A Symmetry-centric Perspective on the Geometry
of the String Landscape and the Swampland

Tom Rudelius†

<EMAIL_ADDRESS>

†Department of Mathematical Sciences, Durham University,
Upper Mountjoy Campus, Stockton Rd, Durham DH1 3LE UK

As famously observed by Ooguri and Vafa nearly twenty years ago, scalar field
moduli spaces in quantum gravity appear to exhibit various universal features.
For instance, they seem to be infinite in diameter, have trivial fundamental
group, and feature towers of massive particles that become light in their
asymptotic limits. In this essay, we explain how these features can be
reformulated in more modern language using generalized notions of global
symmetries. Such symmetries are ubiquitous in non-gravitational quantum field
theories, but it is widely believed that they must be either gauged or broken
in quantum gravity. In what follows, we will see that the observations of
Ooguri and Vafa can be understood as consequences of such gauging or breaking.

_Essay written for the Gravity Research Foundation 2024 Awards for Essays on
Gravitation._

## Introduction

Scalar fields play a vital role in the physics of our universe. The most
famous example of this is the Higgs boson, but scalar fields also feature
prominently in models of dark matter, dark energy, and inflation, and in
resolving the strong-CP problem. Scalar fields also lie at the heart of
naturalness puzzles such as the hierarchy problem and the cosmological
constant problem, and thus they may offer a unique window into the inner
workings of the ever-mysterious theory of quantum gravity.

Nearly twenty years ago, the quest to understand the effects of quantum
gravity on scalar field theories led Ooguri and Vafa to publish their seminal
work, “On the Geometry of the String Landscape and the Swampland” [1], which
proposed and provided evidence for five conjectures about scalar fields in
quantum gravity. A counterexample to one of the conjectures was discovered
shortly thereafter [2], but the other four conjectures remain conjectures:
none of them have been proven, yet no counterexamples have been given, either.

At present, most of the evidence for the Ooguri-Vafa conjectures comes from
string theory, which offers a vast landscape of UV-complete scalar field
theories coupled to Einstein gravity. One issue with this line of evidence,
however, is that most of our knowledge of scalar fields in string theory comes
from studies of exactly massless scalar fields, also known as moduli, in
theories with unbroken supersymmetry. As a result, it is unclear whether the
apparently universal features of these theories will persist once
supersymmetry is broken, as it is in our universe.

In this essay, we explain how these conjectures can be reformulated in the
language of generalized global symmetries. Such symmetries make sense even in
non-supersymmetric quantum field theories, so these reformulations help lend
credence to the Ooguri-Vafa conjectures beyond the supersymmetric lamppost
and, hopefully, will ultimately lead us toward their proofs.

In a first course on quantum field theory, students learn about “ordinary”
global symmetries, which transform local operators by elements of a group $G$
in such a way as to keep the Lagrangian invariant. In recent years, however,
it has become clear that such symmetries are just an example of a much broader
zoo of generalized notions of symmetry: high-form symmetries [3] involve
symmetry transformations that act not on local operators of the theory, but
rather on extended operators (line operators, surface operators, etc.).
Higher-group symmetries [4, 5, 6] involve two or more higher-form symmetries
that intertwine into a larger structure. Non-invertible symmetries involve
transformations that are labeled not by group elements, but by elements of a
more general category. Many of the features of ordinary symmetries—charged
operators, symmetry generators, anomalies, gauging, and so on—carry over to
these generalized notions of symmetry as well.

It is widely believed that all of these different types of global symmetries
are forbidden in a consistent theory of quantum gravity [7, 8, 9, 10, 11, 12,
13, 14, 15, 16]. However, such symmetries appear readily in quantum field
theories, which means that any symmetry must be either broken or gauged upon
coupling it to gravity. In what follows, we will discuss each of the Ooguri-
Vafa conjectures in turn, and we will see how they may be viewed as
consequences of gauging or breaking generalized global symmetries.

