# Infinitely many new renormalization group flows of minimal models from non-
invertible symmetries

Yu Nakayama and Takahiro Tanaka

###### Abstract

Based on the study of non-invertible symmetries, we propose there exist
infinitely many new renormalization group flows between Virasoro minimal
models $\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$ induced by
$\phi_{(1,2k+1)}$. They vastly generalize the previously proposed ones $k=I=1$
by Zamolodchikov, $k=1,I>1$ by Ahn and Lässig, and $k=2$ by Dorey et al. All
the other $\mathbb{Z}_{2}$ preserving renormalization group flows sporadically
known in the literature (e.g. $\mathcal{M}(10,3)\to\mathcal{M}(8,3)$ studied
by Klebanov et al) fall into our proposal (e.g. $k=3,I=1$). We claim our new
flows give a complete understanding of the renormalization group flows between
Virasoro minimal models that preserve a modular tensor category with the
$SU(2)_{q-2}$ fusion ring.

YITP-24-92

Infinitely many new renormalization group flows between Virasoro minimal
models from non-invertible symmetries

Takahiro Tanaka and Yu Nakayama

Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa
Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

###### Contents

  1. 1 Introduction
  2. 2 Virasoro minimal models
    1. 2.1 Non-invertible symmetries and topological defect lines
    2. 2.2 Invertible $\mathbb{Z}_{2}$ symmetry
    3. 2.3 Renormalization group invariants
  3. 3 New renormalization group flows
    1. 3.1 Renormalization group flow $\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$
    2. 3.2 Half-integer $k$
  4. 4 Some examples
    1. 4.1 $\mathcal{M}(3k+I,3)\to\mathcal{M}(3k-I,3)$ and anomaly matching
    2. 4.2 $\mathcal{M}(4k+I,4)\to\mathcal{M}(4k-I,4)$ and duality defects
    3. 4.3 $\mathcal{M}(5k+I,5)\to\mathcal{M}(5k-I,5)$
    4. 4.4 Flows to $\mathcal{M}(p,2)$ or $\mathcal{M}(2,q)$
  5. 5 Discussions

##  1 Introduction

While we believe we know everything about the Virasoro minimal models in two-
dimensional conformal field theories, the renormalization group flow between
them, in particular when one of them is non-unitary has been largely unknown.
The Virasoro minimal model $\mathcal{M}(p,q)$ is characterized by two coprime
integers $p$ and $q$, and we can compute all the conformal data for an
arbitrary choice of $(p,q)$. It has been, however, a surprisingly difficult
question to ask if they are connected by a renormalization group flow, given
two randomly chosen integers $(p,q)$ and $(p^{\prime},q^{\prime})$, Our goal
is to answer this question by using the non-invertible symmetries to classify
the flows. Indeed, we will show that non-invertible symmetries give rise to
infinitely many new renormalization group flows between two Virasoro minimal
models.

Our infinitely many new flows $\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$
induced by $\phi_{(1,2k+1)}$ vastly generalize the previously proposed ones
$k=I=1$ by Zamolodchikov [1], $k=1,I>1$ by Ahn [2] and Lässig [3], and $k=2$
by Dorey et al [4].111Dorey et al also studied the flows corresponding to
$k=\frac{1}{2}$ with odd $q$ [4]. Since the preserved non-invertible
symmetries become smaller, we study the generalization of half-integer $k$
separately in section 3.2. It also encodes the other $\mathbb{Z}_{2}$
preserving renormalization group flow sporadically known in the literature
(e.g. $\mathcal{M}(10,3)\to\mathcal{M}(8,3)$ studied by Klebanov et al [5]).
With a slight twist, the renormalization group flows between multi-critical
Lee-Yang fixed points $\mathcal{M}(2,q)$ studied in the literature [6] fall
into our proposal. We claim that our flows give a complete understanding of
the renormalization group flows between minimal models that preserve a modular
tensor category (or more precisely modular fusion category) with the
$SU(2)_{q-2}$ fusion ring.222The fusion ring $SU(2)_{q-2}$ is given by the
fusion rule of the primary operators in the $SU(2)$ WZW model at level $q-2$.
It is essentially angular momentum addition of
$J=0,\frac{1}{2},\cdots,\frac{q-2}{2}$ with a $q$ dependent “cap”:
$J_{1}+J_{2}+J_{3}\leq q-2$. Given the fusion ring, the modular tensor
category is further classified by the solutions of the Pentagon identity. The
relation between the solutions of the Pentagon identity and the
renormalization group invariants we will discuss can be found in [7].

In our discussions, the non-invertible symmetries or categorical
symmetries,333Recent applications of non-invertible symmetries in two-
dimensional conformal field theories include [7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35].  which are realized by topological defect lines in two-dimensional
conformal field theories, will play an analog of the ’t Hooft anomaly.
Similarly to the ’t Hooft anomaly, their properties are preserved under the
renormalization group flows. When the renormalization group invariants
obtained from the non-invertible symmetries are different, they cannot be
connected by the renormalization group flow that preserves the symmetries. We
will show that our proposed renormalization group flows are not only
consistent with the constraint but also give the actual flows between two
minimal models with the same preserved non-invertible symmetries.

Some of our results may appeal to more physical intuitions. For instance, the
preserved non-invertible symmetries in
$\mathcal{M}(p,4)\to\mathcal{M}(p^{\prime},4)$ flows are given by the
$\mathbb{Z}_{2}$ Tambara-Yamagami modular tensor category in math terms or the
duality defect line in physics terms. We know from mathematics that there are
two distinct quantum dimensions consistent with the $\mathbb{Z}_{2}$ Tambara-
Yamagami (or Ising) fusion rule. We also know from physics that the quantum
dimensions are renormalization group invariants. Accordingly, there exist two
intrinsically different duality defects in two-dimensional conformal field
theories. Our new renormalization group flows know them and the flows are
completely separated as long as the duality is preserved. One physical
application of such a flow was to identify the fate of the non-supersymmetric
Yukawa fixed point in two dimensions [36]. As a vast generalization, our new
results should give us a new map to explore renormalization group flows in
two-dimensional quantum field theories.

The organization of our paper is as follows. In section 2, we give a review of
Virasoro minimal models and topological defect lines. In section 3, we present
our main claim of the new renormalization group flows and give supporting
evidence from the study of the renormalization group invariants associated
with the non-invertible symmetries. In section 4, we study several physically
interesting examples of our new renormalization group flows. In section 5, we
conclude with some discussions for future directions.

##  2 Virasoro minimal models

Let us first state our conventions of Virasoro minimal models. We specify the
Virasoro minimal model $\mathcal{M}(p,q)$ by two coprime integers $p$ and $q$.
While our convention is more or less the same as the one in the yellow book
[37], one notable exception is they always assume $p>q$ in the yellow book,
but we take $q$ to be a fixed integer, and we investigate the renormalization
group flow that changes $p$. For instance, the well-known renormalization
group flow from the tricritical Ising model to the critical Ising model is
$\mathcal{M}(5,4)\to\mathcal{M}(3,4)$ in our paper rather than
$\mathcal{M}(5,4)\to\mathcal{M}(4,3)$ as in the yellow book.

The central charge of $\mathcal{M}(p,q)$ is

$\displaystyle c=\bar{c}=1-6\dfrac{(p-q)^{2}}{pq}$ (1)

and it has $\frac{(p-1)(q-1)}{2}$ (chiral) primary operators $\phi_{(r,s)}$
where $r$ and $s$ are the Kac indices whose ranges are $1\leq r<q,1\leq s<p$.
Two primary operators $\phi_{(r,s)}$ and $\phi_{(q-r,p-s)}$ are identified:
they have the same conformal weight

$\displaystyle h_{r,s}=h_{q-r,p-s}=\dfrac{(pr-qs)^{2}-(p-q)^{2}}{4pq}.$ (2)

The fusion rule of the primary operators is given by

$\displaystyle\phi_{(r,s)}\times\phi_{(m,n)}=\sum_{\begin{subarray}{c}k=1+\lvert
r-m\rvert\\\ k+r+m=1\mod
2\end{subarray}}^{\min(r+m-1,2q-1-r-m)}\sum_{\begin{subarray}{c}l=1+\lvert
s-n\rvert\\\ l+s+n=1\mod 2\end{subarray}}^{\min(s+n-1,2p-1-s-n)}\phi_{(k,l)}.$
(3)

For example, we can choose a fundamental domain $\Gamma$ of the Kac table,
which specifies $\frac{(p-1)(q-1)}{2}$ distinct Kac indices as

$\displaystyle\Gamma=\\{(r,s)\ |\ 1\leq r\leq q-1,\ 1\leq s\leq p-1,\
pr+qs<pq\\}.$ (4)

See Figure 1.

$r$$s$$q$$p$$q-1$$1$$1$

$\to$

$\Gamma$ Figure 1: The Kac table and its fundamental domain $\Gamma$ of
$\mathcal{M}(p,q)$.

The Virasoro minimal models are further classified by the modular invariant
partition functions. In this paper, we focus on those with A-series modular
invariant partition functions (a.k.a. A-series minimal models). Since most of
the renormalization group flows that we will discuss preserve the
$\mathbb{Z}_{2}$ symmetry, which can be used to relate the A-series and
D-series by orbifolding (when it can be gauged), most of the following
discussions apply to the D-series minimal models. Exceptions are half-integer
$k$ flow discussed in section 3.2 because the deformation may not exist in
D-series. Furthermore, when we can gauge the $\mathbb{Z}_{2}$ symmetry, we
have fermionic minimal models [38, 39] as well. Our discussions also apply to
their cases. Again, exceptions are half-integer $k$ flow discussed in section
3.2. Our discussions do not directly apply to the E-series minimal models,
however, and we need separate considerations. Some studies of the topological
defect lines in E-series minimal models in view of the renormalization group
flows can be found in [7, 36].

###  2.1 Non-invertible symmetries and topological defect lines

In two-dimensional quantum field theories, non-invertible symmetries are
synonyms of topological defect lines. In Virasoro minimal models with A-series
modular invariant partition functions, it is widely believed that all the
topological defect lines are given by the Verlinde lines.444To avoid trivial
counterexamples, we here exclude space-time symmetries such as Poincaré
transformation or space-time parity. Verlinde lines $L_{(r,s)}$ have the same
label $(r,s)$ as the (chiral) primary operators $\phi_{(r,s)}$ of the same
theory, so $\frac{(p-1)(q-1)}{2}$ of them are independent.

The action of $L_{(r,s)}$ on a primary state
$\ket{\phi_{(\rho,\sigma)}}=\phi_{(\rho,\sigma)}\ket{0}$ is given by

$\displaystyle
L_{(r,s)}\ket{\phi_{(\rho,\sigma)}}=\dfrac{S_{(r,s),(\rho,\sigma)}}{S_{(1,1),(\rho,\sigma)}}\ket{\phi_{(\rho,\sigma)}},$
(5)

where $S_{(r,s),(\rho,\sigma)}$ is modular S-matrix. The explicit expression
of the S-matrix is

$\displaystyle
S_{(r,s),(\rho,\sigma)}=2\sqrt{\dfrac{2}{pq}}(-1)^{1+s\rho+r\sigma}\sin\Bigl{(}\pi\dfrac{p}{q}r\rho\Bigr{)}\sin\Bigl{(}\pi\dfrac{q}{p}s\sigma\Bigr{)},$
(6)

which is real and symmetric.

As a special case, the action of $L_{(r,s)}$ on the vacuum
$\ket{0}=\ket{\phi_{(1,1)}}$ is given by

$\displaystyle
L_{(r,s)}\ket{0}=d_{(r,s)}\ket{0}=\dfrac{S_{(r,s),(1,1)}}{S_{(1,1),(1,1)}}\ket{0},$
(7)

where the eigenvalue $d_{(r,s)}$ is called quantum dimension of $L_{(r,s)}$.
The salient property of the quantum dimension is that it is a renormalization
group invariant when $L_{(r,s)}$ commutes with the deforming operators as we
will see.

The Verlinde lines satisfy the same fusion rule as (chiral) primary operators:

$\displaystyle L_{(r,s)}\times L_{(m,n)}$ $\displaystyle=\sum
N^{(k,l)}_{(r,s)(m,n)}L_{(k,l)}$ (8)
$\displaystyle=\sum_{\begin{subarray}{c}k=1+\lvert r-m\rvert\\\ k+r+m=1\mod
2\end{subarray}}^{\min(r+m-1,2q-1-r-m)}\sum_{\begin{subarray}{c}l=1+\lvert
s-n\rvert\\\ l+s+n=1\mod 2\end{subarray}}^{\min(s+n-1,2p-1-s-n)}L_{(k,l)}.$
(9)

The fusion coefficients $N^{(k,l)}_{(r,s)(m,n)}$ are related to the modular
S-matrix by the Verlinde formula:

$\displaystyle N_{ab}^{c}=\sum_{d}\frac{S_{ad}S_{bd}S_{dc}}{S_{0d}},$ (10)

where we collectively denote $(r,s)$ by $a$ etc. This short-hand collective
notation is sometimes used hereafter.

