Symmetry TFTs and Anomalies of Non-Invertible Symmetries

Justin Kaidi1, Emily Nardoni2, Gabi Zafrir3, and Yunqin Zheng2,4

1 | Department of Physics,
---|---
| University of Washington, Seattle, WA, 98195, USA
2 | Kavli Institute for the Physics and Mathematics of the Universe,
| University of Tokyo, Kashiwa, Chiba 277-8583, Japan
3 | Simons Center for Geometry and Physics,
| Stony Brook University, Stony Brook, NY 11794-3636, USA
4 | Institute for Solid State Physics,
| University of Tokyo, Kashiwa, Chiba 277-8581, Japan

It is known that the ’t Hooft anomalies of invertible global symmetries can be
characterized by an invertible TQFT in one higher dimension. The analogous
statement remains to be understood for non-invertible symmetries. In this note
we discuss how the linking invariants in a non-invertible TQFT known as the
Symmetry TFT (SymTFT) can be used as a diagnostic for ’t Hooft anomalies of
non-invertible symmetries. When the non-invertible symmetry is non-
intrinsically non-invertible, and hence the SymTFT is a Dijkgraaf-Witten
model, the linking invariants can be computed explicitly. We illustrate this
proposal through the examples of the abelian Higgs model in 2d, as well as
adjoint QCD and ${\mathcal{N}}=4$ super Yang-Mills in 4d. We also comment on
how the ’t Hooft anomalies of non-invertible symmetries impose new constraints
on the dynamics.

###### Contents

  1. 1 Introduction and summary
    1. 1.1 Anomalies of invertible symmetries
    2. 1.2 Anomalies of non-invertible symmetries
  2. 2 Non-invertible symmetries and anomalies in 2d
    1. 2.1 SymTFT of invertible symmetries
    2. 2.2 Anomaly of invertible symmetry from linking invariants
    3. 2.3 Non-invertible symmetries from gauging
    4. 2.4 Non-invertible defects from SymTFT
    5. 2.5 Anomalies of the non-invertible symmetries
    6. 2.6 Application: Abelian Higgs Model
  3. 3 Non-invertible symmetries and anomalies in 4d
    1. 3.1 SymTFT of invertible symmetries
    2. 3.2 Anomaly of invertible symmetry from linking invariants
    3. 3.3 Non-invertible symmetries from gauging
    4. 3.4 Non-invertible defects from SymTFT
    5. 3.5 Anomalies of non-invertible symmetries
    6. 3.6 Application 1: Adjoint QCD
    7. 3.7 Application 2: $\mathcal{N}=4$ SYM
  4. A Linking numbers
  5. B $\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ Dijkgraaf-Witten model in 3d
    1. B.1 Line operators and fusion rules
    2. B.2 Linking numbers between lines
  6. C $\mathbb{Z}_{2MN}^{(0)}\times\mathbb{Z}_{N}^{(1)}$ Dijkgraaf-Witten model in 5d
    1. C.1 Extended operators and fusion rules
    2. C.2 Linking numbers between extended operators

##  1 Introduction and summary

Non-invertible symmetries have a long history of study in two dimensions [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], but it was not until fairly recently that they were realized in
higher-dimensional theories [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]. One of the main goals in the study of non-invertible
symmetries is to obtain a detailed understanding of their dynamical
consequences. For standard, invertible symmetries, it is well known that ‘t
Hooft anomalies provide important constraints on the low-energy physics of the
theory, and one may optimistically hope that the notion of a ‘t Hooft anomaly
can be extended to non-invertible symmetries as well. The purpose of this note
is to discuss certain easy-to-compute quantities which can probe the existence
of ’t Hooft anomalies of non-invertible symmetries, and to illustrate them
using examples in two and four dimensions.

###  1.1 Anomalies of invertible symmetries

Let us begin by briefly reviewing the familiar case of ’t Hooft anomalies for
invertible symmetries; see e.g. [82, 83] for an overview. Given a
$d$-dimensional quantum field theory with an invertible global symmetry $G$ on
a manifold $X_{d}$, we define the ’t Hooft anomaly to be the obstruction to
gauging $G$. To see how this obstruction arises, we couple the theory to a
background $G$ gauge field $A$. Under background field transformations, the
partition function transforms as

$Z[X_{d},A]\to Z[X_{d},A]\,e^{2\pi i\int_{X_{d}}f(A,g)}~{},$ (1.1)

where $g$ is the gauge transformation parameter, and $f$ is some local
functional of $g$ and $A$. Gauging $G$ means choosing a representative of $A$
in each gauge orbit (i.e. fixing a gauge) and then summing over such
representatives. When the phase $\mathrm{exp}\\{{2\pi i\int_{X_{d}}f(A,g)}\\}$
is non-trivial and cannot be cancelled by modifying the partition function by
local counterterms built from background fields, summing over the
representative $A$ in the gauge orbit is ambiguous, i.e. the resulting
partition function depends on the choice of representatives. Hence the gauged
theory is ill-defined, and there is an obstruction to gauging $G$.

In the modern understanding of anomalies (as explained in e.g. [84, 85]), the
’t Hooft anomaly of $G$ is naturally described by an invertible field theory
in one higher dimension, defined on a manifold $X_{d+1}$ whose boundary is
$X_{d}$. This goes under the name of _anomaly inflow_ [86]. Concretely, the
anomaly inflow theory is an invertible TQFT with action
$2\pi\int_{X_{d+1}}\omega(A)$, such that under background gauge
transformations it changes by $\omega(A)\to\omega(A)-df(A,g)$. Hence the
combination

$Z[X_{d},A]\,e^{2\pi i\int_{X_{d+1}}\omega(A)}$ (1.2)

is gauge invariant. The possible inflow actions $\omega(A)$ are classified by
bordisms equipped with a map to the classifying space of $G$, and with a
suitable spacetime structure.111Note that there are bordism invariants which
cannot be written as the integral of a local functional of $A$, e.g. the Arf
invariant.  For convenience, we will refer to this invertible TQFT as the
Anomaly TFT (AnomTFT). Conversely, the AnomTFT can be taken to define the ’t
Hooft anomaly of the invertible symmetry.

When the symmetry $G$ is a finite group, there is an alternative perspective.
In this case, turning on a background gauge field $A$ amounts to inserting a
network of $G$ symmetry defects, and gauge transformations amount to local
deformations of the defect network. In 2d the defects are topological lines,
and the local deformations of the topological lines are characterized by the
$F$-symbols, as shown in Figure 1. Since gauging $G$ amounts to summing over
all gauge field configurations up to gauge transformations, it is also
equivalent to first fixing a triangulation of the spacetime manifold and then
summing over all possible defect configurations on the dual lattice of the
triangulation. When the partition function is not invariant under a local re-
triangulation of the spacetime manifold (for instance by performing Pachner
moves), then the resulting partition function is subject to an ambiguity, and
hence there is an obstruction to gauging. Such an ambiguity arises when the
$F$-symbols in Figure 1 belong to a non-trivial group cohomology class in
$H^{3}(G,U(1))$.

$L_{g}$$L_{h}$$L_{k}$$L_{x}$$L_{\ell}$

$\,\,=\,\,\sum_{L_{y}\in L_{h}\times L_{k}}(F_{g,h,k}^{\ell})^{x}_{y}$
$L_{g}$$L_{h}$$L_{k}$$L_{y}$$L_{\ell}$

Figure 1: Definition of the $F$-symbols, which generalize ‘t Hooft anomalies
for non-invertible symmetries in $(1+1)$d. The sum is over all $L_{y}$
appearing in the fusion of $L_{h}$ with $L_{k}$. When the defects are
invertible, the sum on the right hand side only contains one term.

###  1.2 Anomalies of non-invertible symmetries

We now proceed to the case of non-invertible symmetries. For simplicity, we
will assume that the non-invertible symmetries are finite. As for invertible
symmetries, it is possible to define gauging of non-invertible symmetries by
fixing a triangulation of the spacetime manifold and summing over all possible
non-invertible defect configurations.222The gauging of non-invertible
symmetries in higher dimensions involves additional subtleties and a complete
description remains unavailable at present. An example of such a subtlety is
whether one should include all possible condensation defects in the sum above.
Fortunately, the following discussion will not depend on these subtleties.
Furthermore, non-invertible symmetries can also have non-trivial ’t Hooft
anomalies, in the sense of obstructions to gauging. The presence of non-
trivial ’t Hooft anomalies means that after summation the resulting partition
function depends on the choice of triangulation (i.e. it is not “gauge
invariant”), and hence is ill-defined.

We may now ask about the anomaly inflow mechanism for non-invertible
symmetries. It is natural to suspect that the anomalies for non-invertible
symmetries can again be cancelled by coupling to a bulk invertible TQFT.
However, because the notion of background fields for non-invertible symmetries
is still poorly understood, a systematic understanding of this AnomTFT is
lacking. We will not shed light on this issue here. Instead, we point out that
while the AnomTFT of non-invertible symmetries is poorly understood, the
Symmetry TFT (SymTFT)[87, 88, 89, 90, 30, 91, 92, 57, 60, 93, 94, 95, 21, 57,
58] is relatively well-understood (at least in some cases), and can be used as
a partial diagnostic for the existence of ‘t Hooft anomalies.

Before proceeding, let us mention that there is already a definitive test for
anomalies of non-invertible symmetries in 2d QFTs, introduced in [15, 16]. In
particular, those references showed that the presence of an anomaly is
equivalent to the lack of a so-called “fiber functor,” i.e. a module category
with a single simple object. While this gives a complete characterization of
anomalies in 2d, determining in practice whether a fiber functor exists is not
always straightforward. Furthermore, the notion of fiber functor is more
complicated in higher dimensions, where one has to consider fusion higher-
categories. Our construction in this note provides an alternative,
occasionally more practical, but strictly weaker characterization of anomalies
of non-invertible symmetries.

SymTFT
---
Topo.b.c.Dyn.b.c. AnomTFT
---
Figure 2: Any $d$-dimensional QFT can be expanded into a $(d+1)$-dimensional
slab with the SymTFT living inside of it, with a topological boundary
condition on the left boundary and a dynamical boundary condition on the right
boundary.

####  1.2.1 Symmetry TFT

We now describe a sufficient condition for the presence of an anomaly for a
non-invertible symmetry, making use of the SymTFT. To set the stage, let us
recall that for any $d$-dimensional QFT with non-invertible symmetry described
by a higher fusion category ${\mathcal{C}}$, one can expand the system into a
$(d+1)$-dimensional slab with the SymTFT living inside it; see Figure 2. The
SymTFT is given by (the higher-categorical generalization of) the Turaev-Viro
theory of ${\mathcal{C}}$, whose topological operators are given by (the
higher-categorical generalization of) the Drinfeld center of ${\mathcal{C}}$,
i.e. ${\mathcal{Z}}({\mathcal{C}})$. The left boundary is a topological
Dirichlet boundary condition, while the right boundary is a dynamical boundary
condition. Shrinking the slab by colliding the two boundaries reproduces the
original $d$-dimensional QFT. See e.g. [60, 6] for further details.

$a$$x$$b$$c$$y$ Figure 3: Bulk topological operators terminating on the
topological boundary. We refer to $a$ as “terminable” and $b$ as “non-
terminable.” $a$$a^{\prime}$$\longrightarrow$$a$$a^{\prime}$ Figure 4:  Two
topological operators that are terminable on the topological boundary should
have trivial linking invariant in the bulk.

Conversely, given any $(d+1)$-dimensional TQFT in a slab with topological
operators denoted by ${\mathcal{Z}}$, together with a topological boundary
condition (associated with a Lagrangian subalgebra ${\mathcal{A}}$) on the
left as well as a dynamical boundary condition on the right, shrinking the
slab defines a $d$-dimensional QFT whose global symmetry is determined by
${\mathcal{Z}}$ and the topological boundary condition ${\mathcal{A}}$. To see
this more concretely, let us place the topological operators in the bulk
orthogonal to the topological boundary, as shown in Figure 3.

When the bulk $p$-dimensional topological operator labeled by $a$ belongs to
the Lagrangian algebra ${\mathcal{A}}$ associated with the topological
boundary (red plane), $a$ can terminate on the topological boundary at a
$(p-1)$-dimensional locus $x$; see the left panel of Figure 3. This implies
that the linking invariant between two bulk operators
$a,a^{\prime}\in{\mathcal{A}}$ must be trivial. To see why, we consider the
two configurations shown in Figure 4, which differ by a braiding in the bulk.
Since the boundary condition is topological, one can freely move the boundary
locus of $a^{\prime}$ around $a$, and hence the two configurations are
topologically equivalent. Thus the linking invariants between $a$ and
$a^{\prime}\in{\mathcal{A}}$ must be trivial.

When the $p$-dimensional topological operator labeled by $b$ does not belong
to ${\mathcal{A}}$, the $(p-1)$-dimensional locus $y$ is further attached to a
$p$-dimensional topological operator $c$ sitting within the topological
boundary. Here $y$ is a morphism between the bulk operator $b$ and the
boundary operator $c$, and we can denote the bulk operator $b$ as a pair
$(c,y)$;333In general, this labeling is not unique. For a given
$b\in{\mathcal{B}}$, there may exist multiple $c\in{\mathcal{C}}$ on the
boundary such that they are connected by some morphisms. We will encounter
such examples in the main text below.  see the right panel of Figure 3. The
topological operators $c$ are expected to form a higher fusion category
${\mathcal{C}}$ describing the global symmetry of the $d$-dimensional QFT
obtained by shrinking the slab. We will denote the set of bulk operators which
can end on the boundary operator $c$ as ${\mathcal{B}}_{c}$. We refer to the
operators in ${\mathcal{A}}$ as “terminable,” and to those in
${\mathcal{B}}_{c}$ (for non-identity $c$) as “non-terminable.”

####  1.2.2 Probing the anomaly via linking invariants

We would now like to consider a non-invertible symmetry whose defects are
given by ${\mathcal{C}}$. Below we will assume that ${\mathcal{C}}$ is
anomaly-free, meaning that it is gaugable. As reviewed above, gauging the
symmetry means that we sum over all possible configurations of ${\mathcal{C}}$
defects in the $d$-dimensional spacetime. We now discuss what this assumption
implies in terms of the topological operators in the bulk SymTFT.

Assuming that the symmetry is anomaly-free, we may gauge ${\mathcal{C}}$ and
then expand the $d$-dimensional ${\mathcal{C}}$-gauged QFT to a
$(d+1)$-dimensional slab. The SymTFT of the gauged theory is the same as the
SymTFT of the ungauged theory, and in particular the topological operators
within the slab are still given by ${\mathcal{Z}}({\mathcal{C}})$. However,
the left topological boundary condition corresponding to the
${\mathcal{C}}$-gauged theory is different from the one before. Because
gauging means summing over all possible ${\mathcal{C}}$ defect configurations,
the left boundary becomes a condensate of the operators in ${\mathcal{C}}$.
Hence there should be at least one topological operator
$b_{c}=(c,y_{c})\in{\mathcal{B}}_{c}$ for each $c\in{\mathcal{C}}$ such that
after gauging it becomes terminable. These operators will be within the
Lagrangian subalgebra of the ${\mathcal{C}}$-gauged topological boundary
condition. By applying the discussion around Figure 4, we see that such
operators must have trivial braiding in the bulk.

We thus have the following main result,

##### Main result:

If the non-invertible symmetry ${\mathcal{C}}$ is anomaly-free (i.e. there is
no obstruction to gauging), then for each $c\in{\mathcal{C}}$ there must be a
choice of morphisms $y_{c}$ denoted by $y_{c}^{*}$ such that the set of
topological operators $\\{b_{c}:=(c,y_{c}^{*})\\}$ in the SymTFT has trivial
linking invariants, i.e. their correlation functions have trivial phases.
Schematically,

${\mathcal{C}}$ is anomaly-free $\displaystyle\Longrightarrow$
$\displaystyle\,\,\begin{matrix}\text{$\exists$ $b_{c}\in{\mathcal{B}}_{c}$
such that the linkings between $b_{c_{1}}$ and $b_{c_{2}}$}\\\ \text{are
trivial for every $c_{1},c_{2}\in{\mathcal{C}}$}\end{matrix}~{}.$ (1.3)

Said in another way, if one cannot find a choice of representatives
$b_{c}\in{\mathcal{B}}_{c}$ such that they have trivial linking invariants,
then the symmetry ${\mathcal{C}}$ is anomalous,

$\displaystyle\begin{matrix}\text{$\nexists$ $b_{c}\in{\mathcal{B}}_{c}$ such
that the linkings between $b_{c_{1}}$ and $b_{c_{2}}$}\\\ \text{are trivial
for every
$c_{1},c_{2}\in{\mathcal{C}}$}\end{matrix}\,\,\,\,\,\Longrightarrow\,\,\,\,\,\text{${\mathcal{C}}$
is anomalous}~{}.$ (1.4)

Let us make a few comments before proceeding to examples:

  1. 1.

The only linking invariants relevant for this note are linking numbers.
However, as reviewed in Appendix A, there can be multiple types of linking
numbers involving different numbers of topological operators.

  2. 2.

Our main result is only a sufficient condition for a non-trivial ’t Hooft
anomaly. Indeed, the condition is not necessary, i.e. even if the linking
numbers among the $b_{c_{i}}$ are trivial, there can still be an obstruction
to gauging ${\mathcal{C}}$.

As an example, consider a (non-invertible) one-form symmetry in 3d, whose
defects are given by lines. We may now consider the linking numbers among
lines in the 4d SymTFT. Let us consider a link with $N$ components, of type
$k$ (see (A.10)). Then the quantities $N$ and $k$ must satisfy

$2\cdot(N-k)+1\cdot k=4~{}.$ (1.5)

As reviewed in Appendix A, when $N=2$ there is only one type of linking
number, with $k=1$; when $N>2$, the allowed linking numbers are labelled by
$k=0,1,...,N-1$. None of these possibilities solves the constraint (1.5), and
we conclude that the anomaly of the 1-form symmetry is not captured by the
linking numbers in the 4d SymTFT. However, interestingly, it is known that the
anomaly of the one-form symmetry in this case is given by the spin of the
defects in 3d [96].

  3. 3.

In two spacetime dimensions, our results are corollaries of the results in
[15]; namely, the non-existence of a fiber functor is a sufficient and
necessary condition for a nontrivial ’t Hooft anomaly, whereas our condition
is only sufficient.

  4. 4.

In the special case in which the non-invertible symmetry is non-intrinsically
non-invertible [60, 35], the associated SymTFT is a Dijkgraaf-Witten (DW)
theory. Computing the linking invariants of operators in a DW theory is a
standard exercise in any dimension. Hence in this case our main result
provides a set of easily computable observables which probe the anomalies of
non-invertible symmetries.

  5. 5.

Using the gauged AnomTFT to probe anomalies has a long history. For example,
in [97] the anomaly of a $\mathbb{Z}_{2}$ symmetry in 2d was detected by
computing the mutual braiding statistics between anyons (i.e. the linking
number involving two magnetic lines) in the double semion TQFT in 3d. Here we
point out that the gauged AnomTFT of the invertible symmetry is none other
than the SymTFT, and that the linking invariants detecting anomalies can
involve multiple operators and go beyond links involving two lines.