## Conjecture 0 and lower-form symmetries

The first conjecture of [1], entitled “Conjecture 0,” reads as follows:

###### Conjecture 0.

Let $\mathcal{M}$ be the moduli space of a quantum gravity theory in $d\geq
4$. $\mathcal{M}$ is parameterized by inequivalent expectation values of
massless scalar fields.

In other words, Conjecture 0 forbids the existence of (continuous) free
parameters in quantum gravity. Coupling constants are determined by vacuum
expectation values of scalar fields; they cannot be tuned to generate distinct
theories of quantum gravity the way the parameters of the standard model can
be tuned to generate different quantum field theories.

Conjecture 0 predates even [1] (hence why it was called Conjecture 0 rather
than Conjecture 1), but recent progress permits an alternative understanding
of Conjecture 0 in terms of generalized global symmetries.

A $p$-form global symmetry in a quantum field theory is a global symmetry for
which the charged operators are $p$-dimensional objects. For instance, Wilson
line operators may be charged under a 1-form global symmetry, while local
operators may be charged under a 0-form global symmetry. Such a symmetry is
labeled by a group $G$, and when $G$ is continuous, the theory may be coupled
to a background $(p+1)$-form gauge field. Gauging the symmetry then amounts to
promoting this background gauge field to a dynamical gauge field by adding a
kinetic term for the field to the Lagrangian and integrating over it in the
path integral.

With this, parameters in quantum field theory can be viewed as $(-1)$-form
global symmetries [17, 18, 19]: they may be coupled to background scalar
fields, and varying these background fields leads to different values of the
parameters. More precisely, a compact, circle-valued background scalar field
leads to a $U(1)$ $(-1)$-form global symmetry, while a noncompact scalar field
leads to an $\mathbb{R}$ $(-1)$-form global symmetry. In this language,
Conjecture 0 amounts to the statement that all continuous $(-1)$-form
symmetries in quantum gravity must be gauge symmetries.

Conjecture 0 can thus be understood as a special case of the more general
conjecture that all symmetries in quantum gravity must be either gauged or
broken. At present, however, the arguments against global symmetries in
quantum gravity deal primarily with ordinary global symmetries and, to a
lesser extent, higher-form global symmetries [13]. Extending these arguments
to the case of $(-1)$-form symmetries remains a task for future study.111See
however [20] for progress in this direction.

## Conjecture 4 and non-invertible higher-form symmetries

###### Conjecture 4.

There is no non-trivial 1-cycle with minimum length within a given homotopy
class in $\mathcal{M}$.

In [1], Ooguri and Vafa argued that a nontrivial fundamental group
$\pi_{1}(\mathcal{M})$ would lead to global symmetry upon compactifying to 1+1
dimensions. If global symmetries are not allowed in 2d theories of quantum
gravity, then this scenario should not be allowed, either.

From a modern perspective, we can recast this argument from Ooguri and Vafa in
terms of $(d-2)$-form symmetries in $d$ dimensions, removing the need to
compactify to 2d. Suppose that our theory features an axion $\theta$ with
periodic identification $\theta\sim\theta+2\pi$ and action

$S=\int\textrm{d}^{d}x\sqrt{-\mathfrak{g}_{d}}\left[-\frac{f^{2}}{2}(\partial_{\mu}\theta)^{2}\right]\,,$
(1)

where $f$ is a constant known as the axion decay constant. The periodicity of
$\theta$ implies that the moduli space $\mathcal{M}$ is isomorphic to $S^{1}$,
which has $\pi_{1}(S^{1})\cong\mathbb{Z}$, and the fact that $f$ is constant
means that there is no way to contract a nontrivial loop in
$\pi_{1}(\mathcal{M})$ to a point, in violation of Conjecture 4.

Relatedly, this theory features a family of codimension-2 surface operators,
each labeled by an integer $n$, such that $\theta$ winds by $2\pi n$ as one
winds around the surface operator in the transverse space. These operators are
charged under a $(d-2)$-form symmetry [3]. Indeed, adopting the perspective of
the previous section, we may think of the axion as a 0-form gauge field for a
$(-1)$-form gauge symmetry, and we may think of these surface operators as ’t
Hooft surfaces for this gauge field, which are charged under a magnetic
$(d-2)$-form symmetry with current $(\star j)_{\mu}=\partial_{\mu}\theta$.

