# Extra-Dimensional Axion Expectations

Matthew Reece
mreece (@g.harvard.edu)
Department of Physics, Harvard University, Cambridge, MA, 02138

(June 12, 2024)

###### Abstract

Axions arising as modes of higher-dimensional gauge fields are known to offer
a compelling solution to the axion quality problem and to naturally arise in
string theory. In this context, it is interesting to ask how we would
interpret an experimental measurement of the axion decay constant $f$. I give
several arguments for, as well as concrete examples in string theory of, the
existence in such a model of an axion string with tension of order
$2\text{p}S_{\mathrm{inst}}f^{2}$, where $S_{\mathrm{inst}}$ is the instanton
action. Furthermore, in models of this type axion strings are typically
fundamental objects (rather than solitons), whose tension is at or above the
fundamental cutoff of the theory. As a result, I argue that for an extra-
dimensional QCD axion, it is likely that the fundamental cutoff scale lies at
most two orders of magnitude above $f$. In addition to these core arguments,
this paper begins with a self-contained introduction to the physics of extra-
dimensional axions and ends with some comments on axion physics in relation to
chiral fermions.

###### Contents

  1. 1 Introduction and Summary
  2. 2 Axions from higher-dimensional gauge fields
    1. 2.1 What is an extra-dimensional axion?
      1. 2.1.1 Geometry and topology
      2. 2.1.2 $p$-form gauge theory on $M$
      3. 2.1.3 Existence of massless 4d axion modes
      4. 2.1.4 Axion kinetic matrix and periodicity
    2. 2.2 Masses for extra-dimensional axions
      1. 2.2.1 Brane instantons
      2. 2.2.2 Gauge theory instantons
      3. 2.2.3 Tree-level masses
    3. 2.3 Axion strings for extra-dimensional axions
    4. 2.4 Qualitative features of extra-dimensional axions
  3. 3 Estimating a UV cutoff
  4. 4 Axion string tension scaling with decay constant
    1. 4.1 Scaling of decay constants with volumes
    2. 4.2 Bogomolnyi bounds and repulsive forces
      1. 4.2.1 Euclidean instantons
      2. 4.2.2 Axion strings
      3. 4.2.3 Interpretation
      4. 4.2.4 Example: exponential ansatz
      5. 4.2.5 Example: SUSY form, power law ansatz
    3. 4.3 Magnetic and electric tension scales
      1. 4.3.1 Magnetic volume scaling argument
      2. 4.3.2 Axion string tension and magnetic monopole scale
  5. 5 Axion string tension in examples
    1. 5.1 “Model-independent” $B^{(2)}$ axion
    2. 5.2 $C^{(4)}$ axions in Type IIB string theory
      1. 5.2.1 Elements of Calabi-Yau geometry
      2. 5.2.2 Translation to physical quantities
      3. 5.2.3 First example: 3-axion, fibered “Swiss cheese” Calabi-Yau
      4. 5.2.4 Second example: 2-axion “Swiss cheese” Calabi-Yau
      5. 5.2.5 Comments on the Calabi-Yau examples
    3. 5.3 Type IIB axion $C_{0}$
  6. 6 Comments on chiral fermions and axions
    1. 6.1 Tree-level masses, revisited
      1. 6.1.1 Monodromy mass versus chiral fermions
      2. 6.1.2 Green-Schwarz mechanism and axions
    2. 6.2 Fermion couplings
  7. 7 Outlook
  8. A Derivation of 2-cycle and 4-cycle relationship in CY 3-folds

##  1 Introduction and Summary

The most well-studied solution to the Strong CP problem is an axion field,
namely a field $\theta(x)$ of period $2\text{p}$ that couples to the gluon
field strength $G$ via the interaction

$S_{\mathrm{axion}}=\frac{k_{G}}{8\text{p}^{2}}\int\theta(x)\operatorname{tr}\left[G(x)\wedge
G(x)\right],$ (1)

for some integer $k_{G}$. If this is the only interaction that violates the
continuous shift symmetry $\theta\mapsto\theta+\text{const.}$, then strong
dynamics generates an effective potential that dynamically relaxes the QCD
theta angle to zero [1, 2, 3, 4, 5, 6]. The original axion model involved
spontaneous breaking of an approximate U(1) global symmetry at the electroweak
scale, and was soon ruled out by data. Shortly thereafter, three distinct
axion scenarios with much weaker interactions, safe from experimental limits
at the time, were proposed. The first two, the KSVZ [7, 8] and DFSZ [9, 10]
models, posit the axion as a pseudo-Nambu-Goldstone boson (PNGB) of an
approximate U(1) (“Peccei-Quinn” or PQ) global symmetry that is spontaneously
broken at a high scale. They differ in whether or not the Standard Model Higgs
and fermion fields are charged under this symmetry (no for KSVZ, yes for
DFSZ). These are the best-known paradigms for axion models. They are widely
taught, and minimal incarnations of each have become standard benchmarks on
experimental sensitivity plots. (The basic KSVZ and DFSZ paradigms can be
realized with a variety of choices of field content, predicting different
values for the axion–photon coupling, so one should not take the standard plot
labels too seriously.)

The third axion scenario, the extra-dimensional axion [11], is the subject of
this paper. It was proposed just a few years after the KSVZ and DFSZ
scenarios, and has very different physics. There is no spontaneous breaking of
a 4d U(1) symmetry to produce a pseudo-Nambu-Goldstone boson. Instead, the
axion is a mode of a higher-dimensional gauge field. These models are well-
motivated, from the bottom-up perspective (they significantly ameliorate the
axion quality problem), from the top-down perspective (the ingredients
automatically arise in string theory), and from more general principles (such
axions appear to play a key role for the consistency of quantum gravity).
Despite the large literature on such models, they are under-appreciated,
particularly in discussions of axion phenomenology and cosmology. For example,
because these models do not have a 4d PQ-breaking phase transition, the often-
discussed post-inflationary axion cosmology scenario does not apply to them.
Furthermore, the physics at the scale of the axion decay constant $f$ is
dramatically different: rather than simply expecting to find some new 4d
fields with masses of order $f$, one expects that $f$ is near the quantum
gravity cutoff scale, where local quantum field theory breaks down entirely.
If an axion particle is discovered and its decay constant is measured, then,
the conclusions we would draw from this measurement are radically different in
the extra-dimensional scenario and in the more widely-studied 4d scenarios.

I have two main goals in this paper. First, I give a general exposition of the
extra-dimensional axion scenario, based on its core ingredients rather than
starting from a specific string theory realization. Much of this exposition
covers familiar territory, though there are certain details (e.g., the
argument for the existence of massless axions in a compactification with
general warping) that I have not seen spelled out explicitly before (though
they are surely known to experts). Second, I explain some expectations for how
various energy scales are related to each other in extra-dimensional axion
models. In particular, I emphasize that in the extra-dimensional scenario
(unlike the KSVZ or DFSZ scenarios) the axion decay constant is likely to
carry information about the fundamental scale where quantum field theory
breaks down. This raises the stakes for an experimental discovery of an axion,
but also points to the need for observational signals that could convincingly
distinguish between a 4d axion and an extra-dimensional one.

Before summarizing the claims of this paper in more detail, I offer a brief
(necessarily incomplete) review of the literature on extra-dimensional axions.
To the best of my knowledge, the extra-dimensional axion was first discussed
in the context of Type I string theory [11], where the necessary ingredients
appeared automatically, rather than being constructed specifically to solve
the Strong CP problem. Such axions received significant attention in studies
of Type I and heterotic string theory [12, 13, 14, 15, 16, 17, 18, 19]. The
potential for string-theoretic axions to ameliorate the axion quality problem
was emphasized in [20]. Later, it was understood that constructions of gauge
fields on D-branes in Type II string theory also automatically contain axion-
like couplings to RR gauge fields [21, 22, 23]. Following the initial wave of
research on moduli stabilization in string theory, two key papers made strides
toward bridging the gap between string theory constructions and axion
phenomenology [24, 25]. A few papers introduced the idea of extra-dimensional
axions into bottom-up phenomenological model-building [26, 27, 28].
Eventually, the fact that string theory can predict a significant number of
axion-like fields (already observed in [11]) became widespread knowledge, with
phenomenological studies emphasizing that the axion decay constant in such a
scenario was likely to be large (with associated cosmological difficulties
[29]) and that the number of axion-like particles could be large [30].
Phenomenological interest, in turn, fed back into string theory studies [31,
32], which have recently made computational advances allowing the exploration
of phenomenological implications of a large number of axion fields (e.g., [33,
34, 35, 36, 37, 38]). Recent work has emphasized that although 4d axion models
produce solitonic axion strings, with PQ symmetry restored in their core, by
contrast axion strings for extra-dimensional axions are not solitons and have
cores that probe deep ultraviolet physics and infinite-distance limits of
moduli space [39, 40, 41, 42, 43]. This leads to qualitative cosmological
differences; in particular, there is no PQ-violating phase transition in which
axion strings would be produced by the Kibble-Zurek mechanism [44, 45]. Other
recent work discussing differences between extra-dimensional and 4d axion
models includes [46, 47, 48, 49]. A recent talk by Kiwoon Choi [50] took a
similar perspective to my own, proposing the terminology “Type I axion” and
“Type II axion” for 4d and extra-dimensional axions respectively and
emphasizing their differences.

Here is a brief summary of the main messages to take away from the paper; some
review well-known facts, while others are novel:

  * •

Extra-dimensional gauge theories automatically produce high-quality axions.
Specifically, given a U(1) $p$-form (antisymmetric $p$-index tensor) gauge
field $C_{\mu_{1}\cdots\mu_{p}}$, there is a 4d axion mode, schematically of
the form
$\int_{\Sigma}C_{\mu_{1}\cdots\mu_{p}}\operatorname{d}\\!{x^{\mu_{1}}}\cdots\operatorname{d}\\!{x^{\mu_{p}}}$,
for each independent, non-torsion, $p$-dimensional cycle $\Sigma$ contained in
the extra dimensions. This is true in complete generality, including in warped
compactifications. The 4d axion is a periodic scalar field that obtains a mass
only through exponentially small instanton effects, corresponding either to
traditional 4d gauge theory instantons or to semiclassical effects from
wrapping extended objects around the extra dimensions.

  * •

There is no Peccei-Quinn phase transition for an extra-dimensional axion. In
these models, the axion is not a PNGB of an anomalous 4d symmetry, except in
the trivial sense that the axion particle can be created by the shift-symmetry
current $\partial_{\mu}\theta(x)$. In particular, the analogue of the PQ
symmetry is already present in the higher dimensional theory and acts
nonlinearly on the gauge field, descending to an approximate shift symmetry on
$\theta$ in 4d. There is no analogue of the usual process by which a complex
scalar gets a VEV as the universe cools, breaking PQ symmetry. Importantly,
this means that extra-dimensional axions intrinsically belong to the “pre-
inflationary axion” class of cosmologies—the PQ symmetry is always already
broken—and the most serious cosmological problem with such models is the axion
isocurvature problem rather than the domain wall problem.

  * •

The axion decay constant is a guide to the fundamental UV cutoff of the
theory. This claim, in quantitative form, is the main novel content of the
paper. I will give several (non-rigorous) arguments and survey several
examples in string theory in support of the claim that, given an axion coupled
to gauge fields with associated instanton action $S_{\mathrm{inst}}$ and an
axion decay constant $f$, we expect to find a (possibly nonminimal) axion
string with core tension near the scale

${\cal T}\approx 2\text{p}S_{\mathrm{inst}}f^{2}.$ (2)

In theories where the instanton expansion is under control, and particularly
for a QCD axion where $S_{\mathrm{inst}}=8\text{p}^{2}/g^{2}$ (for a UV value
of the QCD coupling $g$), this is significantly larger than the naive 4d field
theory estimate ${\cal T}\sim\text{p}f^{2}$. It is also, in general,
significantly lower than the upper bound ${\cal T}\sim
2\text{p}fM_{\mathrm{Pl}}$ allowed by the magnetic axion Weak Gravity
Conjecture (WGC) [51].

This bound is important because, in theories of an extra-dimensional axion, we
expect that the axion strings are fundamental objects (e.g., fundamental
strings, or D-branes wrapped on internal dimensions), and that their tension
scale lies at or above the quantum gravity cutoff $\Lambda_{\textsc{QG}}$ at
which local quantum field theory completely breaks down. If an axion is
discovered and we measured $f$, then, we can estimate that—assuming an extra-
dimensional axion—$\Lambda_{\textsc{QG}}$ is no larger than $\sim 10^{2}f$.
This is a powerful claim, as currently the best that we can say from general
principles is that $\Lambda_{\textsc{QG}}\lesssim
M_{\mathrm{Pl}}/\sqrt{N_{\textsc{SM}}}$ where $N_{\textsc{SM}}$ is the number
of weakly-coupled fields in the Standard Model [52].

  * •

In theories with charged chiral matter, like the Standard Model, it is
difficult to turn on tree-level mass terms that entirely remove a light axion
from the spectrum. There are two types of tree-level mass that an axion can
have: a monodromy mass of the form $\theta F^{(4)}$, and a Stueckelberg mass
of the form $|A-\mathrm{d}\theta|^{2}$. We argue, in the context of a
particular string realization, that the former clashes with the existence of
chiral matter charged under gauge fields the axion couples to. We also argue
that the latter case typically involves not only an extra-dimensional axion,
but a second axion arising from the phase of a charged matter field. This
scenario can interpolate between an extra-dimensional axion and a conventional
4d axion, depending on which linear combination of the two is eaten and which
remains light. These considerations support the robustness of the existence of
a light axion interacting with the Standard Model gauge fields.

The remainder of the paper is organized as follows. In §2, I provide a self-
contained exposition of the general theory of axions arising as zero modes of
$p$-form gauge fields in higher dimensions, as well as the origin of axion-
gauge field couplings, instantons, and axion strings in this setting. The
contents of this section are mostly review of well-known physics. The next
three sections constitute the main original content of the paper. In §3, I
give a simple effective field theory estimate of an upper bound on the UV
cutoff of a theory of an extra-dimensional axion, scaling like the ratio $f/g$
of the axion decay constant to the gauge theory coupling. In §4, I provide
arguments for the estimate (2) for the axion string tension. In §5, I give
explicit examples of axion string tensions computed in string theory that
substantiate this estimate, finding an interesting twist in one case where the
relevant axion string is nonminimal. In §6, I offer a few comments on the
relationship between axion physics and the existence of chiral fermions in the
context of extra-dimensional axions. In particular, I argue that tree-level
mass terms are unlikely to remove light axions from the spectrum. I also
review an argument that, for extra-dimensional axions, the derivative
couplings of axions to Standard Model fermions are likely to be of one-loop
size, in contrast to DFSZ models (where they appear at tree level) and KSVZ
models (where they appear at two loops). In §7, I conclude with a list of
questions and future directions to consider.

##  2 Axions from higher-dimensional gauge fields

###  2.1 What is an extra-dimensional axion?

The basic idea that a zero mode of a higher-dimensional gauge field can give
rise to an axion is likely familiar to many readers. For instance, given an
ordinary ${\mathrm{U(1)}}$ gauge field in a 5d theory compactified on a
circular direction $y\cong y+2\text{p}R$, the field
$\theta(x)\equiv\int_{0}^{2\text{p}R}A_{y}(x,y)\operatorname{d}\\!{y}$ behaves
as a periodic scalar $\theta\cong\theta+2\text{p}$. In this section, I will
derive the general case:

Existence of extra-dimensional axions. Suppose that spacetime has $d=(4+n)$
dimensions in the form of a warped product compactification, $M=X\times_{w}Y$
for a 4-dimensional spacetime $X$ and an $n$-dimensional space $Y$. If there
is a $p$-form gauge field $C^{(p)}$ on $M$, there is a distinct massless,
periodic 4d axion field for every independent non-torsion $p$-cycle in $Y$.

Let’s explain this step by step, with a very quick review of several relevant
concepts in mathematics and physics. Readers who find this difficult to follow
might want to first review portions of my TASI lecture notes [48].

####  2.1.1 Geometry and topology

Throughout this paper, all manifolds are assumed to be oriented. Physically,
this is equivalent to saying that we assume that our theories do not have an
orientation-reversing spacetime symmetry, like parity or CP. This appears to
be true of the world that we live in, at least at low energies, and
mathematically it underlies many assumptions below. It would be interesting to
revisit all of these results in the non-orientable case. (See [53] for a
recent introduction to physics on non-orientable manifolds, aimed at particle
physics applications.) We will also assume that our manifolds are spin, since
we are interested in real-world theories admitting fermions.

A warped compactification $M=X\times_{w}Y$ is the most general
compactification preserving the symmetries of $X$ (e.g., Poincaré symmetry, if
we take $X$ to be 4d Minkowski spacetime). It has a metric

$\operatorname{d}\\!{s}^{2}=w(y)\operatorname{d}\\!{s}^{2}_{X}+\operatorname{d}\\!{s}^{2}_{Y},$
(3)

where $w(y)\geq 0$. The amount of warping is measured by
$\operatorname{d}\\!{w}(y)$. A useful geometric fact is that the Hodge star on
a warped product manifold is a warped product of Hodge stars, i.e., if
$\eta^{(q)}_{X}$ is a $q$-form on $X$ and $\xi_{Y}^{(r)}$ is an $r$-form on
$Y$ then

$\displaystyle\star\left(\eta^{(q)}_{X}\wedge\xi^{(r)}_{Y}\right)=(-1)^{qr}w(y)^{\mathrm{dim}(X)/2-q}\left(\star_{X}\eta^{(q)}_{X}\wedge\star_{Y}\xi^{(r)}_{Y}\right).$
(4)

If we perform a Weyl rescaling of the metric on $Y$, the Hodge star acting on
an $r$-form changes as follows:

$\widetilde{\operatorname{d}\\!{s}^{2}_{Y}}\equiv
f(y)\operatorname{d}\\!{s}^{2}_{Y}\quad\Rightarrow\quad\widetilde{\star_{Y}}\xi^{(r)}_{Y}=f(y)^{n/2-r}{\star_{Y}\xi^{(r)}_{Y}}.$
(5)

The factor of $f(y)^{n/2}$ comes from $\sqrt{|\det\widetilde{g_{Y}}|}$ and the
factor $f(y)^{-r}$ from $r$ factors of $\widetilde{g_{Y}}^{\mu\nu}$.

