When are Duality Defects Group-Theoretical?

Zhengdi Sun1 and Yunqin Zheng2,3

1 | Department of Physics, University of California, San Diego, CA 92093, USA
---|---
2 | Kavli Institute for the Physics and Mathematics of the Universe,
| University of Tokyo, Kashiwa, Chiba 277-8583, Japan
3 | Institute for Solid State Physics,
| University of Tokyo, Kashiwa, Chiba 277-8581, Japan

A quantum field theory with a finite abelian symmetry $G$ may be equipped with
a non-invertible duality defect associated with gauging $G$. For certain $G$,
duality defects admit an alternative construction where one starts with
invertible symmetries with certain ’t Hooft anomaly, and gauging a non-
anomalous subgroup. This special type of duality defects are termed group
theoretical. In this work, we determine when duality defects are group
theoretical, among $G=\mathbb{Z}_{N}^{(0)}$ and $\mathbb{Z}_{N}^{(1)}$ in $2$d
and 4d quantum field theories, respectively. A duality defect is group
theoretical if and only if its Symmetry TFT is a Dijkgraaf-Witten theory, and
we argue that this is equivalent to a certain stability condition of the
topological boundary conditions of the $G$ gauge theory. By solving the
stability condition, we find that a $\mathbb{Z}_{N}^{(0)}$ duality defect in
2d is group theoretical if and only if $N$ is a perfect square, and under
certain assumptions a $\mathbb{Z}_{N}^{(1)}$ duality defect in 4d is group
theoretical if and only if $N=L^{2}M$ where $-1$ is a quadratic residue of
$M$. For these subset of $N$, we construct explicit topological manipulations
that map the non-invertible duality defects to invertible defects. We also
comment on the connection between our results and the recent discussion of
obstruction to duality-preserving gapped phases.

###### Contents

  1. 1 Introduction and summary
    1. 1.1 The problem
    2. 1.2 Main results
  2. 2 Criteria of group-theoretical duality defects
    1. 2.1 Symmetry TFT of duality defects
    2. 2.2 Criteria for group-theoretical duality defects
  3. 3 Group theoretical duality defects in 2d
    1. 3.1 Lagrangian subgroups of 3d $\mathbb{Z}_{N}$ gauge theory
    2. 3.2 $\mathbb{Z}_{2}^{\text{em}}$ stable Lagrangian subgroup
    3. 3.3 SymTFT as a Dijkgraaf-Witten theory
    4. 3.4 Explicit topological manipulation
  4. 4 Group theoretical duality defects in 4d
    1. 4.1 Lagrangian subgroups of 5d $\mathbb{Z}_{N}$ 2-form gauge theory
    2. 4.2 $\mathbb{Z}_{4}^{\text{em}}$ stable Lagrangian subgroup
    3. 4.3 SymTFT as a Dijkgraaf-Witten theory
    4. 4.4 Explicit topological manipulations
  5. 5 Connection with obstruction to duality-preserving gapped phases
    1. 5.1 Duality defects, gauging, and SPT
    2. 5.2 Duality defects, gauging, TQFT, and relation with group theoretical duality defects
  6. A Lagrangian subgroup for $\mathbb{Z}_{N}$ $2$-form gauge theory in 5d
  7. B Lagrangian subgroup with one generator

##  1 Introduction and summary

###  1.1 The problem

##### Duality defects:

A $d$-dimensional quantum field theory (QFT) ${\mathcal{X}}$ with a finite,
non-anomalous, $p$-form abelian global symmetry $G^{(p)}$ has a _non-
invertible duality symmetry_ when ${\mathcal{X}}$ is invariant under gauging
$G^{(p)}$ [1, 2],

$\displaystyle{\mathcal{X}}={\mathcal{X}}/G^{(p)}.$ (1.1)

This in particular requires $p=\frac{d}{2}-1$, so that the gauged theory
${\mathcal{X}}/G^{(p)}$ also has the same $G^{(p)}$ global symmetry.111For
simplicity, we only consider the symmetry of a single form-degree. One can
also consider symmetries with multiple form-degrees, say, a $p$-form symmetry
and a $q$-form symmetry. The theory can also be self-dual under gauging both
symmetries if $d-2=p+q$. See e.g. [3, 4] for such examples. More generally, a
finite invertible symmetry in $d$ dimensional quantum field theory is
described by a higher-group, and gauging it leads to the higher representation
category of a higher group. Self-duality under gauging the higher group
requires its higher representation category coincides with itself. See [5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15] for the recent developments of higher
categorical theory description of global symmetries.  Each finite symmetry is
associated with a topological defect [16], and in the present case the
_duality defect_. The duality defect can be constructed by gauging $G^{(p)}$
on half-space and imposing the Dirichlet boundary condition for $G^{(p)}$ at
the defect locus. It has been shown that the duality defect satisfies _non-
invertible_ fusion rule [1, 2]. By construction, duality defect implements
gauging $G^{(p)}$. Note that there are additional data specifying the duality
defects, such as the (symmetric and non-degenerate) bicharacter and the
Frobenius-Schur indicator. Different QFT ${\mathcal{X}}$ satisfying (1.1) may
yield duality defects with different choices of such additional data.
Throughout this work, for simplicity, we will denote the duality defect
associated with gauging $G^{(p)}$ as _$G^{(p)}$ duality defect_, to emphasize
the underlying invertible symmetry, and leave the dependence of additional
data implicit.

A generic $G^{(p)}$ symmetric QFT does not satisfy (1.1). However, there are
some well-known examples satisfying (1.1), including compact scalars in 2d
with $\mathbb{Z}_{N}^{(0)}$ symmetry [17, 1], Maxwell theories in 4d with
$\mathbb{Z}_{N}^{(1)}$ symmetry [1], ${\mathcal{N}}=4$ super Yang-Mills
theories in 4d with $\mathbb{Z}_{N}^{(1)}$ symmetry [3], etc. For these
theories, showing (1.1) often requires a T-duality in 2d or S-duality in
4d,222One should distinguish the T or S-dualities from the duality
transformation associated with gauging $G^{(p)}$. and these dualities are
usually found only in highly fine-tuned theories. Thus it is highly non-
trivial and interesting to find deformations preserving the duality symmetry,
i.e. the relation (1.1). See [18] for recent discussions on duality-preserving
deformation of the ${\mathcal{N}}=4$ super Yang-Mills theory.

##### An alternative construction:

In [3], an alternative construction of theories with duality symmetry was
proposed for certain $G^{(p)}$. See also [5, 15] for further generalizations.
The idea is to start with a theory ${\mathcal{Y}}$ with invertible symmetries
only, and a mixed anomaly. Gauging a non-anomalous subgroup of ${\mathcal{Y}}$
yields another theory ${\mathcal{X}}$, and the mixed anomaly in
${\mathcal{Y}}$ enforces the existence of duality symmetry in ${\mathcal{X}}$.

Let’s illustrate the idea by an example. Take a 4d QFT ${\mathcal{Y}}$ with
global symmetry $G^{(1)}\times
H^{(0)}=\mathbb{Z}_{2}^{(1)}\times\mathbb{Z}_{4}^{(0)}$. We also assume a
mixed anomaly characterized by the 5d anomaly theory333Only
$\mathbb{Z}_{2}^{(0)}$ normal subgroup of $\mathbb{Z}_{4}^{(0)}$ is anomalous.
But rigorously speaking, (1.2) is well-defined only when $A^{(1)}$ belongs to
$H^{1}(X_{5},\mathbb{Z}_{4})$ (rather than $H^{1}(X_{5},\mathbb{Z}_{2})$) due
to the $1/2$ factor.

$\displaystyle\pi\int_{X_{5}}A^{(1)}\cup\frac{{\mathcal{P}}(B^{(2)})}{2}.$
(1.2)

The anomaly means that the partition function of ${\mathcal{Y}}$ obeys

$\displaystyle{\mathcal{Z}}_{{\mathcal{Y}}}[B^{(2)}]={\mathcal{Z}}_{{\mathcal{Y}}}[B^{(2)}]e^{i\pi\int_{X_{4}}{\mathcal{P}}(B^{(2)})/2}.$
(1.3)

In particular, when $e^{i\pi\int_{X_{4}}{\mathcal{P}}(B^{(2)})/2}$ evaluates
to be a non-trivial phase, the partition function
${\mathcal{Z}}_{{\mathcal{Y}}}[B^{(2)}]$ vanishes (for example due to the
presence of zero modes). Next, we construct the theory ${\mathcal{X}}$ by
gauging $G^{(1)}=\mathbb{Z}_{2}^{(1)}$ of ${\mathcal{Y}}$ and then stacking an
$\mathbb{Z}_{2}^{(1)}$ SPT,444We suppress the overall normalization throughout
the paper. Please refer to [1, 19] for systematic discussions of the overall
normalizations.

$\displaystyle{\mathcal{Z}}_{{\mathcal{X}}}[B^{(2)}]=\sum_{b^{(2)}\in
H^{2}(X_{4},\mathbb{Z}_{2})}{\mathcal{Z}}_{{\mathcal{Y}}}[b^{(2)}]e^{i\pi\int_{X_{4}}b^{(2)}B^{(2)}+i\pi\int_{X_{4}}{\mathcal{P}}(B^{(2)})/2}.$
(1.4)

Combining with (1.3), it is straightforward to check that
${\mathcal{X}}={\mathcal{X}}/\mathbb{Z}_{2}^{(1)}$, i.e.

$\displaystyle{\mathcal{Z}}_{{\mathcal{X}}}[B^{(2)}]=\sum_{b^{(2)}\in
H^{2}(X_{4},\mathbb{Z}_{2})}{\mathcal{Z}}_{{\mathcal{X}}}[b^{(2)}]e^{i\pi\int_{X_{4}}b^{(2)}B^{(2)}}.$
(1.5)

The associated duality defect can be obtained using half-gauging. Hence we
have found an alternative, yet systematic, way to construct theories with
duality symmetry associated with gauging $\mathbb{Z}_{2}^{(1)}$, as well as
the $\mathbb{Z}_{2}^{(1)}$ duality defect.

It is useful to know that the duality symmetry associated with gauging
$\mathbb{Z}_{2}^{(1)}$ admits the above alternative construction. Note that
this construction does not require any detailed dynamical information of
${\mathcal{Y}}$ (such as whether ${\mathcal{Y}}$ is a CFT or a free field
theory). Any ${\mathcal{Y}}$, as long as it has the requested symmetry and
anomaly can be fed into the construction. In particular, because only the
invertible symmetries of the theory ${\mathcal{Y}}$ play a role in the
construction, it is easy to turn on perturbations leaving the symmetry and
anomaly unchanged. After gauging $\mathbb{Z}_{2}^{(1)}$, such symmetric
perturbation in ${\mathcal{Y}}$ becomes a duality-symmetry preserving
perturbation in ${\mathcal{X}}$. Hence it is easy to turn on duality-
preserving deformations, and allows one to study the consequence of duality
symmetry along the RG flow. Moreover, this alternative construction allows one
to uncover new duality defects in gauge theories. For instance, this
construction can be used to show the presence of duality defects in a large
class of gauge theories, including non-invertible time reversal symmetries in
the 4d Yang-Mills theories [3, 5, 20], non-invertible axial symmetries in 4d
QED and QCD [21, 22, 23], etc.

It turns out that for an arbitrary $G^{(p)}$, such an alternative construction
may or may not exist. It is therefore useful to ask for which $G^{(p)}$ such
an alternative construction exists. When it exists, the non-invertible duality
defect in QFT ${\mathcal{X}}$ can be mapped to an invertible defect in QFT
${\mathcal{Y}}$ under a _topological manipulation_ $\xi$, which includes
gauging a non-anomalous subgroup and stacking an SPT etc, and such duality
defect was named _non-intrinsically non-invertible_ [24]. Conversely, a
duality defect which does not admit the alternative construction will be
called intrinsically non-invertible.

On the other hand, for 2d QFTs with a generic finite Abelian symmetry
$G^{(0)}$, the condition of when the alternative construction exists has been
classified in mathematical literatures [25]. The duality defect was named
_group theoretical_ if the alternative construction exists. In this work, we
will follow the math notation and determine for which $G^{(p)}$ the duality
defect is group theoretical.

As the answer in 2d is known, we will first review the results in [25], and
the goal is to present the discussion there using a more physical language,
and pave the way for generalization to higher dimensions. For concreteness, we
focus on the $\mathbb{Z}_{N}^{(0)}$ symmetry, although generalization to more
complicated Abelian symmetries is possible.

We then generalize the 2d discussion and proceed to determine when duality
defects associated with gauging $\mathbb{Z}_{N}^{(1)}$ in 4d QFTs are group
theoretical. A partial list of group theoretical duality defects have been
identified in [2, 26]. Our results not only reproduce the known ones in [2],
but also uncover new ones. As a systematic theory of higher category is less
well-established than those in lower dimensions, we are unable to fully
generalize the proofs in [25] to higher dimensions. Hence the completeness of
our list of group theoretical duality defects only holds under certain
assumptions which we will specify in Sec. 2.

###  1.2 Main results

The key idea to find the group theoretical duality defect is to use the
_Symmetry Topological Field Theory_ (SymTFT) [19, 27, 7, 28, 29, 30, 31, 11,
32].555See also [33, 11, 12, 34, 35, 36] for applications of the SymTFT to the
dynamics of QFTs and lattice models.  A $d$-dimensional QFT with a global
symmetry described by a (higher) fusion category ${\mathcal{C}}$ is equivalent
to a $(d+1)$\- dimensional “sandwich" where in the bulk is a SymTFT describing
the Drinfeld center of ${\mathcal{C}}$, the left $d$-dimensional boundary
condition is a canonical/Dirichlet boundary condition where the defects
labeled by ${\mathcal{C}}$ are supported, and the right boundary condition is
a non-topological boundary condition encoding all the dynamical information of
the QFT. Since the bulk is topological, one can shrink the sandwich by
colliding the left and right boundaries to recover the $d$-dimensional QFT.

One of the advantages of the sandwich construction is that it automatically
encodes the additional data of the duality defect mentioned in Sec. 1.1, i.e.
the bicharacters and the Frobenius-Schur indicator, while they are not
explicit from (1.1).

The sandwich construction also enjoys two interesting properties [19, 27, 7].
First, when the symmetry of the QFT is invertible, the SymTFT is a gauged
anomaly theory, i.e. a Dijkgraaf-Witten theory, whose partition function is

$\displaystyle{\mathcal{Z}}=\sum_{g\in G}e^{i\omega(g)}$ (1.6)

where $G$ is the finite (higher) gauge group of the Dijkgraaf-Witten theory,
$g$ is a (higher form) gauge field valued in $G$,666For simplicity we use
$g\in G$ to represent a $g$ gauge field taking value in $G$, although a more
appropriate way is to write $g\in H^{*}(X_{d+1},G)$. We hope this simplified
notation does not lead to confusion. and $\omega(g)$ is the twist term which
specifies the $G$-anomaly.777There are more restricted definition of the
Dijkgraaf-Witten in the literature, where $G$ is an ordinary group. For more
general $G$, e.g. $G$ is a 2-group, the corresponding 4d TFT is called Yetter
TFT [37]. See [33] for a review of various TFTs in higher dimensions. Here we
denote the $G$ gauge theory with any finite (higher) group $G$ as Dijkgraaf-
Witten.  Second, because all the symmetry defects (or background fields when
the symmetry is invertible) are supported on the left topological boundary,
topological manipulations (including gauging and stacking an invertible phase)
do not change the SymTFT. Combining these two properties, we conclude that _a
duality defect is group theoretical if and only if its SymTFT is a Dijkgraaf-
Witten theory_.

The SymTFT of a duality defect associated with gauging $G^{(p)}$ admits a
convenient construction [19, 36]. One starts with the SymTFT of $G^{(p)}$
symmetry (without a duality defect), which is simply a $G^{(p)}$ $(p+1)$-form
gauge theory in $2p+3$ dimensions. Such a theory admits an electro-magnetic
(EM) exchange symmetry whose order depends the parity of $p$. The SymTFT of
the duality defect (including $G^{(p)}$ symmetry) is obtained by gauging the
EM exchange symmetry of the $G^{(p)}$ $(p+1)$-form gauge theory. As we will
review in Sec. 2, such SymTFT is a Dijkgraaf-Witten theory amounts to the
existence of an EM stable topological boundary condition of the $G^{(p)}$
$(p+1)$-form gauge theory. This latter condition will be explicitly checked in
the following sections.

