Exploring duality symmetries, multicriticality and RG flows at $c=2$

Jeremias Aguilera Damiaa, Giovanni Galatia,
Ondrej Hulikb, and Salvo Mancania

a Physique Théorique et Mathématique and International Solvay Institutes
Université Libre de Bruxelles, C.P. 231, 1050 Brussels, Belgium
b Theoretische Natuurkunde, Vrije Universiteit Brussel
Pleinlaan 2, B-1050 Brussels, Belgium

###### Abstract

In this work, we study the realization of non-invertible duality symmetries
along the toroidal branch of the $c=2$ conformal manifold. A systematic
procedure to construct symmetry defects is implemented to show that all
Rational Conformal Field Theories along this branch enjoy duality symmetries.
Furthermore, we delve into an in-depth analysis of two representative cases of
multicritical theories, were the toroidal branch meets various orbifold
branches. For these particular examples, the categorical data and the defect
Hilbert spaces associated to the duality symmetries are obtained by resorting
to modular covariance. Finally, we study the interplay between these novel
symmetries and the various exactly marginal and relevant deformations,
including some representative examples of Renormalization Group flows where
the infrared is constrained by the non-invertible symmetries and their
anomalies.

###### Contents

  1. 1 Introduction
  2. 2 Review of duality symmetries for the $c=1$ bosonic CFT
  3. 3 $c=2$ Toroidal CFT and Duality symmetries
    1. 3.1 Discrete gauging and duality symmetries
  4. 4 Duality symmetries at multicriticality
    1. 4.1 The quadri-critical point
      1. 4.1.1 Duality symmetry and fusion category
      2. 4.1.2 Modular bootstrap at the quadri-critical point
      3. 4.1.3 Marginal and relevant deformations at the quadri-critical point
    2. 4.2 The bi-critical point
      1. 4.2.1 Duality symmetry and fusion category
      2. 4.2.2 Modular bootstrap at the bi-critical point
      3. 4.2.3 Marginal and relevant deformations at the bi-critical point
  5. 5 Duality symmetries of $SU(3)_{1}$ WZW
  6. A Details on compact bosons
    1. A.1 The self-dual point at $c=1$
    2. A.2 The $SU(3)$ point at $c=2$
  7. B Partition function for RCFTs at $c=2$
  8. C Tambara-Yamagami categories
  9. D Modular functions

##  1 Introduction

Quantum Field Theories in two spacetime dimensions often provide a suitable
arena to explore interesting dynamical aspects while still retaining some
analytical control. This becomes more explicit when the field theory enjoys
conformal invariance, i.e. it is a Conformal Field Theory (CFT). In such a
case, the theory dynamics get highly constrained by the infinite dimensional
Virasoro algebra [1, 2]. Moreover, when the spacetime is taken to be the
Eucliedean torus111Of course, this notion extends to any Riemann surface of
arbitrary genus, with the large diffeomorphisms comprised in the Mapping Class
Group (see for instance [3]). However, considering the two-dimensional torus
will be enough for the purposes of this work., covariance under large
diffeomorphisms, encoded in the modular group, places additional restrictions
on the space of states of the theory. For some purposes, this proves enough to
determine the structure of the Hilbert space or the behavior of some
observables analytically over certain parametric regimes [4], remarkably the
high temperature density of states as pioneered by Cardy [5] (see also [6] for
a novel generalization to higher dimensions). More generally, non-trivial
solutions of these constraints can be found either analytically or numerically
via more refined conformal bootstrap techniques [7, 8, 9, 10, 11]. A special
set of conformal field theories in two dimensions corresponds to the Rational
Conformal Field Theories (RCFT) [9]. A key property of these theories is that
the Hilbert space decomposes into a finite set of representations of some
chiral algebra. For $c<1$, all unitary conformal field theories are of this
kind and the partition function is accounted by a finite set of
representations of the Virasoro algebra. These are the minimal models. For
higher values of the central charge, i.e. $c\geq 1$, certain points on the
conformal manifold meet some extra structure due to the occurrence of
additional, generically higher spin, conserved currents assembling an enhanced
chiral algebra. Notably, the dynamics of these remarkable theories is
described in terms of a finite set of conformal primaries with respect to the
enhanced chiral algebra, hence corresponding to RCFTs. We will focus on
examples of this kind throughout this work.

Concomitantly, rational CFTs played a prominent role in the discovery and
characterization of topological defects, leading to novel forms of global
symmetry, and the development of alternative descriptions in terms of higher
dimensional topological field theories (TQFT) (see e.g. [12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22]). In this work, we will focus on the former aspect,
namely the description of interesting global symmetry structures and the
exploration of their consequences. A comprehensive treatment of the general
formalism and some applications for $c\leq 1$ CFTs can be found in e.g. [18,
19, 20, 23], whereas we will be mainly concerned about examples at higher
values of the central charge, specifically bosonic RCFTs 222Recently, non-
invertible duality symmetries were constructed for a particular subset of
irrational field theories at $c=2$ [24]. We will not consider this type of
theories. at $c=2$. A prominent role is played by duality symmetries. This
type of global symmetry arise in theories featuring self-duality upon gauging
a particular subset of its global symmetry. In particular, the conformal
manifold of $c\geq 1$ CFTs accommodates the action of non-trivial duality
groups (see e.g. [25] for a review), making the occurrence of duality
symmetries quite natural333 Let us mention that duality symmetries are not
exclusive of lower dimensional theories, see for instance [26, 27, 28, 29, 30]
for some constructions and applications in four dimensions.

Due to the fact that the action associated to duality symmetries usually
involves an orbifold by some (invertible)444See the recent work [31, 32] for
some alternative constructions involving gauging non-invertible symmetries.
global symmetry, the fusion realized by its associated topological operators
does not follow a group law. The underlying structure is generically accounted
for by Category Theory. A distinctive feature of conformal field theories in
two dimensions is that the construction of topological defects is highly
conditioned by the modular bootstrap555A priori, these conditions find a
natural generalization in higher dimensions, achieved by putting the theory on
appropriate spacetime manifolds. However, we are not aware of a precise
treatment along these lines for higher dimensional theories., as a natural
consequence of modular covariance [12, 33, 13, 18, 19]. This program not only
enables the determination of consistent topological defects, but also allows
to obtain the additional topological data involved in the underlying
categorical structure [20]. We will apply these tools extensively for the
examples studied in this paper. Remarkably, solutions to the various
constraints imposed by associativity of the OPE and modular covariance belong
to a finite discrete set, a fact referred to as Ocneanu rigidity [34]. This
naturally endows categorical symmetries with a notion of robustness, rendering
well defined Renormalization Group (RG) invariants, such as ordinary
symmetries. Concomitantly, duality symmetries may participate in anomalies
[20, 35, 36, 37, 38, 39], hence leading to interesting constraints on RG flows
triggered by duality symmetric operators. In this work we explicitly show how
anomalous duality symmetries constraint symmetry preserving RG flows.

A special loci pertaining to the conformal manifold of CFTs with central
charge $c\geq 1$ corresponds to the set of multicritical points. These
(generically rational) theories may connect various branches of the conformal
manifold. A prototypical example is the Kosterlitz-Thouless (KT) theory
sitting at the merging point between the circle and the orbifold branches of
the $c=1$ bosonic conformal manifold. In particular, it has been found that
the KT point preserves a series of non-invertible duality symmetries some of
which are further preserved by the exactly marginal deformation spanning the
orbifold branch [20]. Inspired by this kind of analysis, we present a
characterization of the duality symmetry structure featured at certain
multicritical points along the toroidal branch of the conformal manifold of
$c=2$ bosonic CFTs. Contrary to what happens at $c=1$, the richer duality
group at $c=2$ allows for the occurrence of categorical symmetries graded by a
non-commutative structure, as we will illustrate in the examples below (see
[29, 40] for recent examples of non-commutative duality symmetries constructed
in higher dimensions).

Topological defects in CFT are a subset of the conformal defects. On general
grounds, the characterization of conformal defects plays a significant role in
many fields, ranging from the description of finite size effects in critical
systems [41, 42, 43] to the classification of branes in String Theory [44,
45]. Furthermore, recent studies established a connection between self-duality
symmetries in bosonic CFT’s and interesting topological transitions between
fermionic theories by means of a generalized Jordan-Wigner map [46, 47]. By
similar considerations, the structures uncovered in this work may shed light
onto more intricate transitions taking place along the fermionic conformal
manifold at higher values of the central charge. This type of systems have
engendered some interest due to their potential application in designing
quantum computing devices [48]. We hope to explore these aspects in more
detail in the near future.

This paper is organized as follows. In section 2 we briefly review some
properties of the $c=1$ theory, emphasizing its symmetry structure and the
modular properties of its (twisted) torus partition function that will be
important for the discussion of the $c=2$ theory. In section 3 we review
properties of the $c=2$ toroidal branch of the conformal manifold and we
explicitly show that any RCFT point within this branch enjoys non-invertible
duality symmetries arising form the self-duality of these theories under
gaugings of subgroups of their symmetries. To achieve that, we introduce the
generalized metric $\mathcal{E}$ as a useful parametrization of the conformal
manifold. We rephrase the generic condition for the presence of a duality
symmetry defect $\mathcal{D}$ as the existence of an associated matrix
$\mathsf{D}\in O(2,2,\mathbb{Q})$, parametrizing a given topological
manipulation, such that

$(\mathsf{D}^{-1})^{\intercal}\mathcal{E}\mathsf{D}^{-1}=\mathcal{E}\,.$ (1.1)

In section 4 we delve into the detailed analysis of some multicritical points
of the toroidal branch which we dub quadri-critical and bi-critical point,
where the theory possess extra exactly marginal parameters generating orbifold
branches. Being RCFTs, these points have non-invertible duality symmetries
which we analyze in detail, describing the underlying categorical structure
and their consistency conditions coming from the modular bootstrap. We then
describe duality preserving marginal and relevant deformations. In the quadri-
critical point the theory factorize in the product of two $c=1$ RCFT.
Therefore we can explicitly check some of the constraints coming from the
above-mentioned duality symmetry, for instance verifying the ones coming from
the non-invertible ’t Hooft anomalies. On the other hand, in the bi-critical
point the theory is genuinely $c=2$, in the sense that it is not a product of
$c=1$ theories, and some considerations coming from the duality symmetry are
actually non-trivial predictions.

In section 5 we briefly comment on the enhanced symmetry point where the
theory is equivalent to $SU(3)_{1}$ WZW model. As opposed to the $c=1$ case,
this point still enjoys some non-invertible duality symmetries which are not
of the Verlinde type and that we briefly analyze. We end with some appendices
containing some technical details.

##  2 Review of duality symmetries for the $c=1$ bosonic CFT

We begin by summarizing the main aspects concerning the dynamics of non-
invertible duality symmetries in bosonic $c=1$ CFT’s, for a comprehensive
analysis see [20, 18]. These theories are realized in terms of a compact
scalar $\phi\sim\phi+2\pi$ with action

$S=\frac{R^{2}}{4\pi}\int{\rm d}^{2}x\,d\phi\wedge\star d\phi$ (2.1)

where $R$ is the radius of the target space circle. We chose the conventions
in which the self-dual radius is $R=1$ and the circle branch in the conformal
manifold is parametrized by $R\geq 1$. The connected piece666In addition,
there are three disconnected exceptional orbifold points pertaining to the
$c=1$ conformal manifold. See [20] (section 5) for a description of the
categorical structures featured by these theories. of the conformal manifold
comprises an additional branch, namely the orbifold branch, parametrized by an
exactly marginal parameter $R_{orb}\geq 1$. These two branches merge at $R=2$
($R_{orb}=1$), the Kosterlitz-Thouless (KT) point [49], which is then a
multicritical point, see Figure 1.

$R$$R_{orb.}$$R=2$KT pt.$R=1$$SU(2)_{1}$ pt. Figure 1: Pictorial
representation of the $c=1$ conformal manifold.

Theories in this conformal manifold admit a $T$-dual description in terms of a
dual variable $\widetilde{\phi}$ with radius $\widetilde{R}=R^{-1}$. In terms
of the chiral left (right) moving fields $X$ ($\overline{X}$), the scalar
field and its dual have the following decomposition

$\phi=\frac{1}{\sqrt{2}R}(X+\overline{X})\quad,\quad\tilde{\phi}=\frac{R}{\sqrt{2}}(X-\overline{X})$
(2.2)

hence $T$-duality acts on the right movers as777There are actually multiple
consistent definitions of $T$-duality. In this article we chose the one that
acts on the right movers and moreover, defines a (non-anomalous)
${\mathbb{Z}}_{2}$ action at the self-dual point. For a detailed discussion
about alternative choices see [50]. $\overline{X}\to-\overline{X}$.

A generic point along the circle branch features a $U_{n}(1)\times U_{w}(1)$
symmetry , associated to shifts of $\phi$ and $\widetilde{\phi}$
respectively888Equivalently, there are two chiral symmetries
$U(1)\times\overline{U(1)}$ acting respectively on $X$ and $\overline{X}$..
The spectrum of (genuine) conformal primaries along the circle branch contains
vertex operators of the form999Throughout this paper, normal ordering is
implicit in the definition of vertex operators.

$V_{n,w}=e^{in\phi}e^{iw\widetilde{\phi}}=e^{ipX}e^{i\overline{p}\overline{X}}\quad,\quad
p=\frac{1}{\sqrt{2}}\left(\frac{n}{R}+wR\right)\quad,\quad\overline{p}=\frac{1}{\sqrt{2}}\left(\frac{n}{R}-wR\right)$
(2.3)

with $n,w\in\mathbb{Z}$. The conformal dimension and spin reads

$h=\frac{1}{2}p^{2}\quad,\quad\overline{h}=\frac{1}{2}\overline{p}^{2}\quad,\quad\Delta=h+\overline{h}=\frac{1}{2}\left(\frac{n^{2}}{R^{2}}+w^{2}R^{2}\right)\quad,\quad
s=h-\overline{h}=nw\,.$ (2.4)

In this presentation, $n$ and $w$ span the charges of $V_{n,w}$ under the
$U(1)_{n}$ and $U(1)_{w}$ respectively and by Dirac quatization condition we
have $nw\in\mathbb{Z}$. Upon gauging a ${\mathbb{Z}}_{N}\subset U_{n}(1)$, the
theory gets mapped to one at $R^{\prime}=\frac{R}{N}$. This is so due to the
fact that the effect of the gauging is twofold. On the one hand, it projects
the spectrum to invariant states, namely the ones with $n=Nn^{\prime}$
($n^{\prime}\in{\mathbb{Z}}$). In addition, it also incorporates twisted
sectors, namely Virasoro multiplets corresponding to vertex operators with
fractional winding number $w=\frac{w^{\prime}}{N}$
($w^{\prime}\in{\mathbb{Z}}$ mod $N$). Remarkably, the incorporation of the
twisted sectors is required by modular invariance in two dimensions, see for
instance [1]. One can retrieve the charge lattice to its original form by
rescaling $n\to n^{\prime}=\frac{n}{N}$ and $w\to w^{\prime}=Nw$, hence
leading to the aforementioned rescaling of the radius. Analogously, gauging
${\mathbb{Z}}_{N}\times{\mathbb{Z}}_{M}\subset U_{n}(1)\times U_{w}(1)$ leads
to $R^{\prime}\to\frac{M}{N}R$. Notice that the latter combined gauging is
only consistent for ${\rm gcd}(N,M)=1$ due to the mixed anomaly involving the
two $U(1)$ symmetries101010 By the same token, operators of the form $V_{n,0}$
and $V_{0,w}$ are not mutually local except for $nw\in 2{\mathbb{Z}}$. See
Appendix A.. In the following, we will use the symbol $\sigma_{N,M}$ to denote
this kind of operations.

At the special values $R=\sqrt{N/M}$ with $N,M\in{\mathbb{Z}}$, the chiral
algebra is enhanced to $U(1)_{2K}$ ($K=NM$) by additional holomorphic
conserved currents $J_{\pm}=e^{\pm i\sqrt{2K}X}$ of spin $K$. In particular,
for $N=M=1$, this corresponds to the self-dual point, where the additional
currents have spin one and combine into an $SU(2)_{1}$ chiral algebra (see
Appendix A). The theory at any of these special points becomes rational and
the Hilbert space decompose into a finite number of representations of the
chiral algebra $U(1)_{2K}$. At $R=\sqrt{N}$, the partition function is given
by a diagonal modular invariant in terms of the characters of such
representations. More generally, the partition function on the Euclidean torus
with complex structure $\tau$ takes the form

$Z=\sum_{a=0}^{2M-1}\sum_{b=0}^{2N-1}\chi_{Na+Mb}\overline{\chi}_{Na-
Mb}\quad,$ (2.5)

with the chiral characters defined as

$\chi_{k}(\tau)\sim\chi_{-k}(\tau)\sim\chi_{k+2K}(\tau)=\frac{1}{\eta(\tau)}\sum_{r\in{\mathbb{Z}}}q^{\frac{1}{4K}(k+2Kr)^{2}}\quad,\quad
q=e^{2\pi i\tau}\,.$ (2.6)

Each block in the sum (2.5) accounts for the contribution of primary states
with $U(1)_{n}\times U(1)_{w}$ charges satisfying $Mn=Na+MN(r+r^{\prime})$ and
$Nw=Mb+MN(r-r^{\prime})$, for $r,r^{\prime}\in{\mathbb{Z}}$ 111111More
generally, by equating the conformal weights of the operators involved in the
chiral characters of $U(1)_{2MN}$ with their expressions in terms of the
global charges one finds
$\chi_{k}\overline{\chi}_{k^{\prime}}\,\,\Rightarrow\,\,Mn=\frac{k+k^{\prime}}{2}+MN(r+r^{\prime})\quad,\quad
Nw=\frac{k-k^{\prime}}{2}+MN(r-r^{\prime})$ where the integers $r,r^{\prime}$
correspond to each term in the sums defining the characters. .

Interestingly, the circle branch theories at $R=\sqrt{N/M}$ feature non-
trivial duality symmetries [20, 18, 16]. These are found by combining
$T$-duality with gauging ${\mathbb{Z}}_{N}\times{\mathbb{Z}}_{M}\subset
U_{m}(1)\times U_{w}(1)$, i.e. $T\circ\sigma_{N,M}$. In fact, it is
straightforward to check that these two combined actions leave the radius
invariant. Away from the self-dual point, the defects implementing these
global symmetries are non-invertible and, together with the generator
$\eta$($\tilde{\eta}$) of the invertible
${\mathbb{Z}}_{N}$(${\mathbb{Z}}_{M}$), describe a Tambara-Yamagami (TY)
category with fusion

${\cal
D}^{2}=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}\eta^{n}\tilde{\eta}^{m}\quad,\quad\eta^{N}=\tilde{\eta}^{M}=1\quad,\quad{\cal
D}\eta=\eta{\cal D}={\cal D}\tilde{\eta}=\tilde{\eta}{\cal D}={\cal D}$ (2.7)

where ${\cal D}$ denotes the defect implementing the $T\circ\sigma_{N,M}$
duality symmetry. The TY category described by the above objects is also
characterized by additional topological data, namely a bicharacter $\chi_{ab}$
($a,b\in{\mathbb{Z}}_{N}\times{\mathbb{Z}}_{M}$) and Frobenius-Schur (FS)
indicator $\epsilon\in\\{-1,1\\}$. We refer the reader to appendix C for a
more detailed description of these categories. When acting on vertex
operators, the action reads121212More generally, there is a continuous set of
consistent duality defects constructed by stacking ${\cal D}$ with invertible
operators implementing chiral rotations and/or charge conjugation. See [20]
for a detailed discussion on this point.

${\cal
D}\,:\,\,V_{n,w}\,\to\,\sqrt{K}(-1)^{nw}V_{\tfrac{Nw}{M},\tfrac{Mn}{N}}\quad,\quad
n\in N{\mathbb{Z}}\,,\,w\in M{\mathbb{Z}}$ (2.8)

and mapping it to a non-genuine operator otherwise. Note the occurrence of the
$\sqrt{K}$ prefactor associated to the quantum dimension of ${\cal D}$. The
additional phase $(-1)^{nm}$ arises due to the fact that momentum and winding
modes, which are exchanged by $T$-duality, are not mutually local (see
appendix A).

