Wormholes in the axiverse,
and the species scale

Luca Martucci1,2, Nicolò Risso1,2, Alessandro Valenti1,2,3 and Luca Vecchi2

1 _Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli
Studi di Padova_

2 _INFN Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy_

3 _Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056
Basel, Switzerland_

###### Abstract

We analyze a large class of four-dimensional ${\cal N}=1$ low-energy
realizations of the axiverse satisfying various quantum gravity constraints.
We propose a novel upper bound on the ultimate UV cutoff of the effective
theory, namely the species scale, which only depends on data available at the
two-derivative level. Its dependence on the moduli fields and the number $N$
of axions matches expectations from other independent considerations. After an
assessment of the regime of validity of the effective field theory, we
investigate the non-perturbative gravitational effects therein. We identify a
set of axionic charges supported by extremal and non-extremal wormhole
configurations. We present a universal class of analytic wormhole solutions,
explore their deformations, and analyze the relation between wormhole energy
scales and species scale. The connection between these wormholes and a special
subclass of BPS fundamental instantons is discussed, and an argument in favor
of the genericity of certain axion-dependent effective superpotentials is
provided. We find a lower bound increasing with $N\gg 1$ on the Gauss-Bonnet
coefficient, resulting in an exponential suppression of non-extremal wormhole
effects. Our claims are illustrated and tested in concrete string theory
models.

###### Contents

  1. 1 Introduction
  2. 2 ${{\cal N}}=1$ axiverse models
    1. 2.1 Axions in ${\cal N}=1$ EFTs
    2. 2.2 Semi-topological couplings to gravity
    3. 2.3 Dual formulation and EFT strings
  3. 3 EFT regime of validity
    1. 3.1 Saxionic cones, fundamental instantons and strings
    2. 3.2 Perturbative domain and saxionic convex hull
    3. 3.3 Quantum gravity bounds
    4. 3.4 UV mass scales
  4. 4 String theory models
    1. 4.1 F-theory/type IIB orientifold models
      1. 4.1.1 Model 1: ${\mathbb{P}}^{3}$
      2. 4.1.2 Model 2: $\mathbb{P}^{1}$ fibration over $\mathbb{P}^{2}$
    2. 4.2 Heterotic models
      1. 4.2.1 Energy scales in heterotic models
  5. 5 $SO(4)$-symmetric wormhole configurations
    1. 5.1 Euclidean action
    2. 5.2 Effective one-dimensional action and equations of motion
    3. 5.3 Extremal wormholes and fundamental BPS instantons
    4. 5.4 EFT instantons
    5. 5.5 A non-standard BPS bound
  6. 6 Wormholes in the ${\cal N}=1$ axiverse
    1. 6.1 The homogeneous solution
    2. 6.2 Perturbative conditions and relevant scales
    3. 6.3 Non-homogeneous generalization
    4. 6.4 Marginally degenerate case
    5. 6.5 Examples in string theory models
      1. 6.5.1 Wormholes in F-theory models
      2. 6.5.2 Wormholes in heterotic models
      3. 6.5.3 Wormholes in M-theory models
  7. 7 Low-energy implications
    1. 7.1 Large $N$ suppressions and Gauss-Bonnet
    2. 7.2 Extremal wormholes and F-terms
    3. 7.3 Wormhole D-terms
    4. 7.4 Marginally degenerate wormholes and superpotentials
    5. 7.5 Implications for purely axionic models
  8. 8 Conclusions
  9. A Naive Dimensional Analysis
  10. B Matching in heterotic/F-theory dual models
    1. B.1 F-theory side
    2. B.2 Heterotic side
  11. C Other tests of the species scale bound
    1. C.1 Another F-theory model: $\mathbb{P}^{1}$ fibration over $\mathbb{F}_{p}$
    2. C.2 Energy scales in the $\mathbb{P}^{1}$ over $\mathbb{F}_{p}$ model
    3. C.3 Asymptotic tests of our species scale bound
      1. C.3.1 EFT string limits with $p_{a}e^{a}=0$
      2. C.3.2 EFT string limits with $p_{a}e^{a}\geq 1$
  12. D Wormholes and supersymmetry
    1. D.1 Extremal BPS wormholes
    2. D.2 Non-extremal wormholes
    3. D.3 Marginally degenerate wormholes

##  1 Introduction

From a purely bottom-up perspective, light axions can provide a solution to
several open problems in particle phenomenology: they are well known to offer
an elegant solution to the strong CP problem [1, 2, 3], to be interesting dark
matter candidates [4] and possibly drive inflation [5]. Remarkably, since the
early eighties it was recognized that string theory offers a natural framework
for such particles, see for instance [6] for an overview and references to the
older literature. In fact, the low-energy description of string theory
generically features a number of moduli fields; some of these are axions,
spinless particles carrying shift symmetries that are broken solely by non-
perturbative effects. In generic string theory models the number $N$ of low-
energy axions can easily be in the hundreds, if not thousands. A setup with
$N\gg 1$ axions can lead to an array of interesting phenomenological
signatures and is often referred to as axiverse [7]. A recent review and an
updated list of references on the string theory axiverse can be found in [8].

Despite their genericity, stringy axions can be phenomenologically relevant
only if their potential interactions are very small. The non-perturbative
effects, expected to break the axionic shift symmetries in generic string
constructions, must somehow be more suppressed than naively expected. Non-
perturbative effects can belong to two qualitatively different classes. The
first class has a genuinely UV nature and, in string theory models, typically
includes worldsheet and brane instantons. We will refer to them as fundamental
instantons. These would appear to any effective field theory (EFT) observer
simply as bare symmetry-breaking local operators. Yet, non-perturbative
effects may also emerge within the EFT itself. Besides the familiar gauge
theory instantons, in the presence of axions the second class includes axionic
wormholes [9] (for a review and more references on aspects relevant for the
present paper, see [10]). Given the variety of non-perturbative contributions
available in these theories, and their genericity, we should ask: How likely
is it to find light axions in axiverse models compatible with quantum gravity?
More generally, how and under which conditions are non-perturbative
corrections to their potentials suppressed?

The answer to the first question might simply turn out to be a mere
probabilistic argument: the number of axions in the axiverse is so large that
it becomes statistically likely that at least a few of them remain light. A
more interesting possibility is perhaps that some structural property of
quantum gravity lies behind it, though. Yet, to assess the viability of this
option one must at least partially address the second question. In particular,
we must improve our understanding of the string axiverse and all non-
perturbative effects therein. Encouraging results have been obtained in
particular in [11, 12], where it was shown that in a large class of IIB models
with $N\gg 1$ axions non-perturbative stringy corrections are suppressed in an
unexpectedly strong way.

The present paper makes a further contribution to this subject. We adopt the
formalism of [13, 14, 15] and consider four-dimensional effective field
theories with an arbitrary, possibly large, number $N$ of axions and minimal
${\cal N}=1$ supersymmetry. For simplicity we take the minimal field content
that is sufficient to describe the low-energy limit of an axiverse, but our
framework can be easily generalized by including additional degrees of
freedom, if needed. We focus on fundamental axions, which originate directly
from the UV completion of the EFT, rather than arising as (approximate) Nambu-
Goldstone bosons of some linearly realized accidental compact symmetry within
the EFT. The structure of our theories is constrained by quantum gravity and
string theory in a number of non-trivial ways; we will make heavy use of those
constrains. Two are the main objectives of our work. The first is a clear
identification of the regime of validity and of the relevant quantum gravity
scales of such effective theories. The second is a characterization of the
non-perturbative gravitational effects within that setup.

After an introduction of our EFTs (Section 2), in Section 3 we present a
careful analysis of their regime of validity both in energy and field space.
An interesting result is a lower bound of order $N$ on the (field dependent)
coefficient of the Gauss-Bonnet operator. We also present a detailed
discussion of the highest possible UV cutoff of our theory, namely the species
scale [16, 17, 18, 19]. The determination of the species scale is by itself a
very active area of research especially in the context of the Swampland
program (see [20, 21, 22, 23] for reviews). Our contribution is the proposal
of a new upper bound on that quantity. Our upper bound is fully determined by
EFT data already available at the two-derivative level and has the advantage
of having a clear physical interpretation and being radiatively stable. We
subsequently test our lower bound on the coefficient of the Gauss-Bonnet
operator and our upper bound on the species scale in a number of explicit
string theory models in Section 4.

