This term can be thought of as a sort of “discrete $\theta$-parameter.” To
make this connection explicit, we can rewrite the action (189) in a more
suggestive way as

$S=\frac{iN}{4\pi
p}\int\left(dA_{1}+pB_{2}\right)\wedge\left(dA_{1}+pB_{2}\right)-\frac{iN}{4\pi
p}\int dA_{1}\wedge dA_{1}~{}.$ (190)

The fact that the torsion term gives rise to the traditional “$\theta$-term”
for the $U(1)$ gauge field $A_{1}$, is the reason why $p$ is sometimes
referred to as a discrete $\theta$-parameter.

Our goal is to understand the effect of this additional term on the
generalized global symmetries and associated Wilson operators. To this end, we
first show that the integer parameter $p$ must satisfy the conditions:

$\text{(i)}~{}\frac{Np}{2}\in{\mathbb{Z}}~{},\qquad\text{(ii)}~{}p\sim
p+2N~{}.$ (191)

To see the first condition, first note that including the discrete torsion
term requires that the standard gauge symmetries of the theory (136) be
modified to:

$\displaystyle\begin{split}&B_{2}\longmapsto B_{2}+d\lambda_{1}~{},\hskip
28.45274pt\oint\lambda_{1}=2\pi{\mathbb{Z}}~{},\\\ &A_{1}\longmapsto
A_{1}+d\lambda_{0}-p\lambda_{1}~{},\quad\quad\lambda_{0}\sim\lambda_{0}+2\pi~{}.\end{split}$
(192)

Here we see that adding the torsion term causes the 1-form gauge field $A_{1}$
to transform (with charge $p$) under the 1-form gauge symmetry. Note that
these modified transformation properties are consistent with the $U(1)$ gauge
transformations of $A_{1}$ when $p$ is an integer.

Now notice that under a 1-form gauge transformation the action shifts as

$\displaystyle\begin{split}\delta S=&\frac{iN}{2\pi}\int d\lambda_{1}\wedge
dA_{1}-\frac{iNp}{2\pi}\int d\lambda_{1}\wedge d\lambda_{1}\\\ =&2\pi
i\int\left(\frac{d\lambda_{1}}{2\pi}\right)\wedge\left(N\frac{dA_{1}}{2\pi}\right)-2\pi
i\left(\frac{Np}{2}\right)\int\left(\frac{d\lambda_{1}}{2\pi}\right)\wedge\left(\frac{d\lambda_{1}}{2\pi}\right)~{}.\end{split}$
(193)

This means that the theory is only invariant under 1-form gauge transformation
provided $\frac{Np}{2}\in{\mathbb{Z}}$.232323Note that on a spin manifold
$\int\left(\frac{d\lambda_{1}}{2\pi}\right)\wedge\left(\frac{d\lambda_{1}}{2\pi}\right)\in
2{\mathbb{Z}}~{},$ (194) and thus $p$ can take any integer value. This follows
from the Wu formula which states:
$\int_{M_{4}}\left[\frac{\lambda_{1}}{2\pi}\right]\wedge\left[\frac{\lambda_{1}}{2\pi}\right]=\int_{M_{4}}\nu_{2}(TM_{4})\cup\left[\frac{\lambda_{1}}{2\pi}\right]_{\rm
mod2}\;\text{mod}_{2}~{},$ (195) where
$\nu_{2}(TM_{4})=\omega_{2}(TM_{4})+\omega_{1}^{2}(TM_{4})$ is the Wu class
and $\omega_{1}$ and $\omega_{2}$ are the first and second Stiefel-Whitney
class of the tangent bundle $TM_{4}$. For spin manifolds
$w_{2}(TM)=w_{1}(TM)=0$ which proves the identity (194). However, for more
general orientable manifold $\omega_{1}=0$ and $\nu_{2}(TM)$ is in general
non-zero. Thus, on a general non-spin manifold, we find that $p$ can take any
integer value when $N$ is even, but $p\in 2{\mathbb{Z}}$ must be even when $N$
is odd.  To see the second condition, $p\sim p+2N$, first note that the
${\mathbb{Z}}_{N}$ gauge field $B_{2}$ is normalized

$\displaystyle\begin{split}\oint
B_{2}\in\frac{2\pi}{N}{\mathbb{Z}}~{}.\end{split}$ (196)

We can then rewrite the torsion term as

$S_{\scriptscriptstyle\rm SPT}=\frac{iNp}{4\pi}\int B_{2}\wedge
B_{2}=\frac{2\pi
ip}{2N}\int\left(N\frac{B_{2}}{2\pi}\right)\wedge\left(N\frac{B_{2}}{2\pi}\right)~{},$
(197)

so that it is clear the term shifts by an multiple of $2\pi$ when $p\mapsto
p+2N$.

Now let us discuss the effect of the torsion term on the global symmetries.
First, note that when $p=0$, the theory is simply the standard BF theory
discussed in previous sections. For $p=0$, the global symmetry is given by
${\mathbb{Z}}_{N}^{(1)}\times{\mathbb{Z}}_{N}^{(2)}$ with transformation rules

$\displaystyle\begin{split}&A_{1}\longmapsto
A_{1}+\frac{1}{N}\epsilon_{1}~{},\;\;\;\oint\epsilon_{1}=2\pi{\mathbb{Z}}~{},\\\
&B_{2}\longmapsto
B_{2}+\frac{1}{N}\epsilon_{2}~{},\;\;\;\oint\epsilon_{2}=2\pi{\mathbb{Z}}~{},\end{split}$
(198)

When $p\neq 0$, symmetry structure changes. The equation of motion for
$B_{2}$, $dA_{1}+pB_{2}=0$, implies that the Wilson surface operator satisfies

$\left(e^{i\oint_{\Gamma_{2}}B_{2}}\right)^{p}=e^{-i\oint_{\Gamma_{2}}dA_{1}}=1~{}.$
(199)

Additionally, when we introduce a dual gauge field $\tilde{A}_{1}$, the
equation of motion for $A_{1}$ similarly implies that

$\left(e^{i\oint_{\Gamma_{2}}B_{2}}\right)^{N}=e^{-i\oint_{\Gamma_{2}}d\tilde{A}_{1}}=1~{}.$
(200)

Together these conditions mean that the minimal $B_{2}$ Wilson surface
operator $W_{1}(\Gamma_{2})=e^{i\oint_{\Gamma_{2}}B_{2}}$ satisfies242424This
can be shown using Bézout’s identity. Bézout’s identity says that given two
integers $a,b\in{\mathbb{Z}}$ with the greatest common divisor
$d=\text{gcd}(a,b)$, there always exist integers $x,y\in{\mathbb{Z}}$ such
that $ax+by=d$. In our case, any integer $\ell$ can be decomposed as
$\ell=n\;\text{gcd}(N,p)+m$ with
$n\in{\mathbb{Z}},\;m=0,1,\cdots,\text{gcd}(N,p)-1$. Also, by Bézout’s
identity $\text{gcd}(N,p)$ can always been written as $\text{gcd}(N,p)=xN+yp$
for integers $x,y$. This immediately implies (201).

$\left(W_{1}(\Gamma_{2})\right)^{{\rm gcd}(N,p)}=1,$ (201)

where gcd denotes the greatest common divisor.

This means that when $p\neq 0$, the discrete torsion term reduces the number
of Wilson surface operators and hence the the theory has
${\mathbb{Z}}_{\text{gcd}(N,p)}^{(2)}$ 2-form global symmetry with
transformation rules

$\displaystyle\begin{split}B_{2}\longmapsto
B_{2}+\frac{1}{\text{gcd}(N,p)}\epsilon_{2}~{},\qquad A_{1}\longmapsto
A_{1}-\frac{p}{\text{gcd}(N,p)}\tilde{\epsilon}_{1}~{},\end{split}$ (202)

where locally, $d\tilde{\epsilon}_{1}=\epsilon_{2}$. It is a straightforward
exercise to show that the action in (189) is indeed invariant under these
transformations.

Now let us consider the effect of the torsion term on the 1-form global
symmetry. Note that the additional term does not depend on the 1-form field
$A_{1}$ and hence does not directly effect the 1-form symmetry. However, the
naive line operator $e^{i\oint_{\Gamma_{1}}A_{1}}$ is not gauge invariant
under (192). Rather, it can only be made gauge invariant if we dress it with a
surface operator:

$\widetilde{W}=e^{i\oint_{\Gamma_{1}}A_{1}}e^{ip\int_{\Sigma_{2}}B_{2}}~{}.$
(203)

The resulting operator is not a genuine line operator since it is attached to
a surface. Of course, this is consistent with our previous observation (199)
that charge-$p$ Wilson surface is “invisible” since it can end on a line. Note
however, that we can use (200) to construct a gauge invariant (genuine) line
operator in terms of the line operator

$\widetilde{W}^{N}=e^{i\oint_{\Gamma_{1}}\left(NA_{1}-p\tilde{A}_{1}\right)}~{}.$
(204)

This is gcd$(N,p)$ times a minimal (gauge invariant) line operator

$W(\Gamma_{1})=\left(\widetilde{W}\right)^{\frac{N}{\text{gcd}(N,p)}}.$ (205)

which satisfies

$\displaystyle\begin{split}\left(W(\Gamma_{1})\right)^{\text{gcd}(N,p)}=1~{}.\end{split}$
(206)

Therefore, we see that the torsion term reduces the symmetry of both the 1-
and 2-form global symmetries:

$\displaystyle\begin{split}{\mathbb{Z}}_{N}^{(1)}\times{\mathbb{Z}}_{N}^{(2)}~{}\stackrel{{\scriptstyle
p\neq 0}}{{\longrightarrow}}~{}{\mathbb{Z}}_{{\rm
gcd}(N,p)}^{(1)}\times{\mathbb{Z}}_{{\rm gcd}(N,p)}~{}.\end{split}$ (207)

###  3.3 1-Form Center Symmetry in Non-Abelian Yang-Mills Theory

Another important class of theories with discrete higher form symmetry is non-
abelian Yang-Mills theory Aharony:2013hda ; Gaiotto:2014kfa ; Gaiotto:2017yup
; Tachikawa:2017gyf ; Cordova:2019jnf ; Cordova:2019uob . These theories can
also be analyzed by using discrete background gauge fields. For our discussion
let us take $G$ to be a non-abelian, continuous Lie group that is not simply
connected and has a non-trivial center.

In non-abelian Yang-Mills theory, the story of center and magnetic 1-form
symmetry differs from that of $U(1)$ Maxwell theory because non-abelian gauge
theories are interacting. The self-interacting nature of non-abelian Yang-
Mills has many consequences on the dynamics of the theory. One consequence of
the interactions is that the gluons are charged fields and therefore can
screen Wilson lines. However, since the gluons are in the adjoint
representation – which is often not the minimal representation of $G$ – the
gluons can only screen certain classes of Wilson lines. In cases where the
gluons cannot completely screen Wilson lines there is a (often discrete)
1-form center symmetry that measures the conserved charge modulo screening.
This symmetry is classified by the center of the gauge group $Z(G)$.

Additionally, in non-abelian gauge theories there are smooth, dynamical
monopole gauge field configurations that are summed over in the path integral.
Because of this, the monopole lines of the theory can also be dynamically
screened. This can be seen by a stability analysis of the equations of motion
in the monopole background Coleman:1982cx . In cases where the monopole lines
cannot be screened, we say that they are protected by a magnetic 1-form
symmetry which measures their protected magnetic charge. This symmetry is
classified by the fundamental group of the gauge group $\pi_{1}(G)$.

We will now compare and contrast these 1-form electric and magnetic symmetries
in a pair of connected examples: $SU(2)$ and $SO(3)$ non-abelian Yang-Mills
theory.

####  3.3.1 $SU(2)$ vs. $SO(3)$

Here we would like to consider the difference between the center and magnetic
1-form symmetries of $SU(2)$ and $SO(3)$ gauge theories. Since these groups
have the same Lie algebra, they have the same perturbative physics. However,
since $SO(3)=SU(2)/{\mathbb{Z}}_{2}$, the two groups have different center and
fundamental groups:

$\displaystyle\begin{split}&Z(SU(2))={\mathbb{Z}}_{2}\qquad\qquad
Z(SO(3))=1~{},\\\
&\pi_{1}(SU(2))=1~{}\qquad\qquad\pi_{1}(SO(3))={\mathbb{Z}}_{2}~{},\end{split}$
(208)

which means that (according to our proposed classification) the theories will
have different higher form symmetries.

First consider $SU(2)$ gauge theory. Here a Wilson line wrapping a curve
$\gamma$ is defined by a choice of representation $R_{j}$ of spin-$j$ where
$j=0,\,1/2,\,1,\,3/2,\,...$:

$\displaystyle\begin{split}W_{R_{j}}(\gamma)={\rm Tr}_{R_{j}}\,{\cal P}~{}{\rm
exp}\left\\{i\int_{\gamma}A\right\\}~{}.\end{split}$ (209)

These Wilson lines can be dynamically screened by the dynamical gluons.
However, since gluons are in the adjoint representation (i.e. the spin-1
representation), they can only screen Wilson lines of integer spin. We can see
this by introducing a charged adjoint scalar field and using it to break
$SU(2)\to U(1)$. In this phase, particles charged under the spin-$j$
representation have abelian charges $q\in 2{\mathbb{Z}}+2j$ so that the gluons
(charge $\pm 2$) can only partially screen Wilson lines of half-integral spin
(odd charge).

Now recall that the center of $SU(2)$ is $Z(SU(2))={\mathbb{Z}}_{2}$. The
representations of half-integral spin are the representations that transform
non-trivially under this ${\mathbb{Z}}_{2}$. Therefore, detecting whether or
not a Wilson line can be screened is equivalent to measuring its charge under
the center ${\mathbb{Z}}_{2}\subset SU(2)$. Thus, we see $SU(2)$ Yang-Mills
theory has a ${\mathbb{Z}}_{2}^{(1)}$1-form center symmetry that measures the
spin of Wilson lines modZ.

In $SU(2)$ gauge theory there are no unprotected monopoles (see Brandt:1979kk
; Coleman:1982cx  for details). This implies that $SU(2)$ gauge theory has no
magnetic 1-form symmetry, matching with the fact that $\pi_{1}(SU(2))=1$.

Now consider $SO(3)$ gauge theory. Here the Wilson lines are classified by
representations of spin $j=0,1,2,...$. These Wilson lines can all be screened
by gluons due to the fact that the minimal Wilson line is the same
representation as the gluons. Therefore there is no center symmetry – matching
with the fact that $Z(SO(3))=1$.

However, due to the fact that $SO(3)=SU(2)/{\mathbb{Z}}_{2}$, the theory has
stable monopole lines. In order to see this, let us first consider monopoles
in $SU(2)$ gauge theory. Here, the monopoles have magnetic charge given by
$n\sigma^{3}$ for $n\in{\mathbb{Z}}$. This can be seen by noting that near the
monopole, the magnetic field goes like

$\displaystyle\begin{split}\vec{B}\underset{r\to
0}{\sim}B_{0}\frac{1}{2r^{2}}\hat{r}\quad,\quad
B_{0}=\alpha\sigma^{3}~{}.\end{split}$ (210)

An explicit gauge field configuration can be written in two patches
(northern/southern-hemisphere) covering the asymptotic 2-sphere:

$\displaystyle\begin{split}A_{N/S}=B_{0}\frac{\pm
1-\cos(\theta)}{2}d\phi~{},\end{split}$ (211)

where $\pm$ is correlated to the northern/southern hemisphere. In order for
this to describe a good monopole configuration, the two patches have to be
related by a gauge transformation: $g=e^{iB_{0}\phi}$. We require that this
gauge transformation be well defined, which implies

$\displaystyle\begin{split}g(\phi+2\pi)=g(\phi)\quad\Rightarrow\quad e^{2\pi
iB_{0}}=\mathds{1}_{SU(2)}~{}.\end{split}$ (212)

This leads to the quantization condition $B_{0}=n\sigma^{3}$ where
$n\in{\mathbb{Z}}$. As shown in Brandt:1979kk ; Coleman:1982cx , these
monopoles are not stable to emission by $W$-bosons.

However, in $SO(3)$ gauge theory $-\mathds{1}_{SU(2)}\sim\mathds{1}_{SO(3)}$.
This means that $SO(3)$ magnetic monopoles can now have half-integral magnetic
charges: $B_{0}=\frac{n}{2}\sigma^{3}$. As it turns out, monopoles with
magnetic charge $B_{0}=\frac{1}{2}\sigma^{3}$ _are_ stable to gluon emission.
This gauge transformation $g(\phi)=e^{\frac{1}{2}\sigma^{3}\phi}$ is the non-
trivial representative of $\pi_{1}(SO(3))$. Because $g(\phi)$ is in a non-
trivial topological class, heuristically it cannot “decay” by gluon
fluctuations and hence the corresponding monopole is stable. Therefore, we see
that $SO(3)$ gauge theory has a $\to{\mathbb{Z}}_{2}^{(1)}$ magnetic symmetry
that measures the magnetic charge $mod_{\mathbb{Z}}$.

####  3.3.2 General Center and Magnetic 1-Form Symmetry

The story for $SU(2)$ and $SO(3)$ gauge theory is in fact general. Consider a
non-abelian gauge group $G$ that has a center $Z$. Let us also define $G_{\rm
ad}=G/Z$ – this is sometimes called the adjoint form of the group – and
${\widehat{G}}$ which is the simply connected cover $G={\widehat{G}}/Z$. In
general, the center symmetry of $G$ Yang-Mills theory is described by $Z$ and
its 1-form magnetic symmetry is given by $\pi_{1}(G)$.

In fact, there is a parallel structure between the center and magnetic 1-form
symmetries:

Center Symmetry | Gauge Group | Magnetic Symmetry
---|---|---
“$\pi_{1}(G)\times Z$” | ${\widetilde{G}}$ | 1
$Z$ | $G={\widetilde{G}}/\pi_{1}(G)$ | $\pi_{1}(G)$
1 | $G_{ad}=G/Z$ | “$\pi_{1}(G)\times Z$”

Here we use “ ” to indicate that this is only true as sets. In general,
$Z(G)=Z$ and $\pi_{1}(G)$ can descend from a simply connected group
${\widetilde{G}}$ where $Z({\widetilde{G}})$ is an extension of $Z$ by
$\pi_{1}(G)$. Similarly, $G_{\rm ad}=G/Z$ will have a fundamental group that
is an extension of $\pi_{1}(G)$ by $Z$. For more details see Appendix B.

####  3.3.3 Background Gauge Fields

Now we would like to discuss how to use background gauge fields to study the
center and magnetic 1-form symmetry of non-abelian Yang-Mills.

