Exploring Non-Invertible Symmetries in Free Theories

Pierluigi Niro, Konstantinos Roumpedakis, Orr Sela

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

Symmetries corresponding to local transformations of the fundamental fields
that leave the action invariant give rise to (invertible) topological defects,
which obey group-like fusion rules. One can construct more general
(codimension-one) topological defects by specifying a map between gauge-
invariant operators from one side of the defect and such operators on the
other side. In this work, we apply such construction to Maxwell theory in four
dimensions and to the free compact scalar theory in two dimensions. In the
case of Maxwell theory, we show that a topological defect that mixes the field
strength $F$ and its Hodge dual $\star F$ can be at most an $SO(2)$ rotation.
For rational values of the bulk coupling and the $\theta$-angle we find an
explicit defect Lagrangian that realizes values of the $SO(2)$ angle $\varphi$
such that $\cos\varphi$ is also rational. We further determine the action of
such defects on Wilson and ’t Hooft lines and show that they are in general
non-invertible. We repeat the analysis for the free compact scalar $\phi$ in
two dimensions. In this case we find only four discrete maps: the trivial one,
a $\mathbbm{Z}_{2}$ map $d\phi\rightarrow-d\phi$, a $\mathcal{T}$-duality-like
map $d\phi\rightarrow i\star d\phi$, and the product of the last two.

###### Contents

  1. 1 Introduction
    1. 1.1 Generalities
    2. 1.2 Defect Lagrangian
    3. 1.3 Ordinary Symmetries
    4. 1.4 Organization
  2. 2 Topological defects in 4d Maxwell theory
    1. 2.1 $\sin\varphi=0$
    2. 2.2 $\sin\varphi\neq 0$
  3. 3 Action on Wilson and ’t Hooft lines
  4. 4 Topological defects in 2d scalar theory
  5. A Defects from gauging and dualities
  6. B Action of surface defects in Chern-Simons theory
  7. C $\mathcal{S}$ defect at integer coupling
  8. D Action of generic defects on lines
  9. E Fusion of $\mathcal{T}$ defects

##  1 Introduction

###  1.1 Generalities

Symmetries are a central tool in the study of Quantum Field Theory. In the
last decade the existence of symmetries has become synonymous to the existence
of topological operators, i.e. operators whose position can be deformed
infinitesimally without affecting correlation functions. Usual symmetries are
generated by codimension-one topological operators satisfying a group-like
fusion rule. This point of view has led to two important generalizations of
symmetries. First, the $p$-form symmetries [1], which are generated by
topological operators of codimension $p+1$, and by now have become a standard
tool for studying QFTs. Second, the so-called non-invertible symmetries,
associated to operators that do not satisfy a group-like fusion rule, and are
the subject of this work.

Non-invertible symmetries are long known to exist in two-dimensional rational
CFTs since the work of Verlinde [2] (see also [3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14]). It was recently realized that non-invertible symmetries also
exist in non-conformal theories in two dimensions [15], as well as in QFTs in
higher dimensions [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30], and have turned out to be useful in various contexts [31, 32, 33, 34, 35,
36, 37, 38, 39, 40]. However, so far in the literature the existence of non-
invertible symmetries has been shown either by using indirect methods or by
using methods that are particular to certain classes of theories. One such
method requires the presence of a higher-form symmetry which can be gauged
along a surface giving rise to a so-called condensation defect [21, 41, 30].
Although condensation defects can be constructed in any QFT with a higher-form
symmetry, such defects do not act on local operators. Another method for
constructing non-invertible defects makes use of dualities [17, 19, 29, 28].
Although such duality defects can act on local operators, their construction
requires the existence of dualities, which is not a generic feature of QFTs.

In the case of ordinary symmetries, for every transformation that leaves the
action invariant there is a topological operator that generates that
transformation (assuming there is no anomaly). For continuous $p$-form
symmetries, Noether’s theorem provides an explicit way to construct that
operator as

$U_{g=e^{i\alpha}}[\Sigma_{(d-p+1)}]=\exp\left(i\alpha\int_{\Sigma_{(d-p-1)}}\star
J\right)~{},$ (1)

where $g$ is the element of the symmetry group, $\Sigma$ is a
codimension-$(d-p-1)$ surface, and $J$ is the conserved ($p+1$)-form current
($d\star J=0$). A natural question is whether there is an analog of the above
criterion for non-invertible symmetries, which would provide a more systematic
way to identify them, as well as to construct the corresponding topological
operators explicitly.

In this work, although we do not attempt to provide a general criterion for
the existence of non-invertible symmetries, we put forward a constructive way
to analyze them in free theories, such as the four-dimensional Maxwell theory
and the two-dimensional theory of a compact scalar.

In the 4d Maxwell theory, we consider generic defects that act by mixing the
field strength $F$ and its Hodge dual $\star F$, and we show that such defects
are topological when they act as an $SO(2)$ rotation on these operators. We
give a Lagrangian description of these defects for arbitrary rational values
of the coupling constants $e^{4}$ and $\theta/\pi$, realizing the $SO(2)$
rotations with angle $\varphi$ such that $\cos\varphi$ is a rational number.
Despite the fact that the action on the local operators $F$ and $\star F$ is
invertible, the action on Wilson and ’t Hooft lines is generically non-
invertible.

In a similar fashion, we consider topological defects in the 2d free compact
scalar theory, which act by mixing $d\phi$ and $\star d\phi$. We show that in
this case we only find four types of allowed transformations, corresponding to
the trivial one, the $\mathbb{Z}_{2}$, a $\mathcal{T}$-duality-like
transformation for rational values of the squared radius of the target space,
and the product of the last two.

Our construction may turn out to be mostly useful for free theories, but we
hope that it will also be useful towards developing a criterion for the
existence of non-invertible symmetries.

###  1.2 Defect Lagrangian

In this work, we focus on codimension-one defects in free theories, which we
construct and study using an explicit Lagrangian description. Let us begin by
considering a theory with Lagrangian $\mathcal{L}$, and a defect along a
surface $S$ that splits the spacetime into two regions, such that $M=S^{+}\cup
S^{-}$ with $\partial S^{-}=-\partial S^{+}=S$. For simplicity, we choose $S$
to be an infinite flat surface at $x=0$ and denote collectively the fields on
the left and right sides of the defect by $\Phi_{L}$ and $\Phi_{R}$,
respectively. The bulk Lagrangian $\mathcal{L}$ on both sides is the same as
we are considering a defect in a single theory and not an interface between
two different theories. On the defect itself, the fields from the two sides
are related in a non-trivial way determined by the choice of defect Lagrangian
$\mathcal{L}_{S}$. To see this, let us examine the full action in the presence
of a defect

$S=\int_{x<0}\mathcal{L}(\Phi_{L})+\int_{x=0}\mathcal{L}_{S}(\Phi_{L},\Phi_{R},b)+\int_{x>0}\mathcal{L}(\Phi_{R})\,,$
(2)

where $b$ denotes additional dynamical fields living on the defect. Then, the
equations of motion for $\Phi_{L}$, $\Phi_{R}$, and $b$ on the defect are

$\begin{split}\frac{\partial\mathcal{L}(\Phi_{L})}{\partial
d\Phi_{L}}\bigg{\rvert}_{S}+\frac{\partial\mathcal{L}_{S}(\Phi_{L},\Phi_{R},b)}{\partial\Phi_{L}}\bigg{\rvert}_{S}-(-1)^{p_{\Phi}}\,d\,\frac{\partial\mathcal{L}_{S}(\Phi_{L},\Phi_{R},b))}{\partial
d\Phi_{L}}\bigg{\rvert}_{S}&=0\,,\\\
\frac{\partial\mathcal{L}(\Phi_{R})}{\partial
d\Phi_{R}}\bigg{\rvert}_{S}-\frac{\partial\mathcal{L}_{S}(\Phi_{L},\Phi_{R},b)}{\partial\Phi_{R}}\bigg{\rvert}_{S}+(-1)^{p_{\Phi}}\,d\,\frac{\partial\mathcal{L}_{S}(\Phi_{L},\Phi_{R},b))}{\partial
d\Phi_{R}}\bigg{\rvert}_{S}&=0\,,\\\
\frac{\partial\mathcal{L}_{S}(\Phi_{L},\Phi_{R},b)}{\partial
b}\bigg{\rvert}_{S}-(-1)^{p_{b}}\,d\,\frac{\partial\mathcal{L}_{S}(\Phi_{L},\phi_{R},b))}{\partial
db}\bigg{\rvert}_{S}&=0\,,\end{split}$ (3)

where $p_{\Phi}$ and $p_{b}$ are the form degrees of $\Phi_{L,R}$ and $b$,
respectively. Note that the first term in the first two equations results from
integrating by parts the variation of the bulk Lagrangian, while all the other
terms correspond to the variation of the defect Lagrangian. We see that for a
given theory with Lagrangian $\mathcal{L}$, the above relations (3) between
$\Phi_{L}$ and $\Phi_{R}$ are determined by the defect Lagrangian
$\mathcal{L}_{S}$. These relations do not necessarily correspond to a
transformation which is a symmetry of the action, acting on the fundamental
fields locally. In the latter case, that leads to an invertible defect.
However, in general the map between $\Phi_{L}$ and $\Phi_{R}$ defines a non-
invertible defect.

Not every defect Lagrangian $\mathcal{L}_{S}$ leads to a topological defect.
For this, the energy-momentum tensor has to be conserved in the presence of
the defect. To see what this implies for $\mathcal{L}_{S}$, let us first
rewrite the action as

$S=\int_{M}\big{(}\mathcal{L}(\Phi_{L})\Theta(-x)+\mathcal{L}_{S}(\Phi_{L},\Phi_{R},b)\delta(x)+\mathcal{L}(\Phi_{R})\Theta(x)\big{)}\,,$
(4)

where $\Theta(x)$ and $\delta(x)$ are the usual theta and delta functions,
respectively. We can now compute the total energy-momentum tensor from the
variation of (4) with respect to the metric. Assuming that the defect
Lagrangian does not depend on the metric, in order for the energy-momentum
tensor to be conserved, it should obey

$n^{\mu}(T_{\mu\nu}(\Phi_{L})-T_{\mu\nu}(\Phi_{R}))\big{|}_{S}=0\,.$ (5)

This condition is satisfied only for specific relations between $\Phi_{L}$ and
$\Phi_{R}$. Hence, the relations (3) serve as a constraint to be imposed on
$\mathcal{L}_{S}$ to obtain a topological defect.111Under the assumption that
the defect Lagrangian does not depend on the metric, the condition (5) is
equivalent to the vanishing of the displacement operator, defined by
$D_{\mu}(\sigma^{a})=\frac{\delta S}{\delta X^{\mu}(\sigma^{a})}$, where
$X^{\mu}(\sigma^{a})$ is the defect embedding and $\sigma^{a}$ the coordinates
parametrizing the defect. See e.g. [42] for a recent discussion.

In this note, instead of starting with a given defect Lagrangian, our strategy
will be first to determine the relations between $\Phi_{L}$ and $\Phi_{R}$ on
the defect such that equation (5) is satisfied. Once we find such relations,
we have a candidate for a topological defect (which can be either invertible
or not) that generates such action on local operators. Then, our goal will be
to give an explicit construction of such topological defect by finding a
defect Lagrangian that leads to those relations through (3).