Having firmly established our framework and its perturbative regime, we can
next move to wormholes. We begin in Section 5 recalling the derivation of
$O(4)$-invariant wormhole configurations in Euclidean space, and presenting an
analysis of their regime of validity. In particular, we discuss the relation
between extremal wormholes and fundamental BPS instantons, and subsequently
introduce the notion of EFT instanton [14]. Section 6 is dedicated to non-
extremal wormholes. We identify a universal class of _homogeneous_ wormhole
solutions, which involve all $N$ axions and their supersymmetric saxionic
partners, taking advantage of crucial inputs from string theory. Their
perturbative domain is analyzed, and a universal constraint relating the
minimal radius of the wormhole throat to the species scale is pointed out. We
later show how these homogeneous solutions are instrumental in understanding
some properties of more general solutions. We argue that a peculiar role is
played by _marginally degenerate_ non-extremal wormholes, unveiling a
suggestive analogy between this type of wormholes and the fundamental EFT
instantons carrying the same charges, which will be explored in more detail
subsequently. The general discussion is illustrated and tested, also through
numerical simulations, in concrete string theory models.

The physical implications of the non-perturbative effects associated to the
configurations discussed in Section 6 are analyzed in Section 7. We argue that
extremal wormholes can induce potentially relevant symmetry-breaking effective
superpotentials, as well as higher derivative F-terms [24, 25], at low
energies. On the other hand, non-extremal non-degenerate wormholes can at most
induce corrections to the Kähler potential. However, such D-terms come with
associated Coleman’s $\alpha$-parameters [26], which are undetermined in the
EFT, and so their physical relevance cannot be firmly established within our
framework. String theory experience and quantum gravity arguments strongly
suggest [27, 28] that the $\alpha$-parameters should be determined by the UV
completion, possibly in terms of other dynamical fields, but the actual
mechanism by which such determination might occur remains a mystery.
Interestingly, the physics of a specific subclass of non-extremal wormholes,
which we call _marginally degenerate_ , might shed light on this puzzling open
problem. Indeed, various considerations indicate that marginally degenerate
wormholes represent the low-energy manifestation of fundamental EFT instantons
and, as such, should be capable of inducing effective superpotentials at low
energies, generalizing a mechanism first pointed out in [29, 30]. If correct,
this conclusion in turn implies that the $\alpha$-parameters of marginally-
degenerate wormholes must necessarily be fixed by the UV-complete description
of the fundamental EFT instantons. Perhaps something similar might happen to
the $\alpha$-parameters of non-degenerate wormholes as well. In Section 7 we
also discuss how our results are inherited by non-supersymmetric scenarios UV-
completed by an ${\cal N}=1$ axiverse. Our conclusions are presented in
Section 8.

Our work is complemented by a few appendices. Naive Dimensional Analysis is
proposed in Appendix A as a guide to estimate the factors of $2\pi$ appearing
in some of the formulas of the paper. Some details on dual heterotic/F-theory
models, which we use in Section 4, are presented in Appendix B. Additional
evidence of the validity of our new bound on the species scale is given in
Appendix C. Appendix D discusses in some detail the wormhole fermionic zero-
modes and their implications on the structure of the wormhole-induced low-
energy effective operators.

##  2 ${{\cal N}}=1$ axiverse models

The basic assumption underlying our work is the existence of an effective
four-dimensional field theory (EFT) with a possibly large number $N$ of light
axions. We will focus on _fundamental_ axions, namely periodic axions like
those that typically originate from string theory, and that cannot be regarded
as angular components of some elementary field in four dimensions.

Because in quantum gravity (QG) global symmetries are expected to be at most
approximate, the global shift symmetries that prevent our axions to acquire
large masses must ultimately be broken. It is then crucial to identify a
concrete and realistic general framework in which the axionic shift symmetries
can be considered exact up to small corrections dictated by QG. Such a
framework is provided by the ${\cal N}=1$ setup outlined in [13, 14, 15]. The
associated EFTs emerge from large classes of string theory models and
naturally take into account the relevant QG constraints.

In Section 2.1 we review the basic setup of [13, 14, 15]. We focus on the
leading two-derivative approximation but, for reasons that will become clearer
later, also keep an eye on the (semi-)topological higher-derivative couplings
to gravity, and in particular on the Gauss-Bonnet interaction (see Section
2.2). Section 2.3 reformulates the EFT in a dual 2-form language in
preparation of the subsequent sections.

###  2.1 Axions in ${\cal N}=1$ EFTs

Let us start recalling the minimal structure of an ${\cal N}=1$ effective
theory involving $N$ _periodic_ axions $a^{i}$, $i=1,\ldots,N$ and their
bosonic partners. Without loss of generality we normalize the $a^{i}$’s so as
to have unit periodicity:

$a^{i}\simeq a^{i}+1\,.$ (2.1)

The “angular” variables $\theta^{i}$ often used to denote axions are related
to our fields via $\theta^{i}=2\pi a^{i}$. By supersymmetry the axions combine
with saxions $s^{i}$ into complex fields

$t^{i}=a^{i}+{\rm i}s^{i}\simeq t^{i}+1\,,$ (2.2)

which represent the bottom components of corresponding ${\cal N}=1$ chiral
multiplets. The EFT in general contains other fields. For simplicity we will
ignore them and just consider $t^{i}$ plus gravity. Our main conclusions do
not depend on this assumption.

The exact gauge symmetry (2.1) combined with supersymmetry constrains
significantly the EFT. The only manifestly supersymmetric non-derivative
couplings of the axions can be either semi-topological or functions of
$e^{2\pi iq_{i}t^{i}}$ with $q_{i}\in{\mathbb{Z}}$, and of their complex
conjugate, exponentially suppressed by the saxions. We will discuss the semi-
topological couplings shortly and postpone an analysis of the exponentially
suppressed instanton-like corrections to the following subsection, where we
also provide a quantitative definition of the perturbative regime in which
such corrections can be considered small.

Up to semi-topological couplings and instanton-like effects, the EFT is
invariant under arbitrary constant shifts of the axions. At the two-derivative
level and in Lorentzian signature, the contribution of the terms involving
only gravity and $t^{i}$ to the most general ${\cal N}=1$ shift-symmetric
action is

$\frac{1}{2}M_{\text{\tiny P}}^{2}\int R*1-\frac{1}{2}M_{\text{\tiny
P}}^{2}\int{\cal G}_{ij}(s)\left({\rm d}s^{i}\wedge*{\rm d}s^{j}+{\rm
d}a^{i}\wedge*{\rm d}a^{j}\right)$ (2.3)

where ${\cal G}_{ij}(s)$ is a symmetric positive matrix function of the saxion
fields and we have omitted appropriate Gibbons-Hawking boundary term. By
supersymmetry, the kinetic terms of the scalars are specified by a Kähler
potential $K$. Within our perturbative regime, $K$ depends on the complex
fields $t^{i}$ only through their saxionic component $s^{i}=-\frac{{\rm
i}}{2}(t^{i}-\bar{t}^{i})$, i.e. $K=K({s})$ (a possible dependence on
additional spectator multiplets is ignored), via the relation

${\cal G}_{ij}\equiv\frac{1}{2}\frac{\partial^{2}K}{\partial s^{i}\partial
s^{j}}.$ (2.4)

We can actually say more about our EFT if we take into account additional non-
trivial inputs from the UV completion. Indeed, within the perturbative regime
we are considering (to be more precisely defined later), for a large class of
string theory models the Kähler potential reads

$K({s})=-\log P({s})\,,$ (2.5)

where $P({s})$ is a positive homogeneous function,
$P(\lambda{s})=\lambda^{n}P({s})$ and $n$ is an integer ranging from $1$ to
$7$ – see Sections 4 and 6 for explicit examples. As discussed in [13, 14] the
perturbative structure (2.5) conforms with various formulations of the weak
gravity conjecture [31] and the distance conjecture [32] in the present
setting. The homogeneity of $P$ and the relation (2.5) will play a crucial
role in some of the subsequent sections.

