compat=1.0.0 MnLargeSymbols’164 MnLargeSymbols’171

# Axions, Higher-Groups, and Emergent Symmetry

###### Abstract

Axions, periodic scalar fields coupled to gauge fields through the instanton
density, have a rich variety of higher-form global symmetries. These include a
two-form global symmetry, which measures the charge of axion strings. As we
review, these symmetries typically combine into a higher-group, a kind of non-
abelian structure where symmetries that act on operators of different
dimensions, such as points, lines, and strings, are mixed. We use this
structure to derive model independent constraints on renormalization group
flows that realize theories of axions at long distances. These give universal
inequalities on the energy scales where various infrared symmetries emerge.
For example, we show that in any UV completion of axion-Yang-Mills, the energy
scale at which axion strings can decay is always larger than the mass scale of
charged particles.

###### Contents

  1. 1 Introduction
  2. 2 Symmetry Structure in Models of Axions
    1. 2.1 Higher-Form Symmetry
      1. 2.1.1 Higher-Group Global Symmetry
    2. 2.2 The Symmetries of Axion-Maxwell Theory
      1. 2.2.1 Coupling to Background Fields
      2. 2.2.2 Extension to Axion-QED
    3. 2.3 The Symmetries of Axion-Yang-Mills
      1. 2.3.1 Background Fields for Axion Yang-Mills
      2. 2.3.2 Extension to Axion-QCD
  3. 3 Higher-Groups and Constraints on Emergence
    1. 3.1 Inequalities for Axion-Yang-Mills
      1. 3.1.1 Pecci-Quinn Example
    2. 3.2 Inequalities for Axion-QCD
    3. 3.3 Inequalities for Axion Electrodynamics
      1. 3.3.1 Adjoint Higgsing Example

##  1 Introduction

Axions are are ubiquitous in models of beyond the standard model physics and
string theory, and have been a central focus of a variety of experiments. (See
e.g. [1, 2, 3] and references therein.) One of the key phenomena in the
physics of axions is the existence of string-like excitations. Let $a$ denote
the periodic axion field with periodicity $2\pi f$ where $f$ is the axion
decay constant. Around a string of charge $n$ the axion field winds as

$\oint da=2\pi nf~{}.$ (1.1)

In a low energy model of axions, the strings are non-dynamical defects with
infinitesimal thickness. At higher energies these theories have a rich array
of new degrees of freedom. In particular, the axion is often revealed to be
the angular part of a complex field $\varphi$ which is liberated in the
ultraviolet. Relatedly, at higher energies the strings are brought to life as
dynamical excitations of the field theory. For instance at the scale set by
the string tension $T^{1/2}_{\mathrm{string}}$, strings may nucleate from the
vacuum.

Thus, quantum field theories with axions in their long distance physics are
intrinsically equipped with an approximate energy scale $E_{\mathrm{string}}$
above which the string quantum number is no longer conserved. This scale is
less than that set by the string tension, but in perturbative models of axions
these scales are often far separated. One of our goals in the following will
be to derive model independent constraints on the scale $E_{\mathrm{string}}$
applicable to any UV completion of axion physics.

It is fruitful to describe aspects of axion strings in the language of
generalized global symmetries [4]. In this terminology, the periodic scalar
field $a$ gives rise to a conserved current of a generalized two-form global
symmetry $U(1)^{(2)}$:111A continuous $p$-form global symmetry has a conserved
current which is a $(p+1)$-form. The charged objects are extended operators of
dimension $p$. Familiar ordinary global symmetries are described by the
special case when $p=0$.

$J_{\mu\nu\rho}\sim\varepsilon_{\mu\nu\rho\sigma}\partial^{\sigma}a~{},$ (1.2)

which is tautologically conserved. The axion strings described above carry the
charge associated to this current and equation (1.1) expresses the Ward
identity of a current in the presence of a charged object. The current defined
by (1.2) should be distinguished from the frequently discussed current
$J_{\alpha}=\partial_{\alpha}a$ for the shift symmetry of the axion. The
latter is spontaneously broken by the choice of vacuum in the theory (with $a$
the associated Goldstone mode). Meanwhile the higher-dimensional analog of the
Coleman-Mermin-Wagner theorem [4, 5] implies that the symmetry associated to
$J_{\mu\nu\rho}$ is never spontaneously broken at long distances.222Recall
from [6], that in two spacetime dimensions a continuous global symmetry cannot
be spontaneously broken because the associated Goldstone boson would
necessarily have an unphysical logarithmic two-point function. The statement
above is the direct dimensional uplift to four spacetime dimensions.

The hallmark of an axion is its coupling to gauge fields. Consider for
instance axion-Yang-Mills with gauge group $SU(N)$ and action:

$S=\frac{1}{2}\int
da\wedge*da+\frac{1}{g^{2}}\int\mathrm{Tr}(F\wedge*F)-\frac{i}{8\pi^{2}f}\int
a\mathrm{Tr}(F\wedge F)~{}.$ (1.3)

In addition to the two-form symmetry $U(1)^{(2)}$ described above, this theory
also has a discrete one-form symmetry $\mathbb{Z}_{N}^{(1)}$. The objects
charged under this discrete symmetry are Wilson lines, and the conserved
quantum number is the charge under the $\mathbb{Z}_{N}$ center of the gauge
group. This is preserved by interactions because, in the absence of additional
matter, the dynamical gluons can only screen adjoints and other
representations of vanishing charge. This one-form symmetry is also intimately
connected to confinement: at long distances it is preserved or broken
precisely when the gauge theory is in a confined or deconfined phase
respectively [4].

Thus, the presence of the one-form symmetry $\mathbb{Z}_{N}^{(1)}$, endows UV
completions of axion-Yang-Mills with another important energy scale,
$E_{\mathrm{screen}},$ above which the one-form symmetry is broken.
Physically, this is the energy scale where matter fields charged under $SU(N)$
appear and hence can screen general Wilson lines. Below in section 3 we will
show that, independent of the details of the ultraviolet physics, there is an
inequality on these symmetry breaking scales $E_{\mathrm{string}}$ and
$E_{\mathrm{screen}}$:

$E_{\mathrm{screen}}\mathrel{\mathstrut\smash{\ooalign{\raise
2.5pt\hbox{$<$}\cr\lower 2.5pt\hbox{$\sim$}}}}E_{\mathrm{string}}~{}.$ (1.4)

Of course, since they are defined by emergent symmetries, the energy scales
above are in general only approximately meaningful. Far below $E_{i}$ the
corresponding quantum number is preserved up to small effects, while far above
$E_{i}$ the symmetry is badly broken. We thus interpret (1.4) as forbidding a
parametric separation of scales violating the inequality. In other words,
$E_{\mathrm{screen}}\gg E_{\mathrm{string}}$ is not possible in any
renormalization group trajectory, which flows at long distances to axion-Yang-
Mills described by the action (1.3).

To argue for the inequality (1.4) and other similar bounds, we first study in
detail the symmetry structure of theories of axions in section 2. As we
describe, the various symmetries of these models do not factorize into a
simple product of the symmetries of each form degree. Instead, a more precise
analysis shows that they form a mixed structure known as a “higher group”
global symmetry, which is somewhat analogous to a non-abelian symmetry group
composed of symmetries of different degrees.

Higher-group global symmetry is an analog of the Green-Schwarz mechanism [7]
for global symmetries (as opposed to its more common application when all
gauge fields are dynamical) [8, 9]. This symmetry structure frequently appears
in gauge theories and in topological quantum field theories when symmetries of
differing form degrees are present [10, 11, 12]. In the context of models of
axions, the presence of these interesting symmetry structures has been noted
for non-abelian gauge groups in [13, 14], and recently explored in abelian
gauge theories in [15, 16], and our presentation in section 2 draws closely
from these references. Our analysis leading to inequalities on energy scales
using higher-group symmetry follows closely the logic of [11], and more
broadly the spirit of [17].

One straightforward way to make the presence of this higher-group transparent
is by coupling the theory to background gauge fields for the global
symmetries. For example, in axion Yang-Mills the $U(1)^{(2)}$ two-form
symmetry that measures the charges of strings, has a natural source which is a
three-form gauge field $A^{(3)}$ with gauge-invariant field strength
$G^{(4)}$, while the $\mathbb{Z}_{N}^{(1)}$ one-form symmetry that measures
the charges of Wilson lines is sourced by a two-form gauge field $B^{(2)}$
which has periods that are $N$-th roots of unity. The higher-group structure
derived below then implies that the field strength $G^{(4)}$ is:

$G^{(4)}=dA^{(3)}+\frac{N(N-1)}{4\pi}B^{(2)}\wedge B^{(2)}~{}.$ (1.5)

Notice in particular that the presence of the one-form symmetry background
induces non-trivial (and in general fractional) $G^{(4)}$ flux. This mixing
between sources for global symmetries is the signature of higher-group global
symmetry.

In section 3 we derive the inequality (1.4) using the flux formula (1.5). We
then investigate the bound in the context of weakly-coupled KSVZ-type models.
In such theories axions and the symmetry structure above emerge at low-
energies after breaking an anomalous Peccei-Quinn symmetry [18, 19, 20, 21].
As we show, the inequality (1.4) implies constraints on ultraviolet coupling
constants which are indeed true in theories that flow to axion-Yang-Mills in
the IR.

In section 3 we also discuss constraints on axion-Maxwell theories. We
consider the action:

$S=\frac{1}{2}\int da\wedge*da+\frac{1}{2g^{2}}\int
F\wedge*F-\frac{iK}{8\pi^{2}f}\int aF\wedge F~{},$ (1.6)

where above $K\in\mathbb{Z}$ is an integral coupling constant. In particular,
for $|K|>1$ we find a non-trivial higher-group global symmetry and resulting
constraints on emergent symmetry scales. The symmetries involved are of two
types. First, there is the continuous $U(1)^{(2)}$ string symmetry of the
axion and the $U(1)_{m}^{(1)}$ magnetic symmetry of the photon that measures
the magnetic charges of ’t Hooft lines. There is also a discrete
$\mathbb{Z}_{K}^{(0)}$ shift-symmetry of the axion and a discrete
$\mathbb{Z}_{K}^{(1)}$ one-form symmetry of the photon that measures the
charges of Wilson lines modulo $K$. In addition to results similar to (1.4) we
also derive the general constraint

$\min\\{E_{\mathrm{shift}},~{}E_{\mathrm{screen}}\\}\mathrel{\mathstrut\smash{\ooalign{\raise
2.5pt\hbox{$<$}\cr\lower 2.5pt\hbox{$\sim$}}}}E_{\mathrm{magnetic}}~{},$ (1.7)

where above, $E_{\mathrm{shift}}$ is the scale where the shift symmetry of the
axion is violated, $E_{\mathrm{screen}}$ is the scale at which Wilson lines
can be screened by charged matter, and $E_{\mathrm{magnetic}}$ is the scale of
emergence of the $U(1)$ gauge field defining the photon, above which $*F$,
ceases to define a conserved current. We verify (1.7) in a simple example and
find that the implied constraints on coupling constants are indeed necessary
for self-consistency of our analysis.

