mu}(x,y,t)j_{\nu}(0)\rangle\supset\sigma_{H}\epsilon_{\mu\nu\lambda}\partial^{\lambda}\delta^{3}(x,y,t)~{},$
(3.20)

where $\sigma_{H}$ is the quantum Hall conductance in the unit of $e^{2}/h$
for electron charge $e$ and Planck constant $h$. The formula follows from
taking the functional derivative in the quadratic Chern-Simons response [50].
We note that while the full two-point function requires a Wilson line of the
$\mathbb{Z}_{2}$ gauge field connecting the two points after gauging the
charge conjugation symmetry, the contact term is not modified. Thus the Hall
response is well-defined for the non-invertible cosine symmetry, and it is
given by the response before gauging the charge conjugation symmetry.

####  3.3.1 $O(2)$ Chern-Simons theory

We now show that the theory above has a fractional quantum quantum Hall
response $\sigma_{H}$ for non-invertible continuous symmetry that cannot be
obtained from the spin of a vison.

Let us start with $U(1)_{2k}$ theory for integer $k>1$. We can enrich the
theory with $U(1)$ symmetry by identifying the vison with the charge-one
Abelian anyon $v$. We can read off the Hall conductance as
$\sigma_{H}=1/(2k)$, from the spin of $v$.

Now, let us gauge the charge conjugation symmetry in the minimal way. In
general, gauging a 0-form symmetry in (2+1)D comes with a choice of an
$H^{3}(K,U(1))$ class. Physically, this corresponds to stacking with a $K$ SPT
before gauging. For simplicity, we will only consider gauging without any
additional SPT in this example.

The charge conjugation symmetry transforms $a\rightarrow a^{-1}$ for a generic
Abelian anyon $a$, so the gauged theory becomes a non-Abelian topological
order. The gauge symmetry is enlarged to be $U(1)\rtimes\mathbb{Z}_{2}=O(2)$,
i.e. the theory becomes $O(2)_{2k}$ Chern-Simons theory. The new particle
content is

  * •

Charge $Q\neq 0,k$: these anyons become non-Abelian anyons with quantum
dimension two, given by $Q\oplus(-Q)$.

  * •

Charge $Q=0$: the vacuum anyon and the $\mathbb{Z}_{2}$ symmetry defects lead
to four anyons: the trivial anyon, the Abelian boson $W$ given by the
$\mathbb{Z}_{2}$ Wilson line, and anyons $\xi_{0},\chi_{0}$ of spin
$\frac{1}{16}$ and $\frac{9}{16}$ respectively. The quantum dimension of
$\xi_{0}$ and $\chi_{0}$ is $\sqrt{k}$. Therefore, these are non-Abelian for
$k>1$.

  * •

Charge $Q=k$: this anyon and $\mathbb{Z}_{2}$ symmetry defects together with
this anyon also lead to four anyons: the Abelian anyons $a_{1},a_{2}$ with
spin $\frac{k}{4}$, and anyons $\xi_{k},\chi_{k}$ of spin $\frac{1}{16}$ and
$\frac{9}{16}$ respectively. The quantum dimension of $\xi_{k},\chi_{k}$ is
also $\sqrt{k}$, so these are also non-Abelian for $k>1$.

In the new theory $O(2)_{2k}$ the only Abelian anyons have spin $0,k/4$ mod 1,
so one would not be able to detect the fractional quantum Hall conductance
$\sigma_{H}=1/(2k)$ for $k>1$ from looking at the spins of the abelian anyons.
It is instead related to the spin of the non-Abelian anyon $v\oplus v^{-1}$.
Thus the non-invertible cosine symmetry $O(2)/\mathbb{Z}_{2}$ gives rise to a
Hall conductance proportional to the spin of non-Abelian anyon, rather than an
abelain one.

We remark that when $k=1$, $O(2)_{2}\leftrightarrow U(1)_{8}$ is an Abelian
TQFT [51, 52]. Thus even in Abelian TQFTs there can be fractionalized non-
invertible coset symmetry.

####  3.3.2 Gapless edge modes

Here we consider a (1+1)D edge state of the above FQH state with cosine
symmetry. We argue that the nonzero electric Hall conductance (3.20) enforces
a gapless edge mode if the boundary preserves the cosine symmetry. Before
giving a proof for this statement, we first need to clarify what we mean by a
symmetry-preserving edge state, since there can be multiple definitions for it
which bifurcate for non-invertible symmetries [53]. In this paper, symmetry-
preserving boundary condition means a boundary condition that satisfies the
following two requirements:666Following the terminology of [53], this amounts
to requiring that the boundary condition is both strongly and weakly
symmetric.

  1. 1.

The boundary state is invariant the symmetry action. That is, the boundary
state $\ket{\mathcal{B}}$ supported on a closed 2d space is an eigenstate of
the symmetry operators $\mathcal{D}$:
$\mathcal{D}\ket{\mathcal{B}}\propto\ket{\mathcal{B}}$. This is equivalent to
requiring that the type of the boundary condition is invariant under pushing
the symmetry defect onto the boundary.

  2. 2.

The symmetry defect can terminate at the boundary. Its endpoint is topological
and defines the symmetry defect at the boundary theory.

Let us show that the cosine symmetry-preserving boundary must carry
$\sigma_{H}=0$.

We take a symmetry-preserving gapped boundary state $|\mathcal{B}\rangle$.
First, the invariance of the boundary state on a torus $\ket{\mathcal{B}}$
under the cosine symmetry defect $\tilde{U}_{\theta=0}$ implies that the
Wilson line $W$ of the $\mathbb{Z}_{2}$ charge conjugation symmetry must be
condensed at the boundary. This can be seen by noticing that
$\tilde{U}_{\theta=0}$ is a condensation defect of $W$ due to (3.14), which
acts on a torus $T^{2}_{xy}$ by an operator
$(1+W(\gamma_{x})+W(\gamma_{y})+W(\gamma_{x})W(\gamma_{y}))$.

Meanwhile, the boundary state $\ket{\mathcal{B}}$ on a torus contains a state
$\ket{1}$ labeled by a trivial anyon 1, and has the form of
$\ket{\mathcal{B}}=\ket{1}+Z_{W}\ket{W}+...$ with some non-negative integer
$Z_{W}$. One can then immediately see that the state
$\tilde{U}_{\theta=0}\ket{\mathcal{B}}$ contains the state $\ket{W}$ with some
positive coefficient, which implies that $W$ is condensed at the boundary once
we require $\tilde{U}_{\theta=0}\ket{\mathcal{B}}\propto\ket{\mathcal{B}}$.

Therefore, it must be possible to obtain the gapped boundary must by first
condensing $W$ in the theory $\mathcal{T}/\mathbb{Z}_{2}$. This brings the
theory back to an original FQH state $\mathcal{T}$ with
$U(1)\rtimes\mathbb{Z}_{2}$ symmetry. If there is a gapped boundary, then we
can further condense Lagrangian algebra anyons in $\mathcal{T}$. In other
words, without loss of generality, we can describe the boundary state
$\ket{\mathcal{B}}$ of $\mathcal{T}/\mathbb{Z}_{2}$ by first acting the
operator $\mathcal{D}_{\mathrm{Rep}(\mathbb{Z}_{2})}$ on some gapped boundary
state $\ket{\mathcal{B}^{\prime}}$ of the theory $\mathcal{T}$,

$\displaystyle\ket{\mathcal{B}}=\mathcal{D}_{\mathrm{Rep}(\mathbb{Z}_{2})}\ket{\mathcal{B}^{\prime}}$
(3.21)

where $\mathcal{D}_{\mathrm{Rep}(\mathbb{Z}_{2})}$ is an interface operator
obtained by half-gauging (see section 3.2 for discussions). The gapped
boundary $\mathcal{B}$ is obtained by pushing the defect
$\mathcal{D}_{\mathrm{Rep}(\mathbb{Z}_{2})}$ in parallel onto the gapped
boundary $\mathcal{B}^{\prime}$.

Now let us consider a cosine symmetry defect $\tilde{U}_{\theta}$ terminating
at the gapped boundary $\mathcal{B}$. Since the $\tilde{U}_{\theta}$ is
transformed into $U_{\theta}$ or $U_{-\theta}$ by crossing through the
interface $\mathcal{D}_{\mathrm{Rep}(\mathbb{Z}_{2})}$, one can see that the
$U(1)$ defects $U_{\theta}$ have to end at the gapped boundary
$\mathcal{B}^{\prime}$, meaning that the gapped boundary
$\mathcal{B}^{\prime}$ preserves the $U(1)$ symmetry. This enforces
$\sigma_{H}=0$.

###  3.4 More example of fractionalized coset symmetry: $S_{3}$ gauge theory

Let us consider $\mathbb{Z}_{3}$ gauge theory enriched by $U(1)$ symmetry,
where the fractionalization is given by braiding with $v=em$. Since $v$
carries the spin $h_{v}=1/3$, this topological order has the Hall response
$\sigma_{H}=2/3$. If we gauge the charge conjugation symmetry, we get $S_{3}$
gauge theory with the Hall response for the cosine symmetry.