###  2.2 Semi-topological couplings to gravity

Eq. (2.3) just represents the leading two-derivative term in our EFT. In
general one should also allow the presence of higher dimensional interactions
suppressed by some mass scale $M_{\text{\tiny UV}}$ that depends on the UV
completion of the EFT and in general on the EFT scalar fields. On the other
hand, from the Wilsonian viewpoint any EFT is associated with a (field
independent) cutoff scale $\Lambda$, which defines the upper bound on the
allowed momentum scales ($p\leq\Lambda$) and must obey $\Lambda\ll
M_{\text{\tiny UV}}$. Note that $M_{\text{\tiny UV}}$ in general depends on
saxions $s^{i}$, $M_{\text{\tiny UV}}=M_{\text{\tiny UV}}(s)$, and then the
condition $\Lambda\ll M_{\text{\tiny UV}}(s)$ will in general restrict the
field space region in which the EFT is valid. But how can we know what
$M_{\text{\tiny UV}}(s)$ is without knowing the UV completion of the EFT? We
will come back to this important question in Section 3.4. For the moment we
observe that imposing $\Lambda\ll M_{\text{\tiny UV}}$ one may naively presume
that the effect of all higher dimensional operators can be safely neglected.
However, a very special class of higher-dimensional operators may be
unsuppressed at low scales. These are the _topological terms_ which can be
obtained as integrals of total derivative operators. Because of their nature,
they do not affect the equations of motion nor induce particle vertices.
Nevertheless, they can contribute to the on-shell action and therefore impact
semiclassical calculations.

In a four-dimensional gravitational context, a well-known example of such a
topological term is provided by the integral of the Gauss-Bonnet (GB) operator

$E_{\text{\tiny
GB}}\equiv\frac{1}{32\pi^{2}}\left(R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^{2}\right)\,,$
(2.6)

which can indeed be locally written as a total covariant derivative. In an
${{\cal N}}=1$ SUSY framework, this operator originates from a superspace
combination of the form [33, 34, 35, 36]

$\left(\int{\rm d}^{4}x\,{\rm d}^{2}\theta\,2{\cal E}\,f\,{\cal
W}^{\alpha\beta\gamma}{\cal
W}_{\alpha\beta\gamma}+\text{c.c.}\right)+\text{D-terms}\,,$ (2.7)

where ${\cal W}^{\alpha\beta\gamma}$ is the Weyl chiral superfield, $f$ is a
holomorphic function of chiral superfields and the D-terms take a specific
form which will not be relevant in the following – see [37] for more details
in the present context. If $f$ is constant then (2.7) is topological, but in
our context can in general depend on the chiral fields $t^{i}$. More
precisely, the F-term appearing in (2.7) includes both a coupling of ${\rm
Im}f$ to the bosonic Weyl density and of ${\rm Re}f$ to the Pontryagin form
$\mathop{\rm tr}({\cal R}\wedge{\cal R})$, while the D-terms provide the
$R^{mn}R_{mn}$ and $R^{2}$ counterterms which combine with the Weyl density to
give the GB operator (2.6).

Consistency with (2.1) implies that, in addition to a constant, $f(t)$ can
contain a linear combination $\tilde{C}_{i}t^{i}$ for some real constants
$\tilde{C}_{i}$, plus possible exponentially suppressed instanton-like
corrections which will be ignored. Adopting the same normalization conventions
of [37], the $\tilde{C}_{i}t^{i}$ contribution in (2.7) gives a GB term

$\int{\rm d}^{4}x\,\sqrt{-g}\,\gamma(s)\,E_{\text{\tiny GB}}\,,$ (2.8)

with

$\gamma(s)\equiv\frac{\pi}{6}\tilde{C}_{i}s^{i}\,,$ (2.9)

and a Pontryagin term proportional to $\tilde{C}_{i}\int a^{i}\mathop{\rm
tr}({\cal R}\wedge{\cal R})$. Taking into account the precise numerical
factors and imposing various consistency conditions on the Pontryagin
operator, one finds that the constants $\tilde{C}_{i}$ must be integrally
quantized [37]: $\tilde{C}_{i}\in\mathbb{Z}$.111In complete analogy,
supersymmetry fixes linear couplings of (s)axions to vector fields to take the
form  $-\frac{1}{8\pi}\int C_{i}s^{i}\mathop{\rm
tr}(F\wedge*F)-\frac{1}{8\pi}\int C_{i}a^{i}\mathop{\rm tr}(F\wedge F)\,.$
(2.10) We will study only solutions with trivial gauge configurations, and so
the above couplings are not of primary interest here. See section 2 of [37]
for a more detailed discussion of these quasi-topological terms in the present
setting. Note that in presence of boundaries (2.8) must be supplemented by
Gibbons-Hawking-like boundary terms, which will be explicitly discussed in
Section 5.

The coefficients of the D-terms appearing in (2.7) are instead not protected
by holomorphy and hence supersymmetry is not enough to provide robust
information about them. In particular, they can in principle have a more
complicated dependence on $s^{i}$, and moreover be affected by radiative
corrections. Fortunately, Ricci squared terms are also basis-dependent, in the
sense that re-defining the metric one can always trade them for operators
involving derivatives of $s^{i}$. Hence, without loss of generality, we can
choose a field basis in which the non-derivative saxions couplings to
curvature squared operators reduce to the GB term (2.8).

The coefficient of the GB operator receives non-perturbative as well as
perturbative corrections. The former are negligible in our setup (see Section
3.1). The latter are of two types. By supersymmetry, radiative contributions
to $\gamma$ are exhausted by a constant 1-loop correction $\propto
N\log\Lambda$. Yet, a more subtle correction to the GB appears in our
scenario. This is because, strictly speaking, the standard manifestly
supersymmetric formulation [38] is not automatically in the Einstein frame,
which instead we used in (2.3). In order to pass to the Einstein frame a Weyl
rescaling $\Phi\to e^{\beta K}\Phi$ of all fields, with $\beta$ some number,
is necessary. Such transformation is anomalous and brings a non-manifestly
supersymmetric correction to the coefficient of the GB term of the parametric
form $\sim N\ln e^{\beta K}=\beta NK$. 222Similarly, the re-definition of the
metric necessary to remove the saxion couplings to $R_{ab}R^{ab}$ and $R^{2}$
may induce a Weyl anomaly, but that does not carry an $\sim N$ enhancement and
is hence parametrically smaller. Despite the $\propto N$ nature of these two
perturbative corrections, however, we will see in Section 3.3 that they are
both subleading compared to (2.9) in any tractable framework. Therefore (2.8)
and (2.9) provide an accurate approximation of the GB term.

The above GB term is singled out from the infinite set of higher-derivative
interactions by its quasi-topological nature. For example, after stabilization
of the saxions (2.8) (and supplemented by the boundary terms discussed in
Section 5), it becomes a purely topological term that does not alter the
axions’ equations of motion but nevertheless contributes to the on-shell
action of topologically non-trivial space-times. In particular it plays an
important role in wormhole physics, as we will see. At the perturbative level,
the topological nature of GB is connected to the absence of ghosts, which is
why string theory effective actions of any dimension seem to favor it, so to
speak, over other higher curvature terms [39].