Since our discussion so far was entirely classical, one might wonder whether
the defects constructed in this way are also realized in the quantum theory.
If they are not, that would be the analog of an ABJ anomaly for generic
topological defects. For a general Lagrangian we cannot provide a sufficient
condition that guarantees that they survive quantum corrections. Below, we
analyze usual (group-like) symmetries using this approach and show that in the
presence of an anomaly the defect is still topological, but it becomes an
interface between two different theories. Nevertheless, in the example we are
considering, namely Maxwell theory, defects of the type we discuss can also be
constructed by gauging discrete subgroups of the one-form symmetry combined
with $SL(2,\mathbbm{Z})$ duality transformations on half-space (see appendix
A). This can be done at the quantum level and shows that such defects exist in
the quantum theory as well.

###  1.3 Ordinary Symmetries

Let us examine how ordinary symmetries (continuous or not) are described in
this Lagrangian approach. Consider a symmetry of the action, transforming the
fundamental fields as

$\Phi\rightarrow U(\Phi)\,.$ (6)

We then schematically define the corresponding defect as

$S=\int_{S^{-}}L(\Phi_{L})+\int_{S}b\wedge(\Phi_{L}-U(\Phi_{R}))+\int_{S^{+}}L(\Phi_{R})\,,$
(7)

where $b$ is a Lagrange-multiplier field that enforces the relation
$\Phi_{L}|_{S}=U(\Phi_{R})|_{S}$. To see how the defect acts on operators, let
us insert it inside a correlation function. Then, operator insertions on the
left side of the defect will depend on $\Phi_{L}$, while operators on the
right will depend on $\Phi_{R}$. We can now trivialize the defect Lagrangian
by redefining the bulk fields only on one side of the defect. For example, we
can redefine

$\Phi_{R}\rightarrow U^{-1}(\Phi_{R})\,,$ (8)

which will transform all the operator insertions constructed from $\Phi_{R}$
on the right side of the defect. If there is no anomaly (see below) the defect
action now becomes

$\int_{S}b\wedge(\Phi_{L}-\Phi_{R})\,,$ (9)

which is simply the trivial defect identifying $\Phi_{L}$ and $\Phi_{R}$ on
it. Hence, the insertion of (7) is equivalent to transforming the operators on
one side of the defect, leading to the usual Ward identities.

Notice that to trivialize the defect, we assumed that (8) can be done at the
quantum level, which requires the absence of anomalies. Instead, if there is
an anomaly then the redefinition (8) will produce a bulk term of the form

$\int_{S^{+}}\mathcal{A}(\Phi_{R})\,,$ (10)

where $\mathcal{A}(\Phi_{R})$ is the anomaly coming from the transformation of
the functional measure in the path integral. In this case, the defect becomes
an interface between two different theories, one with and one without the term
(10) in the action.

###  1.4 Organization

This work is organized as follows. In section 2, we consider 4d Maxwell theory
and study transformations acting linearly on $F$ and $\star F$. We show that
such transformations can be at most an $SO(2)$ rotation in order for the
corresponding defect to be topological. For rational values of the bulk
coupling constant $e^{4}$ and $\theta/\pi$, we show that there is a defect
Lagrangian which realizes rational values of $\cos\varphi$, where $\varphi$ is
the $SO(2)$ rotation angle.

In section 3, we study the action of the above defects on line operators. We
show that, in general, a loop operator dragged across the defect becomes a
disc operator, which is determined by the relation between left and right
fields imposed on the defect. We then turn to the more interesting case of
line operators piercing the defect. In this case, the operators at the
junction are monopole operators and the (generically non-invertible) action on
lines is determined by their quantum numbers.

In section 4, we repeat the discussion of section 2 in the case of the 2d
compact scalar theory, by considering transformations that mix $d\phi$ and
$\star d\phi$. This case is not as rich as the Maxwell one and we only find
four discrete types of transformations: the trivial one, a $\mathbb{Z}_{2}$, a
$\mathcal{T}$-duality-like transformation, and the product of the last two.

Several appendices contain additional material, which makes contact with
previous results in the literature and supplements the main text. In appendix
A, we show how the defects we construct for Maxwell theory in section 2 can
also be constructed using gauging and dualities. In appendix B, we review the
action of condensation defects on line operators in 3d Chern-Simons theory
using the approach discussed in this note. In appendix C, we show how the
action on lines in Maxwell theory of a particular defect we investigated in
section 3 fits with the dualization procedure discussed in [43], and how it
can be analyzed using it. In appendix D, we explain in more detail how a
generic defect of the type considered in section 2 acts on lines. Finally, in
appendix E we calculate the fusion of two $\mathcal{T}$-duality-like defects
using their Lagrangian description.

##  2 Topological defects in 4d Maxwell theory

Consider the Maxwell theory with a theta term in four dimensions, with a
Euclidean Lagrangian

$\mathcal{L}(e^{2},\theta;A)=\frac{1}{4\pi e^{2}}F\wedge\star
F+\frac{i\theta}{8\pi^{2}}F\wedge F\,,$ (11)

where $F=dA$ is the field strength. The symmetry of the theory includes a
$\mathbb{Z}_{2}$ charge conjugation symmetry and two continuous one-form
symmetries $U(1)_{e}$ and $U(1)_{m}$ which measure the electric and magnetic
charges

$n=\int_{\Sigma_{2}}\frac{\tilde{F}}{2\pi}\,,\qquad
m=\int_{\Sigma_{2}}\frac{F}{2\pi}~{},$ (12)

of Wilson and ’t Hooft lines

$W_{n}(\gamma)=e^{in\oint_{\gamma}A}\,,\qquad
H_{m}(\gamma)=e^{-im\oint_{\gamma}\tilde{A}}\,,$ (13)

where we have defined the dual field strength

$\tilde{F}=d\tilde{A}=-\frac{i}{e^{2}}\star F+\frac{\theta}{2\pi}F~{}.$ (14)

In addition, Maxwell theory enjoys $\mathcal{S}$-duality which exchanges $F$
and $\tilde{F}$ while acting on the coupling as $\tau\rightarrow-1/\tau$,
where $\tau=\frac{\theta}{2\pi}+\frac{i}{e^{2}}$. At the self-dual point
$\tau=i$, it becomes a global symmetry exchanging $F$ and $\star F$. In the
following, we study more general (non-invertible) topological defects
implementing this transformation away from the self-dual point, which we refer
to as $\mathcal{S}$ defects. Such defects have already been constructed in
[17] for integer values of $i\tau$, and we will discuss below how to realize
them also for generic rational values of $e^{4}$ and $\theta/\pi$. In fact,
our goal will be to study topological defects in Maxwell theory that implement
even more general transformations between $F$ and $\star{F}$. Let us emphasize
that in this discussion we consider defects in the same theory and not
interfaces between different theories. Hence, the coupling $\tau$ is the same
on both sides of the defect.

As explained in section 1, we define a codimension-one defect by the
action222Due to the $2\pi$ periodicity of $\theta$, one could in principle
allow for a $2\pi\mathbb{Z}$ difference between the left and the right theta
angles. However, such a difference can be reabsorbed in integer shifts of the
levels of the 3d Chern-Simons terms for the $A_{R}$ and $A_{L}$ fields on the
defect. Since such terms will be present in our defect Lagrangian description,
we assume that $\theta_{L}=\theta_{R}$ without loss of generality.

$S=\int_{S^{-}}\mathcal{L}(e^{2},\theta;A_{L})+\int_{S^{+}}\mathcal{L}(e^{2},\theta;A_{R})+\int_{S}\mathcal{L}_{S}(A_{L},A_{R},b)\,.$
(15)

Before we specify the defect Lagrangian, let us ask what is the most general
transformation that acts linearly on $F$ and $\star F$. To this end, we
consider a transformation of the form

$\displaystyle F_{L}|_{S}$ $\displaystyle=\alpha\,F_{R}|_{S}+\beta\;i\star
F_{R}|_{S}\,,$ (16) $\displaystyle i\star F_{L}|_{S}$
$\displaystyle=\gamma\,F_{R}|_{S}+\delta\;i\star F_{R}|_{S}\,,$ (17)

where $\alpha$, $\beta$, $\gamma$ and $\delta$ are real parameters. Assuming
that the defect Lagrangian does not depend on the metric, for the defect to be
topological the energy-momentum tensor

$T_{\mu\nu}(A)=\frac{1}{2\pi
e^{2}}\left(-F_{\mu\alpha}F^{\alpha}_{\,\nu}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right),$
(18)

has to satisfy (5). Using this explicit expression for $T_{\mu\nu}$, we get
that the transformation generated by a topological defect is at most an
$SO(2)$ rotation,

$\begin{pmatrix}F_{L}|_{S}\\\ i\star
F_{L}|_{S}\end{pmatrix}=\begin{pmatrix}\cos\varphi&\sin\varphi\\\
-\sin\varphi&\cos\varphi\end{pmatrix}\begin{pmatrix}F_{R}|_{S}\\\ i\star
F_{R}|_{S}\end{pmatrix}\,.$ (19)

The special cases $\varphi=\pi$ and $\varphi=\pi/2$ correspond to the
$\mathbb{Z}_{2}$ and $\mathcal{S}$ defects, respectively. Another subset of
the above defects with a rational $\cos\varphi=-\frac{\theta}{2\pi}$ and
$\frac{1}{e^{2}}=\sin\varphi$ has previously been constructed in [28] by
gauging a subgroup of the one-form symmetry accompanied with a duality
transformation in half-space. In addition, the defect corresponding to
$\tau=e^{2\pi i/3}N$ (where $N\in\mathbb{N}$) and $\varphi=-2\pi/3$, which is
an example of a triality defect, has been constructed in [19] using a similar
method. We next turn to discuss explicit realizations of the defect Lagrangian
$\mathcal{L}_{S}$ for rational values of $\cos\varphi$, $e^{4}$ and
$\theta/\pi$ (we discuss the construction of such defects using gauging and
dualities in appendix A). It would be interesting if it is possible to realize
the whole $SO(2)$ transformations with also irrational values of
$\cos\varphi$. We leave that for future work.

###  2.1 $\sin\varphi=0$

Let us first consider two special values for $\varphi$, namely $\varphi=0$
corresponding to the trivial $SO(2)$ element ($F_{L}|_{S}=F_{R}|_{S}$, $\star
F_{L}|_{S}=\star F_{R}|_{S}$), and $\varphi=\pi$ corresponding to the non-
trivial $\mathbb{Z}_{2}$ element ($F_{L}|_{S}=-F_{R}|_{S}$, $\star
F_{L}|_{S}=-\star F_{R}|_{S}$).

There are multiple non-trivial defects corresponding to the trivial $SO(2)$
element. In the literature these are known as condensation defects [21, 19].
Although they act trivially on local operators, they act non-trivially on non-
local operators like Wilson and ’t Hooft lines. In these cases the gauge
fields on the two sides of the bulk differ only by a discrete gauge field
(with zero field strength) and (5) is trivially satisfied. They arise from
gauging a non-anomalous333In order for a discrete subgroup of a one-form
symmetry to be gaugable along a codimension-one surface, it needs to be free
from 1-anomalies [21]. discrete subgroup of the electric or magnetic one-form
symmetry [17, 29, 19, 28].

