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On Triality Defects in 2d CFT

Da-Chuan Lu and Zhengdi Sun

Department of Physics, University of California, San Diego, CA 92093, USA

We consider the triality fusion category discovered in the $c=1$ Kosterlitz-
Thouless theory [1]. We analyze this fusion category using the tools from the
group theoretical fusion category and compute the simple lines, fusion rules
and $F$-symbols. We then studied the physical implication of this fusion
category including deriving the spin selection rule, computing the asymptotic
density of states of irreducible representations of the fusion category
symmetries, and analyzing its anomaly and constraints under the
renormalization group flow. There is another set of $F$-symbols for the fusion
categories with the same fusion rule known in the literature [2]. We find
these two solutions are different as they lead to different spin selection
rules. This gives a complete list of the fusion categories with the same
fusion rule by the classification result in [3].

###### Contents

  1. 1 Introduction
    1. 1.1 Outline of the Paper
    2. 1.2 Future Directions
  2. 2 Review on TDLs and their applications in 2d CFT
    1. 2.1 TDLs in 2d CFT
    2. 2.2 Modular bootstrap with TDLs
    3. 2.3 Asymptotic density of states
  3. 3 Non-intrinsic triality fusion category as group theoretical fusion category
    1. 3.1 A brief review of the triality category discovered in [4, 1]
    2. 3.2 Group theoretical fusion category
    3. 3.3 Simple lines
    4. 3.4 Fusion rules
    5. 3.5 $F$-symbols
  4. 4 Physical implication of the group theoretical triality fusion categories
    1. 4.1 Spin selection rules
    2. 4.2 Asymptotic density of states
    3. 4.3 Constraints on RG flow
  5. 5 Example: $c=1$ Compact boson at Kosterlitz-Thouless point
    1. 5.1 A lightning review of $c=1$ compact boson and the triality defect
    2. 5.2 Spectrum of triality defect and twisted partition functions on torus
    3. 5.3 Constructing more triality lines from the known ones
  6. 6 More Triality Fusion Categories
    1. 6.1 The classification of triality fusion category
    2. 6.2 F-symbols of intrinsic triality fusion category
    3. 6.3 Spin selection rules
      1. 6.3.1 Determine the triality fusion category from spin selection rules
  7. A Details on the compact boson partition function
  8. B Group theory convention
  9. C Detail on the basis in $\operatorname{Hom}_{A}(M\otimes_{A}N,L)$

##  1 Introduction

Global symmetry is an important guideline for constructing and analyzing
quantum field theories. In modern language, the global symmetries, whether
continuous or discrete, can be represented as invertible topological operators
supported on a codimensions-1 (codim-1) surface. Any correlation functions
with the topological surface operator insertion are invariant under the small
deformation of the codim-$1$ surface where the topological operator is
supported. As the result, the topological operator commute with the stress-
energy tensor. For a continuous symmetry given by a conserved current
$j_{\mu}(x)$, the corresponding topological surface operator is constructed as
$e^{i\alpha\oint_{\Sigma_{d-1}}d^{d-1}x\,n^{\mu}(x)j_{\mu}(x)}$ and the
invariance under the small deformation then follows from the conservation
equation $\partial_{\mu}j^{\mu}(x)=0$.

Several generalizations have been made from this point of view, which leads to
the concept of generalized global symmetries [5]. For instance, the support of
the invertible topological operator can be generalized to codim-$p$ surfaces
for $p>1$, and the corresponding symmetries are called $(p-1)$-form symmetries
[5, 6]. The standard $0$-form symmetries can interact non-trivially with the
$1$-form symmetries, and this leads to the structure of $2$-group symmetries
[7, 8, 9]. Another direction of generalization is to study the topological
operators which are not invertible. For instance, the non-invertible $0$-form
symmetries in 2-dimensional field theory can be described by the fusion
categories, and the topological operators correspond to the objects in the
fusion categories. Therefore, these non-invertible symmetries are also called
categorical symmetries. There is a lot of progress in the study of non-
invertible symmetries in dimension $d\geq 3$ recently [10, 11, 12, 13, 14, 15,
16, 17]. Of course, the categorical $0$-form symmetries can interact non-
trivially with the categorical higher form symmetries, this leads to the
concept of higher categorical symmetries [18].

For a more detailed review and a more complete list of references on the
development of the generalized global symmetries and their applications from
the high energy physics perspective and the condensed matter physics
perspective, we refer the readers to the two reviews [9, 19].

There are two classes of non-invertible symmetries [17, 20]. Consider a field
theory $\mathcal{T}$ with non-invertible symmetries $\mathcal{C}$, if there
exists some topological manipulation $\phi$ such as finite gauging on the
theory $\mathcal{T}$ such that the theory $\phi(\mathcal{T})$ contains only
invertible symmetries $\phi(\mathcal{C})$ and the constraints on the RG flows
of the theory $\mathcal{T}$ from the non-invertible symmetries $\mathcal{C}$
can be completely determined from the constraints on the RG flows from the
invertible symmetries $\phi(\mathcal{C})$ of the theory $\phi(\mathcal{T})$,
we refer the non-invertible symmetries $\mathcal{C}$ as non-intrinsic [21].
Otherwise, the non-invertible symmetries are called intrinsic. A simple
example of non-intrinsic invertible symmetries is the
$\operatorname{Rep}(S^{3})$ symmetries in the 2-dimensional field theory,
acquired from the topological manipulation of gauging the $S^{3}$ global
symmetries. (For example, see [22].)

The study of codim-$1$ non-invertible topological operators (also known as
topological defect lines, or TDLs) in 2-dimensional CFT has a long history
[23, 24, 25, 26, 27, 28, 29, 30, 31], focusing on their connection to boundary
CFT, twisted partition function on various 2-manifold, orbifolds and symmetry
TFT. Recent studies not only focus on searching for interesting non-invertible
TDLs [32, 4, 1, 33, 34, 35], but also extending the applications of the TDLs,
including their lattice construction [36, 37], constraints for 2d modular
bootstrap [38, 39, 40, 41, 42, 43], constraints on the RG flow [4, 44, 45,
46], gauging non-invertible symmetries [22, 47] etc.

Perhaps the most well-studied non-invertible TDL is the duality line $N$ under
the $\mathbb{Z}_{2}$ gauging. The duality line $N$ and two lines
$\mathds{1},\eta$ in the $\mathbb{Z}_{2}$ symmetries form the Ising fusion
category with the fusion rule

$\eta N=N\eta=N,\quad N^{2}=I+\eta.$ (1.1)

The Ising fusion category has been extensively studied in the literature [32,
48, 38, 36, 37, 24, 25, 4, 1]. The existence of the duality line $N$ implies
the conformal field theory is invariant under the $\mathbb{Z}_{2}$ gauging.
One possible generalization is of course to consider the duality line $N$
under $A$-gauging where $A$ is a finite Abelian group. The corresponding
fusion category is known as the Tambara-Yamagami Fusion category
$\mathcal{T}(A,\chi,\epsilon)$ [49, 50] where $\chi$ is a symmetric
bicharacter of the group $A$ and $\epsilon=\pm 1$ is the Frobenius-Schur
indicator. The $\mathcal{T}(A,\chi,\epsilon)$ with the same $A$ but different
$\chi$ and $\epsilon$ satisfies the same fusion rules, yet they are different
fusion categories distinguished by the $F$-symbols (also known as the crossing
kernels $\mathcal{K}$ in [44]), which measures the difference between two
different ways of resolving the crossing of two TDLs. The $F$-symbols reduce
to the familiar group anomaly measured by $H^{3}(G,U(1))$ when considering
only invertible TDLs. For instance, there are two types of Ising fusion
categories with different FS indicators $\epsilon=\pm 1$ and they can also be
distinguished from the so-called spin selection rules [44]. The studies of the
Tambara-Yamagami fusion category symmetries in the physics literature include
[32, 38, 44, 4, 1].

Another generalization of the Ising fusion category is the triality fusion
category and recently is studied in [4, 1]. The simple TDLs contain symmetry
operators which generate $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ global
symmetries, as well as a triality line $\mathcal{L}_{Q}$ and its orientation
reversal $\mathcal{L}_{\overline{Q}}$, satisfying the fusion rule 111In
general, the invertible symmetries in a triality fusion category does not have
to be $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. For simplicity, however, the
notion of the triality fusion category would specifically mean the case where
the invertible symmetries are $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$.
generalizing (1.1)

$\mathcal{L}_{Q}\times\mathcal{L}_{\overline{Q}}=\sum_{g\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}}g,\quad\mathcal{L}_{Q}\times\mathcal{L}_{Q}=2\mathcal{L}_{\overline{Q}},\quad
g\times\mathcal{L}_{Q}=\mathcal{L}_{Q}\times g=\mathcal{L}_{Q},\quad
g\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}.$ (1.2)

The triality fusion category is obtained by gauging the $\mathbb{Z}_{2}$
global symmetry in $A_{4}$ global symmetry of the $SU(2)_{1}$ theory. In
general, consider a 2-dimensional theory $\mathcal{T}$ with the global
symmetry $G$ and anomaly $\omega\in H^{3}(G,U(1))$. One can gauge a non-
anomalous subgroup $H\subset G$ and additional data which is the discrete
torsion $\psi\in H^{2}(H,U(1))$ needs to be specified in this gauging process.
The fusion category that describes the categorical symmetries of the gauged
theory $\mathcal{T}/H$ is called the group theoretical fusion category, and it
is denoted by $\mathcal{C}(G,\omega,H,\psi)$. In this language, the triality
fusion category in the KT theory is $\mathcal{C}(A_{4},1,\mathbb{Z}_{2},1)$.

In this paper, we study this triality fusion category using the tools from the
group theoretical fusion category. We compute the spectrum of simple TDLs,
their fusion rules, and the $F$-symbols by using the description of the group
theoretical fusion category in terms of bimodules [51].

We then study the physical implication of the triality fusion category. We
derive the spin selection rules from the $F$-symbols we acquire. We also
derive the Cardy formula for densities of states using the result in [40].
Since these triality fusion categories are group-theoretical, their
constraints on the RG flow can be determined by the group $G$ and the anomaly
$\omega$ as well as the anomaly free subgroup $H$ (with the discrete torsion
$\psi$) that we are gauging. An interesting feature is that, even if the
finite symmetry $G$ is anomaly free (that is, $\omega$ is trivial), the
resulting group theoretical fusion categories can be anomalous.222The authors
thank for the referee for pointing out a mistake on this in the draft. We will
describe a general criteria for the group theoretical fusion category to be
anomaly free in the section 4.3. In general, this means that finite non-
intrinsic non-invertible 0-form symmetries are completely characterized by
group theoretical fusion categories in a 2-dimensional bosonic theory. We then
consider the $c=1$ KT theory as an example, and compute its twisted partition
function explicitly and show the results indeed agree with our general
analysis.

It is a natural question to ask if there are more allowed $F$-symbols with the
same fusion relations. Just like the TY-categories characterizing the duality
can have distinct FS indicators with $\epsilon=\pm 1$, the possible FS
indicators of the triality fusion category are given by $\alpha=e^{2\pi
ik/3}\in\mathbb{Z}_{3}$ with $k=0,1,2$. One can see this from the fact that
there are $\mathbb{Z}_{3}$ phase rotations of the $F$-symbols which preserve
the pentagon equations and are not gauge transformations. Physically, this
means one can construct triality fusion category with different FS indicator
$\alpha^{\prime}$ from a known category with FS indicator $\alpha$ by taking
the theory $\mathcal{T}$ with the triality defect $\mathcal{L}_{Q}$, staking a
decoupled theory $\mathcal{T}^{\prime}$ with an anomalous $\mathbb{Z}_{3}$
global symmetry generated by $\eta$, and defining a new triality line
$\mathcal{L}_{Q}^{\prime}\equiv\mathcal{L}_{Q}\eta$. From the group
theoretical fusion category point of view, to acquire triality fusion
categories with the FS indicator $\alpha=e^{2\pi ik/3}$ means we are gauging
the non-anomalous $\mathbb{Z}_{2}$ symmetries in $A_{4}$ but now with non-
trivial ’t Hooft anomaly for $A_{4}$. The anomaly for $A_{4}$ is classified by
$H^{3}(A_{4},U(1))\simeq\mathbb{Z}_{6}$ and let’s denote its generator as
$\omega_{0}$. The triality fusion categories with different FS indicators are
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2},1)$ for $k=0,1,2$.

However, just taking into account the FS indicators does not enumerate all the
possible triality fusion categories. Indeed, another set of $F$-symbols is
computed in the condensed matter literature [2] and it is natural to ask if
these $F$-symbols lead to the same fusion categories as ours. We show that the
two sets of fusion categories are different and can be distinguished using
spin selection rules. Then this enumerates all the possible inequivalent
$F$-symbols for the triality fusion categories according to the classification
result in [3].

###  1.1 Outline of the Paper

The goal of the paper is to provide a description of group theoretical
triality fusion categories
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2},1)$ in terms of the
bimodules of the $\mathbb{Z}_{2}$ group algebra, and use it to compute the
fusion rule and $F$-symbols. The simplicity of the group theoretical fusion
category is that the objects naturally have a $\mathbb{C}$-vector spaces
structure, therefore every data we need can be described using linear
algebras.333Notice that there is no $\mathbb{C}$-vector spaces structure on
objects in a generic fusion category $\mathcal{C}$. For finite $0$-form
symmetries in a bosonic theory, non-intrinsic non-invertible symmetries form
group theoretical fusion categories. This means non-intrinsic non-invertible
symmetries naturally have $\mathbb{C}$-vector space structure while the
intrinsic non-invertible symmetries do not. It would be interesting to check
if there’s any physical understanding or implication of this difference. We
also pointed out there are triality fusion categories that are not group
theoretical, therefore are intrinsic. We show that whether the triality fusion
category is intrinsic or non-intrinsic can be determined from the spins of the
defect Hilbert space of the triality line.

In section 2, we briefly review the TDLs in CFT. In section 2.1, we introduce
the basic notions relate to the TDLs in 2d CFT. In section 2.2, we review the
modular bootstrap program, and describe how to relate the twisted partition
function computes the action of TDL $\mathcal{L}$ on Hilbert space
$\mathcal{H}$ and to the twisted partition function computes the states of the
defect Hilbert space $\mathcal{H}_{\mathcal{L}}$. Then, in section 2.3, we
briefly review the result in [40] on the asymptotic density of states in
various Hilbert spaces, which will be useful for us later.

In section 3, we describe how to understand the triality fusion categories
discovered in [4, 1] as group theoretical fusion category
$\mathcal{C}(A_{4},1,\mathbb{Z}_{2},1)$. In section 3.1, we briefly review the
triality fusion category, which is acquired from gauging $\mathbb{Z}_{2}$
subgroup in $A_{4}$. Then, we introduce the notion of group theoretical fusion
category in section 3.2 and show the triality fusion category in the KT theory
can be described as $\mathcal{C}(A_{4},1,\mathbb{Z}_{2},1)$. We then begin to
describe the data of this triality fusion category from the
$\mathcal{C}(A_{4},1,\mathbb{Z}_{2},1)$ using the language of bimodules. In
the rest of this section, we give an explicit calculation of the spectrum of
simple TDLs, their fusion rules, and the $F$-symbols for
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2},1)$ using bimodules.

In section 4, we use the above result to derive physical consequences. In
section 4.1, we compute the spin selection rules following the techniques in
[44]. Since $\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2},1)$ is acquired
from gauging the $\mathbb{Z}_{2}$ subgroup in a theory with global symmetry
$A_{4}$, we can match the irreducible representations (irreps) of the fusion
ring of $\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2},1)$ to different
sectors in $\mathcal{T}$. This allows us to derive the Cardy-like formulas for
different irreps of $\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2},1)$ in
the section 4.2. Finally in the section 4.3, we study the anomaly of the group
theoretical fusion categories $\mathcal{C}(G,\omega,H,\psi)$ by analyzing its
possible symmetric gapped phases444Here, by $\mathcal{C}$-symmetric gapped
phase we mean the Hamiltonian or Lagrangian preserves the
$\mathcal{C}$-symmetry while it could be spontaneously broken etc. By
$\mathcal{C}$-symmetric trivial gapped phase, we mean there’s a unique ground
state in this phase.. We find that while the symmetric gapped phases of
$\mathcal{C}(G,\omega,H,\psi)$ are in canonical bijection with the symmetric
gapped phases of $\operatorname{Vec}_{G}^{\omega}$, this bijection does not
preserve the number of the ground states. From the analysis of the possible
symmetric gapped phases of
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2},1)$ we conclude that they
are all anomalous. We then give an anomaly free condition for the group
theoretical fusion category in general. This means that the finite non-
intrinsic non-invertible symmetries are completely characterized by group
theoretical fusion categories for 2-dimensional bosonic field theory.