More generally, a sigma model with target space $\mathcal{M}$ will have
codimension-2 surface operators labeled by conjugacy classes of
$\pi_{1}(\mathcal{M})$ [21]. These operators are charged under a $(d-2)$-form
symmetry whose elements are labeled by representations of
$\pi_{1}(\mathcal{M})$. If $\pi_{1}(\mathcal{M})$ is abelian, then the
symmetry group is simply the Pontryagin dual of $\pi_{1}(\mathcal{M})$, and it
is an abelian group. If $\pi_{1}(\mathcal{M})$ is non-abelian, however, then
$(d-2)$-form symmetry in question is known as a non-invertible symmetry, as
its elements do not fuse according to a simple group multiplication law
($g*h=gh$), but rather according to the tensor product of representations of
$\pi_{1}(\mathcal{M})$ ($\rho_{i}\times\rho_{j}=\sum_{k}C_{ij}^{k}\rho_{k}$).

To break this symmetry, we must (a) promote the decay constant $f$ to a
dynamical field (as also required by Conjecture 0 above), and (b) demand that
this decay constant vanishes at some point in the closure of field space,
$\overline{\mathcal{M}}$. The simplest (and most common) way to do this is to
complete the axion into some version of the abelian Higgs model [22, 23, 24,
25]:

$S=\int\textrm{d}^{d}x\sqrt{-\mathfrak{g}_{d}}\left[-\frac{1}{2}(\partial_{\mu}\phi^{\dagger})(\partial^{\mu}\phi)-\frac{\lambda}{4}\left(|\phi|^{2}-\frac{\mu^{2}}{\lambda}\right)^{2}\right]\,.$
(2)

At low energies, $\theta\equiv\arg(\phi)$ behaves like an axion with decay
constant $f=\langle|\phi|\rangle=\mu/\sqrt{\lambda}$, and there is an emergent
$(d-2)$-form symmetry. However, at higher energies, this symmetry is broken,
as winding number is not a conserved quantity at the origin $\phi=0$. In this
model, the point $\phi=0$ lies at finite distance and is part of the moduli
space, so $\pi_{1}(\mathcal{M})$ is trivial, there is no $(d-2)$-form
symmetry, and Conjecture 4 is satisfied.

However, within quantum gravity there exists another type of axion, which does
not arise as the angular mode of a complex scalar field. Such an axion is
called a “fundamental axion” [26], and its decay parameter $f$ vanishes only
at an infinite-distance boundary of the moduli space.. As a result, winding a
fundamental axion around its fundamental domain yields a nontrivial element of
$\pi_{1}(\mathcal{M})$, and breaking the associated $(d-2)$-symmetry requires
the existence of dynamical, finite-energy axion vortices, which (like the
aforementioned surface operators) induce monodromy of the axion field as one
winds around their core [27, 28]. One example of this is the D7-brane of Type
IIB string theory, which induces a monodromy of the Type IIB axion $C_{0}$
upon winding around the core of the D7-brane.

As in the case of the abelian Higgs model, the decay parameter $f$ of a
fundamental axion is controlled by the vev of a “saxion” scalar field:

$S=-\frac{1}{2}\int
d^{d}x\sqrt{-\mathfrak{g}_{d}}\left[g(f)(\partial_{\mu}f)^{2}+f^{2}(\partial_{\mu}\theta)^{2}\right]\,,$
(3)

where we have assumed that the metric on moduli space $g_{ff}=g(f)$ does not
depend on $\theta$, and we may further assume that the spacetime metric
$\mathfrak{g}_{d}$ is flat.222Adding an Einstein-Hilbert term
$S_{EH}=-\frac{1}{2}\int d^{d}x\sqrt{-\mathfrak{g}_{d}}\mathcal{R}_{d}$ and
promoting gravity to a dynamical field, the Einstein equations are solved by
the metric
$ds_{d}^{2}=\eta^{(d-2)}_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2\Omega(r)}(dr^{2}+r^{2}d\varphi^{2})$
for a suitable choice of the warp factor $\Omega(r)$. The inclusion of this
warp factor has no effect on our results in this section, as we explain
further in an appendix.