###  2.2 Invertible $\mathbb{Z}_{2}$ symmetry

All the Virasoro minimal models with A-series modular invariance (except when
$p$ or $q$ is $2$) possess a special topological defect line $L_{(q-1,1)}$
corresponding to the invertible $\mathbb{Z}_{2}$ symmetry. It is invertible
because the fusion rule (see e.g. (9)) gives $L_{(q-1,1)}\times
L_{(q-1,1)}=1$.

Under the action of $L_{(q-1,1)}$, $\mathbb{Z}_{2}$ charge of the primary
operators $\phi_{(r,s)}$ can be summarized as

$\displaystyle\begin{split}r-1\ &\mod 2\ \ (\text{when $q$ is even})\\\ s-1\
&\mod 2\ \ (\text{when $p$ is even})\\\ r+s-2\ &\mod 2\ \ (\text{when $q$ and
$p$ are both odd}).\end{split}$ (11)

See e.g. [3]. The $\mathbb{Z}_{2}$ symmetry is compatible with the fusion rule
of the primary operators and operator identifications. If we gauge the
$\mathbb{Z}_{2}$ symmetry (or orbifold it in the conformal field theory
language), A-series minimal models are exchanged with the D-series minimal
models unless it is anomalous.

At this point, we note that the $\mathbb{Z}_{2}$ symmetry is anomalous when
$p$ and $q$ are both odd. One way to see this is to compute the quantum
dimensions of $L_{(q-1,1)}$, which is $-1$ rather than $+1$. As we will review
in the next subsection, the quantum dimension is a renormalization group
invariant and it gives a selection rule of the renormalization group flow as
the ’t Hooft anomaly. We can also compute the spin contents of the defect
Hilbert space, which will be again reviewed in the next subsection, and see
that it is $\pm\frac{1}{4}$ rather than $0,\pm\frac{1}{2}$. As discussed in
[7] this is a clear signal of the ’t Hooft anomaly. Let us also recall that
there is no D-series modular invariant partition function when $p$ and $q$ are
both odd. This means that we cannot gauge the $\mathbb{Z}_{2}$ invertible
symmetry and it is consistent with the existence of the anomaly.

In this paper, we focus on the renormalization group flow that preserves the
$\mathbb{Z}_{2}$ invertible symmetry. To be more precise, our renormalization
group flow will preserve a larger modular tensor category with the
$SU(2)_{q-2}$ fusion ring. This modular tensor category includes the
$\mathbb{Z}_{2}$ invertible symmetry but gives extra $q-3$ non-invertible
symmetries. In the next subsection, we will review how the preserved non-
invertible symmetries will constrain the renormalization group flows.

###  2.3 Renormalization group invariants

The salient feature of the topological defect lines regarded as non-invertible
symmetries is that they give renormalization group invariants when they are
preserved along the flow. Let us first discuss the condition that the
topological defect lines are preserved along the renormalization group flow in
the general setup.

Suppose we are at the ultraviolet conformal field theory with a topological
defect line $L_{a}$. We deform the theory by a relevant operator $\phi_{b}$.
More generally we have to consider a collection of relevant operators
$\\{\phi_{b}\\}$ to reach the desired infrared fixed point. In such a case, we
assume that the following conditions are met with all the operators we need.
Now, we say that $L_{a}$ is preserved along the renormalization group flow
when they commute with $\phi_{b}$: $L_{a}\phi_{b}=\phi_{b}L_{a}$ on the
cylinder. More explicitly, they must satisfy

$\displaystyle L_{a}\phi_{b}\ket{\Phi}=\phi_{b}L_{a}\ket{\Phi}$ (12)

on any state $\ket{\Phi}$ on the cylinder. In unitary conformal field
theories, it is sufficient to check it on the vacuum
$\ket{0}=\ket{\phi_{(1,1)}}$ [7], but in non-unitary conformal field theories,
we may have to study all the states. We will, however, explicitly see that in
our discussions we do not gain any new constraint by studying all the states
rather than the vacuum alone.

In the mathematics language, topological defect lines give a realization of
the modular tensor category. As a modular tensor category, the physical
properties of the topological defect lines are severely constrained. It is
believed that the finite-rank modular tensor categories have no continuous
parameters (so-called rigidity). Therefore, when the topological defect lines
are preserved, we expect that the properties of the topological defect lines
as a modular tensor category should be preserved by regarding the
renormalization group flow as a continuous deformation.

Some properties of the topological defect lines are directly related to the ’t
Hooft anomaly i.e. an obstruction to gauge the symmetry, in which case it is
not necessary to assume the non-existence of the continuous deformation of the
modular tensor category to deduce the renormalization group invariance of the
modular tensor category. It is simply the ’t Hooft anomaly matching.

In this paper, we focus on two such properties. One is the quantum dimensions
of topological defect lines and the other is the spin contents of the defect
Hilbert space. See e.g. [40, 41, 42, 43] for studies of these properties in
various examples.

We have already introduced the quantum dimensions of the topological defect
lines $L_{a}$ in (7). The renormalization group invariance of the quantum
dimensions can be understood as follows. Recall that the fusion of the
topological defect lines is

$\displaystyle L_{a}\times L_{b}=\sum_{c}N_{ab}^{c}L_{c},$ (13)

where $N_{ab}^{c}$ is the fusion coefficients.

Let us now take the vacuum expectation value of this equality: $\langle
L_{a}\rangle:=\langle 0|L_{a}|0\rangle$, where
$|0\rangle=|\phi_{(1,1)}\rangle$ on the cylinder. The topological defect lines
are topological, and we can place them far away. Then we may assume that the
expectation value is factorized555Potentially, this argument can fail because
we are applying the idea developed in unitary quantum field theories to non-
unitary conformal field theories. It, however, turns out that the quantum
dimensions that we can compute explicitly in non-unitary minimal models do
satisfy the constraints obtained from this cluster decomposition ansatz.

$\displaystyle\langle L_{a}\rangle\times\langle
L_{b}\rangle=\sum_{c}N_{ab}^{c}\langle L_{c}\rangle.$ (14)

The solutions of these quadratic equations turn out to be discrete, so it is
invariant under the continuous renormalization group flows.

As the simplest example, let us consider the $\mathbb{Z}_{2}$ invertible
symmetry with the fusion rule $L_{(q-1,1)}\times L_{(q-1,1)}=L_{(1,1)}(=1)$.
By taking the vacuum expectation value, we can immediately see that $\langle
L_{(q-1,1)}\rangle=\pm 1$. The sign choice corresponds to the anomaly of the
$\mathbb{Z}_{2}$ invertible symmetry.666Whether one can choose the sign
depends on the constraint from the other fusion rules. In Appendix C of [36],
we can find the discussion of why the minus sign is still consistent with the
$\mathbb{Z}_{2}$ selection rule in correlation functions. In this case, the
renormalization group invariance of the quantum dimension is equivalent to the
’t Hooft anomaly matching.

The other renormalization group invariants we will discuss are the spin
contents of the defect Hilbert space associated with the preserved topological
defect line $L_{a}$. To introduce the notion of the defect Hilbert space and
the spin contents, we begin with the torus partition function with the
insertion of $L_{a}$ in the spatial direction.

$\displaystyle Z_{L_{a}}(\tau,\bar{\tau})$
$\displaystyle=\mathrm{Tr}\left(L_{a}q^{L_{0}-c/24}\bar{q}^{\bar{L}_{0}-c/24}\right)$
(15)
$\displaystyle=\sum_{b}\frac{S_{ab}}{S_{0b}}\chi_{b}(\tau)\bar{\chi}_{b}(\bar{\tau}),$
(16)

where $q=e^{2\pi i\tau}$ and $\chi_{b}$ is the Virasoro character of the
primary operator $\phi_{b}$.

Let us perform the S-modular transformation ($\tilde{\tau}=-\tau^{-1}$) to
regard the partition function as the sum over the defect Hilbert space
$\mathcal{H}_{L_{a}}$:

$\displaystyle Z_{L_{a}}(\tau,\bar{\tau})$
$\displaystyle=\sum_{b,c,d}\frac{S_{ab}}{S_{0b}}S_{bc}S_{bd}\chi_{c}(\tilde{\tau})\bar{\chi}_{d}(\bar{\tilde{\tau}})$
(17)
$\displaystyle=\sum_{c,d}N_{cd}^{a}\chi_{c}(\tilde{\tau})\bar{\chi}_{d}(\bar{\tilde{\tau}})$
(18)
$\displaystyle=\mathrm{Tr}_{\mathcal{H}_{L_{a}}}\left(\tilde{q}^{L_{0}-c/24}\bar{\tilde{q}}^{\bar{L}_{0}-c/24}\right),$
(19)

where we have used the Verlinde formula (10). In the last line, the
topological defect line is inserted in the time direction, and the
quantization condition on the spatial circle is modified. The spin contents of
the defect Hilbert space $\mathcal{H}_{L_{a}}$ are defined by the set of
“Lorents spin” $\\{h_{c}-h_{d}\ (\mathrm{mod}\ \mathbb{Z})\\}$ evaluated over
non-zero $N_{cd}^{a}$.

Let us now argue that the spin contents of preserved topological defect lines
are renormalization group invariants (see e.g. [7]). The idea is that the
topological defect lines are topological and the relevant deformation
preserves the $U(1)$ rotational symmetry of the quantum field theory. It
implies that the twisted quantization condition on the defect Hilbert space
cannot change in a discrete manner. More precisely, in [41, 44] it is claimed
that the spin content of the infrared theory must be a subset of the
ultraviolet theory. The spin contents of the infrared theory can be a subset
because the states might become heavy and decouple.

##  3 New renormalization group flows

In this section, based on the study of non-invertible symmetries, we propose
infinitely many new renormalization group flows in Virasoro minimal models. We
give some formal checks of the agreement of renormalization group invariants
under the flow. Examples will be given in the next section.

###  3.1 Renormalization group flow
$\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$

The main claim of our paper is there exist infinitely many renormalization
group flows $\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$ induced by the
$\phi_{(1,2k+1)}$ deformation, which satisfy the constraint from the non-
invertible symmetries. We further claim that the renormalization group flow is
complete when we preserve a modular tensor category with the $SU(2)_{q-2}$
fusion rule that includes the $\mathbb{Z}_{2}$ invertible symmetry. By
complete, we mean that all the possible flows that preserve a modular tensor
category with the $SU(2)_{q-2}$ fusion ring can be represented by a repeated
use of our proposed flows $\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$. For
concreteness, we assume $k$ and $I$ are both integers in this subsection. The
generalization for non-integer $k$ (and $I$) will be discussed in the next
subsection. The half-integer case does not preserve the modular tensor
category with the $SU(2)_{q-2}$ fusion ring.

We first show that the $\phi_{(1,2k+1)}$ deformation preserves the
$SU(2)_{q-2}$ subcategory $L_{(r,1)}$ $(i=1,\cdots,q-1)$ out of the non-
invertible symmetries of the undeformed theory $\mathcal{M}(kq+I,q)$. For this
purpose, as reviewed in section 2.3, we should check $L_{(r,1)}$ and
$\phi_{(1,2k+1)}$ commute on the cylinder.