In this paper, we focus on the duality defects associated with gauging
$G^{(p)}=\mathbb{Z}_{N}^{(0)}$ in 2d, and $G^{(p)}=\mathbb{Z}_{N}^{(1)}$ in
4d. We determine for which $N$ together with the choice of bicharacters and
the Frobenius-Schur indicator, the duality defect is group theoretical by
examining when the EM stable topological boundary condition of
$\mathbb{Z}_{N}^{(0)}$ (or $\mathbb{Z}_{N}^{(1)}$) gauge theory exists in 3d
(or 5d). We find the following results:

#####  $\mathbb{Z}_{N}^{(0)}$ duality defects in 2d:

The $\mathbb{Z}_{N}^{(0)}$ duality defect is group theoretical if and only if
$N$ is a perfect square.

#####  $\mathbb{Z}_{N}^{(1)}$ duality defects in 4d:

On spin manifolds, the $\mathbb{Z}_{N}^{(1)}$ duality defect is group
theoretical if and only if $N=L^{2}M$ where $-1$ is a quadratic residue of
$M$.

The above conclusion holds for arbitrary choices of additional data, including
the bicharacters and the Frobenius-Schur indicators.

The 2d results were already proven in a mathematics reference [25]. In the
main text, for both 2d and 4d cases, we will show the if direction by
explicitly demonstrating the SymTFT to be a Dijkgraaf-Witten, and also working
out the explicit topological manipulation. In particular, the special case of
$L=1$ in 4d was known in [2], and our result shows that there are new cases
for $L>1$. However, unlike the 2d case where the only if direction is
proven,888In [19], the authors used the SymTFT to give a physical derivation
of the only if direction in 2d. The observation there is that the SymTFT of
the duality defect has line operators of quantum dimension $\sqrt{N}$. Note
that all line operators in any bosonic Dijkgraaf-Witten theory are of integer
quantum dimension [38, 39, 40], the SymTFT can be Dijkgraaf-Witten only if $N$
is a perfect square.  for the 4d case the only if direction remains a
conjecture.

Note that we assumed the 4d spacetime to be a spin manifold. On non-spin
manifolds, the criteria for odd $N$ remains the same. However, for even $N$,
the situation is more complicated, and we will comment on them in the main
text.

The organization of the paper is as follows. In Sec. 2, we discuss the general
strategy to determine when a duality defect is group theoretical. This section
is largely based on [25], presented in a way that applies to higher dimensions
as well. Sec. 3 and Sec. 4 are in parallel, which discuss the group
theoretical duality defects in 2d and 4d respectively. In both sections, we
first identify the group theoretical $N$’s using the stable topological
boundary condition following the strategy of Sec. 2. Then for each $N$, we
explicitly demonstrate that the SymTFT is a Dijkgraaf-Witten theory, and also
propose the explicit topological manipulation which maps the duality defect to
the invertible defect. In Sec. 5, we comment on the relation between group
theoretical duality defects and obstructions to duality preserving gapped
phases.

##  2 Criteria of group-theoretical duality defects

In this section, we study in general when the duality defects are group
theoretical. A rigorous mathematical discussion of this problem in 2d was
already available in [25]. The goal of this section is to translate the
discussion to a more physicist-friendly language. We also try to present the
discussion in a way applicable to 4d.

###  2.1 Symmetry TFT of duality defects

As reviewed in the introduction, the SymTFT is a useful tool to identify
whether a fusion category is group theoretical. We thus first review the
properties of the SymTFT of duality defects, focusing on
$\mathbb{Z}_{N}^{(d/2-1)}$ symmetry in $d$ dimensions, for $d=2,4$. We will
follow the discussion in [19].

Consider a $d$ dimensional QFT ${\mathcal{X}}$ with a non-anomalous
$\mathbb{Z}_{N}^{(d/2-1)}$ symmetry. Let’s denote its partition function as
${\mathcal{Z}}_{{\mathcal{X}}}[B^{(d/2)}]$. Any such theory can be expanded
into a $d+1$ dimensional slab, as shown in Fig. 1. In the bulk of the slab,
there is a $d+1$ dimensional $\mathbb{Z}_{N}$ $d/2$-form gauge theory, whose
Lagrangian is

$\displaystyle{\mathcal{L}}=\frac{2\pi}{N}\widehat{b}^{(d/2)}\delta
b^{(d/2)}.$ (2.1)

This is the SymTFT of the $\mathbb{Z}_{N}^{(d/2-1)}$ symmetry. On the left
boundary, there is a Dirichlet topological boundary condition/state obtained
by setting the electric field $b^{(d/2)}$ to background value,

$\displaystyle\bra{\text{Dir}(B^{(d/2)})}=\sum_{b^{(d/2)}\in
H^{d/2}(X_{d},\mathbb{Z}_{N})}\delta(b^{(d/2)}-B^{(d/2)})\bra{b^{(d/2)}}.$
(2.2)

On the right boundary, there is a dynamical boundary encoding all the
information of the QFT ${\mathcal{X}}$, where the boundary state is

$\displaystyle\ket{{\mathcal{X}}}=\sum_{b^{(d/2)}\in
H^{d/2}(X_{d},\mathbb{Z}_{N})}{\mathcal{Z}}_{{\mathcal{X}}}[b^{(d/2)}]\ket{b^{(d/2)}}.$
(2.3)

Shrinking the slab amounts to colliding the boundary states (2.1) and (2.2),
which reproduces the partition function
${\mathcal{Z}}_{{\mathcal{X}}}[B^{(d/2)}]$.

${\mathcal{Z}}_{{\mathcal{X}}}[B^{(d/2)}]$ $\mathbb{Z}_{N}$ $d/2$-form
---
gauge theory
$\langle\text{Dir}(B^{(d/2)})|$$|{\mathcal{X}}\rangle$ Figure 1: A $d$
dimensional QFT ${\mathcal{X}}$ with a non-anomalous
$\mathbb{Z}_{N}^{(d/2-1)}$ symmetry can be expanded into a $d+1$ dimensional
slab, where the bulk of the slab is the SymTFT of the
$\mathbb{Z}_{N}^{(d/2-1)}$ symmetry—$\mathbb{Z}_{N}$ $d/2$-form gauge theory,
the left boundary is a topological boundary condition setting the electric
2-form gauge field to classical background value $B^{(2)}$, and the right
boundary is a non-topological dynamical boundary condition.

We further require the QFT ${\mathcal{X}}$ to be invariant under gauging
$\mathbb{Z}_{N}^{(d/2-1)}$, i.e.
${\mathcal{X}}={\mathcal{X}}/\mathbb{Z}_{N}^{(d/2-1)}$, so that the symmetry
of ${\mathcal{X}}$ contains not only $\mathbb{Z}_{N}^{(d/2-1)}$ but also self-
duality. To obtain the SymTFT of the full symmetry, we start with the
$\mathbb{Z}_{N}$ $d/2$-form gauge theory in $d+1$ dimensions and gauge the
electro-magnetic (EM) exchange symmetry. This symmetry basically exchanges
$b^{(d/2)}$ and $\widehat{b}^{(d/2)}$, to be more precise,

$\displaystyle b^{(d/2)}\to
u\widehat{b}^{(d/2)},\qquad\widehat{b}^{(d/2)}\to(-1)^{d/2+1}vb^{(d/2)},$
(2.4)

where $uv=1$ mod $N$. By Chinese Remainder Theorem, both $u$ and $v$ are
coprime with $N$. The minus sign means that the EM exchange symmetry is
$\mathbb{Z}_{2}^{\text{em}}$ when $d=2$ mod $4$, and is
$\mathbb{Z}_{4}^{\text{em}}$ when $d=0$ mod $4$.999When $N=2$ and $d=0$ mod
$4$, the EM exchange symmetry is still $\mathbb{Z}_{2}^{(0)}$ because
$b^{(d/2)}=-b^{(d/2)}$ mod $2$. Below for simplicity the EM symmetry will be
denoted as $\mathbb{Z}_{\chi}^{\text{em}}$, with $\chi=4/2^{[d/2]_{2}}$. Note
that when gauging $\mathbb{Z}_{\chi}^{\text{em}}$, there is a freedom of
choosing the discrete theta term labeled by $\epsilon\in
H^{d+1}(\mathbb{Z}_{\chi}^{\text{em}},U(1))$. When $d=2$, different choices of
$u$ (and hence $v$) correspond to bicharacters of the Tambara-Yamagami fusion
category $\text{TY}(\mathbb{Z}_{N}^{(0)},u,\epsilon)$, while different choices
of the discrete theta term $\epsilon$ correspond to the Frobenius-Schur
indicator [41, 36, 42]. For simplicity, we will adopt the same notation in
$d=4$ as well. In short,

$\displaystyle\text{SymTFT of duality symmetry }=\text{ $\mathbb{Z}_{N}$
$d/2$-form gauge theory}/(\mathbb{Z}_{\chi}^{\text{em}})_{u,\epsilon}.$ (2.5)

###  2.2 Criteria for group-theoretical duality defects

We proceed to use the SymTFT to determine when the $\mathbb{Z}_{N}^{(d/2-1)}$
duality defect is group theoretical. The schematic idea is shown in Fig. 2. We
first recall, as discussed in the introduction, that a duality defect is group
theoretical if and only if its SymTFT is a Dijkgraaf-Witten theory. Using
(2.5), the problem boils down to showing that the $\mathbb{Z}_{N}$ $d/2$-form
gauge theory $/(\mathbb{Z}_{\chi}^{\text{em}})_{u,\epsilon}$ is a Dijkgraaf-
Witten theory. This completes the first two arrows on the top of Fig. 2.

$\mathbb{Z}_{N}^{(d/2-1)}$ duality defect is group theoretical
$\mathbb{Z}_{N}$ $d/2$-form gauge
theory$/(\mathbb{Z}_{\chi}^{\text{em}})_{u,\epsilon}$ is Dijkgraaf-Witten
$\exists$ Lagrangian subcategory ${\mathcal{S}}$ in $\mathbb{Z}_{N}$
$d/2$-form gauge theory$/(\mathbb{Z}_{\chi}^{\text{em}})_{u,\epsilon}$
$\exists\mathbb{Z}_{\chi}^{\text{em}}$ stable Lagrangian subgroup
${\mathcal{S}}_{\text{pre}}$ in $\mathbb{Z}_{N}$ $d/2$-form gauge theory
Figure 2: Idea of determining when a $\mathbb{Z}_{N}^{(d/2-1)}$ duality defect
is group theoretical. We will argue (at the physics level of rigor) for the
solid arrows, while for the dashed arrow, our argument is based on an
assumption that is only proved in $d=2$.

Below, we argue for the remaining arrows. Sec. 2.2.1 establishes the two
arrows in the middle of Fig. 2. Sec. 2.2.2 and Sec. 2.2.3 discusses the upward
and dashed downward arrow at the bottom, respectively.

####  2.2.1 Dijkgraaf-Witten $=$ existence of Lagrangian subcategory

Let’s start with the observation that a SymTFT (2.5) is Dijkgraaf-Witten if
and only if there exist a set of topological operators $S_{\alpha}$’s such
that

  1. 1.

$S_{\alpha}$’s form a (higher) representation category of some symmetry group
$G$, and for simplicity we schematically denote them as
$\operatorname{Rep}(G)$. 101010The higher representation category, e.g.
$2\operatorname{Rep}(G)$, has been discussed recently in the context of
generalized symmetries and generalized charges in [43, 6, 13, 10, 9, 15, 11].
In this section, we will schematically denote both the ordinary and higher
representation categories as $\operatorname{Rep}(G)$.

  2. 2.

$S_{\alpha}$’s are gaugable and gauging them leads to an invertible theory.
111111Gauging ${\mathcal{S}}$ amounts to first form a algebra object
${\mathcal{A}}$ from $S_{\alpha}$’s, and insert a mesh of ${\mathcal{A}}$ in
the path integral. For 3d TFT, an algebra such that after gauging it leads to
an invertible theory is known as Lagrangian algebra, which is shown to be a
gaugable algebra, whose quantum dimension is the total quantum dimension of
the TFT. In higher dimension, the theory of Lagrangian algebra is less well
established, and we will simply define the Lagrangian algebra to be gaugable
(i.e. can consistently insert a mesh of it) and has the property that gauging
it would lead to an invertible theory. See [44] for a recent discussion of the
Lagrangian algebra of 4d TFTs. Mathematically rigorous discussions on gauging
a fusion 2-category can be found in [45, 4].

Let’s denote the subcategory whose objects are $S_{\alpha}$ as the _Lagrangian
subcategory_ ${\mathcal{S}}$. One direction of this claim is obvious: in the
Dijkgraaf-Witten theory with gauge group $G$ and cocycle $\omega$, obviously
the set of Wilson operators form $\operatorname{Rep}(G)$. Hence the first
property is satisfied. Furthermore, $\operatorname{Rep}(G)$ is gaugable
because it is the quantum symmetry from gauging $G$ of the $G$ symmetric
invertible theory (i.e. $G$-SPT) $\omega$. This further means that gauging
$\operatorname{Rep}(G)$ leads to an invertible theory. This shows the second
property, and completes the only if direction, i.e. the $\downarrow$ in the
middle of Fig. 2.

For the if direction, suppose a SymTFT has a set of gaugable topological
operators labelled by $\operatorname{Rep}(G)$ and gauging them leads to an
invertible theory whose partition function is a phase $e^{i\omega}$, then the
quantum symmetry is $G$ and we can gauge $G$ to recover the original SymTFT.
This ensures the original SymTFT is Dijkgraaf-Witten. This completes the if
direction, i.e. the $\uparrow$ in the middle of Fig. 2.

The above definition of the “Lagrangian subcategory" was motivated by the
results of 3d TFT. In the context of 3d TFT, the set of anyons satisfying the
above conditions form a fusion subcategory known as the Lagrangian subcategory
$L$ [25, 46]. It satisfies the following two conditions

  1. 1.

$L$ is of the form $\operatorname{Rep}(G)$ equipped with the standard
symmetric braiding for some finite group $G$. The subcategory of this form is
called _Tannakian_.

  2. 2.

$L=L^{\prime}$ where $L^{\prime}$ is the centralizer defined as the fusion
subcategory (of the entire braided modular tensor category of the 3d TFT)
which contains all the anyon $a$ having trivial braiding with every anyon in
$L$.

Notice that the first condition ensures the 1-form symmetries generated by
$\operatorname{Rep}(G)$ is gaugable, and to gauge it we can consider
condensing the algebra $A_{G}$ which is the regular representation in
$\operatorname{Rep}(G)$. The second condition implies every line operator
outside $L$ is charged non-trivially under the 1-form symmetry generated by
the lines in $L$, hence condensing $L$ would project out every line operator
and end up with an invertible theory. In other words, the algebra $A_{G}$ is
Lagrangian. We reformulate the second condition such that the statement works
for higher dimension as well. For this reason, we will also name the higher
dimensional generalization of $L$, i.e. $\mathcal{S}$, as the Lagrangian
subcategory.

Given a Lagrangian algebra $A_{G}$ as the regular representation in
$\operatorname{Rep}(G)$, we can consider half-gauging it to engineer a gapped
boundary for the SymTFT. In the half space with invertible theory, there’s
apparently a quantum $G$ symmetry whose symmetry defects can terminate on the
gapped boundary and the intersections are the $G$-symmetry defects on the
gapped boundary.

####  2.2.2 Up arrow at the bottom of Fig. 2

We proceed to argue for the up arrow at the bottom of Fig. 2, i.e.

(2.6)

Here we denote the Lagrangian subgroup of the $\mathbb{Z}_{N}$ $d/2$-form
gauge theory (without gauging $\mathbb{Z}_{\chi}^{\text{\text{em}}}$) as
${\mathcal{S}}_{\text{pre}}$, where the subscript pre is to distinguish it
from the Lagrangian subcategory ${\mathcal{S}}$ of the
$\mathbb{Z}_{\chi}^{\text{em}}$ gauged theory (2.5). Since $\mathbb{Z}_{N}$
$d/2$-form gauge theory is an abelian TQFT, its boundary condition is
specified by the Lagrangian (higher) subgroup [47]. The operators in the
Lagrangian subgroup automatically form a $d/2$ representation category, i.e.
${\mathcal{S}}_{\text{pre}}=\operatorname{Rep}(G)$, for certain finite group
$G$. Thus the Lagrangian subgroup is automatically a Lagrangian subcategory.
For this reason, we will use the Lagrangian subgroup to denote the Lagrangian
subcategory of an Abelian TQFT throughout this paper.