By requiring that the gradient energy of the vortex solution remains finite as
$\theta$ winds around its fundamental domain, we see that the parameter $f$
must vanish at the core of the string, ensuring that Conjecture 4 is
satisfied. Thus, we see that Conjecture 4 is intimately connected to the
presence of axion vortex solutions, which is in turn related to the absence of
$(d-2)$-form symmetries in quantum gravity.

In fact, the requirement of finite-energy vortex solutions imposes even
tighter constraints on the form of the effective action (3). Assuming a
spherically symmetric ansatz $f=f(r)$, $\theta=\varphi$ near the core of the
vortex,333Note that in the example of the D7-brane, this spherical symmetry
eventually breaks down some distance from the core of the string, but it is
valid near the core [29, 30]. where $(r,\varphi)$ are the polar coordinates of
the transverse plane, the requirements of (a) infinite-distance and (b) finite
energy imply $h(r)\equiv\sqrt{g(f(r))}f^{\prime}(r)\sim r^{-1}|\log
r/r_{0}|^{-\gamma}$, where $\frac{1}{2}<\gamma\leq 1$. Further imposing the
equations of motion implies $f(r)=rh(r)$ up to a constant of integration. In
terms of the proper field distance $\Delta$, this implies that the decay
constant $f$ vanishes in the limit $r\rightarrow 0$ as

$f\sim\left\\{\begin{array}[]{cc}\Delta^{\frac{-\gamma}{1-\gamma}}&\frac{1}{2}<\gamma<1\\\
\exp(-\alpha\Delta)&\gamma=1\end{array}\right.,$ (4)

for some constant $\alpha$.444The $\gamma=1$ case is common in string
theory—for instance, the D7-brane in Type IIB string theory has $\gamma=1$,
$\alpha=\sqrt{2}$. In other words, $f$ must vanish more quickly than
$1/\Delta$, and it can vanish no more quickly than an exponential in the
proper field distance $\Delta$.

The upshot of this is that the effective action for the saxion-axion system is
constrained by the requirement that dynamical vortex solutions exist; these
vortices are in turn required in order to break the $(d-2)$-form symmetry
associated with a nontrivial $\pi_{1}(\mathcal{M})$. The fact that $f$
vanishes near the core of the string ensures that Conjecture 4 is satisfied,
and it often plays an important role in satisfying Conjecture 3, as we now
explain.

## Conjecture 3 and axion vortices

###### Conjecture 3.

The scalar curvature near the points at infinity is non-positive. (It is
strictly negative if the dimension of the moduli space is greater than 1.)

Conjecture 3 is the one Ooguri-Vafa conjecture which has been falsified: in
[2], Trenner and Wilson found examples of Calabi-Yau manifolds whose scalar
curvature is positive in certain asymptotic limits. More recently, it was
argued that the positive curvature in these examples is related to the
existence of a light sector that decouples from gravity in the infinite-
distance limit [31]. Nonetheless, Conjecture 3 is valid in many examples of
moduli spaces in string theory where such a decoupled sector does not exist,
and thus it points us toward common, if not entirely universal, features of
quantum gravity.

Infinite-distance limits in quantum gravity typically feature a flat slice of
moduli space parametrized by one or more noncompact scalar fields. In
addition, they have a number of compact scalar fields, or axions, whose decay
parameters depend on the noncompact scalar fields. These compact directions in
moduli space are responsible for the negative curvature of the moduli space.
In the simplest case of a single axion coupled to a noncompact saxion scalar
field, this negative curvature can be seen from the analysis above.

