Tameness, Strings, and the Distance Conjecture

Thomas W. Grimm, Stefano Lanza, Chongchuo Li

_Institute for Theoretical Physics, Utrecht University_

_Princetonplein 5, 3584 CE Utrecht, The Netherlands_

###### Abstract

The Distance Conjecture states that an infinite tower of modes becomes
exponentially light when approaching an infinite distance point in field
space. We argue that the inherent path-dependence of this statement can be
addressed when combining the Distance Conjecture with the recent Tameness
Conjecture. The latter asserts that effective theories are described by tame
geometry and implements strong finiteness constraints on coupling functions
and field spaces. By exploiting these tameness constraints we argue that the
region near the infinite distance point admits a decomposition into finitely
many sectors in which path-independent statements for the associated towers of
states can be established. We then introduce a more constrained class of tame
functions with at most polynomial asymptotic growth and argue that they
suffice to describe the known string theory effective actions. Remarkably, the
multi-field dependence of such functions can be reconstructed by one-
dimensional linear test paths in each sector near the boundary. In four-
dimensional effective theories, these test paths are traced out as a discrete
set of cosmic string solutions. This indicates that such cosmic string
solutions can serve as powerful tool to study the near-boundary field space
region of any four-dimensional effective field theory. To illustrate these
general observations we discuss the central role of tameness and cosmic string
solutions in Calabi-Yau compactifications of Type IIB string theory.

###### Contents

  1. 1 Introduction
  2. 2 The Distance Conjecture and the Tameness Conjecture
    1. 2.1 Distance Conjecture and path dependence
    2. 2.2 Tameness Conjecture
    3. 2.3 A special class of tame functions
    4. 2.4 Taming the Distance Conjecture
  3. 3 Test strings and tame functions
    1. 3.1 Cosmic and axion strings
    2. 3.2 Strings as probes for polynomially and monomially tamed functions
    3. 3.3 Test strings and the Distance Conjecture
  4. 4 Tameness in Type IIB EFTs
    1. 4.1 Type IIB effective field theories
    2. 4.2 Type IIB complex structure sector and Hodge theory
    3. 4.3 The polynomially tamed behavior of the Hodge inner product
    4. 4.4 Tame EFT couplings and the Distance Conjecture in Type IIB EFTs
  5. 5 Conclusions
  6. A A primer on monomially and polynomially tamed functions
    1. A.1 Restricted analytic functions
    2. A.2 Generalities of monomially and polynomially tamed functions
    3. A.3 Characterization of monomially and polynomially tamed functions
    4. A.4 Properties of monomially and polynomially tamed functions
  7. B Monomial bounds for polynomially tamed functions
  8. C The Hodge inner product growth
    1. C.1 Lightning review of asymptotic Hodge theory
    2. C.2 Proof of Theorem 1

##  1 Introduction

It is widely believed that only a limited subset of quantum field theories can
be regarded as effective field theories (EFTs) arising from a UV-complete
quantum gravity theory. Singling out the criteria that EFTs have to satisfy in
order to admit a UV quantum gravity completion is the core of the _Swampland
program_ (see, for example, [1, 2] for reviews on the program). In recent
years several of these criteria have been proposed and collected within the
so-called ‘Swampland conjectures’. The Swampland conjectures put constraints
on many crucial features of the EFTs, such as the gauge sector, the allowed
symmetries, or the spectrum of objects that the EFT can describe. Such
conjectures are typically formulated independently, but several relations have
subsequently led to the formation of an interconnected web. Eventually one
might thus hope that the Swampland conjectures can be reduced to a set of
basic principles that the quantum nature of gravity imposes on the effective
descriptions. The aim of this work is to further combine some of the Swampland
conjectures and thereby clarify their statements. In particular, we will
investigate how the constraints set by the Distance Conjecture [3] can be more
generically addressed by employing the recently formulated Tameness Conjecture
[4].

The Distance Conjecture [3] poses constraints on the explorable field space of
any effective field theory. Consider an effective field theory with a set of
scalar fields, known as moduli, that are not subjected to any scalar
potential. The field space of these moduli can be highly nontrivial and, in
particular, can admit boundaries at which the field space metric degenerates.
In turn, we can distinguish the boundary points by using the length of the
minimal geodesic distance that is required to reach them, which can either be
finite or infinite from a regular field space point. The family of infinite
distance points are central in the Distance Conjecture, which asserts that in
any consistent EFT of quantum gravity, an infinite tower of exponentially
light states becomes relevant near the infinite distance point. In other
words, infinite distance limits in an EFT should be viewed as an artifact of
the effective description and cannot be reached within the same EFT due to the
emergence of additional light states.

The obstructions that the Distance Conjecture suggests are path independent:
no matter how the infinite distance locus is reached, the effective
description is eventually rendered invalid close to an infinite distance
point. However, it is known from many examples [5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21] that the details on how this happens precisely
generically depend on the path and that the physical interpretation can change
from one path to another. For example, along different paths leading to the
same infinite distance point, the leading tower of light states invalidating
the EFT might well be different. We are thus facing a number of questions that
are crucial in order to understand the physics emerging in the near-boundary
region: (1) Given an infinite tower of states invalidating the EFT along a
given path, along which other paths does this tower remain relevant? (2) How
many families of infinite tower of states are relevant when considering all
paths? (3) Is there a special set of paths that allows us to probe the physics
near a given boundary? In particular, we might wonder if there is only a
finite number of towers realizing the Distance Conjecture that can be
systematically explored to characterize the boundary. As is clear from the
outset, these pivotal questions can be answered only by knowing universal
features that the EFTs display close to any field space boundary.

The recent Tameness Conjecture [4] suggests a novel way to constrain field
spaces, parameter spaces, and coupling functions of any EFT compatible with
quantum gravity. While motivated originally by the finiteness conditions on
EFTs arising in string theory, it proposes to use a general mathematical
framework, known as tame or o-minimal geometry, within which any effective
field theory should be formulated and studied. In particular, the Tameness
Conjecture notes that couplings should not be arbitrary functions of the
moduli fields or parameters that enter the EFT, but rather should be
sufficiently _tame_ functions. As will become clearer in Section 2.2, this
means that the couplings have to be _definable_ in a so-called o-minimal
structure. Such o-minimal structures were originally introduced in the context
of model theory, which is part of mathematical logic, but later found
application in various areas of mathematical research. In particular, we will
be interested in their relation with topology (see [22] for an introductory
reference on the subject). Remarkably, assuming that the couplings are
definable in an o-minimal structure imposes sets of functional constraints and
allows us to address path-dependency questions arising in the Distance
Conjecture.

While the tameness of EFT couplings is a general property, it becomes
particularly powerful when considering their functional form in the near-
boundary region of the moduli space. In fact, the definability in an o-minimal
structure allows one to infer general properties about the growth or fall-off
of the EFT couplings towards the field space boundary. As a first step, we
need to specify the o-minimal structure in which the EFT couplings are
defined. In [4] it was proposed that the o-minimal structure that is relevant
for stringy EFTs is $\mathbb{R}_{\rm an,exp}$. This structure appears in many
geometric applications and allows for defining the exponential function and
all restricted analytic functions. We will introduce $\mathbb{R}_{\rm an,exp}$
in more detail below. In this work we will further restrict the structure in
which the EFT couplings are defined, in order to give more precise statements
about their growth towards the field space boundary. In fact, for concrete
applications in stringy EFTs, not all functions definable in the o-minimal
structure $\mathbb{R}_{\rm an,exp}$ seem to arise. To constrain the allowed
set of functions further we will be following [23] and introduce special
families of tame functions in $\mathbb{R}_{\rm an,exp}$, that we name
_monomially tamed_ and _polynomially tamed_. Roughly speaking, given a
subregion $\mathcal{U}$ that touches the boundary of moduli space, monomially
tamed functions are those which display a leading monomial behavior in the
moduli fields along any path in $\mathcal{U}$ towards the field space
boundary. Instead, a polynomially tamed functions in $\mathcal{U}$ are finite
sums of such monomials for which one cannot single out a leading monomial
behavior in $\mathcal{U}$; rather, they exhibit different leading behaviors
according to the chosen path that leads to the field space boundary, as
depicted in Figure 1. We will propose that in most of the concrete stringy
EFTs, EFT couplings can be ascribed to these two families of functions. EFTs
that stem from Type IIB ten-dimensional string theory compactified on a
Calabi-Yau threefold, eventually supplemented by an orientifold projection,
provide evidence for this claim. For instance, as we will show in Section 4,
the couplings involving the vector multiplet sector within 4D $\mathcal{N}=2$
Type IIB EFTs are fully determined by Hodge inner products. As demonstrated in
[23] that Hodge inner products are polynomially tamed near the boundaries of
the complex structure moduli space. Consequently, the field space metric, the
gauge coupling functions, the masses and physical charges of D3-particles are
also polynomially or monomially tamed.

Figure 1: A function $f$ may exhibit different leading asymptotic behaviors
along different paths approaching the infinite field distance boundary. Here
is depicted a function $f$ that displays the different leading behaviors
$f_{1}$, $f_{2}$ and $f_{3}$ along, respectively, the red, purple and blue
paths connecting a regular point $\varphi_{0}$ of the moduli space to the
near-boundary point $\varphi$.

The aforementioned restricted notion of tameness of the EFT couplings has a
remarkable implication: important information about the physics emerging
towards the field space boundaries can be understood by only examining what
happens along certain special paths that stretch towards the boundary. To
single out such paths we note that this reduction to one-dimensional slices
was central in [23] where it was shown that the relevant curves are linear
paths with rational slopes. Following this work, we will indeed show that, if
the EFT couplings are mutually bounded along these test paths, such bounds can
be extended in a wider region of the moduli space. This feature will allow us
to give a recipe to concretely test the Distance Conjecture: it is enough that
the Distance Conjecture is obeyed along these special curves, and it holds in
a wider region of the moduli space path independently.

Thus, such curves serve as _test paths_ and can be employed to diagnose
pathologies that the effective description may develop. Such linear test paths
constitute a very restricted family of all the possible paths that may drive
the fields towards the boundary. Can we get a phenomenological understanding
of why such test paths are ‘special’? The curves that in [23] have been
employed to test the behavior of monomially and polynomially tamed functions
can be viewed as the backreactions induced by cosmic string solutions. As
shown in the seminal work [24] and recently explored in [18, 25], along BPS
cosmic string solutions the moduli fields develop a linear backreaction when
regions too close to the singularity are probed. Therefore, it is enough to
show that a massless tower of states emerges along all the allowed string
backreactions, and we know that such a pathology is spread throughout a wider
region of the moduli space. These observations fit well with the Distant
Axionic String Conjecture proposed in [18, 25]. This conjecture postulates
that, in four-dimensional EFTs, any infinite distance limit is characterized
by the appearance of a tensionless axion string, namely a string that is
magnetically coupled to an axion. In [25], such a claim was checked in a large
class of minimally supersymmetric examples by showing that, along the BPS
backreactions of axion strings, the emergence of an infinite tower of states
and consequent lowering of the EFT cutoff are always accompanied by an axion
string that becomes tensionless. We will here show that these statements are
enough to prove that axion strings do indeed emerge along any path that drives
the moduli towards infinite distance limits.

This work is articulated as follows. In Section 2 we review the Distance
Conjecture and the Tameness Conjecture, and we illustrate how the latter can
help addressing some features of the former. Indeed, therein, special families
of tame functions are introduced, namely the monomially and polynomially tamed
functions. We will then show that, assuming that EFT couplings are described
by such special tame functions, the Distance Conjecture can be rephrased in a
path-independent fashion. In Section 3 we show that, provided that EFT
couplings are either monomially or polynomially tamed, crucial information
about the near-boundary physics can be obtained by examining how the EFT
couplings behave on certain special curves. We will further show how, at the
EFT level, such curves can be regarded as the backreaction of cosmic strings,
which then serve as candidate objects to test the EFT couplings. In Section 4
we provide evidence for the statements made in the previous sections.
Specifically, we prove that monomially and polynomially tamed functions
dictate the couplings entering the vector multiplet sector of the EFTs
obtained after compactifying Type IIB string theory on Calabi-Yau three-folds.
The appendices contain important mathematical results that are used throughout
the main text. Indeed, Appendix A collects some properties that the monomially
and polynomially tamed functions exhibit. In Appendix B, following [23], we
show how polynomially tamed functions can be bounded by monomially tamed
functions by studying their behavior on curves. Finally, in Appendix C we
prove that the Hodge inner products, that dictate the couplings of the Type
IIB EFTs studied in Section 4, grow as polynomially or monomially tamed
functions.

##  2 The Distance Conjecture and the Tameness Conjecture

In this section we first introduce the Distance Conjecture [3]. As we will
see, the Distance Conjecture predicts how any effective description gets
broken when approaching an infinite distance point in field space due to the
emergence of an infinite tower of states that become light at a certain rate.
However, this statement does not make precise assertions about either the
specific path that leads to the infinite distance point or the number
different towers becoming massless towards infinite distance. Here we will
introduce a framework, the one of ‘tame geometry’, within which both issues
can be addressed in full generality. Developing on the recently formulated
Tameness Conjecture [4], we will illustrate that, if the couplings
distinguishing the effective field theory are sufficiently ‘tamed’ (in a sense
that will become clear in Section 2.2), one can formulate the Distance
Conjecture in a path independent fashion.

###  2.1 Distance Conjecture and path dependence

Let us begin by recalling the original statement of the Distance Conjecture
[3]. Consider a $D$-dimensional effective theory with a number of real scalar
fields $\varphi^{i}$ spanning a field space $\mathcal{M}$. The effective field
theory under consideration is assumed to include Einstein gravity coupled to
some scalar fields $\varphi^{i}$, $i=1,\ldots,n$ via an action of the form

$S^{(D)}=M_{\rm P}^{D-2}\int{\rm
d}^{D}x\sqrt{-g}\Big{(}\frac{1}{2}R-\frac{1}{2}G_{ij}\partial_{\mu}\varphi^{i}\partial^{\mu}\varphi^{j}+\ldots\Big{)}\
,$ (2.1)

where $R$ denotes the $D$-dimensional Ricci scalar, $G_{ij}$ is the field
space metric on $\mathcal{M}$ and the dots indicate additional couplings that
potentially also include the scalars $\varphi^{i}$. Let us denote the shortest
geodesic distance between two points $\varphi,\varphi_{0}\in\mathcal{M}$ by
$d(\varphi,\varphi_{0})$. The statement of the Distance Conjecture [3] can be
split into two parts:

  * (1)

Let $\mathcal{M}$ be a moduli space of dimension at least one, i.e. assume
that there are scalar fields $\varphi^{i}$ that are not subjected to a scalar
potential. For any point $\varphi\in\mathcal{M}$ and any positive number $C$
there exists another point $\varphi_{0}\in\mathcal{M}$ such that
$d(\varphi,\varphi_{0})>C$. This implies that the space $\mathcal{M}$ cannot
be compact and that it admits at least one boundary point $\varphi_{\rm
b}\in\partial\mathcal{M}$ which is at infinite distance from any point of
$\mathcal{M}$.

  * (2)

When approaching an infinite distance point $\varphi_{\rm
b}\in\partial\mathcal{M}$ with $d(\varphi,\varphi_{\rm b})\rightarrow\infty$,
there exists an infinite tower of states that becomes exponentially light.
More precisely, consider a $\varphi\in\mathcal{M}$ for
$d(\varphi,\varphi_{0})$ sufficiently large, the masses of the states at
$\varphi$ compared with the masses at $\varphi_{0}$ behave as 111Notice that
the the masses $M_{n}(\varphi)$ are assumed to be computed for canonically
normalized fields.

$M_{n}(\varphi)\sim M_{n}(\varphi_{0})e^{-\lambda d(\varphi,\varphi_{0})}\ ,$
(2.2)

with $\lambda$ an unspecified real parameter. In refined versions of this
conjecture it is claimed that $\lambda$ is $\mathcal{O}(1)$ [26, 27].

