$-2\pi\frac{(N-1)\ell^{2}}{2N}\int_{5d}{\mathcal{P}}\left(\frac{dA}{N}\right)\cup
A~{}.$ (5.30)

Written in terms of $A^{\prime}=\frac{2\pi}{2N}A$, this is

$-2\pi\frac{(N-1)\ell^{2}}{2N}\int_{5d}\left(\frac{dA^{\prime}}{\pi}\right)^{2}\frac{2N}{2\pi}A^{\prime}=\frac{(-N+1)\ell^{2}}{\pi^{2}}\int_{5d}A^{\prime}dA^{\prime}dA^{\prime}~{}.$
(5.31)

Thus changing the fractionalization classes reproduces the jump of the anomaly
as in (5.21).

###  5.4 Domain walls

In this section we make a few comments about the matching of the
$\mathbb{Z}_{2n}$ anomaly by the low-energy effective theory. We consider
$\mathcal{N}=1$ super-Yang-Mills, that is, adjoint QCD with some gauge group
$G_{\text{gauge}}$. In this case, the axial symmetry is $\mathbb{Z}_{2h}$,
where $h=T(\text{adj})$ is the Coxeter number of $G_{\text{gauge}}$.

It is a well-known fact that $3+1d$ $\mathcal{N}=1$ super-Yang-Mills with
simply-connected gauge group is gapped and confining, and that the infrared
consists of $h$ discrete vacua as a result of the spontaneous breaking of the
$\mathbb{Z}_{2h}$ symmetry down to $\mathbb{Z}_{2}^{F}$. These vacua are
separated by domain walls, which carry non-trivial $2+1d$ degrees of freedom.
Specifically, the wall separating $p\in\mathbb{Z}_{h}$ units of vacua supports
a $2+1d$ $\mathcal{N}=1$ super-Yang-Mills theory with the same gauge group and
a Chern-Simons level $k=h/2-p$ [106, 92]. It is these degrees of freedom what
matches the $\mathbb{Z}_{2h}$ anomaly as computed in the ultraviolet.

For simplicity, here we will restrict ourselves to the parity-symmetric wall
$p=h/2$ that exists when $h$ is even. This wall is special since the CPT wall
construction [66] implies that the unitary axial symmetry in the bulk is
transmuted to an anti-unitary symmetry on the wall. In this case, it is very
easy to show that the ultraviolet anomaly is matched by the $3d$ degrees of
freedom.

The time-reversal symmetric wall corresponds to the $\mathbb{Z}_{2}$ subgroup
of $\mathbb{Z}_{h}$, i.e., to a spontaneously broken $g^{2}=(-1)^{F}$
symmetry. In the bulk, anomalies for such unitary symmetry are classified by
$\mathbb{Z}_{a}\times\mathbb{Z}_{b}$, where $a,b$ are given by (5.4) at $n=2$,
namely $a=16$ and $b=1$. In other words, the anomalies take values in
$\mathbb{Z}_{16}$. On the wall, this a time-reversal symmetry, whose anomalies
are also classified by $\mathbb{Z}_{16}$. It is easy to show that these
anomalies indeed match.

The $\mathbb{Z}_{16}$ anomaly in the bulk is given by (5.6) with $n=2$, to
wit,

$\nu_{a}=\frac{11q_{3}-5q_{1}}{6}\mod 16\,.$ (5.32)

If we let $n_{\pm}$ be the number of fermions that transform with sign $\pm
1$, then $q_{k}=n_{+}-n_{-}$, and the anomaly simplifies to

$\nu_{a}=n_{+}-n_{-}\mod 16\,,$ (5.33)

which precisely matches the time-reversal anomaly $\nu_{\mathsf{T}}$ of the
wall, namely the time-reversal anomaly of $3d$ adjoint QCD.

The non-parity-symmetric walls require a more careful study that we leave to
future work. In this case, non-invertible topological defects are potentially
important, as well as junctions of domain walls. See [107, 108, 70, 109, 110,
111, 112, 113, 114, 115, 116] for related work in various dimensions.

## Acknowledgement

We thank M. Cheng, J. Davighi, C.-M. Jian, N. Seiberg, T. Senthil and M. Yu
for discussions. We thank Maissam Barkeshli, Anton Kapustin, Nathan Seiberg,
Shu-Heng Shao, Yuya Tanizaki, Juven Wang and Cenke Xu for comments on a draft.
P.-S. H. is supported by the Simons Collaboration on Global Categorical
Symmetries. D.D. and J.G. acknowledge support by Perimeter Institute for
Theoretical Physics. Research at Perimeter Institute is supported in part by
the Government of Canada through the Department of Innovation, Science and
Economic Development Canada and by the Province of Ontario through the
Ministry of Colleges and Universities. Any opinions, findings, and conclusions
or recommendations expressed in this material are those of the authors and do
not necessarily reflect the views of the funding agencies.

##  Appendix A Congruence Vector of Simple Lie Algebras

In this appendix we quote the value of $\rho\cdot v\mod|Z(G_{\text{gauge}})|$
for the classical Lie groups, where $\rho=(1,1,\dots,1)$ is the Weyl vector.

##### Unitary group $\boldsymbol{SU(N)}$.

The congruence vector is $v=(1,2,\dots,N-1)$. Thus,

$\rho\cdot v=\frac{1}{2}N(N-1)\mod N$ (A.1)

which is nonzero when $N$ is even.