First, let us discuss center symmetry. Consider the example of $SU(N)$ gauge
theory. Center symmetry allows us to shift the dynamical gauge field $a_{1}$
(with field strength $f_{2}$) by a ${\mathbb{Z}}_{N}$ gauge field
$A_{{\mathbb{Z}}_{N}}$ corresponding to the center ${\mathbb{Z}}_{N}\subset
SU(N)$. We can see that this transformation implements a ${\mathbb{Z}}_{N}$
action on Wilson lines:

$\displaystyle\begin{split}W_{R}(\gamma)=Tr_{R}~{}{\rm
exp}\left\\{i\int_{\gamma}a_{1}\right\\}\longmapsto Tr_{R}~{}{\rm
exp}\left\\{i\int_{\gamma}a_{1}+i\int_{\gamma}A_{{\mathbb{Z}}_{N}}\right\\}=e^{iI_{N}(R)\int_{\gamma}A_{{\mathbb{Z}}_{N}}}W_{R}(\gamma)~{},\end{split}$
(213)

where $I_{N}(R)$ is the $N$-ality of the representation $R$ (i.e. the charge
of $R$ thought of as a ${\mathbb{Z}}_{N}\subset SU(N)$ representation so
$N$-ality is the number of boxes in a Young tableau mod $N$). Here we see that
the ${\mathbb{Z}}_{N}$-action comes from the fact that
$e^{i\int_{\gamma}A_{{\mathbb{Z}}_{N}}}\in{\mathbb{Z}}_{N}$.

In order to use our knowledge of ${\mathbb{Z}}_{N}$ gauge theory, we can lift
the $SU(N)$ gauge theory to a $U(N)$ gauge theory. Here the idea is that we
can realize $SU(N)$ gauge theory as a constrained $U(N)$ gauge theory, but in
$U(N)$ gauge theory, the shift by the ${\mathbb{Z}}_{N}$ gauge field
$A_{{\mathbb{Z}}_{N}}$ lifts to a shift by the $U(1)\subset U(N)$ gauge field
$\hat{A}_{{\mathbb{Z}}_{N}}$ where we can easily analyze the symmetry using
background gauge fields.

So, let us first lift our $SU(N)$ gauge theory to a $U(N)$ gauge theory which
has a gauge field and field strength $\tilde{a}_{1},\tilde{f}_{2}$. The
projection from $U(N)\to SU(N)$ gauge theory can be achieved by introducing a
2-form Lagrange multiplier ${\cal F}_{2}\in\Omega^{2}(M_{4})$ so that the full
action can be written as

$\displaystyle\begin{split}S=\int\frac{1}{g^{2}}{\rm
Tr}[\tilde{f}_{2}\wedge\ast\tilde{f}_{2}]+\frac{i}{2\pi}\int{\cal
F}_{2}\wedge{\rm Tr}[\tilde{f}_{2}]~{}.\end{split}$ (214)

Here, shifting by the ${\mathbb{Z}}_{N}$ gauge field $A_{{\mathbb{Z}}_{N}}$ is
equivalent to shifting by a $U(1)$ gauge field:

$\displaystyle\begin{split}\tilde{a}_{1}\longmapsto\tilde{a}_{1}+\hat{A}_{{\mathbb{Z}}_{N}}\quad\Rightarrow\quad\tilde{f}_{2}\longmapsto\tilde{f}_{2}+{\widehat{F}}_{{\mathbb{Z}}_{N}}~{},\end{split}$
(215)

where ${\widehat{F}}_{\mathbb{Z}_{N}}$ is the field strength of the $U(1)$
gauge field $\hat{A}_{\mathbb{Z}_{N}}$. However, this shift is not invariant
under a choice of lift of $A_{\mathbb{Z}_{N}}$ to $\hat{A}_{\mathbb{Z}_{N}}$:

$\displaystyle\begin{split}{\widehat{F}}_{{\mathbb{Z}}_{N}}\longmapsto{\widehat{F}}_{{\mathbb{Z}}_{N}}+\Lambda_{2}\quad,\quad\tilde{f}_{2}\longmapsto\tilde{f}_{2}+\Lambda_{2}\mathds{1}_{N}\quad,\quad{\frac{\Lambda_{2}}{2\pi}}\in
H^{2}(M_{4};{\mathbb{Z}})~{}.\end{split}$ (216)

To make the action invariant, we need to introduce a ${\mathbb{Z}}_{N}$
background gauge field $B_{2}$ that similarly transforms

$\displaystyle\begin{split}B_{2}\longmapsto B_{2}+\Lambda_{2}~{},\end{split}$
(217)

and substitute $\tilde{f}_{2}\longmapsto\tilde{f}_{2}-B_{2}\mathds{1}_{N}$ in
the action:

$\displaystyle\begin{split}S=\int\frac{1}{g^{2}}{\rm
Tr}[(\tilde{f}_{2}-B_{2}\mathds{1}_{N})\wedge\ast(\tilde{f}_{2}-B_{2}\mathds{1}_{N})]+\frac{i}{2\pi}\int{\cal
F}_{2}\wedge{\rm Tr}[\tilde{f}_{2}-B_{2}\mathds{1}_{N}]~{}.\end{split}$ (218)

Now notice that the Lagrange multiplier term has fixed the $U(1)$ part of the
$U(N)$ gauge field to be a ${\mathbb{Z}}_{N}$ gauge field that is fixed by the
choice of $B_{2}$. Due to the fact that we can realize $U(N)=\frac{SU(N)\times
U(1)}{{\mathbb{Z}}_{N}}$, we see that the ${\mathbb{Z}}_{N}$ gauge field in
the $U(1)$ factor is actually identified with a ${\mathbb{Z}}_{N}$ gauge
factor along the center ${\mathbb{Z}}_{N}\subset SU(N)$.

Mathematically, turning on the background gauge field for ${\mathbb{Z}}_{N}$
center symmetry is equivalent to lifting the $SU(N)$ gauge theory (where we
sum over $SU(N)$ bundles with connection in the path integral) to a
$SU(N)^{c}$ gauge theory where we sum over $SU(N)^{c}$ bundles with fixed
fractional flux $B_{2}$. Here we use the notation $SU(N)^{c}$ to denote the
$SU(N)$ analog of lifting a $Spin(N)$ bundle to a $Spin(N)^{c}$ bundle (which
is somewhat standard in geometry).252525We would like to thank G. Moore (and
N. Seiberg) for comments clarifying this point. See Appendix A.4 for more
details.

Now we can understand how to introduce a background gauge field for the
magnetic 1-form symmetry of $G$. Let us start with ${\widetilde{G}}$ gauge
theory where $\pi_{1}({\widetilde{G}})=1$. $G$ gauge theory differs by taking
the quotient of the gauge group ${\widetilde{G}}$ by $\pi_{1}(G)\subset
Z({\widetilde{G}})$. At the level of the gauge theory, taking the quotient of
the gauge group is equivalent to gauging 1-form center symmetry
$\pi_{1}(G)\subset Z({\widetilde{G}})$. This can be seen by noting that
gauging $\Gamma\subset Z$ projects out states that are electrically charged
under $Z$ while adding states that have fractional magnetic charge. This is
analogous to the story of orbifolding 2D CFT’s where we project out charged
states while adding twisted sectors. See DiFrancesco:1997nk ; Ginsparg:1988ui
for related reviews.

Formally, gauging $\pi_{1}(G)\subset Z({\widetilde{G}})$ can be achieved by
summing over all possible $B_{2}$ corresponding to $\pi_{1}(G)\subset
H^{2}(M_{d};Z({\widetilde{G}}))$. At the level of the path integral, we can
express this as

$\displaystyle\begin{split}Z_{G}=\sum_{[B_{2}]}Z_{{\widetilde{G}}}[B_{2}]~{},\end{split}$
(219)

where $Z_{{\widetilde{G}}}[B_{2}]$ is the path integral of the
${\widetilde{G}}$ gauge theory with background field $B_{2}$ turned on.

We can now formally introduce a new background gauge field $B_{2}^{(m)}$ as
the discrete gauge field that is “dual” to $B_{2}$:

$\displaystyle\begin{split}Z_{G}=\sum_{[B_{2}]}Z_{{\widetilde{G}}}[B_{2}]e^{{\frac{in}{2\pi}}\int
B_{2}^{(m)}\cup
B_{2}}=\sum_{[B_{2}]}Z_{{\widetilde{G}}}[B_{2}]e^{{\frac{in}{2\pi}}\int\hat{B}_{2}^{(m)}\wedge\hat{B}_{2}}~{},\end{split}$
(220)

where we have written $\hat{B}_{2}$ as an integral lift of the discrete
cohomology class $B_{2}$ and $n=|\pi_{1}(G)|$. Here, $B_{2}^{(m)}$ becomes the
1-form background gauge field for the new magnetic 1-form symmetry associated
to $\pi_{1}(G)$. Here we see explicitly that taking the quotient of the gauge
group exchanges center symmetry for magnetic 1-form symmetry.

Remark Note that including charged matter fields can “break” center symmetry.
The physical reason is that in addition to gluons, dynamical charged matter
fields can also screen Wilson lines. Thus, adding matter fields that are
charged under the center of the gauge group can lead to additional screening
of Wilson lines, thereby reducing the dynamically conserved charge.

####  3.3.4 Instantons and Center Symmetry

An important consequence of center symmetry is that turning on a background
gauge field has a dramatic effect on instanton physics Aharony:2013hda ;
Gaiotto:2017yup ; Cordova:2019uob .

Let us consider the $\theta$-term in $SU(N)$ Yang-Mills theory. Here we can
turn on a background gauge field $B_{2}$ for the ${\mathbb{Z}}_{N}$ center
symmetry as discussed in the previous section. The $\theta$-term now reads

$\displaystyle\begin{split}S_{\theta}=\frac{i\theta}{8\pi^{2}}\int{\rm
Tr}[(\tilde{f}_{2}-B_{2}\mathds{1}_{N})\wedge(\tilde{f}_{2}-B_{2}\mathds{1}_{N})]~{}.\end{split}$
(221)

We can now expand this out to separate out the background gauge field
component from the dynamical field:

$\displaystyle\begin{split}S_{\theta}=\frac{i\theta}{8\pi^{2}}\int\left({\rm
Tr}[\tilde{f}_{2}\wedge\tilde{f}_{2}]-NB_{2}\wedge
B_{2}\right)~{},\end{split}$ (222)

where we used the constraint ${\rm Tr}[\tilde{f}_{2}-B_{2}\mathds{1}_{N}]=0$.
Now we can express the first term as an integral class (i.e. the second Chern
class):

$\displaystyle\begin{split}c_{2}(\tilde{f}_{2})=\frac{{\rm
Tr}[\tilde{f}_{2}\wedge\tilde{f}_{2}]-{\rm Tr}[\tilde{f}_{2}]\wedge{\rm
Tr}[\tilde{f}_{2}]}{8\pi^{2}}~{},\end{split}$ (223)

since $\tilde{f}_{2}$ is simply a $U(N)$ gauge field that we are interpreting
in a special way. After trivial manipulation, we find that the $\theta$-term
can be written as

$\displaystyle\begin{split}S_{\theta}&=\frac{i\theta}{8\pi^{2}}\int\left({\rm
Tr}[\tilde{f}_{2}\wedge\tilde{f}_{2}]-{\rm Tr}[\tilde{f}_{2}]\wedge{\rm
Tr}[\tilde{f}_{2}]\right)+\frac{i\theta}{8\pi^{2}}N(N-1)\int B_{2}\wedge
B_{2}\\\ &=i\theta\left(\int c_{2}(\tilde{f}_{2})+N(N-1)\int\frac{B_{2}\wedge
B_{2}}{8\pi^{2}}\right)~{}.\end{split}$ (224)

Now, since $N\oint{\frac{B_{2}}{2\pi}}\in{\mathbb{Z}}$, we see that turning on
a background gauge field for ${\mathbb{Z}}_{N}$ center symmetry shifts the
instanton number by a $1/N$-fractional amount. In fact, it is often standard
to define the integral class $w_{2}=N\frac{B_{2}}{2\pi}\in
H^{2}(M_{d};{\mathbb{Z}})$ to emphasize the fact that the new term is
fractional

$\displaystyle\begin{split}S_{\theta}=i\theta\left(n+\frac{(N-1)}{N}\int\frac{w_{2}\wedge
w_{2}}{2}\right)\quad,\quad\int\frac{w_{2}\wedge
w_{2}}{2}\in{\mathbb{Z}}~{}.\end{split}$ (225)

As we discussed, mathematically, turning on a ${\mathbb{Z}}_{N}$ background
gauge field is equivalent to lifting the $SU(N)$ gauge theory to a $SU(N)^{c}$
bundles with fixed fractional flux: $B_{2}$. Our computation here shows that
lifting to $SU(N)^{c}$ bundles in this way results in summing over gauge
fields that have a fixed fractional flux and hence a fractional instanton
number:

$\displaystyle\begin{split}I_{inst}=\frac{(N-1)}{N}\int\frac{w_{2}\wedge
w_{2}}{2}~{}{\rm mod}_{\mathbb{Z}}~{}.\end{split}$ (226)

See Gaiotto:2014kfa  for more details.

There is also a generalization of 1-form center symmetry story to the case
where there are combined actions of gauge and global symmetries that act
trivially on all the matter in the theory. In this case, one can only turn on
combined fractional fluxes in the gauge and global symmetry groups which is
sometimes called BCU-, or CFU- background Anber:2021iip ; Anber:2019nze . This
idea is is important for the analysis of anomalies in gauge theories as well
as for the idea of charge fractionalization Wang:2018qoy ; Brennan:2022tyl ;
Delmastro:2022pfo ; Barkeshli:2014cna .

###  3.4 Discrete $\theta$-Angles

Here we would like to demonstrate a way in which discrete symmetries can be
implemented to further probe non-abelian gauge theories with center symmetry.

We consider gauge theories with gauge groups that have a trivial center. These
theories, while defined in their own right, can also be obtained by taking the
gauge theory of the simply connected cover and then gauging the center
symmetry.

A simple case is $G=SU(N)$ where $Z(SU(N))={\mathbb{Z}}_{N}$. The centerless
form of the group is then $PSU(N)=SU(N)/{\mathbb{Z}}_{N}$. If we write the
partition function of $SU(N)$ gauge theory with $\theta$-term and background
gauge field $B_{2}$ for the center symmetry as $Z_{SU(N)}[B_{2};\theta]$, then
we can write down the partition function of $PSU(N)$ gauge theory as

$\displaystyle\begin{split}Z_{PSU(N)}[\theta]=\sum_{[B_{2}]\in
H^{2}(M_{d};{\mathbb{Z}}_{N})}Z_{SU(N)}[B_{2};\theta]~{}.\end{split}$ (227)

In general, one is allowed to include $B_{2}$-dependent (gauge-invariant)
phases to the partition function that differentiate between different
topological sectors. In general, these phases are simply added by

$\displaystyle\begin{split}Z_{PSU(N)}=\sum_{[B_{2}]\in
H^{2}(M_{d};{\mathbb{Z}}_{N})}Z_{SU(N)}[B_{2};\theta]\;e^{{\frac{ipN}{2\pi}}\int\frac{B_{2}\wedge
B_{2}}{2}}~{},\end{split}$ (228)

where $p=0,...,N-1$. These terms are called _discrete theta angles_.

Note that this term is the only possible phase we can add to the partition
function that is only dependent on $B_{2}$ and is gauge invariant. Note that
the quantization of $p$ also comes from the demanding that the phase is gauge
invariant (which in this case means independent of choice of integer lift
$B_{2}\to B_{2}+\Lambda_{2}$ where ${\frac{\Lambda_{2}}{2\pi}}\in
H^{2}(M_{d};{\mathbb{Z}})$).

However, we know that there is an intimate connection between the instanton
number and the proposed phase we wish to add. In particular, due to the
identification

$\displaystyle\begin{split}I=\frac{1}{8\pi^{2}}\int{\rm Tr}[F\wedge
F]=N(N-1)\int\frac{B_{2}\wedge B_{2}}{8\pi^{2}}~{}{\rm
mod}_{\mathbb{Z}}~{},\end{split}$ (229)

we see that shifting $\theta\to\theta+2\pi n$ shifts the phase

$\displaystyle\begin{split}Z_{SU(N)}[B_{2};\theta+2\pi
n]=Z_{SU(N)}[B_{2};\theta]e^{-{\frac{inN}{2\pi}}\int\frac{B_{2}\wedge
B_{2}}{2}}~{}.\end{split}$ (230)

So, we see that in the case of $PSU(N)$ gauge theory, we can generate all of
the possible phases in the partition function by simply shifting $\theta$.