Let us start with the fusion rule (3). We have

$\displaystyle\phi_{(1,2k+1)}\times\phi_{(\rho,\sigma)}=\sum_{\begin{subarray}{c}l=1+\lvert
2k+1-\sigma\rvert\\\ l+\sigma=0\mod
2\end{subarray}}^{\min(2k+\sigma,2(kq+I)-2-2k-\sigma)}\phi_{(\rho,l)}=\sum_{l\in\Lambda}\phi_{(\rho,l)}$
(20)

in $\mathcal{M}(kq+I,q)$, where we have defined $\Lambda$ as

$\displaystyle\Lambda=\\{\ l\ |\ 1\leq
l\leq\min(2k+\sigma,2(kq+I)-2-2k-\sigma),\ l+\sigma=0\ \ \mathrm{mod}\ 2\
\\},$ (21)

and then we can compare the action of $L_{(r,1)}\phi_{(1,2k+1)}$ and
$\phi_{(1,2k+1)}L_{(r,1)}$ on arbitrary primary states
$\ket{\phi_{(\rho,\sigma)}}$:

$\displaystyle L_{(r,1)}\phi_{(1,2k+1)}\ket{\phi_{(\rho,\sigma)}}$
$\displaystyle=L_{(r,1)}\phi_{(1,2k+1)}\phi_{(\rho,\sigma)}\ket{0}=\sum_{l\in\Lambda}L_{(r,1)}\phi_{(\rho,l)}\ket{0}$
$\displaystyle=\sum_{l\in\Lambda}L_{(r,1)}\ket{\phi_{(\rho,l)}}=\sum_{l\in\Lambda}\dfrac{S_{(r,1),(\rho,l)}}{S_{(1,1),(\rho,l)}}\ket{\phi_{(\rho,l)}}$
(22) $\displaystyle\phi_{(1,2k+1)}L_{(r,1)}\ket{\phi_{(\rho,\sigma)}}$
$\displaystyle=\dfrac{S_{(r,1),(\rho,\sigma)}}{S_{(1,1),(\rho,\sigma)}}\phi_{(1,2k+1)}\ket{\phi_{(\rho,\sigma)}}=\dfrac{S_{(r,1),(\rho,\sigma)}}{S_{(1,1),(\rho,\sigma)}}\phi_{(1,2k+1)}\phi_{(\rho,\sigma)}\ket{0}$
$\displaystyle=\sum_{l\in\Lambda}\dfrac{S_{(r,1),(\rho,\sigma)}}{S_{(1,1),(\rho,\sigma)}}\phi_{(\rho,l)}\ket{0}=\sum_{l\in\Lambda}\dfrac{S_{(r,1),(\rho,\sigma)}}{S_{(1,1),(\rho,\sigma)}}\ket{\phi_{(\rho,l)}}.$
(23)

Because $l+\sigma$ is even in the above sum,

$\displaystyle\left.\dfrac{S_{(r,1),(\rho,l)}}{S_{(1,1),(\rho,l)}}\middle/\right.\dfrac{S_{(r,1),(\rho,\sigma)}}{S_{(1,1),(\rho,\sigma)}}$
$\displaystyle=\left.(-1)^{(r-1)l}\dfrac{\sin(\pi\frac{kp+I}{p}r\rho)}{\sin(\pi\frac{kp+I}{p}\rho)}\middle/\right.(-1)^{(r-1)\sigma}\dfrac{\sin(\pi\frac{kp+I}{p}r\rho)}{\sin(\pi\frac{kp+I}{p}\rho)}$
$\displaystyle=1$ (24)

and hence $L_{(1,1)},L_{(2,1)},\cdots,L_{(q-2,1)},L_{(q-1,1)}$ commute with
$\phi_{(1,2k+1)}$. This means that $SU(2)_{q-2}$ subcategory generated by
$L_{(r,1)}$ is preserved under the renormalization group flow induced by the
$\phi_{(1,2k+1)}$ deformation. In particular, we stress that $L_{(q-1,1)}$
generates the invertible $\mathbb{Z}_{2}$ symmetry of the minimal models, so
the renormalization group flow is $\mathbb{Z}_{2}$ symmetric.

In the above discussions, we realize that $\phi_{(1,2k+1)}$ can be replaced
with $\phi_{(1,2l+1)}$ for any integer $l$ such that $\phi_{(1,2l+1)}$ is in
the Kac table. The physical significance of choosing $l=k$ is that then
$\phi_{(1,2k+1)}$ will be the least relevant deformation within
$\\{\phi_{(1,2l+1)}\\}$ if $I<k$. When $k>1$, we typically have to introduce
the other more relevant deformation $\phi_{(1,2l+1)}$ (with $l<k$) and fine-
tune the deformation parameters to get to the critical point.777Generally we
have to fine-tune $k-1$ parameters. When $I>k$, we need fewer fine-tunings to
reach the fixed point dictated by the proposed flow.

Let us now show that the quantum dimensions of all the preserved topological
defect lines match under the proposed renormalization group flows
$\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$.888The matching of the quantum
dimensions when $k=1$ and $k=2$, corresponding to the previously known
renormalization group flows, was discussed in [40, 43]. The proof follows by
the direct calculation. The quantum dimension $d_{(r,1)}$ of $L_{(r,1)}$ in
$\mathcal{M}(kq+I,q)$ is
$(-1)^{r+1}\sin(\pi\frac{kq+I}{q}r)/\sin(\pi\frac{kq+I}{q})$ and that in
$\mathcal{M}(kq-I,q)$ is
$(-1)^{r+1}\sin(\pi\frac{kq-I}{q}r)/\sin(\pi\frac{kq-I}{q})$. Its ratio is

$\displaystyle\dfrac{\sin(\pi\frac{kq+I}{q}r)}{\sin(\pi\frac{kq-I}{q}r)}\dfrac{\sin(\pi\frac{kq-I}{q})}{\sin(\pi\frac{kq+I}{q})}=\dfrac{\sin(\pi\frac{I}{q}r)}{\sin(-\pi\frac{I}{q}r)}\dfrac{\sin(-\pi\frac{I}{q})}{\sin(\pi\frac{I}{q})}=1.$
(25)

We can further argue that the constraint from the quantum dimensions is
sufficiently non-trivial. We observe that because $kq+(\pm I+q)=(k+1)q\pm I$,
$I$ takes a value in $1\leq I\leq q-1$, and $kq\pm I$ and $q$ are coprime, our
proposed flows can be classified by $\varphi(q)$ kinds of $I$, where
$\varphi(q)$ is Euler function. Now we want to see that each connected flows
have different quantum dimensions. For this purpose, let us examine the
quantum dimensions of $L_{(2,1)}$ in $\mathcal{M}(p,q)$ in particular; the
constraint from the other preserved topological defect lines is typically not
as strong. The explicit evaluation of the formula (7) gives
$d_{(2,1)}=-2\cos\left(\frac{p}{q}\pi\right)$ and we immediately observe
$d_{(2,1)}$ takes a distinct value for different $p$ ($\mathrm{mod}\ q$),
given $q$. Thus, the quantum dimensions do distinguish our renormalization
group flows.

From the general argument in section 2.3, under the new renormalization group
flow, the spin contents of the preserved line $L_{(r,1)}$ $(r=1,\cdots,q-1)$
should remain the same or become a subset. While it may be non-trivial to
check the agreement of all the spin contents in the most generic cases, one
can systematically check the agreement in the following way.

We first realize that $N_{(s,m)(t,n)}^{(r,1)}\chi_{(s,m)}\bar{\chi}_{(t,n)}$
is non-zero only when $m=n$. Furthermore, the fusion coefficients necessary
for us do not depend on $m$, so we can first determine the non-zero entry of
$N_{(s,m)(t,n)}^{(r,1)}=\delta_{mn}N_{(s,1)(t,1)}^{(r,1)}$ from $SU(2)_{q-2}$
fusion rule, and then we can compute the spin of the defect Hilbert space from
(2). In this way, it is possible to systematically compute the spin contents
of $L_{(r,1)}$ and see the agreement under the proposed renormalization group
flow. We will show some examples in section 4.

Now let us show that the spin contents of $\mathcal{H}_{L_{(2,1)}}$ in
$\mathcal{M}(p,q)$ are preserved under the proposed renormalization group
flows. We first observe that the only non-zero fusion coefficients are
$N_{(t,n)(t+1,m)}^{(2,1)}$ and $N_{(t+1,m)(t,m)}^{(2,1)}$. By computing
$\pm(h_{(t+1,m)}-h_{(t,m)})$, we find the spin contents of
$\mathcal{H}_{L_{(2,1)}}$ are $\pm\frac{p}{q}\frac{1+2t}{4}$ (mod
$\frac{1}{2}\mathbb{Z}$) with $t=1,\cdots,q-2$. Thus, the spin contents of
$\mathcal{H}_{L_{(2,1)}}$ in $\mathcal{M}(kq+I,q)$ contain those in
$\mathcal{M}(kq-I,q)$.

We further want to see that the spin contents of $\mathcal{H}_{L_{(2,1)}}$
distinguish the renormalization group flows as the quantum dimensions. We
observe that $\frac{p}{q}\frac{1+2t}{4}$ never equals to
$\frac{p}{q}\frac{1}{4}$ mod $\frac{\mathbb{Z}}{2}$, so
$\frac{p}{q}\frac{1}{4}$ is always missing in the spin content. (Indeed, if
this were true, we would have $\frac{p}{q}l=\mathbb{Z}$, but since $p$ and $q$
are coprime, it should be a contradiction.) On the other hand, the missing
ones (except $\frac{1}{4q}$ and $\frac{1}{2q}$) always appear in $p=2q-1$ or
$p=2q-2$ as long as $q>3$, so the spin contents of $p=2q-1$ and $p=2q-2$ are
different from the other $p$’s (mode $q$). We can repeat the analysis with the
other $p=2q-s$ with $s=3,4,\cdots,q-1$ to realize that they all have different
spin contents. The exceptional but easier $q=3$ case can be treated separately
(see section 4.1).

The renormalization group invariants cannot tell which of
$\mathcal{M}(kq+I,q)$ or $\mathcal{M}(kq-I,q)$ is the ultraviolet theory. To
address this question, let us mention two evidence that $\mathcal{M}(kq+I,q)$
is the ultraviolet theory. The first evidence is the $c_{\mathrm{eff}}$
theorem [45]. While $\mathcal{M}(p,q)$ can be non-unitary, it is believed that
as long as the deformation preserves the $\mathcal{PT}$-symmetry (see e.g.
[46] for a review), the renormalization group flow shows the monotonicity with
respect to the effective central charge $c_{\mathrm{eff}}=1-\frac{6}{pq}$
along the flow. If we accept the $c_{\mathrm{eff}}$ theorem (with the
assumption our flows preserve the $\mathcal{PT}$-symmetry), we may conclude
$\mathcal{M}(kq+I,q)$ is the ultraviolet theory.999One caveat here is that we
have not verified our renormalization group flow preserves the $\mathcal{PT}$
symmetry. It is an interesting question to see if the $\mathcal{PT}$ symmetry
can be realized as a topological defect line.

The other evidence comes from the number of (singlet) relevant deformations
[47]. It was argued there that along the renormalization group flow, the
number of (singlet) relevant deformations should decrease. Intuitively, if we
deform the theory by relevant perturbation, the deformed theory becomes less
critical and we expect less fine-tuning is needed to reach the criticality. In
[48], it was put on firmer ground by studying the topology of the
renormalization group flow. Again once we accept this conjecture, we realize
$\mathcal{M}(kq+I,q)$ is the ultraviolet theory because it has a larger number
of relevant singlet operators. Note also that if our constraint from the non-
invertible symmetries did not exist, we could have found counterexamples of
the conjecture. For instance in the would-be flow
$\mathcal{M}(7,4)\to\mathcal{M}(5,4)$, the number of relevant singlet
operators did not decrease. We have, however, forbidden the flow by matching
the non-invertible symmetries. In this discussion, it is important to note
that by singlet, we mean the operators that commute with all the non-
invertible symmetries $\\{L_{(r,1)}\\}$ whose fusion ring is $SU(2)_{q-2}$
than just those that commute with the $\mathbb{Z}_{2}$ invertible symmetry.

Here are some historical comments. In a classic paper [1], Zamolodchikov
studied the renormalization group flow $k=I=1$, which turns out to be
integrable [49]. This case is most physically interesting because it describes
the flow between unitary minimal models. Subsequently, Ahn [2] and Lässig [3]
proposed the generalization to the non-unitary minimal models independently,
corresponding to $k=1,I>1$. Later, Dorey, Dunning and Tateo [4] (see also [50,
51, 52]) studied the case with $k=2$ from the viewpoint of non-linear integral
equations. Our approach, based on the non-invertible symmetries, also provides
us with new insights into the previously known flows.

###  3.2 Half-integer $k$

Our proposed renormalization group flow
$\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$ induced by $\phi_{(1,2k+1)}$
formally makes sense even if $k$ is a half-integer, where $I$ takes a half-
integer or integer depending on $q$. The main difference is that $2k+1$ is
even so not all the topological defect lines $L_{(r,1)}$ commute with
$\phi_{(1,2k+1)}$, but only the subcategory spanned by $L_{(2\ell+1,1)}$,
where $\ell$ is an integer, is preserved. This restriction can be explicitly
seen in (24), where for half-integer $k$, $l+\sigma$ is odd (rather than even
for integer $k$ case discussed there), so we have to assume $r$ is an odd
integer. This subcategory has the fusion ring of $SO(3)_{[\frac{q}{2}]-1}$.