In section 5, we first review the Kosterlitz-Thouless (KT) theory and its
triality fusion categories discovered in [4, 1]. We then compute the twisted
partition function of the triality defect and show the spins in the defect
Hilbert space $\mathcal{H}_{\mathcal{L}_{Q}}$ indeed match the spin selection
rules in the section 5.2. Since the KT theory has
$\mathbb{Z}_{3}\times\mathbb{Z}_{3}\subset(U(1)^{\tilde{\theta}}\times
U(1)^{\tilde{\phi}})\rtimes\mathbb{Z}_{2}$ global symmetries, it is
interesting to check if one can construct new triality line via
$\mathcal{L}_{Q}\eta$, where $\eta$ generates the one of the $\mathbb{Z}_{3}$
symmetry. We show explicitly this is not possible by checking that the line
$\mathcal{L}_{Q}\eta$ does not satisfy the fusion rule in the section 5.3.

In section 6, we consider other triality fusion categories. We first review
the classification of the triality fusion categories in [3] in the section
6.1. We then list the $F$-symbols for the intrinsic triality fusion categories
computed in [2] in the section 6.2. In the section 6.3, We then compute the
spin selection for the triality line $\mathcal{L}_{Q}$ for the intrinsic
triality fusion categories and show that the triality fusion categories can be
distinguished by the spins of the states in the
$\mathcal{H}_{\mathcal{L}_{Q}}$. We conclude the paper with a comment on when
the spin selection rules should be saturated.

###  1.2 Future Directions

We outline some of the future directions to explore.

#### Exploring generic group theoretical fusion categories

In this paper, we mainly focus on describing the group theoretical fusion
categories $\mathcal{C}(A_{4},1,\mathbb{Z}_{2}^{\sigma},1)$. Since our
approach can be easily generalized to other group theoretical fusion
categories, it would be interesting to systematically explore the group
theoretical fusion categories using Mathematica, for example, computing the
simple TDLs, their fusion rules, and the $F$-symbols, as well as the physical
implications such as the spin selection rules, classification of
$\mathcal{C}$-symmetric gapped phases, and solutions of general modular
bootstrap equations, etc. Group theoretical fusion categories provide a
systematic way of constructing $N$-ality fusion categories in 2-dimension.
Some of these will be done in the upcoming work by the authors [52].

#### Intrinsic triality defects

In this paper, we only study the spin selection rules of the intrinsic
triality fusion categories, which is sufficient to distinguish them from the
non-intrinsic ones. Unlike the group theoretical fusion category symmetries,
it’s harder to find examples of CFT with non-intrinsic triality symmetries. A
possible candidate is the bosonization of the theory of 8 Majorana fermions
discussed in [53]. Furthermore, it would be interesting to understand the
anomaly and in general, the possible symmetric gapped phases, which are
described to module categories $\mathcal{M}$ over the fusion category
$\mathcal{C}$. This can be done using the techniques developed in [54]. More
generally, a class of intrinsic $N$-ality fusion categories is constructed in
[3] and it would be interesting to answer the above questions for these as
well.

Note added: By the time we are about to post this paper, a nice paper [55]
generalizing the result in [40] appears and their general result can be used
to produce our results in the section 4.2. Also, the three papers [56, 57, 58]
appear on the arXiv the same day as the authors post the first version of the
draft on arXiv, which generalizes the idea of gauging non-normal or non-
Abelian subgroup will lead to non-invertible symmetries to higher form and
higher group symmetries in higher dimensions.555For understanding the non-
invertible symmetries from holographic point of view, see recent papers [59,
60, 61, 62].

##  2 Review on TDLs and their applications in 2d CFT

In this section, we give a brief review of TDLs in 2d CFT to fix the
convention, and readers who are interested in a more detailed review should
look at [44, 22]. We also review results on modular bootstraps and Cardy-like
formulas which will be useful later.

###  2.1 TDLs in 2d CFT

In 2-dimensional conformal field theory, the symmetry defects of the ordinary
symmetry are line operators and they are the very first examples of TDLs. For
symmetry $G$ of a CFT, the corresponding TDLs are denoted as
$\mathcal{L}_{g},g\in G$. The juxtaposition of two such TDLs satisfies the
group algebra, $\mathcal{L}_{g}\times\mathcal{L}_{h}=\mathcal{L}_{gh}$. The
TDL corresponds to the identity $\mathds{1}$ in the group $G$ is the identity
line which we will also denote as $\mathds{1}\equiv\mathcal{L}_{\mathds{1}}$.
Since every group element has its inverse, these TDLs also have their inverse
in the sense that there exists another line such that they fuse into a single
identity line $\mathds{1}$, i.e. they are invertible TDLs.

However, TDLs are ubiquitous in the CFT, and the way they fuse is beyond the
group algebra, more generally satisfying the fusion algebra. These TDLs then
generate the non-invertible symmetries or the fusion category symmetries,
since the TDLs may not have their inverse. The Kramer-Wannier duality line in
the Ising CFT and most of the Verlinde lines in rational CFT are examples of
the non-invertible TDLs.

The TDL $\mathcal{L}$ can be deformed locally without changing the correlation
functions $\langle\mathcal{L}\cdots\rangle$ where $\cdots$ denotes any other
operator insertions. The topological nature of these TDLs imply that they
commute with stress energy tensor, therefore can be algebraically expressed
as,

$[L_{n},\mathcal{L}_{a}]=[\bar{L}_{n},\mathcal{L}_{a}]=0,$ (2.1)

where $L_{n},\bar{L}_{n}$ are the generators of the Virasoro algebra. When the
TDL $\mathcal{L}_{a}$ acts on a state $\ket{\phi}$ in Hilbert space
$\mathcal{H}$, the resulting state $\mathcal{L}_{a}\ket{\phi}$ is still in
$\mathcal{H}$.

When two TDLs are brought close to each other, their juxtaposition satisfies
the fusion rule,

$\mathcal{L}_{a}\times\mathcal{L}_{b}=\bigoplus_{c}N_{ab}^{c}\mathcal{L}_{c}$
(2.2)

where $N_{ab}^{c}$ is the fusion multiplicity which can only be non-negative
integers. More specifically, the TDLs can join at the point-like junction,
which is equipped with a Hilbert space. For example, the fusion of TDLs
$\mathcal{L}_{1},\mathcal{L}_{2}$ into $\mathcal{L}_{3}$ corresponds to a
vector space with dimension depending on the fusion multiplicity, we dub the
vector space as fusion space and denote by
$V_{\mathcal{L}_{1},\mathcal{L}_{2}}^{\mathcal{L}_{3}}$. Correspondingly, the
TDL $\mathcal{L}_{3}$ can split into $\mathcal{L}_{1},\mathcal{L}_{2}$, whose
vector space is split space and denoted by
$V_{\mathcal{L}_{3}}^{\mathcal{L}_{1},\mathcal{L}_{2}}$.

Figure 1: The fusion vertex of TDLs $\mathcal{L}_{1},\mathcal{L}_{2}$ fuse
into $\mathcal{L}_{3}$ and split vertex of TDLs $\mathcal{L}_{3}$ splits into
$\mathcal{L}_{1},\mathcal{L}_{2}$. The red cross marks the last leg, which
determines the ordering of each vertex.

More complicated fusion/split process can be decomposed into the fusion/split
space with 3 TDLs. However, the decomposition is not unique but under the
isomorphism,
$\bigoplus_{\mathcal{L}_{5}}V_{\mathcal{L}_{5}}^{\mathcal{L}_{1},\mathcal{L}_{2}}\otimes
V_{\mathcal{L}_{4}}^{\mathcal{L}_{5},\mathcal{L}_{3}}\cong\bigoplus_{\mathcal{L}_{6}}V_{\mathcal{L}_{6}}^{\mathcal{L}_{2},\mathcal{L}_{3}}\otimes
V_{\mathcal{L}_{4}}^{\mathcal{L}_{1},\mathcal{L}_{6}}$. This is called
$F$-move and can be written diagrammatically as shown in Figure 2. Notice that
here we use the notion of $F$-symbol, which is same as the crossing kernels
$\mathcal{K}$ in [44] up to the flipping of some of the orientation of the
TDLs.

Figure 2: The fusion of three TDLs has two different ways, but they are
related by the $F$-move and characterized by the $F$-symbol defined in this
diagram.

The $F$-symbols are constrained by the self-consistent conditions when
applying $F$-moves to the split process with $5$ TDLs. Two sequences of
$F$-moves start with the same state and end with the same state should be
equivalent. These consistent conditions diagrammatically shown in Figure 3
yield the pentagon equations on the $F$-symbols,

$\displaystyle\sum_{\delta}\left[F_{e}^{fcd}\right]_{\left(g,\beta,\gamma\right)\left(l,\nu,\delta\right)}\left[F_{e}^{abl}\right]_{\left(f,\alpha,\delta\right)\left(k,\mu,\lambda\right)}$
$\displaystyle=$
$\displaystyle\sum_{h,\sigma,\psi,\rho}\left[F_{g}^{abc}\right]_{\left(f,\alpha,\beta\right)\left(h,\psi,\sigma\right)}\left[F_{e}^{ahd}\right]_{\left(g,\sigma,\gamma\right)\left(k,\rho,\lambda\right)}\left[F_{k}^{bcd}\right]_{\left(h,\psi,\rho\right)\left(l,\nu,\mu\right)}.$
(2.3)

Figure 3: The upper 2 $F$-moves and the lower 3 $F$-moves yield the same
diagram, which gives the pentagon equation in (2.1).

For example, the invertible symmetry $G$ with anomaly $\omega$ is described by
the category of $G$-graded vector spaces, denoted by
$\operatorname{Vec}_{G}^{\omega}$. The simple objects are $1-$dimensional
$\mathbb{C}$-vector space labeled by $g\in G$, and physically they correspond
to the TDLs which generates the $g$-action. The fusion rule is
$\mathcal{L}_{g}\times\mathcal{L}_{h}=\mathcal{L}_{gh}$. The $F$-symbols of
this category are $U(1)$ phase factors (rather than generic complex numbers in
order to preserve the normalization), $\omega(g,h,k)\equiv
F^{\mathcal{L}_{g},\mathcal{L}_{h},\mathcal{L}_{k}}_{\mathcal{L}_{ghk}}$ for
$\mathcal{L}_{1,2,3}=\mathcal{L}_{g,h,k}$ as in Figure 2. These $U(1)$ phase
factors satisfy the pentagon equations,

$\omega(g,h,kl)\omega(gh,k,l)=\omega(h,k,l)\omega(g,hk,l)\omega(g,h,k),$ (2.4)

which is the cocycle condition for the 3-cocycle $\omega:G\times G\times
G\rightarrow U(1)$.

Notice that one can consider shifting the basis in $V^{g,h}_{gh}$ by a phase
$\beta(g,h)$, which is an element in $C^{2}(G,U(1))$. This basis change does
not change the physics and should be understood as a gauge transformation of
the $F$-symbols. The $F$-symbol changes as

$\omega(g,h,k)\rightarrow\omega(g,h,k)\frac{\beta(g,hk)\beta(h,k)}{\beta(g,h)\beta(gh,k)},$
(2.5)

and is interpreted as changing $\omega$ by an exact element in the set of
3-coboundaries $Z^{3}(G,U(1))\subset C^{3}(G,U(1))$. Therefore, inequivalent
$\operatorname{Vec}_{G}^{\omega}$’s are labelled by a finite group $G$ and its
anomaly $\omega\in H^{3}(G,U(1))=C^{3}(G,U(1))/Z^{3}(G,U(1))$.

###  2.2 Modular bootstrap with TDLs

By utilizing the modular transformation properties of the partition functions
or correlation functions of a CFT on a torus with complex structure $\tau$,
one can extract many useful information of the CFT [63, 64, 65, 40, 38, 39,
41, 42, 43, 66, 67]. Here, we are interested in studying the modular
properties of the CFT partition functions with networks of TDLs inserted on
the torus, which is called the twisted partition function on the torus. We
will adopt the convention from [1] on the twisted partition function over
torus, as in Figure 4.

Figure 4: Convention on the twisted partition function
$Z_{\mathcal{L}_{1}\mathcal{L}_{2}}^{\mathcal{L}_{3}}(\tau,\overline{\tau})$.

To reveal the physical meaning of
$Z_{\mathcal{L}_{1},\mathcal{L}_{2}}^{\mathcal{L}_{3},\mu,\nu}(\tau,\overline{\tau})$,
we first consider the simple case where $\mathcal{L}_{2}=\mathds{1}$ and
$\mathcal{L}_{3}=\mathcal{L}_{1}=\mathcal{L}$. For convenience, we will
sometimes abbreviate this partition function as
$Z_{\mathcal{L}}(\tau,\overline{\tau})$. Because the topological defect
$\mathcal{L}$ is inserted along the time direction, it should be interpreted
as a defect along the spatial direction, leading to a different Hilbert space
on the spatial circle $S^{1}$, and we will denote this Hilbert space as the
defect Hilbert space $\mathcal{H}_{\mathcal{L}}$.

$Z_{\mathcal{L}}(\tau,\overline{\tau})=Z^{\mathcal{L},1,1}_{\mathcal{L},\mathds{1}}(\tau,\overline{\tau})=\Tr_{\mathcal{H}_{\mathcal{L}}}q^{L_{0}-\frac{c}{24}}\overline{q}^{\overline{L}_{0}-\frac{c}{24}}.$
(2.6)

For instance, if $\mathcal{L}$ is a $\mathbb{Z}_{2}$ symmetry defect $\eta$,
then the defect Hilbert space $\mathcal{H}_{\mathcal{L}}$ is the
$\mathbb{Z}_{2}$ twisted Hilbert space acquired by imposing the twisted
boundary condition.

As an another simple example, let’s consider instead $\mathcal{L}_{1}=I$ and
$\mathcal{L}_{2}=\mathcal{L}_{3}=\mathcal{L}$. Similarly, we will sometimes
abbreviate this partition function as $Z^{\mathcal{L}}(\tau,\overline{\tau})$.
In this configuration, the TDL $\mathcal{L}$ is inserted on an equal time
slice, therefore should be interpreted as a quantum operator
$\widehat{\mathcal{L}}$ on the Hilbert space $\mathcal{H}$:

$Z^{\mathcal{L}}(\tau,\overline{\tau})\equiv
Z^{\mathcal{L},1,1}_{\mathds{1},\mathcal{L}}(\tau,\overline{\tau})=\Tr_{\mathcal{H}}\widehat{\mathcal{L}}q^{L_{0}-\frac{c}{24}}\overline{q}^{\overline{L}_{0}-\frac{c}{24}}.$
(2.7)

More generally, if we consider the twisted partition function
$Z_{\mathcal{L}_{1},\mathcal{L}_{2}}^{\mathcal{L}_{3},\mu,\nu}(\tau,\overline{\tau})$,
we should interpret $\mathcal{L}_{2}$ as a quantum operator
$(\widehat{\mathcal{L}}_{2})_{\mathcal{L}_{3},\mu,\nu}$ acting on the defect
Hilbert space $\mathcal{H}_{\mathcal{L}_{1}}$, where the subscript
$(\mathcal{L}_{3},\mu,\nu)$ indicates that for different intermediate lines
$\mathcal{L}_{3}$ and different vertex labels $\mu,\nu$ when the fusion
multiplicities are greater than $1$, we will have a different operator in
general.

$Z_{\mathcal{L}_{1},\mathcal{L}_{2}}^{\mathcal{L}_{3},\mu,\nu}(\tau,\overline{\tau})=\Tr_{\mathcal{H}_{\mathcal{L}_{1}}}(\widehat{\mathcal{L}}_{2})_{\mathcal{L}_{3},\mu,\nu}q^{L_{0}-\frac{c}{24}}\overline{q}^{\overline{L}_{0}-\frac{c}{24}}.$
(2.8)

Under the modular transformation, the twisted partition function
$Z^{\mathcal{L}_{3},\mu,\nu}_{\mathcal{L}_{1},\mathcal{L}_{2}}(\tau,\overline{\tau})$
transforms as the Figure 5,

$Z^{\mathcal{L}_{3},\mu,\nu}_{\mathcal{L}_{1},\mathcal{L}_{2}}\left(-\frac{1}{\tau},-\frac{1}{\overline{\tau}}\right)=\sum_{\mathcal{L}_{4},\rho,\sigma}\left[F^{\mathcal{L}_{2}\mathcal{L}_{1}\overline{\mathcal{L}}_{2}}_{\mathcal{L}_{1}}\right]_{(\mathcal{L}_{3},\mu,\nu),(\overline{\mathcal{L}}_{4},\rho,\sigma)}Z^{\mathcal{L}_{4},\sigma,\rho}_{\mathcal{L}_{2},\overline{\mathcal{L}}_{1}}(\tau,\overline{\tau}).$
(2.9)

Figure 5: Generalized modular bootstrap equations from $S$ modular
transformation of the twisted partition function
$Z^{\mathcal{L}_{3},\mu,\nu}_{\mathcal{L}_{1},\mathcal{L}_{2}}(\tau,\overline{\tau})$.

As a simple example, we may consider taking $\mathcal{L}_{1}$ to be the
identity line. Then the $F$-symbol is always trivial in this case (for
example, see [44]) and $\mu,\nu=1$ as the fusion is always one dimensional.
Then we have

$Z_{\mathcal{L}}\left(-\frac{1}{\tau},-\frac{1}{\overline{\tau}}\right)\equiv
Z_{\mathds{1},\mathcal{L}}^{\mathcal{L},1,1}\left(-\frac{1}{\tau},-\frac{1}{\overline{\tau}}\right)=Z^{\mathcal{L}}_{\mathcal{L},\mathds{1}}(\tau,\overline{\tau})\equiv
Z^{\mathcal{L}}(\tau,\overline{\tau}).$ (2.10)

As one can see, the $S$ modular transformation relates the partition functions
that count states the defect Hilbert space $\mathcal{H}_{\mathcal{L}}$ to the
partition functions that compute the action of the TDL $\mathcal{L}$ on the
Hilbert space $\mathcal{H}$. This allows us to derive interesting Cardy-like
formula which we will now review in the next subsection.