An example of a condensation defect [21, 19] for arbitrary values of $e^{2}$
and $\theta$ is realized by the defect Lagrangian

$\mathcal{L}_{S}=\frac{i\kappa}{2\pi}a\wedge(dA_{R}-dA_{L})~{},$ (20)

where we have introduced an auxiliary $U(1)$ gauge field $a$ living only on
the defect, and $\kappa\in\mathbb{Z}$ for gauge invariance (without loss of
generality we can choose $\kappa$ to be positive). The equation of motion for
$a$ enforces the condition $F_{L}|_{S}=F_{R}|_{S}$, while the equations of
motion for the bulk gauge fields read

$\frac{\kappa}{2\pi}da=-\frac{i}{2\pi e^{2}}\star
F_{R}|_{S}+\frac{\theta}{4\pi^{2}}F_{R}|_{S}=-\frac{i}{2\pi e^{2}}\star
F_{L}|_{S}+\frac{\theta}{4\pi^{2}}F_{L}|_{S}\,.$ (21)

Combining them, we also get $\star F_{L}|_{S}=\star F_{R}|_{S}$. Integrating
the above equation over a closed two-surface on the defect we get

$\kappa\,m=n_{R}=n_{L}\,,$ (22)

where the integer $m$ measures the flux of $a$ and $n_{R,L}$ are the one-form
electric charges defined in (12). Note that for $\kappa\neq 1$ this imposes
the constraint on the charges $n_{R,L}$ to be multiples of $\kappa$. As we
will see in section 3, this implies that although the defect acts trivially on
local operators it actually induces a non-trivial transformation on line
operators. On the other hand, for $\kappa=1$ equation (22) does not impose any
constraint and indeed in this case the defect is trivial.

Similarly, the non-trivial $\mathbbm{Z}_{2}$ element of the $SO(2)$ can be
realized by the defect Lagrangian

$\mathcal{L}_{S}=\frac{i\kappa}{2\pi}a\wedge(dA_{R}+dA_{L})\,.$ (23)

For $\kappa=1$, it induces the usual (invertible) charge conjugation symmetry.
For $\kappa\neq 1$ it is a product of the charge conjugation and a
condensation defect. To see this, consider two defects on two surfaces $S$ and
$S+\delta S$, one generating the charge conjugation and the other a
condensation defect as in (20). The combined defect action is given by

$\frac{i}{2\pi}\int_{S}a_{1}\wedge(dA_{I}+dA_{L})+\frac{i\kappa}{2\pi}\int_{S+\delta
S}a_{2}\wedge(dA_{R}-dA_{I})\,,$ (24)

where $A_{I}$ is the gauge field in the strip between the two surfaces. Taking
the limit $\delta S\rightarrow 0$ we get

$\int_{S}\left(\frac{i}{2\pi}a_{1}\wedge(dA_{I}+dA_{L})+\frac{i\kappa}{2\pi}a_{2}\wedge(dA_{R}-dA_{I})\right),$
(25)

and after redefining $a_{1}\rightarrow a_{1}+\kappa a_{2}$ and
$A_{I}\rightarrow A_{I}-A_{L}$ we find

$\int_{S}\left(\frac{i}{2\pi}a_{1}\wedge
dA_{I}+\frac{i\kappa}{2\pi}a_{2}\wedge(dA_{R}+dA_{L})\right).$ (26)

The first term corresponds to a trivial decoupled TQFT and can be dropped.
Hence, we see that the product of the usual charge conjugation defect and the
condensation defect (20) is equal to (23).

###  2.2 $\sin\varphi\neq 0$

Let us now construct defects that generate transformations with
$\sin\varphi\neq 0$. Those transformations arise as boundary conditions which
have to be imposed on the theory in order to satisfy a well-defined
variational principle. Hence, we need to impose that if we vary
$A_{L,R}\rightarrow A_{L,R}+\delta A_{L,R}$ the corresponding variation of the
defect action cancels the contribution coming from the bulk, when (19) are
satisfied. The bulk variation reads

$\delta S_{bulk}=\int_{S}\left(\frac{1}{2\pi e^{2}}\delta A_{L}\wedge\star
dA_{L}+\frac{i\theta}{4\pi^{2}}\delta A_{L}\wedge dA_{L}-\frac{1}{2\pi
e^{2}}\delta A_{R}\wedge\star dA_{R}-\frac{i\theta}{4\pi^{2}}\delta
A_{R}\wedge dA_{R}\right)\,,$ (27)

where we have ignored the contributions on $S^{+}$ and $S^{-}$, which lead to
the bulk equations of motion $d\star F_{L}=d\star F_{R}=0$. Using (19) to
express $\star F_{L}$ and $\star F_{R}$ in terms of $F_{L}$ and $F_{R}$ we get

$\delta S_{bulk}=\int_{S}\left(-\frac{iN_{L}}{2\pi}\,\delta A_{L}\wedge
dA_{L}-\frac{iN_{R}}{2\pi}\delta A_{R}\wedge dA_{R}+\frac{iN}{2\pi}(\delta
A_{L}\wedge dA_{R}+\delta A_{R}\wedge dA_{L})\right),$ (28)

where

$N_{L}=\frac{1}{e^{2}\tan\varphi}-\frac{\theta}{2\pi}\,,\qquad
N_{R}=\frac{1}{e^{2}\tan\varphi}+\frac{\theta}{2\pi}\,,\qquad
N=\frac{1}{e^{2}\sin\varphi}\,.$ (29)

Naively, a defect Lagrangian whose variation cancels (28) is

$\mathcal{L}_{S}=\frac{iN_{L}}{4\pi}A_{L}\wedge
dA_{L}+\frac{iN_{R}}{4\pi}A_{R}\wedge dA_{R}-\frac{iN}{2\pi}A_{L}\wedge
dA_{R}\,.$ (30)

However, gauge invariance requires $N_{L}$, $N_{R}$, and $N$ to be integers.
Together with the relations (29), this implies that only some special values
of the angle $\varphi$ can be realized for given values of the bulk couplings
$e^{2}$ and $\theta$.

In the rest of this section, we attempt to realize any rational value of
$N_{L}$, $N_{R}$, and $N$. First, let us split $N_{L}$, $N_{R}$, and $N$ as

$N_{L}=k_{L}+\frac{p_{L}}{q_{L}}\,,\qquad
N_{R}=k_{R}+\frac{p_{R}}{q_{R}}\,,\qquad N=k+\frac{p_{N}}{q_{N}}\,,$ (31)

where $k_{L},k_{R}$ and $k$ are integers chosen such that
$\frac{p_{i}}{q_{i}}\in[0,1)$, and gcd$(p_{i},q_{i})=1$ for every $i=L,R,N$.
The integer part of the variation (28) can be canceled by (30) with the
$N_{i}$’s replaced by their integer parts $k_{i}$. In order to realize the
non-integer part of the defect Lagrangian we need to introduce additional
dynamical fields living on the defect. One can easily write down a Lagrangian
with the right variation, but a generic choice a priori might lead to a non-
simple defect, such as a direct sum of more elementary defects.

To avoid this, we make use of the minimal TQFT $\mathcal{A}^{q,p}$ in three
dimensions with one-form symmetry $\mathbb{Z}_{q}$ and anomaly labeled by $p$
($p\sim p+q$) [44]. Because of the anomaly, when the $\mathcal{A}^{q,p}$
theory is coupled to a background two-form gauge field $B$ for the one-form
symmetry, it is not invariant under background gauge transformations
$B\rightarrow B+d\lambda$. For an infinitesimal transformation, it changes as

$\int_{M}\mathcal{A}^{q,p}[B+d\lambda]=\int_{M}\mathcal{A}^{q,p}[B]-\frac{iqp}{2\pi}\int_{M}\lambda\wedge
B\,.$ (32)

In our case, we can couple the bulk Maxwell theory to an
$\mathcal{A}^{q,p}[B]$ TQFT on the defect by activating its background
$\mathbb{Z}_{q}$ gauge field $B$ using the bulk field strengths $dA_{L}$ and
$dA_{R}$. More specifically, we can choose

$B=\frac{l\;dA_{L}+r\;dA_{R}}{q}\,,$ (33)

for some integers $l$ and $r$.

One can show that coupling to a single minimal TQFT is not enough to realize
all the rational values of $e^{4}$ and $\theta/\pi$. Instead, we couple the
theory to two minimal TQFTs and consider the defect Lagrangian

$\mathcal{L}_{S}=\frac{ik_{L}}{4\pi}A_{L}\wedge
dA_{L}+\frac{ik_{R}}{4\pi}A_{R}\wedge dA_{R}-\frac{ik}{2\pi}A_{L}\wedge
dA_{R}+\mathcal{L}_{\mathcal{A}}\,,$ (34)

with

$\mathcal{L}_{\mathcal{A}}=\mathcal{A}^{q_{1},p_{1}}\left[\frac{l_{1}\,dA_{L}+r_{1}\,dA_{R}}{q_{1}}\right]+\mathcal{A}^{q_{2},p_{2}}\left[\frac{l_{2}\,dA_{L}+r_{2}\,dA_{R}}{q_{2}}\right]\,,$
(35)

where $l_{i},r_{i}$ are integers and $p_{i}$ is defined modulo $q_{i}$, with
gcd$(p_{i},q_{i})=1$ for $i=1,2$. The parameters in the above action are
related to the $N_{i}$’s by

$\begin{split}N_{L}&=k_{L}-\frac{p_{1}}{q_{1}}l_{1}^{2}-\frac{p_{2}}{q_{2}}l_{2}^{2}\,,\\\
N_{R}&=k_{R}-\frac{p_{1}}{q_{1}}r_{1}^{2}-\frac{p_{2}}{q_{2}}r_{2}^{2}\,,\\\
N&=k+\frac{p_{1}}{q_{1}}l_{1}r_{1}+\frac{p_{2}}{q_{2}}l_{2}r_{2}\,.\end{split}$
(36)

In this way, any rational value for the $N_{i}$’s is allowed. We can further
invert equations (29) to get

$\displaystyle\frac{\theta}{2\pi}$ $\displaystyle=\frac{N_{R}-N_{L}}{2}\,,$
(37) $\displaystyle\frac{1}{e^{4}}$
$\displaystyle=N^{2}-\left(\frac{N_{R}+N_{L}}{2}\right)^{2}\,.$ (38)

For a theory with given rational coupling constants $e^{4}$ and $\theta/\pi$,
the above can be viewed as constraints on the allowed values of parameters in
(35). Given the above constraints, one can construct defects that realize
elements of the $SO(2)$ in (17), labeled by an angle $\varphi$ that satisfies

$\cos\varphi=\frac{N_{R}+N_{L}}{2N}\,.$ (39)

It is easy to see that there are several choices of the parameters in (35)
(and hence several defects) that lead to the same $SO(2)$ transformation (for
fixed bulk couplings). However, this does not mean that such defects act the
same on non-local operators. For instance, one can find examples of two
defects $D_{\varphi}$ and $D^{\prime}_{\varphi}$ that give rise to the same
angle $\varphi$ and are related by a condensation defect $\mathcal{C}$. This
means that $D_{\varphi}=(Z)\;\mathcal{C}\times D^{\prime}_{\varphi}$, where
$Z$ is the partition function of a decoupled TQFT. This suggests that
different choices of the parameters leading to the same $SO(2)$ angle are
related by fusing with a condensation defect. However, we were not able to
show this for a general angle nor that fusing with a generic condensation
defect produces another defect of the form (35).

As an example, let us consider the case of the $\mathcal{S}$ defect
$\varphi=\pi/2$. Given arbitrary rational values of $e^{2}$ and $\theta/\pi$,
such a defect is realized by taking $N_{R}=-N_{L}=\frac{\theta}{2\pi}$ and
$N=1/e^{2}$. Another special example is the case where $\theta=0$. In this
case we have $N_{R}=N_{L}\equiv M$ and

$\cos\varphi=\frac{M}{N}\,,\qquad\frac{1}{e^{4}}=N^{2}-M^{2}\,.$ (40)

Given an arbitrary rational value for $e^{4}$ (since any rational can be
written as the difference of two squared rationals $N$ and $M$), all $\varphi$
rotations satisfying the equations above can be realized. In particular, the
$\mathcal{S}$ defect discussed above can be realized at $\theta=0$ by choosing
$M=0$ and $N=1/e^{2}$.