However, this is not the case for a generic gauge group. For other gauge
groups, shifting $\theta$ does not generate all such possible phases. In the
following table we enumerate the relations for the classical Lie groups (
(6.13) in Aharony:2013hda ):

$\widetilde{G}$ | Center | Background Fields |  $I$ modZ | Extra Phases
---|---|---|---|---
$SU(N)$ | ${\mathbb{Z}}_{N}$ | ${\frac{B_{2}}{2\pi}}=\frac{1}{N}w_{2}$ | $\frac{N-1}{N}\int\frac{w_{2}\wedge w_{2}}{2}$ | none
$Sp(2N)$ | ${\mathbb{Z}}_{2}$ | ${\frac{B_{2}}{2\pi}}=\frac{1}{2}w_{2}$ | always $\in{\mathbb{Z}}$ | $\frac{1}{2}\int\frac{w_{2}\wedge w_{2}}{2}$
$Sp(2N+1)$ | ${\mathbb{Z}}_{2}$ | ${\frac{B_{2}}{2\pi}}=\frac{1}{2}w_{2}$ | $\frac{1}{2}\int\frac{w_{2}\wedge w_{2}}{2}$ | none
$Spin(2N+1)$ | ${\mathbb{Z}}_{2}$ | ${\frac{B_{2}}{2\pi}}=\frac{1}{2}w_{2}$ | always $\in{\mathbb{Z}}$ | $\frac{1}{2}\int\frac{w_{2}\wedge w_{2}}{2}$
$Spin(8N)$ | ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ |  ${\frac{B_{2}}{2\pi}}=\frac{1}{2}w_{2}$, $\frac{{\tilde{B}}}{2\pi}=\frac{1}{2}\tilde{w}_{2}$ | $\frac{1}{2}\int w_{2}\wedge\tilde{w}_{2}$ |  $\frac{1}{2}\int\frac{w_{2}\wedge w_{2}}{2}$, $\frac{1}{2}\int\frac{\tilde{w}_{2}\wedge\tilde{w}_{2}}{2}$
$Spin(8N+4)$ | ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ |  ${\frac{B_{2}}{2\pi}}=\frac{1}{2}w_{2}$, $\frac{{\tilde{B}}}{2\pi}=\frac{1}{2}\tilde{w}_{2}$ | $\frac{1}{2}\int\left(\frac{w_{2}\wedge w_{2}}{2}+\frac{\tilde{w}_{2}\wedge\tilde{w}_{2}}{2}\right)$ |  $\frac{1}{2}\int\frac{w_{2}\wedge w_{2}}{2}$, $\frac{1}{2}\int w_{2}\wedge\tilde{w}_{2}$
$Spin(8N+2)$ | ${\mathbb{Z}}_{4}$ | ${\frac{B_{2}}{2\pi}}=\frac{1}{4}w_{2}$ | $\frac{1}{4}\int\frac{w_{2}\wedge w_{2}}{2}$ | none
$Spin(8N+6)$ | ${\mathbb{Z}}_{4}$ | ${\frac{B_{2}}{2\pi}}=\frac{1}{4}w_{2}$ | $-\frac{1}{4}\int\frac{w_{2}\wedge w_{2}}{2}$ | none
$\widetilde{E_{6}}$ | ${\mathbb{Z}}_{3}$ | ${\frac{B_{2}}{2\pi}}=\frac{1}{3}w_{2}$ | $\frac{2}{3}\int\frac{w_{2}\wedge w_{2}}{2}$ | none
$\widetilde{E_{7}}$ | ${\mathbb{Z}}_{2}$ | ${\frac{B_{2}}{2\pi}}=\frac{1}{2}w_{2}$ | $\frac{1}{2}\int\frac{w_{2}\wedge w_{2}}{2}$ | none
$E_{8}$ | trivial | none | always $\in{\mathbb{Z}}$ | none

This tells us that pure gauge theories with gauge group $Sp(2N)$, $SO(2N+1)$,
$SO(8N)$, $SO(8N+4)$ all have phases we can turn on in the path integral that
are independent of the usual $\theta$-angle. These extra phases are in a sense
independent discrete $\theta$-angles. For more details see Aharony:2013hda ;
Hsin:2020nts .

###  3.5 Comment on Dual Higher Form Global Symmetries

We would now like to comment on the role of dual higher form global symmetries
and their relation to the anomaly of $p$-form gauge theories. Here we will
compare and contrast the dual symmetry for continuous vs. discrete symmetries.

We say that a pair of continuous higher form symmetries are dual when their
currents are related by Hodge duality. This occurs in the following situation.
Let us consider the case of a conserved (co-closed) $(p+1)$-form current
$J_{p+1}$ that is associated to a $p$-form global symmetry. We say that there
is a dual $(d-p-1)$-form current ${\tilde{J}}_{d-p-1}$ if $J_{p+1}$ is also
closed:

$\displaystyle\begin{split}dJ_{p+1}=0\quad\Rightarrow\quad
d\ast{\tilde{J}}_{d-p-1}=0\quad,\quad{\tilde{J}}_{d-p-1}=\ast
J_{p+1}~{}.\end{split}$ (231)

This current corresponds to a $(d-p-2)$-form global symmetry which we call the
“dual” global symmetry. It follows that dual continuous global symmetries are
related as:

$\displaystyle\begin{split}\text{Continuous:~{}~{}}p\text{-form
symmetry}~{}\longleftrightarrow~{}(d-p-2)\text{-form symmetry}~{}.\end{split}$
(232)

Due to the fact that $dJ_{p+1}=d\ast J_{p+1}=0$, we can locally express
$J_{p+1}$ in terms of a $p$-form field $A_{p}$ that satisfies:

$\displaystyle\begin{split}J_{p+1}=dA_{p}\quad,\quad d\ast
dA_{p}=0~{},\end{split}$ (233)

which are simply the equations of motion for $p$-form electromagnetism. We
then see that $J_{p+1}$ and $J_{d-p-1}$ are the currents corresponding to dual
electric and magnetic higher form symmetries and thus carry an anomaly of the
type (113).

In this setting, EM duality can be implemented by coupling $A_{p}$ to a
dynamical $(d-p-2)$-form $U(1)^{(d-p-3)}$ gauge field $\tilde{A}_{d-p-2}$ via

$\displaystyle\begin{split}S=...+{\frac{i}{2\pi}}\int dA_{p}\wedge
d\tilde{A}_{d-p-2}~{}.\end{split}$ (234)

This process is allowed and well defined since it is gauge invariant. Now we
can see that the duality is induced at the level of the equations of motion:

$\displaystyle\begin{split}d\left(\ast
dA_{p}-{\frac{d\tilde{A}_{d-p-2}}{2\pi}}\right)=0~{}\Longrightarrow~{}{\frac{d\tilde{A}_{d-p-2}}{2\pi}}=\ast
dA_{p}~{}.\end{split}$ (235)

Then by integrating out the gauge field $A_{p}$, we find a theory of the dual
gauge field $\tilde{A}_{d-p-2}$.262626 In the case of $4d$ $U(1)$ Maxwell
theory, this duality relates the field strength $dA_{1}$ to the field strength
of a dual gauge field $d\tilde{A}_{1}=\ast dA_{1}$ and integrating out $A_{1}$
in exchange for $\tilde{A}_{1}$ implements $U(1)$ electric-magnetic duality.
Now the duality relating the $p$\- and $(d-p-2)$-form global symmetries can be
understood as the exchanging the symmetry that measures the charges of Wilson
lines of $A_{p}$ for the symmetry that measures the charges of Wilson lines of
$\tilde{A}_{d-p-2}$.

There is a similar story for discrete global symmetries. Here, pairs of dual
symmetries are related by

$\displaystyle\begin{split}\text{Discrete:~{}~{}}p\text{-form
symmetry}~{}\longleftrightarrow~{}(d-p-1)\text{-form symmetry}~{}.\end{split}$
(236)

Note that the degree of the dual symmetry is different from that of the
continuous case. The reason is because continuous gauge theories have
2-derivative actions while discrete gauge theories have 1-derivative actions.
To see this, we can identify dual symmetries as in the continuous case by
performing the discrete version of EM-duality.

Let us consider a ${\mathbb{Z}}_{N}$ $p$-form discrete gauge field which we
lift to a $U(1)$ gauge field $\hat{A}_{p}$. Now let us introduce a dual
${\mathbb{Z}}_{N}$ $(d-p-1)$ form gauge field that we lift to $U(1)$:
$\hat{B}_{d-p-1}$. We can again implement the duality between these two by
adding the gauge invariant coupling272727 Here we see that this is gauge
invariant under ${\mathbb{Z}}_{N}$ gauge transformations: $A_{p}\to
A_{p}+d\lambda_{p-1}$ and $B_{d-p-1}\to B_{d-p-1}+d\Lambda_{d-p-2}$. Since
${\frac{d\Lambda}{2\pi}},\ {\frac{d\lambda}{2\pi}}$, ${\frac{NA_{p}}{2\pi}}$,
and ${\frac{NB_{d-p-1}}{2\pi}}$ are all integer quantized, the shift
$\displaystyle\begin{split}\delta S={\frac{iN}{2\pi}}\int d\lambda_{p-1}\wedge
dB_{d-p-1}\in 2\pi i{\mathbb{Z}}~{},\end{split}$ (237) means that the
exponentiated action is gauge invariant.

$\displaystyle\begin{split}S=...+{\frac{iN}{2\pi}}\int\hat{A}_{p}\wedge
d\hat{B}_{d-p-1}~{}.\end{split}$ (238)

Note that in contrast to the case of continuous gauge symmetries, the discrete
quantization of discrete gauge fields allows us to write down a gauge
invariant coupling between a $p$-form and a $(d-p-1)$-form which means that
the dual to a ${\mathbb{Z}_{N}}^{(p)}$ discrete $p$-form symmetry is a
${\mathbb{Z}_{N}}^{(d-p-1)}$ $(d-p-1)$-form global symmetry. As in the case of
continuous electric-magnetic duality, dual discrete symmetries will carry the
anomaly of discrete gauge theory (233).

###  3.6 Mixed 0-Form/1-Form Anomaly

We would now like to discuss a general class of mixed anomaly between 0-form
and 1-form global symmetries in QCD-like gauge theories.

Consider a $4d$ $SU(N)$ non-abelian gauge theory coupled to some collection of
charged fermions $\\{\psi_{i}\\}$. Now let us consider an abelian flavor
symmetry $G_{0}$ that rotates the fermions:

$\displaystyle\begin{split}\psi_{i}\longmapsto
e^{iq_{i}\alpha}\psi_{i}~{}.\end{split}$ (239)

Here we can take the rotation parameter $\alpha$ to be either discrete or
continuous. In the case where there is an ABJ anomaly for $G_{0}$, the chiral
rotation above generates a term in the action

$\displaystyle\begin{split}\frac{ik\alpha}{8\pi^{2}}\int{\rm Tr}[F\wedge
F]~{},\end{split}$ (240)

for some $k\in{\mathbb{Z}}$.

Now we see that the action is invariant under shifts of $\alpha$ if
$\alpha\in\frac{2\pi}{k}$. Because of this, we say that the symmetry is broken
to ${\mathbb{Z}}_{k}\cap G_{0}$ by the ABJ anomaly. In the case where
${\mathbb{Z}}_{k}\subset G_{0}$, it simply breaks to ${\mathbb{Z}}_{k}$. Let
us assume that this is indeed the case so that the ABJ anomaly breaks
$G_{0}\to{\mathbb{Z}}_{k}$.

Let us take the fermions to transform in in representations of $G$ so that the
center symmetry is preserved. In this case we can turn on a background gauge
field for the center symmetry ${\frac{B_{2}}{2\pi}}=\frac{1}{N}w_{2}$, where
$w_{2}\in H^{2}(M_{4};{\mathbb{Z}})$. As we have discussed, turning on this
background gauge field for the center symmetry of $SU(N)$ changes the
quantization of the instanton number:

$\displaystyle\begin{split}\frac{1}{8\pi^{2}}\int{\rm Tr}[F\wedge
F]=\frac{N-1}{N}\int\frac{w_{2}\wedge w_{2}}{2}\quad{\rm
mod}_{\mathbb{Z}}~{}.\end{split}$ (241)

This fractional quantization changes the standard periodicity of the
$\theta$-angle from $2\pi$ to to $2\pi N$.

This has a nontrivial implication on the $G_{0}$ ABJ anomaly. As with the
$\theta$-angle, when the background gauge field $B_{2}$ is turned on, the
effective periodicity of $\alpha$ is modified to $2\pi N/k$. This means that
under the ${\mathbb{Z}}_{k}$ transformation, the partition function shifts by
a phase given by:

$\displaystyle\begin{split}\delta_{\alpha}Z[w_{2}]={\rm
exp}\left\\{ik\alpha\frac{N-1}{N}\int\frac{w_{2}\wedge
w_{2}}{2}\right\\}\,Z[w_{2}]~{}.\end{split}$ (242)

Explicitly, this means that the ${\mathbb{Z}}_{k}$ symmetry is broken in the
fractional instanton background. Since this phase is only dependent on the
background gauge field and transformation parameters of a global symmetry, we
can interpret the phase as a mixed 0-form/1-form ’t Hooft anomaly between
${\mathbb{Z}}_{k}$ and 1-form center symmetry. If the gauge group were
$PSU(N)$ instead, then the fractional instantons would be summed over in the
path integral and the anomalous phase (242) becomes an ABJ anomaly which
breaks ${\mathbb{Z}}_{k}$ to its anomalous subgroup
${\mathbb{Z}}_{k}\to{\mathbb{Z}}_{\text{gcd}(k,N)}$.

###  3.7 The Time Reversal/Center Symmetry Anomaly in Yang-Mills Theory

Let us consider $SU(N)$ Yang-Mills theory in $4d$ . We take the action to
include a $\theta$-angle:

$\displaystyle\begin{split}S=\int\frac{1}{2g^{2}}{\rm Tr}[F\wedge\ast
F]+\frac{i\theta}{8\pi^{2}}{\rm Tr}[F\wedge F]~{}.\end{split}$ (243)

One important feature of this theory is that the $\theta$-term generically
breaks time reversal symmetry $T$. Effectively, $T$ acts on $\theta$ as

$\displaystyle\begin{split}T:~{}~{}\theta\longmapsto-\theta~{}.\end{split}$
(244)

Now, because the instanton number is integer quantized in $SU(N)$ gauge
theory, $\theta$ is $2\pi$-periodic, $\theta\sim\theta+2\pi$. This implies
that $T$-symmetry is preserved when $\theta=0,\pi$ since $\pi=-\pi+2\pi$.

As we have discussed many times by this point, $SU(N)$ Yang-Mills additionally
has a 1-form center symmetry that measures $N$-ality of the representation of
Wilson lines. As in previous discussions, we can turn on a background gauge
field $\frac{B_{2}}{2\pi}=\frac{1}{N}w_{2}$ for the center symmetry. Again
this changes the quantization of the instanton number

$\displaystyle\begin{split}\frac{1}{8\pi^{2}}\int{\rm Tr}[F\wedge
F]=\frac{N-1}{N}\int\frac{w_{2}\wedge w_{2}}{2}\quad{\rm
mod}_{\mathbb{Z}}~{}.\end{split}$ (245)

The effect of this fractionalization of the instanton number is that it
changes the periodicity of $\theta$ to $2\pi N$. Now we see that $\theta=\pi$
is no longer symmetric under $T$ (i.e. $\pi\neq-\pi$ mod2πN). Rather, we now
see that the partition function now transforms under $T$-symmetry as

$\displaystyle\begin{split}\delta_{T}Z[w_{2}]={\rm exp}\left\\{2\pi
i\frac{N-1}{N}\int\frac{w_{2}\wedge
w_{2}}{2}\right\\}\,Z[w_{2}]~{}.\end{split}$ (246)

Since this phase is only dependent on background gauge fields, we can
interpret this phase as a mixed ’t Hooft anomaly between time reversal and
center symmetry Gaiotto:2017yup .

Because ’t Hooft anomalies are preserved under RG flows, this implies that
non-abelian Yang-Mills theory cannot flow to a trivially gapped theory at
$\theta=\pi$. Much of the analysis of Gaiotto:2017yup  is then dedicated to
investigating the implications of this anomaly on what we know about the phase
structure of the low energy effective theory. For related topics, also see
Komargodski:2017keh ; Gaiotto:2017tne ; Gomis:2017ixy ; Cordova:2017kue ;
Cordova:2018acb .

##  4 Higher Group Symmetries

As it turns out, higher form global symmetries fit into a more general
symmetry structure called higher group symmetry. Formally, higher groups can
be realized as an $n$-category – generalizing the structure of a group
Baez:2004in ; Baez:2010ya . However, in physics it is perhaps more useful to
think of higher group global symmetry as the symmetry structure that arises
when higher form symmetries of different degrees mix together.

Let us consider a theory with a collection of $p_{i}$-form global symmetries
where

$\displaystyle\begin{split}p_{1}<p_{2}<...<p_{n}~{}.\end{split}$ (247)

Now introduce background gauge fields for the $p_{i}$-form global symmetries,
$A_{i}$, which transform under the standard background gauge transformations

$\displaystyle\begin{split}\delta_{i}A_{i}=d\Lambda_{p_{i}}~{}.\end{split}$
(248)

We say that the theory has a _higher group global symmetry_ – in particular a
$(p_{n}+1)$-group global symmetry, which we denote $\mathbb{G}^{(p_{n}+1)}$ –
if the theory is not invariant under (248), but rather is only invariant under
a more general type of background gauge transformation:

$\displaystyle\begin{split}\delta A_{i}=d\Lambda_{p_{i}}+\sum_{j\leq
i}\Lambda_{p_{j}}\wedge\alpha_{j}^{(i)}(\\{A_{j}\\})+\\{{\rm
Schwinger~{}terms}\\}~{},\end{split}$ (249)

where $\alpha_{j}^{(i)}$ is a $(p_{i}-p_{j})+1$-form that is dependent on
background gauge fields $A_{j}$ for $j<i$ and Schwinger terms are terms that
are non-linear in the background gauge transformation parameters (but again we
demand that these only depend on $\\{A_{j},\Lambda_{p_{j}}\\}$ for $j<i$).

It is important to note that the background gauge fields $A_{i}$ only
transform non-trivially under the background gauge transformations of the
$p_{j}$-form global symmetries of lower degree ($p_{j}<p_{i}$). This fact
leads to two important consequences Cordova:2018cvg ; Brennan:2020ehu .

Breaking Patterns for $\mathbb{G}^{(p_{n}+1)}$

First let us consider the possible breaking patterns of the above higher group
symmetry. Let us consider the (spontaneous) breaking of the top $p_{n}$-form
global symmetry. Since the theory only preserves the modified transformations
as in (249), any background transformation that couples to a non-trivial
$\alpha^{(n)}_{j}$ will generate a non-trivial $A_{n}$. However, since the
$p_{n}$-form global symmetry is broken, this implies that _all_ background
gauge transformations that couple to $A_{n}$ and their associated symmetries
are broken. Therefore, breaking the $p_{n}$-form symmetry generically breaks
the entire $p_{n}+1$-group $\mathbb{G}^{(p_{n}+1)}$ symmetry.

A simple generalization of the above analysis shows that breaking the
$p_{i}$-form global symmetry breaks all $p_{j}$-form symmetries for
$p_{j}<p_{i}$. This leads to a nested structure of higher group symmetries

$\displaystyle\begin{split}\mathbb{G}^{(p_{n}+1)}=\mathbb{G}^{(p_{n}+1)}_{p_{1}}\subset\mathbb{G}^{(p_{n}+1)}_{p_{2}}\subset...\subset\mathbb{G}^{(p_{n}+1)}_{p_{n}}~{},\end{split}$
(250)

where $\mathbb{G}^{(p_{n}+1)}_{p_{i}}$ is the $(p_{n}+1)$-group whose lowest
degree symmetry is a $p_{i}$-form global symmetry.

Because of this nested $(p_{n}+1)$-group structure, the allowed breaking
patterns are of the form

$\displaystyle\begin{split}\mathbb{G}^{(p_{n}+1)}\longrightarrow\mathbb{G}^{(p_{n}+1)}_{p_{i}}\longrightarrow\mathbb{G}^{(p_{n}+1)}_{p_{j<i}}\longrightarrow\mathbb{G}^{(p_{n}+1)}_{p_{k<j}}\longrightarrow...\end{split}$
(251)

Emergent $\mathbb{G}^{(p_{n}+1)}$ Symmetry

The nested higher group structure also has an effect on the emergence of
higher group symmetries along RG flows. To see this, let us first consider a
related phenomenon with ordinary group symmetries.