Finally, we also discuss analogs of (1.4) and (1.7) in theories with axions
and charged matter fields that have higher-group global symmetry and hence the
ensuing consequences.

##  2 Symmetry Structure in Models of Axions

In this section we review the basic definitions of higher-form global symmetry
following [4], and higher-group global symmetry following [11, 12]. We then
present the symmetries and couplings to background gauge fields in models of
axions. In the case of axions coupled to abelian gauge fields these symmetries
have been previously studied in [15, 16]. In the case of axions coupled to
non-abelian gauge fields, our analysis follows from results in [13, 14].

###  2.1 Higher-Form Symmetry

In general, a $p$-form global symmetry is a symmetry that acts on extended
operators of dimension $p$ [4]. Thus, ordinary global symmetries, which act on
point operators, correspond to the case $p=0$. Meanwhile one-form symmetries
act on line operators, and two-form symmetries act on surface operators. We
will encounter all of these symmetries below. Higher-form symmetries form a
group (necessarily abelian if $p>0$) that we denote $G^{(p)}$. In the simplest
case of continuous $G^{(p)}$, there is a conserved current with $p+1$
antisymmetric indices:

$\partial^{\mu^{1}}J_{\mu_{1}\cdots\mu_{p+1}}=0~{}.$ (2.1)

It is frequently convenient to work with the hodge dual $*J$ which is
therefore closed, $d*J_{p+1}=0.$

As usual when discussing the implications of symmetry it is fruitful to
introduce background gauge fields that act as sources for the conserved
currents. For a $p$-form global symmetry the appropriate background is a
$(p+1)$-form gauge field $A^{(p+1)}$. In the simplest case of a continuous
$G^{(p)}$ symmetry, the background gauge field couples in a standard way to
the conserved current:

$S\supset i\int A^{(p+1)}\wedge\ast J_{p+1}~{}.$ (2.2)

Current conservation means that (up to possible ’t Hooft anomalies) the theory
is invariant under background gauge transformations of the source:

$A^{(p+1)}\longmapsto A^{(p+1)}+d\Lambda^{(p)}~{},$ (2.3)

where $\Lambda^{(p)}$ is locally a $p$-form gauge parameter. The normalization
of the coupling (2.2) between the source and the current is fixed by charge
quantization. In our conventions $G^{(p)}$ charges, which are measured on any
closed surface $\Sigma_{d-p-1}$ of dimension $(d-p-1)$, are integrally
quantized:

$\int_{\Sigma_{d-p-1}}*J_{p+1}\in\mathbb{Z}~{}.$ (2.4)

In particular, this means that the fluxes and gauge parameters of the
background fields are also quantized:

$\int_{\Sigma_{p+2}}\frac{dA^{(p+1)}}{2\pi}\in\mathbb{Z}~{},\hskip
36.135pt\int_{\Sigma_{p+1}}\frac{d\Lambda^{(p)}}{2\pi}\in\mathbb{Z}~{}.$ (2.5)

Alternatively, we can also describe symmetry through the symmetry group
operators. For an element $g\in G^{(p)},$ the associated operator is denoted
$U_{g}(\Sigma_{d-p-1})$ and is defined on any closed $\Sigma_{d-p-1}$. In the
case of a continuous symmetry these are given by exponentiated integrals of
the current

$U_{g}(\Sigma_{d-p-1})=e^{i\lambda\int_{\Sigma_{d-p-1}}*J_{p+1}}~{},$ (2.6)

where $g=e^{i\lambda}$. The primary advantage of the symmetry group operators
is that the continue to exist even when the symmetry is discrete.
Mathematically these symmetry defect operators can be thought of as Poincaré
dual to the background fields. They describe flat background gauge fields,
i.e. those with vanishing field strength.

####  2.1.1 Higher-Group Global Symmetry

Higher-group global symmetry is a natural possibility in theories with
symmetries of differing form degrees. As in the discussion above, there are
various related point of view in terms of currents, symmetry group operators,
and background fields.

Perhaps the simplest context where this mixing occurs is in situations where
all symmetries in question are continuous and so we can reduce to statements
about the conserved currents. In this case we can find contact terms in the
operator product expansion of the generic form:

$d*J_{p_{1}+1}(x)J_{p_{2}+1}(y)\sim\partial^{k}\delta^{(d)}(x-y)J_{p_{3}+1}(x)~{},$
(2.7)

where $\partial^{k}$ is a $k$-th derivative and $p_{3}=p_{1}+p_{2}+k$. Note in
particular that, since $k\geq 0,$ the fusion always produces symmetries of
form degree at least $p_{1}+p_{2}$ on the right-hand-side. Such fusion laws
are similar in spirit to the more familiar non-abelian current algebra of
ordinary (0-form) global symmetries. In that context, the currents are
conserved at separated points, but at coincident points one encounters a
contact term in the divergence controlled by the Lie algebra structure
constants $f^{abc}$. Equation (2.7) is similar except that now the fusion
involves currents of differing form degrees and there are derivatives on the
delta function. For instance, an example of this structure, explored in detail
in [11], occurs in multiflavor massless QED, where the zero-form flavor
symmetry forms a higher-group with the one-form magnetic symmetry with current
$J_{2}=*F$. Other examples are described in detail in [17].

In the applications below, we will often be interested in higher-groups where
the symmetries involved are discrete. In this case we read off the structure
from the symmetry group operators $U_{g}(\Sigma)$ described above following
[12]. To begin with, we recall from [4] that the group multiplication laws in
$G^{(p)}$ are encoded by an operator product algebra of the symmetry group
operators. For instance in the abstract notation (2.6) above we have

$U_{g_{1}}(\Sigma_{d-p-1})U_{g_{2}}(\Sigma_{d-p-1})=U_{g_{1}g_{2}}(\Sigma_{d-p-1})~{},$
(2.8)

which generalizes the additive nature of conserved charges. We can also view
this as a rule for what happens when the defects $U_{g_{1}}$ and $U_{g_{2}}$
intersect: they produce $U_{g_{1}g_{2}}$.

The fusion rule (2.8) encodes only the simplest possible collision of symmetry
group operators. More generally we must permit multiple defects to intersect
in generic configurations. When this happens we can encounter defects of
various form degrees leading to a higher-group global symmetry. An example
taken from [12] is shown in Figure 1. When the resulting algebra of symmetry
defects involves non-trivial intersections with form degrees which are not all
equal we say there is higher-group global symmetry.

${\bf g}$${\bf h}$${\bf k}$${\bf g}{\bf h}{\bf k}$${\bf g}{\bf h}$${\bf h}{\bf
k}$$\beta({\bf g},{\bf h},{\bf k})$ Figure 1: A junction where zero-form
symmetry defects of type ${\bf g}$, ${\bf h}$, ${\bf k}$, ${\bf g}{\bf h}{\bf
k}\in G$ meet in codimension three. This configuration is generic in spacetime
dimension three and above. The junctions of three codimension-one defects are
in red, and their intersection is the black point. At the codimension-three
intersection, a one-form symmetry defect $\beta({\bf g},{\bf h},{\bf k})$
emanates, signaling the 2-group symmetry. In $d$ dimensions, all objects span
the remaining $d-3$ dimensions.

Finally, we can also describe higher-group global symmetry in terms of
background fields, which is the method that we employ below. We consider a
collection of symmetries and background fields $A^{(1)},A^{(2)},\cdots
A^{(\ell)}$ that are sources for the higher-form symmetries of different
degrees as in (2.2). Each such background field is subject to a gauge
transformation as in (2.3), but now we allow for mixing of the gauge
transformation between background fields of differing form degree:

$A^{(p+1)}\longmapsto
A^{(p+1)}+d\Lambda^{(p)}+\sum_{k<p}\alpha_{k}(A^{(i)})\Lambda^{(k)}+\text{Schwinger~{}terms}~{}.$
(2.9)

Here, the sum is over $k<p$ mirroring the restrictions on $p_{3}$ in equation
(2.7). Meanwhile the Schwinger terms above refers to possible pieces of the
gauge transformation rules that are non-linear in the gauge parameters. This
sort of mixed gauge transformation rule involving forms of differing degrees
is familiar from the Green-Schwarz mechanism [7]. In that context one
encounters such mixed transformations for dynamical fields whereas here it
occurs for background fields and signifies a higher-group global symmetry.
Crucially, as we see in examples below, this means that the gauge invariant
fluxes for background fields are modified to include Chern-Simons-like
corrections constructed out of the other gauge fields.

###  2.2 The Symmetries of Axion-Maxwell Theory

Let us now describe in more detail the symmetry structure of axion-Maxwell
theory, i.e. an axion coupled to an abelian gauge field. We first ignore the
coupling between $a$ and the gauge fields and then incorporate modifications
due to the interactions. The symmetries and higher-group structure present in
these theories has also been discussed in [15, 16].

The axion field $a$ has two natural symmetries: A $U(1)^{(0)}$ shift symmetry
and a $U(1)^{(2)}$ two-form symmetry which acts on strings. The associated
currents are respectively:333The normalization of $J_{1}$ can be checked by
noting that the charged operators are ${\rm exp}(ina/f)$ for integer $n$.

$*J_{1}=if*da~{},\hskip 36.135pt*J_{3}=\frac{1}{2\pi f}da~{}.$ (2.10)

The choice of vacuum for $a$ spontaneously breaks the shift symmetry.
Meanwhile the two-form symmetry is always preserved. The action coupled to
background fields $(A^{(1)},A^{(3)})$ is written as:

$S=\frac{1}{2}\int(da-fA^{(1)})\wedge*(da-fA^{(1)})+\frac{i}{2\pi f}\int
A^{(3)}\wedge da~{}.$ (2.11)

In particular, the coupling of the axion to $A^{(1)}$ is standard for a
Goldstone mode and under gauge transformations of $A^{(1)}$, $a$ also shifts
to ensure the action is invariant.444In the presence of both $A^{(1)}$ and
$A^{(3)}$ the action is not fully invariant under background gauge
transformations but instead has a mixed ’t Hooft anomaly characterized by an
inflow action  $S_{\mathrm{inflow}}=\frac{i}{2\pi}\int_{X}dA^{(1)}\wedge
A^{(3)}~{},$ (2.12) where $X$ is a five-dimensional manifold with boundary the
physical spacetime $Y$. See e.g. [22, 23] and references therein for an
overview of anomaly inflow.