##  3 Action on Wilson and ’t Hooft lines

$L_{R}$$S$$\longrightarrow$

$L_{L}$$S$

Figure 1: A Wilson or a ’t Hooft loop $L$ dragged across a non-invertible
defect may become an improperly quantized disc operator. The red arrow denotes
the orientation of the surface.

In this section, we study the action of the defects, constructed in section 2,
on line operators. Let us begin by first considering what happens when one
drags a loop operator across a defect, as depicted in figure 1. In our
conventions the dual gauge field $\tilde{A}$ is defined by (see (14))

$d\tilde{A}=-\frac{i}{e^{2}}\star dA+\frac{\theta}{2\pi}dA\,,$ (41)

allowing us to write Wilson and ’t Hooft loops, with integer charges $n$ and
$m$ respectively, as disc operators in the following way:

$\displaystyle W_{n}(\gamma)$
$\displaystyle=e^{in\oint_{\gamma}A}=e^{in\int_{D}dA}~{},$ (42) $\displaystyle
H_{m}(\gamma)$
$\displaystyle=e^{-im\oint_{\gamma}\tilde{A}}=e^{-im\int_{D}d\tilde{A}}=e^{-\frac{m}{e^{2}}\int_{D}\star
dA-\frac{im\theta}{2\pi}\int_{D}dA}~{},$ (43)

where $\partial D=\gamma$. We can now use the sewing conditions (19) to
determine the result of dragging such a loop operator across the defect from
the right to the left. The defect transforms a loop
$W_{n}(\gamma)H_{m}(\gamma)$ into a disc operator

$\begin{split}W_{n}H_{m}\rightarrow
W_{mN+\frac{\left(n-mN_{R}\right)N_{L}}{N}}H_{\frac{mN_{R}-n}{N}}\,,\end{split}$
(44)

where the $N$’s are given in terms of the bulk couplings $e^{2}$ and $\theta$
and the $SO(2)$ angle $\varphi$ as in (29), and are generically rational
numbers. In the above, improperly quantized Wilson and ’t Hooft loops should
be viewed as disc operators. The fact that a loop can transform into a disc
operator reflects the non-invertible nature of the defect, as already
discussed in [17, 19, 29].

$L_{R}$$L_{L}$$S$$\phi$ Figure 2: Piercing a defect $S$ by a line operator
$L$. At the junction, there is a monopole operator $\phi$ as required by gauge
invariance.

The more interesting case is when a line operator is piercing a defect, as in
figure 2. In this case, a line with one-form symmetry charges $n_{R}$ and
$m_{R}$ (see (12)) that pierces the defect from the right is transformed into
a line with charges $n_{L}$ and $m_{L}$ from the left. As we discuss below,
using the equations of motion on the defect one can find the relation between
these charges, and determine the action $S\cdot L$ of the defect $S$ on a line
$L$.

At the vertex in figure 2, there is a junction operator that changes the
incoming line from the right to the line exiting to the left. From the three-
dimensional TQFT perspective, we will see that such junctions correspond to
monopole operators for the fields $A_{L}$ and $A_{R}$, as well as for other
possible dynamical gauge fields living only on the defect. Monopole operators
are in general not gauge invariant, and need to be attached to Wilson lines.
The lines in figure 2 are exactly what is needed in order to render the whole
configuration gauge invariant.

The action of a defect on lines can be determined by identifying the monopole
operators living on it and their quantum numbers. Then, for each monopole
operator, examining which combinations of lines should be attached from the
two sides for gauge-invariance gives the action of defect on lines. If for an
incoming line there is no monopole operator with the right quantum numbers
that can transform it to an exiting line, the resulting action of the defect
should be interpreted as zero. On the other hand, in some cases there may be
more than one choice of monopole operators that can convert an incoming line
$L_{R}$. In such cases, the result of the action of the defect on the line is
given by the sum over all the possible such lines $L_{L}$. An example in which
this happens is 3d Chern-Simons theory, which we review in appendix B.

Let us illustrate the above in the case where the defect Lagrangian is given
by

$\mathcal{L}_{S}=\frac{iN_{L}}{4\pi}A_{L}\wedge
dA_{L}+\frac{iN_{R}}{4\pi}A_{R}\wedge dA_{R}-\frac{iN}{2\pi}A_{L}\wedge
dA_{R}\,,$ (45)

for integer $N_{L}$, $N_{R}$, and $N$. The equations of motion on the defect
for $A_{L}$ and $A_{R}$ are

$\frac{\widetilde{F}_{L}}{2\pi}{\Big{|}}_{S}=N\frac{F_{R}}{2\pi}{\Big{|}}_{S}-N_{L}\frac{F_{L}}{2\pi}{\Big{|}}_{S}\,,\qquad\frac{\widetilde{F}_{R}}{2\pi}{\Big{|}}_{S}=N_{R}\frac{F_{R}}{2\pi}{\Big{|}}_{S}-N\frac{F_{L}}{2\pi}{\Big{|}}_{S}\,.$
(46)

When integrated along a two-dimensional surface on the defect around the
junction, we obtain the following relations between the corresponding charges:

$n_{L}=Nm_{R}-N_{L}m_{L}\,,\qquad n_{R}=N_{R}m_{R}-Nm_{L}\,.$ (47)

We can solve for the left charges in terms of the right ones and get the
following action:

$S\cdot
W_{n}H_{m}=\begin{cases}W_{mN+\frac{(n-mN_{R})N_{L}}{N}}H_{\frac{mN_{R}-n}{N}}\quad&\text{if
}N|n-mN_{R}\,,\\\ 0\quad&\text{otherwise}\,.\end{cases}$ (48)

Note that the first equation above formally coincides with (44). However, in
(44) the $N$’s take generic rational values and fractional charges are
allowed, since improperly quantized loop operators are actually disc
operators. Instead, in (48) we consider integer $N$’s and only integer charges
for the lines are allowed, as required by gauge invariance.

Let us now examine equation (48) from the point of view of the monopole
operators living on the defect. We have two such basic monopoles,
$e^{i\phi_{L}}$ and $e^{i\phi_{R}}$, where $\phi_{L}$ and $\phi_{R}$ are the
dual photons of $A_{L}$ and $A_{R}$ on the defect, respectively. Each such
monopole extends in the corresponding bulk to a ’t Hooft line (with the same
magnetic flux). From the defect Lagrangian (45) we see that each monopole
transforms non-trivially under $U(1)_{L}$ and $U(1)_{R}$ gauge
transformations. As a result, by attaching suitable Wilson lines for $A_{L}$
and $A_{R}$ to make them gauge invariant we exactly end up with (48). Let us
finally comment that the monopoles $e^{i\phi_{L}}$ and $e^{i\phi_{R}}$ arise
naturally when considering the dual frame in the bulk, as we explain in
appendix C.

In some cases, there might be additional dynamical fields living on the defect
apart from $A_{L}$ and $A_{R}$. In such cases, the corresponding monopole
operators also have to be taken into account. As a simple example, let us
consider the Lagrangian (20) describing a condensation defect (for $\kappa\neq
1$). In this case, there is also the defect gauge field $a$ and equation (22)
requires that in order to get a non-zero action, the charges of the lines on
the left and on the right do not only have to be equal ($n_{R}=n_{L}$) but are
also a multiple of $\kappa$. This extra requirement, coming from having a non-
trivial monopole associated with $a$, leads to a non-trivial action on lines.

To illustrate this better, let us consider another example with a defect
Lagrangian of the type discussed in section 2.2. Consider the defect
Lagrangian

$\mathcal{L}_{S}=\mathcal{A}^{q,1}\left[\frac{rdA_{R}+ldA_{L}}{q}\right]-\frac{ik}{2\pi}A_{L}\wedge
dA_{R}=\frac{iq}{4\pi}a\wedge
da+\frac{i}{2\pi}a\wedge\left(rdA_{R}+ldA_{L}\right)-\frac{ik}{2\pi}A_{L}\wedge
dA_{R}\,,$ (49)

where $a$ is a dynamical $U(1)$ gauge field living on the defect and $k$ is an
integer. The equations of motion for $A_{L}$, $A_{R}$ and $a$ on the defect
yield the following relations between the charges

$n_{L}=-lm_{a}+km_{R}\,,\qquad n_{R}=rm_{a}-km_{L}\,,\qquad
qm_{a}+rm_{R}+lm_{L}=0\,,$ (50)

where $m_{a}$ is the magnetic flux of the monopole of $a$. Solving as before
for the left charges in terms of the right ones, we obtain the following
action of the defect on lines:

$S\cdot
W_{n}H_{m}=\begin{cases}W_{mN+\frac{(n-mN_{R})N_{L}}{N}}H_{\frac{mN_{R}-n}{N}}\qquad&\text{if
}lr+kq\,|\,ln-krm\text{ and }qn+r^{2}m\,,\\\
0\qquad&\text{otherwise}\,,\end{cases}$ (51)

where $N_{L}$, $N_{R}$ and $N$ are given by

$N_{L}=-\frac{p}{q}l^{2}\,,\qquad N_{R}=-\frac{p}{q}r^{2}\,,\qquad
N=k+\frac{p}{q}lr\,.$ (52)

Notice that the condition for having a non-zero defect action written in the
first line of (51) takes into account the requirement that the monopole flux
$m_{a}$ is an integer,444Indeed, in this case $m_{a}=\frac{ln-krm}{lr+kq}$ .
which (as in the case of the condensation defect) adds an extra restriction
compared to only demanding that the charges of the left lines are integer.
Indeed, the relation between the left and right charges in (51) corresponds to
the sewing conditions (19), however using them alone will miss those lines
that are in fact annihilated by the defect. As an example, let us consider the
defect with $q=8$, $l=4$, $r=2$, $N=1$ and its action on the line
$W_{3}H_{2}$. Naively, using (51) while ignoring the specified conditions on
the right-hand side (or using only the sewing conditions) it might seem
consistent that the result of the defect action is given by the left line
$H_{2}$. However, in this case $m_{a}=1/2$ and this combination of lines is
annihilated by the defect.

In the case of a generic defect of the type discussed in section 2.2 (see
equation (34)), the action on lines is given by (see appendix D for details)

$S\cdot
W_{n}H_{m}=\begin{cases}W_{mN+\frac{(n-mN_{R})N_{L}}{N}}H_{\frac{mN_{R}-n}{N}}\qquad&\text{if
}d\,|\,w,\,h,\,u_{1},\text{ and }u_{2}\,,\\\
0\qquad&\text{otherwise}\,,\end{cases}$ (53)

where $N_{L}$, $N_{R}$ and $N$ are given in (36) and are generically rational,
and

$\begin{split}d&=kq_{1}q_{2}+l_{1}p_{1}q_{2}r_{1}+l_{2}p_{2}q_{1}r_{2}\,,\\\
u_{i}&=l_{i}nq_{i+1}-l_{i}mk_{R}q_{i+1}-mkq_{i+1}r_{i}-l_{i+1}mp_{i+1}r_{1}r_{2}+l_{i}mp_{i+1}r_{i+1}^{2}\,,\\\
h&=-nq_{1}q_{2}+mk_{R}q_{1}q_{2}-mp_{1}q_{2}r_{1}^{2}-mp_{2}q_{1}r_{2}^{2}\,,\\\
w&=q_{1}q_{2}\left(nk_{L}+mk^{2}-mk_{L}k_{R}\right)+p_{2}q_{1}\left(2ml_{2}kr_{2}+mk_{L}r_{2}^{2}-l_{2}^{2}n+l_{2}^{2}mk_{R}\right)\\\
&+p_{1}q_{2}\left(l_{1}^{2}mk_{R}+2l_{1}mkr_{1}+mk_{L}r_{1}^{2}-l_{1}^{2}n\right)-mp_{1}p_{2}\left(l_{1}r_{2}-l_{2}r_{1}\right)^{2}\,,\end{split}$
(54)

where the subscript $i=1,2$ should be regarded modulo 2. As expected, we note
that the result of equation (53) is the same as in (48), but for the (more
restrictive) conditions on the various integer coefficients for having a non-
zero defect action.