If we restrict ourselves to these topological defect lines, all the
discussions above can be repeated. In particular, the consistency of the
proposed renormalization group flow with the preserved non-invertible symmetry
holds. For instance, the invariance of the quantum dimensions of $L_{(r,1)}$
can be checked as in (25), but with half-integer $k$, the ratio becomes
$(-1)^{r+1}$. Thus only the preserved lines with $r=2\ell+1$ have the same
quantum dimensions as claimed. The strongest constraint from the spin contents
comes from $L_{(3,1)}$ (rather than $L_{(2,1)}$ which does not exist). In
$\mathcal{M}(p,q)$, the spin contents of $\mathcal{H}_{L_{(3,1)}}$ are
$\\{0,\pm\frac{p}{q}(\ell+1)\\}$ ($\ell=1,\cdots,q-3$), and we can check it is
consistent with our flow $\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$ with
half-integer $k$.

Some specific renormalization group flows have been studied in the literature.
The particular case with $k=\frac{1}{2}$ was discussed in Dorey et al [4], but
note $(p,q)$ there was swapped compared with our unified notation so that
their $\phi_{(1,2)}$ is our $\phi_{(2,1)}$. The non-invertible symmetries, in
this previously known case, were studied in [43]. Again, we generalize their
works by letting $k$ be an arbitrary half-integer.

Let us comment on the fate of the $\mathbb{Z}_{2}$ invertible symmetry under
the half-integer $k$ flow. When $k$ is a half-integer and $q$ is odd,
$\phi_{(1,2k+1)}$ is odd under the $\mathbb{Z}_{2}$ invertible symmetry, so
the preserved modular tensor category does not include $L_{(q-1,1)}$.
Nevertheless, after the flow, we do recover the entire modular tensor category
with the $SU(2)_{q-2}$ fusion ring, including the invertible $\mathbb{Z}_{2}$
symmetry. They must be emergent symmetries. While this must be the case since
this half-integer $k$ flow allows us to change the (non-)anomalous nature of
the $\mathbb{Z}_{2}$ symmetry, the microscopic details of the renormalization
group flow may be of interest and needs further study.

##  4 Some examples

In this section, we study explicit examples of
$\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$ flows for $q=3,4,5$, emphasizing
physical interpretations. As a potentially exceptional case, we then examine
$q=2$ separately.

###  4.1 $\mathcal{M}(3k+I,3)\to\mathcal{M}(3k-I,3)$ and anomaly matching

In this case, our proposal with integer $k$ boils down to the conclusion that
there exists distinct renormalization group flows among $\mathcal{M}(2l+1,3)$
and among $\mathcal{M}(2l,3)$. The only non-trivial topological defect line
preserved by the $\phi_{(1,2k+1)}$ deformation is $L_{(2,1)}$ that generates
the invertible $\mathbb{Z}_{2}$ symmetry. The constraint from the quantum
dimensions as well as spin contents will be equivalent to the ’t Hooft anomaly
matching. We recall that the $\mathbb{Z}_{2}$ invertible symmetry is anomalous
in $\mathcal{M}(2l+1,3)$ and non-anomalous in $\mathcal{M}(2l,3)$ so they
cannot mix under the $\mathbb{Z}_{2}$ preserving renormalization group flow.

While the quantum dimensions of $L_{(2,1)}$ can be directly computable from
(7), the spin contents of $\mathcal{H}_{L_{(2,1)}}$ can be computed by the
algorithm mentioned in section 2.3. As argued there, non-zero entry of
$N_{(s,m)(t,n)}^{(2,1)}$ implies $m=n$ and $(s,t)=(2,1)\ \mathrm{or}\ (1,2)$,
which have the opposite spins. From the dimensions of primary operators (2),
we find the spin of the defect Hilbert space $\mathcal{H}_{L_{(2,1)}}$ is
given by $\pm(h_{(2,m)}-h_{(1,m)})=\pm\frac{1}{4}(p-2m)$ with $1\leq m\leq
p-1$. The spin contents are therefore $\\{\pm\frac{1}{4}\\}$ when $p$ is odd
and $\\{0,\pm\frac{1}{2}\\}$ when $p$ is even.

We have summarized the quantum dimensions and spin contents of
$\mathcal{M}(p,3)$ models in Table 1 and Table 2. Note that for the anomalous
$\mathbb{Z}_{2}$ invertible symmetry, the quantum dimension is $-1$ and the
spin content is $\\{\pm\frac{1}{4}\\}$, which is consistent with the general
argument [7].

$p$ | 4 | 5 | 7 | 8 | 10 | 11 | 13 | 14 | $\cdots$
---|---|---|---|---|---|---|---|---|---
$d_{(2,1)}$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $\cdots$
Table 1: The quantum dimensions of $L_{(2,1)}$ in $\mathcal{M}(p,3)$ series. $p$ | 4 | 5 | 7 | 8 | 10 | 11 | 13 | 14 | $\cdots$
---|---|---|---|---|---|---|---|---|---
$\mathcal{H}_{L_{(2,1)}}$ | $0,\pm\frac{1}{2}$ | $\pm\frac{1}{4}$ | $\pm\frac{1}{4}$ | $0,\pm\frac{1}{2}$ | $0,\pm\frac{1}{2}$ | $\pm\frac{1}{4}$ | $\pm\frac{1}{4}$ | $0,\pm\frac{1}{2}$ | $\cdots$
Table 2: The spin contents of $\mathcal{H}_{L_{(2,1)}}$ in $\mathcal{M}(p,3)$
series.

We would like to note that in the literature [53, 5], one of the flows in this
class $\mathcal{M}(10,3)\to\mathcal{M}(8,3)$ induced by $\phi_{(1,7)}$
($k=3,I=1$) was used to study the Landau-Ginzburg-like Lagrangian
constructions of non-unitary minimal models. The further flow
$\mathcal{M}(8,3)\to\mathcal{M}(4,3)$ induced by $\phi_{(1,5)}$ ($k=2,I=2$)
can be straightforwardly realized in their Landau-Ginzburg-like Lagrangian.
Another flow $\mathcal{M}(7,3)\to\mathcal{M}(5,3)$ induced by $\phi_{(1,5)}$
($k=2,I=1$) was studied in [54] from the truncated conformal space approach.

Let us briefly discuss the case with half-integer $k$. With the half-integer
$k$, $\phi_{(1,2k+1)}$ deformation does not preserve any non-trivial
topological defect lines in $\mathcal{M}(p,3)$, so the constraint is none. We,
however, observe that some of the predicted flow such as
$\mathcal{M}(10,3)\overset{\phi_{(1,6)}}{\longrightarrow}\mathcal{M}(5,3)\overset{\phi_{(1,4)}}{\longrightarrow}\mathcal{M}(4,3)$
may give an interesting Lagrangian description of $\mathcal{M}(5,3)$, given
the Lagrangian description of $\mathcal{M}(10,3)$ is available. Indeed, the
flow has some theoretical interest because of the (dis)appearance of the
anomalous $\mathbb{Z}_{2}$ symmetry of $\mathcal{M}(5,3)$ in the middle of the
renormalization group flow.

###  4.2 $\mathcal{M}(4k+I,4)\to\mathcal{M}(4k-I,4)$ and duality defects

Here, under the $\phi_{(1,2k+1)}$ deformations with integer $k$, the preserved
topological defect lines are $L_{(1,1)},L_{(2,1)}$ (sometimes called
$\mathcal{N}$) and $L_{(3,1)}$ (sometimes called $\eta$) and generate the
$\mathbb{Z}_{2}$ Tambara-Yamagami modular tensor category. There are
$\varphi(4)=2$ distinct renormalization group flows predicted from our
proposal.

Let us first study the quantum dimensions of $L_{(2,1)}$ and $L_{(3,1)}$, as
shown in Table 3. As discussed in section 3, $d_{(2,1)}$ distinguishes the two
renormalization group flows inside $\mathcal{M}(p,4)$. Since the
$\mathbb{Z}_{2}$ invertible symmetry is non-anomalous, $d_{(3,1)}=1$ for all
$p$, and it is not useful as a constraint on the renormalization group flow.

$p$ | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17
---|---|---|---|---|---|---|---|---
$d_{(2,1)}$ | $\sqrt{2}$ | $\sqrt{2}$ | $-\sqrt{2}$ | $-\sqrt{2}$ | $\sqrt{2}$ | $\sqrt{2}$ | $-\sqrt{2}$ | $-\sqrt{2}$
$d_{(3,1)}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
Table 3: The quantum dimensions of $L_{(2,1)}=\mathcal{N}$ and
$L_{(3,1)}=\eta$ in $\mathcal{M}(p,4)$ series.

We can also compute the spin contents of $L_{(2,1)}$ and $L_{(3,1)}$. Since
the topological defect line $L_{(3,1)}$ generates the non-anomalous
$\mathbb{Z}_{2}$ invertible symmetry, the spin contents must be
$\\{0,\pm\frac{1}{2}\\}$ as can be checked easily, so we focus on $L_{(2,1)}$.
The result of the explicit computation is summarized in Table 4.

$p$ | $h_{s,m}-h_{t,n}$ | the spin contents
---|---|---
3 | $\pm\frac{1}{16},\pm\frac{7}{16}$ | $\pm\frac{1}{16},\pm\frac{7}{16}$
5 | $\pm\frac{1}{16},\pm\frac{7}{16},\pm\frac{9}{16},\pm\frac{17}{16}$ | $\pm\frac{1}{16},\pm\frac{7}{16}$
7 | $\pm\frac{3}{16},\pm\frac{5}{16},\pm\frac{11}{16},\pm\frac{13}{16},\pm\frac{19}{16},\pm\frac{27}{16}$ | $\pm\frac{3}{16},\pm\frac{5}{16}$
9 | $\pm\frac{3}{16},\pm\frac{5}{16},\pm\frac{11}{16},\pm\frac{13}{16},\pm\frac{19}{16},\pm\frac{21}{16},\pm\frac{29}{16},\pm\frac{37}{16}$ | $\pm\frac{3}{16},\pm\frac{5}{16}$
11 | $\pm\frac{1}{16},\pm\frac{7}{16},\pm\frac{9}{16},\pm\frac{15}{16},\pm\frac{17}{16},\pm\frac{23}{16},\pm\frac{25}{16},\pm\frac{31}{16},\pm\frac{39}{16},\pm\frac{47}{16}$ | $\pm\frac{1}{16},\pm\frac{7}{16}$
13 | $\pm\frac{1}{16},\pm\frac{7}{16},\pm\frac{9}{16},\pm\frac{15}{16},\pm\frac{17}{16},\pm\frac{23}{16},\pm\frac{25}{16},\pm\frac{31}{16},\pm\frac{33}{16},\pm\frac{41}{16},\pm\frac{49}{16},\pm\frac{57}{16}$ | $\pm\frac{1}{16},\pm\frac{7}{16}$
15 | $\pm\frac{3}{16},\pm\frac{5}{16},\pm\frac{11}{16},\pm\frac{13}{16},\pm\frac{19}{16},\pm\frac{21}{16},\pm\frac{27}{16},\pm\frac{29}{16},\pm\frac{35}{16},\pm\frac{37}{16},\pm\frac{43}{16},\pm\frac{51}{16},\pm\frac{59}{16},\pm\frac{67}{16}$ | $\pm\frac{3}{16},\pm\frac{5}{16}$
17 | $\pm\frac{3}{16},\pm\frac{5}{16},\pm\frac{11}{16},\pm\frac{13}{16},\pm\frac{19}{16},\pm\frac{21}{16},\pm\frac{27}{16},\pm\frac{29}{16},\pm\frac{35}{16},\pm\frac{37}{16},\pm\frac{43}{16},\pm\frac{45}{16},\pm\frac{53}{16},\pm\frac{61}{16},\pm\frac{69}{16},\pm\frac{77}{16}$ | $\pm\frac{3}{16},\pm\frac{5}{16}$

Table 4: The values of $h_{s,m}-h_{t,n}$ when ${N}_{(s,m)(t,n)}^{(2,1)}\neq 0$
and the spin content of the defect Hilbert space $\mathcal{H}_{\mathcal{N}}$
for minimal model $\mathcal{M}(p,4)\ (p=3,5,\cdots,17)$. The spin contents can
be obtained by taking mod 1.