A topological defect implementing a symmetry in quantum field theory must obey
certain consistency conditions. In two dimensions, these are imposed by
modular invariance or, more generally, modular covariance, finding their
natural implementation in terms of the modular bootstrap. More precisely,
torus partition functions twisted by topological defects along non-trivial
cycles are mapped to each other by the modular group $SL(2,{\mathbb{Z}})$

$Z_{({\cal D},{\widetilde{\cal
D}})}(\gamma\cdot\tau)=e^{i\phi(\gamma)}Z_{({\cal\widetilde{D}}^{-b}{\cal
D}^{d},{\cal\widetilde{D}}^{a}{\cal
D}^{-c})}(\tau)\quad,\quad\gamma\cdot\tau=\frac{a\tau+b}{c\tau+d}$ (2.9)

where we allowed for the occurrence of an anomalous phase $\phi(\gamma)$. This
phase is not going to play any role in the following, hence being omitted from
now on. Whenever negative exponents occur in the right hand side of (2.9),
those should be interpreted as the insertion of the orientation reversed
defect, as the inverse might not exist. An important implication of the above
is depicted in the following diagram:

$Z_{(1,\mathcal{D})}\;=\;\raisebox{-35.00005pt}{ \leavevmode\hbox
to100.66pt{\vbox to86.36pt{\pgfpicture\makeatletter\hbox{\hskip
0.5pt\lower-0.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{85.35828pt}\pgfsys@lineto{85.35828pt}{85.35828pt}\pgfsys@lineto{85.35828pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
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}{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@lineto{85.35828pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{89.19128pt}{39.26248pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mathcal{D}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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to99.86pt{\pgfpicture\makeatletter\hbox{\hskip 0.5pt\lower-13.99933pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{85.35828pt}\pgfsys@lineto{85.35828pt}{85.35828pt}\pgfsys@lineto{85.35828pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{0.0pt}\pgfsys@lineto{42.67914pt}{85.35828pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{38.8597pt}{-10.66632pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mathcal{D}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\;\equiv\;Z_{(\mathcal{D},1)}$
(2.10)

where, in the last equality, we emphasized the fact that a consistent
topological defect should lead to a well defined twisted Hilbert space. More
precisely, being topological, it commutes with the stress tensor, hence the
defect Hilbert space must have a natural decomposition in terms of Virasoro
characters

$Z_{(\mathcal{D},1)}=\sum_{h,\overline{h}}n_{h,\overline{h}}\mathcal{V}_{h}\overline{\mathcal{V}}_{\overline{h}}$
(2.11)

where, by the above considerations, the coefficients $n_{h,\overline{h}}$ must
be positive integers. This becomes a key test under which any proposed
symmetry defect should be contrasted, as we will show in many examples in this
paper. Moreover, similar applications of (2.9) enable extracting more refined
topological data associated to a given symmetry structure [18, 19, 20, 51, 52,
53, 54].

As an instructive example, we apply the above analysis to the defect
implementing the symmetry action (2.8). Similar computations can be found in
[20], though we are not aware of an explicit application to the non-diagonal
case. By the action of $T$-duality, namely $\overline{X}\to-\overline{X}$, the
insertion of the duality defect ${\cal D}$ along the spatial cycle projects
into the $\overline{p}=0$ subspace. This condition sets $Na-Mb=0$, hence the
twisted partition function gets contributions only from the identity
characters, spanned by the states $|p=\sqrt{2MN}r,\overline{p}=0,{\bf
N},\overline{\bf N}\rangle$, $r\in{\mathbb{Z}}$. Here ${\bf N}$
($\overline{\bf N}$) denote, as usual, the number of (anti-)holomorphic
oscillator modes. In addition, $T$-duality induces a $(-1)$ phase for states
with an odd number of right-moving oscillator modes [50]. Finally, there is
the phase factor in (2.8) due to the non-locality between winding and momentum
modes. Putting all together one obtains

$Z_{(1,{\cal
D})}(\tau)=\frac{\sqrt{NM}}{|\eta(\tau)|^{2}}\left(\sum_{r\in{\mathbb{Z}}}(-1)^{MNr}q^{MNr^{2}}\right)\vartheta_{4}(2\overline{\tau})\,,$
(2.12)

where we already performed the sum over the oscillator modes. See appendix D
for a list of the relevant modular functions and their modular properties.

For the case of even $MN$ we get

$Z_{(1,{\cal
D})}=\frac{\sqrt{NM}}{|\eta|^{2}}\vartheta_{3}(2NM\tau)\vartheta_{4}(2\overline{\tau})\,\xrightarrow{\tau\to-\frac{1}{\tau}}\,Z_{({\cal
D},1)}(\tau)=\frac{1}{2|\eta|^{2}}\vartheta_{3}\left(\tfrac{\tau}{2NM}\right)\vartheta_{2}\left(\tfrac{\overline{\tau}}{2}\right)\,,$
(2.13)

whereas for odd $MN$ one gets instead

$Z_{(1,{\cal
D})}=\frac{\sqrt{NM}}{|\eta|^{2}}\vartheta_{4}(2NM\tau)\vartheta_{4}(2\overline{\tau})\,\xrightarrow{\tau\to-\frac{1}{\tau}}\,Z_{({\cal
D},1)}(\tau)=\frac{1}{2|\eta|^{2}}\vartheta_{2}\left(\tfrac{\tau}{2NM}\right)\vartheta_{2}\left(\tfrac{\overline{\tau}}{2}\right)\,.$
(2.14)

It can be easily checked that both expressions comply with the condition shown
in (2.10), hence leading to a well defined trace over a defect Hilbert space.

##  3 $c=2$ Toroidal CFT and Duality symmetries

The conformal manifold of $c=2$ bosonic CFT’s is significantly richer than the
one for $c=1$ described in the previous section. In particular, the connected
non-exceptional component comprises 28 non-equivalent branches, meeting at
various multicritical regions. We refer the reader to [55] for a comprehensive
overview. A distinguished branch is the toroidal branch, where the target
space corresponds to a torus with complex structure $\tau$ and Kahler modulus
$\rho$, hence spanning a four-dimensional variety. This branch also
accommodates a collection of special points with enhanced global symmetry, as
we will comment momentarily.

Before delving into particular examples, let us review some aspects pertaining
to generic theories along the $c=2$ toroidal branch (for more details we refer
to [25, 55]). The field content of these theories consists on two compact
scalar fields $\phi^{i}\sim\phi^{i}+2\pi$ ($i=1,2$) with action

$S=\frac{1}{4\pi}\int\delta^{\mu\nu}G_{ij}\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}+\frac{i}{4\pi}\int\epsilon^{\mu\nu}B_{ij}\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}\,,$
(3.1)

where $G,B$ are respectively the metric and Kalb-Ramond fields parametrizing
the target space two-torus

$G=R^{2}\begin{pmatrix}1&\tau_{1}\\\
\tau_{1}&\tau_{1}^{2}+\tau_{2}^{2}\end{pmatrix}\qquad B=\begin{pmatrix}0&b\\\
-b&0\end{pmatrix}\,.$ (3.2)

It is customary to repackage the geometric data in terms of the complex
structure and Kahler modulus

$\tau=\tau_{1}+i\tau_{2}\quad,\quad\rho=\rho_{1}+i\rho_{2}=b+i\sqrt{G}\quad,\quad\left(\sqrt{G}=R^{2}\tau_{2}\right)\,.$
(3.3)

These two complex variables parametrize the $4$-dimensional toroidal branch of
the $c=2$ conformal manifold. Motion along this branch is generated by the
four exactly marginal operators of the form
$\partial\phi^{i}\partial\phi^{j}$. Note that when $\tau_{1}=\rho_{1}=0$ the
theory factorizes as a tensor product of two $c=1$ compact bosons at radii
$R_{1}=R$ and $R_{2}=R\tau_{2}$.

These theories admit equivalent descriptions in terms of dual variables
$\widetilde{\phi}^{i}\sim\widetilde{\phi}^{i}+2\pi$. In addition, the real
fields $\phi^{i}$, $\widetilde{\phi}^{i}$ admit the following decomposition
into left and right movers

$\begin{split}&\phi^{1}=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\big{[}\tau_{2}(X^{1}+\overline{X}^{1})-\tau_{1}(X^{2}+\overline{X}^{2})\big{]}\\\
&\phi^{2}=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}(X^{2}+\overline{X}^{2})\\\
&\widetilde{\phi}^{1}=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\big{[}\rho_{2}(X^{1}-\overline{X}^{1})-\rho_{1}(X^{2}+\overline{X}^{2})\big{]}\\\
&\widetilde{\phi}^{2}=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\big{[}(\rho_{1}\tau_{2}+\rho_{2}\tau_{1})X^{1}+(\rho_{1}\tau_{2}-\rho_{2}\tau_{1})\overline{X}^{1}+(\rho_{2}\tau_{2}-\rho_{1}\tau_{1})X^{2}-(\rho_{1}\tau_{1}+\rho_{2}\tau_{2})\overline{X}^{2}\big{]}\,.\end{split}$
(3.4)

At a generic point along this branch, the theory possess a continuous global
symmetry of the form

$U(1)_{\bf n}\times U(1)_{\bf w}\equiv U(1)_{n_{1}}\times U(1)_{n_{2}}\times
U(1)_{w_{1}}\times U(1)_{w_{2}}$ (3.5)

with $U_{n_{i}}(1):\,\phi^{i}\to\phi^{i}+\alpha^{i}$ and
$U_{w_{i}}(1):\,\widetilde{\phi}^{i}\to\widetilde{\phi}^{i}+\widetilde{\alpha}^{i}$,
($\alpha^{i},\widetilde{\alpha}^{i}\in[0,2\pi)$). Note that the exactly
marginal operators spanning the toroidal branch are naturally neutral under
these symmetries. The spectrum of charged conformal primaries is therefore
determined by four integer charges ${\bf n}=(n_{1},n_{2})^{T}$ and ${\bf
w}=(w_{1},w_{2})^{T}$

$V_{{\bf n},{\bf w}}=e^{i{\bf n}^{T}{\boldsymbol{\phi}}}e^{i{\bf
w}^{T}\widetilde{\boldsymbol{\phi}}}\,,$ (3.6)

where $\boldsymbol{\phi}=\left(\phi^{1},\,\phi^{2}\right)$,
$\widetilde{\boldsymbol{\phi}}=(\widetilde{\phi}^{1},\,\widetilde{\phi}^{2})$.
Equivalently, these primaries are in one to one correspondence with sites in
the even, self-dual, integer charge lattice with signature $(2,2)$ spanned by
the following left- and right-moving momenta

$\displaystyle{\bf p}$
$\displaystyle=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\left(\begin{array}[]{c}n_{1}\tau_{2}+\rho_{1}\tau_{2}w_{2}+\rho_{2}(w_{1}+w_{2}\tau_{1})\\\
n_{2}-\tau_{1}n_{1}-\rho_{1}(w_{1}+w_{2}\tau_{1})+\rho_{2}\tau_{2}w_{2}\end{array}\right)$
(3.9) $\displaystyle\overline{\bf p}$
$\displaystyle=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\left(\begin{array}[]{c}n_{1}\tau_{2}+\rho_{1}\tau_{2}w_{2}-\rho_{2}(w_{1}+w_{2}\tau_{1})\\\
n_{2}-\tau_{1}n_{1}-\rho_{1}(w_{1}+w_{2}\tau_{1})-\rho_{2}\tau_{2}w_{2}\end{array}\right)\,.$
(3.12)

In terms of these momenta, the conformal dimensions read

$\begin{split}&h=\frac{1}{2}(p_{1}^{2}+p_{2}^{2})=\frac{1}{4}\left[n+(G+B)w\right]^{\intercal}G^{-1}\left[n+(G+B)w\right]\\\
&\overline{h}=\frac{1}{2}(\overline{p}_{1}^{2}+\overline{p}_{2}^{2})=\frac{1}{4}\left[n-(G-B)w\right]^{\intercal}G^{-1}\left[n-(G-B)w\right]\,\end{split}$
(3.13)

and the spins $s=h-\overline{h}=n_{i}w_{i}\in\mathbb{Z}$ are integrally
quantized since the theory is bosonic.

There are several criteria for defining rationality of a generic CFT. A
natural one states that a theory is rational if its Hilbert space decompose
into a finite number of non-equivalent irreducible representations of a
certain (generally enhanced) chiral algebra [56, 57]131313More precisely, in
such a case we say that the theory is rational with respect to such enhanced
chiral algebra, even if it might not be rational with respect to Virasoro.
These are the prototypical examples that occur at $c\geq 1$..

Analogously to the $c=1$ theories at $R^{2}\in\mathbb{Q}$, there are
distinguished points in the $c>1$ conformal manifold where the theory becomes
rational. At $c=2$, the criterion for rationality of a toroidal CFT described
by (3.1) can be rephrased in terms of geometric properties of its charge
lattice. Alternatively, it can be proven that a $c=2$ toroidal CFT is rational
if $G\in GL(2,\mathbb{Q})$ and $B\in Skew(2,\mathbb{Q})$. The several criteria
just mentioned can be proven to be equivalent. Geometrically, it has been
shown in [58] that rationality is also achieved if the target space torus
admits Complex Multiplication141414Given a torus
$\mathcal{T}^{2}=\mathbb{C}/\Lambda$ determined by some lattice $\Lambda$,
$\mathcal{T}^{2}$ admits Complex Multiplication if there exists a complex
number $z\in\mathbb{C}-\mathbb{R}$ such that $z\Lambda\subset\Lambda$.. The
latter condition implies that the moduli $\tau,\rho$ satisfy

$\tau\,,\,\rho\,\in{\mathbb{Q}}\left(\sqrt{D}\right)\;,$ (3.14)

where ${\mathbb{Q}}\left(\sqrt{D}\right)$ denotes the imaginary quadratic
number field, for some negative integer $D$. The latter is obtained from the
field of the rational numbers by introducing $\sqrt{D}$, i.e.
$x\in{\mathbb{Q}}\left(\sqrt{D}\right)$ then $x=x_{1}+x_{2}\sqrt{D}$ with
$x_{1},x_{2}\in\mathbb{Q}$. The field structure of
$\mathbb{Q}\left(\sqrt{D}\right)$ then naturally descends from the one of
$\mathbb{Q}$. In particular, if $\tau$ belongs to
${\mathbb{Q}}\left(\sqrt{D}\right)$, then there exist some integers $a$, $b$
and $c$ such that

$a\tau^{2}+b\tau+c=0\quad,\quad a,b,c\in{\mathbb{Z}}\quad,\quad{\rm
gcd}(a,b,c)=1$ (3.15)

hence

$\tau=-\frac{b}{2a}+\frac{i}{2a}\sqrt{-D}\quad,\quad D=b^{2}-4ac$ (3.16)

and similarly for $\rho$. Even if the equations for both complex variables may
have different integer coefficients, they must have the same discriminant $D$
for the theory to be an RCFT. The condition $D<0$ is implied in order for the
solutions to correspond to a physical theory, i.e. $\tau_{2},\rho_{2}>0$.
151515As an illustrative example, consider the case of a product of two $c=1$
RCFTs at $R_{1}=\sqrt{N_{1}/M_{1}}$ and $R_{2}=\sqrt{N_{2}/M_{2}}$, for which
$\tau=iR_{1}/R_{2}$, $\rho=iR_{1}R_{2}$. This situation trivially fits into
the class described above since  $N_{2}M_{1}\tau^{2}+N_{1}M_{2}=0\quad,\quad
M_{1}M_{2}\rho^{2}+N_{1}N_{2}=0\quad,\quad D=-N_{1}N_{2}M_{1}M_{2}\;.$ It can
be proven that the set of RCFTs is dense within the toroidal branch, similarly
to the rational theories along the $c=1$ circle branch.

An RCFT in the toroidal branch features an extended chiral algebra of the form

$u(1)^{2}_{K_{L}}\times u(1)^{2}_{K_{R}}$ (3.17)

where $K_{L,R}$ are some matrices constructed out of $G$ and $B$ at each
particular rational point. We refer to appendix B for some details on the
determination of these matrices. The partition function of a rational theory
is accounted for by a finite modular invariant combination of characters of
(3.17). Furthermore, the (anti)holomorphic representations of the chiral
algebra are in one to one correspondence with the sites in the following
lattices

${\mathcal{L}}_{L,R}\equiv\left\\{{\boldsymbol{\lambda}_{L,R}}\in{\mathbb{Z}}^{2}\,,\,{\boldsymbol{\lambda}_{L,R}}\sim{\boldsymbol{\lambda}_{L,R}}+K_{L,R}{\bf
v}\,,\,{\bf v}\in{\mathbb{Z}}^{2}\right\\}\,.$ (3.18)

The partition function then reads

$Z=\sum_{\vec{\lambda}_{L}\in{\cal
D}_{L}}\chi_{{\boldsymbol{\lambda}}_{L}}\overline{\chi}_{\widehat{\omega}{\boldsymbol{\lambda}}_{L}}$
(3.19)

where $\hat{\omega}:\mathcal{L}_{L}\to\mathcal{L}_{R}$ is a particular group
isomorphism. We refer to appendix B for a detailed account on the construction
of the above partition function. The characters corresponding to the
representations labelled ${\boldsymbol{\lambda}_{L,R}}$ read161616We denote
$|v|^{2}_{M}=v^{i}M_{ij}v^{j}$.

$\chi_{{\boldsymbol{\lambda}}_{L}}(\tau)=\frac{1}{\eta(\tau)^{2}}\sum_{{\bf
l}\in{\mathbb{Z}}^{2}}q^{\frac{1}{2}|{\boldsymbol{\lambda}}_{L}+K_{L}{\bf
l}|^{2}_{K_{L}^{-1}}}\quad,\quad\bar{\chi}_{{\boldsymbol{\lambda}}_{R}}(\bar{\tau})=\frac{1}{\eta(\bar{\tau})^{2}}\sum_{{\bf
r}\in{\mathbb{Z}}^{2}}\bar{q}^{\frac{1}{2}|{\boldsymbol{\lambda}}_{R}+K_{R}{\bf
r}|^{2}_{K_{R}^{-1}}}\,.$ (3.20)

Furthermore, the partition function of the theory is given by a diagonal
modular invariant (i.e. $\widehat{\omega}=1$) whenever there exists a duality
frame such that that $\tau=fa\rho$ with $a$ the same as in (3.15) and $f$ some
positive integer [58].

Within the toroidal branch, there are special (generically rational) points
where the global symmetry gets enhanced to higher rank chiral algebra, hence
corresponding to enhanced symmetry points. More precisely, the algebras
realized at these special loci at $c=2$ are of the A-type of maximal rank 2,
i.e. $su(2)$, $su(2)^{2}$ or $su(3)$. Theories featuring these algebras have
been classified (see for instance [55])

$\displaystyle(\tau,\rho)$ $\displaystyle=(i,i)\;:\qquad
SU(2)^{2}\times\overline{SU(2)}^{2}\;,$ $\displaystyle(\tau,\rho)$
$\displaystyle=(e^{i2\pi/3},e^{i2\pi/3})\;:\qquad
SU(3)\times\overline{SU(3)}\;,$ $\displaystyle(\tau,\rho)$
$\displaystyle=(\tau,\tau)\neq\\{i,e^{i2\pi/3}\\}\;,\,\tau_{1}=\\{0,\frac{1}{2}\\}\;:\qquad
SU(2)\times U(1)\times\overline{SU(2)}\times\overline{U(1)}\;,$
$\displaystyle(\tau,\rho)$
$\displaystyle=(\tau,\tau)\;,\tau_{1}\neq\\{0,\frac{1}{2}\\}\;:\qquad
U(1)\times U(1)\times\overline{SU(2)}\times\overline{U(1)}\;,$
$\displaystyle(\tau,\rho)$
$\displaystyle=(\tau,-\overline{\tau})\;,\tau_{1}\neq\\{0,\frac{1}{2}\\}\;:\qquad
SU(2)\times U(1)\times\overline{U(1)}\times\overline{U(1)}\;.$ (3.21)

Note that the last two lines above correspond to continuous sets of theories,
hence comprising both rational and irrational theories.

