# Composing parafermions: a construction of $Z_{N}$ fractional quantum Hall
systems and a modern understanding of confinement and duality

Yoshiki Fukusumi  Division of Physics and Applied Physics, Nanyang
Technological University, Singapore 637371.

###### Abstract

In this work, we propose a modern view of the integer spin simple currents
which have played a central role in discrete torsion. We reintroduce them as
nonanomalous composite particles constructed from $Z_{N}$ parafermionic field
theories. These composite particles have an analogy with the Cooper pair in
the Bardeen-Cooper-Schrieffer theory and can be interpreted as a typical
example of anyon condensation. Based on these $Z_{N}$ anomaly free composite
particles, we propose a systematic construction of the cylinder partition
function of $Z_{N}$ fractional quantum Hall effects (FQHEs). One can expect
realizations of a class of general topological ordered systems by breaking the
bulk-edge correspondence of the bosonic parts of these FQH models. We also
give a brief overview of various phenomena in contemporary condensed matter
physics, such as $SU(N)$ Haldane conjecture, general gapless and gapped
topological order with respect to the quantum anomaly defined by charges of
these simple currents and bulk and boundary renormalization group flow.
Moreover, we point out an analogy between these FQHEs and 2d quantum gravities
coupled to matter, and propose a $Z_{N}$ generalization of supersymmetry known
as ”fractional supersymmetry” in the composite parafermionic theory and study
its analogy with quark confinement. Our analysis gives a simple but general
understanding of the contemporary physics of topological phases in the form of
the partition functions derived from the operator formalism.

###### pacs:

73.43.Lp, 71.10.Pm

##  I Introduction

Recently, a general construction for fractional quantum Hall (FQH) states is
proposed by Schoutens and Wen in Schoutens and Wen (2016), by using the
$Z_{N}$ simple currents or $Z_{N}$ parafermions in a typical caseFateev and
Zamolodchikov (1985). Their construction is quite simple and universal, just
by assuming the existence of the $Z_{N}$ simple currents and imposing single-
valuedness of the wavefunctions for the FQH systems. Hence, in principle, one
can construct an arbitrary FQH wavefunction started from any conformal field
theory (CFT) with $Z_{N}$ simple currents. Especially for $SU(N)_{K}$ Wess-
Zumino-Witten (WZW) models, realizations of such $Z_{N}$ FQH states have been
proposed in Fuji and Lecheminant (2017) by using the coupled wire
constructionKane et al. (2002); Teo and Kane (2014); Sagi and Oreg (2014);
Imamura et al. (2019). Therefore, one can say that the existence of the bulk
wavefunction of $Z_{N}$ FQHE has been established. However, only with their
constructions, there still exist at least two difficulties to obtain the
candidates of protected edge modes which can define a characteristic property
of the topological phase of matterLevin (2013). The first is the difficulty to
obtain multipoint correlation functions of CFT which corresponds to FQHE
wavefunctions. Without some particular structure of the CFT, the calculation
for the multipoint correlation functions needs analysis of higher-order
ordinal differential equations or corresponding Dotsenko-Fateev
integralDotsenko and Fateev (1984, 1985); Estienne and Santachiara (2009);
Estienne et al. (2010, 2012). The second is the difficulty to construct the
braiding and modular property of CFTs which result in the Lagrangian
subalgebra Levin (2013) of the CFTs in the cylinder geometry. In this work, we
propose a systematic way to overcome the second difficulty by considering the
simple charges of the $Z_{N}$ simple currents in CFT which is a generalization
of Lieb-Schultz-Mattis(LSM) type anomalyKong and Wen (2014); Cho et al.
(2017); Lanzetta and Fidkowski (2022); Li et al. (2022).

The simple charge of an operator in conformal field theoryFuchs (1997);
Schweigert et al. (2000), which corresponds to the symmetry charge of the
corresponding operator in the lattice model, is one of the most important
structures to study renormalization group (RG) flow under symmetry and gauging
finite group symmetryBhardwaj and Tachikawa (2018); Tachikawa (2020). At first
sight, these two operations, analyzing the RG flow and gauging symmetry seem
different. However, as can be seen in the development of conformal field
theory and Haldane conjectureHaldane (1983a, b); Cabra et al. (1998);
Lecheminant (2015); Lajkó et al. (2017); Wamer et al. (2020); Yao and Furusaki
(2020), these physical and mathematical structures should have some common
underlying structures. For example, $SU(2)_{K}$ WZW model, where $K$ is the
level of the model corresponding to the spin $K/2$ in the spin chain, is
represented as the composition of $Z_{K}$ parafermion and $U(1)$ theory,
$Z_{K}\times U(1)$. Hence by gapping the parafermion part, one can obtain
$U(1)$ theory but the anomaly of $Z_{2}$ symmetry in the $SU(2)_{K}$ model
becomes LSM type anomaly. In the lattice model, this results in the
connectivity problem to $SU(2)_{1}$ point for higher spin $K/2$ Heisenberg XXZ
chain and the structure $K\ (\text{mod.}2)$ appearAlcaraz and Moreo (1992);
Alcaraz and Martins (1989); Cabra et al. (1998). In this case, only the model
with $K=1\ (\text{mod.}2)$ can be connected to $SU(2)_{1}$ point. On the other
hand, the $SU(2)_{K}$ model has a $Z_{2}$ simple current, and this becomes
nonanomalous for $K=0\ (\text{mod}.2)$ and one can obtain fermionic
representation by gauging this $Z_{2}$ symmetry coming from $Z_{2}$ simple
current in that caseHsieh et al. (2020); Fukusumi et al. (2021)111It should be
noted that there exists nonanomalous $Z_{2}$ symmetry generated by the charge
conjugation in the $Z_{K}$ parafermion model. Historically, this nonanomalous
symmetry appear as the $Z_{2}$ symmetry of $D_{K}=Z_{2}\times Z_{K}$
representation of the Fateev-Zamolodchikov parafermion modelFateev and
Zamolodchikov (1985). More recently, it has typically appeared as (Majorana)
fermionization of three state Potts model in Hsieh et al. (2020). Because of
anomaly matching of $SU(2)_{K}=Z_{K}\times U(1)$ model, the $Z_{2}$ anomaly of
the model survives by gapping $Z_{K}$ parafermion. It should be noted that
this charge conjugation can be outside of primary fields in the original
$Z_{K}$ model. In that case, one cannot use the terminology ”simple current”
as the symmetry operator in a rigorous sense.. Hence, these two concepts, RG
flow and gauging finite group symmetry are related by the quantum anomaly and
the fermionic representation of the model.

Roughly speaking, compared with an anomaly free case, an anomalous theory
tends to become gapless because an anomaly of a symmetry restricts the
permitted perturbative operators in Hamiltonian more strongly by the symmetry
charge. In this sense, the gaplessness can be protected by symmetry in an
anomalous theory. In an anomaly free case, because there exists less anomaly
obstruction, the (mass) condensations occur and the system tends to become
gapped. This is a simple understanding of the Haldane conjecture by
considering $Z_{2}$ anomaly. Of course, this is a rough statement, and for
justification in the specific models, the RG analysis with the table of the
possible operators (possibly, both in bulk and boundary) in the theories is
necessary. For an simplest example, one can consider that the spin $1/2$
anisotropic Heisenberg XXZ chain, which is $Z_{2}$ anomalous, becomes gapped
under the strong anisotropy, whereas the higher spin integer anisotropic
Heisenberg XXZ chain becomes gapless under the weak anisotropyAlcaraz and
Moreo (1992); Alcaraz and Martins (1989).

More recently, the defect analog of the RG flow, called RG domain wall, has
been proposed and applied to several important classes of modelsGaiotto
(2012); Poghosyan and Poghosyan (2015); Brunner and Schmidt-Colinet (2016);
Stanishkov (2016); Poghosyan and Poghossian (2022). Putting this framework and
anomaly analysis together, one can estimate the RG connectivity of the models
by considering coupling the RG-connected systems and their simple charges of
the product of simple currents in the original theories. Related proposals
with an emphasis on the nontrivial conformal defects and analysis with the
quantum impurity problems can be seen in Kimura and Murata (2014, 2015);
Quella et al. (2007). Moreover, this may give a unified understanding also for
the boundary of $2+1$ dimensional topologically ordered systems. By stacking
two topologically ordered systems and considering the simple charge of the
product theory, one can obtain the equivalence relations, so-called Witt
equivalent in the contemporary mathematical and theoretical condensed
physicsKong and Zhang (2022). For example, this aspect related to topological
(nonprotected or condensed) boundary conditions has been discussed in Kaidi et
al. (2021). Hence, in other words, the Haldane conjecture and the
classification problems of the protected edge modes of topologically ordered
states can be understood in the language of the anomaly matching condition of
the simple currents. Here, we note the importance to start the RG analysis of
the connected systems with enough symmetries or central charges where the
symmetry acts on site.

On the other hand, there exists another approach relating the Haldane
conjecture and the structure of CFT, by investigating modular properties of
the orbifolded model and symmetry action to BCFTFuruya and Oshikawa (2017);
Numasawa and Yamaguch (2018); Kikuchi and Zhou (2019). Their analysis is based
on the modular $T$ and $S$ property of the systems under the symmetry action
and can be understood as a classification based on the t’Hooft anomaly.
Unfortunately, without some modifications, these approaches cannot produce the
$K(\text{mod}.N)$ periodicity of $SU(N)_{K}$ systems Yao et al. (2019) which
is expected from the $Z_{N}$ anomalies from the $Z_{N}$ simple currents. In
this sense, there exists some difference between the t’Hooft anomaly and
anomaly of simple current.

So far, these two approaches, the RG domain wall interpretation of topological
ordered systems and the modular property of the orbifolded theory seem
different. In this work, we give a unified way to understand these phenomena
by constructing the anomaly free composite $Z_{N}$ objects which have first
appeared in the analysis of discrete torsion as the ”integer spin simple
current” by Schellekens and Gato-RiveraGato-Rivera and Schellekens (1992);
Schellekens (1990)222For the review of this aspect of CFT with emphasis on the
simple current, see Fuchs (1997),for example.. For this purpose, we first
start from the $Z_{N}$ fractional quantum Hall states, because the
mathematical procedure which corresponds to ”gauging finite group symmetry”
appears naturallyBhardwaj and Tachikawa (2018); Tachikawa (2020). This leads
to the emergence of the modular $S$ invariant which has been expected to
survive as protected edge modesFukusumi (a, b); Ino (1998); Milovanovic and
Read (1996); Cappelli and Zemba (1997); Levin (2013). By considering the
imaginary gapping operation (or gauging) of multicomponent bosonic parts, one
can expect to obtain a class of general topological orderCheng et al. (2022);
Cheng and Jian (2022). Similar proposals can be seen several condensed matter
works 333See for example Kong and Zhang (2022); Chatterjee and Wen (2022b) and
reference therein., but by interpreting bulk edge correspondence as
$CFT_{D}/BQFT_{D+1}$ correspondence, and its bulk RG flow as
$CFT_{D}/BTQFT_{D+1}$ correspondence, one can expect a possibility of
generalization of our analysis to both gapless and gapped models in general
space-time dimensions $D$ or $D+1$. By gapping modular $S$ noninvariant parts,
one may obtain modular $S$ invariant parts as candidates for protected edge
modes. As we will show, this distinction between modular $S$ invariant part
(or low energy part) and noninvariant part (or high energy part) for anomaly
free models naturally leads to a dual model description of $Z_{N}$ models,
which is analogous to fractional supersymmetryAhn et al. (1990); Durand
(1992); Mohammedi (1995); Isaev et al. (2002); Rausch de Traubenberg and
Slupinski (2000); OBrien and Fendley (2020); Mong et al. (2014); Perez et al.
(1996). Moreover, in our discussion, one can understand the appearance of
nonlocal CFT as a consequence of gauging anomalous symmetries.

The rest of the manuscript is organized as follows. In section II, we revisit
the integer spin simple currents and introduce them as composite particles,
”composite parafermion”, which can serve as a building block of the cylinder
partition function of $Z_{N}$ FQHE. The anomaly freeness of these particles
and its implication to the modular property are discussed. In section III, we
introduce the extension of our analysis for nonunitary theories and introduce
a new mechanism, Galois shuffle cancelation. We introduce the case for
$Z_{2}$, but applying the technique of the previous section and this section,
one can construct the candidates of the protected edge modes of general
nonunitary theories. In section IV, we propose a new duality in the composite
parafermion model, which can be interpreted as a kind of fractional
supersymmetry. Its analogy with quark confinement is shown. Moreover, we also
discuss a possible interpretation of the controversy around $\nu=5/2$
statesWang et al. (2017); Mross et al. (2018) with respect to composite
Majorana fermionization and quantum anomaly. In section V, we revisit the
anomalous parafermionization in Yao and Furusaki (2020) with emphasis on
anomaly and chirality. The appearance of chirality or mirror symmetry breaking
under the gauging anomalous symmetry is discussed, and its implications in a
2+1 dimensional system is shown. In section VI, we give a concise
understanding of $SU(N)$ Haldane conjecture based on composite
parafermionization. In section VII, we give a concluding remark, with emphasis
on the analogy between FQHE and 2d quantum gravity coupled to matter. In the
appendix, we show some detailed calculations in the main texts and also
introduce the 1+1 dimensional analog of models with composite parafermion and
confinement in the main text.

