ACFI-T22-07

Sharpening the Distance Conjecture
in Diverse Dimensions

Muldrow Etheredgea, Ben Heidenreicha, Sami Kayab, Yue Qiua,

and Tom Rudeliusb

aDepartment of Physics, University of Massachusetts, Amherst, MA 01003 USA
bDepartment of Physics, University of California, Berkeley, CA 94720 USA

The Distance Conjecture holds that any infinite-distance limit in the scalar
field moduli space of a consistent theory of quantum gravity must be
accompanied by a tower of light particles whose masses scale exponentially
with proper field distance $\|\phi\|$ as $m\sim\exp(-\lambda\|\phi\|)$, where
$\lambda$ is order-one in Planck units. While the evidence for this conjecture
is formidable, there is at present no consensus on which values of $\lambda$
are allowed. In this paper, we propose a sharp lower bound for the lightest
tower in a given infinite-distance limit in $d$ dimensions: $\lambda\geq
1/\sqrt{d-2}$. In support of this proposal, we show that (1) it is exactly
preserved under dimensional reduction, (2) it is saturated in many examples of
string/M-theory compactifications, including maximal supergravity in
$d=\text{4 -- 10}$ dimensions, and (3) it is saturated in many examples of
minimal supergravity in $d=\text{4 -- 10}$ dimensions, assuming appropriate
versions of the Weak Gravity Conjecture. We argue that towers with
$\lambda<1/\sqrt{d-2}$ discussed previously in the literature are always
accompanied by even lighter towers with $\lambda\geq 1/\sqrt{d-2}$, thereby
satisfying our proposed bound. We discuss connections with and implications
for the Emergent String Conjecture, the Scalar Weak Gravity Conjecture, the
Repulsive Force Conjecture, large-field inflation, and scalar field potentials
in quantum gravity. In particular, we argue that if our proposed bound applies
beyond massless moduli spaces to scalar fields with potentials, then
accelerated cosmological expansion cannot occur in asymptotic regimes of
scalar field space in quantum gravity.

###### Contents

  1. 1 Introduction
  2. 2 Dimensional Reduction
    1. 2.1 Winding Modes and Kaluza-Klein Monopoles
    2. 2.2 A Convex Hull Condition
  3. 3 A Reexamination of Supergravity in Four Dimensions
  4. 4 Top-Down Evidence in Maximal Supergravity
    1. 4.1 Ten Dimensions
    2. 4.2 Nine Dimensions
    3. 4.3 Eight Dimensions
    4. 4.4 Seven Dimensions
    5. 4.5 Six, Five, and Four Dimensions
  5. 5 Bottom-Up Evidence in Minimal Supergravity
    1. 5.1 Five Dimensions
    2. 5.2 Six Dimensions
    3. 5.3 Seven Dimensions
    4. 5.4 Eight Dimensions
    5. 5.5 Nine Dimensions
  6. 6  Discussion
    1. 6.1 Scalar Field Potentials
    2. 6.2 Large-Field Inflation
    3. 6.3 Axions and a Tower Scalar Weak Gravity Conjecture
    4. 6.4 Black Holes and Repulsive Forces
    5. 6.5 The Emergent String Conjecture and an Upper Bound on $\lambda$
  7. A Top Down Evidence in Maximal Supergravity for Six, Five, and Four dimensions
    1. A.1 Symplectic Conventions
    2. A.2 U-Duality Results
    3. A.3 Six Dimensions
    4. A.4 Five Dimensions
    5. A.5 Four Dimensions

##  1 Introduction

In recent years, the search for universal features of theories of quantum
gravity has produced an enormous body of research. This search is hindered
significantly by the complicated nature of quantum gravity, and as a result
our understanding of quantum gravity beyond the weakly coupled, supersymmetric
context is very limited.

Within this world of darkness and confusion, the Distance Conjecture shines as
a beacon of clarity and hope. This conjecture, originally referred to as
“Conjecture 2” by Ooguri and Vafa in their seminal work Ooguri:2006in , deals
with asymptotic limits of moduli spaces of quantum gravity theories, which are
parametrized by vacuum expectation values of massless scalar fields. Exactly
massless scalar fields ordinarily require an infinite degree of fine-tuning,
so they are expected to appear only in theories with eight or more
supercharges, where their masses are protected by supersymmetry. Furthermore,
asymptotic limits of moduli spaces represent weak coupling limits of quantum
gravity. This means that the claims of the Distance Conjecture, in its most
conservative formulation, are restricted to the weakly coupled, supersymmetric
regime of quantum gravity, where our understanding is extensive and claims can
be tested with a relatively high level of rigor.111Already in their original
paper Ooguri:2006in , Ooguri and Vafa conjectured that the Distance Conjecture
should apply beyond massless moduli spaces to include scalar fields with
potentials as well. We will discuss scalar fields with potentials further in
§6. So far, the Distance Conjecture has passed all of these tests.

Morally speaking, the Distance Conjecture stipulates the existence of a tower
of exponentially light states in every infinite-distance limit of scalar field
moduli space. A more precise statement is as follows:

###### The Distance Conjecture.

Let $\mathcal{M}$ be the moduli space of a quantum gravity theory in $d\geq 4$
dimensions, parametrized by vacuum expectation values of massless scalar
fields. Compared to the theory at some point $p_{0}\in\mathcal{M}$, the theory
at a point $p\in\mathcal{M}$ has an infinite tower of particles, each with
mass scaling as

$m\sim\exp(-\lambda\|p-p_{0}\|)\,,$ (1)

where $\|p-p_{0}\|$ is the geodesic distance in $\mathcal{M}$ between $p$ and
$p_{0}$, and $\lambda$ is some order-one number in Planck units $(8\pi
G=\kappa_{d}^{2}=1)$.

This conjecture has been confirmed in a vast array of top-down examples in
string theory, and strong bottom-up arguments for its validity have been given
in the context of effective field theory. However, despite this enormous body
of research into the Distance Conjecture, a simple but crucial question
remains: which values of $\lambda$ are allowed? “Order-one” carries a wide
range of possible interpretations, but if $\lambda$ is too small, the
constraints from the Distance Conjecture on low-energy effective field theory
will be very weak. As a result, the key open question facing us is to place a
lower bound on the coefficient $\lambda$ appearing in the definition of the
Distance Conjecture (1).

In this paper, we propose a simple answer to this question: in quantum gravity
in $d$ dimensions, any infinite-distance limit in moduli space features at
least one tower which satisfies the Distance Conjecture with

$\boxed{\lambda\geq\frac{1}{\sqrt{d-2}}\,.}$ (2)

Note that this does not preclude the existence of additional, heavier towers
in this infinite-distance limit with $\lambda<1/\sqrt{d-2}$.

This proposed bound may come as a surprise to Distance Conjecture
connoisseurs, since there are a number of examples of towers in 4d theories
that scale as $m\sim\exp(-\|p-p_{0}\|/\sqrt{6})$, naively satisfying the
Distance Conjecture with $\lambda=1/\sqrt{6}$ yet violating our proposed bound
(2). Indeed, this has led a number of authors to single out
$\lambda_{\textrm{min}}=1/\sqrt{(d-1)(d-2)}$ as the minimal value for
$\lambda$ in $d$ dimensions Grimm:2018ohb ; Andriot:2020lea ; Gendler:2020dfp
; Lanza:2020qmt ; Bedroya:2020rmd ; Lanza:2021qsu . However, in what follows,
we will not only explain why towers with $\lambda=1/\sqrt{(d-1)(d-2)}$ are
ubiquitous in supergravity theories and string compactifications (namely, they
arise upon dimensional reduction as Kaluza-Klein zero modes of towers of
particles in the parent $(d+1)$-dimensional theory), but we will also argue
that such towers are always accompanied in their appropriate limits in moduli
space by even lighter towers (namely, Kaluza-Klein towers) that satisfy the
Distance Conjecture with $\lambda=\sqrt{(d-1)/(d-2)}$ and in turn satisfy our
proposed bound (2). As mentioned above, the bound (2) must therefore be
understood as a bound on the _lightest_ tower of exponentially light states in
a given infinite-distance limit of moduli space, not as a bound on _all_
towers of exponentially light particles in this limit. Indeed, there is no
possible nontrivial lower bound on the coefficient $\lambda$ which applies to
all such exponentially light particles in a given infinite-distance
limit.222For example, consider a theory with two canonically normalized
massless scalar fields $\rho$, $\phi$ and a tower of massive particles whose
masses scale as $m\sim\exp(-\lambda_{\rho}\rho)$. Take $\epsilon>0$ to be an
arbitrarily small positive number. We then find an arbitrarily small
coefficient $\lambda<\epsilon$ by considering the $s\rightarrow\infty$
infinite-distance limit of a geodesic $(\rho(s),\phi(s))=(\delta s,s)$ for
$\delta\equiv\epsilon/(2\lambda_{\rho})$ (i.e., the limit
$\rho,\phi\rightarrow\infty$ with $\rho/\phi\equiv\delta$ fixed). This
scenario readily occurs, for instance, in circle compactification of Type II
string theory to nine dimensions, where $\phi$ is the dilaton and $\rho$ is
the radion. Our bound (2) will nonetheless be satisfied in this scenario
provided there is a second tower whose masses decay at a faster rate in this
limit.

Our proposed bound (2) is closely related to the Scalar Weak Gravity
Conjecture Palti:2017elp ; Lee:2018spm ; Andriot:2020lea , which we define as
follows:333 In its original formulation Palti:2017elp , the Scalar Weak
Gravity Conjecture merely requires the existence of one particle of mass $m$
satisfying the bound
$\frac{g^{ij}\partial_{i}m\partial_{j}m}{m^{2}}\geq\gamma\kappa_{d}^{2}\,,$
(3) where $g^{ij}$ is the inverse metric on moduli space and $\gamma$ is some
order-one coefficient. For $\gamma\geq(d-3)/(d-2)$, this bound implies that
the attractive force mediated by massless scalar fields is greater in
magnitude that the force of gravity. However, there is no compelling argument
as to why the force mediated by massless scalar fields should be greater than
the gravitational force, and this proposal cannot be true in non-
supersymmetric systems like the world we live in.

###### The Scalar Weak Gravity Conjecture.

Given a massless scalar field modulus $\phi$ in a quantum gravity theory in
$d$ spacetime dimensions, there necessarily exists a particle of mass $m$
satisfying

$\frac{(\partial_{\phi}m)^{2}}{g_{\phi\phi}m^{2}}\geq{\lambda_{\text{min}}^{2}}{\kappa_{d}^{2}}\equiv\frac{\kappa_{d}^{2}}{d-2}\,,~{}~{}~{}\text{
with }\partial_{\phi}m<0\,,$ (4)

where $g_{\phi\phi}$ is the $\phi\phi$ component of the metric on scalar
moduli space, so that
$\frac{1}{2}g_{\phi\phi}\textrm{d}\phi\wedge\star\textrm{d}\phi$ is the
kinetic term for $\phi$ in the action.

This is a natural extension of the ordinary Weak Gravity Conjecture
Arkanihamed:2006dz , which holds that given a 1-form gauge field $A$ with
coupling constant $g_{A}$, there must exist a particle of mass $m$ and
quantized charge $q_{A}\in\mathbb{Z}$ satisfying

$\frac{g_{A}^{2}q_{A}^{2}}{m^{2}}\geq\gamma_{d}^{2}\kappa_{d}^{2}\,,$ (5)

where $\gamma_{d}$ is an order-one number fixed by the black hole extremality
bound. Note that the condition $\partial_{\phi}m<0$ is needed to ensure that
the mass of the particle is decreasing as $\phi$ increases, since there is no
analog of charge conjugation for scalar charges $\mu_{\phi}=\partial_{\phi}m$
the way there is for gauge charges.

By varying over all possible 1-form gauge fields $A$ in the theory, one finds
that the Weak Gravity Conjecture as stated in (5) is equivalent to the convex
hull condition of Cheung:2014vva . Similarly, by varying over all massless
scalar fields $\phi$ in the theory, (4) may be viewed as an analog of the
convex hull condition for scalar charges rather than gauge charges, as
proposed previously in Calderon-Infante:2020dhm . More precisely, given a
particle of mass $m$, we define the scalar charge-to-mass vector as

$\zeta_{i}\equiv\frac{1}{\kappa_{d}}\frac{\partial}{\partial\phi^{i}}\log
m\,,$ (6)

where the differentiation is performed with the $d$-dimensional Planck mass
held fixed. The length of a scalar charge-to-mass vector is determined by
contracting with the inverse of the scalar kinetic matrix,
$|\vec{\zeta}|\equiv\sqrt{g^{ij}\zeta_{i}\zeta_{j}}$. With these definitions,
the Scalar Weak Gravity Conjecture defined above is equivalent to the
statement that the convex hull generated by all of the $\vec{\zeta}$-vectors
contains a ball of radius $\lambda_{\textrm{min}}=1/\sqrt{d-2}$ centered at
the origin of the scalar charge-to-mass vector space, as illustrated in Figure
1.

Figure 1: The convex hull condition. The Scalar Weak Gravity Conjecture
requires the convex hull (blue, shaded) generated by the scalar charge-to-mass
vectors $\zeta_{i}$ of the particles of the theory (red dots) to contain a
ball of radius $1/\sqrt{d-2}$ centered at the origin (gray, shaded).

The coefficient $1/(d-2)$ appearing in (4) has been carefully chosen to
coincide with the value of $\lambda_{\textrm{min}}^{2}$ we have proposed in
(2), since for $\phi\rightarrow\infty$, a tower of particles of mass
$m=m_{0}\exp(-\lambda\phi)$ will have
$g^{\phi\phi}(\partial_{\phi}m)^{2}/m^{2}=\lambda^{2}$. In light of this,
there is a simple analogy between the various conjectures highlighted in this
paper: just as the ordinary Weak Gravity Conjecture requires a single particle
satisfying the bound (5) and the tower Weak Gravity Conjecture requires a
whole tower of particles satisfying that bound, so too does the Scalar Weak
Gravity Conjecture require a single particle satisfying the bound (4), while
the Distance Conjecture requires a whole tower of particles satisfying this
latter bound. Our paper could therefore be viewed equally well as a sharpening
of the Distance Conjecture or as a sharpening of the Scalar Weak Gravity
Conjecture.

There is one important difference between the Distance Conjecture and the
Scalar Weak Gravity Conjecture, however: the Scalar Weak Gravity Conjecture is
a bound on the scalar charges with respect to _all_ massless scalars in the
theory, including compact scalar fields known as axions. The Distance
Conjecture, on the other hand, says very little about couplings to compact
scalar fields, since it constrains infinite displacements in moduli space, and
a compact scalar field can have only a finite displacement. Thus, despite
significant overlap between the two conjectures, neither the Distance
Conjecture nor the Scalar Weak Gravity Conjecture implies the other: the
former requires an infinite tower of exponentially light particles while the
latter only requires a finite number, but the latter requires (4) to be
satisfied even if $\phi$ is an axion, while the former does not constrain
couplings to axions. Nonetheless, in what follows, we will see strong evidence
that both conjectures are satisfied in supergravity theories with a
coefficient $\lambda_{\textrm{min}}=1/\sqrt{d-2}$.

We provide several lines of evidence in support of our proposal (2). To begin,
in §2 we argue that the bound is exactly preserved under dimensional
reduction: a simple theory that saturates the Distance Conjecture with
$\lambda\geq 1/\sqrt{D-2}$ in $D$ spacetime dimensions will, after dimensional
reduction, saturate the Distance Conjecture with $\lambda=1/\sqrt{d-2}$ in
$d=D-1$ dimensions. This feature of preservation under dimensional reduction
is not a necessity: there is nothing wrong in principle with a bound that is
saturated in $D$ dimensions yet satisfied comfortably after dimensional
reduction to $d$ dimensions. However, many of the most rigorously tested
quantum gravity conjectures–including the absence of global symmetries
Hawking:1974sw ; Banks:2010zn , the Weak Gravity Conjecture Arkanihamed:2006dz
and the Repulsive Force Conjecture Palti:2017elp ; Heidenreich:2019zkl –are
exactly preserved under dimensional reduction Banks:2010zn ;
Heidenreich:2015nta ; Heidenreich:2019zkl , and a number of more speculative
conjectures may be sharpened by demanding such preservation as well
Rudelius:2021oaz ; Montero:2021otb . The preservation of (2) under dimensional
reduction therefore offers a tantalizing hint that
$\lambda_{\text{min}}=1/\sqrt{d-2}$ is indeed the correct value for a lower
bound on $\lambda$, but this hypothesis merits further testing.

In §3-5, we thus carry out tests of our proposal in supergravity theories and
string/M-theory compactifications to 4 – 10 dimensions. These tests may be
carried out in one of two ways. The “top-down” approach begins with a
particular infinite-distance limit of an explicit string/M-theory
compactification, computes the exponential decay of masses of towers of light
particles with increasing field distance, and compares the coefficient to
$\lambda_{\textrm{min}}=1/\sqrt{d-2}$. This method is difficult for general
Calabi-Yau compactifications, though a number of examples have already been
considered in the four-dimensional context. In §3, we review these 4d examples
and argue that they are consistent with our proposed bound $\lambda\geq
1/\sqrt{d-2}$.

In §4, we carry out further top-down checks of our proposed bound by
considering maximal supergravities in $d=\text{4 -- 10}$ dimensions, which
arise from M-theory compactified on $T^{n}$. Using U-duality Hull:1994ys , we
show that our bound $\lambda\geq 1/\sqrt{d-2}$ is saturated in all of these
examples. We further argue that this bound is saturated in type I and
heterotic string theory in 10 dimensions. In all of these cases where we have
checked, whenever the bound is saturated, there is a string scale with an
associated tower of string oscillator modes saturating the bound.

In §5, we employ a “bottom-up” approach (which was previously used in
Gendler:2020dfp ) to determine the coefficient $\lambda_{\textrm{min}}$ in
minimal supergravity in 5 – 9 dimensions. This approach proceeds by examining
the behavior of the gauge couplings for the 1-form and 2-form gauge fields in
the theory and then invoking appropriate versions of the Weak Gravity
Conjecture. More precisely, the tower Weak Gravity Conjecture
Heidenreich:2015nta ; Heidenreich:2016aqi ; Andriolo:2018lvp  holds that as
the gauge coupling $g_{A}$ for some 1-form gauge field $A_{1}$ tends to zero,
there will be a tower of light states with masses bounded in Planck units as
$m\lesssim g_{A}$. If $g_{A}$ decays exponentially in some asymptotic limit of
moduli space as $g_{A}\sim\exp(-\lambda\|p-p_{0}\|)$, therefore, the tower
Weak Gravity Conjecture will immediately imply that the Distance Conjecture is
satisfied with that same coefficient $\lambda$. Similarly, the Weak Gravity
Conjecture for a 2-form gauge field $B_{2}$ says that as the gauge coupling
$g_{B}$ tends to zero, there will be a charged string whose tension in Planck
units is bounded as $T_{\textrm{string}}\lesssim g_{B}$. If this string is a
fundamental string, meaning that its core probes quantum gravity physics in
the deep ultraviolet Reece:2018zvv  (see also Dolan:2017vmn ), then it will
give rise to a tower of string oscillator modes beginning at the string scale
$M_{\textrm{string}}\equiv\sqrt{2\pi
T_{\textrm{string}}}\lesssim\sqrt{g_{B}}$. If $g_{B}$ decays exponentially in
some asymptotic limit of moduli space as
$g_{B}\sim\exp(-2\lambda\|p-p_{0}\|)$, therefore, the Weak Gravity Conjecture
will immediately imply that the Distance Conjecture is satisfied with the
coefficient $\lambda$.