Furthermore, the gauged AnomTFT arises naturally in the context of brane
constructions in string theory via anomaly inflow [98, 99, 100, 101, 102]. In
this context, the anomalous variation of the effective 10 or 11d action
cancels the anomalies for degrees of freedom on the worldvolume of the branes,
and the dimensional reduction of the topological terms in the 10 or 11d
effective action yields the AnomTFT via the descent procedure. The SymTFT can
be seen to arise from the dimensional reduction of the same topological terms
[30], and it was recently explained [47] how to use this inflow perspective to
obtain the SymTFT for non-invertible symmetries in the context of holography.

####  1.2.3 Simple examples

It is useful to illustrate the above main result in two simple examples:
namely the cases of ${\mathcal{C}}=\text{Vec}_{\mathbb{Z}_{2}}^{\omega}$ with
$\omega=0,1$ in 2d, i.e. invertible $\mathbb{Z}_{2}$ symmetry in 2d without an
anomaly ($\omega=0$) and with an anomaly ($\omega=1$).

#####  $\omega=0$:

In this case the SymTFT is a $\mathbb{Z}_{2}$ gauge theory without DW twist.
This SymTFT has four line operators denoted by $1,e,m,$ and $f:=e\times m$. We
start with the topological boundary condition associated with the Lagrangian
algebra ${\mathcal{A}}=1\oplus e$. In this case the line $e$ is terminable,
and the set of non-terminable lines is given by $\\{m,f\\}$. In particular,
both $m$ and $f$ are attached to a boundary line $\mathsf{m}\in{\mathcal{C}}$,
and we can write $m=(\mathsf{m},x_{m})$ and $f=(\mathsf{m},x_{f})$, where
$x_{m},x_{f}$ are appropriate morphisms between $m,f$, and $\mathsf{m}$,
respectively. The subset ${\mathcal{B}}_{\mathsf{m}}$ thus contains both $m$
and $f$. To see whether $\mathsf{m}$ is anomalous, we choose an element in
${\mathcal{B}}_{\mathsf{m}}$, i.e. either $m$ or $f$, and compute the linkings
of it with itself. These are known to be trivial, and hence we don’t find an
obstruction to gauging. This is consistent with the known fact that for a
$\mathbb{Z}_{2}$ gauge theory, $\mathsf{m}$ is anomaly-free.

#####  $\omega=1$:

In this case the SymTFT is a $\mathbb{Z}_{2}$ twisted gauge theory, known as
the double semion model. This theory has four line operators $1,b,s,$ and
$\bar{s}:=b\times s$. We start with the topological boundary condition
associated with the Lagrangian algebra ${\mathcal{A}}=1\oplus b$. In this case
$b$ is terminable and the set of non-terminable lines is $\\{s,\bar{s}\\}$.
Both $s$ and $\bar{s}$ are attached to a boundary line
$\mathsf{s}\in{\mathcal{C}}$, and we can write $s=(\mathsf{s},x_{s})$ and
$\bar{s}=(\mathsf{s},x_{\bar{s}})$, where $x_{s},x_{\bar{s}}$ are appropriate
morphisms between $s,\bar{s},$ and $\mathsf{s}$ respectively. The subset
${\mathcal{B}}_{\mathsf{s}}$ thus contains $\bar{s}$ and $s$. To see whether
$\mathsf{s}$ is anomalous, we choose an element of
${\mathcal{B}}_{\mathsf{s}}$, i.e. either $s$ or $\bar{s}$, and compute the
linking invariants of that representative with itself. In the double semion
theory, it is known that the Hopf linking between $s$ and itself, as well as
the Hopf linking between $\bar{s}$ and itself, is non-trivial. In particular,
unlinking produces a factor of $-1$. Hence by our main result, the anomaly of
$\mathsf{s}$ must be non-trivial, as expected.

In the remainder of this note, we will apply our main result to two families
of examples, one in two dimensions (Section 2) and the other in four
dimensions (Section 3). For simplicity, we will restrict to non-intrinsically
non-invertible symmetries constructed via the strategy of [26], in which case
the SymTFT is just a DW theory. In these cases the anomaly is in a sense
“obvious,” since it follows from the self-anomaly of an invertible symmetry
before gauging, but it is nevertheless a good illustration of our strategy.
The computation of the linkings in the corresponding 3d and 5d DW theories is
outlined in Appendices B and C, respectively. We further use the anomaly of
the non-invertible symmetries to make statements regarding the low-energy
dynamics of the abelian Higgs models in 2d, together with adjoint QCD and
${\mathcal{N}}=4$ super Yang-Mills (SYM) in 4d.

##  2 Non-invertible symmetries and anomalies in 2d

In this section we illustrate the discussion in the introduction by means of a
simple 2d example. Our starting point will be a 2d quantum field theory
${\mathcal{X}}$ with an invertible
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ zero-form
global symmetry and ’t Hooft anomalies specified by the 3d AnomTFT,

$\displaystyle\int_{X_{3}}\pi\left(ABC+A\beta A\right)~{},$ (2.1)

written in terms of the background fields $A,B,C$. The operation
$\beta:H^{1}(X_{3},\mathbb{Z}_{2})\to H^{2}(X_{3},\mathbb{Z}_{2})$ is the
Bockstein map associated with the central extension
$1\to\mathbb{Z}_{2}\to\mathbb{Z}_{4}\to\mathbb{Z}_{2}\to 1$, and acts on
$\mathbb{Z}_{2}$-valued cocycles as $\beta=\delta/2$. We will denote the
collection of topological line defects of
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ as
${\mathcal{C}}$.

In order to obtain a theory with a non-invertible symmetry, we gauge
$\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$, which is possible since neither
of these symmetries has a self-anomaly, nor do they have a mixed anomaly with
one another. By the general results of [26], this gauging changes
$\mathbb{Z}_{2}^{A}$ into a non-invertible symmetry. It is natural to expect
that the self-anomaly for the invertible $\mathbb{Z}_{2}^{A}$ symmetry before
gauging leads to a self-anomaly for the non-invertible symmetry after gauging.
We will now show that this is the case by relating it to non-trivial linking
invariants in the SymTFT.

###  2.1 SymTFT of invertible symmetries

##### The SymTFT:

Let us begin by discussing the SymTFT for the theory ${\mathcal{X}}$ with
invertible symmetry. As reviewed in the introduction, a 2d QFT can be expanded
into a 3d slab filled with the SymTFT. The SymTFT is a gauged version of the
AnomTFT given in (2.1), i.e. it is a Dijkgraaf-Witten (DW) TQFT with DW twist
term specified by the AnomTFT,

$\displaystyle\int_{X_{3}}\pi\left(\widehat{a}\delta a+\widehat{b}\delta
b+\widehat{c}\delta c+abc+\frac{1}{2}a\delta a\right)~{}.$ (2.2)

Here all the dynamical fields $\widehat{a},a,...$ are $\mathbb{Z}_{2}$-valued
1-cochains. Integrating out $\widehat{a}$ enforces $a$ to be a
$\mathbb{Z}_{2}$ cocycle, and likewise for $b$ and $c$. Such a DW theory has
been studied in [103, 104] and is reviewed in detail in Appendix B. For our
purposes here, we note that this theory has 22 genuine line operators, among
which eight are invertible and generated by the three independent invertible
Wilson line operators

$\displaystyle U_{a}(M_{1})=e^{i\pi\oint_{M_{1}}a}~{},\hskip
28.45274ptU_{b}(M_{1})=e^{i\pi\oint_{M_{1}}b}~{},\hskip
28.45274ptU_{c}(M_{1})=e^{i\pi\oint_{M_{1}}c}~{}.$ (2.3)

The remaining 14 operators are non-invertible, each having quantum dimension
2. Three of them are given by the magnetic line operators

$\begin{split}\widehat{U}_{a}(M_{1})&\sim\sum_{\phi_{2},\phi_{3}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}\widehat{a}+i\pi\oint_{M_{1}}(-\phi_{2}c+\phi_{3}b+\phi_{2}\delta\phi_{3})}~{},\\\
\widehat{U}_{b}(M_{1})&\sim\sum_{\phi_{3},\phi_{1}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}\widehat{b}+i\pi\oint_{M_{1}}(-\phi_{3}a+\phi_{1}c+\phi_{3}\delta\phi_{1})}~{},\\\
\widehat{U}_{c}(M_{1})&\sim\sum_{\phi_{1},\phi_{2}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}\widehat{c}+i\pi\oint_{M_{1}}(-\phi_{1}b+\phi_{2}a+\phi_{1}\delta\phi_{2})}~{},\\\
\end{split}$ (2.4)

and the others are obtained by fusion between these three and the invertible
line operators. See (B.6) for a complete list. Throughout this note, since we
are mainly interested in the phases of partition and correlation functions, we
will not specify the overall real positive normalization factors and use
$\sim$ to denote equality up to such an overall normalization. Note that in
the definition of the magnetic line operators we have introduced a 1d TQFT on
the worldline of the defect in order to maintain gauge invariance; since the
TQFT is a $(0+1)$d $\mathbb{Z}_{2}$ gauge theory, the quantum dimension is 2.
It is this fact which leads to their non-invertibility. The total quantum
dimension is $D_{\text{tot}}=\sqrt{8\cdot 1^{2}+14\cdot 2^{2}}=8$. See
Appendix B for more details.

##### Dirichlet boundary condition:

To recover the theory ${\mathcal{X}}$, the left topological boundary of the
SymTFT should be taken to be the Dirichlet boundary condition for all three
$\mathbb{Z}_{2}$ symmetries. This sets the dynamical fields $a,b,c$ equal to
background fields $A,B,C$, respectively. In terms of boundary states, this can
be written as

$\displaystyle\bra{D_{a,b,c}}=\sum_{a,b,c\in
C^{1}(X_{2},\mathbb{Z}_{2})}\bra{a,b,c}\,\delta(a-A)\delta(b-B)\delta(c-C)~{}.$
(2.5)

This Dirichlet boundary condition means that all the invertible lines in (2.3)
become trivial on the boundary, and can also terminate perpendicularly on the
boundary. We denote the set of all invertible lines as
${\mathcal{A}}_{\text{inv}}$, where the subscript indicates that the
topological boundary condition gives rise to invertible symmetries. The sum of
the quantum dimension of all the objects in ${\mathcal{A}}_{\text{inv}}$ is
$D_{\text{inv}}=8$, matching the total quantum dimension
$D_{\text{inv}}=D_{\text{tot}}$.

To see how this boundary condition gives rise to an invertible
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ symmetry,
we consider placing each of the 22 bulk line operators orthogonal to the
boundary as shown in Figure 3. Without loss of generality, we may set all the
background fields $A,B,C=0$. All invertible lines belong to
${\mathcal{A}}_{\text{inv}}$ by definition, and hence can terminate on the
boundary, whereas all of the non-invertible lines cannot end. For example,
when the non-invertible line $\widehat{U}_{a}$ touches the Dirichlet boundary,
both $b$ and $c$ become trivial (and consequently $\phi_{2}$ and $\phi_{3}$
also become trivial) but the magnetic portion
$e^{i\pi\oint_{M_{1}}\widehat{a}}$ is not trivialized on the boundary, and
instead extends along the boundary; see the right panel of Figure 3. In other
words, there are eight boundary lines collectively denoted as
${\mathcal{C}}_{\text{inv}}$, generated by

$\displaystyle\widehat{U}_{a}(M_{1}|_{\partial})=e^{i\pi\oint_{M_{1}|_{\partial}}\widehat{a}}~{},\hskip
28.45274pt\widehat{U}_{b}(M_{1}|_{\partial})=e^{i\pi\oint_{M_{1}|_{\partial}}\widehat{b}}~{},\hskip
28.45274pt\widehat{U}_{c}(M_{1}|_{\partial})=e^{i\pi\oint_{M_{1}|_{\partial}}\widehat{c}}~{},$
(2.6)

where $M_{1}|_{\partial}$ represents a line on the boundary. As all of these
boundary lines are invertible and satisfy $\mathbb{Z}_{2}$ fusion rules, upon
shrinking the slab they generate a
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$
invertible global symmetry. We denote the boundary symmetry generators by
$U_{A},U_{B},$ and $U_{C}$, respectively.

##### Dynamical boundary condition:

On the other hand, the dynamical boundary condition on the right captures the
dynamics of the 2d theory, and in particular is non-topological. In terms of
boundary states, it can be written as

$\displaystyle\ket{{\mathcal{X}}}=\sum_{a,b,c\in
C^{1}(X_{2},\mathbb{Z}_{2})}Z_{{\mathcal{X}}}[X_{2};a,b,c]\ket{a,b,c}~{}.$
(2.7)

Shrinking the 3d slab amounts to taking the inner product between the bra
(2.5) and the ket (2.7), which recovers the 2d partition function
$Z_{\mathcal{X}}[X_{2};A,B,C]$.

###  2.2 Anomaly of invertible symmetry from linking invariants

Before discussing non-invertible symmetries, let us review how the anomaly
(2.1) of the invertible symmetry
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ can be
probed via linking invariants in the SymTFT, following the general discussion
in the introduction.

Note that both $\widehat{U}_{a}$ and $\widehat{U}_{a}U_{a}$ are non-
terminable, ending on the boundary line $\widehat{U}_{a}|_{\partial}$. Thus
these two lines live in the same subset ${\mathcal{B}}_{\widehat{a}}$. In a
similar way, the 14 non-invertible lines in the bulk can be grouped into 7
subsets,

$\displaystyle{\mathcal{B}}_{\widehat{a}}$
$\displaystyle=\\{\widehat{U}_{a},\,\widehat{U}_{a}U_{a}\\}~{},$
$\displaystyle{\mathcal{B}}_{\widehat{b}}$
$\displaystyle=\\{\widehat{U}_{b},\,\widehat{U}_{b}U_{b}\\}~{},$
$\displaystyle{\mathcal{B}}_{\widehat{c}}$
$\displaystyle=\\{\widehat{U}_{c},\,\widehat{U}_{c}U_{c}\\}~{},$
$\displaystyle{\mathcal{B}}_{\widehat{ab}}$
$\displaystyle=\\{\widehat{U}_{ab},\,\widehat{U}_{ab}U_{a}\\}~{},$
$\displaystyle{\mathcal{B}}_{\widehat{ac}}$
$\displaystyle=\\{\widehat{U}_{ac},\,\widehat{U}_{ac}U_{a}\\}~{},$
$\displaystyle{\mathcal{B}}_{\widehat{bc}}$
$\displaystyle=\\{\widehat{U}_{bc},\,\widehat{U}_{bc}U_{b}\\}~{},$
$\displaystyle{\mathcal{B}}_{\widehat{abc}}$
$\displaystyle=\\{\widehat{U}_{abc},\,\widehat{U}_{abc}U_{a}\\}~{}.$ (2.8)

Following the discussion in the introduction, in order to probe the anomaly,
we must check whether it is possible to choose one representative from each
subset such that all of the linking invariants are trivial. If for every
choice of representatives the linking invariants are non-trivial, then the
anomaly must be non-trivial as well.

As reviewed in Appendix A, there are only two types of linking invariants to
consider in this context: the linking number involving two loops, and the
type-0 linking number involving three loops. The former can be probed by the
Hopf link, and the latter by the Borromean rings. We first consider the
linking configuration involving two identical representatives
$\widehat{U}_{a}(M_{1})U^{p_{a}}_{a}(M_{1})$, where $p_{a}=0,1$ denotes the
two choices of representatives in ${\mathcal{B}}_{\widehat{a}}$. This is
expected to probe the self-anomaly for $\mathbb{Z}_{2}^{A}$. A straightforward
computation described in Appendix B gives

$\displaystyle\begin{split}\braket{\widehat{U}_{a}(M_{1})U^{p_{a}}_{a}(M_{1})\widehat{U}_{a}(M_{1}^{\prime})U^{p_{a}}_{a}(M_{1}^{\prime})}&\sim(-1)^{\text{Link}(M_{1},M_{1}^{\prime})}~{},\hskip
28.45274ptp_{a}=0,1\,,\end{split}$ (2.9)

where $\sim$ means equal up to a real positive normalization. We see that no
matter which representative of $\mathcal{B}_{\widehat{a}}$ we choose, the Hopf
link configuration gives a non-trivial sign. By the main result in the
introduction, we conclude that the invertible $\mathbb{Z}_{2}^{A}$ symmetry is
anomalous. Indeed, from the calculation of linking invariants in Appendix B,
this non-trivial linking invariant is seen to follow directly from the
Dijkgraaf-Witten term $\frac{1}{2}a\delta a$ in (2.2), which in turn follows
from the anomaly $A\beta A$ in (2.1). Hence the non-trivial linking invariant
detects the anomaly $A\beta A$.

We may also consider linking configurations involving three components. We
begin by choosing representatives from the subsets
${\mathcal{B}}_{\widehat{a}},{\mathcal{B}}_{\widehat{b}}$, and
${\mathcal{B}}_{\widehat{c}}$, which we denote by
$\widehat{U}_{a}U_{a}^{p_{a}},\widehat{U}_{b}U_{b}^{p_{b}}$, and
$\widehat{U}_{c}U_{c}^{p_{c}}$. Their correlation function is

$\displaystyle\begin{split}&\braket{\widehat{U}_{a}(M_{1})U_{a}^{p_{a}}(M_{1})\widehat{U}_{b}(M_{1}^{\prime})U_{b}^{p_{b}}(M_{1}^{\prime})\widehat{U}_{c}(M_{1}^{\prime\prime})U_{c}^{p_{c}}(M_{1}^{\prime\prime})}\\\
&\sim(-1)^{\text{Link}(M_{1},M_{1}^{\prime},M_{1}^{\prime\prime})_{0}}(-1)^{(p_{a}+p_{b})\text{Link}(M_{1},M_{1}^{\prime})+(p_{a}+p_{c})\text{Link}(M_{1},M_{1}^{\prime\prime})+(p_{b}+p_{c})\text{Link}(M_{1}^{\prime},M_{1}^{\prime\prime})}~{}.\end{split}$
(2.10)

If we take $M_{1},M_{1}^{\prime},M_{1}^{\prime\prime}$ to form the Borromean
rings, for which the linking number between any pair of loops vanish and
$(-1)^{\text{Link}(M_{1},M_{1}^{\prime},M_{1}^{\prime\prime})_{0}}=-1$, then
the above correlation function is non-trivial for any $p_{a},p_{b},p_{c}$. By
the main result in the introduction, we then find that the invertible
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ symmetry
is anomalous. Indeed, since the non-trivial type-0 linking number between
three loops follows from the $abc$ term in the DW theory (2.2), which in turn
follows from the anomaly $ABC$ in (2.1), this configuration detects the
anomaly $ABC$.

###  2.3 Non-invertible symmetries from gauging

We have just seen how the non-trivial ‘t Hooft anomalies of invertible
symmetries can be detected using the linkings of operators in the SymTFT. We
now turn to the case of non-invertible symmetries, for which the AnomTFT is
not well-understood. As explained above, it is possible to obtain a theory
$\widehat{\mathcal{X}}$ with non-invertible symmetries by starting with the
theory ${\mathcal{X}}$ and gauging the
$\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ symmetry. Let us briefly review
the properties of these non-invertible symmetries, before discussing their
anomalies. To begin, note that the partition function of
$\widehat{\mathcal{X}}$ is

$\displaystyle
Z_{\widehat{{\mathcal{X}}}}[B,C]=\frac{1}{|H^{0}(X_{2},\mathbb{Z}_{2})|^{2}}\sum_{b,c\in
H^{1}(X_{2},\mathbb{Z}_{2})}Z_{{\mathcal{X}}}[b,c]\,e^{i\pi\int_{X_{2}}bC+cB}~{},$
(2.11)

where we have turned off the $\mathbb{Z}_{2}^{A}$ background field in the
partition function. Our goal is to show that the topological defect for
$\mathbb{Z}_{2}^{A}$ becomes a non-invertible topological defect.