##  II Anomaly free $Z_{N}$ simple currents

In this section, we introduce the anomaly free $Z_{N}$ simple current. For $N$
odd, this has already been proposed as integer spin simple current and is
widely known because of its relation to discrete torsionGato-Rivera and
Schellekens (1992); Schellekens (1990). For $N$ even, as we will see, there
exist two types of objects and they seem less well-known in the condensed
matter theory 444After submitting the manuscript, a work which revisits this
anomaly free obeject appeared Kikuchi (2022)..

First, let us assume the existence of $Z_{N}$ simple current generated by $J$
with minimal conformal weight $h_{J}$. The simple currents are represented as
$J_{I}=J^{I}$ for $I=1,2,...N-1$ with conformal dimension $h_{J_{I}}$ and they
act like a group to the other fields in the theory. Then, the anomaly free
object, which we call ”composite parafermionic theory” satisfies the following
conditions,

  * •

For $N$ odd, $h_{J_{I}}=\text{integer}$ for all $I$

  * •

For $N$ even, $h_{J_{I}}=\text{half-integer}$ for all $I$ odd and
$h_{J_{I}}=\text{integer}$ for all $I$ even, or $h_{J_{I}}=\text{integer}$ for
all $I$

It should be noted that with these conditions, the simple currents satisfy,

$Q_{J_{I}}(J_{I^{\prime}})=0\ (\text{mod.}1),\text{for all}\ I,I^{\prime}$ (1)

where $Q_{J}(\alpha)=h_{\alpha}+h_{J}-h_{J\times\alpha}$ is the simple charge
of an operator $\alpha$ (with comformal dimension $h_{\alpha}$) under the
simple current $J$ (with confromal dimension $h_{J}$). To avoid complications,
we do not specify the $(\text{mod}.1)$ structure of the simple charges in the
following discussion. This is the generalized version of anomaly free
condition in the LSM theoremYao et al. (2019). In other words, the simple
currents are untwisted from each other and they can condensate each other
without anomaly obstructions in the anomaly free case. This phenomenon can be
thought of as a $Z_{N}$ generalized version of BCS cooper pair creation known
as anyon condensationBardeen et al. (1957); Kong (2013); Neupert et al.
(2016); Eliëns et al. (2014); Burnell (2018). To emphasize these objects
satisfy the $Z_{N}$ fusion rule, we call them ”composite parafermions” whereas
they do not satisfy the parafermi statistics. In the later section, we also
give the picture of these composite parafermions as composite particles of
anomalous parafermions, known as the Fateev-Zamolodchikov parafermionsFateev
and Zamolodchikov (1985).

In the following discussion, we mainly consider characters of CFTs
parametrized by the modular parameter $\tau$ and its complex conjugate
$\overline{\tau}$. We mainly consider the following two modular
transformations,

$\displaystyle T:\tau\rightarrow\tau+1,$ (2) $\displaystyle
S:\tau\rightarrow-\frac{1}{\tau},$ (3)

and study the invariance and noninvariance of the partition functions under
these transformations. As has been discussed in Fukusumi (a, b); Ino (1998);
Milovanovic and Read (1996); Cappelli and Zemba (1997), one can expect the
modular $S$ invariant parts of the total partition function can appear under
bulk gapping operation with bulk-edge correspondence in cylinder geometry for
2+1 dimensional systems.

Here we note the partition function of the $Z_{N}$ composite parafermionic
model constructed from bosonic partition function,
$Z_{M}=\sum_{i,p}\chi_{i,p}\overline{\chi}_{i,-p}+\sum_{a}|\chi_{a}|^{2}$where
$i$ labels the $Z_{N}$ noninvariant sectors corresponding to the character
$\chi_{i,p}$ and $\\{p\\}^{N-1}_{p=0}$ labels its parafermionic parity which
takes an integer (mod.$N$) and $a$ labels the $Z_{N}$ invariant sectors
corresponding to the character $\chi_{a}$ . The composite parafermionic
partition function can be characterized by the simple charge of each sector,
$\\{Q_{I}\\}_{I=1}^{N-1}$, and is summarized as in the following form 555The
possible value of $\\{Q_{I}\\}_{I=1}^{N-1}$ is restricted by the charge
conjugate condition and detail of the models in general, but we do not specify
the possible value to avoid complications of the discussion,

$Z_{\\{Q_{I}\\}}=\sum_{i\in\\{Q_{J_{I}}(i)=Q_{I}\\}}|\sum_{p}\chi_{i,p}|^{2}+N\sum_{a\in\\{Q_{J_{I}}(a)=Q_{I}\\}}|\chi_{a}|^{2},$
(4)

where we have used the straightforward generalization of the discussion in
Hsieh et al. (2020). Because of anomaly freeness, $Q_{J_{I}}(i,p)$ does not
depend on the parafermionic parity. Hence we have dropped the index of $p$ for
the simple charges. Each charge sector may correspond to the twisted boundary
conditions of the composite parafermionic chain, as we will show in the
appendix. In unitary CFTs, one can obtain the sector with $Q_{I}=0$ for all
$I$ as modular $S$ invariant and this plays a central role in the following
construction of the $Z_{N}$ FQHEs.

###  II.1 $Z_{N}$ fractional quantum Hall effect

For a simple application, we study the fractional quantum Hall effect
constructed from the $Z_{N}$ composite parafermion. We combine together the
approaches in Schoutens and Wen (2016); Milovanovic and Read (1996); Ino
(1998); Fukusumi (b, a). First, let us introduce the partition function of
multicomponent bosonic fields $\varphi_{I}$ indexed by integer $q_{I}$,
$I=1,2,...,N-1$ as,

$\displaystyle Z_{mb}$
$\displaystyle=\prod_{I=1}^{N-1}\sum_{r_{I}=0}^{q_{I}-1}\sum_{m_{I},\overline{m}_{I}}\frac{1}{|\eta(x)|^{2}}x^{\frac{(m_{I}q_{I}+r_{I})^{2}}{2q_{I}}}\overline{x}^{\frac{(\overline{m}_{I}q_{I}+r_{I})^{2}}{2q_{I}}},$
(5) $\displaystyle x$ $\displaystyle=e^{2\pi i\tau},\overline{x}=e^{2\pi
i\overline{\tau}},$ (6)

where $\eta(x)=x^{-1/24}\prod_{n=1}^{\infty}(1-x^{n})$ is the eta function and
$\tau$ and $\overline{\tau}$ are modular parameter corresponding to each edge,
and $r_{I}$ corresponds to the number of quasihole. It should be noted that
the $m_{I}$ and $\overline{m}_{I}$ correspond to the number of particles at
each edge and take all integer values. In the following discussion, we take
the summation indices $r_{I}$, $m_{I}$ and $\overline{m}_{I}$ the same as in
Eq.(5) without a specific caution.

The partition function (5) corresponds to the direct product of the partition
functions of Laughlin states with the filling factor
$\\{1/q_{I}\\}^{N-1}_{I=1}$ and satisfies the modular $S$ and $T$ invariance.
One can easily generalize the situation with other multicomponent bosonic
models, such as the model with a general $K$ matrix.

For the later discussion, we introduce the following character,

$\theta^{m_{I}}_{\frac{r}{q_{I}}}=\frac{1}{\eta(x)}x^{\frac{(m_{I}q_{I}+r_{I})^{2}}{2q_{I}}},$
(7)

Here we introduce the coupling of the multicomponent bosonic model and the CFT
with composite parafermion. The general electronic objects generated by
$Z_{N}$ simple currents are
$\\{J_{I}e^{i\sqrt{q_{I}}\varphi_{I}}\\}^{N-1}_{I=1}$. Again, the simple
charges of the operator indexed by $\alpha$ are,

$Q_{J_{I}}(\alpha)=h_{J_{I}}+h_{\alpha}-h_{J_{I}\alpha}.$ (8)

Corresponding to these twists, one has to redefine the chiral character
$r_{I}\rightarrow r_{I}+Q_{J_{I}}(\alpha)$ for each primary field
$\phi_{\alpha}$. This modification is analogous to the gravitational dressing
or Knizhnik-Polyakov-Zamolodchikov (KPZ) relation in 2d quantum gravity
coupled to matterDavid (1988); Knizhnik et al. (1988); Distler and Kawai
(1989), and FQHE and 2d quantum gravity coupled to matter share the basic
theoretical structure, the single-valuedness of the correlation function. In
other words, FQHE is a gauge theory analogous to string theory, and this
similarity can be interpreted as a generalized (or relaxed) version of
gauge/gravity duality. For the later discussion, we denote the character
$\phi_{\alpha}$ without such twist, i.e. $Q_{J_{I}}(\alpha)=0$ for all $I$ as
”untwisted”. Besides these simple charges and the twists, one has to introduce
$Z_{N}$ parity coming from the coupling of the models. Hence the resulting
partition function becomes complicated when one only assumes the bulk-edge
correspondence. However, because of the anomaly freeness, one can represent
the resulting partition function with modular $T$ or $T^{2}$ invariance as,

$Z=Z^{\text{untw}}+Z^{\text{tw}},$ (9)

where $Z^{\text{untw}}$ and $Z^{\text{tw}}$ correspond to the untwisted and
twisted part of the theory classified by the simple charges $Q_{J_{I}}$. The
explicit form of the above partition function is shown in appendix.

The detailed characters are summarized as,

$\displaystyle\Xi^{p^{\prime}}_{i,p,\\{r_{I}\\}}$
$\displaystyle=\chi_{i,p}\prod_{I}\sum_{m_{I};\sum_{I}m_{I}+p=p^{\prime}(\text{mod}.N)}\theta^{m_{I}}_{\left(r_{I}+Q_{J_{I}}\left(\phi_{i,p}\right)\right)/q_{I}},$
(10) $\displaystyle\Xi_{a,\\{r_{I}\\}}$
$\displaystyle=\chi_{a}\prod_{I}\sum_{m_{I}}\theta^{m_{I}}_{\left(r_{I}+Q_{J_{I}}\left(\phi_{i,p}\right)\right)/q_{I}},$
(11)

where ${i,p}$ and $a$ label $Z_{N}$ noninvariant and $Z_{N}$ invariant states
respectively, and $p$ is the parafermionic parity coming from parafermionic
representation as we have introduced in the previous section. For the $Z_{N}$
invariant primary fields $\phi_{a}$ in the bosonic represntation, one can
consider parafermionization as
$\phi_{a}=\left(\sum_{p=0}^{N-1}\phi_{a,p}\right)/\sqrt{N}$. As in the $Z_{2}$
case, there exists gauge choices $\\{\nu_{p}\\}_{p=0}^{N-1}$ as
$\phi_{a}=\left(\sum_{p=0}^{N-1}\omega^{\nu_{p}}\phi_{a,p}\right)/\sqrt{N}$
where $\omega$ is $N$th root of unity, $\omega=\text{exp}\left(2\pi
i/N\right)$, but we have taken the simplest gauge that gives the nonnegative
integer fusion rule. In analogy with $Z_{2}$ case, we call $\phi_{a,p}$ as
$Z_{N}$ semion and we have assumed a counting rule which is a natural
generalization as in Milovanovic and Read (1996); Ino (1998). This may need
further numerical tests as in Jolicoeur et al. (2014), because the modular
property can change under this counting. The similar counting problem in
$Z_{2}$ case has been discussed in Ino (2000); Lou et al. (2021), for example.

By gapping fractional flux sectors, one can obtain the modular $S$ invariant
partition function as,

$\begin{split}Z&\rightarrow Z^{\text{untw}}\\\
&=\left(\sum_{i,p\in\text{untwisted}}|\sum_{p=0}^{N-1}\chi_{i,p}|^{2}+N\sum_{a\in\text{untwisted}}|\chi_{a}|^{2}\right)Z_{mb}.\end{split}$
(12)

This is nothing but the product of Eq. (5) and untwisted part of Eq.(4), and
satisfies the modular $S$ invariance.

Hence, for the $Z_{N}$ FQHE constructed from the composite parafermionic
theories, there exists no obstruction to obtain bulk gappability from their
modular $S$ property. In other words, we have obtained the candidates of the
protected edge modes in the cylinder geometry. Next, we show the corresponding
BCFT structure of the theories which has a close relationship with the bulk
topological properties and the topological entanglementFukusumi (a); Qi et al.
(2012); Das and Datta (2015). For simplicity, we drop off the indices coming
from multibosonic parts in the rest of this section.

In this composite parafermionic representation, one can easily extend the
analysis of fermionic BCFT in Fukusumi et al. (2021); Ebisu and Watanabe
(2021) to anomaly free parafermionic BCFT. The results are,

$\displaystyle|i\rangle_{\text{PF}}$
$\displaystyle=\sum_{p=0}^{N-1}|i,p\rangle_{\text{Cardy}},$ (13)
$\displaystyle|a,p\rangle_{\text{PF}}$
$\displaystyle=|a\rangle_{\text{Cardy}}\otimes|p\rangle,$ (14)

where $i,p$ labels $Z_{N}$ noninvariant states and $a$ labels $Z_{N}$
invariant states in the bosonic fusion rule. We have introduced $|p\rangle$ as
in the fermionic case to label $Z_{N}$ parafermionic parityFukusumi et al.
(2021). Hence by mapping from the bulk degeneracy problem to the tunnel
problemWen (1990), and by applying Cardy’s conjecture Cardy (2017) to this
situation, one can obtain the bulk topological degeneracies by considering the
degeneracies of the untwisted particles, $\\{\phi_{i,p}\\}_{p=0}^{N-1}$ or
$Z_{N}$ semion $\\{\phi_{a,p}\\}_{p=0}^{N-1}$666However, the way choosing the
appropriate sector or the type of anyon may depend on the detail of the
interactions corresponding to the detail of models. Some ambiguity of the bulk
topological degeneracies may correspond to thisWatanabe et al. (2022). It may
also be related to subtle relations between modular invariants and BCFTs
Gannon (2002).. The $Z_{N}$ semions only appear as untwisted states when
composite parafermion becomes integer spin simple current.