##  4 Topological defects in 2d scalar theory

In this section, we repeat our discussion in the case of the 2d compact scalar
$\phi\sim\phi+2\pi$ at radius $R$. The topological defects we consider in this
section have been previously studied in various works [45, 46, 12, 19, 17],
and our goal here is to parallel our discussion of Maxwell theory and see how
the 2d compact scalar fits in this picture.

The Euclidean Lagrangian is given by

$\mathcal{L}(R;\phi)=\frac{R^{2}}{4\pi}d\phi\wedge\star d\phi\,.$ (55)

The theory has the discrete $\mathbb{Z}_{2}$ zero-form symmetry
$\phi\rightarrow-\phi$ and two continuous $U(1)$ zero-form symmetries
associated to constant shifts of the scalar and its dual. The corresponding
charges are

$m=\int_{\gamma}\frac{d\tilde{\phi}}{2\pi}\,,\qquad
w=\int_{\gamma}\frac{d\phi}{2\pi}\,,$ (56)

where the dual scalar is defined by

$d\tilde{\phi}=-iR^{2}\star d\phi\,.$ (57)

We refer to them as $U(1)_{m}$ and $U(1)_{w}$, where the subscripts stand for
momentum and winding.

This theory also enjoys $\mathcal{T}$-duality which exchanges $d\phi$ and
$d\tilde{\phi}$, while acting on the radius as $R\rightarrow 1/R$. At the
self-dual point $R=1$, it becomes a global symmetry exchanging $d\phi$ and
$i\star d\phi$. In the rest of this section, we refer to a defect implementing
this transformation (while leaving the radius invariant) as $\mathcal{T}$
defect.

As in the Maxwell case, we define a codimension-one defect (i.e. a line) by
the action

$S=\int_{S^{-}}\mathcal{L}(R;\phi_{L})+\int_{S^{+}}\mathcal{L}(R;\phi_{R})+\int_{S}\mathcal{L}_{S}(\phi_{L},\phi_{R},b)\,.$
(58)

Our goal is to investigate defects that mix $d\phi$ and $i\star d\phi$ as

$\displaystyle d\phi_{L}|_{S}$
$\displaystyle=\alpha\,d\phi_{R}|_{S}+\beta\;i\star d\phi_{R}|_{S}\,,$ (59)
$\displaystyle i\star d\phi_{L}|_{S}$
$\displaystyle=\gamma\,d\phi_{R}|_{S}+\delta\;i\star d\phi_{R}|_{S}\,,$ (60)

where $\alpha$, $\beta$, $\gamma$ and $\delta$ are real parameters. As in the
Maxwell case, in order to have a topological defect we need to impose the
condition (5), where now the energy-momentum tensor is given by

$T_{\mu\nu}(\phi)=\frac{R^{2}}{2\pi}\left(\partial_{\mu}\phi\,\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\partial_{\alpha}\phi\,\partial^{\alpha}\phi\right)\,.$
(61)

In contrast with the Maxwell theory, in this case we only get four discrete
solutions. They are as follows:

  * •

The trivial defect ($d\phi_{L}|_{S}=d\phi_{R}|_{S}$, $\star
d\phi_{L}|_{S}=\star d\phi_{R}|_{S}$) for an arbitrary value of $R^{2}$,

$\mathbbm{1}:\quad\frac{i}{2\pi}\varphi(d\phi_{L}-d\phi_{R})~{},$ (62)

where $\varphi$ is a compact scalar field living on the defect. We can also
multiply this defect Lagrangian by an integer $\kappa$, as in the case of the
condensation defect in Maxwell theory (see equation (20)). However, in this
case, the defect becomes non-simple and it is rather a direct sum of
$U(1)_{m}$ defects (see appendix E). This is an example where naively choosing
a defect Lagrangian might a priori look like giving rise to a non-invertible
defect, but it actually turns out to be a direct sum of invertible ones.

  * •

The $\mathbb{Z}_{2}$ defect ($d\phi_{L}|_{S}=-d\phi_{R}|_{S}$, $\star
d\phi_{L}|_{S}=-\star d\phi_{R}|_{S}$) for an arbitrary value of $R^{2}$,

$\mathbb{Z}_{2}:\quad\frac{i}{2\pi}\varphi(d\phi_{L}+d\phi_{R})~{},$ (63)

where $\varphi$ again is a compact scalar field living on the defect.
Similarly to the trivial defect, multiplying the above Lagrangian by an
integer makes the defect non-simple. Note that it is straightforward to
generalize the Lagrangian construction of the trivial and the $\mathbb{Z}_{2}$
defects for a free scalar theory in arbitrary dimension.

  * •

The $\mathcal{T}$ defect ($d\phi_{L}|_{S}=i\star d\phi_{R}|_{S}$) which is
described by the Lagrangian

$\mathcal{T}:\quad\frac{iN_{1}}{2\pi}\phi_{L}d\varphi_{1}+\frac{iN_{2}}{2\pi}\phi_{R}d\varphi_{2}-\frac{iN_{12}}{2\pi}\varphi_{1}d\varphi_{2}\,,$
(64)

where $\varphi_{1}$ and $\varphi_{2}$ are two compact scalars living on the
defect, and the $N_{i}$’s are integers. This realizes the $\mathcal{T}$ defect
for rational values of the radius squared:

$R^{2}=\frac{N_{1}N_{2}}{N_{12}}\in\mathbb{Q}\,.$ (65)

We see that there are different values of the $N_{i}$’s leading to the same
radius. To avoid overcounting defects we can just make the choice $N_{1}=1$,
with $N_{2}$ and $N_{12}$ coprime. The case with $R^{2}\in\mathbb{N}$ was
previously considered in [17], and corresponds to the choice $N_{12}=1$ which
leads to the defect Lagrangian

$\frac{i}{2\pi}\phi_{L}d\varphi_{1}+\frac{iN_{2}}{2\pi}\phi_{R}d\varphi_{2}-\frac{i}{2\pi}\varphi_{1}d\varphi_{2}\,.$
(66)

Performing the change of variables
$\varphi_{1}\rightarrow\varphi_{1}+N_{2}\phi_{R}$ and
$\varphi_{2}\rightarrow\varphi_{2}-\phi_{L}$ we end up with

$\frac{iN_{2}}{2\pi}\phi_{L}d\phi_{R}-\frac{i}{2\pi}\varphi_{1}d\varphi_{2}\,,$
(67)

which is the same defect Lagrangian considered in [17] plus a trivial TQFT.

Let us stress again that this defect exchanges $d\phi$ and $i\star d\phi$, but
leaves the bulk theory invariant (and hence $R$ is the same on both sides). In
appendix E we compute the fusion of two $\mathcal{T}$ defects in the case of
integer $R^{2}=N$. For $N=1$ it is invertible, while for $N\neq 1$ we show
that $\mathcal{T}\times\mathcal{T}$ is a sum of $U(1)_{m}$ defects [45].

  * •

Finally, we also have the product $\mathbb{Z}_{2}\times\mathcal{T}$ defect
($d\phi_{L}|_{S}=-i\star d\phi_{R}|_{S}$) for $R^{2}\in\mathbb{Q}$.

## Acknowledgements

We would like to thank Thomas Dumitrescu for multiple discussions. We are
grateful to Lorenzo Di Pietro, Luigi Tizzano, Sahand Seifnashri, and Shu-Heng
Shao for comments on the manuscript. OS would like to thank Gabi Zafrir, and
PN would like to thank Edoardo Lauria and Jeremias Aguilera Damia for related
discussions. Our work is supported by the Mani L. Bhaumik Institute for
Theoretical Physics. PN is supported by a DOE Early Career Award under DE-
SC0020421.

##  Appendix A Defects from gauging and dualities

In this appendix we show how to construct defects of the type considered in
section 2.2 by combining gauging of discrete subgroups of the one-form
symmetry with elements of the $SL(2,\mathbbm{Z})$ duality, in the spirit of
the recent literature [17, 29, 28, 19]. As we discussed in section 3, such
defects act on the one-form electric and magnetic charges as

$\begin{pmatrix}n\\\
m\end{pmatrix}\rightarrow\begin{pmatrix}\frac{N_{L}}{N}&N-\frac{N_{L}N_{R}}{N}\\\
-\frac{1}{N}&\frac{N_{R}}{N}\end{pmatrix}\begin{pmatrix}n\\\
m\end{pmatrix}\,,$ (68)

where the $N$’s are generally rational. Notice that the above matrix has unit
determinant and therefore it is an element of $SL(2,\mathbbm{Q})$. In the
following, we show that such transformation can be also obtained by a sequence
of $SL(2,\mathbbm{Z})$ duality transformations along with gauging of discrete
subgroups of the one-form symmetry.

We can decompose the above matrix as

$U\cdot G\cdot T=\begin{pmatrix}1&0\\\ y&1\end{pmatrix}\begin{pmatrix}z&0\\\
0&z^{-1}\end{pmatrix}\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}~{},$ (69)

where

$x=\frac{N^{2}}{N_{L}}-N_{R}\equiv\frac{p_{x}}{q_{x}}\,,\qquad
y=-\frac{1}{N_{L}}\equiv\frac{p_{y}}{q_{y}}\,,\qquad
z=\frac{N_{L}}{N}\equiv\frac{p_{z}}{q_{z}}~{},$ (70)

with $p_{i},q_{i}\in\mathbbm{Z}$ such that gcd$(p_{i},q_{i})=1$. A
$G$-transformation is just the gauging of a
$\mathbbm{Z}_{q}^{e}\times\mathbbm{Z}_{p}^{m}$ subgroup of the $U(1)_{e}\times
U(1)_{m}$ one-form symmetry. To see this, consider for simplicity only the
gauging of a $\mathbbm{Z}_{q}^{e}$ electric one-form symmetry. After the
gauging we have a $U(1)/\mathbbm{Z}_{q}$ gauge theory with the same coupling
$\tau$, but whose gauge field $\tilde{A}$ has fluxes that are multiples of
$\frac{1}{q}$. This $U(1)/\mathbbm{Z}_{q}$ theory can then be rewritten as a
$U(1)$ theory by redefining $A^{\prime}=q\tilde{A}$, thus changing the
coupling to $\tau^{\prime}=\tau/q^{2}$. In this procedure, due to the
rescaling, note that the line with charges $(q,0)$ in the original theory
before gauging maps to the line with charges $(1,0)$. We therefore see that
gauging a $\mathbbm{Z}_{q}^{e}$ subgroup of the electric one-form symmetry
provides a map from a $U(1)$ gauge theory with coupling $\tau$ to another one
with coupling $\tau/q^{2}$, and the charges map as