Using this bottom-up approach, we find that the bound $\lambda\geq
1/\sqrt{d-2}$ is saturated in certain infinite-distance limits in moduli space
in $d=\text{5 -- 9}$ dimensions. All of these limits are emergent string
limits, meaning that some 2-form gauge coupling vanishes, and the Weak Gravity
Conjecture implies a charged string whose oscillator modes satisfy the
Distance Conjecture with $\lambda=1/\sqrt{d-2}$. In contrast, the other
infinite-distance limits we consider in these theories involve a 1-form gauge
field whose gauge coupling vanishes in the limit, and the tower implied by the
tower Weak Gravity Conjecture instead satisfies $\lambda>1/\sqrt{d-2}$ with
room to spare. In fact, in all the cases we encounter, these towers have the
scaling behavior expected of Kaluza-Klein towers. Thus, both our top-down and
bottom-up analyses lend strong support to the Emergent String Conjecture
Lee:2019wij ; Lee:2019xtm , which holds that every infinite-distance limit
must be either an emergent string limit or a decompactification limit, and
they further suggest that only the emergent string limits may saturate our
proposed bound $\lambda\geq 1/\sqrt{d-2}$.

In §6, we conclude our analysis with brief discussion of open questions and
applications of our proposed bound (2). Notably, we point out in §6.1 that the
lower bound $\lambda\geq 1/\sqrt{d-2}$ implies a low UV cutoff on effective
field theory in any infinite-distance limit. If this bound applies to scalar
fields with a potential and not merely massless moduli (as conjectured in
Ooguri:2006in ; Klaewer:2016kiy ), then this low cutoff leads to an upper
bound on scalar potentials in asymptotic limits of scalar field space
Hebecker:2018vxz ; Andriot:2020lea ; Bedroya:2020rmd . Assuming the Emergent
String Conjecture applies to such limits, we show that the resulting bound
forbids accelerated expansion of the universe in asymptotic regions of scalar
field space Obied:2018sgi , which agrees with the strong asymptotic de Sitter
Conjecture of Rudelius:2021oaz ; Rudelius:2021azq  and suggests that
quintessence, like de Sitter, can persist for only a finite period of time in
quantum gravity.444Note that our results do not forbid eternal inflation in
the interior of scalar field space, which may occur even if every de Sitter
vacuum is metastable and every period of quintessence ends after a finite
period of time. For further discussion on this point, see §6.1.

This low cutoff may also present a problem for large-field inflation models
with $|\Delta\phi|\gtrsim 10M_{\textrm{Pl}}$, as we discuss in §6.2. In §6.3,
we comment on an extension to periodic scalar fields, also known as axions,
and in §6.4 we consider implications of our bound for black holes in
supergravity and the Repulsive Force Conjecture Palti:2017elp ;
Heidenreich:2019zkl . In §6.5, we elaborate on connections to the Emergent
String Conjecture and the possibility of an upper bound on the coefficient
$\lambda$.

##  2 Dimensional Reduction

In this section, we use dimensional reduction to isolate three special values
of the Distance Conjecture parameter $\lambda$.

We begin with an Einstein-dilaton action in $D=d+1$ dimensions,

$S=\int\textrm{d}^{D}x\sqrt{-g}\left(\frac{1}{2\kappa_{D}^{2}}{\cal
R}_{D}-\frac{1}{2}(\nabla\hat{\phi})^{2}\right)\,.$ (7)

Here and in what follows, we often use $\hat{\cdot}$ to indicate that the
scalar field $\cdot$ is canonically normalized. We then consider the
dimensional reduction ansatz:

$ds^{2}={\mathrm{e}}^{-\frac{\rho(x)}{d-2}}d{\hat{s}}^{2}(x)+{\mathrm{e}}^{\rho(x)}dy^{2},$
(8)

where $y\cong y+2\pi R$.

Suppose that there is a tower of particles in $D$ dimensions with masses that
scale in the limit $\hat{\phi}\rightarrow\infty$ as

$m_{\text{part}}^{(D)}\sim\exp(-\kappa_{D}\lambda_{D}\hat{\phi})\,,$ (9)

Upon reduction, the tower of particles reduces to a tower of particles with
masses that scale as

$m_{\text{part}}^{(d)}\sim\exp\left(-\kappa_{d}\lambda_{D}\hat{\phi}-\frac{\kappa_{d}}{\sqrt{(d-1)(d-2)}}\hat{\rho}\right)\,,$
(10)

where we have defined the canonically normalized radion field
$\hat{\rho}=\frac{2}{\kappa_{d}}\sqrt{\frac{d-2}{d-1}}\rho$. Defining another
canonically normalized field

$\hat{\phi}^{\prime}=\frac{1}{\lambda_{D}^{2}+\frac{1}{(d-1)(d-2)}}\left(\lambda_{D}\hat{\phi}+\frac{1}{\sqrt{(d-1)(d-2)}}\hat{\rho}\right)\,,$
(11)

we may also write this as

$m_{\text{part}}^{(d)}\sim\exp\left(-\kappa_{d}\left(\lambda_{D}^{2}+\frac{1}{(d-1)(d-2)}\right)^{1/2}\hat{\phi}^{\prime}\right)\equiv\exp\left(-\kappa_{d}\lambda_{d}\hat{\phi}^{\prime}\right)\,.$
(12)

There is also a tower of Kaluza Klein modes for the graviton with masses that
scale as

$m_{\text{KK}}^{(d)}\sim\exp\left(-\kappa_{d}\sqrt{\frac{{d-1}}{{d-2}}}\hat{\rho}\right)\,.$
(13)

From the above analysis, we note three special values of the parameter
$\lambda_{d}$:

  1. 1.

$\lambda_{d}=1/\sqrt{(d-1)(d-2)}$. This is the coefficient of the radion
$\hat{\rho}$ for the dimensionally reduced tower in (10). In other words, as
$\hat{\rho}\rightarrow\infty$, this tower of states will become massless, with
masses decaying exponentially as
$m\sim\exp(-\lambda_{d}\kappa_{d}\hat{\rho})$.

  2. 2.

$\lambda_{d}=\sqrt{(d-1)/(d-2)}$. This is the coefficient of the radion
$\hat{\rho}$ for the Kaluza Klein modes in (13). Note that this value is
always larger than the first value of $\lambda_{d}$, so these Kaluza Klein
modes will become massless more quickly than the dimensionally reduced modes

  3. 3.

$\lambda_{d}=1/\sqrt{d-2}$. This value is distinguished by the fact that for
$\lambda_{D}=1/\sqrt{D-2}$, (12) gives $\lambda_{d}=1/\sqrt{d-2}$. In other
words, this value of $\lambda_{d}$ is _exactly preserved_ under dimensional
reduction.555More generally, $\lambda_{d}^{2}=\frac{1}{d-2}+\beta$ is exactly
preserved under dimensional reduction, but we will see many examples below
that saturate this bound with $\beta=0$, leading us to single out this
particular value from the rest. Preservation under dimensional reduction has
proven to be a useful tool for sharpening various quantum gravity
conjectures–see Heidenreich:2015nta ; Heidenreich:2019zkl ; Rudelius:2021oaz ;
Montero:2021otb  for examples.

Of the three distinguished values, the first is the smallest. It is therefore
tempting to conjecture that this is the “correct” minimum value of $\lambda$
in the Distance Conjecture, i.e.,

$\lambda\geq\frac{1}{\sqrt{(d-1)(d-2)}}\,.$ (14)

This bound was, in fact, proposed in Andriot:2020lea ; Gendler:2020dfp ,
motivated by the work of e.g. Grimm:2018ohb .

However, we note something interesting in the example above: although towers
of states saturating the value $\lambda={1}/{\sqrt{(d-1)(d-2)}}$ appear
naturally in dimensional reduction, they are accompanied in this context by
Kaluza Klein towers, which saturate the stronger bound

$\lambda\geq\sqrt{(d-1)/(d-2)}\,.$ (15)

Thus, every decompactification limit from $d$ to $D=d+1$ dimensions seems to
introduce a tower of particles satisfying the Distance Conjecture with a
coefficient $\lambda\geq\sqrt{(d-1)/(d-2)}$.

However, not every infinite-distance limit is a decompactification limit, and
the bound $\lambda\geq\sqrt{(d-1)/(d-2)}$ is not satisfied in general.
According to the Emergent String Conjecture Lee:2019xtm , every infinite-
distance limit that is not a decompactification limit is an emergent string
limit, in which a charged fundamental string becomes tensionless
asymptotically and a tower of string states become light. As we will see
below, the tension of a fundamental string scales as
$T_{\textrm{string}}\sim\exp(-2\kappa_{d}\hat{\phi}/\sqrt{d-2})$ for
$\hat{\phi}$ the canonically normalized dilaton in $d$ dimensions, which means
that the tower of light string states satisfies the Distance Conjecture with
coefficient $\lambda_{d}=1/\sqrt{d-2}$. This is nothing but the third
distinguished value of $\lambda$, which we saw was exactly preserved under
dimensional reduction. This leads us finally to conjecture

$\lambda\geq\frac{1}{\sqrt{d-2}}$ (16)

as the correct, sharpened version of the Distance Conjecture, as stated
previously in (2).

It is instructive to see how (16) is satisfied in the dimensional reduction
example considered above if we set $\lambda_{D}\geq 1/\sqrt{D-2}$. Upon
dimensional reduction, we find a theory with two towers of states: one tower
of Kaluza-Klein modes, and one that descends from the tower of particles in
$D$ dimensions. By the definition in (6), these have $\vec{\zeta}$-vectors in
the $(\hat{\phi},\hat{\rho})$ basis given by

$\vec{\zeta}_{\textrm{KK}}^{(d)}=\begin{pmatrix}0\\\
\sqrt{\frac{d-1}{d-2}}\end{pmatrix}\qquad\text{and}\qquad\vec{\zeta}_{\text{part}}^{(d)}=\begin{pmatrix}\lambda_{D}\\\
\frac{1}{\sqrt{(d-1)(d-2)}}\end{pmatrix},$ (17)

which may be computed from the mass formulas in (10), (13). For
$\lambda_{D}=1/\sqrt{D-2}$, the vector $\vec{\zeta}_{\text{part}}^{(d)}$ has
magnitude $\lambda_{d}=1/\sqrt{d-2}$ and therefore saturates our proposed
bound in the limit $\phi^{\prime}\rightarrow\infty$. The vector
$\vec{\zeta}_{\text{KK}}^{(d)}$ has magnitude
$\lambda_{\textrm{KK}}=\sqrt{(d-1)/(d-2)}$ and satisfy our bound comfortably
in the limit $\rho\rightarrow\infty$. In the intermediate regime
$\rho,\phi\rightarrow\infty$ with fixed $\rho/\phi\geq 1/\sqrt{d-2}$, these
Kaluza-Klein modes will still satisfy our bound $\lambda\geq 1/\sqrt{d-2}$,
saturating this bound for $\lambda_{D}=1/\sqrt{D-2}$ when
$\rho,\phi\rightarrow\infty$ with fixed $\rho/\phi=1/\sqrt{d-2}$.

This is illustrated pictorially in Figure 2. The part of the convex hull
generated by $\vec{\zeta}_{\textrm{KK}}^{(d)}$ and
$\vec{\zeta}_{\textrm{part}}^{(d)}$ remains outside the ball of radius
$1/\sqrt{d-2}$ and is tangent to the ball at the point
$\vec{\zeta}_{\textrm{part}}^{(d)}$ provided $\lambda_{D}=1/\sqrt{D-2}$. This
means that the Distance Conjecture will be satisfied with $\lambda\geq
1/\sqrt{d-2}$ everywhere along this line segment, i.e., in any infinite-
distance limit with $\rho,\phi\rightarrow\infty$, $\rho/\phi\geq
1/\sqrt{d-2}$.

Figure 2: Dimensional reduction and the convex hull condition. The gray region
is the ball of radius $1/\sqrt{d-2}$ centered at the origin. The vector
$\vec{\zeta}_{\text{KK}}=(0,\sqrt{(d-1)/(d-2)})$ comes from the Kaluza Klein
modes, and the vector
$\vec{\zeta}_{\text{part}}=(1/\sqrt{d-1},1/\sqrt{(d-1)(d-2)})$ comes from
particles with scalar charges in the $D$-dimensional theory that saturate the
proposed bound $\lambda_{\text{min}}^{(D)}=1/\sqrt{D-2}$. Since the convex
hull of these vectors contains the ball of $1/\sqrt{d-2}$, the Distance
Conjecture will be satisfied with a coefficient $\lambda\geq 1/\sqrt{d-2}$ in
every direction $\rho,\phi\rightarrow\infty$ with $\rho/\phi\geq
1/\sqrt{d-2}$.

One additional feature of this line segment is worth noting, as it further
distinguishes the value $\lambda_{d}=1/\sqrt{d-2}$. Namely, when
$\lambda_{D}\leq 1/\sqrt{D-2}$, the point on this line segment closest to the
origin is simply the endpoint $\vec{\zeta}_{\text{part}}^{(d)}$. In contrast,
when $\lambda_{D}>1/\sqrt{D-2}$, the point on this line segment closest to the
origin lies in the interior of the line segment.

A very similar analysis applies to a more general compactification from
$D=d+n$ dimensions to $d$ dimensions, for $n\geq 1$. Such a compactification
has MReece

$\displaystyle m_{\textrm{KK}}^{(d)}$
$\displaystyle\sim\exp\left(-\kappa_{d}\sqrt{\frac{{n+d-2}}{{n(d-2)}}}\hat{\rho}\right)\,,$
(18) $\displaystyle m_{\textrm{part}}^{(d)}$
$\displaystyle\sim\exp\left(-\lambda_{D}\phi-\kappa_{d}\sqrt{\frac{n}{{(n+d-2)(d-2)}}}\hat{\rho}\right)\,.$
(19)

This leads to

$\vec{\zeta}_{\textrm{KK}}^{(d)}=\begin{pmatrix}0\\\
\left(\frac{n+d-2}{n(d-2)}\right)^{1/2}\end{pmatrix}\qquad\text{and}\qquad\vec{\zeta}_{\text{part}}^{(d)}=\begin{pmatrix}\lambda_{D}\\\
\left(\frac{n}{(n+d-2)(d-2)}\right)^{1/2}\end{pmatrix}.$ (20)

As in the $n=1$ case above, the vector $\vec{\zeta}_{\text{part}}^{(d)}$ lies
on the ball of radius $\lambda_{\textrm{min}}=1/\sqrt{d-2}$ for
$\lambda_{D}=1/\sqrt{D-2}$, and the line between this vector and
$\vec{\zeta}_{\text{KK}}^{(d)}$ is tangent to this ball, as depicted in Figure
2. In the limit $n\rightarrow\infty$, the vectors
$\vec{\zeta}_{\text{part}}^{(d)}$ and $\vec{\zeta}_{\text{KK}}^{(d)}$
coalesce, and $\lambda_{\textrm{KK}}\rightarrow 1/\sqrt{d-2}$.

###  2.1 Winding Modes and Kaluza-Klein Monopoles

Although the picture in Figure 2 is suggestive, it is not complete. In order
satisfy our bound (2) in all infinite-distance limits, we must ensure that the
entire ball of radius $\lambda_{\textrm{min}}=1/\sqrt{d-2}$ is contained in
the convex hull of the $\vec{\zeta}$-vectors. So far in our discussion of
dimensional reduction, we have only considered limits in which the dilaton
$\phi$ and the radion $\rho$ tend to $+\infty$. What about the opposite
limits, in which one or both of these fields tend to $-\infty$?

In these cases, satisfying the Distance Conjecture typically requires new
ingredients beyond what we have considered so far in this section. In string
theory, the strong coupling limit $\phi\rightarrow-\infty$, in which the
dilaton diverges, is typically equivalent to a weak coupling limit in a dual
frame. This is seen most clearly in the case of Type IIB string theory in 10
dimensions or Type II string theory compactified on a circle to 9 dimensions,
which we will review below.

In the limit $\rho\rightarrow-\infty$, the radius of the dimensional reduction
circle vanishes. Here, the Kaluza-Klein tower becomes heavy, but a tower of
light charged particles appears from string winding modes. In particular,
wrapping a string of tension $T$ in $D$ dimensions around the dimensional
reduction circle will produce a tower of charged particles in $d$ dimensions
whose masses scale with the canonically-normalized radion $\hat{\rho}$ as

$m\sim\exp\left(\frac{d-3}{\sqrt{(d-1)(d-2)}}\kappa_{d}\hat{\rho}\right)\,.$
(21)

For $d\geq 5$, these winding modes will satisfy our proposed bound (2). For
$d=4$, we have $\lambda=1/\sqrt{6}$, so these winding modes do not satisfy our
proposed bound.

However, in $d=4$, we expect another tower of light particles to appear in the
$\rho\rightarrow\infty$ limit: Kaluza-Klein monopoles. Our dimensional
reduction ansatz produces a Kaluza-Klein gauge field $A^{\text{KK}}_{1}$ in
$d$ dimensions with gauge coupling

$\frac{1}{e_{\textrm{KK}}^{2}}=\frac{R^{2}}{2\kappa_{d}^{2}}{\mathrm{e}}^{2\sqrt{(d-1)/(d-2)}\kappa_{d}\hat{\rho}}\,.$
(22)

In the limit $\hat{\rho}\rightarrow-\infty$, the magnetic gauge coupling
$g_{\textrm{KK}}\equiv 2\pi/e_{\textrm{KK}}$ vanishes as
$g_{\textrm{KK}}\sim\exp(-\sqrt{(d-1)/(d-2)}\kappa_{d}|\hat{\rho}|)$. In 4d,
the tower Weak Gravity Conjecture applied to the electromagnetic dual gauge
field therefore implies a tower of Kaluza-Klein monopoles with $m\lesssim
g_{\textrm{KK}}/\kappa_{4}\Rightarrow\lambda\geq\sqrt{3/2}$, so the tower of
Kaluza-Klein monopoles satisfies our proposed bound (2).

In 5d, the Kaluza-Klein monopole is a string. This string will be charged
magnetically under the Kaluza-Klein gauge field, and its tension scales as

$T_{\textrm{string}}\lesssim
1/(e_{\textrm{KK}}\kappa_{5})\sim\exp\biggl{(}\frac{2}{\sqrt{3}}\kappa_{5}\hat{\rho}\biggr{)}\,.$
(23)

The oscillator modes of this string will produce a tower of massive particles
beginning at the scale $M_{\text{string}}=\sqrt{2\pi
T_{\text{string}}}\lesssim\exp(\kappa_{5}\hat{\rho}/\sqrt{3})$, satisfying the
bound $\lambda\geq 1/\sqrt{d-2}$ in the small radius limit
$\hat{\rho}\rightarrow-\infty$.

We expect, therefore, that in the small radius limit
$\hat{\rho}\rightarrow-\infty$ of a dimensional reduction, the Distance
Conjecture with $\lambda\geq 1/\sqrt{d-2}$ will be satisfied by winding string
modes in $d\geq 5$ and by Kaluza-Klein monopoles in $d=4,5$. Indeed,
T-dualities in string theory suggest that the small radius limit of a theory
is likely equivalent to a large radius limit in another duality frame, so it
is unsurprising that our proposed bound is satisfied in each of these two
limits.