Now let us consider an abelian gauge field $C$ with gauge invariant field
strength $F=dC$. We first focus on the case without matter and later
generalize to include charged matter fields. First, in the case of free
Maxwell theory without coupling to $a$, there are 1-form symmetries
$U(1)_{e}^{(1)}\times U(1)_{m}^{(1)}$, where the subscripts indicate the
common association of these as “electric” and “magnetic” respectively. The
associated conserved currents are:

$*J_{2,e}=\frac{i}{g^{2}}*F~{},\hskip 36.135pt*J_{2,m}=\frac{F}{2\pi}~{}.$
(2.13)

It is straightforward to couple these symmetries to background two-form gauge
fields $(B_{e}^{(2)},B_{m}^{(2)})$:

$S=\frac{1}{2g^{2}}\int(F-B_{e}^{(2)})\wedge*(F-B_{e}^{(2)})+i\int
B_{m}^{(2)}\wedge\frac{F}{2\pi}~{}.$ (2.14)

Under gauge transformations of $B_{e}^{(2)},$ the dynamical gauge field $F$
also shifts indicating that the photon is the Goldstone mode for the electric
1-form symmetry.

We now couple the axion to the Maxwell field leading to the action (without
background fields):

$S=\frac{1}{2}\int da\wedge*da+\frac{1}{2g^{2}}\int
F\wedge*F-\frac{iK}{8\pi^{2}f}\int aF\wedge F~{},$ (2.15)

Here $K\in\mathbb{Z}$ is a discrete coupling constant. It is quantized to
ensure that the action respects periodicity of the axion. For some purposes
below it is also useful to write the action in way that more manifestly
respects both the periodicity of the action and gauge invariance. This can be
done by choosing an auxiliary five-manifold $X$ whose boundary is the physical
four-dimensional spacetime $Y$ and writing

$\frac{iK}{8\pi^{2}f}\int_{Y}aF\wedge F=\frac{iK}{8\pi^{2}f}\int_{X}da\wedge
F\wedge F~{},$ (2.16)

where now the quantized coupling $K$ ensures that the action does not depend
on the choice of $X$. For further mathematical details about the definition of
the axion interaction see [14].

The interaction term in the action leads to a modification of the equations of
motion. It is convenient to describe this in terms of the currents introduced
in equation (2.10) and (2.13):

$d*J_{3}=0~{},\hskip 14.45377ptd*J_{2,m}=0~{},\hskip
14.45377ptd*J_{1}=\frac{K}{8\pi^{2}}F\wedge F~{},\hskip
14.45377ptd*J_{2,e}=-\frac{K}{4\pi^{2}f}d(aF)~{}.$ (2.17)

The non-zero right-hand-side of $d*J_{1}$ and $d*J_{2,e}$ above implies that
in axion electrodynamics, these symmetries are broken. In fact, the as we will
see these symmetries are broken to discrete groups controlled by $K$:

$U(1)^{(0)}\longrightarrow\mathbb{Z}_{K}^{(0)}~{},\hskip
14.45377ptU(1)_{e}^{(1)}\longrightarrow\mathbb{Z}_{K}^{(1)}~{}.$ (2.18)

In particular, in the case of minimal coupling, $K=1$, these symmetries are
completely broken.

To demonstrate the above, we need to define topological operators generating
discrete transformations. Those generating the discrete $\mathbb{Z}_{K}^{(0)}$
axion shift symmetry are given by:

$U_{\ell}(\Sigma_{3})={\rm exp}\left(\frac{2\pi
i\ell}{K}\int_{\Sigma_{3}}*J_{1}-\frac{K}{8\pi^{2}}C\wedge F\right)~{},\hskip
14.45377pt\ell\in\mathbb{Z}~{}.$ (2.19)

By the equations of motion, the argument of the integral is formally closed.
However, since it involves the dynamical gauge field $C$ explicitly it is not
manifestly gauge invariant. The quantization condition $\ell\in\mathbb{Z}$
ensures that the Chern-Simons correction term is well-defined.

Similarly, the symmetry operators for the discrete electric one-form symmetry
$\mathbb{Z}_{K}^{(1)}$ are:

$U_{\ell}(\Sigma_{2})={\rm exp}\left(\frac{2\pi
i\ell}{K}\int_{\Sigma_{2}}*J_{2,e}+\frac{K}{4\pi^{2}f}a\,F\right)~{},\hskip
14.45377pt\ell\in\mathbb{Z}~{}.$ (2.20)

Again the equations of motion (2.17) imply that the integrand is closed so
that the operator is topological and the quantization $\ell\in\mathbb{Z}$
implies that the operator is well-defined under the shift redundancy of the
axion $a\rightarrow a+2\pi f$.

####  2.2.1 Coupling to Background Fields

We now explain the coupling of axion electrodynamics to background fields. To
do so, we must introduce the appropriate backgrounds for the discrete
symmetries $\mathbb{Z}_{K}^{(0)}$ and $\mathbb{Z}_{K}^{(1)}$. These are
discrete gauge fields. We represent them by standard gauge fields obeying
constraints:

$KA^{(1)}=d\lambda^{(0)}~{},\hskip 14.45377ptKB^{(2)}_{e}=d\lambda^{(1)}~{},$
(2.21)

where $\lambda^{(i)}$ above are gauge parameters with the periods of
$d\lambda^{(i)}$ valued in $2\pi\mathbb{Z}$. Thus the background fields are
flat gauge fields with holonomies that are $K$-th roots of unity.
Mathematically one can also view the gauge invariant data in these background
gauge fields in terms of cohomology classes. Under gauge transformation:

$\displaystyle A^{(1)}\longmapsto A^{(1)}+d\Lambda^{(0)}~{},$
$\displaystyle\lambda^{(0)}\longmapsto\lambda^{(0)}+K\Lambda^{(0)}~{},$ (2.22)
$\displaystyle B_{e}^{(2)}\longmapsto B_{e}^{(2)}+d\Lambda_{e}^{(1)}~{},$
$\displaystyle\lambda^{(1)}\longmapsto\lambda^{(1)}+K\Lambda_{e}^{(1)}~{}.$

Thus the invariant information is encoded in

$\left[\frac{K}{2\pi}A^{(1)}\right]\in H^{1}(Y,\mathbb{Z}_{K})~{},\hskip
14.45377pt\left[\frac{K}{2\pi}B_{e}^{(2)}\right]\in
H^{2}(Y,\mathbb{Z}_{K})~{},$ (2.23)

where the square brackets indicate the gauge equivalence class, and $Y$ is
spacetime.

At the linearized level, it is straightforward to couple the symmetries to
background fields by introducing couplings of the form (2.2). At the non-
linear level this is more subtle due to the periodicity of the axion and the
appearance of Chern-Simons terms. A simple strategy is to recall that the
axion $a$ and the gauge field $C$ must still shift under the discrete
symmetries. To account for the possible non-linear mixing between background
fields we write the coupling to the $U(1)^{(2)}$ two-form symmetry of the
axion and $U(1)^{(1)}$ magnetic symmetry of the gauge field in terms of field
strengths:

$S\supset\frac{i}{2\pi f}\int a\,G^{(4)}-\frac{i}{2\pi}\int H^{(3)}\wedge
C~{},$ (2.24)

where $G^{(4)}$ and $H^{(3)}$ are gauge invariant field strengths:

$G^{(4)}=dA^{(3)}+\cdots~{},\hskip 14.45377ptH^{(3)}=dB^{(2)}_{m}+\cdots~{},$
(2.25)

and the omitted terms above indicate possible corrections that are non-linear
in the background fields and will be determined below. Additionally, we find
it convenient to use the five-dimensional presentation of (2.16).

With these preliminaries the action coupled to all background fields is
simply:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{Y}(da-
fA^{(1)})\wedge*(da-
fA^{(1)})+\frac{1}{2g^{2}}\int_{Y}(F-B_{e}^{(2)})\wedge*(F-B_{e}^{(2)})$
$\displaystyle+$ $\displaystyle\frac{i}{2\pi
f}\int_{Y}a\,G^{(4)}-\frac{i}{2\pi}\int_{Y}H^{(3)}\wedge
C-\frac{iK}{8\pi^{2}f}\int_{X}(da-
fA^{(1)})\wedge(F-B_{e}^{(2)})\wedge(F-B_{e}^{(2)})~{}.$

Written as above, it is transparent that the theory is invariant under all
background gauge transformation and enjoys the correct linear couplings to
backgrounds.555Here we are ignoring shifts by terms involving only background
fields i.e. ’t Hooft anomalies discussed below. However, superficially it
would seem to depend on the explicit choice of five-manifold $X$. Demanding
that this dependence drop our for all terms that involve dynamical fields
fixes the non-linear form of the field strengths in (2.25) to
be:666Mathematically, $\frac{1}{4\pi}B^{(2)}_{e}\wedge B^{(2)}_{e}$ is the
Pontryagin square of $B_{e}$ embedded into a continuous variables as in
(2.21).

$G^{(4)}=dA^{(3)}+\frac{K}{4\pi}B^{(2)}_{e}\wedge B^{(2)}_{e}~{},\hskip
14.45377ptH^{(3)}=dB_{m}^{(2)}+\frac{K}{2\pi}A^{(1)}\wedge B_{e}^{(2)}~{}.$
(2.27)

Notice that in general when $B^{(2)}_{e}$ and $A^{(1)}$ are activated, the
fluxes $G^{(4)}$ and $H^{(3)}$ are fractional. Moreover, the non-linear terms
above mean that the gauge transformations of background fields must also
contain non-linear corrections as in the Green-Schwarz mechanism [7]. In
particular this means that while the gauge transformations for $A^{(1)}$ and
$B_{e}^{(2)}$ are standard:

$A^{(1)}\longmapsto A^{(1)}+d\Lambda^{(0)}~{},\hskip
14.45377ptB^{(2)}_{e}\longmapsto B^{(2)}_{e}+d\Lambda^{(1)}_{e}~{},$ (2.28)

those of the other background fields contain contain non-linear terms:777Below
we omit possible total derivatives. As usual in descent these can be absorbed
into choices of local counterterms.