As derived in section 3, for $\mathcal{M}(p,4)$, the general formula for
$h_{s,m}-h_{t,n}$ and the spin content of $\mathcal{H}_{L_{(2,1)}}$ can be
derived:

$\displaystyle
h_{s,m}-h_{t,n}:\Bigl{\\{}\pm\dfrac{3p-8m}{16},\pm\dfrac{5p-8m}{16}\ |\ 1\leq
m\leq p-1\Bigr{\\}},$ (26) $\displaystyle\text{the spin contents of}\
\mathcal{H}_{L_{(2,1)}}:\begin{cases}\pm\dfrac{1}{16},\pm\dfrac{7}{16}&(p=3,5\
\text{mod}\ 8)\\\ \pm\dfrac{3}{16},\pm\dfrac{5}{16}&(p=1,7\ \text{mod}\
8)\end{cases}$ (27)

This is compatible with (6.5) in [7] on the spin content of the duality defect
line. With this agreement in mind, let us argue that $L_{(2,1)}$ is indeed the
duality defect line.

As is well-known in the critical Ising model $\mathcal{M}(3,4)$ and the
tricritical Ising model $\mathcal{M}(5,4)$, the $\mathbb{Z}_{2}$ Tambara-
Yamagami modular tensor category has a physical interpretation as the duality
defect line. We would like to offer a similar interpretation to the general
$\mathcal{M}(p,4)$ Virasoro minimal models.

First, let us review the duality defect in more generality [55, 56, 57].
Consider a quantum field theory $\mathcal{Q}$ on a $d$-dimensional spacetime
with a non-anomalous $\mathbb{Z}_{N}$ $q$-form global symmetry. We divide the
spacetime into two regions and gauge the $\mathbb{Z}_{N}$ global symmetry in
half of the spacetime. Then, we can construct a codimension-one topological
interface between the original theory $\mathcal{Q}$ and its gauged theory
$\mathcal{Q}/\mathbb{Z}_{N}$. If the original theory $\mathcal{Q}$ is
invariant under gauging the $\mathbb{Z}_{N}$ global symmetry

$\displaystyle\mathcal{Q}\cong\mathcal{Q}/\mathbb{Z}_{N},$ (28)

the interface becomes the duality defect in the original theory $\mathcal{Q}$.

When do we expect the duality defect in this so-called half-space gauging
construction? Note that the gauged theory has
$\widehat{\mathbb{Z}_{N}}\cong\mathbb{Z}_{N}$ $(d-q-2)$-form dual (or quantum)
symmetry, so the ungauged theory and the gauged theory may have the same
symmetry as in (28) if we require

$\displaystyle q=\dfrac{d-2}{2}.$ (29)

Our cases discussed in this paper correspond to $(d,q)=(2,0)$.

Given a duality defect in the ultraviolet conformal field theory, we want to
understand its fate under the renormalization group flow. Let us assume that
the duality procedure commutes with the deformation of the ultraviolet
conformal field theory.101010This assumption is stronger than just assuming
the deformation preserves $\mathbb{Z}_{2}$ invertible symmetry. For the
difference, see the example of half-integer $k$ flows below. Then, we should
have the duality defect with the same property in the infrared as well. Such
an existence should provide us with a strong constraint on the renormalization
group flow.

We want to argue that the topological defect line $L_{(2,1)}$ in
$\mathcal{M}(p,4)$ is the duality defect line. Let us consider general
$\mathcal{M}(p,q)$ with the A-series modular invariant partition functions
$Z_{A_{q-1},A_{p-1}}$. Let us further assume the case when $p$ or $q$ is even
(but not $2$) so that the $\mathbb{Z}_{2}$ symmetry is non-anomalous and can
be gauged. The partition function of the $\mathbb{Z}_{2}$ gauged theory can be
obtained by the orbifold technique:

$\displaystyle
Z_{\mathcal{M}(p,q)/\mathbb{Z}_{2}}=\begin{cases}Z_{D_{q/2+1},A_{p-1}}&\text{when
$q=0$ mod 2}\\\ Z_{A_{q-1},D_{p/2+1}}&\text{when $p=0$ mod 2}.\end{cases}$
(30)

The right-hand side is generically a non-diagonal modular invariant partition
function of D-series, but the exception is when $q=4$ (or $p=4$). Because
$A_{3}\cong D_{3}$, A-series minimal models $\mathcal{M}(p,q)$ are invariant
under gauging $\mathbb{Z}_{2}$ global symmetry if and only if $p$ or $q$ is 4.
In other words, $\mathcal{M}(4,q)$ is self-dual under the $\mathbb{Z}_{2}$
gauging and it has a duality defect line $\mathcal{N}$. Moreover, we can show
that the duality defect line $\mathcal{N}$ satisfies the $\mathbb{Z}_{2}$
Tambara-Yamagami fusion rule such as $\eta\times\eta=1$,
$\eta\times\mathcal{N}=\mathcal{N}\times\eta=\mathcal{N}$ and
$\mathcal{N}\times\mathcal{N}=1+\eta$. This is nothing but the fusion rule of
the topological defect lines $L_{(2,1)}$ and $L_{(3,1)}$ preserved under the
renormalization group flow $\mathcal{M}(4k+I,4)\to\mathcal{M}(4k-I,4)$ induced
by $\phi_{(1,2k+1)}$.

After seeing how the duality line plays a significant role in the
renormalization group flows of $\mathcal{M}(4k+I,4)\to\mathcal{M}(4k-I,4)$
when $k$ is an integer, let us briefly discuss the case when $k$ is a half-
integer. Here, $\phi_{(1,2k+1)}$ does not commute with $L_{(2,1)}$ and the
only preserved symmetry is $L_{(3,1)}$. Accordingly, there are no obstructions
for the renormalization group flows between any pair of $p$ in
$\mathcal{M}(p,4)$. The simplest example would be $\mathcal{M}(3,4)$ to
“$\mathcal{M}(1,4)$” induced by $\phi_{(1,2)}(=\epsilon)$, which is a massive
flow. Physically, if we change the temperature of the Ising model, the
Kramers–Wannier duality is broken and the criticality will be lost. See [4]
for an interpretation of “$\mathcal{M}(1,4)$” from the viewpoint of the
integral equations.

$\mathcal{M}(3,4)$$\mathcal{M}(5,4)$$\mathcal{M}(7,4)$$\mathcal{M}(9,4)$$\mathcal{M}(11,4)$$\mathcal{M}(13,4)$$\mathcal{M}(15,4)$$\mathcal{M}(17,4)$

$\times$

$\times$

$\times$

$(k,I)=(1,1)$$(k,I)=(3,1)$$(k,I)=(2,3)$$(k,I)=(2,1)$$(k,I)=(4,1)$$(k,I)=(3,3)$$\cdots$
Figure 2: $\mathcal{M}(p,4)$ are classified in terms of the spin contents and
the quantum dimensions of the duality defect $\mathcal{N}$. They constrain the
renormalization group flow like this figure. Dotted arrows are possible in the
half-integer $k$ flow which does not preserve the duality defect lines.

To conclude this subsection, we have summarized the renormalization group
flows between $\mathcal{M}(p,4)$ in Figure 2. As a physical application beyond
the classic tri-critical Ising to critical Ising flow
$\mathcal{M}(5,4)\to\mathcal{M}(3,4)$, the fermionic version of the
renormalization group flow $\mathcal{M}(11,4)$ to $\mathcal{M}(5,4)$ was
discussed in [36] to understand the fate of the non-supersymmetric Yukawa
fixed point.

###  4.3 $\mathcal{M}(5k+I,5)\to\mathcal{M}(5k-I,5)$

We repeat our analysis on the renormalization group flows
$\mathcal{M}(5k+I,5)\to\mathcal{M}(5k-I,5)$ induced by $\phi_{(1,2k+1)}$ that
preserve a modular tensor category with the $SU(2)_{3}$ fusion ring, assuming
$k$ is an integer. Here, we have $\varphi(5)=4$ distinct flows that cannot
mix. The ’t Hooft anomaly matching of the $\mathbb{Z}_{2}$ invertible symmetry
distinguishes $\mathcal{M}(p,5)$ with even $p$ and odd $p$, but the other non-
invertible symmetries give a finer classification.

The quantum dimensions of preserved non-invertible symmetries are given in
Table 5. Here, $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio. As discussed
in section 3, not only all the quantum dimensions are consistent with the
proposed renormalization group flow, but $L_{(2,1)}$ gives the strongest
constraint such that it fully distinguishes our proposed renormalization group
flows.

$p$ | 2 | 3 | 4 | 6 | 7 | 8 | 9 | 11 | $\cdots$
---|---|---|---|---|---|---|---|---|---
$d_{(2,1)}$ | $-\phi^{-1}$ | $\phi^{-1}$ | $\phi$ | $\phi$ | $\phi^{-1}$ | $-\phi^{-1}$ | $-\phi$ | $-\phi$ | $\cdots$
$d_{(3,1)}$ | $-\phi^{-1}$ | $-\phi^{-1}$ | $\phi$ | $\phi$ | $-\phi^{-1}$ | $-\phi^{-1}$ | $\phi$ | $\phi$ | $\cdots$
$d_{(4,1)}$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $-1$ | $\cdots$
Table 5: The quantum dimensions of $L_{(r,1)}$ in $\mathcal{M}(p,5)$ series.

Let us also compute the spin contents of the preserved topological defect
lines. On $\mathcal{H}_{L_{(2,1)}}$, the non-zero fusion coefficients are
$N_{(1,m)(2,m)}^{(2,1)}$, $N_{(2,m)(1,m)}^{(2,1)}$, $N_{(2,m)(3,m)}^{(2,1)}$,
$N_{(3,m)(2,m)}^{(2,1)}$, $N_{(3,m)(4,m)}^{(2,1)}$, and
$N_{(4,m)(3,m)}^{(2,1)}$, so we have

$\displaystyle\left\\{\pm\left(\frac{3p}{20}-\frac{m}{2}\right),\pm\frac{1}{4}(p-2m),\pm\left(\frac{7p}{20}-\frac{m}{2}\right)\right\\},$
(31)

where $1\leq m\leq p-1$. Similarly on $\mathcal{H}_{L_{(3,1)}}$, from the non-
zero fusion coefficients $N_{(1,m)(3,m)}^{(3,1)}$, $N_{(3,m)(1,m)}^{(3,1)}$,
$N_{(2,m)(4,m)}^{(3,1)}$, $N_{(4,m)(2,m)}^{(3,1)}$, $N_{(2,m)(2,m)}^{(3,1)}$,
and $N_{(3,m)(3,m)}^{(3,1)}$, we have

$\displaystyle\left\\{\pm\left(\frac{2p}{5}-m\right),\pm\left(\frac{3p}{5}-m\right),0\right\\}.$
(32)

Finally, on $\mathcal{H}_{L_{(4,1)}}$, from the non-zero fusion coefficients
$N_{(1,m)(4,m)}^{(4,1)}$, $N_{(4,m)(1,m)}^{(4,1)}$, $N_{(2,m)(3,m)}^{(4,1)}$,
and $N_{(3,m)(2,m)}^{(4,1)}$, we have

$\displaystyle\left\\{\pm\frac{3}{4}(p-2m),\pm\frac{1}{4}(p-2m)\right\\}.$
(33)

The results are summarized in Table 6. We note that the constraints from the
quantum dimensions and constraints from the spin contents are the same.

$p$ | 2 | 3 | 4 | 6 | 7 | 8 | 9 | 11
---|---|---|---|---|---|---|---|---
$\mathcal{H}_{L_{(2,1)}}$ | $0,\pm\frac{1}{5}$ | $\pm\frac{1}{20},\pm\frac{1}{4},\pm\frac{9}{20}$ | $0,\pm\frac{1}{10},\pm\frac{2}{5},\pm\frac{1}{2}$ | $0,\pm\frac{1}{10},\pm\frac{2}{5},\pm\frac{1}{2}$ | $\pm\frac{1}{20},\pm\frac{1}{4},\pm\frac{9}{20}$ | $0,\pm\frac{1}{5},\pm\frac{3}{10},\pm\frac{1}{2}$ | $\pm\frac{3}{20},\pm\frac{1}{4},\pm\frac{7}{20}$ | $\pm\frac{3}{20},\pm\frac{1}{4},\pm\frac{7}{20}$
$\mathcal{H}_{L_{(3,1)}}$ | $0,\pm\frac{1}{5}$ | $0,\pm\frac{1}{5}$ | $0,\pm\frac{2}{5}$ | $0,\pm\frac{2}{5}$ | $0,\pm\frac{1}{5}$ | $0,\pm\frac{1}{5}$ | $0,\pm\frac{2}{5}$ | $0,\pm\frac{2}{5}$
$\mathcal{H}_{L_{(4,1)}}$ | $0$ | $\pm\frac{1}{4}$ | $0,\pm\frac{1}{2}$ | $0,\pm\frac{1}{2}$ | $\pm\frac{1}{4}$ | $0,\pm\frac{1}{2}$ | $\pm\frac{1}{4}$ | $\pm\frac{1}{4}$

Table 6: The spin contents of $\mathcal{H}_{L_{(r,1)}}$ in $\mathcal{M}(p,5)$
series.