A Lagrangian subgroup ${\mathcal{S}}_{\text{pre}}$ is
$\mathbb{Z}_{\chi}^{\text{\text{em}}}$ stable means that
$\mathbb{Z}_{\chi}^{\text{\text{em}}}$ is a group endomorphism of
${\mathcal{S}}_{\text{pre}}$, i.e. under
$\mathbb{Z}_{\chi}^{\text{\text{em}}}$, ${\mathcal{S}}_{\text{pre}}$ is mapped
to ${\mathcal{S}}_{\text{pre}}$ itself, while each object in
${\mathcal{S}}_{\text{pre}}$ might transform non-trivially. Suppose
${\mathcal{S}}_{\text{pre}}$ is $\mathbb{Z}_{\chi}^{\text{\text{em}}}$ stable,
we can recover a $\mathbb{Z}_{\chi}^{\rho}$ automorphism acting the group $G$
itself from this. In the $\mathbb{Z}_{\chi}^{\text{\text{em}}}$ gauged theory,
we can then construct a fusion subcategory of the form
$\operatorname{Rep}(G\rtimes_{\rho}\mathbb{Z}_{\chi})$, generated by
$\mathbb{Z}_{\chi}$ orbit of $\operatorname{Rep}(G)$ and the quantum
$\mathbb{Z}_{\chi}$ symmetry defect $K$. Indeed, an irreducible representation
in $G\rtimes_{\rho}\mathbb{Z}_{\chi}$ is either constructed as a direct sum of
two irreducible representations of $G$ related by $\mathbb{Z}_{\chi}$, or a
tensor product between a $\mathbb{Z}_{\chi}$-invariant irreducible
representation of $G$ and a representation of the $\mathbb{Z}_{\chi}^{\rho}$.

It remains to show that
${\mathcal{S}}^{\prime}=\operatorname{Rep}(G\rtimes_{\rho}\mathbb{Z}_{\chi})$
is gaugable and that gauging it leads to an invertible theory. This is the
case because gauging the entire
$\operatorname{Rep}(G\rtimes_{\rho}\mathbb{Z}_{\chi})$ can be achieved by
sequential gauging where we first gauge $\mathbb{Z}_{\chi}^{(d-2)}$ generated
by the quantum $\mathbb{Z}_{\chi}$ defect $K$ to recover
$\operatorname{Rep}(G)$ and then gauge the anomaly free
$\operatorname{Rep}(G)$ which will leads to the trivial theory. Hence
$\operatorname{Rep}(G\rtimes_{\rho}\mathbb{Z}_{\chi})$ is the Lagrangian
subcategory of the $\mathbb{Z}_{\chi}^{\text{em}}$ gauged theory. This
completes the proof.

We comment that the $\mathbb{Z}_{\chi}^{\text{\text{em}}}$ stable Lagrangian
subgroup is equivalent to $\mathbb{Z}_{\chi}^{\text{\text{em}}}$ stable
boundary condition or state of the $\mathbb{Z}_{N}$ $d/2$-form gauge theory.
Given a Lagrangian subgroup consisting of the operators $S_{\alpha}$, where
$\alpha\in{\mathcal{I}}$ and ${\mathcal{I}}$ is the index set labeling the
simple object in ${\mathcal{S}}_{\text{pre}}$, the topological boundary state
$\ket{\psi_{{\mathcal{I}}}}$ is determined via
$S_{\alpha}\ket{\psi}=\ket{\psi}$ for all $\alpha\in{\mathcal{I}}$. Note that
the boundary state is unique, and can be constructed by gauging the Lagrangian
subgroup on half space. Denote the symmetry operator of
$\mathbb{Z}_{\chi}^{\text{\text{em}}}$ as $U$, stable Lagrangian subcategory
means $US_{\alpha}U^{-1}=S_{\beta}$ where $\beta\in{\mathcal{I}}$. It follows
that

$\displaystyle
S_{\beta}U\ket{\psi}=US_{\alpha}U^{-1}U\ket{\psi}=US_{\alpha}\ket{\psi}=U\ket{\psi}$
(2.7)

meaning that $U\ket{\psi}$ is stabilized by the same Lagrangian subgroup. By
uniqueness, we have $U\ket{\psi}=\ket{\psi}$, which means the boundary state
$\ket{\psi}$ is also $\mathbb{Z}_{\chi}^{\text{\text{em}}}$ stable.

In Sec. 3 and Sec. 4, we will enumerate all the Lagrangian subgroups of the
$\mathbb{Z}_{N}$ $d/2$-form gauge theory and classify when they are
$\mathbb{Z}_{\chi}^{\text{\text{em}}}$ stable. When there exist stable
Lagrangian subgroups, we will explicitly show that gauging the
$\mathbb{Z}_{\chi}^{\text{\text{em}}}$ symmetry leads to Dijkgraaf-Witten
theory, and also find explicit topological manipulations that map the duality
defect to an invertible defect. Such an explicit construction gives a
practical proof of the up arrows, from the bottom to the top, in Fig. 2.

####  2.2.3 Dashed down arrow at the bottom of Fig. 2

We proceed to show the dashed down arrow at the bottom of Fig. 2, i.e.

(2.8)

We use the dashed arrow to emphasize that it is proven only in $d=2$ [25],
while for higher $d$, we are only able to generalize the results in [25] under
an assumption which we specify below.

Suppose the $\mathbb{Z}_{N}$ $d/2$-form gauge theory
$/(\mathbb{Z}_{\chi}^{\text{em}})_{u,v,\epsilon}$ contains a Lagrangian
subcategory ${\mathcal{S}}=\operatorname{Rep}(G)$. Since the theory comes from
gauging $\mathbb{Z}_{\chi}^{\text{\text{em}}}$, there must be an
$\mathbb{Z}_{\chi}^{\text{\text{em}}}$ bosonic topological line operator $K$
corresponding to the quantum $\mathbb{Z}_{\chi}$ $(d-1)$-form symmetry of the
gauged theory. Clearly, the subcategory generated by $K$ is of the form
$\operatorname{Rep}(\mathbb{Z}_{\chi}^{K})$. Below we will make a key
assumption:

##### Key assumption:

$\operatorname{Rep}(\mathbb{Z}_{\chi}^{K})$ is a subcategory of
${\mathcal{S}}=\operatorname{Rep}(G)$.

Since $K$ obeys the $\mathbb{Z}_{\chi}$ fusion rule, it must be labelled by an
order $\chi$ representation of $G$. Equivalently, we can describe $K$ as a
surjective homomorphism $G\rightarrow\mathbb{Z}_{\chi}$, and denote the kernel
of this homomorphism as $H$ then we have the following short exact sequence:

$0\rightarrow H\rightarrow G\rightarrow\mathbb{Z}_{\chi}\rightarrow 0.$ (2.9)

After gauging $\mathbb{Z}_{\chi}^{K}$ by condensing the $K$-line, the
Lagrangian subcategory $\operatorname{Rep}(G)$ then becomes
$\operatorname{Rep}(H)$. To see $\operatorname{Rep}(H)$ is Lagrangian, notice
that gauging $H$ is the second step in the sequential gauging, which
corresponds to gauge $\operatorname{Rep}(G)$ in the original theory.
Therefore, gauging $\operatorname{Rep}(H)$ must lead to an invertible theory
and $\operatorname{Rep}(H)$ is Lagrangian. It is also clear that
$\operatorname{Rep}(H)$ must be stable under the $0$-form
$\mathbb{Z}_{\chi}^{\text{\text{em}}}$ symmetry from condensing $K$. Hence, up
to the assumption that the Lagrangian subcategory $\operatorname{Rep}(G)$
contains $\operatorname{Rep}(\mathbb{Z}_{\chi}^{K})$ as a subcategory, we
argued that (2.8) holds.

When $d=2$, however, there is a theorem guarantees that the key assumption is
true: we can always find a $\operatorname{Rep}(G)$ such that it contains the
$K$-line[48, 49]. Given a Tannakian subcategory
$\operatorname{Rep}(H)$,121212Note that $H$ here is a generic group and should
not be confused with the $H$ in (2.9). in general it may be contained as a
subcategory in another Tannakian subcategory $\operatorname{Rep}(H^{\prime})$.
A Tannakian subcategory which is not properly131313Given two categories
${\mathcal{C}}$ and ${\mathcal{D}}$, ${\mathcal{D}}$ is properly contained in
${\mathcal{C}}$ if ${\mathcal{D}}$ is a subcategory of ${\mathcal{C}}$ but
${\mathcal{D}}$ is not the same as ${\mathcal{C}}$. contained in another
Tannakian subcategory is called maximal [48, 49]. Notice that a Lagrangian
subcategory is automatically a maximal Tannakian subcategory. Since a
Tannakian subcategory describes gaugable 1-form symmetries, one can consider
condense those anyons to get a smaller TFT. In [48, 49], it was shown that
condensing maximal Tannakian subcategory would lead to equivalent TFT which
doesn’t depend on the choice of the maximal Tannakian subcategory.141414The
TFT acquired from the gauging is called the core of the original TFT, which
seems to give a measure on the intrinsically non-invertibleness of a symmetry
TFT. Using this, we can start with the Tannakian subcategory
$\operatorname{Rep}(\mathbb{Z}_{2}^{K})$ and find the maximal Tannakian
subcategory $\operatorname{Rep}(G)$ containing it. Then, by the above theorem,
condensing $\operatorname{Rep}(G)$ would lead to trivial theory and therefore
$\operatorname{Rep}(G)$ is Lagrangian. This completes the proof for the
necessary condition for $d=2$. For higher dimensions, we do not know if the
analog of such theorem exists, so (2.8) is only a conjecture.

##  3 Group theoretical duality defects in 2d

The SymTFT of $\mathbb{Z}_{N}^{(0)}$ duality defects is a 3d $\mathbb{Z}_{N}$
gauge theory with $\mathbb{Z}_{2}^{\text{\text{em}}}$ symmetry gauged. It has
been shown in [25], which we revisited in Sec. 2, that the duality defect is
group theoretical if and only if the 3d $\mathbb{Z}_{N}$ gauge theory admits
$\mathbb{Z}_{2}^{\text{\text{em}}}$ stable topological boundary condition. In
this section, we first review the topological boundary condition of the 3d
$\mathbb{Z}_{N}$ gauge theories, and determine when a
$\mathbb{Z}_{2}^{\text{\text{em}}}$ stable topological boundary condition is
allowed. We then explicitly show that for those allowed $N$ the SymTFT is a
Dijkgraaf-Witten theory, and find explicit topological manipulations under
which the duality defects are mapped to invertible defects.

###  3.1 Lagrangian subgroups of 3d $\mathbb{Z}_{N}$ gauge theory

The action of the $\mathbb{Z}_{N}$ gauge theory is

$\displaystyle{\mathcal{L}}=\frac{2\pi}{N}\widehat{a}\delta a$ (3.1)

where $\widehat{a},a$ are both $\mathbb{Z}_{N}$ cochains. It has a
$\mathbb{Z}_{2}^{\text{\text{em}}}$ exchange symmetry

$\displaystyle a\to u\widehat{a},\qquad\widehat{a}\to va,$ (3.2)

with $uv=1$ mod $N$. The topological lines are

$\displaystyle L_{(e,m)}(\gamma)=e^{\frac{2\pi
ie}{N}\oint_{\gamma}a}e^{\frac{2\pi im}{N}\oint_{\gamma}\widehat{a}}.$ (3.3)

The topological boundary conditions of an Abelian TQFT are classified by the
Lagrangian subgroups. The Lagrangian subgroup ${\mathcal{A}}$ consists of $N$
topological line operators $L_{(e,m)}$ with the following conditions,

  1. 1.

$L_{(e,m)}$ has trivial topological spin, i.e. $e^{i2\pi em/N}=1$.

  2. 2.

Any two lines $L_{(e,m)}$ and $L_{(e^{\prime},m^{\prime})}$ in the Lagrangian
subgroup ${\mathcal{A}}$ have trivial mutual braiding, i.e. $e^{2\pi
i(em^{\prime}+e^{\prime}m)/N}=1$.

  3. 3.

Any other line operator that does not belong to ${\mathcal{A}}$ braids non-
trivially with at least one line in ${\mathcal{A}}$.

We remark that the first condition automatically implies the second condition,
since the mutual braiding phase $B_{(e,m),(e^{\prime},m^{\prime})}$ between
two lines $L_{(e,m)}$ and $L_{(e^{\prime},m^{\prime})}$ is determined by their
self spins $\theta_{(e,m)}$ as
$B_{(e,m),(e^{\prime},m^{\prime})}=\theta_{(e+e^{\prime},m+m^{\prime})}/(\theta_{(e,m)}\theta_{(e^{\prime},m^{\prime})})$.
However, in order to contrast with the analogue condition in higher
dimensions, we still present the second property explicitly. The third
property is guaranteed by the fact that there are $N$ lines in
${\mathcal{A}}$, which we will verify at the end of this subsection.

To enumerate all possible Lagrangian subgroups, we first assume that a
particular line $L_{(e,m)}$ belongs to ${\mathcal{A}}$, hence $L_{(ke,km)}$
also belongs to ${\mathcal{A}}$ due to group structure, for any
$k\in\mathbb{Z}$.

The trivial topological spin means $e$ and $m$ can not be simultaneously
coprime with $N$. This means that there must exist a $k<N$ such that
$L_{(ke,km)}$ is a purely electric line $L_{(p^{\prime},0)}$ or purely
magnetic line $L_{(0,q^{\prime})}$. So any Lagrangian subgroup contains at
least one non-trivial purely electric or magnetic line. Without loss of
generality, we assume that a electric line $L_{(p^{\prime},0)}$ belongs to
${\mathcal{A}}$, where $1<p^{\prime}<N$. Under multiplication, $L_{(p,0)}$
also belongs to ${\mathcal{A}}$, where $p=\gcd(p^{\prime},N)$. Note that
$L_{(p,0)}$ is the purely electric line with the _smallest_ electric charge.
For convenience, we also denote $q=N/p$. Summarizing the above, we have shown
that ${\mathcal{A}}$ contains $q$ purely electric lines among $N$ lines in
total

$\displaystyle L_{(kp,0)},\qquad k=0,1,...,q-1.$ (3.4)

Suppose $L_{(e,m)}$ also belongs to ${\mathcal{A}}$ (which is independent of
the $L_{(e,m)}$ in the previous paragraph), with $m\neq 0$ mod $N$. Trivial
mutual braiding with purely electric lines requires $pm=0$ mod $N$, or
equivalently

$\displaystyle m=sq$ (3.5)

for _certain_ $s\in\mathbb{Z}$. Denoting $t=\gcd(s,p)$, it follows that under
multiplication, $L_{(e^{\prime},tq)}$ also belongs to ${\mathcal{A}}$, for
certain $e^{\prime}$. Trivial topological spin of $L_{(e^{\prime},tq)}$
requires

$\displaystyle te^{\prime}=0\text{ mod }p.$ (3.6)

Note that $tq$ is the minimal magnetic charge mod $N$ in the orbit
$L_{(ke,ksq)}$ for different $k\in\mathbb{Z}$. Supplementing
$L_{(e^{\prime},tq)}$ to the set (3.4), we find the following lines in the
Lagrangian subgroup ${\mathcal{A}}$,

$\displaystyle L_{(kp+k^{\prime}e^{\prime},k^{\prime}tq)},\qquad
k=0,1,...,q-1,\qquad k^{\prime}=0,1,...,p/t-1.$ (3.7)

Thus we find $qp/t=N/t$ lines in the Lagrangian subgroup. Since the Lagrangian
subgroup contains $N$ lines, unless $t=1$, the above set does not contain
enough lines to form a Lagrangian subgroup. However, since the charge lattice
is two dimensional, any missing line should be generated by the two generators
$L_{(p,0)}$ and $L_{(e^{\prime},tq)}$ (note that they are not linearly
dependently when $t>1$), which is a contradiction. This implies that $t=1$,
and by (3.6), $e^{\prime}=0$ mod $p$.151515Another way to see $t=1$ is as
follows. We prove by contradiction. Assume $t>1$, and we would like to find an
operator that is not generated by $L_{(p,0)}$ and $L_{(e^{\prime},tq)}$. By
(3.6), $e^{\prime}=(p/t)x$ for $x\in\mathbb{Z}$. When $x=0$ mod $t$,
$e^{\prime}=0$ mod $p$, and one can compose $L_{(e^{\prime},tq)}$ with
$L_{(p,0)}$ to get $L_{(0,tq)}$. Clearly, the generator $L_{(0,q)}$ can not be
generated by $L_{(p,0)}$ and $L_{(0,tq)}$. When $x\neq 0$ mod $t$, we can
assume $0<x<t$, i.e. $0<e^{\prime}<p$ without loss of generality. We can then
consider $L_{(0,tq)}$, which has the trivial topological spin and the trivial
mutual braiding with both $L_{(p,0)}$ and $L_{(e^{\prime},tq)}$ thanks to
(3.6). Because $0<e^{\prime}<p$, $L_{(0,tq)}$ is not generated by $L_{(p,0)}$
and $L_{(e^{\prime},tq)}$. In both cases, we find at least one operator that
can be added into the Lagrangian subgroup, showing that $L_{(p,0)}$ and
$L_{(e^{\prime},tq)}$ do not generate the full Lagrangian subgroup. When $t=1$
however, (3.6) shows $e^{\prime}=0$ mod $p$, and clearly $L_{(p,0)}$ and
$L_{(e^{\prime},q)}$ generate the entire Lagrangian subgroup. In conclusion,
the Lagrangian subgroup is completely specified by $p$ where $p|N$, which is
generated by $L_{(p,0)}$ and $L_{(0,N/p)}$, i.e.