###  2.3 Dual formulation and EFT strings

In order to make contact with the QG structures highlighted in [13, 14, 37],
it is convenient to recall the basic features of the dual formulation, in
which the axions are traded for two-form potentials ${\cal B}_{2,i}$ with
corresponding field-strengths ${\cal H}_{3,i}={\rm d}{\cal
B}_{2,i}=-M_{\text{\tiny P}}^{2}{\cal G}_{ij}*{\rm d}a^{i}$. This duality
transformation can be completed into a full supersymmetric duality which
trades the $t^{i}$ chiral multiplets for corresponding linear multiplets [40].
Following [41], with the notation of [13, 14], the linear multiplets have as
bottom components the dual saxions $\ell_{i}$, which are related to the
saxions $s^{i}$ by

$\ell_{i}=-\frac{1}{2}\frac{\partial K}{\partial s^{i}}\,.$ (2.11)

The kinetic terms are specified by the kinetic potential

${\cal F}=K+2\ell_{i}s^{i}\,,$ (2.12)

which must be considered as a function of the dual saxions $\ell_{i}$ (and of
the spectator fields). Note that $K$ and ${\cal F}$ are defined up to an
arbitrary constant. The leading order action (2.3) is equivalently re-written
as

$\frac{1}{2}M_{\text{\tiny P}}^{2}\int R*1-\frac{1}{2}M_{\text{\tiny
P}}^{2}\int{\cal G}^{ij}{\rm d}\ell_{i}\wedge*{\rm
d}\ell_{j}-\frac{1}{2M_{\text{\tiny P}}^{2}}\int{\cal G}^{ij}{\cal
H}_{3,i}\wedge*{\cal H}_{3,j}\,,$ (2.13)

where

${\cal G}^{ij}\equiv-\frac{1}{2}\frac{\partial^{2}{\cal
F}}{\partial\ell_{i}\partial\ell_{j}}$ (2.14)

is the inverse matrix of (2.4). Furthermore the inverse of the relation (2.11)
is given by

$s^{i}=\frac{1}{2}\frac{\partial{\cal F}}{\partial\ell_{i}}\,.$ (2.15)

The dualization from the axions $a^{i}$ to the two-forms ${\cal B}_{2,i}$
produces also a boundary term, to be added to (2.13),

$-{\rm i}\int_{\partial{\cal M}}a^{i}{\cal H}_{3,i}\,,$ (2.16)

which may be relevant in evaluating the on-shell actions.

The field-strengths ${\cal H}_{3,i}$ satisfy the Bianchi identity

${\rm d}{\cal H}_{3,i}=-\frac{1}{96\pi}\tilde{C}_{i}\mathop{\rm tr}({\cal
R}\wedge{\cal R})\,.$ (2.17)

In fact, for our purposes one can consistently neglect the Pontryagin four-
form appearing in (2.17), since it will be identically vanishing in all the
configurations that we will explore. Because in the following discussions
gauge fields will play no role, in (2.17) we have not included their
contributions, which is dual to the axionic terms appearing in (2.10) (see for
instance Section 3.2 of [37]). Eq. (2.17) can also be corrected by the
localized contribution of fundamental instantons, of the type considered in
Section 5.3.

Note that by the homogeneity of $P({s})$ (see Eq. (2.5)) we have
$\ell_{i}s^{i}=\frac{n}{2}$ and then, omitting an irrelevant additional
constant, the dual saxion kinetic potential (2.12) takes the form

${\cal F}({\ell})=\log\tilde{P}({\ell})\,,$ (2.18)

where

$\tilde{P}({\ell})\equiv\frac{1}{P({s}({\ell}))}\,$ (2.19)

is a homogeneous function of degree $n$:

$\tilde{P}(\lambda{\ell})=\lambda^{n}\tilde{P}({\ell})\,.$ (2.20)

Similarly, the (semi-)topological couplings to gravity can be written as in
(2.8)-(2.9) provided we interpret $s^{i}$ as a function of the dual saxions,
as dictated by (2.15).

In the present setting it is natural to consider strings carrying magnetic
axionic charges $e^{i}\in\mathbb{Z}$, around which $a^{i}\rightarrow
a^{i}+e^{i}$. If such strings are BPS, their tension is completely fixed by
supersymmetry [41]:

${\cal T}_{\bf e}=M^{2}_{\text{\tiny P}}e^{i}\ell_{i}\equiv M^{2}_{\text{\tiny
P}}\langle{\bm{\ell}},{\bf e}\rangle\,,$ (2.21)

where we have introduced the index-free notation $\langle{\bm{\ell}},{\bf
e}\rangle\equiv\ell_{i}e^{i}$, which will be largely used in the
following.333See [13, 14] for a thorough discussion on the precise
interpretation of the field-dependence of (2.21). The formula (2.21) imposes a
non-trivial constraint $\langle{\bm{\ell}},{\bf e}\rangle\geq 0$ on the
charges as well as the dual saxions. We will elaborate on this constraint in
the next section, when a domain for the saxions and dual saxions is
identified.

##  3 EFT regime of validity

The EFT described in the previous section has a limited domain of validity.
First, as any low-energy description, it has an associated derivative
expansion and thus a maximal UV cutoff. Second, any such EFT can reliably
describe the low-energy limit of string theory models only in a limited domain
in field space. Furthermore, in the spirit of the Swampland Program,
compatibility with QG/string theory imposes some non-trivial constraints on
the EFT structure, in addition to the standard QFT ones. In this section we
will discuss these aspects in some detail. Specifically, in Section 3.1 we
quantify the size of the QG effects that violate the axionic shift-symmetry
and introduce the concept of saxionic cone. In Section 3.2 we provide an
unambiguous definition of the domain of validity of our EFT _in field space_.
In Section 3.3 it is then shown that, within this perturbative domain and in
the large $N$ limit, the coefficient of the Gauss-Bonnet interaction is
subject to an interesting lower bound. The domain of validity of our EFT _in
momentum space_ is finally analyzed in Section 3.4.

Before embarking in our detailed analysis it is useful to make some simple
consideration based on dimensional analysis, which is sufficient to
qualitatively understand how the EFT domain of validity may be controlled by
the saxions $s^{i}$. If we momentarily restore the powers of $\hbar$ and
insist that $\tilde{C}_{i}$ in (2.8) – see also $C_{i}$ in (2.10) – be truly
dimensionless integers, one infers that $s^{i}$ and $a^{i}$ have dimension
$[s^{i}]=[a^{i}]=[\hbar]$. Because quantum corrections come with powers of
$\hbar$, one realizes that

$\alpha_{*}\equiv\frac{1}{s_{*}},$ (3.1)

with $s_{*}$ denoting some appropriate positive linear combination of the
saxions $s^{i}$, represents a sort of “fine structure constant” of our theory.
Accordingly, in order for our EFT to make sense one should require the
dimensionless “loop counting parameter” in four-dimensional theories to be
small, namely

$\frac{\alpha_{*}\hbar}{2\pi}\equiv\frac{\hbar}{2\pi s_{*}}\ll
1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left(\frac{\hbar\ell_{*}}{2\pi}\ll
1\right)\,,$ (3.2)

where $(2\pi)^{-1}$ arises from the usual four-dimensional loop
factor.444Recall that the structure constant is a coupling squared divided by
$2\pi$. Throughout the paper we will keep track of the “geometric” factors of
$2\pi$ but ignore factors of order unity (see Appendix A), and
$2\pi\alpha_{*}$ will represent our “coupling squared”. The analogous
expansion in $\hbar/(2\pi a_{*})$ simply cannot appear because of the
approximate shift symmetry. As a simple example, which is obvious from the
effective field theorist’s viewpoint, the gauge coupling squared appearing in
(2.10) takes the form $2\pi\alpha_{*}=2\pi/s_{*}$ with $s_{*}=C_{i}s^{i}$, and
so (3.2) represents the standard perturbative regime for the gauge theory.
What the effective field theorist cannot know, however, is that conditions of
the form (3.2) are in fact instrumental in computing EFTs from string theory
models and, for instance, disguise large volume or weak string coupling
expansions. As an example, of crucial importance for the present paper is the
form (2.5) of the Kähler potential: such form holds only to first
approximation in an appropriate large-saxion expansion and is expected to
receive both perturbative and non-perturbative corrections. In view of our
dimensional analysis argument, it should not come as a surprise that
perturbative corrections are controlled by $\alpha_{*}/(2\pi)$ whereas the
non-perturbative effects that break the axionic shift symmetries are of order
$~{}e^{-\frac{2\pi}{\alpha_{*}\hbar}}=e^{-\frac{2\pi s_{*}}{\hbar}}$, and are
hence exponentially suppressed by a requirement of the form (3.2).