##### Symplectic group $\boldsymbol{Sp(N)}$.

The congruence vector is $v=(1,2,\dots,N)$. Thus,

$\rho\cdot v=\frac{1}{2}N(N+1)\mod 2$ (A.2)

which is nonzero when $N=1,2\mod 4$.

##### Orthogonal group $\boldsymbol{SO(N)}$.

The congruence vectors depend on $N\mod 4$, which we consider in turn.

$\bullet$ In $SO(2n+1)$, the congruence vector is $v=(0,0,\dots,1)$. Thus,

$\rho\cdot v=1\mod 2$ (A.3)

which is always nonzero.

$\bullet$ In $SO(4n+2)$, the congruence vector is
$v=(2,4,6,\dots,-2n+1,-2n-1)$. Thus,

$\rho\cdot v=-2n\mod 4$ (A.4)

which is nonzero when $n$ is odd.

$\bullet$ In $SO(4n)$, the congruence vectors are $v_{1}=(0,0,\dots,1,1)$ and
$v_{2}=(1,2,3,\dots,1-n,-n)$. Thus,

$\displaystyle\rho\cdot v_{1}$ $\displaystyle=0\mod 2$ (A.5)
$\displaystyle\rho\cdot v_{2}$ $\displaystyle=-n\mod 2$

and the latter is nonzero when $n$ is odd.

These results are summarized in table (2.15).

##  Appendix B One-Form Symmetry Anomaly in QCD0+1

Consider QCD in $0+1d$ with simple gauge group $G_{\text{gauge}}$ and a
fermion in some representation $R$. The one-form symmetry $\Gamma$ is the
subgroup of the center of gauge group that acts trivially on the
representation $R$. We would like to compute the ’t Hooft anomaly of the one-
form symmetry.

Since the background for the one-form symmetry changes the $G_{\text{gauge}}$
gauge field into $G_{\text{gauge}}/\Gamma$ gauge field, to address the
question we can compute the partition function of the free fermion theory in
the presence of a background gauge field for $G_{\text{gauge}}/\Gamma$. While
the partition function with background $G_{\text{gauge}}$ is well-defined (and
thus the QCD0+1 with the corresponding gauge group is free of gauge
anomalies), the partition function with background $G_{\text{gauge}}/\Gamma$
can have ambiguities, which is the anomaly of the one-form symmetry $\Gamma$
in QCD0+1 with gauge group $G_{\text{gauge}}$. Such anomaly arises if and only
if the symmetry $G_{\text{gauge}}/\Gamma$ is realized projectively on the
Hilbert space of the free fermions.

Consider the embedding of $\mathfrak{g}_{\text{gauge}}$ inside
$\mathfrak{so}(\dim(R))$ via the representation $R$. This embedding is defined
by the branching rule

$\displaystyle\mathfrak{so}(\dim(R))$ $\displaystyle\supseteq\
\mathfrak{g}_{\text{gauge}}$ (B.1) $\displaystyle\ydiagram 1$
$\displaystyle\mapsto R\,.$

Let us first turn on a holonomy $U$ of the background gauge field that takes
value in $Spin(\dim(R))$. Since the Hilbert space realizes a spinor
representation $s$ of $Spin(\dim(R))$, the partition function can be expressed
in terms of the character of such spinor representation:

$\displaystyle Z_{\pm}(U)$
$\displaystyle=\chi_{s}(U)\pm\chi_{\bar{s}}(U),\qquad$
$\displaystyle\dim(R)\text{ even}$ (B.2) $\displaystyle Z_{+}(U)$
$\displaystyle=\sqrt{2}\chi_{s}(U),\qquad$ $\displaystyle\dim(R)\text{ odd}$
$\displaystyle Z_{-}(U)$ $\displaystyle=0,\qquad$ $\displaystyle\dim(R)\text{
odd}\,.$

where $\pm$ denotes the spin structure (Ramond or Neveu-Schwarz), and $s$ the
spinor representation of $Spin(\dim(R))$ (and $\bar{s}$ the conjugate
representation). When we restrict the holonomy $U$ to take value in
$G_{\text{gauge}}$, the partition function is

$\displaystyle Z_{\pm}(U)$
$\displaystyle=\chi_{s(R)}(U)\pm\chi_{\bar{s}(R)}(U),\qquad$
$\displaystyle\dim(R)\text{ even}$ (B.3) $\displaystyle Z_{+}(U)$
$\displaystyle=\sqrt{2}\chi_{s(R)}(U),\qquad$ $\displaystyle\dim(R)\text{
odd}$ $\displaystyle Z_{-}(U)$ $\displaystyle=0,\qquad$
$\displaystyle\dim(R)\text{ odd}\,.$

where $s(R)\in\operatorname{Rep}(G_{\text{gauge}})$ is the image of
$s\in\operatorname{Rep}(Spin(\dim(R)))$ under the embedding (B.1),

$\displaystyle\mathfrak{so}(\dim(R))$ $\displaystyle\supseteq\
\mathfrak{g}_{\text{gauge}}$ (B.4) $\displaystyle s$ $\displaystyle\mapsto
s(R)\,.$

(In general, $s(R)$ is a reducible representation even if $R$ is irreducible.)
The partition function without insertion transforms under the center $\Gamma$
according to the transformation of $s(R)$. Thus in the $G_{\text{gauge}}$
gauge theory, the one-form symmetry has anomaly if and only if $s(R)$
transforms non-trivially under $\Gamma$ in the center of $G_{\text{gauge}}$.
The anomaly can be described by the bulk term

$\int_{2d}\langle q(s(R)),B\rangle~{},$ (B.5)

where $q(s(R))\in\text{Hom}(\Gamma,U(1))$ is the one-form charge of $s(R)$
under $\Gamma$ in the center of $G_{\text{gauge}}$, and $B$ is the background
two-form gauge field for the $\Gamma$ one-form symmetry.