The topology of $Y$ can be characterized by homology classes (cycles) in
$H_{p}(Y,\mathbb{Z})$. These cycles correspond to integer linear combinations
of equivalence classes of $p$-dimensional submanifolds without boundary,
modulo boundaries. Linear combinations of spaces may seem counterintuitive, so
one can also think of these classes in terms of integrals over those spaces,
with the equivalence
$\int_{\Sigma^{(p)}}\sim\int_{\Sigma^{(p)}}+\int_{\partial\Lambda^{(p+1)}}$
holding (due to Stokes’s theorem) whenever the form that we integrate is
closed. We choose a basis of $p$-cycles $[\Sigma^{(p)}_{i}]$ for the torsion-
free part of $H_{p}(Y,{\mathbb{Z}})$, where torsion-free means that there is
no integer $n\neq 0$ such that $n[\Sigma^{(p)}]=0$. The number of independent
torsion-free cycles is also known as the $p^{\textrm{th}}$ Betti number
$b_{p}(Y)$. There are corresponding (non-torsion) cohomology classes
$[\omega^{(p)}_{i}]\in H^{p}(Y,{\mathbb{Z}})$, which are equivalence classes
of closed differential forms ($\operatorname{d}\\!{\omega}=0$) which give
integers when integrated over cycles in $H_{p}(Y,{\mathbb{Z}})$, modulo the
addition of exact forms ($\omega\sim\omega+\operatorname{d}\\!{\lambda}$). We
can choose our bases in homology and cohomology to be related, such that

$\int_{\Sigma^{(p)}_{i}}\omega^{(p)}_{j}=\delta^{i}_{j}.$ (6)

This integral is independent of the choice of differential form
$\omega^{(p)}_{j}$ from the cohomology class $[\omega^{(p)}_{j}]$ (because
$\partial\Sigma^{(p)}_{i}=0$) and of the choice of submanifold
$\Sigma^{(p)}_{i}$ from the homology class $[\Sigma^{(p)}_{i}]$ (because
$\operatorname{d}\\!{\omega}^{(p)}_{j}=0$).

Each cohomology class contains a unique harmonic representative, i.e., there
is precisely one $\omega_{p}\in[\omega_{p}]$ such that

${\operatorname{d}\\!{\star}_{Y}\omega_{p}}=0.$ (7)

(A form is harmonic if and only if $\operatorname{d}\\!{\omega}=0$ and
$\operatorname{d}\\!{\star}_{Y}\omega=0$.) Notice that the definition of
harmonic depends on a choice of metric, via $\star_{Y}$.

####  2.1.2 $p$-form gauge theory on $M$

We’ll consider a theory with a $p$-form U(1) gauge field

$C^{(p)}\equiv\frac{1}{p!}C_{\mu_{1}\cdots\mu_{p}}\operatorname{d}\\!{x}^{\mu_{1}}\wedge\cdots\wedge\operatorname{d}\\!{x}^{\mu_{p}}.$
(8)

As usual, when we introduce a gauge field, we require invariance under local
gauge transformations $C^{(p)}\mapsto
C^{(p)}+\operatorname{d}\\!{\lambda}^{(p-1)}$ for any $(p-1)$-form $\lambda$.
However, we also have to allow for the case where $\lambda^{(p-1)}$ is not a
single-valued form, i.e., when it has some sort of “winding” around extra
dimensions. We can make this more precise using the cohomology classes
introduced above. First, it is useful to introduce Wilson loops, as a probe
for the right notion of gauge invariance in the theory.

Associated to the gauge field is a family of Wilson-loop operators: given a
U(1) charge $q\in{\mathbb{Z}}$ and a $p$-dimensional submanifold $\Xi^{(p)}$,
we have an operator

$W_{q}(\Xi^{(p)})\equiv\exp\left[{\mathrm{i}}q\int_{\Xi^{(p)}}C^{(p)}\right].$
(9)

The Wilson loop operator is not topological; it depends on the actual geometry
of $\Xi^{(p)}$. (This is familiar in 4d gauge theory, where we commonly
discuss perimeter or area laws for Wilson loops.) Nonetheless, it is useful to
decompose the homology class of $\Xi^{(p)}$ in our chosen basis,
$[\Xi^{(p)}]=\sum_{j}m_{j}\Sigma_{j}^{(p)},m_{j}\in{\mathbb{Z}}$. Gauge
transformations of $C^{(p)}$ preserve all of the Wilson loops. These include
the local gauge transformations $C^{(p)}\mapsto
C^{(p)}+\operatorname{d}\\!{\lambda}^{(p-1)}$. However, they also include the
“large” gauge transformations,

$C^{(p)}\mapsto
C^{(p)}+2\text{p}\sum_{i=1}^{b_{p}(Y)}n^{i}\omega^{(p)}_{i},\quad
n^{i}\in{\mathbb{Z}}.$ (10)

Under this operation, we have

$W_{q}(\Xi^{(p)})\mapsto
W_{q}(\Xi^{(p)})\exp\left[{\mathrm{i}}q\sum_{j}\left(m_{j}\int_{\Sigma^{(p)}_{j}}2\text{p}\sum_{i}n^{i}\omega^{(p)}_{i}\right)\right]=W_{q}(\Xi^{(p)})\exp\left[2\text{p}{\mathrm{i}}q\sum_{j}m_{j}n^{j}\right]=W_{q}(\Xi^{(p)}),$
(11)

using (6). These large gauge transformations are the higher-form analogue of
familiar winding gauge transformations in the case of ordinary 1-form U(1)
gauge fields. They are important for our story because they descend to
periodicity conditions on the 4d axions.

Readers for whom these “large” gauge transformations are unfamiliar are
recommended to think through the case of a 5d compactification on a circle
$S^{1}$ parametrized by a coordinate $y\cong y+2\text{p}R$, where the homology
class $\Sigma^{(1)}$ corresponding to the circle is dual to the class
$[\operatorname{d}\\!{y}/(2\text{p}R)]\in H^{1}(S^{1},{\mathbb{Z}})$ in
cohomology. Despite the notation, $\operatorname{d}\\!{y}$ is not exact,
because $y$ is not a well-defined (single-valued) function on the circle. This
is the prototypical example of a U(1) gauge transformation with winding.

####  2.1.3 Existence of massless 4d axion modes

We consider the standard kinetic term for our $p$-form gauge field in $d=4+n$
dimensions,

$-\int_{M}\frac{1}{2e_{p}^{2}}\Phi(x,y)\operatorname{d}\\!{{C^{(p)}}}(x,y)\wedge\star\operatorname{d}\\!{{C^{(p)}}}(x,y).$
(12)

where $e_{p}$ is the $p$-form gauge coupling and $\Phi(x,y)$ is a scalar
modulus field. We assume $\Phi(x,y)\geq 0$ to avoid instabilities arising from
wrong-sign kinetic terms. In our conventions, $C_{\mu_{1}\cdots\mu_{p}}$ has
mass dimension $p$, so ${C^{(p)}}$ is dimensionless,
$\star\operatorname{d}\\!{{C^{(p)}}}$ has dimension $2p-d+1$, and $e_{p}$ has
mass dimension $p+1-\frac{d}{2}$. The equation of motion of ${C^{(p)}}$, in
the limit that only the kinetic term matters, is simply

$\operatorname{d}\\!{\left(\Phi{\star}{\operatorname{d}\\!{{C^{(p)}}}}\right)}=0.$
(13)

Furthermore, we assume that there is a background solution in which
$\Phi=\Phi(y)$ is independent of $x$.

Now, we consider an ansatz for a 4d axion field perturbing around this
background. Specifically, we consider

${C^{(p)}}(x,y)=\theta(x)\omega^{(p)}(y),$ (14)

where $\theta$ is a scalar function on $X$ and $\omega^{(p)}$ is a $p$-form on
$Y$. Then

$\operatorname{d}\\!{{C^{(p)}}}(x,y)=\operatorname{d}\\!{\theta}(x)\wedge\omega^{(p)}(y)+\theta(x)\operatorname{d}\\!{\omega}^{(p)}(y).$
(15)

We are interested in searching for massless axion fields, for which the
kinetic term (12) should depend only on $\operatorname{d}\\!{\theta}(x)$ and
not $\theta(x)$. We see that this motivates searching for solutions where
$\operatorname{d}\\!{\omega}^{(p)}=0$, i.e., where $\omega^{(p)}$ is a closed
$p$-form on $Y$, so that we can drop the second term in (15).

Next, we require that our ansatz obeys the equation of motion (13) for the
higher-dimensional theory. We have

$\displaystyle\operatorname{d}\\!{\left(\Phi{\star}{\operatorname{d}\\!{{C^{(p)}}}}\right)}$
$\displaystyle=\operatorname{d}\\!{\left[\Phi(y){\star}{\left(\operatorname{d}\\!{\theta(x)}\wedge\omega^{(p)}(y)\right)}\right]}=\operatorname{d}\\!{\left[{\star}\left(\operatorname{d}\\!{\theta}\wedge(\Phi(y)\omega^{(p)}(y))\right)\right]}$
$\displaystyle=(-1)^{p}\operatorname{d}\\!{\left[w(y){\star_{X}}{\operatorname{d}\\!{\theta}}\wedge{\star_{Y}}{(\Phi(y)\omega^{(p)}(y))}\right]}$
$\displaystyle=(-1)^{p}\left\\{w(y)\Phi(y)(\operatorname{d}\\!{\star_{X}}{\operatorname{d}\\!{\theta}})\wedge{\star_{Y}}{\omega^{(p)}}-{\star_{X}}{\operatorname{d}\\!{\theta}}\wedge\operatorname{d}\\!{\left[w(y)\Phi(y){\star_{Y}{\omega^{(p)}}}\right]}\right\\}=0.$
(16)

In moving from the first line to the second we used (4). In order for the
final line to match the expected 4d equation of motion of a massless scalar
field, $\operatorname{d}\\!{\star_{X}}{\operatorname{d}\\!{\theta}}=0$, we
should require that

$\operatorname{d}\\!{\left[w(y)\Phi(y){\star_{Y}{\omega^{(p)}}}\right]}=0.$
(17)

In the case $w(y)=\Phi(y)=1$, this is just the condition that $\omega^{(p)}$
be a harmonic form. (More precisely, it’s the condition that $\omega^{(p)}$ is
co-closed, but because $\omega^{(p)}$ was already assumed to be closed, this
is equivalent to being harmonic.) As noted above, there is always a unique
choice of such a form in every cohomology class. But suppose we have
nontrivial warping $w(y)$ or a nontrivial modulus profile $\Phi(y)$ in the
extra dimensions. It is still true that a unique solution to (17) exists for a
given cohomology class; we will call such a representative a warped harmonic
form. (Here and below I will neglect $\Phi(y)$ in my terminology by saying
“warped” instead of “warped and modulated,” for brevity.)

One way to show that a unique warped harmonic form exists is to simply repeat
the usual argument in the unwarped case (e.g., [54] for details, or [55, §7.9]
for a summary) with a minor modification. First, we define a symmetric,
positive definite warped inner product on $r$-forms on $Y$ by

$(\xi,\eta)=\int_{Y}w(y)\Phi(y)\,\xi(y)\wedge\star_{Y}\eta(y),$ (18)

that is, we weight the usual definition by the (positive) factor
$\Phi(y)w(y)$. Then we can define an adjoint exterior derivative operator
taking $r$-forms on $Y$ to $(r-1)$-forms on $Y$, via

$\mathrm{d}^{\dagger}\xi^{(r)}\equiv(-1)^{n(r+1)}\frac{1}{w(y)\Phi(y)}\star_{Y}\operatorname{d}\\!{\left(w(y)\Phi(y){\star_{Y}}\xi^{(r)}\right)}.$
(19)

This is an adjoint with respect to our inner product, in the sense that
$(\operatorname{d}\\!{\xi},\eta)=(\xi,\mathrm{d}^{\dagger}\eta)$ for an
$(r-1)$-form $\xi$ and $r$-form $\eta$. We then define the warped Laplacian
$\Delta=\mathrm{d}^{\dagger}\mathrm{d}+\mathrm{d}\mathrm{d}^{\dagger}$. The
usual Hodge decomposition theorem then follows by precisely the usual logic,
with our modified definitions. A warped harmonic form with $\Delta\omega=0$ is
closed and warped co-closed, i.e., $\mathrm{d}^{\dagger}\omega=0$, which means
it obeys the condition (17).

If $n\neq 2p$, we can give a simple alternative argument. Define an auxiliary
metric, which is a Weyl rescaling of our original metric:

$\widehat{\operatorname{d}\\!{s}_{Y}}^{2}\equiv\left[w(y)\Phi(y)\right]^{2/(n-2p)}\operatorname{d}\\!{s}_{Y}^{2}.$
(20)

By (5), we see that the condition (17) is the same as the condition

$\operatorname{d}\\!{\left(\widehat{\star_{Y}}\omega^{(p)}\right)}=0.$ (21)

In other words, we can find the desired form $\omega^{(p)}$ for our extra-
dimensional ansatz by choosing the unique representative in its cohomology
class that is harmonic with respect to the auxiliary metric (20) rather than
the original metric on $Y$.

####  2.1.4 Axion kinetic matrix and periodicity

Now that we have seen how to write down an ansatz for a $d$-dimensional
${C^{(p)}}$ field that leads to a 4-dimensional axion $\theta$, let’s consider
the general case where we have a collection of such axions. Given a basis of
cohomology classes for $H^{p}(Y,{\mathbb{Z}})$ as in (6), let
$\widehat{\omega}^{(p)}_{i}(y)\in[\omega^{(p)}_{i}]$ be the unique
representative in its class that is a warped harmonic form as in (17). Then we
consider the ansatz

${C^{(p)}}(x,y)=\sum_{i=1}^{b_{p}(Y)}\theta^{i}(x)\widehat{\omega}^{(p)}_{i}(y).$
(22)

Substituting this into the kinetic term (12), together with the choice
$\Phi=\Phi(y)$ (independent of $x$), we obtain the four-dimensional kinetic
term

$\displaystyle S_{\mathrm{kin}}$
$\displaystyle=-\int_{X}\frac{1}{2}\kappa_{ij}\operatorname{d}\\!{\theta}^{i}(x)\wedge\star_{X}\operatorname{d}\\!{\theta}^{j}(x),\qquad\text{where}$
$\displaystyle\kappa_{ij}$
$\displaystyle\equiv\frac{1}{e_{p}^{2}}\left(\widehat{\omega}^{(p)}_{i},\widehat{\omega}^{(p)}_{j}\right)=\frac{1}{e_{p}^{2}}\int_{Y}w(y)\Phi(y)\,\widehat{\omega}^{(p)}_{i}\star_{Y}\widehat{\omega}^{(p)}_{j}.$
(23)

In $S_{\mathrm{kin}}$ we have left the sum over
$i,j\in\\{1,\ldots,b_{p}(Y)\\}$ implicit. The warping $w(y)$ and the modulus
$\Phi(y)$ appear only through the definition of the warped inner product (18)
and the choice of the warped harmonic representatives
$\widehat{\omega}^{(p)}_{i}$. The matrix $\kappa_{ij}$ has mass dimension $2$.

The redundancy (10) under “large” gauge transformations becomes precisely a
periodicity constraint on the 4d axions:

$\theta^{i}(x)\cong\theta^{i}(x)+2\text{p}n^{i},\quad n^{i}\in{\mathbb{Z}}.$
(24)

In other words, our conventions are such that every axion field is
$2\text{p}$-periodic; we should think of (24) as a gauge transformation on the
axion fields. This is the only remnant of the $p$-form gauge invariance in the
4d axion theory. In particular, the Wilson loop operator
$W_{q}(\Sigma^{(p)}_{i})$ matches to the 4d local operator
$\exp({\mathrm{i}}q\theta^{i})$, which is gauge invariant. We can interpret
the diagonal entries of the kinetic matrix as squared decay constants,
$f_{i}\equiv\sqrt{\kappa_{ii}}$, such that the period of the canonically
normalized axion field is $2\text{p}f_{i}$. The off-diagonal entries are
kinetic mixing parameters [56].

Finally, we note that we can also think of the individual 4d axion fields in
terms of integrals of the higher-dimensional gauge field over $p$-cycles,

$\theta^{i}=\int_{\Sigma^{(p)}_{i}}{C^{(p)}}.$ (25)

However, this is slightly imprecise, because it is only independent of the
choice of $\Sigma^{(p)}_{i}$ in its homology class when ${C^{(p)}}$ is
constructed only from closed forms on $Y$.

###  2.2 Masses for extra-dimensional axions

So far, we have seen that the theory of an extra-dimensional $p$-form gauge
field ${C^{(p)}}$ gives rise to 4d axions that are massless, if we only
consider the kinetic term. In general, there should also be effects that give
rise to a small 4d mass. These must come from terms in the full theory that
involve ${C^{(p)}}$ without derivatives acting on it. Such terms are highly
restricted, by gauge invariance. They come in two types: couplings to objects
that carry a ${C^{(p)}}$ gauge charge, and Chern-Simons interactions of
${C^{(p)}}$ with itself or with other gauge fields.