Let us stress that the first part of the conjecture solely restricts the
geometry of $\mathcal{M}$, while the second part delivers a more precise
statement about what happens to the effective theory when approaching an
infinite distance point $\varphi_{\rm b}$. According to the conjecture, any
effective theory with fixed energy scale $\Lambda$ eventually breaks down near
$\varphi_{\rm b}$, since at this point an infinite tower of massless degrees
of freedom ought to be included. This can be also stated by noting that there
is a quantum gravity cut-off $\Lambda_{\rm QG}$ associated to the infinite
tower of states that becomes exponentially small compared with $\Lambda$ when
approaching $\varphi_{\rm b}$. Effective field theories are thus only valid
for finite scalar field excursions measured by the shortest geodesic distance.
Instead, the second part of the conjecture asserts that the masses of the
states constituting an infinite tower need to fall-off exponentially in the
geodesic distance. However, no information is provided about the path along
which one approaches the boundary point $\varphi_{\rm b}$. Starting at some
point $\varphi_{0}$ in the field space $\mathcal{M}$ we can consider any path
$\gamma:[0,1)\rightarrow\mathcal{M}$ with one end being at
$\gamma(0)=\varphi_{0}$ and the other end reaching towards the point
$\varphi_{\rm b}$. The considered points $\varphi$ appearing in (2.2) are, by
construction, on the path $\gamma([0,1))$ sufficiently close to $\varphi_{\rm
b}$ such that $d(\varphi,\varphi_{0})$ is large, as depicted in Figure 2. The
path $\gamma$ can be very complicated and there is no a priori assertion that
it has to be a geodesic. Nevertheless the distance $d(\varphi,\varphi_{0})$
appearing in the exponential fall-off in (2.2) is asserted to be the geodesic
distance.

Figure 2: Starting from a point $\varphi_{0}$, a point $\varphi$ close to the
boundary $\varphi_{\rm b}$ can be reached via a path $\gamma$ that be either a
geodesic or non-geodesic.

Evidence for the Distance Conjecture has been collected by studying various
effective theories arising from string theory (see, for example, [5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]). In particular, in the
study of the vector multiplet sector of $\mathcal{N}=2$ effective theories
arising from Type IIA and Type IIB on Calabi-Yau threefolds, the existence of
a whole network of infinite distance points has been established in [5, 28]
and a candidate tower of states has been identified for many of the associated
limits. More precisely, it has been argued in [5, 28] that in many limits a
tower of states with charges $q_{n}=T^{n}q_{0}$ can be constructed by acting
with an appropriately chosen monodromy symmetry $T$ on some seed charge
$q_{0}$. It was a crucial aspect of [5, 28] to show that the tower constructed
in this way was actually satisfying the desired behavior for the Distance
Conjecture in whole sectors of near the infinite distance point. This apparent
feature is not predicted by the Distance Conjecture and we will explain in the
remainder of this section what is the underlying reason for this local
universality of the towers. It will turn out that it can be related to the
tameness of the masses of the states and hence is fundamentally linked with
the Tameness Conjecture [4] reviewed in subsection 2.2.

Any explicit check of the Distance Conjecture can be generically split into
two steps: (1) identifying an infinite tower of states that become massless
towards the infinite distance point and (2) showing that the masses of said
states fall-off as suggested by (2.2). Note that it can be notoriously
difficult to perform these steps along any path $\varphi(s)\equiv\gamma(s)$ in
$\mathcal{M}$, especially when considering higher-dimensional field spaces
$\mathcal{M}$. In order to test the conjecture we need to ensure that along
each path we retain the behavior

$\frac{M_{n}(\varphi)}{M_{n}(\varphi_{0})}\sim e^{-\lambda
d(\varphi,\varphi_{0})}\ ,\qquad\qquad
M_{1}(\varphi)<M_{2}(\varphi)<M_{3}(\varphi)<\ldots,$ (2.3)

where $M_{n},n=1,2,...$ denote the masses of the states in the infinite tower.
The Distance Conjecture hereby does not give an answer to the following
questions:

  * •

Does one need to find a different tower for each individual path? How far can
one deform a path and still use the same tower to satisfy the conjecture?

  * •

How many different towers of states are becoming massless at an infinite
distance point?

It is clear that any restriction on the considered paths might lead to an
incomplete picture and prevent us from answering these questions. As it
stands, the Distance Conjecture might require us to construct a gigantic set
of towers of states in a path-dependent way. As we will argue in Section 2.4
the above questions can be answered when additionally enforcing the Tameness
Conjecture. This will open the possibility to only study a special set of
paths and then use the resulting insights to infer information about all paths
that leads to the infinite distance points.

There are various motivations that would lead one to only consider a special
set of paths. Firstly, for practical purposes one could restrict to only
geodesics when moving to the infinite distance point. Then can directly
replace $d(\varphi_{0},\varphi)$ in (2.3) with the length of the considered
curve. Secondly, on a more fundamental level, the proper identification of the
light states can be more apparent along a set of ‘special paths’. In
particular, in the recent study of the Distant Axionic String Conjecture [25]
a concrete suggestion for the tower of states was made when considering
certain linear paths as we will review in more detail in Section 3. In a
nutshell, the conjecture suggests that each infinite distance limit can be
understood by examining strings in the effective theory that couple to the
axion-like scalar emerging in the limit. Namely, one investigates the
backreaction of a string on the scalar fields of the effective theory. Along
the radial coordinate transverse to the string, some of these fields acquire a
nontrivial profile, and they are driven to boundary in the near-core region of
the string. This renders such axion strings powerful tools to investigate the
physics emerging towards any infinite-distance field space boundary. From a
field theoretical viewpoint, as stressed in [18], the string backreaction can
be regarded as an RG flow of the coupling when changing the energy scale. This
interpretation allows one to map the spacetime backreaction onto the fields of
the theory as a path within the moduli space. Concretely, this implies the
following. Let us consider a two-dimensional field space $\mathcal{M}$ with
complex coordinate $z$ and assume that the infinite distance point is at $z=0$
on $\partial\mathcal{M}$. Locally, we can model the near boundary region by
considering the punctured disk, i.e. the unit disk $|z|<1$ with the center
$z=0$ removed. As showed in [25] and further summarized in Section 3.1, axion
strings generate a backreaction that can be mapped to the lines with
arg$(z)=\theta_{0}$ in the punctured disk as depicted in Figure 3. For each
line a tower of states arises from the oscillation modes of the string. While
not necessarily the lightest tower, it gives a candidate set of states that
can be used in the Distance Conjecture. It is then important to show that
these states remain relevant away from these special paths. It will be one of
the goals of Section 3 to show that this exactly happens if an appropriate
tameness condition is imposed on the couplings of the effective theory.

Figure 3: On the left is depicted a path with fixed arg$(z)=\theta_{0}$
towards $z=0$ in the punctured unit disk $\Delta^{*}=\\{0<|z|<1\\}$. Such a
path is in correspondence with the backreaction of an axion string, that is
depicted on the right: the axion string, whose core is depicted in green,
induces a backreaction on $z$, depicted in blue, in the $(r,\theta)$-plane
transverse to the string; along the radial coordinate $r$, ${\rm Im}\,z\to 0$,
as happens along the red path at fixed arg$(z)$.

Let us close this section by noticing that the original form of the conjecture
assumes that $\mathcal{M}$ is a moduli space with no scalar potentials for the
$\varphi^{i}$. It was subsequently suggested in [26] to apply this conjecture
to theories with scalar potentials and much recent research [27, 29, 30, 15,
31, 32] has focused on clarifying this possibility. Such an application
immediately leads to a puzzle, since then the notion of field space
$\mathcal{M}$ depends on the energy scale $\Lambda$ of the effective theory.
Lowering the energy scale might require us to integrate out massive scalars,
thereby reducing the original field space $\mathcal{M}$ to a subspace
$\widehat{\mathcal{M}}\subset\mathcal{M}$. The value
$d_{\widehat{\mathcal{M}}}(\varphi,\varphi_{0})$ of the shortest geodesic
distance between $\varphi,\varphi_{0}\in\widehat{\mathcal{M}}$ will in general
differ from $d_{\mathcal{M}}(\varphi,\varphi_{0})$ measured in $\mathcal{M}$
and we always have

$d_{\mathcal{M}}(\varphi,\varphi_{0})\leq
d_{\widehat{\mathcal{M}}}(\varphi,\varphi_{0})\
,\qquad\varphi,\varphi_{0}\in\widehat{\mathcal{M}}\subset\mathcal{M}\ ,$ (2.4)

(see Figure 4 for a pictorial representation). This inequality implies that
even if we assume that the Distance Conjecture is satisfied in $\mathcal{M}$,
it might well be the case that it is violated in $\widehat{\mathcal{M}}$. To
see this, imagine that we have found a tower of states with masses $M_{n}$
that become exponentially light with the geodesic distance $d_{\mathcal{M}}$
in $\mathcal{M}$ with a rate specified by (2.3). Now, performing the
computation in $\widehat{\mathcal{M}}$ the fall-off of the tower should be
determined by $d_{\widehat{\mathcal{M}}}$. However, generically the masses
$M_{n}$ of the above tower does not generically scale as $M_{n}\sim
e^{-\lambda_{n}d_{\widehat{\mathcal{M}}}}$, for
$e^{-\lambda_{n}d_{\widehat{\mathcal{M}}}}$ falls off faster than
$e^{-\lambda_{n}d_{\mathcal{M}}}$. In fact, for the Distance Conjecture to be
satisfied in $\widehat{\mathcal{M}}$ we have to find a new tower of states
that becomes massless faster and there is no reason for this tower to coincide
with the same tower of state that realizes the Distance Conjecture in
$\mathcal{M}$. In other words, assuming the Distance Conjecture in
$\mathcal{M}$ does not imply the Distance Conjecture in
$\widehat{\mathcal{M}}$, if this smaller field space is obtained by a general
potential.

Figure 4: On the left is depicted the near-boundary region of a two-
dimensional moduli space $\mathcal{M}$. On the right, the constraints
introduced by the scalar potential, depicted as the blue surface, reduce the
moduli space to the one-dimensional space $\widehat{\mathcal{M}}$, depicted as
the red line, obtained as the intersection of the moduli space $\mathcal{M}$
and the potential constraint surface. Consider two points $\varphi_{0}$ and
$\varphi$ in $\widehat{\mathcal{M}}$: while they can be linked via a geodesic
path in $\mathcal{M}$ (depicted in blue on the left picture, and with a blue,
dotted line on the right), in $\widehat{\mathcal{M}}$ they can be linked only
along the one-dimensional space $\widehat{\mathcal{M}}$, along the dark-red
path, and such a path may be non-geodesic in $\mathcal{M}$.

Therefore, it may then appear that the Distance Conjecture loses its
predictability when the effective description displays a nontrivial potential
for the moduli. However, one can address the issue from another perspective:
if a scalar potential is present, the moduli fields $\varphi^{i}$ are
constrained to follow given paths in the wider space $\mathcal{M}$, which are
here flat directions of the potential. It is then crucial to have an
understanding of whether the Distance Conjecture holds also for such non-
geodesic paths. This issue was explored on different grounds also in [32],
where it was proposed that the Distance Conjecture can be viewed as a
constraint on the scalar potential: namely, the flat directions that the
potential allows need to be such that the Distance Conjecture is realized
along those as well. In what follows, we will see that the general tame
structure of any consistent EFT allows us to investigate the realization of
the Distance Conjecture also along such non-geodesic paths.

###  2.2 Tameness Conjecture

Previously we have explained that the Distance Conjecture alone does not allow
us to develop a very detailed picture of what happens near an infinite
distance point. However, in the following we argue that the situation changes
if we invoke another recent conjecture, the Tameness Conjecture, that
implements finiteness constraints into the structure of the effective theory.
In essence, this conjecture states that all effective theories that can be
consistently coupled to quantum gravity can be defined using ‘tame geometry’,
which is a currently very active field of mathematics linking geometry and
logic. More precisely, we recap stating that effective theories that can be
consistently coupled to quantum gravity admit the following properties:

  * (1)

All effective theories valid below a fixed finite energy cut-off scale are
labelled by a definable parameter space and must have scalar field spaces and
coupling functions that are definable in an o-minimal structure.

  * (2)

The relevant o-minimal structure is $\mathbb{R}_{\rm an,exp}$.

Here we note that part (2) gives a strengthening of the conjecture, since it
specifies an o-minimal structure denoted by $\mathbb{R}_{\rm an,exp}$. Note
that the conjecture is very far reaching, since it asserts the existence of a
general parameter space and constrains its properties. As we will explain
momentarily, it thereby excludes that this parameter space can contain
infinite discrete components such as a lattice. It was stressed in [4] that
this finiteness condition is deeply linked to the coupling to gravity.
Furthermore, the conjecture restricts valid field spaces and every coupling
function varying over the parameter space and field space. In particular, we
will apply this condition to the masses and distances appearing in the
Distance Conjecture.

To explain the conjecture, we first have to define what we mean by an
o-minimal structure $\mathcal{S}$. The basic idea hereby is to introduce sets
of subsets of $\mathbb{R}^{n}$ denoted by $\mathcal{S}_{n}$, for all
$n=1,2,...$, that form a structure and then add a tameness constraint making
it into an o-minimal structure. The collection of all $\mathcal{S}_{n}$,
$n=1,2,...$ are called $\mathcal{S}$-definable sets, or definable sets if the
o-minimal structure has been specified before. The conditions on the sets
$\mathcal{S}_{n}$ are as follows: (1) $\mathcal{S}_{n}$ contains the zero-set
of any polynomial in $n$ variables; (2) $\mathcal{S}_{n}$ is closed under
finite intersections, finite unions, and complements; (3) the Cartesian
product of a set in $\mathcal{S}_{n}$ and a set in $\mathcal{S}_{m}$ is in
$\mathcal{S}_{m+n}$; and (4) linear projections
$\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n}$ applied to a set in
$\mathcal{S}_{n+1}$ give a set in $\mathcal{S}_{n}$. Finally, the tameness
condition is then stated as:

  * •

The $\mathcal{S}$-definable sets in $\mathbb{R}$ are the finite unions of
points and intervals.222Note that these intervals can be closed or open and be
of finite or infinite length.

It is a remarkable fact that this tameness property solely imposed on the
subsets of the real line $\mathbb{R}$ constrains the space of allowed sets in
all $\mathcal{S}_{n}$ so significantly that strong finiteness properties can
be inferred. This is rooted in the projection property, which implies that any
projection to a real line of a higher-dimensional definable set should lead
only finite unions of points and intervals. We depict a definable and a non-
definable set in $\mathbb{R}^{2}$ in Fig. 5. It is then natural to introduce
$\mathcal{S}$-definable maps $\varphi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$
by requiring that their graph, which is a subset of $\mathbb{R}^{n+m}$, is
definable in $\mathcal{S}_{n+m}$. Using $\mathcal{S}$-definable sets and
definable functions one can then define an $\mathcal{S}$-definable topological
space and a definable manifold by requiring that it admits definable atlas
with appropriate definable transition functions [22]. In fact, one can use
this as a starting point for introducing many other geometric structures by
adding the definability criterium as an additional constraint. The resulting
geometry framework is provides that the aforementioned tame geometry.

Figure 5: On the left, an example of tamed function with a finite set of
zeros. On the right, an example of non-tamed function, with an infinite number
of zeros.

It is a remarkable fact that there exist multiple examples of o-minimal
structures that extend the simplest structure, denoted by $\mathbb{R}_{\rm
alg}$, generated by polynomial equations only. $\mathbb{R}_{\rm alg}$ is
hereby obtained by collecting all zero-sets $P(x_{1},...,x_{n})=0$ in
$\mathbb{R}^{n}$ and completing this set by including their unions,
complements, intersections, and projections. The resulting set of semi-
algebraic sets is the smallest o-minimal structure. The strategy to find non-
minimal examples is to carefully extend the set of functions that is used to
define the definable sets. There are two o-minimal structures that will be of
importance in this work and we will introduce in some detail in the following.

O-minimal structure $\boldsymbol{\mathbb{R}}_{\textbf{an}}$: This o-minimal
structure is obtained by extending $\mathbb{R}_{\rm alg}$ by also considering
zero-sets of so-called restricted analytic functions. More precisely, we also
include subsets of $\mathbb{R}^{n}$ defined by the equations
$P(x_{1},...,x_{n},f_{1},...,f_{n})=0$, where $P$ is a polynomial and
$f_{i}(x_{1},...,x_{n})$ are restricted analytic functions. Before defining
restricted analytic functions, let us recall that an analytic function defined
on a domain is a function that coincides with its own Taylor series on that
domain. Analytic functions are necessarily smooth, but the converse is not
true. Roughly speaking, restricted analytic functions are restrictions of
analytic functions to smaller domains. More precisely, such functions are all
restrictions $f|_{B(R)}$ of functions $f$ that are analytic on a ball
$B(R_{0})$ of finite radius $R_{0}$ to a ball $B(R)$ of strictly smaller
radius $R<R_{0}$. Applying this definition to the punctured disk $\Delta^{*}$
introduced in Fig. 3, we realize that a restricted analytic function $f$ is
more constrained than an analytic function when examining its behavior near
the puncture, i.e. the point $z=0$. By definition, such an $f$ needs to come
from a function that is also analytic at the puncture and hence implies that
$f$ cannot ‘go wild’ when approaching the puncture. As an example of an
analytic but not restricted analytic function consider $f(z)=1/z$ over the
punctured disk $\Delta^{*}$. This function is analytic over $\Delta^{*}$, but
its singularity at $z=0$ forbids it being restricted analytic over
$\Delta^{*}$. Note, however, that $1/z$ is still definable in $\mathbb{R}_{\rm
an}$, since its graph can be given by an algebraic equation.