As one can see in Eq. (12), the partition function naturally contains the
subalgebra structure of fusion rule as the author has observed in the
fermionic caseFukusumi (a), called Lagrangian subalgebra as,

$\phi_{\alpha}^{\text{untw}}\times\phi_{\alpha^{\prime}}^{\text{untw}}=\sum_{\beta}N_{\alpha,\alpha^{\prime}}^{\beta}\phi_{\beta}^{\text{untw}},$
(16)

where $\alpha$, $\alpha^{\prime}$, and $\beta$ are labeling untwisted fields
(or extracting untwisted fields labeled by $\\{i,p\\}_{p=0}^{N-1}$ and
$\\{a,p\\}_{p=0}^{N-1}$), and $N_{\alpha\alpha^{\prime}}^{\beta}$ is the
fusion matrix of the theory. Moreover, the partition function Eq. (12)
naturally represents the type of particles $\\{i,p\\}^{N-1}_{p=0}$ or
$\\{a,p\\}^{N-1}_{p=0}$. By considering BCFTs, one can extract topological
degeneracies corresponding to each type $i$ and $a$.

Our construction can be applied to all anomaly free models constructed from
the bulk-edge correspondence in principle. A typical example is $SU(N)_{K}$
FQHE constructed by Gepner parafermionFuji and Lecheminant (2017). Hence we
only introduce several examples and their implication in other aspects of
physics in the next subsections.

###  II.2 Coupling three states Potts three times

One of the simplest models which are in scope of the composite parafermionic
$Z_{N}$ FQHE is the $Z_{3}$ model constructed by coupling three state Potts
models three times and considering simple current construction as inSchoutens
and Wen (2016)777Recently, such products of cyclic groups has captured some
attentions of condensed matter physicistsTopchyan et al. (2022); Dupont et al.
(2021).. Each Potts model has six primary fields, noted as
$\\{I,\psi_{1},\psi_{2},\epsilon,\sigma_{1},\sigma_{2}\\}$, and $\psi_{1}$ and
$\psi_{2}$ are the $Z_{3}$ simple currents. Where we have used the notation
that parafermionic parity is identified in the lower index of the fields. The
nontrivial fusion rule of each Potts CFT which is outside of $Z_{3}$ symmetry
applications is,

$\displaystyle\epsilon\times\epsilon=I+\epsilon.$ (17)

This is nothing but the fusion rule of Fibonacci anyon.

In this coupled model, there exist $6^{3}=216$ primary fields.
$J_{1}=\psi_{1}^{(1)}\psi_{1}^{(2)}\psi_{1}^{(3)}$ is a typical choice of the
composite parafermion where the upper index corresponds to each copy of three
state Potts CFT. The partition function by bulk edge correspondence contains a
lot of fields, but if one considers the untwisted part which corresponds to
protected edge modes, one can easily write down the cylinder partition
function of the $Z_{3}$ FQHE model as,

$Z^{\text{untw}}_{(Z_{3})^{3}}=\sum_{p_{\alpha^{(1)}}=0,p_{\alpha^{(2)}}+p_{\alpha^{(3)}}=0}|\sum_{i=0}^{2}\chi_{J^{i}\alpha^{(1)}\alpha^{(2)}\alpha^{(3)}}|^{2}Z_{mb}$
(18)

where $\alpha^{(i)}$ takes
$\\{I,\psi_{1},\psi_{2},\epsilon,\sigma_{1},\sigma_{2}\\}$ resepctively under
the condition $\\{p_{\alpha^{(1)}}=0,p_{\alpha^{(2)}}+p_{\alpha^{(3)}}=0\\}$,
where $p_{\alpha^{(i)}}$ represents the parity of the field $a^{(i)}$. It
contains $(2\times 6\times 2)\times 3=72$ primary fields coming from coupled
Potts part of CFT.

In this model, there exists the other anomaly free $Z_{3}$ simple currents
with total parafermionic parity $1$,
$J_{1}^{\prime}=\psi_{1}^{(1)}\psi_{1}^{(2)}\psi_{2}^{(3)}$ and their
permutations by the upper indices. The partition function for this case is,

$Z^{\text{untw}}_{(Z_{3})^{3}}=\sum_{p_{\alpha^{(1)}}=0,p_{\alpha^{(2)}}-p_{\alpha^{(3)}}=0}|\sum_{i=0}^{2}\chi_{J^{i}\alpha^{(1)}\alpha^{(2)}\alpha^{(3)}}|^{2}Z_{mb}$
(19)

Similarly, one can consider anomaly free $Z_{N}$ coupled Potts model and the
corresponding FQHE by coupling Potts model $KN$ times for odd $N$ and $KN/2$
times for even $N$ where $K$ is an arbitrary positive integer. As one can
easily see, this type of coupled model has a gigantic number of primary
fields, approximately $N^{ANK}\prod_{I}q_{I}$ where $A$ is a constant
depending on the model.

###  II.3 $SU(N)$ spinon-holon state from $SU(N)_{K}$ Wess-Zumino-Witten model

In $SU(N)_{K}$ Wess-Zumino-Witten models, the $Z_{N}$ simple currents $J_{I}$
have the following conformal dimensions,

$h_{J_{I}}=\frac{KI(N-I)}{2N},I=1,...,N-1$ (20)

Hence $SU(N)_{K}$ WZW models have the composite parafermionic representation
when $K=0\ (\text{mod}.N)$, and one can consider gapped FQHE by coupling $N-1$
multicomponent bosons. This can be thought of as a generalization of spinon-
holon states proposed in Ino (1998). It is interesting to note that even when
$K$ does not satisfy the composite parafermion condition, it may be possible
to construct modular $S$ invariant partition functions by extending the method
in Ino (1998). However, these partition functions break modular $T$ and
$T^{2}$ invariance and they may not have the corresponding lattice model in
1+1 dimensional quantum systems with local interaction. Related discussion
under the thin torus limit of anomalous (gapless) FQHE model can be seen in
Papić (2014).

##  III Anomaly and Galois shuffle cancelation in coupled nonunitary models

In this section, we discuss a possibility to generalize our discussion to
nonunitary CFTs. For simplicity, we concentrate on a $Z_{2}$ symmetric case.
In Davenport et al. (2013), the spin singlet Gaffnian state which is a natural
generalization of the Gaffnian FQH state has been proposed. Their model can be
described by the product of nonunitary minimal models, $M(3,r+2)\times
M(r,2r+1)$ and chiral Lattinger liquid. Interestingly, the product theory is
closely related to the RG domain wall in Gaiotto (2012) when $r=3$. In this
section, we revisit the structure of this type of modelswith the view of
anomaly defined by the simple charge coming from the operators and monodromy
charge coming from modular $S$ property of quantum states. Consequently, we
introduce the notion of Galois shuffle cancelation (shuffle cancelation in
short) which represents the cancelation of anomalous monodromy charge.

In nonunitary $Z_{2}$ anomalous rational minimal model $M(p,q)$, there exist
two remarkable properties:

  * •

The conformal dimension $h_{J}$ of $Z_{2}$ simple current $J$ takes
$h_{J}=(2k+1)/4$ and becomes anomalous where $k$ is an integer.

  * •

The simple charge $Q_{J}(\alpha)$ of the arbitrary operator $\phi_{\alpha}$
does not match with the monodromy charge
$Q^{\text{mon}}_{J}(\phi_{\alpha})=Q_{J}(\phi_{\alpha})-Q_{J}(\sigma)$ of the
same operator where the $\sigma$ is the operator with lowest conformal
dimension and $Q_{J}(\sigma)=\text{half-integer}$.

The former property defines the anomalous $Z_{2}$ symmetry and the latter
defines the nonunitarity of the model. The later property of the shuffle
between the simple charge and the monodromy charge has been called Galois
shuffleGannon (2003). We name the operator $\sigma$ as shuffle generator, for
convenience. In considering FQHE, the former becomes relevant to consider
modular $T$ property or locality of the model and the latter becomes relevant
to consider modular $S$ property or stability of the edge modesIno (1998).
Here we mostly discuss the later property. In FQHE, whereas the wavefunction
is constructed by considering correlation functions and corresponding simple
charge, the modular property coming from gauging $Z_{2}$ symmetry structure is
defined by the monodromy charge. This difference between monodromy charge (or
the symmetry charge of the states) and simple charge (or the symmetry charge
of operators) has been studied extensively in Gannon (2003) and stressed at
least in two different fields, mathematical physics as in Harvey et al. (2020)
and earlier analysis of Haldane-Rezayi modelMilovanovic and Read (1996); Ino
(1998).

Here, let us introduce the coupling of the two anomalous nonunitary conformal
field theories $M_{1}$, $M_{2}$ with $Z_{2}$ simple current $J^{(1)}$ and
$J^{(2)}$, and with Galois shuffle by the lowest conformal dimension fields
$\sigma^{(1)}$ and $\sigma^{(2)}$ respectively. In this setting, the coupled
theory has a nonanomalous $Z_{2}$ simple current $J^{(1)}J^{(2)}$. Moreover,
the shuffle property of the product theory is canceled as we show in the
following.

One can calculate the monodromy charge of the operator
$\phi_{\alpha^{(1)}}\phi_{\beta^{(2)}}$ in the product theory as,

$\begin{split}&Q_{J_{1}J_{2}}^{\text{mon}}(\phi_{\alpha^{(1)}}\phi_{\beta^{(2)}})\\\
&Q_{J^{(1)}J^{(2)}}(\phi_{\alpha^{(1)}}\phi_{\beta^{(2)}})-Q_{J_{1}J_{2}}(\sigma^{(1)}\sigma^{(2)})\\\
&=Q_{J^{(1)}}(\phi_{\alpha^{(1)}})-Q_{J^{(1)}}(\sigma^{(1)})+Q_{J_{2}}(\phi_{\beta^{(2)}})-Q_{J^{(2)}}(\sigma^{(2)})\\\
&=Q_{J^{(1)}}(\phi_{\alpha^{(1)}})+Q_{J^{(2)}}(\phi_{\beta^{(2)}})\\\
&=Q_{J_{1}J_{2}}(\phi_{\alpha^{(1)}}\phi_{\beta^{(2)}}),\end{split}$ (21)

where we have used the shuffle property
$Q_{J^{(1)}}(\sigma^{(1)})+Q_{J^{(2)}}(\sigma^{(2)})=0$. Hence the product
theory $M_{1}\times M_{2}$ shows cancelation of anomaly and Galois shuffle.

Because of this cancelation of anomaly and shuffle, one can construct modular
$S$ invariant fermionic theory and the corresponding fermionic FQHE as in
usual nonanomalous unitary CFT by taking $J^{(1)}J^{(2)}$ as $Z_{2}$ simple
currentCappelli and Viola (2011); Fukusumi (a). Hence, at least from this
bulk-edge correspondence, the spin singlet Gaffnian state can be gapped out
under preserving bulk-edge correspondence, whereas its gaplessness has been
proposed in Davenport et al. (2013). Further study and construction of bulk
perturbation in various geometries in the lattice models are necessary to show
their bulk gap or gaplessness. One can easily apply the same discussion to
more general $Z_{N}$ models which can include both unitary and nonunitary CFT.

##  IV Duality and confinement of composite parafermion models coming from
anomalous parafermions

In this section, we propose a (hopefully new) $Z_{N}$ duality which is a
$Z_{N}$ generalization of supersymmetry known as fractional supersymmetry
Mohammedi (1995) or composite parafermionic T duality as in the $Z_{2}$
modelsFukusumi (a). As a simple model, we study the anomaly cancelation of
$SU(3)_{1}$ WZW model and three state Potts model.

In $SU(3)_{1}$ WZW model, there exist three pimary fields,
$\\{I,j_{1},j_{2}\\}$ with conformal dimension
$h_{I}=0,h_{j_{1}}=1/3,h_{j_{2}}=1/3$ where $j_{1}$ and $j_{2}$ are the
$Z_{3}$ simple currents.

As in the previous subsections, one can construct the two sets of composite
parafermions, $\\{I,\psi_{1}j_{1},\psi_{2}j_{2}\\}$ or
$\\{I,\psi_{2}j_{1},\psi_{1}j_{2}\\}$. Each representation has $6$ primary
fields as untwisted sectors and one can expect that they appear as protected
edge modes of the $Z_{3}$ FQHE constructed from these composite parafermions.