Let us pause here to emphasize a parallel between the large radius limit and
the small radius limit of a dimensional reduction from $D=5$ to $d=4$
dimensions. In the large radius limit $\hat{\rho}\rightarrow\infty$, we found
one tower with $\lambda=1/\sqrt{6}$, which came from the Kaluza Klein zero
modes of a tower of particles in the parent theory in $D=5$ dimensions. We
also found another tower with $\lambda=\sqrt{3/2}$, which came from the
Kaluza-Klein modes of the graviton. Similarly, in the small radius limit
$\hat{\rho}\rightarrow-\infty$, we found one tower with $\lambda=1/\sqrt{6}$,
which came from winding string modes, and another tower with
$\lambda=\sqrt{3/2}$, which came from Kaluza-Klein monopoles. The Emergent
String Conjecture suggests this is not an accident: the
$\hat{\rho}\rightarrow-\infty$ limit should correspond to a decompactification
limit in a dual frame, so the towers of winding modes and Kaluza-Klein
monopoles are respectively identified with towers of 5d particles and Kaluza-
Klein modes in this dual frame. More generally, while towers with
$\lambda=1/\sqrt{6}$ (which do not satisfy our proposed bound (2)) seem to be
common in infinite-distance limits in 4d, the Emergent String Conjecture
strongly suggests that these limits will feature even lighter towers with
$\lambda=\sqrt{3/2}$ (which do satisfy the bound (2)). In the following
section, we will see compelling evidence that this expectation is borne out in
4d supergravity theories and string compactifications.

To conclude this subsection, we consider winding modes and wrapped branes in
more general compactifications. Consider a reduction from $D=d+n$ dimensions
to $d$ dimensions, and suppose that a $(P-1)$-brane wraps $k$ dimensions of
the internal geometry, yielding a $(p-1)$-brane in $d$ dimensions (here,
$P=p+k$). The tension of the resulting $(p-1)$-brane will then scale with the
canonically normalized radion in the small volume limit
$\hat{\rho}\rightarrow\infty$ as MReece

$T_{p}\sim\exp\left(-\sqrt{\frac{d-2}{n(n+d-2)}}\frac{pn-k(d-2)}{d-2}\kappa_{d}\hat{\rho}\right)\,.$
(24)

Let us now examine several special cases of this formula. First, we consider
the case $p=1$, which corresponds to a particle in $d$ dimensions, for which
the tension $T$ is simply the mass. We further suppose that this particle
arises from wrapping a brane over the entire $n$-dimensional compactification
manifold. Plugging in $p=1$, $k=n$, we have

$m\sim\exp\left(\sqrt{\frac{n(d-2)}{n+d-2}}\frac{d-3}{d-2}\kappa_{d}\hat{\rho}\right)~{}~{}\Rightarrow~{}~{}\lambda=\sqrt{\frac{n(d-2)}{n+d-2}}\frac{d-3}{d-2}\,,$
(25)

which satisfies $\lambda\geq 1/\sqrt{d-2}$ for $n\geq 2$, $d\geq 4$. In other
words, our proposed bound (2) will be satisfied by $n$-branes wrapping a
compactification $n$-manifold in the small volume limit for $n>1$.

Next, we consider the case $p=2$, $n=k=1$, which corresponds to a 2-brane
wrapping a circle to produce a string in $d$ dimensions. This gives

$T_{\text{string}}\sim\exp\left(\frac{d-4}{\sqrt{(d-1)(d-2)}}\kappa_{d}\hat{\rho}\right)\,.$
(26)

This string will give rise to a tower of string oscillator modes at the mass
scale $M_{\text{string}}=\sqrt{2\pi T_{\text{string}}}$, which satisfy the
Distance Conjecture with a coefficient of
$\lambda=\frac{d-4}{2\sqrt{(d-1)(d-2)}}$. This saturates the bound
$\lambda\geq 1/\sqrt{d-2}$ for $d=10$. Indeed, this describes the weak
coupling limit of the Type IIA superstring, which may be realized as the small
radius limit of an M2-brane wrapped on a circle.

###  2.2 A Convex Hull Condition

We now combine the ingredients above in a simple yet illustrative toy model of
dimensional reduction, showing how the Distance Conjecture and the Scalar Weak
Gravity Conjecture may be satisfied with a coefficient
$\lambda_{\text{min}}^{(d)}=1/\sqrt{d-2}$ under the assumption that they are
satisfied in $D=d+1$ dimensions with
$\lambda_{\text{min}}^{(D)}=1/\sqrt{D-2}$.

We consider a $D$-dimensional theory with a single modulus, the dilaton
$\phi$, and we suppose that this theory has two kinds of strings, where one
kind of strings has a tension which scales with the dilaton by
$T_{\text{string}}^{+}\sim\exp\left(2\lambda_{\text{string}}^{(D)}\kappa_{D}\phi\right)$,
and the other kind of strings has a tension which scales with the dilaton by
$T_{\text{string}}^{-}\sim\exp\left(-2\lambda_{\text{string}}^{(D)}\kappa_{D}\phi\right)$,
where $\lambda_{\text{string}}^{(D)}=1/\sqrt{D-2}=1/\sqrt{d-1}$. This behavior
occurs, for instance, in Type IIB string theory, where S-duality switches the
strong and weak coupling limits $\phi\rightarrow\pm\infty$. The string
oscillator modes of these respective strings form towers with masses which
scale with the dilaton as
$m_{\text{string}}^{(D)}(\phi)\sim\exp(\pm\lambda_{\text{string}}^{(D)}\kappa_{D}\phi)$,
and thus they saturate the Distance Conjecture and the Scalar Weak Gravity
Conjecture in $D$ dimensions.

We may then calculate the minimal radius $\lambda_{\text{min}}^{(d)}$ by
examining the convex hull of all the $\vec{\zeta}^{(d)}$-vectors, as discussed
in the introduction. This convex hull is generated by the
$\vec{\zeta}^{(d)}$-vectors for (a) the Kaluza-Klein modes and (b) the string
winding modes. By our discussion earlier in this section, these have
$\vec{\zeta}^{(d)}$-vectors in the $(\hat{\phi},\hat{\rho})$ basis given by

$\vec{\zeta}_{\text{KK}}^{(d)}=\begin{pmatrix}0\\\
\sqrt{\frac{d-1}{d-2}}\end{pmatrix},\qquad\vec{\zeta}_{\text{wind}}^{(d)}=\begin{pmatrix}\pm\frac{2}{\sqrt{d-1}}\\\
-\frac{d-3}{\sqrt{(d-1)(d-2)}}\end{pmatrix}.$ (27)

When $d=4$, we must include an additional point corresponding to the Kaluza-
Klein monopoles:

$\vec{\zeta}_{\text{mon}}^{(4)}=\begin{pmatrix}0\\\
-\sqrt{\frac{3}{2}}\end{pmatrix}\,.$ (28)

The string oscillator modes lie on the boundary of the convex hull. They have

$\zeta_{\text{string}}^{(d)}=\begin{pmatrix}\pm\frac{1}{\sqrt{d-1}}\\\
\frac{1}{\sqrt{(d-1)(d-2)}}\end{pmatrix}\,.$ (29)

When $d=5$, we may include an additional point corresponding to the string
oscillator modes for the Kaluza-Klein monopole string:

$\vec{\zeta}_{\text{mon. str.}}^{(5)}=\begin{pmatrix}0\\\
-\frac{1}{\sqrt{3}}\end{pmatrix}\,.$ (30)

(a) $d>5$

(b) $d=5$

(c) $d=4$

Figure 3: The convex hull condition in a toy model of dimensional reduction.
The gray region in both figures is the ball of radius $1/\sqrt{d-2}$ centered
at the origin. The vector
$\vec{\zeta}_{\text{KK}}^{(d)}=(0,\sqrt{(d-1)/(d-2)})$ comes from the Kazula-
Klein modes; the vectors $\vec{\zeta}_{\text{string}}=(\pm
1/\sqrt{d-1},1/\sqrt{(d-1)(d-2)})$ come from string oscillation modes in the
$D$-dimensional theory, and the vectors $\vec{\zeta}_{\text{wind}}=(\pm
2/\sqrt{d-1},-(d-3)/\sqrt{(d-1)(d-2)}$ come from winding states. In the $d=5$
case in 3(b), there is additionally the
$\vec{\zeta}_{\text{mon}}^{(5)}$-vector coming from the string oscillator
modes of the Kaluza-Klein monopole string. In the $d=4$ case in 3(c), the
vector $\vec{\zeta}_{\text{mon}}^{(4)}=(0,-\sqrt{3/2})$ comes from the Kaluza-
Klein monopole and is needed for the ball of radius $1/\sqrt{d-2}$ to be
contained in the convex hull.

For $d\geq 4$, the convex hulls of the $\vec{\zeta}^{(d)}$-vectors have a
minimal radius $\lambda_{\text{min}}^{(d)}$ of precisely

$\lambda_{\text{min}}^{(d)}=\frac{1}{\sqrt{d-2}}\,,$ (31)

as shown in Figure 3. Thus, the bound $\lambda\geq 1/\sqrt{d-2}$ is saturated
in $d$ dimensions, just as it was in $D$ dimensions. Whenever it is saturated,
there is always a tower of string oscillator modes with
$\lambda=1/\sqrt{d-2}$, which either descend from string oscillator modes in
$D$ dimensions or else come from the Kaluza-Klein monopole string in $d=5$.

It is interesting to note that the string oscillator modes are not generators
of the convex hull in these examples, since they always saturate the bound
$\lambda\geq 1/\sqrt{d-2}$. Instead, the convex hull is generated by a
combination of Kaluza-Klein modes, winding modes, and Kaluza-Klein monopoles.
More generally, if the Emergent String Conjecture is true, then any generator
$\vec{\zeta}^{(d)}$ of the convex hull must either correspond to a tower of
string oscillator modes or else a tower of Kaluza-Klein modes in some duality
frame. By (18), this means there is only a finite set of possibilities for the
length of such a generator:

$|\vec{\zeta}^{(d)}|=\frac{1}{\sqrt{d-2}}~{}~{}\text{ or
}~{}~{}|\vec{\zeta}^{(d)}|=\sqrt{\frac{n+d-2}{n(d-2)}}\,,$ (32)

for some $n\geq 1$.

In our brief analysis in this section, we have notably ignored the possibility
of axions. If the $D$-dimensional theory includes 1-form gauge fields, then
the $d$-dimensional theory will have axions, and the electric charge-to-mass
ratios in the $D$-dimensional theory determine the axion-charge components of
the $\vec{\zeta}^{(d)}$-vectors of the $d$-dimensional theory. The Weak
Gravity Conjecture and Repulsive Force Conjectures in the higher dimensional
theory then have important consequences for the axion components of the
$\vec{\zeta}^{(d)}$-vectors. We defer to future research a thorough
investigation of these consequences.

##  3 A Reexamination of Supergravity in Four Dimensions

So far, nearly all studies of the Distance Conjecture coefficient $\lambda$
have taken place within the context of four-dimensional supergravity
Grimm:2018ohb ; Blumenhagen:2018nts ; Joshi:2019nzi ; Erkinger:2019umg ;
EnriquezRojo:2020hzi ; Ashmore:2021qdf . Multiple works have suggested
$\lambda_{\textrm{min}}=1/\sqrt{6}$ as the minimal bound in four dimensions
Grimm:2018ohb ; Andriot:2020lea ; Gendler:2020dfp , and several examples have
been previously claimed to saturate this bound. Such examples naively violate
our proposed bound $\lambda\geq 1/\sqrt{d-2}$, so before we provide evidence
for this bound in higher dimensions, we must first address this apparent
contradiction between our proposal and previous claims in the literature.

As we saw in the previous section, the coefficient $\lambda=1/\sqrt{6}$
appears readily in decompactification limits: it is the scaling behavior
expected for the Kaluza-Klein zero modes of a tower of particles in $D=5$
dimensions after compactification to $d=4$ dimensions. Such towers are always
accompanied in these decompactification limits by towers of Kaluza-Klein modes
with $\lambda=\sqrt{3/2}$, which ensure consistency with our proposed bound
(2) on the lightest tower of particles in any infinite-distance limit. In the
remainder of this section, we argue that this situation is generic: explicit
examples with $\lambda=1/\sqrt{6}$ discussed previously in the literature
always feature lighter towers with $\lambda=\sqrt{3/2}$, so they in fact
represent examples in support of our bound, not counterexamples to it.

The first extensive discussion of the coefficient $\lambda=1/\sqrt{6}$
appeared in Grimm:2018ohb , which examined the behavior of towers of massive
particles in asymptotic regions of scalar field space in four dimensions. In
one-dimensional vector multiplet moduli spaces of Type IIB compactifications,
they argued that asymptotic limits are characterized by one of three possible
values for $\lambda$:

$\lambda=\frac{1}{\sqrt{6}}\,,\frac{1}{\sqrt{2}}\,,1\,.$ (33)

These values can be determined from the type of singularity that occurs in the
infinite-distance limit of complex structure moduli space, which can be
classified using the theory of mixed Hodge structures Grimm:2018ohb ;
Corvilain:2018lgw ; Grimm:2018cpv ; Grimm:2019ixq . This classification,
however, only implies the existence of some exponentially light tower with
coefficient $\lambda$ given by one of the values above. It does not imply that
this is the only exponentially light tower, nor that it is the lightest such
tower. Thus, while infinite-distance limits in this classification
corresponding to $\lambda=1/\sqrt{2}$ and $\lambda=1$ necessarily satisfy our
bound (2), limits with $\lambda=1/\sqrt{6}$ require further analysis.

Such an analysis was recently carried out in the supergravity context in
Gendler:2020dfp , which pointed out that for $\mathcal{N}=2$ theories with a
single vector multiplet, there are three possible forms for the perturbative
part of the prepotential, which in turn lead to two distinct infinite-distance
limits. Reference Gendler:2020dfp  considered one such prepotential in detail:

$\mathcal{F}=-\frac{(X^{1})^{3}}{X^{0}}\,.$ (34)

After setting the axion vevs to vanish, this prepotential leads to a gauge
kinetic matrix of the form

$a_{IJ}=\mathrm{diag}({\mathrm{e}}^{\sqrt{6}\kappa_{4}\hat{\rho}},3{\mathrm{e}}^{\sqrt{2/3}\kappa_{4}\hat{\rho}})\,,$
(35)

for $\hat{\rho}$ a canonically normalized scalar field. In the limit
$\hat{\rho}\rightarrow\infty$, assuming the tower Weak Gravity Conjecture, we
therefore expect two towers of charged particles, one for each gauge field,
with

$\lambda=\frac{1}{\sqrt{6}}\,,~{}\lambda=\sqrt{\frac{3}{2}}\,,$ (36)

respectively. This reflects precisely the behavior we expect: the existence of
a tower with $\lambda=1/\sqrt{6}<1/\sqrt{2}$ is accompanied by another tower
with $\lambda\geq 1/\sqrt{2}$.

Indeed, these values of $\lambda$ match precisely with what we expect from
dimensional reduction of $\mathcal{N}=1$ supergravity with no massless vector
multiplets in five dimensions, which has one 1-form gauge field but no scalar
fields. Dimensionally reducing to 4d, a tower of superextremal charged
particles with $\lambda_{5}=0$ will reduce to a tower of charged particles
with $\lambda_{4}=1/\sqrt{6}$ as in (10), and there will be Kaluza Klein
towers with $\lambda_{4}=\sqrt{3/2}$ as in (13). These indeed match the values
in (36). We see here that although there is a tower of particles saturating
the bound $\lambda\geq 1/\sqrt{(d-1)(d-2)}$ in the limit
$\hat{\rho}\rightarrow\infty$, there is an even lighter tower of particles
that satisfies our proposed bound, $\lambda\geq 1/\sqrt{d-2}$.

This behavior also fits nicely with the results of Lanza:2020qmt ;
Lanza:2021qsu ; Heidenreich:2021yda , which considered the behavior of axion
strings666Recall that an axion string is a string charged magnetically under
an axion $\theta$, so that $\theta\rightarrow\theta+2\pi$ as one circles the
core of the string. in infinite-distance limits in moduli space. Reference
Lanza:2021qsu  argued that any such limit corresponds to the tensionless limit
of an axion string, and the mass of the lightest tower of particles scales (in
Planck units) as either $m^{2}\sim T_{\textrm{string}}$, $m^{2}\sim
T_{\textrm{string}}^{2}$, or $m^{2}\sim T_{\textrm{string}}^{3}$ in this
limit. According to Heidenreich:2021yda , the large radius limit of pure 5d
supergravity with a nontrivial Chern-Simons coupling on a circle features two
gauge fields $A_{1}$ and $A_{1}^{\textrm{KK}}$ with gauge couplings $e$ and
$e_{\textrm{KK}}$ satisfying the relation $e\sim e_{\textrm{KK}}^{1/3}$, and
consistency with various forms of the Weak Gravity Conjecture implies an axion
string whose tension scales as $T_{\textrm{string}}\sim e^{2}$. The tower Weak
Gravity Conjecture for $A_{1}^{\textrm{KK}}$ then implies a tower of particles
whose masses scale as $m_{\textrm{KK}}^{2}\sim e_{\textrm{KK}}^{2}\sim
T_{\textrm{string}}^{3}$, in agreement with Lanza:2021qsu . This tower is, of
course, simply the Kaluza Klein tower with $\lambda=\sqrt{3/2}$, whereas the
tower with $\lambda=1/\sqrt{6}$ is the tower of oscillator modes for the axion
string.

The presence of a tower with $\lambda=\sqrt{3/2}$ accompanying a tower with
$\lambda=1/\sqrt{6}$ has been observed not only in supergravity and Kaluza-
Klein theory, but also in UV complete string compactifications. In particular,
Joshi:2019nzi  found that near an “M-point” of a Calabi-Yau compactification
of Type IIA string theory, there exists a tower of light D0-branes with
$\lambda=\sqrt{3/2}$ accompanied by a tower of light D2-branes with
$\lambda=1/\sqrt{6}$.777Note that the conventions of Joshi:2019nzi  differ
from ours by a factor of $\sqrt{2}$:
$\lambda_{\textrm{them}}=\sqrt{2}\lambda_{\textrm{us}}$, as previously noted
in Andriot:2020lea . This scaling behavior is unsurprising, as this limit may
be viewed as a decompactification of the M-theory circle of Type IIA.

In that same paper, Joshi:2019nzi  studied towers of particles that occur near
the “s-point” of a Calabi-Yau geometry they called $X_{2,2,2,2}$. In Section
6.2.3. of that paper, they found that the masses of the light particles are
given in terms of a pair of integers $x$, $y$ by

$m(x,y)=m_{0}|y|e^{-\kappa_{4}\hat{\phi}/\sqrt{6}}+m_{1}|2iy\log(4)-x|e^{-\kappa_{4}\hat{\phi}\sqrt{3/2}}+O(e^{-\kappa_{4}\hat{\phi}\sqrt{5/2}})\,,$
(37)

where $m_{0}$, $m_{1}$ are constants, and the s-point corresponds to the limit
$\hat{\phi}\rightarrow\infty$. For $|y|\neq 0$, therefore, there is a tower of
light particles indexed by $y\in\mathbb{Z}$ with $\lambda=1/\sqrt{6}$. For
$|y|=0$, however, the first term vanishes, and there is a tower of light
particles indexed by $x\in\mathbb{Z}$ with $\lambda=\sqrt{3/2}$. Again, these
towers have the expected scaling behavior for a decompactification limit to
five dimensions.