There are eight anyons in the $S_{3}$ gauge theory (see e.g. [54]), which are
usually denoted by $A,B,C,D$,$E,F,G,H$. Each anyon is associated with a
conjugacy class of $S_{3}$ and an irreducible representation of the
centralizer of a representative element from the conjugacy class:

  * •

$\\{1\\}$: the centralizer is $S_{3}$ with the trivial ($A$), sign ($B$), and
two dimensional ($C$) irreps.

  * •

$\\{s,sr,sr^{2}\\}$: the centralizer is $\\{1,s\\}$ with the trivial ($D$) and
sign ($E$) irreps.

  * •

$\\{r,r^{2}\\}$: the centralizer is $\\{1,r,r^{2}\\}$ with the trivial ($F$),
$e^{2\pi i/3}$ ($G$) and $e^{4\pi i/3}$ ($G$) irreps.

The $T$ matrix is $\mathrm{diag}(1,1,1,-1,1,1,e^{2\pi i/3},e^{4\pi i/3})$. $A$
and $B$ correspond to the vacuum of the $\mathbb{Z}_{3}$ gauge theory with and
without the charge conjugation charge respectively. $C$ corresponds to the
orbit $(e,e^{2})$ under charge conjugation. $D$ and $E$ correspond to
$\mathbb{Z}_{2}$ charge conjugation defects with and without the charge
conjugation charge. $F,G,$ and $H$ correspond to the orbits
$(m,m^{2}),(em,e^{2}m^{2}),$ and $(e^{2}m,em^{2})$. Note that the $S_{3}$
gauge theory does not have an Abelian anyon with the spin $1/3$, so the Hall
conductance $\sigma_{H}=2/3$ cannot be accounted for by ordinary
fractionalization using Abelian anyons.

Based on the example discussed in section 3.4, we can describe the cosine
symmetry defect of $S_{3}$ gauge theory as a sandwich of gapped interfaces of
$S_{3}$ gauge theory. As discussed in section 3.2, this perspective allows us
to describe the junction of the symmetry defects of cosine symmetry. The
junction of the cosine symmetry defects that corresponds to fusion outcome
$\tilde{U}_{[\theta]}\times\tilde{U}_{[\theta^{\prime}]}\to\tilde{U}_{[\theta+\theta^{\prime}]}$
can be realized by the junction of invertible symmetry $U(1)$ defects among
$\theta,\theta^{\prime},\theta+\theta^{\prime}$ decorated with non-invertible
defects, see Figure 3. Similarly, the junction for the fusion outcome
$\tilde{U}_{[\theta]}\times\tilde{U}_{[\theta^{\prime}]}\to\tilde{U}_{[\theta-\theta^{\prime}]}$
can also be obtained from the junction of invertible symmetry $U(1)$ defects
among $\theta,-\theta^{\prime},\theta-\theta^{\prime}$.

The symmetry fractionalization of a non-Abelian anyon $G=(em,e^{2}m^{2})$ in
$S_{3}$ gauge theory is illustrated in Figure 3. When the line operator of the
non-Abelian anyon $G$ crosses through the junction of cosine symmetries, the
junction non-trivially acts on the anyon $G$, which is regarded as the
symmetry fractionalization.

Unlike the standard fractionalization of the invertible symmetry, the
fractional charge of the cosine symmetry is given by a superposition of
opposite $U(1)$ fractional charges. It reflects Figure 3: the fractional
charge depends on whether it splits into $em$ or $e^{2}m^{2}$ with the
opposite $U(1)$ charge when the $S_{3}$ gauge theory is condensed into a
$\mathbb{Z}_{3}$ gauge theory.

Figure 3: The symmetry fractionalizaton of the non-Abelian anyon
$G=(em,e^{2}m^{2})$. This anyon splits into $em$, $e^{2}m^{2}$ of
$\mathbb{Z}_{3}$ gauge theory and they carry the opposite fractional charge.
As a result, the fractional charge carried by $G$ is a diagonal matrix
$\mathrm{diag}(1/3,2/3)$ rather than a number.

##  4 Lattice Model with Fractionalized Non-Invertible Symmetry

In this section, we demonstrate the properties of the non-invertible symmetry
fractionalization with a microscopic lattice model. We consider the (2+1)D
quantum double model on a square lattice, which is a standard lattice model
effectively described by a finite $G$ gauge theory. We find that the quantum
double model with the gauge group $G=S_{3}$ has a fractionalized cosine
symmetry.

###  4.1 Cosine symmetry in $S_{3}$ quantum double model

##### Warm-up: $\mathbb{Z}_{3}$ gauge theory with $U(1)$ symmetry

Let us start with $\mathbb{Z}_{3}$ toric code with $U(1)$ symmetry in (2+1)D.
We consider a square lattice with a $\mathbb{Z}_{3}$ qudit on each edge, with
the Hamiltonian

$\displaystyle H_{\mathbb{Z}_{3}}=-\sum_{v}A_{v}-\sum_{p}B_{p}$ (4.1)

where $v$ and $p$ denote the vertex and plaquette respectively. Specifically,
each term is given by

$\displaystyle
A_{v}=\frac{1}{3}\left(1+{X}_{N(v)}{X}_{E(v)}{X}^{\dagger}_{W(v)}{X}^{\dagger}_{S(v)}+{X}^{\dagger}_{N(v)}{X}^{\dagger}_{E(v)}{X}_{W(v)}{X}_{S(v)}\right)~{},$
(4.2) $\displaystyle
B_{p}=\frac{1}{3}\left(1+Z_{01}Z_{13}Z^{\dagger}_{02}Z^{\dagger}_{23}+Z^{\dagger}_{01}Z^{\dagger}_{13}Z_{02}Z_{23}\right)~{}.$
(4.3)

These two terms are illustrated in Figure 4.

We define the $\mathbb{Z}_{3}$ gauge field $a$ on links satisfying
$e^{\frac{2\pi i}{3}a_{e}}=Z_{e}$. The state with $B_{p}=1$ corresponds to the
flat $\mathbb{Z}_{3}$ gauge field $da=0$. $A_{v}=1$ corresponds to the Gauss
law constraint of the $\mathbb{Z}_{3}$ gauge field, so the ground state
Hilbert space is described by the $\mathbb{Z}_{3}$ gauge theory.

The $U(1)$ global symmetry $U_{\theta}$ is defined as the action on the state

$\displaystyle
U_{\theta}\ket{a}=\exp\left(i[\theta]_{2\pi}\int\frac{d\hat{a}}{3}\right)\ket{a}.$
(4.4)

where $\hat{a}$ is the integral lift of $a$. This $U(1)$ symmetry exhibits the
symmetry fractionalization. When we write the electric particle of the
$\mathbb{Z}_{3}$ gauge theory as labeled by $e$, the Abelian anyon
$e^{\eta(\theta_{1},\theta_{2})}$ is decorated at the junction of $U(1)$
transformations $\theta_{1},\theta_{2},$ and $\theta_{1}+\theta_{2}$ mod
$2\pi$.

Figure 4: The edges nearby a vertex $v$ and plaquette $p$.

##### Cosine symmetry of $S_{3}$ gauge theory

We then consider the $S_{3}$ quantum double model in (2+1)D. The local Hilbert
space on each edge has 6 dimensions, whose basis states $\\{\ket{g}\\}$ are
labeled by group elements $g\in S_{3}$. The Hamiltonian is given by

$\displaystyle H_{S_{3}}=-\sum_{v}A_{v}-\sum_{p}B_{p}$ (4.5)

with each term given by

$\displaystyle A_{v}=\frac{1}{|G|}\sum_{g\in
G}\overrightarrow{X}_{N(v)}^{g}\overrightarrow{X}_{E(v)}^{g}\overleftarrow{X}_{W(v)}^{g^{-1}}\overleftarrow{X}_{S(v)}^{g^{-1}},\quad
B_{p}=\delta_{g_{01}g_{13}g^{-1}_{02}g^{-1}_{23},0}~{}.$ (4.6)

Here we defined the Pauli $X$ like operators as

$\displaystyle\overrightarrow{X}^{g}\ket{h}=\ket{gh},\quad\overleftarrow{X}^{g^{-1}}\ket{h}=\ket{hg^{-1}}~{}.$
(4.7)

It is convenient to label $g\in S_{3}$ by a pair $g=(a,b)$ with
$a\in\mathbb{Z}_{3},b\in\mathbb{Z}_{2}$, satisfying the group multiplication
law

$\displaystyle(a_{1},b_{1})\times(a_{2},b_{2})=(a_{1}+(-1)^{b_{1}}a_{2},b_{1}+b_{2}).$
(4.8)

$a,b$ are regarded as $\mathbb{Z}_{3}$ and $\mathbb{Z}_{2}$ gauge fields
respectively.

Let us define the operators acting like Pauli $X,Z$ operators on the $a,b$
fields:

$\displaystyle Z^{a}\ket{a,b}=e^{\frac{2\pi i}{3}a}\ket{a,b},\quad
X^{a}\ket{a,b}=\ket{a+1,b}$ (4.9)

$\displaystyle Z^{b}\ket{a,b}=(-1)^{b}\ket{a,b},\
\overrightarrow{X}^{b}\ket{a,b}=\ket{-a,b+1},\
\overleftarrow{X}^{b}\ket{a,b}=\ket{a,b+1}$ (4.10)

$\overrightarrow{X}^{b},\overleftarrow{X}^{b}$ corresponds to the left and
right action of $(0,1)\in S_{3}$ on $(a,b)$ respectively.