In spite of being interesting by themselves, the enhanced symmetry points play
a crucial role in the characterization of the conformal manifold. In
particular, it has been shown that all multicritical theories, i.e. where
different branches of the conformal manifold intersect, can be obtained from
non-trivial quotients of theories with enhanced symmetry [59, 60]. Along these
lines, the additional exactly marginal deformations featured by multicritical
theories are easily constructed in terms of conserved currents at enhanced
symmetry points (see appendix A for an example of this construction at the
$c=1$ KT point). In this work, we will mainly focus on two particular examples
of multicritical theories, namely the quadri-critical and the bi-critical
points, which are obtained by taking certain $\mathbb{Z}_{2}$ and
$\mathbb{Z}_{3}$ quotients of the $SU(2)^{2}$ and the $SU(3)$ theories
respectively (see figure 2 for a pictorial representation of the points we are
interested in).

$(w,\alpha)$$(w,w)$$\mathbb{Z}_{3}^{s}$$SU(3)_{1}$$(i,i)$$\mathbb{Z}_{2}^{s}$$SU(2)^{2}_{1}$$(i,2i)$$(\tau,\rho)$
Figure 2: Pictorial representation of a slice of the toroidal branch
containing two multicritical points. Here $w=e^{\frac{2\pi i}{3}}$ while
$\alpha=-\frac{1}{2}+3\frac{\sqrt{3}}{2}$ as in the main text. The red and
blue lines represent respectively the gauging of the $\mathbb{Z}_{3}^{s}$ and
$\mathbb{Z}_{2}^{s}$ shift symmetries defined in (4.2) and (4.1). At the
$4$-critical point $(i,2i)$ three orbifold branches (one 4d and two 2d) join
while at the bi-critical point $(w,\alpha)$ we get just one 2d orbifold
branch.

The duality groups along the $c>1$ toroidal branch are significantly larger
than the $T$-duality at $c=1$, hence making room for richer structures of
duality symmetries featured at special points. In particular, for $c=2$ the
duality group is of the form

${\rm O}(2,2,{\mathbb{Z}})\simeq{\rm P}\left({\rm
SL}(2,{\mathbb{Z}})_{\tau}\times{\rm
SL}(2,{\mathbb{Z}})_{\rho}\right)\rtimes\left({\mathbb{Z}}_{2}^{M}\times{\mathbb{Z}}_{2}^{I}\right)\,,$
(3.22)

with the first two factors $SL(2,{\mathbb{Z}})_{\tau,\rho}$ acting separately
on $\tau$ and $\rho$ via

$\displaystyle\tau\to\frac{a\tau+b}{c\tau+d}\quad$
$\displaystyle,\quad\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL(2,{\mathbb{Z}})_{\tau}$
$\displaystyle\rho\to\frac{a^{\prime}\rho+b^{\prime}}{c^{\prime}\rho+d^{\prime}}\quad$
$\displaystyle,\quad\begin{pmatrix}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\end{pmatrix}\in SL(2,{\mathbb{Z}})_{\rho}\,.$ (3.23)

These two actions are quotiented by the central element
$(\tau,\rho)\sim(-\tau,-\rho)$. The remaining two ${\mathbb{Z}}_{2}$ actions
correspond to mirror symmetry and target space inversion respectively

${\mathbb{Z}}_{2}^{M}\,:\,(\tau,\rho)\,\to\,(\rho,\tau)\quad,\quad{\mathbb{Z}}_{2}^{I}\,:\,(\tau,\rho)\,\to\,(-\overline{\tau},-\overline{\rho})\,.$
(3.24)

The various orbifold branches in the $c=2$ conformal manifold generically
preserve a subgroup of the duality group (3.22) of the toroidal branch (see
for instance [55]).

The four dimensional representation of the duality group (3.22) has a natural
action by conjugation over the generalized metric $\mathcal{E}$ (see e.g. [61,
62]). The latter corresponds to an alternative description of these theories
in terms of a $4\times 4$ matrix acting on the vector
${\boldsymbol{\Phi}}\equiv({\boldsymbol{\phi}},\widetilde{\boldsymbol{\phi}})^{T}$.
More precisely, in terms of the metric $G$ and antisymmetric tensor $B$, the
generalized metric reads

${\cal E}=\left(\begin{array}[]{cc}G-B\cdot G^{-1}\cdot B&-B\cdot G^{-1}\\\
G^{-1}\cdot B&G^{-1}\end{array}\right)\,.$ (3.25)

For a given element $\mathsf{O}\in O(2,2,\mathbb{Z})$ acting on
${\boldsymbol{\Phi}}$ as
${\boldsymbol{\Phi}}^{\prime}=\mathsf{O}{\boldsymbol{\Phi}}$, the parameters
of the theory get transformed according to

$\mathsf{O}\,:\,\mathcal{E}\,\to\,\mathcal{E}^{\prime}=(\mathsf{O}^{-1})^{\intercal}\mathcal{E}\mathsf{O}^{-1}\;.$
(3.26)

In this formulation, the generators of the duality group (3.22) admit the
following four dimensional representation

$\displaystyle\begin{pmatrix}a&-b&0&0\\\ -c&d&0&0\\\ 0&0&d&c\\\
0&0&b&a\end{pmatrix}\quad$ $\displaystyle{\rm for}\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{Z})_{\tau}$
$\displaystyle\begin{pmatrix}d^{\prime}&0&0&c^{\prime}\\\
0&d^{\prime}&-c^{\prime}&0\\\ 0&-b^{\prime}&a^{\prime}&0\\\
b^{\prime}&0&0&a^{\prime}\end{pmatrix}\quad$ $\displaystyle{\rm
for}\quad\begin{pmatrix}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\end{pmatrix}\in SL(2,\mathbb{Z})_{\rho}$ (3.27)
$\displaystyle M=\begin{pmatrix}0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\
0&0&0&1\end{pmatrix}\in\mathbb{Z}_{2}^{M}\quad$ $\displaystyle,\quad
I=\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{pmatrix}\in\mathbb{Z}_{2}^{I}$

As we will discuss further below in this section, this description of the
theory enables to construct a matrix representation of topological
manipulations, and more generally duality symmetries. Precisely, to a given
defect $\mathcal{D}$ implementing such a manipulation, one associates a
$4\times 4$ matrix $\mathsf{D}\in O(2,2,\mathbb{Q})$ acting on the fields as
described around (3.26). Notice that the coefficients are in $\mathbb{Q}$
instead of $\mathbb{Z}$. This is a consequence of the fact that, as we will
verify in several examples below, duality symmetries usually involve
topological manipulations, such as discrete gauging, which cannot be
represented by a matrix in $O(2,2,\mathbb{Z})$.

###  3.1 Discrete gauging and duality symmetries

Similarly to the $c=1$ case, gauging a non-anomalous subgroup of $U(1)_{\bf
n}^{2}\times U(1)_{\bf w}^{2}$ connects two points in the toroidal branch and
therefore can be represented as a $4\times 4$ matrix $\sigma$ acting on the
4-component vector
$(\boldsymbol{\phi}^{\intercal},\widetilde{\boldsymbol{\phi}}^{\intercal})$,
or equivalently on the charge $(\bf n^{\intercal},\bf w^{\intercal})$, as

$\begin{pmatrix}\boldsymbol{\phi}^{\prime}\\\
\widetilde{\boldsymbol{\phi}}^{\prime}\end{pmatrix}=\sigma\begin{pmatrix}\boldsymbol{\phi}\\\
\widetilde{\boldsymbol{\phi}}\end{pmatrix}\quad\Longleftrightarrow\quad\begin{pmatrix}\boldsymbol{n}^{\prime}\\\
{\boldsymbol{w}}^{\prime}\end{pmatrix}=(\sigma^{-1})^{\intercal}\begin{pmatrix}\boldsymbol{n}\\\
{\boldsymbol{w}}\end{pmatrix}\,.$ (3.28)

Dirac quantization condition after the gauging, implies that

${\bf n}^{\prime\intercal}{\bf
w}^{\prime}\in\mathbb{Z}\quad\Longrightarrow\quad\sigma\in
O(2,2,\mathbb{Q})\,.$ (3.29)

Therefore any gauging of subgroups of $U(1)_{\bf n}^{2}\times U(1)_{\bf
w}^{2}$ can be represented as a matrix $\sigma\in O(2,2,\mathbb{Q})$. This is
a particular example of the matrix representation for topological
manipulations introduced before. We emphasize that the map between gaugings
and matrices in $O(2,2,\mathbb{Q})$ is not bijective: there can be different
matrices representing the same gauging operation171717Equivalently in the c=1,
the group $O(1,1,\mathbb{Q})=\Big{\\{}\begin{pmatrix}p/q&0\\\
0&q/p\end{pmatrix}\;,\;\begin{pmatrix}0&p/q\\\ q/p&0\end{pmatrix}$} represents
all the possible gauging of subgroups
$\mathbb{Z}_{p}\times\mathbb{Z}_{q}\subset U(1)_{n}\times U(1)_{w}$ but the
diagonal and off-diagonal matrices span the same set of operations.. A duality
symmetry defect $\mathcal{D}$ then implies the existence of an
$O(2,2,\mathbb{Q})$ matrix $\mathsf{D}$ which leaves the theory parameters
invariant. This condition can be written, using the generalized metric $\cal
E$, as181818This condition can be easily generalizable to CFTs with integer
central charge $c>2$.

$(\mathsf{D}^{-1})^{\intercal}\mathcal{E}\mathsf{D}^{-1}-\mathcal{E}=0\,.$
(3.30)

At this point it is important to emphasize the distinction between the actual
symmetry operation over the Hilbert space (or equivalently on the conformal
primaries) implemented by $\mathcal{D}$, which does not need to be invertible,
and its matrix representation $\mathsf{D}$ acting on the global charges, for
which an inverse matrix can always be defined. The invariance of the theory
parameters in (3.30) is the manifestation of the self-duality of the theory
under the gauging of a global symmetry subgroup $G$ parameterized by
$\mathsf{D}\in O(2,2,\mathbb{Q})$. Thoughout this paper, will show in several
examples how the actual subgroup $G$ can be extracted from the matrix
$\mathsf{D}$ satisfying (3.30). On the contrary, when $\mathsf{D}\in
O(2,2,\mathbb{Z})$, this corresponds to an invertible symmetry which permutes
the primary operators. The action of the corresponding symmetry defect on the
set of conformal primary operators is

$\mathcal{D}\,:\,\begin{cases}&V_{\bf n,\bf
w}\rightarrow\sqrt{|G|}(-1)^{\alpha}V_{\bf n^{\prime},\bf
w^{\prime}}\qquad\text{ if }n_{i}^{\prime},w_{i}^{\prime}\in\mathbb{Z}\\\
&V_{\bf n,\bf w}\rightarrow\text{ non genuine op.}\qquad\text{ if
}n_{i}^{\prime},w_{i}^{\prime}\not\in\mathbb{Z}\end{cases}$ (3.31)

where $\bf n^{\prime},\bf w^{\prime}$ are the charges obtained upon acting
with the associated matrix as

$\begin{pmatrix}{\bf n}^{\prime}\\\ {\bf
w}^{\prime}\end{pmatrix}=(\mathsf{D}^{-1})^{\intercal}\cdot\begin{pmatrix}{\bf
n}\\\ {\bf w}\end{pmatrix}\;,$ (3.32)

and the phase $\alpha$ comes by imposing the consistency of this action with
locality (see appendix A for more details).

As mentioned before, matrices $\mathsf{D}$ naturally decompose in terms of
duality operations in $O(2,2,\mathbb{Z})$ and topological manipulations
associated to gauging of discrete subgroups of the global symmetry,
represented by matrices $\sigma\in O(2,2,\mathbb{Q})$. Let us momentarily
concentrate on the latter. Given a matrix $\sigma\in O(2,2,\mathbb{Q})$
implementing a gauging of a subgroup $G\subset U(1)_{\bf n}^{2}\times
U(1)_{\bf w}^{2}$, it becomes instrumental to extract the form of such a
subgroup of the global symmetry. In the following, we present an algorithmic
way of achieving that. Consider the set of primary operators

$S^{\sigma}\equiv\left\\{V_{{\bf n},{\bf w}}\,|\,\\{({\bf n},{\bf
w})\\}\xrightarrow{(\sigma^{-1})^{\intercal}}\\{{\bf e}\\}\right\\}$ (3.33)

where ${\bf e}_{a}$ is the set of unital vector charges, namely $({\bf
e}_{a})_{i}=\delta_{i,a}$ ($i,a\in\\{1,\ldots 4\\}$). Such operators are the
minimal gauge-invariant operators, both in the untwisted and twisted Hilbert
space of the ungauged theory, which become genuine operators after the
gauging191919For instance, in the $c=1$ case a gauging of
$\mathbb{Z}_{k}\subset U(1)_{n}$ acts on the charges as
$(n,w)\rightarrow(\frac{n}{k},kw)$. Therefore primaries with charge
$(k,0),(0,\frac{1}{k})$ are mapped to $(1,0),(0,1)$ and they correspond to the
invariant operators under the action of $\mathbb{Z}_{k}$.. A trial version of
the subgroup $G$ is parametrized by five integers $\\{k,a,b,c,d\\}$ such that
its action on a vertex operator reads

$G:\,V_{{\bf n},{\bf w}}\,\to\,V_{{\bf n},{\bf w}}^{\prime}=e^{\frac{2\pi
i}{k}(an_{1}+bn_{2}+cw_{1}+dw_{2})}V_{{\bf n},{\bf w}}\,.$ (3.34)

Being invarinat under the action of $G$, the operators in $S^{\sigma}$ satisfy
the condition

$V_{{\bf n},{\bf w}}\in S^{\sigma}\,\Rightarrow\,V_{{\bf n},{\bf w}}=V_{{\bf
n},{\bf w}}^{\prime}\;.$ (3.35)

Hence, given a matrix $\sigma\in O(2,2,\mathbb{Q})$, determining the set
$S^{\sigma}$ allows to easily derive the action of the symmetry $G$, i.e.
determining $k,a,b,c,d$.202020 Generically, the solution is not unique. More
precisely, the condition (3.35) is equally satisfied by any subgroup $H\subset
G$. However, there is always a solution for which ${\rm gcd}(k,a,b,c,d)=1$,
which we identify with the maximal group.  In order to illustrate this
procedure, we apply it to the following example

$\sigma=\left(\begin{array}[]{cccc}1&1&0&0\\\ 1&-1&0&0\\\
0&0&\tfrac{1}{2}&\tfrac{1}{2}\\\
0&0&\tfrac{1}{2}&-\tfrac{1}{2}\end{array}\right)\quad,\quad(\sigma^{-1})^{T}=\left(\begin{array}[]{cccc}\tfrac{1}{2}&\tfrac{1}{2}&0&0\\\
\tfrac{1}{2}&-\tfrac{1}{2}&0&0\\\ 0&0&1&1\\\ 0&0&1&-1\end{array}\right)$
(3.36)

for which (3.33) corresponds to

$S^{\sigma}=\left\\{V_{1,1,0,0},V_{1,-1,0,0},V_{0,0,\tfrac{1}{2},\tfrac{1}{2}},V_{0,0,\tfrac{1}{2},-\tfrac{1}{2}}\right\\}\,.$
(3.37)

Now, imposing the invariance condition given in (3.35), it is straightforward
to verify that integer solutions for $\\{a,b,c,d\\}$ only exist for $k\in
2\mathbb{Z}$. Finally, the maximal non-trivial solution (see footnote 20)
corresponds to

$k=2\,,\,a=b=1\,,\,c=d=0\,\Rightarrow\,G\equiv\mathbb{Z}_{2}^{D}\subset
U(1)_{n_{1}}\times U(1)_{n_{2}}$ (3.38)

We will encounter the matrix (3.36) again when studying the quadri-critical
point in the next section.

We want now to discuss the existence of non-invertible duality symmetries on
the toroidal branch. Even if the condition (3.30) in principle generates all
the possible $\mathsf{D}\in O(2,2,\mathbb{Q})$ implying a duality symmetry
defect $\mathcal{D}$, explicit solutions of such equation are generically hard
to find. To this aim, we proceed to describe a particular method by which some
solutions can be easily obtained. In particular by performing some topological
manipulation $\mathcal{T}$, namely a combination of gaugings and dualities, we
can connect the theory at $(\tau,\rho)$ to another point
$(\widehat{\tau},\widehat{\rho})$ in the toroidal branch for which some
duality symmetries are manifest. We denote the latter symmetries by
$\widehat{\mathcal{D}}$ and their associated matrix by $\widehat{\mathsf{D}}$.
Therefore the original theory will automatically enjoy a non-invertible
symmetry defect $\mathcal{D}$ corresponding to a matrix $\mathsf{D}$
constructed as

$\mathsf{D}=\mathsf{T}\,\widehat{\mathsf{D}}\,\mathsf{T}^{-1}\,,$ (3.39)

where $\mathsf{T}\in O(2,2,\mathbb{Q})$ is the matrix representation of the
topological manipulation $\mathcal{T}$ used to connect the points
$(\tau,\rho)$ and $(\widehat{\tau},\widehat{\rho})$ in the conformal manifold.
For instance, a very convenient choice of $(\widehat{\tau},\widehat{\rho})$ is
one for which the theory factorizes in terms of two $c=1$ RCFT’s, namely
$\widehat{\tau}_{1}=\widehat{\rho}_{1}=0$. In particular, one can show that
such a factorized theory enjoy duality symmetries coming from the separate
gaugings of subgroups of $U(1)_{n_{1}}\times U(1)_{w_{1}}$ and
$U(1)_{n_{2}}\times U(1)_{w_{2}}$.212121This is a trivial consequence of the
existence of duality symmetries in any $c=1$ RCFT as shown in section 2.

Consider a generic $c=2$ RCFT with

$\tau=\frac{p}{q}+i\tau_{2}\quad,\quad\rho=\frac{p^{\prime}}{q^{\prime}}+i\rho_{2}\,\qquad,\qquad
p,q,p^{\prime},q^{\prime}\in\mathbb{Z}\,.$ (3.40)

where $\tau_{2}$ and $\rho_{2}$ are determined in terms of the rationality
condition reviewed previously in this section. For any $p,q\in\mathbb{Z}$, we
can always perform a gauging $\sigma^{*}$ that makes
$\tau_{1},\rho_{1}\in\mathbb{Z}$. Indeed this can be achieved by the gauging
of $\mathbb{Z}_{n_{i},w_{i}}\subset U(1)_{n_{i},w_{i}}$with

$\frac{w_{1}}{n_{1}}=\frac{pq^{\prime}}{qp^{\prime}}\quad,\quad\frac{w_{2}}{n_{2}}=qq^{\prime}\,.$
(3.41)

This gauging brings the theory at $(\tau,\rho)$ to a theory with

$\tau^{\prime}=qp^{\prime}+i\frac{q^{2}p^{\prime}}{p}\tau_{2}\quad,\quad\rho^{\prime}=pq^{\prime}+i\frac{q^{2}p^{\prime}}{p}\rho_{2}$
(3.42)

which, by applying the dualities
$T_{\rho}^{pq^{\prime}}T_{\tau}^{qp^{\prime}}$, is equivalent to the
factorized point

$\widehat{\tau}=i\frac{q^{2}p^{\prime}}{p}\tau_{2}\quad,\quad\widehat{\rho}=i\frac{q^{2}p^{\prime}}{p}\rho_{2}\,.$
(3.43)

Duality symmetry defects $\widehat{\mathcal{D}}$ of the factorized theory
above are easily detected by the logic exposed in section 2 and the
topological manipulation just described reads
$\mathcal{T}=T_{\rho}^{pq^{\prime}}T_{\tau}^{qp^{\prime}}\sigma^{*}$.