Here we comment on the duality of the model. For simplicity, let us consider
the situation when the composite parafermion is
$\\{I,\psi_{1}j_{1},\psi_{2}j_{2}\\}$. Then we can obtain the set of twisted
and untwisted partition functions corresponding to the simple charge
$Q_{1}=Q_{j_{1}\psi_{1}}=0,\ 1/3,\ 2/3$ as,

$\displaystyle Z_{Q_{1}=0}$
$\displaystyle=|\chi_{I}+\chi_{j_{1}\psi_{1}}+\chi_{j_{2}\psi_{2}}|^{2}+|\chi_{\epsilon}+\chi_{j_{1}\sigma_{1}}+\chi_{j_{2}\sigma_{2}}|^{2},$
(22) $\displaystyle Z_{Q_{1}=1/3}$
$\displaystyle=|\chi_{j_{1}}+\chi_{j_{2}\psi_{1}}+\chi_{\psi_{2}}|^{2}+|\chi_{j_{1}\epsilon}+\chi_{j_{2}\sigma_{1}}+\chi_{\sigma_{2}}|^{2},$
(23) $\displaystyle Z_{Q_{1}=2/3}$
$\displaystyle=|\chi_{j_{2}}+\chi_{\psi_{1}}+\chi_{j_{1}\psi_{2}}|^{2}+|\chi_{j_{2}\epsilon}+\chi_{\sigma_{1}}+\chi_{j_{1}\sigma_{2}}|^{2},$
(24)

where we have used the charge conjugate condition and dropped off the label
coming from $Q_{2}=Q_{j_{2}\psi_{2}}$.

As can be seen from these partition functions, one can obtain this set of
partition functions starting from the untwisted partition function
$Z_{Q_{1}=0}$ by shifting the parity or simple charge by the operator $j_{1}$.
This duality generated by the anomalous parafermionic parity shift or simple
charge shift can be thought of as a kind of $Z_{N}$ version of supersymmetry,
called fractional supersymmetryAhn et al. (1990); Durand (1992); Mohammedi
(1995); Isaev et al. (2002); Rausch de Traubenberg and Slupinski (2000);
OBrien and Fendley (2020); Mong et al. (2014). One can name this general
duality as ”composite parafermionic T duality”. In the existing literature,
the anomalous $Z_{N}$ particles with fractional spin and the associated
currents have been considered as a characteristic property of fractional
supersymmetric models. However, the composite parafermionic T duality is a
consequence of anomaly freeness. Related anomaly free representation can be
seen, for example in Perez et al. (1996). By applying the general construction
of $Z_{3}$ FQHE in the cylinder geometry, one can observe that the only
untwisted sector can survive as the edge modes and the other twisted sectors
go to the bulk excitations. In this understanding, it should be stressed that
the anomalous particle, $j_{i}$, $\psi_{i}$, and $\sigma_{i}$ do not appear
without anomaly cancellation or ”paring”. This is analogous to quark
confinment, because such anomalous particles (or ”quarks”) go to high energy
states under the bulk gapping process. Hence by applying the bosonization to
our model, the anomalous particle may appear, and the fractional supersymmetry
in the literature can appear naturally. In this sense, it seems appropriate to
interpret our composite parafermionic T duality as a more primitive version of
fractional symmetry.

One can apply a similar analysis to more general $Z_{N}$ models by coupling
anomalous $Z_{N}$ models. For the simplest model let us introduce the
partition function of $\\{SU(N)_{1}\\}^{N}$ model with
$J_{I}=\prod_{k}j^{(k)}_{I}$ as composite parafermion where the upper index
$(k)$ specifies each copy of $SU(N)_{1}$. In this model, each $SU(N)_{1}$ has
only simple currents and identity fields, $I,j_{1},...,j_{N-1}$ as primary
fields. The untwisted partition function is,

$Z^{\text{untw}}=\sum_{\sum
p_{\alpha^{(i)}}=0}\left|\sum_{I=0}^{N-1}\chi_{J_{I}\prod_{k=2}^{N}\alpha^{(k)}}\right|^{2}.$
(26)

where $p_{\alpha^{(k)}}$ is the parafermionic parity of the operator
$\alpha^{(k)}$ and $\alpha^{(k)}$ takes $I,j_{1},...,j_{N-1}$ respectively.
The twisted part can be obtained by adding $j^{(1)}_{1}$ recursively for
example. Hence one can observe the analogy of quark confinement even in this
simple model.

Figure 1: Analogy between quark confinement and FQHE

It may be interesting to consider the relation between $Z_{N}$ duality in the
general coset CFTs and such $Z_{N}$ composite parafermionic T duality of the
coupled models. For the simples example, let us consider $SU(2)_{1}\times
SU(2)_{1}=M(3,4)\times SU(2)_{2}$. The equivalence is ensured by coset
construction of the Ising model. Related discussion can be seen in Thorngren
and Wang (2021), for example. The primary fields of the lefthand side are the
tensor product of two $SU(2)_{1}$ whose primary fields are $I,\phi_{1}$, with
conformal dimension $h_{I}=0,h_{\phi_{1}}=1/4$, and its fusion rule is just
$Z_{2}\times Z_{2}$. The anomaly free $Z_{2}$ is generated by $I$ and
$\phi^{(1)}_{1}\phi^{(2)}_{1}$ where the upper index is labeling the copy of
$SU(2)_{1}$. The other two primary fields $\phi^{(1)}_{1}$, $\phi^{(2)}_{1}$
are anomalous. Hence these anomalous $Z_{2}$ field generates fermionic T
duality. The NS and R sectors of fermionic partition functions are obtained
as,

$\displaystyle Z_{NS}$
$\displaystyle=|\chi_{0}+\chi_{\phi^{(1)}_{1}\phi^{(2)}_{1}}|^{2}$ (27)
$\displaystyle Z_{R}$
$\displaystyle=|\chi_{\phi^{(1)}_{1}}+\chi_{\phi^{(2)}_{1}}|^{2}$ (28)

On the other hand, in the representation $M(3,4)\times SU(2)_{2}$, the fusion
rule of $M(3,4)$ and $SU(2)_{2}$ are the same Tambara-Yamagami category. The
$Z_{2}$ duality operator has conformal dimension $h_{\sigma}=1/16$ and
$h_{\phi_{1}}=3/16$ and the product of these fields, $\sigma\phi_{1}$ has
conformal dimension $h_{\sigma\phi_{1}}=1/4$. Hence one can observe the
product of the two $Z_{2}$ Kramers-Wannier duality becomes anomalous $Z_{2}$
parity shift. This picture is consistent with the intuitive understanding of
semionic statistics of duality operators. We expect this mechanism is
ubiquitous and gives some understanding of RG and anomaly in the way analogous
to ”particle” or ”quark” physics. In appendix, we show a general construction
of such a quark confinement-like model in 1+1 dimension from bosonic CFT and
the corresponding lattice model.

As another example, one can consider anomaly analysis of the double semion
model, which is equivalent to the fermionized fusion rule of $M(3,4)$ model.
In this model there exist four fields $\\{I,\psi,e,m\\}$ with the conformal
dimensions $h_{I}=0,h_{\psi}=1/2,h_{e}=h_{m}=1/16$. Except for the identity
operator, all fields in this theory can be thought of as $Z_{2}$ simple
current, but $e$ and $m$, known as semion, have anomalous conformal dimension
$1/16$. Hence one can expect to obtain anomaly free theory coupling this
semion model $16$ times888To obtain the theory with integer or half integer
simple charge, $4$ semion might be enough. Hence there may exist $4$-fold
periodicity, more generally. This analysis can be extended to other $Z_{2}$
and $Z_{N}$ models.. This may give an intuitive understanding of Kitaev’s
$16$-fold periodicityKitaev (2006).

###  IV.1 Hierarchy from coupled Majorana fermion and its implication on bulk
and boundary flows

Figure 2: Bulk and boundary RG interpretation of CFT/BQFT correspondence and
its ambiguities. In a general topological order, one can attach different CFTs
connected under RGs at the edge theoretically. This nonuniqueness of edge
theory may result in difficulty in observing edge CFT in experimental
settings.

Interestingly, as we have discussed, $M(3,4)$ and $SU(2)_{2}$ models have the
same fusion rule. Moreover, it is well-known that there exists three
components Majorana fermionic representation for $SU(2)_{2}$ WZW model. It
coincides with a generalized anomaly cancelation mechanism which is analogous
to the periodicity of spin-$S$ $SU(2)$ Heisenberg model in Oshikawa (1992). As
a generalization, one can consider five component fermionic theory. In this
line, the corresponding central charges of the FQHE model constructed from
$M(3,4)$, $SU(2)_{2}$, and $\\{M(3,4)\\}^{5}$ take $1/2+1$, $3/2+1$, $5/2+1$
and this seems to correspond to the controversy about experimental and
numerical observation of Moore-Read states in Mross et al. (2018); Wang et al.
(2017). It might be interesting to note that the number $5$ is the smallest
number which becomes coprime with the “conductor”, $48$ of $M(3,4)$ in Harvey
et al. (2020). Hence following the approaches in Harvey et al. (2020),
$\\{M(3,4)\\}^{5}$ may have the same fusion rule with $M(3,4)$ by considering
its orbifolding999 Interestingly, the Hecke operation which is a central
structure of their construction is similar to Renyfication which is closely
related to the entanglement of models. With respect to RG, it may be
interesting to consider their construction and cristalization of operators
under Renyification Sté phan et al. (2009).. It should be noted that all these
theories can be connected to each other without anomaly and spin statistics
obstruction. In bulk and boundary RG, it may indicate a nontrivial relation
between CFT which constructs the bulk wavefunctions (with central charge
$c^{\prime}$) and appears at the boundary with central charge
$c^{\prime\prime}$ probably with $c^{\prime\prime}>c^{\prime}$. If one starts
from $D+1$ dimensional bulk-edge correspondence or $CFT_{D}/BQFT_{D+1}$
correspondence with sufficiently large central charge $c$, or with sufficient
degrees of freedom, one can safely explain the relation between these CFTs as
the consequence of bulk and boundary RG flow, with $c>c^{\prime}$,
$c>c^{\prime\prime}$ (FIG. 2). However, it may be difficult to determine this
$c$ started from a lattice model or experimental setting whereas it seems
possible to construct such bulk and boundary RG flow purely from QFT. This
difficulty is coming from the absence of a general criterion to determine the
sufficiency of degrees of freedoms101010As the author has stressed in Fukusumi
and Iino (2020), $BQFT_{D+1}$ can contain unstable boundary conditions which
are outside of Cardy’s condition. Hence further studies of bulk and boundary
RG flows are necessary to establish a general theory of protected edge modes.

##  V Anomalous parafermionization

In this section, we introduce a field theoretic analog of parafermionization
of $1+1$ dimensional quantum systems in the existence of $Z_{N}$ anomaly and
its implication to $2+1$ dimensional systems constructed by the bulk-edge
correspondence. We mainly follow the approach by Yao and FurusakiYao and
Furusaki (2020) and express their results in a more unified way by using
gauging. Studies of $Z_{2}$ anomalous symmetry can be seen in several related
works Hao et al. (2022); Chatterjee and Wen (2022a, b).

First, let us introduce the paraspin variables at each site $l$ satisfying the
following relations,

$\sigma^{N}_{l}=\tau^{N}_{l}=1,\ \sigma_{l}^{\dagger}=\sigma_{l}^{-1},\
\tau_{l}^{\dagger}=\tau_{l}^{-1},\
\sigma_{l}\tau_{l}=\omega\tau_{l}\sigma_{l},$ (29)

where $\omega=\text{exp}(2\pi i/N)$ A typical example is the $Z_{N}$ Fateev-
Zamolochikov spin chain and its parafermionization Albertini (1994) which can
be described by $Z_{N}$ parafermion CFTFateev and Zamolodchikov (1985). This
type of parafermionic model has captured the attentions of condensed matter
physicists as a new type of topological phase. These models have been expected
to have a possibility to realize the topological quantum computation because
of the existence of nonabelian anyons in the theoriesSantos and Hughes (2017);
Alicea and Fendley (2016).

The spin chain Hamiltonian of $Z_{N}$ Fateev Zamolochikov model with chain
length $L$ is,

$H_{FZ}=-\sum_{l=1}^{L}\sum_{n=1}^{N-1}\frac{1}{\text{sin}\left(n\pi/N\right)}\left(\sigma^{-n}_{l}\sigma^{n}_{l+1}+\tau_{l}\right)$
(30)

Then, let us introduce the order and disorder fields in the $Z_{N}$ models
following the existing works, Yao and Furusaki (2020). The disorder operator
is

$\mu_{l}=\prod_{j=1}^{l-1}\tau_{j},$ (31)

Then, one can define the parafermionic operator as,

$\displaystyle\gamma_{2l-1}$ $\displaystyle=\sigma_{l}\mu_{l}$ (32)
$\displaystyle\gamma_{2l}$ $\displaystyle=\omega^{(N-1)/2}\sigma_{l}\mu_{l}$
(33)

There exists the other choice of disorder operator Mong et al. (2014) as,

$\mu^{\prime}_{l}=\prod_{j=L}^{l}\tau_{j},$ (34)

This gives the other parafermionization,

$\displaystyle\gamma_{2l}^{\prime}$ $\displaystyle=\sigma_{l}\mu^{\prime}_{l}$
(35) $\displaystyle\gamma_{2l-1}^{\prime}$
$\displaystyle=\omega^{(N-1)/2}\sigma_{l}\mu^{\prime}_{l}$ (36)

These parafermionizations result in different chirality,

$\displaystyle\gamma_{l}\gamma_{l^{\prime}}=\omega^{\text{sgn}(l-l^{\prime})}\gamma_{l}\gamma_{l^{\prime}},$
(37)
$\displaystyle\gamma^{\prime}_{l}\gamma^{\prime}_{l^{\prime}}=\omega^{-\text{sgn}(l-l^{\prime})}\gamma^{\prime}_{l}\gamma^{\prime}_{l^{\prime}},$
(38)

Figure 3: Chirality from Fradkin-Kadanoff transformations

Because of the anomaly, the two parafermionizations do not give the same
algebraic relations whereas the models have been constructed from the
identical model. Because of this mismatch, the chiralities of the theories
appear. One can observe this more generally by considering the parafermionic
partition functions in Yao and Furusaki (2020), and their non-invariance under
the mirror transformation $\tau\leftrightarrow\overline{\tau}$.