The vertex term of the $S_{3}$ quantum double model can then be expressed as

$\displaystyle A_{v}=A_{v}^{a}A_{v}^{b}$ (4.11)

where $A_{v}^{a}$ is the vertex term of the $\mathbb{Z}_{3}$ toric code
expressed by $X^{a}$, and

$\displaystyle
A_{v}^{b}=\frac{1+\overrightarrow{X}_{N(v)}^{b}\overrightarrow{X}_{E(v)}^{b}\overleftarrow{X}_{W(v)}^{b}\overleftarrow{X}_{S(v)}^{b}}{2}$
(4.12)

We then define the projection operator

$\displaystyle\mathcal{D}^{b}=\prod_{e}\left(\frac{1+Z^{b}_{e}}{2}\right),$
(4.13)

and the projection operator

$\displaystyle\Pi^{b}=\prod_{v}A^{b}_{v}$ (4.14)

Now we are ready to describe the cosine symmetry of the $S_{3}$ quantum double
model (4.5). The generator of the cosine symmetry is given by

$\displaystyle\tilde{U}_{\theta}=2^{|v|}\cdot\Pi^{b}U_{\theta}\mathcal{D}^{b}$
(4.15)

where $|v|$ is the number of vertices. This generates a non-invertible
symmetry of $S_{3}$ gauge theory. $\mathcal{D}^{b}$ condenses the Wilson line
from the gauged charge conjugation symmetry, resulting in a $\mathbb{Z}_{3}$
gauge theory. $U_{\theta}$ is an invertible defect in $\mathbb{Z}_{3}$ gauge
theory, and $\Pi^{b}$ then brings the theory back to $S_{3}$ gauge theory. We
note that the form of the operator $\tilde{U}_{\theta}$ is aligned with the
sandwich form (3.7); the operators $\Pi^{b}$ on the left, $\mathcal{D}^{b}$ on
the right of (4.16) correspond to half-gauging defects
${D}_{\mathrm{Rep}(\mathbb{Z}_{2})}$,
$\overline{D}_{\mathrm{Rep}(\mathbb{Z}_{2})}$ at a time slice. For the purpose
of computing the fusion rule of operators below, we consider the low energy
subspace where $\Pi^{b}=1$ is satisfied. Within this subspace,
$\tilde{U}_{\theta}$ is expressed as

$\displaystyle\tilde{U}_{\theta}=2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta}\mathcal{D}^{b}\Pi^{b}~{}.$
(4.16)

This allows us to derive the fusion rules for the coset symmetry in the
absence of electric particle excitations for charge conjugation symmetry. If
there are such excitations, the fusion rules can be modified to reflect the
fractionalization of the symmetry on the excitations. We note that for
invertible symmetries, the symmetry action on a pair of conjugate excitations
is the same as the action on the vacuum, since the projective phases cancel.
Here, when the symmetry is non-invertible, we need to be more careful about
the distinction.

Below let us derive the properties of the operator:

  * •

When $\theta=0$,

$\displaystyle\tilde{U}_{\theta=0}=2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}\Pi^{b}$
(4.17)

Here, note that $\mathcal{D}^{b}$ can be rewritten as the sum of open string
operators

$\displaystyle\mathcal{D}^{b}=\frac{1}{2^{|e|}}\sum_{C\in
C_{1}(M,\mathbb{Z}_{2})}\left(\prod_{e\subset C}Z^{b}_{e}\right)$ (4.18)

where the sum of $C$ is over all possible 1-chains of the square lattice, and
$|e|$ is the number of edges. When projected onto the states with
$A_{v}^{b}=1$, only the closed strings $C$ survives, since the product of
$Z_{e}^{b}$ is a line operator of the Abelian anyon in $S_{3}$ gauge theory
and it excites $A^{b}_{v}$ at its ends. We then have

$\displaystyle\tilde{U}_{\theta=0}=\Pi^{b}\left(2^{\chi-1}\sum_{C\in
H^{1}(M,\mathbb{Z}_{2})}\left(\prod_{e\in C}Z_{e}^{b}\right)\right)\Pi^{b},$
(4.19)

where $\chi=|v|-|e|+|p|$ is the Euler characteristic of the lattice. This is a
condensation operator for the Abelian boson of $S_{3}$ gauge theory that is
the Wilson line from gauging the charge conjugation symmetry. See Refs. [55,
28] for other examples of condensation operators on (2+1)D lattice models.

  * •

Let us then derive the fusion rules of the operators. One can derive

$\displaystyle\tilde{U}_{\theta}\times\tilde{U}_{\theta^{\prime}}=\tilde{U}_{\theta+\theta^{\prime}}+\tilde{U}_{\theta-\theta^{\prime}}$
(4.20)

To see this, it is convenient to find a simple expression of the operator
$\mathcal{D}^{b}\Pi^{b}\mathcal{D}^{b}$. Since the operator
$\overrightarrow{X}_{N(v)}^{b}\overrightarrow{X}_{E(v)}^{b}\overleftarrow{X}_{W(v)}^{b}\overleftarrow{X}_{S(v)}^{b}$
shifts $b$ fields $b\to b+d\hat{v}$, most of the product of these vertex terms
are projected out by $\mathcal{D}^{b}$. We can get

$\displaystyle\mathcal{D}^{b}\Pi^{b}\mathcal{D}^{b}=\mathcal{D}^{b}\left(\prod_{v}A_{v}^{b}\right)\mathcal{D}^{b}=\frac{1}{2^{|v|}}\mathcal{D}^{b}\left(1+V_{\mathbb{Z}_{2}}^{C}\right)\mathcal{D}^{b}$
(4.21)

with

$\displaystyle
V_{\mathbb{Z}_{2}}^{C}=\prod_{v}\overrightarrow{X}_{N(v)}^{b}\overrightarrow{X}_{E(v)}^{b}\overleftarrow{X}_{W(v)}^{b}\overleftarrow{X}_{S(v)}^{b}$
(4.22)

One can see that $V_{\mathbb{Z}_{2}}^{C}$ leaves $b$ fields invariant, while
it acts by charge conjugation on the $\mathbb{Z}_{3}$ gauge field $a$,

$\displaystyle V_{\mathbb{Z}_{2}}^{C}\ket{\\{a\\}}=\ket{\\{-a\\}}.$ (4.23)

The fusion rule of $\tilde{U}_{\theta}$ is derived by

$\displaystyle\begin{split}\tilde{U}_{\theta}\times\tilde{U}_{\theta^{\prime}}&=2^{|2v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta}\mathcal{D}^{b}\Pi^{b}\mathcal{D}^{b}U_{\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\\\
&=2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta}\mathcal{D}^{b}\left(1+V_{\mathbb{Z}_{2}}^{C}\right)\mathcal{D}^{b}U_{\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\\\
&=2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta}\left(1+V_{\mathbb{Z}_{2}}^{C}\right)U_{\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\\\
&=2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta}\left(U_{\theta^{\prime}}+U_{-\theta^{\prime}}V_{\mathbb{Z}_{2}}^{C}\right)\mathcal{D}^{b}\Pi^{b}\\\
&=2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta+\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}+2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta-\theta^{\prime}}V_{\mathbb{Z}_{2}}^{C}\mathcal{D}^{b}\Pi^{b}\\\
&=2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta+\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}+2^{|v|}\cdot\Pi^{b}\mathcal{D}^{b}U_{\theta-\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\\\
&=\tilde{U}_{\theta+\theta^{\prime}}+\tilde{U}_{\theta-\theta^{\prime}}\end{split}$
(4.24)

###  4.2 Fractionalization of cosine symmetry in $S_{3}$ quantum double model

##### Warm-up: Fractionalization of $U(1)$ symmetry in $\mathbb{Z}_{3}$ gauge
theory

Let us consider a state with anyon excitations, and suppose that an anyon $x$
is separated from other excitations. Let us pick a subsystem $R$ that contains
the $x$ excitation. The fusion rule of the $U(1)$ symmetry generators support
at $R$ is given by

$\displaystyle U_{\theta}(R)\times
U_{\theta^{\prime}}(R)=U_{\theta+\theta^{\prime}}(R)\times W_{e}(\partial
R)^{\eta(\theta,\theta^{\prime})},$ (4.25)

where
$\eta(\theta,\theta^{\prime})=\frac{[\theta]_{2\pi}+[\theta^{\prime}]_{2\pi}-[\theta+\theta^{\prime}]_{2\pi}}{2\pi}$.
Here, $W_{e}(\partial R)^{\eta(\theta,\theta^{\prime})}$ is a closed string
operator for $e^{\eta(\theta,\theta^{\prime})}$, that wraps around the
boundary of $R$. Note that when the closed string operator acts on a state
with no anyon excitations, it gets absorbed into the ground state. However,
when there is an anyon excitation in $R$, the string operator produces a
braiding phase between the anyon and $e^{\eta(\theta,\theta^{\prime})}$. This
implies that the difference between the action of $U_{\theta}\times
U_{\theta^{\prime}}$ and $U_{\theta+\theta^{\prime}}$ on the anyon $x$ is
given by the fractional phase

$\displaystyle\begin{split}U_{\theta}(R)\times
U_{\theta^{\prime}}(R)\ket{x,\dots}&=U_{\theta+\theta^{\prime}}(R)\times
W_{e}(\partial R)^{\eta(\theta,\theta^{\prime})}\ket{x,\dots}\\\
&=U_{\theta+\theta^{\prime}}(R)\times\exp\left(2\pi
iq_{x}\eta(\theta,\theta^{\prime})\right)\ket{x,\dots}\end{split}$ (4.26)

where $\exp(2\pi iq_{x})=M_{x,e}$, and $M_{x,e}$ denotes the mutual braiding
between two particles $x,e$. The phase proportional to the fractional charge
$q_{x}$ manifests the symmetry fractionalization on the anyon $x$.