As an application of this result, let us show that fermions in complex
representations do not contribute to the anomaly of the one-form symmetry.
When $R$ is complex, the branching rules are

$\displaystyle s(R)$
$\displaystyle\mapsto\bigoplus_{j=0}^{\lfloor\dim(R)/2\rfloor}\wedge^{2j}R$
(B.6) $\displaystyle\bar{s}(R)$
$\displaystyle\mapsto\bigoplus_{j=0}^{\lceil\dim(R)/2\rceil-1}\wedge^{2j+1}R\,.$

Of course, $\wedge^{n}R$ is not charged under $\operatorname{ker}(R)$ for any
$n$, and therefore complex representations never lead to an anomaly for one-
form symmetry.

Anomalies are only present in the case of real representations. As a simple
illustration, consider adjoint QCD. Here the embedding becomes
$\mathfrak{so}(\dim G_{\text{gauge}})\supset\mathfrak{g}_{\text{gauge}}$, and
the spinor representation branches as

$s\mapsto 2^{\lceil r/2\rceil-1}\rho\,,$ (B.7)

where $\rho$ is the Weyl vector of $\mathfrak{g}_{\text{gauge}}$ and
$r=\operatorname{rank}(\mathfrak{g}_{\text{gauge}})$. Therefore, the one-form
symmetry of adjoint QCD is anomalous if and only if $\rho$ is charged under
it, in agreement with section 2.1.

An equivalent way to phrase this discussion is in terms of how exactly
$G_{\text{gauge}}$ embeds into $Spin(\dim(R))$. We sketch the argument for
adjoint $SU(N)$ below.

###  B.1 $SU(N)$ gauge theory with an adjoint fermion

Consider $SU(N)$ gauge theory with an adjoint fermion in $0+1d$. We would like
to understand whether the $\mathbb{Z}_{N}$ one-form symmetry is anomalous.

This is equivalent to asking whether the $PSU(N)$ zero-form symmetry that acts
on $N^{2}-1$ free Majorana fermions is anomalous. In other words, whether the
Hilbert space of the free fermions transforms as projective representation of
$PSU(N)$ symmetry. When $N$ is odd, there is an even number of Majorana
fermions and a properly $\mathbb{Z}_{2}$-graded Hilbert space; when $N$ is
even, we add one extra singlet Majorana fermion to form a well-defined Hilbert
space.

The discussion divides into two cases: $N$ even and $N$ odd.

####  B.1.1 Even $N$

When $N$ is even, we add one additional Majorana fermion that is singlet under
$PSU(N)$ to obtain a well-defined Hilbert space. We quantize the fermions,
which form the Clifford algebra of rank $N^{2}$. The Hilbert space forms the
spinor representation of $Spin(N^{2})$ symmetry. The states generated by the
$N^{2}-1$ Majorana fermions form spinor of $Spin(N^{2}-1)$. We have the
embedding

$Spin(N^{2}-1)\subset SU(N)/\mathbb{Z}_{N/2}~{},$ (B.8)

where both sides have $\mathbb{Z}_{2}$ center. Thus the spinor representation
decomposes into $SU(N)$ representation with non-trivial $N$-ality, and thus
the $PSU(N)$ symmetry acts projectively on the Hilbert space. This means the
$PSU(N)$ symmetry is anomalous, with the anomaly

$\frac{2\pi(N/2)}{N}\int w_{2}^{PSU(N)}=\pi\int w_{2}^{PSU(N)}~{}.$ (B.9)

We note that this implies that the Hilbert space of $SU(N)$ gauge theory with
adjoint fermion is empty (since the vacuum carries non-trivial $SU(N)$
representation) unless considering the twisted Hilbert space with insertion of
Wilson line of $N$-ality $N/2$ in the time direction.

In the $SU(N)$ gauge theory with adjoint fermion, this implies the one-form
symmetry is anomalous. In terms of the one-form symmetry gauge field $B_{2}$,
the anomaly is

$\pi\int B_{2}~{}.$ (B.10)

####  B.1.2 Odd $N$

When $N$ is odd, $N^{2}-1$ is even, and there is a well-defined Hilbert space.
Let us quantize the fermions, which give Clifford algebra of rank $N^{2}-1$.
The Hilbert space forms the spinor representations of $Spin(N^{2}-1)$. Let us
study how the Hilbert space transforms under $SU(N)$ symmetry. Since $SU(N)$
for odd $N$ does not have a $\mathbb{Z}_{2}$ subgroup center, the spinor
representation of $Spin(N^{2}-1)$ decompose into $SU(N)$ representation with 0
$N$-ality, and thus we have the embedding

$Spin(N^{2}-1)\supset PSU(N)~{}.$ (B.11)

Thus the $PSU(N)$ symmetry acts linearly on the Hilbert space, and it does not
have an anomaly. Thus we conclude that when $N$ is odd, the $\mathbb{Z}_{N}$
one-form symmetry in $SU(N)$ gauge theory with one adjoint fermion is non-
anomalous.

##  Appendix C $U(1)$ Gauge Theory in 1+1d

In this appendix we use one-form symmetry to discuss the anomaly of
$\mathbb{Z}_{2}$ charge conjugation symmetry in $U(1)$ gauge theory in $1+1d$
for different fractionalization classes.