For the chiral parafermionic model, one can summarize the partition function
characterized by the simple charge $\\{Q_{I}\\}_{I=1}^{N-1}$ as,

$\begin{split}&Z_{\\{Q_{I}\\}}\\\
&=\sum_{i,p\in\\{Q_{J_{I}}(i,p)=Q_{I}\\}}\left(\chi_{i,p}\right)\left(\sum_{p^{\prime}}\overline{\chi}_{i,p^{\prime}}\right),\end{split}$
(39)

where we have assumed the bosonic modular invariant as
$\sum_{i,p}\chi_{i,p}\overline{\chi}_{i,-p}$ and this does not have $Z_{N}$
invariant sector denoted as $\\{a\\}$ in the previous sections because of the
quantum anomaly. By replacing $\tau$ and $\overline{\tau}$, one can obtain the
antichiral parafermionic model111111It might be interesting to note that
summation of the chiral and antichiral partition functions satisfy the usual
modular $S$ property as in the $Z_{2}$ caseFukusumi (b), and corresponds to
bosonic twisted partition function generated by Verlinde line. This may be
related to anomalous topological order, for example in Kong and Wen (2014),
but it is out of the scope of the present work..

In this sense, the space inversion symmetry or mirror symmetry of the
parafermionic theory is broken by the anomaly. The action of mirror operation
in rational CFT has been studied in Brunner and Hori (2004) for example, and
our observation may clarify the relationship between chirality and anomaly.
This nonuniqueness has appeared in Mong et al. (2014) for example, but we
cannot find the works discussing its relation to the nonuniqueness of
partition function and its interpretation by quantum anomaly except for the
author’s other related workFukusumi (b). This nonuniqueness may be trivial to
some extent by considering the meaning of quantum anomaly in a textbook or
historical aspects of quantum anomalyHarvey (2005). However, we would
emphasize its importance in considering the modular property and locality of
the theory. In the following discussion of this section, we demonstrate the
appearance of such nonuniqueness in the $2+1$ dimensional system constructed
by the bulk-edge correspondence.

Here, we see what happens by constructing $Z_{3}$ FQHE from $SU(3)_{1}$ model
which can be considered as the $SU(3)$ extension of the spinon-holon states in
Ino (1998). We assume the simple bosonic part as in the previous section,
labeled by the parameter $q_{1}$ and $q_{2}$. In the $SU(3)_{1}$ model, there
exist two simple current, $j_{1}$ and $j_{2}$ as we have introduced in the
previous section. We also assume the total $Z_{N}$ parity is given by the
summation of Laughlin parts and $SU(3)_{1}$ part. Corresponding to these
simple currents, the characters of the total system can be modified as,

$\displaystyle\Xi_{0}^{p}$
$\displaystyle=\chi_{0}\sum_{m_{1}+m_{2}=p(\text{mod}.3.)}\theta_{\frac{r_{1}}{q_{1}}}^{m_{1}}\theta_{\frac{r_{2}}{q_{2}}}^{m_{2}},$
(40) $\displaystyle\Xi_{j_{1}}^{p}$
$\displaystyle=\chi_{j_{1}}\sum_{m_{1}+m_{2}=p-1(\text{mod}.3.)}\theta_{\frac{r_{1}+1/3}{q_{1}}}^{m_{1}}\theta_{\frac{r_{2}+2/3}{q_{2}}}^{m_{2}},$
(41) $\displaystyle\Xi_{j_{2}}^{p}$
$\displaystyle=\chi_{j_{2}}\sum_{m_{1}+m_{2}=p-2(\text{mod}.3.)}\theta_{\frac{r_{1}+2/3}{q_{1}}}^{m_{1}}\theta_{\frac{r_{2}+1/3}{q_{2}}}^{m_{2}},$
(42)

where the upper index $p$ labels the parafermionic parity as in the previous
sections. These characters are constructed by considering the parity shift
operation to bosonic parts. Hence the total partition function with modular
$T$ invariance can be obtained as,

$Z_{T-\text{inv}}=\sum_{p,\overline{p}}\Xi^{p}_{0}\overline{\Xi}^{\overline{p}}_{0}+\Xi^{p}_{j_{1}}\overline{\Xi}^{\overline{p}}_{j_{1}}+\Xi^{p}_{j_{2}}\overline{\Xi}^{\overline{p}}_{j_{2}},$
(43)

However, in this case, there exist other parafermionic parity shifts by
operator $j_{1}$ and fractional flux quantum. First, let us assume the
parafermionic parity $0$ sector as,

$Z^{0}=\sum_{p,\overline{p},p+\overline{p}=0}\Xi^{p}_{0}\overline{\Xi}^{\overline{p}}_{0}+\Xi^{p}_{j_{1}}\overline{\Xi}^{\overline{p}}_{j_{1}}+\Xi^{p}_{j_{2}}\overline{\Xi}^{\overline{p}}_{j_{2}}$
(44)

By applying parafermionic parity shift to the chiral part,

$\displaystyle Z^{1}$
$\displaystyle=\sum_{p,\overline{p},p+\overline{p}=1}\Xi^{p}_{j_{1}}\overline{\Xi}^{\overline{p}}_{0}+\Xi^{p}_{j_{2}}\overline{\Xi}^{\overline{p}}_{j_{1}}+\Xi^{p}_{0}\overline{\Xi}^{\overline{p}}_{j_{2}},$
(45) $\displaystyle Z^{2}$
$\displaystyle=\sum_{p,\overline{p},p+\overline{p}=2}\Xi^{p}_{j_{2}}\overline{\Xi}^{\overline{p}}_{0}+\Xi^{p}_{0}\overline{\Xi}^{\overline{p}}_{j_{1}}+\Xi^{p}_{j_{1}}\overline{\Xi}^{\overline{p}}_{j_{2}}$
(46)

Apparently, these two partition functions break modular $T$ invariance. Hence
the summation $\sum Z^{p}$ becomes different from the modular $T$ invariant
Eq. (43). As one can see, the antichiral parafermionic parity shift also
generates different sets of partition functions.

Even in this model, it might be possible to obtain a modular $S$ invariant
partition as in $Z_{2}$ caseIno (1998), but this breaks modular $T$ and
$T^{2}$ invariance, and it may result in nonlocal CFT. Hence, the gauging of
anomalous symmetry leads to nonlocal models, and this may prevent the
condensation of the model. This seems to result in bulk gaplessness under
bulk-edge correspondence, but further numerical tests of lattice models are
required. For example, by taking the thin torus limit, we expect that an
anomalous FQHE may result in a model with nonlocal interaction as in Papić
(2014), because of the nonlocal nature of anomalous symmetryHao et al. (2022);
Chatterjee and Wen (2022a, b)

##  VI $SU(n)$ Haldane conjecture and composite parafermionization

In this section, we revisit $SU(N)$ Haldane conjecture, with emphasis on
quantum anomaly and composite parafermion condensation. Related discussion can
be seen in Yao et al. (2019) with emphasis on LSM anomaly and in Lecheminant
(2015) with emphasis on integrability and coset representations.

As has been proposed in Fuji and Lecheminant (2017); Lecheminant (2015), there
exists mapping from $\\{SU(N)_{1}\\}^{K}$ to $SU(N)_{K}$. Remarkablely, the
mutual simple charges from the simple currents
$\\{\prod_{k=1}^{K}j_{n}^{(k)}\\}^{N-1}_{n=1}$ in the $\\{SU(N)_{1}\\}^{K}$
and those of $\\{J_{I}\\}_{I=1}^{N-1}$ in $SU(N)_{K}$ have the same structure.
This can be interpreted as anomaly matching and this should survive under RG
flow. Hence, by analyzing the simple theory $\\{SU(N)_{1}\\}^{K}$ constructed
by simply taking products of $SU(N)_{1}$, one can expect some information of
RG connectivity of CFT with a smaller central charge, and this corresponds to
the $SU(N)$ Haldane conjecture (FIG.5). The commutativity of the diagram in
FIG.5 can be interpreted as Haldane conjecture. Under the RG flows of the red
arrows, the simple currents anomaly which can be considered as the on-site
symmetry charges, gradually become the LSM-like anomalies. Hence the RG
connectivity of $\\{U(1)\\}^{N-1}$ model with some anisotropy parameters as in
XXZ chain, can be changed depending on simple charge anomaly of the product
theory with a larger central charge. Typically, one may expect to observe the
connectivity problem to $SU(N)_{1}$ WZW modelsYao et al. (2019). It should be
worth noting that this type of product theory has appeared in the analysis of
discrete torsion theoryGato-Rivera and Schellekens (1992), and it has a close
relation with integrable perturbations known as $\lambda$ deformationGeorgiou
and Sfetsos (2017, 2019).

Figure 4: RG flows of $SU(N)$ models. Assocativity of red arrows and blue
arrows corresponds to the Haldane conjecture.

Because of the $Z_{N}$ anomaly matching, one expects
$K-K^{\prime}(\text{mod}.N)$ periodicity naively. Especially, one can
interprete the connectivity of $K=K^{\prime}(\text{mod}N)$ as composite
parafermion condensation of $\\{SU(N)_{1}\\}^{K-K^{\prime}}$ model.

The above analysis is coming from anomaly matching. Related discussion (and
diagram) can be seen in several works, Cabra et al. (1998); Lecheminant
(2015); Cheng and Seiberg (2022), for example. The RG domain wall
interpretation of the model, $\\{SU(N)_{1}\\}^{K+K^{\prime}}$ predicts another
periodicity $K+K^{\prime}\ (\text{mod}.N)$. Interestingly, this periodicity is
different from the above anomaly matching argument.

Here we comment on the structure of coset representations in Fuji and
Lecheminant (2017). The mapping from $\\{SU(N)_{1}\\}^{K}$ to the following
coset model is,

$\\{SU(N)_{1}\\}^{K}\sim\\{U(1)\\}^{N-1}\times\frac{SU(N)_{K}}{\\{U(1)\\}^{N-1}}\times\frac{SU(K)_{N}}{\\{U(1)\\}^{K-1}}$
(47)

where $\frac{SU(N)_{K}}{\\{U(1)\\}^{N-1}}$ and
$\frac{SU(K)_{N}}{\\{U(1)\\}^{K-1}}$ are known as $Z_{K}$ Gepner parafermion
and $Z_{N}$ Gepner parafermion respectivelyGepner (1987). It should be noted
that one can expect $\frac{SU(N)_{K}}{\\{U(1)\\}^{N-1}}$ as $Z_{N}$ anomaly
free and $\frac{SU(K)_{N}}{\\{U(1)\\}^{K-1}}$ as $Z_{K}$ anomaly free as in
the $SU(2)$ caseCabra et al. (1998). Interestingly, the Gepner parafermion
part contains $Z_{K}$ simple current which is not evident from the original
product theory.

The RG flow of this model to $SU(N)_{K}$ can be expressed as,

$\begin{split}&\\{U(1)\\}^{N-1}\times\frac{SU(N)_{K}}{\\{U(1)\\}^{N-1}}\times\frac{SU(K)_{N}}{\\{U(1)\\}^{K-1}}\\\
&\rightarrow\\{U(1)\\}^{N-1}\times\frac{SU(N)_{K}}{\\{U(1)\\}^{N-1}}\sim
SU(N)_{K}\end{split}$ (48)

Hence the $Z_{N}$ anomaly of the model coming from
$\frac{SU(K)_{N}}{\\{U(1)\\}^{K-1}}$ should be transfered to
$\\{U(1)\\}^{N-1}$ part. It is also interesting to note that the level-rank
duality of $SU(N)_{K}$ and $SU(K)_{N}$ explains the emergence of the $Z_{K}$
symmetry which was not evident from the original product theory
$\\{SU(N)_{1}\\}^{K}$. By gapping the parafermion part, one may obtain the
$SU(N)$ Haldane conjecture. Unfortunately, this final step seems to require
some complicated calculations. In principle, by generalizing the
fermionization procedure in Alvarez-Gaume et al. (1986); Yao and Fukusumi
(2019), or directly implementing the momentum shift property of LSM operator
in $SU(N)_{1}\sim\\{U(1)\\}^{N-1}$ CFT, one can represent the total picture in
CFT determined by the anomaly. In this approach, one can obtain the table of
the possible conformal dimensions of perturbations which depend on the
generalized Luttinger parameter for $\\{U(1)\\}^{N-1}$ CFT determined by the
anisotropy parameters of the $SU(N)$ lattice model.

Figure 5: Summary of matching and cancellation flows. Similar to anomaly
analysis by considering the simple charges, one may be possible to analyze the
Galois shuffle property by considering the monodromy charges.

Consequently, there exists at least two periodicity
$K-K^{\prime}(\text{mod.}N)$ and $K+K^{\prime}(\text{mod.}N)$ for
$\\{SU(N)_{1}\\}^{K}$ model121212By considering the level-rank duality of
$SU(N)_{K}$, one can obtain $N+N^{\prime}(\text{mod.}K)$ and
$N-N^{\prime}(\text{mod.}K)$ periodicity of $SU(N)_{K}$ model by fixing $K$.
We thank Yuan Yao for the helpful comments on this aspect.. These two
periodicities come from slightly different mechanisms, and we call the
corresponding integrable flow for $K-K^{\prime}=0(\text{mod.}N)$ and
$K+K^{\prime}=0(\text{mod.}N)$ as ”anomaly matching flow” and ”anomaly
cancellation flow ” respectively.