To see this, we define two topological manipulations: a gauging of the
$\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ symmetry denoted by $\sigma$, and
a stacking with a 2d $\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$ SPT denoted
by $\tau$. Concretely,

$\displaystyle\begin{split}Z_{\sigma{\mathcal{X}}}[B,C]&=\frac{1}{|H^{0}(X_{2},\mathbb{Z}_{2})|^{2}}\sum_{b,c\in
H^{1}(X_{2},\mathbb{Z}_{2})}Z_{{\mathcal{X}}}[b,c]\,e^{i\pi\int_{X_{2}}bC+cB}~{},\\\
Z_{\tau{\mathcal{X}}}[B,C]&=Z_{{\mathcal{X}}}[B,C]\,e^{i\pi\int_{X_{2}}BC}~{}.\end{split}$
(2.12)

Note that the right-hand side of (2.11) is precisely the partition function of
$\sigma{\mathcal{X}}$. To see the non-invertible symmetry of
$\widehat{{\mathcal{X}}}$, we start with $\widehat{{\mathcal{X}}}$ and perform
a $\tau\sigma\tau$ transformation followed by a $\mathbb{Z}_{2}^{A}$
transformation $g$. It is straightforward to check that
$\widehat{{\mathcal{X}}}$ is invariant under $g\tau\sigma\tau$ (up to an Euler
counterterm):

$\displaystyle\begin{split}Z_{g\tau\sigma\tau\widehat{{\mathcal{X}}}}[B,C]&=\frac{1}{|H^{0}(X_{2},\mathbb{Z}_{2})|^{4}}\sum_{b,c,\widetilde{b},\widetilde{c}\in
H^{1}(X_{2},\mathbb{Z}_{2})}Z_{{\mathcal{X}}}[b,c]\,e^{i\pi\int_{X_{2}}b\widetilde{c}+c\widetilde{b}+\widetilde{b}\widetilde{c}+\widetilde{b}C+\widetilde{c}B+BC+bc}\\\
&=\frac{|H^{1}(X_{2},\mathbb{Z}_{2})|}{|H^{0}(X_{2},\mathbb{Z}_{2})|^{4}}\sum_{b,c\in
H^{1}(X_{2},\mathbb{Z}_{2})}Z_{{\mathcal{X}}}[b,c]\,e^{i\pi\int_{X_{2}}(b+B)(c+C)+BC+bc}\\\
&=\chi^{-1}[X_{2}]\frac{1}{|H^{0}(X_{2},\mathbb{Z}_{2})|^{2}}\sum_{b,c\in
H^{1}(X_{2},\mathbb{Z}_{2})}Z_{{\mathcal{X}}}[b,c]\,e^{i\pi\int_{X_{2}}bC+cB}=\chi^{-1}[X_{2}]Z_{\widehat{{\mathcal{X}}}}[B,C]~{}.\\\
\end{split}$ (2.13)

In the first equality, we have used the mixed anomaly (2.1), which implies
that under a global $\mathbb{Z}_{2}^{A}$ transformation $g$ the partition
function of ${\mathcal{X}}$ acquires a phase $Z_{{\mathcal{X}}}[b,c]\to
Z_{{\mathcal{X}}}[b,c]e^{i\pi\int_{X_{2}}bc}$. In the last line we have used
the definition of the Euler counterterm
$\chi[X_{2}]:=|H^{0}(X_{2},\mathbb{Z}_{2})|^{2}/|H^{1}(X_{2},\mathbb{Z}_{2})|$,
assuming $X_{2}$ is a closed manifold. Ignoring the Euler counterterm, we then
have

$\displaystyle g\tau\sigma\tau\widehat{{\mathcal{X}}}=\widehat{{\mathcal{X}}}$
(2.14)

and hence we see that $g\tau\sigma\tau$ is a symmetry of ${\mathcal{X}}$.
Since the symmetry involves gauging, i.e. a $\sigma$ operation, the symmetry
is non-invertible, as can be confirmed by explicitly calculating the fusion
rules following [26, 27]. Moreover, since the operation $g\tau\sigma\tau$ is
obtained by dressing the $\mathbb{Z}_{2}^{A}$ transformation $g$ with a
twisted gauging $\tau\sigma\tau$, we conclude that the invertible
$\mathbb{Z}_{2}^{A}$ symmetry in ${\mathcal{X}}$ becomes a non-invertible
symmetry implementing $g\tau\sigma\tau$ upon gauging
$\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$.

$M_{1}|_{0}$$X_{2}^{<0}$$X_{2}^{\geq 0}$ Figure 5: Decomposition of $X_{2}$
along a neck. The interface is located at $x=0$.

Let us construct the non-invertible operator implementing $g\tau\sigma\tau$
explicitly. We start by dividing the spacetime into two parts, as shown in
Figure 5. The left part $X_{2}^{<0}$ and right part $X_{2}^{>0}$ share a
common boundary $M_{1}|_{0}$. The subscript/superscript indicates the value of
the local coordinate around the neck. The defect implementing
$g\tau\sigma\tau$ is obtained by placing the $\mathbb{Z}_{2}^{A}$ defect
$g(M_{1}|_{0})$ at $M_{1}|_{0}$ and further acting by $\tau\sigma\tau$ on only
half of the spacetime, say $X_{2}^{>0}$, with Dirichlet boundary conditions at
$M_{1}|_{0}$ as shown in Figure 6. For simplicity, we have turned off the
background fields $B,C$. Note that the $g(M_{1}|_{0})$ defect alone is not
gauge invariant, but rather only the combination

$\displaystyle g(M_{1}|_{0})\cdot\exp\left(i\pi\int_{X_{2}^{\geq
0}}bc\right)~{}$ (2.15)

is. To interpret this as a line operator, we introduce a 1d TQFT supported on
$X_{2}^{\geq 0}$ to cancel the bulk $X_{2}^{\geq 0}$ dependence. Supposing
that the gauge transformations are $b\to b+\delta\beta$ and $c\to
c+\delta\gamma$, the 1d TQFT can be chosen to be
$i\pi(-\phi_{2}c+\phi_{3}b+\phi_{2}\delta\phi_{3})$, where $\phi_{2,3}$ are
scalars supported only on the line $\gamma|_{0}$ with the gauge transformation
$\phi_{2}\to\phi_{2}+\beta$ and $\phi_{3}\to\phi_{3}+\gamma$. Thus the
combination

${\mathcal{N}}_{A}(M_{1}|_{0})=\frac{1}{|C^{0}(M_{1}|_{0},\mathbb{Z}_{2})|}\sum_{\phi_{2},\phi_{3}\in
C^{0}(M_{1}|_{0},\mathbb{Z}_{2})}g(M_{1}|_{0})\cdot\exp\left(i\pi\int_{M_{1}|_{0}}-\phi_{2}c+\phi_{3}b+\phi_{2}\delta\phi_{3}\right)$
(2.16)

is a genuine topological line operator, generating the transformation
$g\tau\sigma\tau$. Following the prescription in [27, 29] (and treated more
carefully in [60]) we find the fusion rules

$\displaystyle\begin{split}{\mathcal{N}}_{A}\times{\mathcal{N}}_{A}&=1+U_{b}+U_{c}+U_{b}U_{c}~{},\\\
{\mathcal{N}}_{A}\times
U_{b}&=U_{b}\times{\mathcal{N}}_{A}={\mathcal{N}}_{A}~{},\\\
{\mathcal{N}}_{A}\times
U_{c}&=U_{c}\times{\mathcal{N}}_{A}={\mathcal{N}}_{A}~{},\\\ U_{b}\times
U_{b}&=U_{c}\times U_{c}=1~{},\end{split}$ (2.17)

where $U_{a,b,c}=e^{i\pi\oint_{M_{1}|_{0}}a,b,c}$ are the generators for the
quantum $\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$
symmetry. These fusion rules coincide with those of the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ Tambara-Yamagami fusion category. Note
that the discussion in this section does not depend on the existence of the
self-anomaly of $\mathbb{Z}_{2}^{A}$ (i.e. the term $\pi A\beta A$) because we
turned off its background field $A$ throughout.

$g$$Z_{\widehat{\mathcal{X}}}$$\sum_{b,c}Z_{\widehat{\mathcal{X}}}[b,c]e^{i\pi\int_{X_{2}^{\geq
0}}bc}$$x=0$$x$ Figure 6: The $g\tau\sigma\tau$ defect is obtained by placing
$\mathbb{Z}_{2}^{A}$ defect $g$ at $M_{1}|_{0}$ and performing
$\tau\sigma\tau$ on half of the spacetime $X_{2}^{\geq 0}$.

###  2.4 Non-invertible defects from SymTFT

We next review how non-invertible symmetry defects are realized in the SymTFT.
Because the theory $\widehat{{\mathcal{X}}}$ is obtained by gauging a non-
anomalous finite symmetry in ${\mathcal{X}}$, the SymTFT is the same for both
${\mathcal{X}}$ and $\widehat{{\mathcal{X}}}$. Hence the SymTFT is still given
by the DW theory in (2.2); the only difference is the topological boundary
condition on the left. For ${\mathcal{X}}$, the relevant topological boundary
condition was the Dirichlet boundary condition such that all of
$U_{a},U_{b},U_{c}$ could terminate. For $\widehat{{\mathcal{X}}}$, the
relevant boundary condition is given by Neumann boundary conditions for $b$
and $c$ and Dirichlet boundary conditions for $a$, or equivalently the
Dirichlet boundary condition for $a,\widehat{b}$, and $\widehat{c}$. In terms
of boundary states, this is

$\displaystyle\bra{D_{a},N_{b,c}}=\sum_{a,b,c\in
C^{1}(X_{2},\mathbb{Z}_{2})}\bra{a,b,c}\delta(a-A)\,e^{i\pi\int_{X_{2}}bC+cB}~{}.$
(2.18)

Under this new boundary condition, the lines
$U_{a},\widehat{U}_{b},\widehat{U}_{c}$, and the composite $\widehat{U}_{bc}$
can terminate.444To see this, we note that the Dirichlet boundary condition
for $a$ implies $\phi_{1}=0$ on the boundary. Hence all three terms in the 1d
TQFT of $\widehat{U}_{b}$ and $\widehat{U}_{c}$ are trivial. Together with the
Dirichlet boundary condition for $\widehat{b},\widehat{c}$, we conclude that
$\widehat{U}_{b}$, $\widehat{U}_{c}$ are terminable on the new boundary.  We
thus have ${\mathcal{A}}_{\text{noninv}}=1\oplus
U_{a}\oplus\widehat{U}_{b}\oplus\widehat{U}_{c}\oplus\widehat{U}_{bc}$.
Indeed, the total quantum dimension of ${\mathcal{A}}_{\text{noninv}}$ is
$1+1+2+2+2=8$, matching the total quantum dimension $D_{\text{tot}}$ as
required for a Lagrangian algebra. As another consistency check, all the
operators in ${\mathcal{A}}_{\text{noninv}}$ have trivial linking invariants
in the bulk.

There are 5 lines belonging to ${\mathcal{A}}_{\text{noninv}}$, and the
remaining 17 lines belong to the complement of
${\mathcal{A}}_{\text{noninv}}$. This includes 6 invertible lines and 11 non-
invertible lines. We now discuss how these lines connect with boundary lines.
As usual, it is useful to organize the bulk lines into subsets labelled by the
boundary lines they can end on. We first list the results,

(2.24)

We now explain the above results by means of a few representative examples.
First consider the bulk line $U_{b}$. Since $b$ obeys Neumann boundary
conditions, it cannot terminate on the topological boundary. Instead, when
placed orthogonal to the boundary as shown in Figure 3, it bends at the
junction and becomes a boundary line $U_{b}|_{\partial}$, which is the
$\mathbb{Z}_{2}^{B}$ symmetry defect of the 2d QFT.

A slightly more non-trivial example is the bulk line $\widehat{U}_{a}$.
Because $b$ and $c$ obey Neumann boundary conditions, the 1d TQFT survives in
the boundary line $\widehat{U}_{a}|_{\partial}$. Hence it becomes a non-
invertible symmetry defect of the 2d QFT $\widehat{{\mathcal{X}}}$. Note that
the 1d TQFT is precisely the one appearing in the definition of the non-
invertible defect ${\mathcal{N}}_{A}$, c.f. (2.16).

Here, we also see an example where the same bulk line can terminate on
different boundary lines. To see this, we consider the bulk line
$\widehat{U}_{bc}U_{b}$. Since $\widehat{U}_{bc}$ is terminable, only
$U_{b}|_{\partial}$ survives on the boundary. On the other hand, the bulk line
$\widehat{U}_{bc}U_{c}$ is attached to $U_{c}|_{\partial}$ on the boundary.
Note that the two bulk lines $\widehat{U}_{bc}U_{b}$ and
$\widehat{U}_{bc}U_{c}$ are actually the same on a closed circle, and hence
the same bulk line can terminate on two different boundary lines.

We find that there are five line operators,
$1,U_{b}|_{\partial},U_{c}|_{\partial},U_{bc}|_{\partial}$, and
$\widehat{U}_{a}|_{\partial}$ on the boundary. These operators are in obvious
one-to-one correspondence with the symmetry operators in Section 2.3;
concretely, we have the map

(2.30)

The fusion rules among
$U_{b}|_{\partial},U_{c}|_{\partial},U_{bc}|_{\partial},$ and
$\widehat{U}_{a}|_{\partial}$ can also be straightforwardly computed, and
coincide with those in (2.17).

###  2.5 Anomalies of the non-invertible symmetries

We may now finally turn to the question of anomalies of the non-invertible
symmetry of $\widehat{{\mathcal{X}}}$. Since there is currently no notion of a
background field for non-invertible symmetries, the meaning of the AnomTFT is
unclear. In Section 2.2 we reviewed how the SymTFT can be used to probe the
anomalies of invertible symmetries, following the general discussion in the
introduction. In this section, we will use the SymTFT to probe the anomalies
of the non-invertible symmetry reviewed in Section 2.3 and 2.4.

As before, the idea is to pick representatives from each subset
${\mathcal{B}}_{b},{\mathcal{B}}_{c},{\mathcal{B}}_{bc}$, and
${\mathcal{B}}_{\widehat{a}}$, and then to compute the linking invariants
among them. Since the original, invertible symmetry $U_{a}|_{\partial}$ had a
self-anomaly before gauging, it is natural to expect that the corresponding
non-invertible symmetry ${\mathcal{N}}_{A}=\widehat{U}_{a}|_{\partial}$ also
has a self-anomaly. To detect this, we choose a representative from the subset
${\mathcal{B}}_{\widehat{a}}$ and compute the linking number between the
representative and itself. In fact, it is straightforward to check that for
any choice of representative, the linking number is always non-trivial,

$\displaystyle\begin{split}\braket{{\mathcal{O}}(M_{1}){\mathcal{O}}(M^{\prime}_{1})}&\sim(-1)^{\text{Link}(M_{1},M_{1}^{\prime})}~{},\hskip
28.45274pt\forall~{}~{}{\mathcal{O}}\in{\mathcal{B}}_{\widehat{a}}~{},\end{split}$
(2.31)

which can (for instance) be detected by a Hopf link. In particular, the self-
linkings of ${\mathcal{O}}=\widehat{U}_{a}$ and $\widehat{U}_{a}U_{a}$ were
already computed to detect the invertible $\mathbb{Z}_{2}^{A}$ self-anomaly in
Section 2.2. Using the main result in the introduction, we thus conclude that
the non-invertible symmetry ${\mathcal{N}}_{A}$ in the 2d QFT
$\widehat{{\mathcal{X}}}$ enjoys a non-trivial ’t Hooft anomaly.

We give two remarks before moving on to an example:

  1. 1.

The merit of using linking invariants to quantify ’t Hooft anomalies is that
the linking invariants (especially when the SymTFT is Dijkgraaf-Witten) are
relatively easy to compute, and can be easily generalized to higher
dimensions.

  2. 2.

In [27, 29, 78], a powerful sufficient condition for a non-invertible symmetry
implementing $g\tau\sigma\tau$ to have non-trivial anomaly was proposed.
Concretely, suppose a QFT $\widehat{{\mathcal{X}}}$ has an invertible global
symmetry $G$, and furthermore that $\widehat{{\mathcal{X}}}$ is invariant
under a sequence of topological manipulations $f(\sigma,\tau,\rho)$, where
$\rho$ rescales the background fields, e.g. $B\to kB$ for certain $k$. Then in
general $\widehat{{\mathcal{X}}}$ has a non-invertible symmetry implementing
$f(\sigma,\tau,\rho)$. The authors of [27, 29] showed that
$f(\sigma,\tau,\rho)$ is anomalous if one cannot find a $G$-SPT such that it
is invariant under $f(\sigma,\tau,\rho)$. However, this condition is not
necessary. Namely, the existence of an $f(\sigma,\tau,\rho)$-invariant $G$-SPT
does not imply that $f(\sigma,\tau,\rho)$ is anomaly-free. Indeed, in our
example of $G=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, there is a 2d
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT which is invariant under
$g\tau\sigma\tau$, whose topological action is

$\displaystyle\pi\int_{X_{2}}BC~{}.$ (2.32)

Here $g$ acts trivially on (2.32). Hence the condition in [27, 29, 78] would
not detect the anomaly, whereas our current condition does (although as
emphasized in the introduction, our condition is still only sufficient, and
not necessary). The non-trivial linking invariant (2.31) dictates additional
structures for the junction between the non-invertible defects, and these are
not satisfied by (2.32) [15].

###  2.6 Application: Abelian Higgs Model

We close this section by giving a concrete example of the discussion above. We
will take the theory ${\mathcal{X}}$ to be the 2d Abelian Higgs Model (AHM)
with $N_{f}=2$ complex scalars charged under the dynamical gauge group $U(1)$,
and with a non-trivial theta term $\theta=\pi$. The action is

$\displaystyle
S_{{\mathcal{X}}}=\int_{X_{2}}\sum_{i=1}^{2}|D_{a}\phi_{i}|^{2}+m^{2}\sum_{i=1}^{2}|\phi_{i}|^{2}+\lambda\left(\sum_{i=1}^{2}|\phi_{i}|^{2}\right)^{2}+\frac{1}{2e^{2}}da*da+\frac{\pi}{2\pi}da~{}.$
(2.33)

Let us discuss the symmetries and anomalies of this theory.