Consider a theory with a 0-form global symmetry $G$ which is a group extension
of $H$ by an abelian group $A$:

$\displaystyle\begin{split}1\longrightarrow A\longrightarrow G\longrightarrow
H\longrightarrow 1~{}.\end{split}$ (252)

Such an extension is determined by a function $c:H\times H\to A$ with certain
nice properties (classified by $c\in H^{2}(H,A)$).282828Technically, an
extension of $H$ by an abelian group $A$ (called a central extension) is
classified by two pieces of data: 1.) an action of $H$ on $A$ given by
$\omega:H\to Aut(A)$ and 2.) a twisted cohomology class $c\in
H^{2}_{\omega}(H,A)$. In the case where $\omega$ is trivial,
$H^{2}_{\omega}(H,A)=H^{2}(H,A)$ and the extension is given by the data above.
See GmooreGroupTheory  for more details.  See Dijkgraaf:1989pz ;
GmooreGroupTheory  for details.

Let us realize $G$ as a set $G=A\times H$. The non-triviality of $c$ implies
that the multiplication of $h_{1},h_{2}\in H$ generates an element of $A$:

$\displaystyle\begin{split}(1,h_{1})\cdot(1,h_{2})=(c(h_{1},h_{2})\,,\,h_{1}h_{2})~{}.\end{split}$
(253)

This means that $H$ is not a subgroup of $G$.

Therefore, along an RG flow, a theory cannot have $H$-symmetry without also
(or first) developing $A$-symmetry. This is clear because if $A$ is not a
symmetry, the multiplication of two elements of $H\subset G$ can generate an
element of $A$ which implies $H$ also cannot be a symmetry. Therefore, along
an RG flow we find a hierarchy of scales

$\displaystyle\begin{split}E_{A}\gtrsim E_{H}~{},\end{split}$ (254)

where $E_{A,H}$ are the scales at which $A,H$-symmetries emerge.

There is a completely analogous story for higher group symmetries. Let us
consider an RG flow which starts at a UV symmetry with no higher form symmetry
that flows in the IR to a theory with a $p_{n}$-group global symmetry in the
IR. Since the $\mathbb{G}^{(p_{n})}_{p_{i}}$ obey a nested structure as in
(251), we see that along an RG flow there must also be a hierarchy of scales

$\displaystyle\begin{split}...\gtrsim E_{p_{i+1}\text{-}form}\gtrsim
E_{p_{i}\text{-}form}\gtrsim E_{p_{i-1}\text{-}form}\gtrsim...~{},\end{split}$
(255)

where $E_{p_{i}\text{-}form}$ is the energy scale at which the $p_{i}$-form
symmetry emerges.

The hierarchy of energy scales clearly gives one a great deal of information
about the structure of the theory along the RG flow. Since symmetries often
emerge when dynamical fields are either integrated out or when dynamical
extended objects (such as strings or domain walls) become stable, the
hierarchy of energy scales can often be translated into a statement about
different coupling constants in theories that develop higher group global
symmetry in the IR.

Note that since the above bounds are not strict, the proper physical
interpretation is that the bound cannot be parametrically violated:

$\displaystyle\begin{split}E_{p_{i+1}\text{-}form}\not{\ll}E_{p_{i}\text{-}form}~{}.\end{split}$
(256)

We will now give a couple of examples of higher group global symmetries. For
more uses of higher group global symmetries in phenomenological settings, see
Cordova:2022qtz ; Hidaka:2020iaz ; Hidaka:2020izy ; Brennan:2020ehu ;
Brennan:2023kpw .

###  4.1 Example: 2-Groups

A 2-group is the simplest example of a higher group global symmetry. They
consist of a 0-form global symmetry that mixes with a 1-form global symmetry
Baez:2004in ; Baez:2010ya ; Gukov:2013zka ; Kapustin:2013uxa ; Cordova:2018cvg
; Benini:2018reh . Here we will take an example that is a continuous 2-group
(i.e. made up of continuous 0- and 1-form global symmetries) although more
generally we can also consider 2-groups that are composed of discrete higher
form symmetries.

Let us consider the example of a $U(1)$ 0-form global symmetry that mixes with
a $U(1)$ 1-form global symmetry in $4d$ . Here we have a background gauge
field $A_{1}$ (with curvature $F_{2}$) for the 0-form global symmetry and
$B_{2}$ (with curvature $H_{3})$ for the 1-form global symmetry. Their
transformation properties obey

$\displaystyle\begin{split}\delta A_{1}=d\lambda_{0}\quad,\quad\delta
B_{2}=d\Lambda_{1}-{\frac{\kappa}{2\pi}}\lambda_{0}F_{2}~{}.\end{split}$ (257)

Theories that have 2-group global symmetry are often realized as theories that
have a 1-form global symmetry and an anomalous 0-form global symmetry whose
6-form anomaly polynomial can be realized as

$\displaystyle\begin{split}{{\cal I}}_{6}=i\kappa\int\frac{F_{2}\wedge
F_{2}\wedge\ast J_{2}}{4\pi}~{},\end{split}$ (258)

where $J_{2}$ is the current for the 1-form global symmetry. In this case, the
anomalous transformation of the partition function is of the form

$\displaystyle\begin{split}\delta_{\lambda_{0}}Z={\rm
exp}\left\\{{\frac{i\kappa}{2\pi}}\int\lambda_{0}F\wedge\ast
J_{2}\right\\}Z~{}.\end{split}$ (259)

However, this anomalous phase can be canceled by modifying the standard
transformation of the background gauge field $B_{2}$ to (257) due to the
coupling

$\displaystyle\begin{split}S=...+i\int B_{2}\wedge\ast J_{2}~{}.\end{split}$
(260)

Note that this cancellation is simply an application of the Green-Schwartz
mechanism Green:1984sg .

####  4.1.1 Concrete Example: QED

Let us consider a concrete theory that has a 2-group global symmetry. Let us
take $U(1)_{g}$ gauge theory with dynamical gauge field $a_{g}$ (with
curvature $f_{g}$) coupled to 4 Weyl fermions $\\{\psi_{i}^{\pm}\\}$. In
addition to being charged under the gauge group $U(1)_{g}$, the fermions also
transform under a global flavor symmetry $U(1)_{f}$:

Field | $U(1)_{g}$ | $U(1)_{f}$
---|---|---
$\psi_{1}^{+}$ | +1 | +1
$\psi_{2}^{+}$ | +1 | $-1$
$\psi_{1}^{-}$ | $-1$ | +$q$
$\psi_{2}^{-}$ | $-1$ | $-q$

The most general possible anomaly is of the form

$\displaystyle\begin{split}{{\cal
I}}_{6}=\frac{1}{(2\pi)^{2}}\int\Big{[}\frac{\kappa_{g^{3}}}{3!}f_{g}\wedge
f_{g}\wedge f_{g}+\frac{\kappa_{g^{2}f}}{2!}f_{g}\wedge f_{g}\wedge
F_{2}+\frac{\kappa_{gf^{2}}}{2!}f_{g}\wedge F_{2}\wedge
F_{2}+\frac{\kappa_{f^{3}}}{3!}F_{2}\wedge F_{2}\wedge
F_{2}\Big{]}~{}.\end{split}$ (261)

By a standard computation we can compute the coefficients to be

$\displaystyle\begin{split}\kappa_{g^{3}}=\kappa_{f^{3}}=\kappa_{g^{2}f}=0\quad,\quad\kappa_{gf^{2}}=2(1-q^{2})~{}.\end{split}$
(262)

Let us assume that $|q|\neq 1$ so that $U(1)_{f}$ has an ’t Hooft anomaly:

$\displaystyle\begin{split}{{\cal I}}_{6}=\frac{2(1-q^{2})}{8\pi^{2}}\int
f_{g}\wedge F_{2}\wedge F_{2}~{}.\end{split}$ (263)

This leads to the anomalous variation of the path integral by inflow

$\displaystyle\begin{split}\delta_{\lambda_{0}}Z={\rm
exp}\left\\{\frac{2(1-q^{2})i}{(2\pi)^{2}}\int\lambda_{0}F_{2}\wedge
f_{g}\right\\}Z~{}.\end{split}$ (264)

In addition to the (anomalous) $U(1)_{f}$ 0-form global symmetry, the theory
also has a 1-form magnetic global symmetry $\ast J_{2}=\frac{1}{2\pi}f_{g}.$
This allows us to rewrite the anomalous variation as

$\displaystyle\begin{split}\delta_{\lambda_{0}}Z={\rm
exp}\left\\{{\frac{2(1-q^{2})i}{2\pi}}\int\lambda_{0}F_{2}\wedge\ast
J_{2}\right\\}Z~{}.\end{split}$ (265)

This symmetry can be coupled to a background gauge field $B_{2}$ by adding to
the action

$\displaystyle\begin{split}S=...+i\int B_{2}\wedge\ast J_{2}~{},\end{split}$
(266)

which allows us to cancel the anomaly by modifying the transformation
properties of $B_{2}$:

$\displaystyle\begin{split}\delta
B_{2}=d\Lambda_{1}-{\frac{2(1-q^{2})}{2\pi}}\lambda_{0}F_{2}~{}.\end{split}$
(267)

This is the 2-group global symmetry of the theory.

####  4.1.2 Spontaneous Symmetry Breaking

Now we can ask what happens when we spontaneously break 2-group global
symmetry Cordova:2018cvg . Let us consider the case where we have a 0-form
global symmetry $G^{(0)}$ and a $U(1)^{(1)}$ 1-form global symmetry that make
up a 2-group that is spontaneously broken. From what we know, the spontaneous
breaking of $G^{(0)}$ gives us a $G^{(0)}$-valued Goldstone boson and the
spontaneously broken $U(1)^{(1)}$ 1-form symmetry gives us a $U(1)$ gauge
field.

Let us denote the Goldstone boson $\chi_{a}$ and the $U(1)$ gauge field
$a_{1}$. This theory now has a 0-form global shift symmetry of $\chi_{a}$ and
a dynamical Goldstone gauge field $a_{1}$ (with curvature $f_{2})$. The action
can then be written as

$\displaystyle\begin{split}S=\int{\rm Tr}[d\chi\wedge\ast
d\chi]+\frac{1}{2g^{2}}f_{2}\wedge\ast f_{2}~{}.\end{split}$ (268)

This theory now has a 0-form shift symmetry of $\chi$ and a 1-form magnetic
symmetry of $f_{2}$. Let us introduce background gauge fields $A_{1}$ and
$B_{2}$ respectively. Now the background gauge fields couple to the theory as

$\displaystyle\begin{split}S=\int{\rm Tr}[(d\chi-A_{1})\wedge\ast(d\chi-
A_{1})]+\frac{1}{2g^{2}}f_{2}\wedge\ast f_{2}+2\pi iB_{2}\wedge
f_{2}-\frac{i\kappa}{4\pi^{2}}{\rm Tr}[\chi F_{2}]\wedge f_{2}~{}.\end{split}$
(269)

Now we see that the theory is invariant under the 2-group transformations

$\displaystyle\begin{split}\chi\to\chi+\lambda\quad,\quad A_{1}\to
A_{1}+d\lambda\quad,\quad B\to B+d\Lambda_{1}-{\frac{\kappa}{2\pi}}\lambda
F_{2}~{}.\end{split}$ (270)

This indeed reproduces the current equation

$\displaystyle\begin{split}dj_{3}={\frac{\kappa}{2\pi}}F_{2}\wedge
J_{2}~{},\end{split}$ (271)

where now

$\displaystyle\begin{split}j_{3}=\ast(d\chi-
A_{1})-\frac{i\kappa}{4\pi^{2}}d\chi\wedge f_{2}~{}.\end{split}$ (272)

####  4.1.3 Implication on IR Theory

Now let us consider the case where the above theory is the IR limit of some UV
theory which we _assume has no 2-group global symmetry_. By assumption, such a
UV completion must break the 0-form flavor symmetry and 1-form magnetic
symmetry by dynamical objects. In particular, the breaking of 1-form magnetic
symmetry requires a theory with dynamical monopoles (as in a non-abelian gauge
theory).

So let us take the UV theory to be an $SU(2)$ gauge theory (which has no
magnetic 1-form symmetry) coupled to an adjoint Higgs field. We can now embed
the fermions in the UV theory as two Weyl fermions in the fundamental
representation. If we introduce a potential for the Higgs field so that it
develops a vev, the theory will flow in the IR to a $U(1)$ gauge theory with 2
Weyl fermions of charge +1 and 2 Weyl fermions of charge $-1$. Now there is an
emergent symmetry $U(1)_{f}\subset SU(4)$ as considered in Section 4.1.1 which
mixes with the 1-form global symmetry to make up the 2-group. This symmetry is
broken in the IR by the fact that it does not commute with the $SU(2)$ gauge
symmetry.

$\displaystyle\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
134.83391pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\text{UV: }SU(2)~{}\text{ gauge theory, }\Phi\text{
adjoint scalar }\Psi_{1},\Psi_{2}\text{ fund
Weyl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-28.79443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\big{\langle}Tr[\Phi]^{2}\big{\rangle}=v^{2}\Rightarrow
SU(2)\to
U(1)~{},~{}\Psi_{1,2}\to\psi_{1,2}^{\pm}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-71.35553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\text{IR: }U(1)\text{ gauge theory with
}\psi_{1,2}^{\pm}\text{ Weyl with
2-group}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$ (273)

The fact that 2-group symmetry emerges in the IR implies that constraints on
the above RG flow (and indeed any RG flow that ends at our IR theory in
question). The constraint is

$\displaystyle\begin{split}E_{1\text{-}form}\gtrsim
E_{0\text{-}form}~{},\end{split}$ (274)

where $E_{p\text{-}form}$ is the energy scale where the $p$-form symmetry
emerges.

In our above proposed UV completion $E_{1\text{-}form}$ is the scale where
monopoles become stable which is determined by the scale at which non-abelian
gauge symmetry is broken. This is the energy scale needed to “unwind” the
gauge field, which is the scale necessary to activate dynamical non-abelian
degrees of freedom:

$\displaystyle\begin{split}E_{1\text{-}form}=m_{W}=gv~{},\end{split}$ (275)

where $g$ is the gauge coupling.

The scale $E_{0\text{-}form}$ is the scale where the $U(1)_{f}$ flavor
symmetry emerges which also occurs when we spontaneously break gauge symmetry.
Here, the flavor symmetry is violated by non-abelian interactions at tree
level – i.e. scattering with $W$-bosons. This means that the energy scale
$E_{0\text{-}form}$ is also given by

$\displaystyle\begin{split}E_{0\text{-}form}=m_{W}=gv~{}.\end{split}$ (276)

We then see that the hierarchy constraint from emergent 2-group symmetry is
trivially satisfied since

$\displaystyle\begin{split}E_{1\text{-}form}=E_{0\text{-}form}=m_{W}=gv~{}.\end{split}$
(277)

###  4.2 Example: 3-Groups

A 3-group is the next simplest example of a higher group global symmetry. It
consists of 0-form, 1-form, and 2-form global symmetries that mix together.
Let us consider the case where the three symmetry groups
$G^{(0)},G^{(1)},G^{(2)}$ are all $U(1)$. Now introduce background gauge
fields $A_{1},B_{2},C_{3}$. The most general transformations of these fields
are of the form292929Here we restrict to terms that are gauge invariant (i.e.
not dependent on the gauge fields except through their field strength).

$\displaystyle\begin{split}\delta A_{1}&=d\lambda_{0}~{},\\\ \delta
B_{2}&=d\Lambda_{1}+k\lambda_{0}F_{1}~{},\\\ \delta
C_{3}&=d\Lambda_{2}+\alpha\,d\lambda_{0}\wedge
F_{1}+\beta_{1}\,\Lambda_{1}\wedge B_{2}+\beta_{2}\,\Lambda_{1}\wedge
d\Lambda_{1}+\kappa_{1}\,\Lambda_{1}\wedge F_{2}+...\end{split}$ (278)

Due to the complexity of these transformation properties, it is perhaps
clearer to illustrate the structure of the 3-group with an example.

####  4.2.1 Axion-Yang-Mills

Here we will take a slightly different example in which $G^{(0)}$ is trivial,
and $G^{(1)}={\mathbb{Z}_{N}}^{(1)}$.

Let take $4d$ $SU(N)$ axion-Yang-Mills Brennan:2020ehu ; Cordova:2019uob ;
Hidaka:2020iaz ; Hidaka:2020izy . This theory is described by the action

$\displaystyle\begin{split}S=\frac{1}{2}\int da\wedge\ast
da+\frac{1}{g^{2}}\int{\rm Tr}[F\wedge\ast F]-\frac{i}{8\pi^{2}f_{a}}\int
a\,{\rm Tr}[F\wedge F]~{},\end{split}$ (279)

where $a\sim a+2\pi f_{a}$.

This theory has two higher form global symmetries: ${\mathbb{Z}}_{N}^{(1)}$
1-form center symmetry and $U(1)^{(2)}$ 2-form axion string symmetry. Here
${\mathbb{Z}}_{N}^{(1)}$ is the standard center symmetry that shifts the gauge
field by a ${\mathbb{Z}}_{N}$ gauge field. The $U(1)^{(2)}$ magnetic string
symmetry has a current given by $\ast J_{3}=\frac{1}{2\pi f_{a}}da$. As
discussed in Section 2.3.1, this symmetry counts the winding charge of the
axion field.

We can now couple the theory to background fields $B_{2}$, $C_{3}$ for
${\mathbb{Z}}_{N}^{(1)}$ and $U(1)^{(2)}$:

$\displaystyle\begin{split}S=&\frac{1}{2}\int da\wedge\ast da+\frac{i}{2\pi
f_{a}}\int a\,dC_{3}+\frac{1}{g^{2}}\int{\rm
Tr}[(F-B_{2})\wedge\ast(F-B_{2})]\\\ &-\frac{i}{8\pi^{2}f_{a}}\int a\,{\rm
Tr}[(F-B_{2})\wedge(F-B_{2})]+\int\varphi\,{\rm Tr}[F-B_{2}]~{}.\end{split}$
(280)

Note that here we have chosen to integrate the coupling of $C_{3}$ to $\ast
J_{3}$ by parts.