$\displaystyle{\mathcal{A}}_{p}=\\{L_{(xp,yN/p)}|x\in\mathbb{Z}_{N/p},y\in\mathbb{Z}_{p}\\}.$
(3.8)

Finally, we show the third property of the Lagrangian subgroup, that for any
line $L_{(e,m)}$ not within ${\mathcal{A}}_{p}$, it must braid non-trivially
with at least one element in ${\mathcal{A}}_{p}$. We prove by contradiction.
Suppose there exists $L_{(e,m)}\notin{\mathcal{A}}_{p}$ which braids trivially
with every element in ${\mathcal{A}}_{p}$. The assumption implies
$eyN/p+mxp=0$ mod $N$ for any $x,y$. We first take $y=0$, then $mxp=0$ mod $N$
for any $x$ implies $m=0$ mod $N/p$. Similarly, by taking $x=0$, then
$eyN/p=0$ mod $N$ for any $y$ implies $e=0$ mod $p$. Thus
$L_{(e,m)}\in{\mathcal{A}}_{p}$, which contradicts with the assumption. This
completes the proof.

###  3.2 $\mathbb{Z}_{2}^{\text{em}}$ stable Lagrangian subgroup

We further classify which topological boundary condition is
$\mathbb{Z}_{2}^{\text{\text{em}}}$ stable. The
$\mathbb{Z}_{2}^{\text{\text{em}}}$ symmetry (3.2) maps the charges $(e,m)$ to
$(vm,ue)$. The generators of the Lagrangian subgroup ${\mathcal{A}}_{p}$ are
mapped to

$\displaystyle L_{(p,0)}\to L_{(0,up)},\qquad L_{(0,N/p)}\to L_{(vN/p,0)}.$
(3.9)

$\mathbb{Z}_{2}^{\text{\text{em}}}$ stability implies the above
$\mathbb{Z}_{2}^{\text{\text{em}}}$ of the generators also belong to
${\mathcal{A}}_{p}$, i.e.

$\displaystyle L_{(0,up)}=L_{(xp,yN/p)},\qquad L_{(vN/p,0)}=L_{(zp,wN/p)}.$
(3.10)

The above implies $x\in(N/p)\mathbb{Z}$, $w\in p\mathbb{Z}$, and

$\displaystyle up=yN/p\text{ mod }N,\qquad vN/p=zp\text{ mod }N.$ (3.11)

Combining the two conditions, we find $zy=1$ mod $p$ as well as $zy=1$ mod
$N/p$. This in particular implies $y$ and $z$ are coprime with both $p$ and
$N/p$. On the other hand, the first condition in (3.11) implies
$p=(vy+p\alpha)N/p$ for certain integer $\alpha$. However, because $y$ is
coprime with $p$ and $v$ is coprime with $N$ (hence $p$), $(vy+p\alpha)$ is
also coprime with $p$. So this condition can be satisfied only when $(N/p)|p$.
Similarly, the second equality in (3.11) implies $p|(N/p)$. Combining the two
conditions, we find

$\displaystyle p=N/p$ (3.12)

which means $N=p^{2}$ is a perfect square. By the results in Sec. 2, we
conclude that the $\mathbb{Z}_{N}^{(0)}$ duality defect is group theoretical
if and only if $N$ is a perfect square, for any choice of $u,v$ (i.e. the
bicharacter). Since the choice of the Frobenius-Schur indicator $\epsilon$
does not enter the above discussion, the group-theoretical condition is also
independent of $\epsilon$ as well. We emphasize that this result has been
already proven in [25], and hope that our discussion is more accessible to
physicists.

###  3.3 SymTFT as a Dijkgraaf-Witten theory

In Sec. 3.2, we showed that a $\mathbb{Z}_{N}$ duality defect is group
theoretical if and only if $N$ is a perfect square, for any choice of
$u,v,\epsilon$. In this section, we would like to explicitly show that the
SymTFT for a perfect square $N$ is indeed a Dijkgraaf-Witten theory.

Since $N=p^{2}$, the $\mathbb{Z}_{N}$ cochains $a,\widehat{a}$ can be
rewritten as

$\displaystyle
a=p\widehat{b}+c^{\prime},\qquad\widehat{a}=p\widehat{c}+b^{\prime}$ (3.13)

where $\widehat{c}$, $\widehat{b}$ are $\mathbb{Z}_{p}$ cochains, and
$b^{\prime}$, $c^{\prime}$ are $\mathbb{Z}_{N}$ cochains. In terms of these
new variables, (3.1) is rewritten as

$\displaystyle{\mathcal{L}}=\frac{2\pi}{N}(p\widehat{c}+b^{\prime})\delta(p\widehat{b}+c^{\prime})=\frac{2\pi}{p}b\delta\widehat{b}+\frac{2\pi}{p}\widehat{c}\delta
c+\frac{2\pi}{N}b^{\prime}\delta c^{\prime}.$ (3.14)

In the second equality, we dropped $2\pi\widehat{b}\delta\widehat{c}$ since it
belongs to $2\pi\mathbb{Z}$. On the right hand side, the first two terms are
the standard BF couplings where $b=b^{\prime}$ mod $p$ and $c=c^{\prime}$ mod
$p$, while the last term is the DW twist term. Thus the $\mathbb{Z}_{N}$ gauge
theory without a DW term is equivalent to $\mathbb{Z}_{p}\times\mathbb{Z}_{p}$
gauge theory with a DW term.

The advantage of recasting (3.1) into (3.14) is that the electric-magnetic
exchange symmetry exchanging $a$ and $\widehat{a}$ becomes a symmetry that
only exchanges among the electric fields $b,c$, and among the magnetic fields
$\widehat{b},\widehat{c}$, separately. But the electric and magnetic fields do
not mix under $\mathbb{Z}_{2}^{\text{\text{em}}}$. Concretely, the (3.2) acts
on the $\mathbb{Z}_{p}$ cochains via

$\displaystyle b\to vc,\qquad c\to ub,\qquad\widehat{b}\to
u\widehat{c},\qquad\widehat{c}\to v\widehat{b}.$ (3.15)

Thus gauging (3.2) of (3.1) is equivalent to gauging (3.15) of (3.14). From
the latter, it is almost by definition that the gauged theory is Dijkgraaf-
Witten, where the definition is reviewed in Sec. 1.2.

We remark that the new variables introduced in (3.13) is motivated from the
Lagrangian subgroup derived in Sec. 3.2. The Lagrangian subgroup is generated
by $e^{\frac{2\pi i}{N}p\oint a}=e^{\frac{2\pi i}{p}\oint c}$ and
$e^{\frac{2\pi i}{N}p\oint\widehat{a}}=e^{\frac{2\pi i}{p}\oint b}$, and we
require that these operators are closed under $\mathbb{Z}_{2}^{\text{em}}$
transformation. This is obvious from (3.15).

Let’s derive the Lagrangian from gauging (3.15) of (3.14). The first step is
to couple to the $\mathbb{Z}_{2}^{\text{em}}$ background field $x$, and sum
over all the flat configuration of $x$, i.e. gauge
$\mathbb{Z}_{2}^{\text{em}}$. Coupling to the $\mathbb{Z}_{2}^{\text{em}}$
background field amounts to changing the ordinary differential operator
$\delta$ to the twisted differential operator $\delta_{x}$, and ordinary cup
product $\cup$ to twisted cup product $\cup_{x}$. See [50, App. A] for a
review of twisted cochains, differentials and cup products. In components, the
$\mathbb{Z}_{2}^{\text{em}}$ gauged Lagrangian on a 3-simplex $(ijkl)$ is

$\begin{split}{\mathcal{L}}^{\text{gauged}}_{ijkl}=\frac{2\pi}{p}\widehat{\mathbf{b}}_{ij}^{T}K^{x_{ij}}\left(K^{x_{jk}}\mathbf{b}_{kl}-\mathbf{b}_{jl}+\mathbf{b}_{jk}\right)+\frac{\pi}{N}{\mathbf{b}^{\prime}}_{ij}^{T}\sigma^{x}K^{x_{ij}}\left(K^{x_{jk}}\mathbf{b}^{\prime}_{kl}-\mathbf{b}^{\prime}_{jl}+\mathbf{b}^{\prime}_{jk}\right)+\pi\epsilon
x_{ij}x_{jk}x_{kl}\end{split}$ (3.16)

where

$\displaystyle K=\begin{pmatrix}0&v\\\
u&0\end{pmatrix},\qquad\mathbf{b}=\begin{pmatrix}b\\\
c\end{pmatrix},\qquad\mathbf{b}^{\prime}=\begin{pmatrix}b^{\prime}\\\
c^{\prime}\end{pmatrix}$ (3.17)

and $x$ is the dynamical, flat, $\mathbb{Z}_{2}^{\text{em}}$ gauge field. The
last term is the twist one can add upon gauging $\mathbb{Z}_{2}$, whose
coefficient $\epsilon$ is related to the FS indicator $(-1)^{\epsilon}$. The
action is invariant under the gauge transformations

$\begin{split}\mathbf{b}_{ij}\to
K^{-\gamma_{i}}(\mathbf{b}_{ij}+K^{x_{ij}}\mathbf{\beta}_{j}-\mathbf{\beta}_{i}),\quad\widehat{\mathbf{b}}_{ij}\to(K^{T})^{-\gamma_{i}}(\widehat{\mathbf{b}}_{ij}+K^{x_{ij}}\widehat{\mathbf{\beta}}_{j}-\widehat{\mathbf{\beta}}_{i}),\quad
x_{ij}\to x_{ij}+\gamma_{j}-\gamma_{i}\end{split}.$ (3.18)

Summing over $\widehat{\mathbf{b}}$ constrains ${\mathbf{b}}$ to be a twisted
cocycle (i.e. it is twisted-flat), in components,
$(\delta_{x}\mathbf{b})_{jkl}=K^{x_{jk}}\mathbf{b}_{kl}-\mathbf{b}_{jl}+\mathbf{b}_{jk}=0$
mod $p$. Thus the partition function is independent of how the
$\mathbb{Z}_{p}$ fields $\mathbf{b}$ are lifted to $\mathbb{Z}_{N}$ fields
$\mathbf{b}^{\prime}$. The full partition function is

$\displaystyle{\mathcal{Z}}_{\text{gauged}}=\sum_{(\mathbf{b},x)}\prod_{ijkl}\exp\left(\frac{\pi
i}{N}\mathbf{b}_{ij}^{T}\sigma^{x}K^{x_{ij}}\left(K^{x_{jk}}\mathbf{b}_{kl}-\mathbf{b}_{jl}+\mathbf{b}_{jk}\right)+\pi
i\epsilon x_{ij}x_{jk}x_{kl}\right).$ (3.19)

The pair $(\mathbf{b},x)$ with the above flatness condition is the gauge field
for the gauge group
$(\mathbb{Z}_{p}\times\mathbb{Z}_{p})\rtimes\mathbb{Z}_{2}$, with the
$\mathbb{Z}_{2}$ exchanging the two $\mathbb{Z}_{p}$’s.161616The special case
of $N=4,p=2$ was discussed in a lattice model in [51]. The partition function
is clearly of the form of DW, for any choice of $u,v,\epsilon$.

###  3.4 Explicit topological manipulation

We finally proceed to find an explicit topological manipulation that maps the
duality defect to an invertible defect.

Suppose a 2d QFT ${\mathcal{X}}$ is self-dual. This means

$\displaystyle{\mathcal{Z}}_{{\mathcal{X}}}[A]=\sum_{a\in\mathbb{Z}_{N}}{\mathcal{Z}}_{{\mathcal{X}}}[a]e^{\frac{2\pi
iv}{N}\int_{M_{2}}aA}.$ (3.20)

The summation $a\in\mathbb{Z}_{N}$ means summing over 1-cochain $a$ valued in
$\mathbb{Z}_{N}$, with flatness condition $\delta a=0$ mod $N$, modulo gauge
transformations. The parameter $v$ (which is coprime with $N$) paramaterizes
different ways of gauging $\mathbb{Z}_{N}$. By gauging $\mathbb{Z}_{N}$ on
half of the spacetime, we get a duality defect ${\mathcal{N}}$.

To motivate the topological manipulation, we first note that the QFT
${\mathcal{X}}$ corresponds the SymTFT being $\mathbb{Z}_{N}$ gauge theory and
Dirichlet boundary condition of $a$. Under the decomposition (3.13), the
Dirichlet boundary condition of $a$ translates to the Dirichlet boundary
condition of $c$ and Dirichlet boundary condition of $\widehat{b}$. Given a
Dijkgraaf-Witten theory as the SymTFT, the invertible symmetry corresponds to
the Dirichlet boundary condition of electric fields i.e. $b,c$ only, so the
topological manipulation should transform the above topological boundary
condition to the Dirichlet boundary condition of both $b$ and $c$. Concretely,
in terms of the boundary states defined in Sec. 2.1,

$\displaystyle\bra{A}\to\sum_{\widehat{b}\in\mathbb{Z}_{p}}\bra{p\widehat{b}+C}e^{\frac{2\pi
i}{N}\int_{X_{2}}\widehat{b}B}$ (3.21)

where $\bra{A}$ is the Dirichlet boundary state with
$\mathbb{Z}_{N}=\mathbb{Z}_{p^{2}}$ background field $A$, the second factor on
the RHS is the boundary term coming from the integration by parts of the first
term in (3.14), so that it has the standard BF coupling
$\frac{2\pi}{p}\widehat{b}\delta b$. We also set the electric fields $b,c$ in
the SymTFT (3.14) to background fields $B,C$ respectively. Taking the inner
product between the new topological boundary state (3.21) and the dynamical
boundary state (2.3), we get the new partition function

$\displaystyle{\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[B,C]=\sum_{\widehat{b}\in\mathbb{Z}_{p},\delta\widehat{b}=-\beta
C}{\mathcal{Z}}_{{\mathcal{X}}}[p\widehat{b}+C]e^{\frac{2\pi
i}{p}\int_{M_{2}}\widehat{b}B}.$ (3.22)

The summation is over all $\widehat{b}$ with the constraint
$\delta\widehat{b}=-\beta C$, modulo gauge transformations.171717$\beta$ is
the Bockstein homomorphism $\beta:H^{1}(M_{2},\mathbb{Z}_{p})\to
H^{2}(M_{2},\mathbb{Z}_{p})$. The constraint comes from the flatness condition
of $A$ before topological manipulation, which descends to the flatness
condition of $p\widehat{b}+C$. The constraint also enforces that the exponent
$\frac{2\pi i}{p}\int_{M_{2}}\widehat{b}B$ is not gauge invariant, and one way
to make it gauge invariant is to introduce a 3d bulk[52],

$\displaystyle\frac{2\pi i}{p}\int_{M_{3}}B\beta C.$ (3.23)

This shows that after gauging, there is a mixed anomaly between two
$\mathbb{Z}_{p}$ symmetries.