Since this is the first example of the appearance of $p=2$ as in the flow
$\mathcal{M}(8,5)\to\mathcal{M}(2,5)$ induced by $\phi_{(1,3)}$, let us
comment on it. We will give more detailed discussions in section 4.4. Unlike
in $\mathcal{M}(p,5)$ with $p>2$, in $\mathcal{M}(2,5)$, $L_{(4,1)}$ is
identified with $L_{(1,1)}$ and $L_{(3,1)}$ is identified with $L_{(2,1)}$, so
we have only half of the preserved non-invertible symmetries in the infrared
compared with the ones in the ultraviolet. These symmetries may be
spontaneously broken or disappear. Note, in particular, that The
$\mathbb{Z}_{2}$ invertible symmetry generated by $L_{(4,1)}$ is non-
anomalous, so it can simply disappear. Understanding the fate of these
symmetries requires dynamical analysis which is beyond the scope of our paper,
but further discussions follow in section 4.4.

Let us finally make a brief discussion on the half-integer $k$ flow. The
preserved topological defect lines are $L_{(1,1)}$ and $L_{(3,1)}$, which form
the fusion rule of $\mathcal{M}(2,5)$ (Lee-Yang fusion ring). It is also
called the Fibonacci fusion ring (see the appearance of the golden ratio
above). There exist two distinct renormalization group flows depending on
$d_{(3,1)}$, which is consistent with the classification of the modular tensor
category with the Fibonacci fusion ring. The $\mathbb{Z}_{2}$ invertible
symmetry is broken by the deformation, but it reappears as an emergent
symmetry. This is how we can change the (non-)anomalous nature of the
$\mathbb{Z}_{2}$ invertible symmetry along the renormalization group flow.

###  4.4 Flows to $\mathcal{M}(p,2)$ or $\mathcal{M}(2,q)$

As far as the non-invertible symmetry goes, there is nothing wrong with our
flows $\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$ even when $q=2$ or $kq-I=2$.
A closer look tells us, however, the situation is more subtle essentially
because there is no $\mathbb{Z}_{2}$ invertible symmetry in $\mathcal{M}(p,q)$
when either $p$ or $q$ is $2$. From the viewpoint of the topological defect
lines, the would-be $\mathbb{Z}_{2}$ topological defect line $L_{(q-1,1)}$ is
identified with the identity defect line $L_{(1,1)}$. In other words, there
exist no charged primary operators under the $\mathbb{Z}_{2}$ invertible
symmetry.

By taking $q=2$, our proposal suggests the existence of the renormalization
group flow $\mathcal{M}(2k+I,2)\to\mathcal{M}(2k-I,2)$ induced by
$\phi_{(1,2k+1)}$. The case with $I=1$, however, may sound problematic because
$\phi_{(1,2k+1)}$ is outside of the Kac table. The most reasonable
interpretation is that the flow exists but the primary operator that induces
the renormalization group flow is a certain fine-tuned combination of
$\phi_{(1,l)}$ with $l=1,2,\cdots,k$.111111Under the identification, this is
equivalent to taking $l=1,3,\cdots,2k-1$. Indeed, the numerical analysis based
on the truncated conformal space approach shows the existence of such flows
(for small $k$) [6, 58]. Of course, we could be simply agnostic about these
flows because the argument based on the non-invertible symmetries does not
give any constraint at all here, but we are tempted to unify them within our
proposal.

Another subtle case is $q\neq 2$ but $kq-I=2$. In this case, the question is
about the fate of the $\mathbb{Z}_{2}$ invertible symmetry. We start with the
ultraviolet theory with a $\mathbb{Z}_{2}$ invertible symmetry and we add the
deformation that preserves it. In the infrared limit, we end up with the
theory that does not possess any $\mathbb{Z}_{2}$ invertible symmetry. What
happened to the symmetry? One possibility is it is spontaneously broken.
Another possibility is all the charged operators are decoupled in the infrared
limit. The latter is possible because the $\mathbb{Z}_{2}$ symmetry here is
non-anomalous.

In the massive flow example of $\mathcal{M}(4,3)\to\mathcal{M}(2,3)$ induced
by $\phi_{(1,3)}$ (formally $k=1$, $I=1$ flow), where a Lagrangian description
is available, both can happen depending on the sign of the deformation.
Regarding $\mathcal{M}(4,3)$ as a critical $\phi^{4}$ theory, adding
$m^{2}\phi^{2}$ with positive $m^{2}$ gives decoupling of the $\mathbb{Z}_{2}$
charged field $\phi$, and with negative $m^{2}$ it gives spontaneous breaking
of the $\mathbb{Z}_{2}$ symmetry.

Physically $\mathcal{M}(2,q)$ can be realized in the multi-critical Lee-Yang
fixed point in two dimensions. So far, the canonical construction of the
(multi-critical) Lee-Yang fixed point was to start with the $\mathbb{Z}_{2}$
symmetric theory, say Ising model, and add the (imaginary) $\mathbb{Z}_{2}$
breaking interaction, say pure imaginary magnetic field, to get to the fixed
point. This new construction without explicitly breaking the $\mathbb{Z}_{2}$
symmetry is novel and worth studying further.

##  5 Discussions

In this paper, based on the study of non-invertible symmetries, we have
proposed there exist infinitely many new renormalization group flows between
Virasoro minimal models $\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$ induced by
$\phi_{(1,2k+1)}$. They vastly generalize the previously proposed
renormalization group flows in the literature.

One should be able to check our new renormalization group flows by using the
conformal perturbation theory or truncated conformal space approach [59, 60,
61]. Here, let us simply observe that the deformation operator
$\phi_{(1,2k+1)}$ is more relevant when the separation of the renormalization
group flow given by $I$ is larger. This agrees with our intuitive picture; the
more relevant deformations, the longer the renormalization group trajectory
becomes. It should also agree with the difference of the central charge at the
fixed points. More recent approaches to the renormalization group flow based
on the boundary states and interfaces can be found in [62, 63, 64], and they
may be useful to understand our new renormalization group flows better.

For a future direction, it is interesting to study the integrable structure of
our new renormalization group flows
$\mathcal{M}(kq+I,q)\to\mathcal{M}(kq-I,q)$. As initiated in [49, 65] the
flows with $I=1$ (or $I=\frac{1}{2})$ can be described by the Thermodynamic
Bethe Ansatz [66, 67, 68, 69, 70, 71, 72]. When $I>1$, there seem to be no
known descriptions in terms of the Thermodynamic Bethe Ansatz, but the
integral equations that describe the effective central charge along the flow
were proposed [4]. It should be interesting to generalize these works to our
new renormalization group flows.

Another direction to be explored further is a concrete realization of our new
renormalization group flows in statistical models or quantum field theories.
It is known that any (non-unitary) minimal models have a statistical model
realization based on the Restricted Solid-On-Solid (RSOS) model [73, 74, 75,
76] (see also [77, 78] for the related spin chain construction). It is
interesting to understand the nature of the anomalous $\mathbb{Z}_{2}$
symmetries, duality defects as well as general non-invertible symmetries
preserved by our renormalization group flow.

As for the quantum field theory realization, we are still in our infancy even
more than fifty years after the birth of conformal field theories. This is
because the quantum field theory descriptions (e.g. Landau-Ginzburg theory
[79] or gauge theory [80]) for non-unitary minimal models are largely unknown.
We, nevertheless, note that $2d$-$4d$ correspondence [81] gives realizations
of certain (but probably not all) non-unitary chiral algebra from
supersymmetric conformal field theories in four dimensions. We hope our
renormalization group flow may become a breakthrough in this direction.

Concerning the explicit constructions of minimal models from known quantum
field theories, while our discussions focus on the infrared flow, it seems
interesting to look at the flow in the opposite way as an ultraviolet flow
[82, 83]. Since the ultraviolet flow is a non-renormalizable deformation, we
may need some other input such as an integrability to pursue, but once done,
it should give a strong hint toward the quantum field theory descriptions of
the ultraviolet theory whose “Lagrangian description” were unknown.

Why do we crave Lagrangian descriptions of the Virasoro minimal models? We
wish to point out here that once the Lagrangian descriptions are available, we
may generalize these models in higher dimensions. When we obtain the higher
dimensional fixed points, we can ask whether the constraints on the
renormalization flow from the non-invertible symmetries can also be uplifted.
Since we have less explicit constructions of non-invertible symmetries in
higher dimensions, this is an outstanding question to be pursued.

## Acknowledgments

TT is supported by JSPS KAKENHI Grant Number JP24KJ1500. YN is in part
supported by JSPS KAKENHI Grant Number 21K03581. YN would like to thank C. Ahn
and K. Kikuchi for the correspondence and discussions.