Now we see that turning on the background gauge field $B_{2}$ causes the axion
coupling to be ill-defined. In other words, the fact that the instanton number
is now fractional

$\displaystyle\begin{split}\frac{N}{8\pi^{2}}\int{\rm
Tr}[(F-B_{2})\wedge(F-B_{2})]\in{\mathbb{Z}}~{},\end{split}$ (281)

implies that the action is no longer invariant under $a\to a+2\pi f_{a}$:

$\displaystyle\begin{split}\delta_{a}S=-\frac{2\pi N(N-1)}{8\pi^{2}}\int
B_{2}\wedge B_{2}\neq 2\pi in~{},\end{split}$ (282)

where $n\in{\mathbb{Z}}$. Since this is a gauge transformation that makes the
axion a periodic scalar, we would say that the 1-form symmetry is explicitly
broken.

However, the offending variation of the action can be eliminated by modifying
the background gauge transformations for $C_{3}$. In particular, we can
elevate its field strength

$\displaystyle\begin{split}G_{4}=dC_{3}+\frac{N(N-1)}{4\pi}B_{2}\wedge
B_{2}~{},\end{split}$ (283)

which we demand is gauge invariant under gauge transformations of
${\mathbb{Z}}_{N}^{(1)}$ and $U(1)^{(2)}$. This implies that the fields
transform under the 3-group background as transformations

$\displaystyle\begin{split}B_{2}&\longmapsto B_{2}+d\Lambda_{1}~{},\\\
C_{3}&\longmapsto C_{3}+d\Lambda_{2}-\frac{N(N-1)}{2\pi}\Lambda_{1}\wedge
B_{2}-\frac{N(N-1)}{4\pi}\Lambda_{1}\wedge d\Lambda_{1}~{}.\end{split}$ (284)

These transformations indicate a 3-group symmetry structure without a 0-form
global symmetry.

This theory has several variations that include modifying the axion coupling
and adding charged matter fields which all have more interesting/complicated
3-group symmetries. For further details see Brennan:2020ehu .

####  4.2.2 Constraints on RG Flow

We can now infer information about UV completions of our theory by using the
constraints from emergent 3-group symmetry along RG flows that end at axion-
Yang-Mills:

$\displaystyle\begin{split}E_{2\text{-}form}\gtrsim
E_{1\text{-}form}~{}.\end{split}$ (285)

Let us examine this hierarchy of scales in a simple UV completion. Take the UV
theory to be a $SU(N)$ gauge theory with a pair of Weyl fermions $\psi_{\pm}$
in conjugate (fundamental/anti-fundamental) representations and an uncharged
complex scalar field $\varphi(x)$. Let us take the action to be

$\displaystyle\begin{split}S=\int d^{4}x\left\\{\frac{1}{g^{2}}{\rm
Tr}[F\wedge\ast
F]+i\bar{\psi}_{\pm}\not{D}\psi_{\pm}+\frac{1}{2}d\varphi\wedge\ast
d\bar{\varphi}-V(\varphi)+\lambda\bar{\varphi}\psi_{+}\psi_{-}+\lambda\varphi\bar{\psi}_{-}\bar{\psi}_{+}\right\\}~{},\end{split}$
(286)

where we take the potential

$\displaystyle\begin{split}V(\varphi)=m^{2}(|\varphi|^{2}-f_{a}^{2})^{2}~{}.\end{split}$
(287)

This theory has a classical $U(1)$ global symmetry which we will denote
$U(1)_{PQ}$ under which the matter fields have charges

Field |  $U(1)_{PQ}$ charge
---|---
$\psi_{\pm}$ | +1
$\varphi$ | +2

This symmetry suffers from an ABJ anomaly so that the rotation

$\displaystyle\begin{split}\psi_{\pm}\to e^{i\alpha}\psi_{\pm}~{},\end{split}$
(288)

generates a term

$\displaystyle\begin{split}\frac{i\alpha}{8\pi^{2}}\int{\rm Tr}[F\wedge
F]~{},\end{split}$ (289)

in the action.

Now let us flow to the IR. Along the flow, the scalar field will obtain a vev
$\big{\langle}|\varphi|\big{\rangle}=f_{a}$. Below the scale $\sqrt{m}f_{a}$,
the scalar field is classically unable to reach the local maximum of the
scalar potential so that the field is effectively valued in an interval$\times
S^{1}$ and the axion strings become stable. Thus, in the effective theory at
energies $E<\sqrt{m}f_{a}$, it is appropriate to describe the field as

$\displaystyle\begin{split}\varphi=(f_{a}+\rho)e^{ia/f_{a}}~{},\end{split}$
(290)

where $\rho,a$ are dynamical degrees of freedom.

Now the Yukawa term becomes

$\displaystyle\begin{split}{\cal L}_{Y}=\lambda
f_{a}e^{ia/f_{a}}\psi_{+}\psi_{-}+...\end{split}$ (291)

We can then perform a chiral rotation of the $\psi_{\pm}$ to move the coupling
to $a(x)$ to an axion term

$\displaystyle\begin{split}S_{ax}=\frac{i}{8\pi^{2}f_{a}}\int a\,{\rm
Tr}[F\wedge F]~{}.\end{split}$ (292)

Now, the Yukawa term has generated a mass term for the fermions where
$m_{\psi}=\lambda f_{a}$. Thus, if we flow to the IR below the mass scale of
the radial mode $\rho$ and the fermions $\psi_{\pm}$, we can trivially
integrate out the massive fields $\rho,\psi$. Then, the theory flows to pure
axion-Yang-Mills which has an emergent 3-group symmetry.

Now we can again consider the 3-group constraint (285). Here the 2-form
symmetry is emergent when the axion strings becomes stable. As we discussed
above, this occurs when the scalar field $\varphi$ becomes effectively
interval$\times S^{1}$-valued at $E_{2\text{-}form}\sim\sqrt{m}f_{a}$.
Additionally, the 1-form center symmetry emerges when the fundamental fermion
fields are integrated out which allows us to identify
$E_{1\text{-}form}=m_{\psi}=\sqrt{m}f_{a}$.

Now by putting these together, we see that the bound from emergent 3-group
symmetry implies

$\displaystyle\begin{split}\sqrt{m}\gtrsim\lambda~{}.\end{split}$ (293)

As it turns out this bound coincides with the bound from EFT. As shown in
Brennan:2020ehu  when $\sqrt{m}\ll\lambda$ the fermion contribution to the
scalar potential is large and outside of the control of EFT.

##  5 Non-Invertible Symmetries

As we have stressed, the notion of global symmetries in quantum theories is
best phrased in terms of the topological operators of the theory. Thus far, we
have focused on the theories where the topological operators have additional
special properties such as obeying a group multiplication law. However,
topological operators more generally are described by the story of
“categorical” global symmetries. In general, categorical symmetries do not
obey a group like structure. Because of this, we often call the symmetries
which are not group-like or higher group-like _non-invertible_ global
symmetries.

There is a lot of mathematical structure that is used to describe the general
categorical symmetries – this structure gets more and more complicated in
higher and higher dimensions. Here it is our goal to avoid category theory,
but the subject is large and incredibly rich/interesting! We direct the
interested reader to Bhardwaj:2023ayw ; Bartsch:2023wvv ; Schafer-
Nameki:2023jdn ; Bhardwaj:2022lsg ; Bhardwaj:2022yxj ; Bartsch:2022ytj ;
Bartsch:2022mpm ; Freed:2022qnc ; Freed:2022iao ; Kaidi:2022cpf ; Choi:2022jqy
; Choi:2022zal ; Choi:2022fgx ; Choi:2022rfe ; Cordova:2022ieu ;
Bhardwaj:2022kot ; Bhardwaj:2022maz  for more details.

Here we would like to describe a couple constructions of non-invertible
symmetries that arise in Lagrangian theories. The main idea of a non-
invertible symmetry is the following. Consider a set ${\cal C}$ to which we
associate topological defect operators ${\cal D}_{x}$ for $x\in{\cal C}$. As
we have discussed, if ${\cal D}_{x}$ were group-like symmetry defect
operators, their multiplication would obey some group structure: ${\cal
D}_{x_{1}}\times{\cal D}_{x_{2}}={\cal D}_{x_{1}x_{2}}$. However, the fusion
of two defect operators can obey a more general categorical structure which
can take the form

$\displaystyle\begin{split}{\cal D}_{x_{1}}\times{\cal
D}_{x_{2}}=\begin{cases}\sum_{i}{\cal D}_{x_{i}}\\\ {\cal T}\otimes{\cal
D}_{x_{1}x_{2}}\\\ \quad\vdots\end{cases}\end{split}$ (294)

for example, where here ${\cal T}$ is the partition function of some $3d$
TQFT.

The linchpin for constructions in Lagrangian theories of non-invertible
symmetries is the construction of what are called condensation defects. The
idea and terminology name for condensation defects comes from condensed
matter. The general idea is that in a $d$-dimensional QFT, we can gauge a
global symmetry along a fixed $p$-dimensional hypersurface. This allows us to
define a larger class of topological defects in the theory – and therefore
allows us to construct a larger class of global symmetries.

This idea of condensation defects is a bit esoteric, so let us relate this
idea to something that is perhaps more familiar. The idea of condensation
defects is somewhat similar to the idea of having a defect with a non-trivial
world volume theory such as axion strings or monopoles. However, for
condensation defects the defect theory is a gauge theory that couples to the
bulk via a conserved current. In particular, we can think of condensation
defects as theories which flow to a TQFT in the IR that couples to the bulk
theory only via a gauge-current interaction.

In this section, we plan to first discuss the construction of condensation
defects. Then we will go on to discuss two examples of non-invertible
symmetries in Lagrangian theories: non-invertible chiral symmetries and non-
invertible 1-form symmetries in axion-electromagnetism.

###  5.1 Condensation Defects

Let us consider a $d$-dimensional theory with a $p$-form global symmetry
$G^{(p)}$. For the moment let us assume that the symmetry has no obstructions
to being gauged – i.e. no ABJ anomaly or ‘t Hooft anomaly only involving
itself. The symmetry $G^{(p)}$ can be gauged in the full space time. As we
have discussed this can be formally treated by summing over all possible
insertions of the associated symmetry defect operators in the path integral.

In our theory, we can consider removing a co-dimension $q$ hypersurface
$\Sigma^{(q)}$. We can then introduce a “defect” QFT that lives on this
$\Sigma^{(q)}$ that couples to the bulk theory. This theory by virtue of
coupling to the bulk theory admits an action of the symmetry $G^{(p)}$. This
can be seen by taking charged operators in the bulk theory and dragging them
to the defect – this leads to operators in the defect theory that must also
necessarily be charged under the action of the bulk $G^{(p)}$. However, we are
also allowed to couple to QFTs where $G^{(p)}$ is gauged on $\Sigma^{(q)}$. In
this case, the result of dragging a charged operator to $\Sigma^{(q)}$ will
lead to charged operator in the defect theory which will in general be dressed
by a Wilson operator to make a gauge invariant operator. When we are allowed
to gauge the symmetry $G^{(p)}$ on any $\Sigma^{(q)}$, we say that the theory
is _$q$ -gaugeable_.

Formally, we can think about gauging a $q$-gaugeable $p$-form symmetry
$G^{(p)}$ along a co-dimension-$q$ hypersurface $\Sigma^{(q)}$ by summing over
the insertion of all $(d-q-p-1)$-dimensional symmetry defect operators
restricted to $\Sigma^{(q)}$. Practically, we can think of gauging $G^{(p)}$
along $\Sigma^{(q)}$ by introducing an associated gauge field $B_{p+1}$ that
only lives on $\Sigma^{(q)}$. Since these $G^{(p)}$ symmetry defect operators
restricted to $\Sigma^{(q)}$ are not the same as the bulk $G^{(p)}$ symmetry
defect operators, the restricted $G^{(p)}$ can have fewer ‘t Hooft anomalies
than the bulk $G^{(p)}$. This allows some symmetries to be $q$-gaugeable for
$q>p+1$ even though the bulk $G^{(p)}$ symmetry may have obstructions to its
gauging. In essence, restricting to a co-dimension-$q$ hypersurface can
sometimes trivialize anomalies of the bulk $G^{(p)}$ symmetry.

As in the ordinary procedure of gauging of any symmetry, we are allowed to add
a phase that weights the different gauge sectors – i.e. add a non-trivial
action. For our purposes, we will restrict to the case where we only consider
phases that are topological (this will be important for constructing non-
invertible symmetry defect operators). This class of local actions are given
by gauge-invariant Chern-Simons-type actions of $B_{p+1}$. These Chern-Simons-
type actions are also referred to as $(d-q)$-dimensional SPT phases that are
described by cohomology classes $\eta\in H^{q}(B^{p}G,U(1))$.

Formally, we can think of the condensation defects as operators we insert into
the path integral. Formally, we can write these as

$\displaystyle\begin{split}{\cal C}_{\eta}(\Sigma^{(q)})=\frac{1}{{\cal
N}}\sum_{B_{p+1}}{\rm exp}\left\\{i\int\eta[B_{p+1}]\right\\}~{},\end{split}$
(295)

where here the sum enforces the gauging, $\eta[B_{p+1}]$ is the Chern-Simons-
type action, and ${\cal N}$ is a normalization factor that captures the space
of gauge inequivalent $B_{p+1}$.

####  5.1.1 Example: 1-Gauging ${\mathbb{Z}}_{N}^{(1)}$ in 4d

Let us illustrate these ideas by showing how to 1-gauge a 1-form
${\mathbb{Z}}_{N}^{(1)}$ global symmetry in a $4d$ theory. Here, we will have
in mind the case where we gauge a ${\mathbb{Z}}_{N}^{(1)}$ magnetic symmetry
of a gauge theory.

Let us fix a $3d$ hypersurface $\Sigma^{(1)}$, and consider 1-gauging
${\mathbb{Z}}_{N}^{(1)}$ on $\Sigma^{(1)}$. On $\Sigma^{(1)}$, the magnetic
1-form global symmetry is generated by line operators (the same way that $3d$
Chern-Simons theory has a 1-form symmetry whose lines are both the charged
operators and symmetry defect operators). We can then 1-gauge the
${\mathbb{Z}}_{N}^{(1)}$ by coupling to a ${\mathbb{Z}}_{N}$ 1-form gauge
field $v_{1}$ that only lives on $\Sigma^{(1)}$. In the case where we are
1-gauging a ${\mathbb{Z}}_{N}^{(1)}$ magnetic symmetry, we can couple this
gauge field to the bulk symmetry as

$\displaystyle\begin{split}{\cal C}_{0}(\Sigma^{(1)})=\int[dv_{1}]\,e^{i\int
v_{1}\wedge\frac{f}{2\pi}}~{},\end{split}$ (296)

where here the path integral over $v_{1}$ is restricted to gauge fields on
$\Sigma^{(1)}$.

We can then couple the condensation defect by a SPT phase – i.e. Chern-Simons
term – so that we get an expression of the form

$\displaystyle\begin{split}{\cal C}_{N}(\Sigma^{(1)})=\int[dv_{1}]~{}{\rm
exp}\left\\{\frac{iN}{4\pi}\int v_{1}\wedge dv_{1}+\frac{i}{2\pi}\int
v_{1}\wedge f\right\\}~{}.\end{split}$ (297)

In the case where $p=1$, this particular choice of condensation defect is an
example of a minimal $3d$ ${\mathbb{Z}}_{N}$ TQFT which in the literature is
often denoted ${\cal A}^{N,1}(f)$ Hsin:2018vcg . This also describes the $3d$
fractional quantum Hall state at filling fraction $\nu=1/N$.

###  5.2 Example: Non-Invertible Chiral Symmetry in $4d$

In this section, we discuss non-invertible chiral symmetries appearing in $4d$
gauge theories with fermions. As we will see, these symmetries arise in the
study of the Standard Model and may have meaningful applications to BSM model
building. These symmetries were first introduced in Cordova:2022ieu ;
Choi:2022jqy  where it was used in the study of pions and axions.
Subsequently, non-invertible symmetries have also been applied to explain the
smallness of the neutrino mass Cordova:2022fhg  and further study axion
theories in Choi:2022rfe .

Let us take $4d$ QED: $U(1)$ gauge theory with a Dirac fermion $\psi$ of
charge $1$. This has an action

$\displaystyle\begin{split}S=\int\frac{1}{2g^{2}}f\wedge\ast
f+i\bar{\psi}\not{D}\psi~{},\end{split}$ (298)

where $f=dA$ is the dynamical field strength. This theory has a $U(1)_{A}$
chiral symmetry that acts as303030In the limit where the gauge coupling
vanishes, $g\to 0$, the free Dirac theory has $U(1)\times U(1)$ global
symmetry: $U(1)$ rotation for each Weyl fermions in the Dirac fermion
$\psi=(\chi,\epsilon\psi^{*})^{T}$. The vector-like combination is free of
cubic anomaly and is the gauged $U(1)$ in $4d$ QED. The orthogonal combination
is $U(1)_{A}$.

$\displaystyle\begin{split}\psi\longmapsto
e^{i\alpha\gamma_{5}}\psi\quad,\quad
j_{A}^{\mu}(x)=\bar{\psi}\gamma^{\mu}\gamma^{5}\psi(x)~{}.\end{split}$ (299)

However, the $U(1)_{A}$ chiral symmetry has an ABJ anomaly which is given by:

$\displaystyle\begin{split}d\ast j_{A}=\frac{2}{8\pi^{2}}f\wedge
f~{}.\end{split}$ (300)

This means that if we try to construct a symmetry defect operator for
$U(1)_{A}$

$\displaystyle\begin{split}U_{g}(\Sigma_{3})={\rm
exp}\left\\{i\alpha\oint_{\Sigma_{3}}\ast j_{A}\right\\}~{},\end{split}$ (301)

we see that the operator is not topological

$\displaystyle\begin{split}U_{g}(\Sigma_{3})=U_{g}(\Sigma_{3}^{\prime})\times{\rm
exp}\left\\{2i\alpha\int_{\Gamma_{4}}\frac{f\wedge
f}{8\pi^{2}}\right\\}~{},\end{split}$ (302)

where $\Gamma_{4}$ is a 4-manifold that fills in the space between
$\Sigma_{3},\Sigma_{3}^{\prime}$:
$\partial\Gamma_{4}=\Sigma_{3}\cup\overline{\Sigma_{3}^{\prime}}$ where
$\overline{\Sigma_{3}^{\prime}}$ is the orientation reversal of
$\Sigma^{\prime}$.