As we have discussed in section 3, one can consider similar matching and
cancellation for Galois shuffle in RG connective theories. Let us introduce
the two theory $M_{1}$ with simple currents $J_{I}^{(1)}$ and shuffle
generator $\sigma$, and $M_{2}$ with simple current $J_{I}^{(2)}$ and the
shuffle generator $\sigma^{\prime}$. Then, one can express the (hopefully
integrable) flow with
$Q_{J_{I}^{(1)}}(\sigma)-Q_{J_{I}^{(2)}}(\sigma^{\prime})=0$ as ”shuffle
matching flow” and that with
$Q_{J_{I}^{(1)}}(\sigma)+Q_{J_{I}^{(2)}}(\sigma^{\prime})=0$ as ”shuffle
cancellation flow”. Hopefully, this may give a new criterion to establish a
classification of nonunitary theories.

In summary, we have discussed the importance of the RG analysis of CFT
described by the products of simple theories possibly with large central
charge, with emphasis on anomaly131313In this sense, studying CFTs with large
central charge or gigantic symmetries are reasonable.. The essence of our
proposal is simple in that RG started from sufficient bulk and boundary
degrees of freedom should explain a class of nontrivial symmetry-related
phenomena, such as $SU(N)$ Haldane conjecture.

##  VII Conclusion

In this work, we have constructed topologically ordered systems with simple
current anomaly free theory. The cylinder partition functions constructed in
this work satisfy the necessary and sufficient conditions for the stability in
the existing literature almost automatically. It may be worth stressing that
we have never used the category theory at least directly except for assuming
the bosonic charge conjugated partition functions. We hope this work may give
a new bridge to connect experimentalists in condensed matter physics and
theorists in all related areas such as high energy physics and mathematical
physics.

We have viewed several modern aspects of the topological phase, such as
confinement, fractional supersymmetry, and non-locality, with emphasis on the
quantum anomaly in the simple models constructed by the bulk-edge
correspondence. We assume $CFT_{D}/BQFT_{D+1}$ correspondence known as bulk-
edge correspondence and its RG flow to $CFT_{D}/TQFT_{D+1}$ correspondence.
This proposition is implicitly assumed by a lot of existing literature,
typically in the name of $AdS/CFT$ or $AdS/CMT$ correspondence. So far,
because we have constructed everything almost automatically, there appears a
simple question. We have assumed the protectedness of edge modes, but it is
not clear whether this can be shown by using a traditional RG argument, in the
language of bulk and boundary RG flow. Hence for the future problem, as the
author has emphasized in the related workFukusumi (a), it may become important
to check the extent of the protectedness by using field theory and numerical
simulation of lattice models. For example, developments of the truncated
conformal space approach (TCSA) in higher dimensional CFT with boundary seem
necessaryJames et al. (2018). However, the work for higher dimensional CFT is
limitedHogervorst et al. (2015). Moreover, there exist some open problems for
the appropriate definition of the operators at the boundary even in lower
dimensions. For example, one cannot use the Kac identification or other useful
property of the bulk local operatorsGori and Viti (2017). This type of
difficulty has never captured sufficient attention in the fields, but to study
edge modes of topological phase by using RG, it seems impossible to avoid this
difficulty.

As a concluding remark, we emphasize the analogy with FQHE and string theory,
known as 2d quantum gravity coupled to matter. In FQHE, the most essential
part of the construction is the single-valuedness of the wavefunction. This
restricts the building structure of CFT correlation functions as a
wavefunction and results in the nontrivial emergence of fractional flux
quantum. Historically, the single-value condition appeared in the study of
strong interaction and non QFT (or non-Lagrangian) aspects appear naturally.
As we have discussed in the series of previous worksFukusumi (b, a), there
exist common properties between FQHE and 2d quantum gravity coupled to
matterPolyakov (1981). We summarize the point in the following figure (FIG.
6). The appearance of fractional flux quantum in FQHE system corresponds to
the appearance of gravitational dressing by the KPZ relationDavid (1988);
Knizhnik et al. (1988); Distler and Kawai (1989). The correspondence between
the fusion rules in the original matter CFT and 2d gravity coupled to matter
is called ground ring Lian and Zuckerman (1991a, b); Kutasov et al. (1992),
and this corresponds to the abelian property of the multicomponent bosons in
FQHE. There exist related works which point out connections between FQHE and
string theories by using the holography principle Fujita et al. (2009); Ryu
and Takayanagi (2010); Bayntun et al. (2011); Karch et al. (2010); Melnikov et
al. (2013); Wu and Wu (2015); Mezzalira and Parnachev (2016). One can say that
our argument is a more direct approach to construct the corresponding systems
in operator formalism in quantum method whereas these works use the Lagrangian
formalism which corresponds to higher dimensional classical methods.

Figure 6: Analogy between 2d quantum gravity couple to matter and FQHE. FQHE
can be considered as a gauge theory analogous to string theory, in the context
of gauge/gravity duality.

In statistical mechanics, 2d quantum gravity coupled to matter is interpreted
as coupling of matter and random surfaceWatabiki (1993). Hence one can
interpret gravity as random objects, which can be realized in a probabilitic
model. For example, one can find several classes of statistical mechanical
models corresponding to this gravity, such as whole-plane or backward Schramm-
Loewner evolutionDuplantier et al. (2012); Katori and Koshida (2020),
Laplacian growthAlekseev (2019), log-correlated random energy modelCao et al.
(2017), and Gaussian multiplicative chaosRhodes and Vargas (2013). Hence, if
there exists a similar interpretation for the 2+1 dimensional system
constructed from 2d quantum gravity coupled to matter as for FQHE, one can
expect applications of such models to condensed matter systems with impurities
or random potential for example. Similar approaches can be seen in the random
Dirac fermion model related to localization problem Kogan et al. (1996);
Bhaseen et al. (2001); Bhaseen (2000). Moreover, at first sight, this coupling
to gravity automatically cancels the anomaly of simple currents even when the
original matter CFT is anomalous. Hence further studies of 2d quantum gravity
constructed from a composite parafermionic theory may give a unified
understanding and construction of general topological ordered systems.

##  VIII Acknowlegement

The author thanks Yuji Tachikawa and Yunqin Zheng for the fruitful
collaboration in the previous project. He also thanks Yang Bo for the
discussion and collaboration with the other related projects. He also thanks
Yuan Yao for the stimulating discussions about parafermionization and Yohei
Fuji for introducing the basic aspects of $Z_{N}$ fractional quantum Hall
effects, and Shunsuke Furukawa for the discussion about Renyification. He also
thanks Ken Kikuchi for the discussions and his comments on the manuscript.

##  Appendix A Summary of parafermionic partition function

In this section, we summarize the partition functions which we have introduced
in the main text. First, let us introduce the $Z_{N}$ model with charge
conjugated modular invariant,

$Z=\sum_{i,p}\chi_{i,p}\overline{\chi}_{i,-p}+\sum_{a}|\chi_{a}|^{2}$ (49)

where $\\{i,p\\}_{p=0}^{N-1}$ is the label of the $Z_{N}$ noninvariant states,
and $\\{a\\}$ is that of the $Z_{N}$ invariant states and $\chi$
($\overline{\chi}$) is the corresponding chiral (antichiral) character with
the modular parameter $\tau$ ($\overline{\tau}$).

In the anomaly free case, the composite parafermionic partition function for
each simple charge sector $\\{Q_{I}\\}_{I=1}^{N-1}$ where
$\\{Q_{I}\\}_{I=0}^{N-1}$ corresponds to the simple charge
$\\{Q_{J_{I}}(\alpha)\\}_{I=1}^{N-1}$ for the operator labeled $\alpha$ under
$Z_{N}$ simple current $\\{J_{I}\\}_{I=1}^{N-1}$ is expressed as,

$Z_{\\{Q_{I}\\}}=\sum_{i\in\\{Q_{J_{I}}(i)=Q_{I}\\}}|\sum_{p}\chi_{i,p}|^{2}+N\sum_{a\in\\{Q_{J_{I}}(a)=Q_{I}\\}}|\chi_{a}|^{2}.$
(50)

By considering the fractional flux quantum insertion corresponding to above
$\\{Q_{I}\\}_{I=1}^{N-1}$ and assigning modular $T$ or $T^{2}$ invariance, one
can obtain the $Z_{N}$ FQHE partition functions in the cylinder geometry in
the main section. The total (modular noninvariant) partition function is

$Z=\sum_{\\{Q_{I}\\}}\left(\sum_{\\{r_{I}\\}}\left(\sum_{i}|\sum_{p}\Xi_{i,p,\\{r_{I}\\}}|^{2}+N\sum_{a}|\Xi_{a,\\{r_{I}\\}}|^{2}\right)\right)$
(51)

where we have taken the same notations in the main text in section II.1, and
introduced a character
$\Xi_{i,p,\\{r_{I}\\}}=\sum_{p^{\prime}}\Xi_{i,p,\\{r_{I}\\}}^{p^{\prime}}$.

In an anomalous case, the chiral parafermionization gives the following
partition function,

$Z_{\\{Q_{I}\\}}=\sum_{i,p\in\\{Q_{J_{I}}(i,p)=Q_{I}\\}}\left(\chi_{i,p}\right)\left(\sum_{p^{\prime}}\overline{\chi}_{i,p^{\prime}}\right),$
(52)

by replacing $\tau\leftrightarrow\overline{\tau}$, one can obtain the
partition function corresponding to antichiral parafermionization.

Here, we comment on two straightforward generalizations of our approach.
First, by replacing simple charge with monodromy charge, one can easily obtain
the parafermionic partition function for nonunitary theories. Second, by
considering operator counting of the bosonic model, one can obtain a similar
form of parafermionization in general.

##  Appendix B Comments on subalgebra structure in fusion rule

As we have shown, the introduction of simple charges of the fields naturally
extracts the subalgebra structure of the fusion rule of CFT. In our
construction, there exists the other fundamental structure, parafermionic
parity. For simplicity, let us focus on the $Z_{2}$ case. In this fermionic
theory, the fermionic parity of an operator is even or odd. Hence assuming
that this parity is well-defined, one can obtain the set of the parity even
fields as a subalgebra, because it satisfies,

$\phi_{\alpha}^{\text{even}}\times\phi_{\alpha^{\prime}}^{\text{even}}=\sum_{\beta}N_{\alpha,\alpha^{\prime}}^{\beta}\phi_{\beta}^{\text{even}}$
(53)

As one can obtain untwisted part of the fusion rule by the parafermionization,
it might be interesting to consider whether it is possible to obtain this
subalgebra by considering the generalized gauging141414In finalizing this
work, we have noticed related literature considering the similar (or possibly
the same) operation Runkel et al. (2022).. To distinguish parity odd and even
states, it might be useful to consider parity operators $(-1)^{F}$,
$(-1)^{\overline{F}}$, $(-1)^{F+\overline{F}}$ where $F$ (or $\overline{F}$)
is chiral (or antichiral) fermionic parity operator. As a simple example, we
note the appearance of Fibonacci anyon, with fusion rule
$\tau\times\tau=I+\tau$, from the parafermionic parity 0 sector of the three
state Potts model and from the untwisted fermionic parity 0 sector of the
tricritical Ising model. One can systematically obtain such subalgebra
structures by considering the simple charge and the parafermionic parity of
the model.

In the existing literature, there exist several proposals to insert
topological symmetry operators, but we comment here on other more traditional
interpretation based on Hamiltonian formalism. As the author has discussed in
Fukusumi and Iino (2020), the insertion of the topological symmetry operator
can be understood as the insertion of generalized twisted boundary condition
or integral of motion (We denote the topological symmetry operators as $Q_{s}$
and $Q_{t}$ corresponding to each direction). This interpretation can be seen
in the study of Temperley-Lieb algebra description of lattice models and the
corresponding CFTsBelletête et al. (2020). It should be noted, these
operations themselves cannot be treated in a local manner in quantum
systemsRoy and Saleur (2022, 2021). Whereas this insertion of topological
symmetry operator in the Lagargian formalism is established, there exists
another ”gauging” operation like in lattice gauge field theory,

$H^{\prime}=H_{CFT}+gQ_{t}+\text{h.c.c.}.$ (54)

By taking the large $|g|$ limit, one can obtain topological sectors of CFT
corresponding to the eigenvalue of the integral of motions $Q_{t}$. This can
be interpreted as a generalization of Gauss law in the lattice gauge models.
One can easily generalize this analysis to higher-form objects in general
dimensions. It might be interesting whether this ”gauging” can reproduce other
types of local models, such as anyonic modelsFeiguin et al. (2007).