####  2.2.1 Brane instantons

Let’s begin by considering objects electrically charged under $C^{(p)}$. These
are conventionally referred to as $(p-1)$-branes, and have a $p$-dimensional
worldvolume. A brane of charge $q$ under ${C^{(p)}}$ couples via a term
$\int_{\Gamma^{(p)}}q\,{C^{(p)}}$ on the worldvolume ${\Gamma^{(p)}}$. (This
is the dynamical version of a Wilson line operator, and is invariant under
large “winding” gauge transformations for the same reason the Wilson loop is.)
To isolate the axion modes in ${C^{(p)}}$, we use the ansatz (22), so this
integral is only nonzero if $\Gamma^{(p)}$ extends only along internal
directions within $Y$. But this means that it is completely localized in 4d
spacetime $X$, i.e., it is an instanton from the viewpoint of the 4d theory.
These are often referred to as “Euclidean brane instantons,” because the
$(p-1)$-brane worldvolume has been wrapped on a cycle $\Gamma^{(p)}$ in $Y$,
and only has spatial (Euclidean) directions and does not extend in the time
direction.

How large are the effects of a Euclidean brane instanton? For this we need to
consider not just the coupling to ${C^{(p)}}$ but the other leading term in
the brane worldvolume action, namely, the tension of the brane, which gives a
contribution to the action that is linear in the volume:

$S_{\textrm{brane}}=\int_{\Gamma^{(p)}}\left({\cal
T}_{(p)}\,{\star_{\Gamma}}1+{\mathrm{i}}q\,{C^{(p)}}\right).$ (26)

Here $\star_{\Gamma}1$ is the volume form on the submanifold $\Gamma^{(p)}$,
and ${\cal T}_{(p)}$ is the brane’s tension. The ${\mathrm{i}}$ comes from a
Euclidean continuation. A dilute gas of these Euclidean brane instantons then
gives rise to an effective axion potential in the infrared. Representing the
cycle in terms of our basis of $p$-cycles as
$[\Gamma^{(p)}]=\sum_{j}m_{j}[\Sigma^{(p)}_{j}]$, the simplest estimate is

$V_{\textrm{eff}}^{\textrm{(brane)}}(\\{\theta^{i}\\})=\left(M^{4}{\mathrm{e}}^{-{\cal
T}_{(p)}\,\mathrm{Vol}(\Gamma^{(p)})}{\mathrm{e}}^{{\mathrm{i}}qm_{j}\theta^{j}}+\mathrm{h.c.}\right),$
(27)

where $M$ has units of mass, and one’s first expectation might be that it
corresponds to an ultraviolet cutoff energy like the compactification scale at
which the $d$-dimensional theory matches to a 4d effective theory of axions.
In fact, there could be further suppression: for instance, in a supersymmetric
theory the Euclidean brane instantons may generate an effective superpotential
or an effective Kähler potential, and the eventual size of the potential could
then be suppressed by supersymmetry-breaking parameters. Even without
supersymmetry, there may be fermion zero modes associated with these
instantons, so that what is generated is an ’t Hooft vertex rather than a
potential. Independent of such subtleties, we expect to always have the
exponential suppression by the brane tension times the volume of the cycle the
brane wraps, which we refer to as the instanton action:

$S_{\mathrm{inst}}={\cal T}_{(p)}\,\mathrm{Vol}(\Gamma^{(p)}).$ (28)

The exponential suppression of the axion potential is very powerful. It
implies that, if the volumes are somewhat large in units of whatever physical
scale sets the brane tension, these contributions to the axion potential can
be very small. Any effect that can generate an axion potential must see the
entire cycle over which ${C^{(p)}}$ is integrated to obtain a given axion
mode, and such nonlocal effects are naturally suppressed. In this way, extra-
dimensional axion models effectively take the logarithm of the axion quality
problem, compared to models of axions as 4d pseudo-Nambu-Goldstone bosons
where one finds power-law suppression factors instead of exponential
suppression factors.

####  2.2.2 Gauge theory instantons

The axion that we’re most interested in is the QCD axion, with coupling (1) to
gluons. More generally, axions might have similar couplings to any 4d gauge
fields. Unlike in models of axions as 4d pseudo-Nambu-Goldstone bosons, where
such couplings generally arise from integrating out fermions at one loop, in
extra-dimensional models we expect them to arise from dimensional reduction of
higher-dimensional Chern-Simons couplings.

Specifically, we consider that the 4d gauge fields are zero modes of higher-
dimensional gauge fields, so that (for example) the 4d gluons arise from an
SU(3) gauge theory in $4+p$ dimensions. We do not necessarily assume that
$d=4+p$; rather, these gauge fields can be localized on $(p+3)$-branes
wrapping a $p$-cycle $\Gamma^{(p)}$ within $Y$, with
$[\Gamma^{(p)}]=\sum_{j}m_{j}[\Sigma^{(p)}_{j}]$. Along these branes, we have
an action defined on the manifold $N=X\times_{w}\Gamma^{(p)}$

$S_{p+3}=\int_{N}\left(-\frac{1}{2g_{p}^{2}}\operatorname{tr}[G(x,y)\wedge\star_{N}G(x,y)]+\frac{k_{G}}{8\text{p}^{2}}{C^{(p)}}(x,y)\wedge\operatorname{tr}[G(x,y)\wedge
G(x,y)]\right).$ (29)

Here $G(x,y)=\operatorname{d}\\!{A}(x,y)$ is the 2-form field strength of an
SU(3) gauge field and $g_{p}$ is a gauge coupling of mass dimension $-p/2$.
The Chern-Simons term is not gauge invariant; the requirement that
$\exp({\mathrm{i}}S_{p+3})$ is well-defined under ${C^{(p)}}$ gauge
transformations implies that $k_{G}\in{\mathbb{Z}}$.

The 4d gluon fields arise by simply taking $G(x,y)=G(x)$ to be a
$y$-independent, 4d two-form field strength. From (4) with $q=2$, we see that
warping plays no role in this case. We obtain the 4d action

$S_{\text{gauge}}=\int_{X}\left(-\frac{1}{2g^{2}}\operatorname{tr}[G(x)\wedge\star_{X}G(x)]+\frac{k_{G}}{8\text{p}^{2}}\sum_{j}m_{j}\theta^{j}(x)\operatorname{tr}[G(x)\wedge
G(x)]\right),$ (30)

where

$\frac{1}{g^{2}}=\frac{1}{g_{p}^{2}}\int_{\Gamma^{(p)}}\star_{N}1=\frac{1}{g_{p}^{2}}\mathrm{Vol}(\Gamma).$
(31)

Thus, we see that the higher-dimensional Chern-Simons coupling gives rise to
(properly quantized) couplings of axions to gauge fields in 4d. As usual,
these gauge fields then contribute to an axion potential via exponentially
small 4d instanton effects, which take the form

$V_{\textrm{eff}}^{\textrm{(gauge)}}(\\{\theta^{i}\\})=\left(M^{4}{\mathrm{e}}^{-8\text{p}^{2}/g^{2}}{\mathrm{e}}^{{\mathrm{i}}km_{j}\theta^{j}}+\mathrm{h.c.}\right).$
(32)

This is very similar to (27), with the brane charge $q$ replaced by the Chern-
Simons level $k_{G}$. The similarity is even stronger if we take into account
the volume scaling of $1/g^{2}$ in (31), so that we can identify the brane
tension ${\cal T}$ with $8\text{p}^{2}/g_{p}^{2}$.

This is no accident. In fact, BPST Yang-Mills instanton solutions lift to
$(4+p)$ dimensions, where they are not localized in spacetime but are
extended, dynamical objects with $p$-dimensional worldvolumes (taking the
solution to be constant along $p$ directions), and these indeed have tension
${\cal T}=8\text{p}^{2}/g_{p}^{2}$. To obtain the 4d instanton, we wrap the
worldvolume of such an object over the cycle $\Gamma^{(p)}$. In this sense,
the Yang-Mills instantons and the Euclidean brane instantons discussed in
§2.2.1 are not so distinct. Indeed, in string-theoretic examples it is known
that small Yang-Mills instantons are precisely the same objects as Euclidean
branes [57, 21, 58].

Of course, once we match to the 4d theory, we have the usual infrared physics
of QCD. At low energies the description in terms of a dilute gas of instantons
breaks down as the instantons become large and overlapping, so (32) is no
longer accurate. In this regime, one can compute the effective axion
potential, which has $O(1)$ deviations in shape from a cosine potential, using
the chiral Lagrangian [5].

####  2.2.3 Tree-level masses

So far everything has worked nicely: higher-dimensional gauge fields
automatically give rise to 4d axions. These can have the necessary couplings
to 4d gauge fields to solve the Strong CP problem (or to play a role in other
phenomenologically interesting scenarios), arising from higher-dimensional
Chern-Simons terms. Euclidean branes can give rise to additional contributions
to axion potentials, but these are exponentially small and in some cases are
in fact identical to small gauge theory instantons, so the axion quality
problem is inherently mild.

However, if an axion has a different type of mass term that is not
exponentially small, this would give the axion a large mass, decoupling it
from the low-energy theory and eliminating the solution to the Strong CP
problem. Because the axion originates from a higher-dimensional gauge field,
it is relatively protected against such effects, but there are two mechanisms
by which it could obtain a tree-level mass. We summarize these by saying the
axion can eat or be eaten.

The first case is a monodromy mass for the axion [59, 60, 61, 62] (also see
[20, 63, 64, 65, 66] for a discussion of the standard axion potential in a
similar formalism). In four dimensions, it comes from a coupling of the form
$\frac{n}{2\text{p}}\theta F^{(4)}$, where
$F^{(4)}=\operatorname{d}\\!{C}^{(3)}$ is the four-form field strength of a
3-form gauge field (which has no propagating degrees of freedom). If $F^{(4)}$
has a standard quadratic kinetic term, this is equivalent to a quadratic mass
term $\frac{1}{2}m^{2}f^{2}\theta^{2}$. Such a term appears to violate the
$\theta\mapsto\theta+2\text{p}$ gauge invariance. In fact, what happens is
that the potential has infinitely many branches, related by monodromy. One can
dualize $F^{(4)}$ to a “zero-form field strength” which is simply an integer
$n$; the potential is actually a function of the combination
$(\theta-2\text{p}n)$, with a gauge invariance
$\theta\mapsto\theta+2\text{p},n\mapsto n+1$. This is essentially just the
familiar physics of a particle on a circle in quantum mechanics. We could say
that the axion eats the integer (or the fictitious $(-1)$-form field whose
field strength is the integer) to become massive. Alternatively, in the
magnetic frame, the $B^{(2)}$ field dual to the axion is eaten by $C^{(3)}$.

The monodromy mass is the axion analogue of a $BF$ term. It could arise from a
term of the form
$\frac{n}{2\text{p}}{C^{(p)}}\wedge\operatorname{d}\\!{B}^{(3-p+m)}$
integrated over $m\geq p$ extra dimensions. This is just another type of
Chern-Simons term, but involving two gauge field factors instead of one gauge
field factor integrated over a charged object’s worldline (in §2.2.1) or three
gauge field factors (in §2.2.2). Such a term could also arise from a cubic
Chern-Simons term of the form
$\frac{1}{4\text{p}^{2}}{C^{(p)}}\wedge\operatorname{d}\\!{B}^{(3-p+m)}\wedge\operatorname{d}\\!{A}^{(q)}$
when $A^{(q)}$ has flux over $q+1$ additional extra dimensions, or similarly
from Chern-Simons terms with even more fields and more fluxes.

The second case is a Stueckelberg mass for an ordinary 1-form gauge field,
which eats an axion to become massive. This mass originates from the axion
kinetic term proportional to $|\mathrm{d}\theta-qA^{(1)}|^{2}$, where
$q\in{\mathbb{Z}}$ is a ${\mathrm{U(1)}}$ charge associated with the gauge
transformation $A^{(1)}\mapsto A^{(1)}+\mathrm{d}\lambda$,
$\theta\mapsto\theta+q\lambda$. This is dual to a $BF$ term of the form
$\frac{q}{2\text{p}}B^{(2)}\wedge\mathrm{d}A^{(1)}$, with $B^{(2)}$ the 2-form
gauge field dual to the axion. This could again be dualized to a frame in
which the field $B^{(2)}$ eats the magnetic dual gauge field
$\widetilde{A}^{(1)}$.

Because we are interested in light axions, we will assume that the fields we
focus on do not obtain tree-level masses of either type. This is natural,
since such a mass is proportional to an integer that can simply be zero (and
is not renormalized). In §6.1, we will revisit the possibility of tree-level
axion masses and discuss their relationship with chiral fermions, making more
model-dependent assumptions.

###  2.3 Axion strings for extra-dimensional axions

Up to this point, the higher-dimensional physics we have discussed gives rise
to an effective field theory of light axions just like that in models of
axions as 4d pseudo-Nambu-Goldstone bosons. However, the physics at higher
energies can be dramatically different in these different types of models. One
key difference lies in the physics of axion strings, which are dynamical
objects such that the axion field $\theta(x)$ winds around its target space as
the spatial position $x$ winds around the string. In a weakly-coupled 4d model
like the classic KSVZ or DFSZ models, an axion string can be constructed as a
classical solution to the equations of motion, with the classical Peccei-Quinn
symmetry restored in the core of the string. For extra-dimensional axions,
there is no 4d classical Peccei-Quinn symmetry, and correspondingly the axion
string’s core probes deep UV physics, as recently emphasized in [39, 40, 41,
42, 43, 44, 45].

An axion string carries a magnetic charge under the axion, in the sense that
$\oint\operatorname{d}\\!{\theta}=2\text{p}n$ where the integral is over a
circle that links with the string worldsheet, just as
$\int_{\Sigma}F=2\text{p}n$ for an ordinary gauge field strength integrated
over a sphere linking a magnetic monopole worldline. In fact, an axion string
originates as a magnetic monopole for ${C^{(p)}}$. Specifically, a magnetic
monopole for a $p$-form gauge field in $d$ spacetime dimensions is an object
with a $(d-p-2)$-dimensional worldvolume. In the compactified theory, we can
wrap this worldvolume over $n-p$ internal dimensions to obtain a string in 4d.

In the extra-dimensional theory, then, the magnetic $(d-p-3)$ brane wraps an
$(n-p)$-dimensional submanifold $\Gamma^{(n-p)}$ and extends along a
worldsheet $\Lambda\subset X$. Poincaré duality associates the homology class
$[\Gamma^{(n-p)}]\in H_{n-p}(Y,{\mathbb{Z}})$ with a cohomology class
$[\gamma^{(p)}]\in H^{p}(Y,{\mathbb{Z}})$. This is defined so that, for any
closed $(n-p)$-form $\xi^{(n-p)}$, we have
$\int_{Y}\gamma^{(p)}\wedge\xi^{(n-p)}=\int_{\Gamma^{(n-p)}}\xi^{(n-p)}$.
(Loosely speaking, it can be thought of as a Dirac delta function localizing
fields to $\Gamma^{(n-p)}$.) If we decompose the Poincaré dual class in our
chosen basis, $[\gamma^{(p)}]=\sum_{i}n^{i}[\omega^{(p)}_{i}]$, then in 4d we
have an axion string with

$\oint_{C}\operatorname{d}\\!{\theta}^{i}=2\text{p}n^{i}\,\mathrm{Link}(C,\Lambda),$
(33)

where $\mathrm{Link}(C,\Lambda)\in{\mathbb{Z}}$ is the linking number of a
closed curve $C$ with the worldsheet $\Lambda$. This means that $\theta^{i}$
winds $n^{i}$ times around the string. In particular, to obtain a minimally-
charged axion string for which a particular axion $\theta^{i}$ winds once
around the string, we should wrap the magnetic brane on the cycle Poincaré
dual to $[\omega^{(p)}_{i}]$. By (6), this is a cycle that has intersection
number $\delta^{i}_{j}$ with the cycle $[\Sigma^{(n-p)}_{j}]$.

An axion string has a tension that is determined by the product of the
magnetic brane tension in the $d$-dimensional theory and the warped volume of
the wrapped cycle:

${\cal T}_{(2)}={\cal
T}_{(d-p-2)}\int_{\Gamma^{(n-p)}}w(y)\,{\star_{\Gamma^{(n-p)}}1}={\cal
T}_{(d-p-2)}\mathrm{Vol}_{w}(\Gamma^{(n-p)}),$ (34)

where we have defined the “warped volume” $\mathrm{Vol}_{w}$ in the last step.
The tension (34) should be thought of as the tension of the string core. The
winding of the axion around the string contributes, at larger distances, an
infrared divergent contribution ${\cal
T}_{\textsc{IR}}\approx\text{p}f^{2}\log(r_{\textsc{IR}}/r_{\mathrm{core}})$.

###  2.4 Qualitative features of extra-dimensional axions

Extra-dimensional axions are appealing because they render the axion quality
problem much more mild. The QCD contribution to the axion potential is always
exponentially small, since the QCD scale arises from dimensional
transmutation. In 4d axion models, this competes with effects from PQ-breaking
local operators that are suppressed by powers of $f/\Lambda_{\textsc{QG}}$.
Here $\Lambda_{\textsc{QG}}$ denotes the fundamental cutoff scale at which
gravity becomes strong and all global symmetries are badly broken, which is at
most the Planck scale. Such models require new gauge symmetries to forbid
dangerous operators up to a high dimension (see, e.g., [67, 68, 69, 70, 71,
72, 73, 74, 75, 20]). In the extra-dimensional case, on the other hand, terms
in the axion potential inherently involve nonlocal effects from extended
objects stretched across the extra dimensions. These are always exponentially
small. This is not to say that the problem has been trivialized. One still
must check that the resulting small potential is small enough that the Strong
CP problem is solved; see, e.g., discussions in [25, 37].