O-minimal structure $\boldsymbol{\mathbb{R}}_{\textbf{an,exp}}$: This
o-minimal structure will be central for the discussion of the upcoming
sections. It is obtained by further extending the former ${\mathbb{R}}_{\rm
an}$ so as to include real exponentials. Specifically,
${\mathbb{R}}_{\rm{an,exp}}$ can be understood as the subsets of
$\mathbb{R}^{n}$ described by the equation
$P(x_{1},\ldots,x_{n},f_{1},\ldots,f_{m},e^{x_{1}},\ldots,e^{x_{n}})=0$,
where, as above, $P$ is a polynomial and $f_{i}(x_{1},...,x_{n})$ are
restricted analytic functions.

It is worth stressing that the choice of the domain is crucial in order to
correctly tell whether the function is definable in a given o-minimal
structure. The complex exponential $e^{z}:\mathbb{C}\to\mathbb{C}$ is not
definable in general. In fact, rewriting $e^{z}=e^{r+2\pi{\bf
i}\phi}=e^{r}(\cos(2\pi\phi)+{\bf i}\sin(2\pi\phi))$, $e^{z}$ has an infinite,
discrete set of zeros for $r,\phi\in\mathbb{R}$. Only if we further reduce the
domain of $\phi$, $e^{z}$ is definable. In most of the applications below,
$\phi$ is an axion, whose fundamental domain is bounded, say
$0+\varepsilon<\phi<1-\varepsilon$, with $\varepsilon\ll 1$. Under such an
assumption, $e^{z}$ is definable in ${\mathbb{R}}_{\rm{an,exp}}$.

Let us now explicitly see how the Tameness Conjecture reflects on the
couplings that effective field theories are allowed to display. To this end,
let us consider a generic effective field theory, focusing on the bosonic
spectrum only, valid up to the energy cutoff $\Lambda_{\text{\tiny{EFT}}}$ and
coupled to Einstein gravity. We assume that the EFT is endowed with:

  * •

a set of real parameters $\lambda^{\kappa}$, $\kappa=1\,\ldots,k$, spanning a
subset of $\mathbb{R}^{k}$; in concrete EFT models obtained from compactifying
a higher-dimensional string theory, such parameters may stem either from
compactification data, or from the integration of fields more massive than
$\Lambda_{\text{\tiny{EFT}}}$;

  * •

a set of real scalar fields $\varphi^{a}$, $a=1,\ldots,m$, locally
parametrizing a (pseudo-)moduli space $\mathcal{M}_{\lambda}$, fibered over
the parameter space; thus, the local field space metric
$G_{ab}(\varphi,\lambda)$ generically depends not only on the fields
$\varphi^{a}$, but also on the parameters $\lambda^{\kappa}$;

  * •

a set of abelian $p$-form gauge fields $A^{\mathcal{I}}_{p_{\mathcal{I}}}$,
$\mathcal{I}=1,\ldots,N$, with field strengths
$F^{\mathcal{I}}_{p_{\mathcal{I}}+1}={\rm
d}A^{\mathcal{I}}_{p_{\mathcal{I}}}$; the associated gauge kinetic function
$f_{\mathcal{I}\mathcal{J}}(\varphi,\lambda)$ is allowed to depend on both the
moduli fields $\varphi^{a}$ and the parameters $\lambda^{\kappa}$.

A prototypical, generic $D$-dimensional action built out of these ingredients
would include the following terms333Notice that, in our conventions, a
$p$-form gauge field $A_{p}$ has zero mass dimensions.

$\displaystyle S^{(D)}$ $\displaystyle=\int\Big{(}\frac{1}{2}M^{D-2}_{\rm
P}R*1-\frac{1}{2}M^{D-2}_{\rm P}G_{ab}(\varphi,\lambda){\rm
d}\varphi^{a}\wedge*{\rm d}\varphi^{b}$ (2.5)
$\displaystyle\qquad\qquad-M^{D-2(p_{\mathcal{I}}+1)}_{\rm
P}f_{\mathcal{I}\mathcal{J}}(\varphi,\lambda)F^{\mathcal{I}}_{p_{\mathcal{I}}+1}\wedge*F^{\mathcal{J}}_{p_{\mathcal{J}}+1}-V(\varphi,\lambda)+\ldots\Big{)}\,.$

Here, $V(\varphi,\lambda)$ denotes the scalar potential which the moduli
fields $\varphi^{a}$ are subjected to, and eventually higher-derivatives terms
can be included to the action. Moreover, the action (2.5) is not required to
be supersymmetric. Then, the Tameness Conjecture is a restriction on all the
couplings entering (2.5) – i.e. the field space metric
$G_{ab}(\varphi,\lambda)$, the gauge kinetic matrix
$f_{\mathcal{I}\mathcal{J}}(\varphi,\lambda)$, the scalar potential
$V(\varphi,\lambda)$, etc.

Concretely, let us consider a generic coupling $g$. Here we neglect additional
structures dressing the coupling, namely we assume the coupling $g$ to be a
scalar coupling, without any index. Then, the coupling can be regarded as map
$g(\varphi,\lambda)$, with domain in $\mathbb{R}^{m+k}$ and values in
$\mathbb{R}$. As stated above, requiring that the coupling is definable in a
given o-minimal structure implies that the graph of the function
$g(\varphi,\lambda)$, namely the set of points
$(\varphi,\lambda,g(\varphi,\lambda))$, has to be definable in
$\mathcal{S}_{m+k+1}$. According to the strong version of the Tameness
Conjecture, the o-minimal structure in which the EFT couplings ought to be
defined is $\mathbb{R}_{\rm{an,exp}}$. Therefore, according to the explanation
above, the coupling $g(\varphi,\lambda)$ can be understood as originating from
a locus built as follows. We first introduce a set of an arbitrary number of
_auxiliary_ variables, $x_{q}$, $q=1,\ldots,l$, which do not enter the EFT
either as couplings or fields, and we assume that, at a first stage, the
coupling $g$ is a function of these auxiliary variables as well,
$g(\varphi,\lambda,x)$. Then, the coupling $g$ originates from the locus:

$\displaystyle\exists\;x_{1},\ldots,x_{l}:$ $\displaystyle
P_{i}(\varphi,\lambda,x,g,f_{1},\ldots,f_{m},e^{\varphi},e^{\lambda},e^{x},e^{g})=0\,,$
(2.6) $\displaystyle
Q_{j}(\varphi,\lambda,x,g,f_{1},\ldots,f_{m},e^{\varphi},e^{\lambda},e^{x},e^{g})>0\,,$

where $P_{i}$, $Q_{j}$ are polynomials and $f_{i}(\varphi,\lambda,x,g)$ are
restricted analytic functions in the variables
$(\varphi^{a},\lambda^{\kappa},x_{q},g(\varphi,\lambda,x))$. The dependence of
the coupling $g(\varphi,\lambda,x)$ on the auxiliary variables $x_{q}$, may be
necessary in order to identify the locus (2.6). However, by solving the
defining equations of the locus (2.6) for $x_{q}$, one can eliminate the
explicit dependence of $g$ on $x_{i}$. We stress that the definition of the
locus (2.6) is, in principle, sensitive to change of parametrizations of the
fields $\varphi^{a}$ and the parameters $\lambda^{\kappa}$. For instance,
redefining the fields as $\varphi^{a}=e^{t^{a}}$, the locus (2.6) is not
described by polynomials in $t^{a}$. Thus, we shall assume that there exists
an appropriate parameterization $(\varphi,\lambda)$ of a patch within
$\mathcal{M}_{\lambda}$ such that the coupling $g$ can be computed as the
locus (2.6). Moreover, it is worth noticing that the finiteness of the
constraints and the form of (2.6) needed to specify a coupling definable in
$\mathbb{R}_{\rm an,exp}$ is a deep mathematical result that is related to
model completeness [33].

Let us motivate the appearance of $e^{g}$ and the auxiliary variable $x$ in
(2.6) via two simple examples. First, take the logarithmic function
$g(\varphi)=\log(\varphi)$. It is straightforward to check that the graph of
the logarithm is $\mathbb{R}_{\mathrm{an,exp}}$-definable, which can also be
written as the vanishing locus of the polynomial
$P(\varphi,e^{g})=\varphi-e^{g}$ with the help of $e^{g}$. To see the
necessity of the auxiliary variables, take the double exponential function
$g(\varphi)=\exp(\exp(\varphi))$. In order to show the definability of the
double exponential, denote $\Gamma(\exp)$ the graph of the exponential
function, and note that the graph of the double exponential function is simply
a coordinate projection of the definable set
$(\mathbb{R}\times\Gamma(\exp))\cap(\Gamma(\exp)\times\mathbb{R})$, hence the
double exponential is $\mathbb{R}_{\mathrm{an,exp}}$-definable. This fact is
expressed in (2.6) by introducing an auxiliary variable $x$, and considering
the locus of the polynomials
$P_{1}(g,e^{x})=g-e^{x},P_{2}(x,e^{\varphi})=x-e^{\varphi}$. It is worth
noting that in our first example, the (global) logarithmic function does not
belong to the class of restricted analytic or exponential functions that
define the $\mathbb{R}_{\mathrm{an,exp}}$-structure. In fact, it can be
shown[34, Corollary 4.7] that every $\mathbb{R}_{\mathrm{an,exp}}$-definable
function can be written piece-wisely as polynomials of compositions of
restricted analytic, exponential444It is worth mentioning that by [34, (4.9)],
the exponential function can be replaced by compositions of the reciprocal,
and $n$-th roots for all positive integer $n$., and logarithmic functions.

Examples of couplings that are definable $\mathbb{R}_{\rm an,exp}$ include
polynomials in either the $\varphi^{a}$ and the parameters $\lambda^{\kappa}$.
However, also exponential functions of the form $e^{p(\varphi,\lambda)}$,
where $p(\varphi,\lambda)$ is a generic polynomial in
$(\varphi^{a},\lambda^{\kappa})$, are definable in $\mathbb{R}_{\rm an,exp}$.
Instead, the Tameness Conjecture hinders the appearance of _any_ periodic
coupling defined in $\mathbb{R}^{n}$. Other notable examples of functions
definable in ${\mathbb{R}}_{\rm{an,exp}}$ include the period integrals of
Calabi-Yau manifolds. This important result, proved in [23], allows for
delivering nontrivial evidences of the aforementioned strong version of the
Tameness Conjecture, as we will see in section 4. Therein we will investigate
the four-dimensional EFTs obtained after compactifying Type IIB string theory
on a Calabi-Yau manifold, or an orientifold projection thereof. In the former
case, the couplings of the vector multiplet sectors are fully determined by
the period integrals, thus residing in ${\mathbb{R}}_{\rm{an,exp}}$ [4]; in
the latter case, the couplings of a chiral multiplet sector are definable in
${\mathbb{R}}_{\rm{an,exp}}$, for they are also specified by the periods only.

###  2.3 A special class of tame functions

As we have seen, the Tameness Conjecture constrains the form of the couplings
that appear in any effective field theory that is consistent with quantum
gravity. However, the constraints imposed appear to be rather mild, and the
loci (2.6) lead to a great variety of couplings. However, in concrete stringy
EFTs we typically face couplings of a more restricted form than the ones
delivered by the loci (2.6).

In order to better characterize the couplings, we first need to specify the
region of $\mathcal{M}_{\lambda}$ in which the EFT is defined. The effective
description under investigation is assumed to be well-controlled: for
instance, one can assume that the EFT is defined in regimes of small string
coupling, or large internal volume. Typically, the corners in which the EFT is
well-defined are near-boundary regions of $\mathcal{M}_{\lambda}$, for some
given values of the parameters $\lambda^{\kappa}$. We will denote
$\mathcal{E}$ such an asymptotic field space region in (the cover of)
$\mathcal{M}$, and we assume that it can be parametrized by a set of $m$ real
local coordinates $\varphi^{a}$. Within $\mathcal{E}$ the real fields
$\varphi^{a}$ span different domains, and it is convenient to split them
accordingly. Thus, we introduce real local coordinates $s^{i}>1$ and
$\phi^{\alpha}$, such that the field space boundary is at
$s^{1},...,s^{n}\rightarrow\infty$. The residual $m-n$ fields $\phi^{\alpha}$
are assumed to stay bounded $|\phi^{\alpha}|<\delta$. In other words, we
identify the asymptotic region as described by the following

$\mathcal{E}=\\{|\phi^{\alpha}|<\delta,\ \ s^{1},s^{2},\ldots,s^{n}>1\\}\,.$
(2.7)

Loosely speaking, the fields $s^{i}$ spanning a non-compact domain may be
assumed to be those which tell information about the validity of the EFT: as
the boundary $s^{i}\to\infty$ is approached, the corrections that would modify
the effective description become more and more negligible. On the other hand,
the fields $\phi^{\alpha}$ lying in a compact domain are not expected to make
the EFT depart from its regime of validity for any value
$|\phi^{\alpha}|<\delta$. To get a feeling for this parametrization, let us
mention a few well-known examples. The simplest case is a circle
compactification in which we have one real field $s$ parametrizing the radius
of the circle. The decompactification limit $s\rightarrow\infty$ is in the
open interval $\mathcal{E}=\\{s>1\\}$. A more involved example is provided by
the Kähler moduli space of some Calabi-Yau manifold. In this case
$s^{i},\phi^{\alpha}$ can be identified with the set of Kähler moduli arising
by expanding the Kähler form as
$J=s^{i}\omega_{i}+\phi^{\alpha}\omega_{\alpha}$. If one considers the
complexified Kähler form with the NS-NS B-field, the new scalars will be part
of the $\phi^{\alpha}$. While these examples might be instructive, we note
that the following discussion is general.

Now consider an EFT defined in the domain (2.7), and consider any coupling
$g(s,\phi,\lambda)$ that the EFT is endowed with. The dependence of the
coupling $g(s,\phi,\lambda)$ on the fields $(s,\phi)$ is expected to come from
two different contributions, namely

$g(s,\phi,\lambda)=g_{\text{pert}}(s,\phi,\lambda)+g_{\text{non-
pert}}(s,e^{-s},\phi,\lambda)\,.$ (2.8)

Here $g_{\text{pert}}(s,\phi,\lambda)$ encapsulates the perturbative
contributions to $g(s,\phi,\lambda)$ and, in the chosen parametrization,
$g_{\text{pert}}(s,\phi,\lambda)$ is expected to behave as a rational function
in the fields $s^{i}$ for large $s^{i}$. Instead, $g_{\text{non-
pert}}(s,e^{-s},\phi,\lambda)$ collects all the non-perturbative corrections
to the coupling and, as such, $g_{\text{non-pert}}$ is exponentially
suppressed in the non-compact fields $s^{i}$. Couplings such as (2.8) are
endowed with a crucial property: _given the subregion of the moduli space (
2.7) and fixed parameters $\lambda^{\kappa}$, any coupling with the structure
(2.8) can be upper bounded by a monomial in the fields $s^{i}$_. These special
class of tame functions, which are definable in $\mathbb{R}_{\rm an,exp}$, are
indeed enough to fully characterize the couplings of most of the known stringy
EFTs. Following [23], these special tame functions can be split into two
families, that we call _monomially tamed_ and _polynomially tamed_
functions.555Notice that in [23] these families of tame functions are referred
to as _roughly monomial_ and _roughly polynomial_ , respectively. Here, we
just limit ourselves to their (loose) definition and main properties, and we
refer to Appendix A for additional details:

  * •

Monomially tamed functions: We consider a function
$g:\mathcal{U}\rightarrow\mathbb{R}$, with $\mathcal{U}\subseteq\mathcal{E}$ a
given subset of the asymptotic region (2.7). We require that on $\mathcal{U}$
the coupling $g$ can be written as

$g(s,\phi)=\sum_{\mathbf{m}}\rho_{\mathbf{m}}(e^{-s^{i}},\phi^{\alpha})(s^{1})^{m_{1}}\cdots(s^{n})^{m_{n}}\,,$
(2.9)

where $\rho_{\mathbf{m}}$ are coefficient functions that we require to be
restricted analytic on $\mathcal{U}$ and are labelled by a multi-index
$\mathbf{m}=(m_{1},...,m_{n})$ with integer $m_{i}$. This expansion is
restricted further by requiring that $g$ behaves as

$g(s)\sim(s^{1})^{k_{1}}\cdots(s^{n})^{k_{n}}\qquad\text{on}\quad\mathcal{U}\
,$ (2.10)

for some integers $k_{1},\ldots,k_{n}$. Here and in the following the symbol
$\sim$ should be read as a boundedness statement, i.e. we write $f\sim h$ is
there exists $C_{1},C_{2}$, such that $C_{1}h<f<C_{2}h$, with $C_{1}$, $C_{2}$
real numbers having the same sign. In particular, the condition (2.10) implies
that near the infinite distance point there is leading monomial term in the
expansion (2.9) for every path $s^{i}\rightarrow\infty$. Thus, clearly, (2.10)
gives a strong constraint on the functional form of $g$: in fact, the
restricted analytic functions $\rho_{\mathbf{m}}(e^{-s^{i}},\phi^{\kappa})$
that appear in the expansion (2.9) need to guarantee that the double bound
$C_{1}(s^{1})^{k_{1}}\cdots(s^{n})^{k_{n}}<g<C_{2}(s^{1})^{k_{1}}\cdots(s^{n})^{k_{n}}$,
for some $C_{1}$, $C_{2}$ has to hold throughout the region $\mathcal{U}$.