##### Invertible symmetries and anomalies:

The AHM has an $SU(2)/\mathbb{Z}_{2}=SO(3)$ global symmetry; the
$\mathbb{Z}_{2}$ quotient is included since the $\mathbb{Z}_{2}$ normal
subgroup of $SU(2)$ actually belongs to the $U(1)$ gauge group. Below, we will
focus on only the $\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ subgroup of
$SO(3)$. This means that we are allowed to turn on interactions that
explicitly break $SO(3)$ while preserving its
$\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ subgroup. To show how the
$\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ subgroup acts on the scalars, we
consider the scalar bilinears,

$\displaystyle
n^{\alpha}=\sum_{i,j=1}^{2}\phi^{\dagger}_{i}\tau^{\alpha}_{ij}\phi_{j}~{},\hskip
28.45274pt\alpha=1,2,3\,,$ (2.34)

which transform as vectors under $SO(3)$ and which are $U(1)$ gauge
invariant.555Here $\tau^{\alpha}$ are the Pauli matrices, not to be confused
with the topological manipulation $\tau$ in (2.12).  Then the
$\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ symmetry acts on $n^{\alpha}$ as

$\displaystyle\begin{split}\mathbb{Z}_{2}^{x}:\hskip
28.45274pt&(n^{1},n^{2},n^{3})\to(n^{1},-n^{2},-n^{3})~{},\\\
\mathbb{Z}_{2}^{y}:\hskip
28.45274pt&(n^{1},n^{2},n^{3})\to(-n^{1},n^{2},-n^{3})~{}.\\\ \end{split}$
(2.35)

Apart from the flavor rotation symmetry, there is also a charge conjugation
symmetry $\mathbb{Z}_{2}^{C}$ acting on the scalars and $U(1)$ gauge field as
[105, 106, 107]

$\displaystyle\mathbb{Z}_{2}^{C}:\hskip
28.45274pt\phi_{i}\to(i\tau^{2}\phi)_{i}~{},\hskip 28.45274pta\to-a~{}.$
(2.36)

Note that although $\mathbb{Z}_{2}^{C}$ acts on the scalar as
$C^{2}\phi_{i}=-\phi_{i}$, and hence $C^{2}=(-1)^{2j}$ where $j$ is the
$SU(2)$ isospin, all the gauge invariant local operators have integer $SU(2)$
isospin, and therefore $C^{2}=1$ on all gauge invariant operators. The
$\mathbb{Z}_{2}^{C}$ symmetry acts on the $SO(3)$ vector $n^{\alpha}$ as

$\displaystyle\mathbb{Z}_{2}^{C}:\hskip
28.45274pt(n^{1},n^{2},n^{3})\to(-n^{1},n^{2},-n^{3})~{},$ (2.37)

which coincides with the action of $\mathbb{Z}_{2}^{y}$ on the scalar
bilinears. However, note that $\mathbb{Z}_{2}^{C}$ additionally acts on the
$U(1)$ gauge field.

As we have mentioned, we will allow ourselves to add interactions to the
action such as

$\displaystyle
V=\int_{X_{3}}\xi_{1}(n^{1})^{2}+\xi_{2}(n^{2})^{2}+\xi_{3}(n^{3})^{2}+\xi_{123}n^{1}n^{2}n^{3}+\dots~{}$
(2.38)

which preserve the
$\mathbb{Z}_{2}^{C}\times\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ symmetry
while explicitly breaking the $SO(3)$ symmetry.

The anomaly of the
$\mathbb{Z}_{2}^{C}\times\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ symmetry
of the AHM has been studied in detail in [108, 107, 106]. In the presence of
the $SO(3)$ symmetry, the anomaly is found to be

$\displaystyle\int_{X_{3}}\pi Cw_{2}^{SO(3)}+\pi C\beta C\,,$ (2.39)

where $C$ is the background field of $\mathbb{Z}_{2}^{C}$. The mixed anomaly
$\pi Cw_{2}^{SO(3)}$ is interpreted as a Lieb-Schultz-Mattis (LSM) type
anomaly, since $\mathbb{Z}_{2}^{C}$ can be interpreted as the $\mathbb{Z}_{2}$
reduction of the translation symmetry when the AHM is realized as the low-
energy description of an antiferromagnetic spin chain in the UV. The self-
anomaly $\pi C\beta C$ was referred to as an “emergent anomaly” in [108],
since it is beyond LSM-type anomalies; indeed, the translation symmetry
$\mathbb{Z}$ of the UV spin chain clearly does not have an anomaly, and the
anomaly only appears when $\mathbb{Z}\to\mathbb{Z}_{2}$ at low energies.666To
see that translation itself does not have a self-anomaly, we consider
deforming the antiferromagnetic spin chain Hamiltonian
$H_{\text{AFM}}=\sum_{i}\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{y}_{i}\sigma^{y}_{i+1}+\sigma^{z}_{i}\sigma^{z}_{i+1}$
by a transverse field term $-h\sum_{i}\sigma^{x}_{i}$ with large $h$. This
term preserves translation symmetry and drives the theory to a trivially
gapped phase.  We should note that recently the $\mathbb{Z}_{2}^{C}$ in the
low-energy theory has been dubbed an _emanant symmetry_ (to be distinguished
from an emergent symmetry) since it descends from a UV symmetry [109]; the UV
spin-chain manifestation of this self-anomaly was identified there as
well.777We thank Shu-Heng Shao for discussions on this point. When we
explicitly break the $SO(3)$ symmetry of the AHM by adding the potential
(2.38), the anomaly (2.39) reduces to

$\displaystyle\int_{X_{3}}\pi CA^{x}A^{y}+\pi C\beta C~{},$ (2.40)

where $A^{x}$ and $A^{y}$ are the background fields of $\mathbb{Z}_{2}^{x}$
and $\mathbb{Z}_{2}^{y}$ respectively. This anomaly is precisely the one in
(2.1) which we have studied extensively throughout this section.

##### Dynamical constraints on the
$\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ gauged AHM:

Because our discussions in Sections 2.2 through 2.5 are based purely on the
symmetries and anomalies of the theory, we may immediately apply our results
to the AHM. In particular, if we gauge the
$\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ symmetry of the perturbed AHM
(whose action is $S_{{\mathcal{X}}}+V$) the resulting theory will have a non-
invertible charge conjugation symmetry. In addition, from the results of
Section 2.5, this non-invertible charge conjugation symmetry is anomalous, and
measurable via the linking number between two non-invertible charge
conjugation defects in the SymTFT.

The presence of the anomalous non-invertible symmetry has the following
dynamical consequences:

  1. 1.

Firstly, independent of the anomaly, the non-invertible symmetry itself
forbids some terms that might otherwise have been radiatively generated. If
one were not aware of the existence of the non-invertible symmetry, then after
gauging $\mathbb{Z}_{2}^{x}\times\mathbb{Z}_{2}^{y}$ one would naively
conclude that $\mathbb{Z}_{2}^{C}$ was explicitly broken, and that there was
no obstruction for terms such as $\eta_{1}n^{1}+\eta_{3}n^{3}+...$ charged
under $\mathbb{Z}_{2}^{C}$ to be radiatively generated. In other words, not
including such terms in the action would be “unnatural.” However, when the
non-invertible symmetry is accounted for, such operators are not allowed, and
naturalness is restored.

  2. 2.

Secondly, the presence of the anomaly forbids the system from flowing to a
trivially gapped phase, no matter what perturbation is turned on (as long as
it is uncharged under the non-invertible symmetry, and hence under
$\mathbb{Z}_{2}^{C}$ before gauging).

##  3 Non-invertible symmetries and anomalies in 4d

We now repeat the discussion of the previous section in a very similar four-
dimensional setup. Our starting point will be a 4d spin quantum field theory
${\mathcal{X}}$ with an invertible
$\mathbb{Z}_{2MN}^{(0)}\times\mathbb{Z}_{N}^{(1)}$ global symmetry, where the
superscripts indicate that they are respectively zero- and one-form
symmetries. We will denote the background fields as $A^{(1)}$ and $B^{(2)}$
respectively. We further assume that the theory has a ’t Hooft anomaly
specified by the 5d AnomTFT,

$\displaystyle\begin{split}\int_{X_{5}}\left(\frac{2\pi}{2N}A^{(1)}{\mathcal{P}}(B^{(2)})+\frac{\pi(N^{2}-1)}{6N}A^{(1)}\beta
A^{(1)}\beta A^{(1)}\right)~{},&\hskip 28.45274ptN\in 2\mathbb{Z}\\\
\int_{X_{5}}\left(\frac{2\pi}{N}\frac{1+N}{2}A^{(1)}B^{(2)}B^{(2)}+\frac{\pi(N^{2}-1)}{6N}A^{(1)}\beta
A^{(1)}\beta A^{(1)}\right)~{},&\hskip 28.45274ptN\in 2\mathbb{Z}+1\\\
\end{split}$ (3.1)

which is an invertible TQFT. Here ${\mathcal{P}}$ is the Pontryagin square
operation, while $\beta$ is the Bockstein map associated with the exact
sequence $1\to\mathbb{Z}_{2MN}\to\mathbb{Z}_{(2MN)^{2}}\to\mathbb{Z}_{2MN}\to
1$, acting on $\mathbb{Z}_{2MN}$-valued cocycles as $\beta=\delta/(2MN)$. The
coefficients are chosen to match with our two examples: (1) adjoint QCD and
(2) ${\mathcal{N}}=4$ SYM. The zero-form symmetry being $\mathbb{Z}_{2MN}$ is
a simplifying assumption, and one can in principle discuss more general
symmetries and anomalies.

In order to obtain a theory with non-invertible symmetry, we now gauge the
$\mathbb{Z}_{N}^{(1)}$ one-form symmetry, which is possible since it does not
have a self-anomaly. This gauging changes $\mathbb{Z}_{2MN}^{(0)}$ into a non-
invertible symmetry. As in the two-dimensional case studied in the previous
section, the self-anomaly of $\mathbb{Z}_{2MN}^{(0)}$ before gauging will lead
to an anomaly for the non-invertible symmetry after gauging. Our analysis will
again proceed by studying the structure of the SymTFT.

###  3.1 SymTFT of invertible symmetries

##### The SymTFT:

Let us begin by discussing the SymTFT for the theory ${\mathcal{X}}$ with
invertible symmetry. As reviewed in the introduction, a 4d QFT can be expanded
into a 5d slab in which the SymTFT lives. The SymTFT is a gauged version of
the AnomTFT given in (3.1), i.e. it is a DW TQFT. The SymTFT for even $N$ is

$\displaystyle\int_{X_{5}}\left(\frac{2\pi}{2MN}\widehat{a}^{(3)}\delta
a^{(1)}+\frac{2\pi}{N}\widehat{b}^{(2)}\delta
b^{(2)}+\frac{2\pi}{2N}a^{(1)}b^{(2)}b^{(2)}+\frac{\pi(N^{2}-1)}{6N}a^{(1)}\frac{\delta
a^{(1)}}{2MN}\frac{\delta a^{(1)}}{2MN}\right)~{}.$ (3.2)

Here all the fields $a^{(1)},\widehat{a}^{(3)},b^{(2)}$, and
$\widehat{b}^{(2)}$ are dynamical, with the superscripts indicating their form
degrees. Note that we suppressed the higher cup products in the
SymTFT.888Throughout this work we suppress $\cup_{1}$ terms and ignore the
fact that cochains are not super-commutative. To justify our treatment here,
one may replace all cochains by differential forms by adjusting their
normalizations, e.g. treating $\frac{2\pi}{4}a^{(1)}$ as a $2\pi$-periodic
one-form. In the differential form formalism, all fields are super-
commutative.  For odd $N$, the third term in (3.2) should be replaced by
$\frac{2\pi}{N}\frac{1+N}{2}a^{(1)}b^{(2)}b^{(2)}$. As the discussions of even
and odd $N$ are similar, we will focus on the case of even $N$ below.

The spectrum of topological operators in the DW theory (3.2) is studied in
Appendix C, and we simply summarize the results here. First there are Wilson
lines and Wilson surfaces,

$\displaystyle U_{a}(M_{1})=e^{\frac{2\pi
i}{2MN}\oint_{M_{1}}a^{(1)}}~{},\hskip 28.45274ptU_{b}(M_{2})=e^{\frac{2\pi
i}{N}\oint_{M_{2}}b^{(2)}}~{},$ (3.3)

which satisfy the obvious invertible fusion rules $U_{a}^{2MN}=1$ and
$U_{b}^{N}=1$. There are also non-invertible magnetic surface and 3-volume
operators,

$\displaystyle\begin{split}\widehat{U}_{a}(M_{3})&\sim\sum_{\phi^{(1)}\in
C^{1}(M_{3},\mathbb{Z}_{N})}e^{\frac{2\pi
i}{2MN}\oint_{M_{3}}\left(\widehat{a}^{(3)}-M\phi^{(1)}\delta\phi^{(1)}+2M\phi^{(1)}b^{(2)}\right)}~{},\\\
\widehat{U}_{b}(M_{2})&\sim\sum_{\begin{subarray}{c}\phi^{(0)}\in
C^{0}(M_{2},\mathbb{Z}_{2MN})\\\ \phi^{(1)}\in
C^{1}(M_{2},\mathbb{Z}_{N})\end{subarray}}e^{{2\pi i\over
N}\oint_{M_{2}}\left(\widehat{b}^{(2)}-\phi^{(0)}b^{(2)}-\phi^{(1)}a^{(1)}+\phi^{(1)}\delta\phi^{(0)}\right)}~{},\\\
\end{split}$ (3.4)

where the TQFTs attached to the naive magnetic operators $e^{\frac{2\pi
i}{2MN}\oint_{M_{3}}\widehat{a}^{(3)}}$ and $e^{\frac{2\pi
i}{N}\oint_{M_{2}}\widehat{b}^{(2)}}$ are needed to cancel the gauge non-
invariance under $a^{(1)}\to a^{(1)}+\delta\alpha^{(0)}$ and $b^{(2)}\to
b^{(2)}+\delta\beta^{(1)}$. The fields $\phi^{(0)}$ and $\phi^{(1)}$ live only
on the worldvolumes of the topological operators and transform as
$\phi^{(0)}\to\phi^{(0)}+\alpha^{(0)}$ and
$\phi^{(1)}\to\phi^{(1)}+\beta^{(1)}$. The TQFTs on the operator worldvolumes
render the operators non-invertible. The non-invertibility can also be seen
from the fusion rules

$\displaystyle\begin{split}\widehat{U}_{a}(M_{3})\times\overline{\widehat{U}}_{a}(M_{3})&\sim\sum_{\phi^{(1)}\in
H^{1}(M_{3},\mathbb{Z}_{N})}e^{{2\pi i\over
N}Q(\text{PD}(\phi^{(1)}))}e^{{2\pi i\over
N}\oint_{\text{PD}(\phi^{(1)})}b^{(2)}}~{},\\\
\widehat{U}_{b}(M_{2})\times\overline{\widehat{U}}_{b}(M_{2})&\sim\sum_{\begin{subarray}{c}\phi^{(0)}\in
H^{0}(M_{2},\mathbb{Z}_{2MN})\\\ \phi^{(1)}\in
H^{1}(M_{2},\mathbb{Z}_{N})\end{subarray}}e^{{2\pi i\over
N}\oint_{\text{PD}(\phi^{(0)})}b^{(2)}+{2\pi i\over
N}\oint_{\text{PD}(\phi^{(1)})}a^{(1)}}~{},\end{split}$ (3.5)

where
$Q(\text{PD}(\phi^{(1)})):=\frac{1}{N}\int_{M_{3}}\phi^{(1)}\delta\phi^{(1)}$
is the triple intersection number in $M_{3}$, and $\text{PD}(\phi)$ is the
Poincaré dual of $\phi$.

##### Dirichlet boundary condition:

To obtain the theory ${\mathcal{X}}$, the topological boundary of the SymTFT
should be the Dirichlet boundary condition for both $a^{(1)}$ and $b^{(2)}$.
This sets the dynamical fields $a^{(1)}$ and $b^{(2)}$ to background fields
$A^{(1)}$ and $B^{(2)}$, respectively. The corresponding boundary state is

$\displaystyle\bra{D_{a^{(1)},b^{(2)}}}=\sum_{\begin{subarray}{c}a^{(1)}\in
C^{1}(X_{4},\mathbb{Z}_{2MN})\\\ b^{(2)}\in
C^{2}(X_{4},\mathbb{Z}_{N})\end{subarray}}\bra{a^{(1)},b^{(2)}}\,\delta(a^{(1)}-A^{(1)})\delta(b^{(2)}-B^{(2)})~{}.$
(3.6)

This Dirichlet boundary condition means that all the invertible line and
surface operators in the bulk become trivial on the boundary, and can also
terminate perpendicularly on the boundary. The set of all invertible operators
is denoted by ${\mathcal{A}}_{\text{inv}}$. To see how the boundary condition
gives rise to invertible $\mathbb{Z}_{2MN}^{(0)}\times\mathbb{Z}_{N}^{(1)}$
symmetries, we consider placing the bulk operators orthogonal to the boundary
as shown in Figure 3. Since both $a^{(1)}$ and $b^{(2)}$ become trivial on the
boundary, the TQFTs on both operator volumes are trivialized, and consequently
the boundary operators are invertible, i.e.

$\displaystyle\widehat{U}_{a}(M_{3}|_{\partial})=e^{\frac{2\pi
i}{2MN}\oint_{M_{3}|_{\partial}}\widehat{a}^{(3)}}~{},\hskip
28.45274pt\widehat{U}_{b}(M_{2}|_{\partial})=e^{\frac{2\pi
i}{N}\oint_{M_{2}|_{\partial}}\widehat{b}^{(2)}}~{}.$ (3.7)

We denote the collection of the above boundary operators as well as the
condensation defects constructed out of them as
${\mathcal{C}}_{\text{inv}}$.999Although $\widehat{U}_{a}(M_{3}|_{\partial})$
and $\widehat{U}_{b}(M_{3}|_{\partial})$ are both invertible operators, the
condensation defects constructed from them can be non-invertible. The
subscript on ${\mathcal{C}}_{\text{inv}}$ means that the simple defects (which
are not the sum of other defects of the same dimension) in
${\mathcal{C}}_{\text{inv}}$ that are not condensation defects are all
invertible.  Since $\widehat{U}_{a}$ and $\widehat{U}_{b}$ are respectively
codimension-one and -two operators on the 4d boundary, they generate a
$\mathbb{Z}_{2MN}^{(0)}\times\mathbb{Z}_{N}^{(1)}$ invertible symmetry after
shrinking the slab.

##### Dynamical boundary condition:

The dynamical boundary condition on the right captures the dynamics of the 4d
theory, and in particular is non-topological. The corresponding boundary state
is given by

$\displaystyle\ket{{\mathcal{X}}}=\sum_{\begin{subarray}{c}a^{(1)}\in
C^{1}(X_{4},\mathbb{Z}_{2MN})\\\ b^{(2)}\in
C^{2}(X_{4},\mathbb{Z}_{N})\end{subarray}}Z_{{\mathcal{X}}}[X_{4};a^{(1)},b^{(2)}]\ket{a^{(1)},b^{(2)}}~{}.$
(3.8)

Shrinking the 5d slab and taking the inner product between the two boundary
states (3.6) and (3.8) reproduces the 4d partition function
$Z_{{\mathcal{X}}}[X_{4};A^{(1)},B^{(2)}]$.

###  3.2 Anomaly of invertible symmetry from linking invariants

Proceeding in parallel with Section 2, we now review how to use the linking
invariants to diagnose the anomaly of the invertible
$\mathbb{Z}_{2MN}^{(0)}\times\mathbb{Z}_{N}^{(1)}$ symmetry. We begin by
grouping the bulk defects into subsets, such that all defects in a given
subset can end on the same boundary operator. For convenience, we denote the
condensation of an operator $U$ on the manifold $\Sigma$ as
$\text{Cond}_{\Sigma}(U)$. We will suppress the $\Sigma$ when the manifold
dependence is not emphasized. Then the operators in the SymTFT can be
organized into the following subsets

$\displaystyle{\mathcal{B}}_{\widehat{a}}=\\{\widehat{U}_{a},\,\widehat{U}_{a}\text{Cond}(U_{a}),\,\widehat{U}_{a}\text{Cond}(U_{b}),\,...\\}~{},\hskip
28.45274pt{\mathcal{B}}_{\widehat{b}}=\\{\widehat{U}_{b}U_{b}^{p},\,\widehat{U}_{b}\text{Cond}(U_{a}),\,...\\}~{},$
(3.9)

where the $...$ represents stacking $\widehat{U}_{a,b}$ with other
condensation operators—with or without discrete torsions—and $p=0,...,N-1$.
For simplicity we will not discuss the condensation defects of non-invertible
operators, and consequently will not discuss the condensation defects of
boundary operators (3.7). Following the discussion in the introduction, in
order to probe the anomaly, we must check whether it is possible to choose one
representative from each subset such that all of the linking invariants are
trivial. If such a choice of representatives does not exist, then the anomaly
for ${\mathcal{C}}_{\text{inv}}$ is non-trivial.