###  2.3 Asymptotic density of states

As an application of the modular bootstrap reviewed above, one can derive the
asymptotic density of states of a CFT [63, 66]. Intuitively, the modular
bootstrap equation,

$Z^{\mathcal{L}}\left(-\frac{1}{\tau},-\frac{1}{\overline{\tau}}\right)=Z_{\mathcal{L}}(\tau,\overline{\tau}),$
(2.11)

relates the high-temperature limit of the partition function $Z^{\mathcal{L}}$
which computes the action of the TDL $\mathcal{L}$ on $\mathcal{H}$ to the
low-temperature limit of the partition function $Z_{\mathcal{L}}$ which
computes the spectrum of the defect Hilbert space $\mathcal{H}_{\mathcal{L}}$
and vice versa. Since in the low-temperature limit, the partition function is
always dominated by the ground state, (2.11) essentially allows us to
determine the partition function $Z^{\mathcal{L}}$ and $Z_{\mathcal{L}}$ in
the high-temperature limit. In the high-temperature limit, states with
different energy would contribute equally to the partition sum, knowing
$Z_{\mathcal{L}}$ in such limit would allow us to derive an approximation of
the density of states for the defect Hilbert space
$\mathcal{H}_{\mathcal{L}}$. This idea is made into a rigorous mathematical
statement in [40] using the techniques in [66], and we will only mention the
result below.

Let $F_{\mathcal{L}}(\Delta)$ denote the total number of states with scaling
dimension $\Delta^{\prime}<\Delta$ in the defect Hilbert space. By asymptotic
density of states, we mean a continuous function
$\rho_{0,\mathcal{L}}(\Delta)$ which approximates the actual density of states
in the sense that

$F_{\mathcal{L}}(\Delta)=\int_{0}^{\Delta}d\Delta^{\prime}\,\rho_{0,\mathcal{L}}(\Delta^{\prime})+O(\Delta^{-1/2}),\quad\Delta\rightarrow\infty.$
(2.12)

Let

$\rho_{0}(\Delta)=\left(\frac{c}{48\Delta^{3}}\right)^{1/4}\exp\left[2\pi\sqrt{\frac{c\Delta}{3}}\right],$
(2.13)

it is shown in [40] that the asymptotic density of states
$\rho_{0,\mathcal{L}}$ of the defect Hilbert space $\mathcal{H}_{\mathcal{L}}$
is simply given by

$\rho_{0,\mathcal{L}}(\Delta)=\langle 0|\mathcal{L}|0\rangle\rho_{0}(\Delta),$
(2.14)

where $|0\rangle$ is the ground state and $\langle 0|\mathcal{L}|0\rangle$ is
the quantum dimension of the TDL $\mathcal{L}$.

Furthermore, let’s consider the theory with finite global symmetry $G$. Then
the states in the Hilbert space will organize into irreducible representations
of $G$. For simplicity, we assume $G$ acts faithfully. Recall that a
particular type of irreducible representation $\alpha$ can be counted by
$\frac{1}{|G|}\sum_{g\in G}\chi_{\alpha}(g)^{*}\text{Tr}g$ in a reducible
representation of $G$ where $\chi_{\alpha}(g)$ is the character function of
the irrep $\alpha$ and $|G|$ is the order of the group, then the partition
function counts the number of irrep $\alpha$ in the Hilbert space is given by

$\frac{1}{|G|}\sum_{g\in G}\chi_{\alpha}(g)^{*}Z^{g}(\tau,\overline{\tau}),$
(2.15)

whose high-temperature limit is known since each $Z^{g}$’s high-temperature
limit is known. Moreover, the assumption that $G$ acts faithfully implies that
the ground state in the defect Hilbert space has $h+\overline{h}>0$, therefore
(2.15) is dominated by
$Z^{\mathds{1}}(\tau,\overline{\tau})=Z(\tau,\overline{\tau})$ in the high
temperature limit. This allows us to derive the following result. Every
irreducible representation has to appear in the Hilbert space $\mathcal{H}$
and they also have a Cardy-like growth. Specifically, we can consider the
asymptotic growth $\rho_{0,\alpha}(\Delta)$ of the occurrence of a particular
irrep $\alpha$ of $G$ as a function of the scaling dimension $\Delta$ takes
the form,

$\rho_{0,\alpha}(\Delta)=\frac{d_{\alpha}}{|G|}\rho_{0}(\Delta).$ (2.16)

This result will be useful for us later.

##  3 Non-intrinsic triality fusion category as group theoretical fusion
category

As pointed out in [1], the origin of the triality fusion rule in the KT theory
arises from gauging $\mathbb{Z}_{2}$ subgroup of $A_{4}\subset SO(4)$ in the
$SU(2)_{1}$ theory. Generically, gauging a subgroup $H$ which is not normal in
$G$ or not Abelian leads to non-invertible symmetries [68]. In this section,
we first review the construction of triality acquired in [4, 1]. We then
discuss how to understand the group theoretical triality fusion category using
the mathematical tools of bimodule categories in [51]. Specifically, we
describe how to understand and compute simple TDLs, fusion rules and
$F$-symbols.

###  3.1 A brief review of the triality category discovered in [4, 1]

The origin of the triality fusion category discovered in [4, 1] is a result of
gauging $\mathbb{Z}_{2}$ subgroup in a theory with $A_{4}$ global symmetry.
Notice that $A_{4}$ is an order $12$ group and can be presented as

$A_{4}=\langle\sigma,\eta,q|q^{3}=\sigma^{2}=\eta^{2},q\sigma
q^{-1}=\sigma\eta=\eta\sigma,q\eta q^{-1}=\sigma\rangle.$ (3.1)

This means we can think of $A_{4}$ containing a
$\mathbb{Z}_{2}^{\sigma}\times\mathbb{Z}^{\eta}_{2}$ subgroup, and the
conjugation by the $\mathbb{Z}^{q}_{3}$ generator $q$ will permute the three
$\mathbb{Z}_{2}$ generators in
$\mathbb{Z}^{\sigma}_{2}\times\mathbb{Z}^{\eta}_{2}$. After gauging the
$\mathbb{Z}_{2}^{\sigma}$ symmetry, the $\mathbb{Z}_{2}^{\eta}$ subgroup will
survive since it commutes with the $\mathbb{Z}_{2}^{\sigma}$ subgroup,
together with the quantum $\mathbb{Z}_{2}$ which we will denote as
$\mathbb{Z}_{2}^{\hat{\sigma}}$, they form the
$\mathbb{Z}_{2}^{\hat{\sigma}}\times\mathbb{Z}_{2}^{\eta}$ invertible
symmetries in the gauged theory. The symmetry operator $q$ does not commute
with $\sigma$, therefore is not gauge invariant, and will not appear as a
genuine topological line operator in the gauged theory. However, the linear
combination

$q+\sigma q\sigma$ (3.2)

does commute with $\sigma$ and will survive as a genuine topological line
operator in the gauged theory, which we will denote as $\mathcal{L}_{Q}$.
However, this operator is not invertible, as it has quantum dimension $2$.
Similarly, its orientation reversal $\mathcal{L}_{\overline{Q}}$ relates to
the gauge-invariant linear combination $q^{-1}+\sigma q^{-1}\sigma$. It is
pointed out in [4, 1] that $\mathcal{L}_{Q},\mathcal{L}_{\overline{Q}}$
together with the $\mathbb{Z}_{2}^{\hat{\sigma}}\times\mathbb{Z}_{2}^{\eta}$
forms the triality fusion categories with the fusion rules,

$\displaystyle g\times\mathcal{L}_{Q}=\mathcal{L}_{Q}\times
g=\mathcal{L}_{Q},\quad
g\times\mathcal{L}_{\overline{Q}}=\mathcal{L}_{\overline{Q}}\times
g=\mathcal{L}_{\overline{Q}},$ (3.3)
$\displaystyle\mathcal{L}_{Q}\times\mathcal{L}_{Q}=2\mathcal{L}_{\overline{Q}},\quad\mathcal{L}_{\overline{Q}}\times\mathcal{L}_{\overline{Q}}=2\mathcal{L}_{Q},\quad\mathcal{L}_{\overline{Q}}\times\mathcal{L}_{Q}=\mathcal{L}_{Q}\times\mathcal{L}_{\overline{Q}}=\sum_{g\in\mathbb{Z}_{2}^{\hat{\sigma}}\times\mathbb{Z}_{2}^{\eta}}g.$

The existence of the triality fusion category implies the theory is invariant
under the $\mathbb{Z}_{2}^{\hat{\sigma}}\times\mathbb{Z}_{2}^{\eta}$ gauging,
but $\mathbb{Z}_{2}^{\hat{\sigma}}\times\mathbb{Z}_{2}^{\eta}$ charge
assignments will be permuted. This can be understood as the following. Gauging
the $\mathbb{Z}_{2}^{\hat{\sigma}}$ quantum symmetry gives back the original
theory with $A_{4}$ symmetry, then gauging $\mathbb{Z}_{2}^{\eta}$ symmetry
will give the same theory acquired from gauging the $\mathbb{Z}_{2}^{\sigma}$
symmetry, since $\mathbb{Z}_{2}^{\eta}$ and $\mathbb{Z}_{2}^{\sigma}$ are
related by the conjugation of $q$. However, this will change the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ charge assignment in the gauged theory
since we are gauging different but equivalent $\mathbb{Z}_{2}$’s.

###  3.2 Group theoretical fusion category

Let’s consider the fusion category $\operatorname{Vec}_{G}^{\omega}$, where
$G$ is a finite group and $\omega\in H^{3}(G,U(1))$. Physically, this fusion
category describes the global symmetry $G$ with anomaly $\omega$ for
2-dimensional field theory. The simple elements are labeled by group element
$g\in G$ and the fusion rule is simply the product of the group elements. The
$F$-symbols are given by

$F_{l}^{ghk}=\omega(g,h,k).$ (3.4)

If we consider a finite subgroup $H$ in $G$, then the anomaly-free condition
for $H$ is that $\omega$ as a function from $G^{3}$ to $U(1)$ restricting to
$H^{3}$ is trivial, that is,

$\omega(h_{1},h_{2},h_{3})=1,\quad\forall\,h_{i}\in H.$ (3.5)

When $H$ is anomaly free, we can consider gauging the subgroup $H$ and we have
to choose a discrete torsion $\psi\in H^{2}(G,U(1))$. The resulting fusion
category which describes the global symmetry in the gauged theory is denoted
as $\mathcal{C}(G,\omega,H,\psi)$ and this class of fusion category is called
the group theoretical fusion category. In particular,
$\operatorname{Vec}_{G}^{\omega}=\mathcal{C}(G,\omega,\mathbb{Z}_{1},1)$.

Generically, when $H$ is not a normal subgroup of $G$ or is not Abelian, then
the resulting group theoretical fusion category is not of the form
$\operatorname{Vec}_{G}^{\omega}$, meaning it contains non-invertible simple
lines. For example, gauging the $\mathbb{Z}_{2}$ symmetry of $S^{3}$ with
$\omega=1$ leads to the fusion category $\operatorname{Rep}(S^{3})$ [22] and
gauging the $\mathbb{Z}_{2}$ symmetry of $A_{4}$ with $\omega=1$ leads to the
triality fusion category [4, 1].

Below, we will describe how to understand and compute the simple TDLs, fusion
rules, and $F$-symbols of the triality fusion category from the data of the
group theoretical fusion category. It would be helpful to point out that
although the $F$-symbols for the group theoretical fusion category can be
computed by the more general method given in [47] by considering the tube
algebra $\mathbf{Tub}_{\mathcal{C}}(\mathcal{M})$, it is more convenient to
compute the $F$-symbols using our approach. For instance, the calculation of
the $F$-symbol for $\mathcal{C}(A_{4},1,\mathbb{Z}_{2},1)$ using [47] requires
to construct an explicit Artin-Wedderburn isomorphism from the tube algebra
$\mathbf{Tub}_{\operatorname{Vec}_{A_{4}}}(\mathcal{M})$ to a direct sum of
several matrix algebras, but since the tube algebra
$\mathbf{Tub}_{\operatorname{Vec}_{A_{4}}}(\mathcal{M})$ has a very large
dimension $432$ in this case, it’s computationally hard to explicitly
construct the Artin-Wedderburn isomorphism.

###  3.3 Simple lines

As pointed out in [51], the simple TDLs in the gauged theory are described by
indecomposable $A-A$ bimodules in ${\operatorname{Vec}}_{G}^{\omega}$, where
$A$ is the group algebra of $H$ twisted by some 2-cocycle of $\psi\in
H^{2}(H,U(1))$, corresponding to a choice of discrete torsion. In the case of
$SU(2)_{1}$ theory, we take $G=A_{4}$ and $G$ is anomaly free so that
$\omega=1\in H^{3}(G,U(1))$. $H=\mathbb{Z}_{2}\subset G$ and there’s no non-
trivial discrete torsion for $\mathbb{Z}_{2}$, so we can choose $\psi$ to be
trivial as well. Hence, $A$ is simply the group algebra of $\mathbb{Z}_{2}$,
namely a 2-dimensional vector space over $\mathbb{C}$ with a basis
$\\{1,\sigma\\}$ equipped with multiplication $1^{2}=\sigma^{2}=1$ and $\sigma
1=1\sigma=\sigma$ which is the group multiplication of $\mathbb{Z}_{2}$.

In the special case of group theoretical fusion category, by $A-A$ bimodule
$M$, we mean a $\mathbb{C}$-vector space $M$ together a left multiplication
$A\times M\rightarrow M$ denoted as $am$ for $a\in A,m\in M$ and a right
multiplication $M\times A\rightarrow M$ denoted as $ma$ for $a\in A,m\in M$
such that $\forall a_{i}\in A$ and $m\in M$:

$a_{1}(a_{2}m)=(a_{1}a_{2})m\equiv
a_{1}a_{2}m,\quad(ma_{1})a_{2}=m(a_{1}a_{2})\equiv
ma_{1}a_{2},\quad(a_{1}m)a_{2}=a_{1}(ma_{2})\equiv a_{1}ma_{2}.$ (3.6)

The indecomposable $A-A$ bimodule $M$ is of the following form. Consider a
double coset $HgH$ of $H$ in $G$, and let $H^{g}$ denote the little group of
$HgH$:

$H^{g}=\\{h\in H|\exists h^{\prime}\in H\,s.t.\,hgh^{\prime}=g\\}\subset H.$
(3.7)

As one can check from the above definition, the little group $H^{g^{\prime}}$
does not depend on the choice of representative $g^{\prime}\in HgH$ up to
isomorphism.

Each indecomposable $A-A$ bimodule is labelled by a double coset $HgH$ and an
irreducible representation $\rho$ (potentially twisted by some 2-cocycle of
the little group) of the little group of an arbitrary element in the double
coset $HgH$. More specifically, given $(HgH,\rho)$, an indecomposable $A-A$
bimodule $M_{HgH}^{\rho}$ is a vector space over $\mathbb{C}$ such that

$M_{HgH}^{\rho}=\bigoplus_{g^{\prime}\in HgH}M_{g^{\prime}}^{\rho}.$ (3.8)

To describe the action of $A$ on $M_{HgH}^{\rho}$, we only need to describe
the action of the basis $\\{h\\}_{h\in H}$ on $M_{HgH}^{\rho}$. And
multiplication of $h$ on the left induces an isomorphism from
$M_{g^{\prime}}^{\rho}$ to $M_{hg^{\prime}}^{\rho}$ while the multiplication
of $h$ on the right induces an isomorphism from $M_{g^{\prime}}^{\rho}$ to
$M_{g^{\prime}h}^{\rho}$. This implies $M_{g^{\prime}}^{\rho}$’s are
isomorphic to each other. Consider $M_{g}^{\rho}$ and we find for $h\in
H^{g}$, multiplying $h$ on the left and multiplying the corresponding
$h^{\prime}$ on the right leads to an isomorphism between $M_{g}^{\rho}$ and
itself (since $hgh^{\prime}=g$), this is why $H^{g}$ is called the little
group. $M_{g}^{\rho}$ is the vector space of the irreducible representation
$\rho$ of $H^{g}$. Notice that in the case where the group algebra $A$ is
twisted by non-trivial 2-cocycle $\psi$, the composition of two left(right)
multiplications on $M_{g}^{\rho}$ can composite non-trivially. But in the case
of the gauging $\mathbb{Z}_{2}$ symmetry in $A_{4}$ with no anomaly, such data
is trivial. The fusion of the TDLs in the gauged theory is described by the
tensor product of $A-A$ bimodule over the algebra $A$.