$\begin{pmatrix}n\\\ m\end{pmatrix}\rightarrow\begin{pmatrix}\frac{1}{q}&0\\\
0&q\end{pmatrix}\begin{pmatrix}n\\\ m\end{pmatrix}\,.$ (71)

Similarly, the gauging of a $\mathbbm{Z}_{p}^{m}$ magnetic one-form symmetry
results in the rescaling $\tau\rightarrow p^{2}\tau$ and the map on charges

$\begin{pmatrix}n\\\ m\end{pmatrix}\rightarrow\begin{pmatrix}p&0\\\
0&\frac{1}{p}\end{pmatrix}\begin{pmatrix}n\\\ m\end{pmatrix}\,.$ (72)

A $T$-transformation is a combination of two $G$-transformations and a
$\mathcal{T}$-duality,555A $T$-transformation can also be obtained by stacking
SPT phases and gauging a $\mathbbm{Z}_{q}^{m}\in U(1)_{m}$ magnetic one-from
symmetry, an operation usually referred to as $CT^{(k)}S^{(k)}T^{(k)}$ gauging
(see e.g. [29]). as it can be seen from

$\begin{pmatrix}1&\frac{p}{q}\\\
0&1\end{pmatrix}=\begin{pmatrix}\frac{1}{q}&0\\\
0&q\end{pmatrix}\begin{pmatrix}1&pq\\\ 0&1\end{pmatrix}\begin{pmatrix}q&0\\\
0&\frac{1}{q}\end{pmatrix}\,,$ (73)

while a $U$-transformation can be obtained from $T$ using
$\mathcal{S}$-duality as follows

$\begin{pmatrix}1&0\\\ y&1\end{pmatrix}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}1&-y\\\ 0&1\end{pmatrix}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\,.$ (74)

Most importantly, we see that (68) can be obtained from a series of
transformations which leave the coupling $\tau$ invariant, since

$\tau\quad\rightarrow\quad\frac{\frac{N_{L}}{N}\tau+N-\frac{N_{L}N_{R}}{N}}{-\frac{1}{N}\tau+\frac{N_{R}}{N}}=\tau~{},$
(75)

if

$\tau=\frac{\theta}{2\pi}+\frac{i}{e^{2}}=\frac{N_{R}-N_{L}}{2}+i\sqrt{N^{2}-\left(\frac{N_{R}+N_{L}}{2}\right)^{2}}~{},$
(76)

as in equation (38). Hence, we conclude that one can construct defects that
generate an $SO(2)$ rotation on $F$ and $\star F$ with
$\cos\varphi\in\mathbbm{Q}$, by combining the gauging of discrete subgroups of
the one-form symmetry on half-space with elements of the $SL(2,\mathbbm{Z})$
duality group.

##  Appendix B Action of surface defects in Chern-Simons theory

In this appendix, we review the action of condensation defects on lines in
Chern-Simons theory and we study their action on line operators in the spirit
of section 3. We consider their Lagrangian description as in [21] and we
analyze the scalar operators which serve as junctions between Wilson lines on
the left and right of the defect. These operators give a nice interpretation
for the action of the defect on lines. In particular, if there are multiple
allowed junction operators then the action of the defect on lines is not
simple and results in a sum of several terms.

In [21], it was shown that topological surfaces in $U(1)_{2N}$ Chern-Simons
theory are labeled by a divisor of $N=nm$ and can be described by the action

$S_{CS}={2iN\over 4\pi}\int_{S^{-}}A_{L}\wedge dA_{L}+{2iN\over
4\pi}\int_{S^{+}}A_{R}\wedge dA_{R}+\int_{S}\left({in\over
2\pi}d\phi\wedge(A_{L}-A_{R})+{iN\over 2\pi}\int_{S}A_{L}\wedge A_{R}\right),$
(77)

which is invariant under the gauge transformations

$A_{L}\rightarrow A_{L}+d\alpha_{L}\,,\qquad A_{R}\rightarrow
A_{R}+d\alpha_{R}\,,\qquad\phi\rightarrow\phi+m(\alpha_{L}+\alpha_{R})\,.$
(78)

The equations of motion on the defect are

$dA_{L}|_{S}=dA_{R}|_{S}\,,\qquad d\phi=m(A_{L}+A_{R})|_{S}\,.$ (79)

Considering the second equation, the line integral of the left-hand side
around the junction measures the winding of $\phi$

$\oint_{\gamma}\frac{d\phi}{2\pi}=\lambda\in\mathbb{Z}\,,$ (80)

while the integral of the right-hand side measures the one-form charges of
left and right Wilson lines, leading to

$\lambda=m\left(\frac{q_{L}}{2N}+\frac{q_{R}}{2N}\right)~{}.$ (81)

Using the fact that $N=nm$ we arrive at

$q_{L}+q_{R}=2n\lambda~{},$ (82)

where $\lambda=0,\dots,m-1$ since the charges in the bulk are defined modulo
$2N=2nm$. For each $\lambda$ we have an operator sitting at the junction which
converts a line coming from the right to a line exiting to the left of the
defect. We can then write the action on lines as

$S_{n}\cdot W_{q}=\sum_{\begin{subarray}{c}\lambda=0\\\
q-n\lambda=0~{}\text{mod}~{}m\end{subarray}}^{m-1}W_{2n\lambda-q}\,.$ (83)

This is in agreement with the result of [21] (see equation 6.16), where it was
shown that the action of a defect on a Wilson line of charge $q$ is given by

$S_{n}\cdot W_{q}=\sum_{\begin{subarray}{c}k=0\\\
mk+q=0~{}\text{mod}~{}n\end{subarray}}^{n-1}W_{q+2mk}\,.$ (84)

An equivalent way to determine the action on lines is by considering the
scalar operators on the defect that can sit at the junction. Apart from the
operator $e^{ik\phi}$ there is also another scalar living on the defect,
namely the twist operator $e^{i\lambda\bar{\phi}}$ around which $\phi$ has
winding $\lambda$, as in (80).666One way to analyze this operator is by
dualizing $\phi$ in (77). The Lagrangian becomes  $\int_{S}\left({in\over
2\pi}b\wedge(A_{L}-A_{R})-{i\over 2\pi}\bar{\phi}\,db+{iN\over
2\pi}A_{L}\wedge A_{R}\right)\,,$ (85) and the additional gauge transformation
for $\bar{\phi}$ reads
$\bar{\phi}\rightarrow\bar{\phi}+n(\alpha_{L}-\alpha_{R})$.  Because of the
third term in (77) the twist operator $e^{i\lambda\bar{\phi}}$ is not gauge
invariant and should be attached to a Wilson line of $A_{L}$ of charge $n$,
and to a Wilson line of $A_{R}$ of the same charge. In order to transform a
Wilson line $W_{R}$ coming from the right into a Wilson line $W_{L}$ exiting
to the left in a gauge-invariant way, we need a gauge invariant junction on
$S$. If we consider the operator $\exp(ik\phi+i\lambda\bar{\phi})$ at the
junction (see figure 3),

$W_{R}=e^{iq_{R}\int A_{R}}$$W_{L}=e^{iq_{L}\int
A_{L}}$$S$$e^{ik\phi+i\lambda\bar{\phi}}$ Figure 3: Gauge-invariant
configuration of Wilson lines and twist operators on a defect $S$ in
$U(1)_{2N}$ Chern-Simons theory.

gauge invariance requires

$q_{L}-km-\lambda n=0\,,\qquad q_{R}+km-\lambda n=0\,.$ (86)

This implies

$q_{L}=q_{R}+2mk\,,$ (87)

where $k=0,1,\dots,n-1$ labels the ways to convert a right Wilson line into a
left Wilson line. This leads to (84) or, equivalently, to (83) if in (86) we
eliminate $k$ instead of $\lambda$.

##  Appendix C $\mathcal{S}$ defect at integer coupling

Let us examine the action on lines of the $\mathcal{S}$ defect constructed in
[17], using the dual-frame description of one of the bulk sides. We consider
the action at $\theta=0$ and $1/e^{2}=N\in\mathbb{N}$, which reads777Notice
that there is a sign difference in the defect Lagrangian in (88) compared to
the one in [17].

$S=\frac{N}{4\pi}\int_{S^{-}}dA_{L}\wedge\star
dA_{L}-\frac{iN}{2\pi}\int_{S}A_{L}\wedge
dA_{R}+\frac{N}{4\pi}\int_{S^{+}}dA_{R}\wedge\star dA_{R}\,,$ (88)

and choose to dualize one of the sides, say the right one. Following [43], we
rewrite (88) as

$\begin{split}S=&\frac{N}{4\pi}\int_{S^{-}}dA_{L}\wedge\star
dA_{L}-\frac{iN}{2\pi}\int_{S}A_{L}\wedge(dA_{R}-G_{R})-\frac{i}{2\pi}\int_{S}\phi_{R}dG_{R}\\\
&+\frac{N}{4\pi}\int_{S^{+}}(dA_{R}-G_{R})\wedge\star(dA_{R}-G_{R})-\frac{i}{2\pi}\int_{S^{+}}V_{R}\wedge
dG_{R}\,,\end{split}$ (89)

where $V_{R}$ and $G_{R}$ are one-form and two-form fields, respectively. In
the presence of the defect, we are also forced to introduce a scalar field
$\phi_{R}$ living on the defect in order to preserve the gauge transformations

$\begin{split}A_{L}\rightarrow A_{L}+d\alpha_{L}\;,\qquad&A_{R}\rightarrow
A_{R}+b_{R}\;,\qquad G_{R}\rightarrow G_{R}+db_{R}\,,\\\ V_{R}\rightarrow
V_{R}+d\widetilde{\alpha}_{R}\;&,\qquad\phi_{R}\rightarrow\phi_{R}-N\alpha_{L}+\widetilde{\alpha}_{R}\,.\end{split}$
(90)

This scalar is the dual photon of $A_{R}$ on the defect, and $e^{i\phi_{R}}$
is the corresponding monopole operator. Notice that this monopole transforms
non-trivially also under the gauge group $U(1)_{L}$, as is evident both from
(88) and (90). We can now gauge fix $A_{R}$ to zero, and arrive at

$\begin{split}S=&\frac{N}{4\pi}\int_{S^{-}}dA_{L}\wedge\star
dA_{L}+\frac{iN}{2\pi}\int_{S}A_{L}\wedge
G_{R}-\frac{i}{2\pi}\int_{S}\phi_{R}dG_{R}\\\
&+\frac{N}{4\pi}\int_{S^{+}}G_{R}\wedge\star
G_{R}-\frac{i}{2\pi}\int_{S^{+}}V_{R}\wedge dG_{R}\,.\end{split}$ (91)

The equation of motion for $G_{R}$ on the defect is

$V_{R}|_{S}=NA_{L}|_{S}+d\phi_{R}\,.$ (92)

Using this equation, we can see that a Wilson line for $V_{R}$ (or
equivalently a ’t Hooft line for $A_{R}$) with charge $p$ (for
$p\in\mathbb{Z}$) coming from the right of the defect is transformed into a
Wilson line with charge $pN$ from the left, matching (48).