###  B.1 Ising chain and gauging

As a simplest example for the gauging Eq. (54), let us consider the transverse
Ising model at criticality and the corresponding partition function by CFT.
The lattice Hamiltonian of the model is,

$H_{\text{Ising}}=-\sum_{l}\left(\sigma_{l}\sigma_{l+1}+\tau_{l}\right).$ (55)

The $Z_{2}$ symmetry operation which corresponds to topological defect
$Q_{\epsilon}$ for Ising CFT is,

$Q_{\epsilon}=\prod_{l}\tau_{l},$ (56)

This commute with Hamiltonian and one can consider the spectrum of each charge
sector with eigenvalue $\pm 1$. The partition function of the original
Hamiltonian with periodic boundary condition or twisted boundary condition is,

$\displaystyle Z_{\ ,+}$
$\displaystyle=|\chi_{0}|^{2}+|\chi_{\epsilon}|^{2}+|\chi_{\sigma}|^{2},$ (57)
$\displaystyle Z_{\ ,-}$
$\displaystyle=\chi_{0}\overline{\chi_{\epsilon}}+\chi_{\epsilon}\overline{\chi_{0}}+|\chi_{\sigma}|^{2},$
(58)

It should be noted that each sector can be labeled by the eigenvalue of
$Q_{\epsilon}$ or simple charge in the main section. Hence the partition
function can be decomposed as,

$\displaystyle Z_{+,+}$ $\displaystyle=|\chi_{0}|^{2}+|\chi_{\epsilon}|^{2},$
(59) $\displaystyle Z_{+,+}$ $\displaystyle=|\chi_{\sigma}|^{2},$ (60)
$\displaystyle Z_{+,-}$
$\displaystyle=\chi_{0}\overline{\chi_{\epsilon}}+\chi_{\epsilon}\overline{\chi_{0}},$
(61) $\displaystyle Z_{-,-}$ $\displaystyle=|\chi_{\sigma}|^{2},$ (62)

where $Z_{\ ,\pm}=Z_{+,\pm}+Z_{-,\pm}$.

Here we introduce the ”gauged” Hamiltonian,

$H_{\text{Ising}}+gQ_{\epsilon}$ (63)

By taking $|g|$ large limit, splitting by simple charge becomes large and the
partition function $Z_{\pm,\pm}$ and $Z_{\pm,\mp}$ appear. There exist
following remarkable relations,

$\displaystyle Z_{NS}$
$\displaystyle=Z_{+,+}+Z_{+,-}=|\chi_{0}+\chi_{\epsilon}|^{2},$ (64)
$\displaystyle Z_{R}$ $\displaystyle=Z_{-,+}+Z_{-,-}=2|\chi_{\sigma}|^{2}.$
(65)

The summation $Z_{\pm,+}+Z_{\pm,-}$ was taken corresponding to the Gauss law.
This is analogous to the various models by considering $Z_{2}$ gauging, and
one can see the nonlocal aspects of the gauged theory as an addition of
$Q_{\epsilon}$ started from a conformal field theory. Moreover, as we have
discussed in the main text and the related paper by the authorFukusumi (a),
the energy splitting by $g$ and duality (or supersymmetry) between NS and R
sector is analogous to quark confinement. In this section, we have mainly used
the operator formalism, but this appearance of Gauss law in the model can be
considered as a Hamiltonian and discrete symmetry analog of the established
proposals of chiral gauging in the Lagrangian formalismFaddeev (1984); Hosono
(1989); Fujiwara et al. (1988)

By considering the $Z_{N}$ symmetry operator and the lattice realization of
composite parafermions, one can easily generalize the discussion above.

##  Appendix C Composite parafermion on the lattice

In analogy with CFT, one can construct lattice composite parafermion by
coupling the lattice parafermion $N$ times in general $N$ and $N/2$ times for
$N$ even with the same chirality. First, let us consider the general case, and
assume $Z_{N}$ parafermionic lattice systems $\\{\gamma_{i}\\}$ with $NL$
site. Then, one can define the following lattice composite parafermion

$\Gamma_{i}=\gamma_{i}\times\gamma_{i+L}\times...\times\gamma_{i+(N-1)L}=\prod_{j=0}^{N-1}\gamma_{i+jL}$
(66)

where $i$ takes $i=1$ to $i=L$. This is a lattice analog of the composite
parafermion with the following algebraic relation,

$\displaystyle\Gamma_{i}\Gamma_{j}=\Gamma_{j}\Gamma_{i},$ (67)
$\displaystyle\Gamma_{i}^{N}=1,\Gamma_{i}^{\dagger}=\Gamma_{i}^{-1},$ (68)

For $N$ even case, by coupling the parafermion $N/2$ times, one can obtain the
similar relations,

$\displaystyle\Gamma_{i}\Gamma_{j}+\Gamma_{j}\Gamma_{i}=0,$ (69)
$\displaystyle\Gamma_{i}^{N}=1,\Gamma_{i}^{\dagger}=\Gamma_{i}^{-1}.$ (70)

For simplicity, let us observe the nonuniqueness of composite
parafermionization for $N=2$ and study its relation to $SU(2)_{K}$ Haldane
conjecture. In $SU(2)_{K}$ model, there exists periodicty $K(\text{mod}.4)$
for statistcs of the $Z_{2}$ simple current, and for $K=0(\text{mod}.2)$
corresponds to anomaly free case. Hence we introduce $K=2S$ where $S$ is an
integer and corresponds to the spin of the Heisenberg spin chain. In this
setting there exists another periodicity $S(\text{mod}2)$ related to hidden
$Z_{2}\times Z_{2}$ symmetryOshikawa (1992). Then, one can interpret $S$ as
the number of Majorana fermions because of anomaly matching, this corresponds
to the above two different types of composite fermionization. As we have
stressed in the main text, the nontrivial hidden symmetries can be more
evident by considering the theory with a larger central charge.

For a future problem, it might be interesting to study the phase diagram of
systems which is described by such composite particles. For example, our
formulation may give an understanding of the lattice fractional supersymmetric
model in Mong et al. (2014). We expect the appearance of fractional spin
particles by twisted boundary conditions, which are commonly used in the
bosonic Coulomb gas representation, as we have discussed in the main text.