##### Fractionalization of cosine symmetry in $S_{3}$ gauge theory

We will see that the cosine symmetry $\tilde{U}_{\theta}$ induces the symmetry
fractionalization of the anyon $(m,m^{2})$ in the $S_{3}$ gauge theory. The
anyon $(m,m^{2})$ is created by an open string operator on the line
$\hat{\gamma}_{p,p^{\prime}}$ terminating at the plaquettes $p,p^{\prime}$.
Given that each anyon carries the quantum dimension 2, the dimension of the
Hilbert space with a pair of anyon excitations has 4 dimensions and is spanned
by the states $\ket{(j,k)}$. Note that this 4 dimensional Hilbert space is
technically not physical because the individual states cannot be distinguished
by gauge invariant operators; $j,k\in\\{1,2\\}$ labels the internal basis
states of non-Abelian anyon at $p,p^{\prime}$ respectively. The symmetry
fractionalization can be seen by acting the symmetry generators at the region
$R$ that contains the plaquette $p$ but not $p^{\prime}$ (see Figure 6). We
explicitly compute the action of cosine symmetry in the presence of anyon
excitations, and show that

$\displaystyle\begin{split}&\tilde{U}_{\theta}(R)\times\tilde{U}_{\theta^{\prime}}(R)\ket{(j,k)}\\\
&=\left[\cos\left(\frac{2\pi}{3}\eta(\theta,\theta^{\prime})+\frac{1}{3}[\theta+\theta^{\prime}]_{2\pi}\right)+\cos\left(\frac{2\pi}{3}\eta(\theta,-\theta^{\prime})+\frac{1}{3}[\theta-\theta^{\prime}]_{2\pi}\right)\right]\left(\ket{(1,k)}+\ket{(2,k)}\right)\end{split}$
(4.27)

and

$\displaystyle\begin{split}&\left[\tilde{U}_{\theta+\theta^{\prime}}(R)+\tilde{U}_{\theta-\theta^{\prime}}(R)\right]\ket{(j,k)}\\\
&=\left[\cos\left(\frac{1}{3}[\theta+\theta^{\prime}]_{2\pi}\right)+\cos\left(\frac{1}{3}[\theta-\theta^{\prime}]_{2\pi}\right)\right]\left(\ket{(1,k)}+\ket{(2,k)}\right)\end{split}$
(4.28)

By comparing the above two expressions, one can see that the fusion rule
$\tilde{U}_{\theta}\times\tilde{U}_{\theta^{\prime}}=\tilde{U}_{\theta+\theta^{\prime}}+\tilde{U}_{\theta-\theta^{\prime}}$
is modified by the fractional charge proportional to
$\frac{2\pi}{3}\eta(\theta,\pm\theta^{\prime})$. This implies that the non-
invertible cosine symmetry is fractionalized on a non-Abelian anyon
$(m,m^{2})$.

There are two points special about fractionalization of cosine symmetry:

  * •

The fusion of cosine symmetries
$\tilde{U}_{\theta}\times\tilde{U}_{\theta^{\prime}}$ splits into two fusion
outcomes $\tilde{U}_{\theta+\theta^{\prime}}$ or
$\tilde{U}_{\theta-\theta^{\prime}}$. Depending on the fusion outcome, the
effect of symmetry fractionalization becomes different; in the former case it
depends on $\eta(\theta,\theta^{\prime})$, while in the latter
$\eta(\theta,-\theta^{\prime})$.

  * •

Once we focus on one fusion outcome $\tilde{U}_{\theta\pm\theta^{\prime}}$,
the symmetry fractionalization appears as the difference between
$\tilde{U}_{\theta}\times\tilde{U}_{\theta^{\prime}}$ and
$\tilde{U}_{\theta\pm\theta^{\prime}}$ in the cosine of the fractional $U(1)$
charge carried by an anyon.777We note that cosine symmetry acts on the state
$\ket{(1,k)}+\ket{(2,k)}$ by an overall normalization given by cosine of
fractional $U(1)$ charge. In particular, the above fusion rule can annihilate
the states with specific choice of $\theta,\theta^{\prime}$. One could
normalize the state $\ket{(1,k)}+\ket{(2,k)}$ to suppress the effect of
fractionalization with specific $\theta,\theta^{\prime}$, but one cannot
choose a normalization which eliminates the fractionalization of all symmetry
operators.

Below, we confirm (4.27), (4.28). First, we have

$\displaystyle\begin{split}\tilde{U}_{\theta}(R)\times\tilde{U}_{\theta^{\prime}}(R)&=2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}U_{\theta}U_{\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\right]_{R}+2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}U_{\theta}U_{-\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\right]_{R}\\\
\end{split}$ (4.29)

See Figure 5 for details of the operators $\Pi^{b},\mathcal{D}^{b},U_{\theta}$
defined on the region $R$.

Let us describe an open string operator for the anyon $(m,m^{2})$ in $S_{3}$
gauge theory. We consider a straight line of the dual lattice
$\hat{\gamma}_{p,p^{\prime}}$ with length $L$ terminating at two plaquettes
$p$, $p^{\prime}$. Then, the string operator
$W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})$ is labeled by $(j,k)$,
where $j,k\in\\{1,2\\}$ labels the basis for the Hilbert space with anyons
with quantum dimension 2. The Hilbert with a pair of anyons $(m,m^{2})$ has 4
dimensions, and its basis is given by
$\ket{(j,k)}=W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\ket{\mathrm{GS}}$.

The string operator is then given by [45]

$\displaystyle\begin{split}W_{(m,m^{2})}^{(1,1)}(\hat{\gamma}_{p,p^{\prime}})&=\left(\prod_{j=0}^{L}(X_{\hat{e}_{j}}^{a})^{\prod_{k=0}^{j-1}Z^{b}_{e_{k}}}\right)\times\frac{1+\prod_{k=0}^{L-1}Z^{b}_{e_{k}}}{\sqrt{2}}\\\
W_{(m,m^{2})}^{(1,2)}(\hat{\gamma}_{p,p^{\prime}})&=\left(\prod_{j=0}^{L}(X_{\hat{e}_{j}}^{a})^{\prod_{k=0}^{j-1}Z^{b}_{e_{k}}}\right)\times\frac{1-\prod_{k=0}^{L-1}Z^{b}_{e_{k}}}{\sqrt{2}}\\\
W_{(m,m^{2})}^{(2,1)}(\hat{\gamma}_{p,p^{\prime}})&=\left(\prod_{j=0}^{L}(X_{\hat{e}_{j}}^{a\dagger})^{\prod_{k=0}^{j-1}Z^{b}_{e_{k}}}\right)\times\frac{1-\prod_{k=0}^{L-1}Z^{b}_{e_{k}}}{\sqrt{2}}\\\
W_{(m,m^{2})}^{(2,2)}(\hat{\gamma}_{p,p^{\prime}})&=\left(\prod_{j=0}^{L}(X_{\hat{e}_{j}}^{a\dagger})^{\prod_{k=0}^{j-1}Z^{b}_{e_{k}}}\right)\times\frac{1+\prod_{k=0}^{L-1}Z^{b}_{e_{k}}}{\sqrt{2}}\\\
\end{split}$ (4.30)

where we label the edges by numbers starting with 0 at the plaquette $p$, and
terminating with $L$ at the plaquette $p^{\prime}$. $\hat{e}_{j}$ is an edge
of $\hat{\gamma}_{p,p^{\prime}}$ cutting the edges, and $e_{k}$ is an edge of
${\gamma}_{p,p^{\prime}}$. See Figure 6 (b) for an illustration.

Let us consider a line operator
$W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})$ starting at a plaquette
$p$ inside the region $R$. The termination $p^{\prime}$ is away from the
region $R$. The state with an anyon $(m,m^{2})$ at the plaquette $p$ is given
by acting any of the four line operators
$W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})$ on the ground state. See
Figure 6 (a) for the geometry.

Figure 5: The region $R$ consists of a set of plaquettes inside a disk. The
edges of the plaquettes in $R$ are represented by orange lines, and the
vertices of plaquettes in $R$ are represented by purple dots. Then, $\Pi^{b}$
is the product of $A_{v}^{b}$ over the purple vertices $v$, $\mathcal{D}^{b}$
is the product of $(1+Z^{b})/2$ over the orange edges $e$, and $U_{\theta}$ is
the integral $\exp(i[\theta]_{2\pi}\int d\hat{a}/3)$ over the orange region
$R$. Note that $\Pi^{b}$ also acts on thick black edges, since the operators
$A_{v}^{b}$ on the boundary of $R$ act on these vertices.  Figure 6: The anyon
created by an open line operator $\hat{\gamma}_{p}$.