This simple computation shows that, because of (3.39), _any_ $c=2$ RCFT enjoy
duality symmetries which are more manifest in a factorzied point connected to
the original RCFT via topological manipulations. We will illustrate this
procedure when studying some particular examples in the next section. Before
concluding, we emphasize that the duality symmetries which are manifest in the
factorized point generically do not span the entire set of duality symmetries
in a given RCFT. For instance there can be symmetries coming from non-diagonal
gauging in the factorized point which are, however, more manifest if we look
at different points connected by topological manipulations. We will encounter
an example of this kind when we look at the bi-critical point in section
4.2.1.

##  4 Duality symmetries at multicriticality

As mentioned in the previous section, the intricate structure of the bosonic
$c=2$ conformal manifold allows for several special loci where various
orbifold branches intersect. Moreover, the theories sitting at these
multicritical loci are generically rational. Examples of this kind will be the
focus of this section, putting particular emphasis on the interplay between
non-invertible duality symmetries and several deformations. Regarding the
latter, we will study the additional exactly marginal deformations arising at
multicriticality, together with some examples of RG flows triggered by duality
preserving relevant operators.

We begin from the points on the conformal manifold of $c=2$ theories where the
global symmetry is enhanced. These loci are given in (3) and we focus on the
following:

$\displaystyle(\tau,\rho)$ $\displaystyle=(i,i)\;:\qquad
SU(2)^{2}\times\overline{SU(2)}^{2}\;,$ $\displaystyle(\tau,\rho)$
$\displaystyle=(e^{i2\pi/3},e^{i2\pi/3})\;:\qquad
SU(3)\times\overline{SU(3)}\;.$ (4.1)

Orbifolding by the $\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$ symmetries of the
$SU(2)^{2}$ point and the $SU(3)$ point respectively, one obtains two
multicritical points, where four and two branches of the conformal manifold
interesect. See figure 2 for a pictorial representation of the conformal
manifold. Therefore, these are respectively a quadri-critical and a bi-
critical points. These two multicritical points are respectively located in
the toroidal branch at $(\tau,\rho)=(i,2i)$ and $(\tau,\rho)=(\omega,\alpha)$
with $\omega=e^{\frac{2\pi i}{3}}$, $\alpha=\omega+i\sqrt{3}$. The reason
behind this choice is the fact that they exemplify many of the properties of
$c=2$ RCFTs with non-invertible duality symmetries. At the quadri-critical,
point the theory is factorized into two copies of $c=1$, while at the bi-
critical point the $B$-field takes non-zero value and the theory is genuinely
$c=2$.

###  4.1 The quadri-critical point

We begin by looking at the point $(\tau,\rho)=(i,i)$, featuring an
$SU(2)^{2}\times\overline{SU(2)}^{2}$ enhanced global symmetry. We review some
features of this point in Appendix A.1. The theory is factorized into two
copies of $c=1$, as it can be seen already in the holomorphic decomposition of
the fields

$\displaystyle\phi^{a}=\frac{1}{\sqrt{2}}\left(X^{a}+\overline{X}^{a}\right)\;,\quad\widetilde{\phi}^{a}=\frac{1}{\sqrt{2}}\left(X^{a}-\overline{X}^{a}\right)\;.$
(4.2)

Each factor enjoys two $\mathbb{Z}_{2}$ symmetries, i.e. the shift
$\phi^{a}\to\phi^{a}+\pi$ and the reflection $\phi^{a}\to-\phi^{a}$. On the
chiral fields, these act as a shift by $\sqrt{2}\pi/2$ and a sign flip. One
can then identify four (non-anomalous) $\mathbb{Z}_{2}$ symmetries at
$(\tau,\rho)=(i,i)$, which in the chiral basis take the following form

$\displaystyle{\mathbb{Z}}_{2}^{s}\;:$
$\displaystyle\quad\left(X^{a},\;\overline{X}^{a}\right)\,\to\,\left(X^{a}+\sqrt{2}\pi/2,\;\overline{X}^{a}+\sqrt{2}\pi/2\right)\;,$
$\displaystyle{\mathbb{Z}}_{2}^{orb}\;:$
$\displaystyle\quad\left(X^{a},\;\overline{X}^{a}\right)\,\to\,\left(-X^{a},\;-\overline{X}^{a}\right)\;,$
$\displaystyle{\mathbb{Z}}_{2}^{R_{1}}\;:$
$\displaystyle\quad\left(X^{1},\;\overline{X}^{1}\right)\,\to\,\left(X^{1}+\sqrt{2}\pi/2,\;\overline{X}^{1}+\sqrt{2}\pi/2\right)$
(4.3)
$\displaystyle\quad\left(X^{2},\;\overline{X}^{2}\right)\,\to\,\left(-X^{2},\;-\overline{X}^{2}\right)\;,$
$\displaystyle{\mathbb{Z}}_{2}^{R_{2}}\;:$
$\displaystyle\quad\left(X^{1},\;\overline{X}^{1}\right)\,\to\,\left(-X^{1}+\sqrt{2}\pi/2,\;-\overline{X}^{1}+\sqrt{2}\pi/2\right)\;,$
$\displaystyle\quad\left(X^{2},\;\overline{X}^{2}\right)\,\to\,\left(X^{2}+\sqrt{2}\pi/2,\;\overline{X}^{2}+\sqrt{2}\pi/2\right)\;.$

The $\mathbb{Z}_{2}^{s}$ shift symmetry corresponds to a simultaneous
translation in field space. The $\mathbb{Z}_{2}^{orb}$ implements a reflection
on both fields, analogous to the $c=1$ case (see appendix A.1). The remaining
$\mathbb{Z}_{2}$ symmetries implement combinations of the previous actions.
For more details, see [55]222222The reader should be aware that we are using a
slightly different notation than in [55].. As usual, the orbifold by the
$\mathbb{Z}_{2}^{s}$ shift symmetry leads to another theory in the toroidal
branch. The remaining $\mathbb{Z}_{2}$ actions define three independent
orbifold branches. Furthermore, the four actions just described are mapped to
each other by the $SU(2)^{2}\times\overline{SU(2)}^{2}$ global symmetry.
Therefore, performing the quotient by any of them leads to four different
descriptions of the same point of the conformal manifold, where the toroidal
and the three orbifold branches meet.

Note that the orbifold by the $\mathbb{Z}_{2}^{s}$ symmetry is implemented
precisely by the matrix in Eq. (3.36). The resulting theory sits on the
toroidal branch at $(\tau,\rho)=\left(i,i/2\right)$, and by acting with
$S_{\rho}\,:\,\rho\to-1/\rho$, the theory is mapped to
$\left(\tau,\rho\right)=\left(i,2i\right)$. One therefore identifies the
theory at $\left(\tau,\rho\right)=\left(i,2i\right)$ with the quadri-critical
point. The overall map of the charges read

$n_{1}^{\prime}=\frac{n_{1}+n_{2}}{2}\;,\>n_{2}^{\prime}=\frac{n_{1}-n_{2}}{2}\;,\>w_{1}^{\prime}=w_{1}+w_{2}\;,\>w_{2}^{\prime}=w_{1}-w_{2}\;,$
(4.4)

where the non-primed symbols denote the original charges at the enhanced
symmetry point.

The theory at the quadri-critical point is itself rational and, moreover,
corresponds to a diagonal modular invariant since $\rho=2\tau$ [58]. This
property is also manifest by the fact that this theory can be regarded as a
product of two rational $c=1$ theories at radius equal to $\sqrt{2}$. Even if
many of the properties of the global symmetries featured at
$(\tau,\rho)=(i,2i)$ naturally descend from its product structure, we will see
that there are still some salient features pertaining to the $c=2$ theory,
such as the non-Abelian fusion satisfied by the (non-invertible) duality
symmetries. This is just a consequence of the fact that the latter symmetries
descend from a non-Abelian subgroup of the $c=2$ duality group (3.22) which is
not a product of two $c=1$ dualities.

####  4.1.1 Duality symmetry and fusion category

As we have already discussed in the general case, RCFT points in the toroidal
branch enjoy duality symmetries, obtained by composing gaugings and dualities
in different ways. We now discuss some of the duality symmetries present in
the quadri-critical point $(\tau,\rho)=(i,2i)$. Since it is a factorized point
(i.e. it can be decomposed as a product of two $c=1$ theories) with
$\rho_{2},\tau_{2}\in\mathbb{Z}$, it generically hosts duality symmetries
coming from diagonal gaugings, of the form232323For sake of notational
simplicity, from now on we we will omit the symbol $\circ$ denoting
composition dualities and topological manipulations. For instance
$M\sigma_{2,1}\equiv M\circ\sigma_{2,1}$.

$\displaystyle\mathcal{D}_{A}=M\sigma_{2,1}\to\mathsf{D}_{A}=\begin{pmatrix}0&0&\tfrac{1}{2}&0\\\
0&1&0&0\\\ 2&0&0&0\\\
0&0&0&1\end{pmatrix}\;,\;\mathcal{D}_{B}=S_{\rho}M\tilde{\sigma}_{2,1}\to\mathsf{D}_{B}=\begin{pmatrix}0&0&0&-\frac{1}{2}\\\
1&0&0&0\\\ 0&-2&0&0\\\ 0&0&1&0\end{pmatrix}\;\;\;$ (4.5)

together with the invertible duality symmetries $S_{\tau}$ and $I$ that do not
involve any gauging. The matrices above represent the action of the associated
operators over the $({\boldsymbol{\phi}},\widetilde{\boldsymbol{\phi}})$. In
(4.5) we introduced the following notation for the diagonal gauging

$\displaystyle\sigma_{N,M}$ $\displaystyle\equiv{\rm
gauging}\;\mathbb{Z}_{N}\times\mathbb{Z}_{M}\subset U(1)_{n_{1}}\times
U(1)_{w_{1}}$ (4.6) $\displaystyle\widetilde{\sigma}_{N,M}$
$\displaystyle\equiv{\rm gauging}\;\mathbb{Z}_{N}\times\mathbb{Z}_{M}\subset
U(1)_{n_{2}}\times U(1)_{w_{2}}\,.$

and their corresponding matrix representation reads

$\sigma_{N,M}\to\begin{pmatrix}\frac{N}{M}&0&0&0\\\ 0&1&0&0\\\
0&0&\frac{M}{N}&0\\\
0&0&0&1\end{pmatrix}\quad,\quad\widetilde{\sigma}_{N,M}\to\begin{pmatrix}1&0&0&0\\\
0&\frac{N}{M}&0&0\\\ 0&0&1&0\\\ 0&0&0&\frac{M}{N}\end{pmatrix}\,,$ (4.7)

As emphasized before, this set of duality symmetries does not capture the full
set of symmetries of this point; for instance duality symmetries coming from
non-diagonal gaugings can be present242424It was also recently noticed in [31]
that duality defects coming from gauging non-invertible symmetries can be
present in $2$d CFT.. It would be nice to develop a systematic method capable
of identifying all duality symmetries, along the lines of [63].

Given a set of duality symmetries closed under fusion, we can discuss the
fusion algebra and the underlying fusion category (see appendix C for some
details on this type of categorical symmetries). Since they are symmetries
coming from self-dualities, the fusion algebra will be graded by (a subgroup
of) the underlying $O(2,2,\mathbb{Z})$ duality group at $c=2$. This grading
fixes the generic structure of the fusion algebra to be

$\mathcal{D}_{\alpha}\times\mathcal{D}_{\beta}=\mathcal{D}_{\alpha\beta}\times\mathcal{C}_{0}\,,$
(4.8)

where $\mathcal{D}_{\alpha\beta}$ is the composed duality symmetry whose
matrix representation is obtained by multiplying the matrix representation of
$\mathcal{D}_{\alpha}$ and $\mathcal{D}_{\beta}$ and $\mathcal{C}_{0}$ is some
combination of topological lines generating a subgroup of the invertible
$U(1)^{2}_{\mathbf{n}}\times U(1)^{2}_{\mathbf{w}}$ symmetry. The occurrence
of the gauging $\sigma_{\alpha}$ in the definition of $\mathcal{D}_{\alpha}$
ensures that this topological defect absorbs all the lines generating the
gauged symmetry $G_{\alpha}\subset U(1)^{2}_{\mathbf{n}}\times
U(1)^{2}_{\mathbf{w}}$, namely

$\mathcal{D}_{\alpha}\times\eta_{\alpha}=\eta_{\alpha}\times\mathcal{D}_{\alpha}=\mathcal{D}_{\alpha}\qquad\forall\alpha\in
G_{\alpha}\subset U(1)^{2}_{\mathbf{n}}\times U(1)^{2}_{\mathbf{w}}\,.$ (4.9)

The fusion of $\mathcal{D}_{\alpha}$ with its orientation reversal can be
uniquely fixed by imposing the consistency with the above property of
absorption, finding

$\mathcal{D}_{\alpha}\times\overline{\mathcal{D}}_{\alpha}=\sum\limits_{\alpha\in
G}\eta_{\alpha}\qquad G_{\alpha}\subset U(1)^{2}_{\mathbf{n}}\times
U(1)^{2}_{\mathbf{w}}\,.$ (4.10)

Let us now delve into the specific examples presented in (4.5). From the
gauging matrices used to define those defects we find

$\eta_{a,0}\widetilde{\eta}_{b,0}\times\mathcal{D}_{A}=\widetilde{\eta}_{b,0}\times\mathcal{D}_{A}\quad,\quad\eta_{a,0}\widetilde{\eta}_{b,0}\times\mathcal{D}_{B}=\eta_{a,0}\times\mathcal{D}_{B}\,,$
(4.11)

where $\eta_{a,0}$($\eta_{0,b}$) generate $\mathbb{Z}_{2}\subset U(1)_{n_{1}}$
($\mathbb{Z}_{2}\subset U(1)_{w_{1}}$) while $\widetilde{\eta}_{a,0}$
($\widetilde{\eta}_{0,b}$) generate $\mathbb{Z}_{2}\subset U(1)_{n_{2}}$
($\mathbb{Z}_{2}\subset U(1)_{w_{2}}$). Here $a,b\in\mathbb{Z}_{2}$. Because
of the grading structure we also need to include additional duality defects

$\mathcal{D}_{C}=S_{\rho}\sigma_{2,1}\widetilde{\sigma}_{2,1}\quad,\quad\mathcal{D}_{\tau
C}=S_{\tau}S_{\rho}\sigma_{2,1}\widetilde{\sigma}_{2,1}\quad,\quad C\,,$
(4.12)

where $C$ is the topological defect implementing charge conjugation and the
new duality defects satisfy

$\eta_{a,0}\widetilde{\eta}_{b,0}\times\mathcal{D}_{C}=\mathcal{D}_{C}\quad,\quad\eta_{a,0}\widetilde{\eta}_{b,0}\times\mathcal{D}_{\tau
C}=\mathcal{D}_{\tau C}\,.$ (4.13)

Applying the generic rules described above we find the fusion between defects
and their inverse to be

$\mathcal{D}_{C}\times\overline{\mathcal{D}}_{C}=\sum\limits_{a,b=0,1}\eta_{a,0}\widetilde{\eta}_{b,0}\quad,\quad\mathcal{D}_{A}\times\overline{\mathcal{D}}_{A}=\sum\limits_{a=0,1}\eta_{a,0}\quad,\quad\mathcal{D}_{B}\times\overline{\mathcal{D}}_{B}=\sum\limits_{b=0,1}\widetilde{\eta}_{b,0}\,$
(4.14)

while some fusion rules between different duality symmetries read

$\begin{split}&\mathcal{D}_{C}\times\mathcal{D}_{A}=\mathcal{D}_{B}\sum\limits_{a=0,1}\eta_{a,0}\;\;,\;\;\mathcal{D}_{A}\times\mathcal{D}_{C}=S_{\tau}\mathcal{D}_{A}\sum_{b=0,1}\widetilde{\eta}_{b,0}\\\
&\mathcal{D}_{A}\times\mathcal{D}_{B}=S_{\tau}\sum_{a,b}\eta_{a,0}\widetilde{\eta}_{b,0}\;\;,\;\;\mathcal{D}_{B}\times\mathcal{D}_{A}=\mathcal{D}_{C}\,,\end{split}$
(4.15)

together with other ones that can be easily derived. We notice that the
category is non-commutative, as as a consequence of the non-Abelian
$O(2,2,\mathbb{Z})$ grading252525Non-commutative categorical symmetries
appeared before in the literature in $4$d (see e.g. [29, 40]) and also in $2$d
$c=2$ theories (see e.g. [24]).. In order to describe the non-commutative
grading more explicitly, we introduce the group with five generators
$\\{s,d_{c},d_{a},i,c\\}$ satisfying the following multiplication law

$\displaystyle s^{2}=d_{c}^{2}=c\quad,\quad d_{a}^{2}=c^{2}=i^{2}=1\quad$
$\displaystyle,\quad isi=s^{3}\quad,\quad id_{c}i=d_{c}^{3}$ $\displaystyle
asa=d_{c}\quad$ $\displaystyle,\quad ad_{c}a=s\;\;.$ (4.16)

It can be easily verified that the fusion of the category described before is
graded by the non-commutative group (4.16) by making the following assignment
of (non-)invertible defects to the generators

$S_{\tau}\to s\;\;,\;\;\mathcal{D}_{C}\to d_{c}\;\;,\;\;\mathcal{D}_{A}\to
d_{a}\;\;,\;\;I\to i\;\;,\;\;C\to c\;,$ (4.17)

with the remaining defects corresponding to products of the ones above.

As already emphasized, the quadri-critical point is the product of two $c=1$
CFT at $R=\sqrt{2}$. Therefore the partition function can be written as

$Z_{c=2}[\tau=i,\rho=2i]=\sum_{m,n\in\mathbb{Z}_{4}}\chi_{m}\chi_{n}\overline{\chi}_{-m}\overline{\chi}_{-n}=Z_{c=1}[R=\sqrt{2}]Z_{c=1}[R=\sqrt{2}]$
(4.18)

and we can trivially prove that the partition function of the $c=2$ theory at
the quadri-critical point is self-dual under gauging of $\mathbb{Z}_{2}\subset
U(1)_{n_{1}}$ and $\mathbb{Z}_{2}\subset U(1)_{n_{2}}$, implying the existence
of the defects described above (see appendix C).

####  4.1.2 Modular bootstrap at the quadri-critical point

As a further check that the above structure is actually consistent, we compute
the twisted partition functions and make use of modular covariance. We find
that all the elements introduced above lead to well defined defect Hilbert
spaces, hence complying with (2.11). For sake of the extension of this
exposition, we do not include all the examples but just a few illustrative
ones and comment on the rest.

Regarding the defect $\mathcal{D}_{A}$, it can be explicitly checked that its
action emulates the action of $T$-duality in one of the sectors of the
factorized theory, namely $\overline{p}_{1}\to-\overline{p}_{1}$, hence the
computation works out exactly as at $c=1$ and $R=\sqrt{2}$ [20]. Similarly,
target space inversion $I$ implements a reflection in one of the two sectors,
hence clearly leading to a well defined defect on the product theory.