Let us examine in more detail what this configuration looks like. We begin
with a ’t Hooft line $\exp(ip\int V_{R})$ coming from the right and ending on
the defect. Under the gauge transformations (90) it transforms by a phase
$\exp(ip\widetilde{\alpha}_{R})$, which can in turn be canceled by inserting
the monopole operator $\exp(-ip\phi_{R})$ at its end. However, this
configuration is still not gauge invariant and transforms by the phase
$\exp(ipN\alpha_{L})$ which is then canceled by attaching the Wilson line
$\exp(ipN\int A_{L})$ exiting the defect to the left. The same is true also in
the other direction, and a Wilson line with charge $pN$ piercing the defect
from the left is transformed into a ’t Hooft line with charge $p$ from the
right (where now the monopole at the junction is $\exp(ip\phi_{R})$). From
this description, it is clear that if the charge of this Wilson line is not a
multiple of $N$, it cannot pierce the defect in a gauge invariant way and is
therefore annihilated by it. Notice also that a similar dualization can be
done to the left side, reproducing the full picture in (48) which in this case
is given by

$\mathcal{S}\cdot W_{n}H_{m}=\begin{cases}W_{mN}H_{-n/N}\quad&\text{if
}N|n\,,\\\ 0\quad&\text{otherwise}\,.\end{cases}$ (93)

##  Appendix D Action of generic defects on lines

In this appendix, we derive the result (53) for the action of a generic defect
on lines. Consider the full action

$S=\int_{S^{-}}\left(\frac{1}{4\pi e^{2}}F_{L}\wedge\star
F_{L}+\frac{i\theta}{8\pi^{2}}F_{L}\wedge
F_{L}\right)+\int_{S^{+}}\left(\frac{1}{4\pi e^{2}}F_{R}\wedge\star
F_{R}+\frac{i\theta}{8\pi^{2}}F_{R}\wedge
F_{R}\right)+\int_{S}\mathcal{L}_{S}(A_{L},A_{R})\,,$ (94)

where $\mathcal{L}_{S}$ is the generic defect Lagrangian (34), explicitly
given by

$\begin{split}\mathcal{L}_{S}=&\frac{ik_{L}}{4\pi}A_{L}\wedge
dA_{L}+\frac{ik_{R}}{4\pi}A_{R}\wedge dA_{R}-\frac{ik}{2\pi}A_{L}\wedge
dA_{R}\\\
&+\mathcal{A}^{q_{1},p_{1}}\left[\frac{l_{1}\,dA_{L}+r_{1}\,dA_{R}}{q_{1}}\right]+\mathcal{A}^{q_{2},p_{2}}\left[\frac{l_{2}\,dA_{L}+r_{2}\,dA_{R}}{q_{2}}\right].\end{split}$
(95)

For generic values of the parameters, there is no simple description for the
above defect Lagrangian. Nevertheless, the $\mathcal{A}^{q,p}$ TQFT can be
realized on the boundary of a $3+1$ dimensional model. In the following, we
make use of the folding trick and rewrite the bulk theory (94) as a theory
only on the right of the defect $S^{+}$ with a boundary on $S$. We then use
the aforementioned $3+1$ dimensional description of the defect TQFT. For
simplicity, we consider a flat boundary at $x=0$.

We can replace the left side of the defect by its $x$-reversal living on
$S^{+}$. Then, the bulk action is given by

$S=\int_{S^{+}}\left(\frac{1}{4\pi e^{2}}F_{R}\wedge\star
F_{R}+\frac{i\theta}{8\pi^{2}}F_{R}\wedge F_{R}+\frac{1}{4\pi
e^{2}}\overline{F}_{L}\wedge\star\overline{F}_{L}-\frac{i\theta}{8\pi^{2}}\overline{F}_{L}\wedge\overline{F}_{L}\right)+\int_{S}\mathcal{L}_{S}(A_{L},A_{R})\,.$
(96)

where $\overline{F}=d\overline{A}$ satisfies

$\overline{F}|_{S}=F|_{S},\quad\star\overline{F}|_{S}=-\star F|_{S}~{}.$ (97)

The minimal $\mathcal{A}^{q,p}$ TQFT can be realized on the boundary of a
$3+1$ dimensional system [44] with action

$\int_{S^{+}}\left(\frac{iq}{2\pi}b\land dc+\frac{iqh}{4\pi}b\land
b+\frac{i}{2\pi}b\land dA\right)\,,$ (98)

where $b$ is a two-form gauge field with Dirichlet boundary conditions on $S$,
$c$ is a one-form gauge field constraining $b$ to be a $\mathbb{Z}_{q}$ gauge
field, and $h$ is the inverse of $p$ mod $q$. The above is equivalent to the
combined action

$-\int_{S^{+}}\frac{ip}{4\pi q}dA\land
dA+\int_{S}\mathcal{A}^{q,p}\left[dA/q\right]\,.$ (99)

Hence, we can replace (94) by

$S=\int_{S^{+}}\left(\frac{1}{4\pi e^{2}}F_{R}\wedge\star
F_{R}+\frac{i}{4\pi}\left(\frac{\theta}{2\pi}-N_{R}\right)F_{R}\wedge
F_{R}+\frac{1}{4\pi
e^{2}}\overline{F}_{L}\wedge\star\overline{F}_{L}-\frac{i}{4\pi}\left(\frac{\theta}{2\pi}+N_{L}\right)\overline{F}_{L}\wedge\overline{F}_{L}\right)\\\
+\int_{S^{+}}\frac{iN}{2\pi}\overline{F}_{L}\wedge
F_{R}+\int_{S^{+}}\left(\frac{iq_{1}}{2\pi}b_{1}\land
dc_{1}+\frac{iq_{1}h_{1}}{4\pi}b_{1}\land
b_{1}+\frac{i}{2\pi}b_{1}\land\left(l_{1}\overline{F}_{L}+r_{1}F_{R}\right)\right)\\\
+\int_{S^{+}}\left(\frac{iq_{2}}{2\pi}b_{2}\land
dc_{2}+\frac{iq_{2}h_{2}}{4\pi}b_{2}\land
b_{2}+\frac{i}{2\pi}b_{2}\land\left(l_{2}\overline{F}_{L}+r_{2}F_{R}\right)\right)\,,$
(100)

where $N_{L}$, $N_{R}$, and $N$ are defined as in (36). Now, the equations of
motion for $A_{L}$, $A_{R}$, and $b_{i}$ on the boundary $S$ (where $b_{i}=0$
due to the Dirichlet boundary conditions) are, respectively

$\frac{\widetilde{F}_{L}}{2\pi}{\Big{|}}_{S}+N_{L}\frac{F_{L}}{2\pi}{\Big{|}}_{S}-N\frac{F_{R}}{2\pi}{\Big{|}}_{S}=0\,,\qquad\frac{\widetilde{F}_{R}}{2\pi}{\Big{|}}_{S}+N\frac{F_{L}}{2\pi}{\Big{|}}_{S}-N_{R}\frac{F_{R}}{2\pi}{\Big{|}}_{S}=0\,,$
(101)

and

$q_{1}\frac{dc_{1}}{2\pi}{\Big{|}}_{S}+l_{1}\frac{F_{L}}{2\pi}{\Big{|}}_{S}+r_{1}\frac{F_{R}}{2\pi}{\Big{|}}_{S}=0\,,\qquad
q_{2}\frac{dc_{2}}{2\pi}{\Big{|}}_{S}+l_{2}\frac{F_{L}}{2\pi}{\Big{|}}_{S}+r_{2}\frac{F_{R}}{2\pi}{\Big{|}}_{S}=0\,.$
(102)

The first two equations are analogous to the case with integer $N$’s given in
(46). They are just a consequence of the sewing conditions, and imply

$n_{L}=Nm_{R}-N_{L}m_{L}\,,\qquad n_{R}=N_{R}m_{R}-Nm_{L}\,.$ (103)

In addition, the equations in (102) give extra conditions securing integer
monopole fluxes $m_{c_{1}}$ and $m_{c_{2}}$

$q_{1}m_{c_{1}}+l_{1}m_{L}+r_{1}m_{R}=0\,,\qquad
q_{2}m_{c_{2}}+l_{2}m_{L}+r_{2}m_{R}=0\,.$ (104)

Overall, combining equations (103) and (104) we obtain the following action of
a generic defect of the type discussed in section 2.2 on lines:

$S\cdot
W_{n}H_{m}=\begin{cases}W_{mN+\frac{(n-mN_{R})N_{L}}{N}}H_{\frac{mN_{R}-n}{N}}\qquad&\text{if
}d\,|\,w,\,h,\,u_{1}\text{ and }u_{2}\,,\\\
0\qquad&\text{otherwise}\,,\end{cases}$ (105)

where

$\begin{split}d&=kq_{1}q_{2}+l_{1}p_{1}q_{2}r_{1}+l_{2}p_{2}q_{1}r_{2}\,,\\\
u_{i}&=l_{i}nq_{i+1}-l_{i}mk_{R}q_{i+1}-mkq_{i+1}r_{i}-l_{i+1}mp_{i+1}r_{1}r_{2}+l_{i}mp_{i+1}r_{i+1}^{2}\,,\\\
h&=-nq_{1}q_{2}+mk_{R}q_{1}q_{2}-mp_{1}q_{2}r_{1}^{2}-mp_{2}q_{1}r_{2}^{2}\,,\\\
w&=q_{1}q_{2}\left(nk_{L}+mk^{2}-mk_{L}k_{R}\right)+p_{2}q_{1}\left(2ml_{2}kr_{2}+mk_{L}r_{2}^{2}-l_{2}^{2}n+l_{2}^{2}mk_{R}\right)\\\
&+p_{1}q_{2}\left(l_{1}^{2}mk_{R}+2l_{1}mkr_{1}+mk_{L}r_{1}^{2}-l_{1}^{2}n\right)-mp_{1}p_{2}\left(l_{1}r_{2}-l_{2}r_{1}\right)^{2}\,.\end{split}$
(106)

Here, the subscript $i=1,2$ should be regarded modulo 2.

##  Appendix E Fusion of $\mathcal{T}$ defects

In [45] (see also [12]) it was shown that the $\mathcal{T}$ defect is non-
invertible for $N\neq 1$ and squares to a sum of $U(1)_{m}$ defects. Let us
now compute its fusion rule (we consider here the case of integer $R^{2}=N$)
using the defect Lagrangian.