Similarly, EnriquezRojo:2020hzi  found towers with $\lambda=\sqrt{3/2}$ and
$1/\sqrt{6}$ in a Type IIB Calabi-Yau orientifold compactification. This
example is especially interesting because it features $\mathcal{N}=1$
supersymmetry in four dimensions, so the scalar field in question is not even
a massless modulus. We will elaborate on the application of the Distance
Conjecture to massive scalar fields below in §6.

The authors of Lanza:2021qsu  studied the spectrum of charged particles and
strings in a model of Type IIA string theory compactified on a Calabi-Yau
threefold $X$ given by a particular $\mathbb{P}^{1}$ fibration over
$\mathbb{P}^{2}$. In one limit, they found an asymptotically tensionless
string with $T_{\textrm{string}}\sim\exp(-\sqrt{2/3}\kappa_{4}\hat{\rho})$ and
a tower of Kaluza-Klein modes for the Calabi-Yau threefold $X$ with
$m\sim\exp(-\kappa_{4}\hat{\rho}/\sqrt{6})$. From our discussion in §2, this
scaling behavior is precisely what is to be expected for M-theory compactified
first on the Calabi-Yau $X$ and then on $S^{1}$, where $\hat{\rho}$ is the
canonically normalized radion of the $S^{1}$. From an M-theory perspective,
the tower of Kaluza-Klein modes for $X$ with $\lambda=1/\sqrt{6}$ in the
$\hat{\rho}\rightarrow\infty$ limit will be accompanied in four dimensions by
a tower of light Kaluza-Klein modes for the M-theory circle with
$\lambda=\sqrt{3/2}$. From a Type IIA perspective, these latter Kaluza-Klein
modes will be D0-branes, which were not considered in the analysis of
Lanza:2021qsu .

Reference Gendler:2020dfp  also considered a 4d $\mathcal{N}=2$ theory with
two vector multiplets and a prepotential of the form

$\mathcal{F}=-\frac{1}{6X^{0}}TS^{2}\,.$ (38)

They argued that every infinite-distance limit has a tower satisfying
$\lambda\geq 1/\sqrt{2}$, and this bound is in fact saturated in certain
directions in scalar field space. This matches precisely with our proposed
bound.

To our knowledge, our discussion in this section has now addressed every
reference in the literature to a Distance Conjecture tower in four dimensions
with an exact coefficient $\lambda<1/\sqrt{2}$. We have seen that all such
examples feature even lighter towers with $\lambda\geq 1/\sqrt{2}$, and a
number of examples in fact saturate this bound. The only potential
counterexamples left to discuss are numerical examples from e.g.
Blumenhagen:2018nts ; Erkinger:2019umg , which are listed in Table 3 of
Andriot:2020lea . These examples involve an averaging procedure, so the
associated values of $\lambda$ are known only within a range,
$\lambda\in[\lambda_{-},\lambda_{+}]$. Some of these ranges have
$\lambda_{-}<1/\sqrt{2}$, indicating a possible violation of our proposed
bound (2), but all of them have $\lambda_{+}>1/\sqrt{2}$, which is consistent
with our bound. In fact, the central values
$\lambda_{0}=(\lambda_{-}+\lambda_{+})/2$ across all the examples range from
$\lambda_{0}=0.7668$ to $0.8638$, tantalizingly close to our proposed minimum
value $\lambda_{\text{min}}=1/\sqrt{2}\approx 0.7071$. Thus, more precise
studies of these examples could lead to either a counterexample to our bound
or remarkable evidence in favor of it, but we leave this to future study.

##  4 Top-Down Evidence in Maximal Supergravity

In these sections, we explicitly compute $\lambda_{\text{min}}$ in maximal
supergravity in four to ten dimensions. In each dimension, we find that the
Distance Conjecture and Scalar Weak Gravity Conjecture are satisfied in all
directions of moduli space by a tower of particles with
$\lambda\geq\lambda_{\text{min}}=1/\sqrt{d-2}$. For each dimension, this bound
is in fact saturated in one or more directions, and in all cases that we have
checked (namely, for $d\geq 7$), these directions correspond to emergent
string limits featuring a tower of string oscillator modes with
$\lambda=1/\sqrt{d-2}$.

###  4.1 Ten Dimensions

There are three types of 10d supergravity for us to consider: Type I, Type
IIA, and Type IIB. The relevant parts of their actions (in Einstein frame)
take the form Polchinski:1998rr :

$\displaystyle S_{\textrm{IIA}}$
$\displaystyle=\frac{1}{2\kappa_{10}^{2}}\left(\int-\frac{1}{2}\textrm{d}\sigma\wedge\star\textrm{d}\sigma-\frac{1}{2}e^{\sigma}H_{3}\wedge\star
H_{3}-\frac{1}{2}e^{-3\sigma/2}F_{2}\wedge\star F_{2}\right)$ (39)
$\displaystyle S_{\textrm{IIB}}$
$\displaystyle=\frac{1}{2\kappa_{10}^{2}}\left(\int-\frac{1}{2}\textrm{d}\sigma\wedge\star\textrm{d}\sigma-\frac{1}{2}e^{\sigma}H_{3}\wedge\star
H_{3}\right)$ (40) $\displaystyle S_{\textrm{I}}$
$\displaystyle=\frac{1}{2\kappa_{10}^{2}}\left(\int-\frac{1}{2}\textrm{d}\sigma\wedge\star\textrm{d}\sigma-\frac{1}{2}e^{\sigma}H_{3}\wedge\star
H_{3}-\frac{1}{g_{A}^{2}}e^{\sigma/2}\textrm{Tr}(F_{2}\wedge\star
F_{2})\right)\,.$ (41)

Let us consider each of these in turn, beginning with Type IIA. Here, in the
limit $\sigma\rightarrow\infty$, the gauge coupling for the 2-form $B_{2}$
scales as $g_{B}\sim e^{-\sigma/2}$. Assuming that a fundamental string
charged under $B_{2}$ satisfies the Weak Gravity Conjecture bound $T\lesssim
g_{B}\kappa_{10}$, its string oscillator modes will scale as

$M_{\textrm{string}}\sim\sqrt{2\pi
T}\sim\exp(\kappa_{10}\hat{\sigma}/\sqrt{8})\,,$ (42)

where we have defined $\hat{\sigma}=\sigma/(\sqrt{2}\kappa_{10})$ to be the
canonically normalized dilaton. Thus, this tower of string oscillator modes
will satisfy the Distance Conjecture with

$\lambda=1/\sqrt{8}\,,$ (43)

saturating our proposed bound $\lambda\geq 1/\sqrt{d-2}$. Of course, such a
string is not merely a hypothetical entity: this is simply the IIA
superstring.

In the other infinite-distance limit, $\sigma\rightarrow-\infty$, the gauge
field $A_{1}$ will become weakly coupled as
$g_{A}\sim\exp(3\sigma/4)\sim\exp(\sqrt{9/8}\kappa_{10}\hat{\sigma})$. The
tower Weak Gravity Conjecture then implies a tower of light particles with
masses beginning at the scale $g_{A}\kappa_{10}$, which satisfy the Distance
Conjecture with a coefficient of

$\lambda=\sqrt{\frac{9}{8}}\,.$ (44)

This matches the value expected for Kaluza Klein modes (13), and indeed it
points towards the well-known fact that this tower of charged particles in
Type IIA string theory (D0-branes) is in fact a Kaluza Klein tower for
M-theory on a circle.

Next, let us consider Type IIB supergravity. The limit
$\sigma\rightarrow+\infty$ is identical to the Type IIA case, and the bound
$\lambda\geq 1/\sqrt{d-2}$ is saturated by oscillator modes of the Type IIB
superstring. The limit $\sigma\rightarrow-\infty$ is mysterious from the
perspective of the supergravity action we have written above, as the gauge
field $B_{2}$ becomes strongly coupled. However, here we invoke the well-known
S-duality of the Type IIB superstring, which implies that the strong coupling
limit of one Type IIB superstring is the weak coupling limit of a different
Type IIB superstring. The bound $\lambda\geq 1/\sqrt{d-2}$ will therefore be
saturated in this limit also. More generally, taking into account the axion
$C_{0}$, the convex hull condition for the $\vec{\zeta}$-vectors will be
satisfied in every direction in scalar field space by towers of string
oscillator modes for $(p,q)$-strings, as the corresponding vectors
$\vec{\zeta}_{p,q}$ densely fill in the sphere of radius
$\lambda=1/\sqrt{d-2}$. We see that the duality web plays a crucial role here
in satisfying our proposed bound.

Finally, we have Type I supergravity. Here, the limit
$\sigma\rightarrow\infty$ once again introduces a tower of string oscillator
modes saturating the bound $\lambda\geq 1/\sqrt{d-2}$, by the same calculation
as in Type IIA. The strong coupling limit $\sigma\rightarrow-\infty$ is more
mysterious, and again we must use known details of the string duality web.
Three different string theories have Type I supergravity as their low energy
limit: Type I string theory, $SO(32)$ heterotic string theory, and
$E_{8}\times E_{8}$ heterotic string theory. The first two of these are
S-dual, so the $\sigma\rightarrow-\infty$ limit of Type I string theory
corresponds to the $\sigma\rightarrow+\infty$ limit of $SO(32)$ heterotic
string theory, and vice versa. Thus, each of these limits will saturate the
bound $\lambda\geq 1/\sqrt{d-2}$ as well. The $\sigma\rightarrow-\infty$ limit
of $E_{8}\times E_{8}$ heterotic string theory, on the other hand, corresponds
to M-theory on an interval separating two Hořava-Witten walls. Here, there is
a tower of Kaluza Klein modes with $\lambda=\sqrt{9/8}$, as in the
decompactification limit of Type IIA string theory to M-theory above. Once
again, we see that string dualities ensure that our proposed Distance
Conjecture bound is satisfied.

###  4.2 Nine Dimensions

In $d=9$ dimensional maximal supergravity, coming from M-theory on $T^{2}$,
there are three moduli, all originating from the eleven-dimensional graviton.
To determine their couplings, we reduce the $D=11$ dimensional Einstein-
Hilbert action with the ansatz

$ds^{2}_{D}=\|g\|^{-\frac{1}{d-2}}g_{\mu\nu}dx^{\mu}dx^{\nu}+g_{mn}dy^{m}dy^{n}\,,$
(45)

where $m,n$ index the $k=D-d$ compact directions, $\mu,\nu$ index the $d$
noncompact directions, $y^{m}\cong y^{m}+2\pi R$, and $\|g\|=\det g_{mn}$. It
is convenient to decompose $g_{mn}$ in terms of volume and shape parameters
$U$ and $\tau=\tau_{1}+i\tau_{2}$, respectively, where

$g_{mn}=e^{U}\frac{1}{\tau_{2}}\begin{pmatrix}1&\tau_{1}\\\
\tau_{1}&|\tau|^{2}\end{pmatrix}.$ (46)

With this, the Einstein-moduli sector of the dimensionally reduced action is

$S_{9}=\frac{1}{2\kappa_{9}^{2}}\int
d^{9}x\sqrt{-g}\left(\mathcal{R}-\frac{9}{14}(\partial
U)^{2}-\frac{(\partial\tau_{1})^{2}+(\partial\tau_{2})^{2}}{2\tau_{2}^{2}}\right)\,.$
(47)

Working in modified nine-dimensional Planck units where
$2\kappa_{9}^{2}=(2\pi)^{6}$ for convenience, the spectrum of 1/4 BPS
particles is

$m_{p,q,w}=\frac{|p+\tau
q|}{\sqrt{\tau_{2}}R}e^{-\frac{9}{14}U}+R^{4/3}|w|e^{\frac{6}{7}U}\,,$ (48)

where $p,q\in\mathbb{Z}$ are the Kaluza-Klein charges and $w\in\mathbb{Z}$ is
the M2 brane winding charge. These particles are $1/2$ BPS when either $w=0$
or $p=q=0$.

At a particular point in moduli space, the canonically normalized moduli are

$\hat{\phi}^{a}=(\hat{U},\hat{\tau}_{1},\hat{\tau}_{2})=\frac{1}{\kappa_{9}}\Biggl{(}\sqrt{\frac{9}{14}}U,\frac{\tau_{1}}{\sqrt{2}\langle\tau_{2}\rangle},\frac{\tau_{2}}{\sqrt{2}\langle\tau_{2}\rangle}\Biggr{)}\,,$
(49)

where $\langle\tau_{2}\rangle$ is the value of $\tau_{2}$ at the point in
question (not including its fluctuations). The canonically normalied scalar
charge-to-mass vectors
$\zeta_{a}=\frac{1}{\kappa_{9}}\frac{\partial}{\partial\hat{\phi}^{a}}\log
m_{p,q,w}$ are then

$\displaystyle\zeta_{\hat{U}}$
$\displaystyle=\frac{4\sqrt{\tau_{2}}e^{\frac{3U}{2}}R^{7/3}|w|-3|p+\tau
q|}{\sqrt{14}\left(|p+\tau
q|+\sqrt{\tau_{2}}e^{\frac{3U}{2}}R^{7/3}|w|\right)},$ (50a)
$\displaystyle\zeta_{\hat{\tau}_{1}}$
$\displaystyle=\frac{\sqrt{2}q\tau_{2}(p+\tau_{1}q)}{|p+\tau q|\left(|p+\tau
q|+\sqrt{\tau_{2}}e^{\frac{3U}{2}}R^{7/3}|w|\right)},$ (50b)
$\displaystyle\zeta_{\hat{\tau}_{2}}$
$\displaystyle=\frac{q^{2}\tau_{2}^{2}-(p+\tau_{1}q)^{2}}{\sqrt{2}|p+\tau
q|\left(|p+\tau q|+\sqrt{\tau_{2}}e^{\frac{3U}{2}}R^{7/3}|w|\right)}\,,$ (50c)

for the $1/4$ BPS particles.

Figure 4: The convex hull in 9d maximal supergravity. The
$\vec{\zeta}$-vectors of BPS states densely fill the shaded cone, with $1/2$
BPS states at the base and apex and $1/4$ BPS states in between. The ball of
radius $1/\sqrt{d-2}=1/\sqrt{7}$ lies entirely within the cone, touching it
along the dashed red circle, which is itself densely populated by the
oscillator modes of $1/2$ BPS fundamental strings.

As illustrated in Figure 4, these $\vec{\zeta}$-vectors lie on a cone. At the
tip of the cone lie the 1/2 BPS states with $p=q=0$ and $w\neq 0$,
corresponding to an M2 brane wrapped $w$ times on $T^{2}$:

$\vec{\zeta}_{9}^{\text{winding}}=\left(\sqrt{\frac{8}{7}},0,0\right).$ (51)

The base of the cone is populated by the 1/2 BPS Kaluza-Klein modes with $w=0$
but nonzero $p$ or $q$:

$\vec{\zeta}_{9}^{\text{KK}}\in\left\\{\left(-\frac{3}{\sqrt{14}},\zeta_{\tau_{1}},\zeta_{\tau_{2}}\right)\in\mathbb{R}^{3}:\zeta_{\tau_{1}}^{2}+\zeta_{\tau_{2}}^{2}=\frac{1}{2}\right\\}.$
(52)

The remaining 1/4 BPS states lie along the cone somewhere between its tip and
circular base. From this example, we see that the convex hull generated by the
1/2 BPS states contains the convex hull generated by the 1/4 BPS states.

The purely winding $\vec{\zeta}_{9}^{\text{winding}}$-vectors are a distance
$\sqrt{\frac{8}{7}}$ from the origin, and so too are the Kaluza Klein mode
$\vec{\zeta}_{9}^{\text{KK}}$-vectors. Thus, the points on the cone closest to
the origin lie on a circle halfway between the base and apex. The radius of
this circle is $\lambda_{\text{min}}=1/\sqrt{d-2}$, saturating our proposed
bound.

Note that wrapping an M2 brane on the $(p,q)$ cycle of the torus gives rise to
$1/2$ BPS fundamental strings of tension

$T_{\textrm{string}}=\frac{R^{1/3}}{2\pi}\frac{|p+\tau
q|}{\sqrt{\tau_{2}}}e^{\frac{3U}{14}}\,.$ (53)

The corresponding string oscillator modes of mass $m\propto\sqrt{2\pi
T_{\textrm{string}}}$ densely populate the $\lambda=1/\sqrt{d-2}$ circle where
the bound is saturated. Thus, when our bound is saturated, there is an
associated string scale, as before.

###  4.3 Eight Dimensions

Reducing the M-theory effective action

$S_{11}=\frac{1}{2\kappa_{11}^{2}}\int
d^{11}x\sqrt{-g}\left(\mathcal{R}-\frac{1}{2}|F_{4}|^{2}\right)-\frac{1}{12\kappa_{11}^{2}}\int
C_{3}\wedge F_{4}\wedge F_{4}\,,\qquad F_{4}=dC_{3}\,,$ (54)

with the ansatz (45), we obtain the Einstein-moduli sector in $d=8$
dimensions,

$\displaystyle S_{d}$ $\displaystyle=\frac{1}{2\kappa_{d}^{2}}\int
d^{d}x\sqrt{-g}\bigg{(}\mathcal{R}-\frac{1}{4}\left(g^{mm^{\prime}}g^{nn^{\prime}}+\frac{1}{d-2}g^{mn}g^{m^{\prime}n^{\prime}}\right)\partial
g_{mn}\cdot\partial g_{m^{\prime}n^{\prime}}$
$\displaystyle\mathrel{\phantom{=}}-\frac{1}{2\times
3!}g^{mm^{\prime}}g^{nn^{\prime}}g^{pp^{\prime}}(\partial C_{mnp}\cdot\partial
C_{m^{\prime}n^{\prime}p^{\prime}})\bigg{)}\,,$ (55)

where $g_{mn}=e^{U}\hat{g}_{mn}$, $g^{mn}$ is its matrix inverse, and
$C_{mnp}$ are the components of $C_{3}$ along the compact directions.

The BPS states are characterized by the integral Kaluza-Klein momenta $N_{m}$
as well as the integral M2 brane wrapping numbers $W^{mn}=-W^{nm}$ around the
various cycles of the three-torus, and 1/2 BPS states exist when
$N_{m}W^{mn}=0$. In terms of the rescaled quantities

$n_{m}=\frac{N_{m}}{R}\,,\qquad w^{mn}=R^{\frac{d-5}{3}}W^{mn}\,,$ (56)

the 1/2 BPS mass formula is Obers:1998fb

$m^{2}=\|g\|^{-\frac{1}{d-2}}\left(Z_{m}Z^{m}+\frac{1}{2}{w}_{mn}{w}^{mn}\right),\quad
Z_{m}={n}_{m}+\frac{1}{2}C_{mnp}{w}^{np},\quad Z_{n}{w}^{mn}=0,$ (57)

again in modified 8d Planck units $2\kappa_{d}^{2}=(2\pi)^{d-3}$, where
indices are raised and lowered using $g_{mn}$ and $\|g\|=\det g_{mn}$.