For simplicity, we will consider only operators of spherical topology, which
means that all the condensation operators in each subset are trivialized.
Hence the two subsets in (3.9) simplify significantly, and we have
${\mathcal{B}}_{\widehat{a}}=\\{\widehat{U}_{a}\\}$ and
${\mathcal{B}}_{\widehat{b}}=\\{\widehat{U}_{b}U_{b}^{p},\,p=0,...,N-1\\}$. We
first check the linking invariants among two components. As discussed in
Appendix A, the only possible linking invariants are between two surface
operators. It is easy to check that there exists a representative
$\widehat{U}_{b}$ such that the correlation function
$\braket{\widehat{U}_{b}(M_{2})\widehat{U}_{b}(M_{2}^{\prime})}$ has a trivial
phase when both $M_{2}$ and $M_{2}^{\prime}$ are $S^{2}$. This is consistent
with the fact that there is no anomaly of the form $B^{(2)}\beta B^{(2)}$.

We next consider the linking invariants among three components. There are
three types of such linking invariants referred to as type 0, type 1, and type
2 in Appendix A, and we will find that the type 0 and type 2 invariants are
non-trivial in the present case. The type 0 linking involves one 3-volume
operator and two surface operators, and we may compute the following
correlation function

$\displaystyle\braket{\widehat{U}_{a}(M_{3})\widehat{U}_{b}(M_{2}^{\prime})U^{p}_{b}(M_{2}^{\prime})\widehat{U}_{b}(M_{2}^{\prime\prime})U^{p}_{b}(M_{2}^{\prime\prime})}\sim
e^{-\frac{2\pi
i}{N}\text{Link}(M_{3},M^{\prime}_{2},M^{\prime\prime}_{2})_{0}}\,,$ (3.10)

where we have assumed that $M_{2}^{\prime}$ and $M_{2}^{\prime\prime}$ are not
Hopf linked. We see that the right-hand side is nontrivial, and is independent
of the choice of representative. This non-trivial type 0 linking invariant
captures the $A^{(1)}B^{(2)}B^{(2)}$ mixed anomaly in (3.1).

On the other hand, the type 2 linking involves three 2-volume operators, and
hence there is only one choice of correlation function,

$\displaystyle\braket{\widehat{U}_{a}(M_{3})\widehat{U}_{a}(M^{\prime}_{3})\widehat{U}_{a}(M^{\prime\prime}_{3})}\sim
e^{-{i\pi(N^{2}-1)\over
4N^{3}M^{2}}\text{Link}(M_{3},M^{\prime}_{3},M^{\prime\prime}_{3})_{2}}~{}.$
(3.11)

This captures the non-trivial $A^{(1)}\beta A^{(1)}\beta A^{(1)}$ anomaly in
(3.1).

###  3.3 Non-invertible symmetries from gauging

We have just seen how the non-trivial ’t Hooft anomalies of invertible
symmetries can be detected via the linkings of operators in the SymTFT. We now
turn to the case of non-invertible symmetries. We consider non-invertible
symmetries which are obtained by gauging the $\mathbb{Z}_{N}^{(1)}$ one-form
symmetry of the QFT ${\mathcal{X}}$. Let us denote the
$\mathbb{Z}_{N}^{(1)}$-gauged QFT by $\widehat{{\mathcal{X}}}$, with partition
function given by

$\displaystyle
Z_{\widehat{{\mathcal{X}}}}[B^{(2)}]=\frac{|H^{0}(X_{4},\mathbb{Z}_{N})|}{|H^{1}(X_{4},\mathbb{Z}_{N})|}\sum_{b^{(2)}\in
H^{2}(X_{4},\mathbb{Z}_{N})}Z_{{\mathcal{X}}}[b^{(2)}]e^{\frac{2\pi
i}{N}\int_{X_{4}}b^{(2)}B^{(2)}}~{}.$ (3.12)

Here we have turned off the $\mathbb{Z}_{2MN}^{(0)}$ background field in the
partition function. The fact that $\widehat{{\mathcal{X}}}$ has a non-
invertible symmetry has been discussed in [26, 27, 34, 60], and we now briefly
review this fact here.

We first define two topological manipulations: gauging $\mathbb{Z}_{N}^{(1)}$
and stacking with a $\mathbb{Z}_{N}^{(1)}$ SPT, denoted by $\sigma$ and $\tau$
respectively. Concretely, we have

$\displaystyle\begin{split}Z_{\sigma{\mathcal{X}}}[B^{(2)}]&=\frac{|H^{0}(X_{4},\mathbb{Z}_{N})|}{|H^{1}(X_{4},\mathbb{Z}_{N})|}\sum_{b^{(2)}\in
H^{2}(X_{4},\mathbb{Z}_{N})}Z_{{\mathcal{X}}}[b^{(2)}]e^{\frac{2\pi
i}{N}\int_{X_{4}}b^{(2)}B^{(2)}}~{},\\\
Z_{\tau{\mathcal{X}}}[B^{(2)}]&=\begin{cases}Z_{{\mathcal{X}}}[B^{(2)}]e^{\frac{2\pi
i}{2N}\int_{X_{4}}{\mathcal{P}}(B^{(2)})}~{},&N\in 2\mathbb{Z}\\\
Z_{{\mathcal{X}}}[B^{(2)}]e^{\frac{2\pi
i}{N}\frac{1+N}{2}\int_{X_{4}}B^{(2)}B^{(2)}}~{},&N\in 2\mathbb{Z}+1\\\
\end{cases}~{}.\end{split}$ (3.13)

To see the non-invertible symmetry of $\widehat{{\mathcal{X}}}$, we note that
it is invariant under $gC\tau\sigma\tau$, where $g$ is a global
$\mathbb{Z}_{2MN}^{(0)}$ transformation and $C$ is the charge conjugation
operation. To see this concretely, for $N\in 2\mathbb{Z}$ we have

$\displaystyle\begin{split}Z_{gC\tau\sigma\tau\widehat{{\mathcal{X}}}}[B^{(2)}]&=\frac{|H^{0}(X_{4},\mathbb{Z}_{N})|^{2}}{|H^{1}(X_{4},\mathbb{Z}_{N})|^{2}}\sum_{b^{(2)},\widetilde{b}^{(2)}}Z_{{\mathcal{X}}}[b^{(2)}]e^{\frac{2\pi
i}{N}\int_{X_{4}}b^{(2)}\widetilde{b}^{(2)}+\frac{{\mathcal{P}}(\widetilde{b}^{(2)})}{2}-\widetilde{b}^{(2)}B^{(2)}+\frac{{\mathcal{P}}(B^{(2)})}{2}+\frac{{\mathcal{P}}(b^{(2)})}{2}}\\\
&=\chi^{\frac{1}{2}}[X_{4}]\,\frac{|H^{0}(X_{4},\mathbb{Z}_{N})|}{|H^{1}(X_{4},\mathbb{Z}_{N})|}\sum_{b^{(2)}}Z_{{\mathcal{X}}}[b^{(2)}]e^{\frac{2\pi
i}{N}\int_{X_{4}}b^{(2)}B^{(2)}}=\chi^{\frac{1}{2}}[X_{4}]\,Z_{\widehat{{\mathcal{X}}}}[B^{(2)}]~{},\end{split}$
(3.14)

where the Euler counterterm is given by
$\chi[X_{4}]:=\frac{|H^{0}(X_{4},\mathbb{Z}_{N})|^{2}|H^{2}(X_{4},\mathbb{Z}_{N})|}{|H^{1}(X_{4},\mathbb{Z}_{N})|^{2}}$.
In short, we have

$\displaystyle
gC\tau\sigma\tau\widehat{{\mathcal{X}}}=\widehat{{\mathcal{X}}}~{}.$ (3.15)

The defect implementing $gC\tau\sigma\tau$ can be constructed from half-space
gauging, following [27, 60]. Another equivalent construction is to start with
the 3d $\mathbb{Z}_{2MN}^{(0)}$ defect, and decorate on the defect worldvolume
some TQFT to cancel the worldvolumn $\mathbb{Z}_{N}^{(1)}$ anomaly [26]. Both
methods give rise to the following non-invertible 3-volume defect,

$\displaystyle{\mathcal{N}}_{A}(M_{3})\sim\sum_{\phi^{(1)}\in
C^{1}(M_{3},\mathbb{Z}_{N})}g(M_{3})\cdot\exp\left(\frac{2\pi
i}{2N}\int_{M_{3}}-\phi^{(1)}\delta\phi^{(1)}+2\phi^{(1)}b^{(2)}\right)\,,$
(3.16)

where the gauge transformation of $\phi^{(1)}$ is
$\phi^{(1)}\to\phi^{(1)}+\beta^{(1)}$. The fusion rules are

$\displaystyle\begin{split}{\mathcal{N}}_{A}(M_{3})\times\overline{{\mathcal{N}}}_{A}(M_{3})&\sim\sum_{M_{2}\in
H_{2}(M_{3},\mathbb{Z}_{N})}(-1)^{Q(M_{2})}U_{b}(M_{2})~{},\\\
{\mathcal{N}}_{A}(M_{3})\times
U_{b}(M_{2})&={\mathcal{N}}_{A}(M_{3})~{},\end{split}$ (3.17)

where $U_{b}(M_{2})=e^{\frac{2\pi i}{N}\int_{M_{2}}b^{(2)}}$ is the defect for
the quantum $\mathbb{Z}_{N}^{(1)}$ symmetry in $\widehat{{\mathcal{X}}}$,
$Q(M_{2}):=\frac{1}{N}\int_{M_{3}}\text{PD}(M_{2})\delta\text{PD}(M_{2})$ is
the triple intersection number in $M_{3}$, and $\text{PD}(M_{2})$ is the
Poincaré dual of $M_{2}$.

We conclude that $\widehat{{\mathcal{X}}}$ has a non-invertible symmetry whose
defects satisfy non-invertible fusion rules. This symmetry follows from
$\mathbb{Z}_{2MN}^{(0)}$ in ${\mathcal{X}}$. The above derivation is
insensitive to the presence of the self-anomaly $A^{(1)}\beta A^{(1)}\beta
A^{(1)}$, as we turned off the background field $A^{(1)}$ throughout.

###  3.4 Non-invertible defects from SymTFT

We now review how the non-invertible symmetry defects are realized in the
SymTFT. Gauging only changes the topological boundary condition, and the
SymTFT of $\widehat{{\mathcal{X}}}$ remains the same as ${\mathcal{X}}$, i.e.
it is still given by (3.2). For ${\mathcal{X}}$, we saw that the topological
boundary condition was the Dirichlet boundary condition for $a^{(1)}$ and
$b^{(2)}$. After gauging $\mathbb{Z}_{N}^{(1)}$, the relevant topological
boundary condition becomes the Neumann boundary condition for $b^{(2)}$ and
the Dirichlet boundary condition for $a^{(1)}$, or equivalently the Dirichlet
boundary condition for both $a^{(1)}$ and $\widehat{b}^{(2)}$. In terms of
boundary states, this is

$\displaystyle\bra{D_{a^{(1)}},N_{b^{(2)}}}=\sum_{a^{(1)},b^{(2)}}\bra{a^{(1)},b^{(2)}}\,\delta(a^{(1)}-A^{(1)})e^{\frac{2\pi
i}{N}\int_{X_{4}}b^{(2)}B^{(2)}}~{}.$ (3.18)

What bulk operators are terminable on the new topological boundary? Clearly
the topological line $U_{a}$ can still terminate. Moreover, due to the
Dirichlet boundary condition for $\widehat{b}^{(2)}$, the non-invertible
surface $\widehat{U}_{b}$ can terminate as well. Indeed, since
$a|_{\partial}=0$ we have $\phi^{(0)}|_{\partial}=0$, and the TQFT on its
worldvolume is trivialized. We denote the collection of bulk operators that
are terminable as
${\mathcal{A}}_{\text{noninv}}=\\{U_{a},\widehat{U}_{b},...\\}$, where $...$
represents the operators constructed from $U_{a},\widehat{U}_{b}$ such as
their condensation defects.

By definition, all other bulk operators besides those in
${\mathcal{A}}_{\text{noninv}}$ should be attached to boundary operators when
intersecting the new topological boundary. Under this new boundary condition,
the topological surface $U_{b}$ would be attached to a boundary surface
operator $U_{b}|_{\partial}$. The topological 3-volume operator
$\widehat{U}_{a}$ remains non-invertible when moved to the boundary and
becomes a boundary operator $\widehat{U}_{a}|_{\partial}$—in particular, the
worldvolume TQFT is not trivialized. The boundary operators are collectively
denoted as
${\mathcal{C}}_{\text{noninv}}=\\{U_{b}^{p}|_{\partial},\widehat{U}_{a}^{q}|_{\partial},...\\}$
where $...$ are again various condensation defects. The bulk operators can be
organized into the following subsets,

${\mathcal{B}}_{b}=\\{U_{b},\,U_{b}\widehat{U}_{b}^{p},\,U_{b}\text{Cond}(U_{a}),...\\}~{},\hskip
22.76228pt{\mathcal{B}}_{\widehat{a}}=\\{\widehat{U}_{a},\,\widehat{U}_{a}\text{Cond}(U_{a}),\,\widehat{U}_{a}\text{Cond}(\widehat{U}_{b}),...\\}~{}.$
(3.19)

Assuming all defects to be of spherical topology, the above subsets simplify
significantly to

$\displaystyle{\mathcal{C}}_{\text{noninv}}=\\{U_{b}^{p}|_{\partial},\widehat{U}_{a}^{q}|_{\partial}\\}~{},\hskip
28.45274pt{\mathcal{B}}_{b}=\\{U_{b},U_{b}\widehat{U}_{b}^{p}\\}~{},\hskip
28.45274pt{\mathcal{B}}_{\widehat{a}}=\\{\widehat{U}_{a}\\}~{}.$ (3.20)

We finally note that the invertible boundary operator $U_{b}|_{\partial}$ and
the non-invertible boundary operator $\widehat{U}_{a}|_{\partial}$ are
precisely the generators of the invertible $\mathbb{Z}_{N}^{(1)}$ one-form
symmetry and the non-invertible zero-form symmetry of the 4d QFT
$\widehat{{\mathcal{X}}}$ after shrinking the slab, i.e. we have

(3.24)

###  3.5 Anomalies of non-invertible symmetries

We finally turn to the question of anomalies of non-invertible symmetry of
$\widehat{{\mathcal{X}}}$, using the linking invariants among the operators in
the SymTFT. Since the QFT ${\mathcal{X}}$ before gauging has a
$\mathbb{Z}_{2MN}^{(0)}$ self-anomaly and the $\mathbb{Z}_{2MN}^{(0)}$
generators become the non-invertible defects of the QFT
$\widehat{{\mathcal{X}}}$ after gauging, it is natural to expect that the non-
invertible symmetry also has a non-trivial ’t Hooft anomaly. To see this, we
pick a representative in ${\mathcal{B}}_{\widehat{a}}$ and compute the type 2
linking number. Since we assume spherical topology, there is only one choice
of representative $\widehat{U}_{a}$, and indeed the linking number is non-
trivial,

$\displaystyle\braket{\widehat{U}_{a}(M_{3})\widehat{U}_{a}(M^{\prime}_{3})\widehat{U}_{a}(M^{\prime\prime}_{3})}\sim
e^{-{i\pi(N^{2}-1)\over
4N^{3}M^{2}}\text{Link}(M_{3},M^{\prime}_{3},M^{\prime\prime}_{3})_{2}}~{}.$
(3.25)

This non-trivial linking invariant diagnoses the non-trivial ’t Hooft anomaly
of the non-invertible symmetry ${\mathcal{N}}_{A}$ in
$\widehat{{\mathcal{X}}}$.

The two remarks at the end of Section 2.5 still apply. Namely, the advantage
of this approach is that linking invariants for the DW theories are easy to
compute, though a full understanding of the anomaly requires knowledge of the
fusion higher-category. See [72, 68] for recent developments in higher
categories from the generalized symmetry point of view. Moreover, the anomaly
of the non-invertible symmetry detected here cannot be probed using the method
of [27, 29], although our condition is still only a sufficient condition for
the anomaly to be non-trivial.

###  3.6 Application 1: Adjoint QCD

We close this section with two examples. We first take ${\mathcal{X}}$ to be
$SU(N_{c})$ gauge theory with $N_{f}$ Weyl fermions in the adjoint
representation of $SU(N_{c})$. The action is

$\displaystyle
S_{{\mathcal{X}}}=\int_{X_{4}}\left(-\frac{1}{2g^{2}}\mathrm{Tr}f\wedge\star
f+\sum_{i=1}^{N_{f}}i\overline{\psi}_{i}\not{D}_{a}\psi_{i}\right)~{},$ (3.26)

where $D_{a}$ is the covariant derivative with adjoint indices (suppressed).

##### Invertible symmetries and anomalies:

Because the fermions are in the adjoint representation, this theory has an
electric $\mathbb{Z}_{N_{c}}^{(1)}$ one-form symmetry. Classically, there is
also a $U(1)$ zero-form symmetry which acts on the fermion by a phase,
$\psi_{i}\to e^{i\alpha}\psi_{i}$. It is well-known that the $U(1)$ is broken
by $SU(N_{c})$ instantons down to $\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$. Thus the
total symmetry is
$\mathbb{Z}_{N_{c}}^{(1)}\times\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$. This symmetry
has a non-trivial ’t Hooft anomaly [110, 55], specified by the 5d AnomTFT,

$\displaystyle\begin{split}\int_{X_{5}}\left(\frac{2\pi}{2N_{c}}A^{(1)}{\mathcal{P}}(B^{(2)})+\frac{\pi(N_{c}^{2}-1)}{6N_{c}}A^{(1)}\beta
A^{(1)}\beta A^{(1)}\right)~{},&\hskip 28.45274ptN_{c}\in 2\mathbb{Z}\\\
\int_{X_{5}}\left(\frac{2\pi}{N_{c}}\frac{1+N_{c}}{2}A^{(1)}B^{(2)}B^{(2)}+\frac{\pi(N_{c}^{2}-1)}{6N_{c}}A^{(1)}\beta
A^{(1)}\beta A^{(1)}\right)~{},&\hskip 28.45274ptN_{c}\in 2\mathbb{Z}+1\\\
\end{split}$ (3.27)

where $A^{(1)}$ and $B^{(2)}$ are the $\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$ and
$\mathbb{Z}_{N_{c}}^{(1)}$ background fields respectively. Thus, the anomaly
of adjoint QCD is precisely the anomaly (3.2) we have been discussing
throughout this section, upon replacing $(N,M)\to(N_{c},N_{f})$. All of the
discussions so far thus apply straightforwardly.

We will now discuss some of the dynamical implications of the anomaly. Before
doing so, let us comment that the theory (3.26) actually has a larger symmetry
than just $\mathbb{Z}_{N_{c}}^{(1)}\times\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$; for
example, there is a flavor rotation symmetry $SU(N_{f})$. We will not require
that these additional symmetries be preserved below, and in particular we will
allow for perturbations that explicitly break $SU(N_{f})$ as long as
$\mathbb{Z}_{N_{c}}^{(1)}\times\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$ is preserved.