In particular, consider once again the action (3). In terms of the canonically
normalized saxion field $\rho$, the metric on moduli space is given by
$g_{ij}=\text{diag}(1,f(\rho)^{2})$, and the scalar curvature for this metric
is given by $R=-2{f^{\prime\prime}(\rho)}/{f}$. In the infinite-distance limit
at the core of the axion vortex, we have $\rho\rightarrow\infty$,
$f\rightarrow 0$, $f>0$. Setting $\Delta=\rho$ in (21), we see that
$f^{\prime\prime}(\rho)>0$ in this limit, so indeed $R<0$.

Thus, at least in this toy model, the presence of axion strings is closely
related to the negative curvature of the moduli space in the asymptotic limit.
More generally, it has been conjectured that every infinite-distance limit of
a 4d effective field theory is realized at the core of some axion string [32].
This offers a possible explanation as to why negative curvature is so
prevalent in quantum gravity moduli spaces, though again we stress that it is
not quite universal.

Finally, let us remark that—as discussed in [1]—the negative curvature in
asymptotic limits of moduli space is closely related to the finiteness of
their volume.555In [33], Vafa noted that moduli spaces in quantum gravity of
dimension greater than one seem to have finite volume. In the example above,
the volume of moduli space in the infinite-distance limit is given by the
integral $\text{vol}(\mathcal{M})=\int d\omega=\int d\rho d\theta f(\rho)$,
where $\omega$ is the volume form on moduli space. Plugging in the scaling of
$f(\rho)$ from (21), we see that the lower bound $\gamma>1/2$ is precisely the
condition that the volume of the $\rho$-$\theta$ moduli space remains finite
in the limit $\rho\rightarrow\infty$.

## Conjecture 1 and Dynamical Cobordisms

###### Conjecture 1.

Choose any point $p_{0}\in\mathcal{M}$. For any positive $T$, there is another
point $p\in\mathcal{M}$ such that $||p-p_{0}||>T$.

In other words, Conjecture 1 holds that the diameter of moduli space is
infinite.

In the axion vortex example discussed above, the infinite diameter of moduli
space was related to the presence of dynamical axion vortices. These vortices
are responsible for breaking a (possibly non-invertible) higher-form symmetry.
In modern language, they ensure that the codimension-2 operators charged under
the would-be symmetry can end on operators of codimension-3, which implies
that the winding number around these codimension-2 operators is no longer a
valid conserved charge.

This represents just one example of a more general lesson: breaking a higher-
form global symmetries can be understood as the statement that certain
extended operators of dimension $p$ can end on operators of dimension $p-1$.
Mathematically speaking, this “endability” is related to the notion of
cobordism. Two manifolds $M$, $N$ of dimension $d$ are said to be cobordant if
there exists a $(d+1)$-dimensional manifold whose boundary is the disjoint
union $M\sqcup N$. This defines an equivalence relation between the manifolds
$M$ and $N$, and the equivalence classes can be imbued with a group structure,
with disjoint union as the group multiplication law.

Within quantum field theory, the statement that an extended operator can end
on a lower-dimension operator is tantamount to the statement that any operator
which links it must be trivial in cobordism, and it in turn implies that that
the endable operator cannot be charged under a global symmetry. As a result,
the statement that there are no global higher-form symmetries in quantum
gravity is more or less equivalent to the statement that all cobordism groups
in quantum gravity are trivial [34].

With this, the axion vortex solutions discussed above are just one example of
a more general class of “dynamical cobordism defects,” [35, 36, 37, 38, 39]
which (a) trivialize cobordism groups by allowing certain extended operators
to end, and (b) involve a scalar field which runs to infinite distance at
their core. As explained in [37], D-branes in string theory can be understood
as examples of such dynamical cobordism defects.

These cobordism defects suggest a close relationship between infinite-distance
limits in scalar field space and the absence of nontrivial cobordism groups.
Making this relationship precise is tricky because there are also examples of
cobordism defects in string theory that do not involve infinite-distance
limits in field space—most famously, the Horava-Witten wall can be understood
as a cobordism defect of M-theory [34], even though M-theory has no massless
moduli at all. By clarifying the conditions under which infinite-distance
limits are required for the existence of cobordism defects, one may be able to
put Conjecture 1 on firmer footing.