We now list the indecomposable bimodule in
$\mathcal{C}(A_{4},1,\mathbb{Z}_{2}^{\sigma},1)$ and compute its fusion rule,
and show indeed we reproduce the fusion rule of the triality fusion category.
There are 4 double cosets of
$H=\mathbb{Z}_{2}=\\{(),(12)(34)\\}\equiv\\{1,\sigma\\}$ in $A_{4}$:666The $q$
used in (3.1) is the cycle $(143)$.

$\displaystyle I=\\{(),(12)(34)\\}\equiv\\{1,\sigma\\},\quad
J=\\{(13)(24),(14)(23)\\}\equiv\\{\eta,\eta\sigma\\},\quad$ (3.9)
$\displaystyle
Q=\\{(143),(124),(132),(234)\\},\quad\overline{Q}=\\{(134),(142),(123),(243)\\}.$

The little group for the first two double cosets is $H$ itself while for the
last two double cosets are trivial. Hence, for the first two double cosets
$I,J$, we need to label the irreducible representation $\pm$ of
$H=\mathbb{Z}_{2}$ as well. Hence, we find the following $6$ indecomposable
$A-A$ bimodules,

$\displaystyle M_{I}^{\pm}=M_{1}^{\pm}\oplus M_{\sigma}^{\pm},\quad
M_{J}^{\pm}=M_{\eta}^{\pm}\oplus M_{\sigma\eta}^{\pm},$ (3.10) $\displaystyle
M_{Q}=M_{(143)}\oplus M_{(124)}\oplus M_{(132)}\oplus M_{(234)},$
$\displaystyle M_{\overline{Q}}=M_{(134)}\oplus M_{(142)}\oplus
M_{(123)}\oplus M_{(243)}.$

Notice that each $M_{g}^{\rho}$ appears in the direct sum is a 1-dimensional
vector space, so we will choose a basis vector $m_{g}^{\rho}$ for each
$M_{g}^{\rho}$. We choose the action of $h\in A$ on $m_{g}^{\rho}$’s as,

$1m_{g}^{\rho}=m_{g}^{\rho}1=m_{g}^{\rho},\quad\sigma m_{g}^{\pm}=\pm
m^{\pm}_{\sigma g},\quad m_{g}^{\pm}\sigma=m_{g\sigma}^{\pm}.$ (3.11)

As one can check, for $M_{I}^{-}$ and $M_{J}^{-}$, the multiplication of
$\sigma$ on left and right simultaneously does generate non-trivial action of
$\mathbb{Z}_{2}$ on $M_{g}^{-}$ for $g\in I,J$. The bimodules $M_{I}^{\pm}$
correspond to the quantum $\mathbb{Z}_{2}$ symmetry in the gauged theory,
while $M_{J}^{+}$ generates the unbroken $\mathbb{Z}_{2}$ symmetry in $A_{4}$.
Together they form the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry in the
gauged theory. $M_{Q}$ and $M_{\overline{Q}}$ are identified with the triality
line $\mathcal{L}_{Q}$ and $\mathcal{L}_{\overline{Q}}$.

To conclude, we point it out there is a natural dual module
$\operatorname{Hom}_{A}(M_{HgH}^{\rho},A)$ for each bimodule $M_{HgH}^{\rho}$,
which is isomorphic to $M_{Hg^{-1}H}^{\rho^{\dagger}}$.777Generically, for an
$A-B$ bimodule $M$, one can either consider the dual being
$\operatorname{Hom}_{A}(M,A)$ or $\operatorname{Hom}_{B}(M,B)$, which are both
$B-A$ bimodules. Here, since we are considering $A-A$ bimodule, we can
consider either choice and the results should be isomorphic. Physically this
corresponds to taking the orientation reversal of the TDL.

###  3.4 Fusion rules

The fusion of the TDLs in the gauged theory corresponds to the tensor product
of bimodules over the algebra $A$. Generically, let $M$ and $N$ be $A-A$
bimodules, then

$M\otimes_{A}N=M\otimes N/\sim$ (3.12)

where the first tensor product is the tensor product of vector spaces and
$\sim$ is the equivalence relation

$(ma)\otimes n\sim m\otimes(an),\quad a\in A,m\in M,n\in N.$ (3.13)

The $M\otimes_{A}N$ is naturally an $A-A$ bimodule and we can then decompose
it as a direct sum over indecomposable $A-A$ bimodules.

Using this, one can compute the tensor product of bimodules explicitly and
acquire the fusion rule. There is an additional rule one needs to know is that
the grading of $M_{g}^{\rho}\otimes_{A}M_{g^{\prime}}^{\rho^{\prime}}$ is
$gg^{\prime}$.

We will not list all the calculations but provide several examples to help the
reader understand the procedure.

To start, let’s consider the fusion between $M_{I}^{+}$ with a generic
$M_{HgH}^{\rho}$. The tensor product $M_{I}^{+}\otimes M_{HgH}^{\rho}$ has a
basis $\\{m_{1}^{+}\otimes m_{g^{\prime}}^{\rho},m_{\sigma}^{+}\otimes
m_{g^{\prime}}^{\rho}\\}_{g^{\prime}\in HgH}$. After the identification with
the equivalence relation, we denote the equivalence class as
$m_{1}^{+}\otimes_{A}m_{g^{\prime}}^{\rho}=\rho(\sigma)m_{\sigma}^{+}\otimes_{A}m_{\sigma
g^{\prime}}^{\rho}$. The resulting bimodule $M_{I}^{+}\otimes_{A}M_{g}^{\rho}$
is isomorphic to $M_{g}^{\rho}$ itself where
$m_{1}^{+}\otimes_{A}m_{g}^{\rho}\simeq m_{g}^{\rho}$. We can check the left
action of $\sigma$ on $m_{1}^{+}\otimes_{A}m_{g}^{\rho}$ gives the correct
sign for $\rho=-$:

$\sigma(m_{1}^{+}\otimes_{A}m_{g}^{\rho})=(\sigma
m_{1}^{+})\otimes_{A}m_{g}^{\rho}=m_{\sigma}^{+}\otimes_{A}m_{g}^{\rho}=(m_{1}^{+}\sigma)\otimes_{A}m_{g}^{\rho}=m_{1}^{+}\otimes_{A}(\sigma
m_{g}^{\rho})=\rho(\sigma)m_{1}^{+}\otimes_{A}m_{\sigma g}^{\rho},$ (3.14)

where $\rho(g)$ will produce the desired sign when $\rho=-$. Hence, we find
the fusion rule $M_{I}^{+}\otimes_{A}M_{HgH}^{\rho}=M_{HgH}^{\rho}$.

As another example, let’s consider the fusion between $M_{Q}$ and $M_{Q}$.
There are 16 basis vectors in $M_{Q}\otimes M_{Q}$ and after modding out the
equivalence relation we are left with 8 basis vectors. The grading suggests
there are two copies of $M_{\overline{Q}}$ and we take them to be

$\displaystyle m_{(143)}\otimes_{A}m_{(143)}\simeq m_{(134),1},\quad
m_{(143)}\otimes_{A}m_{(124)}\simeq m_{(123),1},$ (3.15) $\displaystyle
m_{(132)}\otimes_{A}m_{(143)}\simeq m_{(142),1},\quad
m_{(132)}\otimes_{A}m_{(124)}\simeq m_{(243),1},$ $\displaystyle
m_{(134)}\otimes_{A}m_{(234)}\simeq m_{(142),2},\quad
m_{(143)}\otimes_{A}m_{(132)}\simeq m_{(243),2},$ $\displaystyle
m_{(132)}\otimes_{A}m_{(132)}\simeq m_{(123),2},\quad
m_{(132)}\otimes_{A}m_{(234)}\simeq m_{(134),2},$

where the subscript $m_{g,i},i=1,2$ indicates which copies of
$M_{\overline{Q}}$.

As the final example, let’s consider the fusion between $M_{Q}$ and
$M_{\overline{Q}}$. There are 16 basis vectors in $M_{Q}\otimes
M_{\overline{Q}}$ and after modding out the equivalence relation there are
only 8 left. After rewrite them as different linear combinations, we find they
generate $M_{I}^{+}\oplus M_{I}^{-}\oplus M_{J}^{+}\oplus M_{J}^{-}$ where

$\displaystyle\frac{m_{(132)}\otimes_{A}m_{(123)}+m_{(143)}\otimes_{A}m_{(134)}}{\sqrt{2}}\simeq
m_{1}^{+},\quad\frac{m_{(132)}\otimes_{A}m_{(134)}+m_{(143)}\otimes_{A}m_{(123)}}{\sqrt{2}}\simeq
m_{\sigma}^{+},$ (3.16)
$\displaystyle\frac{m_{(132)}\otimes_{A}m_{(123)}-m_{(143)}\otimes_{A}m_{(134)}}{\sqrt{2}}\simeq
m_{1}^{-},\quad\frac{m_{(132)}\otimes_{A}m_{(134)}-m_{(143)}\otimes_{A}m_{(123)}}{\sqrt{2}}\simeq
m_{\sigma}^{-},$
$\displaystyle\frac{m_{(132)}\otimes_{A}m_{(243)}+m_{(143)}\otimes_{A}m_{(142)}}{\sqrt{2}}\simeq
m_{\eta}^{+},\quad\frac{m_{(132)}\otimes_{A}m_{(142)}+m_{(143)}\otimes_{A}m_{(243)}}{\sqrt{2}}\simeq
m_{\sigma\eta}^{+},$
$\displaystyle\frac{m_{(132)}\otimes_{A}m_{(243)}-m_{(143)}\otimes_{A}m_{(142)}}{\sqrt{2}}\simeq
m_{\eta}^{-},\quad\frac{m_{(132)}\otimes_{A}m_{(142)}-m_{(143)}\otimes_{A}m_{(243)}}{\sqrt{2}}\simeq
m_{\sigma\eta}^{-}.$

One can check the above identification is consistent with the action of
$\sigma$. For instance,

$\sigma
m_{1}^{-}\simeq\sigma\frac{m_{(132)}\otimes_{A}m_{(123)}-m_{(143)}\otimes_{A}m_{(134)}}{\sqrt{2}}=\frac{m_{(143)}\otimes_{A}m_{(123)}-m_{(132)}\otimes_{A}m_{(134)}}{\sqrt{2}}\simeq-
m_{\sigma}^{-}.$ (3.17)

$\otimes_{A}$ | $M_{I}^{+}$ | $M_{I}^{-}$ | $M_{J}^{+}$ | $M_{J}^{-}$ | $M_{Q}$ | $M_{\overline{Q}}$
---|---|---|---|---|---|---
$M_{I}^{+}$ | $M_{I}^{+}$ | $M_{I}^{-}$ | $M_{J}^{+}$ | $M_{J}^{-}$ | $M_{Q}$ | $M_{\overline{Q}}$
$M_{I}^{-}$ | $M_{I}^{-}$ | $M_{I}^{+}$ | $M_{J}^{-}$ | $M_{J}^{+}$ | $M_{Q}$ | $M_{\overline{Q}}$
$M_{J}^{+}$ | $M_{J}^{+}$ | $M_{J}^{-}$ | $M_{I}^{+}$ | $M_{I}^{-}$ | $M_{Q}$ | $M_{\overline{Q}}$
$M_{J}^{-}$ | $M_{J}^{-}$ | $M_{J}^{+}$ | $M_{I}^{-}$ | $M_{I}^{+}$ | $M_{Q}$ | $M_{\overline{Q}}$
$M_{Q}$ | $M_{Q}$ | $M_{Q}$ | $M_{Q}$ | $M_{Q}$ | $M_{\overline{Q}}\oplus M_{\overline{Q}}$ | $M_{I}^{+}\oplus M_{I}^{-}\oplus M_{J}^{+}\oplus N_{J}^{-}$
$M_{\overline{Q}}$ | $M_{\overline{Q}}$ | $M_{\overline{Q}}$ | $M_{\overline{Q}}$ | $M_{\overline{Q}}$ | $M_{I}^{+}\oplus M_{I}^{-}\oplus M_{J}^{+}\oplus N_{J}^{-}$ | $M_{Q}\oplus M_{Q}$
Table 1: Result of tensor product of $A-A$ bimodules.

The rest of the tensor products can be computed as above and we list the
result in Tab. 1. With the identifications between bimodules and TDLs given
previously, we reproduce the fusion rule of the triality fusion category.

###  3.5 $F$-symbols

The advantage of the description using bimodules is that we can easily compute
the $F$-symbols for the resulting fusion category. The local fusion of two
TDLs labeled by bimodules $M,N$ into the TDL labeled by bimodule $L$ can be
identified as the vector space of bimodule homomorphism from $M\otimes_{A}N$
to $L$:

$\operatorname{Hom}_{A-A}(M\otimes_{A}N,L).$ (3.18)

Here, a $A-A$ bimodule homomorphism $\phi\in\operatorname{Hom}_{A-A}(M,N)$
from $A-A$ bimodule $M$ to $A-A$ bimodule $N$ is a $\mathbb{C}$-linear map
from $M$ to $N$ such that

$a\phi(m)a^{\prime}=\phi(ama^{\prime}),\quad\forall a,a^{\prime}\in A,m\in M.$
(3.19)

We can then consider choose a basis for each vector space
$\operatorname{Hom}_{A-A}(M\otimes_{A}N,L)$ and use the Greek letter
$\mu,\nu,\cdots=1,2,\cdots$ to label the basis vectors.

For instance, let’s consider the fusion $M_{1}^{+}\otimes_{A}M_{g}^{\rho}$
studied previously. Since $M_{1}^{+}\otimes_{A}M_{g}^{\rho}\simeq
M_{g}^{\rho}$, there is only one fusion channel and
$\operatorname{Hom}_{A-A}(M\otimes_{A}N,L)$ is a 1-dim vector space. The
isomorphism

$m_{1}^{+}\otimes_{A}m_{g}^{\rho}\simeq m_{g}^{\rho}$ (3.20)

is a basis vector for $\operatorname{Hom}_{A-A}(M\otimes_{A}N,L)$.

As an another example, let’s consider $M_{Q}\otimes_{A}M_{Q}\simeq
M_{\overline{Q}}\oplus M_{\overline{Q}}$. In this case, $M_{\overline{Q}}$
appears twice in the direct sum, therefore
$\operatorname{Hom}_{A-A}(M_{Q}\otimes_{A}M_{Q},M_{\overline{Q}})$ would be
2-dimensional. The choice of identifications in (3.15) leads a basis in
$\operatorname{Hom}_{A-A}(M_{Q}\otimes_{A}M_{Q},M_{\overline{Q}})$, which are
given by two $A-A$ bimodule homomorphism $\phi_{1},\phi_{2}$ from
$M_{Q}\otimes_{A}M_{Q}\rightarrow M_{\overline{Q}}$ :

$\phi_{1}:\begin{pmatrix}m_{(143)}\otimes_{A}m_{(143)}\\\
m_{(143)}\otimes_{A}m_{(124)}\\\ m_{(132)}\otimes_{A}m_{(124)}\\\
m_{(132)}\otimes_{A}m_{(143)}\end{pmatrix}\mapsto\begin{pmatrix}m_{(134)}\\\
m_{(123)}\\\ m_{(243)}\\\ m_{(142)}\end{pmatrix},$ (3.21)

and,

$\phi_{2}:\begin{pmatrix}m_{(143)}\otimes_{A}m_{(234)}\\\
m_{(143)}\otimes_{A}m_{(132)}\\\ m_{(132)}\otimes_{A}m_{(132)}\\\
m_{(132)}\otimes_{A}m_{(234)}\end{pmatrix}\mapsto\begin{pmatrix}m_{(142)}\\\
m_{(243)}\\\ m_{(123)}\\\ m_{(134)}\end{pmatrix}.$ (3.22)

Finally, let’s consider the fusion $M_{Q}\otimes_{A}M_{\overline{Q}}\simeq
M_{I}^{+}\oplus M_{I}^{-}\oplus M_{J}^{+}\oplus M_{J}^{-}$. Since there are 4
modules that appear in the direct sum and each only appears once, there are 4
$\operatorname{Hom}$ spaces and each has dimension $1$. For instance,
$\operatorname{Hom}_{A-A}(M_{Q}\times_{A}M_{\overline{Q}},M_{I}^{-})$ is the
space of projection maps from $M_{Q}\otimes_{A}M_{\overline{Q}}$ to
$M_{I}^{-}$. Our identification in (3.16) also determines a basis vector
(which is a projection map) in
$\operatorname{Hom}_{A-A}(M_{Q}\times_{A}M_{\overline{Q}},M_{I}^{-})$, given
by the following $A-A$ bimodule homomorphism,

$\begin{pmatrix}m_{(132)}\otimes_{A}m_{(123)}\simeq
m_{(234)}\otimes_{A}m_{(243)}\\\ m_{(143)}\otimes_{A}m_{(134)}\simeq
m_{(124)}\otimes_{A}m_{(142)}\\\ m_{(143)}\otimes_{A}m_{(123)}\simeq
m_{(124)}\otimes_{A}m_{(243)}\\\ m_{(132)}\otimes_{A}m_{(134)}\simeq
m_{(234)}\otimes_{A}m_{(142)}\end{pmatrix}\mapsto\begin{pmatrix}\frac{1}{\sqrt{2}}m_{1}^{-}\\\
-\frac{1}{\sqrt{2}}m_{1}^{-}\\\ -\frac{1}{\sqrt{2}}m_{\sigma}^{-}\\\
\frac{1}{\sqrt{2}}m_{\sigma}^{-}\end{pmatrix}$ (3.23)

where unlisted elements are mapped to $0$.