We will then proceed to illustrate the computation for the non-invertible
defects ${\cal D}_{B}$ and ${\cal D}_{C}$. Making use of (3.32) and (3.9), the
action of $\mathcal{D}_{B}$ maps

${\bf p}\to R_{B}{\bf p}\quad,\quad\overline{\bf
p}\to\overline{R}_{B}\overline{\bf p}\quad,\quad R_{B}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\quad,\quad\overline{R}_{B}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,.$ (4.19)

In addition, it turns out that the phase $\alpha_{B}({\bf n},{\bf w})$
determined by equation (A.9) is trivial once we impose the transformed
operator to be local262626In fact, the action of ${\cal D}_{B}$ leads to the
map of charges
$\\{n_{1},n_{2},w_{1},w_{2}\\}\to\\{-2w_{2},n_{1},-\tfrac{n_{2}}{2},w_{1}\\}$.
Therefore, the resulting operator corresponds to a genuine local operator only
for $n_{2}=2\mathbb{Z}$. Moreover, plugging the previous map of charges into
equation (A.9) leads to the solution $\alpha_{B}({\bf n},{\bf w})=n_{2}w_{2}$
mod 2 for the phase entailing mutual locality. The latter clearly becomes
trivial for $n_{2}$ even.. Due to (4.19), when computing the partition
function twisted by the insertion of $\mathcal{D}_{B}$ along the spatial
cycle, the sum gets contributions only from states with $p_{1}=p_{2}=0$,
$\overline{p}_{1}=\overline{p}_{2}=2n$ ($n\in\mathbb{Z}$). Finally, in order
to perform the sum over oscillator modes, we consider a basis over which
$R_{B}$ ($\overline{R}_{B}$) act diagonally with eigenvalues $\pm i$ ($\pm
1$). Putting all together we get

$\displaystyle Z_{(1,{\mathcal{D}}_{B})}$
$\displaystyle=\sqrt{2}\frac{q^{-\frac{1}{12}}\overline{q}^{-\frac{1}{12}}\sum_{n\in\mathbb{Z}}\overline{q}^{4n^{2}}}{\prod_{m=1}^{\infty}(1+iq^{m})(1-iq^{m})(1+\bar{q}^{m})(1-\bar{q}^{m})}=\sqrt{2}\frac{\vartheta_{4}(4\tau)\vartheta_{4}(2\bar{\tau})\vartheta_{3}(8\bar{\tau})}{\eta(2\tau)\eta(\bar{\tau})^{2}}$
$\displaystyle\xrightarrow{\tau\to-\frac{1}{\tau}}\,\,Z_{({\mathcal{D}}_{B},1)}$
$\displaystyle=\frac{\vartheta_{2}(\tau/4)\vartheta_{2}(\bar{\tau}/2)\vartheta_{3}(\bar{\tau}/8)}{4\eta(\tau/2)\eta(\bar{\tau})^{2}}=\frac{1}{|\eta(\tau)|^{4}}\left(q^{\frac{3}{32}}\overline{q}^{\frac{1}{16}}+2q^{\frac{3}{32}}\overline{q}^{\frac{1}{8}}+\ldots\right)\;.$
(4.20)

We therefore verify that $Z_{({\mathcal{D}}_{B},1)}$ has a consistent
interpretation as a trace over the defect Hilbert space.

Now we turn to the defect $\mathcal{D}_{C}$ implementing the following action

${\bf p}\to R_{C}{\bf p}\quad,\quad\overline{\bf
p}\to\overline{R}_{C}\overline{\bf p}\quad,\quad
R_{C}=-\overline{R}_{C}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\,.$ (4.21)

In addition, the states get multiplied by the $\mathbb{Z}_{2}$ phase
$\alpha_{C}={\bf n}\cdot{\bf w}$. Given the transformation (4.21), the twisted
partition function gets contribution from states in the identity multiplet
$p_{a}=\overline{p}_{a}=0$ ($a=1,2$). Following similar steps as the ones
described above, we get

$\displaystyle Z_{(1,{\mathcal{D}}_{C})}$
$\displaystyle=2\frac{q^{-\frac{1}{12}}\overline{q}^{-\frac{1}{12}}}{\prod_{m=1}^{\infty}(1+iq^{m})(1-iq^{m})(1+i\bar{q}^{m})(1-i\bar{q}^{m})}=2\frac{\vartheta_{4}(4\tau)\vartheta_{4}(4\bar{\tau})}{|\eta(2\tau)|^{2}}$
(4.22)
$\displaystyle\xrightarrow{\tau\to-\frac{1}{\tau}}\,\,Z_{({\mathcal{D}}_{C},1)}$
$\displaystyle=\frac{\vartheta_{2}(\tau/4)\vartheta_{2}(\bar{\tau}/4)}{|\eta(\tau/2)|^{2}}=\frac{4}{|\eta(\tau)|^{4}}\left[q^{\tfrac{3}{32}}\overline{q}^{\tfrac{3}{32}}+q^{\tfrac{3}{32}}\overline{q}^{\tfrac{3}{32}}\left(q^{\tfrac{1}{4}}+\overline{q}^{\tfrac{1}{4}}\right)+\ldots\right]\;,$

hence corroborating that $\mathcal{D}_{C}$ leads to a consistent defect
Hilbert space.

We end this subsection by considering the case of $\mathcal{D}_{\tau C}$ whose
action maps $p_{a}\to-p_{a}$ ($a=1,2$) while leaving the right movers
invariant. The $\mathbb{Z}_{2}$ phase again obtains $\alpha_{\tau C}={\bf
n}\cdot{\bf w}$. By comparison with the $c=1$ case reviewed in section 2, one
notices that this action can be regarded as a composition of two (left)
$T$-dualities on each $c=1$ factor (composed with a gauging of the
corresponding ${\mathbb{Z}}_{2}$ subgroups), hence leading to

$\displaystyle Z_{(1,{\cal D}_{\tau C})}$
$\displaystyle=2\frac{\vartheta_{4}(2\tau)^{2}\vartheta_{3}(4\overline{\tau})^{2}}{|\eta(\tau)|^{4}}$
(4.23) $\displaystyle\xrightarrow{\tau\to-\frac{1}{\tau}}\,\,Z_{({\cal
D}_{\tau C},1)}$
$\displaystyle=\frac{\vartheta_{2}(\tau/2)^{2}\vartheta_{3}(\overline{\tau}/4)^{2}}{4|\eta(\tau)|^{4}}=\frac{1}{|\eta(\tau)|^{4}}\left(q^{\frac{1}{8}}+4q^{\frac{1}{8}}\overline{q}^{\frac{1}{8}}+4q^{\frac{1}{8}}\overline{q}^{\frac{1}{4}}\ldots\right)$

Note the important role played by the quantum dimension here. If it was
absent, the final result would be

$\frac{\vartheta_{2}(\tau/2)^{2}\vartheta_{3}(\overline{\tau}/4)^{2}}{8|\eta(\tau)|^{4}}\sim\frac{1}{2}q^{\frac{1}{8}}+\ldots$
(4.24)

clearly spoiling the interpretation of the twisted partition function as a
trace over a Hilbert space. Of course, the positive outcome of the modular
bootstrap analysis for ${\cal D}_{C}$ and ${\cal D}_{\tau C}$ naturally
implies the consistency of the invertible defect implementing $S_{\tau}$.

####  4.1.3 Marginal and relevant deformations at the quadri-critical point

We proceed to study the interplay between the symmetry structure presented in
the previous sections and the various marginal and relevant deformations of
the theory. As a multicritical point, one expects finding additional
independent exactly marginal deformations which span the orbifold branches,
besides the ones of the form $\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}$
which span the four dimensional toroidal branch. The strategy consists in
constructing the set of marginal operators of the form
$J^{a}_{i}\overline{J}_{j}^{b}$, where $a,b=1,2$ and $i,j=+,-$, and in terms
of the chiral fields

$\displaystyle J_{\pm}^{1}$ $\displaystyle=V_{\pm 1,0,\pm 1,0}=e^{\pm
i\sqrt{2}X^{1}}\;,\qquad\overline{J}_{\pm}^{1}=V_{\pm 1,0,\mp 1,0}=e^{\pm
i\sqrt{2}\,\overline{X}^{1}}\;,$ $\displaystyle J_{\pm}^{2}$
$\displaystyle=V_{0,\pm 1,0,\pm 1}=e^{\pm
i\sqrt{2}X^{2}}\;,\qquad\overline{J}_{\pm}^{2}=V_{0,\pm 1,0,\mp 1}=e^{\pm
i\sqrt{2}\,\overline{X}^{2}}\;,$ $\displaystyle J_{3}^{a}$
$\displaystyle=\frac{i}{\sqrt{2}}\partial
X^{a}\;,\qquad\overline{J}_{3}^{a}=\frac{i}{\sqrt{2}}\overline{\partial}\,\overline{X}^{a}\;.$
(4.25)

At the enhanced symmetry point, they are all equivalent to
$J_{3}^{a}\overline{J}_{3}^{b}$ by the $SU(2)^{2}\times\overline{SU(2)}^{2}$
global symmetry. We then construct invariant combinations under the
$\mathbb{Z}_{2}$ automorphisms and charge conjugation. The action on the
currents is

$\displaystyle{\mathbb{Z}}_{2}^{s}\;:$
$\displaystyle\;\left(J_{3}^{a},\;J^{a}_{\pm}\right)\,\to\,\left(J_{3}^{a},\;-J^{a}_{\pm}\right)\;,$
$\displaystyle{\mathbb{Z}}_{2}^{orb}\;:$
$\displaystyle\;\left(J_{3}^{a},\;J^{a}_{\pm}\right)\,\to\,\left(-J_{3}^{a},\;J^{a}_{\mp}\right)\;,$
$\displaystyle{\mathbb{Z}}_{2}^{R_{1}}\;:$
$\displaystyle\;\left(J_{3}^{1},\;J^{1}_{\pm}\right)\,\to\,\left(J_{3}^{1},\;-J^{1}_{\pm}\right)\;,$
$\displaystyle\;\left(J_{3}^{2},\;J^{2}_{\pm}\right)\,\to\,\left(-J_{3}^{2},\;J^{2}_{\mp}\right)\;,$
$\displaystyle{\mathbb{Z}}_{2}^{R_{2}}\;:$
$\displaystyle\;\left(J_{3}^{1},\;J^{1}_{\pm}\right)\,\to\,\left(-J_{3}^{1},\;J^{1}_{\mp}\right)\;,$
$\displaystyle\;\left(J_{3}^{2},\;J^{2}_{\pm}\right)\,\to\,\left(J_{3}^{2},\;-J^{2}_{\pm}\right)\;.$
(4.26)

Given the $\mathbb{Z}_{2}$-invariant operators, the exactly marginal operators
are further combinations of these, whose OPE with themselves has no simple
poles [49]. Finally, the exactly marginal operators are mapped to the quadri-
critical point by application of the map (4.4).

For the case of $\mathbb{Z}_{2}^{orb}$, the four exactly marginal operators

$\displaystyle\mathcal{O}_{ab}$
$\displaystyle=J_{+}^{a}\overline{J}_{+}^{b}+J_{-}^{a}\overline{J}_{-}^{b}+J_{+}^{a}\overline{J}_{-}^{b}+J_{-}^{a}\overline{J}_{+}^{b}\;,a\,b=1,2\;,$
(4.27)

are mapped to the quadri-critical points as

$\mathbb{Z}_{2}^{orb}\text{
branch}\begin{cases}\mathcal{O^{\prime}}_{11}&=V_{0,0,+1,-1}+V_{+2,-2,0,0}+V_{0,0,-1,+1}+V_{-2,+2,0,0}\;,\\\
\mathcal{O^{\prime}}_{12}&=V_{2,0,0,-1}+V_{0,-2,+1,0}+V_{-2,0,0,+1}+V_{0,+2,-1,0}\;,\\\
\mathcal{O^{\prime}}_{21}&=V_{-2,0,0,-1}+V_{0,-2,-1,0}+V_{+2,0,0,+1}+V_{0,+2,+1,0}\;,\\\
\mathcal{O^{\prime}}_{22}&=V_{0,0,-1,-1}+V_{-2,-2,0,0}+V_{0,0,+1,+1}+V_{+2,+2,0,0}\;,\end{cases}$

where they span the 4-dimensional $\mathbb{Z}_{2}^{orb}$ orbifold branch.

Similarly, and considering the action in Eq. (4.1), the deformations into the
$\mathbb{Z}_{2}^{R_{1}}$ branch are generated by the following
$\mathbb{Z}_{2}^{R_{1}}$-invariant exactly marginal operators

$\mathbb{Z}_{2}^{R_{1}}\text{
branch}\begin{cases}\mathcal{O^{\prime}}_{A}^{R_{1}}&=V_{+2,0,0,-1}-V_{0,-2,+1,0}+V_{-2,0,0,+1}-V_{0,+2,-1,0}\;,\\\\[7.0pt]
\mathcal{O^{\prime}}_{B}^{R_{1}}&=V_{-2,0,0,-1}-V_{0,+2,+1,0}+V_{+2,0,0,+1}-V_{0,-2,-1,0}\;,\end{cases}$

while for $\mathbb{Z}_{2}^{R_{2}}$, we get the following

$\mathbb{Z}_{2}^{R_{2}}\text{
branch}\begin{cases}\mathcal{O^{\prime}}_{A}^{R_{2}}&=V_{0,0,+1,-1}-V_{+2,-2,0,0}+V_{0,0,-1,+1}-V_{-2,+2,0,0}\;,\\\\[7.0pt]
\mathcal{O^{\prime}}_{B}^{R_{2}}&=V_{0,0,-1,-1}-V_{-2,-2,0,0}+V_{0,0,+1,+1}-V_{+2,+2,0,0}\;.\end{cases}$

Both the $\mathbb{Z}_{2}^{R_{1}}$ and $\mathbb{Z}_{2}^{R_{2}}$ orbifold
branches are 2-dimensional.

The duality defects discussed in the previous subsection are preserved by some
combinations of the exactly marginal operators at the quadri-critical
point272727Note that, given their charges, none of these vertex operators
acquires a phase under the action of the duality symmetries described in the
previous subsection.. For the four operators $\mathcal{O^{\prime}}_{ab}$
parametrizing the $4$-dimensional $\mathbb{Z}_{2}^{orb}$ orbifold branch, we
get

$\displaystyle\mathcal{O^{\prime}}_{11}+\mathcal{O^{\prime}}_{12}\;,\quad\mathcal{O^{\prime}}_{21}+\mathcal{O^{\prime}}_{22}\qquad\text{preserved
by }$ $\displaystyle\qquad\mathcal{D}_{A}\;,$
$\displaystyle\mathcal{O^{\prime}}_{11}+\mathcal{O^{\prime}}_{21}\;,\quad\mathcal{O^{\prime}}_{12}+\mathcal{O^{\prime}}_{22}\qquad\text{preserved
by }$ $\displaystyle\qquad\mathcal{D}_{B}\;,$
$\displaystyle\mathcal{O^{\prime}}_{11}+\mathcal{O^{\prime}}_{22}\;,\quad\mathcal{O^{\prime}}_{12}+\mathcal{O^{\prime}}_{21}\qquad\text{preserved
by }$ $\displaystyle\qquad S_{\tau}\;,\;I\;,\;\mathcal{D}_{C}\;,$
$\displaystyle\mathcal{O^{\prime}}_{11}\;,\;\mathcal{O^{\prime}}_{12}\;,\;\mathcal{O^{\prime}}_{21}\;,\;\mathcal{O^{\prime}}_{22}\qquad\text{preserved
by }$ $\displaystyle\qquad\mathcal{D}_{\tau C}\;.$ (4.28)

In particular $\mathcal{D}_{\tau C}$ is preserved along the full
$\mathbb{Z}_{2}^{orb}$ orbifold branch. In addition, there are two extra non-
invertible symmetries $\mathcal{D}_{A,B}$ which are preserved along certain
$2$-dimensional submanifolds of this particular orbifold branch.

Regarding the $\mathbb{Z}_{2}^{R_{1}}$, $\mathbb{Z}_{2}^{R_{2}}$ branches the
invariant combinations are

$\displaystyle\mathcal{O^{\prime}}_{A}^{R_{1}}+\mathcal{O^{\prime}}_{B}^{R_{1}}\qquad\text{preserved
by }$ $\displaystyle\qquad I\;,\;\mathcal{D}_{C}\;,$
$\displaystyle\mathcal{O^{\prime}}_{A}^{R_{1}}-\mathcal{O^{\prime}}_{B}^{R_{1}}\qquad\text{preserved
by }$ $\displaystyle\qquad S_{\tau}\;,$
$\displaystyle\mathcal{O^{\prime}}_{A}^{R_{2}}+\mathcal{O^{\prime}}_{B}^{R_{2}}\qquad\text{preserved
by }$ $\displaystyle\qquad S_{\tau}\;,\;I\;,$
$\displaystyle\mathcal{O^{\prime}}_{A}^{R_{2}}-\mathcal{O^{\prime}}_{B}^{R_{2}}\qquad\text{preserved
by }$ $\displaystyle\qquad\mathcal{D}_{C}\;.$ (4.29)

Interestingly, we find that $\mathcal{D}_{A}$ and $\mathcal{D}_{\tau
B}=S_{\tau}^{-1}MS_{\tau}\circ\widetilde{\sigma}_{2,1}$ act as

$\displaystyle\mathcal{D}_{A}\;:$
$\displaystyle\quad\mathcal{O^{\prime}}_{A}^{R_{1}}\to\mathcal{O^{\prime}}_{A}^{R_{2}}\;,$
$\displaystyle\quad\mathcal{O^{\prime}}_{B}^{R_{1}}\to\mathcal{O^{\prime}}_{B}^{R_{2}}\;,$
$\displaystyle\mathcal{D}_{\tau B}\;:$
$\displaystyle\quad\mathcal{O^{\prime}}_{A}^{R_{1}}\to-\mathcal{O^{\prime}}_{A}^{R_{2}}\;,$
$\displaystyle\quad\mathcal{O^{\prime}}_{B}^{R_{1}}\to-\mathcal{O^{\prime}}_{B}^{R_{2}}\;,$
(4.30)

thus they effectively exchange the two orbifold branches.

#### Relevant deformations

We now proceed with the analysis of the relevant deformations of the quadri-
critical point. Classifying them by their dimension, we have

$(h,\overline{h})$ |  $(\mathbf{n},\mathbf{w})$
---|---
$\left(\frac{1}{8}\,,\,\frac{1}{8}\right)$ |  $(\pm 1,0,0,0)\;,\;(0,\pm 1,0,0)$
$\left(\frac{1}{4}\,,\,\frac{1}{4}\right)$ |  $(\pm 1,\pm 1,0,0)\;,\;(\pm 1,\mp 1,0,0)$
$\left(\frac{1}{2}\,,\,\frac{1}{2}\right)$ |  $(\pm 2,0,0,0)\;,\;(0,\pm 2,0,0)\;,\;(0,0,\pm 1,0)\;,\;(0,0,0,\pm 1)$
$\left(\frac{5}{8}\,,\,\frac{5}{8}\right)$ |  $(0,\pm 1,\pm 1,0)\;,\;(\pm 2,\mp 1,0,0)\;,\;(\pm 1,0,0,\pm 1)\;,\;(\pm 1,0,0,\mp 1)$ $(\pm 2,\pm 1,0,0)\;,\;(\pm 2,\mp 1,0,0)\;,\;(\pm 1,\pm 2,0,0)\;,\;(\pm 1,\mp 2,0,0)$

Some non-invertible defects at the quadri-critical point commute with
particular combinations of these operators. We thus proceed to analyse some
representative cases of duality-preserving RG flows, triggered by some of the
relevant deformations presented in the above table. Given the factorized
structure quadri-critical theory, these RG flows can be studied by building up
on the intuition acquired at $c=1$ (see [20] for some examples of the latter).
This enables to check some non-trivial constraints that are a direct
consequence of the presence of non-invertible duality symmetries.