## Conjecture 2 and approximate global symmetries

###### Conjecture 2.

Compared to the theory at $p_{0}\in\mathcal{M}$, the theory at $p$ with
$||p-p_{0}||>T$ has an infinite tower of light particles starting with mass of
the order of ${\mathrm{e}}^{-\alpha T}$ for some $\alpha>0$ In the
$T\rightarrow\infty$ limit, the number of extra light particles of mass less
than a fixed mass scale becomes infinite.

Also known as the “Distance Conjecture,” this conjecture may be related to the
breaking of approximate global symmetries in quantum gravity.

A free, massless scalar field $\phi$ has an exact shift symmetry. Coupling
this scalar field to a fermion $\psi$ with a Yukawa-like coupling $\lambda
m_{0}(\phi)\bar{\psi}\psi$, this symmetry is restored in the limit
$\lambda\rightarrow 0$. Thus, for small $\lambda$, we have an approximate
global symmetry.

Naively, this suggests that a weakly coupled scalar field theory necessarily
leads to an approximate symmetry. However, this approximate symmetry will be
badly broken provided there are a sufficient number of light fields coupled
weakly to the scalar, as these run in loops and drive the scalar field toward
strong coupling in the UV. As shown in [40] (based on earlier calculations in
[41, 42, 43]), the requirement that the symmetry is badly broken within the
low-energy EFT is equivalent (up to order-one factors) to the statement that
the fermion masses $\langle m_{0}(\phi)\rangle$ decay exponentially with
proper field distance, .

At present, this connection between Conjecture 2 and the breaking of
approximate global symmetries is suggestive but rough: there is no precise
argument against approximate global symmetries in quantum gravity, and it is
not clear how to extend the argument of [40] to towers of string oscillation
modes, which also decay exponentially in infinite-distance limits but are not
well-described by low-energy EFT. Making these arguments more precise would be
an important step toward a proof of Conjecture 2.

.

## Conclusions

In this essay, we have seen that generalized notions of global symmetries are
closely related to all of the Ooguri-Vafa conjectures. The requirement that
these symmetries must be gauged or broken motivates many observed features of
scalar field theories in quantum gravity, and it helps to shed light on these
theories beyond the supersymmetric lamppost of string theory.

Yet at the same time, many open questions remain. What limitations does
quantum gravity place on approximate global symmetries, and why? What can
statements about asymptotic regimes in scalar field spaces tell us about our
own universe, which likely lives somewhere in the interior? Can holography,
quantum information theory, and pure mathematics help shed further light on
scalar fields in quantum gravity? At the heart of these questions lie answers
to some of the most fundamental mysteries of our universe.

## Acknowledgements

It is a pleasure to thank Iñaki García-Etxebarria, Ben Heidenreich, Jacob
McNamara, Matthew Reece, Mykhaylo Usatyuk, and Timo Weigand for useful
discussions. This work was supported in part by STFC through grant
ST/T000708/1.

##  Appendix A Axion vortices

In this appendix, we include additional details of the calculations on axion
vortices in the section on Conjecture 4. We begin with the action of (3), plus
an additional Einstein-Hilbert term:

$S=-\frac{1}{2}\int
d^{d}x\sqrt{-\mathfrak{g}_{d}}\left[-\mathcal{R}_{d}+g(f)(\partial_{\mu}f)^{2}+f^{2}(\partial_{\mu}\theta)^{2}\right]\,.$
(5)

Once again, we have assumed that the metric on moduli space $g_{ff}=g(f)$ does
not depend on $\theta$. We take a metric ansatz of the form,

$ds_{d}^{2}=ds_{d-2}^{2}+e^{2\Omega(r)}(dr^{2}+r^{2}d\varphi^{2}),$ (6)

where $\Omega(r)$ is the warp factor and
$ds_{d-2}^{2}=\eta_{\mu\nu}^{(d-2)}dx^{\mu}dx^{\nu}$ is taken to be flat,
$(d-2)$-dimensional Minkowski space. With this, the action (5) becomes