We claim that (3.22) defines the desired topological manipulation mapping the
self duality under gauging $\mathbb{Z}_{N}$ to an invertible symmetry. To see
this, we perform a self-duality transformation on the right hand side of
(3.22). Concretely, we substitute (3.20) to the right hand side of (3.22), and
get

$\displaystyle\begin{split}{\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[B,C]&=\sum_{\begin{subarray}{c}a\in\mathbb{Z}_{N},\widehat{b}\in\mathbb{Z}_{p},\\\
\delta\widehat{b}=-\beta
C\end{subarray}}{\mathcal{Z}}_{{\mathcal{X}}}[a]e^{\frac{2\pi
i}{N}\int_{M_{2}}p\widehat{b}B+va(p\widehat{b}+C)}.\end{split}$ (3.24)

Summing over $\widehat{b}$ enforces $a=uB$ mod $p$, which is equivalent to
$a=pc+uB$ mod $N$. The flatness condition of $a$ enforces $\delta c=-u\beta
B$. Substituting this solution to the partition function, the exponent becomes
$\frac{2\pi i}{N}\int_{M_{2}}v(pc+uB)C=\frac{2\pi
i}{p}\int_{M_{2}}vcC+\frac{2\pi i}{N}BC$. The last term is a counter term,
which can be absorbed to the bulk anomaly SPT action (3.23) by exchanging
$B\leftrightarrow C$, i.e. $\frac{2\pi i}{N}\int_{M_{2}}BC+\frac{2\pi
i}{p}\int_{M_{3}}B\beta C=\frac{2\pi i}{p}\int_{M_{3}}C\beta B$. Combining the
above, (3.24) finally becomes

$\displaystyle{\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[B,C]=\sum_{c\in\mathbb{Z}_{p},\delta
c=-u\beta B}{\mathcal{Z}}_{{\mathcal{X}}}[pc+uB]e^{\frac{2\pi
i}{p}\int_{M_{2}}vcC}={\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[vC,uB].$
(3.25)

This shows that the self duality symmetry in ${\mathcal{X}}$ is mapped to an
invertible $\mathbb{Z}_{2}$ symmetry that simply maps the background fields as

$\displaystyle B\to vC,\qquad C\to uB$ (3.26)

which is indeed consistent with the transformation (3.15) in the SymTFT.

##  4 Group theoretical duality defects in 4d

We generalize the discussion of duality defects in 2d to duality defects in
4d. The discussion is overall in parallel with Sec. 3, but there are
additional subtleties which we highlight.

###  4.1 Lagrangian subgroups of 5d $\mathbb{Z}_{N}$ 2-form gauge theory

The action of the $\mathbb{Z}_{N}$ 2-form gauge theory is

$\displaystyle{\mathcal{L}}=\frac{2\pi}{N}\widehat{b}\delta b,$ (4.1)

where $b,\widehat{b}$ are both $\mathbb{Z}_{N}$ 2-cochains. It has a
$\mathbb{Z}_{4}^{\text{em}}$ exchange symmetry

$\displaystyle b\to u\widehat{b},\qquad\widehat{b}\to-vb,$ (4.2)

where $uv=1$ mod $N$. Comparing with (3.2), the additional minus sign comes
from the change of form degree of the gauge fields, which consequently makes
the exchange symmetry to be order four, rather than order two.181818When
$N=2$, due to $b=-b$ mod $2$, $\mathbb{Z}_{4}^{\text{em}}$ reduces to
$\mathbb{Z}_{2}^{\text{em}}$.  The topological surfaces are

$\displaystyle S_{(e,m)}(\sigma)=e^{\frac{2\pi
ie}{N}\oint_{\sigma}b}e^{\frac{2\pi im}{N}\oint_{\sigma}\widehat{b}}.$ (4.3)

One significance is that the topological surface operator can be modified by a
local counter term $e^{\frac{2\pi ik}{2N}{\mathcal{P}}(\sigma)}$ for even $N$,
and $e^{\frac{2\pi i}{N}\frac{N+1}{2}\braket{\sigma,\sigma}}$ for odd $N$,
where we ${\mathcal{P}}(\sigma)$ is the Pontryagin square of $\sigma\in
H_{2}(X,\mathbb{Z}_{N})$. In the absence of such counter term, the surface
operators are not closed under fusion,
$S_{(e,m)}(\sigma)S_{(e^{\prime},m^{\prime})}(\sigma)=e^{\frac{2\pi
i}{N}me^{\prime}\braket{\sigma,\sigma}}S_{(e+e^{\prime},m+m^{\prime})}(\sigma)$.

A full classification of topological boundary conditions of a generic 5d TQFT
is still under development. However, for 5d $\mathbb{Z}_{N}$ 2-form gauge
theory, its topological boundary conditions are expected to be classified by
the Lagrangian subgroup of its surface operators, with additional data
specifying the topological refinements [47]. We will take this as an
assumption throughout the rest of the paper. In particular, the surface
operators within the Lagrangian subgroup are closed under fusion.191919One way
to understand the closedness under fusion is as follows. Assuming two
operators $\widetilde{S}_{(e,m)}$ and
$\widetilde{S}_{(e^{\prime},m^{\prime})}$ are within the Lagrangian
subalgebra, where $\widetilde{S}_{(e,m)}$ is related to $S_{(e,m)}$ by
stacking a counter term specified in the previous paragraph. This means that
the associated boundary state $\ket{{\mathcal{B}}}$ is stabilized by both of
them. $\widetilde{S}_{(e,m)}\ket{{\mathcal{B}}}=\ket{{\mathcal{B}}}$,
$\widetilde{S}_{(e^{\prime},m^{\prime})}\ket{{\mathcal{B}}}=\ket{{\mathcal{B}}}$.
This means that their product
$\widetilde{S}_{(e,m)}\widetilde{S}_{(e^{\prime},m^{\prime})}$ also stabilizes
the boundary state $\ket{{\mathcal{B}}}$, hence belongs to the Lagrangian
subalgebra. Now, let $(e+e^{\prime},m+m^{\prime})=(0,0)$ mod $N$, hence
$\widetilde{S}_{(e,m)}\widetilde{S}_{(e^{\prime},m^{\prime})}$ is at most a
phase specified by the counter term. If the counter term is non-trivial, the
boundary state must vanish. Thus we should carefully choose the counter term
such that $\widetilde{S}_{(e,m)}\widetilde{S}_{(e^{\prime},m^{\prime})}=1$
whenever $(e+e^{\prime},m+m^{\prime})=(0,0)$ mod $N$. This is ensured if we
demand the operators $\widetilde{S}_{(e,m)}$ are closed under fusion for
arbitrary $(e,m)$, i.e.
$\widetilde{S}_{(e,m)}\widetilde{S}_{(e^{\prime},m^{\prime})}=\widetilde{S}_{(e+e^{\prime},m+m^{\prime})}$.
The Lagrangian subgroup ${\mathcal{A}}$ consists of $N$ topological surface
operators $S_{(e,m)}$ with the following conditions,

  1. 1.

$S_{(e,m)}$ has trivial topological spin. This is automatically satisfied for
any $(e,m)$ [47].

  2. 2.

Any two surfaces $S_{(e,m)}$ and $S_{(e^{\prime},m^{\prime})}$ in the
Lagrangian subgroup ${\mathcal{A}}$ have trivial mutual braiding, i.e.
$e^{\frac{2\pi i}{N}(em^{\prime}-me^{\prime})}=1$.

  3. 3.

Any other surface operator that does not belong to ${\mathcal{A}}$ braids non-
trivially with at least one line in ${\mathcal{A}}$.

Since the trivial self-braiding condition is automatically satisfied, the
Lagrangian subgroup in 5d is less constrained than that in 3d, hence the
structure of ${\mathcal{A}}$ is richer. For example, we will see that the
Lagrangian subgroups can be generated by an arbitrary single surface
$\widetilde{S}_{(e,m)}$ as long as $\gcd(e,m,N)=1$, while in the 3d
$\mathbb{Z}_{N}$ gauge theory we should further require it to be a self-boson
$em=0$ mod $N$ which is much more constraining.

Like in 3d $\mathbb{Z}_{N}$ gauge theory, the third condition is guaranteed
when there are $N$ topological surface operators in ${\mathcal{A}}$. We will
verify this at the end of this subsection.

Below, we classify the topological boundary condition of $\mathbb{Z}_{N}$
2-form gauge theory by classifying its Lagrangian subgroups. We first derive
the classification at the level of charges in Sec. 4.1.1, by assuming that the
surface operators within it are closed fusion, i.e.

$\displaystyle\widetilde{S}_{(e,m)}(\sigma)\widetilde{S}_{(e,m)}(\sigma^{\prime})$
$\displaystyle=$ $\displaystyle\widetilde{S}_{(e,m)}(\sigma+\sigma^{\prime}),$
(4.4)
$\displaystyle\widetilde{S}_{(e,m)}(\sigma)\widetilde{S}_{(e^{\prime},m^{\prime})}(\sigma)$
$\displaystyle=$
$\displaystyle\widetilde{S}_{(e+e^{\prime},m+m^{\prime})}(\sigma),$ (4.5)

where $\widetilde{S}_{(e,m)}=S_{(e,m)}e^{iK_{e,m}\braket{\sigma,\sigma}}$ with
$K_{e,m}$ determined by the closeness under fusion. We then discuss the phase
factors in Sec. 4.1.2, where we will find for each charge $(e,m)$ there can be
distinct topological refinements, as emphasized in [47].

####  4.1.1 Classifying Lagrangian subgroup: charges

We first classify the charges of the surface operators in the Lagrangian
subgroup, following the discussion in Sec. 3.1. In App. A we provide an
alternative derivation of the Lagrangian subgroups.

Suppose there are pure electric surface operators in the Lagrangian subgroup,
and the one with _minimal_ electric charge is $\widetilde{S}_{(p,0)}$, where
$p|N$ and $1\leq p\leq N$.202020 When $p=N$, there is no pure electric
operator. It is clear that there are additional operators
$\widetilde{S}_{(\ell^{\prime},sN/p)}$ having trivial mutual braiding with
$\widetilde{S}_{(p,0)}$, hence can be supplemented into the Lagrangian
subgroup. Denoting $t=\gcd(s,p)$, it follows that
$\widetilde{S}_{(\ell,tN/p)}$ also belongs to the Lagrangian subgroup, where
$\ell=x\ell^{\prime}$ mod $p$ and $sx=t$ mod $p$.

We proceed to show that $t=1$ for $\widetilde{S}_{(p,0)}$ and
$\widetilde{S}_{(\ell,tN/p)}$ to generate the entire Lagrangian subgroup. To
see this, we denote $\gcd(t,\ell)=\tilde{\ell}$. It follows that
$\widetilde{S}_{(\ell/\tilde{\ell},tN/p\tilde{\ell})}$ also has trivial
braiding with the above two generators, and should belong to the Lagrangian
subgroup, meaning that it can be expended by the generators. This is possible
only when $\tilde{\ell}=1$, meaning $t$ and $\ell$ are coprime. Then we fuse
$p/t$ copies of the second generator to get $\widetilde{S}_{(\ell p/t,0)}$,
hence should be generated by the first generator, i.e. $\ell p/t=px$ for an
integer $x$. This means $\ell=xt$, in other words, $\gcd(\ell,t)=t=1$, as
desired. When $t=1$, indeed $\widetilde{S}_{(p,0)}$ and
$\widetilde{S}_{(\ell,N/p)}$ generate $(N/p)\times p=N$ surfaces, which is the
size of the Lagrangian subgroup. In summary, Lagrangian subgroups are
specified by $(\ell,p)$, generated by $\widetilde{S}_{(p,0)}$ and
$\widetilde{S}_{(\ell,N/p)}$, i.e.

$\displaystyle{\mathcal{A}}_{(\ell,p)}=\\{\widetilde{S}_{(xp+y\ell,yN/p)}|x\in\mathbb{Z}_{N/p},y\in\mathbb{Z}_{p}\\}.$
(4.6)

We finally verify the third property of the Lagrangian subgroup, that for any
surface $S_{(e,m)}$ not within ${\mathcal{A}}_{(\ell,p)}$, it must braid non-
trivially with at least one element in ${\mathcal{A}}_{(\ell,p)}$. We again
prove by contradiction. Suppose there exists
$S_{(e,m)}\notin{\mathcal{A}}_{(\ell,p)}$ which braids trivially with every
element in ${\mathcal{A}}_{(\ell,p)}$. The assumption means

$\displaystyle ey\frac{N}{p}-m(xp+y\ell)=0\text{ mod }N,\qquad\forall
x\in\mathbb{Z}_{N/p},y\in\mathbb{Z}_{p}.$ (4.7)

Setting $y=0$, we get $mxp=0$ mod $N$ for any $x\in\mathbb{Z}_{N/p}$, which
means $m=0$ mod $N/p$. Let’s denote $m=\widehat{m}N/p$. We can alternatively
set $x=0$, then $(eN/p-m\ell)y=0$ mod $N$ for any $y$, meaning
$e-\widehat{m}\ell=0$ mod $p$. Let’s then denote
$e=p\widehat{e}+\ell\widehat{m}$. So

$\displaystyle(e,m)=(p\widehat{e}+\ell\widehat{m},\widehat{m}N/p)=(p,0)\widehat{e}+(\ell,N/p)\widehat{m}\in{\mathcal{A}}_{(\ell,p)}.$
(4.8)

This contradicts with the assumption that $(e,m)$ is not within
${\mathcal{A}}_{(\ell,p)}$. Hence the third property in the Lagrangian
subgroup is verified.

####  4.1.2 Classifying Lagrangian subgroup: including phases

In Sec. 4.1.1, we have determined the charges in the Lagrangian subgroups. In
this section, we explicitly construct the surface operators, in particular
specify the counter terms within $\widetilde{S}_{(e,m)}$. The most general
ansatz is

$\displaystyle\widetilde{S}_{(e,m)}(\sigma)=S_{(e,0)}(\sigma)S_{(0,m)}(\sigma)e^{iK_{e,m}\braket{\sigma,\sigma}}.$
(4.9)

Closedness under fusion requires (4.4) and (4.5). In the above, $(e,m)$
belongs to $k(p,q)$ for one generator case, and $(px+\ell y,yN/p)$ for two
generators case. We discuss these two cases separately.

We substitute $(e,m)=(xp+y\ell,yN/p)$ with $x=0,1,...,N/p-1$ and
$y=0,1,...,p-1$. Similarly,
$(e^{\prime},m^{\prime})=(x^{\prime}p+y^{\prime}\ell,y^{\prime}N/p)$. However,
$(e+e^{\prime},m+m^{\prime})=((x+x^{\prime})p+(y+y^{\prime})\ell,(y+y^{\prime})N/p)$
should be more precisely written as

$\displaystyle(e^{\prime},m^{\prime})=[x+x^{\prime}+(y+y^{\prime}-[y+y^{\prime}]_{p})\frac{\ell}{p}]_{N/p}(p,0)+[y+y^{\prime}]_{p}(\ell,N/p),$
(4.10)

where by $[x]_{p}$ is the mod $p$ function taking value in $0,\cdots,p-1$.
Then (4.4) yields

$\displaystyle
K_{xp+yLr,yN/p}=\begin{cases}\frac{2\pi}{N}\frac{N+1}{2}(xp+y\ell)yN/p\text{
mod }2\pi,&N\text{ odd }\\\ \frac{2\pi}{2N}(xp+y\ell)yN/p+\pi J_{x,y}\text{
mod }2\pi,&N\text{ even }\\\ \end{cases}.$ (4.11)

Substituting (4.11) into (4.5), we find that for odd $N$ (4.11) is
automatically satisfied, while for even $N$, $J_{x,y}$ should satisfy

$\begin{split}&\frac{\ell}{p}((y+y^{\prime})^{2}-[y+y^{\prime}]_{p}^{2})+xy+x^{\prime}y^{\prime}-[x+x^{\prime}+(y+y^{\prime}-[y+y^{\prime}]_{p})\ell/p]_{N/p}[y+y^{\prime}]_{p}\\\
&=J_{[x+x^{\prime}+(y+y^{\prime}-[y+y^{\prime}]_{p})\ell/p]_{N/p},[y+y^{\prime}]_{p}}-J_{x,y}-J_{x^{\prime},y^{\prime}}\text{
mod }2.\\\ \end{split}$ (4.12)

It is not straightforward to explicitly solve for $J_{x,y}$ analytically.
However, by numerically solving (4.12) for $N\leq 100$, we confirmed that the
solution exist for all even $N$’s: there are four solutions when $p,N/p,\ell$
are all even, and two solutions otherwise. In particular, from the numerical
results, we find that when $p,N/p,\ell$ are all even, the four solutions are
of the form $J_{x,y}=xy+c_{1}x+c_{2}y$ mod $2$, where $c_{1},c_{2}=0,1$
parameterize the four solutions. Another situation where we are able to solve
$J_{x,y}$ is when the Lagrangian subgroup is generated by only one surface
operator, which we will discuss in App. B.

The situation significantly simplifies when we restrict the 4d boundary of the
5d SymTFT to be spin manifolds. In this case, the $\pi J_{x,y}$ term in (4.11)
can be ignored due to $\braket{\sigma,\sigma}=0$ mod $2$.

###  4.2 $\mathbb{Z}_{4}^{\text{em}}$ stable Lagrangian subgroup

We proceed to examine when the Lagrangian subalgebra is stable under
$\mathbb{Z}_{4}^{\text{em}}$. Recall that under $\mathbb{Z}_{4}^{\text{em}}$,
the gauge field transforms as (4.2). So $S_{(e,0)}(\sigma)\to
S_{(0,ue)}(\sigma)$, and $S_{(0,m)}(\sigma)\to S_{(-vm,0)}(\sigma)$.