Consider the $U(1)$ gauge theory with $\theta=\pi$

$\frac{1}{2e^{2}}da\star da+\frac{1}{2}da~{}.$ (C.1)

The theory has $\Gamma=U(1)$ one-form symmetry that shifts $a$, and
$\mathbb{Z}_{2}$ charge conjugation symmetry that acts as $a\rightarrow-a$.

Let us compute the anomaly for the symmetries. We turn on background $B_{2}$
for the $U(1)$ one-form symmetry, and $B_{1}$ for the $\mathbb{Z}_{2}$ charge
conjugation symmetry. This replaces $da$ by

$D_{B_{1}}a-B_{2}=(d-2B_{1})a-B_{2}~{}.$ (C.2)

Thus the theta term contributes the bulk dependence

$\int\frac{1}{2}\left(2B_{1}da-dB_{2}\right)=\int
B_{1}B_{2}-\frac{1}{2}dB_{2}=-\frac{1}{2}\int D_{B_{1}}B_{2}~{}.$ (C.3)

This is a mixed anomaly between the one-form symmetry and charge conjugation
zero-form symmetry.

There are two fractionalization classes for the $\mathbb{Z}_{2}$ charge
conjugation 0-form symmetry, classified by
$H^{2}(B\mathbb{Z}_{2},\mathbb{Z}_{2})=\mathbb{Z}_{2}$:

  * •

$B_{2}=0$, then the zero-form symmetry is not anomalous.

  * •

$B_{2}=\pi B_{1}^{2}$, then the zero-form symmetry has an anomaly

$\pi\int B_{1}^{3}~{}.$ (C.4)

This agrees with the discussion in section 3.

##  Appendix D Fermionic $\mathbb{Z}_{2}$ Gauge Theory with $\mathbb{Z}_{2}$
Symmetry in 1+1d

Let us consider $\mathbb{Z}_{2}$ gauge theory in $1+1d$, viewed as a fermionic
theory, and enriched with unitary $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
symmetry, where $\mathbb{Z}_{2}^{F}$ is the fermion parity symmetry.

The action without any background gauge fields is

$\pi\int a_{0}\cup\delta b_{1}~{}.$ (D.1)

If we integrate out $b_{1}$, this sets $\delta a_{0}=0$ mod 2. Such condition
will be modified in the presence of background gauge fields.

The theory has $\mathbb{Z}_{2}$ one-form symmetry that acts on the Wilson
line, and different symmetry fractionalizations in the fermionic world admit a
classification using the spin bordism group

$\text{Hom}(\Omega^{spin}_{2}(B\mathbb{Z}_{2}),\mathbb{Z}_{2})~{}.$ (D.2)

The group is $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, generated by the Arf
invariant, and the quadratic function $\frac{1}{2}q_{\rho}(A)$, where we
denote the spin structure by $\rho$, and the $\mathbb{Z}_{2}$ background gauge
field by $A$. If we denote the symmetry fractionalization by
$(m,n)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, then the path integral is
modified to be

$Z=\int Da_{0}e^{\frac{2\pi
im}{8}\text{Arf}(\text{PD}_{2d}(a_{0}))}e^{n\frac{\pi
i}{2}\int_{\text{PD}_{2d}(\phi_{0})}q_{\rho}(A)}~{},$ (D.3)

where $\text{PD}_{2d}(\phi_{0})$ is the two-cycle that is Poincaré dual to
$\phi_{0}$. It is closed in $\mathbb{Z}_{2}$ homology, since
$\delta\phi_{0}=0$ mod 2, and thus the above partition function is well-
defined. We note that $\text{PD}_{2d}(\phi_{0})$ is orientable,434343 For a
$2d$ surface $D$ embedded in 2d orientable spacetime $M$, $TM|=TD+ND$, but
$ND=0$ since it has zero dimension, and thus $TM|=TD$, and for orientable
$TM$, $TD$ is also orientable. and thus $q_{\rho}$ is even, and Arf is a
multiple of $4$ in the above normalization, and the phases in the path
integral are signs. We can also include
$(-1)^{\ell\int_{\text{PD}(\phi_{0})}A\cup A}$ for $\ell\in\mathbb{Z}_{2}$,
which is trivial for $\delta\phi_{0}=0$.

The theory also has $\mathbb{Z}_{2}$ zero-form symmetry that acts on the
$\mathbb{Z}_{2}$ vortex point operator. Let us consider the $\mathbb{Z}_{2}$
unitary symmetry that acts on such operators. Then, in the presence of
background $A$, $\phi_{0}$ is no longer closed in $\mathbb{Z}_{2}$,

$\delta\phi_{0}=A~{},$ (D.4)

and $\text{PD}_{2d}(\phi_{0})$ has a boundary that is non-trivial in
$\mathbb{Z}_{2}$ homology

$\partial\text{PD}_{2d}(\phi_{0})=\text{PD}_{2d}(d\phi_{0})=\text{PD}_{2d}(A)~{}.$
(D.5)

In such a case, the partition function is no longer well-defined. This is a
manifestation of an anomaly for the $\mathbb{Z}_{2}$ symmetry. One way to cure
the problem is by introducing a $3d$ bulk, and attaching the boundary surface
to another surface in the bulk that anchors on the boundary along
$\text{PD}_{2d}(A)$, to form a closed surface, so the phases in the partition
function evaluated on the 2-cycle $\Sigma_{\text{2-cycle}}$ can be defined,
where

$\Sigma_{\text{2-cycle}}=\text{PD}_{2d}(\phi_{0})\cup\overline{\Sigma_{\text{bulk}}},\quad\partial\Sigma_{\text{bulk}}=\text{PD}_{2d}(A)~{}.$
(D.6)

For instance, if $2d$ spacetime has trivial topology and we take the bulk
direction to be the $z$ coordinate, then such a bulk surface can be a cigar
with base $\text{PD}_{2d}(A)$ and extends along the $z$ direction.