At this point, it is expedient to specialize more particularly to the case
$d=8$. We define $w_{m}=\frac{1}{2}\varepsilon_{mnp}w^{np}$ where
$\varepsilon_{mnp}=\pm 1$ is the Levi-Civita symbol on the $k=3$ compact
directions. Then in terms of $C$ such that $C_{mnp}=\varepsilon_{mnp}C$, the
1/2 BPS mass formula becomes

$m^{2}=\|g\|^{-\frac{1}{6}}((\vec{n}+C\vec{w})^{2}+\|g\|\vec{w}^{2})\,,\qquad\varepsilon^{mnp}n_{n}w_{p}=0,$
(58)

using the notation $\vec{v}^{2}=v_{p}v^{p}$. Furthermore, the shortening
condition $\varepsilon^{mnp}n_{n}w_{p}=0$ implies that $\vec{n}$ and $\vec{w}$
are proportional, so $\vec{w}=\frac{1}{r}\vec{n}$ for some $r\in\mathbb{R}$.
Thus,

$m^{2}=\|g\|^{-\frac{1}{6}}\bigl{[}(r+C)^{2}+\|g\|\bigr{]}\frac{\vec{n}^{2}}{r^{2}}\,.$
(59)

In terms of the moduli

$\phi^{a}=\frac{1}{2\kappa_{8}}\begin{pmatrix}g_{mn}\\\ C\end{pmatrix},$ (60)

we obtain

$\displaystyle\vec{\zeta}(\theta,\hat{n})$
$\displaystyle=\begin{pmatrix}\zeta^{mn}\\\
\zeta\end{pmatrix}=\begin{pmatrix}\frac{1}{2}\cos(\theta)g^{mn}+\frac{1}{3}g^{mn}-\hat{n}^{m}\hat{n}^{n}\\\
\sin(\theta)\end{pmatrix},$ (61a)
$\displaystyle\vec{\zeta}\cdot\vec{\tilde{\zeta}}$
$\displaystyle=G^{ab}\zeta_{a}\tilde{\zeta}_{b}=\zeta^{mn}\tilde{\zeta}_{mn}-\frac{1}{9}\zeta^{m}_{m}\tilde{\zeta}^{n}_{n}+\frac{1}{2}\zeta\tilde{\zeta},$
(61b)

where $\theta=2\arctan(r+C)$, $\hat{n}=\vec{n}/\sqrt{\vec{n}^{2}}$, $G^{ab}$
is the inverse metric on moduli space read off from the effective action (55),
and we set $\langle\|g\|\rangle=1$ (i.e., $\langle U\rangle=0$) after taking
the moduli derivatives.

Notice that $\vec{\zeta}(\theta,\hat{n})=\vec{N}(\hat{n})+\vec{T}(\theta)$
decomposes into pieces $\vec{N}(\hat{n})$ and $\vec{T}(\theta)$ depending only
on $\hat{n}$ and $\theta$, respectively, where

$\displaystyle\vec{N}(\hat{n})$
$\displaystyle\mathrel{:=}\begin{pmatrix}\frac{1}{3}g^{mn}-\hat{n}^{m}\hat{n}^{n}\\\
0\end{pmatrix},$ $\displaystyle\vec{T}(\theta)$
$\displaystyle\mathrel{:=}\begin{pmatrix}\frac{1}{2}\cos(\theta)g^{mn}\\\
\sin(\theta)\end{pmatrix},$ (62) and $\displaystyle|\vec{N}(\hat{n})|^{2}$
$\displaystyle=\frac{2}{3}\,,\qquad\vec{N}(\hat{n})\cdot\vec{T}(\theta)=0\,,$
$\displaystyle|\vec{T}(\theta)|^{2}$
$\displaystyle=\frac{1}{2}\,,\qquad\vec{\zeta}^{2}(\theta,\hat{n})=\frac{7}{6}\,.$
(63)

Thus, the convex hull of 1/2 BPS $\vec{\zeta}$ vectors is the product of the
radius $1/\sqrt{2}$ circle around the origin traced out by $\vec{T}(\theta)$
with the convex hull traced out by $\vec{N}(\hat{n})$ in the remaining five
directions orthogonal to this circle. In particular, $\lambda_{\text{min}}$
for the overall convex hull is the smaller of $1/\sqrt{2}$
($\lambda_{\text{min}}$ for the circle $\vec{T}(\theta)$) and
$\lambda_{\text{min}}$ for the convex hull of the $\vec{N}(\hat{n})$.

To determine the latter, we first outline a general strategy for obtaining
$\lambda_{\text{min}}$ from a set of $\vec{\zeta}$-vectors that we will use
repeatedly. Suppose that $\Delta$ is the convex hull generated by a set of
points $\mathcal{B}=\\{\vec{e}_{i}\\}$. To find $\lambda_{\text{min}}$ for
$\Delta$, it is sufficient to find

$\lambda(\hat{n})=\max_{\vec{e}\in\mathcal{B}}(\vec{e}\cdot\hat{n}),$ (64)

and then minimize over different choices of $\hat{n}$. Alternatively, for each
direction $\hat{n}$, we can choose $\vec{\pi}\propto\hat{n}$ such that

$\vec{e}_{i}\cdot\vec{\pi}\leq 1$ (65)

for all $\vec{e}_{i}\in\mathcal{B}$, where _at least one_ $\vec{e}_{i}$
saturates the bound. Then $|\vec{\pi}|=\frac{1}{\lambda(\hat{n})}$ and we find
$\lambda_{\text{min}}$ by _maximizing_ $|\vec{\pi}|$ as we vary $\hat{n}$.

Applying the latter method to the case at hand, consider the vector space of
traceless, symmetric $k\times k$ matrices with the inner product
$\vec{M}\cdot\vec{N}=\text{Tr}(MN)$, with $\mathcal{B}$ consisting of those of
the form

$E^{mn}=\hat{e}^{m}\hat{e}^{n}-\frac{1}{k}\delta^{mn}$ (66)

for any unit vector $\hat{e}^{m}$. Choosing any direction $\hat{\Pi}^{mn}$, we
use an $O(k)$ transformation to diagonalize $\hat{\Pi}^{mn}$, so that

$\Pi^{mn}=\mathrm{diag}(\lambda_{1},\dots,\lambda_{k}),\qquad\sum_{i=1}^{k}\lambda_{i}=0.$
(67)

We impose $\text{Tr}(\Pi^{mn}E_{mn})=\hat{e}_{m}\Pi^{mn}\hat{e}_{n}\leq 1$
with at least one unit vector giving equality. This is nothing but the
condition $\lambda_{i}\leq 1$ with at least one equality, i.e.,

$\Pi^{mn}=\mathrm{diag}(1,\lambda_{2},\dots,\lambda_{k}),\qquad\lambda_{2},\dots,\lambda_{k}\leq
1,\qquad\sum_{i=2}^{k}\lambda_{i}=1.$ (68)

Now we want to scan over directions to maximize $\Pi^{2}$, per the strategy
explained above. Note that for two variables $x$ and $y$ with a fixed total
$t$, $x^{2}+y^{2}=x^{2}+(t-x)^{2}$ is _minimized_ when $x=y=t/2$, and
_increases_ as the difference between $x$ and $y$ increases (in either
direction). Thus, for any pair of $\lambda_{i}$’s, we can increase $\Pi^{2}$
by increasing their difference while maintaining
$\sum_{i=2}^{k}\lambda_{i}=-1$, until we saturate one of the inequalities
$\lambda_{2},\dots,\lambda_{k}\leq 1$. Thus, there is always a way to increase
$\Pi^{2}$ unless _all but one_ of the $\lambda_{i}$’s is equal to 1, i.e., the
longest $\Pi^{mn}$ will be of the form

$\Pi^{mn}_{\text{max}}=\mathrm{diag}(1,\dots,1,-(k-1))\quad\Rightarrow\quad\Pi^{2}_{\text{max}}=(k-1)+(k-1)^{2}=k(k-1),$
(69)

and so

$\lambda_{\text{min}}=\frac{1}{\sqrt{k(k-1)}}=\frac{1}{\sqrt{6}},$ (70)

since $k=3$ in the case of interest. The convex hull of the $\vec{N}(\hat{n})$
vectors is illustrated schematically in Figure 5.

Figure 5: A schematic representation of the convex hull of the
$\vec{N}(\hat{n})$ vectors. Only the moduli arising from the diagonal
components of $\hat{g}_{mn}$ are depicted and likewise only the points
corresponding to $\hat{n}=(1,0,0)$, $\hat{n}=(0,1,0)$ and $\hat{n}=(0,0,1)$
are shown,

Since $1/\sqrt{6}<1/\sqrt{2}$, we conclude that
$\lambda_{\text{min}}=1/\sqrt{6}=1/\sqrt{d-2}$ for 8d maximal SUGRA, again
saturating our bound. As before, there are fundamental strings arising from
wrapping M2 branes on the various one-cycles of $T^{3}$, with tension

$T_{\text{string}}=\frac{\|g\|^{-1/6}}{2\pi}\sqrt{g_{mn}W^{m}W^{n}}\,,$ (71)

where the integers $W^{m}$ describe the cycle in question. The corresponding
string oscillator modes $m\propto\sqrt{2\pi T_{\textrm{string}}}$ have
$\vec{\zeta}$ vectors

$\vec{\zeta}(\hat{W})=\begin{pmatrix}-\frac{1}{6}g^{mn}+\frac{1}{2}\hat{W}^{m}\hat{W}^{n}\\\
0\end{pmatrix},\\\ $ (72)

and indeed these are precisely the directions in which our bound was
saturated. Thus, there is a string scale associated to each such direction, as
before.

###  4.4 Seven Dimensions

The 7d Einstein-moduli action is still given by (55), now with $d=7$.
Likewise, the $1/2$ BPS mass formula (57) is still valid, except that an
additional shortening condition $w^{[mn}w^{pq]}=0$ (trivial for $k<4$) now
comes into play. Defining $v_{mn}=\frac{1}{2}\varepsilon_{mnpq}w^{pq}$ and
$C^{q}$ such that $C_{mnp}=\epsilon_{mnpq}C^{q}$, we obtain

$m^{2}=\|g\|^{-\frac{1}{5}}\left(Z_{m}Z^{m}+\frac{\|g\|}{2}{v}_{mn}{v}^{mn}\right),\quad
Z_{m}={n}_{m}+v_{mq}C^{q},$ (73)

with the $1/2$ BPS shortening conditions

$\varepsilon^{mnpq}Z_{n}v_{pq}=0,\qquad\varepsilon^{mnpq}v_{mn}v_{pq}=0.$ (74)

Thus, in terms of the moduli

$\phi^{a}=\frac{1}{2\kappa_{7}}\begin{pmatrix}g_{mn}\\\ C^{m}\end{pmatrix},$
(75)

we obtain

$\displaystyle\vec{\zeta}$ $\displaystyle=\begin{pmatrix}\zeta^{mn}\\\
\zeta_{m}\end{pmatrix}=\begin{pmatrix}-\frac{1}{5}\delta^{mn}+\frac{\delta^{mn}v^{2}-Z^{m}Z^{n}-v^{mp}v^{n}\,_{p}}{Z^{2}+v^{2}}\\\
2\frac{Z^{n}v_{nm}}{Z^{2}+v^{2}}\end{pmatrix},$ (76a)
$\displaystyle\vec{\zeta}\cdot\vec{\tilde{\zeta}}$
$\displaystyle=\zeta_{mn}\tilde{\zeta}^{mn}-\frac{1}{9}\zeta^{m}_{m}\tilde{\zeta}^{n}_{n}+\frac{1}{2}\zeta_{m}\tilde{\zeta}^{m}.$
(76b)

where $Z^{2}\mathrel{:=}Z_{m}Z^{m}$,
$v^{2}\mathrel{:=}\frac{1}{2}v_{mn}v^{mn}$, and we choose a basis where
$g_{mn}=\delta_{mn}$ for simplicity. Note that for $Z_{m}\neq 0$ the $1/2$ BPS
shortening conditions can be solved by $v_{mn}=Z_{m}u_{n}-Z_{n}u_{m}$ for some
$u_{m}$ that can be chosen to be orthogonal to $Z_{m}$ without loss of
generality. One can easily check that this leads to $\vec{\zeta}^{2}=6/5$,
which also holds when $Z_{m}=0$.

To find the convex hull of the $1/2$ BPS states, we simplify the above
expressions by encapsulating $\zeta^{mn}$ and $\zeta_{m}$ into a single
$5\times 5$ symmetric traceless matrix

$\zeta^{ab}=\left(\begin{array}[]{cc}\zeta^{mn}-\frac{1}{3}\zeta^{p}_{p}\delta^{mn}&\frac{1}{2}\zeta^{n}\\\
\frac{1}{2}\zeta^{m}&\frac{1}{3}\zeta^{m}_{m}\end{array}\right),$ (77)

so that

$\zeta_{ab}\tilde{\zeta}^{ab}=\zeta_{mn}\tilde{\zeta}^{mn}-\frac{1}{9}\zeta^{m}_{m}\tilde{\zeta}^{n}_{n}+\frac{1}{2}\zeta_{m}\zeta^{m}=\vec{\zeta}\cdot\vec{\tilde{\zeta}}\,.$
(78)

For the $1/2$ BPS states, we find explicitly:

$\zeta^{ab}=\begin{pmatrix}\frac{2}{5}\delta^{mn}-\frac{Z^{m}Z^{n}+v^{mp}v^{n}\,_{p}}{Z^{2}+v^{2}}&\frac{Z^{n}v_{nm}}{Z^{2}+v^{2}}\\\
\frac{Z^{n}v_{nm}}{Z^{2}+v^{2}}&\frac{2}{5}-\frac{Z^{2}}{Z^{2}+v^{2}}\end{pmatrix}.$
(79)

Defining the $5\times 5$ antisymmetric matrix,

$V_{ab}=\begin{pmatrix}v_{mn}&-Z_{m}\\\ Z_{n}&0\end{pmatrix},$ (80)

we see that the above is the same as

$\zeta^{ab}=\frac{2}{5}\delta^{ab}-\frac{V^{ac}V^{b}\,_{c}}{V^{2}},\qquad
V^{2}\equiv\frac{1}{2}V_{ab}V^{ab},$ (81)

where the $1/2$ BPS shortening conditions are now

$\varepsilon^{abcde}V_{bc}V_{de}=0.$ (82)

The simplicity of these expressions is a manifestation of U-duality, where the
7d U-duality group is $SL(5,\mathbb{Z})$, which enhances to $SL(5,\mathbb{R})$
in the low-energy effective action (ignoring charge quantization).

The shortening condition (82) implies that $V_{ab}=X_{a}Y_{b}-X_{b}Y_{a}$ has
rank two. In particular, choosing $X_{a}$ and $Y_{a}$ to be orthogonal without
loss of generality,

$\Pi^{a}_{b}=\frac{V^{ac}V_{bc}}{V^{2}}=\hat{X}^{a}\hat{X}_{b}+\hat{Y}^{a}\hat{Y}_{b},$
(83)

is a rank-two projection matrix, i.e., satisfying $\Pi^{2}=\Pi$ and
$\operatorname{Tr}\Pi=2$. Thus,

$\zeta^{ab}=\frac{2}{5}\delta^{ab}-\Pi^{ab}\qquad\Rightarrow\qquad\zeta_{ab}\zeta^{ab}=\text{Tr}\left[\frac{4}{25}\mathbf{1}+\frac{1}{5}\Pi\right]=\frac{4}{5}+\frac{2}{5}=\frac{6}{5}.$
(84)

Now consider an arbitrary symmetric traceless matrix $P^{ab}$, and diagonalize

$P^{ab}=\left(\begin{array}[]{ccc}\lambda_{1}&&\\\ &\ddots&\\\
&&\lambda_{5}\end{array}\right),\qquad\sum_{i=1}^{5}\lambda_{i}=0,\qquad\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{5}.$
(85)

We have

$P\cdot\zeta=-\sum_{a}\lambda_{a}\Pi_{aa}.$ (86)

The rank-two projector is constrained by $\Pi_{aa}\leq 1$ (no sum) and
$\sum_{a}\Pi_{aa}=2$, so

$P\cdot\zeta\geq-\lambda_{1}-\lambda_{2},$ (87)

and the minimum value is achieved for $\Pi_{ab}=\text{diag}(1,1,0,0,0)$. Thus,
we set $\lambda_{1}+\lambda_{2}=1$ to obtain $P\cdot\zeta\geq-1$ with the
bound saturated in at least one direction. We then have:

$\displaystyle\begin{aligned}
P^{ab}&=\text{diag}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}),\qquad\lambda_{1}+\lambda_{2}=1,\\\
\lambda_{3}+\lambda_{4}+\lambda_{5}&=-1,\qquad\min(\lambda_{1,2})\geq\max(\lambda_{3,4,5}).\end{aligned}$
(88)

We want to maximize
$P^{ab}P_{ab}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}+\lambda_{4}^{2}+\lambda_{5}^{2}$
subject to these constraints. Recall that for fixed $x+y=t$, $x^{2}+y^{2}$
_increases_ as the difference between $x$ and $y$ increases. Thus, for fixed
$\lambda_{1,2}$ it is optimal to saturate the bound
$\min(\lambda_{1,2})\geq\max(\lambda_{3,4,5})$ with two out of three of
$\lambda_{3}$, $\lambda_{4}$ and $\lambda_{5}$, so that (again taking
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{5}$ without loss of
generality)

$P^{ab}=\text{diag}(1-\lambda_{2},\lambda_{2},\lambda_{2},\lambda_{2},-1-2\lambda_{2}),\qquad\frac{1}{2}\geq\lambda_{2}\geq-\frac{1}{3}.$
(89)

Now we just have to maximize

$P_{ab}P^{ab}=(1-\lambda_{2})^{2}+3\lambda_{2}^{2}+(1+2\lambda_{2})^{2},\qquad\text{for}\qquad\frac{1}{2}\geq\lambda_{2}\geq-\frac{1}{3}.$
(90)

The maximum value occurs at $\lambda_{2}=1/2$, i.e., for

$P^{ab}=\text{diag}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-2\right).$
(91)

We find $P^{2}_{\text{max}}=\frac{4}{4}+4=5$, hence
$\lambda_{\text{min}}=\frac{1}{\sqrt{5}}=\frac{1}{\sqrt{d-2}}$, once again
saturating our bound.