## References

  * [1] A. B. Zamolodchikov,  Sov. J. Nucl. Phys. 46, 1090 (1987).
  * [2] C.-r. Ahn,  Phys. Lett. B 294, 204 (1992), hep-th/9202028.
  * [3] M. Lassig,  Phys. Lett. B 278, 439 (1992).
  * [4] P. Dorey, C. Dunning, and R. Tateo,  Nucl. Phys. B 578, 699 (2000), hep-th/0001185.
  * [5] I. R. Klebanov, V. Narovlansky, Z. Sun, and G. Tarnopolsky,  JHEP 02, 066 (2023), 2211.07029.
  * [6] M. Lencsés, A. Miscioscia, G. Mussardo, and G. Takács,  JHEP 02, 046 (2023), 2211.01123.
  * [7] C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin,  JHEP 01, 026 (2019), 1802.04445.
  * [8] Y.-H. Lin and S.-H. Shao,  J. Phys. A 54, 065201 (2021), 1911.00042.
  * [9] D. Aasen, P. Fendley, and R. S. K. Mong,  (2020), 2008.08598.
  * [10] R. Thorngren and Y. Wang,  JHEP 07, 051 (2024), 2106.12577.
  * [11] T.-C. Huang, Y.-H. Lin, K. Ohmori, Y. Tachikawa, and M. Tezuka,  Phys. Rev. Lett. 128, 231603 (2022), 2110.03008.
  * [12] M. Buican, A. Dymarsky, and R. Radhakrishnan,  JHEP 03, 017 (2023), 2112.12162.
  * [13] I. M. Burbano, J. Kulp, and J. Neuser,  JHEP 10, 186 (2022), 2112.14323.
  * [14] C.-M. Chang, J. Chen, and F. Xu,  SciPost Phys. 15, 216 (2023), 2208.02757.
  * [15] Y.-H. Lin, M. Okada, S. Seifnashri, and Y. Tachikawa,  JHEP 03, 094 (2023), 2208.05495.
  * [16] D.-C. Lu and Z. Sun,  JHEP 02, 173 (2023), 2208.06077.
  * [17] J. Kaidi, K. Ohmori, and Y. Zheng,  Commun. Math. Phys. 404, 1021 (2023), 2209.11062.
  * [18] M. Cheng and N. Seiberg,  SciPost Phys. 15, 051 (2023), 2211.12543.
  * [19] A. Chatterjee, W. Ji, and X.-G. Wen,  (2022), 2212.14432.
  * [20] Y.-H. Lin and S.-H. Shao,  Phys. Rev. D 107, 125025 (2023), 2302.13900.
  * [21] J. L. Jacobsen and H. Saleur,  JHEP 12, 090 (2023), 2305.05746.
  * [22] Y. Choi, B. C. Rayhaun, Y. Sanghavi, and S.-H. Shao,  Phys. Rev. D 108, 125005 (2023), 2305.09713.
  * [23] B. Haghighat and Y. Sun,  (2023), 2306.16555.
  * [24] N. Seiberg and S.-H. Shao,  SciPost Phys. 16, 064 (2024), 2307.02534.
  * [25] A. Antinucci, F. Benini, C. Copetti, G. Galati, and G. Rizi,  (2023), 2308.11707.
  * [26] Z. Duan, Q. Jia, and S. Lee,  JHEP 11, 206 (2023), 2309.01913.
  * [27] J. Chen, B. Haghighat, and Q.-R. Wang,  (2023), 2309.01914.
  * [28] Y. Nagoya and S. Shimamori,  JHEP 12, 062 (2023), 2309.05294.
  * [29] M. Sinha, F. Yan, L. Grans-Samuelsson, A. Roy, and H. Saleur,  (2023), 2310.19703.
  * [30] Y. Choi, D.-C. Lu, and Z. Sun,  JHEP 01, 142 (2024), 2310.19867.
  * [31] O. Diatlyk, C. Luo, Y. Wang, and Q. Weller,  JHEP 03, 127 (2024), 2311.17044.
  * [32] C. Cordova and G. Rizi,  (2023), 2312.17308.
  * [33] N. Seiberg, S. Seifnashri, and S.-H. Shao,  SciPost Phys. 16, 154 (2024), 2401.12281.
  * [34] C. Copetti, L. Cordova, and S. Komatsu,  (2024), 2403.04835.
  * [35] A. Chatterjee, O. M. Aksoy, and X.-G. Wen,  (2024), 2405.05331.
  * [36] Y. Nakayama and K. Kikuchi,  JHEP 03, 240 (2023), 2212.06342.
  * [37] P. Di Francesco, P. Mathieu, and D. Senechal,  Conformal Field TheoryGraduate Texts in Contemporary Physics (Springer-Verlag, New York, 1997).
  * [38] V. B. Petkova,  Int. J. Mod. Phys. A 3, 2945 (1988).
  * [39] C.-T. Hsieh, Y. Nakayama, and Y. Tachikawa,  Phys. Rev. Lett. 126, 195701 (2021), 2002.12283.
  * [40] K. Kikuchi,  (2021), 2109.02672.
  * [41] K. Kikuchi,  (2022), 2204.03247.
  * [42] K. Kikuchi,  (2022), 2207.06433.
  * [43] K. Kikuchi,  (2022), 2207.10095.
  * [44] K. Kikuchi,  (2022), 2209.00016.
  * [45] O. A. Castro-Alvaredo, B. Doyon, and F. Ravanini,  J. Phys. A 50, 424002 (2017), 1706.01871.
  * [46] C. M. Bender and D. W. Hook,  (2023), 2312.17386.
  * [47] S. Gukov,  JHEP 01, 020 (2016), 1503.01474.
  * [48] S. Gukov,  Nucl. Phys. B 919, 583 (2017), 1608.06638.
  * [49] A. B. Zamolodchikov,  JETP Lett. 46, 160 (1987).
  * [50] M. J. Martins,  Phys. Rev. Lett. 69, 2461 (1992), hep-th/9205024.
  * [51] M. J. Martins,  Nucl. Phys. B 394, 339 (1993), hep-th/9208011.
  * [52] F. Ravanini, M. Stanishkov, and R. Tateo,  Int. J. Mod. Phys. A 11, 677 (1996), hep-th/9411085.
  * [53] L. Fei, S. Giombi, I. R. Klebanov, and G. Tarnopolsky,  Phys. Rev. D 91, 045011 (2015), 1411.1099.
  * [54] O. Delouche, J. Elias Miro, and J. Ingoldby,  SciPost Phys. 16, 105 (2024), 2312.09221.
  * [55] Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao,  Phys. Rev. D 105, 125016 (2022), 2111.01139.
  * [56] Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao,  Commun. Math. Phys. 402, 489 (2023), 2204.09025.
  * [57] W. Cui, B. Haghighat, and L. Ruggeri,  (2024), 2406.09261.
  * [58] M. Lencsés, A. Miscioscia, G. Mussardo, and G. Takács,  (2024), 2404.06100.
  * [59] V. P. Yurov and A. B. Zamolodchikov,  Int. J. Mod. Phys. A 5, 3221 (1990).
  * [60] V. P. Yurov and A. B. Zamolodchikov,  Int. J. Mod. Phys. A 6, 4557 (1991).
  * [61] G. Feverati, K. Graham, P. A. Pearce, G. Z. Toth, and G. Watts,  J. Stat. Mech. 0803, P03011 (2008), hep-th/0612203.
  * [62] D. Gaiotto,  JHEP 12, 103 (2012), 1201.0767.
  * [63] J. Cardy,  SciPost Phys. 3, 011 (2017), 1706.01568.
  * [64] A. Konechny,  JHEP 11, 004 (2023), 2306.13719.
  * [65] A. B. Zamolodchikov,  Nucl. Phys. B 358, 524 (1991).
  * [66] A. B. Zamolodchikov,  Nucl. Phys. B 342, 695 (1990).
  * [67] A. B. Zamolodchikov,  Nucl. Phys. B 358, 497 (1991).
  * [68] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov,  Commun. Math. Phys. 177, 381 (1996), hep-th/9412229.
  * [69] G. Feverati, E. Quattrini, and F. Ravanini,  Phys. Lett. B 374, 64 (1996), hep-th/9512104.
  * [70] H. Kausch, G. Takacs, and G. Watts,  Nucl. Phys. B 489, 557 (1997), hep-th/9605104.
  * [71] P. A. Pearce, L. Chim, and C.-r. Ahn,  Nucl. Phys. B 601, 539 (2001), hep-th/0012223.
  * [72] P. A. Pearce, L. Chim, and C. Ahn,  Nucl. Phys. B 660, 579 (2003), hep-th/0302093.
  * [73] V. Pasquier,  Nucl. Phys. B 295, 491 (1988).
  * [74] T. Nakanishi,  Nucl. Phys. B 334, 745 (1990).
  * [75] G. Takacs,  Nucl. Phys. B 489, 532 (1997), hep-th/9604098.
  * [76] G. Takacs and G. Watts,  Nucl. Phys. B 642, 456 (2002), hep-th/0203073.
  * [77] D. Bianchini and F. Ravanini,  Jounral of Physics A (2015), 1509.04601.
  * [78] R. Couvreur, J. L. Jacobsen, and H. Saleur,  Phys. Rev. Lett. 119, 040601 (2017), 1611.08506.
  * [79] A. B. Zamolodchikov,  Sov. J. Nucl. Phys. 44, 529 (1986).
  * [80] D. Delmastro, J. Gomis, and M. Yu,  JHEP 02, 157 (2023), 2108.02202.
  * [81] C. Beem and L. Rastelli,  JHEP 08, 114 (2018), 1707.07679.
  * [82] C. Ahn and A. LeClair,  JHEP 08, 179 (2022), 2205.10905.
  * [83] C. Ahn and Z. Bajnok,  (2024), 2407.06582.


# Monkeypox Skin Lesion Detection Using Deep Learning Models: A Feasibility
Study

Shams Nafisa Ali1, Md. Tazuddin Ahmed1, Joydip Paul1, Tasnim Jahan1, S. M.
Sakeef Sani1,
Nawsabah Noor2, Taufiq Hasan1, _Senior Member, IEEE_ 1mHealth Lab, Department
of Biomedical Engineering, Bangladesh University of Engineering and Technology
2Popular Medical College, Dhaka, Bangladesh. Email<EMAIL_ADDRESS>
###### Abstract

The recent monkeypox outbreak has become a public health concern due to its
rapid spread in more than 40 countries outside Africa. Clinical diagnosis of
monkeypox in an early stage is challenging due to its similarity with
chickenpox and measles. In cases where the confirmatory Polymerase Chain
Reaction (PCR) tests are not readily available, computer-assisted detection of
monkeypox lesions could be beneficial for surveillance and rapid
identification of suspected cases. Deep learning methods have been found
effective in the automated detection of skin lesions, provided that sufficient
training examples are available. However, as of now, such datasets are not
available for the monkeypox disease. In the current study, we first develop
the “Monkeypox Skin Lesion Dataset (MSLD)” consisting skin lesion images of
monkeypox, chickenpox, and measles. The images are mainly collected from
websites, news portals, and publicly accessible case reports. Data
augmentation is used to increase the sample size, and a 3-fold cross-
validation experiment is set up. In the next step, several pre-trained deep
learning models, namely, VGG-16, ResNet50, and InceptionV3 are employed to
classify monkeypox and other diseases. An ensemble of the three models is also
developed. ResNet50 achieves the best overall accuracy of $82.96(\pm 4.57\%)$,
while VGG16 and the ensemble system achieved accuracies of $81.48(\pm 6.87\%)$
and $79.26(\pm 1.05\%)$, respectively. A prototype web-application is also
developed as an online monkeypox screening tool. While the initial results on
this limited dataset are promising, a larger demographically diverse dataset
is required to further enhance the generalizability of these models.

###### Index Terms:

Computer-aided diagnosis, Skin lesion detection, Monkeypox, Deep learning.

##  I Introduction

As the world recovers from the COVID-19 pandemic, the recent multi-country
outbreak of monkeypox has raised concerns in global communities. The World
Health Organization (WHO) declared that the outbreak poses a moderate risk to
global public health and has stopped short of declaring it a public health
emergency. However, healthcare organizations such as World Health Network
(WHN) expressed a heightened concern [1] and highlighted the need for
immediate and concerted global action against the disease.

Monkeypox is a zoonotic disease from the genus Orthopoxvirus. It closely
resembles chickenpox, measles, and smallpox regarding clinical features [2].
The minor differences in the skin rash of these diseases, coupled with the
relative rarity of monkeypox have made the early diagnosis of this condition
very challenging for healthcare professionals. On the contrary, the
confirmatory PCR test is also not widely available. Although the case fatality
ratio has been reported to be 3–6% for the recent outbreak [2], early
detection of monkeypox, corresponding contact tracing, and isolation are
essential to limit the community transmission of the virus. In this scenario,
AI-based automated computer-aided systems may substantially limit its global
spread.

Figure 1: A flow-diagram of the proposed monkeypox detection system. A
prototype web-app is developed that incorporates the developed deep learning
models to detect monkeypox from skin lesions uploaded by users.

In recent years, the multi-faceted applications of deep learning (DL),
particularly the variations of Convolutional Neural Networks (CNNs), have
revolutionized different domains of medical science due to their superior
learning capability [3, 4]. When trained with a large number of data, these
deep networks can process images in different layers, automatically extracting
salient features and learning to identify the optimal representations for
specified tasks [5]. However, the requirement for large amounts of data and
time-consuming training with dedicated computational resources limits the
applicability of DL-based frameworks [6]. While using accelerators (e.g., GPU,
TPU) resolves the time and resource-related issues, the dataset-related
concerns persist due to the difficulty of obtaining unbiased, homogeneous
medical data. Data augmentation [7] is a well-known method of increasing the
dataset size by generating additional samples through slight modifications of
existing data. In case of scarcity of data, transfer learning [6] is also a
commonly used technique. This method utilizes a CNN model pre-trained on a
large dataset (e.g., ImageNet) and transfers its knowledge for context-
specific learning on a different, comparatively smaller dataset.

Currently, there is no publicly available monkeypox skin lesion dataset for
developing automated detection algorithms. There are impediments considering
privacy and validity concerns. Moreover, the high prevalence of monkeypox in
the under-developed African regions may introduce a bias in the dataset
because of very high inter-class similarity and intra-class variability. In
this paper, we first introduce the “Monkeypox Skin Lesion Dataset
(MSLD)”111https://github.com/mHealthBuet/Monkeypox-Skin-Lesion-Dataset, an
openly accessible dataset containing web-scrapped images of different body
parts (face, neck, hand, arm, leg) of patients with monkeypox and non-
monkeypox (measles, chickenpox) cases. We also present a DL-based preliminary
feasibility study leveraging transfer learning involving VGG16 [8], ResNet50
[9] and InceptionV3 [10] architectures to explore the potential of deep
learning models for the early detection of the monkeypox disease.
Additionally, we have soft-launched a web-app222https://monkey-pox-detector-
mhealthlab.herokuapp.com/ built on the open-source streamlit framework that is
capable of analysing the uploaded image and predicting whether a subject is a
potential monkeypox suspect and he/she should consult a physician on an urgent
basis or not. Our current working pipeline is graphically represented in Fig.
1.

Figure 2: The Monkeypox lesion through its various stages: (a) early vesicle,
(b) small pustule ($\oslash~{}2$mm), (c) umbilicated pustule
($\oslash~{}3$-$4$mm), (d) ulcerated lesion ($\oslash~{}5$mm), (e) crusting of
a mature lesion, (f) partially removed scab.

##  II Background

Monkeypox virus, an enveloped double-stranded DNA virus, is one of the members
of the Poxviridae family, which is closely related to the variola virus [2].
Various animal species, including squirrels, Gambian pouched rats, dormice,
and non-human primates, were first identified as the natural hosts of
monkeypox virus [2]. The first human case of monkeypox was confirmed in the
Democratic Republic of Congo (DRC) in 1970, and subsequently, it has been
endemic in the tropical rainforests of west and central Africa [2].