$\displaystyle A^{(3)}$ $\displaystyle\longmapsto$ $\displaystyle
A^{(3)}+d\Lambda^{(2)}-\frac{K}{2\pi}B_{e}^{(2)}\wedge
d\Lambda_{e}^{(1)}-\frac{K}{4\pi}\Lambda_{e}^{(1)}\wedge
d\Lambda_{e}^{(1)}~{},$ (2.29) $\displaystyle B_{m}^{(2)}$
$\displaystyle\longmapsto$ $\displaystyle
B^{(2)}_{m}+d\Lambda^{(1)}_{m}+\frac{K}{2\pi}A^{(1)}\wedge\Lambda_{e}^{(1)}-\frac{K}{2\pi}B_{e}^{(2)}\,\Lambda^{(0)}+\frac{K}{2\pi}d\Lambda^{(0)}\wedge\Lambda_{e}^{(1)}~{}.$
(2.30)

These mixed gauge transformations mean that the symmetries of axion-Maxwell
theory form a non-trivial higher-group. Specifically, in the terminology of
section 2.1.1 this is a three-group. As noted in section 1 this means that at
loci where the one-form symmetry defects intersect we find localized $G^{(4)}$
flux, while at the loci where the defects of $\mathbb{Z}_{K}^{(0)}$ and
$\mathbb{Z}_{K}^{(1)}$ intersect we find $H^{(3)}$ flux. For more details on
this algebra of symmetry defects see also [16].

To conclude this section, let us also note that we can immediately also
determine the ’t Hooft anomaly from the action (2.2.1). As usual the anomaly
is conveniently encoded in an five-dimensional local topological action for
the background fields which encodes the anomaly via inflow:

$S_{\mathrm{inflow}}=\frac{i}{2\pi}\int_{X}G^{(4)}\wedge
A^{(1)}+\frac{i}{2\pi}\int_{X}H^{(3)}\wedge B_{e}^{(2)}~{}.$ (2.31)

This is the appropriate generalization of the separate anomalies of Maxwell
theory and the axion theory.

####  2.2.2 Extension to Axion-QED

Let us now discuss the extension of this analysis to theories with matter
fields. We include $N_{F}$ massive fermion flavors of electric charge one.
(Similar analysis applies to theories with charged bosons.) The action is
thus:

$S=\frac{1}{2}\int da\wedge*da+\frac{1}{2g^{2}}\int
F\wedge*F-\frac{iK}{8\pi^{2}f}\int aF\wedge F+\int
i\overline{\Psi}\not{D}\Psi+i\overline{\tilde{\Psi}}\not{D}\tilde{\Psi}+(m\tilde{\Psi}\Psi+c.c)~{},$
(2.32)

where $\Psi$ and $\tilde{\Psi}$ are Weyl fermions of opposite electric charge
and the flavor indices on the fermions are suppressed.

The $\mathbb{Z}_{K}^{(0)}$ shift symmetry of the axion is present as before,
as is the $U(1)^{(1)}_{m}$ magnetic one-form symmetry and the $U(1)^{(2)}$
two-form axion string symmetry. However, the presence of the charged matter
fields means that Wilson lines can be screened and hence the
$\mathbb{Z}_{K}^{(1)}$ one-form symmetry is broken. Instead, in this model we
claim that there is a new zero-form flavor global symmetry given by

$\frac{SU(N_{f})}{\mathbb{Z}_{L}}^{(0)}~{},\hskip
36.135ptL=\mathrm{gcd}(K,N_{f})~{}.$ (2.33)

The fact that there is a flavor symmetry, which locally (i.e. at the Lie
algebra level) is of the form $SU(N_{f})$ is obvious from the presence of
multiple flavors and the invariant mass. What remains to be explained is the
$\mathbb{Z}_{L}$ quotient above.

The precise meaning of the quotient is that all operators are acted on
trivially by the $\mathbb{Z}_{L}$ subgroup of the center of $SU(N_{f})$. If we
look at the sector of local operators that we can build out of $\Psi$’s alone
this is clear: every gauge invariant local operator contains equal numbers of
$\Psi$’s and $\overline{\Psi}$’s and hence is neutral under the entire
$\mathbb{Z}_{N_{f}}$ center of $SU(N_{f})$.

To see that the quotient is in fact only by $\mathbb{Z}_{L}$ and not
$\mathbb{Z}_{N_{f}}$ we examine an axion domain wall across which $a$ jumps by
$2\pi f$. The coupling between $a$ and $F$ then leads to a Chern-Simons theory
at level $K$ along the wall. In particular, this means that monopole operators
embedded in the wall carry electric charge $-K$. We can dress such operators
with $K$ modes of $\Psi$ to obtain a gauge invariant field
configuration.888See [24] and references therein for a review of the quantum
numbers of monopole operators. This extended configuration of operators
transforms with charge $K$ under the central $\mathbb{Z}_{N_{f}}$ subgroup of
$SU(N_{f})$. This means that while $\mathbb{Z}_{N_{f}}$ in general acts non-
trivially on such a dressed monopole, any $\mathbb{Z}_{L}$ subgroup of
$\mathbb{Z}_{N_{f}}$ where $L$ divides $K$ acts trivially. The largest such
$L$ is $\mathrm{gcd}(K,N_{f}).$

We now proceed to couple the theory to background fields for these global
symmetries. The background fields are:

$\mathbb{Z}_{K}^{(0)}\longleftrightarrow A^{(1)}~{},\hskip
14.45377pt\frac{SU(N_{f})}{\mathbb{Z}_{L}}^{(0)}\longleftrightarrow
X^{(1)}~{},\hskip 14.45377ptU(1)^{(1)}_{m}\longleftrightarrow
B^{(2)}_{m}~{},\hskip 14.45377ptU(1)^{(2)}\longleftrightarrow A^{(3)}~{}.$
(2.34)

An important role will be played by the $\mathbb{Z}_{L}$ quotient of the
flavor symmetry above. This quotient means that there are more possible
backgrounds available than with the naive $SU(N_{f})$ global symmetry alone.
The additional backgrounds have a topological invariant, the second Steifel-
Whitney class $w_{2}\in H^{2}(Y,\mathbb{Z}_{L})$, which one can view as a
discrete magnetic flux. In many ways, this flux plays an analogous role to the
electric one-form symmetry background field in the theory without matter. To
emphasize this, we define a dependent background field $B^{(2)}_{e}$

$B^{(2)}_{e}\equiv\frac{2\pi w_{2}}{L}~{},$ (2.35)

where $B^{(2)}_{e}$ behaves as a discrete $\mathbb{Z}_{L}$-valued two-form
gauge field as in (2.21). Importantly, when $B_{e}^{(2)}$ is non-zero, the
fluxes of the dynamical gauge field become fractional as:

$\int_{\Sigma_{2}}(F-B_{e}^{(2)})\in 2\pi\mathbb{Z}~{}.$ (2.36)

This is because the $\mathbb{Z}_{L}$ subgroup defining the quotient in (2.33)
is also a subgroup of dynamical $U(1)$ gauge group. So in configurations with
discrete magnetic flavor flux, the effective gauge group is
$U(1)/\mathbb{Z}_{L}$ and the dynamical flux must also be fractional.

With these preliminaries, we can write the action coupled to backgrounds as:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{Y}(da-
fA^{(1)})\wedge*(da-
fA^{(1)})+\frac{1}{2g^{2}}\int_{Y}F\wedge*F+\int_{Y}i\overline{\Psi}(\not{D}+\not{X}^{(1)})\Psi+i\overline{\tilde{\Psi}}(\not{D}-\not{X}^{(1)})\tilde{\Psi}$
(2.37) $\displaystyle+$ $\displaystyle(m\tilde{\Psi}\Psi+c.c)+\frac{i}{2\pi
f}\int_{Y}G^{(4)}\wedge a-\frac{i}{2\pi}\int_{Y}H^{(3)}\wedge
C-\frac{iK}{8\pi^{2}f}\int_{X}(da-fA^{(1)})\wedge F\wedge F~{}.$

Demanding as before that the action depends only on the dynamical variables in
the physical four-dimensional spacetime results in the gauge invariant field
strengths:

$G^{(4)}=dA^{(3)}+\frac{K}{4\pi}B_{e}^{(2)}\wedge B_{e}^{(2)}~{},\hskip
36.135ptH^{(3)}=dB_{m}^{(2)}+\frac{K}{2\pi}A^{(1)}\wedge B^{(2)}_{e}~{}.$
(2.38)

Thus, as before we find a higher-group structure, now involving the flavor
symmetry, whenever $L=\gcd(K,N_{f})$ is larger than one. We also find an ’t
Hooft anomaly in this theory with inflow action again given by (2.31).

###  2.3 The Symmetries of Axion-Yang-Mills

We now discuss axion-Yang-Mills. For simplicity, we restrict ourselves to
gauge group $SU(N)$. The higher-group symmetries we uncover follow directly
from the results of [13, 14]. To begin let us describe symmetries of the Yang-
Mills sector independent of the axion. This theory has a one-form symmetry
$\mathbb{Z}_{N}^{(1)}$ related to the center of the gauge group [4].
Physically this symmetry is present because the only dynamical fields are the
adjoint gluons which are neutral under the center of the gauge group. Thus,
one cannot screen Wilson lines in representations which are charged under the
center of the gauge group. This leads to a quantum number, conserved modulo
$N,$ that is carried by Wilson lines. In other words a one-form symmetry. In
many ways this $\mathbb{Z}_{N}^{(1)}$ is a discrete analog of the
$U(1)^{(1)}_{e}$ symmetry of Maxwell theory. By contrast, the $U(1)^{(1)}_{m}$
symmetry has no discrete avatar in $SU(N)$ Yang-Mills.999This situation is
changed for other global forms of the gauge group. For instance in
$SU(N)/\mathbb{Z}_{N}$ Yang-Mills, there is no one-form charge carried by the
Wilson lines, but instead the ’t Hooft lines are charged under a
$\mathbb{Z}_{N}^{(1)}$.