It is worth remarking that one could extend the definition of monomially tamed
function to a tame function obeying the more general

$g(s)\sim(s^{1})^{\alpha_{1}}\cdots(s^{n})^{\alpha_{n}}\qquad\text{on}\quad\mathcal{U}\
,$ (2.11)

with $\alpha_{1},\ldots,\alpha_{n}\in\mathbb{R}$. However, (2.11) may be
reduced to (2.9) by redefining $s^{i}\to(s^{i})^{\frac{k_{i}}{\alpha_{i}}}$.

  * •

Polynomially tamed functions: Similarly to a monomially tamed function, a
polynomially tamed function $f:\mathcal{U}\rightarrow\mathbb{R}$ exhibits the
finite expansion

$f(s,\phi)=\sum_{\mathbf{m}}\rho_{\mathbf{m}}(e^{-s^{i}},\phi^{\alpha})(s^{1})^{m_{1}}\cdots(s^{n})^{m_{n}}\,,$
(2.12)

with $\rho_{\mathbf{m}}(e^{-s^{i}},\phi^{\alpha})$ restricted analytic.
However, in contrast with the monomially tamed functions, (2.10) does not hold
generically in $\mathcal{U}$. Thus, being less constrained, polynomially tamed
functions are more general than the monomially tamed ones. Indeed, due to the
properties of the restricted analytic functions (see Appendix A.3), one can at
most upper bound a polynomially tamed function by a monomial as

$f(s,\phi)\prec(s^{1})^{N_{1}}\cdots(s^{n})^{N_{n}}\,,$ (2.13)

for some set of integers $N_{1},\ldots,N_{n}$. Here we have introduced the
symbol ‘$\prec$’, that has to be understood as follows: given two positive
definite functions $f$ and $g$, $f\prec g$ if there exists a real, positive
constant $C$ such that $f<Cg$ for every point in $\mathcal{U}$. It is however
worth stressing that the bound (2.13) might be too rough, and in Section 3.2
we will provide a recipe in order to refine (2.13).

Therefore, the most evident difference between monomially and polynomially
tamed functions is in their growth properties within the subset $\mathcal{U}$.
On the one hand, owing to (2.10), monomially tamed functions display a
definite growth in $\mathcal{U}$, fixed by the set of integers
$k_{1},\ldots,k_{n}$. Said differently, the leading growth of monomially tamed
functions on _every_ path leading to the field space boundary is univocally
fixed by the single set of integers $k_{1},\ldots,k_{n}$. On the other hand, a
polynomially tamed function does not exhibit a clear leading term in the whole
$\mathcal{U}$. Only at most in some subsets of $\mathcal{U}$, one can single
out a leading monomial term in the polynomially tamed expansions, and there
are cases where such a leading monomial term does not exist. Thus, as a
result, the growth of a polynomially tamed function is generically path
dependent within $\mathcal{U}$. Moreover, it is worth stressing that whether a
function is monomially tamed may depend on the choice of the domain
$\mathcal{U}$: as an example, considering a smaller subset
$\hat{\mathcal{U}}\subset\mathcal{U}$, a polynomially tamed function might
become monomially tamed, as

$f(s,\phi)\sim(s^{1})^{N_{1}}\cdots(s^{n})^{N_{n}}\qquad\text{in
$\hat{\mathcal{U}}\subset\mathcal{U}$}.$ (2.14)

Thus, in such a case, the polynomially tamed function exhibit a definite
behavior in the smaller subset $\hat{\mathcal{U}}$. It is worth stressing
however, that a subset $\hat{\mathcal{U}}\subset\mathcal{U}$ such that a
polynomially tamed function reduces to a monomially tamed one may not exist.
In fact, if $f\prec(s^{1})^{N_{1}}\cdots(s^{n})^{N_{n}}$ holds strictly
everywhere in $\mathcal{U}$, its leading growth cannot be monomial. Therefore,
whenever a leading growth can be identified within a subset
$\hat{\mathcal{U}}\subset\mathcal{U}$, it has to be of the form

$f(s,\phi)\sim\rho(e^{-s^{i}},\phi^{\alpha})(s^{1})^{N_{1}}\cdots(s^{n})^{N_{n}}\qquad\text{in
}\ \hat{\mathcal{U}}\subset\mathcal{U}\,,$ (2.15)

with $\rho(e^{-s^{i}},\phi^{\alpha})$ a restricted analytic function. An
example of one-variable polynomially tamed function that is not monomially
tamed is $f(s)=e^{-s}s$ defined over $\mathcal{E}=\\{s>1\\}$: while $f\prec s$
in $\mathcal{E}$, $f\sim s$ does not hold.

Indeed, the behavior of any given polynomially tamed function might be a
guiding principle in order to better characterize the region $\mathcal{U}$.
Consider a general polynomially tamed function $f$. Here, for the sake of
generality, we shall assume that the polynomially tamed function is defined
over the full asymptotic region $\mathcal{E}$ defined in (2.7). Our aim is to
_partition_ the asymptotic region in smaller subsets where the function $f$
exhibits simple behaviors. First, recall that a partition of a set
$\mathcal{E}$ consists of a collection of disjoint subsets
$\mathcal{U}_{\mathsf{A}}\subset\mathcal{E}$, with $\mathsf{A}=1,\ldots,N$,
which covers $\mathcal{E}$:

$\bigcup\limits_{\mathsf{A}}\,\mathcal{U}_{\mathsf{A}}=\mathcal{E}\,,\quad\textrm{and}\quad\mathcal{U}_{\mathsf{A}}\cap\mathcal{U}_{\mathsf{B}}=\varnothing\,,\textrm{
if }{\mathsf{A}}\neq{\mathsf{B}}\,.$ (2.16)

As a simple example of partition of the asymptotic region, one can cover the
subsets of $\mathcal{E}$ as follows. We define the subsets

$\tilde{\Sigma}_{I}=\\{|\phi^{\alpha}|<\delta,\ \
s^{1}>s^{2}>\cdots>s^{n}>1\\}\,,$ (2.17)

where the index $I$ labels all permutations of the $s^{i}$ in the hierarchy
and we have only displayed the simplest permutation with $I=(1,2,\ldots,n)$.
Picking a region (2.17) singles out a specific ordering for the sizes of the
fields $s^{i}$ (see Figure 6 for a pictorial, two-dimensional representation
representation). Then, a partition of $\mathcal{E}$ is realized as the union
of all the sets (2.17) and the $(2^{n-1}-1)$ loci
$\\{s^{1}=s^{2}>s^{3}>\cdots>s^{n}>1\\},\ldots,\\{s^{1}=s^{2}=\cdots=s^{n}>1\\}$.

Figure 6: An example of moduli space $\mathcal{M}$ parametrized by two fields
$s^{1}$ and $s^{2}$. The region close to the boundary point $\phi_{\rm b}$,
reached as $s^{1},s^{2}\to\infty$, can be covered by two sets,
$\tilde{\Sigma}_{12}=\\{s^{1}>s^{2}>1\\}$ and
$\tilde{\Sigma}_{21}=\\{s^{2}>s^{1}>1\\}$, and the locus
$\tilde{\sigma}=\\{s^{1}=s^{2}>1\\}$.

Generally, how the partition of $\mathcal{E}$ is carried is arbitrary.
However, we choose to partition $\mathcal{E}$ in compliance with the
_monotonicity theorem_ 666Strictly speaking, we should be using the regular
cell decomposition in [22, Exercise (2.19)]. We abuse the terminology and call
it monotonicity theorem to emphasize the monotonic property of the function
$f$ on each of the subset $\mathcal{U}_{\mathsf{A}}$. (see, for instance, [22,
Chapter 3]). The theorem states the following: for each definable function
$f:\mathcal{E}\to\mathbb{R}$, there is a partition of $\mathcal{E}$ in finite
subsets $\mathcal{U}_{A}$ such that the restriction
$f|_{\mathcal{U}_{\mathsf{A}}}$, for each $s^{i}$, is either strictly
increasing, strictly decreasing or constant.777Recall that a function $f$ is
strictly increasing (decreasing) for a given field direction $s^{i}$ if, for
any given $s^{i}_{(1)}<s^{i}_{(2)}$, $f(s^{i}_{(1)})<f(s^{i}_{(2)})$
($f(s^{i}_{(1)})>f(s^{i}_{(2)})$).

While given a polynomially tamed function one can always find a partition such
that, on each of its subsets, the function behaves as predicted by the
monotonicity theorem, it is however important to stress that this does not
solve the path dependence issue that we raised above. In fact, albeit on each
subset $\mathcal{U}_{\mathsf{A}}$, a polynomially tamed function has a
definite growth behavior, it does not necessarily display a leading term. We
will then assume that, beyond the partition above, a given polynomially tamed
function $f$ may induce a finer partition
$\\{\hat{\mathcal{U}}_{\mathsf{A}}\\}$ such that, on each of these subsets
$\hat{\mathcal{U}}_{\mathsf{A}}$, $f$ has a definite leading behavior:

$f(s,\phi)\sim\rho(e^{-s^{i}},\phi^{\alpha})(s^{1})^{N_{1}}\cdots(s^{n})^{N_{n}}\quad\text{in
}\ \hat{\mathcal{U}}_{\mathsf{A}}\subset\mathcal{U}\,.$ (2.18)

Clearly, if $\rho(e^{-s^{i}},\phi^{\alpha})\sim 1$ on
$\hat{\mathcal{U}}_{\mathsf{A}}$, the restriction
$f|_{\hat{\mathcal{U}}_{\mathsf{A}}}$ is monomially tamed. The leading
behavior (2.18) can be one of the terms appearing in the general expansion
(2.12); as such, also this finer partition is expected to be composed by a
finite number of subsets.

###  2.4 Taming the Distance Conjecture

In light of the newly introduced terminology and characterization of the
asymptotic region, let us now come back to the original questions that we
posed about the Distance Conjecture. Similarly to the strategy in the Tameness
Conjecture, our basic idea is to single out a set of functions that can occur
in the Distance Conjecture and then give us a more complete picture of the
physics in the infinite distance limit. In particular, in this section, we
will assume that the Distance Conjecture holds in the near-boundary region of
the field space, and we will investigate how the EFT obstructions predicted by
the Distance Conjecture may occur in an EFT that is characterized by tame
couplings. As discussed in the previous section, it is natural to reduce the
class of functions drawn from the o-minimal structure $\mathbb{R}_{\rm
an,exp}$ in the study of concrete stringy EFTs. Indeed, in relation to the
Distance conjecture we will now make concrete assertions about the functional
dependence of the geodesic distance $d(s,\phi)$ (here and below we suppress
the initial point $s_{0},\phi_{0}$) and the masses $M_{n}(s,\phi)$, that make
use of the monomially and polynomially tamed functions only. We will split the
discussion in the three main issues that concur in the realization of the
Distance Conjecture. Note that we will initially not restrict to any
supersymmetric setting and therefore the following discussion might look
rather complicated at first.

Reducing path-dependency of the infinite tower of states. The first ingredient
that we need in order to realize the Distance Conjecture is an infinite tower
of states, with masses $M_{n}$, that become massless as the infinite distance
singularity is approached. Since the masses $M_{n}$ are physical couplings we
require that they are tame. In particular, employing the arguments of Section
2.3, we require that

$M_{n}(s,\phi)\ \text{is polynomially tamed in $\mathcal{E}$}\ .$ (2.19)

Indeed, in Section 4 we will show that the hypothesis (2.19) holds for a large
class of stringy EFTs. In the following we will limit ourselves to the
analysis of the consequences of such an assertion. As stressed in the previous
section, polynomially tamed functions do not display a definite leading term
throughout $\mathcal{E}$. Therefore, the masses $M_{n}$ might exhibit a
different fall-off or growth according to the chosen path. However, one can
partition the set $\mathcal{E}$ into smaller subsets where $M_{n}(s,\phi)$ has
a definite growth. As a preliminary step, let us notice that, if
$M_{n}(s,\phi)$ is a proper candidate tower that realizes the Distance
Conjecture in a subset $\mathcal{U}\subset\mathcal{E}$, it is necessary that
$M_{n}(s,\phi)$ is strictly decreasing on $\mathcal{U}$. Thus, such a set
$\mathcal{U}$ might be a single subset of the partition induced by the
monotonicity theorem, or a union of different subsets thereof. Additionally,
combining the requirement that $M_{n}(s,\phi)$ is polynomially tamed and that
the states become massless near the boundary, we demand

$M_{n}(s,\phi)\prec(s^{1})^{N_{1}}\cdots(s^{n})^{N_{n}}\qquad\text{on}\quad\mathcal{U}\,,$
(2.20)

with $N_{1},\ldots,N_{n}\in\mathbb{Z}_{<0}$. Thus, the condition (2.20)
guarantees that the masses $M_{n}(s,\phi)$ obey $M_{n}(s,\phi)\to 0$ along
_any_ path in $\mathcal{U}$ that approaches the infinite distance singularity.

But what happens in the asymptotic regions outside $\mathcal{U}$? If outside
$\mathcal{U}$ the masses $M_{n}$ do not obey (2.20), they may not become
massless, and thus the tower $M_{n}$ cannot serve as a candidate for realizing
the Distance Conjecture in $\mathcal{E}\mathbin{\mathchoice{\hbox{
\leavevmode\hbox to3.6pt{\vbox to6.6pt{\pgfpicture\makeatletter\hbox{\hskip
0.3pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\hbox{
\leavevmode\hbox to3.6pt{\vbox to6.6pt{\pgfpicture\makeatletter\hbox{\hskip
0.3pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\hbox{
\leavevmode\hbox to2.45pt{\vbox to4.45pt{\pgfpicture\makeatletter\hbox{\hskip
0.22499pt\lower-0.22499pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.45pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{2.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\hbox{
\leavevmode\hbox to1.9pt{\vbox to3.4pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{1.5pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\mathcal{U}$.
However, different infinite towers of states may be available, with masses
$M_{n}^{(a)}$, where the index $a$ labels the tower. The towers are chosen in
such a way that the masses of the constituting states becoming massless in
some near-boundary region are strictly decreasing and

$M_{n}^{(a)}(s,\phi)\prec(s^{1})^{N^{(a)}_{1}}\cdots(s^{n})^{N^{(a)}_{n}}\qquad\text{on}\quad\mathcal{U}^{(a)}\subset\mathcal{E}\,,$
(2.21)

where $N^{(a)}_{1},\ldots,N^{(a)}_{n}\in\mathbb{Z}_{<0}$. The emergence of any
infinite tower of states as the infinite distance singularity is approached
requires that with such sets $\mathcal{U}^{(a)}$ we can cover the whole
asymptotic region, namely $\bigcup\limits_{a}\mathcal{U}^{(a)}=\mathcal{E}$
(see figure 7 for a representation).

Figure 7: Schematic depiction of how the field space region around the
boundary $\phi_{\rm b}$ in such a way that, in any subset $\mathcal{U}^{(a)}$,
there exists an infinite tower of states with masses $M_{n}^{(a)}$ obeying
(2.21). Notice that the dimensions of the subsets $\mathcal{U}^{(a)}$ do not
have to be the same.