If we consider two bulks, the choice of bulk surface $\Sigma_{\text{bulk}}$
depends on $\text{PD}_{3d}(A)$, which is a surface in the bulk (note it may be
an unorientable surface embedded in an orientable bulk, if $A$ has a non-
trivial bundle in the bulk). When the bulk surface is unorientable, we need to
refine the Arf invariant into ABK invariant. Then the phases depend on the
bulk by

$e^{\frac{2\pi im}{8}\text{ABK}(\text{PD}_{3d}(A))}e^{n\frac{\pi
i}{2}\int_{\text{PD}_{3d}(A)}q_{\rho}(A)}(-1)^{\ell\int_{\text{PD}_{3d}(A)}A\cup
A}~{},$ (D.7)

where the first term is the partition function for the root phase of the
$\mathbb{Z}_{8}$ invertible phases in $2+1d$ with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry and zero thermal Hall
conductance. The second term can be written as an odd-level Chern-Simons term
for the gauge field $A$. The third term is the partition function for the
bosonic SPT phase with $\mathbb{Z}_{2}$ symmetry. The above decomposition
represents the Arf layer, $\psi$ layer and bosonic layer in [58]. In
particular, $m,n$ now have extended range, with the identification

$(m+2,n,\ell)\sim(m,n+1,\ell),\quad(m,n+2,\ell)\sim(m,n,\ell+1)~{}.$ (D.8)

##  Appendix E $\mathbb{Z}_{2}$ Gauge Theory with Two-group Symmetry in 2+1d

###  E.1 Intrinsic symmetries and their anomalies

Untwisted $\mathbb{Z}_{2}$ gauge theory in $2+1d$ has
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ one-form symmetry generated by the
electric and magnetic line operators. The theory also has unitary
$\mathbb{Z}_{2}^{\mathsf{C}}$ zero-form symmetry that exchanges the electric
and magnetic lines. Let us denote the background for zero-form symmetry by
$B_{1}$, the background for the diagonal $\mathbb{Z}_{2}$ one-form symmetry
generated by the dyon by $B_{2}$, and the background for the $\mathbb{Z}_{2}$
one-form symmetry generated by the Wilson line by $B_{2}^{\prime}$. They obey
the twisted cocycle condition

$dB_{2}=B_{1}\cup B_{2}^{\prime},\quad dB_{2}^{\prime}=0,\quad dB_{1}=0~{}.$
(E.1)

To understand this relation, we note that in the presence of backgrounds
$B_{2}^{\prime},B_{1}$, there must also be a non-trivial background for
$B_{2}$ due to the action of the zero-form symmetry on the one-form symmetry,
and this is captured by the first relation.

More generally, we can consider $Spin(4k)_{1}$ Chern-Simons theory for integer
$k$, which has $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ one-form symmetry
generated by the lines in the spinor and cospinor representation, and
$\mathbb{Z}_{2}$ charge conjugation 0-from symmetry that exchanges the two
lines. The Toric Code $\mathbb{Z}_{2}$ gauge theory corresponds to $k=0$.

In addition, the theory has $\mathbb{Z}_{2}^{\mathsf{T}}$ time-reversal that
does not permute the line operators. Let us turn on background field by
placing the theory on an unorientable manifold with first Stiefel-Whitney
class $w_{1}$. Then the twisted cocycle condition is modified to be [117]

$dB_{2}=B_{1}\cup\left(B_{2}^{\prime}+w_{1}\cup B_{1}\right)~{}.$ (E.2)

We note that under a $\mathbb{Z}_{2}$ one-form transformation for
$B_{2}^{\prime}$, $B_{2}^{\prime}\rightarrow B_{2}^{\prime}+d\lambda_{1}$, the
background $B_{2}$ is shifted by $B_{2}\rightarrow
B_{2}+B_{1}\cup\lambda_{1}$, where we used $dB_{1}=0$. When later we focus on
unitary $\mathbb{Z}_{2}^{\mathsf{C}}$ zero-form symmetry or time-reversal
symmetry that acts as charge conjugation, this implies that the non-trivial
fractionalization classes only come from shifting $B_{2}^{\prime}$ but not
$B_{2}$.

The zero-form and one-form symmetries have an anomaly, described by the bulk
term [117]

$\displaystyle S_{\text{bulk SPT}}$
$\displaystyle=\pi\int\mathcal{P}(B_{2})+\pi\int
B_{2}\cup(B_{2}^{\prime}+w_{1}\cup B_{1})+\pi\int w_{1}\cup B_{1}\cup B_{2}$
(E.3) $\displaystyle\quad+\pi\int
B_{1}\cup\left[(B_{1}\cup(B_{2}^{\prime}+w_{1}\cup
B_{1}))\cup_{2}(B_{2}^{\prime}+w_{1}\cup
B_{1})+B_{1}\cup(B_{2}^{\prime}+w_{1}\cup B_{1})\right]$ (E.4)
$\displaystyle\quad+\frac{\pi}{2}\int B_{1}^{2}\cup(B_{2}^{\prime}+w_{1}\cup
B_{1})~{},$ (E.5)

where in the last term we take a lift of $B_{1}$ to $\mathbb{Z}_{4}$ cochain
that takes value in $0,1$.