Because extra-dimensional axion models have no 4d PQ symmetry breaking, the
axion in such models is not a PNGB in the traditional sense. There is a
symmetry current that creates an axion particle from the vacuum and that
generates a shift of the axion field in the low energy theory, but it is
simply the operator $\partial_{\mu}\theta$, which does not arise from a more
fundamental current for a symmetry acting linearly on fields. In the higher-
dimensional setting, it arises from the field strength of the gauge field that
gave rise to the axion. Thus, extra-dimensional axions are only PNGBs in the
sense that the photon is a Nambu-Goldstone boson [76]. The symmetry always
acts nonlinearly, by shifting the field.

Because there is no PQ symmetry breaking, an extra-dimensional axion is not
associated with a cosmological phase transition [44, 48, 45]. The frequently
studied post-inflation axion scenario, in which the axion field is randomized
at the PQ phase transition and cosmic strings are produced, which later
annihilate axion domain walls at the QCD phase transition, simply does not
apply to extra-dimensional axions. Instead, these models behave like the
conventional “pre-inflation scenario.” This is often described as a scenario
where the PQ phase transition happened before inflation, but we do not know if
there was a pre-inflation epoch in cosmology, nor do we need one. The
important point is that at any time that the universe can be approximated as
four-dimensional, there is a light axion field in the spectrum of the theory,
which is not the phase of a complex scalar that can be trapped at the origin
of field space. In cases where it can be paired with a radial saxion or
modulus to form a complex scalar field, the point in field space at which the
axion decay constant vanishes lies at infinite distance [39, 40], so
cosmological dynamics never traps the modulus at such a point. The would-be PQ
symmetry is always already broken.

The existence of axions arising from higher dimensional gauge fields is, as we
have seen in §2.1, a topological property of the theory. In particular, there
is no need for additional structure like supersymmetry to explain why an
extra-dimensional axion is light: it is inherently exponentially light,
provided it does not obtain a tree-level mass from the (again, topological)
terms discussed in §2.2.3. Topological quantities are reassuringly robust, but
most of the information that is actually useful for phenomenology is
geometric, not topological: to compute the values of gauge couplings (31),
axion decay constants or kinetic mixing parameters (2.1.4), instanton actions
(28), or axion string tensions (34), we need to understand the geometry of the
extra dimensions. Geometry is, unfortunately, much more difficult than
topology. However, in special cases, the volumes of certain cycles in the
higher dimensions are calibrated: a minimal volume submanifold in a given
topological class $H_{p}(Y,{\mathbb{Z}})$ has its volume given by integrating
a special $p$-form called a calibration over the cycle. This occurs, for
example, when $Y$ is a Kähler manifold, $p=2k$ is even, and the $p$-cycle is
holomorphic; in that case, the calibration is $J^{\wedge k}$, where $J$ is the
Kähler form. We will make use of this to study concrete examples in §5.2
below. Another example is when $Y$ is a Calabi-Yau $n$-fold, $p=n$ is half the
real dimension of $Y$, and the $p$-cycle is special Lagrangian; in that case,
the calibration is $\mathrm{Re}\,\Omega$, where $\Omega$ is the holomorphic
$n$-form on $Y$. Although calibrated cycles are a mathematical tool to make
the computation of volumes easier, it turns out that the conditions under
which they are known to exist are commonly also conditions under which
compactifications on $Y$ are supersymmetric. This is a motivation for studying
supersymmetric theories that is distinct from phenomenological considerations
like the hierarchy problem, or field-theoretic arguments for using
supersymmetry to control QFTs in a strongly-coupled regime. Here,
supersymmetry provides a geometrical toolkit to compute the axion effective
action in otherwise intractable extra-dimensional settings.

##  3 Estimating a UV cutoff

In this section, we give a simple estimate of an upper bound on the UV cutoff
of the theory, above which the extra-dimensional Chern-Simons description
breaks down. This claim was advertised, but not explained, at the end of my
TASI lecture notes [48]; this paper (belatedly) provides the argument. The
idea is simply that the axion-gluon coupling is non-renormalizable, and so
generally signals a UV cutoff on the theory. We could try to estimate this
directly from the higher-dimensional Chern-Simons term appearing in the brane
action (29), but this is complicated by the fact that $C^{(p)}$ in general
propagates in additional directions transverse to the brane, so the proper way
to normalize the field on the brane to read off the cutoff scale is not
obvious. Instead, we will work directly with the 4d effective action, but take
into account the fact that the axion $\theta$ couples to an entire tower of
Kaluza-Klein modes $g_{n}$ of the gluon. The number of these modes scales like
the volume of the $p$-cycle we integrate over to obtain a 4d axion, in units
of the cutoff. These modes all contribute a loop correction to the axion
propagator, as shown in Fig. 1.

Figure 1: Loop corrections to the axion propagator from the sum over loops of
gluon Kaluza-Klein modes.

We estimate that loops correct the momentum-space propagator of the
canonically normalized axion $\hat{\theta}$ as

$\langle\hat{\theta}(q)\hat{\theta}(-q)\rangle=\frac{1}{q^{2}+{\mathrm{i}}\varepsilon}\frac{1}{1+\Pi(q)},\quad\text{where}\quad\Pi(q)\sim\frac{1}{16\text{p}^{2}}\left(\frac{g^{2}}{8\text{p}^{2}f}\right)^{2}\,N_{\textsc{KK}}(q)\,q^{2}+\text{other}.$
(35)

Here $N_{\textsc{KK}}(q)$ denotes the number of Kaluza-Klein modes with mass
below the scale $q$, and “other” refers to any other effects that we have not
included by summing over gluon loops. At the UV cutoff $\Lambda$, we should
have $\Pi(\Lambda)\sim 1$, so in particular (barring fine-tuned
cancellations), we expect that the piece we have estimated should be $\lesssim
1$. From matching the higher-dimensional theory to the 4d theory, we expect
that $N_{\textsc{KK}}(q)\sim 1/g^{2}$, and hence we obtain a bound of the form
$\Lambda\lesssim f/g$. Let us work this out more carefully.

We assume that the gauge kinetic term in $p+4$ dimensions scales like an
appropriate power of the cutoff, times a prefactor $\zeta$ that may be large
but which we might expect to be typically of order one:

$\frac{1}{g_{p}^{2}}=\zeta\Lambda^{p}.$ (36)

For example, for a gauge coupling on a D-brane in string theory, if we
identify $\Lambda$ with the string scale $M_{s}$, then
$\zeta=\frac{1}{(2\text{p})^{p+1}g_{s}}$. According to (31), the 4d gauge
coupling is given by $1/g^{2}=\mathrm{Vol}(\Gamma)/g_{p}^{2}$. The number of
KK modes on a $p$-cycle up to a scale $q$ is given by Weyl’s law,
$N_{\textsc{KK}}(q)\sim\frac{1}{(4\text{p})^{p/2}\Gamma(\frac{p}{2}+1)}\mathrm{Vol}(\Gamma)q^{p}$.
Putting the pieces together, we obtain a cutoff

$\Lambda\lesssim\frac{f}{2g}\left[\zeta(4\text{p})^{6+p/2}\Gamma(1+p/2)\right]^{1/2}.$
(37)

Although this was a fairly crude estimate, the point that we would like to
emphasize is that it is proportional to $f/g$, with a proportionality constant
that we might expect to be $O(1)$ in many models. For example, in the string
theory setting, the proportionality factor $\zeta$ could be made very large
only at the expense of going to very small $g_{s}$ (which makes string modes
much lighter than the Planck scale). More generally, small $g_{p}$ is expected
to correlate with a low cutoff on the higher-dimensional theory, due to the
Weak Gravity Conjecture [51, 77].

This UV cutoff is one at which the Chern-Simons description of the axion-gluon
interaction breaks down. In particular, it is not simply associated with a
breakdown of the 4d EFT and the need for a higher-dimensional description. We
expect that it is a fundamental ultraviolet scale of the theory. We will see
below that in many string theory examples, a cutoff arises at or below a scale
$O(1)f/g$, independent of $g_{s}$, so the argument we have given here is
likely to be overly conservative. In particular, we will see that the scale
$O(1)f/g$ has a concrete physical interpretation: it is the mass scale at
which excitations of the axion string appear.

##  4 Axion string tension scaling with decay constant

In this section, we will argue that it is reasonable to expect the scale
corresponding to the axion string tension to be near, but slightly above, that
of the decay constant. More precisely, in a variety of models there is an
approximate relation of the form

${\cal T}_{(2)}\approx 2\text{p}\,S_{\mathrm{inst}}\,f^{2}.$ (38)

Here we focus on a single axion; ${\cal T}_{(2)}$ is the tension of a minimal
axion string, $S_{\mathrm{inst}}$ is the action of a single instanton for
$\theta$ (equal to $8\text{p}^{2}/g^{2}$ when it arises as a Yang-Mills
instanton for a gauge theory with coupling $g$), and $f$ is the axion decay
constant.

A few words about the interpretation of the estimate (38) are in order. First,
the tension of an axion string is infrared divergent, as discussed in §2.3.
The estimate refers only to the tension of the core of the string. In most of
the examples we will study, this arises as the (well-defined) tension of a
higher-dimensional brane, multiplied by the volume of a cycle wrapped by that
brane. Second, the $\approx$ sign here is somewhat ambiguous. In a very wide
range of examples, it is really a $\sim$, that is, the estimate (38) holds up
to an $O(1)$ factor. However, in many examples, it is even better: it holds
exactly, at least at leading order in various weak-coupling or large-volume
limits, due to BPS conditions in the higher-dimensional theory giving rise to
the 4d axion theory.

This estimate generally lies in between two other guesses one might have. The
first, based on the intuition from 4d field theory axions, is that ${\cal
T}_{(2)}\sim\text{p}f^{2}$. This sets the coefficient of the infrared
logarithm, so it would make little sense for ${\cal T}_{(2)}$ to be taken to
be smaller than this. The second is that ${\cal T}_{(2)}\sim
2\text{p}fM_{\mathrm{Pl}}$. The latter estimate is motivated by the magnetic
axion Weak Gravity Conjecture, which holds that ${\cal T}_{(2)}\lesssim
2\text{p}fM_{\mathrm{Pl}}$ [51].111The axion WGC is somewhat degenerate, from
the viewpoint of the black hole extremality arguments originally motivating
the WGC. However, an argument based on black hole physics for the necessity of
cosmic strings in quantum gravity with an axion was given in [78, 79]. This
bound is in fact saturated in many simple examples (we will discuss some in
§5), which has led to suggestions that the estimate ${\cal T}_{(2)}\sim
2\text{p}fM_{\mathrm{Pl}}$ works quite generally for extra-dimensional axions,
at least in the absence of warping (see, e.g., the recent [45]). However, we
will also see examples where it fails, whereas (38) holds up in a wider range
of examples.

Clearly, we would expect any general relationship among axion string tensions,
decay constants, and instanton actions to be modified in a setting with
multiple axions that mix with each other through an off-diagonal kinetic
matrix $\kappa_{ij}$ as in (2.1.4). In this section, we focus on a single
axion, hoping that the QCD axion is relatively isolated from mixing effects in
theories that solve the Strong CP problem. In §5.2 below, we will encounter a
class of multi-axion examples in which an axion string around which
$\theta^{i}$ winds $w^{i}$ times has tension $2\text{p}w^{i}\kappa_{ij}S^{j}$.
This is one candidate for a more general estimate in the multi-axion case.
Interestingly, in the examples that we study in §5, if we read off the value
of $f$ and $S_{\mathrm{inst}}$ from any single axion, we find an axion string
with tension of order (38), although it may not have minimal winding under the
axion of interest and may also have nonzero winding of additional axions. This
is reminiscent of known 4d field theory examples, where the minimal-tension
axion string is often not the minimal-winding axion string (see, e.g., [72,
80]).

We will now discuss several general, but heuristic, arguments for why we might
expect (38) to hold. In §5, we consider specific UV complete examples (in
string theory) where we can explicitly verify the statement.

###  4.1 Scaling of decay constants with volumes

Let’s first consider a simple geometrical argument. Suppose that physics in
our $d$-dimensional theory is dominated by a single length scale $\ell$ (e.g.,
the string scale, in string theory, or the 11d Planck scale in M-theory). We
will also suppose that warping is not important, that dimensionless moduli
like $\Phi$ in extra dimensions take order-one values, and that all higher-
dimensional couplings and tensions like $e_{p}$ or ${\cal T}_{(p)}$ are order-
one in units of $\ell$. We are not claiming these assumptions are universally
valid, just using them as a starting point to explore our expectations.

We focus on a single axion $\theta$ arising from ${C^{(p)}}$ dimensionally
reduced on a $p$-cycle $[\Sigma^{(p)}]$, corresponding to a warped harmonic
form ${\widehat{\omega}}^{(p)}$, and intersecting a dual cycle
$[\Gamma^{(n-p)}]$. Then our formulas from §2 imply:

$\displaystyle{\cal T}_{(2)}$
$\displaystyle\sim\ell^{-2+p-n}\,\mathrm{Vol}(\Gamma^{(n-p)}),$ $\displaystyle
S_{\mathrm{inst}}$ $\displaystyle\sim\ell^{-p}\,\mathrm{Vol}(\Sigma^{(p)}),$
$\displaystyle f^{2}$
$\displaystyle\sim\ell^{2p-n-2}\int_{Y}{\widehat{\omega}}^{(p)}\wedge\star_{Y}{\widehat{\omega}}^{(p)}.$
(39)

Here $\Sigma^{(p)}$ and $\Gamma^{(n-p)}$ are actual submanifolds, not just
homology classes; we expect that these are (at least locally) volume-
minimizing representatives of the class. However, at the level of the current
scaling argument, such details are not very important. Now,
${\widehat{\omega}}^{(p)}$ is dimensionless, whereas
$\star{\widehat{\omega}}^{(p)}$ has mass dimension $2p-n$. By Poincaré
duality, we have

$\int_{Y}{\widehat{\omega}}^{(p)}\wedge\star_{Y}{\widehat{\omega}}^{(p)}=\int_{\Gamma^{(n-p)}}\star_{Y}{\widehat{\omega}}^{(p)}.$
(40)

How can we estimate this? Assuming that no curvatures are particularly large,
we can make a crude assumption that the answer is not very sensitive to our
precise choice of representative from a cohomology class. Thus, we think of
$[{\widehat{\omega}}^{(p)}]$ as a function localized on ${\Gamma^{(n-p)}}$,
smeared along the $p$ transverse directions and multiplied by their
differentials
$\operatorname{d}\\!{x}^{1}\wedge\cdots\wedge\operatorname{d}\\!{x}^{p}$, such
that the total weight is $1$ when integrated over $\Sigma^{(p)}$. Then
$\star_{Y}{\widehat{\omega}}^{(p)}$ looks like the same localized, smeared
function, but now with the differentials along $\Gamma^{(n-p)}$,
$\operatorname{d}\\!{x}^{p+1}\wedge\cdots\wedge\operatorname{d}\\!{x}^{n}$.
Thus, its integral now acquires a factor of the volume of $\Gamma^{(n-p)}$,
but is suppressed by the volume of $\Sigma^{(p)}$ because we are no longer
integrating over the transverse directions ${\widehat{\omega}}^{(p)}$ was
smeared over. Thus, we estimate

$\int_{Y}{\widehat{\omega}}^{(p)}\wedge\star_{Y}{\widehat{\omega}}^{(p)}\sim\frac{\mathrm{Vol}(\Gamma^{(n-p)})}{\mathrm{Vol}(\Sigma^{(p)})}.$
(41)

We will see that, despite the blithe assumptions that went into it, this
estimate actually works in at least a significant class of examples.222In some
cases, such as product manifolds, it can be exact; examples are discussed in,
e.g., [25]. In some cases in the literature an estimate of the form
$\mathrm{Vol}(Y)/\mathrm{Vol}(\Sigma^{(p)})^{2}$ is assumed, implicitly or
explicitly; this estimate agrees with (41) for product manifolds (or
fibrations) but can be much less accurate in other cases. It implies

${\cal T}_{(2)}\sim S_{\mathrm{inst}}\,f^{2}.$ (42)

This argument is (clearly) too crude to produce a factor like $2\text{p}$.
This factor will be better motivated by other arguments and examples below.
Although we have omitted warping from our argument, it is plausible that it
does not change the relationship because it affects the left- and right-hand
sides in compatible ways. The warping enters via a single factor of $w(y)$ in
the integrand defining ${\cal T}_{(2)}$ in (34), and also in the integrand
defining $f^{2}$ in (2.1.4). In the simplest example of a warped interval,
this expectation holds up because the axion wavefunction peaks in the region
of greatest warping (smallest $w(y)$) and the axion string falls into this
region as well; see [45] for calculations in this case.

###  4.2 Bogomolnyi bounds and repulsive forces

One way to try to quantify the instanton action and axion string tension is to
perform an explicit semiclassical calculation within the low-energy effective
theory. For an axion minimally coupled to gravity, the corresponding Euclidean
instanton solutions [81, 82] and axion string solutions [83] are well-studied,
but singular, and the solutions are not expected to capture the physics of the
core of the object accurately. In the core of both objects, a scalar radial
mode or saxion field is expected to be excited in UV completions. In our
examples, such moduli generally arise as modes of the metric in the extra
dimensions, which determine the volumes of the cycles used to compute the
instanton action, as in (28), and the axion string tension, as in (34). The
coupled equations for the axion-saxion system are often better behaved, and in
any case, capture the physics of the instanton or string core more accurately.
In this subsection, we will study classical solutions to the equations of
motion for an axion together with a saxion $\phi$, coupled through a
$\phi$-dependent axion decay constant $f(\phi)$. (See also [84] for some
related recent work.)