Non-perturbative effects may be due to fundamental instantons beyond the EFT
or by physics within the EFT, e.g. gauge instantons or wormholes. Wormholes
will be discussed at length in the following sections while gauge instantons,
not being directly related to QG aspects, will not be considered in the
present paper. In the following subsection we will instead focus on
fundamental instantons, since they encode non-trivial information on the UV
completion of the theory and turn out to strongly characterize the EFT
structure. From now on we will go back to the more conventional $\hbar=1$
units.

###  3.1 Saxionic cones, fundamental instantons and strings

The contribution of point-like fundamental instantons are ubiquitous in string
theory compactifications, in which they are typically associated to Euclidean
branes wrapping internal cycles – see for instance [42] for a review. Their
effects show up in the four-dimensional EFT defined at its highest possible UV
cutoff as shift-symmetry breaking local operators. Imposing that these are
sufficiently small is what defines our perturbative regime. Among all
fundamental instantons, the BPS ones — preserving $\frac{1}{2}$ of the bulk
supersymmetry — are expected to be the most relevant.555Experience with
supersymmetric instantons suggests that the action $S_{\rm inst}$ of a
possible non-BPS fundamental instanton carrying charges $q_{i}$ obeys a BPS
bound $S_{\text{non-BPS}}>S_{\text{BPS}}$. On the other hand, the axion form
of the weak gravity conjecture [31] suggests the possible existence of non-BPS
instantons violating such bound – see for instance [43, 44] and appendix B of
[12] for related discussions in string theory contexts. We nevertheless expect
such possible violations not to affect the following considerations. These
carry a set of quantized axionic charges $q_{i}\in\mathbb{Z}$ and contribute
to the effective action by terms proportional to $e^{2\pi{\rm i}q_{i}a^{i}}$.
By holomorphy, BPS instantons generate terms proportional to $e^{2\pi{\rm
i}q_{i}t^{i}}$. For each BPS instanton of charges $q_{i}$ there exists an
anti-instanton of charges $-q_{i}$ preserving the opposite $\frac{1}{2}$
supersymmetry and contributing by terms proportional to $e^{-2\pi{\rm
i}q_{i}\bar{t}^{i}}$. Hence, BPS and anti-BPS instantons combine and
contribute to the effective action via operators proportional to

$e^{-2\pi\langle{\bf q},{\bm{s}}\rangle}\,,$ (3.3)

where we are again using the index-free pairing introduced in (2.21):

$\langle{\bf q},{\bm{s}}\rangle\equiv q_{i}s^{i}.$ (3.4)

In our notation $\langle\,.\,,\,.\,\rangle$ is the canonical pairing between
the elements of dual vector spaces $V_{\mathbb{R}}$ and $V^{*}_{\mathbb{R}}$,
and corresponding dual lattices $V_{\mathbb{Z}}\subset V_{\mathbb{R}}$ and
$V^{*}_{\mathbb{Z}}\subset V^{*}_{\mathbb{R}}$. One can introduce an integral
basis $\\{{\bf v}_{i}\\}^{N}_{i=1}$ of generators of $V_{\mathbb{Z}}$ and the
dual basis $\\{{\bf w}^{i}\\}^{N}_{i=1}$ of generators of
$V^{*}_{\mathbb{Z}}$, such that $\langle{\bf w}^{i},{\bf
v}_{j}\rangle=\delta^{i}_{j}$. The saxions $s^{i}$ and the charges $q_{i}$ are
the components of the vectors ${\bm{s}}=s^{i}{\bf v}_{i}\in V_{\mathbb{R}}$
and ${\bf q}=q_{i}{\bf w}^{i}\in V^{*}_{\mathbb{R}}$ respectively. The set of
all BPS instanton charges is denoted by

${\cal C}_{\rm I}=\\{\text{set of BPS instanton charges ${\bf q}$}\\}\subset
V^{*}_{\mathbb{Z}}\,.$ (3.5)

Given two BPS instantons of charge vectors ${\bf q}_{1}$ and ${\bf q}_{2}$,
being mutually BPS, they can be superimposed to form a BPS instanton of charge
vector ${\bf q}_{1}+{\bf q}_{2}$. Hence ${\cal C}_{\rm I}$ can be regarded as
discrete convex cone, generated by a set of “elementary” BPS instanton
charges.

The combination $2\pi\langle{\bf q},{\bm{s}}\rangle$ appearing in (3.3)
represents the real part of the BPS instanton Euclidean action, and must be
positive. Hence the saxions necessarily take values in the saxionic
cone:666The definition of saxionic cone of [13, 14] slightly differs from
(3.6), in that it does not include the boundary faces at which $\langle{\bf
q},{\bm{s}}\rangle=0$ for some ${\bf q}$, and then some instanton action
degenerates. We include these faces since we will anyway restrict ourselves to
more interior regions – see Section 3.2.

$\Delta\equiv\\{{\bm{s}}\in V_{\mathbb{R}}|\langle{\bf q},{\bm{s}}\rangle\geq
0,\ \forall{\bf q}\in{\cal C}_{\rm I}\\}\,.$ (3.6)

This is a convex cone, whose prototypical example is provided by the Kähler
cone in heterotic or type IIA string compactifications on Calabi-Yau spaces,
where ${\cal C}_{\rm I}$ represents the cone of effective curves which can be
wrapped by world-sheet instantons. These and other string theory realizations
will be more explicitly discussed in the following.

The magnetic axionic string charges $e^{i}\in\mathbb{Z}$ introduced around
(2.21) specify an element ${\bf e}=e^{i}{\bf v}_{i}$ of $V_{\mathbb{Z}}$. We
call EFT strings those strings associated to charge vectors ${\bf e}$
belonging to the set [13, 14, 15]

${\cal C}_{\rm S}^{\text{\tiny EFT}}=V_{\mathbb{Z}}\cap\Delta-\\{{\bf
0}\\}\,.$ (3.7)

According to this definition, one may regard the saxionic cone $\Delta$ as
being generated by the EFT string charges. We will assume $\Delta$ to be
polyhedral – see [14] for a more detailed discussion on this assumption – and
hence to be generated by a finite number of elementary EFT string charges,
i.e. charges ${\bf e}\in{\cal C}_{\rm S}^{\text{\tiny EFT}}$ that cannot be
written as a sum with positive integral coefficients of other elements of
${\cal C}_{\rm S}^{\text{\tiny EFT}}$. (We will extend this terminology to BPS
strings and BPS instantons in an obvious way.) As we will see, EFT strings
will play a key role in the following discussions.

We can now define the dual saxionic domain ${\cal P}$ as the closure of the
image of $\Delta$ under the transform (2.11):

${\cal P}=\overline{\Big{\\{}{\bm{\ell}}=\ell_{i}{\bf w}^{i}\in
V^{*}_{\mathbb{R}}\Big{|}\ell_{i}=-\frac{1}{2}\frac{\partial K}{\partial
s^{i}}\Big{|}_{{\bm{s}}\in\Delta}\Big{\\}}}.$ (3.8)

While the general structure of ${\cal P}$ can be a priori complicated, if $K$
takes the form (2.5) then ${\cal P}$ becomes conical. Indeed, if
${\bm{\ell}}\in{\cal P}$ is the image of ${\bm{s}}\in\Delta$ then
$\lambda{\bm{\ell}}$ is the image of $\lambda^{-1}{\bm{s}}$, for any
$\lambda>0$. Since also $\lambda^{-1}{\bm{s}}$ belongs to $\Delta$, then
$\lambda{\bm{\ell}}$ belongs to ${\cal P}$. (On the other hand, ${\cal P}$ is
not necessarily convex.) Note that by consistency any BPS (non-necessarily
EFT) string tension (2.21) must be positive in the interior of ${\cal P}$, and
the condition $\langle{\bm{\ell}},{\bf e}\rangle\geq 0$ for any
${\bm{\ell}}\in{\cal P}$ can be taken as defining condition of the BPS string
charges ${\bf e}\in{\cal C}_{\rm S}$, that is:777We recall that, given two
dual vector spaces $V_{\mathbb{R}}$ and $V^{*}_{\mathbb{R}}$ and any subset
$I\subset V_{\mathbb{R}}$, its dual cone $I^{\vee}$ is by definition the set
of elements ${\bm{a}}\in V^{*}_{\mathbb{R}}$ such that
$\langle{\bm{a}},{\bm{b}}\rangle\geq 0$ for any ${\bm{b}}\in I$.