To couple the $SU(N)$ theory to an appropriate background field we follow the
procedure described in [25]. The background is again represented by a
constrained gauge field $B^{(2)}$ obeying

$NB^{(2)}=d\lambda^{(1)}~{},\hskip
14.45377pt\left[\frac{N}{2\pi}B^{(2)}\right]\in H^{2}(Y,\mathbb{Z}_{N})~{}.$
(2.39)

We first promote the dynamical gauge field to be a $U(N)$ connection (for
simplicity we denote this by the same symbol $C$ now valued in the adjoint of
$U(N)$). We then introduce a Lagrange multiplier field $\varphi$ which
constrains the trace of $C$ to be given by the background source:

$S\supset\frac{1}{2\pi}\int\varphi(\mathrm{Tr}(F)-NB^{(2)})~{}.$ (2.40)

Thus on the solutions to the equation of motion, the abelian part of the field
strength is fixed by $B^{(2)}$ and so $F=B^{(2)}\mathbf{1}+\tilde{F}$, with
$\mathbf{1}$ the $N\times N$ identity matrix and $\tilde{F}$ traceless. As in
the Maxwell example, $C$ shifts under the one-form symmetry. This means that
under gauge transformations:

$B^{(2)}\longmapsto B^{(2)}+d\Lambda^{(1)}~{},\hskip
14.45377pt\lambda^{(1)}\longmapsto\lambda^{(1)}+N\Lambda^{(1)}~{},\hskip
14.45377ptC\longmapsto C+\Lambda^{(1)}\mathbf{1}~{},$ (2.41)

so the interaction (2.40) is invariant under background gauge transformations.
Notice in particular that if $B^{(2)}$ vanishes, $F$ is traceless and the
remaining background gauge redundancy can be used to shift $C$ by an arbitrary
flat $U(1)$ gauge field. Thus, we are reduced to the original $SU(N)$ theory.
More generally, a $U(N)$ Yang-Mills theory with the interaction (2.40) has the
same number of dynamical degrees of freedom as $SU(N)$ Yang-Mills.

With these remarks it is straightforward to write the full action coupled to
$B^{(2)}$ as:

$S=\int\frac{\varphi}{2\pi}(\mathrm{Tr}(F)-NB^{(2)})+\frac{1}{g^{2}}\mathrm{Tr}\left[(F-B^{(2)}\mathbf{1})\wedge*(F-B^{(2)}\mathbf{1})\right]+\frac{i\theta}{8\pi^{2}}\mathrm{Tr}\left[(F-B^{(2)}\mathbf{1})\wedge(F-B^{(2)}\mathbf{1})\right]~{}.$
(2.42)

As it will be important in the following, let us examine the $\theta$ term in
more detail. We recall the formula for the integral second Chern number of a
$U(N)$ gauge field:

$c_{2}=\frac{1}{8\pi^{2}}\int\mathrm{Tr}(F)\wedge\mathrm{Tr}(F)-\mathrm{Tr}(F\wedge
F)~{},\hskip 21.68121ptc_{2}\in\mathbb{Z}$ (2.43)

Then, using the constraint arising from (2.40) we have:

$\displaystyle\frac{i\theta}{8\pi^{2}}\mathrm{Tr}\left[(F-B^{(2)}\mathbf{1})\wedge(F-B^{(2)}\mathbf{1})\right]$
$\displaystyle=$
$\displaystyle\frac{i\theta}{8\pi^{2}}\int\left[\left(\mathrm{Tr}(F\wedge
F)-\mathrm{Tr}(F)\wedge\mathrm{Tr}(F)\right)+N(N-1)B^{(2)}\wedge
B^{(2)}\right]$ (2.44) $\displaystyle=$ $\displaystyle-i\theta
c_{2}+\frac{i\theta N(N-1)}{8\pi^{2}}\int B^{(2)}\wedge B^{(2)}~{}.$

In particular, since $B^{(2)}$ has periods that are $N$-th roots of unity, we
see that in the presence of background $B^{(2)},$ the periodicity of $\theta$
is enlarged to $\theta\sim\theta+2\pi N$. This effect will be responsible for
the intricate symmetry structure present in axion Yang-Mills.101010Here, we
are restricting ourselves to spacetimes which are spin, appropriate to later
applications involving fermions. More generally, on a non-spin manifold the
periodicity of $\theta$ is $2\pi N$ for $N$ odd and $4\pi N$ for $N$ even.

It is clarifying to describe the manipulations above using characteristic
classes. We are making use of the fact that $U(N)\cong\frac{U(1)\times
SU(N)}{\mathbb{Z}_{N}}$. The $\mathbb{Z}_{N}$ quotient means that the allowed
gauge bundles of $U(N)$ are more general than products of $U(1)$ bundles and
$SU(N)$ bundles. Instead, a $U(N)$ bundle can be described as a
$U(1)/\mathbb{Z}_{N}$ bundle and a $SU(N)/\mathbb{Z}_{N}$ bundle subject to a
constraint. This means that the $U(1)$ flux may be fractional (an $N$-th root
of unity) while the discrete magnetic flux of $SU(N)/\mathbb{Z}_{N}$,
mathematically, the second Steifel-Whitney class $w_{2}\in
H^{2}(Y,\mathbb{Z}_{N}),$ may be non-zero as long as these quantities are
related as:

$\frac{1}{2\pi}\mathrm{Tr}(F)=w_{2}=\left[\frac{N}{2\pi}B^{(2)}\right]\in
H^{2}(Y,\mathbb{Z}_{N})~{}.$ (2.45)

In particular, as is well-known, bundles of $SU(N)/\mathbb{Z}_{N}$ have
fractional instanton number which is precisely encoded by formula
(2.44).111111As in (2.27), we can express this in terms of the Pontryagin
square, with the fractional part of the instanton number being given by
$\frac{N-1}{2N}\mathcal{P}(w_{2})$.

####  2.3.1 Background Fields for Axion Yang-Mills

We now describe the symmetries of axion-Yang-Mills. The analysis is similar to
that of axion-Maxwell theory. In the absence of background fields, the action
is

$\frac{1}{2}\int
da\wedge*da+\frac{1}{g^{2}}\int\mathrm{Tr}(F\wedge*F)-\frac{iK}{8\pi^{2}f}\int
a\mathrm{Tr}(F\wedge F)~{},$ (2.46)

where above, the coupling constant $K$ is a positive integer. There are three
symmetry groups of interest: $\mathbb{Z}_{K}^{(0)}$, $\mathbb{Z}_{N}^{(1)}$,
and $U(1)^{(2)}$.

We now activate background fields and account for possible non-linear
corrections to the gauge transformations and fluxes. As before, this is most
clear by writing the axion term as a coupling on a five-manifold $X$ with
boundary the physical spacetime $Y$:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{Y}(da-
fA^{(1)})\wedge*(da-
fA^{(1)})+\int_{Y}\frac{\varphi}{2\pi}(\mathrm{Tr}(F)-NB^{(2)})+\frac{i}{2\pi
f}\int_{Y}a\,G^{(4)}$ $\displaystyle+$
$\displaystyle\frac{1}{g^{2}}\int_{Y}\mathrm{Tr}\left[(F-B^{(2)}\mathbf{1})\wedge*(F-B^{(2)}\mathbf{1})\right]-\frac{iK}{8\pi^{2}f}\int_{X}(da-
fA^{(1)})\mathrm{Tr}\left[(F-B^{(2)}\mathbf{1})\wedge(F-B^{(2)}\mathbf{1})\right]~{}.$

Demanding that, modulo possible anomalies, the action depends only on the
physical spacetime $Y$ implies that the gauge invariant field strength must
be:

$G^{(4)}=dA^{(3)}+\frac{KN(N-1)}{4\pi}B^{(2)}\wedge B^{(2)}~{}.$ (2.48)

In particular, this means that while the gauge transformations of $A^{(1)}$
and $B^{(2)}$ are standard, the gauge transformation rules for $A^{(3)}$ are
more interesting:

$A^{(3)}\longmapsto
A^{(3)}+d\Lambda^{(2)}-\frac{KN(N-1)}{2\pi}\Lambda^{(1)}\wedge
B^{(2)}-\frac{KN(N-1)}{4\pi}\Lambda^{(1)}\wedge d\Lambda^{(1)}~{}.$ (2.49)

Again, this is an example of a higher-group global symmetry.

We can also readily read off the anomaly inflow action

$S_{\mathrm{inflow}}=\frac{i}{2\pi}\int_{X}G^{(4)}\wedge A^{(1)}~{}.$ (2.50)

####  2.3.2 Extension to Axion-QCD

As a final model of axions, we consider axions coupled to Yang-Mills fields
and $N_{f}$ fundamental quarks, all with equal mass. The results follow
straightforwardly from the previous examples and our treatment is brief. The
action is given by:

$\frac{1}{2}\int
da\wedge*da+\frac{1}{g^{2}}\int\mathrm{Tr}(F\wedge*F)-\frac{iK}{8\pi^{2}f}\int
a\mathrm{Tr}(F\wedge F)+\int
i\overline{\Psi}\not{D}\Psi+i\overline{\tilde{\Psi}}\not{D}\tilde{\Psi}+(m\tilde{\Psi}\Psi+c.c)~{},$
(2.51)

where the gauge and flavor indices on the fermions $\Psi$ are suppressed. The
presence of fundamental matter means that the one-form symmetry of axion-Yang-
Mills is broken. However, there is a new zero-form flavor symmetry
$\frac{U(N_{f})}{\mathbb{Z}_{N}}^{(0)}$ arising from rotating the quarks,
where the $\mathbb{Z}_{N}$ quotient is because it is identified with the
center of the gauge group. In summary the symmetry is:

$\mathbb{Z}_{K}^{(0)}\times\frac{U(N_{f})}{\mathbb{Z}_{N}}^{(0)}\times
U(1)^{(2)}~{}.$ (2.52)

We will now show that these symmetries are extended into a non-trivial higher
group whenever $N$ does not divide $K$. The key feature is that when global
symmetry backgrounds that make use of the $\mathbb{Z}_{N}$ quotient are
activated, the flux of the dynamical Yang-Mills field is also fractional.