####  4.2.1 Euclidean instantons

The instanton solutions that we study are Euclidean solutions to the coupled
equations for the scalar modulus $\phi$ and the two-form gauge field $B^{(2)}$
(with field strength $H^{(3)}$) dual to the axion, with action

$I_{E}=\int\left(\frac{1}{2}g(\phi)^{2}H^{(3)}\wedge\star
H^{(3)}+\frac{1}{2}h(\phi)^{2}\mathrm{d}\phi\wedge\star\mathrm{d}\phi\right).$
(43)

We take $g(\phi),h(\phi)>0$. We denote the asymptotic value of the field
$\phi$ at $r\to\infty$ by $\phi_{*}$. Then we read off from the action that
the axion decay constant is

$f=\frac{1}{2\text{p}g(\phi_{*})}.$ (44)

An instanton solution is magnetically charged under $B^{(2)}$, so it has (for
minimal instanton number)

$\int_{S^{3}}H^{(3)}=2\text{p},$ (45)

when integrated over any 3-sphere surrounding the instanton core. We can find
a minimum-action solution using the Bogomolnyi trick:

$I_{E}=\frac{1}{2}\int\left|h(\phi)\mathrm{d}\phi\mp g(\phi){\star
H^{(3)}}\right|^{2}\pm\int h(\phi)g(\phi)\mathrm{d}\phi\int_{S^{3}}H^{(3)}.$
(46)

Thus, we obtain an instanton solution by solving

$h(\phi)\mathrm{d}\phi=\pm g(\phi){\star
H^{(3)}}=\mp\frac{g(\phi)}{\text{p}r^{3}}\mathrm{d}r,$ (47)

with the sign chosen to obtain a well-behaved solution (depending on the
choice of $g$ and $h$). We introduce the quantity

$S(\phi)=2\text{p}\int^{\phi}_{\phi_{0}}h(\phi^{\prime})g(\phi^{\prime})\mathrm{d}\phi^{\prime},$
(48)

with $\phi_{0}=\lim_{r\to 0}\phi(r)$, and then we see that for the Bogomolnyi
solution we have

$S_{\mathrm{inst}}=|S(\phi_{*})|=2\text{p}\left|\int^{\phi_{*}}_{\phi_{0}}h(\phi)g(\phi)\mathrm{d}\phi\right|.$
(49)

Remarkably, these solutions persist when we include gravity. Without $\phi$,
there are gravitational instanton solutions (solutions to the coupled
equations for the metric and the axion) that are singular in their core. With
a scalar modulus $\phi$, gravitational solutions can be instantons or two-
sided wormholes, and there is a special case—the extremal gravitational
instanton—for which the metric is exactly flat. To find such solutions, we
should either use the action involving the field $B^{(2)}$, or we must use a
wrong-sign axion kinetic term (or allow imaginary axion field values). There
is a large literature on this topic; some key references include [81, 82, 85,
86, 87, 88] and recent discussions in the context of the Weak Gravity
Conjecture include [89, 90]. While there is some debate as to the correct
order-one coefficient in the axion WGC, one plausible answer is that it is
determined by the extremal instantons with flat metric, i.e., by the action
$S(\phi_{*})$ determined by (49).

####  4.2.2 Axion strings

We can find an axion string solution in the limit when gravity is negligible.
We work in coordinates where the vortex worldvolume stretches along directions
$t,y$, the axion winds around an angular direction $\varphi$, and the radial
distance from the axion core is measured by $r$. The metric, then, is

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}y^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\varphi^{2}.$
(50)

The solution carries magnetic charge under the axion $\theta$, with
$\theta=\varphi$. In this case, we work directly with $\theta$ instead of its
dual $B$-field, and the kinetic term depends on the function $f(\phi)\equiv
2\text{p}/g(\phi)$. The directions $t,y$ play no role in the solutions we are
interested in, so we can factor them out: we introduce the notation $\ast$ for
the Hodge star in the $(r,\varphi)$ directions only, and then

$I_{E}=-\int\mathrm{d}t\wedge\mathrm{d}y\wedge\left[\frac{1}{2}f(\phi)^{2}\mathrm{d}\theta\wedge{\ast\mathrm{d}\theta}+\frac{1}{2}h(\phi)^{2}\mathrm{d}\phi\wedge{\ast\mathrm{d}\phi}\right]\equiv\int\mathrm{d}t\wedge\mathrm{d}y\wedge{\cal
I},$ (51)

and we can apply the Bogomolnyi trick to ${\cal I}$:

${\cal I}=-\left[\frac{1}{2}\int|h(\phi)\mathrm{d}\phi\mp
f(\phi){\ast\mathrm{d}\theta}|^{2}\pm\int
h(\phi)f(\phi)\mathrm{d}\phi\int_{S^{1}}\mathrm{d}\theta\right].$ (52)

We obtain the axion string solution by solving

$h(\phi)\mathrm{d}\phi=\pm
f(\phi){\ast\mathrm{d}\theta}=\mp\frac{f(\phi)}{r}\mathrm{d}r,$ (53)

choosing the sign so that the solution is well-behaved in the core of the
string. Due to the infrared divergence associated with axion strings, the
solution will generically not be well-behaved as $r\to\infty$. Instead, we
impose an infrared cutoff at some value $r_{*}$ with a chosen
$\phi(r_{*})=\phi_{*}$, and take the tension computed from the solution with
$0\leq r\leq r_{*}$ to be the “core tension” ${\cal T}_{(2)}$. We define

$\mathcal{T}(\phi)=2\text{p}\int_{\phi_{0}}^{\phi}h(\phi^{\prime})f(\phi^{\prime})\mathrm{d}\phi^{\prime},$
(54)

and so estimate

${\cal
T}_{(2)}=|\mathcal{T}(\phi_{*})|=2\text{p}\left|\int_{\phi_{0}}^{\phi_{*}}h(\phi)f(\phi)\mathrm{d}\phi\right|=\left|\int_{\phi_{0}}^{\phi_{*}}\frac{h(\phi)}{g(\phi)}\mathrm{d}\phi\right|.$
(55)

Unlike the case of instantons, axion strings coupled to gravity do not admit a
flat metric. Various aspects of axion string solutions with gravity have been
discussed in, e.g., [83, 91, 92, 39, 42]. As we will mention below, in the
supersymmetric case they are somewhat better understood.

####  4.2.3 Interpretation

We have derived a Euclidean instanton action and an axion string tension. They
are both given by integrals over $\phi$, and the integrands are related by

$\frac{\mathrm{d}{\cal
T}(\phi)}{\mathrm{d}\phi}=\pm\frac{1}{2\text{p}g(\phi)^{2}}\frac{\mathrm{d}S(\phi)}{\mathrm{d}\phi}.$
(56)

The relative minus sign can arise if the solution is such that, for example,
$\phi_{*}<\phi(r\to 0)$ for the instanton solution but $\phi_{*}>\phi(r\to 0)$
for the string solution, so the orientation of the ${\cal T}(\phi)$ and
$S(\phi)$ integrals differs. On the other hand, the relationship (38) is
equivalent to

$|{\cal T}(\phi_{*})|\approx\frac{1}{2\text{p}g(\phi_{*})^{2}}|S(\phi_{*})|.$
(57)

Thus, to the extent that the integral of the relation (56) is dominated near
$\phi=\phi_{*}$, we see that it lends support to (38).

The relationship (56) can be understood as a no-force condition. Indeed, it is
expected in quantum gravity that there exists an object with any given gauge
charge that is (at least marginally) self-repulsive. In other words, for such
an object, the repulsive force mediated by the gauge interaction is stronger
than the combined force of gravity and attractive scalar forces acting on the
object. This Repulsive Force Conjecture (RFC) is one particular variant of the
Weak Gravity Conjecture [51, 93, 94]. The RFC refers specifically to the
leading-power, long-range force mediated by massless fields. For codimension 2
objects like axion strings, a remarkable simplification arises: gravity itself
does not mediate such a force. This is in contrast to Newtonian gravity, which
would lead us to expect a logarithmic potential separating two massive
objects. Instead, in general relativity, codimension 2 objects backreact on
spacetime to produce a flat space with a deficit angle [95]. Thus, the RFC for
axion strings simply requires that the repulsive axion-mediated force at least
compensate the attractive saxion-mediated force, a condition that (for a
single saxion field) becomes

$(2\text{p}f(\phi))^{2}\geq\frac{1}{h(\phi)^{2}}\left(\frac{\partial{\cal
T}}{\partial\phi}\right)^{2}.$ (58)

In the case that this inequality is saturated, it together with (49) implies
exactly (56).

####  4.2.4 Example: exponential ansatz

Suppose that $h(\phi)=1$ (i.e., a canonically normalized modulus $\phi$) and

$g(\phi)=g{\mathrm{e}}^{-b\phi},$ (59)

where without loss of generality we take $\phi_{*}=0$ and $b>0$. The instanton
solution is given by

${\mathrm{e}}^{b\phi(r)}=1+\frac{bg}{2\text{p}r^{2}}.$ (60)

This has the desired property that $\phi(r)\to 0$ at $r\to\infty$, whereas
$\phi(r)\to\infty$ at $r\to 0$. In particular, in the core of the instanton,
$g(\phi)\to 0$. Thus, we find that

$S_{\mathrm{inst}}=2\text{p}\int_{0}^{\infty}g(\phi)\mathrm{d}\phi=\frac{2\text{p}g}{b}.$
(61)

The vortex solution is given by

${\mathrm{e}}^{b\phi(r)}=\frac{2\text{p}g}{b\log(r_{0}/r)}.$ (62)

This gives a well-defined, real value of $\phi$ for $r<r_{0}$. At larger
distances the solution doesn’t make sense, but we only care about the answer
up to some infrared cutoff $r_{*}<r_{0}$ at which $\phi=\phi_{*}=0$. In the
vortex core, $\phi\to-\infty$. Thus we have

${\cal
T}=\int^{0}_{-\infty}\frac{1}{g}{\mathrm{e}}^{b\phi}\mathrm{d}\phi=\frac{1}{bg}.$
(63)

We have $f=\frac{1}{2\text{p}g}$, so

$2\text{p}S_{\mathrm{inst}}f^{2}=2\text{p}\frac{2\text{p}g}{b}\left(\frac{1}{2\text{p}g}\right)^{2}=\frac{1}{bg}={\cal
T}.$ (64)

Thus, the relationship (38) holds precisely, for solutions saturating the
Bogomolnyi bound, in these examples.

####  4.2.5 Example: SUSY form, power law ansatz

In a supersymmetric theory, $\theta$ is partnered with a real modulus $\phi$
in a complex field $\phi-\frac{{\mathrm{i}}}{2\text{p}}\theta$. Thus, the
kinetic terms for $\phi$ and $\theta$ are related via
$f(\phi)^{2}=\frac{1}{4\text{p}^{2}}h(\phi)^{2}$. Let us choose the ansatz

$f(\phi)=M\phi^{\gamma}\quad\Rightarrow\quad
g(\phi)=\frac{1}{2\text{p}M}\phi^{-\gamma},\quad
h(\phi)=2\text{p}f(\phi)=\frac{1}{g(\phi)}.$ (65)

The instanton solution with the asymptotic modulus $\phi_{*}$ is given by:

$\phi(r)=\begin{cases}\left(\phi_{*}^{1+2\gamma}+\frac{1}{8(1+2\gamma)M^{2}\text{p}^{3}r^{2}}\right)^{1/(1+2\gamma)},&\gamma>-\frac{1}{2},\\\
\phi_{*}\exp\left[-\frac{1}{8M^{2}\text{p}^{3}r^{2}}\right],&\gamma=-\frac{1}{2},\\\
\left(\phi_{*}^{1+2\gamma}+\frac{1}{8|1+2\gamma|M^{2}\text{p}^{3}r^{2}}\right)^{1/(1+2\gamma)}.&\gamma<-\frac{1}{2}.\end{cases}$
(66)

We have $\phi(r)\to\infty$ as $r\to 0$ for the case $\gamma>-1/2$, and
$\phi(r)\to 0$ as $r\to 0$ for $\gamma\leq-1/2$. In particular, this means
that $g(\phi)\to 0$ in the instanton core for $\gamma>0$ and for
$\gamma\leq-1/2$, but $g(\phi)\to\infty$ in the instanton core for
$-1/2<\gamma<0$. Because $h(\phi)g(\phi)=1$ for this example, for cases where
$\phi\to 0$ in the instanton core we have

$S(\phi_{*})=2\text{p}\int^{\phi_{*}}_{0}h(\phi)g(\phi)\mathrm{d}\phi=2\text{p}\phi_{*}.$
(67)

This makes sense, since we expect instanton effects to have the form
$\exp(-S+{\mathrm{i}}\theta)$, and holomorphy dictates that this must match
$\exp(-2\text{p}(\phi-{\mathrm{i}}\theta/(2\text{p})))$. In the cases where
$\phi\to\infty$ in the instanton core, we obtain an unbounded action. We
conclude that the case $\gamma>-1/2$ is somehow unphysical, and indeed we do
not know of UV complete examples where such exponents appear and the power law
ansatz remains valid at large $\phi$; see §5.2.3 for some related discussion.

In this case, the axion string solution is simple, as
$f(\phi)/h(\phi)=1/(2\text{p})$ is constant. Hence for the string we have

$\phi(r)=\frac{1}{2\text{p}}\log(r_{0}/r).$ (68)

Notice that this will be true for any axion string obeying the SUSY ansatz,
not just the power law $f(\phi)$. We have $\phi(r)\to\infty$ in the string
core. This means that $f(\phi)\to 0$ as $r\to 0$ if $\gamma<0$, and
$f(\phi)\to\infty$ as $r\to 0$ if $\gamma>0$. The string tension is

${\cal
T}=\int_{\phi_{*}}^{\infty}\left(2\text{p}M\right)^{2}\phi^{2\gamma}\mathrm{d}\phi=\frac{(2\text{p}M)^{2}}{|1+2\gamma|}\phi_{*}^{1+2\gamma},$
(69)

if $\gamma<-\frac{1}{2}$. If $\gamma=-1/2$, the integral is logarithmically
divergent in the string core. Thus, we focus on the case $\gamma<-\frac{1}{2}$
where both the instanton action and string tension are finite. In this limit,
we have

$2\text{p}S_{\mathrm{inst}}f^{2}=(2\text{p})^{2}\phi_{*}^{1+2\gamma}M^{2}=|1+2\gamma|{\cal
T}.$ (70)

Thus, the relationship (38) holds up to the constant factor $|1+2\gamma|$,
which is likely to be $O(1)$ in examples. In particular, in the case
$\gamma=-1$, the canonically normalized modulus is proportional to $\log\phi$,
and in terms of this field we obtain an exponential coupling to the axion as
in §4.2.4. In this case $|1+2\gamma|=1$, consistent with our earlier result
(64).

In the SUSY context, solutions including dynamical gravity can be found, but
they do not have the flat metric (50). Instead, the plane transverse to the
string can be given a complex coordinate $z$ and the metric takes the form
$-\mathrm{d}t^{2}+\mathrm{d}y^{2}+a(z,\overline{z})\mathrm{d}z\,\mathrm{d}\overline{z}$,
where $a(z,\overline{z})$ is determined by the Kähler potential evaluated on
the solution for $\phi$ and $\theta$ [91]. In this context, the relation (56)
is equivalent to equation (3.7) of [42]. Their analysis singles out the
$\gamma=-1$ case (canonically normalized modulus proportional to $\log\phi$)
because they focus on scenarios in which the $\phi\to\infty$ limit is
physical, and assume that the magnetic axion WGC bound is asymptotically
saturated.

###  4.3 Magnetic and electric tension scales

We can think of axion strings as magnetically charged objects for axions, and
instantons as electrically charged objects. The analogue of “tension” for an
instanton is the action. Then we can interpret (38) as a relationship between
the ratio of magnetic and electric tension scales and the axion coupling
constant. It suggests a general statement: given a $p$-form gauge field with
coupling constant $g_{\mathrm{el}}$, whose magnetic dual is a $(d-p-2)$-form
gauge field with coupling constant
$g_{\mathrm{mag}}=2\text{p}/g_{\mathrm{el}}$, the ratio of the tension scale
${\cal T}_{\mathrm{mag}}$ of magnetically charged objects to the tension scale
${\cal T}_{\mathrm{el}}$ of electrically charged objects is

$\frac{{\cal T}_{\mathrm{mag}}}{{\cal
T}_{\mathrm{el}}}\approx\frac{g_{\mathrm{mag}}}{g_{\mathrm{el}}}.$ (71)

One reason that this statement is plausible is that it is manifestly invariant
under exchanging the roles of electric and magnetic quantities. In the axion
case, we identify ${\cal T}_{\mathrm{mag}}$ with the axion string tension,
${\cal T}_{\mathrm{el}}$ with the instanton action $S$, and
$g_{\mathrm{el}}=1/f$, and (71) is precisely (38).