The symmetry fractionalization on $(m,m^{2})$ can be seen by acting symmetry
generators $\tilde{U}_{\theta}(R)$ in the presence of the anyon.

$\displaystyle\begin{split}\tilde{U}_{\theta}(R)\times\tilde{U}_{\theta^{\prime}}(R)\ket{(j,k)}&=2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}U_{\theta}U_{\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\right]_{R}W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\ket{\mathrm{GS}}\\\
&+2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}U_{\theta}U_{-\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\right]_{R}W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\ket{\mathrm{GS}}\end{split}$
(4.31)

The projector $\Pi_{b}=\prod_{v}A_{v}$ behaves as an identity operator away
from the anyon excitations, but $A_{v}$ can change the internal state of the
anyon $\ket{(j,k)}$ when $v$ is located at the position of the anyon. More
concretely, the star operator
$\overrightarrow{X}_{N(v)}^{b}\overrightarrow{X}_{E(v)}^{b}\overleftarrow{X}_{W(v)}^{b}\overleftarrow{X}_{S(v)}^{b}$
acts by charge conjugation on the internal state of the non-Abelian anyon. One
can explicitly check that

$\displaystyle\begin{split}\left[\Pi^{b}\right]_{R}W^{(1,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\ket{\mathrm{GS}}&=\frac{1}{2}\left(W^{(1,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})+W^{(2,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\right)\ket{\mathrm{GS}}\\\
\left[\Pi^{b}\right]_{R}W^{(2,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\ket{\mathrm{GS}}&=\frac{1}{2}\left(W^{(1,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})+W^{(2,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\right)\ket{\mathrm{GS}}\\\
\end{split}$ (4.32)

After the wave function is projected by the operator $[\mathcal{D}^{b}]_{R}$,
the $\mathbb{Z}_{2}$ gauge field satisfies $Z^{b}=1$ inside the region $R$.
According to the expression in (4.30), the expression of string operator
inside the region $R$ is fixed as

$\displaystyle
W^{(1,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})=\prod_{j}X_{\hat{e}_{j}}^{a}\quad\text{when
restricted to the region $R$.}$ (4.33) $\displaystyle
W^{(2,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})=\prod_{j}X_{\hat{e}_{j}}^{a\dagger}\quad\text{when
restricted to the region $R$.}$ (4.34)

With this in mind, one can rewrite the expression as

$\displaystyle\begin{split}&\tilde{U}_{\theta}(R)\times\tilde{U}_{\theta^{\prime}}(R)\ket{(j,k)}\\\
&=2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}U_{\theta}U_{\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\right]_{R}\cdot\frac{1}{2}\left(W^{(1,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})+W^{(2,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\right)\ket{\mathrm{GS}}\\\
&+2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}U_{\theta}U_{-\theta^{\prime}}\mathcal{D}^{b}\Pi^{b}\right]_{R}\frac{1}{2}\left(W^{(1,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})+W^{(2,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\right)\ket{\mathrm{GS}}\\\
&=2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}\right]_{R}\sum_{j}\left[\frac{1}{2}\exp\left(\frac{ij}{3}([\theta]_{2\pi}+[\theta^{\prime}]_{2\pi})\right)W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\right]\ket{\mathrm{GS}}\\\
&+2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}\right]_{R}\sum_{j}\left[\frac{1}{2}\exp\left(\frac{ij}{3}([\theta]_{2\pi}+[-\theta^{\prime}]_{2\pi})\right)W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\right]\ket{\mathrm{GS}}\\\
\end{split}$ (4.35)

One can then see that

$\displaystyle\begin{split}2^{|v|}\cdot\left[\Pi^{b}\mathcal{D}^{b}\right]_{R}W^{(j,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\ket{\mathrm{GS}}=\left(W^{(1,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})+W^{(2,k)}_{(m,m^{2})}(\hat{\gamma}_{p,p^{\prime}})\right)\ket{\mathrm{GS}}\end{split}$
(4.36)

so we have

$\displaystyle\begin{split}&\tilde{U}_{\theta}(R)\times\tilde{U}_{\theta^{\prime}}(R)\ket{(j,k)}\\\
&=\cos\left(\frac{1}{3}([\theta]_{2\pi}+[\theta^{\prime}]_{2\pi})\right)\left(\ket{(1,k)}+\ket{(2,k)}\right)+\cos\left(\frac{1}{3}([\theta]_{2\pi}+[-\theta^{\prime}]_{2\pi})\right)\left(\ket{(1,k)}+\ket{(2,k)}\right)\end{split}$
(4.37)

which produces (4.27). Meanwhile, the action of
$\tilde{U}_{\theta+\theta^{\prime}}(R)$,
$\tilde{U}_{\theta-\theta^{\prime}}(R)$ are evaluated as

$\displaystyle\begin{split}&\tilde{U}_{\theta+\theta^{\prime}}(R)\ket{(j,k)}=\cos\left(\frac{1}{3}[\theta+\theta^{\prime}]_{2\pi}\right)\left(\ket{(1,k)}+\ket{(2,k)}\right),\\\
&\tilde{U}_{\theta-\theta^{\prime}}(R)\ket{(j,k)}=\cos\left(\frac{1}{3}[\theta-\theta^{\prime}]_{2\pi}\right)\left(\ket{(1,k)}+\ket{(2,k)}\right)\\\
\end{split}$ (4.38)

which produces (4.28).

##  5 General Coset Symmetry Fractionalization and Bulk TQFT

###  5.1 General coset symmetry and its fractionalization in (2+1)D
topological order

We now describe the general framework for describing fractionalization of non-
invertible coset symmetry $G/K$ in (2+1)D topological orders. We start with a
(2+1)D TQFT with 0-form global symmetry $G$ and gauge the non-normal subgroup
$K$. $G$ can either be a continuous or discrete group. The original TQFT is
described by a modular tensor category $\mathcal{C}$, whose objects
$\mathrm{Obj}(\mathcal{C})$ corresponds to the set of anyons. When $G$ is not
a connected group, the $G$ symmetry can act on anyons by permuting their
labels according to different connected components $\pi_{0}(G)$ ,
$\rho_{g}:a\to^{g}a$ for $a\in\mathrm{Obj}(\mathcal{C}),g\in G$. The $G$
symmetry action on the TQFT is then characterized by the symmetry
fractionalization data $\\{U,\eta\\}$ (see Figure 7).

Let us gauge the non-normal $K$ subgroup of the modular tensor category
$\mathcal{C}$. The gauged theory is described by a $K$-equivalentization of
the category $\mathcal{C}$, which we denote as $\mathcal{C}/K$. Algebraically,
the gauging procedure to obtain $\mathcal{C}/K$ is performed in two steps [1].
The first step is to include the vortices of the symmetry group $K$, which is
to take the $K$-crossed extension $\mathcal{C}_{K}^{\times}$ of the category
$\mathcal{C}$,

$\displaystyle\mathcal{C}_{K}^{\times}=\bigoplus_{k\in K}\mathcal{C}_{k}$
(5.1)

where a simple object of $\mathcal{C}_{k}$ represents the vortex labeled by
$k\in K$. The second step is to make the $K$ gauge group dynamical, and
include the electric charge of $K$ symmetry. The simple anyon in the gauged
theory is labeled by $([a_{k}],\pi_{a})\in\mathrm{Obj}(\mathcal{C}/K)$, where
$a_{k}\in\mathrm{Obj}(\mathcal{C}_{k})$ is a vortex carrying the holonomy
$k\in K$ in the ungauged theory, and $[a_{k}]$ denotes the orbit of $a_{k}$
under the permutation action of $K$. $\pi_{a}$ is an irreducible projective
representation of the gauge group described as follows. Let us write a
subgroup $K_{a}\subset K$ which fixes the label of the anyon $a_{k}$ under its
action. $\pi_{a}$ then satisfies

$\displaystyle\pi_{a}(k)\pi_{a}(k^{\prime})=\eta_{a}(k,k^{\prime})\pi_{a}(kk^{\prime})\quad\text{for
$k,k^{\prime}\in K_{a}$}.$ (5.2)

This is regarded as an electric charge attached to the particle. The above
projective representation is referred to as an $\eta_{a}$-irrep, and its set
is denoted by $\mathrm{Irrep}_{\eta}(K_{a})$.

Figure 7: Anyon lines (black) passing through invertible symmetry defects
$g,h\in G$ and graphical definitions of the $U$ and $\eta$ symbols.

##### Fractionalization of coset symmetry in the gauged theory $\mathcal{C}/K$

As described in section 3.2, the gauged theory $\mathcal{C}/K$ has the coset
symmetry defect expressed by the sandwich

$\displaystyle\tilde{U}_{[g]}=D_{\mathrm{Rep}(K)}\times
U_{g}\times\overline{D}_{\mathrm{Rep}(K)}.$ (5.3)

When the anyon $([a],\pi_{a})$ of $\mathcal{C}/K$ tunnels through the non-
invertible defect $\tilde{U}_{[g]}$, it can be transformed into multiple
choices of anyons. Concretely, $\tilde{U}_{[g]}$ transforms the anyon
according to the channels

$\displaystyle([a],\pi_{a})\to([^{kgk^{-1}}a],\pi^{\prime}_{{}^{kgk^{-1}}a}),\quad\text{for
$k\in
K,\pi^{\prime}_{{}^{kgk^{-1}}a}\in\mathrm{Irrep}_{\eta}(K_{{}^{kgk^{-1}}a})$}$
(5.4)

Intuitively, the way the non-Abelian anyon $([a],\pi_{a})$ is transformed by
$\tilde{U}_{[g]}$ depends on the internal state of the non-Abelian anyon
excitation. This leads to the multiple ways the anyon gets permuted (5.4), as
well as the superposition for the distinct fractional charge of the anyon.