The mixed anomaly (3.27) has an immediate dynamical consequence: adding any
$\mathbb{Z}_{N_{c}}^{(1)}\times\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$ preserving
deformation does not drive the theory (3.26) to a trivially gapped phase. For
example, assuming that $SU(N_{c})$ Yang-Mills without a theta term has a mass
gap, the fermion bilinear mass term
$m\sum_{i=1}^{N_{f}}\epsilon^{\alpha\beta}{\psi}_{i\alpha}\psi_{i\beta}$ with
$m>0$ drives the theory to a trivially gapped phase. Here $\alpha,\beta=1,2$
are the Lorentz spinor indices, and pairing the two spinors via the epsilon
tensor ensures a Lorentz singlet. However, this term explicitly breaks
$\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$ chiral symmetry.

##### Dynamical constraints of $PSU(N_{c})$ adjoint QCD:

Since $\mathbb{Z}_{N_{c}}^{(1)}$ does not have a self-anomaly, it can be
gauged resulting in $PSU(N_{c})$ adjoint QCD. This theory has a
$\mathbb{Z}_{N_{c}}^{(1)}$ quantum one-form symmetry, whose defect is

$\displaystyle U_{b}(M_{2})=e^{\frac{2\pi
i}{N_{c}}\oint_{M_{2}}b^{(2)}}=e^{\frac{2\pi
i}{N_{c}}\oint_{M_{2}}w_{2}^{PSU(N_{c})}}~{}.$ (3.28)

The general results reviewed in this section show that the theory has a non-
invertible codimension-one defect

$\displaystyle{\mathcal{N}}_{A}(M_{3})\sim\sum_{\phi^{(1)}\in
C^{1}(M_{3},\mathbb{Z}_{N_{c}})}g(M_{3})\cdot\exp\left(\frac{2\pi
i}{2N_{c}}\int_{M_{3}}-\phi^{(1)}\delta\phi^{(1)}+2\phi^{(1)}w_{2}^{PSU(N_{c})}\right)$
(3.29)

implementing a twisted gauging $g\tau\sigma\tau$, where $g$ is a
$\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$ global transformation, $\tau$ is a stacking
with a $\mathbb{Z}_{N_{c}}^{(1)}$ invertible phase, and $\sigma$ is a gauging
of $\mathbb{Z}_{N_{c}}^{(1)}$. Moreover, the non-invertible defect
${\mathcal{N}}_{A}$ has a self-anomaly, diagnosed by a non-trivial triple
linking invariant in the bulk.

As is the case for invertible symmetries, the presence of an anomalous non-
invertible symmetry has non-trivial dynamical implications. Indeed, suppose
that we start with $SU(N_{c})$ adjoint QCD and add
$\mathbb{Z}_{N_{c}}^{(1)}\times\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$ symmetric
deformations, then gauge $\mathbb{Z}_{N_{c}}^{(1)}$ to obtain $PSU(N_{c})$
adjoint QCD. We would like to ask whether there is a symmetric perturbation
which triggers a flow from the gauged theory to a trivially gapped phase.

Without knowledge of the non-invertible symmetry, one may have naively
concluded that the $PSU(N_{c})$ theory arising from gauging of the
$\mathbb{Z}_{N_{c}}^{(1)}$ symmetry of the $SU(N_{c})$ theory has only a
$\mathbb{Z}_{2N_{f}}^{(0)}\times{\mathbb{Z}}_{N_{c}}^{(1)}$ symmetry, where
the first component is the quotient subgroup of
$\mathbb{Z}_{2N_{f}N_{c}}^{(0)}$ that is free of mixed anomaly, while the
second component is the quantum symmetry in (3.28). The one-form symmetry is
anomaly-free, while the zero-form symmetry has an anomaly

$\displaystyle\int_{X_{5}}\frac{\pi(N_{c}^{2}-1)}{6}\widetilde{A}^{(1)}\widetilde{\beta}\widetilde{A}^{(1)}\widetilde{\beta}\widetilde{A}^{(1)}$
(3.30)

with $A^{(1)}=N_{c}\widetilde{A}^{(1)}$ and $\widetilde{A}^{(1)}$ the
$\mathbb{Z}_{2N_{f}}^{(0)}$ background field. Here
$\widetilde{\beta}=\delta/(2N_{f})$ is the Bockstein map. The above anomaly
vanishes for certain $(N_{c},N_{f})$, e.g. $N_{c}=5$ and any $N_{f}$, and thus
one may naively conclude that when $N_{c}=5$ and $N_{f}$ is arbitrary, the
theory is free of anomalies and can be deformed to a trivially gapped phase.

However, with our current knowledge of the non-invertible symmetry, we now
know that this is not true. The gauged theory has a non-invertible symmetry
that suffers from a self-anomaly for any $N_{c}>1$ and $N_{f}\geq 1$, and thus
the theory cannot be deformed to a trivially gapped phase.

##### Low-energy dynamics for $N_{f}=1$:

Let us comment on some special features when $N_{f}=1$. $SU(N_{c})$ massless
adjoint QCD with $N_{f}=1$ has enhanced supersymmetry—namely, it becomes
${\mathcal{N}}=1$ $SU(N_{c})$ SYM. The low-energy dynamics of this theory are
known: the $\mathbb{Z}_{2N_{c}}^{(0)}\times\mathbb{Z}_{N_{c}}^{(1)}$ global
symmetry is spontaneously broken to
$\mathbb{Z}_{2}^{(0)}\times\mathbb{Z}_{N_{c}}^{(1)}$ by a gaugino bilinear
condensation, where $\mathbb{Z}_{2}^{(0)}$ is fermion parity. As a
consequence, there are $N_{c}$ supersymmetric vacua. Each vacuum is trivially
gapped, and the domain walls between two different vacua support a Chern-
Simons theory [111] needed to match the mixed anomaly between
$\mathbb{Z}_{2N_{c}}^{(0)}$ and $\mathbb{Z}_{N_{c}}^{(1)}$ in (3.27). Note
that the Chern-Simons theory couples to the bulk only through the
$\mathbb{Z}_{N_{c}}^{(1)}$ background field (rather than a dynamical field),
and hence the domain wall still has invertible fusion rules.

There has been some confusion in the literature about how the Chern-Simons
theory on the domain wall can match the $\mathbb{Z}_{2N_{c}}^{(0)}$ self-
anomaly in (3.27). In [55], it was pointed out that non-invertible defects and
the junctions of the domain walls are potentially important to the resolution
of this confusion. In the remainder of this section, we make this observation
more concrete using the SymTFT.

SymTFT
---
$\bra{D_{a^{(1)},b^{(2)}}}$$\ket{{\mathcal{X}}_{SU(N_{c})}}$RG SymTFT
---
$\bra{D_{a^{(1)},b^{(2)}}}$$\ket{D_{a^{(1)},\widehat{b}^{(2)}}}$ $SU(N_{c})$
---
AdjQCD
$N_{c}$ vacua
---
RG Figure 7: RG flow of $SU(N_{c})$ adjoint QCD. The dynamical boundary
condition flows to a topological boundary condition.

At low energies, the theory flows to a gapped RG fixed point, and hence the
dynamical boundary condition on the right boundary in the SymTFT flows to a
topological boundary condition; see Figure 7. Because it is known that the
$\mathbb{Z}_{2N_{c}}^{(0)}$ zero-form symmetry is spontaneously broken to
$\mathbb{Z}_{2}^{(0)}$ while the $\mathbb{Z}_{N_{c}}^{(1)}$ one-form symmetry
remains unbroken, the topological boundary condition should be such that
$U_{a}$ is terminable while $U_{b}$ is not. Indeed, in this case $U_{a}$ is
terminable on both left and right boundaries, and shrinking the slab gives
rise to a topological local order parameter of $\mathbb{Z}_{2N_{c}}^{(0)}$
labeling the distinct vacua. Likewise, since $U_{b}$ is not terminable on the
right boundary, shrinking the slab does not produce a topological order
parameter for $\mathbb{Z}_{N_{c}}^{(1)}$, and hence it is not spontaneously
broken.

The domain wall supporting a Chern-Simons theory in the SymTFT setup comes
from the operator $\widehat{U}_{a}|_{\partial}$ supported purely on the left
topological boundary. Note that it is an invertible defect on the boundary.
The Chern-Simons theory on the domain wall follows from the TQFT on the
worldvolume of $\widehat{U}_{a}$, where the dynamical field $b^{(2)}$ is
replaced by the background field $B^{(2)}$. Using the discussion in Section
3.5, the fact that the domain walls and their junctions saturate the
$\mathbb{Z}_{2N_{c}}^{(0)}$ self-anomaly in the UV is related to the fact that
the bulk operators in the SymTFT $\widehat{U}_{a}$ have non-trivial triple
linking invariants. Therefore the right panel in Figure 7 reproduces all the
known IR features of adjoint QCD and manifestly matches the anomaly since the
SymTFT and the topological boundary condition on the left boundary are the
same as in the UV. It would be interesting to study the junctions between
domain walls or $\widehat{U}_{a}|_{\partial}$ in detail.

After gauging $\mathbb{Z}_{N_{c}}^{(1)}$, the left topological boundary
condition is changed to $\bra{D_{a^{(1)},\widehat{b}^{(2)}}}$, while the right
boundary condition is unchanged. At low energies, this in particular means
that the $\mathbb{Z}_{N_{c}}^{(1)}$ one-form symmetry can also be
spontaneously broken. Indeed, the vacua in this case are $\mathbb{Z}_{N_{c}}$
discrete gauge theories with $b^{(2)}b^{(2)}$ couplings, where the
$\mathbb{Z}_{N_{c}}^{(1)}$ one form symmetry is (partially) spontaneously
broken.

###  3.7 Application 2: $\mathcal{N}=4$ SYM

We next consider the example of $\mathcal{N}=4$ SYM. This theory possesses an
$SL(2,\mathbb{Z})$ duality group relating equivalent but different-looking
$\mathcal{N}=4$ theories. The dual theories generally differ in both the value
of the complexified coupling constant $\tau_{\mathrm{YM}}$ and the gauge
group. There are however special gauge groups and values of
$\tau_{\mathrm{YM}}$ for which the theory is mapped to itself under some
discrete subgroup of $SL(2,\mathbb{Z})$. In such cases this discrete subgroup
becomes an invertible symmetry of the theory. On the other hand, when
$\tau_{\mathrm{YM}}$ is fixed under a subgroup of $SL(2,\mathbb{Z})$ but the
gauge group changes, we can in general combine the transformation with a
gauging of the one-form symmetry in order to obtain a non-invertible symmetry.
For more information, we refer the reader to [35, 65]. As in the previous
examples, we will focus on non-invertible symmetries which are non-instrinsic.
In the current context, this means that there exists a different global form,
i.e. a theory with the same gauge algebra but either a different gauge group
or a different invertible phase (or both), in which the symmetry becomes
invertible.

We now ask when these non-invertible symmetries are anomalous. It is
convenient to begin by considering the anomaly of the $SL(2,\mathbb{Z})$
duality group of Maxwell theory. This anomaly was originally identified in
[112], and further studied in [113, 114, 115]. In the original reference
[112], it was discovered that the partition function is not invariant under
$SL(2,\mathbb{Z})$ on a curved manifold, and instead acquires a factor
depending on the Euler characteristic $\chi(X_{4})$ and signature
$\mathsf{sig}(X_{4})$ of the 4d manifold $X_{4}$ as follows,

$\displaystyle Z(\tau_{\mathrm{Max}}+1)$ $\displaystyle=$ $\displaystyle
Z(\tau_{\mathrm{Max}})~{},$ $\displaystyle Z(-1/\tau_{\mathrm{Max}})$
$\displaystyle=$ $\displaystyle\tau_{\mathrm{Max}}^{{1\over
4}\left(\chi(X_{4})-\mathsf{sig}(X_{4})\right)}\,\overline{\tau}_{\mathrm{Max}}^{{1\over
4}\left(\chi(X_{4})+\mathsf{sig}(X_{4})\right)}\,Z(\tau_{\mathrm{Max}})~{}.$
(3.31)

This signals a mixed anomaly between gravity and the $SL(2,\mathbb{Z})$
duality symmetries at certain values of the coupling.

We first consider $\tau_{\mathrm{Max}}=i$, where we would expect a
$\mathbb{Z}_{4}$ symmetry generated by
$\mathsf{S}:\tau_{\mathrm{Max}}\mapsto-1/\tau_{\mathrm{Max}}$. According to
the above results, under the modular $\mathsf{S}$ transformation the partition
function transforms as

$\displaystyle\mathsf{S}:\hskip 21.68121ptZ(i)\hskip
14.45377pt\longrightarrow\hskip 14.45377pti^{-{1\over
2}{\mathsf{sig}(X_{4})}}Z(i)~{}.$ (3.32)

Rokhlin’s theorem states that $\mathsf{sig}(X_{4})\in 16\mathbb{Z}$ on spin
manifolds, so in this case the overall factor is trivial. Hence we do not
expect a mixed anomaly on spin manifolds. On the other hand, for
$\tau_{\mathrm{Max}}=e^{2\pi i/3}$ we would expect a $\mathbb{Z}_{3}$ symmetry
generated by $\mathsf{ST}$. In this case we find that

$\displaystyle\mathsf{ST}:\hskip 21.68121ptZ(e^{2\pi i\over 3})\hskip
14.45377pt\longrightarrow\hskip 14.45377pte^{-{\pi i\over
3}\mathsf{sig}(X_{4})}Z(e^{2\pi i\over 3})~{}$ (3.33)

where $\mathsf{T}$ acts as
$\mathsf{T}:\tau_{\mathrm{Max}}\mapsto\tau_{\mathrm{Max}}+1$. The overall
factor here is non-trivial (even on spin manifolds) and hence we conclude that
there is a mixed anomaly.

We have just seen that by coupling to non-trivial gravitational backgrounds
(i.e. $\mathsf{sig}(X_{4})$), one can detect a mixed anomaly between
(subgroups of) $SL(2,\mathbb{Z})$ and gravity at $\tau_{\mathrm{Max}}=e^{2\pi
i/3}$. For the rest of the discussion here, we shall concentrate on this value
of $\tau_{\mathrm{Max}}$. We can now also allow for non-trivial backgrounds of
$SL(2,\mathbb{Z})$, which will allow us to detect self-anomalies of the
$\mathbb{Z}_{3}$ symmetry. In general, such anomalies must take values in
$\Omega^{\text{Spin}}_{5}(B\mathbb{Z}_{3})=\mathbb{Z}_{9}$, and we now ask
which element of $\mathbb{Z}_{9}$ Maxwell theory realizes.

This problem was considered in [115], which claimed that the self-anomaly of
Maxwell theory is given by $56$ times that of a 4d Weyl fermion. The strategy
in that work was to make use of a 6d interpretation, in terms of the
compactification of the 6d rank-1 E-string SCFT. This theory has a moduli
space consisting of two branches referred to as the tensor branch and Higgs
branch. On a generic point of the former one gets a free $(1,0)$ tensor
multiplet, while on a generic point of the latter one gets $29$ free
hypermultiplets. When reduced to 4d on a torus, the former gives Maxwell
theory plus $2$ Weyl fermions and some scalars, while the latter gives $58$
free Weyl fermions plus some scalars. As we can continuously move from the
tensor branch to the Higgs branch, and as anomalies should be invariant under
continuous deformations, we see that the anomaly for the Maxwell theory should
be equal to that of $56$ Weyl fermions. If we assume that the minimal anomaly
is realized by a single Weyl fermion, then we conclude that the duality
anomaly of Maxwell theory realizes the element $56=2\in\mathbb{Z}_{9}$.

One subtlety in the above derivation is how the $SL(2,\mathbb{Z})$ acts on the
fermions. In particular, the 6d picture leads to $\mathsf{S}^{4}$ being
$(-1)^{\mathsf{F}}$, i.e. fermion parity, instead of just $1$. As such, we
should really be considering $[\mathrm{Spin}(3,1)\times
Mp(2,\mathbb{Z})]/\mathbb{Z}^{F}_{2}$ instead of $\mathrm{Spin}(3,1)\times
SL(2,\mathbb{Z})$, where $Mp(2,\mathbb{Z})$ is a double cover of
$SL(2,\mathbb{Z})$ such that $\mathsf{S}^{4}\neq 1$, and $\mathbb{Z}^{F}_{2}$
is the combination of the fermion parity of $\mathrm{Spin}(3,1)$ and
$\mathsf{S}^{4}$ in $Mp(2,\mathbb{Z})$. A simplifying feature is that, as we
are only concerned with the $\mathbb{Z}_{3}$ subgroup at
$\tau_{\mathrm{Max}}=e^{2\pi i/3}$, we can take the two groups to be
$[\mathrm{Spin}(3,1)\times\mathbb{Z}_{6}]/\mathbb{Z}^{F}_{2}$ and
$\mathrm{Spin}(3,1)\times\mathbb{Z}_{3}$, respectively. These groups are
actually equivalent, since starting from the former we can just combine the
generator of $\mathbb{Z}_{3}$ with fermion parity to get $\mathbb{Z}_{6}$.
Furthermore, as we expect fermion parity to be non-anomalous, we expect the
self-anomalies of the two symmetries to be the same. Indeed, the cobordism
groups with both structure groups, evaluated in [116], are equal to
$\mathbb{Z}_{9}$. From now on we shall concentrate on the $\mathbb{Z}_{6}$
version that includes fermion parity, though we expect the results to also
hold for the $\mathbb{Z}_{3}$ version.

Having discussed the anomaly for Maxwell theory, we are now ready to consider
the case of $\mathcal{N}=4$ SYM. To understand this case, we shall use the
fact that $SL(2,\mathbb{Z})$ (and fermion parity) do not act on the moduli
space and hence are preserved at a generic point on this space. At a generic
point, the theory looks like a theory of $r$ free $\mathcal{N}=4$ vector
multiplets, with $r$ being the rank of the gauge group. Each vector multiplet
contains a copy of Maxwell theory, four Weyl fermions, and six scalars, the
latter of which will play no role in our analysis. As we have not broken the
$SL(2,\mathbb{Z})$ by moving out on the moduli space, the anomaly of the
$\mathcal{N}=4$ SCFT should be equal to the anomaly at a generic point in this
space. From our previous results, we conclude that the total anomaly is

$\displaystyle I^{\mathrm{rank}\,\,r\,\,\text{SYM}}=(2+4)\,rI^{\text{4d
fermion}}=6\,rI^{\text{4d fermion}}~{},$ (3.34)

again in terms of the anomaly of a single Weyl fermion. The anomaly identified
is thus a self-anomaly for the $\mathbb{Z}_{6}$ zero-form modular symmetry.
Note that as the anomaly in
$[\mathrm{Spin}(3,1)\times\mathbb{Z}_{6}]/\mathbb{Z}^{F}_{2}$ is mod $9$, this
anomaly actually only depends on $r$ mod $3$.

For the $\mathfrak{a}$-type cases we can arrive at the same result using the
6d approach of [115]. Here we start with the rank-$(r+1)$ E-string theory and
use the fact that on the tensor branch it reduces to the type
$\mathfrak{a}_{r}$ $(2,0)$ theory plus a decoupled $(1,0)$ tensor, while on
the Higgs branch it reduces to $30r-1$ free hypers. The reduction of the
former now gives $\mathcal{N}=4$ SYM with gauge algebra $\mathfrak{a}_{r}$
plus a decoupled $\mathcal{N}=2$ $U(1)$ gauge theory and equating the anomaly
on the two sides yields the desired result.