Since the beginning of 2022, monkeypox outbreaks have been reported in diverse
countries across the globe. Currently, 5135 laboratory-confirmed cases have
been reported from 66 member states, including the Regions of America, Europe,
Africa, Eastern Mediterranean, and Western Pacific (updated 30 June 2022)
[11]. After the rapid spread of the disease in non-endemic countries with no
epidemiological connections to endemic areas, the disease was declared a
moderate global health risk as assessed by the WHO [2].

Monkeypox is usually a self-limited disease; the symptoms last 2 to 4 weeks.
Children are mostly affected severely. The severity of the disease depends on
the extent of virus exposure, the health status of the patient, and the
complications. The virus’s incubation period is in the range of 5 to 21 days
[2]. The invasion period (0-5 days) is characterized by the symptoms of fever,
lymphadenopathy (swollen lymph nodes), myalgia (muscle ache), asthenia
(physical weakness), and severe headache. The rash begins within 1-3 days of
fever onset and is usually noticed on the face, palms of the hands, and soles
of the feet. However, it can also be found on the mouth, eyes, and genitals
[2]. In the skin eruption phase (2-4 weeks), the lesions follow the four-stage
progression; macules (lesions with a flat base) to papules (raised firm
painful lesions) to vesicles (filled with clear fluid) to pustules (filled
with pus) followed by encrustation.

The gold standard diagnosis of monkeypox is done through histopathology and
virus isolation. In addition, a PCR test can help for the confirmation of the
diagnosis [12]. However, when these diagnostic tools are not readily
available, early diagnosis based on clinical examination of the skin lesions
can be beneficial. With the ubiquitous availability of smartphones, AI-based
skin lesion detection systems can assist in diagnosis and bridge the gap in
the healthcare systems.

TABLE I: Distribution of the Monkeypox Skin Lesion Dataset (MSLD)

Class label |  | No. of Original
---
Images
| No. of Unique
---
Patients
| No. of Augmented
---
Images
Monkeypox | 102 | 55 | 1428
Others | 126 | 107 | 1764
Total | 228 | 162 | 3192

##  III Dataset Preparation

Our monkeypox skin lesion dataset is primarily compiled from publicly
available case reports, news portals, and websites through extensive manual
searching. We did not use automatic web-scrappers. In this work, we mainly
focus on distinguishing the ‘Monkeypox’ cases from similar non-monkeypox cases
[2]. Thus, we include skin lesion images of chickenpox and measles as the
‘Others’ class and prepare the dataset for binary classification

All the skin lesion images were verified using Google’s Reverse Image Search
and cross-referenced with other sources. Through a 2-stage screening process,
the out-of-focus, low-resolution, and low-quality images were discarded, and
only the unique images that satisfy the quality requirements were selected.
Next, the images were cropped to their region of interest and resized to
$224\times 224$ pixels while maintaining the aspect ratio.

Figure 3: Sample images from the collected dataset. Images from the
‘Monkeypox’ class are shown on the left panel, while the ‘Others’ class
(measles and chickenpox) images are shown on the right.

We have collected a total number of $228$ images, among which $102$ belongs to
the ‘Monkeypox’ class and the remaining $126$ represents the ‘Others’ class
(i.e., chickenpox and measles). A few representative samples are shown in Fig.
3. Since this is the early phase of the monkeypox outbreak, reliable data are
scarce. Hence, to aid the classification task, several data augmentation
methods, including rotation, translation, reflection, shear, hue, saturation,
contrast and brightness jitter, noise, and scaling, have been applied to
increase the dataset size. Post-augmentation, the number of images increased
by approximately 14-folds, with the classes ‘Monkeypox’ and ‘Others’
consisting of $1428$ and $1764$ images, respectively. The augmented images are
also provided in a separate folder in the dataset to ensure reproducibility.
However, based on their requirement, the users may also apply different types
of image augmentor on the original images. A detailed distribution of the
dataset is provided in Table I.

##  IV Experiments

###  IV-A Pre-trained Models

In this work, we select three well-known CNN architectures namely, VGG16 [8],
ResNet50 [9] and InceptionV3 [10], pre-trained on the ImageNet dataset. These
models are chosen as they have demonstrated excellent classification
performance across different fields of computer vision, including medical
images through transfer learning.

Figure 4: An original monkeypox skin lesion image and its 13 augmented
versions used for training. These created as follows: (1) Random rotation by a
multiple of 90 degrees, (2) Random rotation in the range of -45 to 45 degrees,
(3) Translation, (4) Reflection, (5) Shear, (6) Hue jitter, (7) Saturation
jitter, (8) Brightness jitter, (9) Contrast jitter, (10) Salt and Pepper
Noise, (11) Gaussian Noise, (12) Synthetic Blur and (13) Scaling.

VGG16 is one of the first networks to investigate CNN’s depth to improve
accuracy in large-scale image recognition tasks [8]. It consists of $3\times
3$ convolutional filters applied in several layers. The concept of utilizing
Factorized Convolution enabled feature extraction from increased depth but
simultaneously ensured that the model did not overfit the training data. The
ResNet model was formulated based on the concept of residual learning [9]. It
consists of residual modules where convolution operations are followed by
Batch Normalization and ReLU non-linearity. These blocks enable the inputs to
forward propagate faster and extract features more efficiently. Because of the
“stacked” Inception modules, despite having fewer parameters than VGG16, the
Inception architecture can perform better back-propagation [10]. Since its
first release, several changes have been made to the original architecture,
and improved versions have been released.

###  IV-B Implementation Details

Input images with dimensions $(224,224,3)$ were fed into the selected pre-
trained models. The fully connected layers were removed. Although the
architectures have varying depths, we experimented with keeping different
numbers (4/6/8/12) of bottom layers trainable to ensure homogeneity and better
generalization. After extensive experimentation, we decided to unfreeze the
bottom eight layers. Next, we flatten the backbone model’s output, followed by
three blocks of fully connected (FC) layers, and dropout to the network. The
FC layers had successively $4096$, $1072$ and $256$ nodes while the dropout
factors were respectively $0.3$, $0.2$ and $0.15$. Finally, an FC layer with
two nodes was employed with a softmax activation function for this binary
classification task.

The network architectures were implemented in Keras and were accelerated using
Nvidia K80 GPUs provided by Kaggle notebooks. The batch size was set to 16.
The adaptive learning rate optimizer (Adam) with an initial learning rate of
$10^{-5}$ and binary cross-entropy loss function was employed for training.

Figure 5: An overview of the train-test split used for 3-fold cross-
validation. Data augmentation is used for the training and validation set
only.

###  IV-C Experimental Design

A three-fold cross-validation experiment was conducted. The original images
were split into training, validation, and test set(s) with the approximate
proportion of 70:10:20 while maintaining patient independence. According to
the commonly perceived data preparation practice, the training and validation
images were augmented while the test set contained only the original images
(see Fig. 5). Accuracy, precision, F1-score, sensitivity are used as
performance metrics.

###  IV-D Results

We evaluated the performance of the selected pre-trained models for the 3-fold
cross-validation experiment. The obtained results are summarized in Table II.
While ResNet50 yields the best accuracy ($82.96\pm 4.57\%$), VGG16 shows
competitive performance ($81.48\pm 6.87\%$). We also implement an ensemble of
the 3 models by employing majority voting. Based on averaged performance
across the 3-folds, the ensemble did not show superior results compared to the
best-performing ResNet50 model. However, the ensemble system shows the lowest
standard deviation of the accuracy metric, indicating that its performance is
more consistent across the 3 folds.

The best performing model has been deployed in a prototype web-app. Users can
upload a photo of their skin lesions and receive an initial assessment for
monkeypox (see Fig. 6).

TABLE II: Performance Comparison of Different Deep Learning Models and The
Proposed Ensemble

Network | Accuracy (%) | Precision | Recall | F1 score
---|---|---|---|---
VGG16 | 81.48$\pm$ 6.87 | 0.85 $\pm$ 0.08 | 0.81 $\pm$ 0.05 | 0.83 $\pm$ 0.06
ResNet50 | 82.96$\pm$ 4.57 | 0.87 $\pm$ 0.07 | 0.83 $\pm$ 0.02 | 0.84 $\pm$ 0.03
InceptionV3 | 74.07$\pm$ 3.78 | 0.74 $\pm$ 0.02 | 0.81 $\pm$ 0.07 | 0.78 $\pm$ 0.04
Ensemble | 79.26$\pm$ 1.05 | 0.84 $\pm$ 0.05 | 0.79 $\pm$ 0.07 | 0.81 $\pm$ 0.02

Figure 6: The interface of the online monkeypox screening tool.

##  V Discussion

In this work, we present our preliminary findings of monkeypox skin lesion
detection using well-known deep learning architectures. Although the
classification performance obtained is quite promising, certain constraints
limit the wider applicability of the results. Firstly, the current number of
unique patients in the dataset is limited. This reduces the generalization
capability of these models. If more samples can be included with a better
geographical, racial, and gender distribution, it is possible to achieve a
consistent performance across such variabilities. Secondly, as a initial
approach, we have used the pre-trained weights from the ImageNet dataset for
transfer learning which does not contain any skin lesion images. Thus, the
accuracy and generalizability of the system could be further improved by using
a multi-source dermatoscopic image dataset for pre-training the models.
Furthermore, the dataset is created primarily by web-scraping, which lacks
relevant meta-data that is vital for diagnosis, e.g., patient’s clinical
history, number of days since disease onset, and the disease stage. A more
consorted effort and international collaboration is needed to collect a larger
dataset that can provide generalizable results across different demographics.

##  VI Conclusion

In this study, we have presented the open-source “Monkeypox Skin Lesion
Dataset (MSLD)” for automatic detection of monkeypox from skin lesions and
performed an initial feasibility study using state-of-the-art deep learning
architectures (VGG16, ResNet50, InceptionV3) leveraging the transfer learning
approach. Despite being a small dataset, the promising results obtained after
3-fold cross-validation reveal the potential to use AI-assisted early
diagnosis of this disease. We are hopeful that this dataset will create new
avenues for the researchers in terms of developing remotely deployable
computer-aided diagnostic tools for wide-scale screening and early detection
of monkeypox, especially in cases where traditional testing methods are
unavailable. We also believe that our soft-launched web-app prototype will
assist the monkeypox suspects to conduct preliminary screening from the
comforts of home and enable them to take adequate action in the early stages
of the infection.

## References

  * [1] “The World Health Network Declares Monkeypox A Pandemic - Press Release— June 22, 2022,” 2022, [Online]. Available: https://www.worldhealthnetwork.global/monkeypoxpressrelease.
  * [2] “WHO — Monkeypox Fact Sheet,” 2022, [Online]. Available: https://www.who.int/news-room/fact-sheets/detail/monkeypox.
  * [3] D. Ravì, C. Wong, F. Deligianni, M. Berthelot, J. Andreu-Perez, B. Lo, and G.-Z. Yang, “Deep learning for health informatics,” _IEEE JBHI_ , vol. 21, no. 1, pp. 4–21, 2016.
  * [4] F. Wang, L. P. Casalino, and D. Khullar, “Deep learning in medicine—promise, progress, and challenges,” _JAMA Int. Med._ , vol. 179, no. 3, pp. 293–294, 2019.
  * [5] K. M. Hosny, M. A. Kassem, and M. M. Foaud, “Classification of skin lesions using transfer learning and augmentation with Alex-net,” _PloS one_ , vol. 14, no. 5, p. e0217293, 2019.
  * [6] S. J. Pan and Q. Yang, “A survey on transfer learning,” _IEEE Trans. knowl. data. en._ , vol. 22, no. 10, pp. 1345–1359, 2009.
  * [7] C. Shorten and T. M. Khoshgoftaar, “A survey on image data augmentation for deep learning,” _J. of Big Data_ , vol. 6, no. 1, pp. 1–48, 2019.
  * [8] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _arXiv preprint arXiv:1409.1556_ , 2014.
  * [9] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proc. IEEE CVPR_ , 2016, pp. 770–778.
  * [10] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” in _Proc. IEEE CVPR_ , 2016, pp. 2818–2826.
  * [11] “Monkeypoxmeter: Real time Monkeybox tracker,” 2022, [Online]. Available: https://www.monkeypoxmeter.com/.
  * [12] Centers for Disease Control and Prevention, “Monkeypox — poxvirus,” https://www.cdc.gov/poxvirus/monkeypox/index.html, (Accessed on 06/30/2022).