$S=-\frac{1}{2}\int d^{d-2}xdrd\varphi
re^{2\Omega(r)}\left[-\mathcal{R}_{d}+g(f)(\partial_{\mu}f)^{2}+f^{2}(\partial_{\mu}\theta)^{2}\right]\,,$
(7)

We next take a spherically symmetric ansatz near the core of the string of the
form

$f=f(r)\,,~{}~{}~{}\theta=\varphi\,,$ (8)

where $(r,\varphi)$ are the polar coordinates of the transverse plane. We then
insist that there exists a vortex solution with nontrivial winding and finite
energy per unit length. The energy per unit length from the gradient of the
scalar fields is given by

$\displaystyle\mathcal{E}=\frac{1}{2}\int
rdrd\varphi\left[g(f)f^{\prime}(r)^{2}+\frac{f(r)^{2}}{(2\pi
r)^{2}}\right]\,.$ (9)

Notably, the warp factor does not appear in this expression, as the factor of
$\exp(2\Omega(r))$ from the square root of the determinant has canceled the
factor of $\exp(-2\Omega(r))$ from the inverse metric in the kinetic terms.
The requirement that $\mathcal{E}$ is finite implies that both terms in the
integrand must integrate to a finite value. However, we have also assumed that
the limit $f\rightarrow 0$ is at infinite distance in field space, which means
that

$\displaystyle||\Delta f||=\int dr\sqrt{g(f)}f^{\prime}(r)$ (10)

must diverge. Together, these constraints are not easy to satisfy! Setting
$h(r)=\sqrt{g(f(r))}f^{\prime}(r)$, we have

$\displaystyle\int_{0}^{r_{0}}drh(r)\rightarrow\infty\,,~{}~{}~{}\int_{0}^{r_{0}}rdrh(r)^{2}=\text{finite}\,.$
(11)

It is easy to see that $h(r)$ cannot satisfy these constraints if it simply
scales like a power law near $r=0$; instead, we must have

$\displaystyle h(r)\sim\frac{1}{r|\log r/r_{0}|^{\gamma}}\,,$ (12)

where $\frac{1}{2}<\gamma\leq 1$.

This already fixes the scaling behavior of the canonically-normalized saxion
field, but we can go even further by imposing the equation of motions for the
fields $f$ and $\theta$:

$\displaystyle\partial^{\mu}(rg\partial_{\mu}f)=\frac{1}{2}rg^{\prime}(f)(\partial_{\mu}f)^{2}+fr(\partial_{\mu}\theta)^{2}$
(13) $\displaystyle\partial^{\mu}rf^{2}\partial_{\mu}\theta=0\,.$ (14)

Once again, powers of the warp factor $\exp(\Omega(r))$ have canceled out in
these equations. Assuming the spherically symmetric ansatz (8), the $\theta$
equation of motion is satisfied trivially, while the equation of motion for
$f$ becomes

$g^{\prime}(f)f^{\prime}(r)^{2}+gf^{\prime\prime}(r)+\frac{1}{r}gf^{\prime}(r)=\frac{1}{2}g^{\prime}(f)f^{\prime}(r)^{2}+\frac{f}{r^{2}}\,.$
(15)

Plugging in $h(r)=\sqrt{g}f^{\prime}(r)$, we can massage this equation into
the simple form

$\frac{\partial}{\partial r}(h^{2}r^{2})=\frac{\partial}{\partial
r}(f^{2})\,.$ (16)

So,

$h^{2}r^{2}=f^{2}\,,$ (17)

up to a constant of integration. Using the form of $h(r)$ in (12), we conclude
that

$f(r)\sim|\log r/r_{0}|^{-\gamma}\,,$ (18)

and therefore

$g(f(r))=\frac{h^{2}}{(f^{\prime}(r))^{2}}\sim|\log
r/r_{0}|^{2}~{}~{}~{}\Rightarrow~{}~{}~{}g(f)\sim f^{-2/\gamma}\,.$ (19)