As before, there are BPS fundamental strings arising from M2 branes wrapping
the various one-cycles of $T^{4}$, but now there are also BPS fundamental
strings arising from M5 branes wrapping $T^{4}$; these are characterized by
integer winding numbers $W^{m}$ and $W$, respectively. The two are intermixed
by U-duality, so we define

$U^{a}=\begin{pmatrix}w^{m}+wC^{m}\\\ w\end{pmatrix}\,\qquad\text{where
$w^{m}=R^{-1/3}W^{m}$ and $w=R^{4/3}W$.}$ (92)

Here the relative powers of $R$ and the term involving $C^{m}$ are both fixed
by the fact that in an alternate basis upon which $SL(5,\mathbb{Z})$ naturally
acts,
$V_{ab}=\bigl{(}\begin{smallmatrix}\frac{1}{2}\varepsilon_{mnpq}W^{pq}&-N_{m}\\\
N_{n}&0\end{smallmatrix}\bigr{)}$ and
$U^{a}=\bigl{(}\begin{smallmatrix}W^{m}\\\ W\end{smallmatrix}\bigr{)}$ both
have integral components. Thus, U-duality together with a few easily-analyzed
special cases fixes the tension formula for $1/2$ BPS strings

$T_{\text{string}}=\frac{1}{2\pi}\sqrt{\delta_{ab}U^{a}U^{b}}=\frac{\|g\|^{-\frac{1}{5}}}{2\pi}\sqrt{(w^{m}+wC^{m})^{2}+\|g\|w^{2}}\,,$
(93)

where in the second equality we return to an arbitrary basis where
$g_{mn}\neq\delta_{mn}$, restoring the correct factors of $\|g\|$ by
considering the special cases of wrapped M2 branes and of wrapped M5 branes
with $C_{mnp}=0$. A straightforward calculation then gives

$\zeta^{ab}=-\frac{1}{10}\delta^{ab}+\frac{1}{2}\frac{U^{a}U^{b}}{U^{2}}\,,$
(94)

for the string oscillator modes. These are precisely the directions that
saturated the bound $\lambda\geq 1/\sqrt{d-2}$ above, hence there is a string
scale associated to each such direction as before.

###  4.5 Six, Five, and Four Dimensions

As seen above, U-duality plays an increasingly important role as we compactify
further. In Appendix A we use an approach that incorporates U-duality from the
start to show that the formula $\lambda_{\text{min}}=1/\sqrt{d-2}$ persists
for $d\in\\{4,5,6\\}$.

##  5 Bottom-Up Evidence in Minimal Supergravity

In the previous section, we saw that our proposed bound $\lambda\geq
1/\sqrt{d-2}$ is saturated in maximal supergravity in dimensions $d=\text{4 --
10}$. In this section, we present further evidence that this bound is
saturated in minimal supergravity in diverse dimensions. As the title of the
section suggests, our analysis here proceeds by a bottom-up approach: with
only a couple of exceptions, we will not study UV complete string/M-theory
compactifications. Instead, following the approach of Gendler:2020dfp  in four
dimensions, we study the scaling behavior of gauge couplings in infinite-
distance limits in moduli space. Invoking the tower Weak Gravity Conjecture or
the Weak Gravity Conjecture for strings then implies a tower of light charged
particles/string oscillator modes in the limit of vanishing gauge couplings,
and by working out the scaling of the gauge couplings with proper field
distance, we may in turn determine the scaling of the particle masses in this
limit.

In the remainder of this section, we will find no counterexamples to the bound
$\lambda\geq 1/\sqrt{d-2}$, and we will find many examples in which this bound
is saturated by oscillator modes of a charged string. We will also find many
examples of decompactification limits, in which one tower satisfies the
Distance Conjecture with $\lambda=\sqrt{(d-1)/(d-2)}$ (as expected for a tower
of Kaluza-Klein modes under dimensional reduction), while another satisfies
the bound with $\lambda=1/\sqrt{(d-1)(d-2)}$ (as expected for the Kaluza-Klein
zero modes of a $D$-dimensional tower after dimensional reduction to $d=D-1$
dimensions). This provides support not only for our bound (2), but also for
the Emergent String Conjecture, which holds than any infinite-distance limit
is either a decompactification limit or an emergent string limit.

This approach relies on several important assumptions. First of all, it relies
on the assumption of the tower Weak Gravity Conjecture and the Weak Gravity
Conjecture for strings, but given the vast body of evidence in favor of these
conjectures (see e.g. Heidenreich:2015nta ; Heidenreich:2016aqi ;
Montero:2016tif ; Andriolo:2018lvp ; Lee:2018urn ; Lee:2019tst ;
Klaewer:2020lfg ; Alim:2021vhs ), this seems to be a relatively minor
assumption.

Secondly, this approach relies on the assumption that the tensionless string
which emerges in the weak coupling limit of a 2-form gauge field is a
fundamental string, meaning that its core probes the deep ultraviolet.
Otherwise, one would not expect an infinite tower of string oscillator modes,
but merely a finite tower. This assumption follows from the Emergent String
Conjecture Lee:2019wij  and the Distant Axionic String Conjecture
Lanza:2021qsu , and it is satisfied in many examples in string theory
Lee:2018urn ; Lee:2019xtm ; Lee:2019xtm ; Lanza:2021qsu ; BPSStrings , so it
appears to be a valid assumption.

Finally, and perhaps most significantly, our discussion in this section will
ignore questions of charge quantization, focusing instead on the scaling
behavior of gauging couplings in the classical action in asymptotic limits of
moduli space. Given a gauge kinetic matrix $a_{IJ}$, we may compute the
eigenvalues of this matrix as a function of the moduli in the theory, and we
may say that a gauge coupling vanishes when some eigenvalue of $a_{IJ}$
diverges. However, the eigenvector associated with the gauge coupling is
generically an axion-dependent quantity, so the linear combination of gauge
fields whose coupling vanishes in the infinite-distance limit is a function of
these axions. The presence of these axions does not necessarily present a
problem, since the axions appearing in such a linear combination may be fixed
to particular values in an asymptotic limit. However, for certain values of
the axions, there may not be any particles charged solely under the weakly
coupled gauge field in question, due to charge quantization.

For instance, consider two 1-form gauge fields $A_{1}^{1}$, $A_{1}^{2}$ with
field strengths $F_{2}^{i}=\textrm{d}A_{1}^{i}$ and electric charges quantized
in the $A^{1}$, $A^{2}$ basis as

$\frac{1}{g_{1}^{2}}\oint_{S^{d-2}}\star
F_{2}^{1},\frac{1}{g_{2}^{2}}\oint_{S^{d-2}}\star F_{2}^{2}\in\mathbb{Z}\,.$
(95)

We may then consider the linear combinations

$A_{1}^{+}=\cos(\theta)A_{1}^{1}+\sin(\theta)A_{1}^{2}\,,~{}~{}~{}A_{1}^{-}=-\sin(\theta)A_{1}^{1}+\cos(\theta)A_{1}^{2}\,,$
(96)

with associated gauge couplings

$g_{+}=\cos(\theta)g_{1}+\sin(\theta)g_{2}\,,~{}~{}~{}g_{-}=-\sin(\theta)g_{1}+\cos(\theta)g_{2}\,.$
(97)

Then, the limit $g_{+}\rightarrow 0$ is a weak coupling limit for the gauge
field $A_{1}^{+}$, and if $\tan(\theta)$ is rational, then the tower Weak
Gravity Conjecture implies a tower of light particles, charged under
$A_{1}^{+}$ but not $A_{1}^{-}$, whose masses vanish in the limit.

On the other hand, if $\tan(\theta)$ is irrational, then because of our charge
quantization condition (95), there can be no particles charged under
$A_{1}^{+}$ that are not also charged under $A_{1}^{-}$. As a result, the
tower Weak Gravity Conjecture does not imply a tower of particles whose masses
vanish in the limit $g_{+}\rightarrow 0$, as long as $g_{-}$ remains finite in
this limit.

In Alim:2021vhs ; BPSStrings , it was argued in the context of M-theory
compactifications to $d=5$ dimensions that the $g_{+}\rightarrow 0$ limit with
$\tan(\theta)$ rational corresponds to an infinite-distance limit of moduli
space, where the tower of light particles required by the tower Weak Gravity
Conjecture also satisfies the Distance Conjecture. In contrast, the
$g_{+}\rightarrow 0$ limit with $\tan(\theta)$ irrational corresponds to a
“periodic boundary”: a boundary of moduli space in which a compact scalar
field traverses its fundamental domain many times. Such a boundary is not at
infinite distance in moduli space, so the Distance Conjecture is satisfied
trivially despite the absence of a tower of light particles.

Guided by our understanding of 5d M-theory compactifications, we will assume
throughout this section that the presence of a vanishing gauge coupling
indicates either (a) an infinite-distance limit in moduli space in which the
tower Weak Gravity Conjecture leads to a tower of massless particles charged
solely under the weakly coupled gauge field or (b) a finite-distance, periodic
boundary of moduli space, in which the Distance Conjecture is satisfied
trivially, but there are no particles charged solely under the weakly coupled
gauge field. Confirming this assumption would require us to go beyond low-
energy supergravity and study these systems from a top-down approach in
string/M-theory. However, the absence of any counterexamples to the Distance
Conjecture in known string/M-theory compactifications offers solid
justification for our assumption.

###  5.1 Five Dimensions

In Corvilain:2018lgw ; Heidenreich:2020ptx , it was argued that every
infinite-distance point in vector multiplet moduli space is a point of
vanishing gauge coupling, and vice versa. Thus, to place an upper bound on the
Distance Conjecture coefficient $\lambda$ in five dimensions, we assume that
the tower weak gravity conjecture holds, and we study the scaling of the gauge
couplings at infinite distance in moduli space.

We begin by reviewing relevant aspects of supergravity in five dimensions,
following Alim:2021vhs . At a generic point in vector multiplet moduli space,
the action for the bosonic fields in a gauge theory with $n$ vector multiplets
is given by

$\displaystyle S$ $\displaystyle=\frac{1}{2\kappa_{5}^{2}}\int
d^{5}x\sqrt{-g}\left(\mathcal{R}-\frac{1}{2}\mathfrak{g}_{ij}(\phi)\partial\phi^{i}\cdot\partial\phi^{j}\right)-\frac{1}{2g_{5}^{2}}\int
a_{IJ}(\phi)F^{I}\wedge\star F^{J}$ $\displaystyle+\frac{1}{6(2\pi)^{2}}\int
C_{IJK}A^{I}\wedge F^{J}\wedge F^{K},$ (98)

where $I=0,\ldots,n$, $i=1,\ldots,n$, and
$g_{5}^{2}=(2\pi)^{4/3}(2\kappa_{5}^{2})^{1/3}$. The scalar metric
$\mathfrak{g}_{ij}(\phi)$, the gauge kinetic matrix $a_{IJ}(\phi)$, and the
Chern-Simons couplings $C_{IJK}$ are all determined by a prepotential
$\mathcal{F}[Y]$, which is a cubic in $Y^{I}$. We define
$\mathcal{F}_{I}\equiv\partial_{I}\mathcal{F}$,
$\mathcal{F}_{IJ}\equiv\partial_{I}\partial_{J}\mathcal{F}$ and
$\mathcal{F}_{IJK}\equiv\partial_{I}\partial_{J}\partial_{K}\mathcal{F}$. The
Chern-Simons couplings are determined by $C_{IJK}=\mathcal{F}_{IJK}$, and the
gauge kinetic matrix is given by

$a_{IJ}(\phi)=\mathcal{F}_{I}\mathcal{F}_{J}-\mathcal{F}_{IJ}\,.$ (99)

The vector multiplet moduli space corresponds to the slice $\mathcal{F}=1$.
The metric on vector multiplet moduli space is the pullback of $a_{IJ}$ to
this slice,

$\mathfrak{g}_{ij}=a_{IJ}\partial_{i}Y^{I}\partial_{j}Y^{J}\,.$ (100)

It is useful to work in homogenous coordinates, invariant under
$Y^{I}\rightarrow\lambda Y^{I}$. We may then drop the constraint
$\mathcal{F}=1$ and instead set

$a_{IJ}=\frac{\mathcal{F}_{I}\mathcal{F}_{J}}{\mathcal{F}^{4/3}}-\frac{\mathcal{F}_{IJ}}{\mathcal{F}^{1/3}}\,.$
(101)

In homogenous coordinates, the metric on scalar field space may be written as
Heidenreich:2020ptx

$\mathfrak{g}_{IJ}=\frac{2}{3}\frac{\mathcal{F}_{I}\mathcal{F}_{J}}{\mathcal{F}^{2}}-\frac{\mathcal{F}_{IJ}}{\mathcal{F}}\,.$
(102)

The distance of a path in moduli space, $\gamma=\gamma(s)$,
$s\in[s_{i},s_{f}]$, may then be written in homogenous coordinates as

$\ell=\frac{1}{\sqrt{2}\kappa_{5}}\int_{s_{i}}^{s_{f}}\textrm{d}s\sqrt{\mathfrak{g}_{IJ}\dot{Y}^{I}\dot{Y}^{J}}\,,$
(103)

where $\dot{Y}^{I}=\partial Y^{I}/\partial s$, and the factor
$1/\sqrt{2}\kappa_{5}$ comes from the prefactor in the action (98).

As noted in Heidenreich:2020ptx , a path $\gamma(s)$ approaching an infinite-
distance point $Y_{0}^{i}$ in homogenous coordinates may always be rescaled
via $Y^{I}(s)\rightarrow\lambda Y^{I}(s)$ to ensure that $Y_{0}^{I}$ remains
finite for all $I$, and $Y_{0}^{I}$ is nonzero for at least one $I$. The
condition that $Y_{0}^{i}$ lies at infinite distance then requires that
$\mathfrak{g}_{IJ}$ must diverge in the $Y^{I}\rightarrow Y_{0}^{I}$ limit.
Since $Y^{I}$ was assumed finite, $\mathcal{F}_{I}$ must also be finite, which
by (102) means that $\mathcal{F}$ must vanish in the limit.

We assume that the path is a straight line in homogeneous coordinates:

$Y^{I}=Y_{0}^{I}+sY_{1}^{I}\,,~{}~{}~{}~{}s\in[0,1]\,,$ (104)

such that $Y_{0}^{I}$ lies at infinite distance. Not all infinite-distance
limits take this straight-line form, but such paths offer a useful starting
point for analysis, and we will return to the more general case below. Note
that such straight lines necessarily remain within the moduli space due to the
convexity of the vector multiplet moduli space (in homogeneous coordinates)
for M-theory compactifications to five dimensions Alim:2021vhs . We further
assume that the path remains within a single Kähler cone, i.e., there are no
flop transitions. This assumption can be justified in M-theory
compactifications on Calabi-Yau threefolds if the strong birational cone
conjecture of BPSStrings  holds true. (This conjecture is a strengthening of
the birational cone conjecture of Morrison94 .)

By a suitable redefinition of coordinates, we may in fact set

$Y_{0}^{I}=\delta_{0}^{I},Y_{1}^{I}=\delta_{1}^{I}\,.$ (105)

We may then expand the prepotential along the path near $Y_{0}^{I}$ in powers
of $s$:

$\displaystyle\mathcal{F}$ $\displaystyle=\frac{1}{6}C_{IJK}Y^{I}Y^{J}Y^{K}$
$\displaystyle=\frac{1}{6}C_{IJK}Y_{0}^{I}Y_{0}^{J}Y_{0}^{K}+\frac{1}{2}C_{IJK}Y_{0}^{I}Y_{0}^{J}Y_{1}^{K}s+\frac{1}{2}C_{IJK}Y_{0}^{I}Y_{1}^{J}Y_{1}^{K}s^{2}+\frac{1}{6}C_{IJK}Y_{1}^{I}Y_{1}^{J}Y_{1}^{K}s^{3}$
(106)
$\displaystyle=\frac{1}{6}C_{000}+\frac{1}{2}C_{001}s+\frac{1}{2}C_{011}s^{2}+\frac{1}{6}C_{111}s^{3}\,.$

By the argument above, $\mathcal{F}$ must vanish as $s\rightarrow 0$ if we
assume that the $Y^{I}$ have been rescaled homogeneously so that $Y_{0}^{I}$
is finite for all $I$. This implies $C_{IJK}Y_{0}^{I}Y_{0}^{J}Y_{0}^{K}=0$,
which implies $\mathcal{F}\sim s^{m}$, for $m=1$, $2$, or $3$. It turns out
that $m=3$ is not at infinite distance, so we have only two options to
consider: i) $\mathcal{F}\sim s$ and ii) $\mathcal{F}\sim s^{2}$. For reasons
that will become clear shortly, we will refer to these as decompactification
limits and emergent string limits, respectively.

Decompactification limits: $\mathcal{F}\sim s$
At an asymptotic boundary corresponding to a decompactification limit, we have
$C_{000}=0$ but $C_{001}\neq 0$, so
$\mathcal{F}=\frac{1}{2}C_{001}Y_{0}^{I}Y_{0}^{J}Y_{1}^{K}s+O(s^{2})=\frac{1}{2}C_{001}s+O(s^{2})$.
We then have

$\displaystyle\mathcal{F}_{I}\dot{Y}^{I}=\mathcal{F}_{I}Y_{1}^{I}=\frac{1}{2}C_{IJK}Y_{0}^{I}Y_{0}^{J}Y_{1}^{K}=\frac{1}{s}\mathcal{F}+O(s)\,,$
(107)
$\displaystyle\mathcal{F}_{IJ}\dot{Y}^{I}\dot{Y}^{J}=\mathcal{F}_{IJ}Y_{1}^{I}Y_{1}^{J}=\frac{1}{2}C_{IJK}Y_{0}^{I}Y_{1}^{J}Y_{1}^{K}=O(s^{0})\,.$
(108)

Plugging these equations into (102) and using (103) gives in the limit
$\epsilon\rightarrow 0$,

$\ell(\epsilon)=\frac{1}{\sqrt{2}\kappa_{5}}\int_{\epsilon}^{1}\textrm{d}s\sqrt{\frac{2}{3s^{2}}+O(1/s)}=-\frac{1}{\sqrt{3}\kappa_{5}}\log(\epsilon)+O(\epsilon^{0})\,,$
(109)

so indeed, the point $Y_{0}^{I}$ is at infinite distance.

Meanwhile, from (106), we have

$\displaystyle\mathcal{F}_{0}=C_{001}s+O(s^{2})\,,~{}~{}$
$\displaystyle~{}\mathcal{F}_{1}=\frac{1}{2}C_{001}+O(s)\,,$ (110)
$\displaystyle\mathcal{F}_{00}=C_{001}s+O(s^{2})\,,~{}~{}~{}\mathcal{F}_{01}=C_{001}$
$\displaystyle+O(s)\,,~{}~{}~{}\mathcal{F}_{11}=C_{011}+O(s)\,,$ (111)

which by (101) gives

$a_{00}=(\sqrt{2}C_{001})^{2/3}s^{2/3}+O(s^{5/3})\,,~{}~{}~{}a_{11}=(C_{001}/2)^{2/3}s^{-4/3}+O(s^{5/3})\,,~{}~{}~{}a_{01}=O(s^{2/3})\,.$
(112)

By positive-definiteness of $a_{IJ}$, the scaling of $a_{00}$ and $a_{11}$
with $s$ implies that one eigenvalue of $a_{IJ}$ must scale as
$a_{\textrm{min}}\sim s^{2/3}$ while another scales at least as
$a_{\textrm{max}}\sim s^{-4/3}$ in the limit $s\rightarrow 0$. The eigenvalues
of $a_{IJ}$ are (up to normalization constants) simply the inverse-squares of
the gauge couplings. Thus, employing (109), we have in the limit
$\epsilon\rightarrow 0$:

$g_{\text{min}}\sim
a_{\textrm{max}}^{-1/2}\lesssim\epsilon^{2/3}\sim\exp\left(-\frac{2}{\sqrt{3}}\kappa_{5}\ell\right)\,.$
(113)

If the tower weak gravity conjecture is satisfied, we expect a tower of
particles charged under this gauge field with mass scale $m\lesssim
g_{\text{min}}$. The Distance Conjecture is satisfied in this case with a
coefficient

$\lambda=\frac{2}{\sqrt{3}}\,.$ (114)

This coefficient matches the value $\lambda=\sqrt{(d-1)/(d-2)}$ expected for
Kaluza Klein modes upon dimensional reduction from $D=6$ dimensions, hence
justifying our use of the term “decompactification limit.” Further
justification comes from noting that the vanishing eigenvalue
$a_{\textrm{min}}\sim s^{2/3}$ from (112) implies a diverging gauge coupling
$g_{\textrm{max}}\sim s^{-1/3}$ in the limit $s\rightarrow 0$. The magnetic
Weak Gravity Conjecture for this gauge field then implies a tensionless string
with string oscillator modes beginning at the scale

$M_{\textrm{string}}=\sqrt{2\pi
T_{\textrm{string}}}\sim\epsilon^{1/6}\sim\exp\left(-\frac{1}{2\sqrt{3}}\kappa_{5}\ell\right)\,.$
(115)

This coefficient $1/(2\sqrt{3})$ is precisely the value $1/\sqrt{(d-1)(d-2)}$
observed in (10): it is the expected scaling for the Kaluza-Klein zero modes
of a tower of particles in $D$ dimensions after reduction to $d$ dimensions.
Thus, assuming the tower Weak Gravity Conjecture and the magnetic Weak Gravity
Conjecture for strings in 5d, we find both of the towers expected in a Kaluza-
Klein decompactification to six dimensions.