###  E.2 Unitary $\mathbb{Z}_{2}^{\mathsf{C}}$ symmetry

Let us consider unitary $\mathbb{Z}_{2}^{\mathsf{C}}$ symmetry, with
$w_{1}=0$. The twisted cocycle condition for $B_{2},B_{2}^{\prime}$ becomes

$dB_{2}=B_{1}\cup B_{2}^{\prime}~{}.$ (E.6)

We note that the global transformation $B_{2}^{\prime}\rightarrow
B_{2}^{\prime}+dB_{1}=B_{2}^{\prime}$ induces the equivalence relation
$B_{2}\sim B_{2}+B_{1}^{2}$.

Let us consider the following symmetry fractionalizations:

  * •

$B_{2}^{\prime}=0$: charge conjugation squares to $+1$ on the dyon. Then
$dB_{2}=0$, and the zero-form symmetry does not participate in a non-trivial
two-group with the one-form symmetry.

  * •

$B_{2}^{\prime}=B_{1}^{2}$: charge conjugation squares to $-1$ on the dyon.
Then $dB_{2}=B_{1}^{3}$, the zero-form symmetry participates in a two-group
with the one-form symmetry, with Postnikov class $B_{1}^{3}$.

####  E.2.1 Anomaly

The zero-form and one-form symmetry has an anomaly, described by the bulk term
[117]

$\pi\int{\cal P}(B_{2})+\pi\int B_{2}\cup B_{2}^{\prime}+\pi\int
B_{1}\cup\left((B_{1}\cup B_{2}^{\prime})\cup_{2}B_{2}^{\prime}+B_{1}\cup
B_{2}^{\prime}\right)+\frac{\pi}{2}\int B_{1}^{2}\cup B_{2}^{\prime}~{},$
(E.7)

where in the last term we take a lift of $B_{1}$ to $\mathbb{Z}_{4}$ cochain
that takes value in $0,1$.

Thus for $B_{2}^{\prime}=0$, the zero-form symmetry by itself does not have an
anomaly.

For $B_{2}^{\prime}=B_{1}^{2}$, there is a 2-group anomaly:

$\pi\int{\cal P}(B_{2})+\pi\int B_{2}\cup B_{1}^{2}+\pi\int
B_{1}\cup\left((B_{1}^{3})\cup_{2}B_{1}^{2}\right)+\frac{3\pi}{2}\int
B_{1}^{4}~{}.$ (E.8)

The last term $B_{1}^{4}$ can be canceled by a local $3d$ counterterm, and we
are left with the first three terms.444444 To see the anomaly is well-defined,
we can extend it to $5d$, and use $\pi\int B_{1}^{5}=\pi Sq^{2}(B_{1}^{2})\cup
B_{1}=\pi\int Sq^{2}(B_{1}^{3})=\pi\int d{\cal P}(B_{2})+\pi\int
d\zeta(B_{1},B_{1}^{2})$, where $\zeta(B_{1},B_{1}^{2})$ is the Cartan
coboundary given by the term that depends only on $B_{1}$, and we used $d{\cal
P}(B_{2})=dB_{2}\cup_{1}dB_{2}=Sq^{2}(B_{1}^{3})$ mod 2 with
$dB_{2}=B_{1}^{3}$. Thus the $\mathbb{Z}_{2}$ gauge theory gives an example
with anomalous 2-group symmetry.

###  E.3 Anti-unitary $\mathbb{Z}_{2}$ symmetry that acts as charge
conjugation

Let us consider instead $\mathbb{Z}_{2}$ time-reversal symmetry that acts as
charge conjugation symmetry, i.e. the diagonal $\mathbb{Z}_{2}$ subgroup zero-
form symmetry for
$\mathbb{Z}_{2}^{\mathsf{C}}\times\mathbb{Z}_{2}^{\mathsf{T}}$. In other
words, we restrict to $B_{1}=w_{1}$. Then the twisted cocycle conditions for
$B_{2},B_{2}^{\prime}$ becomes

$dB_{2}=w_{1}\cup\left(B_{2}^{\prime}+w_{1}^{2}\right)~{}.$ (E.9)

We note that the global transformation $B_{2}^{\prime}\rightarrow
B_{2}^{\prime}+dw_{1}=B_{2}^{\prime}$ induces the equivalence relation
$B_{2}\sim B_{2}+w_{1}^{2}$.

Let us consider the following symmetry fractionalization for the
$\mathbb{Z}_{2}$ anti-unitary symmetry:

  * •

$B_{2}^{\prime}=0$: time-reversal squares to $+1$ on the dyon. Then
$dB_{2}=w_{1}^{3}$, and the zero-form symmetry participates in a two-group
with the one-form symmetry, with Postnikov class $w_{1}^{3}$.

  * •

$B_{2}^{\prime}=w_{1}^{2}$: time-reversal squares to $-1$ on the dyon. Then
$dB_{2}=0$, and the zero-form symmetry does not participate in a non-trivial
two-group with the one-form symmetry.