This statement is obviously not true in general, so we should take some care
to interpret it. We suggest that it might apply in scenarios where both
electrically and magnetically charged objects are somehow fundamental, by
which mean they are not solitons. It would not apply, for example, to a
${\mathrm{U(1)}}$ gauge theory arising from higgsing of ${\mathrm{SU}}(2)$, as
then the magnetic monopoles are ’t Hooft-Polyakov solitonic solutions whose
mass scale depends on the higgsing scale, which can be freely varied. It also
does not apply with ${\cal T}_{\mathrm{el}}$ interpreted as the mass of just
any electrically charged object. For example, chiral fermions can carry
electric charge and be exactly massless (and in the real world, the electron
is nearly massless, relative to fundamental scales); if we identified their
mass as the scale $\cal{T}_{\mathrm{el}}$, equation (71) would predict
massless magnetically charged objects, which is obviously not the case in
general. Instead, we should interpret the scale $\cal{T}_{\mathrm{el}}$ as the
scale at which an infinite tower of electrically charged particles appears.
Such towers are expected to always exist, for weakly-coupled gauge theories
[89, 96, 97, 98]. In the case of higher-form gauge fields, the electrically
charged objects are strings or branes that admit a variety of excitations, and
if these objects are not solitons we expect that the tension of a single such
object is already a good proxy for the fundamental electric scale. We will
also assume that the Yang-Mills instanton action is the correct electric
tension scale in the axion case.

A simple motivation for (71) comes from considering the classical self-energy
of the magnetic monopole. That is, we integrate the energy stored in a
monopole’s magnetic field down to some UV cutoff radius $r_{c}$, finding a
magnetic monopole tension

${\cal
T}_{\mathrm{mag}}\sim\frac{2\text{p}}{g_{\mathrm{el}}^{2}}\frac{1}{r_{c}^{p}}.$
(72)

If we identify $r_{c}^{-1}$ with the mass or tension scale associated with a
tower of electrically charged objects, this becomes (71) with the
interpretation of ${\cal T}_{\mathrm{el}}$ that we have discussed above. Such
an identification is reasonable: the tower signals a breakdown of the
effective field theory, in a way that a single particle like an electron does
not.

The monopole self-energy argument that we have just given bears some
resemblance to the Magnetic Weak Gravity Conjecture [51, 77]. It is worth
commenting on how (71) relates to the Weak Gravity Conjecture in general. The
Weak Gravity Conjecture takes the form:

${\cal
T}_{\mathrm{el}}^{2}\leq\frac{1}{\gamma_{\mathrm{el}}}g_{\mathrm{el}}^{2}M_{\mathrm{Pl}}^{D-2},\quad{\cal
T}_{\mathrm{mag}}^{2}\leq\frac{1}{\gamma_{\mathrm{mag}}}g_{\mathrm{mag}}^{2}M_{\mathrm{Pl}}^{D-2},$
(73)

where the coefficients $\gamma_{\mathrm{el}}$ and $\gamma_{\mathrm{mag}}$ are
chosen so that large extremal black holes saturate the inequalities, and
depend in general on scalar moduli couplings to the gauge-field kinetic term.
In most discussions of the WGC, it is assumed that the $\gamma$ factors are
order-one numbers, which is the case in the simplest examples. Let us say that
a state approximately saturates the naive WGC bound if ${\cal T}^{2}\sim
g^{2}M_{\mathrm{Pl}}^{D-2}$. Then we see from (73) that if the electric and
magnetic states both approximately saturate their respective naive WGC bound,
the ratios of magnetic and electric tensions satisfy (71). This is the case
for a large catalogue of standard examples. One way to restate (71) is:

If the electric WGC tower appears at an energy scale parametrically below the
naive WGC scale, then magnetically charged states also appear parametrically
below the naive magnetic WGC scale by the same factor.

One reason that I have phrased this in terms of the naive WGC scale is that I
expect that towers of states always exist that parametrically saturate the WGC
bound (73). Extremal black hole states do, by definition, and we expect
charged states high in a tower to asymptotically become black holes. Hence, my
expectation is that an electrically charged tower appears parametrically below
the naive WGC scale precisely in cases where $\gamma_{\mathrm{el}}\gg 1$, and
approximately saturates the WGC once $\gamma_{\mathrm{el}}$ is taken into
account. Thus, (71) in the context of the Weak Gravity Conjecture leads me to
expect that, in general, $\gamma_{\mathrm{el}}\sim\gamma_{\mathrm{mag}}$. It
is not obvious how to check this for general scalar couplings to gauge fields,
because explicit extremal black hole solutions are not known in the general
case. In the special case of exponential couplings of the form
$\exp(-\alpha\phi)|F_{p+1}|^{2}$, we have explicit solutions (dilatonic black
holes) [99, 100, 101, 102, 103], and it is true that
$\gamma_{\mathrm{el}}=\gamma_{\mathrm{mag}}$ [89]. Such exponential couplings
are expected to appear in asymptotic regions of the string landscape on quite
general grounds [104, 105, 106, 107, 108, 109], but with $\gamma$ of $O(1)$.
In examples where we expect to find $\gamma\gg 1$, such as for moduli
controlling small cycle volumes within a larger compactification, the
couplings are power-law rather than exponential and the black hole solutions
are not known. Nonetheless, we can check explicit examples of tension scales
to see that (71) holds. We find such axion examples in §5.2.3 and (with a
twist) in §5.2.4.

####  4.3.1 Magnetic volume scaling argument

With the argument of this subsection in mind, it is useful to revisit our
volume scaling argument from §4.1. Instead of compactifying $C^{(p)}$ to
obtain the axion $\theta$, we could have considered its magnetic dual field
$C^{(d-p-2)}_{\textsc{M}}$. Under dimensional reduction, this gives rise to
the 2-form gauge field $B^{(2)}$ dual to $\theta$ by integrating over the dual
cycle $[\Gamma^{(n-p)}]$. The prefactor in the kinetic term of $B^{(2)}$ is
$1/\left(2\text{p}f\right)^{2}$, and we obtain an estimate:

$\frac{1}{\left(2\text{p}f\right)^{2}}\propto\int_{Y}\widehat{\gamma}^{(p)}\wedge\star_{Y}\widehat{\gamma}^{(p)},$
(74)

with $[\gamma^{(p)}]$ as a cohomology class defined in §2.3 and
$\widehat{\gamma}^{(p)}$ denoting its warped harmonic representative. The
analogue of (41) is then

$\int_{Y}\widehat{\gamma}^{(p)}\wedge\star_{Y}\widehat{\gamma}^{(p)}\sim\frac{\mathrm{Vol}(\Sigma^{(p)})}{\mathrm{Vol}(\Gamma^{(n-p)})},$
(75)

which is the reciprocal of the estimate in the electric case. This shows that
the volume scaling estimate (41) is consistent with electric-magnetic duality,
whereas other estimates that have appeared in the literature (like
$\mathrm{Vol}(Y)/\mathrm{Vol}(\Sigma^{(p)})^{2}$) are not—they would take on a
drastically different form in a dual frame.

####  4.3.2 Axion string tension and magnetic monopole scale

A theory with axions coupled to gauge fields automatically produces a tower of
particles electrically charged under the gauge field [110], in the form of
closed loops of axion string with excitations of charged modes on the
worldsheet (which must exist by anomaly inflow arguments [111]). The mass
scale $M_{\mathrm{el}}$ of the tower of electrically charged particles is
related to the axion string tension ${\cal T}_{(2)}$ by:

$M_{\mathrm{el}}=\sqrt{2\text{p}{\cal T}_{(2)}}\approx
2\text{p}\sqrt{S_{\mathrm{inst}}}f,$ (76)

using (38). If we then take (71) seriously, and denote the electric coupling
of the gauge theory by $e$, we would expect to find magnetic monopoles at the
mass scale

$M_{\mathrm{mag}}\approx\frac{2\text{p}}{e^{2}}M_{\mathrm{el}}\approx\frac{4\text{p}^{2}}{e^{2}}\sqrt{S_{\mathrm{inst}}}f.$
(77)

In the context of interactions between axions and monopoles, there are two
important length scales, the monopole’s classical radius
$r_{c}=\text{p}/(e^{2}M_{\mathrm{mag}})$ and the axion screening length scale
near the monopole core $r_{0}=e/(8\text{p}^{2}f)$ [112]. In terms of these
scales, we can solve (77) for $S_{\mathrm{inst}}$ and find the estimate

$S_{\mathrm{inst}}^{\mathrm{est.}}\approx\frac{4\text{p}^{2}}{e^{2}}\left(\frac{r_{0}}{r_{c}}\right)^{2}.$
(78)

This can be compared with the actual computation of the instanton action in
the form of a monopole loop with dyonic winding from [113],

$S_{\mathrm{inst}}\approx\frac{4\text{p}^{2}}{e^{2}}\sqrt{\frac{\max(r_{c},r_{0})}{r_{c}}}.$
(79)

We see that these agree precisely when $r_{c}=r_{0}$, i.e., when the axion
field is screened at the classical radius of the magnetic monopole. This is a
nontrivial consistency requirement on the picture that we have sketched, which
need not be true within every effective field theory, but could be a property
of UV-complete examples where both the magnetic monopole and the axion string
arise from branes wrapped on cycles in extra dimensions. This condition
translates to the expectation that

$M_{\mathrm{mag}}\approx\frac{8\text{p}^{3}}{e^{3}}f.$ (80)

For example, if $f\sim 10^{12}\,\mathrm{GeV}$ this predicts that the mass of a
magnetic monopole is around $10^{16}\,\mathrm{GeV}$.

##  5 Axion string tension in examples

In this section, we will show that the expectation (38) holds in several
examples in string theory. In each example, we will also evaluate the ratios
$fS/M_{\mathrm{Pl}}$ and ${\cal T}_{(2)}/(fM_{\mathrm{Pl}})$, which are
predicted to be $\lesssim O(1)$ by the electric and magnetic axion WGC
respectively, and the ratio of the axion decay constant to the string scale,
to see how an experimental measurement of $f$ might shed light on the
fundamental UV cutoff in our universe.

###  5.1 “Model-independent” $B^{(2)}$ axion

One of the original examples of an extra-dimensional axion is the so-called
“model-independent string axion,” which is just the 10d 2-form gauge field
$B^{(2)}$, viewed as a 4d 2-form gauge field that is dual to an axion $\theta$
[11]. (The adjective “model-independent” is, in light of what we now know
about the string theory landscape, no longer very fitting.) In the Type I or
heterotic context, in our conventions, the relevant kinetic terms in the 10d
effective action for $B^{(2)}$ and the $\mathrm{Spin}(32)/{\mathbb{Z}}_{2}$
gauge fields look like333My conventions follow [25], except that their field
strengths have integer fluxes whereas mine have $2\text{p}$ times integer
fluxes.

$\int\left(-\frac{1}{2\text{p}g_{s}^{2}\ell_{s}^{4}}\frac{1}{2}H^{(3)}\wedge\star
H^{(3)}-\frac{1}{4\text{p}g_{s}^{2}\ell_{s}^{6}}\frac{1}{2}\mathrm{tr}(F\wedge\star
F)\right),$ (81)

where $H^{(3)}=\operatorname{d}\\!{B}^{(2)}$, $\ell_{s}\equiv
2\text{p}\sqrt{\alpha^{\prime}}$ is the string length scale, and $g_{s}$ is
the string coupling constant. The axion couples to the gauge fields due to the
modified Bianchi identity

$\frac{1}{2\text{p}}\operatorname{d}\\!{H}^{(3)}=-\frac{1}{16\text{p}^{2}}\mathrm{tr}(F\wedge
F)+\cdots,$ (82)

which is well-known from the Green-Schwarz mechanism [114].

To see that this is a theory of an extra-dimensional axion in our sense,
notice that since $\theta$ is dual to $B^{(2)}$ in four dimensions, it
originates from dimensional reduction of the six-form gauge field $B^{(6)}$
that is dual to $B^{(2)}$ in 10d. The kinetic term for that field is:

$-\int\frac{g_{s}^{2}\ell_{s}^{4}}{2\text{p}}\frac{1}{2}H^{(7)}\wedge\star
H^{(7)}.$ (83)

After dualizing, the axion coupling is more straightforward, because the
modified Bianchi identity becomes a standard Chern-Simons term

$\frac{1}{16\text{p}^{2}}\int B^{(6)}\wedge\mathrm{tr}(F\wedge F).$ (84)

Thus, everything proceeds as in §2, but this case is relatively
straightforward because all of the integrals are over all six of the extra
dimensions.

We assume there is no warping and that the dilaton is constant over the
internal dimensions. The axion comes from reducing $B^{(6)}$ along the 6-cycle
that is simply the entire manifold $Y$. The volume form $\star_{Y}1$ is
harmonic, because $\operatorname{d}\\!{(}1)=0$ and
$\operatorname{d}\\!{(}\star_{Y}1)=0$ as $\star_{Y}1$ is a top form. Hence the
harmonic form normalized to have integral $1$ over $Y$ is
$\omega^{(6)}=(\star_{Y}1)/\mathrm{Vol}(Y)$. The axion string is a fundamental
string, not wrapped on any internal dimensions. Hence we have:

$\displaystyle{\cal T}_{(2)}$ $\displaystyle=\frac{2\text{p}}{\ell_{s}^{2}},$
$\displaystyle S_{\mathrm{inst}}$
$\displaystyle=\frac{8\text{p}^{2}}{g_{\mathrm{YM}}^{2}}=\frac{2\text{p}\mathrm{Vol}(Y)}{g_{s}^{2}\ell_{s}^{6}},$
$\displaystyle f^{2}$
$\displaystyle=\frac{g_{s}^{2}\ell_{s}^{4}}{2\text{p}}\int_{Y}\frac{\star_{Y}1\wedge
1}{\mathrm{Vol}(Y)^{2}}=\frac{g_{s}^{2}\ell_{s}^{4}}{2\text{p}\mathrm{Vol}(Y)}.$
(85)

Putting these together, we find that ${\cal
T}_{(2)}=2\text{p}\,S_{\mathrm{inst}}\,f^{2}$. This is our first example where
this scaling relation holds precisely.

We can also compare these scales to expectations from the electric and
magnetic axion WGC. For this we need the Planck scale, given by

$M_{\mathrm{Pl}}^{2}=4\text{p}\frac{\mathrm{Vol}(Y)}{g_{s}^{2}\ell_{s}^{8}}.$
(86)

Thus we see that in this example

$fS_{\mathrm{inst}}=\frac{\sqrt{2\text{p}\mathrm{Vol}(Y)}}{g_{s}\ell_{s}^{4}}=\frac{1}{\sqrt{2}}M_{\mathrm{Pl}},$
(87)

parametrically saturating the naive axion WGC. We also have

$\frac{1}{2\text{p}f}{\cal T}_{(2)}=\frac{1}{\sqrt{2}}M_{\mathrm{Pl}},$ (88)

parametrically saturating the naive magnetic axion WGC. As discussed in §4.3,
we believe it is not an accident that both the naive electric and magnetic
axion WGCs are parametrically saturated at the same time. In our next
examples, we will see that both fail to be saturated, by a common factor.

Finally, we would like to compare the axion decay constant $f$ to the string
mass scale

$M_{s}\equiv\frac{2\text{p}}{\ell_{s}},$ (89)

which is related to the Planck scale and the dimensionless volume ${\cal
V}=\mathrm{Vol}(Y)/\ell_{s}^{6}$ via

$M_{s}^{2}=\frac{\text{p}g_{s}^{2}}{\cal V}M_{\mathrm{Pl}}^{2}.$ (90)

In terms of the quantity
$\alpha_{\textsc{YM}}=g_{\textsc{YM}}^{2}/(4\text{p})$, we have

$f=\frac{\sqrt{\alpha_{\textsc{YM}}}}{(2\text{p})^{3/2}}M_{s}.$ (91)

Quantitatively, if we take as a reference the GUT value
$\alpha_{\textsc{YM}}\approx 1/25$, we have

$f\approx 0.013M_{s},$ (92)

and the string scale is about a factor of 80 above the axion decay constant.
(The definition of $M_{s}$ varies in the literature; in [25] it is taken to be
$1/\ell_{s}$. The masses of string oscillator states are closer to
$2\text{p}/\ell_{s}$, hence our choice.)

###  5.2 $C^{(4)}$ axions in Type IIB string theory

Many phenomenological investigations of Standard Model-like physics, including
axions, have focused on Type IIB string theory compactified on a 6d Calabi-Yau
orientifold $Y$, with Standard Model gauge fields living on D7 branes that
wrap 4-cycles, which are also known as divisors (complex codimension-1
submanifolds). In this case, axions arise from the bulk gauge field $C^{(4)}$
dimensionally reduced on the 4-cycles. Axion strings are D3 branes wrapped on
dual 2-cycles. Euclidean D3 branes wrapped on the 4-cycles give rise to
instantons.

In this context, the geometry simplifies significantly because the volumes of
interest are all calibrated by a Kähler 2-form $J$. Integrals of $J$, $J\wedge
J$, and $J\wedge J\wedge J$ measure the volumes of 2-cycles, 4-cycles, and the
entire manifold $Y$, respectively. This structure leads to a precise equation
that generalizes our basic scaling relation (38) to the full set of $C^{(4)}$
axions.

For the purposes of this discussion, I will treat the dilaton field as a
constant over the extra dimensions, so that I only have to keep track of
powers of $g_{s}$. This is never truly valid in the presence of D7 branes,
where a proper treatment would take the varying axiodilaton field into account
using the F-theory description. However, it is sufficient for a discussion of
the scaling relations of interest.

####  5.2.1 Elements of Calabi-Yau geometry

Here we collect some key formulas in Calabi-Yau geometry. Readers can find
more detailed pedagogical introductions in, for example, [115, 116, 117]. A
useful reference focused specifically on axions is [33].