Matching the behavior of masses and the geodesic distance. The Distance
Conjecture asserts the specific asymptotic behavior (2.3) for the masses of
the infinite states that break the effective description. However, in general,
it is hard to estimate the behavior of the geodesic distance $d(s,\phi)$.
Still, if the Distance Conjecture has to hold, then the behavior of the
distance $d(s,\phi)$ has to be such that the behavior of $e^{-\lambda
d(s,\phi)}$ is comparable with one of the masses $M_{n}$ of any tower of
states. Therefore, it is enough to minimally assume that

$e^{-\lambda d(s,\phi)}\ \text{is polynomially tamed in $\mathcal{E}$}\ .$
(2.22)

It is interesting to note that this is a tameness condition on the exponential
of the distance rather than the distance itself. In this way the exponential
function does not necessarily need to be definable and the statement of the
Distance Conjecture (2.2) can be formulated even within $\mathbb{R}_{\rm
alg}$, e.g. in highly supersymmetric setting when instanton corrections are
absent [4, 35]. However, if we insist that the distance itself is tame, we
conclude that the statement of the Distance Conjecture (2.2) requires to use,
at least, the o-minimal structure $\mathbb{R}_{\text{exp}}$ in which the
exponential function is definable. In the following we will stay general and
work with $\mathbb{R}_{\text{an,exp}}$ in which, as in
$\mathbb{R}_{\text{exp}}$, the definability of the exponential of $d(s,\phi)$
implies the definability of $d(s,\phi)$.

Now, in order to satisfy (2.3) throughout the asymptotic region $\mathcal{E}$
we proceed as follows. Since $e^{-\lambda d(s,\phi)}$ is polynomially tamed,
we can partition $\mathcal{E}$ in regions $\\{\mathcal{U}_{\mathsf{A}}\\}$
where the $e^{-\lambda d(s,\phi)}$ is upper bounded by a monomial and has a
distinguished leading behavior. Since our analysis concerns infinite distance
singularities, in any of the sets $\mathcal{U}_{\mathsf{A}}$ that contains the
boundary $\phi_{\rm b}$, to which we will refer to as ‘boundary’ subsets, then

$e^{-\lambda d(s,\phi)}\
\prec(s^{1})^{N^{\mathsf{A}}_{1}}\cdots(s^{n})^{N^{\mathsf{A}}_{n}}\qquad\text{on
any boundary}\quad\mathcal{U}_{\mathsf{A}}\,,$ (2.23)

with $N^{\mathsf{A}}_{1},\ldots,N^{\mathsf{A}}_{n}\in\mathbb{Z}_{<0}$. Then,
on any of these boundary subsets, $e^{-\lambda d(s,\phi)}$ is either
monomially tamed, or strictly upper bounded by a monomially tamed function.

Then, if the Distance Conjecture is satisfied, there must exist some towers of
states, subjected to an appropriate partition of $\mathcal{E}$, such that for
each of the subsets $e^{-\lambda d(s,\phi)}\sim M_{n}^{(a)}(s,\phi)$. For
concreteness, take a single subset $\mathcal{U}_{\mathsf{A}}$. On
$\mathcal{U}_{\mathsf{A}}$ we introduce an additional, finer partition that is
now induced by the behavior of the masses of the states constituting the
infinite towers. Namely, we construct a partition
$\\{\mathcal{U}^{(a,k)}_{\mathsf{A}}\\}$ of $\mathcal{U}_{\mathsf{A}}$ such
that, on each of these subsets, the tower $M_{n}^{(a)}$ displays a definite
leading behavior

$M_{n}^{(a)}(s,\phi)\sim\rho_{k}(e^{-s},\phi)(s^{1})^{n^{(a,k)}_{1}}\cdots(s^{n})^{n^{(a,k)}_{n}}\qquad\text{on}\quad\mathcal{U}^{(a,k)}_{\mathsf{A}}$
(2.24)

with $n^{(a,k)}_{i}\in\mathbb{Z}_{<0}$. Then, on each
$\mathcal{U}^{(a,k)}_{\mathsf{A}}$ the Distance Conjecture requires that
$M_{n}^{(a)}(s,\phi)\sim e^{-\lambda d(s,\phi)}$. This, in turn, guarantees
that, on each set $\mathcal{U}^{(a,k)}_{\mathsf{A}}$ the Distance Conjecture
is realized path-independently.

Finiteness of the infinite towers of states. As we have illustrated, the
realization of the Distance Conjecture in the full asymptotic region
$\mathcal{E}$ might require the presence of different towers of states. But
how many towers are required? In general, the answer depends on how the
partition of $\mathcal{E}$ is carried over, by applying first the monotonicity
theorem, and then refining such a partition by singling out the subsets where
the masses have a distinguished leading behavior. However, the underlying tame
structure of the effective field theory couplings guarantees that the subsets
of $\mathcal{E}$ that realize the partition are never infinite in number. We
can then formulate the following statement:

[frametitle= Finiteness of the infinite towers of states ,nobreak=true] In
order for the Distance Conjecture to be realized within a tame EFT, only a
finite number of infinite tower of states is sufficient.

The Distance Conjecture and partial moduli stabilization. The tameness of the
EFT couplings allows for better addressing the issue raised at the end of
Section 2.2 about the realization of the Distance Conjecture within an EFT
with sliding cutoff. First, let us mention that the scalar potential, regarded
as an EFT coupling, is also conjectured to be tame according to the Tameness
Conjecture. As a relevant class of examples, in [36] it was proved that any
flux-induced F-theory scalar potential is tame. As stressed in [4], an
important consequence of the tameness of the scalar potential resides in the
finiteness on the number of vacua that it can deliver. Here we are interested
in the special case where a partial moduli stabilization occurs. Concretely,
let us consider an EFT endowed with the (pseudo-)moduli $\\{\varphi^{a}\\}$,
$a=1,\ldots,m$, spanning the space $\mathcal{M}$, and assume that a scalar
potential $V(\varphi,\lambda)$ fixes some of the field directions such that
the residual moduli space $\widehat{\mathcal{M}}_{\lambda}$ is spanned by the
fields $\\{{\hat{\varphi}}^{\hat{a}}\\}$, ${\hat{a}}=1,\ldots,{\hat{m}}$,
${\hat{m}}<m$. Now, consider any coupling $g(\varphi,\lambda)$ whose graph is
definable in $\mathcal{M}_{\lambda}$. However, the graph of the coupling
$g(\hat{\varphi},\lambda)$ is also definable is
$\widehat{\mathcal{M}}_{\lambda}$: in fact, this is guaranteed by the
properties of the definable sets outlined in Section 2.2 and by regarding the
graph of $g(\hat{\varphi},\lambda)$ as an appropriate projection of the one of
$g(\varphi,\lambda)$.

Let us now specialize to the case of interest, where the considered coupling
$g(\varphi,\lambda)$ is one of those entering the Distance Conjecture, namely
either $e^{-\lambda d}$ or the masses $M_{n}$ of the states constituting the
infinite towers. We assume that the Distance Conjecture is realized over the
moduli space $\mathcal{M}$ and, in particular, that there exists a finite
partition of the near-boundary region of any infinite distance singularity
where the Distance Conjecture is realized in a path independent fashion as
explained above, with the masses constituting the infinite tower of states
falling off as $M_{n}^{(a)}\sim e^{-\lambda d_{\mathcal{M}}}$. Consider now
the subset $\widehat{\mathcal{M}}$ of $\mathcal{M}$ obtained after a partial
moduli stabilization. Since the masses are tame on $\mathcal{M}$, they are
also tame on $\widehat{\mathcal{M}}$. In particular, the partition of the
near-boundary region $\mathcal{E}$ in (2.21) also covers the subset
$\widehat{\mathcal{M}}$. Consequently, the same towers that are candidates for
realizing the Distance Conjecture in $\mathcal{M}$ may also serve as
candidates in order to realize the Distance Conjecture on
$\widehat{\mathcal{M}}$. Additionally, the number of infinite tower of masses
that are necessary to realize the Distance Conjecture is finite in number,
when employing arguments similar to the ones used for $\mathcal{M}$. However,
in general, assuming solely the definability of the couplings in
$\mathbb{R}_{\rm an,exp}$ seems not to be enough to guarantee that the fall-
off of the masses obeys (2.3) in terms of the geodesic distance on
$\widehat{\mathcal{M}}$. In order to realize (2.3) on $\widehat{\mathcal{M}}$
a different partition of the near-boundary region from the one induced by
(2.23)-(2.24) might be needed and it is tempting to speculate that the
Distance Conjecture can inferred for $\widehat{\mathcal{M}}$ if the potential
is moreover polynomially tamed.

In order to give evidence for these statements, let us first recall the
simplest prototype example for the Distance Conjecture and consider a Kaluza-
Klein compactification of a $D$-dimensional theory on a circle with radius
$s$. In the effective $(D-1)$-dimensional theory the radius is a scalar field
with kinetic term

$S^{(D-1)}=M_{\rm P}^{D-3}\int{\rm
d}^{D}x\sqrt{-g}\left(-\frac{1}{s^{2}}\partial_{\mu}s\partial^{\mu}s+\ldots\right)\
.$ (2.25)

The limit $s\rightarrow\infty$ is an infinite distance limit with a
logarithmic growth in the geodesic length,
$d(s,s_{0})=\int^{s}_{s_{0}}\frac{ds^{\prime}}{s^{\prime}}=\text{log}\big{(}\frac{s}{s_{0}}\big{)}$.
Note that the metric $1/s^{2}$ is definable in $\mathbb{R}_{\rm an,exp}$ in
accordance with the Tameness Conjecture. The masses of the Kaluza-Klein states
arising in the limit $s\rightarrow\infty$ are $M_{n}(s)=\frac{n}{s}\sim
n\,e^{-d(s,s_{0})}$ in accordance with the Distance Conjecture. In fact,
$M_{n}(s)$ is also definable in $\mathbb{R}_{\rm an,exp}$ and both the metric
and the masses have a simple polynomial growth

$G_{ss}\sim\frac{1}{s^{2}}\ ,\qquad M_{n}\sim e^{-d(s,s_{0})}\sim\frac{1}{s}\
.$ (2.26)

This asymptotic polynomial behavior is common to all examples in which the
Distance Conjecture has been tested so far. However, it does not resolve the
path-dependence issues that we have raised in Section 2.1. To see that, let us
extend the setting to multiple variables. Concretely, let us consider a four-
dimensional model with three moduli $s^{i}$, $i=1,2,3$, described by the
following action

$S^{(4)}=M_{\rm P}^{2}\int{\rm
d}^{4}x\sqrt{-g}\left(-\sum\limits_{i=1}^{3}\frac{1}{(s^{i})^{2}}\partial_{\mu}s^{i}\partial^{\mu}s^{i}+\cdots\right)\
.$ (2.27)

Such an action stems, for instance, from the compactification of ten-
dimensional string theory on a six-dimensional toroidal orbifold
$T^{6}/\Gamma$, with $\Gamma$ a discrete group; specifically, in Type IIB
EFTs, the fields $s^{i}$ parametrize the imaginary part of the complex
structure moduli, or in Type IIA EFTs the fields $s^{i}$ parametrize the
volume each the volume of an internal $T^{2}$. The geodesic distance between
the field space points $(s^{i}_{0})$ and $(s^{i})$ is

$d=\sqrt{\sum\limits_{i=1}^{3}\left(\log\frac{s^{i}}{s^{i}_{0}}\right)^{2}}\,.$
(2.28)

Thus, again, infinite distance points are reached when any of the fields
$s^{i}\to\infty$. A candidate infinite tower of states realizing the Distance
Conjecture is given by the tower of the Kaluza-Klein modes. The lightest among
the Kaluza-Klein states have the following behavior in terms of the fields
$s^{i}$:

$M_{n}^{i}\sim\frac{M_{\rm P}}{\sqrt{s^{i}}}\,.$ (2.29)

Clearly, which tower of state is relevant depends on the specific choice of
the path that leads to infinite distance. For instance, along paths in which
only a single field $s^{i}\to\infty$, the distance conjecture is realized by
three different tower of states as $M_{n}^{i}\sim e^{-\frac{1}{2}d}$. More
generally, one can consider three different ‘strict’ asymptotic regimes

$(1)\;s^{1}\gg s^{2},s^{3}\,,\qquad(2)\;s^{2}\gg
s^{1},s^{3}\qquad(3)\;s^{3}\gg s^{1},s^{2}\,,$ (2.30)

such that, asymptotically in each sector, $M_{n}^{i}\sim e^{-\frac{1}{2}d}$.
Moreover, provided a redefinition of fields $s^{i}\to(s^{i})^{2}$, the common
leading behavior of both $M_{n}^{i}$ and $e^{-\frac{1}{2}d}$ is monomially
tamed. This simple example shows us a general feature that is crucial to
consider when testing the Distance Conjecture: we need to specify the
asymptotic regime that we are investigating. Such regimes can be captured, for
instance, by the sets $\tilde{\Sigma}_{I}$ defined in (2.17), or smaller
subsets thereof. Once an appropriate subset is chosen, on the one hand,
$e^{-d(s,s_{0})}$ may acquire a simple asymptotic behavior; on the other hand,
the tower of states that could realize the Distance Conjecture organize
_hierarchically_ , and one can then single out the lightest tower, namely the
one that leads to the breaking of the EFT.

In the above selected examples it is clear how one can choose asymptotic
regimes in order to realize the Distance Conjecture. However, in general, it
is hard to determine whether, and to what extent the Distance Conjecture is
realized by any given tower of states: the towers can change if the path that
leads to infinite distance is chosen differently and, in principle,
$e^{-\lambda d(s,s_{0})}$ may exhibit a complicated fall-off that is hard to
match with the fall-off of the masses of the candidate infinite tower of
states. In the following section we will explore what is the minimal
information to tell whether the relation (2.3) can be realized within a
partition of the asymptotic region $\mathcal{E}$ in a path independent way.

##  3 Test strings and tame functions

In this section we propose that one can probe the behavior of four-dimensional
EFT couplings via strings. Indeed, strings backreact on the scalar fields, and
the backreaction solely depends on the charge of the string. Such a
backreaction offers a _test path_ on which to probe the behaviors of EFT
couplings. We will show that the behavior of monomially and polynomially tamed
functions on families of these test paths – or, on the allowed string
backreactions – delivers the minimal information required to characterize the
behavior of the function throughout $\Sigma_{I}$.

Specifically, as we shall see, the leading behavior of any monomially tamed
function is fully determined by how the monomially tamed function grows or
falls off on such test paths. On the other hand, polynomially tamed functions
have a more complicated structure, and cannot be solely determined by
examining their behaviors on string backreactions. However, we will deliver a
recipe to _bound_ polynomially tamed functions by how they behave on string
backreactions.

In the following, to begin with, we will recall how to construct cosmic string
solutions in four-dimensional EFTs, and we will later promote such solutions
as test paths to examine the behavior of monomially and polynomially tamed
functions.

###  3.1 Cosmic and axion strings

Here, we will review the cosmic string solutions first studied in [24, 37] and
later generalized and applied to axion strings in [18, 25]. To begin with,
assume that a local patch within the moduli space $\mathcal{M}$ is
parametrized by $N$ complex coordinates $z^{\mathcal{A}}$.888It is worth
stressing that, at this stage, $z^{\mathcal{A}}$ is _any_ modulus of the EFT.
Singularities can be locally described as the loci

$z^{\alpha}=0\,,\qquad\alpha=1,\ldots,n\,,$ (3.1)

for some $n\leq N$. It is then convenient to split the coordinates
$z^{\mathcal{A}}$ as $z^{\mathcal{A}}=(z^{\alpha},\zeta^{\kappa})$, with
$\kappa=1,\ldots,N-n$. The domain that will be of interest for us, within
which we assume that the EFT is well defined, is

$\mathcal{E}=(\Delta^{*})^{n}\times\Delta^{N-n}\,,$ (3.2)

where we have denoted $\Delta^{N-n}=\\{|\zeta^{\kappa}|<1\\}$, a product of
disks, and $(\Delta^{*})^{n}=\\{0<|z^{\alpha}|<1\\}$, a product of punctured
disks. However, it is more convenient to redefine

$\varphi^{\alpha}=\frac{1}{2\pi\mathbf{i}}\log z^{\alpha}\,,$ (3.3)

so that the singular locus in (3.1) is reached as
$\varphi^{\alpha}\to\mathbf{i}\infty$. Notice that (3.3) relates
$(\Delta^{*})^{n}$ to the upper half plane $\mathcal{H}^{n}$ as:

$\\{0<|z^{\alpha}|<1\\}=(\Delta^{*})^{n}\;\ni\;z^{\alpha}\quad\longleftrightarrow\quad\varphi^{\alpha}\;\in\;\mathcal{H}^{n}=\\{\imaginary\varphi^{\alpha}>0\\}\,$
(3.4)

as depicted in Fig. 8.