Following [14], we can understand this by looking at the gauge and global
group action on the fundamental fermions. The part that acts faithfully is
given by

$\frac{SU(N)\times
U(N_{f})}{\mathbb{Z}_{N}}\cong\frac{SU(N)}{\mathbb{Z}_{N}}\times\frac{SU(N_{f})}{\mathbb{Z}_{N_{f}}}\times\frac{U(1)}{\mathbb{Z}_{M}}~{},\hskip
14.45377ptM=\mathrm{lcm}(N,N_{f})~{},$ (2.53)

and above our convention is that the $U(1)$ factor (under which baryons are
charged) acts on the fundamental fermions with charge one. This means that in
general, the discrete magnetic flux of the dynamical gauge group
$w_{2}(SU(N))\in H^{2}(M,\mathbb{Z}_{N})$ is non-zero, as is that of the
flavor group $w_{2}(SU(N_{f}))\in H^{2}(M,\mathbb{Z}_{N_{f}})$ and the
fractional part of the $U(1)$ flavor flux (denoted by $Z$ below). However they
are related via:

$\int_{\Sigma_{2}}\frac{Z}{2\pi}=\frac{1}{N}\int_{\Sigma_{2}}w_{2}(SU(N))+\frac{1}{N_{f}}\int_{\Sigma_{2}}w_{2}(SU(N_{f}))+s~{},\hskip
14.45377pts\in\mathbb{Z}~{}.$ (2.54)

Similarly, we can also write a formula relating the fractional flux of the
$U(1)$ to the discrete magnetic flux $w_{2}(SU(N_{f}))$ for the flavor bundles
of $U(N_{f})/\mathbb{Z}_{N}$ as:

$N\int_{\Sigma_{2}}\frac{Z}{2\pi}=\frac{N}{N_{f}}\int_{\Sigma_{2}}w_{2}(SU(N_{f}))+\ell~{},\hskip
14.45377pt\ell\in\mathbb{Z}~{}.$ (2.55)

By varying the fluxes of the flavor gauge fields subject to (2.55), we can
achieve any value of the $w_{2}(SU(N))$ determined via (2.54), and therefore a
general fractional instanton number as in (2.44).

To match our previous discussion and notation we introduce a background field
$B^{(2)}$ dependent on the zero-form symmetry as:

$B^{(2)}=\frac{2\pi}{N_{f}}w_{2}(SU(N_{f}))~{}.$ (2.56)

Then the instanton number is fractional with

$\frac{1}{8\pi^{2}}\int\mathrm{Tr}(F\wedge
F)=\frac{N(N-1)}{8\pi^{2}}(Z-B)\wedge(Z-B)~{},$ (2.57)

Hence proceding as before leads to a formula for the gauge invariant four-form
field strength coupling to the axion:

$G^{(4)}=dA^{(3)}+\frac{KN(N-1)}{4\pi}(Z-B)\wedge(Z-B)~{}.$ (2.58)

In particular, the fluxes of $G^{(4)}/2\pi$ are fractional as long as $N$ does
not divide $K$.

##  3 Higher-Groups and Constraints on Emergence

The symmetries that we have discussed in the previous section are examples of
the general phenomena of higher-group global symmetry described in section
2.1.1. Here we use this to deduce model independent inequalities on the scales
of emergent symmetries in models of axions following the general logic of
[11].

###  3.1 Inequalities for Axion-Yang-Mills

Consider as an illustrative example the case of axion-$SU(N)$-Yang-Mills
described in section 2.3. The action is given by (2.46) and the symmetries are
$\mathbb{Z}_{K}^{(0)}\times\mathbb{Z}_{N}^{(1)}\times U(1)^{(2)}$. The key
result is equation (2.48) for the gauge invariant four-form field strength
associated to the symmetry $U(1)^{(2)}$:

$G^{(4)}=dA^{(3)}+\frac{KN(N-1)}{4\pi}B^{(2)}\wedge B^{(2)}~{},$ (3.1)

where $B^{(2)}$ is the background field for the $\mathbb{Z}_{N}^{(1)}$ one-
form global symmetry. In particular notice that when $B^{(2)}$ is activated,
there is in general also a non-trivial four-form flux $G^{(4)}$ (typically
fractional).

Now imagine a renormalization group flow where both the symmetries in question
are emergent. Let $E_{\mathrm{screen}}$ be the scale below which the
$\mathbb{Z}_{N}^{(1)}$ is a good symmetry of the effective field theory, and
$E_{\mathrm{string}}$ the scale below which $U(1)^{(2)}$ is a good symmetry.
Then below $E_{\mathrm{screen}}$ we can couple our theory to general
backgrounds $B^{(2)}$ and hence, since $G^{(4)}$ is also in general activated,
we must also be below $E_{\mathrm{string}}$ so that we can activate this
background as well. Therefore we deduce that for any renormalization group
trajectory which at low energies flows to axion-Yang-Mills, we have model
independent inequalities:

$E_{\mathrm{screen}}\mathrel{\mathstrut\smash{\ooalign{\raise
2.5pt\hbox{$<$}\cr\lower 2.5pt\hbox{$\sim$}}}}E_{string}~{}.$ (3.2)

Of course, as remarked in the introduction, the scales
$E_{\mathrm{screen}},E_{\mathrm{string}}$ are only approximately defined along
the flow. Therefore, we interpret the above as the statement that
$E_{\mathrm{screen}}$ cannot be parametrically larger than
$E_{\mathrm{string}}$.

####  3.1.1 Pecci-Quinn Example

It is useful to make the inequality more precise in the case of an explicit
KSVZ axion model [18, 19], where the the field $a$ emerges from a Pecci-Quinn
mechanism. In the ultraviolet, we have $SU(N)$ gauge theory with $K$ Weyl
fermions in the fundamental ($\psi_{+}$) and anti-fundamental $(\psi_{-}$)
representation (flavor indices suppressed) and an uncharged complex scalar
field $\varphi$. The action is:

$\displaystyle\begin{split}S=\int
d^{4}x\Bigg{\\{}\frac{1}{g^{2}}\,\textrm{Tr}\,[F\wedge*F]+i\bar{\psi}_{\pm}\not{D}\psi_{\pm}+\frac{1}{2}d\varphi\wedge*d\bar{\varphi}-V(\varphi)+\lambda\bar{\varphi}\psi_{+}\psi_{-}+\lambda\varphi\bar{\psi}_{-}\bar{\psi}_{+}\Bigg{\\}}~{}.\end{split}$
(3.3)

We assume the coupling constants above are small, so that semiclassical
analysis is reliable.

At the classical level, there is a $U(1)_{PQ}$ symmetry where the different
fields have charges

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

At the quantum level, this $U(1)_{PQ}$ is broken by an ABJ anomaly. Let the
scalar potential be:

$\displaystyle\begin{split}V(\varphi)=m^{2}(|\varphi|^{2}-f^{2})^{2}~{},\end{split}$
(3.4)

so that $\varphi$ condenses and gives a mass to the fermions through the
Yukawa couplings. When we flow to energies below the mass of the radial mode
of $\varphi$ we will see that this theory is described by the gauge fields and
a pseudo-goldstone mode $a$ which is an axion.

More explicitly, we write the complex scalar field as

$\varphi=\rho e^{ia/f}~{}.$ (3.5)

The radial mode $\rho$ has a mass $m_{\rho}\sim mf$. Below this scale the
action reduces to:

$\displaystyle\begin{split}S=\int
d^{4}x\Bigg{\\{}\frac{1}{g^{2}}\,\textrm{Tr}\,[F\wedge*F]+i\bar{\psi}_{\pm}\not{D}\psi_{\pm}+\frac{1}{2}da\wedge*da+\lambda
fe^{-ia/f}\psi_{+}\psi_{-}+\lambda
fe^{ia/f}\bar{\psi}_{+}\bar{\psi}_{-}\Bigg{\\}}~{}.\end{split}$ (3.6)

Via a chiral rotation of the fermions we can eliminate the $a$-dependence of
the fermion mass term. The fermions are then massive with $m_{\psi}=\lambda f$
and provided this is above the QCD scale so that the model remains weakly
coupled we can integrate them out. However, since the chiral symmetry is
anomalous, this chiral rotation generates an axion coupling between $a$ and
the remaining gauge fields. Thus the final action is that of axion-Yang-Mills.

$\frac{1}{2}\int
da\wedge*da+\frac{1}{g^{2}}\int\mathrm{Tr}(F\wedge*F)-\frac{iK}{8\pi^{2}f}\int
a\mathrm{Tr}(F\wedge F)~{},$ (3.7)

The UV theory has neither a one-form nor a two-form global symmetry. Both are
emergent at low energies and are associated to characteristic energy scales.
Specifically:

  * •

The one-form symmetry is associated to a scale

$E_{\mathrm{screen}}\approx\lambda f~{},$ (3.8)

which is the mass of the fundamental fermions. Indeed above this scale, the
fermions fluctuate and can screen the charged Wilson lines breaking the one-
form symmetry $\mathbb{Z}_{N}^{(1)}.$

  * •

Defining the two-form symmetry scale $E_{\mathrm{string}}$ is more subtle.
There are several scales naturally associated to the emergence of the axion.
First, there is the string tension $T_{\mathrm{string}}^{1/2}=f$.121212 The
tension of the axion string can be estimated by approximating a charge $P$
axion string solution as
$\displaystyle\begin{split}\varphi=f\theta(|x|-\ell_{core})e^{iP\phi}~{},\end{split}$
(3.9) where $\ell_{core}$ is the characteristic size of the string. The
tension can be roughly determined by the potential energy:
$\displaystyle\begin{split}T_{\mathrm{string}}\sim\int
d^{2}x\,m^{2}(|\varphi|^{2}-f^{2})^{2}=\pi
m^{2}f^{4}\ell_{core}^{2}~{}.\end{split}$ (3.10) Then using
$\ell_{core}\sim\frac{1}{m_{\rho}}\sim\frac{1}{mf}$, we find that the tension
goes like $T_{\mathrm{string}}\sim f^{2}$.  There is also the scale
$m_{\rho}\approx mf$ where the radial mode begins to fluctuate. At weak
coupling, $m<<1$ these scales are well separated
$T_{\mathrm{string}}^{1/2}\sim f>>m_{\rho}\sim mf$.

Note that just above the energy $m_{\rho}$, the scalar potential confines
$\varphi$ to valley surrounding the minimum and the scalar field effectively
becomes $S^{1}\times I$-valued where $I$ is an interval. It is this effective
non-trivial topology in the space of fields, i.e. the presence of a circle
factor, that ensures there is a two form-symmetry and a topologically
conserved winding number. In particular just above the scale $m_{\rho}$ the
two-form symmetry persists. As we go to yet higher scales, we eventually reach
the energy $E_{\mathrm{string}}$ where the $\rho$ particle can traverse all
the way to the origin in field space and break the two-form symmetry.
Physically this means that at this scale there are processes that allow
strings to unwind (See Figure 2). This scale is determined by the value of the
potential at the origin $V(0)\sim m^{2}f^{4}$ so that

$E_{\mathrm{string}}\approx\sqrt{m}f~{}.$ (3.11)

At weak-coupling $m<<1$ there is thus a hierarchy:

$\displaystyle\begin{split}T_{\mathrm{string}}^{1/2}=f~{}~{}>>~{}~{}E_{\mathrm{string}}=\sqrt{m}f~{}~{}>>~{}~{}m_{\rho}=mf~{}.\end{split}$
(3.12)

Figure 2: This figure illustrates how the axion string can unwind. In (a), we
show the winding solution for the axion string along a circle of fixed radius.
As we go around the blue circle (a circle linking the axion string (black) in
spacetime) the scalar field $\varphi$ winds around the bottom of the mexican
hat potential in red. (b) shows the excitations of the radial mode which is
activated at the scale $m_{\rho}\sim mf$. (c-e) shows a process by which we
can unwind the scalar field. Here we deform solution in the radial direction
over the top of the potential which costs energy $\sqrt{m}f>>mf$. The
resulting configuration has no winding – indicating the decay of the string.