Let’s consider the $F$-symbol. The local fusions of the diagrams on both sides
of the diagram give $A-A$ bimodule homomorphisms from
$M\otimes_{A}N\otimes_{A}L$ to $G$ in
$\operatorname{Hom}_{A-A}(M\otimes_{A}N\otimes_{A}L,G)$, and the $F$-symbol
can be interpreted as the matrix elements of the linear transformation between
two sets of elements in
$\operatorname{Hom}_{A-A}(M\otimes_{A}N\otimes_{A}L,G)$.

With a choice of basis for each $\operatorname{Hom}_{A}(M\otimes_{A}N,P)$, we
can acquire $F$-symbols which are the matrix elements of the above
transformation. Practically, let’s consider the fixed bimodule $M,N,L,G$ and
choose basis vectors $\phi_{M\otimes_{A}N\rightarrow
P,\mu}\in\operatorname{Hom}_{A}(M\otimes_{A}N,L)$ for every junction space
$\operatorname{Hom}_{A}(M\otimes_{A}N,L)$, then the diagram on the left hand
side leads to an element $\phi_{P\otimes_{A}L\rightarrow
G,\nu}\circ\phi_{M\otimes_{A}N\rightarrow
P,\mu}\in\operatorname{Hom}_{A-A}(M\otimes_{A}N\otimes_{A}L,G)$:

$\phi_{P\otimes_{A}L\rightarrow G,\nu}\circ\phi_{M\otimes_{A}N\rightarrow
P,\mu}:m\otimes_{A}n\otimes_{A}l\mapsto\phi_{P\otimes_{A}L\rightarrow
G,\nu}(\phi_{M\otimes_{A}N,\mu}(m\otimes_{A}n)\otimes_{A}l).$ (3.24)

Similarly, $\phi_{M\otimes_{A}Q\rightarrow
G,\mu}\circ\phi_{N\otimes_{A}L\rightarrow
Q,\nu}\in\operatorname{Hom}_{A-A}(M\otimes_{A}N\otimes_{A}L,G)$ is defined as

$\phi_{M\otimes_{A}Q\rightarrow G,\mu}\circ\phi_{N\otimes_{A}L\rightarrow
Q,\nu}:m\otimes_{A}n\otimes_{A}l\mapsto\phi_{M\otimes_{A}Q\rightarrow
G,\mu}(m\otimes_{A}\phi_{N\otimes_{A}L\rightarrow Q,\nu}(n\otimes_{A}l)).$
(3.25)

The $F$-symbols are just $\mathbb{C}$-numbers such that the following
equations of $A-A$ bimodule homomorphisms from $M\otimes_{A}N\otimes_{A}L$ to
$G$ hold

$\phi_{P\otimes_{A}L\rightarrow G,\nu}\circ\phi_{M\otimes_{A}N\rightarrow
P,\mu}=\sum_{Q,\alpha,\beta}\left[F^{MNL}_{G}\right]_{(P,\mu,\nu),(Q,\alpha,\beta)}\phi_{M\otimes_{A}Q\rightarrow
G,\beta}\circ\phi_{N\otimes_{A}L\rightarrow Q,\alpha}.$ (3.26)

To solve the above equation, we only need to evaluate the $A$-homomorphism on
basis vectors of $M\otimes_{A}N\otimes_{A}L\rightarrow G$, which will produce
a set of linear equations and can be solved quite easily.

As an example, let’s consider taking
$M=N=Q=M_{Q},L=M_{1}^{-},P=G=M_{\overline{Q}}$. This would compute the
$F$-symbol
$\left[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\hat{\sigma}}_{\mathcal{L}_{\overline{Q}}}\right]_{(\mathcal{L}_{\overline{Q}},\mu,1),(\mathcal{L}_{Q},1,\beta)}$,
that is, we want to solve,

$\displaystyle\phi_{M_{\overline{Q}}\otimes_{A}M_{1}^{-}\rightarrow
M_{\overline{Q}},1}\circ\phi_{M_{Q}\otimes_{A}M_{Q}\rightarrow
M_{\overline{Q}},\mu}$ (3.27) $\displaystyle=$
$\displaystyle\sum_{\beta=1,2}\left[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\hat{\sigma}}_{\mathcal{L}_{\overline{Q}}}\right]_{(\mathcal{L}_{\overline{Q}},\mu,1),(\mathcal{L}_{Q},1,\beta)}\phi_{M_{Q}\otimes_{A}M_{Q}\rightarrow
M_{\overline{Q}},\beta}\circ\phi_{M_{Q}\otimes_{A}M_{1}^{-}\rightarrow
M_{Q},1}.$

We consider evaluate the above equation on
$m_{(143)}\otimes_{A}m_{(143)}\otimes_{A}m_{1}^{-}$ and
$m_{(132)}\otimes_{A}m_{(132)}\otimes_{A}m_{1}^{-}$. As an example, we do this
for $m_{(143)}\otimes_{A}m_{(143)}\otimes_{A}m_{1}^{-}$. On the left-hand
side, we have,

$\displaystyle\phi_{M_{\overline{Q}}\otimes_{A}M_{1}^{-}\rightarrow
M_{\overline{Q}},1}\circ\phi_{M_{Q}\otimes_{A}M_{Q}\rightarrow
M_{\overline{Q}},\mu}(m_{(143)}\otimes_{A}m_{(143)}\otimes_{A}m_{1}^{-})$
(3.28) $\displaystyle=$
$\displaystyle\phi_{M_{\overline{Q}}\otimes_{A}M_{1}^{-}\rightarrow
M_{\overline{Q}},1}(\phi_{M_{Q}\otimes_{A}M_{Q}\rightarrow
M_{\overline{Q}},\mu}(m_{(143)}\otimes_{A}m_{(143)})\otimes_{A}m_{1}^{-})$
$\displaystyle=$
$\displaystyle\phi_{M_{\overline{Q}}\otimes_{A}M_{1}^{-}\rightarrow
M_{\overline{Q}},1}(\delta_{\mu,1}m_{(134)}\otimes_{A}m_{1}^{-})$
$\displaystyle=$ $\displaystyle\delta_{\mu,1}m_{(134)},$

and on the right-hand side, we have,

$\displaystyle\phi_{M_{Q}\otimes_{A}M_{Q}\rightarrow
M_{\overline{Q}},\beta}\circ\phi_{M_{Q}\otimes_{A}M_{1}^{-}\rightarrow
M_{Q},1}(m_{(143)}\otimes_{A}m_{(143)}\otimes_{A}m_{1}^{-})$ (3.29)
$\displaystyle=$ $\displaystyle\phi_{M_{Q}\otimes_{A}M_{Q}\rightarrow
M_{\overline{Q}},\beta}(m_{(143)}\otimes_{A}\phi_{M_{Q}\otimes_{A}M_{1}^{-}\rightarrow
M_{Q},1}(m_{(143)}\otimes_{A}m_{1}^{-}))$ $\displaystyle=$
$\displaystyle\phi_{M_{Q}\otimes_{A}M_{Q}\rightarrow
M_{\overline{Q}},\beta}(m_{(143)}\otimes_{A}m_{(143)})$ $\displaystyle=$
$\displaystyle\delta_{\beta,1}m_{(134)}.$

Hence, we get the following 2 linear equations,

$1=\left[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\hat{\sigma}}_{\mathcal{L}_{\overline{Q}}}\right]_{(\mathcal{L}_{\overline{Q}},1,1),(\mathcal{L}_{Q},1,1)},\quad
0=\left[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\hat{\sigma}}_{\mathcal{L}_{\overline{Q}}}\right]_{(\mathcal{L}_{\overline{Q}},2,1),(\mathcal{L}_{Q},1,1)}.$
(3.30)

Similarly, we get another two equations when evaluating on
$m_{(132)}\otimes_{A}m_{(132)}\otimes_{A}m_{1}^{-}$:

$0=\left[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\hat{\sigma}}_{\mathcal{L}_{\overline{Q}}}\right]_{(\mathcal{L}_{\overline{Q}},1,1),(\mathcal{L}_{Q},1,2)},\quad-1=\left[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\hat{\sigma}}_{\mathcal{L}_{\overline{Q}}}\right]_{(\mathcal{L}_{\overline{Q}},2,1),(\mathcal{L}_{Q},1,2)}.$
(3.31)

Solving the four equations, we found,

$\left[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\hat{\sigma}}_{\mathcal{L}_{\overline{Q}}}\right]_{(\mathcal{L}_{\overline{Q}},\mu,1),(\mathcal{L}_{Q},1,\beta)}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}=\sigma^{3}.$ (3.32)

One can solve the rest of the $F$-symbols in a similar way.

Just as the Ising fusion category containing duality line $N$ is determined up
to the FS indicator $\epsilon\in H^{2}(\mathbb{Z}_{2},U(1))$, the fusion
category containing triality defect also has a FS indicator $\alpha\in
H^{3}(\mathbb{Z}_{3},U(1))$. From the $F$-symbol point of view, given a
solution of $F$-symbols for the pentagon equations of the triality fusion
categories, one can generate a new set of solutions [2] by

$\displaystyle
F^{\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}}_{\mathcal{L}_{Q}}\rightarrow
e^{2\pi
m\mathsf{i}/3}F^{\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}}_{\mathcal{L}_{Q}},\quad
F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{\overline{Q}}}\rightarrow
e^{-2\pi
m\mathsf{i}/3}F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{\overline{Q}}},$
(3.33) $\displaystyle
F^{\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{\overline{Q}}}\rightarrow
e^{-2\pi
m\mathsf{i}/3}F^{\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{\overline{Q}}},\quad
F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}\mathcal{L}_{Q}}_{\mathcal{L}_{Q}}\rightarrow
e^{2\pi
m\mathsf{i}/3}F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}\mathcal{L}_{Q}}_{\mathcal{L}_{Q}},$
$\displaystyle
F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}}_{\mathcal{L}_{\overline{Q}}}\rightarrow
e^{2\pi
m\mathsf{i}/3}F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}}_{\mathcal{L}_{\overline{Q}}},\quad
F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}}_{g}\rightarrow
e^{-2\pi
m\mathsf{i}/3}F_{g}^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}}.$

Alternatively, given a triality fusion category, we can construct the triality
fusion category with different FS indicator by stacking the theory with
another theory with anomalous $\mathbb{Z}_{3}$ symmetry $\tilde{\eta}$ and
identify the new triality line as
$\widetilde{\mathcal{L}_{Q}}=\mathcal{L}_{Q}\tilde{\eta}$. If we gauge the
quantum $\mathbb{Z}_{2}$ symmetry, then we would recover the theory with
$\widetilde{A_{4}}$ symmetry but now the $\widetilde{A_{4}}$ symmetry has an
anomaly due to the anomaly of $\mathbb{Z}_{3}$. This implies the triality
fusion category with different FS indicators can be realized by
$\mathbb{Z}_{2}$ gauging of the $A_{4}$ global symmetry with different
anomalies. Indeed, $H^{3}(A_{4},U(1))=\mathbb{Z}_{6}$ and let $\omega_{0}$
denote the generator of $H^{3}(A_{4},U(1))$. The
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ subgroup of $A_{4}$ is anomaly free only
when the anomaly of $A_{4}$ is $\omega_{0}^{2k}$ for $k=0,1,2$ (where we use
the multiplicative notation for $\mathbb{Z}_{6}$), and gauging one of the
$\mathbb{Z}_{2}$ subgroup leads to the triality fusion category with FS
indicator $\alpha=e^{2\pi k\mathsf{i}/3}$. 888Notice that $k=0$ (i.e. the
anomaly of $A_{4}$ is trivial) always leads to the trivial FS indicator
$\alpha=1$. We choose the generator $\omega_{0}$ such that
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2}^{\sigma},1)$ has FS
indicator $\alpha=e^{2\pi\mathsf{i}k/3}$.

Let $\alpha=e^{2\pi\mathsf{i}/3}\in U(1)$, and we list the $F$-symbols below.

$\displaystyle
F^{\mathcal{L}_{Q}g,h}_{\mathcal{L}_{Q}}=F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}g}_{h}=\begin{pmatrix}1&1&1&1\\\
1&1&1&1\\\ 1&-1&1&-1\\\ 1&-1&1&-1\end{pmatrix},\quad
F^{g\mathcal{L}_{Q}h}_{\mathcal{L}_{Q}}=F^{g\mathcal{L}_{\overline{Q}}h}_{\mathcal{L}_{\overline{Q}}}=F^{\mathcal{L}_{\overline{Q}}g\mathcal{L}_{Q}}_{h}=\begin{pmatrix}1&1&1&1\\\
1&1&-1&-1\\\ 1&-1&1&-1\\\ 1&-1&-1&1\end{pmatrix},$ (3.34) $\displaystyle
F^{g,h,\mathcal{L}_{Q}}_{\mathcal{L}_{Q}}=F^{\mathcal{L}_{\overline{Q}},g,h}_{\mathcal{L}_{\overline{Q}}}=\begin{pmatrix}1&1&1&1\\\
1&1&1&1\\\ 1&1&1&1\\\ 1&1&1&1\end{pmatrix},\quad
F^{\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}g}_{h}=F^{g\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}}_{h}=\begin{pmatrix}1&1&1&1\\\
-1&-1&-1&-1\\\ 1&1&1&1\\\ -1&-1&-1&-1\end{pmatrix},$ $\displaystyle
F^{g,h,\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{\overline{Q}}}=\begin{pmatrix}1&1&1&1\\\
1&1&-1&-1\\\ 1&1&1&1\\\
1&1&-1&-1\end{pmatrix},F^{g\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}}_{h}=\begin{pmatrix}1&1&1&1\\\
1&1&-1&-1\\\ 1&1&1&1\\\
-1&-1&1&1\end{pmatrix},F^{\mathcal{L}_{Q}g\mathcal{L}_{\overline{Q}}}_{h}=\begin{pmatrix}1&1&1&1\\\
1&1&-1&-1\\\ 1&-1&1&-1\\\ -1&1&1&-1\end{pmatrix},$
$\displaystyle\left[F^{\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}}_{\mathcal{L}_{Q}}\right]_{g,h}=\frac{\alpha}{2}\begin{pmatrix}1&1&1&-1\\\
-1&-1&1&-1\\\ 1&-1&1&1\\\
-1&1&1&1\end{pmatrix},\quad\left[F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{\overline{Q}}}\right]_{g,h}=\frac{\alpha^{-1}}{2}\begin{pmatrix}1&-1&1&-1\\\
1&-1&-1&1\\\ 1&1&1&1\\\ 1&1&-1&-1\end{pmatrix}.$

The rest of the $F$-symbols are listed in Table 2.