When the 4d spacetime $X_{4}$ is a spin manifold, the counter terms for even
and odd $N$ are given by (4.11) with $J_{x,y}=0$. Under
$\mathbb{Z}_{4}^{\text{em}}$, the generators $\widetilde{S}_{(p,0)}$ and
$\widetilde{S}_{(\ell,N/p)}$ are mapped to $\widetilde{S}_{(0,up)}$ and
$\widetilde{S}_{(-vN/p,u\ell)}$ respectively for both even and odd $N$.
Stability under $\mathbb{Z}_{4}^{\text{em}}$ implies $(0,up)=(xp+y\ell,yN/p)$
mod $N$, and $(-vN/p,u\ell)=(zp+w\ell,wN/p)$ mod $N$. In components, we have

$\displaystyle xp+y\ell$ $\displaystyle=$ $\displaystyle 0\text{ mod }N,$
(4.13) $\displaystyle yN/p$ $\displaystyle=$ $\displaystyle up\text{ mod }N,$
(4.14) $\displaystyle zp+w\ell$ $\displaystyle=$ $\displaystyle-vN/p\text{ mod
}N,$ (4.15) $\displaystyle wN/p$ $\displaystyle=$ $\displaystyle u\ell\text{
mod }N.$ (4.16)

Once the $\mathbb{Z}_{4}^{\text{em}}$ images of the generators are within the
Lagrangian subgroup, it is not hard to show that the
$\mathbb{Z}_{4}^{\text{em}}$ images of all other surface operators in the
Lagrangian subgroup are all within the Lagrangian subgroup as well.

Given $N,u,v$, can we find $p,\ell,x,y,z,w$ such that (4.13)$\sim$(4.16) are
satisfied? Below, we provide an equivalent but simpler-looking criteria of $N$
for which the solutions exists.212121We are grateful to Justin Kaidi for
discussing the following proof. First, multiplying $p$ on both sides of (4.14)
yields $p^{2}=0$ mod $N$ (note that $u,v$ are coprime with $N$). This means

$\displaystyle p^{2}=NM$ (4.17)

for some integer $M$. As a consequence, we have $p=(N/p)M$, which means

$\displaystyle L:=\gcd(p,N/p)=N/p,$ (4.18)

hence we write $p=ML$. Substituting this into (4.17), we have $(ML)^{2}=NM$,
which gives rise to

$\displaystyle N=L^{2}M.$ (4.19)

So far, we only used (4.14). To see the condition of $M$, we multiply $\ell$
on both sides of (4.16),

$\displaystyle\ell^{2}=vw\ell N/p=v(-vN/p-zp)N/p=-(vN/p)^{2}=-v^{2}L^{2}\text{
mod }N$ (4.20)

where in the second equality we used (4.15). Multiplying $u^{2}$ and dividing
$L^{2}$ on both sides, we find

$\displaystyle(u\ell/L)^{2}=-1\text{ mod }M.$ (4.21)

In other words, $\ell=Lrv$ with $r^{2}=-1$ mod $M$. Finally, we note that
(4.13) does not impose further constraints and can be solved by $x=-r$ and
$y=uM$.

So far, we have shown that (4.13)$\sim$(4.16) imply $N=L^{2}M$ where $-1$ is a
quadratic residue of $M$. To show the converse is also true, we simply take

$\displaystyle(N,x,y,z,w,p,\ell)=(L^{2}M,-r,uM,-v\lambda,r,LM,Lrv)$ (4.22)

where $r$ satisfies $r^{2}+1=\lambda M$, and it is straightforward to check
that (4.13)$\sim$(4.16) are satisfied. In summary, we have the criteria:

##### Stability criteria:

When the 4d spacetime is a spin manifold, a $\mathbb{Z}_{4}^{\text{em}}$
stable Lagrangian subgroup exists if and only if $N=L^{2}M$ and for every
choice of bicharacter labeled by $u,v$ with $uv=1$ mod $N$, where $-1$ is a
quadratic residue of $M$. We enumerate such $N$’s in Tab. 1.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39
40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59
60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79
80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99
Table 1: $N$’s that admit $\mathbb{Z}_{4}^{\text{em}}$ stable Lagrangian
subgroups, i.e. $N=L^{2}M$ labeled by colors. The red and blue ones are the
$N$’s with $L=1$ and $L>1$ respectively. They correspond to the Lagrangian
subgroups that are generated by one and two generators.

Let’s make several comments.

  1. 1.

Combining the above criteria with the general discussion in Sec. 2, we claim
that the duality defect associated with gauging $\mathbb{Z}_{N}^{(1)}$ is
group theoretical if and only if $N=L^{2}M$ where $-1$ is a quadratic residue
of $M$. The if direction is proven in Sec. 2, and will be further supported by
explicitly showing the SymTFT is a Dijkgraaf-Witten theory and also explicitly
constructing the topological manipulation for every such $N$. The condition is
also independent of the choice of bicharacters $u,v$ and the higher
dimensional generalization of the Frobenius-Schur indicator $\epsilon$, which
will be further discussed in Sec. 4.3. However the only if direction is less
solid, and remains a conjecture.

  2. 2.

On non-spin manifolds, the criteria for odd $N$ remains the same. For even
$N$, we don’t have the full classification of stable Lagrangian subgroups, but
let’s comment on two special cases. The first case is when the Lagrangian
subgroup is generated by one surface operator, then we are able to show that
the stable Lagrangian subgroup does not exist. We present the details in App.
B. The second case is when $N=L^{2}$ is a perfect square so that
$(p,N/p,\ell)=(L,L,0)$ are all even for even $L$. On top of the stability
condition on charges (4.13)$\sim$(4.16) whose solution is
$(N,x,y,z,w,p,\ell)=(L^{2},0,u,-v,0,L,0)$ for even $L$, there are additional
stability conditions on the topological refinements $J_{x,y}=xy+c_{1}x+c_{2}y$
mod $2$ enforcing $c_{1}=c_{2}$ mod $2$. In Sec. 4.4 we will comment on the
explicit topological manipulation on non-spin manifold corresponding to this
special case.

  3. 3.

When $L=1$, there is only one generator $\widetilde{S}_{(r,1)}$, with
$(vr)^{2}=-1$ mod $N$. This is precisely the condition found in [20, App. C],
as well as in [26] for prime $N$’s, for the $\mathbb{Z}_{N}^{(1)}$ duality
defect to be group theoretical on spin manifolds.

  4. 4.

It is intriguing to note that the same series $N=L^{2}M$ (with $-1$ being a
quadratic residue of $M$) was found in a completely different context in [53],
where the authors showed that a (spin) TQFT with $\mathbb{Z}_{N}^{(1)}$
symmetry $N$ satisfies exactly the same condition. In Sec. 5, we explain such
a coincidence.

###  4.3 SymTFT as a Dijkgraaf-Witten theory

In this subsection, we show explicitly that the SymTFT for the entire
$\mathbb{Z}_{N}$ duality symmetry is a Dijkgraaf-Witten theory when the
$\mathbb{Z}_{4}^{\text{em}}$ stable Lagrangian subgroup exists. From Sec. 3.3,
we learned that for the SymTFT obtained from gauging the electric-magnetic
exchange symmetry of the $\mathbb{Z}_{N}$ gauge theory to be a Dijkgraaf-
Witten theory, we should rewrite the original $\mathbb{Z}_{N}$ gauge theory in
terms of a set of new gauge fields so that the em exchange symmetry acts on
the electric and magnetic gauge fields separately, and does not exchange them.
We will find below how this can be achieved for 5d $\mathbb{Z}_{N}$ 2-form
gauge theory.

##### Odd $N$:

From Sec. 4.2, we found that the $\mathbb{Z}_{4}^{\text{em}}$ stable
Lagrangian subgroup is generated by surface operators $\widetilde{S}_{(LM,0)}$
and $\widetilde{S}_{(Lrv,L)}$ with the constraints that $r^{2}+1=\lambda M$
for some integer $\lambda$. Inspired by the discussion in Sec. 3.3, we
decompose the $\mathbb{Z}_{N}$ 2-form gauge fields into new ones as follows

$\displaystyle(\widehat{b},b)=(LM,0)\widehat{a}+(Lrv,L)\widehat{c}-(\lambda,0)c-(rv,1)a$
(4.23)

where $(\widehat{a},\widehat{c},a,c)$ are
$(\mathbb{Z}_{L},\mathbb{Z}_{LM},\mathbb{Z}_{N},\mathbb{Z}_{N})$ cochains
respectively. We again label the magnetic fields by hatted letters, while the
electric fields by unhatted letters. Under $\mathbb{Z}_{4}^{\text{em}}$, (4.2)
implies

$\displaystyle\begin{split}(\widehat{b},b)\to(-vb,u\widehat{b})&=(0,uLM)\widehat{a}+(-vL,Lr)\widehat{c}-(0,u\lambda)c-(-v,r)a\\\
&=[-(LM,0)r+(Lrv,L)Mu]\widehat{a}+[-(LM,0)\lambda v+(Lrv,L)r]\widehat{c}\\\
&\quad-[-(\lambda,0)r+(vr,1)\lambda u]c-[-(\lambda,0)Mv+(vr,1)r]a\end{split}$
(4.24)

where in the second line we used (4.13)$\sim$(4.16). Comparing (4.23) and
(4.24), we find the transformation of new gauge fields as222222The coefficient
of $a$ and $c$ in (4.24) is designed to ensure that $a,c$ transform in a
conjugate way as $\widehat{a},\widehat{c}$ and also $(\widehat{a},a)$ and
$(\widehat{c},c)$ appear as conjugate fields in the Lagrangian.

$\displaystyle\begin{split}\widehat{a}&\to-r\widehat{a}-\lambda
v\widehat{c},\\\ \widehat{c}&\to uM\widehat{a}+r\widehat{c},\\\ a&\to
ra+u\lambda c,\\\ c&\to-Mva-rc,\\\ \end{split}$ (4.25)

where the electric and magnetic fields do not mix under
$\mathbb{Z}_{4}^{\text{em}}$. Indeed, the generators of the Lagrangian
subgroup $e^{\frac{2\pi ip}{N}\oint b}=e^{-\frac{2\pi i}{L}\oint a}$ and
$e^{\frac{2\pi i}{N}\oint Lrvb+L\widehat{b}}=e^{-\frac{2\pi i\lambda}{LM}\oint
c}$ are all written in terms of the electric fields, hence their generating
set is stable under $\mathbb{Z}_{4}^{\text{em}}$. The Lagrangian of the
$\mathbb{Z}_{N}$ 2-form gauge theory can be rewritten in terms of the new
gauge fields as

$\begin{split}\frac{2\pi}{N}\widehat{b}\delta
b&=-\frac{2\pi}{L}\widehat{a}\delta a+\frac{2\pi\lambda}{LM}\widehat{c}\delta
c+\frac{2\pi\lambda}{L^{2}M}a\delta c+\frac{2\pi}{L^{2}M}\delta(-\lambda
L\widehat{c}c+\frac{N+1}{2}L^{2}rv\widehat{c}\widehat{c}-Lrva\widehat{c}+\frac{N+1}{2}rvaa)\\\
&=-\frac{2\pi}{L}\widehat{a}\delta a+\frac{2\pi\lambda}{LM}\widehat{c}\delta
c+\frac{2\pi\lambda}{L^{2}M}a\delta c+{\mathcal{L}}_{\text{bdy}}.\end{split}$
(4.26)

In the second line, the first two terms are the BF terms that couple electric
fields $(a,c)$ to the magnetic fields $(\widehat{a},\widehat{c})$, while the
third term is the mixed coupling (Dijkgraaf-Witten twist term) that only
depends on the electric fields $a,c$. The last term
${\mathcal{L}}_{\text{bdy}}$ is the boundary term which does not play any role
in this subsection, while it will be crucial in Sec. 4.4. Because the
$\mathbb{Z}_{4}^{\text{em}}$ only exchanges among the electric fields and the
magnetic fields separately, we conclude that after gauging
$\mathbb{Z}_{4}^{\text{em}}$, the theory is still a Dijkgraaf-Witten. The
explicit form the Dijkgraaf-Witten can be analogously constructed as in Sec.
3.3. See also [19]. Concretely, after gauging $\mathbb{Z}_{4}^{\text{em}}$, we
introduce the $\mathbb{Z}_{4}^{\text{em}}$ cocycle $x$ as a dynamical gauge
field. Denote the two component 2-form gauge field

$\displaystyle\mathbf{a}=\begin{pmatrix}a\\\
c\end{pmatrix},\qquad\widehat{\mathbf{a}}=\begin{pmatrix}\widehat{a}\\\
\widehat{c}\end{pmatrix}.$ (4.27)

The gauged $\mathbb{Z}_{N}$ 2-form gauge theory becomes

$\displaystyle\begin{split}{\mathcal{L}}_{ijklmn}^{\text{gauged}}=&\frac{2\pi}{LM}\widehat{\mathbf{a}}^{T}_{ijk}WK^{x_{ik}}(K^{x_{kl}}\mathbf{a}_{lmn}-\mathbf{a}_{kmn}+\mathbf{a}_{kln}-\mathbf{a}_{klm})\\\
&+\frac{\pi\lambda}{N}\mathbf{a}^{T}_{ijk}VK^{x_{ik}}(K^{x_{kl}}\mathbf{a}_{lmn}-\mathbf{a}_{kmn}+\mathbf{a}_{kln}-\mathbf{a}_{klm})+\frac{\pi\epsilon}{2}x_{ij}(\beta
x)_{jkl}(\beta x)_{lmn}\end{split}$ (4.28)

where

$\displaystyle K=\begin{pmatrix}r&u\lambda\\\
-Mv&-r\end{pmatrix},\quad\widehat{K}=\begin{pmatrix}-r&-\lambda v\\\
uM&r\end{pmatrix},\quad W=\begin{pmatrix}-M&0\\\ 0&\lambda\end{pmatrix},\quad
V=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ (4.29)

and $(\beta x)_{jkl}:=\frac{1}{4}(\delta
x)_{jkl}=\frac{1}{4}(x_{kl}-x_{jl}+x_{jk})$ is always an integer valued since
$x$ is a $\mathbb{Z}_{4}$ cocycle. The gauged theory is invariant under gauge
transformations

$\displaystyle\begin{split}\mathbf{a}_{ijk}&\to
K^{-\gamma_{i}}(\mathbf{a}_{ijk}+K^{x_{ij}}\alpha_{jk}-\alpha_{ik}+\alpha_{ij}),\\\
\widehat{\mathbf{a}}_{ijk}&\to\widehat{K}^{-\gamma_{i}}(\widehat{\mathbf{a}}_{ijk}+\widehat{K}^{x_{ij}}\widehat{\alpha}_{jk}-\widehat{\alpha}_{ik}+\widehat{\alpha}_{ij}),\\\
x_{ij}&\to x_{ij}+\gamma_{j}-\gamma_{i},\end{split}$ (4.30)

thanks to the identities $\widehat{K}^{T}WK=W$ and $K^{T}VK=V$. Summing over
$\widehat{\mathbf{a}}$ constrains $\mathbf{a}$ to be a twisted cocycle, i.e.
$\mathbf{a}$ is twisted-flat. In components,
$W(K^{x_{kl}}\mathbf{a}_{lmn}-\mathbf{a}_{kmn}+\mathbf{a}_{kln}-\mathbf{a}_{klm})=0$
mod $LM$. This means that $\mathbf{a},x$ form a non-trivial 2-group. The full
partition function is

${\mathcal{Z}}_{\text{gauged}}=\sum_{\mathbf{a},x}\prod_{ijklmn}\exp\left(\frac{i\pi\lambda}{N}\mathbf{a}^{T}_{ijk}VK^{x_{ik}}(K^{x_{kl}}\mathbf{a}_{lmn}-\mathbf{a}_{kmn}+\mathbf{a}_{kln}-\mathbf{a}_{klm})+\frac{i\pi\epsilon}{2}x_{ij}(\beta
x)_{jkl}(\beta x)_{lmn}\right)$ (4.31)

where we only sum over the gauge fields $\mathbf{a},x$ with the twisted-
flatness condition. The partition function is obviously of the form of
Dijkgraaf-Witten (see Eq. (1.6)).