Figure 8: The near-boundary region around the boundary point, depicted on the
left, can be described by the polydisk $0<|z^{\alpha}|<1$, with the boundary
located at $z^{\alpha}=0$; in turn, via (3.3), this can be mapped to the
upper-half-plane, where the boundary is reached as $s^{\alpha}\to\infty$.

We also split the complex coordinates $\varphi^{\alpha}$ as

$\varphi^{\alpha}=a^{\alpha}+\mathbf{i}s^{\alpha}\,,$ (3.5)

in terms of the real coordinates $a^{\alpha}$, $s^{\alpha}$, with
$s^{\alpha}>0$. As will become clear soon, the real fields $a^{\alpha}$ can be
regarded as _axions_. We will assume that the axions span a compact domain,
and that they are identified as $a^{\alpha}\simeq a^{\alpha}+1$. Their
partners $s^{\alpha}>0$, that build the complex coordinates $\varphi^{\alpha}$
alongside $a^{\alpha}$, will be referred to as _saxions_.

Let us now introduce the class of EFTs that will be under scrutiny. We will
focus on either $\mathcal{N}=2$ or $\mathcal{N}=1$ supersymmetric effective
field theories. We will assume that the moduli $\zeta^{\kappa}$ are fixed at a
specific point $\zeta^{\kappa}_{0}$ within $\Delta^{N-n}$, and we will regard
them as non-dynamical, ‘spectator’ fields. Then, within the domain
$\mathcal{E}$, the sole dynamical complex fields $\varphi^{\alpha}$, for which
we will write down an effective field theory. The fields $\varphi^{\alpha}$
parametrize a local patch of Kähler manifold, and we will denote the
associated Kähler potential as $K$. The effective action describing the
coupling the complex fields $\varphi^{\alpha}$ to gravity is:

$S=M^{2}_{\rm P}\int\left(\frac{1}{2}R*1-K_{\alpha\bar{\beta}}\,{\rm
d}\varphi^{\alpha}\wedge*{\rm d}\bar{\varphi}^{\bar{\beta}}\right)\,.$ (3.6)

The contributions appearing in (3.6) are common to both the $\mathcal{N}=2$
action (see the following (4.4)) and the $\mathcal{N}=1$ action (see (4.14)).
In the former case, the contributions in (3.6) can be obtained from the
bosonic components of $\mathcal{N}=2$ actions describing the interactions of
$n$ vector multiplets and turning off the gauge fields $A^{I}$. In the case of
$\mathcal{N}=1$ actions, (3.6) correspond to the bosonic components of
supergravity actions describing the interaction of $n$ chiral multiples in the
absence of a scalar potential. In this latter case, setting the scalar
potential $V(\varphi^{\alpha})=0$ can be either achieved by turning off
background fluxes and neglecting additional corrections to the scalar
potential or by stabilizing the ‘spectator’ moduli $\zeta^{\kappa}$ in such a
way that $V(\zeta_{0}^{\kappa})=0$.

Cosmic string solutions are solitonic solutions of the equations of motion
that preserve the two-dimensional Poincaré invariance along two spacetime
directions. Namely, let us split the spacetime coordinates as
$(t,x,\xi,\bar{\xi})$, where $\xi$ and $\bar{\xi}$ are complex coordinates
spanning the space directions orthogonal to $(t,x)$. Indeed, introducing the
polar coordinates $(r,\theta)$, one can relate the latter to $\xi$ as
$\xi=re^{i\theta}$. Then, a cosmic string solution is a solution to the
equations of motion stemming from (3.6) by imposing the following metric
ansatz:

${\rm d}s^{2}=-{\rm d}t^{2}+{\rm d}x^{2}+e^{2D}{\rm d}\xi{\rm d}\bar{\xi}\,,$
(3.7)

where $D$ is a warp factor. For simplicity, we will further assume that both
scalar fields $\varphi^{\alpha}$ and the warp factor $D$ depend only on the
coordinates $(\xi,\bar{\xi})$.

It can be shown that the equations of motion for the fields $\varphi^{\alpha}$
leads to [25]

$K_{\bar{\alpha}\beta}\partial\bar{\partial}\varphi^{\beta}+K_{\bar{\alpha}\beta\gamma}\partial\varphi^{\beta}\wedge{\bar{\partial}}\varphi^{\gamma}=0\,,$
(3.8)

where we have introduced $\partial\equiv{\rm d}\xi\wedge\partial_{\xi}$ and
its complex conjugate $\bar{\partial}\equiv{\rm
d}\bar{\xi}\wedge\partial_{\bar{\xi}}$. As shown in [37, 25], the simplest BPS
solutions to (3.8) are either holomorphic profiles obeying

$\bar{\partial}\varphi^{\alpha}=0$ (3.9)

or anti-holomorphic profiles satisfying $\partial\varphi^{\alpha}=0$, along
which half of the bulk supersymmetry is preserved. Here, we will pick the
holomorphic profiles in (3.9) as solutions to (3.8). For completeness, let us
mention that the Einstein equations deliver the following relation between the
warp factor $D$ in (3.7) and the Kähler potential:

$e^{2D}=|f(z)|^{2}e^{-K}\,,$ (3.10)

with $f(z)$ an arbitrary, non-vanishing holomorphic function.

The cosmic string solution (3.9) is agnostic about the specific profile of the
$\varphi^{\alpha}(\xi)$ in terms of the holomorphic coordinate $\xi$. Here, we
are interested in cosmic string solution exhibiting the following monodromy
transformation

$\varphi^{\alpha}\rightarrow\varphi^{\alpha}+e^{\alpha}\,,\qquad\qquad
e^{\alpha}\in\mathbb{Z}\,,$ (3.11)

when encircling a loop around $r=0$ (i.e. $\xi=0$). The integers $e^{\alpha}$
are those that distinguish the monodromy transformation and – as will become
soon clear – can regarded as the _elementary charges_ of the cosmic string
solution. We also stress that, in order for a solitonic solution with the
property (3.11) to be valid, the EFT has to be invariant under the monodromies
(3.11) [38, 39]. Specifically, in the action (3.6), the Kähler metric has to
be invariant under the monodromies (3.11).

The holomorphic solution that realizes (3.11) is

$\varphi^{\alpha}=\varphi^{\alpha}_{0}+\frac{1}{2\pi\mathbf{i}}e^{\alpha}\log\left(\frac{\xi}{\xi_{0}}\right)\,,$
(3.12)

for some constant $\varphi^{\alpha}_{0}$ and $\xi_{0}$.999One may wonder if
the solution (3.12) is a well defined BPS solitonic solution, with finite
energy density. Indeed, it can be checked that the cosmic string solution
(3.12) delivers configurations of finite energy density, saturating the BPS
bound [24, 37, 25]. Indeed, splitting the complex fields $\varphi^{\alpha}$ as
in (3.5) into the real fields $a^{\alpha}$, $s^{\alpha}$, the solution (3.12)
can be recast as

$\displaystyle s^{\alpha}$
$\displaystyle=s_{0}^{\alpha}-\frac{1}{2\pi}\log\left(\frac{r}{r_{0}}\right)e^{\alpha}\,,$
(3.13a) $\displaystyle a^{\alpha}$
$\displaystyle=\frac{\theta}{2\pi}\,e^{\alpha}+a^{\alpha}_{0}\,.$ (3.13b)

This clearly exhibits that the real fields $a^{\alpha}$ experience the
monodromy $a^{\alpha}\rightarrow a^{\alpha}+e^{\alpha}$ after turning around
any loop centered at $r=0$. On the other hand, the fields $s^{\alpha}$ depend
on the radial distance $r=|\xi|$; importantly, as $r\to 0$,
$s^{\alpha}\to\infty$.

Thus, according to the choice of the elementary charges $e^{\alpha}$,
different saxions are driven towards large vevs as $s^{\alpha}\to
e^{\alpha}\times\infty$. In order to organize the string solutions, as in
[25], we further distinguish _elementary_ and _non-elementary flows_. Consider
a basis of BPS charges $\\{e^{\alpha}\\}$. An elementary flow is a solitonic
solution of the kind (3.13) generated by a cosmic string with charge
coinciding with a single basis element $e^{\hat{\alpha}}$ . A non-elementary
flow is a BPS solitonic solution generated by an axion string whose electric
charge is a linear combination of the basis elements $\\{e^{\alpha}\\}$.

The solution (3.12) is rather general: it relies only on the action (3.6),
with the assumption that the axion fields ought to experience the monodromy
(3.11) once we turn around a loop centered at $\xi=0$. In some cases, however,
we can further elaborate about the phenomenological meaning of the solution
(3.12). As is clear from (3.12), the solution exhibits a singularity at
$\xi=0$, and assumes that the singularity is not resolved within the effective
field theory.101010Notice that we are looking for solutions for which the
singularity at $\xi=0$ _cannot_ be resolved within the EFT. Non-singular
solutions at $\xi=0$ can be built, but they require the presence of additional
gauge fields subjected to Higgsing effect. We refer to [40, 41] for details
and concrete examples. Then, the singularity can be understood as
accommodating a codimension-two spacetime defect, which we will identify as an
_axion string_. In order to see this, preliminarily, let us recall that the
cosmic string solution (3.13) exhibits a nontrivial winding for the axion. A
rather simple, but strong assumption that guarantees the invariance of the
action (3.6) under the monodromy (3.13b) is to assume that the Kähler
potential does _not_ depend on the real fields $a^{\alpha}$, that is
$K(\varphi,\bar{\varphi})=K(s)$, implying that also the Kähler metric
appearing (3.6) is solely saxion-dependent. In other words, (3.6) enjoys the
exact continuous shift symmetry $a^{\alpha}\to a^{\alpha}+c^{\alpha}$, with
$c^{\alpha}\in\mathbb{R}$, rendering $a^{\alpha}$ proper ‘axions’, namely
zero-form gauge fields.111111Note that the assumption that
$K(\varphi,\bar{\varphi})=K(s)$ implies that (3.6) exhibits a zero-form global
symmetry, leading to a three-form conserved current. On the other hand,
invariance under the monodromies (3.11) requires invariance only under a
discrete gauge group. See, for instance, [42] for details on the subject.

It was shown in [43, 44, 45] that it is then convenient to rephrase the action
(3.6) in terms of dual variables so defined:

$\ell_{\alpha}=-\frac{1}{2}\frac{\partial K}{\partial s^{\alpha}}\,,\qquad
H_{3\,\alpha}={\rm d}B_{2\,\alpha}-M^{2}_{\rm
P}\,\mathcal{G}_{\alpha\beta}*{\rm d}a^{\beta}\,,$ (3.14)

with

$\mathcal{G}_{\alpha\beta}\equiv\frac{1}{2}\frac{\partial^{2}K}{\partial
s^{\alpha}\partial s^{\beta}}\,.$ (3.15)

Here, the _dual saxions_ $\ell_{\alpha}$ in (3.14) replace the saxions
$s^{\alpha}$ as the real scalar fields enjoying a non-trivial radial flow.
Instead, in (3.14), the axions $a^{\alpha}$ are traded with the gauge two-
forms $B_{2\,\alpha}$ via a standard electro-magnetic duality. It can be shown
that in this dual framework, the action (3.6) reads

$S_{\rm dual}=\int\left(\frac{M^{2}_{\rm P}}{2}R*1-\frac{M^{2}_{\rm
P}}{2}\mathcal{G}^{\alpha\beta}\,{\rm d}\ell_{\alpha}\wedge*{\rm
d}\ell_{\beta}-\frac{1}{2M^{2}_{\rm
P}}\mathcal{G}^{\alpha\beta}\,H_{3\,\alpha}\wedge*H_{3\,\beta}\right)\,,$
(3.16)

with $\mathcal{G}^{\alpha\beta}$ the inverse of (3.15). Focusing on the
bosonic sector only, the dualization of the saxionic and axionic fields in
(3.14) is general. However, in supersymmetric theories the complex scalar
fields $\varphi^{\alpha}$ are accommodated in appropriate multiplets alongside
with their fermionic partners. In $\mathcal{N}=1$ supergravity such as those
examined in Section 4.1.2, $\varphi^{\alpha}$ reside in chiral multiplets
$\Phi^{\alpha}$. It can be shown that the dualization (3.14) can be performed
at levels of multiplets, with the chiral fields $\Phi^{\alpha}$ traded with
linear multiplets $L_{\alpha}$, which accommodate the dual saxions
$\ell_{\alpha}$ and the gauge two-forms $B_{2\,\alpha}$ in their bosonic
components. We refer to [43, 46, 44, 45] for further details.

Since the dual action (3.16) manifestly contains the gauge two-forms
$B_{2\,\alpha}$, we can include the fundamental objects electrically coupled
to the gauge two-forms. These are strings, effectively described by the
following action:

$S_{\rm string}=-\int_{\mathcal{S}}{\rm d}^{2}\xi\sqrt{-h}\,\mathcal{T}_{\rm
str}+e^{\alpha}\int_{\mathcal{S}}B_{2\,\alpha}\,.$ (3.17)

Here we have introduced the coordinates $\xi^{\hat{\imath}}$,
$\hat{\imath}=1,2$, that parametrize the worldsheet $\mathcal{S}$ of the
strings spanning the time direction and one space direction, and
$h_{{\hat{\imath}}{\hat{\jmath}}}$ the induced metric on the string.
Furthermore, $\mathcal{T}_{\rm str}$ denotes the (field-dependent) string
tension, and $e^{\alpha}$ the electric charges of the string. In order for the
effective description not to be broken by the inclusion of the strings, we
need to require

$\mathcal{T}_{\rm str}\geq\Lambda^{2}_{\text{\tiny{EFT}}}\,,$ (3.18)

with $\Lambda_{\text{\tiny{EFT}}}$ the cutoff of the effective field theory.
The condition (3.18) guarantees that the strings are described as
_fundamental_ objects, whose core is not resolved within the EFT, allowing us
to neglect all the stringy oscillatory modes. For this reason, we will call
the strings described by the action (3.17), equipped with (3.18) _fundamental
axion strings_.

We will be specifically interested in BPS strings. In $\mathcal{N}=1$ theories
it can be shown that for fundamental axion strings maximally preserving two
supercharges over their worldvolume the string tension $\mathcal{T}_{\rm str}$
has to be linear in dual saxions $\ell_{\alpha}$ [47, 48, 45, 44]:

$\mathcal{T}_{\rm str}\equiv\mathcal{T}_{\bf e}=M_{\rm
P}^{2}e^{\alpha}\ell_{\alpha}\,,$ (3.19)

supported by the BPS condition

$e^{\alpha}\ell_{\alpha}>0\,.$ (3.20)

We further recall that, by exploiting the dualization (3.14), one can recast
the above string tension in terms of the saxionic fields $s^{\alpha}$. It can
be shown that (3.16), once coupled to (3.17) with (3.19), delivers the cosmic
string solution (3.13), with the fundamental axion string (3.17) conveniently
capturing the singularity at $r=0$.

Before concluding this section, let us remark that the solution (3.13) is not
a complete cosmic string solution covering the full spacetime. In fact, it
breaks down when the saxions $s^{\alpha}=0$. For instance, in models with only
a single complex field $\varphi$, this happens at the radial distance
$\bar{r}=r_{0}e^{\frac{2\pi s_{0}}{e}}$. In [25], in the context of
$\mathcal{N}=1$ EFTs, the distance $\bar{r}$ was regarded as an energy scale
$\bar{\Lambda}=\frac{1}{\bar{r}}$ at which the EFT becomes strongly coupled.
Indeed, therein it was shown at the scale $\bar{\Lambda}$ the axion string
tension (3.19) diverges. This behavior is typical of codimension-two objects
and is ostensibly in contrast with what happens along the backreactions of
objects of codimension strictly greater than two; in fact, for the latter, the
backreaction becomes more and more negligible as the distance from the object
increases. However, it is worth mentioning that, as in [24, 49], one can
‘complete’ the string solution (3.12) so as to encompass regions of strong
coupling. In this work we will not consider such a continuation of the
solution (3.12), for (3.12), with $r<r_{0}$, is already enough to explore the
near boundary region of the moduli space.