$g$ | $1$ | $\hat{\sigma}$ | $\eta$ | $\eta\hat{\sigma}$
---|---|---|---|---
$[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}g}_{\mathcal{L}_{\overline{Q}}}]_{(\mathcal{L}_{\overline{Q}},\mu,1),(\mathcal{L}_{Q},1,\nu)}$ | $\sigma^{0}$ | $\sigma^{3}$ | $\sigma^{1}$ | $\mathsf{i}\sigma^{2}$
$[F^{\mathcal{L}_{Q}g\mathcal{L}_{Q}}_{\mathcal{L}_{\overline{Q}}}]_{(\mathcal{L}_{Q},1,\mu),(\mathcal{L}_{Q},1,\nu)}$ | $\sigma^{0}$ | $\sigma^{3}$ | $\sigma^{1}$ | $-\mathsf{i}\sigma^{2}$
$[F^{g\mathcal{L}_{Q}\mathcal{L}_{Q}}_{\mathcal{L}_{\overline{Q}}}]_{(\mathcal{L}_{Q},1,\mu),(\mathcal{L}_{\overline{Q}},\nu,1)}$ | $\sigma^{0}$ | $\sigma^{3}$ | $\sigma^{1}$ | $-\sigma^{2}$
$[F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}g}_{\mathcal{L}_{Q}}]_{(\mathcal{L}_{Q},\mu,1),(\mathcal{L}_{\overline{Q}},1,\nu)}$ | $\sigma^{0}$ | $\sigma^{3}$ | $\sigma^{1}$ | $\mathsf{i}\sigma^{2}$
$[F^{\mathcal{L}_{\overline{Q}}g\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{Q}}]_{(\mathcal{L}_{\overline{Q}},1,\mu),(\mathcal{L}_{\overline{Q}},1,\nu)}$ | $\sigma^{0}$ | $\sigma^{3}$ | $\sigma^{1}$ | $\mathsf{i}\sigma^{2}$
$[F^{g\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{Q}}]_{(\mathcal{L}_{\overline{Q}},1,\mu),(\mathcal{L}_{Q},\nu,1)}$ | $\sigma^{0}$ | $\sigma^{3}$ | $\sigma^{1}$ | $-\mathsf{i}\sigma^{2}$
$[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\mathcal{L}_{Q}}_{g}]_{(\mathcal{L}_{\overline{Q}},\mu,1),(\mathcal{L}_{\overline{Q}},\nu,1)}$ | $\sigma^{0}$ | $-\sigma^{3}$ | $\sigma^{1}$ | $\mathsf{i}\sigma^{2}$
$[F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}}_{g}]_{(\mathcal{L}_{Q},\mu,1),(\mathcal{L}_{Q},\nu,1)}$ | $\alpha^{-1}\sigma^{0}$ | $-\alpha^{-1}\sigma^{3}$ | $\alpha^{-1}\sigma^{1}$ | $\mathsf{i}\alpha^{-1}\sigma^{2}$
$[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{Q}}]_{(\mathcal{L}_{Q},\mu,\nu),(g,1,1)}$ | $\frac{1}{\sqrt{2}}\sigma^{0}$ | $\frac{-1}{\sqrt{2}}\sigma^{3}$ | $\frac{1}{\sqrt{2}}\sigma^{1}$ | $\frac{-\mathsf{i}}{\sqrt{2}}\sigma^{2}$
$[F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}}_{\mathcal{L}_{\overline{Q}}}]_{(\mathcal{L}_{Q},\mu,\nu),(g,1,1)}$ | $\frac{\alpha}{\sqrt{2}}\sigma^{0}$ | $\frac{\alpha}{\sqrt{2}}\sigma^{3}$ | $\frac{\alpha}{\sqrt{2}}\sigma^{1}$ | $\frac{-\mathsf{i}\alpha}{\sqrt{2}}\sigma^{2}$
$[F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}\mathcal{L}_{Q}}_{\mathcal{L}_{Q}}]_{(g,1,1),(\mathcal{L}_{\overline{Q}},\mu,\nu)}$ | $\frac{\alpha}{\sqrt{2}}\sigma^{0}$ | $\frac{\alpha}{\sqrt{2}}\sigma^{3}$ | $\frac{\alpha}{\sqrt{2}}\sigma^{1}$ | $\frac{\mathsf{i}\alpha}{\sqrt{2}}\sigma^{2}$
$[F^{\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{\overline{Q}}}]_{(g,1,1),(\mathcal{L}_{Q},\mu,\nu)}$ | $\frac{\alpha^{-1}}{\sqrt{2}}\sigma^{0}$ | $\frac{-\alpha^{-1}}{\sqrt{2}}\sigma^{3}$ | $\frac{\alpha^{-1}}{\sqrt{2}}\sigma^{1}$ | $\frac{-\mathsf{i}\alpha^{-1}}{\sqrt{2}}\sigma^{2}$
Table 2: The $F$-symbols of group theoretical triality fusion categories
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2}^{\sigma},1)$ for fusion
multiplicity $2$ where $\alpha=e^{2\pi k\mathsf{i}/3}$. The $\sigma^{i}$
denotes the Pauli $i$ matrix and $\sigma^{0}$ is the $2\times 2$ identity
matrix.

##  4 Physical implication of the group theoretical triality fusion categories

In this section, we derive the physical implication of the group theoretical
triality fusion categories
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2}^{\sigma},1)$. We first
derive the spin selection rules for the triality defect. Then we show how to
match the states in the Hilbert space which transforms in different
irreducible representations of the fusion category symmetry
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2}^{\sigma},1)$ of the theory
$\mathcal{T}/\mathbb{Z}_{2}^{\sigma}$ with the states in the Hilbert space
$\mathcal{H}$ or the defect Hilbert space $\mathcal{H}_{\sigma}$ of the theory
$\mathcal{T}$. This allows us to derive the asymptotic density of states in
$\mathcal{H}$ transforms in different irreducible representations of
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2}^{\sigma},1)$ by applying the
result in [40]. Finally, we study the anomaly of group theoretical triality
fusion categories by analyzing its symmetric gapped phases and describe the
anomaly free conditions for generic group theoretical fusion categories.

###  4.1 Spin selection rules

We now derive the spin selection rules for using the $F$-symbols computed in
section 3.5.

To do this, we first consider the action of
$g\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ on the defect Hilbert space
$\mathcal{H}_{\mathcal{L}_{Q}}$. Following in the convention in (2.8) and
suppressing the $1,1$ indices for the fusion channel (since the multiplicity
is just $1$ in this case), we denote the operator as
$\hat{g}_{\mathcal{L}_{Q}}$. This is depicted in Figure 6.

Figure 6: Symmetry operator $g\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ acts on
the defect Hilbert space $\mathcal{H}_{\mathcal{L}_{Q}}$, which we denote as
$\hat{g}_{\mathcal{L}_{Q}}$.

Notice that the action of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ on
$\mathcal{H}_{\mathcal{L}_{Q}}$ can be twisted by 2-cocycle $\gamma(g,h)\in
Z^{2}(\mathbb{Z}_{2}\times\mathbb{Z}_{2},U(1))$ such that

$\hat{h}_{\mathcal{L}_{Q}}\cdot\hat{g}_{\mathcal{L}_{Q}}=\gamma(h,g)\widehat{hg}_{\mathcal{L}_{Q}}.$
(4.1)

We can compute $\gamma$ from the $F$-symbols via following configuration shown
in the Figure 7.

Figure 7: The calculation of the product of $\hat{g}_{\mathcal{L}_{Q}}$ and
$\hat{h}_{\mathcal{L}_{Q}}$ using $F$-moves.

Under a sequence of $F$-move, we relate the phase $\gamma(g,h)$ to products of
$F$-symbols as following

$\gamma(h,g)=\left(F^{gh\mathcal{L}_{Q}}_{\mathcal{L}_{Q}}\right)^{-1}\left(F^{g\mathcal{L}_{Q}(gh)}_{\mathcal{L}_{Q}}\right)^{-1}F^{\mathcal{L}_{Q}hg}_{\mathcal{L}_{Q}}F^{g\mathcal{L}_{Q}g}_{\mathcal{L}_{Q}}=\begin{pmatrix}1&1&1&1\\\
1&1&-1&-1\\\ 1&-1&1&-1\\\ 1&-1&-1&1\end{pmatrix}.$ (4.2)

We list the eigenvalues of allowed irreducible representations as the
following

$(1,1,1,-1),\quad(1,1,-1,1),\quad(1,-1,1,1),\quad(1,-1,-1,-1).$ (4.3)

Notice that the 2-cocycle $\gamma$ above is cohomologically trivial in the
group cohomology, which is consistent with the fact that we have $4$
1-dimensional irreducible
representations.999$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ only has a single
2-dimensional irreducible representation when twisted by the cohomologically
non-trivial 2-cocycle. Yet we will see its importance when deriving the spin
selection rule for the intrinsic triality defects in later sections.

Next, to derive the spin selection rules, we consider twisted partition
function $Z_{\mathcal{L}_{Q}}(\tau)$ and apply $T$ modular transformation
three times, see Figure 8. This amounts to inserting $e^{6\pi is}$ in the
trace over the defect Hilbert space $\mathcal{H}_{\mathcal{L}_{Q}}$ where $s$
is the spin of the state.

Figure 8: Applying $T^{3}$ to the twisted partition function
$Z_{\mathcal{L}_{Q}}(\tau)$. This is equivalent to insert $e^{6\pi is}$ in the
trace over $\mathcal{H}_{\mathcal{L}_{Q}}$.

We then apply a sequence of $F$-moves to relate $Z_{\mathcal{L}_{Q}}(\tau+3)$
to the action of the symmetry $g\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ on the
defect Hilbert space $\mathcal{H}_{\mathcal{L}_{Q}}$, as shown in Figure 9. We
find the following relation:

$\displaystyle Z_{\mathcal{L}_{Q}}(\tau+3)$
$\displaystyle=\sum_{\begin{subarray}{c}\mu,\nu=1,2,\\\
g\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}\end{subarray}}\left[F^{\mathcal{L}_{\overline{Q}}\mathcal{L}_{Q}\mathcal{L}_{Q}}_{\mathcal{L}_{Q}}\right]_{(\mathds{1},0,0)(\mathcal{L}_{Q},\mu,\nu)}\left[F^{\mathcal{L}_{Q}\mathcal{L}_{Q}\mathcal{L}_{\overline{Q}}}_{\mathcal{L}_{Q}}\right]_{(\mathcal{L}_{\overline{Q}},\mu,\nu)(g,0,0)}Z_{\mathcal{L}_{Q}g}^{\mathcal{L}_{Q}}(\tau)$
(4.4) $\displaystyle=\alpha
Z^{\mathcal{L}_{Q}}_{\mathcal{L}_{Q}\mathds{1}}(\tau).$

This implies the spin $s$ of the states in defect Hilbert space
$\mathcal{H}_{\mathcal{L}_{Q}}$ satisfies the following relation:

$e^{6\pi\mathsf{i}s}=\alpha,$ (4.5)

which implies,

$e^{2\pi\mathsf{i}s}=\begin{cases}&e^{\frac{2\pi\mathsf{i}k}{3}},\quad
k=0,1,2,\quad\text{when}\quad\alpha=1\\\
&e^{\frac{2\pi\mathsf{i}k}{3}+\frac{2\pi\mathsf{i}}{9}},\quad
k=0,1,2,\quad\text{when}\quad\alpha=e^{2\pi\mathsf{i}/3},\\\
&e^{\frac{2\pi\mathsf{i}k}{3}-\frac{2\pi\mathsf{i}}{9}},\quad
k=0,1,2,\quad\text{when}\quad\alpha=e^{-2\pi\mathsf{i}/3}.\end{cases}$ (4.6)

Now, we provide an alternative derivation of the same spin selection rule by
constructing the twisted partition function from the ungauged theory. Consider
a CFT $\mathcal{T}$ with $A_{4}$ global symmetry with the anomaly
parameterized by $\omega_{0}^{2k}\in H^{3}(A_{4},U(1))\simeq\mathbb{Z}_{6}$
where $k=0,1,2$. The $\mathbb{Z}_{2}^{\sigma}\times\mathbb{Z}_{2}^{\eta}$ is
free of anomaly but the $\mathbb{Z}_{3}$ subgroup generated by $q$ has ’t
Hooft anomaly. As pointed out before, by gauging a $\mathbb{Z}_{2}$ symmetry,
we get the non-intrinsic triality fusion category with FS indicator
$\alpha=e^{2\pi\mathsf{i}k/3}$. By the fusion rule of the triality defect
$\mathcal{L}_{Q}$,

$\mathcal{L}_{Q}\times\mathcal{L}_{Q}\times\mathcal{L}_{Q}=\mathcal{L}_{\overline{Q}}\times\mathcal{L}_{\overline{Q}}\times\mathcal{L}_{\overline{Q}}=2\sum_{g\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}}g.$
(4.7)

In the gauged theory $\mathcal{T}/\mathbb{Z}_{2}$, the twisted sector is odd
under the quantum $\mathbb{Z}_{2}$-symmetry, hence the entire twisted sector
is annihilated by the triality defects $\mathcal{L}_{Q}$ or
$\mathcal{L}_{\overline{Q}}$. Then, we can construct the twisted partition
function $Z^{\mathcal{L}_{Q}}_{1,\mathcal{L}_{\overline{Q}}}(\tau)$ of the
gauged theory from the twisted partition function of the original theory as
follows,

$\displaystyle(Z_{\mathcal{T}/\mathbb{Z}_{2}})^{\mathcal{L}_{Q}}_{1,\mathcal{L}_{\overline{Q}}}(\tau)$
(4.8) $\displaystyle=$
$\displaystyle\text{Tr}_{\mathcal{H}_{\mathcal{T}/\mathbb{Z}_{2}}}(\mathcal{L}_{Q}q^{L_{0}-1/24}\overline{q}^{\overline{L}_{0}-1/24})$
$\displaystyle=$
$\displaystyle\text{Tr}_{\mathcal{H}_{\mathcal{T}/\mathbb{Z}_{2},\text{untwisted}}}(\mathcal{L}_{Q}q^{L_{0}-1/24}\overline{q}^{\overline{L}_{0}-1/24})$
$\displaystyle=$ $\displaystyle\text{Tr}_{\mathcal{H}_{\mathcal{T}}}((q+\sigma
q\sigma)\frac{1+\sigma}{2}q^{L_{0}-1/24}\overline{q}^{\overline{L}_{0}-1/24})$
$\displaystyle=$
$\displaystyle\frac{(Z_{\mathcal{T}})^{q}(\tau)+(Z_{\mathcal{T}})^{\sigma
q}(\tau)+(Z_{\mathcal{T}})^{q\sigma}(\tau)+(Z_{\mathcal{T}})^{\sigma
q\sigma}(\tau)}{2}$ $\displaystyle=$
$\displaystyle(Z_{\mathcal{T}})^{q}(\tau)+(Z_{\mathcal{T}})^{\sigma q}(\tau),$

where we used the above twisted partition function only depending on the
conjugacy class. Applying the $S$-modular transformation on both sides, we
find

$(Z_{\mathcal{T}/\mathbb{Z}_{2}})^{\mathcal{L}_{Q}}_{\mathcal{L}_{Q},1}(\tau)=(Z_{\mathcal{T}})_{q}(\tau)+(Z_{\mathcal{T}})_{\sigma
q}(\tau).$ (4.9)

Then, the spin selection rules of the triality defect $\mathcal{L}_{Q}$ are
the same as the symmetry defect $q$ and $\sigma q$ which generates
$\mathbb{Z}_{3}$ symmetries in $A_{4}$. The spin selection rules of
$\mathbb{Z}_{3}$-symmetry defect has been derived in [44], which takes the
form

$s\in\frac{1}{3}\mathbb{Z}+\frac{k}{9}.$ (4.10)

We then find agreement between (4.6) and (4.10).

Figure 9: Here, we start with the configuration which computes the $e^{6\pi
is}$. Under sequence of $F$-moves, we relate it to the action of
$g\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ on the defect Hilbert space
$\mathcal{H}_{\mathcal{L}_{Q}}$.

###  4.2 Asymptotic density of states

In this subsection, we derive several asymptotic density of states for
different sectors in the Hilbert space $\mathcal{H}$ for a theory with the
group theoretical triality category symmetries.

When acting on the Hilbert space $\mathcal{H}$, the 6 simple lines in the
triality fusion category
$I,\hat{\sigma},\eta,\hat{\sigma}\eta,\mathcal{L}_{Q},\mathcal{L}_{\overline{Q}}$
correspond to 6 operators whose product satisfies the fusion rule (6.1), from
which we learn they all commute with each other, hence can be simultaneously
diagonalized. We enumerate eigenvalues for all $6$ possible 1-dim irreducible
representations as follows,

$\begin{matrix}(1,1,1,1,2,2),&(1,1,1,1,2\omega,2\omega^{2}),&(1,1,1,1,2\omega^{2},2\omega),\\\
(1,1,-1,-1,0,0),&(1,-1,1,-1,0,0),&(1,-1,1,-1,0,0).\end{matrix}$ (4.11)

We now derive a formula of the asymptotic density of states for all the $6$
irreps by relating the above 6 irreps to representations of finite group
symmetries in the ungauged theory and utilizing the result in [40], which we
reviewed in section 2.3. Notice that since the asymptotic density of states
for different irreps of $\mathcal{H}$ only depends on the fusion ring
structure of the fusion category, we can derive the asymptotic density of
states from the simplest case when the fusion category is acquired from
gauging $\mathbb{Z}_{2}^{\sigma}$ subgroup of $A_{4}$ with the trivial
anomaly. Then, by relating the above $6$ irreps to the representation of the
$A_{4}$ in the ungauged theory, we can use the result in [40] to get the
result. Let’s first consider the twisted sector in the gauged theory
$\mathcal{T}/\mathbb{Z}_{2}^{\sigma}$, which is odd under the quantum symmetry
$\hat{\sigma}$. In the ungauged theory $\mathcal{T}$, these states correspond
to the $\mathbb{Z}_{2}^{\sigma}$-even states in the defect Hilbert space
$\mathcal{H}_{\mathcal{T},\sigma}$. Since there’s no mixed ’t Hooft anomaly
between $\mathbb{Z}_{2}^{\sigma}$ and $\mathbb{Z}_{2}^{\eta}$, there are well-
defined $\mathbb{Z}_{2}^{\eta}$ charges for states in the defect Hilbert space
$\mathcal{H}_{\mathcal{T},\sigma}$, which are the $\mathbb{Z}_{2}^{\eta}$
charges for states in the gauged theory.

Next, let’s consider the untwisted sector from the $\mathbb{Z}_{2}^{\sigma}$
gauging. This sector is given by the $\mathbb{Z}_{2}^{\sigma}$ even states in
the ungauged theory with the global $A_{4}$ symmetry. Here, we can use the
fact that the triality line acts on the untwisted sector as $q+\sigma
q\sigma$. We will relate the eigenvalues of $q+\sigma q\sigma$ of the states
in the $\mathbb{Z}_{2}^{\sigma}$-invariant space in each irreducible
representation of $A_{4}$. This group has $4$ irreducible representation,
labelled by the dimension
$\mathbf{1},\mathbf{1}_{A},\mathbf{1}_{B},\mathbf{3}$, where $\mathbf{1}$ is
the trivial irrep and $\mathbf{1}_{A}$ and $\mathbf{1}_{B}$ are two irreps
where $\sigma,\eta$ acts as trivially and $q$ has eigenvalues $\omega$ and
$\omega^{2}$ respectively, which means the states in these two irreps are
invariant under the $\mathbb{Z}_{2}^{\sigma}$ and have eigenvalues $2\omega$
and $2\omega^{2}$ under $\mathcal{L}_{Q}\simeq q+\sigma q\sigma$ respectively.
For the three-dimensional representation $\mathbf{3}$, the representation
matrices are given by

$U_{\mathbf{3}}(\sigma)=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{pmatrix},\quad U_{\mathbf{3}}(\eta)=\begin{pmatrix}-1&0&0\\\
0&1&0\\\ 0&0&-1\end{pmatrix},\quad U_{\mathbf{3}}(q)=\begin{pmatrix}0&1&0\\\
0&0&1\\\ 1&0&0\end{pmatrix}.$ (4.12)

As one can check, $\mathbb{Z}_{2}^{\sigma}$-invariant space is
$1$-dimensional, and the operator $\mathcal{L}_{Q}\simeq q+\sigma q\sigma$
annihilates this state.