##### Even $N$:

When the 4d spacetime is a spin manifold, the discussion is almost the same as
above, such as (4.23), (4.24) and (4.25). The only modification is to replace
the factor $\frac{N+1}{2}$ in the Lagrangian (4.26) by $\frac{1}{2}$, and the
self pairing e.g. $aa$ by the Pontryagin square of $a$, i.e.
${\mathcal{P}}(a)$. When the 4d spacetime is a non-spin manifold, in the
special case where the Lagrangian subgroup has only a single generator, we
find in App. B that there is no stable Lagrangian subgroup, hence the SymTFT
obtained by gauging $\mathbb{Z}_{4}^{\text{em}}$ can not be a Dijkgraaf-
Witten. When there are two generators, we do not solve the stability condition
in this paper.

In summary, for all the cases where the stable Lagrangian subgroup exist, we
showed that the SymTFT for the duality symmetry is a Dijkgraaf-Witten theory.

###  4.4 Explicit topological manipulations

We finally proceed to find explicit topological manipulations that map the
duality defect to an invertible defect.

Suppose a 4d QFT with $\mathbb{Z}_{N}^{(1)}$ one-form symmetry is self-dual
under gauging $\mathbb{Z}_{N}^{(1)}$. This means

$\displaystyle{\mathcal{Z}}_{{\mathcal{X}}}[B]=\sum_{b\in\mathbb{Z}_{N}}{\mathcal{Z}}_{{\mathcal{X}}}[b]e^{-\frac{2\pi
i}{N}\int_{M_{4}}vbB}.$ (4.32)

The summation $b\in\mathbb{Z}_{N}$ means summing over 2-cochain $b$ valued in
$\mathbb{Z}_{N}$ with flatness condition $\delta b=0$ mod $N$, modulo gauge
transformations. The idea of identifying the topological manipulation which
maps the duality defect to an invertible defect is the same as in Sec. 3.4.
The self-dual theory ${\mathcal{X}}$ corresponds to Dirichlet boundary
condition of $b$. The desired topological manipulation should map such
boundary condition to a new topological boundary condition where all the new
electric fields have Dirichlet boundary conditions. In the meantime, the
discrete theta terms should be introduced coming from the boundary terms of
the SymTFT ${\mathcal{L}}_{\text{bdy}}$ in (4.26).

##### Odd $N$:

In this case, $N$ has the form $N=L^{2}M$ where $-1$ is a quadratic residue of
$M$. The topological manipulation should map the Dirichlet boundary condition
of $b$ to the Dirichlet boundary conditions of $a,c$ defined via (4.23).
Denoting the background field for the former as $B$, and those for the latter
as $A$ and $C$, we have

$\displaystyle\bra{B}\to\sum_{\widehat{c}\in\mathbb{Z}_{LM}}\bra{L\widehat{c}-A}e^{-\frac{2\pi
i}{LM}\int_{M_{4}}(\lambda C+rvA)\widehat{c}+\frac{2\pi
i}{N}\frac{N+1}{2}\int_{M_{4}}(L^{2}rv\widehat{c}\widehat{c}+rvAA)}.$ (4.33)

The phase factor on the RHS is the ${\mathcal{L}}_{\text{bdy}}$ in (4.26).
This implies that the new partition function obtained by topological
manipulation is

${\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[A,C]=\sum_{\widehat{c}\in\mathbb{Z}_{LM}}{\mathcal{Z}}_{{\mathcal{X}}}[L\widehat{c}-A]e^{-\frac{2\pi
i}{LM}\int_{M_{4}}(\lambda C+rvA)\widehat{c}+\frac{2\pi
i}{N}\frac{N+1}{2}\int_{M_{4}}(L^{2}rv\widehat{c}\widehat{c}+rvAA)}.$ (4.34)

Notice that we start with taking the background gauge field $A,C$ to be
$\mathbb{Z}_{N}$ gauge field for convenience. However, the action (4.34) is
invariant up to terms containing only background fields under the following
transformation:

$\begin{split}A\rightarrow A+LX,\qquad C\rightarrow
C+LMY,\qquad\widehat{c}\rightarrow\widehat{c}+X,\end{split}$ (4.35)

where $X,Y\in\mathbb{Z}_{N}$. Eq. (4.35) shows that $A$ is actually a
$\mathbb{Z}_{L}$ background field and $C$ is a $\mathbb{Z}_{LM}$ background
field.

We proceed to check what does the duality symmetry of ${\mathcal{X}}$ is
mapped to in $\widetilde{{\mathcal{X}}}$. Substituting (4.32) into (4.34), we
find

$\begin{split}{\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[A,C]=\sum_{\widehat{c}\in\mathbb{Z}_{LM},b\in\mathbb{Z}_{N}}{\mathcal{Z}}_{{\mathcal{X}}}[b]e^{-\frac{2\pi
i}{N}\int_{M_{4}}vb(L\widehat{c}-A)-\frac{2\pi i}{LM}\int_{M_{4}}(\lambda
C+rvA)\widehat{c}+\frac{2\pi
i}{N}\frac{N+1}{2}\int_{M_{4}}(L^{2}rv\widehat{c}\widehat{c}+rvAA)}.\end{split}$
(4.36)

We first sum over $\widehat{c}$, where the relevant terms are
$\sum_{\widehat{c}\in\mathbb{Z}_{LM}}e^{-\frac{2\pi
i}{LM}\int_{M_{4}}(vb+\lambda C+rvA)\widehat{c}+\frac{2\pi
i}{M}\frac{N+1}{2}rv\int_{M_{4}}\widehat{c}\widehat{c}}$. This sum is non-
vanishing only when $vb+\lambda C+rvA$ can be divided by $L$. 232323This can
be seen by replacing $\widehat{c}$ with $\widehat{c}+s$ where $s$ is an
integer valued cochain, and demand the sum does not change as $\widehat{c}$ is
a dummy variable.  We thus define $b=Lb^{\prime}-u(\lambda C+rvA)$.
Substituting this into (4.36) and summing over $\widehat{c}$, we get

$\begin{split}{\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[A,C]&=\sum_{\widehat{c},b^{\prime}\in\mathbb{Z}_{LM}}{\mathcal{Z}}_{{\mathcal{X}}}[Lb^{\prime}-u(\lambda
C+rvA)]e^{-\frac{2\pi i}{N}\int_{M_{4}}(-vLb^{\prime}+\lambda
C+rvA)A+\frac{2\pi
i}{N}\frac{N+1}{2}\int_{M_{4}}rvAA+L^{2}rvb^{\prime}b^{\prime}}\\\
&={\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[u\lambda C+rA,-vMA-
rC]e^{-\frac{2\pi i}{N}\frac{N+1}{2}\lambda\int_{M_{4}}(2\lambda MAC+ru\lambda
CC+rvMAA)}.\end{split}$ (4.37)

Therefore we have shown that the duality symmetry of ${\mathcal{X}}$ becomes
an invertible symmetry mapping on the background fields of the new theory
$\widetilde{{\mathcal{X}}}$ as

$\displaystyle A\to u\lambda C+rA\text{ mod }L,\qquad C\to-vMA-rC\text{ mod
}ML.$ (4.38)

This agrees with (4.25).

##### Even $N$:

When the 4d spacetime is a spin manifold, the discussion is almost identical
to the odd $N$ case. The only modifications are to replace $\frac{N+1}{2}$ by
$\frac{1}{2}$, and self pairing, e.g. $BB$, by ${\mathcal{P}}(B)$ in (4.33),
(4.34), (4.36), (4.37). The duality symmetry (4.32) is mapped to an invertible
symmetry (4.38).

When the 4d spacetime is a non-spin manifold, we haven’t classified all the
solutions. However, as mentioned in Sec. 4.2, when $N=L^{2}$ for even $L$,
there is a stable Lagrangian subgroup. Inspired by a similar discussion in
Sec. 3.4, we claim that the topological manipulation is simply gauging the
$\mathbb{Z}_{L}^{(1)}$ subgroup, i.e.
$\widetilde{{\mathcal{X}}}={\mathcal{X}}/\mathbb{Z}_{L}^{(1)}$. Concretely,

$\displaystyle{\mathcal{Z}}_{\widetilde{{\mathcal{X}}}}[A,C]=\sum_{a\in\mathbb{Z}_{L}}{\mathcal{Z}}_{{\mathcal{X}}}[L\widehat{c}-A]e^{-\frac{2\pi
i}{L}\int_{X_{4}}\widehat{c}C}$ (4.39)

where $A,C$ are both $\mathbb{Z}_{L}$ 2-form gauge fields. Substituting (4.32)
into (4.39), we find the background fields $A,C$ are mapped as

$\displaystyle A\to uC\text{ mod }L,\qquad C\to-vA\text{ mod }L~{}.$ (4.40)

Hence when the 4d spacetime is a non-spin manifold, we have at least confirmed
a spacial case where $N=L^{2}$ such that the $\mathbb{Z}_{N}^{(1)}$ duality
defect is group theoretical.

In summary, for all the cases where the stable Lagrangian subgroup exist (on
spin manifolds), we have constructed explicit topological manipulation which
maps the duality symmetry to an invertible symmetry.

##  5 Connection with obstruction to duality-preserving gapped phases

We have observed in the above that for the $\mathbb{Z}_{N}^{(d/2-1)}$ duality
defect to be group theoretical, some number-theoretic condition should be
satisfied, and exactly the same condition has appeared in other contexts,
including whether there exists an SPT or TQFT invariant under gauging
$\mathbb{Z}_{N}^{(d/2-1)}$[1, 2, 53, 36]. In this section, we comment on the
relation between them.

###  5.1 Duality defects, gauging, and SPT

Given a finite abelian group $\mathbb{Z}_{N}^{(d/2-1)}$ and an integer $u$
coprime with $N$ which specifies how to gauge $\mathbb{Z}_{N}^{(d/2-1)}$, one
may look for $\mathbb{Z}_{N}^{(d/2-1)}$ SPTs in $d$ dimensions satisfying the
one of the following closely related properties:

  1. (a)

The $\mathbb{Z}_{N}^{(d/2-1)}$ SPT is equipped with a
$\mathbb{Z}_{N}^{(d/2-1)}$ duality defect with an _unspecified_ Frobenius-
Schur indicator (labeled by $\epsilon$). Note that the bicharacter is
specified by $u$. Equivalently, the $\mathbb{Z}_{N}^{(d/2-1)}$ SPT is
invariant under gauging $\mathbb{Z}_{N}^{(d/2-1)}$;

  2. (b)

The $\mathbb{Z}_{N}^{(d/2-1)}$ SPT is equipped with a
$\mathbb{Z}_{N}^{(d/2-1)}$ duality defect with a _given_ Frobenius-Schur
indicator (labeled by $\epsilon$).

A $\mathbb{Z}_{N}^{(d/2-1)}$-SPT satisfying property (b) automatically
satisfies property (a) since $\mathbb{Z}_{N}^{(d/2-1)}$ duality defect
implements gauging $\mathbb{Z}_{N}^{(d/2-1)}$. However, the converse is not
true: it is possible that for a given coprime pair $(N,u)$, the duality defect
with certain $\epsilon$ may not be realized by any $\mathbb{Z}_{N}^{(d/2-1)}$
SPT, or equivalently, the duality defect may be anomalous for some choice of
$\epsilon$. See [42] for the general criteria of the anomaly in 2d.

In [1, 2, 53, 19, 42], the authors discussed the $G$-SPTs with property (a).
It was found that

  1. 1.

in 2d and among all $N$’s, the only $\mathbb{Z}_{N}^{(0)}$ SPT is a trivial
SPT with partition function ${\mathcal{Z}}=1$. Furthermore, the trivial SPT is
not invariant under gauging $\mathbb{Z}_{N}^{(0)}$ for any coprime pair
$(N,u)$;

  2. 2.

in 4d and among all $N$’s, an $\mathbb{Z}_{N}^{(1)}$ SPT is classified by

$\displaystyle\begin{split}\frac{2\pi
k}{N}\frac{N+1}{2}\int_{X_{4}}B^{(2)}B^{(2)},&\quad k\in\mathbb{Z}_{N},\quad
X_{4}=\text{ spin or non-spin},\\\ \frac{2\pi
k}{2N}\int_{X_{4}}{\mathcal{P}}(B^{(2)}),&\quad\begin{cases}k\in\mathbb{Z}_{2N},&\quad
X_{4}=\text{ non-spin},\\\ k\in\mathbb{Z}_{N},&\quad X_{4}=\text{ spin},\\\
\end{cases}\end{split}$ (5.1)

and there exists an $\mathbb{Z}_{N}^{(1)}$ SPT invariant under gauging
$\mathbb{Z}_{N}^{(1)}$ for a given $u$ if and only if there is an integer $r$
solving the following equation242424In [1] only the case $u=1$ was discussed.
But it is straightforward to check that the same condition holds for any $u$
coprime with $N$.

$\displaystyle r^{2}=-1\text{ mod }\begin{cases}N,&\text{ odd }N\text{ on spin
and non-spin $X_{4}$},\\\ N,&\text{ even }N\text{ on spin }X_{4},\\\
2N,&\text{ even }N\text{ on non-spin }X_{4}.\end{cases}$ (5.2)

Solving (b) is harder. In 2d, for general $G$, a classification of $G$ SPTs
equipped with a $G$ duality defect was achieved in [42], see also [36]. In 3d,
some related recent works on the anomaly of fusion 2-categories can be found
in [45, 4].

###  5.2 Duality defects, gauging, TQFT, and relation with group theoretical
duality defects

Given an abelian group $\mathbb{Z}_{N}^{(1)}$ and an integer $u$ coprime with
$N$ which specifies how to gauge $\mathbb{Z}_{N}^{(1)}$, one may look for
$\mathbb{Z}_{N}^{(1)}$ symmetric 4d TQFTs252525In this subsection, we
constrain our discussion to duality defects in 4d. The reason is that in 2d,
the TQFTs with one ground state on $S^{1}$ spatial manifold is always an SPT,
hence the discussion reduces to Sec. 5.1.  satisfying the one of the following
closely related properties:

  1. (a’)

The $\mathbb{Z}_{N}^{(1)}$ symmetric TQFT (with one ground state on $S^{3}$
spatial manifold) is equipped with a $\mathbb{Z}_{N}^{(1)}$ duality defect
with an _unspecified_ Frobenius-Schur indicator (labeled by $\epsilon$). Note
that the bicharacter is specified by $u$. Equivalently, the
$\mathbb{Z}_{N}^{(1)}$ symmetric TQFT is invariant under gauging
$\mathbb{Z}_{N}^{(1)}$;

  2. (b’)

The $\mathbb{Z}_{N}^{(1)}$ symmetric TQFT (with one ground state on $S^{3}$
spatial manifold) is equipped with a $\mathbb{Z}_{N}^{(1)}$ duality defect
with a _given_ Frobenius-Schur indicator (labeled by $\epsilon$).

The requirement of one ground state on $S^{3}$ spatial manifold ensures that
the $\mathbb{Z}_{N}^{(1)}$ duality symmetry is not spontaneously broken,
however, the $\mathbb{Z}_{N}^{(1)}$ symmetry is allowed to be spontaneously
broken since we don’t require one ground state on other spatial manifolds such
as $T^{3}$. Similar to the discussion in Sec. 5.1, a $\mathbb{Z}_{N}^{(1)}$
symmetric TQFT satisfying property (b’) automatically satisfies property (a’).
However, the converse is not true.

In [53], a classification for (a’), when the 4d manifold is spin, has been
achieved. It was found that a 4d (spin) TQFT is invariant under gauging
$\mathbb{Z}_{N}^{(1)}$ if and only if $N$ has the form

$\displaystyle N=L^{2}M,\qquad\text{$L\in\mathbb{Z}$, and $\exists$ an integer
$r$ solving }r^{2}=-1\text{ mod }M.$ (5.3)

This is precisely the condition for the $\mathbb{Z}_{N}^{(1)}$ duality defects
to be group theoretical.