We begin by considering the case of the most relevant operators, that is with
conformal weights $(h,\overline{h})=\left(\tfrac{1}{8},\tfrac{1}{8}\right)$.
There are two possible independent charge conjugation invariant combinations,
namely

$R_{1/8}^{1}=V_{1,0,0,0}+V_{-1,0,0,0}\sim\cos\phi^{1}\quad,\quad
R_{1/8}^{2}=V_{0,1,0,0}+V_{0,-1,0,0}\sim\cos\phi^{2}$ (4.31)

Each of them preserve a $U(1)_{m}\times U(1)_{w}$ invertible symmetry
participating in a mixed ’t Hooft anomaly of the form

$\mathcal{A}_{3d}=\frac{1}{2\pi}\int AdB\qquad A\in U(1)_{m}\;,\;B\in
U(1)_{w}\,.$ (4.32)

Due to this anomaly the IR fixed point must be gapless. Moreover both $R_{1}$
and $R_{2}$ preserve respectively $D_{\tau
B}=S_{\tau}^{-1}S_{\rho}M\widetilde{\sigma}_{2,1}$ and
$D_{A}=M\sigma_{2,1}$.282828Note that $\mathcal{D}_{\tau
B}=S_{\tau}^{-1}\mathcal{D}_{B}$ and therefore is not an independent defect.
Therefore the IR gapless theory must enjoy such non-invertible defect292929The
non-invertible symmetry cannot decouple from the gapless sector since its
invertible subgroup partecipates in the ’t Hooft anomalies generating the
gapless mode.. In this case, it is straightforward to show that the IR theory
corresponds to the $c=1$ compact boson at $R=\sqrt{2}$, hence exactly
satisfying all the symmetry constraints.

Let us now comment on slightly more involved flows, triggered by the relevant
operators with $(h,\overline{h})=\left(\tfrac{1}{2},\tfrac{1}{2}\right)$ of
the form

$R_{1/2}^{1}\sim\cos(\widetilde{\phi}_{1})+\cos(2\phi_{2})\quad,\quad
R_{1/2}^{2}\sim\cos(\widetilde{\phi}_{2})+\cos(2\phi_{1})\,.$ (4.33)

Both $R_{1/2}^{1,2}$ break $U(1)^{4}$ to a subgroup $U(1)\times
U(1)\times\mathbb{Z}_{2}$ with a $\mathbb{Z}_{2}$ anomaly

$\mathcal{A}_{3d}=\pi i\int a\cup\beta(b)$ (4.34)

where $a,b\in H^{1}(X_{3},\mathbb{Z}_{2})$ are discrete gauge fields for
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\subset U(1)\times
U(1)\times\mathbb{Z}_{2}$ and $\beta$ is the Bockstein homomorphism (see e.g.
[64]). In the case of $R_{1/2}^{1}$, the discrete subgroup corresponds to a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ momentum and winding symmetries acting on
$(\phi_{2},\widetilde{\phi}_{2})$, and vice versa for $R_{1/2}^{2}$. On the
contrary, the relevant combination $R_{1/2}^{1}+R_{1/2}^{2}$ preserves an
anomaly free $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. Therefore there are no
continuous anomalies and the theory can be non-trivially gapped if we deform
with $R_{1/2}^{1,2}$ or even trivially gapped if we consider the combination
$R_{1/2}^{1}+R_{1/2}^{2}$. However both deformations in (4.33) preserve the
non-invertible symmetry generated by $\mathcal{D}_{C}$ and the sum
$R_{1/2}^{1}+R_{1/2}^{2}$ preserves all the duality defects
$\mathcal{D}_{A},\mathcal{D}_{C},\mathcal{D}_{\tau C}$. Since both
$\mathcal{D}_{A}$ and $\mathcal{D}_{\tau B}$ have a non-invertible anomaly,
while $\mathcal{D}_{C}$ is anomaly free (see appendix C and references
therein)303030In particular, we are referring to the anomalies coming from the
TY($\mathbb{Z}_{2})$ subcategory spanned by one of the three non-invertible
defects together with the its corresponding $\mathbb{Z}_{2}$ symmetry., the IR
theory must match such anomalies. We proceed now to study the RG flows

$\displaystyle S=S_{c=2}[(i,2i)]+\mu\int R_{1/2}^{1}$ (4.35) $\displaystyle
S=S_{c=2}[(i,2i)]+\mu\int(R_{1/2}^{1}+R_{1/2}^{2})$ (4.36)

and check explicitly the non trivial constraints coming from the non-
invertible ’t Hooft anomalies. In order to do that it is instructive to recall
that a $c=1$ compact boson at radius $\sqrt{2}$ corresponds to the
bosonization of two free Majorana fermions (equivalently a Dirac fermion).
From this perspective, the relevant deformations $\cos 2\phi$ and
$\cos\widetilde{\phi}$ are mapped to mass terms for the Majorana fermions.
More precisely, in terms of the two Majora components $\chi_{1,2}$, the
bosonization map reads

$\chi_{1}+i\chi_{2}=e^{iX}\quad,\quad\overline{\chi}_{1}+i\overline{\chi}_{2}=e^{i\overline{X}}$
(4.37)

hence, recalling that $2\phi=X+\overline{X}$ and
$\tilde{\phi}=X-\overline{X}$, one finds

$\cos
2\phi\sim\chi_{1}\overline{\chi}_{1}-\chi_{2}\overline{\chi}_{2}\quad,\quad\cos\widetilde{\phi}\sim\chi_{1}\overline{\chi}_{1}+\chi_{2}\overline{\chi}_{2}\,.$
(4.38)

These considerations, when applied to the theory at hand, imply

$\displaystyle\cos(\widetilde{\phi}_{1})+\cos(2\phi_{2})\quad\rightarrow\quad\chi_{1}\overline{\chi}_{1}+\chi_{2}\overline{\chi}_{2}+\chi_{3}\overline{\chi}_{3}-\chi_{4}\overline{\chi}_{4}$
(4.39)
$\displaystyle\cos(\widetilde{\phi}_{1})+\cos(2\phi_{2})+\cos(\widetilde{\phi}_{2})+\cos(2\phi_{1})\quad\rightarrow\quad\chi_{1}\overline{\chi}_{1}+\chi_{3}\overline{\chi}_{3}\,.$
(4.40)

Therefore the IR theory of the RG flow triggered by $R^{1}_{1/2}$ is the
bosonized version of a trivial theory, i.e. a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge theory, compatible with the anomaly
free $TY(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$ symmetry and an anomalous
$\mathbb{Z}_{2}$ invertible symmetry. On the other hand, the RG triggered by
the sum $R^{1}_{1/2}+R^{2}_{1/2}$ leads in the IR to two Majorana fermions in
the fermionized theory. Upon bosonizing each of them separately, we land on
the $c=1$ ${\rm Ising}^{2}$ theory. In the latter theory, the two preserved
non-invertible defects are naturally realized by the Kramers-Wannier duality
defect preserved by each factor.

###  4.2 The bi-critical point

Let us now delve into the more interesting case of the bi-critical point
$(\tau,\rho)=(\omega,\alpha)$ where $\omega=e^{2\pi i/3}$ and
$\alpha=-\frac{1}{2}+i3\frac{\sqrt{3}}{2}$. Note that this theory cannot be
regarded as a product of two $c=1$ theories. As discussed previously, we take
the theory $(\tau,\rho)=(\omega,\omega)$ with enahnced global symmetry
$SU(3)\times\overline{SU(3)}$ as the starting point for the analysis. In this
theory, the holomorphic decomposition of the fields takes the form

$\displaystyle\phi^{1}=\frac{1}{\sqrt{2}}\left(X^{1}+\overline{X}^{1}\right)+\frac{1}{\sqrt{6}}\left(X^{2}+\overline{X}^{2}\right)\;,\quad\phi^{2}=\sqrt{\frac{2}{3}}\left(X^{2}+\overline{X}^{2}\right)\;,$
(4.41)

and enjoys the shift and rotational $\mathbb{Z}_{3}$ symmetries determined by
the following action on the chiral fields

$\displaystyle{\mathbb{Z}}_{3}^{s}\;:$
$\displaystyle\quad\left(X^{1},\;X^{2}\right)\,\to\,\left(X^{1}+2\frac{\sqrt{2}}{3}\pi,\;X^{2}\right)\;,$
$\displaystyle\quad\left(\overline{X}^{1},\;\overline{X}^{2}\right)\,\to\,\left(\overline{X}^{1}+\frac{\sqrt{2}}{3}\pi,\;\overline{X}^{2}-\sqrt{\frac{2}{3}}\pi\right)\;,$
$\displaystyle{\mathbb{Z}}_{3}^{orb}\;:$
$\displaystyle\quad\left(X^{1},\;X^{2}\right)\,\to\,\left(\cos{\frac{2\pi}{3}}X^{1}+\sin{\frac{2\pi}{3}}X^{2},\;-\sin{\frac{2\pi}{3}}X^{1}+\cos{\frac{2\pi}{3}}X^{2}\right)\;,$
(4.42)
$\displaystyle\quad\left(\overline{X}^{1},\;\overline{X}^{2}\right)\,\to\,\left(\cos{\frac{2\pi}{3}}\overline{X}^{1}+\sin{\frac{2\pi}{3}}\overline{X}^{2},\;-\sin{\frac{2\pi}{3}}\overline{X}^{1}+\cos{\frac{2\pi}{3}}\overline{X}^{2}\right)\;.$

and similarly for the antiholomorphic fields. Again, these two
$\mathbb{Z}_{3}$ actions can be mapped to each other by means of global
$SU(3)\times\overline{SU(3)}$ rotations. Consequently, the corresponding
orbifolds lead to two equivalent descriptions of the same theory, leading to a
bi-critical point.

In order to move to the bi-critical point, we gauge the $\mathbb{Z}_{3}^{s}$
symmetry. This operation implements the following map of charges

$\displaystyle n^{\prime}_{1}=\frac{2}{3}n_{1}+\frac{1}{3}n_{2}\;,\quad
n^{\prime}_{2}=-\frac{1}{3}n_{1}+\frac{1}{3}n_{2}\;,\quad
w^{\prime}_{1}=w_{1}+w_{2}\;,\quad w^{\prime}_{2}=-w_{1}+2w_{2}\;$ (4.43)

and $(\tau,\rho)=(\omega,\omega)\to(\omega,\omega/3)$, subsequently performing
the $SL(2,\mathbb{Z})$ transformation $T_{\rho}^{-2}S_{\rho}$, finally landing
at the point $(\tau,\rho)=(\omega,\alpha)$. For these two operations together,
the overall map of the charges results

$\displaystyle n^{\prime}_{1}$
$\displaystyle=\frac{4}{3}n_{1}+\frac{2}{3}n_{2}+w_{1}-2w_{2}\;,\quad
n^{\prime}_{2}=-\frac{2}{3}n_{1}+\frac{2}{3}n_{2}+w_{1}+w_{2}\;,$
$\displaystyle w^{\prime}_{1}$
$\displaystyle=\frac{1}{3}n_{1}-\frac{1}{3}n_{2}\;,\quad
w^{\prime}_{2}=\frac{2}{3}n_{1}+\frac{1}{3}n_{2}\;.$ (4.44)

as it can be easily verified by conjugating the generalized metric with the
corresponding matrix313131Recall that, as described in section 3, for a map
implementing $({\bf n},{\bf w})\to(M^{-1})^{\intercal}({\bf n},{\bf w})$, the
generalized metric transforms accordingly as
$\mathcal{E}\to(M^{-1})^{\intercal}\mathcal{E}M^{-1}$..

####  4.2.1 Duality symmetry and fusion category

As shown in section 3, a generic RCFT can always be connected to a factorized
point via the composition of dualities and gaugings on each factor. In this
case, we choose to move to the factorized point at
$(\tau,\rho)=(i\sqrt{3},i3\sqrt{3})$ by means of the following naminpulation

$T_{\rho}T_{\tau}\sigma_{\widetilde{M}=2}\,:\,(\omega,\alpha)\,\to\,(i\sqrt{3},i3\sqrt{3})\,.$
(4.45)

Duality symmetries pertaining to the the bi-critical theory can be found by
composing the above map with the duality symmetries of the factorized point.
Indeed, following the procedure described in section 3, we find the following
defects and their associated matrices in $O(2,2,\mathbb{Q})$

$\begin{split}&\mathcal{D}_{1}\to\mathsf{D}_{1}=\begin{pmatrix}0&\frac{1}{3}&\frac{1}{3}&0\\\
0&1&0&0\\\ 3&-1&0&0\\\
-1&\frac{1}{3}&\frac{1}{3}&1\end{pmatrix}\quad,\quad\mathcal{D}_{2}\to\mathsf{D}_{2}=\frac{1}{9}\begin{pmatrix}-1&2&-4&-2\\\
-2&1&-2&-4\\\ -28&14&-1&-2\\\ 14&-28&2&1\end{pmatrix}\,.\end{split}$ (4.46)

Another useful point at which additional duality symmetries become more
manifest is the diagonal RCFT $(\tau=w,\rho=3w)$, which is connected to the
bi-critical point via $T_{\rho}^{-1}$

$T_{\rho}^{-1}\,:\,(w,\alpha)\,\to\,(w,3w)\,.$ (4.47)

At this point we have the duality symmetry coming from composing mirror
symmetry $M$ and a gauging of $\mathbb{Z}_{3}$ shift symmetry. However, this
can be checked to coincide with $\mathcal{D}_{1}$ when written in the bi-
critical point. In addition, there is a new duality symmetry coming from the
duality matrix $IMS_{\rho}S_{\tau}$. When mapped back to the bi-critical
point, the action of such a symmetry becomes implemented by the following
matrix

$\mathcal{D}_{3}\to\mathsf{D}_{3}=\begin{pmatrix}1&0&0&0\\\
\frac{1}{3}&0&0&-\frac{1}{3}\\\ -\frac{1}{3}&1&1&\frac{1}{3}\\\
1&-3&0&0\end{pmatrix}\in O(2,2,\mathbb{Q})\,,$ (4.48)

Accordingly, it can be easily verified that these three operations leave the
generalized metric invariant, that is

$(\mathsf{D}_{a}^{-1})^{\intercal}\mathcal{E}\mathsf{D}_{a}^{-1}=\mathcal{E}\quad,\quad
a=1,2,3\;.$ (4.49)

Note that we are omitting the description of the actions implemented by the
above defects in terms of a string of dualities and topological manipulations,
just because these are quite involved and barely illuminating.

We will focus on the three duality symmetries just described. Of course, we
will identify new ones when consider their fusion products below. Before
delving into those aspects, let us first identify the discrete gaugings
leading to the self-dualitites implied by these non-invertible defects. We do
that by following the arguments in Sec. 3.1. We can interpret
$\mathcal{D}_{1,2,3}$ as performing the gauging of the following subgroups of
the global symmetry

$\displaystyle(\mathcal{D}_{1})$
$\displaystyle\quad\mathbb{Z}_{3}^{(1)}\quad:\quad$ $\displaystyle V_{\bf
n,\bf w}\rightarrow e^{\frac{2\pi ik}{3}(n_{1}+w_{2})}V_{\bf n,\bf w}$
$\displaystyle(\mathcal{D}_{2})$
$\displaystyle\quad\mathbb{Z}_{3}^{(2)}\times\mathbb{Z}_{9}\quad:\quad$
$\displaystyle V_{\bf n,\bf w}\rightarrow e^{\frac{2\pi
ik_{1}}{9}(n_{1}-n_{2}+4(w_{1}+w_{2}))}e^{\frac{2\pi
ik_{2}}{3}(n_{2}-w_{1})}V_{\bf n,\bf w}\,,$ (4.50)
$\displaystyle(\mathcal{D}_{3})$
$\displaystyle\quad\mathbb{Z}_{3}^{(2)}\quad:\quad$ $\displaystyle V_{\bf
n,\bf w}\rightarrow e^{\frac{2\pi ik_{2}}{3}(n_{2}-w_{1})}V_{\bf n,\bf w}\,.$

These expressions should not be confused with the action of the corresponding
non-invertible defects on the vertex operators. The latter will be implemented
by the corresponding map of charges implied by the matrices in (4.46), (4.48),
together with the associated quantum dimensions and $\mathbb{Z}_{2}$ phases,
as discussed around (3.31). We will be more explicit about these action when
testing the defects against the modular bootstrap in the next subsection.

We can now discuss the fusion between those defects. As matrices acting on the
generalized metric, all $\mathcal{D}_{1,2,3}$ square to the identity matrix.
Therefore we get

$\mathcal{D}_{1}\times\mathcal{D}_{1}=\sum_{a\in\mathbb{Z}_{3}^{(1)}}\eta_{a}\quad,\quad\mathcal{D}_{2}\times\mathcal{D}_{2}=\sum_{b\in\mathbb{Z}_{3}^{(2)}\times\mathbb{Z}_{9}}\eta_{b}\quad,\quad\mathcal{D}_{3}\times\mathcal{D}_{3}=\sum_{c\in\mathbb{Z}_{3}^{(2)}}\eta_{c}\,,$
(4.51)

so that they separately generate a Tambara-Yamagami category with a
$\mathbb{Z}_{2}$ grading. Moreover fusion between different defects generates
new duality defects, arising from self-duality under gauging subgroups of
$\mathbb{Z}_{3}^{(1)}\times\mathbb{Z}_{3}^{(2)}\times\mathbb{Z}_{9}$. For
instance

$\mathcal{D}_{1}\times\mathcal{D}_{2}=\mathcal{C}_{0}^{(1,2)}\mathcal{D}_{4}$
(4.52)

where $\mathcal{D}_{4}$ is the duality defect constructed by gauging on half-
space the symmetry $\mathbb{Z}^{(1,2)D}_{3}\times\mathbb{Z}_{9}$, where

$\mathbb{Z}_{3}^{(1,2)D}\quad:\quad V_{\bf n,\bf w}\rightarrow e^{\frac{2\pi
ik}{3}(n_{1}-n_{2}+w_{1}+w_{2})}V_{\bf n,\bf w}\,.$ (4.53)

Comparing the symmetries on the l.h.s. and on the r.h.s. of the fusion (4.52)
we can derive the form of $\mathcal{C}_{0}^{(1,2)}$

$\mathcal{C}_{0}^{(1,2)}=\sum_{a\in\mathbb{Z}_{3}^{(1,2)AD}}\eta_{a}\qquad,\qquad\mathbb{Z}_{3}^{(1,2)AD}=\frac{\mathbb{Z}_{3}^{(1)}\times\mathbb{Z}_{3}^{(2)}}{\mathbb{Z}_{3}^{(1,2)D}}\,.$
(4.54)

Similarly one can compute the other fusion rules, like

$\begin{split}\mathcal{D}_{2}\times\mathcal{D}_{3}=\mathcal{D}_{5}\sum_{b\in\mathbb{Z}_{3}^{AD}}\eta_{b}\,,\end{split}$
(4.55)

where $\mathcal{D}_{5}$ comes from the half-gauging of

$\begin{split}\mathbb{Z}_{3}^{(2)}\times\mathbb{Z}_{9}\supset\mathbb{Z}_{9}^{D}\quad:\quad
V_{\bf n,\bf w}\rightarrow e^{\frac{2\pi
ik}{9}(-n_{1}+4n_{2}+2w_{1}+5w_{2})}V_{\bf n,\bf w}\,,\end{split}$ (4.56)

and

$\mathbb{Z}_{3}^{AD}=\frac{\mathbb{Z}_{3}^{(2)}\times\mathbb{Z}_{9}}{\mathbb{Z}_{9}^{D}}\,.$
(4.57)

#### Self-duality under gauging

The full set of non-invertible symmetries together imply that the RCFT at the
bi-critical point is self-dual under the gauging of

$\begin{split}\mathbb{Z}_{3}^{(1)}\quad&:\quad V_{\bf n,\bf w}\rightarrow
e^{\frac{2\pi ik}{3}(n_{1}+w_{2})}V_{\bf n,\bf w}\,,\\\
\mathbb{Z}_{3}^{(2)}\quad&:\quad V_{\bf n,\bf w}\rightarrow e^{\frac{2\pi
ik_{2}}{3}(n_{2}-w_{1})}V_{\bf n,\bf w}\,,\\\ \mathbb{Z}_{9}\quad&:\quad
V_{\bf n,\bf w}\rightarrow e^{\frac{2\pi
ik_{1}}{9}(n_{1}-n_{2}+4(w_{1}+w_{2}))}V_{\bf n,\bf w}\,.\end{split}$ (4.58)

By using the closed form of the torus partition function for RCFTs, we proceed
to explicitly check such self-dualities, thus confirming the above findings.