###  3.2 Strings as probes for polynomially and monomially tamed functions

In this section we illustrate how the cosmic string solutions reviewed in the
previous sections can be used as tools to probe the behavior of monomially and
polynomially tamed functions. However, monomially and polynomially tamed
functions are defined over a given patch of the moduli space, while the cosmic
string solution (3.12) is defined in spacetime. Therefore, as a preliminary
step, we need to translate the backreaction (3.13) into a path drawn by the
saxionic fields $s^{\alpha}$ within the local patch $\mathcal{E}$ in (3.2). To
this end, we preliminarily define the parameters

$\sigma\equiv-\frac{1}{2\pi}\log\left(\frac{r}{r_{0}}\right)\,,\qquad\rho\equiv\frac{\theta}{2\pi}+\mathbf{i}\,\sigma\,,$
(3.21)

so that the backreaction (3.12) can be recast as

$\varphi^{\alpha}(\rho)=\varphi_{0}^{\alpha}+e^{\alpha}\rho\,.$ (3.22)

that is specified by the choice of string charge ${\bf e}=(e^{\alpha})$. Then,
we can map each point $\varphi^{\alpha}(r,\theta)$ along the cosmic string
solution (3.12) to a point $\varphi^{\alpha}(\rho)$ that specifies a vacuum
configuration of the effective field theory. In other words, the cosmic string
solution (3.13a) maps to a linear path within the moduli space. Moreover, by
further completing the above path with the non-dynamical fields
$\zeta^{\kappa}=\zeta^{\kappa}_{0}$, paths in (3.22) can be promoted to paths
in the full domain (3.2).

The paths (3.22) are suitable to explore the near-boundary region of the
moduli space. As $\rho\to\mathbf{i}\infty$ (corresponding to $r\to 0$ from the
spacetime perspective), the saxions $s^{\alpha}$ are driven towards ‘distant’
regions in the north of the upper-half plane in Figure 8. In Section 2.3 we
have additionally stressed that it is convenient to cover the near-boundary
region via the sets

$\Sigma_{I}=\\{0<a^{\alpha}<1,\ \ s^{1}\geq s^{2}\geq\cdots\geq s^{n}\geq
1\\}\,,$ (3.23)

where the index $I$ labels all permutations of the $s^{\alpha}$ in the
hierarchy and we have only displayed the simplest permutation. The paths
(3.22) may indeed be tuned in order to cover only a single among the sets
$\Sigma_{I}$. For concreteness, let us focus on the set $\Sigma_{12\ldots n}$.
Let us assume that the initial values of saxionic and axionic fields are
chosen such that $s^{1}_{0}>s^{2}_{0}>\ldots>s^{n}_{0}>1$ and
$0<a_{0}^{\alpha}<1$, so that $\varphi^{\alpha}(0)\in\Sigma_{12\ldots n}$.
Given $\varphi^{\alpha}(0)$, the flow of the scalar fields is fully determined
by the string charges $e^{\alpha}$, and we need to ensure that these are
chosen such that $\varphi^{\alpha}(\rho)\in\Sigma_{12\ldots n}$ for any
$\rho$. To this end we identify the following lattice of charges121212Notice
that, for $\mathcal{N}=1$ EFTs and when the cosmic string are axion string
solutions, the charge lattice (3.24) can be understood as a sublattice of the
EFT charge lattice $\mathcal{C}^{\text{\tiny{EFT}}}_{\text{\tiny{S}}}$ defined
in [25]. Indeed, the lattice (3.24) can be regarded as the lattice such that
the instanton corrections – whose charges lie in ‘dual’ lattice
$\mathcal{C}_{\text{I}}=\\{{\bf m}\in\mathbb{N}^{n}\\}$ – are negligible in
$\Sigma$.

$\mathcal{C}_{\bf e}=\\{{\bf e}\in\mathbb{N}^{n}\;|\;e^{1}\geq
e^{2}\geq\ldots\geq e^{n}\geq 0\\}\,.$ (3.24)

Then, provided an additional rescaling ${\rm Re}\rho\to\frac{1}{C{\rm
max}\\{e^{\alpha}\\}}{\rm Re}\rho$ for some $C>0$131313Such a rescaling can be
equivalently understood as a rescaling of the axions $a^{\alpha}$ subjected to
the string backreaction., any path
$\varphi^{\alpha}(\rho)=\varphi_{0}^{\alpha}+e^{\alpha}\rho$ with ${\bf
e}\in\mathcal{C}_{\bf e}$ is fully contained in $\Sigma_{12\ldots n}$. By
choosing the string charge ${\bf e}$ and varying the initial values of the
saxions $s^{\alpha}_{0}$ and the fixed values of the axions $a^{\alpha}_{0}$,
one can span the full set $\Sigma_{12\ldots n}$, as pictorially depicted in
Figure 9. We collect all such paths in the set

$\mathcal{P}_{\bf
e}=\\{\varphi^{\alpha}=\varphi_{0}^{\alpha}+e^{\alpha}\rho\,|\,{\bf
e}\in\mathcal{C}_{\bf
e}\,,s^{1}_{0}>\ldots>s^{n}_{0}>1\,,0<a^{\alpha}_{0}<1\\}\in\Sigma_{12\ldots
n}\,.$ (3.25)

Figure 9: On the left, the subregion spanned by $|z^{1}|$ and $|z^{2}|$ within
the polydisk $\\{0<|z^{1}|<1\\}\cup\\{0<|z^{2}|<1\\}$; within this region, the
string backreaction (3.12) traces curves as depicted by the dotted lines; on
the right, are depicted the paths within the saxionic space $(s^{1},s^{2})$,
further partitioned via the sets (3.23).

In the following it will be convenient to consider special subsets of paths in
(3.25) such that some string charges are zero. To this end, we preliminary
define the charge sublattices

$\mathcal{C}_{\bf e}^{(\hat{n})}=\\{{\bf e}\in\mathbb{N}^{n}\;|\;e^{1}\geq
e^{2}\geq\ldots\geq
e^{\hat{n}}>0\,,e^{\hat{n}+1}=\ldots=e^{n}=0\\}\subset\mathcal{C}_{\bf e}\,,$
(3.26)

and the associated families of paths

$\mathcal{P}_{\bf
e}^{(\hat{n})}=\\{\varphi^{\alpha}=\varphi_{0}^{\alpha}+e^{\alpha}\rho\,|\,{\bf
e}\in\mathcal{C}_{\bf
e}^{(\hat{n})}\,,s^{1}_{0}>\ldots>s^{n}_{0}>1\,,0<a^{\alpha}_{0}<1\\}\in\Sigma_{12\ldots
n}\,.$ (3.27)

Clearly, $\bigcup\limits_{i=1}^{n}\mathcal{P}_{\bf e}^{(i)}=\mathcal{P}_{\bf
e}$. In the language of the previous section, paths belonging to
$\mathcal{P}_{\bf e}^{(1)}$, with the only non-null string charge $e^{1}$,
will be referred to as ‘elementary’ paths, while the others, $\mathcal{P}_{\bf
e}^{(\hat{n})}$ with ${\hat{n}}\geq 2$, will be generically called ‘non-
elementary’.

The paths (3.25) play the role of _test paths_ , which carry important
information about the behavior of the EFT couplings. Indeed, below we show how
the paths (3.25) can be employed to determine the behavior of monomially tamed
functions and to bound the behavior of polynomially tamed functions.

The behavior of monomially tamed functions

Monomially tamed functions display the simple structure (2.10). Indeed, their
leading monomial behavior can be inferred solely by how they grow on BPS
cosmic string solutions. Indeed, consider a generic monomially tamed function
$f\sim(s^{1})^{k_{1}}\cdots(s^{n})^{k_{n}}$. In order to probe the behavior of
$f$, one can consider how $f$ grows along the families of paths in (3.27)
varying $\hat{n}$. Concretely, consider first the elementary paths
$\mathcal{P}_{\bf e}^{(1)}$, along which only the field $\varphi^{1}$ is
driven to boundary according to (3.22). Along these elementary paths
$f\sim\sigma^{k_{1}}$. Thus, the growth of $f$ with respect to the saxion
$s^{1}$ is solely determined by how $f$ behaves on the elementary paths
$\mathcal{P}_{\bf e}^{(1)}$. Let us then consider the non-elementary paths
$\mathcal{P}_{\bf e}^{(2)}$ along which only the saxions $s^{1}$ and $s^{2}$
may reach the asymptotic region. Along these paths,
$f\sim\sigma^{k_{1}+k_{2}}$. Thus, having fixed $k_{1}$ from the behavior on
$\mathcal{P}_{\bf e}^{(1)}$, we can then fix $k_{2}$ from how $f$ behaves on
$\mathcal{P}_{\bf e}^{(2)}$. We can then proceed by considering all the
remaining non-elementary paths, and the leading behavior of $f$ would be fully
fixed by how it grows along the linear paths. However, it is worth noticing
that, since $f$ is monomially tamed, fixing the exponents $k_{i}$ by using
linear paths gives information about the leading behavior of $f$ throughout
$\Sigma_{I}$.

We can exploit such observations in order to compare different monomially
tamed functions. In fact, consider two monomially tamed functions

$f\sim(s^{1})^{k_{1}}\cdots(s^{n})^{k_{n}}\,,\qquad
g\sim(s^{1})^{m_{1}}\cdots(s^{n})^{m_{n}}\,,$ (3.28)

and assume that $f\sim g$ on curves in $\Sigma_{I}$. In particular, this holds
on the families of test paths (3.27). By reasoning as above, it is simple to
show that such an ordering is preserved in $\Sigma$, i.e.

$f\sim g\sim(s^{1})^{k_{1}}\cdots(s^{n})^{k_{n}}\qquad\text{on $\Sigma_{I}$.}$
(3.29)

Thus, the growth of monomially tamed functions can be inferred from how they
grow on linear paths only. Of course, inferring the behavior of the restricted
analytic functions that serve as their coefficients would require some finer
arguments.

The behavior of polynomially tamed functions

As displayed in Section 2.4, polynomially tamed functions exhibit a more
involved structure than the monomially tamed functions. Indeed, the behavior
of a polynomially tamed function throughout $\Sigma$ cannot be generically
inferred from how it behaves on a small subset of curve: in fact, the growth
of a polynomially tamed function is path dependent, and the comparison of the
behavior of polynomially tamed functions would be path dependent as well.
However, one can rather compare the behavior of a polynomially tamed function
with a monomially tamed one. In order to do that we introduce an order
relation $f\prec g$ among the polynomially or monomially tamed functions $f$
and $g$. Recall that writing $f\prec g$ means that there exists a positive
constant $C$ such that $f<Cg$ on all $\Sigma_{I}$. Now, one can bound a
polynomially tamed function using a monomially tamed one. This can be achieved
by exploiting the following lemma formulated in [23] and we reformulate as
follows: [frametitle= Curve-reduction lemma for polynomially tamed functions
,nobreak=true] Consider a polynomially tamed function $f$ and a monomially
tamed function $g$. Assume that

$|f|\prec|g|$ (3.30)

on all the linear paths

$\displaystyle\beta_{1}\varphi_{1}+\gamma_{1}=\cdots=\beta_{n_{0}}\varphi_{n_{0}}+\gamma_{n_{0}}\,,$
(3.31)
$\displaystyle\varphi_{n_{0}+1}=\delta_{n_{0}+1},\ldots,\varphi_{n}=\delta_{n}\,,$

for all choices of $1\leq n_{0}\leq n$, complex numbers
$\delta_{n_{0}+1},\ldots,\delta_{n}$ with positive imaginary parts, positive
rational numbers $\beta_{1},\ldots,\beta_{n_{0}}$, and real numbers
$\gamma_{1},\ldots,\gamma_{n_{0}}$. Then

$|f|\prec|g|\qquad\text{on all}\quad\Sigma_{I}\,.$ (3.32)

We refer to Appendix B for the proof of the statement above. Here, instead, we
will give an idea of how the general proof works for the two moduli case. For
simplicity, let us focus on a polynomially tamed function which depends on two
saxions only as

$f(a,s)=\sum\limits_{m_{1},m_{2}}\rho_{m_{1}m_{2}}(a^{1},a^{2})(s^{1})^{m_{1}}(s^{2})^{m_{2}}\,.$
(3.33)

Let us assume that $|f(a,s)|\prec 1$ on all the linear paths in (3.31). We
want to show that $|f(a,s)|\prec 1$ in all the growth sector $\Sigma_{2}$. In
order to achieve this, it is enough to show that

$|f(a,s)|\leq\sum\limits_{m_{1},m_{2}}|\rho_{m_{1}m_{2}}(a^{1},a^{2})|(s^{1})^{m_{1}}(s^{2})^{m_{2}}\prec
1\,.$ (3.34)

However, since $|f(a,s)|\prec 1$ for linear paths, the powers $m_{1}$, $m_{2}$
in which the saxionic fields appear in (2.12) are greatly constrained. In
fact, $|f(a,s)|\prec 1$ has to hold for both the path choices

$\displaystyle|f(a,s)|\prec 1\qquad\text{on}\quad\mathcal{P}^{(1)}_{\bf
e}\qquad\Rightarrow\qquad m_{1}\leq 0\,,$ (3.35) $\displaystyle|f(a,s)|\prec
1\qquad\text{on}\quad\mathcal{P}^{(2)}_{\bf e}\qquad\Rightarrow\qquad
m_{1}+m_{2}\leq 0\,.$

Thus, in turn, (3.34) holds over the full sector $\Sigma_{2}$.

It is then immediate to show that the linear paths (3.22) can be identified
with the test paths (3.22). Assume that $e^{\alpha}\neq 0$ for $\alpha\leq
n_{0}$, then the paths in (3.22) can be written as

$\displaystyle\frac{\varphi^{1}-\varphi^{1}_{0}}{e^{1}}=\frac{\varphi^{2}-\varphi^{2}_{0}}{e^{2}}=\cdots=\frac{\varphi^{n_{0}}-\varphi^{n_{0}}_{0}}{e^{n_{0}}}\,,$
(3.36)
$\displaystyle\varphi^{\alpha}=\varphi^{\alpha}_{0}\qquad\text{for}\quad
n_{0}+1\leq\alpha\leq n\,.$

These coincide with the test paths in (3.31), upon identifying
$\beta_{\alpha}=1/e^{\alpha}$, and choosing ${\rm
Im}\,\varphi^{\alpha}_{0}=e^{\alpha}$ for $\alpha\leq n_{0}$. Notice that,
since $e^{\alpha}$ can be regarded as string charges, $e^{\alpha}$ are
quantized and we may assume that $e^{\alpha}\in\mathbb{Z}$. Thus, the
coefficients $\beta_{\alpha}$ are rational numbers.

Therefore, the above statement can be rephrased as a statement for the string
flows as follows: [frametitle= Curve-reduction lemma using BPS-strings
,nobreak=true] Consider a polynomially tamed function $f$ and a monomially
tamed function $g$. Assume that

$|f|\prec|g|\qquad\text{on}\quad\mathcal{P}^{(i)}_{\bf e}\quad\forall\;i\,,$
(3.37)

then

$|f|\prec|g|\qquad\text{on}\quad\Sigma_{I}\,.$ (3.38)

###  3.3 Test strings and the Distance Conjecture

In Section 2.4 we illustrated how the Distance Conjecture can be expressed in
a path independent fashion by employing monomially and polynomially tamed
functions. We now re-investigate the statements made in Section 2.4 in light
of the findings of the previous sections. In fact, we will display that, in
order to satisfy the Distance Conjecture in a path independent way in a wide
region of the moduli space, it is enough that it satisfied on a subset of
curves in that region.

Preliminarily, let us consider a simpler case than the one examined in Section
2.4. Namely, let us assume that both $e^{-\lambda d(s,s_{0})}$ and the masses
of the candidate infinite tower of states $M_{n}$ are both monomially tamed on
$\Sigma_{I}$, that is:

$\begin{array}[]{l}e^{-\lambda
d(s,s_{0})}\sim(s^{1})^{k_{1}}\cdots(s^{1})^{k_{n}}\\\
M_{n}\sim(s^{1})^{m_{1}}\cdots(s^{1})^{m_{n}}\end{array}\qquad\text{on}\quad\Sigma_{I}\,.$
(3.39)

Thus, we can apply the reasoning of the previous section: if $M_{n}\sim
e^{-\lambda d}$ along curves, then such a relation holds throughout
$\Sigma_{I}$. This guarantees that the distance conjecture holds on
$\Sigma_{I}$ and the emergence of a tower of states with masses such that
$M_{n}\sim e^{-\lambda d(s,s_{0})}$ does not depend on the path taken towards
the field space boundary. Remarkably, as a byproduct of this analysis, this
further shows that a single tower is enough in order to realize the Distance
Conjecture.