SPT $=$ SPT$/\mathbb{Z}_{N}^{(0)}$
---
(a)
Group theoretical
---
$\mathbb{Z}_{N}^{(0)}$ duality defect
Figure 3: For 2d theory: space of $N$ where group theoretical
$\mathbb{Z}_{N}^{(0)}$ duality defects exist (bounded by orange circle), and
where there exists a $\mathbb{Z}_{N}^{(0)}$-SPT invariant under gauging
$\mathbb{Z}_{N}^{(0)}$ (bounded by blue circle). The orange circle bounds the
regime where $N$ is a perfect square, and the blue circle bounds only a point
$N=1$. As a consequence, the cases satisfying (b) also only contains a trivial
point $N=1$.  SPT $=$ SPT$/\mathbb{Z}_{N}^{(1)}$
---
(a)
$\exists$ TQFT equipped with $\mathbb{Z}_{N}^{(1)}$ duality
---
defect with given $\epsilon$ (b’)
TQFT $=$ TQFT$/\mathbb{Z}_{N}^{(1)}$
---
(a’)
$=$ Group theoretical $\mathbb{Z}_{N}^{(1)}$ duality defect Figure 4: For 4d
theory: space of $N$ where group theoretical $\mathbb{Z}_{N}^{(1)}$ duality
defects exist (bounded by orange circle) which coincides with the situation
where there exists a $\mathbb{Z}_{N}^{(1)}$-TQFT invariant under gauging
$\mathbb{Z}_{N}^{(1)}$ (hence also bounded by orange circle), where there
exists a TQFT equipped with a $\mathbb{Z}_{N}^{(1)}$ duality defect with given
$u,v,\epsilon$, i.e. satisfying (b’) (bounded by red circle), and where there
exists a $\mathbb{Z}_{N}^{(1)}$-SPT invariant under gauging
$\mathbb{Z}_{N}^{(1)}$, i.e. satisfying (a) (bounded by blue circle). The
orange circle bounds the regime where $N$ takes the form $L^{2}M$ and $-1$ is
a quadratic residue of $M$. The red circle bounds a possibly smaller regime
undetermined so far. The blue circle bounds the regime where $-1$ is a
quadratic residue of $N$. There is also a possibly smaller regime satisfying
(b) also undetermined so far, which we haven’t drawn in the figure.

To understand why there is such a coincidence, we again use the SymTFT. It
turns out that the SymTFT is a natural set up in discussing 4d TQFTs with the
property (a’) or (b’). Any 4d TQFT with a non-anomalous $\mathbb{Z}_{N}^{(1)}$
global symmetry can be expanded into a 5d slab where in the 5d bulk is a
$\mathbb{Z}_{N}$ 2-form gauge theory (2.1), the left boundary is the Dirichlet
boundary condition (2.2), and the right boundary is another topological
boundary condition (2.3) specified by the 4d TQFT. Since the SymTFT does not
contain genuinue line or point operator, it is clear that after shrinking the
5d slab to get a genuine 4d theory, there isn’t any non-trivial topological
local operator, neither directly coming from point operator in the bulk nor
from compactifying line operators along the shrinked direction. Hence the only
topological local operator is the trivial identity, and the unique ground
state on $S^{3}$ is guaranteed. Furthermore, it is also known [19, 7, 11, 36,
54] that gauging $\mathbb{Z}_{N}^{(1)}$ of the 4d TQFT can be equivalently
achieved by fusing a $\mathbb{Z}_{4}^{\text{em}}$ symmetry defect (constructed
as a condensation defect in [19, 54]) of the 5d $\mathbb{Z}_{N}^{(1)}$ gauge
theory to either of the topological boundary. Requiring (a’), i.e. the TQFT to
be invariant under gauging $\mathbb{Z}_{N}^{(1)}$, amounts to requiring the
right topological boundary state (2.3) to be invariant under fusing with the
$\mathbb{Z}_{4}^{\text{em}}$ symmetry defect. This is precisely the stability
condition in Sec. 4.2, from where we derived the same condition as (5.3).

Solving (b’) is again harder. To achieve a full classification of TQFTs
equipped with a duality defect, we need to consider the SymTFT of the duality
symmetry, i.e. $\mathbb{Z}_{N}^{(1)}$ 2-form gauge theory with
$\mathbb{Z}_{4}^{\text{em}}$ gauged, as in (2.5). Note that the resulting
SymTFT depends on both the choice of bicharacters (see the discussion below
(2.5)), as well as the choice of a discrete theta term of the
$\mathbb{Z}_{4}^{\text{em}}$ gauge field $x$. The choice of discrete theta
term can be understood as a higher dimensional generalization of the Frobenius
Schur indicator as discussed in Sec. 4.3. There is a topological line operator
$K=e^{\frac{i\pi}{2}\oint x}$, which by construction topologically terminates
on the left topological boundary, just as the electric surface operator
$e^{\frac{2\pi i}{N}\oint b}$ does. To ensure that the duality symmetry is not
spontaneously broken (or equivalently there is one ground state on $S^{3}$),
we should demand that the $K$-line can not topologically terminate on the
right topological boundary. This imposes additional constraints for (b’),
which will be left for future study.

We summarize the main results of $\mathbb{Z}_{N}^{(0)}$ duality defects in 2d
in Fig. 3, and those of $\mathbb{Z}_{N}^{(1)}$ duality defects in 4d in Fig.
4. 262626We thank Philip Boyle Smith for the discussions on the Venn diagrams.

## Acknowledgements

We are grateful to Justin Kaidi for collaboration at the early stage of the
project. We also thank Philip Boyle Smith, Justin Kaidi, and Kantaro Ohmori
for useful discussions, and John McGreevy, Ken Intriligator, Justin Kaidi,
Kantaro Ohmori, and Carolyn Zhang for useful comments on the draft. Z.S. is
supported from the US Department of Energy (DOE) under cooperative research
agreement DE-SC0009919, Simons Foundation award No. 568420 (K.I.) and the
Simons Collaboration on Global Categorical Symmetries. Y.Z. is partially
supported by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo.

##  Appendix A Lagrangian subgroup for $\mathbb{Z}_{N}$ $2$-form gauge theory
in 5d

In this appendix, we provide an alternative derivation of the Lagrangian
subgroups for $\mathbb{Z}_{N}$ $2$-form gauge theory in $5$d. Recall the
Lagrangian subgroup consists of maximal number of pairs $(e,m)$ such that any
two pairs $(e,m)$ and $(e^{\prime},m^{\prime})$ satisfy $e^{-\frac{2\pi
i}{N}(em^{\prime}-me^{\prime})\langle\sigma,\sigma^{\prime}\rangle}=1$. The
maximal condition ensures that if we condense all the surface operators in the
Lagrangian subgroup, every surface operator outside the Lagrangian subgroup
braids non-trivially with at least one surface operator in the Lagrangian
subgroup, and hence is projected out.

To proceed, we quote a theorem from [55, Sec. 4.2], which claims that the
Lagrangian subgroups are classified by a subgroup $Q$ of $H=\mathbb{Z}_{N}$
and a symmetric bilinear form $\Psi$ on $Q$, that is, $\Psi:Q\times
Q\rightarrow U(1)$ where

$\Psi(h_{1},h_{2})=\Psi(h_{2},h_{1}),\quad\Psi(h_{1}h_{2},h_{3})=\Psi(h_{1},h_{3})\Psi(h_{2},h_{3}).$
(A.1)

In this case, $Q=\mathbb{Z}_{p}$ for some integer $p$ dividing $N$. To see all
symmetric bilinear forms $\Psi$ on $\mathbb{Z}_{p}$, let $\eta$ denote the
generator of $\mathbb{Z}_{p}$, and from (A.1) we find

$\Psi(\eta^{m},\eta^{n})=\Psi(\eta,\eta)^{mn}.$ (A.2)

Hence, $\Psi$ is completely determined by $\Psi(\eta,\eta)$. First note that
$\Psi(\eta,1)=\Psi(1,\eta)=1$.272727This follows from the second condition in
(A.1), where $h_{1}=h_{2}=1$ and $h_{3}=\eta$.  Taking $m=p$ and $n=1$ in the
above equation, we find $\Psi(\eta,\eta)=e^{\frac{2\pi i\ell}{p}}$ where
$\ell=0,1,\cdots,p-1$. Hence, we will use the $\Psi_{p,\ell}$ to denote the
symmetric bilinear form such that

$\Psi_{p,\ell}(\eta,\eta)=e^{\frac{2\pi i}{p}\ell}.$ (A.3)

The Ref. [55, Sec. 4.2] also specifies how the elements in the Lagrangian
subgroup can be constructed from $(Q,\Psi_{p,\ell})$. Denote an arbitrary
operator with electric and magnetic charge $(x,y)$ as $\alpha^{x}\beta^{y}$,
with $y\in\mathbb{Z}_{N}$ and
$x\in\operatorname{Hom}(\mathbb{Z}_{N},U(1))\simeq\mathbb{Z}_{N}$. We also
define the standard pairing $\alpha(\beta)=e^{\frac{2\pi i}{N}}$. The key
statement is that $\alpha^{x}\beta^{y}$ belongs to the Lagrangian subgroup
specified by $(Q,\Psi_{p,\ell})$ if

  * •

the magnetic charge takes value in $Q$, i.e. $y=\frac{N}{p}y^{\prime}$ with
$y^{\prime}\in Q$;

  * •

the electric charge is constrained by the pairing relation:
$\alpha^{z}(\beta^{y})=\Psi_{p,\ell}(b^{y},b^{z})$ for any
$z=\frac{N}{p}z^{\prime}$ with $z^{\prime}\in Q=\mathbb{Z}_{p}$.

By definition, $\alpha^{z}(\beta^{y})=e^{\frac{2\pi
i}{N}\frac{N}{p}z^{\prime}x}$. Using (A.3), the above condition gives

$\displaystyle z^{\prime}(x-\ell y^{\prime})=0\text{ mod }p,\qquad\forall
z^{\prime}\in\mathbb{Z}_{p}.$ (A.4)

This enforces $x=\ell y^{\prime}+px^{\prime}$ with
$x^{\prime}\in\mathbb{Z}_{N/p}$. Thus the charges are

$\displaystyle(x,y)=(\ell
y^{\prime}+px^{\prime},y^{\prime}N/p)=x^{\prime}(p,0)+y^{\prime}(\ell,N/p),\qquad
x^{\prime}\in\mathbb{Z}_{N/p},\quad y^{\prime}\in\mathbb{Z}_{p}.$ (A.5)

In other words, the charges in the Lagrangian subgroup are generated by

$\left(\ell,N/p\right),\quad\left(p,0\right).$ (A.6)

Notice that the two generators could be linearly dependent in general, but
they nevertheless generated the full Lagrangian algebra.

##  Appendix B Lagrangian subgroup with one generator

In this appendix, we focus on the case where the Lagrangian subgroup of 5d
$\mathbb{Z}_{N}$ 2-form gauge theory is generated by a single surface operator
$\widetilde{S}_{(p,q)}$, with the coprime condition $\gcd(p,q,N)=1$. This
special case has been explored in [47].

$\displaystyle{\mathcal{A}}_{(p,q)}=\\{\widetilde{S}_{(kp,kq)}|k\in\mathbb{Z}_{N},\gcd(p,q,N)=1\\}.$
(B.1)

Because of the coprime condition, it generates $N$ distinct operators
$\widetilde{S}_{(kp,kq)}$ with $k=0,1,...,N-1$. The trivial mutual braiding
condition is clearly satisfied. Different pairs $(p,q)$ may generate the same
Lagrangian subgroup, for instance when $N=5$, $(p,q)=(1,1)$ and
$(p^{\prime},q^{\prime})=(3,3)$ generate the same Lagrangian subgroup
$\\{\widetilde{S}_{(k,k)}|k\in\mathbb{Z}_{5}\\}$. Such redundancy will not be
a problem for our purposes.

To determine the counter term within $\widetilde{S}_{(kp,kq)}$, we substitute
$(e,m)=k(p,q)$ in (4.4), and find

$\displaystyle K_{kp,kq}=\begin{cases}\frac{2\pi}{N}\frac{N+1}{2}k^{2}pq\text{
mod }2\pi,&\text{ odd }N\\\ \frac{2\pi}{2N}k^{2}pq+\pi J_{k}\text{ mod
}2\pi,&\text{ even }N\end{cases}.$ (B.2)

Substituting (B.2) into (4.5), we find that for odd $N$ (4.5) is automatically
satisfied, while for even $N$, $J_{k}$ should satisfy

$\displaystyle J_{k}=kJ\text{ mod }2$ (B.3)

where $J=0$ or $1$. These two solutions are precisely the topological
refinement discussed extensively in [47].

For even $N$, it is useful to extend the range of charges from
$\mathbb{Z}_{N}$ to $\mathbb{Z}_{2N}$, so that shifting the topological
refinement $J\to J+1$ can be replaced by shifting the electric or magnetic
charge by $N$. For simplicity, take $k=1$. Since $\gcd(p,q,N)=1$, $p,q$ can
not be both even. Suppose $p$ is odd. Then $J\to J+1$ can be achieved by
shifting $q\to q+N$.

In summary, when there is a single generator, the Lagrangian subgroup of 5d
$\mathbb{Z}_{N}$ 2-form gauge theory is generated by the surface operator

$\displaystyle\widetilde{S}_{(p,q)}(\sigma)=\begin{cases}S_{(p,0)}(\sigma)S_{(0,q)}(\sigma)e^{\frac{2\pi
i}{N}\frac{N+1}{2}pq\braket{\sigma,\sigma}},&\text{ odd }N\\\
S_{(p,0)}(\sigma)S_{(0,q)}(\sigma)e^{\frac{2\pi
i}{2N}pq{\mathcal{P}}(\sigma)+i\pi J\braket{\sigma,\sigma}},&\text{ even }N\\\
\end{cases}$ (B.4)

where $J=0,1$ specifies the topological refinement in [47] on 4d non-spin
spacetime manifold. When the 4d spacetime manifold is spin, the $J$ dependence
is trivialized.

#####  $\mathbb{Z}_{4}^{\text{em}}$ stable Lagrangian subgroup for odd $N$:

Under $\mathbb{Z}_{4}^{\text{em}}$, it is mapped to
$\widetilde{S}_{(-vq,up)}(\sigma)$, which should also belong to the Lagrangian
subalgebra, if stable. So stable Lagrangian subalgebra implies that there
exists $k$, such that

$\displaystyle k(p,q)=(-vq,up)\text{ mod }N.$ (B.5)

Given (B.5), the $\mathbb{Z}_{4}^{\text{em}}$ image of any other element
$a(p,q)$ is also within the Lagrangian subalgebra, $a(-vq,up)=ka(p,q)\text{
mod }N$. So the Lagrangian subalgebra is $\mathbb{Z}_{4}^{\text{em}}$ stable
if and only if (B.5) is satisfied.

For which $N,u,v$ do there exist $k,p,q$ such that (B.5) holds? We first note
that (B.5) implies

$\displaystyle(k^{2}+1)p=0\text{ mod }N,\qquad(k^{2}+1)q=0\text{ mod }N.$
(B.6)

Further combining with $\gcd(p,q,N)=1$, we have $xp+yq=1\text{ mod }N$.
Multiplying $x$ and $y$ to the above two equations in (B.6), we find
$k^{2}=-1\text{ mod }N$. Conversely, given $k$ satisfying $k^{2}=-1\text{ mod
}N$, we simply take $(p,q)=(vk,u)$ such that $\gcd(p,q,N)=1$ and (B.5) is
satisfied. Thus we have shown that (B.5) holds if and only if

$\displaystyle k^{2}+1=0\text{ mod }N$ (B.7)

holds, for any $u,v$. This corresponds to the special case $L=1$ in Sec. 4.2.
As commented there, (B.7) is precisely the condition where the
$\mathbb{Z}_{N}^{(1)}$ duality defect in 4d with odd $N$ can be mapped to an
invertible defect discussed in [2, App.C], and also in [26] for prime $N$.
Such odd $N$’s belong to the red series listed in Tab. 1.

#####  $\mathbb{Z}_{4}^{\text{em}}$ stable Lagrangian subgroup for even $N$:

As pointed out in Sec. 4.1.2, the generator also depends on the choice of
topological refinement $J=0,1$, and different choices can be packaged by
extending the range of electric and magnetic charges from $\mathbb{Z}_{N}$ to
$\mathbb{Z}_{2N}$, hence we take $p,q\in\mathbb{Z}_{2N}$ below for
convenience. Under $\mathbb{Z}_{4}^{\text{em}}$, the generator
$\widetilde{S}_{(p,q)}(\sigma)$ is mapped to

$\displaystyle\widetilde{S}_{(p,q)}(\sigma)\to\widetilde{S}_{(-vq,up)}(\sigma),$
(B.8)

and the Lagrangian subgroup is $\mathbb{Z}_{4}^{\text{em}}$ stable if and only
if there exists $k$ such that

$\displaystyle(-vq,up)=k(p,q)\text{ mod }2N.$ (B.9)

By applying the same argument as for (B.7), we find that (B.9) is again
equivalent to

$\displaystyle k^{2}=-1\text{ mod }2N.$ (B.10)

There is no solution to this for any even $N$. However, if we restrict to 4d
spin manifolds, different topological refinements are trivialized and the
electric and magnetic charges obey $p\sim p+N$ and $q\sim q+N$, i.e. the
charges are back to $\mathbb{Z}_{N}$ valued. Thus (B.9) reduces to
$(-vq,up)=k(p,q)\text{ mod }N$, which is equivalent to $k^{2}=-1\text{ mod
}N$. This again reproduces the special case $L=1$ in Sec. 4.2 as well as the
results in [2, App. C].

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