Having identified the emergent symmetry scales in the problem, we can now
apply the general inequality (3.2) to deduce

$\displaystyle\begin{split}\sqrt{m}\gtrsim\lambda\Longleftrightarrow
m_{\rho}f\gtrsim m_{\psi}^{2}~{},\end{split}$ (3.13)

where the left-hand-side expresses the constraint in terms of dimensionless
coupling constants, and the right-hand side expresses the same constraint in
terms of the physical masses and tensions in the problem.

It is interesting to explore what happens when the inequality (3.13) is
violated. Since the scales in question are all approximate, we consider a
parametric limit within a weakly-coupled setting where we retain technical
control: $1>>\lambda^{2}>>m$. We will argue that the Pecci-Quinn field theory
(3.3) breaks down.

To begin let us note that the action (3.3) does not define a UV complete
theory: at very high energies the scalar quartic coupling runs to a Landau
pole. We interpret the UV action as an effective theory defined with a cutoff
scale $\Lambda_{UV}>>f.$ We interpret the inequality (3.13) as a statement
about the couplings defined at the scale $\Lambda_{UV}$. Thus, the inequality
is parametrically violated when $\lambda^{2}(\Lambda_{UV})>>m(\Lambda_{UV})$.
In this limit, we will find that the fermions significantly modify the
effective scalar potential.

It is straightforward to compute the effective potential by integrating out
the fermions in a background of constant $\varphi$. (For convenience, one can
combine the pair of Weyl fermions $\psi_{\pm}$ into a single Dirac fermion,
and express $\varphi$ in terms of its real and imaginary parts). Their
contribution to the effective scalar potential is then:

$\displaystyle\Delta V$ $\displaystyle=$ $\displaystyle\int
d\lambda\partial_{\lambda}\,\textrm{Tr}\,\,{\rm
log}(-i\not{D}+\lambda(\varphi_{R}+i\gamma^{5}\varphi_{I}))$ $\displaystyle=$
$\displaystyle\frac{\lambda^{4}NK}{8\pi^{2}}\left(|\varphi|^{4}{\rm
log}\left(\frac{\lambda|\varphi|}{\Lambda_{UV}}\right)-\frac{|\varphi|^{4}}{4}\right)~{}.$

The full effective potential for the theory is therefore:

$\displaystyle\begin{split}V_{eff}=\frac{\lambda^{4}NK}{8\pi^{2}}\left(|\varphi|^{4}{\rm
log}\left(\frac{\lambda|\varphi|}{\Lambda_{UV}}\right)-\frac{|\varphi|^{4}}{4}\right)+m^{2}(|\varphi|^{2}-f^{2})^{2}~{},\end{split}$
(3.15)

where all couplings are evaluated at the scale $\Lambda_{UV}$.

When $\lambda^{2}$ is small compared to $m,$ it is clear that the tree level
potential dominates and one finds a minimum near $\langle|\varphi|\rangle=f$
and the previous analysis leading to axion-Yang-Mills is valid. However, when
the quartic coupling is non-negligible the minimum moves drastically. Working
in the limit where the cutoff scale is large, $\Lambda_{UV}>>\lambda f$, we
find that the implied vacuum is at:

$\langle|\varphi|\rangle\approx\frac{\Lambda_{UV}}{\lambda}{\rm
exp}\left(-\frac{8\pi^{2}m^{2}}{NK\lambda^{4}}\right)~{}.$ (3.16)

In particular when $\lambda^{2}>>m$ so that the inequality (3.13) is violated,
the minimum is larger than the cutoff scale $\Lambda_{UV}$.131313Notice that
in the limit of interest $\lambda^{2}>>m$ the logarithm is small and we expect
the one-loop effective potential to be reliable. When this happens the
effective field theory breaks down and we lose control. Indeed, in formulating
the effective field theory, we have neglected irrelevant operators in the
action suppressed by powers of the cutoff. For instance, terms in the
potential of the form $|\varphi|^{L+4}/\Lambda_{UV}^{L}$ for large $L$ have
been discarded. This analysis is correct provided that the minimum of
$|\varphi|$ is small, but when $|\varphi|$ is of order the cutoff the EFT
analysis breaks down. Thus we see that the inequality (3.13), derived from
general considerations of symmetry, is built in to the consistency of the
effective field theory.

###  3.2 Inequalities for Axion-QCD

We can apply similar logic to $SU(N)$ axion-QCD which has global symmetry
$\mathbb{Z}_{K}^{(0)}\times\left(U(N_{f})/\mathbb{Z}_{N}\right)^{(0)}\times
U(1)^{(2)}$. As found in equation (2.58), the fluxes of $G^{(4)}$ are
necessarily activated (and fractional) provided that we activate general
backgrounds of $\left(U(N_{f})/\mathbb{Z}_{N}\right)^{(0)}$ that make use of
the $\mathbb{Z}_{N}$ quotient. Therefore, we again obtain constraints on the
scales of symmetry emergence for any flow that realizes axion-QCD in the IR:

$E_{\mathbb{Z}_{N}}\mathrel{\mathstrut\smash{\ooalign{\raise
2.5pt\hbox{$<$}\cr\lower 2.5pt\hbox{$\sim$}}}}E_{\mathrm{string}}~{}.$ (3.17)

Here, the scale $E_{\mathrm{string}}$ is again the string stability scale,
while $E_{\mathbb{Z}_{N}}$ is the energy at which operators charged under the
$\mathbb{Z}_{N}\subset U(N_{f})$ appear.

As in our analysis of axion-Yang-Mills inequalities we expect that (3.17) is
enforced by the consistency of any UV effective field theory which flows to
axion-QCD at long distances. As a simple example consider the action:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\Bigg{\\{}\frac{1}{g^{2}}\,\textrm{Tr}\,[F\wedge*F]+i\bar{\psi}_{\pm}\not{D}\psi_{\pm}+\frac{1}{2}d\varphi\wedge*d\bar{\varphi}-m^{2}(|\varphi|^{2}-f^{2})^{2}+\lambda\bar{\varphi}\psi_{+}\psi_{-}+\lambda\varphi\bar{\psi}_{-}\bar{\psi}_{+}$
(3.18) $\displaystyle+$ $\displaystyle
i\bar{\Psi}_{I}\not{D}\Psi^{I}+i\overline{\tilde{\Psi}}^{I}\not{D}\tilde{\Psi}_{I}+(M\tilde{\Psi}_{I}\Psi^{I}+c.c.)\Big{\\}}~{},$

where the fermions $\Psi^{I}$ ($\tilde{\Psi}_{I}$) transform in the
fundamental (anti-fundamental) representation of the gauge group $SU(N)$.

Note that in this EFT the relevant the flavor symmetry is in fact
$U(N_{f})^{(0)}$ not $\left(U(N_{f})/{\mathbb{Z}}_{N}\right)^{(0)}$. The
reason is that we can construct the local, gauge invariant operators

${\cal O}^{I}=\psi_{1}\Psi^{I}~{},\hskip 36.135pt\tilde{\cal
O}_{I}=\psi_{2}\tilde{\Psi}_{I}~{},$ (3.19)

which transform non-trivially under $\mathbb{Z}_{N}$. When we flow to the IR,
the Higgs field gives the PQ-fermions a mass. This reduces the flavor symmetry
$U(N_{f})\longrightarrow U(N_{f})/{\mathbb{Z}}_{N}$. Therefore, we identify
the scale $E_{\mathbb{Z}_{N}}\approx\lambda f$. Meanwhile, following the
discussion of axion-Yang-Mills, the string stability scale is
$E_{\mathrm{string}}\approx\sqrt{m}f$. The inequality (3.17) reduces to
$\lambda\mathrel{\mathstrut\smash{\ooalign{\raise 2.5pt\hbox{$<$}\cr\lower
2.5pt\hbox{$\sim$}}}}\sqrt{m}$ and is derived identically to the analysis of
axion Yang-Mills above. Again, when the inequality is parametrically violated
the PQ fermions make large contributions to the scalar potential and the EFT
breaks down.

###  3.3 Inequalities for Axion Electrodynamics

We can also apply our analysis to axion-electrodynamics. As discussed in
section 2.2, there are several higher group structures to consider when the
axion coupling constant $K>1$. The implications of the higher group extension
of $U(1)^{(2)}$ are identical to those of the previous section: the emergence
scale $E_{\mathrm{string}}$ must be the largest scale of those involved in the
higher group.

Somewhat more novel is the higher group extension of the $U(1)^{(1)}_{m}$
magnetic one-form symmetry which measures the charges of ’t Hooft lines. The
gauge invariant field strength derived in (2.27) and (2.38) is:

$H^{(3)}=dB_{m}^{(2)}+\frac{K}{2\pi}A^{(1)}\wedge B^{(2)}_{e}~{}.$ (3.20)

Here we recall that $A^{(1)}$ is the background field for the
$\mathbb{Z}_{K}^{(0)}$ shift symmetry of the axion. Meanwhile, in axion-
Maxwell theory (with no charged matter) $B^{(2)}_{e}$ is the background for
the electric $\mathbb{Z}_{K}^{(1)}$ one-form symmetry, while it is a composite
due to the quotient in the flavor symmetry
$\left(SU(N_{f})/\mathbb{Z}_{L}\right)^{(0)}$ ($L=\gcd(K,N_{f})$) in the
axion-QED. In particular, we see that if both $A^{(1)}$ and $B^{(2)}_{e}$ are
present then necessarily $H^{(3)}$ fluxes are also present. Following similar
logic to our analysis of axion-YM in section 3.1, this leads to a constraint
on the scales of symmetry emergence.