The pattern of the coset symmetry fractionalization depends on the choices of
the transformation (5.4) labeled by $k\in K$. The effect of crossing the anyon
through the junction of coset symmetry defects is described in Figure 8. The
phase for the symmetry fractionalization $\eta_{{}^{k}a}(g,h)$ depends on the
label of the anyon ${}^{k}a\in\mathrm{Obj}(\mathcal{C})$ inside the sandwich.
This leads to the phenomena that the fractional charge is given by a
superposition of distinct fractional phases, see section 3.4 for an example.

The action of the coset symmetry on the junction of anyons is described in
Figure 9. Here, we introduced the tunneling matrix $M^{\mu\nu}_{\mathcal{D}}$
where the fusion vertex of the anyons $\mu$ is transformed into the new vertex
$\nu$ by crossing through the gapped interface $\mathcal{D}$, see Figure 10.
This quantity has been introduced in [56].

Figure 8: The anyon line $(a,\pi_{a})$ crosses through the junction of the
cosine symmetry defects. The fractional phase $\eta_{{}^{k}a}(g,h)$ appears
depending on the way the anyon tunnels through the defect. Figure 9: The
cosine symmetry defect crosses through the junction of anyons. The phase
factor $U$ appears depending on the way the anyon tunnels through the defect.
$\mu,\nu,\sigma,\rho$ are the labels for the basis of the fusion vertex.
Figure 10: The tunneling matrix $M_{\mathcal{D}}$ is defined by the shift of
correlation function when the gapped interface $\mathcal{D}$ crosses through
the junction of anyons. For simplicity of figure, we have suppressed the label
at the intersection between the anyon and ${\cal D}$.

##### Redundancy in expressing the coset symmetry defect by a sandwich

As discussed in section 3.2, the sandwich of $g\in G$ or $kgk^{-1}\in G$ with
$k\in K$ leads to the same coset symmetry defect. More generally, for a given
network of coset symmetry defects in the spacetime, redefining the sandwich
$g\to k_{0}gk_{0}^{-1}$ for all the defects in the network with a single
element $k_{0}\in K$ gives an another expression of the same network. However,
this redundancy is not manifest in the symmetry fractionalization laws in
Figure 8, 9. Let us investigate the effect of replacing
$g\to^{k_{0}}g:=k_{0}gk_{0}^{-1}$ with $k_{0}\in K$ on the symmetry
fractionalization data. We will confirm that the relabeling $g\to^{k_{0}}g$
leads to the same symmetry fractionalization class of the coset symmetry,
related by a natural isomorphism of the modular tensor category.

In Figure 8, the replacement $g\to^{k_{0}}g$ affects on the phase factor by

$\displaystyle\bigoplus_{k\in K}\eta_{{}^{k}a}(g,h)\ ...\to\bigoplus_{k\in
K}\eta_{{}^{k}a}(^{k_{0}}g,^{k_{0}}h)\ ...$ (5.5)

The above change of the phase factor is rewritten as

$\displaystyle\eta_{{}^{k}a}(^{k_{0}}g,^{k_{0}}h)=\frac{\gamma_{(^{k\overline{k}_{0}}a)}(gh)}{\gamma_{{}^{g^{-1}}(^{k\overline{k}_{0}}a)}(h)\gamma_{(^{k\overline{k}_{0}}a)}(g)}\eta_{(^{k\overline{k}_{0}}a)}(g,h),$
(5.6)

with

$\displaystyle\gamma_{x}(g)=\frac{\eta_{{}^{k_{0}}x}(g,k_{0}^{-1})}{\eta_{{}^{k_{0}}x}(k_{0}^{-1},k_{0}gk_{0}^{-1})}.$
(5.7)

This transformation by $\gamma$ is a natural isomorphism, which corresponds to
redefining the $G$ action on the theory $\mathcal{C}$ by the unitary
$\hat{\Gamma}_{g}\ket{a,b;c}=[\gamma_{a}(g)\gamma_{b}(g)/\gamma_{c}(g)]\ket{a,b;c}$
acting on fusion vertices of the anyons. The natural isomorphism induces the
equivalence of the $G$ symmetry action on the modular tensor category
$\mathcal{C}$. By relabeling $k\to k\overline{k}_{0}$ in the expression (5.5),
the phase factor after the replacement $g\to^{k_{0}}g$ is given by

$\displaystyle\bigoplus_{k\in K}\breve{\eta}_{{}^{k}a}(g,h)\ ...$ (5.8)

where we defined the new symbol $\breve{\eta}$ related by the natural
isomorphism to the original one

$\displaystyle\breve{\eta}_{x}(g,h)=\frac{\gamma_{x}(gh)}{\gamma_{{}^{g^{-1}}x}(h)\gamma_{x}(g)}\eta_{x}(g,h)$
(5.9)

Similarly, under the replacement $g\to^{k_{0}}g$ Figure 9 is transformed into

$\displaystyle\bigoplus_{k,k^{\prime},k^{\prime\prime}\in
K}\sum_{\nu,\sigma}M_{\mathcal{D}}^{\mu\nu}U^{\nu\sigma}_{{}^{k_{0}}g}(^{k}a,^{k^{\prime}}b,^{k^{\prime\prime}}c)M_{\overline{\mathcal{D}}}^{\sigma\rho}\
...$ (5.10)

The above phase factor $U$ is rewritten as

$\displaystyle
U^{\nu\sigma}_{{}^{k_{0}}g}(^{k}a,^{k^{\prime}}b,^{k^{\prime\prime}}c)=\frac{\gamma_{{}^{k\overline{k}_{0}}a}(g)\gamma_{{}^{k^{\prime}\overline{k}_{0}}b}(g)}{\gamma_{{}^{k^{\prime\prime}\overline{k}_{0}}c}(g)}[U_{k_{0}}U_{g}(^{k\overline{k}_{0}}a,^{k^{\prime}\overline{k}_{0}}b,^{k^{\prime\prime}\overline{k}_{0}}c)U_{\overline{k}_{0}}]^{\nu\sigma}$
(5.11)

By relabeling the group elements
$\\{k,k^{\prime},k^{\prime\prime}\\}\to\\{k\overline{k}_{0},k^{\prime}\overline{k}_{0},k^{\prime\prime}\overline{k}_{0}\\}$,
the phase factor in Figure 9 after the replacement $g\to^{k_{0}}g$ is
expressed as

$\displaystyle\bigoplus_{k,k^{\prime},k^{\prime\prime}\in
K}\sum_{\nu,\sigma}\breve{M}_{\mathcal{D}}^{\mu\nu}\breve{U}^{\nu\sigma}_{g}(^{k}a,^{k^{\prime}}b,^{k^{\prime\prime}}c)\breve{M}_{\overline{\mathcal{D}}}^{\sigma\rho}\
...$ (5.12)

where we defined the new tunneling matrix as
$\breve{M}_{\mathcal{D}}=MU_{k_{0}}$,
$\breve{M}_{\overline{\mathcal{D}}}=U_{\overline{k}_{0}}M_{\overline{\mathcal{D}}}$.
$\breve{M}_{\mathcal{D}},\breve{M}_{\overline{\mathcal{D}}}$ correspond to the
tunneling matrices for $\mathcal{D}\times
U_{k_{0}},U_{\overline{k}_{0}}\times\overline{\mathcal{D}}$, which is the same
as $\mathcal{D},\overline{\mathcal{D}}$ due to the fusion rule of defects
(3.10). It is hence expected that $\breve{M}_{\mathcal{D}}={M}_{\mathcal{D}}$,
$\breve{M}_{\overline{\mathcal{D}}}={M}_{\overline{\mathcal{D}}}$, but
detailed analysis to verify this property is left for future work. Also, the
new symbol $\breve{U}$ is related to the original one by the natural
isomorphism

$\displaystyle\breve{U}^{\nu\sigma}_{g}(a,b,c)=\frac{\gamma_{a}(g)\gamma_{b}(g)}{\gamma_{c}(g)}U^{\nu\sigma}_{g}(a,b,c)$
(5.13)

Therefore, the replacement $g\to^{k_{0}}g$ leads to the same symmetry
fractionalization class related by the natural isomorphism of the data
$\\{U,\eta\\}$.

##### Additional corners at junction of sandwich

We remark that the junctions of finite-size sandwich configurations can
contain three extra corners where the exteriors of the sandwich can change,
i.e. additional insertion with boundary-changing topological defects. We will
not consider such junctions in our discussion, and we will take the limit
where the three corners coincide into the same junction in the center. For
coset symmetries, we do not need to consider such junctions because there is a
canonical choice of domain wall. This domain wall is simply
$\mathcal{D}_{\mathrm{Rep}(K)}$, where $K$ is the minimal nonnormal subgroup
(see the discussion in section 3.2).