The Kähler form of a Calabi-Yau threefold $Y$ can be expressed as $J=\sum
t_{i}D^{i}$, where $D^{i}$ is a 2-form Poincaré dual to a divisor
$\Delta_{i}$.444A divisor is a formal integer linear combination of
holomorphic hypersurfaces (which are 4-manifolds) in $Y$. The coefficients
$t_{i}$ are known as Kähler parameters. The volume of $Y$ is a cubic function
of the $t_{i}$,

${\cal V}=\frac{1}{6}n^{ijk}t_{i}t_{j}t_{k},$ (93)

where the $n^{ijk}\in{\mathbb{Z}}$ are integer intersection numbers. The
4-cycle volumes $\tau^{i}$ of these divisors are related to the Kähler
parameters according to

$\tau^{i}=\frac{\partial{\cal V}}{\partial t_{i}}.$ (94)

In the literature, one finds a convention in which the $t$ variables have
raised indices and the $\tau$ variables have lowered indices. Here I am doing
precisely the opposite, to agree with our earlier convention of raised indices
on $\theta^{i}$ (which partners with $\tau^{i}$ in a chiral superfield). I
hope that this does not cause confusion.

The Kähler parameters can be loosely thought of as measuring 2-cycle volumes,
but this isn’t precisely right; for instance, they are sometimes negative. A
more correct statement is the following: there is an allowed set of values of
the $t_{i}$, known as the Kähler cone, for which one can integrate powers of
$J$ over holomorphic subvarieties to obtain positive volumes. Within the
Kähler cone, the $\tau^{i}$ are positive. A 2-cycle $\Gamma$ can be
characterized by its intersection numbers $w^{i}$ with the divisors
$\Delta_{i}$. Certain 2-cycles are holomorphic and have volumes determined by
integrating $J$ over $\Gamma$; in this case, the volume is given by
$w^{i}t_{i}$. The set of $\Gamma$ for which this holds is known as the Mori
cone.

For now we will treat $t$, $\tau$, and ${\cal V}$ as dimensionless numbers,
and restore units when we translate our mathematical result into physics
below. Viewing the $t_{i}$ and thus also ${\cal V}$ as implicit functions of
the $\tau^{i}$, the Kähler potential $k$ is defined as

$k(\\{\tau^{i}\\})=-2\log{\cal V}.$ (95)

From these definitions, it follows that

$t_{i}=\frac{\partial^{2}k}{\partial\tau^{i}\partial\tau^{j}}\tau^{j}{\cal
V}.$ (96)

We give the full argument in Appendix A.

####  5.2.2 Translation to physical quantities

Now, we translate the dimensionless volumes to physical units and explain how
they relate to gauge couplings, axion decay constants, and axion string
tensions. Dimensionless volumes are taken to be measured in units of
$\ell_{s}$, so that ${\cal V}=\mathrm{Vol}(Y)/\ell_{s}^{6}$. Thus, we have

$M_{\mathrm{Pl}}^{2}=\frac{4\text{p}}{g_{s}^{2}\ell_{s}^{2}}{\cal V}.$ (97)

(In our convention, Einstein and string frame metrics are related by
$g_{\mu\nu,E}=\exp(-(\phi-\phi_{0})/2)g_{\mu\nu,S}$ where
$\exp(\phi_{0})=g_{s}$. Thus, to the extent that the dilaton is constant,
volumes are the same in both frames.)

From the DBI action for a U(1) gauge field, with fluxes normalized to be in
$2\text{p}{\mathbb{Z}}$, on a D7 brane wrapped on a 4-cycle
$\Sigma^{(4)}_{i}$, we read off the 4d U(1) gauge coupling $e$ as

$\frac{1}{e^{2}}=\frac{1}{2\text{p}g_{s}}\frac{\mathrm{Vol}(\Sigma^{(4)}_{i})}{\ell_{s}^{4}}=\frac{1}{2\text{p}g_{s}}\tau^{i}.$
(98)

From this we infer that the 4-cycle volume $\tau^{i}$ in units of
$\ell_{s}^{4}$ is packaged with an axion $\theta^{i}\cong\theta^{i}+2\text{p}$
arising from $C^{(4)}$, in a complex scalar field $T^{i}$ that is a component
of an ${\cal N}=1$ chiral multiplet $\bm{T}^{i}$:

$\bm{T}^{i}=\frac{1}{g_{s}}\tau^{i}-{\mathrm{i}}\frac{\theta^{i}}{2\text{p}}+\cdots.$
(99)

The factor of $1/g_{s}$ is sometimes absorbed in the definition of $\tau$, or
into a different convention for Einstein-frame volumes relative to string-
frame volumes. In the nonabelian case, rather than
$\tau/g_{s}=2\text{p}/e^{2}$ we have $\tau/g_{s}=4\text{p}/g^{2}$, for the
conventional normalization of the gauge coupling. In any case, the instanton
action for gauge fields on a 4-cycle (or a Euclidean D3-brane on the same
cycle) is

$S^{i}=2\text{p}\tau^{i}/g_{s}.$ (100)

The shift symmetry of $\theta^{i}$ ensures that the Kähler potential is a
function of $\bm{T}^{i}+\bm{T}^{i\dagger}$, up to nonperturbatively small
corrections. The leading $\bm{T}^{i}$-dependence is captured by the function
$k$ introduced above,

$K(\\{\bm{T}^{i},{\bm{T}^{i}}^{\dagger}\\})=M_{\mathrm{Pl}}^{2}k(\\{g_{s}(\bm{T}^{i}+{\bm{T}^{i}}^{\dagger})/2\\})+\cdots.$
(101)

The scalar kinetic terms take the form

$-\frac{\partial^{2}K}{\partial
T^{i}\partial{T^{\dagger}}^{j}}\partial_{\mu}T^{i}\partial^{\mu}{T^{\dagger}}^{j}=-\frac{1}{4}\frac{\partial^{2}k}{\partial\tau^{i}\partial\tau^{j}}\left[\partial_{\mu}\tau^{i}\partial^{\mu}\tau^{j}+\frac{g_{s}^{2}}{4\text{p}^{2}}\partial_{\mu}\theta^{i}\partial^{\mu}\theta^{j}\right],$
(102)

from which we read off that the kinetic matrix of the axions in the
normalization of (2.1.4) is

$\kappa_{ij}=\frac{g_{s}^{2}}{8\text{p}^{2}}M_{\mathrm{Pl}}^{2}\frac{\partial^{2}k}{\partial\tau^{i}\partial\tau^{j}}.$
(103)

Finally, a D3 brane has tension ${\cal
T}_{(4)}=\frac{2\text{p}}{g_{s}\ell_{s}^{4}}$. We can obtain an axion string
from wrapping a D3 brane on a 2-cycle $\Gamma$ that has intersection numbers
$w^{i}$ with the divisors $\Delta_{i}$. Physically, $w^{i}$ represents the
winding number of the axion $\theta^{i}$ around the string. When $\Gamma$ is
in the Mori cone, this axion string has tension

${\cal T}_{(2)}^{\\{w\\}}=\frac{2\text{p}}{g_{s}\ell_{s}^{2}}w^{i}t_{i}.$
(104)

Substituting (100), (103), (104), and (97) into (96), we find:

${\cal T}_{(2)}^{\\{w\\}}=2\text{p}w^{i}\kappa_{ij}S^{j}.$ (105)

This is a precise multi-axion generalization of (38).

####  5.2.3 First example: 3-axion, fibered “Swiss cheese” Calabi-Yau

Now, we consider a concrete example where there $Y$ has a large overall volume
but also contains a much smaller cycle. For concreteness, we consider the
Calabi-Yau manifold described in §3.1 of [118] (see also [119]), which has 3
axions arising from $C^{(4)}$ reduced on 3 different 4-cycles. The volume
takes the form

${\cal
V}=9t_{1}t_{2}^{2}+\frac{1}{6}t_{3}^{3}=\frac{1}{6}\sqrt{\tau^{1}}\tau^{2}-\frac{\sqrt{2}}{3}\smash{\left(\tau^{3}\right)}^{3/2},$
(106)

with $t_{1}=\frac{1}{6}\frac{\tau^{2}}{\sqrt{\tau^{1}}}$,
$t_{2}=\frac{1}{3}\sqrt{\tau^{1}}$ , and $t_{3}=-\sqrt{2\tau^{3}}$.555In the
notation of [118], our $(t_{1},t_{2},t_{3})$ are their $(t_{b},t_{f},t_{s})$
and our $(\tau^{1},\tau^{2},\tau^{3})$ are their
$(\tau_{f},\tau_{b},\tau_{s})$. Their subscripts are $b$ for big, $f$ for
fiber, and $s$ for small. Their choice uses the same subscripts for $t_{f}$
and $\tau_{f}$, because both are related to the fiber size, but they are not
linked by (94). We take our numerical labels as in (94), so the fiber
corresponds to $t_{2}$ but $\tau^{1}$. Within the Kähler cone, $t_{1},t_{2}>0$
and $t_{3}<0$. This structure reflects a manifold which, on large scales, is a
four-dimensional K3 manifold of volume $\tau^{1}\sim\smash{(t_{2})}^{2}$
fibered over a two-dimensional $\mathbb{P}^{1}$ base of volume $t_{1}$, and
which contains a small, self-intersecting divisor of volume $\tau^{3}$. We
work in the limit ${\cal V}\gg\smash{\left(\tau^{3}\right)}^{3/2}$, where
there is a small hole in a much larger manifold. This is a prototype for a
much larger family of examples which might have multiple such small holes,
some of which could play a role in stabilizing the overall volume modulus
[120] while others could host the Standard Model gauge fields on wrapped D7
branes. We will consider a similar but more subtle example in §5.2.4.

In this example, the axion kinetic terms are given by

$-\frac{1}{2}\left(\frac{g_{s}^{2}}{8\text{p}^{2}}M_{\mathrm{Pl}}^{2}\right)\left[\frac{1}{(\tau^{1})^{2}}\mathrm{d}\theta^{1}\wedge\star\mathrm{d}\theta^{1}+\frac{2}{(\tau^{2})^{2}}\mathrm{d}\theta^{2}\wedge\star\mathrm{d}\theta^{2}+\frac{1}{\sqrt{2\tau^{3}}{\cal
V}}\mathrm{d}\theta^{3}\wedge\star\mathrm{d}\theta^{3}+\cdots\right].$ (107)

The $\cdots$ represents subleading terms, including kinetic mixing among the
axions as well as volume-suppressed contributions to the kinetic terms
displayed. Although the mixing contributions are volume suppressed, they are
not necessarily negligible in the relation (105), because some of the $S^{j}$
are enhanced by large volumes.

In the limit that $\tau^{3}$ goes to zero, the volume factors into the product
of $\sqrt{\tau^{1}}$ and $\tau^{2}$, and so the moduli $\tau^{1}$ and
$\tau^{2}$ each have logarithmic kinetic terms of the form familiar in
infinite distance limits of moduli space (see, e.g., [104]). They mix only
through subleading terms involving $\tau^{3}$. Correspondingly, for the axions
$\theta^{1}$ and $\theta^{2}$, the single-axion relationship (38) holds up to
volume-suppressed terms. Furthermore, in both cases the naive electric and
magnetic axion WGC relations for $fS_{\mathrm{inst}}$ and ${\cal
T}_{(2)}/(2\text{p}f)$ hold: specifically,

$\displaystyle\theta^{1}:\qquad$ $\displaystyle
f_{1}S^{(1)}=\frac{1}{\sqrt{2}}M_{\mathrm{Pl}},\qquad$
$\displaystyle\frac{1}{2\text{p}f_{2}}{\cal
T}_{(2)}^{\\{1,0,0\\}}=\frac{1}{\sqrt{2}}M_{\mathrm{Pl}};$
$\displaystyle\theta^{2}:\qquad$ $\displaystyle
f_{2}S^{(2)}=M_{\mathrm{Pl}},\qquad$
$\displaystyle\frac{1}{2\text{p}f_{2}}{\cal
T}_{(2)}^{\\{0,1,0\\}}=M_{\mathrm{Pl}}.$ (108)

This is no surprise; we expect all of these relationships to come as a package
for axions associated with moduli that have infinite distance limits. Notice,
also, that in both cases the coefficient in the electric and magnetic WGC
relationships is the same, along the lines we argued in §4.3. Here, and
elsewhere, I have put parentheses around the numerical superscript on the
action in places where it might otherwise be confused with an exponent.

There is no infinite-distance limit in moduli space associated with $\tau^{3}$
alone. Because it contributes negatively to the overall volume, we cannot make
$\tau^{3}$ arbitrarily large while holding $\tau^{1}$ and $\tau^{2}$ fixed.
Thus, the kinetic term for $\tau^{3}$ does not have the universal logarithmic
form expected for moduli with asymptotic limits; in fact, $\tau^{3}\to 0$ is a
finite-distance limit. The kinetic term for $\theta^{3}$, proportional to
$1/\sqrt{\tau^{3}}$, is of the power law form discussed in §4.2.5 with
$\gamma=-1/4$. This is the case $\gamma>-1/2$ where we found no physical
Euclidean instanton solutions. However, the analysis of §4.2.5 breaks down
here because the $1/\sqrt{\tau^{3}}$ dependence is only an approximation,
valid when $\tau^{3}\ll\tau^{1,2}$. Relatedly, the kinetic term for $\tau^{3}$
is highly dependent on the values of $\tau^{1}$ and $\tau^{2}$, and mixing
terms are important in the relationship (96). Specifically, we have

$\frac{\partial^{2}k}{\partial\tau^{3}\partial\tau^{1}}\tau^{1}{\cal
V}\approx-\frac{1}{\sqrt{2}}\sqrt{\tau^{3}};\quad\frac{\partial^{2}k}{\partial\tau^{3}\partial\tau^{2}}\tau^{2}{\cal
V}\approx-\sqrt{2\tau^{3}};\quad\frac{\partial^{2}k}{\partial\tau^{3}\partial\tau^{3}}\tau^{3}{\cal
V}\approx\frac{1}{\sqrt{2}}\sqrt{\tau^{3}}.$ (109)

Here $\approx$ indicates that we have dropped volume-suppressed terms. Summing
these up, we obtain $-\sqrt{2\tau^{3}}=t_{3}$, as promised by (96).
Interestingly, if we kept only the diagonal term, we would obtain
parametrically the same answer, but off by a factor of $-1/2$. Thus, if we
focus on the axion $\theta^{3}$ alone, the holomorphic axion string of minimal
winding has $w^{3}=-1$ and hence

${\cal T}_{(2)}^{\\{0,0,-1\\}}\approx 4\text{p}S^{(3)}f_{3}^{2}.$ (110)

Thus, the single-field parametric relation (38) is still a good guide to the
scale of the axion string tension, even though getting the order-one
coefficient in the relationship right requires us to use the multi-field
formula (105). Furthermore, in this case the naive electric and magnetic axion
WGC are both satisfied by a large parametric factor:

$\theta^{3}:\qquad
f_{3}S^{(3)}=\frac{\smash{\left(\tau^{3}\right)}^{3/4}}{2^{3/4}\sqrt{\cal
V}}M_{\mathrm{Pl}},\qquad\frac{1}{2\text{p}f_{3}}{\cal
T}_{(2)}^{\\{0,0,-1\\}}=\frac{2^{1/4}\smash{\left(\tau^{3}\right)}^{3/4}}{\sqrt{\cal
V}}M_{\mathrm{Pl}}.$ (111)

This is consistent with our expectation, from §4.3, that the parametric
coefficients appearing in the electric and magnetic axion WGC relationships
should be the same. In this case, the quantitative match is not perfect, due
to factors of $2^{1/4}$; this is again related to effects of mixing.

Finally, we evaluate the decay constant of $\theta^{3}$:

$f_{3}=\frac{g_{s}M_{\mathrm{Pl}}}{2^{7/4}\text{p}{\cal
V}^{1/2}\smash{\left(\tau^{3}\right)}^{1/4}}=\frac{1}{2^{7/4}\text{p}^{3/2}\smash{\left(\tau^{3}\right)}^{1/4}}M_{s}.$
(112)

Thus, the axion decay constant is near the string scale. In particular, if we
have in mind an axion coupled to the Standard Model and set
$\tau^{3}=g_{s}\alpha_{\textsc{GUT}}^{-1}$ with the conventional GUT value
$\alpha_{\textsc{GUT}}\approx 1/25$, then numerically we have

$f_{3}\approx 0.024g_{s}^{-1/4}M_{s}.$ (113)

The $g_{s}$ dependence is very mild (and $g_{s}$ should not be too small, or
we would not be able to achieve large enough Standard Model couplings while
keeping volumes large enough to trust our calculations), so we conclude that
the string scale in such a model should be around 20 to 40 times the axion
decay constant. The other decay constants, $f_{1}$ and $f_{2}$, have a less
straightforward relationship to the string scale. When the overall volume is
approximately isotropic, in the sense that $\tau^{1}\sim\tau^{2}$, these decay
constants are both near the Kaluza-Klein scale $M_{\textsc{KK}}\sim
M_{\mathrm{Pl}}/{\cal V}^{2/3}$. Taking $\tau^{1}\gg\tau^{2}$ raises $M_{s}$
relative to $f_{1}$, and taking $\tau^{2}\gg\tau^{1}$ raises $M_{s}$ relative
to $f_{2}$. In any case, if we focus on $f_{i}$ while fixing
$\tau^{i}=g_{s}\alpha_{\textsc{GUT}}^{-1}$, the only way to achieve $M_{s}\gg
10^{2}f_{i}$ is to make some other $\tau^{j}$ small in string units, at which
point the calculation is not under control.