The condition that only certain fractionalization class leads to a symmetry
that does not participate in a two-group is also discussed in equation (234)
of [40]. See also [118] for related examples.454545 We note that in the case
of two-group, one can still discuss different fractionalization classes by
shifting the background gauge field, where background $B_{2}$ for the one-form
symmetry and the background $B_{1}$ for the zero-form symmetry satisfies
$d_{\rho}B_{2}=B_{1}^{*}\Theta~{},$ (E.10) for Postnikov class $\Theta$. When
$\Theta$ is non-trivial, the symmetry mixes to form a two-group, this implies
that under a zero-form symmetry transformation with parameter $\lambda$,[21]
$B_{1}\rightarrow B_{1}^{\lambda},\quad B_{2}\rightarrow
B_{2}+\zeta(\lambda,B_{1})~{},$ (E.11) and this gives additional equivalence
relation on the shift $B_{2}\rightarrow B_{2}+B_{1}^{*}\eta$ that implements
different fractionalization classes [21].

####  E.3.1 Anomaly

The zero-form and one-form symmetries have an anomaly, described by the bulk
term [117]

$\displaystyle S_{\text{bulk SPT}}$ $\displaystyle=\pi\int{\cal
P}(B_{2})+\pi\int B_{2}\cup(B_{2}^{\prime}+w_{1}^{2})+\pi\int w_{1}^{2}\cup
B_{2}$ (E.12) $\displaystyle\quad+\pi\int
w_{1}\cup\left[(w_{1}\cup(B_{2}^{\prime}+w_{1}^{2}))\cup_{2}(B_{2}^{\prime}+w_{1}^{2})+w_{1}\cup(B_{2}^{\prime}+w_{1}^{2})\right]$
(E.13) $\displaystyle\quad+\frac{\pi}{2}\int
w_{1}^{2}\cup(B_{2}^{\prime}+w_{1}^{2})~{},$ (E.14)

For $B_{2}^{\prime}=w_{1}^{2}$, where there is no 2-group symmetry, the
anomaly is

$\pi\int{\cal P}(B_{2})+\pi\int w_{1}^{2}\cup B_{2}~{}.$ (E.15)

For $B_{2}^{\prime}=0$, where there is 2-group symmetry, the anomaly is

$\pi\int{\cal P}(B_{2})+\pi\int w_{1}^{2}\cup_{1}w_{1}^{3}+\pi\int
w_{1}\cup\left(w_{1}^{3}\cup_{2}w_{1}^{2}\right)+\frac{3\pi}{2}\int
w_{1}^{4}~{}.$ (E.16)

###  E.4 $\mathbb{Z}_{2}^{\mathsf{T}}$ symmetry that does not permute anyons

We remark that we can also consider the case when the time-reversal symmetry
does not act as charge conjugation. In such a case, we substitute $B_{1}=0$
instead of $B_{1}=w_{1}$. Then $dB_{2}=0$ for all symmetry fractionalization
classes, and the zero-form symmetry does not participate in a non-trivial two-
group with the one-form symmetry. There are four fractional classes, labelled
by $(n,m)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, corresponding to
$B_{2}=nw_{1}^{2}$, $B_{2}^{\prime}=mw_{1}^{2}$, with anomaly

$n(m+1)\pi\int w_{1}^{4}~{}.$ (E.17)

The class $n=1,m=0$ is the eTmT phase [119, 120], where electric and magnetic
particles are both Kramers doublet.

##  Appendix F Two-Groups for Fractionalization Classes of Charge Conjugation

In this appendix we discuss fractionalization classes for $\mathbb{Z}_{2}$
zero-form symmetry that acts as charge conjugation on $\mathbb{Z}_{N}$ or
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ one-form symmetry. We will show that the
zero-form charge conjugation symmetry with different fractionalization classes
participates in different two-groups, for $\mathbb{Z}_{N}$ one-form symmetry
with $N=0$ mod 4 and $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ one-form symmetry.
The discussion applies to all spacetime dimensions.

###  F.1 $\mathbb{Z}_{N}$ one-form symmetry

For odd $N$, there is only one fractionalization class. Therefore let us focus
on even $N$, where there are two fractionalization classes for the charge
conjugation zero-form symmetry. We will investigate whether some
fractionalization class leads to zero-form symmetry that participates in a
two-group.

When $N=2$ mod $4$, $N/2$ is odd,
$\mathbb{Z}_{N}\cong\mathbb{Z}_{N/2}\times\mathbb{Z}_{2}$, and the charge
conjugation symmetry acts non-trivially only on $\mathbb{Z}_{N/2}$, while the
non-trivial fractionalization class lives in the other $\mathbb{Z}_{2}$. Thus
changing the fractionalization classes of the zero-form symmetry does not make
the symmetry participates in a two-group.

Let us turn to $N=0$ mod 4, where $\mathbb{Z}_{N}$ contains $\mathbb{Z}_{4}$
subgroup. The charge conjugation symmetry sends $Q\in\mathbb{Z}_{N}$ to $-Q$.
In particular, for $Q=N/4$ the difference is the non-trivial element in the
$\mathbb{Z}_{2}$ subgroup. We can express the $\mathbb{Z}_{N}$ two-form gauge
field for the one-form symmetry as $\mathbb{Z}_{2}$ gauge field $B$ and
$\mathbb{Z}_{N/2}=\mathbb{Z}_{N}/\mathbb{Z}_{2}$ two-form gauge field
$B^{\prime}$. In the presence of the background $A$ for the charge conjugation
symmetry, they obey

$dB=\frac{dB^{\prime}}{(N/2)}+B^{\prime}\cup A\quad\text{mod }2~{}.$ (F.1)

This implies that the fractionalization class $B^{\prime}=\frac{dA}{2}$ mod
$N/2$ gives zero-form symmetry that participates in a two-group with the
$\mathbb{Z}_{2}$ subgroup one-form symmetry invariant under the charge
conjugation symmetry,

$dB=A\cup\frac{dA}{2}\quad\text{mod }2~{},$ (F.2)

where by suitable redefinition of $B$ this is the same as $dB=A^{3}$ mod 2.