Emergent string limit: $\mathcal{F}\sim s^{2}$
Next, we turn our attention to the other type of boundaries, which have

$\mathcal{F}=\frac{1}{2}C_{IJK}Y_{0}^{I}Y_{1}^{J}Y_{1}^{K}s^{2}+O(s^{3})=\frac{1}{2}C_{001}s^{2}+O(s^{3})\,.$
(116)

We then have

$\displaystyle\mathcal{F}_{I}\dot{Y}^{I}=\mathcal{F}_{I}Y_{1}^{I}=C_{IJK}Y_{0}^{J}Y_{1}^{K}Y_{1}^{I}s+O(s^{2})=\frac{2}{s}\mathcal{F}+O(s^{2})$
(117)
$\displaystyle\mathcal{F}_{IJ}\dot{Y}^{I}\dot{Y}^{J}=\mathcal{F}_{IJ}Y_{1}^{I}Y_{1}^{J}=C_{IJK}Y_{0}^{I}Y_{1}^{J}Y_{1}^{K}=\frac{2}{s^{2}}\mathcal{F}+O(s)\,.$
(118)

Plugging these equations into (102) and using (103) gives in the limit
$\epsilon\rightarrow 0$,

$\ell(\epsilon)=\frac{1}{\sqrt{2}\kappa_{5}}\int_{\epsilon}^{1}\textrm{d}s\sqrt{\frac{2}{3s^{2}}+O(1/s)}=-\frac{1}{\sqrt{3}\kappa_{5}}\log(\epsilon)+O(\epsilon^{0})\,,$
(119)

so indeed, the point $Y_{0}^{I}$ is at infinite distance along the path.

Meanwhile, we have

$\displaystyle\mathcal{F}_{0}=\frac{1}{2}C_{011}s^{2}\,,~{}$
$\displaystyle~{}~{}\mathcal{F}_{1}=C_{011}+O(s^{2})\,,$ (120)
$\displaystyle\mathcal{F}_{00}=0\,,~{}~{}~{}\mathcal{F}_{01}=C_{011}s$
$\displaystyle+O(s^{2})\,,~{}~{}~{}\mathcal{F}_{11}=C_{011}+O(s)\,,$ (121)

so by (101),

$a_{00}=(C_{011}/2)^{2/3}s^{4/3}+O(s^{7/3})\,,~{}~{}~{}a_{11}=(\sqrt{2}C_{011})^{2/3}s^{-2/3}+O(s^{5/3})\,,~{}~{}~{}a_{01}=O(s^{4/3})\,.$
(122)

By positive-definiteness of $a_{IJ}$, the scaling of $a_{00}$ and $a_{11}$
with $s$ implies that one eigenvalue of $a_{IJ}$ must scale as
$a_{\textrm{min}}\sim s^{4/3}$ while another scales at least as
$a_{\textrm{max}}\sim s^{-2/3}$ in the limit $s\rightarrow 0$. The eigenvalues
of $a_{IJ}$ are (up to normalization constants) simply the inverse-squares of
the gauge couplings. Thus, employing (109), the smallest gauge coupling scales
in the limit $\epsilon\rightarrow 0$ as:

$g_{\text{min}}\sim
a_{\textrm{max}}^{-1/2}\lesssim\epsilon^{1/3}\sim\exp\left(-\frac{1}{\sqrt{3}}\kappa_{5}\ell\right)\,.$
(123)

If the tower weak gravity conjecture is satisfied, we expect a tower of
particles charged under this gauge field with mass scale $m\lesssim
g_{\text{min}}$. The Distance Conjecture is satisfied in this case with a
coefficient

$\lambda=\frac{1}{\sqrt{3}}\,,$ (124)

thereby saturating our proposed bound $\lambda\geq 1/\sqrt{d-2}$.

Geometry | Prepotential $\mathcal{F}$ | Asymptotic Boundary | Type
---|---|---|---
Symmetric Flop | $\frac{1}{3}X^{3}+2X^{2}Y$ | $X\rightarrow 0$ | ES
GMSV | $\frac{5}{6}X^{3}+2X^{2}Y$ | $X\rightarrow 0$ | ES
$h^{1,1}=3$ KMV | $\frac{4}{3}X^{3}+\frac{3}{2}X^{2}Y+\frac{1}{2}XY^{2}$ | $X\rightarrow 0$ | Decomp.
$+X^{2}Z+XYZ$ |  $X,Y\rightarrow 0$, $X/Y$ fixed | ES
Table 1: Types of asymptotic boundaries (emergent string or
decompactification) for M-theory compactified on three examples of Calabi-Yau
geometries (the symmetric flop geometry Alim:2021vhs , the Greene-Morrison-
Strominger-Vafa geometry Greene:1995hu ; Greene:1996dh , and the $h^{1,1}=3$
Klemm-Mayr-Vafa geometry Klemm:1996hh ). Further details of these examples can
be found in Section 7 of Alim:2021vhs .

Evidence that this boundary corresponds to an emergent string limit can be
seen by further analyzing the largest gauge coupling of the system, which by
scales with the smallest gauge coupling as

$g_{\text{max}}\sim a_{\textrm{min}}^{-1/2}\kappa_{5}/g_{\text{min}}^{2}\,.$
(125)

In the $g_{\text{min}}\rightarrow 0$ limit, $g_{\text{max}}$ diverges, but the
magnetic Weak Gravity Conjecture suggests that a string charged magnetically
under this gauge field should have a tension bounded above as

$T_{\textrm{string}}\lesssim
1/(\kappa_{5}g_{\text{max}})\sim\epsilon^{2/3}\sim\exp\left(-\frac{2}{\sqrt{3}}\kappa_{5}\ell\right)\,,$
(126)

so we see that indeed, a tensionless string emerges in the limit
$g_{\text{min}}\rightarrow 0$. Furthermore, this string will have a tower of
string oscillator modes beginning at the mass scale
$M_{\text{string}}=\sqrt{2\pi T}\sim g_{\text{min}}/\kappa_{5}$, and it is
natural to identify this with the tower required by tower Weak Gravity
Conjecture for the gauge field with coupling $g_{\text{min}}$.

Indeed, the scaling $\mathcal{F}\sim s^{2}$ implies a nonzero Chern-Simons
coupling of the form $C_{0ij}$ for some $i,j>0$, which by anomaly inflow
Callan:1984sa  implies that a string charged magnetically under $A_{0}$ will
carry electric charge under $A_{i}$ and $A_{j}$. This string is precisely the
emergent string discussed above, which becomes tensionless in the
$g_{\text{min}}\rightarrow 0$ limit BPSStrings . For more details, see
Heidenreich:2021yda ; Kaya:2022edp  for the simple case of a theory with a
single vector multiplet or BPSStrings  for the more general case. Several
examples of asymptotic boundaries were discussed in Section 7 of Alim:2021vhs
. Table 1 classifies each of these boundaries by type.

Finally, we return to important point mentioned above: not every infinite-
distance geodesic in moduli space takes the form of a straight line in
homogeneous coordinates. However, our analysis of these straight line paths
has revealed that they have precisely the towers expected for a
decompactification limit and an emergent string limit, as discussed in §2.
Thus, we expect that any system featuring two or more of these straight-line
boundaries will essentially prove to be just a special case of the dilaton-
radion system studied above. Just as the convex hull condition was satisfied
for the system in §2.2, we expect that the convex hull condition will be
satisfied here, so that towers satisfying our bound $\lambda\geq 1/\sqrt{d-2}$
will appear in any infinite-distance limit in field space, including the non-
straight-line paths which approach an intermediate regime between an emergent
string boundary and a decompactification boundary.

Figure 6: A 5d supergravity theory with no decompactification limits. A theory
with multiple emergent string boundaries but no decompactification limits will
violate the convex hull condition for the Scalar Weak Gravity Conjecture in
some directions in scalar field space, and it will violate our Distance
Conjecture bound $\lambda\geq 1/\sqrt{d-2}$.

One interesting corollary of our analysis from §2.2 is that a single 5d
supergravity theory cannot have multiple (straight-line) emergent string
boundaries unless there is also a decompactification boundary: since the
former saturate our bound (2), the convex hull condition will be violated in
the intermediate regime between these two directions in field space, as shown
in Figure 6. It would be interesting to prove this statement, or at least to
confirm it in examples of 5d supergravity theories arising from M-theory
compactifications. More generally, such compactifications may offer a fertile
testing ground for our proposed bound (2).

###  5.2 Six Dimensions

To begin, we review relevant aspects of 6d supergravity coupled to abelian
gauge fields, following Riccioni:1999xq ; Riccioni:2001bg . A generic 6d
supergravity features one supergravity multiplet, $n_{T}$ tensor multiplets
and $n_{V}$ vector multiplets. It may also include hypermultiplets, but we do
not consider these in what follows. The supergravity multiplet includes the
metric and an anti-self-dual 2-form gauge field, but no scalar field. A tensor
multiplet features one self-dual 2-form gauge field and a scalar field. A
vector multiplet features a 1-form gauge field, but no scalar field.

As a result, a theory with $n_{T}$ tensor multiplets will have $n_{T}+1$
2-form gauge fields and $n_{T}$ scalar fields. These scalar fields parametrize
the tensor multiplet moduli space, which is the coset space
$SO(1,n_{T})/SO(n_{T})$. We may describe these in terms of the $SO(1,n_{T})$
matrix:

$V=\begin{pmatrix}v_{r}&\\\ x_{r}^{M}\end{pmatrix}$ (127)

where $r=0,1,...,n_{T}$ and $M=1,2,...,n_{T}$ . These are subject to the
conditions

$v^{r}v_{r}=1\,,~{}~{}~{}v_{r}x^{M}_{r}=0\,,~{}~{}~{}v_{r}v_{s}-x^{M}_{r}x^{M}_{s}=\eta_{rs}\,,$
(128)

where here, repeated indices are summed, and $r$ and $s$ indices are raised
and lowered via the metric $\eta_{rs}=\text{diag}(-1,1,1,...,1)$.

The gauge kinetic matrix for the tensor fields is then given by

$G=v_{r}v_{s}+x^{M}_{r}x^{M}_{s}\,.$ (129)

and the relevant part of the action is given by

$S=\frac{1}{2\kappa_{6}^{2}}\int\biggl{(}\mathcal{R}-\textrm{d}v^{r}\wedge\star\textrm{d}v_{r}-\frac{1}{2}G_{rs}H_{3}^{r}\wedge\star
H_{3}^{s}-\frac{1}{2}v_{r}c^{rab}F^{a}\wedge\star
F^{b}+\frac{1}{2}c_{r}^{ab}B_{2}^{r}\wedge F_{2}^{a}\wedge
F_{2}^{b}\biggr{)}.$ (130)

To analyze the general case with $n_{T}$ tensor multiples, we begin by taking
$V$ to be the identity matrix and perform boosts and rotations to get any
matrix in $SO(1,n_{T})$. Thus we have

$V=\Lambda\begin{pmatrix}v_{0,r}&\\\ x_{0,r}^{M}\end{pmatrix}$ (131)

where $v_{0}=e_{1}=\begin{pmatrix}1&0&0&...\end{pmatrix}$,
$x^{M}_{0}=e_{M+1}$, and $\Lambda$ is a Lorentz transformation. Without loss
of generality we can write $\Lambda$ as

$\Lambda=\biggl{(}B_{0}(\\{\phi_{k}\\})\biggl{)}\biggl{(}\prod_{i,j\neq
0}R_{ij}(\phi_{ij})\biggl{)}\,,$ (132)

where $R_{ij}$ are rotations on the plane of $i$th and $j$th axis and
$B_{0k}(\\{\phi_{k}\\})$ are boosts along the $k$th direction.

The kinetic matrix for the 2-form gauge fields then takes the form

$\displaystyle G_{rs}=v_{r}v_{s}+x^{M}_{r}x^{M}_{s}$
$\displaystyle=\Lambda^{k}_{r}\Lambda^{l}_{s}\biggl{(}(v_{0})_{k}(v_{0})_{l}+(x^{M}_{0})_{k}(x^{M}_{0})_{l})\biggl{)}=\Lambda^{k}_{r}\Lambda^{l}_{s}\delta_{kl}=(\Lambda^{T}\Lambda)_{rs}$
$\displaystyle=\biggl{(}\prod_{i,j\neq
0}R_{ij}\biggl{)}^{T}(B_{0})^{2}\biggl{(}\prod_{i,j\neq 0}R_{ij}\biggl{)}\,.$
(133)

since $B^{T}=B$ for any boost matrix. Since rotations do not change the
eigenvalues of a matrix, the eigenvalues of $G$ are equal to the eigenvalues
of $B_{0}^{2}$.

To find the eigenvalues of the boost matrix $B_{0}$, we first write it in
terms of the boost generators as

$B_{0}(\\{\phi_{k}\\})=e^{\phi_{i}K_{i}}=e^{A},$ (134)

where $K_{i}$ is the standard generator of boosts in the $i$th direction. The
eigenvalues of $G(\\{\lambda_{i}\\})$ are then given in terms of the
eigenvalues of $A(\\{a_{i}\\})$ as

$\displaystyle
a_{1}=\sqrt{\sum_{i}(\phi_{i})^{2}}=\phi\Rightarrow\lambda_{1}=e^{2\phi}$
$\displaystyle
a_{2}=-\sqrt{\sum_{i}(\phi_{i})^{2}}=-\phi\Rightarrow\lambda_{1}=e^{-2\phi}$
(135) $\displaystyle a_{i}=0\Rightarrow\lambda_{i}=1\text{ for }i>2\,,$

where we defined $\phi=\sqrt{\sum_{i}(\phi_{i})^{2}}$ as the magnitude of the
boost vector. In the limit $\phi\rightarrow\infty$, we see that one of the
eigenvalues of $G_{rs}$ diverges as $e^{2\phi}$, indicating a vanishing 2-form
gauge coupling in this limit.

The boosted vectors $v_{r},x^{M}$ are given by

$v_{r}=Be_{1}=\begin{pmatrix}\cosh(\phi)&\\\
\phi_{1}\frac{\sinh(2\phi)}{\phi}&\\\ \phi_{2}\frac{\sinh(2\phi)}{\phi}&\\\
...&\\\
\phi_{N-1}\frac{\sinh(2\phi)}{\phi}\end{pmatrix}\,,~{}~{}~{}~{}~{}(x^{M})_{r}=Be_{M+1}=\begin{pmatrix}\phi_{M}\frac{\sinh(2\phi)}{\phi}&\\\
\phi_{M}\phi_{1}\frac{\cosh(\phi)-1}{\phi^{2}}\\\ ...&\\\ \frac{\sum_{i\neq
M}(\phi_{i})^{2}+\phi_{M}^{2}\cosh(\phi)}{\phi^{2}}&\\\ ...&\\\
\phi_{M}\phi_{N-1}\frac{\cosh(\phi)-1}{\phi^{2}}\end{pmatrix}.$ (136)

Defining angular variables $r_{k}=\frac{\phi_{k}}{\phi}$, we may rewrite
$v_{r}$ as

$v_{r}=\begin{pmatrix}\cosh(\phi)&\\\ r_{1}\sinh(\phi)&\\\
r_{2}\sinh(\phi)&\\\ ...&\\\ r_{N-1}\sinh(\phi)&\\\ \end{pmatrix}\,.$ (137)

Note that $\phi$ is independent of $r_{i}$, and $\sum_{i}r_{i}^{2}=1$. We keep
$r_{i}$ constant and take the limit $\phi\rightarrow\infty$. With this, the
scalar kinetic term in the Lagrangian (130) takes the form

$-\frac{1}{2\kappa_{6}^{2}}\partial_{\mu}v^{r}\partial^{\mu}v_{r}=-\frac{1}{2\kappa_{6}^{2}}\frac{\partial
v^{r}}{\partial\phi}\frac{\partial
v_{r}}{\partial\phi}\partial_{\mu}\phi\partial^{\mu}\phi+...\,,$ (138)

from which we can read off

$g_{\phi\phi}=\frac{1}{\kappa_{6}^{2}}\frac{\partial
v^{r}}{\partial\phi}\frac{\partial
v_{r}}{\partial\phi}=\frac{1}{\kappa_{6}^{2}}\biggl{(}-\sinh^{2}(\phi)+\sum_{i=1}^{N-1}r_{i}^{2}\cosh^{2}(\phi)\biggl{)}=\frac{1}{\kappa_{6}^{2}}\,.$
(139)

So, $\phi\rightarrow\infty$ is indeed at infinite distance in moduli space,
and the canonically normalized scalar field is given by
$\hat{\phi}=\phi/\kappa_{6}$.

Assuming the Weak Gravity Conjecture is satisfied for the weakly coupled
2-form gauge field $B$ in the limit $\phi\rightarrow\infty$, we expect a
string with tension

$T_{\textrm{string}}\lesssim
g_{B}\sim\exp(-\phi)\sim\exp(-\kappa_{6}\hat{\phi})\,.$ (140)

The oscillator modes of this string will then give rise to a tower of light
particles at the string scale

$M_{\textrm{string}}=\sqrt{2\pi
T_{\textrm{string}}}\lesssim\sqrt{g_{B}}\sim\exp(\kappa_{6}\hat{\phi}/2)\,,$
(141)

so the Distance Conjecture is satisfied with

$\lambda=\frac{1}{2}=\frac{1}{\sqrt{d-2}}\biggl{|}_{d=6}\,,$ (142)

which saturates our proposed bound (2). An analogous computation applies in
the $\phi\rightarrow-\infty$ limit as well. Note that each of these limits
involve weakly coupled 2-form gauge fields, which by the Weak Gravity
Conjecture imply emergent tensionless strings, so once again the scaling
$\lambda=1/\sqrt{d-2}$ is characteristic of an emergent string boundary, as we
found in five dimensions above.

###  5.3 Seven Dimensions

Minimal 7d supergravity Bergshoeff:1985mr  features one supergravity multiplet
and $n$ vector multiplets. The supergravity multiplet has a graviton, one
scalar field, three abelian vector fields, and one 2-form gauge field, while
each vector multiplet has three scalar fields and one vector field. This means
that there are a total of $3n+1$ scalar fields: a dilaton $\sigma$, which
comes from the gravity multiplet, and $3n$ scalars $\phi^{\alpha}$, which come
from the vector multiplets and parametrize the coset $SO(3,n)/(SO(3)\times
SO(n))$.