###  5.2 Non-invertible symmetry from invertible symmetry

A general, gapped domain wall in (2+1)D topological orders can be expressed as
a sandwich, consisting of an invertible domain wall in the middle region of a
possibly different topological order sandwiched by two gapped invertible
interfaces connecting the middle region to the original theory [57, 58]. The
gapped domain walls between two (2+1)D TQFTs $\mathcal{C}_{1},\mathcal{C}_{2}$
is generally described as

$\displaystyle D=D_{\mathcal{A}_{1}}\times
U\times\overline{D}_{\mathcal{A}_{2}},$ (5.14)

where $D_{\mathcal{A}_{1}}$ denotes the gapped interface that condenses the
anyons specified by the condensable algebra $\mathcal{A}_{1}$ of a modular
tensor category $\mathcal{C}_{1}$, and the theory obtained by anyon
condensation is given by a modular tensor category $\tilde{\mathcal{C}}$. $U$
is an invertible symmetry defect of $\tilde{\mathcal{C}}$, and
$\overline{D}_{\mathcal{A}_{2}}$ is (orientation reversal of) the defect that
condenses the algebra $\mathcal{A}_{2}$ of $\mathcal{C}_{2}$ to obtain
$\tilde{\mathcal{C}}$. When $\mathcal{C}_{1}=\mathcal{C}_{2}=\mathcal{C}$, the
above defect $D$ gives a general expression for non-invertible symmetry of the
(2+1)D TQFT $\mathcal{C}$.

Several comments are in order:

  * •

The description of non-invertible coset symmetry $\tilde{U}_{[g]}$ of (2+1)D
TQFT discussed in section 5.1 fits into the form (5.14), where
$\mathcal{C}_{1}=\mathcal{C}_{2}=\mathcal{C}/K$,
$\tilde{\mathcal{C}}=\mathcal{C},\mathcal{A}_{1}=\mathcal{A}_{2}=\mathrm{Rep}(K)$.
$U$ is then taken to be an invertible $G$ symmetry defect of $\mathcal{C}$.

  * •

The expression (5.14) directly implies that the non-invertible symmetry in
(2+1)D bosonic topological order exists if and only if the TQFT contains
condensable bosons. This fact has been pointed out in [56].

  * •

Let us comment on the realization of the operator $D$ in microscopic lattice
models. In the lattice system, the condensed theory $\tilde{\mathcal{C}}$ is
typically defined on a specific subspace $\tilde{\mathcal{H}}$ of the whole
Hilbert space $\mathcal{H}$. The operator $U$ is a unitary acting within the
subspace $\tilde{\mathcal{H}}$. The form of the non-invertible operator
$D=D_{\mathcal{A}_{1}}\times U\times\overline{D}_{\mathcal{A}_{2}}$ is
reminiscent of the singular value decomposition (SVD) of the operator $D$,
where the rank of the operator $D$ is the dimension of the subspace
$\tilde{\mathcal{H}}$. 888To be more precise, the SVD expresses the operator
$D$ in the form $D=D^{\prime}_{1}V\overline{D}^{\prime}_{2}$ where $V$ is a
full rank diagonal matrix acting within the Hilbert space
$\tilde{\mathcal{H}}$. The matrices $D^{\prime}_{1},\overline{D}^{\prime}_{2}$
satisfy
$D^{\prime\dagger}_{1}D^{\prime}_{1}=\overline{D}^{\prime}_{2}\overline{D}^{\prime\dagger}_{2}=\mathrm{id}_{\tilde{\mathcal{H}}}$.
This analogy is precise for the cosine symmetry of $S_{3}$ gauge theory
discussed in section 4. In this case, the subspace $\tilde{\mathcal{H}}$ is
specified by $Z_{e}^{b}=1$ for all edges as well as
$V^{C}_{\mathbb{Z}_{2}}=1$, which is the Hilbert space of $\mathbb{Z}_{3}$
toric code even under charge conjugation action $V^{C}_{\mathbb{Z}_{2}}$. We
have
$D_{\mathcal{A}_{1}}=\Pi^{b}\mathcal{D}^{b},\overline{D}_{\mathcal{A}_{2}}=\mathcal{D}^{b}\Pi^{b},U=U_{\theta}$.
They satisfy
${D}^{\dagger}_{\mathcal{A}_{1}}{D}_{\mathcal{A}_{1}}=\overline{{D}}_{\mathcal{A}_{2}}\overline{{D}}^{\dagger}_{\mathcal{A}_{2}}=\mathrm{id}$
(up to a real positive value) within the subspace $\tilde{\mathcal{H}}$,
ensuring that the cosine symmetry fits into the form of SVD.

In general, for a given non-invertible symmetry operator $D$ commuting with
the Hamiltonian $DH=HD$, let us take an SVD $D=D_{\mathcal{A}_{1}}\times
U\times\overline{D}_{\mathcal{A}_{2}}$. If there exists a Hamiltonian
$H^{\prime}$ in the subspace $\mathcal{H}^{\prime}$ satisfying
$HD_{\mathcal{A}_{1}}=D_{\mathcal{A}_{1}}H^{\prime}$,
$\overline{D}_{\mathcal{A}_{2}}H=H^{\prime}\overline{D}_{\mathcal{A}_{2}}$,
then we have $UH^{\prime}=H^{\prime}U$ and the non-invertible symmetry
operator gives a sandwich expression using an invertible symmetry $U$ and
topological interface operators
$D_{\mathcal{A}_{1}},\overline{D}_{\mathcal{A}_{2}}$. We conjecture that the
non-invertible symmetry operator $D$ in (2+1)D topological order generally
admits SVD into the composition of topological operators (5.14).

###  5.3 Bulk TQFT for finite coset non-invertible symmetry

Let us show that theories with coset symmetry $G/K$ can live on the boundary
of $G$ gauge theory, where we take $G$ to be a finite group.

Let us first consider the case $K=1$. Any quantum system with $G$ symmetry can
live on the boundary of an $G$ SPT phase in the bulk. The $G$ SPT phase has a
topological interface with $G$ gauge theory given by imposing the Dirichlet
boundary condition of the $G$ gauge field, i.e. condensing the electric Wilson
lines. By shrinking the $G$-SPT region, we find the theory can live on the
boundary of a $G$ gauge theory with Dirichlet boundary condition of the $G$
gauge field. When the $G$ symmetry is anomalous, the bulk $G$ gauge theory has
nontrivial topological action $\omega$ describing the anomaly. In
$(D+1)$-dimensional bulk, we can consider $\omega\in H^{D+1}(BG,U(1))$. For
instance, on the $e$-condensed boundary of $\mathbb{Z}_{2}$ gauge theory there
is $\mathbb{Z}_{2}$ global symmetry.999 Such boundary condition is also
recently discussed in [59].

For general subgroup $K$, we can consider imposing a mixed boundary condition
of the $G$ gauge field, where the $G$ gauge group in the bulk is broken to a
subgroup $K$ on the boundary. If the bulk $G$ gauge theory has topological
action $\omega$, the subgroup should satisfy

$\omega|_{K}=-d\alpha~{},$ (5.15)

and the boundary $K$ gauge theory has a topological action $\alpha$. There are
different topological actions on the boundary of $D$ spacetime dimensions,
related by shifting $\alpha\rightarrow\alpha+\nu$ with $d\nu=0$ classified by
$\nu\in H^{D}(BK,U(1))$. We note that the Rep$(K)$ symmetry comes from the
Wilson lines in the bulk $G$ gauge theory parallel to the boundary, while the
magnetic operators of the $G$ gauge theory generate 0-form symmetry on the
boundary.101010 The description of symmetry using bulk TQFT is also discussed
in e.g. [60, 61].  See e.g. [54] for a lattice construction for such bulk-
boundary description. We remark that the condition (5.15) on the subgroup $K$
is also appears in e.g. [62].

####  5.3.1 Anomalies of finite coset non-invertible symmetry

Let us use the bulk TQFT to investigate whether the coset symmetry can be
realized by an invertible phase, following the method of [32, 33]. In other
words, we will study whether or not the 0-form coset symmetry in (2+1)D is
anomalous. If the 0-form symmetry were an invertible symmetry $G$, then we
would be determining whether or not the symmetry enriched topological order
has a nontrivial $H^{4}(G,U(1))$ class. Since the coset symmetry is non-
invertible, its anomaly is not labeled simply by SPTs in (3+1)D.

For coset symmetry $G/K$, we will examine the situation when the bulk TQFT is
the untwisted $G$ gauge theory without any bulk topological action. We can put
the bulk TQFT on an interval, on one end we impose the boundary condition
corresponding to a (2+1)D theory with $G$ gauge group broken down to its non-
normal subgroup $K$. This means that the $G$ Wilson lines that can end at the
boundary are those whose decomposition under the subgroup $K$ contain the
trivial $K$ representation, while the other Wilson lines remain nontrivial on
the boundary (i.e. they are not condensed at the boundary). In addition, the
magnetic operators in the $G$ gauge theory that carry $K$ -holonomy can end on
the boundary, since the boundary has nontrivial $K$ gauge field; the other
magnetic operators cannot end on the boundary.