To summarize our discussion so far, we have found that at the value of the
complex coupling $\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}}$, the modular
$\mathsf{ST}$ symmetry can have a self-anomaly, determined by the rank $r$ of
the gauge group mod 3. When the gauge group in question is left invariant
under the modular $\mathsf{ST}$ symmetry, then this is a standard ‘t Hooft
anomaly in an invertible symmetry. On the other hand, when the gauge group is
not left invariant under $\mathsf{ST}$, then the modular transformation must
be dressed with appropriate discrete gaugings to make it a non-invertible
symmetry, and in the case of non-intrinsic non-invertible symmetries the
anomaly of the invertible symmetry implies an anomaly for the non-invertible
symmetry.

As concrete examples, let us consider the ${\mathcal{N}}=4$ SYM theories with
gauge algebras $\mathfrak{a}_{1}$, $\mathfrak{a}_{2}$, and $\mathfrak{e}_{6}$,
all of which were analyzed in [35]. In the case of $\mathfrak{a}_{1}$, there
are three global forms $SU(2)$, $SO(3)_{+}$, and $SO(3)_{-}$ (up to stacking
with SPT phases), and none of them have an invertible $\mathsf{ST}$ symmetry
at $\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}}$. Thus all of the corresponding
non-invertible $\mathsf{ST}$ symmetries are intrinsically non-invertible, and
we will not say anything about them here. On the other hand, for
$\mathfrak{a}_{2}$, there are four global forms $SU(3)$, $PSU(3)_{0}$,
$PSU(3)_{1}$, and $PSU(3)_{2}$ (again up to SPT phases), and it turns out that
the $PSU(3)_{1}$ theory has an invertible $\mathsf{ST}$ symmetry. Since the
rank $r=2$ is non-zero mod 3, we conclude that this $\mathsf{ST}$ symmetry has
a ‘t Hooft anomaly, and by our general discussions in this paper that the non-
invertible symmetries in the other global variants are also anomalous.
Finally, in the case of $\mathfrak{e}_{6}$, there are again four global forms
up to SPTs, one of which has an invertible $\mathsf{ST}$ symmetry. However, in
this case the rank $r=6$ is zero mod 3, and hence we expect no anomalies for
this symmetry.

Let us close with a bit more detail on the case of $PSU(3)_{1}$, which has a
$\mathbb{Z}_{6}$ self-anomaly given by $6r=3\mod 9$ copies of the anomaly of
the 4d fermion. Note that the 5d anomaly inflow action of the 4d fermion can
be inferred from the $U(1)$ inflow action by restricting $U(1)$ to
$\mathbb{Z}_{6}$ subgroup. Three copies of the 4d fermions have the $U(1)$
anomaly $\frac{3}{24\pi^{2}}\mathsf{A}d\mathsf{A}d\mathsf{A}$. Substituting
$\mathsf{A}=\frac{2\pi}{6}A^{(1)}$, we then get the $\mathbb{Z}_{6}$ self-
anomaly $\frac{2\pi}{12}A^{(1)}\beta A^{(1)}\beta A^{(1)}$. Combining this
with the mixed anomaly with the $\mathbb{Z}_{3}$ one-form symmetry, the total
anomaly is given by101010The $A^{(1)}B^{(2)}B^{(2)}$ anomaly can be checked by
turning off $A^{(1)}$ and computing how the partition function transforms
under a global $\mathbb{Z}_{6}^{(0)}$ transformation. In principle, another
mixed anomaly of the form $B^{(2)}A^{(1)}\beta A^{(1)}$ can also appear.
Although it is interesting to explicitly check it, we will assume that such
term vanishes. If it does not vanish, there exists a choice of symmetry
fractionalization [55] by shifting $B^{(2)}\to B^{(2)}+k\beta A^{(2)}\mod 3$
for some $k$ such that this term is absorbed. As a consequence, the
coefficient of the self anomaly $A^{(2)}\beta A^{(1)}\beta A^{(2)}$ will be
modified.

$\displaystyle\int_{X_{5}}\frac{4\pi}{3}A^{(1)}B^{(2)}B^{(2)}+\frac{2\pi}{12}A^{(1)}\beta
A^{(1)}\beta A^{(1)}$ (3.35)

where $\beta=\delta/6$, the field $A^{(1)}$ is the $\mathbb{Z}_{6}^{(0)}$ one-
form gauge field, and $B^{(2)}$ is the $\mathbb{Z}_{3}^{(1)}$ two-form gauge
field. We can further gauge the $\mathbb{Z}_{3}^{(1)}$ one-form symmetry,
mapping $PSU(3)_{1}$ SYM to $SU(3)$ SYM (coupled to a non-trivial
$\mathbb{Z}_{3}^{(1)}$ SPT). From the results in this section,111111Although
we focused on $\mathbb{Z}_{N}^{(1)}$ one-form symmetry for even $N$, parallel
discussions apply for odd $N$ as well. we conclude that $SU(3)$ SYM has an
anomalous non-invertible symmetry.

## Acknowledgements

We thank Yichul Choi, Linhao Li, Kantaro Ohmori, Sakura Schafer-Nameki, Yuji
Tachikawa, and Zheyan Wan for discussions. We thank Shu-Heng Shao and Yuji
Tachikawa for comments on a draft. JK and GZ thank Kavli IPMU for their
generous hospitality during the inception of this work. EN and YZ are
partially supported by WPI Initiative, MEXT, Japan at IPMU, the University of
Tokyo. GZ is partially supported by the Simons Foundation grant 815892.

##  Appendix A Linking numbers

In this appendix we discuss the linking numbers relevant to the main text.

##### Two component links:

Given two closed manifolds $M_{1}^{(p)}$ and $M_{2}^{(q)}$, we aim to compute
the linking number between them. We denote their Seifert surfaces as
$N_{1}^{(p+1)}$ and $N_{2}^{(q+1)}$ respectively, which means that $\partial
N_{1}^{(p+1)}=M_{1}^{(p)}$ and $\partial N_{2}^{(q+1)}=M_{2}^{(q)}$. The
linking number between $M_{1}^{(p)}$ and $M_{2}^{(q)}$ in spacetime $S^{d}$ is
given by

$\text{Link}(M_{1}^{(p)},M_{2}^{(q)}):=\int_{S^{d}}\text{PD}(N_{1}^{(p+1)})d\text{PD}(N_{2}^{(q+1)})=\text{Int}(N_{1}^{(p+1)},M_{2}^{(q)})$
(A.1)

where PD is the Poincaré dual and $\text{Int}(\cdot,\cdot)$ counts the number
of (oriented) intersection points between the two arguments. Note that for the
integration to be non-trivial, we need to match the degrees

$p+1+q=d~{}.$ (A.2)

As an example, one can consider $p=q=1$ and $d=3$. In this case, the two lines
form a standard link in three-dimensions. In this case it is easy to see that
a non-trivial Hopf link has linking number 1, while an L4a1 link (also known
as Solomon’s knot) has linking number 2.121212See
http://katlas.math.toronto.edu/wiki/L4a1.

One may wonder why we don’t define another type of linking number by
$\int_{S^{d}}\text{PD}(N_{1}^{(p+1)})\text{PD}(N_{2}^{(q+1)})=\text{Int}^{\prime}(N_{1}^{(p+1)},N_{2}^{(q+1)})$
with $p+q+2=d$. The reason is that this intersection number is unstable: it is
possible to move the boundaries $M_{1}^{(p)}$ and $M_{2}^{(q)}$ without
crossing each other such that the intersection number is zero.

##### Three component links:

Unlike the the case with two components, there are multiple types of linking
numbers involving three components. Suppose the three components are
$M_{1}^{(p)},M_{2}^{(q)}$, and $M_{3}^{(r)}$ respectively. We denote their
Seifert surfaces by $N_{1}^{(p+1)},N_{2}^{(q+1)}$, and $N_{3}^{(r+1)}$. There
are three types of linking numbers between $M_{1}^{(p)},M_{2}^{(q)}$, and
$M_{3}^{(r)}$ in spacetime $S^{d}$, denoted by type 0, type 1, and type 2
respectively. The type 0 linking number is given by

$\begin{split}\text{Link}(M_{1}^{(p)},M_{2}^{(q)},M_{3}^{(r)})_{0}&:=\int_{S^{d}}\text{PD}(N_{1}^{(p+1)})\text{PD}(N_{2}^{(q+1)})\text{PD}(N_{3}^{(r+1)})\\\
&=\text{Int}(N_{1}^{(p+1)},N_{2}^{(q+1)},N_{3}^{(r+1)})\end{split}$ (A.3)

where the dimensions should satisfy

$p+1+q+1+r+1=2d~{}.$ (A.4)

The type 1 linking number is given by

$\begin{split}\text{Link}(M_{1}^{(p)},M_{2}^{(q)},M_{3}^{(r)})_{1}&:=\int_{S^{d}}\text{PD}(N_{1}^{(p+1)})\text{PD}(N_{2}^{(q+1)})d\text{PD}(N_{3}^{(r+1)})\\\
&=\text{Int}(N_{1}^{(p+1)},N_{2}^{(q+1)},M_{3}^{(r)})\end{split}$ (A.5)

where the dimensions should satisfy

$p+1+q+1+r=2d~{}.$ (A.6)

Finally, the type 2 linking number is given by

$\begin{split}\text{Link}(M_{1}^{(p)},M_{2}^{(q)},M_{3}^{(r)})_{2}&:=\int_{S^{d}}\text{PD}(N_{1}^{(p+1)})d\text{PD}(N_{2}^{(q+1)})d\text{PD}(N_{3}^{(r+1)})\\\
&=\text{Int}(N_{1}^{(p+1)},M_{2}^{(q)},M_{3}^{(r)})\end{split}$ (A.7)

where the dimensions should satisfy

$p+1+q+r=2d~{}.$ (A.8)

As an example, for $p=q=r=1$ and $d=3$, the Borromean rings have type 0
linking number 1 because the three Seifert surfaces of the three loops
intersect at one point; see Figure 8. Moreover, for $p=q=r=2$ and $d=4$, the
linking configuration with a non-trivial type 1 linking number is associated
with a 3-loop braiding process [117, 118].

$M_{1}^{(1)}$$M_{2}^{(1)}$$M_{3}^{(1)}$ Figure 8: Borromean rings linking
between $M_{1}^{(1)},M_{2}^{(1)}$ and $M_{3}^{(1)}$ .

#####  $N$ component links:

By generalizing the discussion for triple links, it is clear that there are
$N$ types of $N$-component linking numbers for $N>2$. Suppose the $N$
topological operators are of dimension $p_{i}$ for $i=1,...,N$. We can define
the $N$-component link of type $k$ by having $N-k$ Seifert surfaces and $k$
operators intersect at a point,

$\begin{split}&\text{Link}(M_{1}^{(p_{1})},....,M_{N}^{(p_{N})})_{k}:=\\\
&\hskip
14.45377pt\int_{S^{d}}\text{PD}(N_{1}^{(p_{1}+1)})...\text{PD}(N_{N-k}^{(p_{N-k}+1)})d\text{PD}(N_{N-k+1}^{(p_{N-k+1}+1)})...d\text{PD}(N_{N}^{(p_{N}+1)})\\\
&\hskip
36.135pt=\text{Int}(N_{1}^{(p_{1}+1)},...,N_{N-k}^{(p_{N-k}+1)},M_{N-k+1}^{(p_{N-k+1})},...,M_{N}^{(p_{N})})\end{split}$
(A.9)

where the various dimensions should satisfy

$\sum_{i=1}^{N-k}(p_{i}+1)+\sum_{j=N-k+1}^{N}p_{j}=(N-1)d~{}.$ (A.10)

##### Linking invariants between lines in $d=3$:

From the results above, we find that the linking invariants between lines in
$d=3$ are of two types: the linking number involving two loops probed by the
Hopf link, and the type 0 linking number involving three loops probed by the
Borromean rings. To see that there are no other linking numbers with number of
components $N\geq 3$, we use (A.10) and substitute $p_{i}=p_{j}=1$,

$\displaystyle(N-k)\cdot(1+1)+k\cdot 1=(N-1)3\hskip
14.22636pt\Rightarrow\hskip 14.22636ptN=3-k~{}.$ (A.11)

Hence the only solution for $N\geq 3$ is $N=3$ and $k=0$.

##### Linking invariants among surfaces and 3-volumes in $d=5$:

From the results above, we find that the linking invariants involving surfaces
and 3-volumes in $d=5$ are of the following types: the linking number
involving two 2d operators, the type 0 linking number between one 3d operator
and two 2d operators, the type 1 linking number between two 3d operators and a
2d operator, the type 2 linking number between three 3d operators, the type 1
linking number between four 3d operators, and the type 0 linking number
between five 3d operators.

##  Appendix B
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\times\mathbb{Z}_{2}^{C}$
Dijkgraaf-Witten model in 3d

In this appendix we study the properties of the Dijkgraaf-Witten model (2.2).
For convenience we reproduce the action of this theory,

$\displaystyle\int_{X_{3}}\pi\left(\widehat{a}\delta a+\widehat{b}\delta
b+\widehat{c}\delta c+abc+\frac{1}{2}a\delta a\right)~{}.$ (B.1)

All of the gauge fields are $\mathbb{Z}_{2}$ cochains, i.e. elements of
$C^{1}(X_{3},\mathbb{Z}_{2})$. In this appendix we will assume that $X_{3}$
does not have a boundary. The action is invariant under the gauge
transformations [103]

$\displaystyle\begin{split}&a\to a+\delta\alpha,\hskip 28.45274ptb\to
b+\delta\beta,\hskip 28.45274ptc\to c+\delta\gamma~{},\\\
&\widehat{a}\to\widehat{a}+\delta\widehat{\alpha}+\beta c-\gamma
b+\beta\delta\gamma~{},\\\
&\widehat{b}\to\widehat{b}+\delta\widehat{\beta}+\gamma a-\alpha
c+\gamma\delta\alpha~{},\\\
&\widehat{c}\to\widehat{c}+\delta\widehat{\gamma}+\alpha b-\beta
a+\alpha\delta\beta~{}.\end{split}$ (B.2)

Note that the twist term $abc$ is not gauge invariant under the gauge
transformations of $a,b,c$. The gauge transformations of
$\widehat{a},\widehat{b}$, and $\widehat{c}$ are then fixed by requiring gauge
invariance of the full action.

###  B.1 Line operators and fusion rules

The gauge invariant line operators of the theory are as follows. We first have
the obvious invertible Wilson loops,

$\displaystyle U_{a}(M_{1})=e^{i\pi\oint_{M_{1}}a}~{},\hskip
28.45274ptU_{b}(M_{1})=e^{i\pi\oint_{M_{1}}b}~{},\hskip
28.45274ptU_{c}(M_{1})=e^{i\pi\oint_{M_{1}}c}~{},$ (B.3)

which together with their linear combinations yield eight invertible lines.
The magnetic operators are more interesting. Without the DW twist terms, the
naive magnetic lines $e^{i\pi\oint\widehat{a}}$, $e^{i\pi\oint\widehat{b}}$,
and $e^{i\pi\oint\widehat{c}}$ are gauge invariant. However, in the presence
of the DW twist terms these lines are not gauge invariant. To achieve gauge
invariance, one needs to couple them to appropriate 1d TQFTs. The gauge
invariant operators are as follows,

$\displaystyle\begin{split}\widehat{U}_{a}(M_{1})&\sim\sum_{\phi_{2},\phi_{3}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}\widehat{a}+i\pi\oint_{M_{1}}(-\phi_{2}c+\phi_{3}b+\phi_{2}\delta\phi_{3})}~{},\\\
\widehat{U}_{b}(M_{1})&\sim\sum_{\phi_{3},\phi_{1}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}\widehat{b}+i\pi\oint_{M_{1}}(-\phi_{3}a+\phi_{1}c+\phi_{3}\delta\phi_{1})}~{},\\\
\widehat{U}_{c}(M_{1})&\sim\sum_{\phi_{1},\phi_{2}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}\widehat{c}+i\pi\oint_{M_{1}}(-\phi_{1}b+\phi_{2}a+\phi_{1}\delta\phi_{2})}~{},\\\
\widehat{U}_{ab}(M_{1})&\sim\sum_{\phi_{12},\phi_{3}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}(\widehat{a}+\widehat{b})+i\pi\oint_{M_{1}}(-\phi_{3}(a-b)+\phi_{12}c+\phi_{3}\delta\phi_{12})}~{},\\\
\widehat{U}_{bc}(M_{1})&\sim\sum_{\phi_{23},\phi_{1}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}(\widehat{b}+\widehat{c})+i\pi\oint_{M_{1}}(-\phi_{1}(b-c)+\phi_{23}a+\phi_{1}\delta\phi_{23})}~{},\\\
\widehat{U}_{ca}(M_{1})&\sim\sum_{\phi_{31},\phi_{2}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}(\widehat{c}+\widehat{a})+i\pi\oint_{M_{1}}(-\phi_{2}(c-a)+\phi_{31}b+\phi_{2}\delta\phi_{31})}~{},\\\
\widehat{U}_{abc}(M_{1})&\sim\sum_{\phi_{12},\phi_{23}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}(\widehat{a}+\widehat{b}+\widehat{c})+i\pi\oint_{M_{1}}(-\phi_{12}(b-c)+\phi_{23}(a-b)+\phi_{12}\delta\phi_{23})}~{},\end{split}$
(B.4)

where the gauge transformations of the scalars on the lines are given by

$\displaystyle\phi_{i}\to\phi_{i}+\alpha_{i}~{},\hskip
28.45274pt\phi_{ij}\to\phi_{ij}+\alpha_{i}-\alpha_{j}~{},\hskip
28.45274pti,j=1,2,3~{}$ (B.5)

and $\alpha_{1,2,3}=\alpha,\beta,\gamma$ respectively.131313The last operator
can be naively written as
$\widehat{U}_{abc}(M_{1})\sim\sum_{\phi_{1},\phi_{2},\phi_{3}\in
C^{0}(M_{1},\mathbb{Z}_{2})}e^{i\pi\oint_{M_{1}}(\widehat{a}+\widehat{b}+\widehat{c})+i\pi\oint_{M_{1}}(\phi_{2}-\phi_{3})a+(\phi_{3}-\phi_{1})b+(\phi_{1}-\phi_{2})c+\phi_{1}\delta\phi_{2}+\phi_{2}\delta\phi_{3}+\phi_{3}\delta\phi_{1}}~{}.$
One can then define $\phi_{ij}=\phi_{i}-\phi_{j}$. Note that $\phi_{ij}$ are
not completely independent—they are required to sum to zero. By using
$\phi_{2}\delta\phi_{2}=0$, it is possible to rewrite the last three terms
purely in terms of $\phi_{ij}$’s
$\phi_{12}\delta\phi_{23}=\phi_{1}\delta\phi_{2}+\phi_{2}\delta\phi_{3}+\phi_{3}\delta\phi_{1}~{}.$

The magnetic lines in (B.4) can also fuse with the invertible lines (B.3) to
give new lines. For instance, fusing $U_{b}$ with $\widehat{U}_{a}$ amounts to
a global shift $\phi_{3}\to\phi_{3}+1$ in the sum. Since $\phi_{3}$ is summed
over, the result is simply $\widehat{U}_{a}$ itself. This shows the fusion
rule $\widehat{U}_{a}\times U_{b}=\widehat{U}_{a}$. By the same reasoning, one
can also derive $\widehat{U}_{a}\times U_{c}=\widehat{U}_{a}$. However, the
above argument does not apply to fusing $U_{a}$ with $\widehat{U}_{a}$, and
instead $U_{a}\widehat{U}_{a}$ is a new operator. Likewise there are 7 new
operators involving the magnetic operators,

$\displaystyle\begin{split}&\widehat{U}_{a}U_{a}~{},\hskip
28.45274pt\widehat{U}_{b}U_{b}~{},\hskip
28.45274pt\widehat{U}_{c}U_{c}~{},\hskip
28.45274pt\widehat{U}_{ab}U_{a}=\widehat{U}_{ab}U_{b}~{},\hskip
28.45274pt\widehat{U}_{bc}U_{b}=\widehat{U}_{bc}U_{c}~{},\\\
&\widehat{U}_{ca}U_{c}=\widehat{U}_{ca}U_{a}~{},\hskip
28.45274pt\widehat{U}_{abc}U_{a}=\widehat{U}_{abc}U_{b}=\widehat{U}_{abc}U_{c}~{}.\end{split}$
(B.6)

Together with the 8 invertible lines in (B.3), there are a total of $8+7\times
2=22$ line operators in the theory. The fact that there are 22 lines in a
closely related theory was already known in [104, 118, 103].