Following the general construction reviewed in App. B, the chiral algebra at
the bi-critical point can be written as

$u(1)^{2}_{K}\times u(1)^{2}_{K}\quad,\quad K=\left(\begin{array}[]{cc}6&-3\\\
-3&6\end{array}\right)\;.$ (4.59)

Furthermore, the partition function for this RCFT is given by a (diagonal)
modular invariant combination of the chiral algebra characters

$Z=\sum_{{\boldsymbol{\lambda}}\in{\cal
D}}\chi_{\boldsymbol{\lambda}}\overline{\chi}_{\overline{\boldsymbol{\lambda}}}\,,$
(4.60)

where

$\chi_{\boldsymbol{\lambda}}=\frac{1}{\eta(\tau)^{2}}\sum_{{\bf
l}\in{\mathbb{Z}}^{2}}q^{\frac{1}{2}|{\boldsymbol{\lambda}}+K{\bf
l}|^{2}_{K^{-1}}}\quad,\quad\overline{\chi}_{\overline{\boldsymbol{\lambda}}}=\frac{1}{\eta(\overline{\tau})^{2}}\sum_{{\bf
r}\in{\mathbb{Z}}^{2}}\overline{q}^{\frac{1}{2}|{\overline{\boldsymbol{\lambda}}}+K{\bf
r}|^{2}_{K^{-1}}}\,.$ (4.61)

The vector ${\boldsymbol{\lambda}}$ labelling each representation belongs to
the lattice

${\cal
L}\equiv\left\\{{\boldsymbol{\lambda}}\in{\mathbb{Z}}^{2}\,,\,{\boldsymbol{\lambda}}\sim{\boldsymbol{\lambda}}+K{\bf
v}\,,\,{\bf v}\in{\mathbb{Z}}^{2}\right\\}\,.$ (4.62)

The fact that the theory corresponds to a diagonal RCFT becomes evident by
noticing that there is a duality frame in which $\rho=3\tau$ (see equation
(4.47)).

It turns to be useful to rewrite the action of the ${\mathbb{Z}}_{3}^{(1)}$
over the elements comprised in the different characters. This can be done
straightforwardly by using the expressions (B.14) to write the character
labels in terms of the global charges. More precisely

${\mathbb{Z}}_{3}^{(1)}\,:\,\,\chi_{\boldsymbol{\lambda}}\overline{\chi}_{\overline{\boldsymbol{\lambda}}}\,\to\,e^{i\frac{2\pi}{3}(\lambda_{1}+\overline{\lambda}_{1})}\chi_{\boldsymbol{\lambda}}\overline{\chi}_{\overline{\boldsymbol{\lambda}}}=e^{2\pi
i\left({\boldsymbol{\lambda}}+\overline{\boldsymbol{\lambda}}\right)^{T}K^{-1}{\bf
v}_{3}}\chi_{\boldsymbol{\lambda}}\overline{\chi}_{\overline{\boldsymbol{\lambda}}}\,,$
(4.63)

with the generator of the ${\mathbb{Z}}_{3}$ sublattice ${\bf
v}_{3}\equiv(2,-1)^{T}$. Armed with these tools, we can write the partition
function twisted by $(\eta^{a},\eta^{b})$ along the temporal and spatial
cycles respectively, where $\eta$ denotes the generator of
${\mathbb{Z}}_{3}^{(1)}$

$Z_{(\eta^{a},\eta^{b})}=\frac{1}{|{\cal
L}|}\sum_{{\boldsymbol{\lambda}},{\boldsymbol{\mu}},\overline{{\boldsymbol{\mu}}}\in{\cal
L}}e^{i4\pi a{\boldsymbol{\lambda}}^{T}K^{-1}{\bf v_{3}}}e^{i2\pi
b({\boldsymbol{\mu}}+\overline{\boldsymbol{\mu}})^{T}K^{-1}{\bf
v_{3}}}e^{-i2\pi{\boldsymbol{\lambda}}^{T}K^{-1}({\boldsymbol{\mu}}-\overline{\boldsymbol{\mu}})}\chi_{{\boldsymbol{\mu}}}\overline{\chi}_{\overline{\boldsymbol{\mu}}}\;.$
(4.64)

Summing over ${\boldsymbol{\lambda}}$ and using the orthogonality relation
(B.12) leads to

$Z_{(\eta^{a},\eta^{b})}=\sum_{{\boldsymbol{\mu}}\in{\cal L}}e^{i4\pi
b({\boldsymbol{\mu}}-a{\bf v_{3}})^{T}K^{-1}{\bf
v_{3}}}\chi_{{\boldsymbol{\mu}}}\overline{\chi}_{{\boldsymbol{\mu}}-2a{\bf
v_{3}}}\;.$ (4.65)

The partition function of the gauged theory then obtains

$\frac{1}{3}\sum_{a,b\in{\mathbb{Z}}_{3}}Z_{(\eta^{a},\eta^{b})}\;.$ (4.66)

Now, the sum over $b$ imposes the constraint $a=2\mu_{1}$ mod 3, leading to

$\frac{1}{3}\sum_{a,b\in{\mathbb{Z}}_{3}}Z_{(\eta^{a},\eta^{b})}=\sum_{{\boldsymbol{\mu}}\in{\cal
L}}\chi_{{\boldsymbol{\mu}}}\overline{\chi}_{C{\boldsymbol{\mu}}}=\sum_{{\boldsymbol{\mu}}\in{\cal
L}}\chi_{{\boldsymbol{\mu}}}\overline{\chi}_{{\boldsymbol{\mu}}}\quad,\quad
C=\left(\begin{array}[]{cc}-1&0\\\ 1&1\end{array}\right)\,,$ (4.67)

hence establishing the self-duality of the theory under gauging
${\mathbb{Z}}_{3}^{(1)}$. In the last equality, we used that the matrix $C$
preserves the lattice pairing, namely

$C^{T}K^{-1}C=K^{-1}\,.$ (4.68)

A completely analogous computation shows that the theory at
$(\tau,\rho)=(\omega,\alpha)$ is self-dual upon gauging
${\mathbb{Z}}_{3}^{(2)}$ in (4.58).

For sake of completeness, let us briefly comment on the computation that shows
the self-duality of the theory under gauging the ${\mathbb{Z}}_{9}$ in (4.58).
Its action on the characters is

${\mathbb{Z}}_{9}\,:\,\,\chi_{\boldsymbol{\lambda}}\overline{\chi}_{\overline{\boldsymbol{\lambda}}}\,\to\,e^{i\frac{2\pi}{9}(2\lambda_{1}+\lambda_{2}-\overline{\lambda}_{1}-2\overline{\lambda}_{2})}\chi_{\boldsymbol{\lambda}}\overline{\chi}_{\overline{\boldsymbol{\lambda}}}=e^{2\pi
i\left({\boldsymbol{\lambda}}^{T}K^{-1}{\bf
v}_{9}-\overline{\boldsymbol{\lambda}}^{T}K^{-1}\overline{\bf
v}_{9}\right)}\chi_{\boldsymbol{\lambda}}\overline{\chi}_{\overline{\boldsymbol{\lambda}}}\,,$
(4.69)

with ${\bf v}_{9}\equiv(1,0)^{T}$ and $\overline{\bf v}_{9}\equiv(0,1)^{T}$.
Therefore the twisted partition function formally reads

$Z_{(\eta^{a},\eta^{b})}=\frac{1}{|{\mathcal{L}}|}\sum_{{\boldsymbol{\lambda}},{\boldsymbol{\mu}},\overline{{\boldsymbol{\mu}}}\in{\mathcal{L}}}e^{i2\pi
a{\boldsymbol{\lambda}}^{T}K^{-1}({\bf v}_{9}-\overline{\bf v}_{9})}e^{i2\pi
b\left({\boldsymbol{\mu}}^{T}K^{-1}{\bf
v}_{9}-\overline{\boldsymbol{\mu}}^{T}K^{-1}\overline{\bf
v}_{9}\right)}e^{-i2\pi{\boldsymbol{\lambda}}^{T}K^{-1}({\boldsymbol{\mu}}-\overline{\boldsymbol{\mu}})}\chi_{{\boldsymbol{\mu}}}\overline{\chi}_{\overline{\boldsymbol{\mu}}}\,.$
(4.70)

The sum over ${\boldsymbol{\lambda}}$ now sets
$\overline{\boldsymbol{\mu}}={\boldsymbol{\mu}}-a({\bf v}_{9}-\overline{\bf
v}_{9})$. Finally, taking the orbifold and summing over $b\in{\mathbb{Z}}_{9}$
fixes $a=\mu_{1}-\mu_{2}$ mod 9 hence proving the claimed self-duality

$\frac{1}{9}\sum_{a,b\in{\mathbb{Z}}_{3}}Z_{(\eta^{a},\eta^{b})}=\sum_{{\boldsymbol{\mu}}\in{\mathcal{L}}}\chi_{{\boldsymbol{\mu}}}\overline{\chi}_{C^{\prime}{\boldsymbol{\mu}}}=\sum_{{\boldsymbol{\mu}}\in{\mathcal{L}}}\chi_{{\boldsymbol{\mu}}}\overline{\chi}_{{\boldsymbol{\mu}}}\quad,\quad
C^{\prime}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)\,.$ (4.71)

where we made use of the fact that the matrix $C^{\prime}$ also preserves the
lattice pairing.

####  4.2.2 Modular bootstrap at the bi-critical point

As with the previous example, we now proceed to run the modular bootstrap
analysis to confirm that the symmetry operations introduced above define
consistent topological defects.

Let us begin with ${\cal D}_{1}$ which, by plugging its matrix representation
$\mathsf{D}_{1}$ into (3.32), leads to the following map of the $U(1)$ charges

$n_{1}\to
3w_{1}-w_{2}\,\,,\,\,n_{2}\to\frac{n_{1}}{3}+n_{2}-w_{1}+\frac{w_{2}}{3}\,\,,\,\,w_{1}\to\frac{n_{1}+w_{2}}{3}\,\,,\,\,w_{2}\to
w_{2}\,.$ (4.72)

Before continuing, let us determine the phase associated to the action of
${\cal D}_{1}$ on vertex operators. As explained in Appendix A, mutual
locality of correlation functions under the action (4.72) demands

$\displaystyle\alpha_{1}({\bf n}+{\bf n}^{\prime},{\bf w}+{\bf w}^{\prime})+$
$\displaystyle\alpha_{1}({\bf n},{\bf w})+\alpha_{1}({\bf n}^{\prime},{\bf
w}^{\prime})=$ (4.73)
$\displaystyle\frac{n_{1}+w_{2}}{3}(3w^{\prime}_{1}-w^{\prime}_{2})+\frac{n^{\prime}_{1}+w^{\prime}_{2}}{3}(3w_{1}-w_{2})-w_{1}w_{2}^{\prime}-w_{1}^{\prime}w_{2}\,\,{\rm
mod}\,\,2\,.$

Note that, in order to recast the phase to the above form, we imposed
$n_{1}+w_{2}\in 3{\mathbb{Z}}$ and $n^{\prime}_{1}+w^{\prime}_{2}\in
3{\mathbb{Z}}$, hence rendering the resulting vertex operators genuine. As
such, there are multiple solutions, each of them differing by stacking ${\cal
D}_{1}$ with a particular ${\mathbb{Z}}_{2}$ subgroup of the invertible
symmetry. For a reason that will be clear momentarily, we choose the following
solution

$\alpha_{1}({\bf n},{\bf
w})=\frac{n_{1}+w_{2}}{3}(3w_{1}-w_{2})-w_{1}(w_{1}+w_{2})$ (4.74)

Having fixed the phase in the action of the ${\cal D}_{1}$ defect, we proceed
to run the modular bootstrap. The map (4.72) translates in the momentum basis
to

$(p_{1},p_{2},\overline{p}_{1},\overline{p}_{2})\,\to\,(p_{1},p_{2},-\overline{p}_{1},\overline{p}_{2})$
(4.75)

hence implying that only states with $\overline{p}_{1}=0$ contribute to the
twisted partition function $Z_{(1,{\cal D}_{1})}$. In particular, this implies
$n_{1}=3w_{1}-w_{2}$. Moreover, (4.74) trivializes in ${\mathbb{Z}}_{2}$ for
these values of the charges, which is the main reason for choosing this
particular solution. In terms of the chiral algebra characters, and using the
conversion formulas (B.14), the states contributing to the twisted partition
function have ${\boldsymbol{\lambda}}=(0,\lambda_{2})^{T}$ and ${\bf
r}=(r,2r)^{T}$ with $r\in{\mathbb{Z}}$. The basis of independent right moving
characters then reads

${\boldsymbol{\lambda}}+K\cdot{\bf
r}=\left(0\,,\,\lambda_{2}+9r\right)^{T}\,.$ (4.76)

Therefore, the independent values of $\lambda_{2}$ span a ${\mathbb{Z}}_{9}$.
Turning to the holomorphic blocks, the scaling dimensions are given by (${\bf
l}=(l_{1},l_{2})^{T}$)

$\frac{1}{2}({\boldsymbol{\lambda}}+K{\bf
l})^{T}K^{-1}({\boldsymbol{\lambda}}+K{\bf
l})=\frac{3}{4}\left(l_{+}+\frac{\lambda_{2}}{3}\right)^{2}+\frac{9}{4}\left(l_{-}+\frac{\lambda_{2}}{9}\right)^{2}\;.$
(4.77)

where we diagonalized the matrix $K$ in terms of $l_{+}\equiv l_{1}+l_{2}$ and
$l_{-}\equiv-l_{1}+l_{2}$. Consistently with the fact that the theory at the
bi-critical point $(\tau,\rho)=(\omega,\alpha)$ does not factorize, the
$l_{\pm}$ variables are not free within ${\mathbb{Z}}$. In particular, they
must satisfy either $(l_{+},l_{-})=(2l,2l^{\prime})$ or
$(l_{+},l_{-})=(2l+1,2l^{\prime}+1)$ for $l,l^{\prime}\in{\mathbb{Z}}$
323232This can be easily verified by noticing that
$l_{1}=\frac{l_{+}-l_{-}}{2}$ and $l_{2}=\frac{l_{+}-l_{-}}{2}$ have integer
solutions only for $l_{\pm}=$even or $l_{\pm}=$odd. . We then need to sum over
both the even and odd sectors. In addition, states with an odd number of
right-moving oscillator modes come with a $(-1)$ phase, as usual. Summing over
oscillator modes and plugging the solution described above, the twisted
partition function results

$\displaystyle Z_{(1,{\cal D}_{1})}$
$\displaystyle=\frac{\sqrt{3}}{|\eta(\tau)|^{4}}\sum_{\lambda_{2}\in{\mathbb{Z}}_{9}}\left\\{\left(\sum_{l\in{\mathbb{Z}}}q^{3\left(l+\frac{\lambda_{2}}{6}\right)^{2}}\sum_{l^{\prime}\in{\mathbb{Z}}}q^{9\left(l^{\prime}+\frac{\lambda_{2}}{18}\right)^{2}}+\sum_{l\in{\mathbb{Z}}}q^{3\left(l+\frac{\lambda_{2}+3}{6}\right)^{2}}\sum_{l^{\prime}\in{\mathbb{Z}}}q^{9\left(l^{\prime}+\frac{\lambda_{2}+9}{18}\right)^{2}}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.\times\sum_{r}\overline{q}^{9\left(r+\frac{\lambda_{2}}{9}\right)^{2}}\right\\}\vartheta_{4}(2\overline{\tau})\;,$
(4.78)
$\displaystyle=\frac{\sqrt{3}}{|\eta(\tau)|^{4}}\sum_{\lambda_{2}\in{\mathbb{Z}}_{9}}\left\\{\left(\vartheta\left[{\tfrac{\lambda_{2}}{6}\atop
0}\right](6\tau)\,\vartheta\left[{\tfrac{\lambda_{2}}{18}\atop
0}\right](18\tau)+\vartheta\left[{\tfrac{\lambda_{2}+3}{6}\atop
0}\right](6\tau)\,\vartheta\left[{\tfrac{\lambda_{2}+9}{18}\atop
0}\right](18\tau)\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.\times\vartheta\left[{\tfrac{\lambda_{2}}{9}\atop
0}\right](18\overline{\tau})\right\\}\vartheta_{4}(2\overline{\tau})\;.$
(4.79)

Performing the modular $S$ transformation $\tau\to-1/\tau$ and using the
transformation properties listed in appendix D one gets

$\displaystyle Z_{({\cal D}_{1},1)}$
$\displaystyle=\frac{1}{36}\frac{1}{|\eta(\tau)|^{4}}\sum_{\lambda_{2}\in{\mathbb{Z}}_{9}}\left\\{\left(\vartheta\left[{0\atop-\tfrac{\lambda_{2}}{6}}\right]\left(\tfrac{\tau}{6}\right)\,\vartheta\left[{0\atop-\tfrac{\lambda_{2}}{18}}\right]\left(\tfrac{\tau}{18}\right)+\vartheta\left[{0\atop-\tfrac{\lambda_{2}+3}{6}}\right]\left(\tfrac{\tau}{6}\right)\,\vartheta\left[{0\atop-\tfrac{\lambda_{2}+9}{18}}\right]\left(\tfrac{\tau}{18}\right)\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.\times\vartheta\left[{0\atop-\tfrac{\lambda_{2}}{9}}\right]\left(\tfrac{\overline{\tau}}{18}\right)\right\\}\vartheta_{2}\left(\tfrac{\overline{\tau}}{2}\right)\;,$
(4.80)
$\displaystyle\approx\frac{1}{|\eta(\tau)|^{4}}\left(\overline{q}^{\frac{1}{16}}+4q^{\frac{1}{9}}\overline{q}^{\frac{13}{144}}+2q^{\frac{1}{9}}\overline{q}^{\frac{25}{144}}+2q^{\frac{1}{3}}\overline{q}^{\frac{1}{16}}+\ldots\right)\;.$

In going to the last line, we performed the sum over $\lambda_{2}$. As it
becomes evident from the first orders in the expansion above, the defect
Hilbert space has a consistent expansion in terms of Virasoro characters.