In the case of axion Maxwell theory, we let $E_{\mathrm{magnetic}}$ be the
scale of emergence of the $U(1)^{(1)}_{m}$ magnetic one-form symmetry. We also
denote by $E_{\mathrm{shift}}$ the scale where the $\mathbb{Z}_{K}^{(0)}$
shift symmetry emerges, and $E_{\mathrm{screen}}$ the scale where the
$\mathbb{Z}_{K}^{(1)}$ electric one-form symmetry emerges. Then we obtain a
universal inequality:

$\min\\{E_{\mathrm{shift}},~{}E_{\mathrm{screen}}\\}\mathrel{\mathstrut\smash{\ooalign{\raise
2.5pt\hbox{$<$}\cr\lower 2.5pt\hbox{$\sim$}}}}E_{\mathrm{magnetic}}~{}.$
(3.21)

Similarly in axion-QED the role of $E_{\mathrm{electric}}$ is now played by
$E_{\mathbb{Z}_{L}}$, above which there are operators charged under the
$\mathbb{Z}_{L}$ subgroup of the flavor group $SU(N_{f})$. We then have:

$\min\\{E_{\mathrm{shift}},~{}E_{\mathbb{Z}_{L}}\\}\mathrel{\mathstrut\smash{\ooalign{\raise
2.5pt\hbox{$<$}\cr\lower 2.5pt\hbox{$\sim$}}}}E_{\mathrm{magnetic}}~{}.$
(3.22)

####  3.3.1 Adjoint Higgsing Example

As with the inequalities derived for axion-QCD, we expect that in examples
(3.21) and (3.22) are enforced automatically by self consistency. Let us
illustrate this in a concrete flow ending in axion-Maxwell theory.

In the UV we consider a model of $SU(2)$ axion-Yang-Mills coupled to a real
adjoint Higgs field $\Phi$, and potential $V(\Phi)$. The action is:

$\frac{1}{2}\int
da\wedge*da+\frac{1}{g^{2}}\int\mathrm{Tr}(F\wedge*F)-\frac{i}{8\pi^{2}f}\int
a\mathrm{Tr}(F\wedge
F)+\int\mathrm{Tr}\left(D\Phi\wedge*D\Phi\right)-V(\Phi)~{}.$ (3.23)

Note in particular that we have set the integral coupling constant $K=1$
above. This means that this UV action has symmetry $\mathbb{Z}_{2}^{(1)}$
(arising from the center of the gauge group), and $U(1)^{(2)}$, but no
discrete shift symmetry of the axion.

We now consider a potential such that
$\langle\mathrm{Tr}(\Phi^{2})\rangle=v^{2}$ thus higgsing the gauge group from
$SU(2)$ to $U(1)$. Importantly, this means that the minimum instanton number
is multiplied by two: $\mathrm{Tr}(F\wedge F)\rightarrow 2F\wedge F$, where
the right-hand-side is the remaining abelian gauge field. Therefore at low-
energies we find axion-Maxwell theory (plus an irrelevant decoupled scalar
descending from $\Phi$) with discrete coupling constant $K=2$. Therefore, at
low-energies we arrive in a theory with non-trivial higher group structure
extending the emergent $U(1)^{(1)}_{m}$ magnetic one-form symmetry.

It is straightforward to determine the energy scales associated with the
emergent symmetries:

  * •

The $\mathbb{Z}_{2}^{(1)}$ electric symmetry is present in both the UV and IR.
Thus we have $E_{\mathrm{screen}}\rightarrow\infty$.

  * •

The $U(1)^{(1)}_{m}$ magnetic one-form symmetry emerges at the Higgsing scale,
so $E_{\mathrm{magnetic}}\approx v$. One way to understand this is that, in
terms of UV non-abelian fields, the long-distance gauge field is
$\frac{1}{v}\mathrm{Tr}(\Phi F)$. Therefore, the scale $v$ is where its
correlation functions begin to differ from those of an Abelian gauge field.

  * •

The $\mathbb{Z}_{2}^{(0)}$ emergent zero form symmetry is broken in the UV by
instantons that generate a potential for the axion. One can estimate this
potential using an instanton gas approximation

$V(a)\approx g^{4\alpha}v^{4}{\rm
exp}\left(-\frac{8\pi^{2}}{g^{2}}\right)\cos(a/f)~{}.$ (3.24)

Here, $g$ is the coupling evaluated at the scale $v$ of Higgsing, and $\alpha$
is a constant fixed by the $\beta$ function of the theory. This formula arises
by estimating the integral over instanton moduli space where the size modulus
of the instanton is cut off at the scale $gv$ where the W-bosons become
massive. The energy cost of a shift $a\rightarrow a+\pi f$, then sets the
scale $E_{\mathrm{shift}}$:

$E_{\mathrm{shift}}\approx g^{\alpha}v\ {\rm
exp}\left(-\frac{2\pi^{2}}{g^{2}}\right)~{}.$ (3.25)

We can now apply the inequality (3.21):

$E_{\mathrm{shift}}\mathrel{\mathstrut\smash{\ooalign{\raise
2.5pt\hbox{$<$}\cr\lower
2.5pt\hbox{$\sim$}}}}E_{\mathrm{magnetic}}\Longleftrightarrow g^{\alpha}\ {\rm
exp}\left(-\frac{2\pi^{2}}{g^{2}}\right)\mathrel{\mathstrut\smash{\ooalign{\raise
2.5pt\hbox{$<$}\cr\lower 2.5pt\hbox{$\sim$}}}}1~{}.$ (3.26)

Thus, the inequality is simply the statement that at the higgsing scale $v$,
where $g$ is evaluated, the theory is weakly coupled $g(v)<<1$. Of course, if
this is not true, the theory confines before higgsing and our analysis leading
to axion-maxwell theory in the IR breaks down. Therefore (3.21) is automatic
as expected.

## Acknowledgements

We thank T. Dumitrescu, Z. Komargodski, M. Reece, and L.T. Wang for
discussions and comments. T. D. B. is supported by the Mafalda and Reinhard
Oehme Postdoctoral Fellowship in the Enrico Fermi Institute at the University
of Chicago.

## References

  * [1] J. E. Kim and G. Carosi, “Axions and the Strong CP Problem,” Rev. Mod. Phys. 82 (2010) 557–602, arXiv:0807.3125 [hep-ph]. [Erratum: Rev.Mod.Phys. 91, 049902 (2019)].
  * [2] D. J. E. Marsh, “Axion Cosmology,” Phys. Rept. 643 (2016) 1–79, arXiv:1510.07633 [astro-ph.CO].
  * [3] P. Svrcek and E. Witten, “Axions In String Theory,” JHEP 06 (2006) 051, arXiv:hep-th/0605206.
  * [4] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th].
  * [5] E. Lake, “Higher-form symmetries and spontaneous symmetry breaking,” arXiv:1802.07747 [hep-th].
  * [6] S. R. Coleman, “There are no Goldstone bosons in two-dimensions,” Commun. Math. Phys. 31 (1973) 259–264.
  * [7] M. B. Green and J. H. Schwarz, “Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory,” Phys. Lett. B 149 (1984) 117–122.
  * [8] A. Kapustin and R. Thorngren, “Higher symmetry and gapped phases of gauge theories,” arXiv:1309.4721 [hep-th].
  * [9] A. Kapustin and R. Thorngren, “Anomalies of discrete symmetries in various dimensions and group cohomology,” arXiv:1404.3230 [hep-th].
  * [10] Y. Tachikawa, “On gauging finite subgroups,” SciPost Phys. 8 no. 1, (2020) 015, arXiv:1712.09542 [hep-th].
  * [11] C. Córdova, T. T. Dumitrescu, and K. Intriligator, “Exploring 2-Group Global Symmetries,” JHEP 02 (2019) 184, arXiv:1802.04790 [hep-th].
  * [12] F. Benini, C. Córdova, and P.-S. Hsin, “On 2-Group Global Symmetries and their Anomalies,” JHEP 03 (2019) 118, arXiv:1803.09336 [hep-th].
  * [13] N. Seiberg, Y. Tachikawa, and K. Yonekura, “Anomalies of Duality Groups and Extended Conformal Manifolds,” PTEP 2018 no. 7, (2018) 073B04, arXiv:1803.07366 [hep-th].
  * [14] C. Córdova, D. S. Freed, H. T. Lam, and N. Seiberg, “Anomalies in the Space of Coupling Constants and Their Dynamical Applications II,” SciPost Phys. 8 no. 1, (2020) 002, arXiv:1905.13361 [hep-th].
  * [15] Y. Hidaka, M. Nitta, and R. Yokokura, “Higher-form symmetries and 3-group in axion electrodynamics,” Phys. Lett. B 808 (2020) 135672, arXiv:2006.12532 [hep-th].
  * [16] Y. Hidaka, M. Nitta, and R. Yokokura, “Global 3-group symmetry and ’t Hooft anomalies in axion electrodynamics,” arXiv:2009.14368 [hep-th].
  * [17] C. Córdova, T. T. Dumitrescu, and K. Intriligator, “2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories,” arXiv:2009.00138 [hep-th].
  * [18] J. E. Kim, “Weak Interaction Singlet and Strong CP Invariance,” Phys. Rev. Lett. 43 (1979) 103.
  * [19] M. A. Shifman, A. Vainshtein, and V. I. Zakharov, “Can Confinement Ensure Natural CP Invariance of Strong Interactions?,” Nucl. Phys. B 166 (1980) 493–506.
  * [20] R. Peccei and H. R. Quinn, “CP Conservation in the Presence of Instantons,” Phys. Rev. Lett. 38 (1977) 1440–1443.
  * [21] R. Peccei and H. R. Quinn, “Constraints Imposed by CP Conservation in the Presence of Instantons,” Phys. Rev. D 16 (1977) 1791–1797.
  * [22] C. Córdova, D. S. Freed, H. T. Lam, and N. Seiberg, “Anomalies in the Space of Coupling Constants and Their Dynamical Applications I,” SciPost Phys. 8 no. 1, (2020) 001, arXiv:1905.09315 [hep-th].
  * [23] C. Córdova and K. Ohmori, “Anomaly Obstructions to Symmetry Preserving Gapped Phases,” arXiv:1910.04962 [hep-th].
  * [24] C. Córdova, P.-S. Hsin, and N. Seiberg, “Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)$d$,” SciPost Phys. 5 no. 1, (2018) 006, arXiv:1712.08639 [cond-mat.str-el].
  * [25] D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, “Theta, Time Reversal, and Temperature,” JHEP 05 (2017) 091, arXiv:1703.00501 [hep-th].