Consider two $\mathcal{T}$ defects at positions $S$ and $S+\delta S$, and let
$S^{I}$ be the region of spacetime between those defects. The action is then

$\begin{split}S=&\,\frac{N}{4\pi}\int_{S^{-}}d\phi_{L}\wedge\star
d\phi_{L}+\frac{iN}{2\pi}\int_{S}\phi_{L}d\phi_{I}+\frac{N}{4\pi}\int_{S^{I}}d\phi_{I}\wedge\star
d\phi_{I}\\\ &+\frac{iN}{2\pi}\int_{S+\delta
S}\phi_{I}d\phi_{R}+\frac{N}{4\pi}\int_{S^{+}}d\phi_{R}\wedge\star
d\phi_{R}\,,\end{split}$ (107)

and taking the limit $\delta S\rightarrow 0$ we arrive at

$S=\frac{N}{4\pi}\int_{S^{-}}d\phi_{L}\wedge\star
d\phi_{L}+\frac{iN}{2\pi}\int_{S}d\phi_{I}(\phi_{L}-\phi_{R})+\frac{N}{4\pi}\int_{S^{+}}d\phi_{R}\wedge\star
d\phi_{R}~{}.$ (108)

We can rewrite the above as

$S=\frac{N}{4\pi}\int_{S^{-}}d\phi_{L}\wedge\star
d\phi_{L}+\frac{i}{2\pi}\int_{S}d\varphi(\phi_{L}-\phi_{R})+\frac{i}{N}\,\eta\int_{S}d\varphi+\frac{N}{4\pi}\int_{S^{+}}d\phi_{R}\wedge\star
d\phi_{R}~{},$ (109)

where $d\varphi$ is a compact scalar $\varphi\sim\varphi+2\pi$ and
$\eta=0,1,\dots,N-1$ is an integer-valued field. To see that, notice that the
path integral over $\eta$ is just the sum

$\int
D\eta\,e^{\frac{i}{N}\,\eta\int_{S}d\varphi}=\sum_{\eta=0}^{N-1}e^{\frac{i}{N}\eta\int_{S}d\varphi}\,,$
(110)

which imposes

$\int_{S}d\varphi\in 2\pi N\mathbbm{Z}\,.$ (111)

Redefining $\varphi=N\phi_{I}$ we arrive back to (108). All in all, we get
that the fusion of two $\mathcal{T}$ defects is described by the action

$S=\frac{N}{4\pi}\int_{S^{-}}d\phi_{L}\wedge\star
d\phi_{L}+\frac{i}{2\pi}\int_{S}d\varphi\left(\phi_{L}-\phi_{R}+\frac{2\pi}{N}\eta\right)+\frac{N}{4\pi}\int_{S^{+}}d\phi_{R}\wedge\star
d\phi_{R}~{},$ (112)

with $\eta=0,1,\dots,N-1$.

On the other hand, let us consider the insertion of a $U(1)_{m}$ defect with
parameter $\beta$. The action is

$S=\frac{N}{4\pi}\int_{S^{-}\cup S^{+}}d\phi\wedge\star
d\phi+\beta\frac{N}{2\pi}\int_{S}\star d\phi\,,$ (113)

and introducing a Lagrange multiplier $\varphi$ it can be rewritten as

$S=\frac{N}{4\pi}\int_{S^{-}}d\phi_{L}\wedge\star
d\phi_{L}+\beta\frac{N}{2\pi}\int_{S}\star
d\phi_{L}+\frac{i}{2\pi}\int_{S}d\varphi(\phi_{L}-\phi_{R})+\frac{N}{4\pi}\int_{S^{+}}d\phi_{R}\wedge\star
d\phi_{R}\,.$ (114)

Since on $S$ we have the relation $d\phi_{L}=d\phi_{R}$ we can write the
second term using either fields. We can now remove that term by shifting
$\phi_{L}$ by a step function as follows,

$\phi_{L}\rightarrow\phi_{L}-\beta\Theta(S^{+})\,.$ (115)

However, while the change of the first term cancels with the second term, the
third term becomes

$\frac{i}{2\pi}\int_{S}d\varphi\left(\phi_{L}-\phi_{R}-\beta\right)\,.$ (116)

In conclusion, comparing (112) with (116) we see that the product of two
$\mathcal{T}$ defects is a sum of $U(1)_{m}$ defects,

$\mathcal{T}\times\mathcal{T}=\sum_{\eta=0}^{N-1}e^{-\frac{2\pi\eta}{N}\frac{N}{2\pi}\oint\star
d\phi}~{}.$ (117)

Note that for $N=1$ it becomes invertible.

Let us comment that this calculation also explains a statement we made below
equation (62), according to which multiplying (62) by an integer $\kappa$ does
not result in a new simple defect but rather in a sum of $U(1)_{m}$ defects.
Indeed, as shown above, the defect Lagrangian

$\frac{i\kappa}{2\pi}d\varphi(\phi_{L}-\phi_{R})$ (118)

can be written in the form

$\frac{i}{2\pi}\int_{S}d\varphi\left(\phi_{L}-\phi_{R}+\frac{2\pi}{\kappa}\eta\right)\,,$
(119)

where $\eta=0,1,\dots,\kappa-1$ is an integer-valued field. Then, the defect
Lagrangian (119) is just a sum of $U(1)_{m}$ defects,

$\sum_{\eta=0}^{\kappa-1}e^{-\frac{2\pi\eta}{\kappa}\frac{R^{2}}{2\pi}\oint\star
d\phi}~{}.$ (120)

## References

  * [1] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, _Generalized Global Symmetries_ , _JHEP_ 02 (2015) 172, [1412.5148].
  * [2] E. P. Verlinde, _Fusion Rules and Modular Transformations in 2D Conformal Field Theory_ , _Nucl. Phys. B_ 300 (1988) 360–376.
  * [3] V. Petkova and J. Zuber, _Generalized twisted partition functions_ , _Phys. Lett. B_ 504 (2001) 157–164, [hep-th/0011021].
  * [4] V. Petkova and J.-B. Zuber, _Conformal field theories, graphs and quantum algebras_ , vol. 23, pp. 415–435.  2002\.  hep-th/0108236.
  * [5] J. Fuchs, I. Runkel and C. Schweigert, _TFT construction of RCFT correlators 1. Partition functions_ , _Nucl. Phys. B_ 646 (2002) 353–497, [hep-th/0204148].
  * [6] J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, _Kramers-Wannier duality from conformal defects_ , _Phys. Rev. Lett._ 93 (2004) 070601, [cond-mat/0404051].
  * [7] J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, _Duality and defects in rational conformal field theory_ , _Nucl. Phys. B_ 763 (2007) 354–430, [hep-th/0607247].
  * [8] L. Bhardwaj and Y. Tachikawa, _On finite symmetries and their gauging in two dimensions_ , _JHEP_ 03 (2018) 189, [1704.02330].
  * [9] C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, _Topological Defect Lines and Renormalization Group Flows in Two Dimensions_ , _JHEP_ 01 (2019) 026, [1802.04445].
  * [10] W. Ji, S.-H. Shao and X.-G. Wen, _Topological Transition on the Conformal Manifold_ , _Phys. Rev. Res._ 2 (2020) 033317, [1909.01425].
  * [11] R. Thorngren and Y. Wang, _Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases_ , 1912.02817.
  * [12] R. Thorngren and Y. Wang, _Fusion Category Symmetry II: Categoriosities at $c$ = 1 and Beyond_, 2106.12577.
  * [13] E. Sharpe, _Topological operators, noninvertible symmetries and decomposition_ , 2108.13423.
  * [14] I. M. Burbano, J. Kulp and J. Neuser, _Duality Defects in $E_{8}$_, 2112.14323.
  * [15] Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, _Symmetries and strings of adjoint QCD 2_, _JHEP_ 03 (2021) 103, [2008.07567].
  * [16] M. Nguyen, Y. Tanizaki and M. Ünsal, _Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality_, _JHEP_ 03 (2021) 238, [2101.02227].
  * [17] Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam and S.-H. Shao, _Noninvertible duality defects in 3+1 dimensions_ , _Phys. Rev. D_ 105 (2022) 125016, [2111.01139].
  * [18] J. Kaidi, K. Ohmori and Y. Zheng, _Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories_ , _Phys. Rev. Lett._ 128 (2022) 111601, [2111.01141].
  * [19] Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam and S.-H. Shao, _Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions_ , 2204.09025.
  * [20] C. Cordova and K. Ohmori, _Non-Invertible Chiral Symmetry and Exponential Hierarchies_ , 2205.06243.
  * [21] K. Roumpedakis, S. Seifnashri and S.-H. Shao, _Higher Gauging and Non-invertible Condensation Defects_ , 2204.02407.
  * [22] L. Bhardwaj, L. Bottini, S. Schafer-Nameki and A. Tiwari, _Non-Invertible Higher-Categorical Symmetries_ , 2204.06564.
  * [23] J. A. Damia, R. Argurio and L. Tizzano, _Continuous Generalized Symmetries in Three Dimensions_ , 2206.14093.
  * [24] G. Arias-Tamargo and D. Rodriguez-Gomez, _Non-Invertible Symmetries from Discrete Gauging and Completeness of the Spectrum_ , 2204.07523.
  * [25] Y. Hayashi and Y. Tanizaki, _Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly_ , _JHEP_ 08 (2022) 036, [2204.07440].
  * [26] J. Kaidi, G. Zafrir and Y. Zheng, _Non-invertible symmetries of $\mathcal{N}$ = 4 SYM and twisted compactification_, _JHEP_ 08 (2022) 053, [2205.01104].
  * [27] A. Antinucci, G. Galati and G. Rizi, _On Continuous 2-Category Symmetries and Yang-Mills Theory_ , 2206.05646.
  * [28] Y. Choi, H. T. Lam and S.-H. Shao, _Non-invertible Time-reversal Symmetry_ , 2208.04331.
  * [29] Y. Choi, H. T. Lam and S.-H. Shao, _Non-invertible Global Symmetries in the Standard Model_ , 2205.05086.
  * [30] T. Bartsch, M. Bullimore, A. E. V. Ferrari and J. Pearson, _Non-invertible Symmetries and Higher Representation Theory I_ , 2208.05993.
  * [31] B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius and I. Valenzuela, _Non-invertible global symmetries and completeness of the spectrum_ , _JHEP_ 09 (2021) 203, [2104.07036].
  * [32] C. Cordova, K. Ohmori and T. Rudelius, _Generalized Symmetry Breaking Scales and Weak Gravity Conjectures_ , 2202.05866.
  * [33] T.-C. Huang, Y.-H. Lin and S. Seifnashri, _Construction of two-dimensional topological field theories with non-invertible symmetries_ , _JHEP_ 12 (2021) 028, [2110.02958].
  * [34] V. Bashmakov, M. Del Zotto and A. Hasan, _On the 6d Origin of Non-invertible Symmetries in 4d_ , 2206.07073.
  * [35] F. Benini, C. Copetti and L. Di Pietro, _Factorization and global symmetries in holography_ , 2203.09537.
  * [36] J. A. Damia, R. Argurio and E. Garcia-Valdecasas, _Non-Invertible Defects in 5d, Boundaries and Holography_ , 2207.02831.
  * [37] F. Apruzzi, I. Bah, F. Bonetti and S. Schafer-Nameki, _Non-Invertible Symmetries from Holography and Branes_ , 2208.07373.
  * [38] I. n. García Etxebarria, _Branes and Non-Invertible Symmetries_ , 2208.07508.
  * [39] Y.-H. Lin, M. Okada, S. Seifnashri and Y. Tachikawa, _Asymptotic density of states in 2d CFTs with non-invertible symmetries_ , 2208.05495.
  * [40] J. J. Heckman, M. Hübner, E. Torres and H. Y. Zhang, _The Branes Behind Generalized Symmetry Operators_ , 2209.03343.
  * [41] L. Lin, D. G. Robbins and E. Sharpe, _Decomposition, condensation defects, and fusion_ , 2208.05982.
  * [42] G. Cuomo, M. Mezei and A. Raviv-Moshe, _Boundary conformal field theory at large charge_ , _JHEP_ 10 (2021) 143, [2108.06579].
  * [43] E. Witten, _On S duality in Abelian gauge theory_ , _Selecta Math._ 1 (1995) 383, [hep-th/9505186].
  * [44] P.-S. Hsin, H. T. Lam and N. Seiberg, _Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d_ , _SciPost Phys._ 6 (2019) 039, [1812.04716].
  * [45] J. Fuchs, M. R. Gaberdiel, I. Runkel and C. Schweigert, _Topological defects for the free boson CFT_ , _J. Phys. A_ 40 (2007) 11403, [0705.3129].
  * [46] A. Kapustin and M. Tikhonov, _Abelian duality, walls and boundary conditions in diverse dimensions_ , _JHEP_ 11 (2009) 006, [0904.0840].