The $3n$ scalars of the vector multiplets may then be thought of as boosts
$B_{ai}(\phi_{ai})$ in an ambient $\mathbb{R}^{3,n}$, with coordinates
$\\{t^{a}$, $x^{i}\\}$, $a=1,2,3$, $i=1,...,n$. An infinite-distance path in
the moduli space is a one-parameter family of boosts. As in the case of 6d
supergravity above, we suppose that this path takes the simple linear form
$\phi_{ai}=r_{ai}\phi$ for some constant $r_{ai}$. By an appropriate choice of
coordinate axes, we may in fact further assume
$r_{ai}=\delta_{a1}\delta_{i1}$, so the infinite-distance limit is simply the
limit of an infinite boost $B_{11}(\phi\rightarrow\infty)$ in the $t^{1}$,
$x^{1}$ plane of $\mathbb{R}^{3,n}$.

After this convenient choice of coordinates, the relevant part of the 7d
supergravity action is given by Kaya:2022edp :

$\displaystyle S$
$\displaystyle=\frac{1}{2\kappa_{7}^{2}}\int\Big{(}\mathcal{R}-\frac{1}{2}e^{\sigma}\left(F^{1}_{2}\wedge\star
F^{1}_{2}+F^{2}_{2}\wedge\star
F^{2}_{2}\right)-\frac{1}{2}e^{\sigma}\left(e^{2\phi}F^{+}_{2}\wedge\star
F^{+}_{2}+e^{-2\phi}F^{-}_{2}\wedge\star F^{{}_{2}-}\right)$
$\displaystyle-\frac{1}{2}e^{2\sigma}H_{3}\wedge\star
H_{3}-\frac{5}{4}\textrm{d}\sigma\wedge\star\textrm{d}\sigma-\textrm{d}\phi\wedge\star\textrm{d}\phi\Big{)}\,,$
(143)

where

$H_{3}=\textrm{d}B_{2}+\frac{1}{3\sqrt{2}}\left(A^{1}\wedge
F_{2}^{1}+A^{2}_{1}\wedge F_{2}^{2}+\frac{1}{2}(A^{+}_{1}\wedge
F_{2}^{-}+A^{-}_{1}\wedge F_{2}^{+})\right)\,.$ (144)

The limit $\sigma$ is then a weak coupling limit for the 2-form $B_{2}$, and
the Weak Gravity Conjecture implies a tower of string oscillator modes
beginning at the string scale

$M_{\textrm{string}}=\sqrt{2\pi T_{\textrm{string}}}\sim\exp(-\sigma/2)\,.$
(145)

After canonically normalizing the dilaton, this tower the Distance Conjecture
with a coefficient of

$\lambda=\frac{1}{\sqrt{5}}=\frac{1}{\sqrt{d-2}}\,,$ (146)

which saturates our bound (2), as expected for a tower of string oscillator
modes.

Meanwhile, the tower Weak Gravity Conjecture implies a tower of particles in
the weak coupling limits $\sigma\pm 2\phi\rightarrow\infty$ for the gauge
fields $A^{\pm}$. These towers have

$\lambda=\sqrt{\frac{6}{5}}\,,$ (147)

which is precisely what we expect for towers of Kaluza-Klein modes under
dimensional reduction, from (13).

Indeed, the parallels between this system and the one studied in §2 become
even clearer if we define canonically normalized scalar fields:

$\hat{\rho}\equiv\frac{1}{2\kappa_{7}}\sqrt{\frac{5}{6}}\left(\sigma+2\phi\right)\,,~{}~{}~{}\hat{\sigma}_{8}=\frac{1}{2\kappa_{7}}\frac{1}{\sqrt{6}}\left(-5\sigma+2\phi\right)\,.$
(148)

In terms of these fields, the string scale (145) is given by

$M_{\textrm{string}}\sim\exp\left(-\frac{1}{\sqrt{6}}\kappa_{7}\hat{\sigma}_{8}+\frac{1}{\sqrt{30}}\kappa_{7}\hat{\rho}\right)\,.$
(149)

This matches precisely with (10) upon setting $\lambda_{D}=1/\sqrt{D-2}$ and
taking $\hat{\rho}$ to be the radion while $\hat{\sigma}_{8}$ is the 8d
dilaton. Meanwhile, the Weak Gravity Conjecture tower for the gauge field
$A^{+}$ begins at a scale

$m\sim\exp\left(-\kappa_{7}\sqrt{\frac{6}{5}}\hat{\rho}\right)\,.$ (150)

which matches precisely with (13). In other words, the towers required by the
Weak Gravity Conjecture for $B$ and by the tower Weak Gravity Conjecture for
$A^{+}$ have precisely the scaling behavior expected upon dimensional
reduction of an 8d theory with a dilaton $\hat{\sigma}_{8}$ and a Distance
Conjecture tower with $\lambda_{8}=1/\sqrt{6}$. By a similar analysis for the
gauge field $A^{-}$, taking $\phi\rightarrow-\phi$, the same is true for the
limit $\phi\rightarrow-\infty$. As a result, the analysis of §2.2 implies that
the convex hull condition $\lambda\geq 1/\sqrt{5}$ will be satisfied in the
limit $\phi\rightarrow\pm\infty$, $\sigma\rightarrow\infty$ for any value of
the ratio $\phi/\sigma$. Minimal supergravity cannot tell us about the
strongly coupled $\sigma\rightarrow-\infty$ limit, however, so we must invoke
string dualities to cover this case.

Note that our analysis here also provides good evidence for the Emergent
String Conjecture, as all of the infinite-distance limits of moduli space in
seven dimensions we have introduce a light tower of fields whose masses scale
either as Kaluza-Klein modes or string oscillator modes.

###  5.4 Eight Dimensions

The case of 8d supergravity Awada:1985ag  is similar to 7d supergravity. It
features one supergravity multiplet and $n$ vector multiplets. The
supergravity multiplet has a graviton, one scalar field, two abelian vector
fields, and one 2-form gauge field, while each vector multiplet has two scalar
fields and one vector field. This means that there are a total of $2n+1$
scalar fields: a dilaton $\sigma$, which comes from the gravity multiplet, and
$2n$ scalars $\phi^{\alpha}$, which come from the vector multiplets and
parametrize the coset $SO(2,n)/(SO(2)\times SO(n))$.

Similar to 7d supergravity, we can think of the $2n$ scalar fields in the
vector multiplets as boosts $B_{ai}(\phi_{ai})$ in an ambient
$\mathbb{R}^{2,n}$, with coordinates $\\{t^{a}$, $x^{i}\\}$, $a=1,2$,
$i=1,...,n$. An infinite-distance path in the moduli space is a one-parameter
family of boosts. If we suppose that this path takes the simple linear form
$\phi_{ai}=r_{ai}\phi$ for some constant $r_{ai}$, then by appropriate choice
of axes we may set $r_{ai}=\delta_{a1}\delta_{i1}$, so the infinite-distance
limit is simply the limit of an infinite boost $B_{11}(\phi\rightarrow\infty)$
in the $t^{1}$, $x^{1}$ plane of $\mathbb{R}^{2,n}$.

Under this assumption, we may ignore the angular directions in scalar field
space (i.e., axions) and focus our attention on the scaling of gauge couplings
with the dilaton $\sigma$ and the radial mode $\phi$. The relevant part of the
supergravity action then takes the simple form

$\displaystyle S$
$\displaystyle=\frac{1}{2\kappa_{8}^{2}}\int\Big{(}\mathcal{R}-\frac{1}{2}e^{\sigma}\left(F^{1}_{2}\wedge\star
F^{1}_{2}\right)-\frac{1}{2}e^{\sigma}\left(e^{2\phi}F^{+}_{2}\wedge\star
F^{+}_{2}+e^{-2\phi}F^{-}_{2}\wedge\star F^{{}_{2}-}\right)$
$\displaystyle-\frac{1}{2}e^{2\sigma}H_{3}\wedge\star
H_{3}-\frac{3}{2}\textrm{d}\sigma\wedge\star\textrm{d}\sigma-\textrm{d}\phi\wedge\star\textrm{d}\phi\Big{)}\,.$
(151)

This action may be obtained simply by reducing pure supergravity in 9
dimensions and ignoring the axion $\theta$ associated with the holonomy of the
9d gauge field around the circle, setting the axion vev to zero. Consequently,
this system takes precisely the form studied in §2 above: the limit
$\sigma\rightarrow\infty$ is an emergent string limit as the 2-form $B_{2}$
becomes weakly coupled, and the tower of string oscillator modes satisfy the
Distance Conjecture with $\lambda=1/\sqrt{6}=1/\sqrt{d-2}$, saturating our
proposed bound. Meanwhile, the limit $\sigma\pm 2\phi\rightarrow 0$ is a
decompactification limit, associated with a tower of light Kaluza-Klein modes
with $\lambda=\sqrt{7/6}$ charged under the gauge fields $A^{\pm}$. The whole
system is consistent with the Emergent String Conjecture, and by our analysis
in §2, it is consistent with our proposed bound (2) as well.

###  5.5 Nine Dimensions

9d supergravity Gates:1984kr  features one supergravity multiplet and $n$
vector multiplets. The supergravity multiplet has a graviton, a 2-form gauge
field, a 1-form gauge field, and a scalar, while each vector multiplet has a
vector field and a scalar field. This means that there are a total of $n+1$
scalar fields: a dilaton $\sigma$, which comes from the gravity multiplet, and
$n$ scalars $\phi^{\alpha}$, which come from the vector multiplets and
parametrize the coset $SO(1,n)/SO(n)$.

The relevant part of the supergravity action takes the form

$S=\int\frac{1}{2\kappa_{9}^{2}}\Big{(}\mathcal{R}-\frac{1}{2}e^{-2\sigma}H_{3}\wedge\star
H_{3}-\frac{1}{2}e^{-\sigma}a_{IJ}F_{2}^{I}\wedge\star
F^{J}_{2}-g_{\alpha\beta}\textrm{d}\phi^{\alpha}\wedge\star\textrm{d}\phi^{\beta}-\frac{7}{4}\textrm{d}\sigma\wedge\star\textrm{d}\sigma\Big{)}\,.$
(152)

Here, the metric on vector multiplet moduli space is identical to the metric
on tensor multiplet moduli space in 6d supergravity considered above, as the
moduli spaces in question are identical. Likewise, the gauge kinetic term
$a_{IJ}$ is identical to the 2-form gauge kinetic term $G_{rs}$ in 6d
supergravity. Thus, borrowing our results from our 6d analysis, we have that
in an infinite-distance limit of the vector multiplet moduli space
$\phi\rightarrow\infty$, we have eigenvalues of $a_{IJ}$ given by

$\lambda_{1}=e^{2\phi}\,,~{}~{}~{}~{}~{}\lambda_{2}=e^{-2\phi}\,,~{}~{}~{}~{}~{}\lambda_{i}=1\,,~{}i>2\,,$
(153)

and the kinetic term for $g_{\phi\phi}$ is determined to be

$g_{\phi\phi}=1\,.$ (154)

As expected, this system takes the form studied in §2 above: the limit
$\sigma\rightarrow\infty$ is an emergent string limit as the 2-form $B_{2}$
becomes weakly coupled, and the tower of string oscillator modes satisfy the
Distance Conjecture with $\lambda=1/\sqrt{7}=1/\sqrt{d-2}$, saturating our
proposed bound. Meanwhile, the limit $\sigma\pm 2\phi\rightarrow 0$ is a
decompactification limit, associated with a tower of light Kaluza-Klein modes
with $\lambda=\sqrt{8/7}=\sqrt{(d-1)/(d-2)}$ charged under the gauge fields
$A^{\pm}$ associated to the eigenvalues $\lambda_{1}$ and $\lambda_{2}$. The
whole system is consistent with the Emergent String Conjecture, and by our
analysis in §2, it is consistent with our proposed bound (2) as well.

##  6  Discussion

In this paper, we have proposed and provided evidence for a bound on the
minimal value of the coefficient $\lambda$ appearing in the Distance
Conjecture. Whether or not this bound will stand up to further scrutiny is
undoubtedly the most pressing question opened by our work. In the meantime,
however, it is worth considering possible applications of our bound. In the
remainder of this paper, we contemplate possible applications to various
quantum gravity conjectures and cosmology.

###  6.1 Scalar Field Potentials

If indeed our proposed bound $\lambda\geq 1/\sqrt{d-2}$ is correct, the next
item of interest is possible applications of the bound. From a
phenomenological perspective, the Distance Conjecture in its original
formulation is not very interesting: the real world is not supersymmetric, and
there are no massless scalar fields in the Standard Model. In order to connect
the Distance Conjecture to observable physics, therefore, one must subscribe
to its “refined” version Ooguri:2006in ; Klaewer:2016kiy , which holds that a
tower of exponentially light charged particles should exist when _any_ scalar
field is taken to infinite distance, not just a massless modulus. Before
applying our bound to phenomenological context, it is therefore crucial to
determine if the bound $\lambda\geq 1/\sqrt{d-2}$ applies to the refined
version of the Distance Conjecture as well.

In the most optimistic scenario in which the refined Distance Conjecture is
indeed satisfied with $\lambda\geq 1/\sqrt{d-2}$, our conjecture would have
important consequences for scalar field potentials in asymptotic limits of
scalar field space Hebecker:2018vxz . A scalar field displacement $\Delta\phi$
in $d$ dimensions implies a tower of light states beginning at the mass scale
$m\lesssim\exp(-|\Delta\phi|\kappa_{d}/\sqrt{d-2})M_{\textrm{Pl};d}$, which
for $|\Delta\phi|\gg 1/\kappa_{d}$ in turn implies a large number of light
species and a “species bound” cutoff Dvali:2007hz  no larger than

$\Lambda_{\textrm{UV}}\lesssim\exp\left(-\frac{\kappa_{d}|\Delta\phi|}{(d-1)\sqrt{d-2}}\right)M_{\textrm{Pl};d}\,.$
(155)

Demanding that this UV scale lies above the IR Hubble scale
$H\sim\sqrt{V}\kappa_{d}\lesssim\Lambda_{\textrm{UV}}$ thus leads to a bound
on the potential,

$V\lesssim\exp\left(-\frac{2\kappa_{d}|\Delta\phi|}{(d-1)\sqrt{d-2}}\right)M_{\textrm{Pl};d}^{d}\,.$
(156)

This suggests a bound

$\frac{|\nabla V|}{V}\geq\frac{2}{(d-1)\sqrt{d-2}}\kappa_{d}\,$ (157)

in asymptotic regions of scalar field space. In other words, this simple
argument suggests that the asymptotic de Sitter conjecture Obied:2018sgi ;
Hebecker:2018vxz  should be satisfied with a coefficient of
$c=2/((d-1)\sqrt{d-2})$.

However, our work above, combined with the Emergent String Conjecture,
suggests that this result can be strengthened. This conjecture implies that
every infinite-distance limit should be either an emergent string limit or a
decompactification limit. Combined with our results above, this suggests that
the Distance Conjecture should be satisfied in any infinite-distance limit by
either a tower string oscillator modes with $\lambda=1/\sqrt{d-2}$ or a tower
of Kaluza-Klein modes with $\lambda\geq 1/\sqrt{d-2}$. Thus we have

$\textrm{min}(M_{\textrm{string}},m_{\textrm{KK}})\lesssim\exp\left(-\frac{\kappa_{d}|\Delta\phi|}{\sqrt{d-2}}\right)M_{\textrm{Pl};d}\,.$
(158)

At the string scale $M_{\textrm{string}}$, a Hagedorn density of states
appears, and effective field theory breaks down. Consistency of the low-energy
effective field theory therefore requires a bound on the Hubble scale,
$H\lesssim M_{\textrm{string}}$. Similarly, the Hubble scale $H$ must lie
below the Kaluza-Klein scale $m_{\textrm{KK}}=L^{-1}$, ensuring that the
horizon $H^{-1}$ is larger than the size $L$ of the extra dimensions so that
the system can be treated as a $d$-dimensional FRW cosmology. Thus we have a
bound

$H\lesssim\Lambda_{\textrm{UV}}\equiv\textrm{min}(M_{\textrm{string}},m_{\textrm{KK}})\,.$
(159)

A scalar field rolling in an exponential potential
$V\sim\exp(-\lambda\kappa_{d}\phi)M_{\textrm{Pl};d}^{d}$ gives rise at late
times to a Hubble scale $H\sim\sqrt{V}\kappa_{d}$ Tsujikawa:2013fta . Together
with (158) and (159), this implies

$V\lesssim\exp\left(-\frac{2\kappa_{d}|\Delta\phi|}{\sqrt{d-2}}\right)M_{\textrm{Pl};d}^{d}\,,$
(160)

which gives

$\frac{|\nabla V|}{V}\geq\frac{2}{\sqrt{d-2}}\kappa_{d}\,.$ (161)

This bound is precisely the statement of the strong asymptotic de Sitter
Conjecture Rudelius:2021oaz ; Rudelius:2021azq  (see also Obied:2018sgi ;
Bedroya:2019snp ), which ensures that the strong energy condition is satisfied
and forbids accelerated expansion in asymptotic regimes of scalar field space.
This bound is saturated in a number of string theory examples in $d=4$
dimensions Andriot:2020lea , as well as supercritical string theories in
$d\geq 4$ spacetime dimensions Hellerman:2006nx . This result also fits nicely
with the results of Hellerman:2001yi ; Rudelius:2021azq , which proved that
the bound (161) is satisfied for any equidimensional limit at which
supersymmetry is restored.

It is important to note that this bound has been derived under the assumption
of a $d$-dimensional FRW cosmology and therefore depends on the dynamics of
the scalar field. In particular, the bound (161) may be violated at the
expense of decompactification by relaxing the constraint $H^{-1}>L$.
Furthermore, the bound (160) assumes that the path in question is a gradient
flow trajectory along which the scalar field $\phi$ rolls to infinity at late
times. This means, as emphasized in Rudelius:2021oaz ; Rudelius:2021azq , that
the strong asymptotic de Sitter Conjecture (161) bounds the gradient of the
potential $|\nabla V|$ in asymptotic regions of field space rather than the
derivative $|V^{\prime}(\phi)|$ along an arbitrary path in field space. The
two are equal for a gradient flow trajectory, but for a more general path the
former will be larger than the latter, as it receives contributions from
scalar fields orthogonal to the path. Indeed, the bound (161) is false in
asymptotic regions of scalar field space if one replaces $\nabla V$ with the
derivative $V^{\prime}(\phi)$ along an arbitrary geodesic in field space
Andriot:2020lea .888For example, consider a 4d theory with two scalar fields
$\phi$, $\rho$, with action
$\mathcal{L}=-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}(\partial\rho)^{2}-V(\phi)$,
where $V(\phi)\sim V_{0}\exp(-(\sqrt{2}\phi+\sqrt{2/3}\rho)/M_{\textrm{Pl}})$
in asymptotic regions of scalar field space. Along the geodesic
$\rho\rightarrow\infty$, we have
$M_{\textrm{Pl}}|\partial_{\rho}V|/V=\sqrt{2/3}$, which violates the bound
(160). However, this path is not a gradient descent trajectory, so (160) need