${\cal C}_{\rm S}\in V_{\mathbb{Z}}\cap{\cal P}^{\vee}-\\{{\bf 0}\\}\,.$ (3.9)

This implies that different boundary components of $\partial{\cal P}$ can be
associated with the possible vanishing of different BPS string tensions
(2.21). In particular, components of $\partial{\cal P}$ which are at infinite
field distance are detected by the vanishing of some EFT string tensions, i.e.
corresponding to some ${\bf e}\in{\cal C}^{\text{\tiny EFT}}_{\rm S}$ [14]. On
the other hand, finite distance components of $\partial{\cal P}$ could be
associated with (classically) tensionless non-EFT BPS strings, i.e. with ${\bf
e}\in{\cal C}_{\rm S}-{\cal C}^{\text{\tiny EFT}}_{\rm S}$. These tensionless
strings naturally identify a rational polyhedral part of the boundary of
${\cal P}$. There may also be more general finite distance boundaries of
${\cal P}$, not directly associated with tensionless strings. In any case, one
should keep in mind that finite distance boundaries are not so sharply
defined, since around them non-perturbative corrections can a priori become
relevant and may for instance generate strong corrections to the formula
(2.21). Other strongly coupled regions are reached by radially moving away
from the tip of ${\cal P}$ along different directions. If one insists in using
the “bare” kinetic potential (2.18), these appear as infinite distance limits
in which all the BPS tensions (2.21) diverge. However in these limits the
above description breaks down since it assumes that

${\cal T}_{\bf e}<2\pi M^{2}_{\text{\tiny P}}\,,$ (3.10)

in order for the string to have a weak gravitational backreaction [13].888The
$2\pi$ factor is introduced to match the NDA arguments of Appendix A, but can
also be understood recalling that a string of constant tension ${\cal T}$
generates a deficit angle $\Delta\theta={\cal T}/M^{2}_{\text{\tiny P}}$ (see
e.g. [45]) and imposing $\Delta\theta<2\pi$. Therefore, non-perturbative
physics may again completely change the nature of these limits. In any case,
we see how the behavior of the BPS tension (2.21) can be a useful proxy to
qualitative characterize the different boundary components of ${\cal P}$.

###  3.2 Perturbative domain and saxionic convex hull

The realization that saxions must necessarily belong to $\Delta$ is bringing
us closer to a precise definition of the domain of validity of our EFT.
Unfortunately, ${\bm{s}}\in\Delta$ is not sufficient to suppress instantons
nor to ensure the perturbativity requirement suggested in (3.2) holds. In this
subsection we will more precisely identify a perturbative domain with a subset
of $\Delta$, controlled by a single perturbative coupling $\alpha<2\pi$,
analogous to (3.1). We will consider two possible perturbative domains: the
$\alpha$-saxionic convex hull $\hat{\Delta}_{\alpha}$ and the
$\alpha$-stretched saxionic cone $\tilde{\Delta}_{\alpha}$, the latter being
defined in analogy to the stretched Kähler cones introduced in [11]. Since
$\hat{\Delta}_{\alpha}\subseteq\tilde{\Delta}_{\alpha}$, for simplicity in the
rest of the paper we will adopt the more conservative $\hat{\Delta}_{\alpha}$
as our main definition perturbative domain, though most of our conclusions
would clearly hold for $\tilde{\Delta}_{\alpha}$ as well.

Consider the set of all the elementary EFT string charges $\\{{\bf
e}_{A}\\}_{A\in{\cal J}}\subset{\cal C}^{\text{\tiny EFT}}_{\rm S}$, which
generate the entire ${\cal C}^{\text{\tiny EFT}}_{\rm S}$, where ${\cal J}$
denotes the corresponding set of indices. Take any subset ${\cal
J}_{\sigma}\subset{\cal J}$ of $N$ elements, such that the corresponding
elementary charges $\\{{\bf e}_{A}\\}_{A\in{\cal J}_{\sigma}}$ are linearly
independent. Each of these subsets is associated to a regular simplicial cone

$\sigma=\\{\sum_{A\in{\cal J}_{\sigma}}\lambda^{A}{\bf e}_{A}|\lambda^{A}\geq
0\\}.$ (3.11)

For each of these cones we construct a corresponding “$\alpha$-stretched” cone

$\tilde{\sigma}_{\alpha}=\\{\sum_{A\in{\cal J}_{\sigma}}\lambda^{A}{\bf
e}_{A}|\lambda^{A}\geq\frac{1}{\alpha}\\}$ (3.12)

with $\alpha>0$ some small number that represents the largest possible value
of the couplings of the form (3.1). The $\alpha$-saxionic convex hull
$\hat{\Delta}_{\alpha}$, anticipated at the beginning of this subsection, is
defined as the convex hull of all the stretched sub-cones
$\tilde{\sigma}_{\alpha}$, that is

$\hat{\Delta}_{\alpha}=\Big{\\{}{\bm{s}}=\sum_{\sigma}\lambda^{\sigma}{\bm{s}}_{\sigma}\in\Delta\Big{|}\
\lambda^{\sigma}\geq 0,\ \sum_{\sigma}\lambda^{\sigma}=1,\
{\bm{s}}_{\sigma}\in\tilde{\sigma}_{\alpha}\Big{\\}}.$ (3.13)

Intuitively, any element of $\hat{\Delta}_{\alpha}$ can be considered as a
linear average of saxions whose components satisfy $s^{i}\geq 1/\alpha$ in
some basis of elementary EFT strings charges. The perturbative regime can now
be precisely identified by the requirement ${\bm{s}}\in\hat{\Delta}_{\alpha}$.
With a sufficiently small $\alpha$, this definition simultaneously formalizes
the perturbativity requirement (3.2) as well as guarantees the suppression of
non-perturbative corrections.

In order to better understand this latter point, let us relate
$\hat{\Delta}_{\alpha}$ to the $\alpha$-stretched saxionic cone 999We are
adapting the terminology of [11] which may be slightly misleading, since
$\tilde{\Delta}_{\alpha}$ is generically not a cone, but rather a convex
polyhedron.

$\tilde{\Delta}_{\alpha}\equiv\\{{\bm{s}}\in\Delta|\langle{\bf
q},{\bm{s}}\rangle\geq\frac{1}{\alpha},\ \forall{\bf q}\in{\cal C}_{\rm
I}\\}\,.$ (3.14)

This definition is more directly motivated by the non-perturbative corrections
(3.3) (rather than the perturbative ones), since in $\tilde{\Delta}_{\alpha}$
all such corrections are bounded from above by $e^{-2\pi/\alpha}$. Now,
because our $\tilde{\sigma}_{\alpha}$ appearing in (3.13) are clearly subsets
of $\tilde{\Delta}_{\alpha}$, by convexity the same inclusion extends to the
entire saxionic convex hull, so that:

$\hat{\Delta}_{\alpha}\subset\tilde{\Delta}_{\alpha}\subset\Delta\,.$ (3.15)

Hence the condition ${\bm{s}}\in\hat{\Delta}_{\alpha}$ is stronger, though
qualitatively similar, to the condition ${\bm{s}}\in\tilde{\Delta}_{\alpha}$.
(Clearly $\hat{\Delta}_{\alpha}=\tilde{\Delta}_{\alpha}$ if $\Delta$ is a
simplicial cone.) As a result, the non-perturbative corrections to our EFT are
at least suppressed by $e^{-2\pi/\alpha}$ when
${\bm{s}}\in\hat{\Delta}_{\alpha}$.101010Note that, given a saxionic cone
$\Delta$, its boundary can be regarded as the union of conical faces
$\Delta^{\prime}\subset\partial\Delta$ of various codimensions. These may be
associated with corresponding perturbative domains
$\tilde{\Delta}^{\prime}_{\alpha}$ or $\hat{\Delta}^{\prime}_{\alpha}$ (which
are not subsets of $\tilde{\Delta}_{\alpha}$ or $\hat{\Delta}_{\alpha}$).