We summarize the results in the Tab. 3.

Irrep in $\mathcal{H}_{\mathcal{T}/\mathbb{Z}_{2}^{\sigma}}$ in the gauged theory $\mathcal{T}/\mathbb{Z}_{2}$ | Corresponding states in the ungauged theory $\mathcal{T}$
---|---
$(1,1,1,1,2,2)$ | the irrep $\mathbf{1}$ of $A_{4}$ in $\mathcal{H}_{\mathcal{T}}$
$(1,1,1,1,2\omega,2\omega)$ | the irrep $\mathbf{1}_{A}$ of $A_{4}$ in $\mathcal{H}_{\mathcal{T}}$
$(1,1,1,1,2\omega^{2},2\omega^{2})$ | the irrep $\mathbf{1}_{B}$ of $A_{4}$ in $\mathcal{H}_{\mathcal{T}}$
$(1,1,-1,-1,0,0)$ | the $\mathbb{Z}_{2}^{\sigma}$-even state in the $\mathbf{3}$ of $A_{4}$ in $\mathcal{H}_{\mathcal{T}}$
$(1,-1,1,-1,0,0)$ |  $\mathbb{Z}_{2}^{\eta}$-even, $\mathbb{Z}_{2}^{\sigma}$-even states in $\mathcal{H}_{\mathcal{T},\sigma}$
$(1,-1,-1,1,0,0)$ |  $\mathbb{Z}_{2}^{\eta}$-odd, $\mathbb{Z}_{2}^{\sigma}$-even states in $\mathcal{H}_{\mathcal{T},\sigma}$
Table 3: Relating the states in different irreps of triality fusion category
$\mathcal{C}(A_{4},1,\mathbb{Z}_{2}^{\sigma},1)$ in
$\mathcal{T}/\mathbb{Z}_{2}^{\sigma}$ to states in the ungauged theory
$\mathcal{T}$.

The last $3$ irreps of $\mathcal{C}(A_{4},1,\mathbb{Z}_{2}^{\sigma},1)$
corresponds to the $3$ irreps of
$\mathbb{Z}_{2}^{\hat{\sigma}}\times\mathbb{Z}_{2}^{\eta}$ in the gauged
theory $\mathcal{T}/\mathbb{Z}_{2}^{\sigma}$, therefore, we can directly apply
the result in [40] in the gauged theory and find the asymptotic density of
states to be

$\rho_{0,(1,1,-1,-1,0,0)}(\Delta)=\rho_{0,(1,-1,1,-1,0,0)}(\Delta)=\rho_{0,(1,-1,-1,1,0,0)}(\Delta)=\frac{1}{4}\rho_{0}(\Delta),$
(4.13)

where $\rho_{0}(\Delta)$ is defined in (2.13). To determine the asymptotic
density of states in irreps $(1,1,1,1,2\omega^{k},2\omega^{-k})$ we can simply
use the relation in Table 3 and apply the result in [40]. We then find

$\rho_{0,(1,1,1,1,2\omega^{k},2\omega^{-k})}(\Delta)=\frac{1}{12}\rho_{0}(\Delta),\quad
k=0,1,2.$ (4.14)

###  4.3 Constraints on RG flow

To study the constraints on the RG flow, we want to determine if the fusion
category symmetry $\mathcal{C}$ is anomalous, in the sense that if it
obstructs a $\mathcal{C}$-symmetric trivial gapped phase. 101010Here, by
$\mathcal{C}$-symmetric gapped phase we mean the Hamiltonian or Lagrangian
preserves the $\mathcal{C}$-symmetry while it could be spontaneously broken
etc. By $\mathcal{C}$-symmetric trivial gapped phase, we mean there’s a unique
ground state in this phase. As pointed out in [4], module categories
$\mathcal{M}$ over $\mathcal{C}$ are in bijection with $\mathcal{C}$-symmetric
gapped phases such that the ground states are in bijection with the simple
objects in $\mathcal{M}$. Therefore, to check whether a fusion category
symmetry $\mathcal{C}$ has a trivial gapped phase is to check if it has a
module category $\mathcal{M}$ with a single simple object. Equivalently, one
can check if $\mathcal{C}$ admits a fiber functor.

In general, this is not an easy problem. For the case of the group-theoretic
fusion category, this is relatively easy because all the indecomposable module
categories over $\mathcal{C}(G,\omega,H,\psi)$ can be explicitly constructed
as pointed out in [51]. In fact, the indecomposable module categories over
$\mathcal{C}(G,\omega,H,\psi)$ and the indecomposable module categories over
$\operatorname{Vec}_{G}^{\omega}$ are in a canonical bijection. However, as we
will see in a concrete example below that this bijection does not preserve the
number of simple elements in the module category.111111The authors thank the
referee for pointing out a mistake in our draft which leads to this
discussion. Hence, in general, the existence of a module category with a
single simple object over $\operatorname{Vec}_{G}$ does not imply the
existence of the a module category with a single simple object over
$\mathcal{C}(G,1,H,\psi)$. Therefore, the fact that $\operatorname{Vec}_{G}$
has trivial anomaly does not necessarily imply the symmetry
$\mathcal{C}(G,1,H,\psi)$ has a symmetric trivial gapped phase.

Physically, this bijection between symmetric gapped phases of
$\mathcal{C}(G,\omega,H,\psi)$ and $\operatorname{Vec}_{G}^{\omega}$ can be
seen as follows. Consider a theory $\mathcal{T}$ with global symmetry
$\operatorname{Vec}_{G}^{\omega}$ with a relevant operator $O(x)$ preserving
the $\operatorname{Vec}_{G}^{\omega}$-symmetry driving the $\mathcal{T}$ to
some gapped phase $A$. Since $A$ is
$\operatorname{Vec}_{G}^{\omega}$-symmetric, we can consider gauge the anomaly
free subgroup $H$ with discrete torsion $\psi$ to get a
$\mathcal{C}(G,\omega,H,\psi)$-symmetric gapped phase denoted as $A/H_{\psi}$.
On the other hand, since the operator $O(x)$ preserve the entire $G$-symmetry,
it will survive the $H_{\psi}$-gauging and remain a local operator in the
theory $\mathcal{T}/H_{\psi}$ which we get from gauging $H_{\psi}$ in the
theory $\mathcal{T}$. $O(x)$ will trigger the RG flow in the theory
$\mathcal{T}/H_{\psi}$ and the theory would flow to the
$\mathcal{C}(G,\omega,H,\psi)$-symmetric gapped phase $A/H_{\psi}$. In this
way, we construct a map from the symmetric gapped phases of the theories with
$\operatorname{Vec}_{G}^{\omega}$-symmetry to the symmetric gapped phases of
the theories with $\mathcal{C}(G,\omega,H,\psi)$-symmetry. Similar, we can
construct the inverse map by starting with the theories with
$\mathcal{C}(G,\omega,H,\psi)$-symmetry and gauging
$\operatorname{Rep}(H_{\psi})$ symmetry. The process is summarized as follows:

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}\pgfsys<EMAIL_ADDRESS>(4.15)

All the indecomposable module categories over $\mathcal{C}(G,\omega,H,\psi)$
has been explicitly constructed in [51]. It is parameterized by an anomaly
free subgroup $K$ of $G$ (that is, $\omega|_{K\times K\times K}=1$) together
with a 2-cocycle $\psi_{K}$ and we denote the corresponding module category as
$\mathcal{M}(K,\psi_{K})$ and the simple objects in $\mathcal{M}(K,\psi_{K})$
are parameterized by indecomposable $A(H,\psi)-A(K,\psi_{K})$
bimodules121212It is clear from this that the indecomposable module categories
over $\mathcal{C}(G,\omega,H,\psi)$ for different $(H,\psi)$ are in canonical
bijection as they are parameterized by the same pairs $(K,\psi_{K})$. It is
also clear that in general, the number of simple objects does not have be the
same under the bijection., which can be computed using similar techniques as
in the section 3.3. Specifically, the simple objects are parameterized by a
double coset of the form $HgK$ for $g\in G$ and an irreducible projective
representation $\rho$ of the little group $H^{g}:=\\{(h,k)\in H\times
K:hgk=g\\}$ twisted by some effective 2-cocycle $\psi_{g}$.

Now, we are ready to show that the group theoretical triality fusion
categories $\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2}^{\sigma},1)$ do
not admit a trivial symmetric gapped phases therefore is anomalous. To see
this, we only need to show for every possible indecomposable module category
contains more than $1$ simple objects. First, we must choose $K$ such that
there is only a single double coset of the form $\mathbb{Z}_{2}^{\sigma}gK$
and the only possible choices are $K=A_{4}$ as the only subgroup of $A_{4}$
has order $\geq 6$ is $A_{4}$ itself. Notice that $K=A_{4}$ must be anomaly
free which is not true for $k=1,2$, therefore, we immediately conclude that
$\mathcal{C}(A_{4},\omega_{0}^{2k},\mathbb{Z}_{2}^{\sigma},1)$ for $k=1,2$ are
anomalous. For $k=0$, we consider taking the only double coset
$\mathbb{Z}_{2}^{\sigma}\mathds{1}A_{4}$ and the little group $H^{\mathds{1}}$
is $\mathbb{Z}_{2}\simeq\\{(1,1),(\sigma,\sigma)\\}$. However, since
$H^{2}(\mathbb{Z}_{2},U(1))=\mathbb{Z}_{1}$ meaning the effective 2-cocycle
$\psi_{\mathds{1}}$ has to be trivial, there has to be two irreducible
representations for the little group. This means there are two simple objects
in this module category and the ground states are 2-fold degenerate. Hence, we
conclude that $\mathcal{C}(A_{4},1,\mathbb{Z}_{2}^{\sigma},1)$ is also
anomalous.

It is straight forward to generalize the above discussion [51]. The symmetric
trivial gapped phases of the group theoretical fusion category
$\mathcal{C}(G,\omega,H,\psi)$ are parameterized by $(K,\psi_{K})$ such that

  1. 1.

$K$ is an anomaly free subgroup of $G$;

  2. 2.

There is only a single double coset of the form $HgK$;

  3. 3.

The 2-cocycle $\psi_{\mathds{1}}=(\psi|_{H\cap K})(\psi_{K}|_{H\cap K})^{-1}$
is non-degenerate which implies there’s only a single irreducible projective
representation for the little group $H^{\mathds{1}}\simeq H\cap K$.

The existence of the pair $(K,\psi_{K})$ satisfying the above three conditions
can be viewed as the anomaly free condition for the group theoretical fusion
category $\mathcal{C}(G,\omega,H,\psi)$.

##  5 Example: $c=1$ Compact boson at Kosterlitz-Thouless point

In this section, we consider the example of $c=1$ compact boson at the
Kosterlitz-Thouless (KT) point and compute the twisted partition functions of
triality defect in the $c=1$ compact boson at the KT point. We match the spin
selection rule and also show one can not construct a new triality fusion
category by combining the triality defect $\mathcal{L}_{Q}$ with another
generator $\eta$ of the $\mathbb{Z}_{3}$ symmetry in the KT theory.

###  5.1 A lightning review of $c=1$ compact boson and the triality defect

We first briefly review the $c=1$ compact boson following the convention in
[1]. The theory is described by a scalar field $X$ with period $2\pi R$,

$X\simeq X+2\pi R.$ (5.1)

It is convenient to define $2\pi$-periodic field $\theta$ and $2\pi$-periodic
conjugate momentum $\phi$ and introduce the left and right moving fields
$X_{L,R}$,

$\theta=R^{-1}(X_{L}+X_{R}),\quad\phi=R(X_{L}-X_{R})/2.$ (5.2)

The global symmetry at a generic radius $R$ is

$G_{bos}=(U(1)^{\theta}\times U(1)^{\phi})\rtimes\mathbb{Z}_{2}^{C},$ (5.3)

where $U(1)^{\theta}$ and $U(1)^{\phi}$ are the shifting symmetry of $\theta$
and $\phi$ respectively, and the charge conjugation $C$ flips the sign of
$\theta$ and $\phi$ simultaneously.

At a generic radius $R$, the primary local operators in this theory contain
vertex operators,

$V_{n,w}=e^{i\left(\frac{n}{R}+\frac{wR}{2}\right)X_{L}}e^{i\left(\frac{n}{R}-\frac{wR}{2}\right)X_{R}}=e^{in\theta}e^{iw\phi},\quad
n,w\in\mathbb{Z}$ (5.4)

with the scaling dimension,

$(h,\overline{h})=\left(\frac{1}{2}\left(\frac{n}{R}+\frac{wR}{2}\right)^{2},\frac{1}{2}\left(\frac{n}{R}-\frac{wR}{2}\right)^{2}\right),$
(5.5)

together with the normal ordered Schur symmetric polynomials in the $U(1)$
currents $j_{1}=\partial X_{L}$ and
$\overline{j}_{1}=\overline{\partial}X_{R}$ and their derivatives, denoted as,

$j_{n^{2}}\overline{j}_{m^{2}}\quad(h,\overline{h})=(n^{2},m^{2}).$ (5.6)

The spectrum can also be seen from the partition function,

$Z(R)=\frac{1}{|\eta(\tau)|^{2}}\sum_{n,w\in\mathbb{Z}}q^{\frac{1}{2}\left(\frac{n}{R}+\frac{wR}{2}\right)^{2}}\overline{q}^{\frac{1}{2}\left(\frac{n}{R}-\frac{wR}{2}\right)^{2}}.$
(5.7)

For $c=1$ CFT, there are null states in the descendent states when the
Virasoro primary state has scaling dimension $h=\frac{n^{2}}{4}$ for
$n\in\mathbb{Z}$. For a generic $h$, there is no null states in the descendent
states of a Virasoro primary state and the Virasoro character is given by,

$\chi_{h}(\tau)=\frac{q^{h}}{\eta(\tau)}.$ (5.8)

For the primary state with $h=\frac{n^{2}}{4}$ with $n\in\mathbb{Z}$, because
of the null states, its character takes the form,

$\chi_{h=\left(\frac{n}{2}\right)^{2}}(\tau)=\frac{q^{\left(\frac{n}{2}\right)^{2}}-q^{\left(\frac{n}{2}+1\right)^{2}}}{\eta(\tau)}.$
(5.9)

At a generic point of the moduli space, terms with $n\neq 0$ or $m\neq 0$
correspond to characters with primaries $V_{n,m}$’s containing no null states.
However, the term $\frac{1}{\eta(\tau)\overline{\eta}(\overline{\tau})}$ with
$n=m=0$ cannot be a character of Virasoro primary (the identity operator) due
to the appearance of null states, but should correspond to the sum of
characters of primary states and can be seen via the following rewriting:

$\frac{1}{\eta(\tau)\overline{\eta}(\overline{\tau})}=\frac{1}{\eta(\tau)\overline{\eta}(\overline{\tau})}\sum_{n,m=0}^{\infty}(q^{n^{2}}-q^{(n+1)^{2}})(\overline{q}^{m^{2}}-\overline{q}^{(m+1)^{2}}),$
(5.10)

where each term in the sum is a character for the primary operator with
scaling dimension $(h,\overline{h})=(n^{2},m^{2})$, corresponding to the
primary operator $j_{n^{2}}\overline{j}_{m^{2}}$ mentioned above.