## References

  * Schoutens and Wen (2016) K. Schoutens and X.-G. Wen, Phys. Rev. B 93, 045109 (2016), eprint 1508.01111.
  * Fateev and Zamolodchikov (1985) V. A. Fateev and A. B. Zamolodchikov, Sov. Phys. JETP 62, 215 (1985), URL http://www.jetp.ac.ru/cgi-bin/e/index/e/62/2/p215?a=list.
  * Fuji and Lecheminant (2017) Y. Fuji and P. Lecheminant, Physical Review B 95 (2017), URL https://doi.org/10.1103%2Fphysrevb.95.125130.
  * Kane et al. (2002) C. L. Kane, R. Mukhopadhyay, and T. C. Lubensky, Physical Review Letters 88 (2002), URL https://doi.org/10.1103%2Fphysrevlett.88.036401.
  * Teo and Kane (2014) J. C. Y. Teo and C. L. Kane, Physical Review B 89 (2014), URL https://doi.org/10.1103%2Fphysrevb.89.085101.
  * Sagi and Oreg (2014) E. Sagi and Y. Oreg, Physical Review B 90 (2014), URL https://doi.org/10.1103%2Fphysrevb.90.201102.
  * Imamura et al. (2019) Y. Imamura, K. Totsuka, and T. H. Hansson, Physical Review B 100 (2019), URL https://doi.org/10.1103%2Fphysrevb.100.125148.
  * Levin (2013) M. Levin, Physical Review X 3, 021009 (2013), eprint 1301.7355.
  * Dotsenko and Fateev (1984) V. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 240, 312 (1984).
  * Dotsenko and Fateev (1985) V. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 251, 691 (1985).
  * Estienne and Santachiara (2009) B. Estienne and R. Santachiara, J. Phys. A 42, 445209 (2009), eprint 0906.1969.
  * Estienne et al. (2010) B. Estienne, B. A. Bernevig, and R. Santachiara, Phys. Rev. B 82, 205307 (2010), eprint 1005.3475.
  * Estienne et al. (2012) B. Estienne, V. Pasquier, R. Santachiara, and D. Serban, Nucl. Phys. B 860, 377 (2012), eprint 1110.1101.
  * Kong and Wen (2014) L. Kong and X.-G. Wen (2014), eprint 1405.5858.
  * Cho et al. (2017) G. Y. Cho, S. Ryu, and C.-T. Hsieh, Phys. Rev. B 96, 195105 (2017), eprint 1705.03892.
  * Lanzetta and Fidkowski (2022) R. A. Lanzetta and L. Fidkowski (2022), eprint 2207.05092.
  * Li et al. (2022) L. Li, C.-T. Hsieh, Y. Yao, and M. Oshikawa (2022), eprint 2205.11190.
  * Fuchs (1997) J. Fuchs, Lect. Notes Phys. 498, 1 (1997), eprint hep-th/9702194.
  * Schweigert et al. (2000) C. Schweigert, J. Fuchs, and J. Walcher, in _Eotvos Summer School in Physics: Nonperturbative QFT Methods and Their Applications_ (2000), pp. 37–93, eprint hep-th/0011109.
  * Bhardwaj and Tachikawa (2018) L. Bhardwaj and Y. Tachikawa, JHEP 03, 189 (2018), eprint 1704.02330.
  * Tachikawa (2020) Y. Tachikawa, SciPost Phys. 8, 015 (2020), eprint 1712.09542.
  * Haldane (1983a) F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983a).
  * Haldane (1983b) F. D. M. Haldane, Phys. Lett. A 93, 464 (1983b).
  * Cabra et al. (1998) D. C. Cabra, P. Pujol, and C. von Reichenbach, Phys. Rev. B 58, 65 (1998), eprint hep-th/9802014.
  * Lecheminant (2015) P. Lecheminant, Nucl. Phys. B 901, 510 (2015), eprint 1509.01680.
  * Lajkó et al. (2017) M. Lajkó, K. Wamer, F. Mila, and I. Affleck, Nucl. Phys. B 924, 508 (2017), [Erratum: Nucl.Phys.B 949, 114781 (2019)], eprint 1706.06598.
  * Wamer et al. (2020) K. Wamer, M. Lajkó, F. Mila, and I. Affleck, Nucl. Phys. B 952, 114932 (2020), eprint 1910.08196.
  * Yao and Furusaki (2020) Y. Yao and A. Furusaki (2020), eprint 2012.07529.
  * Alcaraz and Moreo (1992) F. C. Alcaraz and A. Moreo, Phys. Rev. B 46, 2896 (1992), URL https://link.aps.org/doi/10.1103/PhysRevB.46.2896.
  * Alcaraz and Martins (1989) F. C. Alcaraz and M. J. Martins, Journal of Physics A: Mathematical and General 22, 1829 (1989), URL https://doi.org/10.1088%2F0305-4470%2F22%2F11%2F023.
  * Hsieh et al. (2020) C.-T. Hsieh, Y. Nakayama, and Y. Tachikawa (2020), eprint 2002.12283.
  * Fukusumi et al. (2021) Y. Fukusumi, Y. Tachikawa, and Y. Zheng, SciPost Phys. 11, 082 (2021), eprint 2103.00746.
  * Gaiotto (2012) D. Gaiotto, JHEP 12, 103 (2012), eprint 1201.0767.
  * Poghosyan and Poghosyan (2015) G. Poghosyan and H. Poghosyan, JHEP 05, 043 (2015), eprint 1412.6710.
  * Brunner and Schmidt-Colinet (2016) I. Brunner and C. Schmidt-Colinet, J. Phys. A 49, 195401 (2016), eprint 1508.04350.
  * Stanishkov (2016) M. Stanishkov, JHEP 08, 096 (2016), eprint 1606.03605.
  * Poghosyan and Poghossian (2022) H. Poghosyan and R. Poghossian, JHEP 08, 307 (2022), eprint 2205.05091.
  * Kimura and Murata (2014) T. Kimura and M. Murata, Nuclear Physics B 885, 266 (2014), eprint 1402.6705.
  * Kimura and Murata (2015) T. Kimura and M. Murata, Journal of High Energy Physics 2015, 72 (2015), eprint 1505.05275.
  * Quella et al. (2007) T. Quella, I. Runkel, and G. M. T. Watts, Journal of High Energy Physics 2007, 095 (2007), eprint hep-th/0611296.
  * Kong and Zhang (2022) L. Kong and Z.-H. Zhang (2022), eprint 2205.05565.
  * Kaidi et al. (2021) J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao (2021), eprint 2107.13091.
  * Furuya and Oshikawa (2017) S. C. Furuya and M. Oshikawa, Phys. Rev. Lett. 118, 021601 (2017), eprint 1503.07292.
  * Numasawa and Yamaguch (2018) T. Numasawa and S. Yamaguch, JHEP 11, 202 (2018), eprint 1712.09361.
  * Kikuchi and Zhou (2019) K. Kikuchi and Y. Zhou (2019), eprint 1908.02918.
  * Yao et al. (2019) Y. Yao, C.-T. Hsieh, and M. Oshikawa, Phys. Rev. Lett. 123, 180201 (2019), eprint 1805.06885.
  * Gato-Rivera and Schellekens (1992) B. Gato-Rivera and A. N. Schellekens, Commun. Math. Phys. 145, 85 (1992).
  * Schellekens (1990) A. N. Schellekens, Phys. Lett. B 244, 255 (1990).
  * Fukusumi (a) Y. Fukusumi, in preparation.
  * Fukusumi (b) Y. Fukusumi, in preparation.
  * Ino (1998) K. Ino, Nucl. Phys. B 532, 783 (1998), eprint cond-mat/9804198.
  * Milovanovic and Read (1996) M. Milovanovic and N. Read, Phys. Rev. B 53, 13559 (1996), eprint cond-mat/9602113.
  * Cappelli and Zemba (1997) A. Cappelli and G. R. Zemba, Nucl. Phys. B 490, 595 (1997), eprint hep-th/9605127.
  * Cheng et al. (2022) M. Cheng, P.-S. Hsin, and C.-M. Jian (2022), eprint 2205.15347.
  * Cheng and Jian (2022) M. Cheng and C.-M. Jian, SciPost Phys. 12, 202 (2022), eprint 2201.07239.
  * Ahn et al. (1990) C. Ahn, D. Bernard, and A. LeClair, Nucl. Phys. B 346, 409 (1990).
  * Durand (1992) S. Durand, Mod. Phys. Lett. A 7, 2905 (1992), eprint hep-th/9205086.
  * Mohammedi (1995) N. Mohammedi, Mod. Phys. Lett. A 10, 1287 (1995), eprint hep-th/9412133.
  * Isaev et al. (2002) A. P. Isaev, Z. Popowicz, and O. Santillan, in _4th International Workshop on Supersymmetry and Quantum Symmetries: 16th Max Born Symposium_ (2002), pp. 131–143, eprint hep-th/0110246.
  * Rausch de Traubenberg and Slupinski (2000) M. Rausch de Traubenberg and M. J. Slupinski, J. Math. Phys. 41, 4556 (2000), eprint hep-th/9904126.
  * OBrien and Fendley (2020) E. OBrien and P. Fendley, SciPost Physics 9 (2020), URL https://doi.org/10.21468%2Fscipostphys.9.6.088.
  * Mong et al. (2014) R. S. K. Mong, D. J. Clarke, J. Alicea, N. H. Lindner, and P. Fendley, J. Phys. A 47, 452001 (2014), eprint 1406.0846.
  * Perez et al. (1996) A. Perez, M. Rausch de Traubenberg, and P. Simon, Nucl. Phys. B 482, 325 (1996), eprint hep-th/9603149.
  * Wang et al. (2017) C. Wang, A. Vishwanath, and B. I. Halperin, Physical Review B (2017).
  * Mross et al. (2018) D. F. Mross, Y. Oreg, A. Stern, G. Margalit, and M. Heiblum, Physical Review Letters 121 (2018), URL https://doi.org/10.1103%2Fphysrevlett.121.026801.
  * Bardeen et al. (1957) J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
  * Kong (2013) L. Kong, Nuclear Physics 886, 436 (2013).
  * Neupert et al. (2016) T. Neupert, H. He, C. von Keyserlingk, G. Sierra, and B. A. Bernevig, Physical Review B 93 (2016), URL https://doi.org/10.1103%2Fphysrevb.93.115103.
  * Eliëns et al. (2014) I. S. Eliëns, J. C. Romers, and F. A. Bais, Phys. Rev. B 90, 195130 (2014), eprint 1310.6001.
  * Burnell (2018) F. Burnell, Annual Review of Condensed Matter Physics 9, 307 (2018), URL https://doi.org/10.1146%2Fannurev-conmatphys-033117-054154.
  * David (1988) F. David, Mod. Phys. Lett. A 3, 1651 (1988).
  * Knizhnik et al. (1988) V. G. Knizhnik, A. M. Polyakov, and A. B. Zamolodchikov, Mod. Phys. Lett. A 3, 819 (1988).
  * Distler and Kawai (1989) J. Distler and H. Kawai, Nucl. Phys. B 321, 509 (1989).
  * Jolicoeur et al. (2014) T. Jolicoeur, T. Mizusaki, and P. Lecheminant, Phys. Rev. B 90, 075116 (2014), eprint 1406.5891.
  * Ino (2000) K. Ino (2000), eprint cond-mat/0011101.
  * Lou et al. (2021) J. Lou, C. Shen, C. Chen, and L.-Y. Hung, JHEP 02, 171 (2021), eprint 2007.10562.
  * Qi et al. (2012) X.-L. Qi, H. Katsura, and A. W. W. Ludwig, Physical Review Letters 108 (2012), URL https://doi.org/10.1103%2Fphysrevlett.108.196402.
  * Das and Datta (2015) D. Das and S. Datta, Phys. Rev. Lett. 115, 131602 (2015), eprint 1504.02475.
  * Ebisu and Watanabe (2021) H. Ebisu and M. Watanabe, Phys. Rev. B 104, 195124 (2021), eprint 2103.01101.
  * Wen (1990) X. G. Wen, Phys. Rev. B 41, 12838 (1990).
  * Cardy (2017) J. Cardy, SciPost Phys. 3, 011 (2017), eprint 1706.01568.
  * Papić (2014) Z. Papić , Physical Review B 90 (2014), URL https://doi.org/10.1103%2Fphysrevb.90.075304.
  * Davenport et al. (2013) S. C. Davenport, E. Ardonne, N. Regnault, and S. H. Simon, Physical Review B 87 (2013), URL https://doi.org/10.1103%2Fphysrevb.87.045310.
  * Gannon (2003) T. Gannon, Nucl. Phys. B 670, 335 (2003), eprint hep-th/0305070.
  * Harvey et al. (2020) J. A. Harvey, Y. Hu, and Y. Wu, J. Phys. A 53, 334003 (2020), eprint 1912.11955.
  * Cappelli and Viola (2011) A. Cappelli and G. Viola, J. Phys. A 44, 075401 (2011), eprint 1007.1732.
  * Thorngren and Wang (2021) R. Thorngren and Y. Wang (2021), eprint 2106.12577.
  * Kitaev (2006) A. Kitaev, Annals Phys. 321, 2 (2006), eprint cond-mat/0506438.
  * Oshikawa (1992) M. Oshikawa, Journal of Physics: Condensed Matter 4, 7469 (1992), URL https://dx.doi.org/10.1088/0953-8984/4/36/019.
  * Hao et al. (2022) J.-X. Hao, W. Li, and Y. Qi (2022), eprint 2211.05618.
  * Chatterjee and Wen (2022a) A. Chatterjee and X.-G. Wen (2022a), eprint 2205.06244.
  * Chatterjee and Wen (2022b) A. Chatterjee and X.-G. Wen (2022b), eprint 2203.03596.
  * Albertini (1994) G. Albertini, Int. J. Mod. Phys. A 9, 4921 (1994), eprint hep-th/9310133.
  * Santos and Hughes (2017) L. H. Santos and T. L. Hughes, Physical Review Letters 118 (2017), URL https://doi.org/10.1103%2Fphysrevlett.118.136801.
  * Alicea and Fendley (2016) J. Alicea and P. Fendley, Ann. Rev. Condensed Matter Phys. 7, 119 (2016), eprint 1504.02476.
  * Brunner and Hori (2004) I. Brunner and K. Hori, JHEP 11, 005 (2004), eprint hep-th/0303135.
  * Harvey (2005) J. A. Harvey (2005), eprint hep-th/0509097.
  * Georgiou and Sfetsos (2017) G. Georgiou and K. Sfetsos, JHEP 11, 078 (2017), eprint 1707.05149.
  * Georgiou and Sfetsos (2019) G. Georgiou and K. Sfetsos, JHEP 03, 094 (2019), eprint 1812.04033.
  * Cheng and Seiberg (2022) M. Cheng and N. Seiberg (2022), eprint 2211.12543.
  * Gepner (1987) D. Gepner, Nucl. Phys. B 290, 10 (1987).
  * Alvarez-Gaume et al. (1986) L. Alvarez-Gaume, G. W. Moore, P. C. Nelson, C. Vafa, and J. b. Bost, Phys. Lett. B 178, 41 (1986).
  * Yao and Fukusumi (2019) Y. Yao and Y. Fukusumi, Phys. Rev. B 100, 075105 (2019), eprint 1902.06584.
  * James et al. (2018) A. J. A. James, R. M. Konik, P. Lecheminant, N. J. Robinson, and A. M. Tsvelik, Reports on Progress in Physics 81, 046002 (2018), URL https://doi.org/10.1088%2F1361-6633%2Faa91ea.
  * Hogervorst et al. (2015) M. Hogervorst, S. Rychkov, and B. C. van Rees, Phys. Rev. D 91, 025005 (2015), eprint 1409.1581.
  * Gori and Viti (2017) G. Gori and J. Viti, Phys. Rev. Lett. 119, 191601 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.119.191601.
  * Polyakov (1981) A. M. Polyakov, Phys. Lett. B 103, 207 (1981).
  * Lian and Zuckerman (1991a) B. H. Lian and G. J. Zuckerman, Commun. Math. Phys. 135, 547 (1991a).
  * Lian and Zuckerman (1991b) B. H. Lian and G. J. Zuckerman, Phys. Lett. B 254, 417 (1991b).
  * Kutasov et al. (1992) D. Kutasov, E. J. Martinec, and N. Seiberg, Phys. Lett. B 276, 437 (1992), eprint hep-th/9111048.
  * Fujita et al. (2009) M. Fujita, W. Li, S. Ryu, and T. Takayanagi, JHEP 06, 066 (2009), eprint 0901.0924.
  * Ryu and Takayanagi (2010) S. Ryu and T. Takayanagi, Phys. Rev. D 82, 086014 (2010), eprint 1007.4234.
  * Bayntun et al. (2011) A. Bayntun, C. P. Burgess, B. P. Dolan, and S.-S. Lee, New J. Phys. 13, 035012 (2011), eprint 1008.1917.
  * Karch et al. (2010) A. Karch, J. Maciejko, and T. Takayanagi, Phys. Rev. D 82, 126003 (2010), eprint 1009.2991.
  * Melnikov et al. (2013) D. Melnikov, E. Orazi, and P. Sodano, JHEP 05, 116 (2013), eprint 1211.1416.
  * Wu and Wu (2015) C. Wu and S.-F. Wu, JHEP 01, 120 (2015), eprint 1409.1178.
  * Mezzalira and Parnachev (2016) A. Mezzalira and A. Parnachev, Nucl. Phys. B 904, 448 (2016), eprint 1512.06052.
  * Watabiki (1993) Y. Watabiki, Prog. Theor. Phys. Suppl. 114, 1 (1993).
  * Duplantier et al. (2012) B. Duplantier, N. T. P. Chi, N. T. T. Nga, and M. Zinsmeister, _The coefficient problem and multifractality of whole-plane sle and lle_ (2012), URL https://arxiv.org/abs/1211.2451.
  * Katori and Koshida (2020) M. Katori and S. Koshida, Journal of Mathematical Physics 61, 083301 (2020), URL https://doi.org/10.1063%2F1.5145357.
  * Alekseev (2019) O. Alekseev, Phys. Rev. E 100, 012130 (2019), eprint 1710.08303.
  * Cao et al. (2017) X. Cao, P. Le Doussal, A. Rosso, and R. Santachiara, Phys. Rev. Lett. 118, 090601 (2017), eprint 1611.02193.
  * Rhodes and Vargas (2013) R. Rhodes and V. Vargas, _Gaussian multiplicative chaos and applications: a review_ (2013), URL https://arxiv.org/abs/1305.6221.
  * Kogan et al. (1996) I. I. Kogan, C. Mudry, and A. M. Tsvelik, Phys. Rev. Lett. 77, 707 (1996), eprint cond-mat/9602163.
  * Bhaseen et al. (2001) M. J. Bhaseen, J. S. Caux, I. I. Kogan, and A. M. Tsvelik, Nucl. Phys. B 618, 465 (2001), eprint cond-mat/0012240.
  * Bhaseen (2000) M. J. Bhaseen (2000), eprint cond-mat/0012420.
  * Fukusumi and Iino (2020) Y. Fukusumi and S. Iino (2020), eprint 2004.04415.
  * Belletête et al. (2020) J. Belletête, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, and T. S. Tavares, _Topological defects in periodic rsos models and anyonic chains_ (2020), URL https://arxiv.org/abs/2003.11293.
  * Roy and Saleur (2022) A. Roy and H. Saleur, Phys. Rev. Lett. 128, 090603 (2022), eprint 2111.04534.
  * Roy and Saleur (2021) A. Roy and H. Saleur (2021), eprint 2111.07927.
  * Feiguin et al. (2007) A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, Phys. Rev. Lett. 98, 160409 (2007), eprint cond-mat/0612341.
  * Faddeev (1984) L. D. Faddeev, Phys. Lett. B 145, 81 (1984).
  * Hosono (1989) S. Hosono, Phys. Rev. D 39, 1743 (1989).
  * Fujiwara et al. (1988) T. Fujiwara, S. Hosono, and S. Kitakado, Mod. Phys. Lett. A 3, 1585 (1988).
  * Kikuchi (2022) K. Kikuchi (2022), eprint 2212.13851.
  * Watanabe et al. (2022) H. Watanabe, M. Cheng, and Y. Fuji (2022), eprint 2211.00299.
  * Gannon (2002) T. Gannon, Nucl. Phys. B 627, 506 (2002), eprint hep-th/0106105.
  * Topchyan et al. (2022) H. Topchyan, V. Iugov, M. Mirumyan, S. Khachatryan, T. Hakobyan, and T. Sedrakyan (2022), eprint 2210.01187.
  * Dupont et al. (2021) M. Dupont, S. Gazit, and T. Scaffidi, Physical Review B 103 (2021).
  * Sté phan et al. (2009) J.-M. Sté phan, S. Furukawa, G. Misguich, and V. Pasquier, Physical Review B 80 (2009), URL https://doi.org/10.1103%2Fphysrevb.80.184421.
  * Runkel et al. (2022) I. Runkel, L. Szegedy, and G. M. T. Watts (2022), eprint 2210.01057.