####  5.2.4 Second example: 2-axion “Swiss cheese” Calabi-Yau

Next, we consider a classic example which has only two axions instead of 3,
and a similar structure to the previous example. Despite initial appearances,
this example is actually less straightforward, because the axion string of
minimal tension turns out to not be the axion string of minimal winding. This
example is a Calabi-Yau with $h^{1,1}=2$, the anticanonical hypersurface in
the weighted projective space $\mathbb{P}_{[1,1,1,6,9]}$. This is a well-
studied CY manifold often discussed in the context of moduli stabilization
scenarios [121, 122, 120]. We will work in the basis chosen in section 4.1 of
the CYTools manual, which discusses many properties of this manifold very
concretely [123].

We denote the Poincaré duals to the two divisors in the chosen basis as
$D^{1}$ and $D^{2}$, and parametrize the Kähler form as
$J=t_{1}D^{1}+t_{2}D^{2}$. The volume is

${\cal V}=\frac{1}{18}\left[t_{1}^{3}-\left(t_{1}-3t_{2}\right)^{3}\right].$
(114)

The Kähler cone, i.e., the range of parameters for which we can compute
positive volumes by integrating powers of $J$ over subvarieties, is given by
imposing the conditions

$t_{1}-3t_{2}\geq 0\quad\text{and}\quad t_{2}\geq 0.$ (115)

Thus we can think of $t_{1}$ as parametrizing the overall volume of the
Calabi-Yau, and $t_{1}-3t_{2}$ as parametrizing the size of a hole in the
Calabi-Yau.

The divisor volumes are computed as

$\displaystyle\tau^{1}=\frac{\partial{\cal V}}{\partial t_{1}}$
$\displaystyle=\frac{1}{2}t_{2}\left(2t_{1}-3t_{2}\right),$
$\displaystyle\tau^{2}=\frac{\partial{\cal V}}{\partial t_{2}}$
$\displaystyle=\frac{1}{2}\left(t_{1}-3t_{2}\right)^{2}.$ (116)

We see that $D^{2}$ is the small divisor, with $\tau^{2}\to 0$ at the edge of
the Kähler cone where $t_{1}\to 3t_{2}$. We could carry out a
$\mathrm{GL}(2,{\mathbb{Z}})$ transformation of our basis of divisors to make
one of the Kähler parameters $t_{i}$ vanish as $t_{1}\to 3t_{2}$, but then the
conjugate $\tau^{i}$ would not vanish in this limit.666In the literature, one
sometimes sees the formula written as ${\cal
V}\sim\tau_{b}^{3/2}-\tau_{s}^{3/2}$ for a big volume $\tau_{b}$ and small
volume $\tau_{s}$, with $\tau_{b}\sim t_{b}^{2}$ and $\tau_{s}\sim t_{s}^{2}$
in terms of big and small 2-cycle volumes $t_{b}$ and $t_{s}$. Such a form can
only be obtained through a non-integral basis choice of divisors, which
obscures the possible allowed charges of instantons and axion strings. The
overall volume can also be expressed as

${\cal
V}=\frac{\sqrt{2}}{9}\left[\left(3\tau^{1}+\tau^{2}\right)^{3/2}-\left(\tau^{2}\right)^{3/2}\right],$
(117)

in terms of the divisor volumes.

An axion string is obtained by wrapping a D3 brane on a curve $C$ that has
intersection numbers $w^{i}$ with the divisors $D^{i}$. The volume of such a
curve is

$\mathrm{Vol}(C)=w^{i}t^{i}.$ (118)

This formula is reliable and computes positive volumes for curves in the Mori
cone, which obey the inequalities

$w^{1}\geq 0\quad\text{and}\quad w^{2}+3w^{1}\geq 0.$ (119)

In particular, there is a small volume curve with $w^{1}=1$, $w^{2}=-3$, and
volume $t_{1}-3t_{2}$, which wraps a 2-cycle in the Calabi-Yau that shrinks to
zero size at the edge of the Kähler cone.

The axion kinetic terms are given by

$-\frac{1}{2}\left(\frac{g_{s}^{2}}{8\text{p}^{2}}M_{\mathrm{Pl}}^{2}\right)\left[\frac{3}{(\tau^{1})^{2}}\mathrm{d}\theta^{1}\wedge\star\mathrm{d}\theta^{1}+\frac{1}{3\sqrt{2\tau^{2}}{\cal
V}}\mathrm{d}\theta^{2}\wedge\star\mathrm{d}\theta^{2}+\frac{2}{(\tau^{1})^{2}}\mathrm{d}\theta^{1}\wedge\star\mathrm{d}\theta^{2}+\cdots\right].$
(120)

The third term is the leading kinetic mixing between the two axions; the
$\cdots$ represents contributions suppressed by powers of the small ratio
$\sqrt{\tau^{2}/\tau^{1}}$.

If we focus on one axion at a time, inspecting the corresponding instanton
action and the tension of a string of minimal winding, we have (up to
subleading terms):

$\displaystyle\theta^{1}:\qquad$ $\displaystyle
f_{1}S^{(1)}=\sqrt{\frac{3}{2}}M_{\mathrm{Pl}},\qquad$
$\displaystyle\frac{1}{2\text{p}f_{1}}{\cal
T}_{(2)}^{\\{1,0\\}}=\sqrt{\frac{3}{2}}M_{\mathrm{Pl}};$
$\displaystyle\theta^{2}:\qquad$ $\displaystyle
f_{2}S^{(2)}=\frac{\smash{\left(\tau^{2}\right)}^{3/4}}{2^{3/4}\sqrt{3{\cal
V}}}M_{\mathrm{Pl}},\qquad$ $\displaystyle\frac{1}{2\text{p}f_{2}}{\cal
T}_{(2)}^{\\{0,1\\}}=\sqrt{\frac{1}{6}}M_{\mathrm{Pl}}.$ (121)

Here we see a breakdown of the pattern we have seen before. For the axion
corresponding to the large volume modulus, the naive electric and magnetic WGC
hold with the same coefficient. For the axion $\theta^{2}$ corresponding to
the small volume modulus, the instanton action is set by the small quantity
$\tau^{2}$, but the corresponding axion string with $w^{1}=0,w^{2}=1$ has a
tension set by $t_{2}$, which is not small. The naive magnetic WGC estimate
holds, but $fS_{\mathrm{inst}}$ is below the naive electric WGC estimate by a
large factor. Our basic scaling estimate (38) breaks down badly in this case.
On the other hand, the axion string with $w^{1}=1$ and $w^{2}=-3$ has tension

$\frac{1}{2\text{p}f_{2}}{\cal
T}_{(2)}^{\\{1,-3\\}}=\frac{3^{1/2}2^{1/4}\smash{\left(\tau^{2}\right)}^{3/4}}{\sqrt{\cal
V}}M_{\mathrm{Pl}}.$ (122)

If we take this minimal tension string with nontrivial $\theta^{1}$ winding to
be the string whose tension should appear in the estimate (38), then the
pattern holds. We have

$\frac{1}{2\text{p}f_{2}^{2}S^{(2)}}{\cal T}_{(2)}^{\\{1,-3\\}}=6.$ (123)

Finally, we evaluate the decay constant of $\theta^{2}$:

$f_{2}=\frac{g_{s}M_{\mathrm{Pl}}}{3^{1/2}2^{7/4}\text{p}{\cal
V}^{1/2}\smash{\left(\tau^{2}\right)}^{1/4}}=\frac{1}{3^{1/2}2^{7/4}\text{p}^{3/2}\smash{\left(\tau^{2}\right)}^{1/4}}M_{s}\approx
0.014g_{s}^{-1/4}M_{s}.$ (124)

As in our other examples, we see that the fundamental string scale can be at
most about two orders of magnitude above the axion decay constant.

####  5.2.5 Comments on the Calabi-Yau examples

In the two examples we have considered, the axions that have a decay constant
parametrically below the naive WGC bound, as in (111), are paired in a chiral
superfield (99) with a modulus $\tau$ describing the volume of a collapsing
cycle within the larger CY manifold. The small-volume limit $\tau\to 0$ lies
at finite distance in field space, but is at the boundary of the Kähler cone.
There are several different ways that a boundary of the Kähler cone can arise
in moduli space [124, 125, 126]. These include: flop walls, at which the
volume of a curve (2-cycle) goes to zero while divisor volumes and the overall
volume of $Y$ remain finite; Zariski walls, at which the volume of a divisor
goes to zero while the overall volume remains finite; and effective cone
walls, where the overall volume of $Y$ goes to zero. Zariski walls come in two
types, depending on whether the divisor collapses to a finite-volume curve or
to a point. Our examples are Zariski walls of the second type: a divisor
collapses to a point, as measured by $\tau$, and it intersects itself along a
curve that also collapses to a point, as measured by a linear combination of
Kähler parameters $t\sim\sqrt{\tau}$. This scaling explains why our examples
obey the basic relation (38): the axion string tension is proportional to $t$,
the instanton action to $\tau\sim t^{2}$, and the square of the decay constant
to $1/\sqrt{\tau}\sim 1/t$. All of these quantities are determined locally, on
the small cycle, independent of the large-scale geometry of the Calabi-Yau.
Notice that to make this scaling argument, we do not need to be near the
boundary of the Kähler cone in the absolute sense, with $\tau\ll 1$; indeed,
we want $\tau=\alpha^{-1}\gg 1$, to reproduce the small Standard Model gauge
couplings. However, we are working near the boundary of the Kähler cone in the
relative sense that $\tau$ is small relative to the size of the overall
Calabi-Yau volume. The twist that we saw in (5.2.4), where the axion string
corresponding to the small curve has nontrivial winding not only for the axion
reduced on the small divisor but for another axion, depends on how the local
physics of the collapsing cycle is embedded in the larger Calabi-Yau, via the
intersection numbers between small and large divisors.

It would be interesting to extend our study to a larger range of Calabi-Yau
examples. One direction to pursue would be a detailed understanding of simple
examples with a small number of moduli, near different types of boundaries of
the Kähler cone. Another would be a statistical survey of a much larger family
of Calabi-Yau manifolds, along the lines of [33, 34, 35, 36, 37, 38]. It would
also be interesting to explore Type IIA examples, where axions arise from
gauge fields integrated over odd-dimensional cycles.

###  5.3 Type IIB axion $C_{0}$

Finally, we consider an interesting edge case: the 4d axion that simply
coincides with the 10d axion $C_{0}$ in Type IIB string theory rather than
arising by integrating a higher-form field over a cycle. In this case, the
instantons already exist in 10d. They are simply D$(-1)$-branes or
D-instantons, which have action

$S_{\mathrm{inst}}=\frac{2\text{p}}{g_{s}}.$ (125)

Axion strings come from D7 branes wrapped on all six internal dimensions, with
4d tension

${\cal T}_{(2)}={\cal T}_{(D7)}\mathrm{Vol}(Y)=\frac{2\text{p}{\cal
V}}{g_{s}\ell_{s}^{2}}.$ (126)

Finally, dimensional reduction of the 10d kinetic term for $C_{0}$ gives

$f^{2}=\frac{1}{2\text{p}\ell_{s}^{8}}\mathrm{Vol}(Y)=\frac{{\cal
V}}{2\text{p}\ell_{s}^{2}}.$ (127)

We see, once again, that in this example ${\cal
T}_{(2)}=2\text{p}S_{\mathrm{inst}}f^{2}$.

The naive axion electric and magnetic WGC formulas also hold in this case:

$fS_{\mathrm{inst}}=\frac{1}{2\text{p}f}{\cal
T}_{(2)}=\frac{1}{\sqrt{2}}M_{\mathrm{Pl}}.$ (128)

Finally, as in the examples above, we can compare the axion decay constant to
the string scale. We have

$f=\frac{\cal V}{(2\text{p})^{3/2}}M_{s}.$ (129)

For our calculations to be under control we should have ${\cal V}\gg 1$, so in
this case the string scale is expected to be at or below the scale $f$.

This case is less directly phenomenologically relevant than the other examples
above. The axion $C_{0}$ couples to gauge fields that live on D3 branes.
However, D3s in the bulk lack chiral matter. To obtain a Standard Model-like
theory with chiral matter, we can place D3 branes at a singularity, but then
we must consider additional moduli localized at the singularity to obtain a
complete picture of the physics.

##  6 Comments on chiral fermions and axions

In this section, we comment on two aspects of axion physics related to chiral
fermions, based on expectations from string theory. First, we argue that
axions are unlikely to acquire a large tree-level mass in theories with chiral
charged fermions. Second, we comment on the expected size of derivative
couplings of an extra-dimensional axion to fermions, which differs from the
size expected in 4d models of either KSVZ or DFSZ type. The remarks in §6.1.1
are original, whereas most of the remainder of this section gives my viewpoint
on results that have been discussed elsewhere.

###  6.1 Tree-level masses, revisited

In §2.2.3 we briefly discussed two possible tree-level masses for axions: a
$\theta F^{(4)}$ term or monodromy mass, and a Stueckelberg mass
$|\mathrm{d}\theta-qA|^{2}$ in which the axion is eaten by an ordinary gauge
field. We will now return to these two possibilities in more specific
contexts. In the monodromy case, we will argue in specific examples that a
$\theta F^{(4)}$ mass term cannot be turned on for an axion that interacts
with a chiral gauge theory like the Standard Model. In the Stueckelberg case,
we point out that typically the large axion mass arises in the context of the
4d Green-Schwarz mechanism, where there are two axions. One linear combination
is eaten, but another remains light. These considerations strengthen the
argument for the existence of a light axion coupled to the Standard Model in
the extra-dimensional scenario.

####  6.1.1 Monodromy mass versus chiral fermions

In Type IIA string theory, a Standard Model-like theory with chiral fermions
can be constructed on intersecting D6 branes (see, e.g., [127, 128] for
reviews). The D6 branes are wrapped on 3-cycles to produce a 4d gauge theory.
The RR form $C^{(3)}$ reduced on a 3-cycle becomes a 4d axion coupling to the
gauge fields on branes wrapping that cycle, and axion strings are D4 branes
wrapped on an intersecting 3-cycle.

In this particular framework, at first glance the axion quality problem
appears to be endangered by the possibility of a tree-level monodromy
mass.777This was pointed out in a footnote in section 7.1 of [129]. The danger
is a 10d Chern-Simons coupling of the form

$\frac{1}{8\text{p}^{2}}\int C^{(3)}\wedge\mathrm{d}C^{(3)}\wedge H^{(3)},$
(130)

where $H^{(3)}=\mathrm{d}B^{(2)}$ is the field strength of the $B$-field under
which the fundamental string is charged. If we integrate the first $C^{(3)}$
over a 3-cycle $\alpha$ to obtain a 4d axion $\theta$, and we turn on a flux
$\frac{1}{2\text{p}}\int_{\beta}H^{(3)}=n\neq 0$ over an intersecting 3-cycle
$\beta$, we obtain an effective 4d monodromy mass
$\frac{n}{2\text{p}}\int\theta\,\mathrm{d}C^{(3)}$.

However, there is an obstruction to turning on the requisite $H^{(3)}$ flux in
chiral gauge theories (like the Standard Model). In this scenario, chiral
charged fermions arise at the intersection of the cycles $\alpha$ and $\beta$,
when additional D6 branes are placed on $\beta$. The gauge field $A^{(1)}$
living on a D-brane has a Stueckelberg coupling to the bulk field $B^{(2)}$,
i.e., its kinetic term has the form

$-\frac{1}{2e^{2}}(F^{(2)}-B^{(2)})\wedge{\star_{\rm D6}{(F^{(2)}-B^{(2)})}}.$
(131)

If we dualize the field $A^{(1)}$ on a D6 brane, we obtain a magnetic dual
gauge field $A_{\textsc{M}}^{(4)}$ with

$\frac{1}{2\text{p}}\mathrm{d}A_{\textsc{M}}^{(4)}=\frac{1}{e^{2}}{\star_{\rm
D6}{\left(F^{(2)}-B^{(2)}\right)}},$ (132)

together with a Chern-Simons term of the form

$\frac{1}{2\text{p}}\int A_{\textsc{M}}^{(4)}\wedge H^{(3)},$ (133)

to reproduce the physics of the Stueckelberg interaction. Consequently, if we
turn on a flux $H^{(3)}$ on $\beta$ and simultaneously wrap a D6 brane on
$\beta$, we obtain a tadpole sourcing the D6β-localized field
$A_{\textsc{M}}^{(4)}$ in the four noncompact dimensions. This is
inconsistent, provided that we do not also introduce further ingredients to
cancel the tadpole, but we expect that doing so would lead to a brane
configuration that somehow reconnects and removes the chiral fermions from the
spectrum. This is the simple version of a story that is more subtle in the
presence of torsion classes [130], and is also similar to physics discussed in
[131, 132, 133].

Let us be slightly more careful about the details. In particular, what if we
turn on $H^{(3)}$ flux on a cycle $\beta$ that intersects $\alpha$ but we
obtain chiral fermions from the intersection of $\alpha$ with a different
cycle $\gamma$? We can work with a symplectic basis of 3-cycles on the Calabi-
Yau, that is, a set of $\alpha_{i}$ and $\beta_{i}$ (with
$i\in\\{0,\ldots,h^{2,1}\\}$) with intersection numbers

$\alpha_{i}\cap\alpha_{j}=0,\quad\beta_{i}\cap\beta_{j}=0,\quad\alpha_{i}\cap\beta_{j}=\delta_{ij}.$
(134)

Without loss of generality, we can choose this basis so that our initial D6α
branes wrap an integer multiple of the cycle $\alpha_{1}$, and our axion
$\theta^{1}$ is obtained by integrating $C^{(3)}$ over this cycle. Now, it is
clear that in order to give $\theta^{1}$ a monodromy mass, we must have