It is worth mentioning at the special radius $R=\sqrt{2}$, the theory becomes
$SU(2)_{1}$, and the global symmetry is enhanced to
$SO(4)=\frac{SU(2)_{L}\times SU(2)_{R}}{\mathbb{Z}_{2}}$. We can represent
this $SO(4)$ in its vector representation, where the basis is given by 4
operators $(\sin\theta,\cos\theta,\sin\phi,\cos\phi)$ [1]. The charge
conjugation is represented as,

$C=\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&1\end{pmatrix}.$
(5.11)

Similarly, the $U(1)_{\theta}$ and $U(1)_{\phi}$ can be represented as,

$R_{\theta}(\alpha)=\begin{pmatrix}\cos\alpha&-\sin\alpha&0&0\\\
\sin\alpha&\cos\alpha&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix},\quad
R_{\phi}(\alpha)=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos\alpha&-\sin\alpha\\\ 0&0&\sin\alpha&\cos\alpha\end{pmatrix}.$ (5.12)

The spectrum of primary operators of the $SU(2)_{1}$ theory can be derived by
decomposing the partition function in terms of characters (5.9) of irreducible
representations of the Virasoro algebra [69, 70], and the details are
presented in the Appendix A. The $SU(2)_{1}$ Hilbert space decomposes as,

$\mathcal{H}_{SU(2)_{1}}=\bigoplus_{\begin{subarray}{c}j,\overline{j}\in\frac{1}{2}\mathbb{Z}_{\geq
0},\\\ j+\overline{j}\in\mathbb{Z}\end{subarray}}V_{j}\otimes
V_{\overline{j}}\otimes H^{Vir}_{j^{2}}\otimes
H^{\overline{Vir}}_{\overline{j}^{2}},$ (5.13)

where by $V_{j}$ ($\overline{V}_{\overline{j}}$) we denote the
spin-$j$(spin-$\overline{j}$) representation of $SU(2)_{L}$ ($SU(2)_{R}$) and
by $H^{Vir}_{j^{2}}$ we denote the Virasoro representation with $h=j^{2}$.
Notice that here $j$ and $\overline{j}$ label the irrep of $SU(2)_{L}$ and
$SU(2)_{R}$ symmetry rather than the affine $SU(2)_{L}$ or $SU(2)_{R}$,
therefore the affine cut-off of $j$ or $\overline{j}$ is not at presence. It
is clear from this decomposition that how the $\frac{SU(2)_{L}\times
SU(2)_{R}}{\mathbb{Z}_{2}}$ acts on the $\mathcal{H}_{SU(2)_{1}}$.

The $\mathbb{Z}_{2}^{C}$ symmetry is free of anomaly, so one could consider
gauging it. The resulting theories are a class of theories also parameterized
by the radius $R$ of the compact boson, and we call the resulting theories the
orbifold branch. The spectrum of Virasoro primaries on the $c=1$ orbifold
branch consists of two sectors, the untwisted sector which contains
$\mathbb{Z}_{2}^{C}$ invariant operators of the corresponding compact boson
theory, and the twisted sector which contains the $\mathbb{Z}_{2}^{C}$
invariant non-local operators ending on the $C$ defect line in the compact
boson theory.

The $\mathbb{Z}_{2}^{C}$-invariant twisted sector is constructed by acting on
the two ground states $\left|\frac{1}{16},\frac{1}{16}\right\rangle_{i}$
$i=1,2$ with even powers of the operators $\alpha_{-n}$ and
$\overline{\alpha}_{-n^{\prime}}$ (where now
$n,n^{\prime}\in\frac{1}{2}+\mathbb{Z}_{\geq 0}$) appearing in the mode
expansion of the compact boson $\phi$ with twisted boundary condition [71],

$\mathcal{H}_{\text{orbifold,
twisted}}=\left\\{\alpha_{-n_{1}}\cdots\alpha_{-n_{l}}\overline{\alpha}_{-n_{l+1}}\cdots\overline{\alpha}_{-n_{2k}}\left|\frac{1}{16},\frac{1}{16}\right\rangle_{j}:n_{i}\in\frac{1}{2}+\mathbb{Z}_{\geq
0}\right\\}.$ (5.14)

The two ground states $\left|\frac{1}{16},\frac{1}{16}\right\rangle_{i}$ are
denoted as $\sigma_{i}$ where $i=1,2$.131313We abuse the notation slightly
here. These $\sigma_{i}$ should be distinguished from $\sigma$ appear in the
previous section, which denotes an element of the $A_{4}$ group. The readers
should be able to distinguish the two based on context. The first two excited
states are primary states given by
$\alpha_{-\frac{1}{2}}\overline{\alpha}_{-\frac{1}{2}}\left|\frac{1}{16},\frac{1}{16}\right\rangle_{i}$
which both have scaling dimensions $(\frac{9}{16},\frac{9}{16})$ and we will
denote the two as $\tau_{i}$ where $i=1,2$.

The untwisted sector contains,

$V_{n,w}^{+}=\frac{V_{n,w}+V_{-n,-w}}{\sqrt{2}}$ (5.15)

which are invariant under the $\mathbb{Z}_{2}^{C}$, as well as the
$\mathbb{Z}_{2}^{C}$ invariant normal-ordered Schur polynomials which are
given by,

$j_{n^{2}}j_{m^{2}},\quad\text{with}\quad m-n\in 2\mathbb{Z}.$ (5.16)

At a generic point of the orbifold branch, there is a $D_{8}=\langle
s,r|s^{2}=r^{4}=(rs)^{2}=1\rangle$ global symmetry, acting on the untwisted
sector as,

$r:(\theta,\phi)\rightarrow(\theta+\pi,\phi+\pi),\quad
s:(\theta,\phi)\rightarrow(\theta,\phi+\pi).$ (5.17)

For the operators in the twisted sector, $D_{8}$ acts as follows. The
generator $s$ exchanges two ground states $\sigma_{i}$ while $r$ acts as

$r:(\sigma_{1},\sigma_{2})\mapsto(i\sigma_{1},-i\sigma_{2})$ (5.18)

and

$r:\alpha_{-n}\mapsto-\alpha_{-n},\quad
r:\overline{\alpha}_{-m}\mapsto-\overline{\alpha}_{-m},\quad
n,m\in\mathbb{Z}_{\geq 0}+\frac{1}{2}.$ (5.19)

This implies $r:(\tau_{1},\tau_{2})\mapsto(i\tau_{1},-i\tau_{2})$. There are
two important $D_{4}$ subgroups of $D_{8}$:

$D_{4}^{A}\equiv\langle r^{2},s\rangle,\quad D_{4}^{B}\equiv\langle
r^{2},sr\rangle.$ (5.20)

An important result we will use later to determine the action of the triality
line $\mathcal{L}_{Q}$ on the twisted sector $\mathcal{H}_{KT,\text{twisted}}$
is that the action of $r^{2}$ acts as $-1$ on the entire twisted sector, which
can be seen from (5.14)(5.18)(5.19). Furthermore, since $r^{2}$ acts trivially
on the entire untwisted sector and acts as $-1$ on the entire twisted sector,
we identify as the generator $\hat{C}$ of the quantum $\mathbb{Z}_{2}$
symmetry from the $\mathbb{Z}_{2}^{C}$ gauging.

###  5.2 Spectrum of triality defect and twisted partition functions on torus

The KT theory can be acquired by gauging the $\mathbb{Z}_{2}^{C}$ symmetry of
the $SU(2)_{1}$ theory, which locates at the intersection point between the
circle branch and the orbifold branch. And in [1], the triality defect
$\mathcal{L}_{Q}$ has been identified with the element
$Q\in\frac{SU(2)_{L}\times SU(2)_{R}}{\mathbb{Z}_{2}}$ global symmetries of
the $SU(2)_{1}$ theory. In the representation of $SO(4)$ we used above, $Q$
can be represented as

$Q=\left(\begin{array}[]{cccc}0&1&0&0\\\ 0&0&-1&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
\end{array}\right).$ (5.21)

However, the symmetry operator $Q$ does not commute with $C$, therefore, in
the gauged theory, the reminiscent of the symmetry operator $Q$ is given by
the triality line $\mathcal{L}_{Q}$, related to $Q$ as

$\mathcal{L}_{Q}=Q+CQC,$ (5.22)

with the fusion rule [1]

$\mathcal{L}_{Q}\times\mathcal{L}_{Q}=2\mathcal{L}_{\overline{Q}},\quad\mathcal{L}_{Q}\times\mathcal{L}_{\overline{Q}}=\sum_{g\in
D_{4B}}g.$ (5.23)

As one can see, the charge conjugation $C$ corresponds to $\sigma\in A_{4}$
and $Q$ corresponds to $q\in A_{4}$ discussed previously and as one can check
using the matrix representation above the minimal subgroup of $SO(4)$
containing $C$ and $Q$ is indeed $A_{4}$.

From the above fusion rule and the irreducible representations of
$D_{4}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ given by

$(1,1,1,1),\quad(1,1,-1,-1),\quad(1,-1,1,-1)\quad(1,-1,-1,1),$ (5.24)

we find the action of $\mathcal{L}_{Q}$ on a state $|\psi\rangle$ in the KT
theory are non-trivial only if $|\psi\rangle$ transforms in the trivial
representation of $D_{4}$.

Now, we determine the action of the triality line $\mathcal{L}_{Q}$ on the
states in the KT theory. For the states in twisted sector, all twisted sector
states transform non-trivially under $\mathbb{Z}_{2}=\langle r^{2}\rangle$,
thus non-trivially under $D_{4B}$ as well. Hence, by the fusion rule (5.23),
the triality operator $\mathcal{L}_{Q}$ must annihilate all the states in the
twisted sector.

The action of $\mathcal{L}_{Q}$ on the untwisted sector can be determined by
(5.22). We simply need to construct the representation matrices of $C$ and $Q$
for $(j,\overline{j})$ irreducible representation of $SO(4)$ global symmetry
in the $SU(2)_{1}$ theory. And the action of $\mathcal{L}_{Q}$ on the
untwisted sector is simply given by $Q+CQC$ restricted on the
$\mathbb{Z}_{2}^{C}$-invariant sector of each $(j,\overline{j})$ irreducible
representation of $SO(4)$.

Knowing the action of $\mathcal{L}_{Q}$ on the KT theory Hilbert space
$\mathcal{H}_{KT}$ allows us to compute the twisted partition function
$(Z_{KT})^{\mathcal{L}_{Q}}$. Since $\mathcal{L}_{Q}$ annihilates the twisted
sector, the twisted partition function
$(Z_{KT})^{\mathcal{L}_{Q}}=\text{Tr}_{\mathcal{H}_{KT}}(\mathcal{L}_{Q}q^{L_{0}-1/24}\overline{q}^{\overline{L}_{0}-1/24})$
can be reduced to the untwisted sector and expressed as the following sum of
the twisted partition function of the $SU(2)_{1}$ theory,

$\displaystyle(Z_{KT})^{\mathcal{L}_{Q}}$
$\displaystyle=\text{Tr}_{\mathcal{H}_{KT}}(\mathcal{L}_{Q}q^{L_{0}-1/24}\overline{q}^{\overline{L}-1/24})$
(5.25)
$\displaystyle=\text{Tr}_{\mathcal{H}_{KT,\text{untwisted}}}(\mathcal{L}_{Q}q^{L_{0}-1/24}\overline{q}^{\overline{L}-1/24})$
$\displaystyle=\text{Tr}_{\mathcal{H}_{SU(2)_{1}}}((Q+CQC)\frac{1+C}{2}q^{L_{0}-1/24}\overline{q}^{\overline{L}_{0}-1/24})$
$\displaystyle=\frac{(Z_{SU(2)_{1}})^{Q}+(Z_{SU(2)})^{CQ}+(Z_{SU(2)_{1}})^{QC}+(Z_{SU(2)_{1}})^{CQC})}{2}.$

To evaluate the twisted partition function in $SU(2)_{1}$ theories, we must
first rewrite the partition function in terms of irreps of
$\frac{SU(2)_{L}\times SU(2)_{R}}{\mathbb{Z}_{2}}$ global symmetries, as in
[69, 70]. As shown in the Appendix A, this is given by,

$Z_{SU(2)_{1}}(\tau,\overline{\tau})=\sum_{\begin{subarray}{c}j,\overline{j}\in\frac{1}{2}\mathbb{Z}_{\geq
0},\\\
j+\overline{j}\in\mathbb{Z}\end{subarray}}(2j+1)(2\overline{j}+1)\chi_{j^{2}}(\tau)\overline{\chi}_{\overline{j}^{2}}(\overline{\tau}).$
(5.26)

Since TDL commutes with the stress energy tensor $T(z)$ and
$\overline{T}(\overline{z})$, we only need to study its action on the
$V_{j}\otimes\overline{V}_{\overline{j}}$. For this purpose, we can represent
the group element $Q,C\in SO(4)=\frac{SU(2)_{L}\times SU(2)_{R}}{2}$ as the
tensor product of representations of $SU(2)_{L}$ and $SU(2)_{R}$, that is,

$Q=Q_{L}\otimes Q_{R},\quad C=C_{L}\otimes C_{R},$ (5.27)

where $Q_{L},C_{L}$ are matrices of a spin-$j$ representation of $SU(2)_{L}$
and $Q_{R},C_{R}$ are matrices of a spin-$\overline{j}$ representation of
$SU(2)_{R}$. This allows us to compute trace easily since the generic form of
the character of $SU(2)$ is well-known. Following the calculation in Appendix
B, we find,

$\displaystyle
Tr_{V_{j}\otimes\overline{V}_{\overline{j}}}Q=Tr_{V_{j}\otimes\overline{V}_{\overline{j}}}(CQC)=(Tr_{V_{j}}Q_{L})(Tr_{\overline{V}_{\overline{j}}}Q_{R})=\frac{\sin((2j+1)\pi/3)}{\sin(\pi/3)}\frac{\sin((2\overline{j}+1)\pi/3)}{\sin(\pi/3)},$
(5.28) $\displaystyle
Tr_{V_{j}\otimes\overline{V}_{\overline{j}}}CQ=Tr_{V_{j}\otimes\overline{V}_{\overline{j}}}(QC)=(Tr_{V_{j}}Q_{L}C_{L})(Tr_{\overline{V}_{\overline{j}}}Q_{R}C_{R})$
$\displaystyle=$
$\displaystyle\frac{\sin((2j+1)2\pi/3)}{\sin(2\pi/3)}\frac{\sin((2\overline{j}+1)2\pi/3)}{\sin(2\pi/3)}$
$\displaystyle=$
$\displaystyle\frac{\sin((2j+1)\pi/3)}{\sin(\pi/3)}\frac{\sin((2\overline{j}+1)\pi/3)}{\sin(\pi/3)},\quad\text{for}\quad(j,\overline{j})\in(\mathbb{Z}_{\geq
0}\oplus\mathbb{Z}_{\geq 0})\cup\left((\frac{1}{2}+\mathbb{Z}_{\geq
0})\oplus(\frac{1}{2}+\mathbb{Z}_{\geq 0})\right).$

The twisted partition function is therefore given by,

$(Z_{KT})^{\mathcal{L}_{Q}}=\frac{8}{3|\eta(\tau)|^{2}}\sum_{\begin{subarray}{c}j,\overline{j}\in\frac{1}{2}\mathbb{Z}_{\geq
0},\\\
j+\overline{j}\in\mathbb{Z}\end{subarray}}\sin(\frac{\pi(2j+1)}{3})\sin(\frac{\pi(2\overline{j}+1)}{3})(q^{j^{2}}-q^{(j+1)^{2}})(\overline{q}^{\overline{j}^{2}}-\overline{q}^{(\overline{j}+1)^{2}}).$
(5.29)

We can then rewrite the partition function over the familiar sum over the
Narain lattice,

$(Z_{KT})^{\mathcal{L}_{Q}}(\tau,\overline{\tau})=\frac{1}{|\eta(\tau)|^{2}}\sum_{n,w\in\mathbb{Z}}\bigg{(}\cos(\frac{2\pi
n}{3})+\cos(\frac{2\pi
w}{3})\bigg{)}q^{(\frac{n+w}{2})^{2}}\overline{q}^{(\frac{n-w}{2})^{2}}.$
(5.30)

Using the $S$-modular transformation, we find,

$(Z_{KT})_{\mathcal{L}_{Q}}(\tau,\overline{\tau})=\frac{1}{|\eta(\tau)|^{2}}\sum_{n,w\in\mathbb{Z}}q^{\frac{(n+w+\frac{1}{3})^{2}}{4}}\overline{q}^{\frac{(n-w+\frac{1}{3})^{2}}{4}}+q^{\frac{(n+w-\frac{1}{3})^{2}}{4}}\overline{q}^{\frac{(n-w+\frac{1}{3})^{2}}{4}}.$
(5.31)

This twisted partition function computes the states in the defect Hilbert
space $\mathcal{H}_{KT,\mathcal{L}_{Q}}$ and is consistent as it has integer
coefficients in the $q$ and $\overline{q}$ expansion.

Then applying $T$-transformation, we find

$\displaystyle(Z_{KT})_{\mathcal{L}_{Q},\mathcal{L}_{\overline{Q}}}^{I}(\tau)$
(5.32) $\displaystyle=$ $\displaystyle(Z_{KT})_{\mathcal{L}_{Q}}(\tau+1)$
$\displaystyle=$
$\displaystyle\frac{1}{|\eta(\tau)|^{2}}\sum_{n,w\in\mathbb{Z}}e^{\frac{2\pi
iw}{3}}q^{\frac{(n+w+\frac{1}{3})^{2}}{4}}\overline{q}^{\frac{(n-w+\frac{1}{3})^{2}}{4}}+e^{\frac{-2\pi
in}{3}}q^{\frac{(n+w-\frac{1}{3})^{2}}{4}}\overline{q}^{\frac{(n-w+\frac{1}{3})^{2}}{4}}.$

This twisted partition function computes the spin of the twisted Hilbert space
$\mathcal{H}_{KT,\mathcal{L}_{Q}}$, which is given by the phase in front of
the $q$ of $\overline{q}$ expansion. And the result is consistent with the
spin selection rule derived in (4.6) for the case where the FS indicator
$\alpha=1$.

Next, we move to compute the twisted partition function of
$\mathcal{L}_{\overline{Q}}$. By the fusion rule
$\mathcal{L}_{\overline{Q}}=\mathcal{L}_{Q}\times\mathcal{L}_{Q}$,