Starting from the anomaly equation (300), one may try to construct a conserved
current

$*\widehat{j}_{5}=*j_{A}-\frac{2}{8\pi^{2}}A\wedge dA~{},$ (303)

and a associated symmetry defect operator

$\widehat{U}_{g}(\Sigma_{3})=\text{exp}\left\\{i\alpha\oint_{\Sigma_{3}}\ast\widehat{j}\right\\}=\text{exp}\left\\{i\alpha\oint_{\Sigma_{3}}\left(\ast
j_{A}-\frac{2}{8\pi^{2}}A\wedge dA\right)\right\\}~{}.$ (304)

The conservation of the new current $d*\widehat{j}_{5}=0$ ensures that this
new symmetry defect operator is in fact topological. The issue with this
construction is that neither the current $\widehat{j}_{5}$ nor the symmetry
defect operator $\widehat{U}_{g}(\Sigma_{3})$ is gauge invariant and hence
they are ill-defined. This exercise, however, provides a useful viewpoint.

The above operator is ill-defined because of the Chern-Simons term which is in
general not gauge invariant for arbitrary $\alpha$. To see this more
explicitly, we recall that the $3d$ Chern-Simons action

$S_{\rm\scriptscriptstyle CS}=\frac{iK}{4\pi}\int A\wedge dA~{},$ (305)

is only gauge-invariant for $K\in 2{\mathbb{Z}}$ (or for $K\in{\mathbb{Z}}$ on
a spin manifold). This can be seen by noting that under a large gauge
transformation $A\to A+\Lambda_{1}$, the action shifts as

$\delta S_{\rm\scriptscriptstyle CS}=\frac{iK}{4\pi}\int\Lambda_{1}\wedge
d\Lambda_{1}+\frac{iK}{4\pi}\int 2\Lambda_{1}\wedge dA~{}.$ (306)

Using $\oint\Lambda_{1}=2\pi{\mathbb{Z}}$, one obtains the quantization
condition for $K$.

Naively, the fact that generically the modified (topological) defect operator
(304) has improperly quantized Chern-Simons level for generic $\alpha\in
U(1)_{A}$ would be interpreted as an explicit violation of the classical
$U(1)_{A}$ symmetry by quantum mechanical effects (reflecting the ABJ
anomaly). One would not expect any useful selection rules imposed by
$U(1)_{A}$ to hold in any quantum process. On the other hand, one may remember
that the abelian instanton does not exist on ${\mathbb{S}}^{4}$ (since
$\pi_{3}(U(1))=\emptyset$) and wonder if above described effect is completely
absent. This observation, suggests that there may be more going on.

To investigate, let us consider a rational angle $\alpha=\frac{2\pi}{N}$. The
Chern-Simons term in $\widehat{U}_{g}(\Sigma_{3})$ is given by

$\tilde{S}_{\rm\scriptscriptstyle FQH}=-\frac{i}{4\pi
N}\oint_{\Sigma_{3}}A\wedge dA.$ (307)

This is known in condensed matter literature as the action for the fractional
quantum hall state in $(2+1)d$ at filling fraction $\nu=\frac{1}{N}$. As it
turns out, there is a gauge-invariant description which is equivalent to this
naively ill-defined action.

The gauge-invariant action for the fractional quantum hall state in $(2+1)d$
at filling fraction $\nu=\frac{1}{N}$ is

$S_{\rm\scriptscriptstyle FQH}=\int_{\Sigma_{3}}\left(\frac{iN}{4\pi}a\wedge
da+\frac{i}{2\pi}a\wedge dA\right)~{},$ (308)

as we have discussed above. The theory of fractional hall state with $\nu=1/N$
is also known as ${\cal A}^{N,1}(f,\Sigma_{3})$ which is the “minimal
${\mathbb{Z}}_{N}$ TQFT” Hsin:2020nts . More generally, for $\alpha=\frac{2\pi
p}{N}$ we can construct a co-dimension 1 topological operator by dressing the
naive symmetry operator a fractional quantum hall state with $\nu=p/N$ (where
$\text{gcd}(N,p)=1$) which is denoted by ${\cal A}^{N,p}(f,\Sigma_{3})$:

$\displaystyle\begin{split}{\cal D}_{q}(\Sigma_{3})={\rm exp}\left\\{2\pi
i\,q\oint\ast j_{A}\right\\}\times{\cal A}^{N,p}(f,\Sigma_{3})\quad,\quad
q=\frac{p}{N}\in\mathbb{Q}/{\mathbb{Z}}~{}.\end{split}$ (309)

The general ${\cal A}^{N,p}(f,\Sigma_{3})$ are somewhat complicated define as
an intrinsically $3d$ theory. It is easier to describe as the boundary state
of a $4d$ TQFT which we will do in the next section.

####  5.2.1 Half-Space Gauging Construction

In this section, we will discuss the construction of the non-invertible chiral
symmetry defect operator (309) by means of “half-space gauging.” This will
clarify the topological nature of the non-invertible SDO and also illustrate
the role of magnetic 1-form symmetry. This latter point is especially
important when one thinks about the _breaking_ of non-invertible symmetry as
discussed in Cordova:2022ieu ; Cordova:2022fhg .

We start by noting that QED has a $U(1)$ magnetic 1-form symmetry as implied
by the Bianchi identity $df=0$. We imagine dividing the space time $M_{4}$
into two pieces by a 3-manifold $\Sigma_{3}$ and let us take
$x\in{\mathbb{R}}$ to be a normal a coordinate direction.

We define the insertion of the fractional quantum Hall state ${\cal
A}^{N,p}(f,\Sigma)$ by gauging ${\mathbb{Z}}_{N}^{(1)}$ subgroup of $U(1)$
1-form magnetic symmetry on the “half” of $M_{4}$ given by $x\geq 0$ where we
add to the action

${\cal A}^{N,p}(f,\Sigma)~{}\Longleftrightarrow~{}\Delta S=\int_{x\geq
0}\left(\frac{i}{2\pi}b_{2}\wedge f+\frac{iN}{2\pi}b_{2}\wedge
dc_{1}+\frac{Nk}{4\pi}b_{2}\wedge b_{2}\right)~{}.$ (310)

Here $pk=1~{}{\rm mod}_{N}$ (i.e. $k$ is the multiplicative inverse of $p$
modulo $N$) and we require $\text{gcd}(k,N)=1$ (which is equivalent to
$\text{gcd}(p,N)=1$). We also impose a topological boundary condition
$b_{2}|_{x=0}=0$. This condition is topological because
$b_{2}|_{x=0}-b_{2}|_{x^{\prime}=0}=db_{2}=0$.

In the half-space TQFT action, the first term describes the coupling of the
1-form magnetic symmetry current to a 2-form background gauge field $b_{2}$,
while the last two terms are the $4d$ ${\mathbb{Z}}_{N}$ BF theory with a SPT
term (see Section 3 for details). Here, $b_{2}$ and $c_{1}$ are dynamical
$U(1)$ 2- and 1-form gauge field which, due to the BF action, effectively
gauge ${\mathbb{Z}}_{N}^{(1)}\subset U(1)_{m}^{(1)}$.

We will now show that this gauging has an effect of shifting the $\theta$
parameter of QED by

$\theta\longmapsto\begin{cases}\theta-\frac{2\pi p}{N}&x\geq 0\\\
\theta&x<0\end{cases}$ (311)

To see this, let us consider the gauge field part of the action with $\theta$
angle:

$\displaystyle\begin{split}S=\int\left(\frac{1}{2g^{2}}f\wedge*f+\frac{i\theta}{8\pi^{2}}f\wedge
f\right)+\int_{x\geq 0}\left(\frac{i}{2\pi}b_{2}\wedge
f+\frac{iN}{2\pi}b_{2}\wedge dc_{1}+\frac{Nk}{4\pi}b_{2}\wedge
b_{2}\right)~{}.\end{split}$ (312)

First note that integrating out $c_{1}$ constrains $b_{2}$ to be a
${\mathbb{Z}}_{N}$ field and leads to a theory described by the action

$\displaystyle\begin{split}S=&\int\left(\frac{1}{2g^{2}}f\wedge*f+\frac{i\theta}{8\pi^{2}}f\wedge
f\right)+\int_{x\geq 0}\left(\frac{i}{2\pi}b_{2}\wedge
f+\frac{Nk}{4\pi}b_{2}\wedge b_{2}\right)\\\
=&\int\left(\frac{1}{2g^{2}}f\wedge*f+\frac{i\theta}{8\pi^{2}}f\wedge
f\right)+\int_{x\geq
0}\left(\frac{iNk}{4\pi}\left(b_{2}+\frac{p}{N}f\right)^{2}-\frac{2\pi
ip}{8\pi^{2}N}f\wedge f\right)~{}.\end{split}$ (313)

We can then perform the Gaussian integral over $b_{2}$ and we arrive at the
QED action with additional term defined on $x\geq 0$:

$\displaystyle\begin{split}S=&\int\left(\frac{1}{2g^{2}}f\wedge*f+\frac{i\theta}{8\pi^{2}}f\wedge
f\right)-\frac{2\pi ip}{8\pi^{2}N}\int_{x\geq 0}f\wedge f~{}.\end{split}$
(314)

We can rewrite this action as

$\displaystyle\begin{split}S=&\int\frac{1}{2g^{2}}f\wedge*f+\frac{i}{8\pi^{2}}\int\left(\theta-\frac{2\pi
ip}{N}\Theta(x)\right)f\wedge f~{},\end{split}$ (315)

where $\Theta(x)$ is the Heaviside step function. Therefore, we see that
inserting the operator ${\cal A}^{N,p}(f,\Sigma_{3})$ effectively shifts the
$\theta$-parameter across $\Sigma_{3}$ as in (311).

From this, we can see that the effect of the half-space gauging is not
independent of the choice of $\Sigma_{3}$ (i.e. the boundary dividing the two
halves of space time). Because of the difference of the (spatially dependent)
effective $\theta$-angle between the two disconnected regions, we can see
explicitly that the effect of deforming the operator ${\cal
A}^{N,p}(f,\Sigma_{3})$ (or in other words the effect on contribution to the
action (310) of deforming the boundary $\Sigma_{3}$) is:

$\displaystyle\begin{split}{\cal A}^{N,p}(f,\Sigma_{3})={\cal
A}^{N,p}(f,\Sigma_{3}^{\prime})\times{\rm exp}\left\\{-\frac{ip}{4\pi
N}\int_{\Gamma_{4}}f\wedge f\right\\}~{},\end{split}$ (316)

where $\Gamma_{4}$ is a 4-manifold that fills in the space between
$\Sigma_{3},\Sigma_{3}^{\prime}$:
$\partial\Gamma_{4}=\Sigma_{3}\cup\overline{\Sigma_{3}^{\prime}}$ where
$\overline{\Sigma_{3}^{\prime}}$ is the orientation reversal of
$\Sigma^{\prime}$.

We now see by comparing the non-topological phase of the naive symmetry defect
operator (302) to that of (316), we see that the operator constructed by
dressing the naive symmetry defect operator by ${\cal A}^{N,p}(f,\Sigma_{3})$

$\displaystyle\begin{split}{\cal D}_{q}(\Sigma_{3})={\rm exp}\left\\{2\pi
i\,q\oint\ast j_{A}\right\\}\times{\cal A}^{N,p}(f,\Sigma_{3})\quad,\quad
q=\frac{p}{N}\in\mathbb{Q}/{\mathbb{Z}}~{},\end{split}$ (317)

is indeed topological and hence defines a (generalized) symmetry for our
theory.

####  5.2.2 Non-Invertibility and Fusion Algebra

While non-invertible symmetry defect operators are topological and hence
define a symmetry, they do not obey the group structure of
$\mathbb{Q}/{\mathbb{Z}}\subset U(1)$ and therefore define a non-invertible
symmetry. We can demonstrate this explicitly by computing the product of
${\cal D}_{1/N}$ and ${\cal D}_{-1/N}$.

If the ${\cal D}_{q}$ obeyed the multiplicative group law of
$\mathbb{Q}/{\mathbb{Z}}$, we would expect that ${\cal D}_{1/N}\times{\cal
D}_{-1/N}=\mathds{1}$. First note that the factors of ${\rm
exp}\left\\{\frac{2\pi i\,p}{N}\oint\ast j_{5}\right\\}$ obey an invertible
multiplication law following from the computations of Section 2. We then need
to compute the product of the $3d$ minimal TQFTs.

In the case of ${\cal D}_{\pm 1/N}$, we can write down ${\cal
A}^{N,1}(f,\Sigma_{3})$ as a simpler condensation operator as in (297):

$\displaystyle\begin{split}{\cal A}^{N,1}(f,\Sigma_{3})={\cal
C}_{N}(\Sigma_{3})=\int[dv_{1}]~{}{\rm exp}\left\\{\frac{iN}{4\pi}\int
v_{1}\wedge dv_{1}+\frac{i}{2\pi}\int v_{1}\wedge f\right\\}~{}.\end{split}$
(318)

We can use this expression to explicitly compute the product

$\displaystyle\begin{split}{\cal D}_{1/N}\times{\cal
D}_{-1/N}=\int[dv_{1}][dv_{2}]~{}{\rm
exp}\left\\{\frac{iN}{4\pi}\int(v_{1}\wedge dv_{1}-v_{2}\wedge
dv_{2})+\frac{i}{2\pi}\int(v_{1}-v_{2})\wedge f\right\\}~{},\end{split}$ (319)

which simplifies by performing a change of variables
$v_{\pm}=\frac{1}{2}(v_{1}\pm v_{2})$:

$\displaystyle\begin{split}{\cal D}_{1/N}\times{\cal
D}_{-1/N}=\int[dv_{\pm}]~{}{\rm exp}\left\\{\frac{iN}{2\pi}\int v_{+}\wedge
dv_{-}+\frac{i}{2\pi}\int v_{-}\wedge f\right\\}\neq\mathds{1}~{},\end{split}$
(320)

which is equal to a type of condensation operator.

We can also see that the $\mathbb{Q}/{\mathbb{Z}}$ group structure is also
broken by computing ${\cal D}_{1/N}\times{\cal D}_{1/N}\neq{\cal D}_{2/N}$:

$\displaystyle\begin{split}{\cal D}_{1/N}\times{\cal D}_{1/N}=e^{\frac{4\pi
i}{N}\int\ast j_{5}}\,\int[dv_{1}][dv_{2}]~{}{\rm
exp}\left\\{\frac{iN}{4\pi}\int(v_{1}\wedge dv_{1}+v_{2}\wedge
dv_{2})+\frac{i}{2\pi}\int(v_{1}+v_{2})\wedge f\right\\}~{},\end{split}$ (321)

which again simplifies by the same change of variables

$\displaystyle\begin{split}{\cal D}_{1/N}\times{\cal D}_{1/N}&=e^{\frac{4\pi
i}{N}\int\ast j_{5}}\,\int[dv_{\pm}]~{}{\rm
exp}\left\\{\frac{iN}{4\pi}\int(v_{+}\wedge dv_{+}+v_{-}\wedge
dv_{-})+\frac{2i}{2\pi}\int v_{+}\wedge f\right\\}\\\ &={\cal
A}^{N,2}(f,\Sigma_{3})\times{\cal D}_{2/N}~{},\end{split}$ (322)

for $N$ odd. Here the equality follows from some non-trivial identities of the
${\cal A}^{N,p}(f)$ TQFTs. We refer the interested reader to Hsin:2018vcg  for
more details on the general computation.

While we have not given all of the details to fully compute the products of
the ${\cal D}_{q}$ operators, it should be clear even from (320) that the
topological operators ${\cal D}_{q}$, while labeled by elements of a set
$\mathbb{Q}/{\mathbb{Z}}$ (which can be given a group structure), define a
non-invertible symmetry.

###  5.3 Example: Non-Invertible Symmetries in Axion-Electromagnetism

Let us now demonstrate another example of a non-invertible symmetry in $4d$:
axion electromagnetism. This theory also contains several types of non-
invertible global symmetries. Let us take the action

$\displaystyle\begin{split}S=\int\frac{1}{2g^{2}}f\wedge\ast f+(\partial
a)^{2}+\frac{i}{8\pi^{2}f_{a}}\int a\,f\wedge f~{},\end{split}$ (323)

where $a\sim a+2\pi f_{a}$ and $f=dA$ is the $U(1)$ field strength. It is
commonly said that the axion, being a periodic boson, has a continuous shift
symmetry that is broken by the axion interaction term. Under a shift $\delta
a=\alpha\,f_{a}$ for $\alpha\in U(1)$, we see that the action shifts

$\displaystyle\begin{split}\delta S=i\alpha\int\frac{f_{2}\wedge
f_{2}}{8\pi^{2}}~{}.\end{split}$ (324)

This variation of the action is identical to that of a chiral symmetry with an
ABJ anomaly. Given our previous discussion of the non-invertible chiral
symmetry, we see that the axion shift symmetry can be restored to a
$\mathbb{Q}/{\mathbb{Z}}$ non-invertible shift symmetry in exactly the same
way. If we define the shift symmetry current as

$\displaystyle\begin{split}\ast J_{1}=\ast da~{},\end{split}$ (325)

then we can define the non-invertible symmetry defect as

$\displaystyle\begin{split}{\cal D}_{q}=e^{2\pi iq\int\ast J_{1}}\times{\cal
A}^{N,p}(f,\Sigma_{3})\quad,\quad
q=\frac{p}{N}\in\mathbb{Q}/{\mathbb{Z}}~{}.\end{split}$ (326)

This result is perhaps obvious if we note that we can realize the axion as the
pseudo-Goldstone mode for an ABJ-anomalous chiral symmetry as in the KSVZ
model:313131 The KSVZ model flows to axion-electromagnetism by noting that
$\phi$-condenses and gives a mass to the fermions. Then, when integrating out
the fermions, the chiral phase mode $\psi\mapsto e^{i\alpha\gamma_{5}/2}\psi$
gives rise to the axion by the fact that the associated symmetry is an ABJ-
anomalous symmetry (note that this symmetry also rotates $\phi\mapsto
e^{-i\alpha}\phi$). In modern language, the non-invertible chiral symmetry
enforces a sort of Goldstone theorem so that the symmetry which is
spontaneously broken by the $\phi$-condensation so that the IR theory has an
axion GarciaEtxebarria:2022jky .

$\displaystyle\begin{split}S_{\rm KSVZ}=\int\frac{1}{2g^{2}}f\wedge\ast
f+(\partial\phi)^{2}+i\bar{\psi}\not{D}\psi-m(|\phi|^{2}-f_{a}^{2})^{2}+i\lambda\phi\psi\psi+c.c.\end{split}$
(327)

Note that in this example the non-invertible chiral symmetry of the UV KSVZ
theory is matched by the non-invertible shift symmetry of the IR axion theory.