Now let us assume the less constraining case in which $e^{-\lambda d}$ is
polynomially tamed on $\Sigma_{I}$. We will additionally assume the existence
of a finite number of tower of states, with masses $M_{n}^{(a)}$ that also
behave as polynomially tamed functions. Analogously to the reasoning of
Section 2.4, one can perform an appropriate partition of the set $\Sigma_{I}$
into a finite number of subsets $\\{\mathcal{U}_{\mathsf{A}}^{(a,k)}\\}$. On
each subset $\mathcal{U}_{\mathsf{A}}^{(a,k)}$, both $e^{-\lambda d}$ and the
tower $M_{n}^{(a)}$ exhibit a leading behavior. If the towers $M_{n}^{(a)}$
are good candidate towers for realizing the Distance Conjecture only two cases
are allowed:

  * •

on $\mathcal{U}_{\mathsf{A}}^{(a,k)}$ both $e^{-\lambda d}$ and the tower
$M_{n}^{(a)}$ are monomially tamed. Then, in
$\mathcal{U}_{\mathsf{A}}^{(a,k)}$, we can apply the same reasoning as above:
if the Distance Conjecture is realized along the one-dimensional curves
(3.25), it is then realized everywhere within
$\mathcal{U}_{\mathsf{A}}^{(a,k)}$;

  * •

on $\mathcal{U}_{\mathsf{A}}^{(a,k)}$ both $e^{-\lambda d}$ and the tower
$M_{n}^{(a)}$ are strictly polynomially tamed, namely $e^{-\lambda d}\prec
g_{1}$ and $M_{n}^{(a)}\prec g_{2}$ for some monomially tamed functions
$g_{1}$ and $g_{2}$, but $e^{-\lambda d}\not\sim g_{1}$ and
$M_{n}^{(a)}\not\sim g_{2}$; in order for the Distance Conjecture to hold on
$\mathcal{U}_{\mathsf{A}}^{(a,k)}$, we can minimally require that $e^{-\lambda
d},M_{n}^{(a)}\prec g$, for some monomially tamed function such that $g\to 0$
for any path leading to the boundary. Such a requirement can be tested via the
curve-reduction lemma using BPS-strings: if $e^{-\lambda d},M_{n}^{(a)}\prec
g$ hold for every axion-string-induced path (3.25), then $e^{-\lambda
d},M_{n}^{(a)}\prec g$ everywhere on $\mathcal{U}_{\mathsf{A}}^{(a,k)}$. This
renders the tower $M_{n}^{(a)}$ a good candidate for realizing the Distance
Conjecture; however, checking that $e^{-\lambda d}\sim M_{n}^{(a)}$ on
$\mathcal{U}_{\mathsf{A}}^{(a,k)}$ goes beyond the scope of the curve-
reduction lemma; as mentioned in Section 2.4 such a check requires to know the
leading behavior (2.18) of $e^{-\lambda d}$ and $M_{n}^{(a)}$ on the subset
$\mathcal{U}_{\mathsf{A}}^{(a,k)}$.

This novel viewpoint strengthens the Distant Axionic String Conjecture
proposed in [25]. The Distant Axionic String Conjecture asserts that any
infinite distance point can be reached as endpoint of an axion string flow. As
reviewed in Section 3.1, axion strings generate the backreactions (3.13) on
the moduli fields; these are mapped to the families (3.27) of paths in the
moduli space. Moreover, as we move towards the field space boundaries along
the paths (3.27), the axion string generating the flow becomes tensionless.
Thus, the Distant Axionic String Conjecture delivers a bottom-up perspective
on the origin of the EFT breaking at any infinite distance limit: the axion
string is the object that, with its infinite tower of oscillatory modes,
generates the infinite tower of states that can be a candidate to realize the
Distance Conjecture. As a signal of such an EFT breaking, in [25], it was
shown that, along the linear backreaction of BPS axion strings (3.27), the
axion string becomes tensionless and the EFT cutoff
$\Lambda_{\text{\tiny{EFT}}}$ has to consistently become smaller and smaller.
In [25] it was further proposed that the EFT cutoff
$\Lambda_{\text{\tiny{EFT}}}$ is bounded by the tensionless axion string as

$\left(\frac{\Lambda_{\text{\tiny{EFT}}}}{M_{\rm
P}}\right)^{2}\sim\left(\frac{\mathcal{T}_{\rm str}}{M_{\rm
P}^{2}}\right)^{\rm w}\qquad\text{along \eqref{CosmStr_solvarphib}}\,,$ (3.40)

for some _scaling weight_ ${\rm w}\in\mathbb{N}_{>0}$ along the paths (3.27).
We can now revisit these statements, generalizing them, in light of the
discussion above.

Generically, it is too strong to assume that (3.40) holds along any path that
leads to infinite distance, for the integrality of the scaling weight is too
restrictive. However, in order for the axion string to signal the EFT
breaking, it is enough that the EFT cutoff is always upper bounded by any
axion string tension as $\Lambda^{2}_{\text{\tiny{EFT}}}\leq\mathcal{T}_{\rm
str}$. Therefore, we can proceed as follows. For any $\Sigma_{I}$ we choose an
EFT cutoff $\Lambda_{\text{\tiny{EFT}}}^{(I)}$ which dictates when the EFT is
broken within the field space subregion $\Sigma_{I}$. The region $\Sigma_{I}$
can be probed via the axion string flows (3.27); let us denote with
$\mathcal{T}_{\rm str}^{\rm min}$ the minimal string tension among all axion
strings that generate the flow. Then, the consistency of the EFT requires that
$(\Lambda_{\text{\tiny{EFT}}}^{(I)})^{2}\leq\mathcal{T}_{\rm str}^{\rm min}$
everywhere in $\Sigma_{I}$. We rephrase this condition as

$(\Lambda_{\text{\tiny{EFT}}}^{(I)})^{2}\prec\mathcal{T}_{\rm str}^{\rm
min}\qquad\text{in $\Sigma_{I}$}\,.$ (3.41)

Such a condition guarantees that the emergence of an axion-string-induced
infinite tower of states is responsible for the EFT breaking along _every_
path that leads to infinite distance.

However, the more general statement (3.41) follows from the scaling behavior
(3.40) provided that the quantities appearing in (3.40) obey certain tameness
conditions. First, we assume that the EFT cutoff scale
$\Lambda_{\text{\tiny{EFT}}}$ is determined by the lightest mass of the
infinite tower of states that emerge at infinite distance. Thus, as in (2.19),
we consider $\Lambda_{\text{\tiny{EFT}}}$ a polynomially tamed function of the
scalar fields. The behavior of the string tension can be inferred from the
very expression (3.19)-(3.14). In stringy EFTs – as we will see in the next
section for Type IIB EFTs – $e^{K(\phi,\bar{\phi})}$ is monomially tamed.
Consequently, as shown in Appendix A, the string tension (3.19), being given
by the derivative of the Kähler potential, is generically polynomially tamed.
However, let us assume here the stricter hypothesis that also the string
tension (3.19) is monomially tamed. For instance, in all the concrete EFT
models considered in [25], the string tension (3.40) is monomially tamed in
any given set $\Sigma_{I}$. Then, we can straightforwardly apply the
conclusions of Section 3.2: whenever (3.40) holds along any string
backreaction spanning $\Sigma_{I}$, then (3.41) follows. This result thus
greatly expands and generalizes the findings of [25], and shows how the
Tameness Conjecture can be used to refine pre-existing Swampland Conjectures.

##  4 Tameness in Type IIB EFTs

The findings of the previous sections are general and far-reaching. The aim of
this section is to give evidence to the claims made above by showing how tame
couplings appear in concrete stringy EFTs. Specifically, we will focus on
four-dimensional EFTs that are obtained after compactifying the ten-
dimensional Type IIB string theory over a Calabi-Yau three-fold. After first
reviewing some salient features of such a family of EFTs, we will introduce
the central objects of our analysis, namely the _Hodge inner products_. The
Hodge inner products determine many of the couplings entering the EFT, such as
the gauge couplings, the scalar potential, or the masses and tensions of
certain BPS objects. We will then illustrate how the EFT couplings so
determined are not only tame functions of the moduli, but they are either
monomially tamed or polynomially tamed functions.

###  4.1 Type IIB effective field theories

To set the ground for the forthcoming sections, here we review some features
of four-dimensional effective field theories that originate from the
compactification of Type IIB string theory over Calabi-Yau three-folds or
orientifolds thereof.

####  4.1.1 Type IIB $\mathcal{N}=2$ effective field theories

We start by outlining some basic features of $\mathcal{N}=2$ four-dimensional
supergravities that are obtained after compactifying the ten-dimensional Type
IIB string theory over a Calabi-Yau three-fold $Y$ [50, 46]. The resulting
$\mathcal{N}=2$ four-dimensional theory is populated by the gravity multiplet,
whose bosonic components are the graviton $g_{\mu\nu}$ and the graviphoton
$A^{0}$, a set of $h^{2,1}$ vector multiplets, accommodating $h^{2,1}$ complex
scalar fields $\phi^{i}$ and $h^{2,1}$ real vector fields $A^{i}$, and
$h^{1,1}+1$ hypermultiplets. Throughout this section, we will disregard the
hypermultiplet sector even though we expect our approach can be extended to
this sector.

The scalar fields $\phi^{i}$, $i=1,\ldots,h^{2,1}$ within the $h^{2,1}$ vector
multiplets are associated to the deformations of complex structure of $Y$ as
follows. Let us introduce a real, integral basis of three-forms
$\gamma_{\mathcal{I}}$, $\mathcal{I}=1,\ldots,2h^{2,1}+2$ of $H^{3}(Y)$. The
complex structure moduli appear in the expansion of the holomorphic three-form
$\Omega$ as

$\Omega=\Pi^{\mathcal{I}}(\phi)\gamma_{\mathcal{I}}={\bm{\Pi}}^{T}(\phi){\bm{\gamma}}\,,\qquad\Pi^{\mathcal{I}}(\phi)=\int_{\Gamma_{\mathcal{I}}}\Omega\
,$ (4.1)

with the _periods_ $\Pi^{\mathcal{I}}(\phi)$ being holomorphic function of the
fields $\phi^{i}$. Here we have introduced a basis $\Gamma_{\mathcal{I}}$ of
three-cycles such that
$\int_{\Gamma_{\mathcal{I}}}\gamma_{\mathcal{J}}=\delta_{\mathcal{J}}^{\mathcal{I}}$.
The decomposition (4.1) is general but, in what follows, it will be useful to
be more specific about the choice of basis $\gamma_{\mathcal{I}}$. In
particular, let us introduce a symplectic basis of three-cycles
$\Gamma^{\mathcal{I}}=(A^{I},B_{J})$ of $Y$, and a dual basis of three-forms
$\gamma_{\mathcal{I}}=(\alpha_{I},\beta^{J})$, with $I,J=1,\ldots,h^{2,1}+1$
such that

$\int_{Y}\alpha_{I}\wedge\beta^{J}=\int_{B_{J}}\alpha_{I}=-\int_{A^{I}}\beta^{J}=\delta_{I}^{J}\,,\qquad\int_{Y}\alpha_{I}\wedge\alpha_{J}=\int_{Y}\beta^{I}\wedge\beta^{J}=0\,.$
(4.2)

Then, the holomorphic three-form $\Omega$ can be expanded in terms of the
symplectic basis as in (4.1) with the periods

${\bm{\Pi}}(\phi)=\begin{pmatrix}\int_{B_{I}}\Omega\\\
-\int_{A^{I}}\Omega\end{pmatrix}=\begin{pmatrix}X^{I}(\phi)\\\
-\mathcal{F}_{I}(\phi)\end{pmatrix}\,,$ (4.3)

where $X^{I}(\phi)$ and $\mathcal{F}_{I}(\phi)$ holomorphic functions of the
complex structure moduli $\phi^{i}$.

The four-dimensional action describing the interactions among the bosonic
components of the $\mathcal{N}=2$ gravity multiplet and the $h^{2,1}$ vector
fields is

$S=\int\left(\frac{1}{2}M^{2}_{\rm P}R*1-M^{2}_{\rm P}K_{i\bar{\jmath}}^{\rm
cs}{\rm d}\phi^{i}\wedge*{\rm d}\bar{\phi}^{\bar{\jmath}}+\frac{1}{4}{\rm
Im}\mathcal{N}_{IJ}F^{I}\wedge*F^{J}+\frac{1}{4}{\rm
Re}\mathcal{N}_{IJ}F^{I}\wedge F^{J}\right)\,,$ (4.4)

Here $R$ is the Ricci scalar and $K_{i\bar{\jmath}}^{\rm
cs}=\partial_{\phi^{i}}\partial_{\bar{\phi}^{\bar{\jmath}}}K^{\rm cs}$ is the
Kähler metric, with the Kähler potential $K^{\rm cs}$ specified by the periods
as

$K^{\rm
cs}(\Phi,\bar{\Phi})=-\log\left(\mathbf{i}\int_{Y}\Omega\wedge\bar{\Omega}\right)=-\log\mathbf{i}\,{\bm{\Pi}}^{T}\eta\bar{\bm{\Pi}}=-\log\mathbf{i}(\bar{X}^{I}\mathcal{F}_{I}-X^{I}\bar{\mathcal{F}}_{I})\,.$
(4.5)

where we have introduced the intersection matrix
$\eta_{\mathcal{I}\mathcal{J}}=\int_{Y}\gamma_{\mathcal{I}}\wedge\gamma_{\mathcal{J}}$,
computed out of the symplectic basis $(\alpha^{I},\beta_{J})$ employing (4.2).

We will assume that there exists a prepotential $\mathcal{F}(X)$: this is a
homogeneous function of degree two in the projective coordinates $X^{I}$,
using which the quantities $\mathcal{F}_{I}$ appearing in (4.3) can be
understood as derivatives of the prepotential
$\mathcal{F}_{I}=\frac{\partial\mathcal{F}}{\partial X^{I}}$. The dynamics of
the abelian gauge fields $A^{I}=(A^{0},A^{i})$, with field strengths
$F^{I}={\rm d}A^{I}$, is dictated by the matrix

$\mathcal{N}_{IJ}=\bar{\mathcal{F}}_{IJ}+2\mathbf{i}\frac{{\rm
Im}\mathcal{F}_{IK}X^{K}{\rm Im}\mathcal{F}_{JL}X^{L}}{{\rm
Im}\mathcal{F}_{MN}X^{M}X^{N}}\,.$ (4.6)

As is clear from (4.4), the matrix (4.6) determines both the gauge couplings
via its imaginary part and may deliver a $\theta$-term via its real part.

Four-dimensional Type IIB EFTs can be populated by extended objects that stem
from higher-dimensional branes wrapped on some internal cycles. Here we will
focus on D3-branes wrapped on internal three-cycles $\Gamma$. Thus, in the
four-dimensional EFT such D3-branes appear as particles, to which we will
refer as ‘D3-particles’. The mass $M_{\bf q}$ of a BPS D3-particle is obtained
from the central charge $\mathcal{Z}_{\bf q}$ as [51, 52]:

$M_{\bf q}=|\mathcal{Z}_{\bf q}|=e^{\frac{K^{\rm
cs}}{2}}\left|\int_{\Gamma_{\bf q}}\Omega\right|=e^{\frac{K^{\rm
cs}}{2}}\left|\int_{Y}q\wedge\Omega\right|\,,$ (4.7)

with ${\bf q}$ the D3-particle _elementary charges_ and $q$ the three-form
Poincaré dual to the three-cycle $\Gamma_{\bf q}$. In the following, it will
be useful to expand the three-form $q$ in the symplectic basis as
$q=\alpha_{I}p^{I}-q_{I}\beta^{I}$. Accordingly, the elementary charge vector
${\bf q}$ can be split

${\bf q}=\begin{pmatrix}p^{I}\\\ -q_{I}\end{pmatrix}\,,\qquad
q_{I},p^{I}\in\mathbb{Z}\,,$ (4.8)

and (4.7) can be recast as

$M_{\bf q}=e^{\frac{K^{\rm cs}}{2}}|{\bf q}^{T}\,\eta\,{\bm{\Pi}}(\phi)|\,.$
(4.9)

We will refer to $q_{I}$ as the D3 elementary _electric_ charges and $p^{I}$
as the D3 elementary _magnetic_ charges. The physical charge of a D3-particle
can be obtained out of the elementary charges ${\bf q}$ as

$\mathcal{Q}^{2}_{\bf q}=\frac{1}{2}\int_{Y}q\wedge\star q=-\frac{1}{2}{\bf
q}^{T}\mathbb{M}{\bf q}\,,$ (4.10)

with the matrix $\mathbb{M}$ that, in the symplectic basis, can be
conveniently rewritten as

$\mathbb{M}=\begin{pmatrix}{\rm Im}\mathcal{N}+{\rm Re}\mathcal{N}({\rm