We remark that another fractionalization class that corresponds to shifting
$B\rightarrow B+dA/2$ does not give rise to a two-group, since $dB$ is not
affected by the shift.

###  F.2 $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ one-form symmetry

Consider $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ one-form symmetry with the
charge conjugation exchanging the two $\mathbb{Z}_{2}$’s. Denote the
background two-form gauge field for the diagonal $\mathbb{Z}_{2}$ generator by
$B$, and the background for one of the $\mathbb{Z}_{2}$ generator by
$B^{\prime}$, and the background for the charge conjugation symmetry by $A$,
then they satisfy

$dB=B^{\prime}\cup A\text{ mod }2,\quad dB^{\prime}=0\text{ mod }2~{}.$ (F.3)

This implies that the fractionalization class $B^{\prime}=A\cup A$ gives zero-
form symmetry that participates in a two-group with the $\mathbb{Z}_{2}$
subgroup one-form symmetry invariant under charge conjugation symmetry,

$dB=A^{3}\text{ mod }2~{}.$ (F.4)

##  Appendix G Twisted Gauge Bundle and Two-Group Symmetry

Let us consider a gauge theory with simply-connected gauge group
$G_{\text{gauge}}$ and matter fields that transform under global symmetry
$\tilde{G}_{\text{global}}$ which commutes with the gauge group (and in
particular, does not act on the center one-form symmetry). Denote by $\Gamma$
the center subgroup of the gauge group that acts trivially on the matter
fields, which gives the one-form symmetry in the gauge theory. The action of a
center subgroup ${\mathcal{Z}}\subset\tilde{G}_{\text{global}}$ is identified
with a transformation in the center of the gauge group. The symmetry of the
classical action is

$\frac{(G_{\text{gauge}}/\Gamma)\times\tilde{G}_{\text{global}}}{\mathcal{Z}}~{}.$
(G.1)

We note that $\mathcal{Z}$ in the gauge group participates in a central
extension

$1\rightarrow\Gamma\rightarrow{\tilde{\mathcal{Z}}}\rightarrow{\cal
Z}\rightarrow 1~{},$ (G.2)

where ${\tilde{\mathcal{Z}}}/\Gamma=\mathcal{Z}$, and ${\tilde{\mathcal{Z}}}$
is a subgroup in the center of $G_{\text{gauge}}$ that is larger than
$\Gamma$. When we turn on a background two-form gauge field $B_{2}$ for the
$\Gamma$ one-form symmetry and the background for the
$G_{\text{global}}=\tilde{G}_{\text{global}}/{\mathcal{Z}}$ zero-form
symmetry, the gauge bundle is twisted to be a
$G_{\text{gauge}}/{\tilde{\mathcal{Z}}}$ bundle. The background for
$G_{\text{global}}$ symmetry can be described by an obstruction class
$w_{2}^{G}$ to lifting the bundle to a $\tilde{G}_{\text{global}}$, and it is
a two-form that takes value in ${\cal Z}$. Concretely, on triple overlaps of
coordinate patches, the transition functions that take value in
$\tilde{G}_{\text{global}}$ only multiply to an element in
${\mathcal{Z}}\subset\tilde{G}_{\text{global}}$ for
$G_{\text{global}}=\tilde{G}_{\text{global}}/{\mathcal{Z}}$ bundles, and the
equivalence class of the element under redefinition of each transition
function by elements in ${\cal Z}$ gives $w_{2}^{G}$. The two-forms
$w_{2}^{G}$ and $B_{2}$ combine into a two-form gauge field for
${\tilde{\mathcal{Z}}}$, and this implies that they obey the constraint

$dB=\text{Bock}(w_{2}^{G})~{},$ (G.3)

where Bock is the Bockstein homomorphism for the short exact sequence (G.2).
This implies that if $w_{2}^{G}$, which is a ${\mathcal{Z}}$-valued two-
cocycle, cannot be lifted into a ${\tilde{\mathcal{Z}}}$-valued two-cocycle,
then the zero-form symmetry combines with the one-form symmetry into a two-
group symmetry, where the Postnikov class that characterized the two-group is
[21, 121]

$\Theta_{3}=\text{Bock}(w_{2}^{G})\in H^{3}(BG_{\text{global}},\Gamma)~{}.$
(G.4)

Examples of such theories with two-group symmetry are discussed in [21, 121].

In particular, consider the case $G_{\text{global}}=\mathbb{Z}_{n}$,
$\Gamma=\mathbb{Z}_{N}$ and $\tilde{G}_{\text{global}}$ is a central extension
of $G_{\text{global}}$ by ${\cal Z}=\mathbb{Z}_{m}$, then

$w_{2}^{G}=r\frac{dA}{n}\text{ mod }m~{}$ (G.5)

where $r$ is an integer that specifies the extension
$\tilde{G}_{\text{global}}$, $A$ is the $G_{\text{global}}$ background gauge
field. Then

$\text{Bock}(w_{2}^{G})=r^{\prime}\frac{dw_{2}^{G}}{m}\text{ mod }N=0~{},$
(G.6)

where $r^{\prime}$ is an integer that specifies the extension
${\tilde{\mathcal{Z}}}$ of ${\mathcal{Z}}$ by $\Gamma$. Thus such zero-form
symmetry does not participate in a two-group.

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