The perturbative saxionic regions
$\tilde{\Delta}_{\alpha},\hat{\Delta}_{\alpha}\subset\Delta$ are associated to
corresponding dual saxionic regions $\tilde{\cal P}_{\alpha},\hat{\cal
P}_{\alpha}\subset{\cal P}$ through the map (2.11). In the case of Kähler
potentials of the form (2.5) the regions dual to
$\tilde{\Delta}_{\alpha},\hat{\Delta}_{\alpha}$ are concentrated around the
dual saxion origin ${\bm{\ell}}=0$ – see Fig. 1 below for a simple but non-
trivial concrete example.

To summarize, by assuming ${\bm{s}}\in\hat{\Delta}_{\alpha}$ (or
${\bm{\ell}}\in\hat{\cal P}_{\alpha}$) we are certain that our EFT (2.3) with
(2.5) provides a reliable low-energy description of a large class of string
theory models up to controllable powers of $\alpha/2\pi\ll 1$ and
$e^{-2\pi/\alpha}$. In particular, the latter exponential factor suppresses
any explicit breaking of the axion shift symmetries. Our expansion parameter
$\alpha$ is nothing but an upper bound on the quantity $\alpha_{*}$ alluded to
in Eq. (3.1). In the following we will therefore assume that
${\bm{s}}\in\hat{\Delta}_{\alpha}$ and for concreteness have in mind
$\alpha\simeq 0.1$ as benchmark value. In fact, $\alpha\simeq 1$ might already
be enough to sufficiently suppress both perturbative and non-perturbative
effects. However, such values of $\alpha$ do not necessarily ensure the
reliability of the leading order expression (2.5). Concretely, in the case of
the Kähler cone of heterotic compactifications, setting $\alpha\simeq 1$
allows for string-size internal cycles, which for instance cast doubts on the
geometric formula corresponding to (2.5) – see Section 4 for more details.
From these considerations $\alpha\simeq 0.1$ seems a more reassuring choice.
Much smaller values face another problem, though, which will be analyzed in
Section 3.4: in the limit $\alpha\rightarrow 0$ the maximal possible UV cutoff
of the EFT tends to zero!

###  3.3 Quantum gravity bounds

In [37] it was shown how quantum consistency in the presence of EFT strings
imposes strong constraints on the structure of the bulk theory. These
constraints crucially involve the constants $\tilde{C}_{i}$ appearing in
(2.9). In particular $\tilde{C}_{i}$ must satisfy the quantization condition
$\langle\tilde{\bf C},{\bf e}\rangle\equiv\tilde{C}_{i}e^{i}\in\mathbb{Z}$,
for any string charge vector ${\bf e}\in V_{\mathbb{Z}}$. More importantly for
us, in [37] it was argued that $\langle\tilde{\bf C},{\bm{s}}\rangle$ enter
some positivity bounds which, in their weakest form, imply that

$\langle\tilde{\bf C},{\bf e}\rangle\geq
0\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall{\bf e}\in{\cal
C}^{\text{\tiny EFT}}_{\rm S}\,.$ (3.16)

Recalling (2.9) and the fact that the EFT string charges generate the saxionic
cone, (3.16) has as a consequence that $\gamma({s})\geq 0$ for any
${\bm{s}}\in\Delta$.

In order to get a stronger lower bound for $\gamma({s})$ we observe that
$\langle\tilde{\bf C},{\bf e}\rangle$ is proportional to the two-dimensional
gravitational anomaly of the EFT string of charge vector ${\bf e}$. Since such
strings break half of the bulk supersymmetry and support a chiral $(0,2)$
world-sheet, they generically have a chiral spectrum with non-vanishing
gravitational anomaly. This means that (3.16) generically translates into the
stricter bound

$\langle\tilde{\bf C},{\bf e}\rangle\in\mathbb{Z}_{\geq
1}\quad~{}~{}~{}~{}\forall\text{ generic }{\bf e}\in{\cal C}^{\text{\tiny
EFT}}_{\rm S}\,.$ (3.17)

Note that in many models (3.17) can be strengthened to $\langle\tilde{\bf
C},{\bf e}\rangle\in 3\mathbb{Z}_{\geq 1}$. This stronger bound holds if the
world-sheet normal bundle $U(1)_{\rm N}$ symmetry is classically preserved
(though generically anomalous at the quantum world-sheet level). This is a
conceivable expectation, which is indeed realized in large classes of string
theory models, such as the F-theory/type IIB ones of Section 4.1. On the other
hand, in [37] it was pointed out that in addition to the standard quantum
$U(1)_{\rm N}$ anomaly there could be classical Green-Schwarz-like terms on
the world-sheet, which signal the existence of an intermediate microscopic
description in terms of a five dimensional ${\cal N}=1$ supergravity (in
presence of possible supersymmetry breaking defects). The five-dimensional
arguments of [46] hence lead to (3.16), and then also (3.17). For instance,
this weaker bound can hold in the $E_{8}\times E_{8}$ heterotic models of
Section 4.2.

Let us now assume that in our models there are $N\gg 1$ (s)axions and an even
larger number of elementary EFT string charge vectors $\\{{\bf
e}_{A}\\}_{A\in{\cal J}}$ (since these generate the generically non-simplicial
$\Delta$). We can then assume (3.17) to be satisfied for all ${\bf e}_{A}$,
since for $N\gg 1$ any non-generic violation of this assumption would affect
our conclusions by negligible corrections. Hence we will assume that

$\langle\tilde{\bf C},{\bf e}_{A}\rangle\in\mathbb{Z}_{\geq 1}\quad\,,$ (3.18)

for any elementary ${\bf e}_{A}\in{\cal C}_{\rm S}^{\text{\tiny EFT}}$. Take
now any regular simplicial sub-cone (3.11) and an element ${\bm{s}}_{\sigma}$
of the corresponding $\alpha$-stretched cone:
${\bm{s}}_{\sigma}\in\tilde{\sigma}_{\alpha}$. We can write
${\bm{s}}_{\sigma}=\frac{1}{\alpha}\sum_{A\in{\cal J}_{\sigma}}{\bf
e}_{A}+{\bm{v}}_{\sigma}$, where the first contribution represents the tip of
${\bm{s}}_{\sigma}$ and ${\bm{v}}_{\sigma}$ is an element of $\sigma$. As a
consequence, the lower bounds (3.17) and (3.18) imply that

$\langle\tilde{\bf C},{\bm{s}}_{\sigma}\rangle=\frac{1}{\alpha}\sum_{A\in{\cal
J}_{\sigma}}\langle\tilde{\bf C},{\bf e}_{A}\rangle+\langle\tilde{\bf
C},{\bm{v}}_{\sigma}\rangle\geq\frac{1}{\alpha}\sum_{A\in{\cal
J}_{\sigma}}\langle\tilde{\bf C},{\bf e}_{A}\rangle\geq\frac{N}{\alpha}\,.$
(3.19)

Consider next a more general point ${\bm{s}}$ of the saxionic convex hull
(3.13). By definition, we can write it as
${\bm{s}}=\sum\lambda^{\sigma}{\bm{s}}_{\sigma}$, with
$\sum_{\sigma}\lambda^{\sigma}=1$ and $\lambda^{\sigma}\geq 0$. According to
(3.19) we thus have

$\langle\tilde{\bf
C},{\bm{s}}\rangle=\sum_{\sigma}\lambda^{\sigma}\langle\tilde{\bf
C},{\bm{s}}_{\sigma}\rangle\geq\frac{N}{\alpha}\sum_{\sigma}\lambda^{\sigma}=\frac{N}{\alpha}\,.$
(3.20)

Hence we conclude that for ${\bm{s}}\in\hat{\Delta}_{\alpha}$ the coefficient
(2.9) of the GB term (2.8) satisfies the lower bound

$\gamma({s})|_{\hat{\Delta}_{\alpha}}\geq\frac{N\pi}{6\alpha}\,.$ (3.21)

This bound may receive $1/N$ corrections due to non-generic violations of
(3.18), but in the large $N$ regime these can be safely neglected. We also
observe that, as stressed above, in many models we could alternatively adopt
$\langle\tilde{\bf C},{\bf e}_{A}\rangle\in 3\mathbb{Z}_{\geq 1}$ instead of
(3.18), obtaining a slightly stronger lower bound
$\gamma({s})|_{\hat{\Delta}_{\alpha}}\geq\frac{N\pi}{2\alpha}$.