We finally consider the fusion rules between the magnetic line operators. Let
us first compute the fusion
$\widehat{U}_{a}\times\widehat{U}_{a}$,141414Further specifying the overall
normalization of $\widehat{U}_{a}$ to be $1/|C^{0}(M_{1},\mathbb{Z}_{2})|$, we
actually find the fusion rule
$\widehat{U}_{a}\times\widehat{U}_{a}=1+U_{b}+U_{c}+U_{bc}$, i.e. the first
and last expression are actually equal.

$\displaystyle\begin{split}\widehat{U}_{a}\times\widehat{U}_{a}&\sim\sum_{\begin{subarray}{c}\phi_{2},\phi_{3},\widetilde{\phi}_{2},\widetilde{\phi}_{3}\\\
\in
C^{0}(M_{1},\mathbb{Z}_{2})\end{subarray}}e^{i\pi\oint_{M_{1}}(-\phi_{2}c+\phi_{3}b+\phi_{2}\delta\phi_{3}-\widetilde{\phi}_{2}c+\widetilde{\phi}_{3}b+\widetilde{\phi}_{2}\delta\widetilde{\phi}_{3})}\\\
&\sim\sum_{\begin{subarray}{c}\phi_{2}^{\prime},\phi_{3}^{\prime},\widetilde{\phi}_{2},\widetilde{\phi}_{3}\\\
\in
C^{0}(M_{1},\mathbb{Z}_{2})\end{subarray}}e^{i\pi\oint_{M_{1}}(-\phi_{2}^{\prime}c+\phi_{3}^{\prime}b+\phi_{2}^{\prime}\delta\phi_{3}^{\prime}+\widetilde{\phi}_{2}\delta\phi_{3}^{\prime}+\phi_{2}^{\prime}\widetilde{\phi}_{3})}\\\
&\sim\sum_{\begin{subarray}{c}\phi_{2}^{\prime},\phi_{3}^{\prime}\in
Z^{0}(M_{1},\mathbb{Z}_{2})\end{subarray}}e^{i\pi\oint_{M_{1}}(-\phi_{2}^{\prime}c+\phi_{3}^{\prime}b)}\sim
1+U_{b}+U_{c}+U_{bc}~{}.\\\ \end{split}$ (B.7)

In the first equality, we used the fact that $\widehat{a}$ is a
$\mathbb{Z}_{2}$ cochain, and hence that the two factors
$e^{i\pi\oint\widehat{a}}$ cancel against each other. Only the contributions
from the attached 1d TQFT survive. Note that since the $\phi_{i}$ only live on
the individual lines, we should distinguish the scalars for each line
separately. In the second equality, we introduced
$\phi_{2}^{\prime}:=\phi_{2}-\widetilde{\phi}_{2},\phi_{3}^{\prime}:=\phi_{3}-\widetilde{\phi}_{3}$,
and as a consequence the fields $\widetilde{\phi}_{2,3}$ became Lagrangian
multiplers. In the third equality we integrated over $\widetilde{\phi}_{2,3}$,
which enforced that $\phi_{2,3}^{\prime}$ were $\mathbb{Z}_{2}$-valued
0-cocycles. In other words, they are constants on $M_{1}$. Finally, summing
over $\phi_{2,3}^{\prime}\in Z^{0}(M_{1},\mathbb{Z}_{2})$ simply amounts to
summing over $\phi_{2,3}^{\prime}\in\\{0,1\\}$, and we obtain four terms as
shown above. As a consistency check, one can stack $U_{b}$ on both sides, and
ones finds that both sides are indeed invariant.

Similar manipulations can be used to find the other fusion rules. Below, we
enumerate the fusion rules up to permutations of $a,b,c$,

$\displaystyle\begin{split}\widehat{U}_{a}\times\widehat{U}_{a}&\sim
1+U_{b}+U_{c}+U_{bc}~{},\\\
\widehat{U}_{a}\times\widehat{U}_{b}&\sim\widehat{U}_{ab}+\widehat{U}_{ab}U_{a}\sim\widehat{U}_{ab}+\widehat{U}_{ab}U_{b}~{},\\\
\widehat{U}_{a}\times\widehat{U}_{ab}&\sim\widehat{U}_{b}+\widehat{U}_{b}U_{b}~{},\\\
\widehat{U}_{c}\times\widehat{U}_{ab}&\sim\widehat{U}_{abc}+\widehat{U}_{abc}U_{a}\sim\widehat{U}_{abc}+\widehat{U}_{abc}U_{b}\sim\widehat{U}_{abc}+\widehat{U}_{abc}U_{c}~{}.\\\
\end{split}$ (B.8)

Specifying the normalization, one finds that all $\sim$ can be replaced by
$=$. We thus find that all magnetic operators
$\widehat{U}_{i},\widehat{U}_{ij},\widehat{U}_{ijk}$ have quantum dimension 2
and are non-invertible.

###  B.2 Linking numbers between lines

We now compute the linking numbers from the correlation functions of the lines
discussed above. A useful simplifying assumption is to take all the lines to
be unknotted circles $S^{1}$, and to take the spacetime to be of trivial
topology $X_{3}=S^{3}$. This allows us to avoid issues of loops wrapping
spacetime cycles; indeed, if a loop were to wrap a non-contractable spacetime
cycle, this could lead to additional phase factors and contaminate the linking
numbers.

The non-trivial linking numbers can be organized into the following classes:

  1. 1.

Linking number between an electric and magnetic line,

$\displaystyle\braket{U_{i}(M_{1})\widehat{U}_{j}(M^{\prime}_{1})}\sim(-1)^{\text{Link}(M_{1},M^{\prime}_{1})}\delta_{ij}$
(B.9)

for $i,j\in\\{a,b,c\\}$. For instance, $M_{1},M_{1}^{\prime}$ forming a Hopf
link can be used to probe this linking number. This linking invariant is a
consequence of the BF couplings in (B.1).

  2. 2.

Linking number between two magnetic lines,

$\displaystyle\braket{\widehat{U}_{i}(M_{1})\widehat{U}_{j}(M^{\prime}_{1})}\sim\begin{cases}(-1)^{\text{Link}(M_{1},M^{\prime}_{1})}&i=j=a\\\
1&\text{otherwise}\end{cases}$ (B.10)

Once again, $M_{1},M_{1}^{\prime}$ forming a Hopf link can be used to probe
this linking number. This is a consequence of the DW twist $\frac{1}{2}a\delta
a$ in (B.1).

  3. 3.

Linking number of type 0 between three magnetic lines,

$\displaystyle\braket{\widehat{U}_{a}(M_{1})\widehat{U}_{b}(M^{\prime}_{1})\widehat{U}_{c}(M^{\prime\prime}_{1})}\sim(-1)^{\text{Link}(M_{1},M^{\prime}_{1},M^{\prime\prime}_{1})_{0}}~{}.$
(B.11)

The lines $M_{1},M_{1}^{\prime},M_{1}^{\prime\prime}$ forming the Borromean
rings can be used to probe this linking number. This is a consequence of the
DW twist $abc$ in (B.1).

All other linking invariants are either trivial or combinations of the above
ones. For example, the Hopf linkings between electric lines
$\braket{{U}_{i}(M_{1}){U}_{j}(M^{\prime}_{1})}$ are all trivial.

The computations are straightforward. Below we derive (B.11) in detail; all
other invariants can be likewise derived. We first compute the partition
function without any line insertions.

$\displaystyle\begin{split}Z(S^{3})&\sim\sum_{\begin{subarray}{c}a,b,c,\widehat{a},\widehat{b},\widehat{c}\\\
\in
C^{1}(S^{3},\mathbb{Z}_{2})\end{subarray}}e^{i\int\pi\left(\widehat{a}\delta
a+\widehat{b}\delta b+\widehat{c}\delta c+abc+\frac{1}{2}a\beta
a\right)}\sim\sum_{\begin{subarray}{c}a,b,c\\\ \in
Z^{1}(S^{3},\mathbb{Z}_{2})\end{subarray}}e^{i\int\pi\left(abc+\frac{1}{2}a\beta
a\right)}\sim 1~{}.\end{split}$ (B.12)

Note that we suppressed all the real positive normalization constants. In the
second $\sim$ we summed over $\widehat{a},\widehat{b},\widehat{c}$, which
enforced $a,b,c$ to be cocycles. In the third $\sim$ we used the fact that the
spacetime is $S^{3}$, and hence that all the flat connections on $S^{3}$ are
gauge equivalent to the trivial connection.

We then compute the correlation function
$\braket{\widehat{U}_{a}(M_{1})\widehat{U}_{b}(M^{\prime}_{1})\widehat{U}_{c}(M^{\prime\prime}_{1})}$,

$\begin{split}&\braket{\widehat{U}_{a}(M_{1})\widehat{U}_{b}(M_{1}^{\prime})\widehat{U}_{c}(M_{1}^{\prime\prime})}=\frac{1}{Z(S^{3})}Z(S^{3},\widehat{U}_{a}(M_{1}),\widehat{U}_{b}(M^{\prime}_{1}),\widehat{U}_{c}(M^{\prime\prime}_{1}))\\\
&\qquad\sim\sum_{\begin{subarray}{c}a,b,c,\widehat{a},\widehat{b},\widehat{c}\\\
\in
C^{1}(S^{3},\mathbb{Z}_{2})\end{subarray}}\sum_{\begin{subarray}{c}\phi_{2},\phi_{3}\\\
\in
C^{0}(M_{1},\mathbb{Z}_{2})\end{subarray}}\sum_{\begin{subarray}{c}\phi^{\prime}_{2},\phi^{\prime}_{3}\\\
\in
C^{0}(M_{1}^{\prime},\mathbb{Z}_{2})\end{subarray}}\sum_{\begin{subarray}{c}\phi^{\prime\prime}_{2},\phi^{\prime\prime}_{3}\\\
\in
C^{0}(M_{1}^{\prime\prime},\mathbb{Z}_{2})\end{subarray}}e^{i\int_{S^{3}}\pi\left(\widehat{a}\delta
a+\widehat{b}\delta b+\widehat{c}\delta c+abc+\frac{1}{2}a\delta
a\right)}e^{i\pi\oint_{M_{1}}\widehat{a}+i\pi\oint_{M_{1}}(-\phi_{2}c+\phi_{3}b+\phi_{2}\delta\phi_{3})}\\\
&\hskip 42.67912pt\times
e^{i\pi\oint_{M_{1}^{\prime}}\widehat{b}+i\pi\oint_{M_{1}^{\prime}}(-\phi^{\prime}_{3}a+\phi^{\prime}_{1}c+\phi^{\prime}_{3}\delta\phi^{\prime}_{1})}e^{i\pi\oint_{M_{1}^{\prime\prime}}\widehat{c}+i\pi\oint_{M_{1}^{\prime\prime}}(-\phi^{\prime\prime}_{1}b+\phi^{\prime\prime}_{2}a+\phi^{\prime\prime}_{1}\delta\phi^{\prime\prime}_{2})}~{}.\end{split}$
(B.13)

Summing over $\widehat{a},\widehat{b},\widehat{c}$ enforces that $\delta
a=-\delta^{\perp}(M_{1})$, $\delta b=-\delta^{\perp}(M_{1}^{\prime})$, and
$\delta c=-\delta^{\perp}(M_{1}^{\prime\prime})$. These also imply that up to
gauge transformation $a=-\delta^{\perp}(M_{2})$,
$b=-\delta^{\perp}(M_{2}^{\prime})$, and
$c=-\delta^{\perp}(M_{2}^{\prime\prime})$, where $M_{1}=\partial M_{2}$,
$M_{1}^{\prime}=\partial M_{2}^{\prime}$, and $M_{1}^{\prime\prime}=\partial
M_{2}^{\prime\prime}$. The correlation function then simplifies to

$\begin{split}&\braket{\widehat{U}_{a}(M_{1})\widehat{U}_{b}(M_{1}^{\prime})\widehat{U}_{c}(M_{1}^{\prime\prime})}\\\
&\qquad\sim\sum_{\begin{subarray}{c}a,b,c\\\ \in
C^{1}(S^{3},\mathbb{Z}_{2})\end{subarray}}\sum_{\begin{subarray}{c}\phi_{2},\phi_{3}\\\
\in
C^{0}(M_{1},\mathbb{Z}_{2})\end{subarray}}\sum_{\begin{subarray}{c}\phi_{3}^{\prime},\phi_{1}^{\prime}\\\
\in
C^{0}(M_{1}^{\prime},\mathbb{Z}_{2})\end{subarray}}\sum_{\begin{subarray}{c}\phi_{1}^{\prime\prime},\phi_{2}^{\prime\prime}\\\
\in
C^{0}(M_{1}^{\prime\prime},\mathbb{Z}_{2})\end{subarray}}e^{i\int_{S^{3}}\pi\left(abc+\frac{1}{2}a\delta
a\right)}e^{i\pi\oint_{M_{1}}(-\phi_{2}c+\phi_{3}b+\phi_{2}\delta\phi_{3})}\\\
&\hskip 42.67912pt\times
e^{i\pi\oint_{M_{1}^{\prime}}(-\phi^{\prime}_{3}a+\phi^{\prime}_{1}c+\phi^{\prime}_{3}\delta\phi^{\prime}_{1})}e^{i\pi\oint_{M_{1}^{\prime\prime}}(-\phi^{\prime\prime}_{1}b+\phi^{\prime\prime}_{2}a+\phi^{\prime\prime}_{1}\delta\phi^{\prime\prime}_{2})}|_{a=-\delta^{\perp}(M_{2}),b=-\delta^{\perp}(M_{2}^{\prime}),c=-\delta^{\perp}(M_{2}^{\prime\prime})}\\\
&\qquad\sim\sum_{\begin{subarray}{c}\phi_{2},\phi_{3}\\\ \in
C^{0}(M_{1},\mathbb{Z}_{2})\end{subarray}}\sum_{\begin{subarray}{c}\phi_{3}^{\prime},\phi_{1}^{\prime}\\\
\in
C^{0}(M_{1}^{\prime},\mathbb{Z}_{2})\end{subarray}}\sum_{\begin{subarray}{c}\phi_{1}^{\prime\prime},\phi_{2}^{\prime\prime}\\\
\in
C^{0}(M_{1}^{\prime\prime},\mathbb{Z}_{2})\end{subarray}}e^{i\int_{S^{3}}\pi\delta^{\perp}(M_{2}\cap
M_{2}^{\prime}\cap M_{2}^{\prime\prime})}\\\ &\hskip 42.67912pt\times
e^{i\pi\int_{S^{3}}\phi_{2}\delta^{\perp}(M_{1}\cap
M_{2}^{\prime\prime})-\phi_{3}\delta^{\perp}(M_{1}\cap
M_{2}^{\prime})+\phi_{2}\delta\phi_{3}\delta^{\perp}(M_{1})}e^{i\pi\int_{S^{3}}\phi_{3}^{\prime}\delta^{\perp}(M_{1}^{\prime}\cap
M_{2})-\phi_{1}^{\prime}\delta^{\perp}(M_{1}^{\prime}\cap
M_{2}^{\prime\prime})+\phi_{3}^{\prime}\delta\phi_{1}^{\prime}\delta^{\perp}(M_{1}^{\prime})}\\\
&\hskip 42.67912pt\times
e^{i\pi\int_{S^{3}}\phi_{1}^{\prime\prime}\delta^{\perp}(M_{1}^{\prime\prime}\cap
M_{2}^{\prime})-\phi_{2}^{\prime\prime}\delta^{\perp}(M_{1}^{\prime\prime}\cap
M_{2})+\phi_{1}^{\prime\prime}\delta\phi_{2}^{\prime\prime}\delta^{\perp}(M_{1}^{\prime\prime})}~{}.\end{split}$
(B.14)

We now simplify the sum over $\phi_{2},\phi_{3}$. The relevant portion is

$\begin{split}&\sum_{\begin{subarray}{c}\phi_{2},\phi_{3}\in
C^{0}(M_{1},\mathbb{Z}_{2})\end{subarray}}e^{i\pi\int_{S^{3}}\phi_{2}\delta^{\perp}(M_{1}\cap
M_{2}^{\prime\prime})-\phi_{3}\delta^{\perp}(M_{1}\cap
M_{2}^{\prime})+\phi_{2}\delta\phi_{3}\delta^{\perp}(M_{1})}\\\ &\hskip
50.58878pt=\sum_{\begin{subarray}{c}\phi_{3}\in
C^{0}(M_{1},\mathbb{Z}_{2})\end{subarray}}e^{i\pi\int_{S^{3}}\phi_{3}\delta^{\perp}(M_{1}\cap
M_{2}^{\prime})}|_{\delta\phi_{3}\delta^{\perp}(M_{1})=\delta^{\perp}(M_{1}\cap
M_{2}^{\prime\prime})}~{}.\end{split}$ (B.15)

The constraint means $\delta\phi_{3}=\delta^{\perp}(M_{2}^{\prime\prime})$.
However, since
$\delta\delta^{\perp}(M_{2}^{\prime\prime})=\delta^{\perp}(M_{1}^{\prime\prime})$,
$\delta^{\perp}(M_{2}^{\prime\prime})$ itself can not be an exact form. Hence
the constraint is solvable only when $M_{1}$ and $M_{1}^{\prime\prime}$ do not
form a Hopf link, i.e. $\delta^{\perp}(M_{1}\cap M_{2}^{\prime\prime})=0\mod
2$. Hence $\delta\phi_{3}=0$, i.e. $\phi_{3}=0,1$ is a constant over $M_{1}$.
Further summing over $\phi_{3}$ constrains $\delta^{\perp}(M_{1}\cap
M_{2}^{\prime})=0\mod 2$, so we need both $M_{1}^{\prime}$ and
$M_{1}^{\prime\prime}$ to have trivial linking number with $M_{1}$ in order to
obtain a non-vanishing correlation function. Further summing over
$\phi_{2}^{\prime},\phi_{3}^{\prime}$ and
$\phi_{2}^{\prime\prime},\phi_{3}^{\prime\prime}$, we find that the
correlation function is non-vanishing only when $M_{1},M_{1}^{\prime}$, and
$M_{1}^{\prime\prime}$ have trivial linking number between any pair, and in
particular they are not mutually Hopf linked. Moreover, the phase of the