We need to determine whether or not there exists a boundary condition to put
on the other side such that no operators can stretch between the two
boundaries. Specifically, we need to determine whether or not there exists a
boundary condition describing a theory with $G$ broken down to $K^{\prime}$
such that

  * •

$K^{\prime}\cap K=1$: This guarantees that there are no nontrivial magnetic
operators stretching between the two boundaries.

  * •

There are no nontrivial representations of $G$ such that its decomposition
under the subgroups $K,K^{\prime}$ simultaneously contains the trivial
representations of $K,K^{\prime}$. In other words, there are no nontrivial
Wilson lines stretching between the two boundaries.

If any of the above conditions is not met, then is no (2+1)D invertible phase
with coset symmetry $G/K$. The discussion can be generalized to higher
spacetime dimensions.

For instance, if $G=S_{3}=\mathbb{Z}_{3}\rtimes\mathbb{Z}_{2}$ and
$K=\mathbb{Z}_{2}$, there are three irreducible representations
$1,\text{sign},\pi$ where $\pi$ is a two-dimensional representations. The
subgroup $K^{\prime}$ that satisfies $K\cap K^{\prime}=1$ is
$K^{\prime}=\mathbb{Z}_{3}$. Only the trivial representation of $S_{3}$
simultaneously reduces to sums containing the trivial representation under
both subgroups $K,K^{\prime}$. Therefore, the second condition above is not
met. As a result, we can conclude that the coset symmetry
$S_{3}/\mathbb{Z}_{2}$ cannot be realized in a trivially gapped theory. In our
example, it is realized in $S_{3}$ gauge theory, which is not trivially
gapped. The same discussion applies to $G=\mathbb{Z}_{N}\rtimes\mathbb{Z}_{2}$
and $K=\mathbb{Z}_{2}$ for $N\geq 3$.

##  6 Application: Non-Invertible Symmetry in Spin Liquids

In a microscopic model with $G$ global symmetry such that on the low energy
states a subgroup $K$ does not act, i.e. there is Gauss law constraint for $K$
on the low energy subspace, the low energy effective theory can be described
by $K$ gauge theory, and the symmetry is $G/K$ that acts projectively as $G$
symmetry on the electric excitations. $G$ is called the projective symmetry
group (PSG), $K$ is called the invariant gauge group (IGG), and $G/K$ is the
global symmetry (called the symmetry group (SG) when $K$ is a normal
subgroup), which is the quotient of PSG by IGG [39].

When $K$ is a normal subgroup, $G/K$ is an ordinary group-like invertible
symmetry, and the above construction is discussed extensively in the
literature. This construction is particularly useful for describing quantum
spin liquids with $K$ gauge theory. For example, [39] discusses
$\mathbb{Z}_{2}$ spin liquids described by taking $K=\mathbb{Z}_{2}$, which is
the center of the $G=SU(2)$ projective symmetry group. Here, we generalize the
construction to the case when $K$ is not a normal subgroup. Then we obtain a
quantum spin liquid enriched with a non-invertible global symmetry $G/K$. The
symmetry is an example of the coset non-invertible symmetry discussed in
section 3 and section 4.

Suppose we start with a trivially gapped system of fields with $G$ projective
symmetry group and gauge a non-normal subgroup $K$. As discussed in section
3.2, the remaining $G/K$ coset global symmetry obeys the fusion rule

$g_{1}K\times g_{2}K=\sum_{k\in K}\left(g_{1}kg_{2}k^{-1}\right)K~{},$ (6.1)

where $g_{i}K\in G/K$ are elements of the coset. Thus the fusion rule of the
coset does not obey a group multiplication law. We note that even when
$g_{2}=g_{1}^{-1}$, the fusion does produce the coset element $K$, since
$g_{1}kg_{1}^{-1}$ in general is not in the non-normal subgroup $K$.

###  6.1 “Projective” symmetry action on Wilson lines

Since we can describe the system starting with a system with $G$ symmetry, the
fields that transform under the $K$ gauge group also transforms under the $G$
symmetry. Thus the gauge non-invariant fields have $G$ symmetry, while the
gauge-invariant operators have $G/K$ symmetry. To see this using the formalism
of section 5, we note that if $K$ is a non-normal subgroup, the action of the
$G$ symmetry can relate a nontrivial $K$ Wilson line with the vacuum line.

For instance, consider a system with
$G=S_{3}=\mathbb{Z}_{3}\rtimes\mathbb{Z}_{2}$ symmetry, and we gauge the
$\mathbb{Z}_{2}$ subgroup that acts on $\mathbb{Z}_{3}$ by charge conjugation.
On the gauge non-invariant fields, the symmetry is $S_{3}$, while the gauge-
invariant operators see $S_{3}/\mathbb{Z}_{2}$ symmetry. To see how the
symmetry acts on the Wilson lines of $\mathbb{Z}_{2}$ gauge theory, let us
decompose the $S_{3}$ representations in terms of $\mathbb{Z}_{2}$
representations. There are two one-dimensional irreducible representations
$1,\text{sgn}$ of $S_{3}$ and one two-dimensional irreducible representation
$\pi$ of $S_{3}$, and they decompose into $\mathbb{Z}_{2}$ irreducible
representations $1_{0},1_{1}$ (the subscript is the $\mathbb{Z}_{2}$ charge
$0,1$ mod 2) as follows:

  * •

$1\rightarrow 1_{0}$, $\text{sgn}\rightarrow 1_{1}$.

  * •

$\pi\rightarrow 1_{0}+1_{1}$.

The decomposition of $\pi$ indicates that the coset symmetry can change the
type of the $\mathbb{Z}_{2}$ Wilson line. In particular, it maps it to the
vacuum line. Note that this cannot be done with any invertible symmetry, which
must preserve braiding properties. This permutation of the Wilson line to the
vacuum line is consistent with the “sandwich” construction for the coset
symmetry given by sandwiching the $S_{3}$ generator by interface that
condenses the $\mathbb{Z}_{2}$ electric charge: the Wilson line can end on the
coset symmetry generator.

###  6.2 Deconfined critical points with non-invertible symmetry

Let us consider two massless scalars that together transform as the two-
dimensional representation of $S_{3}$. Then we gauge the $\mathbb{Z}_{2}$
subgroup symmetry that flips the sign of one of the scalars. The theory is a
$\mathbb{Z}_{2}$ gauge theory with coset symmetry $S_{3}/\mathbb{Z}_{2}$. The
theory has a $\mathbb{Z}_{2}$ 1-form symmetry generated by the Wilson line.
Due to the fractionalization, there is a mixed anomaly between the 1-form
symmetry and the coset symmetry. To see this, we note if we condense the
$\mathbb{Z}_{2}$ electric charge, i.e. the Wilson line becomes trivial, this
implies that the coset symmetry is extended to be the $S_{3}$ symmetry. This
implies that the theory must have deconfined excitations that carry the
anomaly. In particular, there are no interactions that can drive the
$\mathbb{Z}_{2}$ gauge field to confined phase at the critical point.

We note that before gauging the $\mathbb{Z}_{2}$ symmetry, the theory is a
critical theory of massless scalars without deconfined excitations, and
gauging the discrete symmetry modifies the spectrum by projecting out the
$\mathbb{Z}_{2}$ odd local operators while adding a deconfined Wilson line.

##  7 Discussion and Outlook

In this work we investigate the fractionalization of coset non-invertible
symmetry using field theories and lattice models. We show that the non-
invertible defects can be obtained using a sandwich construction: we can build
them out of invertible defects together with condensation defects. We use
operators obtained in this way to explicitly derive fractionalization data on
the lattice for certain examples of non-invertible symmetry fractionalization.

There are several future directions. The framework of the consistency rules
described in section 5 allows us to explore the classification of new quantum
spin liquids enriched by non-invertible coset symmetry. There is much to
explore in the landscape of solutions to those conditions. More generally, it
would be interesting to explore constraints on the fractionalization of other
non-invertible symmetries beyond the coset construction. Our discussion
focuses on theories in (2+1)D, but the defect sandwich construction can be
generalized straightforwardly to higher spacetime dimensions. Defects in
higher dimensions would have additional higher codimension structures. It
would also be interesting to explore new deconfined quantum critical points
with fractionalized excitations for non-invertible symmetries. Another context
where the coset symmetry can arise is in the Higgs phase of $G$ gauge theory
where the gauge group is broken to a non-normal subgroup $K$, which we will
explore in more detail in an upcoming work.

## Acknowledgement

We thank Shu-Heng Shao for discussions. P.-S.H. was supported by Simons
Collaboration of Global Categorical Symmetry, Department of Mathematics King’s
College London. R.K. is supported by JQI postdoctoral fellowship at the
University of Maryland, and by National Science Foundation QLCI grant
OMA-2120757. C.Z. is supported by the Harvard Society of Fellows and the
Simons Collaboration on Ultra Quantum Matter. P.-S. H. and C.Z. are also
supported in part by grant NSF PHY-2309135 to the Kavli Institute for
Theoretical Physics (KITP). P.-S.H. and C.Z. thank Kavli Institute for
Theoretical Physics for hosting the program “Correlated Gapless Quantum
Matter” in 2024 and Perimeter Institute for hosting the conference “Physics of
Quantum Information” in 2024, during which part of the work is completed.

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