Axion electrodynamics also has a non-invertible 1-form symmetry Choi:2022fgx .
Let us consider the electric 1-form global symmetry of free electromagnetism.
It has an associated current

$\displaystyle\begin{split}\ast J_{2}^{(m)}=\frac{1}{g^{2}}\ast
f~{}.\end{split}$ (328)

When coupling to the axion theory, this group-like symmetry is also broken by
the interaction:

$\displaystyle\begin{split}d\ast J_{2}^{(e)}=\frac{1}{4\pi^{2}f_{a}}da\wedge
f~{}.\end{split}$ (329)

As we have discussed previously, this theory also has a conserved $U(1)^{(1)}$
1-form magnetic symmetry and $U(1)^{(2)}$ 2-form axion string symmetry which
have associated conserved currents:

$\displaystyle\begin{split}d\ast J_{2}^{(m)}=d\frac{f}{2\pi}=0\quad,\quad
d\ast J_{3}=d\frac{da}{2\pi f_{a}}=0~{}.\end{split}$ (330)

Now, because the violation of the conservation equation for $J_{2}^{(e)}$ can
be written in terms of currents:323232For the case of axion shift/chiral
symmetry, the conservation equation can also be written as
$\displaystyle\begin{split}d\ast J_{1}=\frac{1}{2}\ast J_{2}^{(m)}\wedge\ast
J_{2}^{(m)}~{},\end{split}$ (331) which is the reason that we are allowed to
“save” the ABJ-anomalous symmetry.

$\displaystyle\begin{split}d\ast J_{2}^{(e)}=\ast J_{3}\wedge\ast
J_{2}^{(m)}~{},\end{split}$ (332)

we can “save” the broken symmetry by going to a non-invertible global
symmetry. In particular, we can construct a topological (non-invertible)
defect operator by modifying the standard $U(1)^{(1)}_{e}$ 1-form electric
symmetry defect operator by coupling to a $2d$ TQFT:

$\displaystyle\begin{split}{\cal D}^{(1)}_{q}(\Sigma_{2})=e^{2\pi iq\int\ast
J_{2}^{(e)}}\times\int[D\phi\,Dc_{1}]~{}{\rm
exp}\left\\{\frac{i}{2\pi}\int\left(N\phi dc_{1}+p\frac{a}{f_{a}}dc_{1}+\phi
f\right)\right\\}~{},\end{split}$ (333)

where $q=\frac{p}{N}\in\mathbb{Q}$, $\phi\sim\phi+2\pi$ is a periodic scalar
field and $c_{1}$ is a dynamical $U(1)$ 1-form gauge field. We can explicitly
check that the multiplication of this topological defect is non-invertible:

$\displaystyle\begin{split}{\cal D}^{(1)}_{1/N}\times{\cal
D}^{(1)\dagger}_{1/N}&=\int[D\phi\,Dc_{1}\,D\phi^{\prime}\,Dc_{1}^{\prime}]~{}{\rm
exp}\left\\{\frac{i}{2\pi}\int\left(N(\phi
dc_{1}-\phi^{\prime}dc_{1}^{\prime}+\frac{a}{f_{a}}(dc_{1}-dc_{1}^{\prime})+(\phi-\phi^{\prime})f\right)\right\\}\\\
&=\int[D\phi_{\pm}\,Dc_{1}^{\pm}]~{}{\rm
exp}\left\\{\frac{i}{2\pi}\int\left(N\phi_{+}dc_{1}^{+}+N\phi_{-}dc_{1}^{-}+2\frac{a}{f_{a}}dc_{1}^{-}+\phi_{-}f\right)\right\\}\\\
&\neq 1\end{split}$ (334)

By comparison to the 0-form non-invertible symmetry, the 1-form non-invertible
symmetry is emergent along the RG flow from the KSVZ model.

The power of the non-invertible symmetries in this theory is that they impose
constraints on UV models. In particular, since the non-invertible symmetries
are good symmetries of the IR theory, they must either be matched in the UV
theory, or they must be emergent. In the case they are emergent, the structure
of the non-invertible symmetries requires that the symmetries associated to
the currents that arise in the conservation current for the non-invertible
symmetry emerge before the ABJ-anomalous current. This implies:

  * •

Non-invertible shift symmetry: This requires that the 1-form magnetic symmetry
emerge at energy scales higher than the classical shift symmetry

  * •

Non-invertible electric symmetry: This requires that the 1-form magnetic
symmetry and winding symmetry emerge at energy scales that are higher than the
scale of emergent classical electric symmetry

Note that the non-invertible 1-form symmetry gives a model-independent
constraint on the UV physics that is compatible with those derived from the
3-group global symmetry. And indeed, the energy heirarchy that results from
the analysis of non-invertible symmetries is more restrictive than that coming
from the 3-group global symmetry:333333Note that when the axion interaction is
of the form  $\displaystyle\begin{split}S_{axion}=\frac{iK}{8\pi^{2}f_{a}}\int
a\,f\wedge f\quad,\quad K\in{\mathbb{Z}}~{},\end{split}$ (335) for general
$K\neq\pm 1$, the 3-group symmetry structure is enhanced to include a
${\mathbb{Z}}_{|K|}^{(0)}$ shift symmetry. This further constrains ${\rm
min}\left\\{E_{\rm shift},E_{\rm electric}\right\\}\lesssim E_{\rm magnetic}$.
This inequality is still consistent (and weaker) with the one obtained from
non-invertible symmetries.

$\displaystyle\begin{split}{\rm 3-group:}&~{}E_{\rm electric}\lesssim E_{\rm
winding}~{},\\\ {\rm Non-Invertible:}&~{}E_{\rm electric}\lesssim{\rm
min}\left\\{E_{\rm winding},E_{\rm magnetic}\right\\}\quad,\quad E_{\rm
shift}\lesssim E_{\rm magnetic}~{}.\end{split}$ (336)

## Acknowledgements

DB would like to thank the University of Chicago Particle Physics group for
the invitation to give a series of group meetings as well as for numerous
comments. SH would like to thank the organizers and attendees of the series of
two “Snail Lectures” supported by the APCTP. We would especially like to thank
Liantao Wang, Seth Koren, Christian Ferko, and Anthony Ashmore for detailed
comments and insight. We would also like to thank Clay Cordova, Thomas
Dumitrescu, Liantao Wang, Ken Intriligator, Kantaro Ohmori, and Seth Koren for
fruitful collaborations on related topics and to Clay Cordova, Thomas
Dumitrescu, and Gregory Moore for explaining many of the finer points of
generalized global symmetries.

TDB is supported by Simons Foundation award 568420 (Simons Investigator) and
award 888994 (The Simons Collaboration on Global Categorical Symmetries) and
SH is supported by the DOE grants DE-SC-0013642 and DE-AC02-06CH11357.

##  Appendix A Differential Forms and Gauge Theory

In this section we will give a very brief review of the mathematical
background required for the study of generalized global symmetries. For a more
complete review see Nakahara:2003nw .

###  A.1 Crash Course on Differential Forms

One mathematical tool we will use again and again is that of differential
forms. Differential forms are a mathematical tool to describe geometric
properties of manifolds and bundles in a coordinate independent and index-free
language.

Differential forms are labeled by a degree $n$ which is a non-negative
integer. We will denote the set of differential $n$-form on a manifold $X$ as
$\Omega^{n}(X)$. The $n$-forms are made out of basic components which are
differential volume elements which themselves can be constructed from
differential line elements $dx$ for some set of coordinates $\\{x\\}$. The
differential 1-forms are of the form

$\displaystyle\begin{split}\omega_{1}=\sum_{i}f_{i}(x)dx_{i}\in\Omega^{1}(X)~{}.\end{split}$
(337)

These are the “dual” of vector fields. By this, we mean that there is a
natural inner product

$\displaystyle\begin{split}\langle\vec{v},\omega\rangle=\sum_{i}v^{i}(x)f_{i}(x)\quad,\quad\vec{v}=\sum_{i}v^{i}(x)\frac{\partial}{\partial
x^{i}}~{}.\end{split}$ (338)

And indeed, there is a map between vector fields and 1-forms (and more
generally anti-symmetric $n$-tensors and $n$-forms) by using the metric

$\displaystyle\begin{split}\vec{v}\longmapsto
v_{1}=\sum_{ij}g_{ij}v^{j}(x)dx_{i}~{}.\end{split}$ (339)

Differential $n$-forms can be constructed out of differential $1$-forms by the
_wedge product_. The wedge product acts on differential line elements as the
anti-symmetric (tensor) product

$\displaystyle\begin{split}dx_{i}\wedge dx_{j}=-dx_{j}\wedge
dx_{i}~{}.\end{split}$ (340)

Analogously, the wedge product of $n$ differential line elements is also their
anti-symmetric (tensor) product. The reason for this is that the
$n$-dimensional differential polytope (i.e. fundamental $n$-volume) has a
volume which is given by the anti-symmetrization of the differential line
elements. For example, the 3-dimensional differential parallelogram has a
volume element which is given by the triple product (i.e. anti-symmetric
product)

$\displaystyle\begin{split}d{\rm Vol}=d\vec{x}_{1}\cdot(d\vec{x}_{2}\times
d\vec{x}_{3})=dx_{1}\wedge dx_{2}\wedge dx_{3}~{}.\end{split}$ (341)

This is because the anti-symmetrization projects out any redundancies that
arises from coordinate systems that are not orthogonal. For example, the
change of coordinates $(x,y,z)\mapsto(x,y+f(x),z+g(x))$ leads to the
differential line elements

$\displaystyle\begin{split}&(dx,dy,dz)\longmapsto(dx\,,\,dy+f^{\prime}(x)dx\,,\,dz+g^{\prime}(x)dx)~{}.\end{split}$
(342)

However, due to the anti-symmetry of the wedge product, the differential
volume 3-form is invariant under the coordinate transformation

$\displaystyle\begin{split}dx\wedge dy\wedge dz\longmapsto dx\wedge dy\wedge
dz~{}.\end{split}$ (343)

Because of this, we can describe the generic $n$-form from the wedge product
of $n$ differential line elements:

$\displaystyle\begin{split}\omega_{n}=\sum_{i_{1},...,i_{n}}f_{i_{1}...i_{n}}(x)~{}dx_{i_{1}}\wedge...\wedge
dx_{i_{n}}~{}.\end{split}$ (344)

Notice that the structure of differential forms allows us to take the
arbitrary wedge product of a $p$-form $\alpha_{p}$ with a $q$-form
$\beta_{q}$:

$\displaystyle\begin{split}\alpha_{p}=\sum_{i_{1},...,i_{p}}\alpha_{i_{1}...i_{p}}(x)~{}dx_{i_{1}}\wedge...\wedge
dx_{i_{p}}\quad,\quad\beta_{p}=\sum_{j_{1},...,j_{q}}\beta_{j_{1}...j_{q}}(x)~{}dx_{j_{1}}\wedge...\wedge
dx_{j_{q}}~{},\\\
\alpha_{p}\wedge\beta_{p}=\sum_{i_{1},...,i_{p}}\sum_{j_{1},...,j_{q}}\alpha_{i_{1}...i_{p}}(x)\beta_{j_{1}...j_{q}}(x)dx_{i_{1}}\wedge...\wedge
dx_{i_{p}}\wedge dx_{j_{1}}\wedge...\wedge dx_{j_{q}}~{}.\end{split}$ (345)

One can show from the anti-symmetry of wedge-product on differential line
elements that

$\displaystyle\begin{split}\alpha_{p}\wedge\beta_{q}=(-1)^{pq}\beta_{q}\wedge\alpha_{p}~{}.\end{split}$
(346)

As we have mentioned, differential forms are naturally integrable quantities.
By construction, we can integrate an $n$-form along a closed $n$-manifold
$\Sigma_{n}$. Given $\omega_{n}$ as in (344), the integral over $\Sigma_{n}$
is given by

$\displaystyle\begin{split}\oint_{\Sigma_{n}}\omega_{n}=\sum_{i_{1},...,i_{n}}\oint_{\Sigma_{n}}f_{i_{1}...i_{n}}(x)~{}dx_{i_{1}}...dx_{i_{n}}~{},\end{split}$
(347)

where here the restriction (i.e. pullback) of $dx_{i_{1}}...dx_{i_{n}}$ to
$\Sigma_{n}$ vanishes except when $dx_{i_{1}}...dx_{i_{n}}$ is proportional to
the volume element on $\Sigma_{n}$. Note additionally, that we can integrate
an $n$-form along a $p$-dimensional manifold when $n>p$ to get an $n-p$ form.
This can be done straight-forwardly by decomposing the differential form
$\omega_{n}$ as

$\displaystyle\begin{split}\omega_{n}=\left(\sum_{i_{1},...,i_{n-p}}f_{i_{1},...,i_{n-p}}(x)dx_{i_{1}}\wedge...dx_{i_{n-p}}\right)\wedge
d{\rm Vol}_{p}(\Sigma_{p})~{}.\end{split}$ (348)

Then the integral is given by

$\displaystyle\begin{split}\oint_{\Sigma_{p}}\omega_{n}=\sum_{i_{1},...,i_{n-p}}\left(\oint
f_{i_{1},...,i_{n-p}}(x)d{\rm
Vol}_{p}(\Sigma_{p})\right)dx_{i_{1}}\wedge...dx_{i_{n-p}}\end{split}$ (349)

Another fundamental concept in differential forms is the _exterior
derivative_. It is the natural generalization of the partial derivative to the
exterior algebra of differential forms – in fact, we have already implicitly
used the exterior derivative. Consider a general differential $n$-form
$\omega_{n}$:

$\displaystyle\begin{split}\omega_{n}=\sum_{i_{1},...,i_{n}}f_{i_{1}...i_{n}}(x)~{}dx_{i_{1}}\wedge...\wedge
dx_{i_{n}}~{}.\end{split}$ (350)

We can compute the exterior derivative by:

$\displaystyle\begin{split}d\omega_{n}=\sum_{i,i_{1},...,i_{n}}\frac{\partial
f_{i_{1}...i_{n}}(x)}{\partial x_{i}}~{}dx_{i}\wedge dx_{i_{1}}\wedge...\wedge
dx_{i_{n}}~{}.\end{split}$ (351)

Importantly, due to the anti-symmetry of the wedge product, the exterior
derivative squares to zero:

$\displaystyle\begin{split}d^{2}\omega_{n}=\sum_{i,j,i_{1},...,i_{n}}\frac{\partial^{2}f_{i_{1}...i_{n}}(x)}{\partial
x_{i}\partial x_{j}}~{}dx_{i}\wedge dx_{j}\wedge dx_{i_{1}}\wedge...\wedge
dx_{i_{n}}~{},\end{split}$ (352)

since partial derivatives commute. Note that these properties hold for general
curved manifolds since all of the machinery of differential forms only uses
partial derivatives – the “magic” comes from the exterior algebra.

One can show from the definition that the exterior derivative obeys the
product rule:

$\displaystyle\begin{split}d(\alpha_{p}\wedge\beta_{q})=d\alpha_{p}\wedge\beta_{q}+(-1)^{p}\alpha_{p}\wedge
d\beta_{q}~{}.\end{split}$ (353)

The exterior derivative is especially useful for differential forms as it
leads to a simple statement of the _Generalized Stokes’ Theorem_.

Generalized Stokes’ Theorem: Let $\Sigma$ be an oriented $(n+1)$-manifold with
boundary $\partial\Sigma$ and let $\omega_{n}$ be a differential $n$ form.
Then the integral of $d\omega_{n}$ is given in terms of the integral of
$\omega_{n}$ on $\partial\Sigma$:

$\displaystyle\begin{split}\int_{\Sigma}d\omega_{n}=\oint_{\partial\Sigma}\omega~{}.\end{split}$
(354)

Here will not prove this theorem. The interested reader can read the details
in Nakahara:2003nw .

Together with the Stokes’ Theorem, the product law for exterior derivative
(353) allows us to integrate differential forms by parts:

$\displaystyle\begin{split}\int_{M}d\alpha_{p}\wedge\beta_{q}=\int_{\partial
M}\alpha_{p}\wedge\beta_{q}+(-1)^{p}\int_{M}\alpha_{p}\wedge
d\beta_{q}~{}.\end{split}$ (355)

Another important manipulation involving differential forms is the _Hodge
star_ operation. In $d$-dimensions on a Riemannian manifold $M$ (i.e. a
manifold with metric) this is a map:

$\displaystyle\begin{split}\ast:\Omega^{n}(M)\longmapsto\Omega^{d-n}(M)~{}.\end{split}$
(356)

This depends on a choice of orientation (i.e. a choice of ordering of the
coordinates $\\{x^{1},...,x^{d}\\}$ and the metric and acts on the
infinitesimal $n$-form as

$\displaystyle\begin{split}\ast\Big{(}dx^{i_{1}}\wedge...\wedge
dx^{i_{n}}\Big{)}=\frac{\sqrt{{\rm
det}\,g}}{(d-n)!}g^{i_{1}j_{1}}...g^{i_{n}j_{n}}\epsilon_{j_{1}...j_{n}}dx^{j_{n+1}}\wedge
dx^{j_{d}}~{},\end{split}$ (357)

where $\epsilon_{1,...,n}$ is the fully antisymmetric Levi-Civita tensor. It
is important that acting on an $n$-form, the Hodge star squares to $\pm$ the
trivial operation:

$\displaystyle\begin{split}\ast\ast\omega_{n}=(-1)^{n(d-n)}s\,\omega_{n}~{},\end{split}$
(358)

where $s=\pm 1$ is the signature of the space time (i.e. $s=1$ Euclidean and
$s=-1$ for Lorentzian). The Hodge star also allows us to compute divergences
of tensors using their associated differential forms. For example, if we take
$v_{1}$ to be the 1-form associated to the vector field $\vec{v}$ as in (339),
then the divergence of $\vec{v}$ is a function (i.e. 0-form) which can be
computed in terms of $v_{1}$ as:

$\displaystyle\begin{split}\vec{\nabla}\cdot\vec{v}=\ast d\ast
v_{1}~{}.\end{split}$ (359)

We leave this as an exercise for the interested reader.

Often, we are interested in understanding which differential forms
$\omega_{n}$ can be written as an exterior derivative
$\omega_{n}=d\alpha_{n-1}$. Due to the fact that $d^{2}=0$, one may be tempted
to try to solve this by classifying $d\omega_{n}=0$. However, on manifolds
with non-trivial topology, there exist solutions to the equation
$d\omega_{n}=0$ where no $\alpha_{n-1}$ such that $\omega_{n}=d\alpha_{n-1}$.
These differential forms are classified by _cohomology_ classes called _de
Rahm Cohomology_.