Thus, the requirement of a spherically-symmetric, finite-energy vortex
solution to the equations of motion near $r=0$ implies that the effective
action takes the form

$S=-\frac{1}{2}\int
d^{d}x\sqrt{-\mathfrak{g}_{d}}\left[\frac{g_{0}}{f^{2/\gamma}}(\partial_{\mu}f)^{2}+f^{2}(\partial_{\mu}\theta)^{2}\right]\,,$
(20)

where $\frac{1}{2}<\gamma\leq 1$.666This same range of possibilities was also
found in [32] in the specific context of BPS strings in 4d supergravity
theories.

This form of the saxion-axion action is ubiquitous in string theory. For
instance, the Type IIB action takes this form [44], with $\gamma=1$,
$\theta=2\pi C_{0}$, $f=(2\sqrt{2}\pi\tau_{2})^{-1}$, and $g_{0}=1/2$. In this
context, the particular choice $g_{0}=1/2$ is required by $SL(2,\mathbb{Z})$
duality.

Finally, let us introduce a canonically normalized saxion field $\rho$ via
$d\rho=-\sqrt{g_{0}}f^{-1/\gamma}df$. With this, the limit $f\rightarrow 0$
corresponds to $\rho\rightarrow\infty$. In terms of the proper field distance
$\Delta=\rho$, the decay constant $f$ therefore vanishes as

$f\sim\left\\{\begin{array}[]{cc}\Delta^{\frac{-\gamma}{1-\gamma}}&\frac{1}{2}<\gamma<1\\\
\exp(-\alpha\Delta)&\gamma=1\end{array}\right.,$ (21)

where $\alpha=\frac{1}{\sqrt{g_{0}}}$. From the limiting cases $\gamma=1/2$
and $\gamma=1$, we find an lower bound and an upper bound on the rate at which
the decay constant $f$ may tend to zero with proper field distance, namely

$f\Delta\rightarrow 0\text{ as
}{\Delta\rightarrow\infty}\,,~{}~{}~{}~{}\exp(-\alpha\Delta)\lesssim f\,.$
(22)

In other words, $f$ must vanish more quickly than $1/\Delta$, and it can
vanish no more quickly than an exponential in the proper field distance
$\Delta$.

Note that the inclusion of gravity had no effect on the calculations above, as
the warp factor did not appear in the gradient energy (9), the field space
distance (10), or the equations of motion (13)-(14). The only significant
effect of including gravity comes from the Einstein field equation
$G_{tt}=T_{tt}$, which fixes the form of the warp factor $\Omega(r)$ near the
core of the vortex $r=0$:

$-e^{-2\Omega}\frac{1}{r}\partial_{r}(r\Omega^{\prime}(r))=e^{-2\Omega}\left[\frac{1}{2}g(f)f^{\prime}(r)^{2}+\frac{f^{2}}{2r^{2}}(\partial_{\varphi}\theta)^{2}\right]\,.$
(23)

Setting $\theta=\varphi$, $h(r)=\sqrt{g}f^{\prime}(r)$, and using the result
$f(r)=rh(r)$ from (17), this becomes

$\frac{1}{r}\partial_{r}(r\Omega^{\prime}(r))=-h(r)^{2}\,.$ (24)

Setting $h(r)\sim\frac{1}{r|\log r/r_{0}|^{\gamma}}$ near $r=0$, we find that
this equation is solved by

$\Omega(r)\sim\left\\{\begin{array}[]{cc}|\log
r/r_{0}|^{2-2\gamma}&\frac{1}{2}<\gamma<1\\\ |\log|\log
r/r_{0}||&\gamma=1\end{array}\right..$ (25)

We see that the warp factor $\Omega(r)$ tends to $+\infty$ as $r\rightarrow
0$, but $\int_{0}^{\epsilon}dr\exp(\Omega(r))$ converges for small
$\epsilon>0$, so the spacetime distance to the vortex core remains finite.

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