We now proceed to bootstrap the defect ${\cal D}_{2}$. We first notice that
the right hand side of the equation (A.9) determining the phase $\alpha_{2}$
is already in $2{\mathbb{Z}}$ once restricted to genuine operators333333More
precisely, ${\cal D}_{2}$ implements an orbifold by the
${\mathbb{Z}}_{3}^{(1)}\times{\mathbb{Z}}_{9}$ symmetry in (4.58). Hence, by
restricting to genuine local operators we mean  $n_{1}-n_{2}+4(w_{1}+w_{2})\in
9{\mathbb{Z}}\quad,\quad n_{2}-w_{1}\in 3{\mathbb{Z}}\,.$ . We can therefore
set $\alpha_{2}({\bf n},{\bf w})=0$. In addition, the action of ${\cal D}_{2}$
flips the sign of both $p_{1}$ and $p_{2}$. The subset of states contributing
to $Z_{(1,{\cal D}_{2})}$ thus have $p_{1}=p_{2}=0$, which in terms of the
characters reads $\lambda_{1}=\lambda_{2}=l_{1}=l_{2}=0$. By manipulations
analogous to the ones described above, one obtains

$\displaystyle Z_{(1,{\cal D}_{2})}$
$\displaystyle=\frac{\sqrt{27}}{|\eta(\tau)|^{4}}\vartheta_{4}(2\tau)^{2}\left(\vartheta_{3}(6\overline{\tau})\vartheta_{3}(18\overline{\tau})+\vartheta_{2}(6\overline{\tau})\vartheta_{2}(18\overline{\tau})\right)$
$\displaystyle\xrightarrow{\tau\to-\frac{1}{\tau}}Z_{({\cal D}_{2},1)}$
$\displaystyle=\frac{1}{4|\eta(\tau)|^{4}}\vartheta_{2}\left(\frac{\tau}{2}\right)^{2}\left(\vartheta_{3}\left(\frac{\overline{\tau}}{6}\right)\vartheta_{3}\left(\frac{\overline{\tau}}{18}\right)+\vartheta_{4}\left(\frac{\overline{\tau}}{6}\right)\vartheta_{4}\left(\frac{\overline{\tau}}{18}\right)\right)$
$\displaystyle\approx\frac{1}{|\eta(\tau)|^{4}}\left(2q^{\frac{1}{8}}+12q^{\frac{1}{8}}\overline{q}^{\frac{1}{9}}+12q^{\frac{1}{8}}\overline{q}^{\frac{1}{3}}+\ldots\right)\,.$
(4.81)

As a final example, consider the defect ${\cal D}_{5}$ which, in particular,
implements the orbifold by the ${\mathbb{Z}}_{9}$ gauging in (4.58). For the
phase required by mutual locality, we choose the following solution

$\alpha_{5}({\bf n},{\bf w})=n_{2}(w_{1}+w_{2})+w_{1}^{2}+w_{2}^{2}$ (4.82)

which turns out to be trivial when evaluated on the states contributing to the
twisted partition function (recall that all possible solutions differ just by
stacking with an invertible ${\mathbb{Z}}_{2}$ symmetry defect). The action of
${\cal D}_{5}$ amounts to the following

${\bf p}\to R\cdot{\bf p}\quad,\quad
R=-\frac{1}{2}\left(\begin{array}[]{cc}1&\sqrt{3}\\\
\sqrt{3}&-1\end{array}\right)$ (4.83)

Note that $R^{2}=1$, hence defining a ${\mathbb{Z}}_{2}$ action in momentum
space (equivalently, ${\cal D}_{5}:{\bf X}\to R\cdot{\bf X}$). In terms of the
characters, the states contributing to the trace have
$\lambda_{2}=-2\lambda_{1}$ $l_{1}=0$ and their holomorphic scaling dimension
read

$\frac{1}{2}\left|({\boldsymbol{\lambda}+K{\bf
l}})\right|^{2}_{K^{-1}}\,\,\xrightarrow[l_{1}=0\,,\,l_{2}=l]{\lambda_{2}=-2\lambda_{1}}\,\,3\left(l-\frac{\lambda_{1}}{3}\right)^{2}\;.$
(4.84)

It becomes evident from the expression above that the independent
representations contributing to the trace are labelled by $\lambda_{1}$ taking
values in ${\mathbb{Z}}_{3}$. On the other hand, the right moving sector can
be dealt with by means of analogous manipulations as in (4.77).

In order to perform the sum over oscillator modes, we notice that the matrix
$R$ is no more than a rotation in field space, followed by flip of one of the
momenta. This can be verified by performing the following change of basis on
the holomorphic fields

${\bf X}\,\to\,{\bf X}^{\prime}=\mathcal{B}\,{\bf
X}\quad,\quad\mathcal{B}=\left(\begin{array}[]{cc}\tfrac{\sqrt{3}}{2}&-\tfrac{1}{2}\\\
\frac{1}{2}&\tfrac{\sqrt{3}}{2}\end{array}\right)$ (4.85)

for which the action of $R$ reduces to $X_{1}^{\prime}\to-X_{1}^{\prime}$.
Hence, by going to the appropriate basis, the summand acquire the usual sign
factor depending on one of the oscillator numbers. Putting all together we end
up with

$\displaystyle Z_{(1,{\cal D}_{5})}$
$\displaystyle=\frac{\sqrt{9}}{|\eta(\tau)|^{4}}\sum_{\lambda_{1}\in{\mathbb{Z}}_{3}}\left\\{\left(\vartheta\left[{-\tfrac{\lambda_{1}}{6}\atop
0}\right](6\overline{\tau})\,\vartheta\left[{-\tfrac{\lambda_{1}}{6}\atop
0}\right](18\overline{\tau})+\vartheta\left[{\tfrac{-\lambda_{1}+3}{6}\atop
0}\right](6\overline{\tau})\,\vartheta\left[{\tfrac{-\lambda_{1}+3}{6}\atop
0}\right](18\overline{\tau})\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.\times\vartheta\left[{-\tfrac{\lambda_{1}}{3}\atop
0}\right](6\tau)\right\\}\vartheta_{4}(2\tau)$ (4.86)

Finally, upon performing a modular $S$ transformation we obtain the trace over
the twisted Hilbert space

$\displaystyle Z_{(1,{\cal D}_{5})}$
$\displaystyle=\frac{1}{12|\eta(\tau)|^{4}}\sum_{\lambda_{1}\in{\mathbb{Z}}_{3}}\left\\{\left(\vartheta\left[{0\atop\tfrac{\lambda_{1}}{6}}\right]\left(\frac{\overline{\tau}}{6}\right)\,\vartheta\left[{0\atop\tfrac{\lambda_{1}}{6}}\right]\left(\frac{\overline{\tau}}{18}\right)+\vartheta\left[{0\atop\tfrac{\lambda_{1}-3}{6}}\right]\left(\frac{\overline{\tau}}{6}\right)\,\vartheta\left[{0\atop\tfrac{\lambda_{1}-3}{6}}\right]\left(\frac{\overline{\tau}}{18}\right)\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.\times\vartheta\left[{0\atop-\tfrac{-\lambda_{1}}{3}}\right]\left(\frac{\tau}{6}\right)\right\\}\vartheta_{2}\left(\frac{\tau}{2}\right)$
(4.87)
$\displaystyle\approx\frac{1}{|\eta(\tau)|^{4}}\left(q^{\frac{1}{16}}+2q^{\frac{1}{16}}\overline{q}^{\frac{1}{9}}+4q^{\frac{7}{48}}\overline{q}^{\frac{1}{9}}+6q^{\frac{7}{48}}\overline{q}^{\frac{1}{3}}+\ldots\right)\,.$

#### Categorical Data

As explained in [20] (see also appendix C) the computation of the twisted
Hilbert space enables to determine the extra categorical data of the Tambara-
Yamagami categorical symmetry, i.e. the non-degenerate bi-character
$\chi(a,b)$ and the Frobenius-Shur indicator $\epsilon$. As in the analysis of
the modular bootstrap, we focus on the duality defects $\mathcal{D}_{1,2}$ and
$\mathcal{D}_{5}$. The data corresponding to the remaining defects can be
derived in a similar manner.

Following the analysis reviewed in appendix C, the F-symbols of the
categorical symmetry imply selection rules for the spins of the states in the
defect Hilbert space. In particular we get

$e^{4\pi is}|\psi\rangle=\frac{\epsilon}{\sqrt{\mathbb{A}}}\sum\limits_{a\in
G}\widehat{\eta}_{a}|\psi\rangle\,,$ (4.88)

where $\widehat{\eta}_{a}$ are the invertible symmetry operators generating
$G\subset U(1)_{\bf n}^{2}\times U(1)_{\bf w}^{2}$ acting on the Hilbert space
twisted by $\mathcal{D}$, with a particular resolution of the $4$-valent
junction (see the discussion around Eq. (C.12))343434The two possible
resolutions are related by the bi-character. and $|\psi\rangle$ are
eigenstates of $\widehat{\eta}_{a}$ in such twisted Hilbert space. By
performing F-moves, it is straightforward to check that

$\widehat{\eta}_{a}\widehat{\eta}_{b}=\chi(a,b)\widehat{\eta_{ab}}$ (4.89)

so that the group generated by $\widehat{\eta}_{a}$ is generically an
extension of $G$. Let us now start by applying these constraints on
$\mathcal{D}_{1}$. In this case the abelian group $G=\mathbb{Z}_{3}^{(1)}$ and
the possible bi-characters are

$\chi_{r}(a,b)=e^{\frac{2\pi i}{3}rab}\qquad r=\pm 1\,.$ (4.90)

Therefore we get

$e^{4\pi is}=\epsilon e^{\pm\frac{i\pi}{3}n^{2}\mp\frac{\pi i}{4}}\qquad r=\pm
1\,,$ (4.91)

which is compatible with the expansion (4.2.2) only for the values
$r=1,\epsilon=1$.

In the case of $\mathcal{D}_{5}$, the abelian group is $\mathbb{Z}_{9}$ and
the possible non-degenerate bi-characters are

$\chi_{r}(a,b)=e^{\frac{2\pi i}{9}rab}\qquad\text{gcd}(9,r)=1\,,$ (4.92)

which imply

$e^{4\pi is}=\epsilon e^{r\frac{\pi in^{2}}{9}}e^{-r\frac{\pi i}{4}}\,.$
(4.93)

Such spin selection rules are compatible with (4.2.2) if $\epsilon=1,r=-1$.

The combined duality defect $\mathcal{D}_{2}$, coming from the self-duality
under $\mathbb{Z}_{3}\times\mathbb{Z}_{9}$ gauging, will therefore have a
diagonal bi-character

$\chi(a,b)=\exp{\left[{2\pi
i\begin{pmatrix}a_{1}&a_{2}\end{pmatrix}\chi\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}}\right]}\quad,\quad\chi=\begin{pmatrix}1/3&0\\\
0&-1/9\end{pmatrix}\,,$ (4.94)

where $\mathbf{a},\mathbf{b}\in\mathbb{Z}_{3}\times\mathbb{Z}_{9}$ and trivial
Frobenius-Shur indicator.

####  4.2.3 Marginal and relevant deformations at the bi-critical point

As in the easier case of the quadri-critical point, in order to construct the
marginal deformations spanning the orbifold branch in the bi-critical point,
it is instructive to analyze symmetries and marginal operators of the $SU(3)$
enhanced symmetry point $(w,w)$. Following the discussion done in appendix
A.2, we find the following holomorphic currents

$\displaystyle I_{\pm}$ $\displaystyle=\pm V_{\pm 1,\mp 1,\pm 1,0}=\pm e^{\pm
i\sqrt{2}X^{1}}\;,\quad\overline{I}_{\pm}=\pm V_{\pm 1,0,\mp 1,0}=\pm e^{\pm
i\sqrt{2}\,\overline{X}^{1}}$ $\displaystyle K_{\pm}$ $\displaystyle=\mp
V_{\pm 1,0,\pm 1,\pm 1}=\mp e^{\pm i\frac{1}{\sqrt{2}}X^{1}\pm
i\sqrt{\frac{3}{2}}X^{2}}\;,\quad\overline{K}_{\pm}=\mp V_{0,\pm 1,\mp 1,\mp
1}=\mp e^{\pm i\frac{1}{\sqrt{2}}\overline{X}^{1}\pm
i\sqrt{\frac{3}{2}}\overline{X}^{2}}\;,$ $\displaystyle U_{\pm}$
$\displaystyle=\pm V_{0,\pm 1,0,\pm 1}=\pm e^{\mp i\frac{1}{\sqrt{2}}X^{1}\pm
i\sqrt{\frac{3}{2}}X^{2}}\;,\quad\overline{U}_{\pm}=\pm V_{\mp 1,\pm 1,0,\mp
1}=\pm e^{\mp i\frac{1}{\sqrt{2}}\overline{X}^{1}\pm
i\sqrt{\frac{3}{2}}\overline{X}^{2}}\;,$ $\displaystyle J^{3}$
$\displaystyle=i\sqrt{2}\partial
X^{1}\;,\quad\overline{J}^{3}=i\sqrt{2}\,\overline{\partial}\,\overline{X}^{1}$
$\displaystyle J^{8}$ $\displaystyle=i\sqrt{2}\partial
X^{2}\;,\quad\overline{J}^{8}=i\sqrt{2}\,\overline{\partial}\,\overline{X}^{2}\;,$
(4.95)

which span the $su(3)$ algebra, along with a similar set of antiholomorphic
currents for $\overline{su(3)}$. The two relevant $\mathbb{Z}_{3}$ symmetries
are [55]:

$\displaystyle\mathbb{Z}_{3}^{s}\;:$ $\displaystyle\qquad
I_{\pm}\;\to\;\omega^{\pm 2}I_{\pm}\;,\quad K_{\pm}\;\to\;\omega^{\pm
1}K_{\pm}\;,\quad U_{\pm}\;\to\;\omega^{\pm 2}U_{\pm}\;,$
$\displaystyle\mathbb{Z}_{3}^{orb}\;:$ $\displaystyle\qquad
I_{\pm}\;\to\;U_{\pm}\;,\quad K_{\pm}\;\to\;I_{\mp}\;,\quad
U_{\pm}\;\to\;K_{\mp}\;.$ (4.96)

At this point, the various marginal operators are all related to
$\\{J_{3}\overline{J}_{3},J_{3}\overline{J}_{8},J_{8}\overline{J}_{3},J_{8}\overline{J}_{8}\\}$
by means of the enhanced $su(3)$ chiral algebra, and the invariant
combinations under both $\mathbb{Z}_{3}^{s}$ and $\mathbb{Z}_{3}^{orb}$ are

$\displaystyle\mathcal{O}_{1}$
$\displaystyle=\mathcal{O}_{A}+\mathcal{O}_{B}+\mathcal{O}_{C}\;,$
$\displaystyle\mathcal{O}_{2}$
$\displaystyle=\mathcal{O}_{A\omega}+\mathcal{O}_{B\omega}+\mathcal{O}_{C\omega}\;$
(4.97)

where

$\displaystyle\mathcal{O}_{A}$
$\displaystyle=I_{+}\overline{I}_{+}+U_{+}\overline{U}_{+}+K_{-}\overline{K}_{-}+h.c.=V_{2,-1,0,0}+V_{-1,2,0,0}+V_{1,1,0,0}+h.c.\;,$
$\displaystyle\mathcal{O}_{B}$
$\displaystyle=I_{+}\overline{K}_{-}+U_{+}\overline{I}_{+}+K_{-}\overline{U}_{+}+h.c.=V_{1,-2,2,1}+V_{1,1,-1,1}+V_{2,-1,1,2}+h.c.\;,$
$\displaystyle\mathcal{O}_{C}$
$\displaystyle=I_{+}\overline{U}_{+}+U_{+}\overline{K}_{-}+K_{-}\overline{I}_{+}+h.c.=V_{0,0,1,-1}+V_{0,0,1,2}+V_{2,0,0,1}+h.c.\;,$
$\displaystyle\mathcal{O}_{A\omega}$
$\displaystyle=\omega^{2}I_{+}\overline{I}_{+}+\omega
U_{+}\overline{U}_{+}+K_{-}\overline{K}_{-}+h.c.\;,$
$\displaystyle\mathcal{O}_{B\omega}$ $\displaystyle=\omega
I_{+}\overline{K}_{-}+U_{+}\overline{I}_{+}+\omega^{2}K_{-}\overline{U}_{+}+h.c.\;,$
$\displaystyle\mathcal{O}_{C\omega}$
$\displaystyle=I_{+}\overline{U}_{+}+\omega^{2}U_{+}\overline{K}_{-}+\omega
K_{-}\overline{I}_{+}+h.c.\;.$ (4.98)

Finally, at the bi-critical point this set of operators is mapped to

$\displaystyle\mathcal{O^{\prime}}_{A}$
$\displaystyle=V_{2,-2,1,1}+V_{0,2,-1,0}+V_{2,0,0,1}+h.c.\;,$
$\displaystyle\mathcal{O^{\prime}}_{B}$
$\displaystyle=V_{0,1,1,0}+V_{1,0,0,-1}+V_{-1,1,1,1}+h.c.\;,$
$\displaystyle\mathcal{O^{\prime}}_{C}$
$\displaystyle=V_{3,0,0,0}+V_{3,-3,0,0}+V_{0,3,0,0}+h.c.\;,$ (4.99)

and similarly for
$\mathcal{O^{\prime}}_{A\omega},\mathcal{O^{\prime}}_{B\omega},\mathcal{O^{\prime}}_{C\omega}$.
Therefore we have the two exactly marginal operators
$\mathcal{O^{\prime}}_{1},\mathcal{O^{\prime}}_{2}$ that span the
$\mathbb{Z}_{3}^{orb}$ branch. However in this case all the non-invertible
duality symmetries are broken by all these marginal deformations.

#### Relevant deformations

The relevant operators at the bi-critical point are the following:

$(h,\overline{h})$ |  $(\mathbf{n},\mathbf{w})$
---|---
$\left(\frac{1}{9}\,,\,\frac{1}{9}\right)$ |  $(\pm 1,0,0,0)\;,\;(0,\pm 1,0,0)\;,\;(\pm 1,\mp 1,0,0)$
$\left(\frac{1}{3}\,,\,\frac{1}{3}\right)$ |  $(\pm 1,\pm 1,0,0)\;,\;(\pm 2,\mp 1,0,0)\;,\;(\pm 1,\mp 2,0,0)$
$\left(\frac{4}{9}\,,\,\frac{4}{9}\right)$ |  $(\pm 2,0,0,0)\;,\;(0,\pm 2,0,0)\;,\;(\pm 2,\mp 2,0,0)$
$\left(\frac{7}{9}\,,\,\frac{7}{9}\right)$ |  $(0,0,0,\pm 1)\,,\,(\pm 1,0,0,\pm 1)\,,\,(0,\pm 1,\mp 1,0)\,,\,(0,0,\pm 1,\pm 1)$ $(0,0,\pm 1,0)\,,\,(\pm 1,\mp 1,\pm 1,\pm 1)\,,\,(\pm 2,\pm 1,0,0)\,,\,(\pm 1,\pm 2,0,0)$ $(\pm 3,\mp 1,0,0)\,,\,(\pm 1,\mp 3,0,0)\,,\,(\pm 3,\mp 2,0,0)\,,\,(\pm 2,\mp 3,0,0)$

As in the quadri-critical point, we can find combinations that preserve some
non-invertible duality symmetry, therefore leading to interesting constraints
on the corresponding RG flow. In this case however, it is more challenging to
determine the IR theory resulting from these duality preserving RG flows.
Therefore all the possible symmetry constraints which we will describe in the
following may lead to useful predictions for the low energy dynamics.

As before, let us give some representative examples which will serve to
highlight some interesting features regarding these RG flows. A simple
relevant deformation which preserves $\mathcal{D}_{1}$ is

$R_{1/9}\sim\cos(\phi_{2})\,.$ (4.100)

Such a deformation preserves $U(1)_{n_{1}}\times U(1)_{w_{1}}\times
U(1)_{w_{2}}$, with the first two factors participating in a continuous mixed
’t Hooft anomaly; therefore we conclude that the IR theory is gapless. Because
of the presence of $\mathcal{D}_{1}$ we also deduce that such gapless theory
must enjoy a non-invertible TY$(\mathbb{Z}_{3})$ symmetry. A very natural
candidate for this IR theory is therefore the $R=\sqrt{3}$ $c=1$ CFT. However,
as opposite to the quadri-critical case, since the bi-critical point is not
factorized, we cannot immediately deduce the IR fate of this process. In order
to determine it, we can however perform a topological manipulation which
brings the theory to a factorized point, perturbed by a genuine relevant
operator. As explained around (4.45) we have

$T_{\rho}T_{\tau}\sigma_{\widetilde{M}=2}\,:\,(\omega,\alpha)\,\to\,(i\sqrt{3},i3\sqrt{3})\,,$
(4.101)

and, by this manipulation, the relevant deformation is mapped to

$T_{\rho}T_{\tau}\sigma_{\widetilde{M}=2}\,:\,\cos(\phi_{2})|_{(\omega,\alpha)}\,\to\,\cos(2\phi_{2})|_{(i\sqrt{3},i3\sqrt{3})}\,.$
(4.102)

Therefore, at $(i\sqrt{3},i3\sqrt{3})$, we find that the sector described by
$\phi_{2}$ flows to a gapped phase with two degenerate vacua, i.e. the
$\mathbb{Z}_{2}\in U(1)_{n_{2}}$ is spontaneously broken. The reason for this
is that the $\mathbb{Z}_{2}$ shift symmetry which is preserved by the
deformation (4.102) at the factorized point is itself involved in a mixed
anomaly with a $\mathbb{Z}_{2}$ subgroup of the winding symmetry. We then
conclude that the IR theory is described by

$(i\sqrt{3},i3\sqrt{3})+\cos(2\phi_{2})\;\xrightarrow[]{\text{RG
flow}}\;\left(c=1\;R=\sqrt{3}\right)\;\times\left(\text{ 2 vacua}\right)$
(4.103)