The space of $n$-forms $\omega_{n}$ on a manifold $M$ that are _closed_ – i.e.
that obey $d\omega_{n}=0$ – that are not _exact_ – i.e. there is no
$(n-1)$-form $\alpha_{n-1}$ s.t. $\omega_{n}=d\alpha_{n-1}$ – are classified
by the cohomology class $H^{n}_{dR}(M)$. This cohomology class is often
written in terms of the set $Z^{n}(M)$ of closed $n$ co-chains (closed
differential $n$-forms) modulo the set $C^{n-1}(M)$ of exact $(n-1)$ co-chains
(i.e. exact $n$-forms):

$\displaystyle\begin{split}H^{n}_{dR}(M)=Z^{n}(M)/C^{n-1}(M)~{}.\end{split}$
(360)

What is useful about these cohomology classes is that they are equivalent to
other constructions of “cohomology classes” that are easier to compute:

$\displaystyle\begin{split}H^{n}_{dR}(M)=H^{n}(M;{\mathbb{R}})~{}.\end{split}$
(361)

We will not construct these cohomology classes here – the interested reader
can see Nakahara:2003nw .

Something we will use for our discussion here is that we can restrict to the
set of differential $n$-forms with _integral periods_
$\Omega^{n}_{\mathbb{Z}}(M)$: i.e. they integrate to an integer on _all_
$n$-dimensional sub-manifolds of $M$. We can then construct an integral-valued
cohomology class

$\displaystyle\begin{split}H^{n}(M;{\mathbb{Z}})=Z^{n}_{\mathbb{Z}}(M)/C^{n-1}_{\mathbb{Z}}(M)~{},\end{split}$
(362)

where the definition follows analogously from before where we have now
restricted to only differential forms in $\Omega_{\mathbb{Z}}^{n}(M)$.

###  A.2 Gauge Theory

Gauge theory is the theory of principal $G$ bundles with connection where $G$
is some Lie group343434Note that discrete groups such as ${\mathbb{Z}}_{N}$
are also technically Lie groups.. Principal $G$ bundles are fibrations of $G$
over some base manifold

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(363)

where $\pi$ is the projection map onto the base manifold.

Here we will describe a construction of $G$-bundles $P$ with connection. Let
us fix $G$ to be a semi-simple Lie group. A $G$-bundle $P$ over $X$ can be
locally as follows. Take a “nice” covering of $X$ by simply connected open
sets $\\{{\cal U}_{\alpha}\\}$ so that $\bigcup_{\alpha}{\cal U}_{\alpha}=X$.
The bundle $P$ locally can be trivialized over each ${\cal U}_{\alpha}$ so
that the fiber of $P$ over ${\cal U}_{\alpha}$ looks like a product:
$\pi^{-1}({\cal U}_{\alpha})=P_{\alpha}\cong G\times{\cal U}_{\alpha}$. The
fibers of $P$ are related between different patches ${\cal U}_{\alpha},{\cal
U}_{\beta}$ by the gauge transformation map:

$\displaystyle\begin{split}g_{\alpha\beta}:{\cal U}_{\alpha\beta}\to
G\quad,\quad(x,p_{\alpha})=(x,g_{\alpha\beta}(x)\cdot p_{\beta})\quad,\quad
x\in{\cal U}_{\alpha\beta}~{},\end{split}$ (364)

where ${\cal U}_{\alpha\beta...}={\cal U}_{\alpha}\cap{\cal U}_{\beta}\cap...$
All fiber bundles obey the “triple intersection” condition:

$\displaystyle\begin{split}g_{\alpha\beta}(x)\cdot g_{\beta\gamma}(x)\cdot
g_{\gamma\delta}(x)=\mathds{1}\qquad x\in{\cal
U}_{\alpha\beta\gamma}~{}.\end{split}$ (365)

Note that this will differ for “twisted” fiber bundles.

A connection on $P$ is formally a rule for comparing the fibers
$P_{x}=\pi^{-1}(x)$ with $P_{y}$ for $x\neq y$. The point is that parallel
transport along around the base manifold can be identified with a tangent
vector field to $X$ and for non-trivial $P$, motion along such a vector field
can induce a translation in the fiber. This translation is itself described by
a vector that is tangent to the fiber $G$ that is a function of the tangent
vector field $X$. Thus, the translation in $G$ is (locally) described by a
$\mathfrak{g}$-valued 1-form which is called the connection $A$.

The intersection condition (364) implies that the connection $A_{\alpha}$
defined on ${\cal U}_{\alpha}$ can be related to the connection on ${\cal
U}_{\beta}$ by the gauge transformation:

$\displaystyle\begin{split}A_{\beta}=g_{\alpha\beta}^{-1}A_{\alpha}g_{\alpha\beta}+g_{\alpha\beta}^{-1}dg_{\alpha\beta}~{}.\end{split}$
(366)

From this connection we can define the covariant derivative which is given by

$\displaystyle\begin{split}D_{A}=d+A\wedge~{}.\end{split}$ (367)

This connection can have a non-trivial curvature (in physics called the field
strength) which is given by the commutator of covariant derivatives (or
alternatively as the covariant derivative of the connection $A$):

$\displaystyle\begin{split}F=dA+A\wedge A~{}.\end{split}$ (368)

###  A.3 Characteristic Classes

It is a result from mathematics that for a fixed $G$, any principal $G$ bundle
over a fixed base manifold $X$ can be constructed by pulling back a $G$-bundle
from a universal space called the classifying space: $BG$.353535 The
classifying space $BG$ can be constructed as follows. Pick a contractible
space $EG$ that has a free $G$ action. $BG$ can then be constructed as the
quotient $EG/G$. Now any principal $G$ bundle $P\to X$ can be constructed by
pulling back the bundle $EG\to BG$:
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(369) As a consequence of this construction, all characteristic classes of the
principal $G$ bundle $P$ are given by the pullback of cohomology classes of
$BG$. So in particular, the cohomology of $BG$ classifies the possible
topology of principal $G$ bundles.

The characteristic classes classify the possible topological invariants of an
associated gauge bundle. As such, they must be measured by gauge invariant
differential forms that are written in terms of the field strength and gauge
field. When $G$ is a simply connected Lie group, the cohomology classes are
given by the gauge-invariant polynomials in $F$ and their powers (called
Chern-classes):363636The Chern-characters are given by the simple polynomials
$\frac{{\rm Tr}[F\wedge...\wedge F]}{n!(2\pi)^{n}}$. Chern-classes are the
reduction of Chern characters by products of lower-dimensional Chern-classes –
they agree for $ch_{1}(F)=c_{1}(F)=\frac{{\rm Tr}[F]}{2\pi}$. For example,
$c_{2}(F)=ch_{2}(F)-\frac{1}{2}\left[c_{1}(F)\right]^{2}$.

$\displaystyle\begin{split}\frac{{\rm Tr}[F\wedge...\wedge
F]}{n!(2\pi)^{n}}\in H^{2n}(BG;{\mathbb{Z}})~{}.\end{split}$ (370)

Other important cases include the case of $U(1)$ which is completely generated
by polynomials of $F$:

$\displaystyle\begin{split}H^{2n}(BU(1);{\mathbb{Z}})={\mathbb{Z}}\left[\frac{F\wedge...\wedge
F}{n!(2\pi)^{n}}\right]~{},\end{split}$ (371)

and ${\mathbb{Z}}_{N}$:373737Here the cohomology classes are
${\mathbb{Z}}_{N}$ valued because the “field strength” is the
${\mathbb{Z}}_{N}$ Bockstein of the gauge field $\beta_{N}(A)$ which is itself
${\mathbb{Z}}_{N}$-valued. The cohomology classes are then generated by
polynomials of $\beta_{N}(A)$.

$\displaystyle\begin{split}H^{2n}(B{\mathbb{Z}}_{N};{\mathbb{Z}})={\mathbb{Z}}_{N}~{}.\end{split}$
(372)

For our purposes, it will be important to consider the case of non-simply
connected Lie groups such as $SO(N)$ or $PSU(N)$. In these cases, we have the
relation

$\displaystyle\begin{split}H_{1}(G;{\mathbb{Z}})\cong
H^{2}(BG;{\mathbb{Z}})~{},\end{split}$ (373)

due to the Universal Coefficient Theorem Hatcher . This means that $G$-gauge
theory has discrete 2-form fluxes that correspond to the first homotopy group
of $G$ which encodes the 1-form magnetic flux sectors.

###  A.4 Twisted Bundles and Center Symmetry

Center symmetry of a $G$-bundle with (abelian) center $Z=Z(G)$ acts by
shifting the gauge field by a $Z$-gauge field. The mathematical description of
this process is described by the “twisting” of the $G$ bundle which is
classified by an element $w_{2}\in H^{2}(BG;Z)$.

Let us pick $G$ to be a Lie group with non-trivial center
$Z={\mathbb{Z}}_{N}$. As we discussed in the previous section, the standard
triple intersection constraint for $G$-bundles is given by:

$\displaystyle\begin{split}g_{\alpha\beta}\circ g_{\beta\gamma}\circ
g_{\gamma\alpha}=\mathds{1}_{G}\quad,\quad g_{\alpha\beta}:U_{\alpha}\cap
U_{\beta}\to G~{}.\end{split}$ (374)

In the case where we are considering principal $G$-bundles that couple to
matter (i.e. associated vector bundles) that is uncharged under
${\mathbb{Z}_{N}}\subset G$, the triple overlap condition can be modified
(i.e. twisted) to

$\displaystyle\begin{split}g_{\alpha\beta}\circ g_{\beta\gamma}\circ
g_{\gamma\alpha}=\omega\quad,\quad\omega\in{\mathbb{Z}_{N}}~{}.\end{split}$
(375)

The standard problem with twisting a bundle is that it will lead to field
configurations with branch cuts. However, since the matter fields are
uncharged under ${\mathbb{Z}_{N}}$, they do not see the twist, there are no
induced branch cuts, and the path integral is still well defined. The effect
of performing this twist is that the “standard” curvature is now shifted by
the discrete flux $w_{2}\in H^{2}(BG;{\mathbb{Z}_{N}})$:

$\displaystyle\begin{split}F_{G}\longmapsto
F_{G}+\frac{1}{N}w_{2}\mathds{1}_{G}~{}.\end{split}$ (376)

This procedure of passing to twisted $G$-bundles is similar to lifting the
$G$-gauge theory (path integral over $G$-bundles) to a $G^{c}$-gauge theory
(path integral over $G^{c}$-bundles). $G^{c}$-bundles can be defined
analogously to $Spin^{c}$-bundles as follows (see GMspinc  for a review of
$Spin^{c}$ structures). A $G^{c}$-bundle is a principal $\frac{G\times
U(1)}{{\mathbb{Z}}_{N}}$-bundle where the triple overlap condition is now
modified to

$\displaystyle\begin{split}g_{\alpha\beta}\circ g_{\beta\gamma}\circ
g_{\gamma\alpha}=(z,z^{-1})\sim(1,1)\quad,\quad g_{\alpha\beta}:U_{\alpha}\cap
U_{\beta}\to\frac{G\times U(1)}{{\mathbb{Z}}_{N}}~{},\end{split}$ (377)

where $z\in{\mathbb{Z}}_{N}\subset U(1)$. Here we see that the $G^{c}$-bundle
is composed of a twisted $G$-bundle and a twisted $U(1)$-bundle which,
combined together, form an untwisted total bundle (due to the
${\mathbb{Z}}_{N}$-quotient):

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0.0pt\hbox{$\textstyle{G^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.8606pt\raise-30.15834pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
40.32588pt\raise-39.99165pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(378)

We can also introduce a connection on $G^{c}$. The effective result of
performing this lift is that the curvature is restricted

$\displaystyle\begin{split}F_{G^{c}}={\omega}_{U(1)}+F_{G},\end{split}$ (379)

where ${\omega}_{U(1)}$ is the curvature of the twisted $U(1)$ and $F_{G}$ is
the curvature of the twisted $G$-bundle. Since $G^{c}$ is a well defined
bundle, $F_{G^{c}}$ is an integer class. This means that the twisted-$U(1)$
and $G$ bundles will have opposite fractional fluxes.

For applications to center symmetry of gauge theories, we can replace the
$U(1)$ factor with ${\mathbb{Z}}_{N}$. The reason is that with a factor of
${\mathbb{Z}}_{N}$, we are studying a
$\frac{G\times{\mathbb{Z}}_{N}}{{\mathbb{Z}}_{N}}\cong G$-bundle which does
not introduce any new degrees of freedom, but merely turns on a fractional
flux along the center of $G$. This can be achieved at the level of the path
integral or at the level of the action by projecting $U(1)\to{\mathbb{Z}_{N}}$
by including some Lagrange multiplier.

Let us review how this works in an example. Consider the case of $G=SU(N)$ and
let us turn on a ${\mathbb{Z}}_{N}^{(1)}$ flux – i.e. twist the $SU(N)$ bundle
by a ${\mathbb{Z}}_{N}$-valued $b_{2}$-class. To do this, we will go to a
$G^{c}$ bundle where

$\displaystyle\begin{split}G^{c}=\frac{SU(N)\times
U(1)}{{\mathbb{Z}}_{N}}\cong U(N)~{}.\end{split}$ (380)

Because of this isomorphism, we can rewrite

$\displaystyle\begin{split}F_{SU(N)}=F_{U(N)}-F_{U(1)}~{},\end{split}$ (381)

where again $F_{U(1)}$ is the field strength where we have twisted by $b_{2}$:

$\displaystyle\begin{split}\oint\frac{F_{U(1)}}{2\pi}=\oint\frac{b_{2}}{2\pi}~{}{\rm
mod}_{\mathbb{Z}}\quad,\quad\oint\frac{b_{2}}{2\pi}\in\frac{1}{N}{\mathbb{Z}}~{}.\end{split}$
(382)

For a $SU(N)$ gauge theory, we can make the substitution above in the action
if we also impose the constraint

$\displaystyle\begin{split}S=...+\frac{i}{2\pi}\int\varphi_{2}\wedge{\rm
Tr}[F_{U(N)}-B_{2}\mathds{1}_{N}]\quad\Longrightarrow\quad{\rm
Tr}[F_{U(N)}]=NB_{2}~{}.\end{split}$ (383)

where here we have fixed $F_{U(1)}$ to be a background gauge field given by a
representative $B_{2}$ of $b_{2}$. This constraint projects
$U(1)\to{\mathbb{Z}_{N}}$ and therefore $U(N)\to SU(N)$ which is twisted by
$b_{2}$.

Using this, we can show that the instanton number becomes fractional when we
turn on a center symmetry flux. Plugging in the shifted field strength, we can
compute

$\displaystyle\begin{split}&\int\frac{{\rm Tr}[F_{SU(N)}\wedge
F_{SU(N)}]}{8\pi^{2}}=\frac{1}{8\pi^{2}}\int{\rm
Tr}[(F_{U(N)}-B_{2}\mathds{1}_{N})\wedge(F_{U(N)}-B_{2}\mathds{1}_{N})]\\\
&=\int\frac{{\rm Tr}[F\wedge F]-{\rm Tr}[F]\wedge{\rm
Tr}[F]}{8\pi^{2}}+N\int\frac{B_{2}\wedge B_{2}}{8\pi^{2}}+\int\frac{{\rm
Tr}[F]\wedge{\rm Tr}[F]}{8\pi^{2}}-\int\frac{B_{2}\wedge{\rm
Tr}[F]}{4\pi^{2}}~{}.\end{split}$ (384)

Here, this first term is the second Chern class of the $U(N)$ connection which
is integer quantized. We then want to impose the constraint (383):

$\displaystyle\begin{split}\int\frac{{\rm Tr}[F_{SU(N)}\wedge
F_{SU(N)}]}{8\pi^{2}}&=N(N-1)\int\frac{B_{2}\wedge B_{2}}{8\pi^{2}}~{}{\rm
mod}_{\mathbb{Z}}=\frac{N-1}{2N}\int w_{2}\wedge w_{2}~{}{\rm
mod}_{\mathbb{Z}}~{}.\end{split}$ (385)

Here we have rescaled $B_{2}=\frac{2\pi}{N}w_{2}$ to make the fractional part
explicit:

$\displaystyle\begin{split}\oint
w_{2}={\mathbb{Z}}\quad\Longrightarrow\quad\oint w_{2}\wedge
w_{2}\in{\mathbb{Z}}~{}.\end{split}$ (386)

##  Appendix B Classifying Center/Magnetic Symmetry

In this appendix we more carefully discuss the relation between the center and
magnetic 1-form symmetries in non-abelian gauge groups and the center,
fundamental group of the gauge group.

Consider a non-abelian gauge group $G$ that has a center $Z$. Let us also
define $G_{\rm ad}=G/Z$ – this is sometimes called the adjoint form of the
group – and ${\widehat{G}}$ which is the simply connected cover
$G={\widehat{G}}/Z$. As we have discussed, the center symmetry of $G$ Yang-
Mills theory is described by $Z$ and its 1-form magnetic symmetry is given by
$\pi_{1}(G)$. The physical picture is the following.

All finite dimensional representations of a compact Lie group $G$ are “highest
weight” representations. This means that every representation $R$ corresponds
to an element of (i.e. a vector in) the weight lattice $\Lambda_{\rm
wt}(G)$.383838Technically, the weight lattice is defined for a Lie algebra
which captures all representations of ${\tilde{G}}$. Here, what we are calling
the “$G$-weight lattice” $\Lambda_{\rm wt}(G)=\Lambda_{\rm char}(G)$ is the
character lattice which classifies representations of $G$:
$\displaystyle\begin{split}\Lambda_{\rm char}(G)={\rm
Hom}(T,U(1))=\left\\{\mu\in\mathfrak{t}^{\ast}~{}|~{}\langle\mu,H\rangle\in{\mathbb{Z}}~{},~{}\forall
H\in\mathfrak{t}~{}{\rm s.t.}~{}e^{2\pi
iH}=\mathds{1}_{G}\right\\}~{},\end{split}$ (387) where $T$ (with Lie algebra
$\mathfrak{t}$) is a Cartan torus of $G$ (with Lie algebra $\mathfrak{g}$). We
hope that this introduction of slightly incorrect terminology is helpful and
not confusing.  The weight lattice sits inside of the weight lattice of the
simply connected group $\Lambda_{\rm wt}(G)\subseteq\Lambda_{\rm
wt}({\tilde{G}})$ and it further contains the weight lattice of the adjoint
form of the group: $\Lambda_{\rm wt}(G_{\rm ad})\subseteq\Lambda_{\rm wt}(G)$.
This is the statement that every representation of $G$ is also a