At the end of Section 2.2 we discussed the possible corrections to the GB
coefficient (2.9). Within the perturbative regime $\alpha\ll 2\pi$ the
radiative effects not included in (2.9) are parametrically smaller than the
one in (3.21), as anticipated there. We can thus conclude with confidence that
in this regime the bound (3.21) is not significantly spoiled by such
corrections. Note that while this result was derived by restricting ${\bm{s}}$
to the $\alpha$-stretched saxionic convex hull, we expect it to qualitatively
hold (up to a possible overall constant) also if we consider the stretched
saxionic cone. We will provide some evidence of this claim in Section 4, where
we will show that (3.21) is actually very conservative in a set of concrete
string theory models.

###  3.4 UV mass scales

Any EFT is associated with a cutoff energy $\Lambda$ above which it is no
longer valid. As in the Wilson’s view of the renormalization group, $\Lambda$
is not in general a physical scale, but rather a conventional definition of
the regime of validity of the description. As anticipated in Section 2.2, the
EFT cutoff must satisfy $\Lambda<M_{\text{\tiny UV}}$, where $M_{\text{\tiny
UV}}$ is some UV mass scale suppressing the irrelevant operators appearing in
the effective action. In the present context, the UV physical mass scales
beyond the EFT do depend on the moduli fields of (2.3), and specifically on
$s^{i}$. As a result, $M_{\text{\tiny UV}}$ not only represents the highest
possible energy above which the momentum expansion ceases to be effective, but
also an implicit constraint on the domain of the saxions through the condition
$M_{\text{\tiny UV}}(s)>\Lambda$.

In this section we would like to identify a proxy for $M_{\text{\tiny UV}}$ in
the specific scenarios introduced in Section 2. We will see that these models
are characterized by two relevant scales: the _tower scale_ [32] and the
_species scale_ [16, 17, 18, 19]. The former identifies the energy threshold
beyond which a four-dimensional EFT should be replaced by a more fundamental
description whereas the latter scale sets the absolute maximum UV cutoff at
which _any_ EFT inevitably breaks down. In the process of discussing these two
relevant scales we will also propose a novel upper bound on the species scale
defined solely in terms of EFT data. In the reminder of the paper we will then
conservatively take the species scale as our proxy for $M_{\text{\tiny UV}}$.

In the perturbative framework outlined in the Subsection 3.2, the weak
coupling limit $\alpha\rightarrow 0$ pushes the entire domain
$\hat{\Delta}_{\alpha}$ to infinite field space distance. According to the
Distance Conjecture [32] (see also [23] for a recent review) this signals the
appearance of towers of massive single-particle states. More precisely,
denoting by $M_{\rm t}$ the tower scale, i.e. the mass of the lightest
particle of such towers, the Distance Conjecture implies that $M_{\rm
t}/M_{\text{\tiny P}}$ decreases exponentially with the field space distance –
see also [47]. Furthermore, the possible nature of such towers, and with it of
the UV completion of our EFT, is significantly restricted by the Emergent
String Conjecture (ESC) [48], which states that (in appropriate duality
frames) those UV towers can be formed by either Kaluza-Klein modes or by
excitation modes of a weakly coupled critical superstring. The former modes
become massless in limits involving decompactifications to higher-dimensional
EFTs, whereas the latter in limits in which the critical string coupling goes
to zero. In the following we will assume the validity of the ESC conjecture
and take

$M_{\rm t}\equiv{\rm min}\left\\{M_{\text{\tiny KK}},M_{\rm s}\right\\},$
(3.22)

where $M_{\text{\tiny KK}}$ and $M_{\rm s}$ are the KK and superstring scales
(as measured by the four-dimensional observer).

In general it is not known how to precisely determine $M_{\rm t}$ without
knowing the details of the UV completion, that is, of the higher dimensional
EFT and/or string theory. Yet, in the present context, useful information is
encoded in the tension (2.21) of EFT strings, as indicated by the Integral
Weight Conjecture (IWC) [14, 15]. Namely, each EFT string charge ${\bf
e}\in{\cal C}_{\rm S}^{\text{\tiny EFT}}$ identifies an infinite distance
saxionic flow ${\bm{s}}={\bm{s}}_{0}+{\bf e}\sigma$, with
$\sigma\rightarrow\infty$, and is associated with an integral scaling weight
$w_{\bf e}\in\\{1,2,3\\}$. Along this EFT string flow ${\cal T}_{\bf e}\sim
M^{2}_{\text{\tiny P}}/\sigma$ and the scaling weight relates this behavior
with the asymptotic scaling of $M_{\rm t}$ as follows:

$M^{2}_{\rm t}\sim\left(\frac{{\cal T}_{\bf e}}{M^{2}_{\text{\tiny
P}}}\right)^{w_{\bf e}}M^{2}_{\text{\tiny P}}\,.$ (3.23)

Note that this relation reveals the scaling behavior in $\sigma$, but does not
allow one to precisely determine $M^{2}_{\rm t}$. Nevertheless, combined with
the ESC, (3.23) contains important pieces of information on the UV nature of
the EFT strings and of the corresponding infinite distance limits. First of
all, any $w_{\bf e}=1$ EFT string in the corresponding EFT string limit must
uplift to a weakly coupled critical superstring. This in turn implies that any
$w_{\bf e}=1$ EFT string flow111111For instance, assuming a dual $E_{8}\times
E_{8}$ string model of the type discussed below in Section 4.2, this limit
corresponds to $s^{0}\sim\sigma\rightarrow\infty$, with $s^{0}$ as in (4.38)
and fixed $s^{a}$. is dual to a ten-dimensional weak string coupling limit
$g_{\rm s}\rightarrow 0$, along which the tower scale $M_{\rm t}$ can be
identified with the mass of the first excited string mode. Thus, for any
elementary EFT string charge ${\bf e}$ of scaling weight $w_{\bf e}=1$, (3.23)
can actually be promoted to the identity $M_{\rm t}^{2}|_{w_{\bf e}=1}=M_{\rm
s}^{2}=2\pi{\cal T}_{\bf e}$. On the other hand, EFT strings with $w_{\bf
e}\geq 2$ cannot be identified with critical strings, and the corresponding
infinite distance limits must correspond to decompactification limits along
which $M_{\rm t}$ corresponds to a Kaluza-Klein (KK) mass scale.

There is another important mass scale that controls the transition away from
the EFT: the species scale $M_{\text{sp}}$ [16, 17, 18, 19] – see also the
recent review [23]. Various definitions of species scale have been given. In
this paper, by $M_{\text{sp}}$ we mean the highest possible scale at which our
gravitational setup admits a reliable (possibly higher dimensional) EFT
description. The ESC allows us to make this concept more concrete:
$M_{\text{sp}}$ is either the lightest string excitation mass or the scale at
which the KK excitations become strongly-coupled, depending on which of the
two is lower. If a weakly coupled string description exists, then the KK
excitations are never strongly coupled and the scale $M_{\text{sp}}$ can be
identified with the critical superstring mass, properly converted to the four-
dimensional frame: $M_{\text{s}}=\sqrt{2\pi{\cal T}_{\rm F1}}$. On the other
hand, if no perturbative stringy description exists then $M_{\text{sp}}$
corresponds to a quantum gravity scale $M_{\text{\tiny QG}}$, at which the
gravitational interactions become strong, which roughly coincides with the
higher-dimensional Planck mass (properly converted to the four-dimensional
frame). In the latter case, a quantitative criterion for determining the
quantum gravity scale, based on “Naive Dimensional Analysis” (NDA), is
discussed in Appendix A and tested on some concrete string theory models in
Section 4.

We will refer to $M_{\text{sp}}={\rm min}\left\\{M_{\text{s}},M_{\text{\tiny
QG}}\right\\}$ as “species scale” to conform to the terminology adopted in
most of the literature. Such terminology originates from more general models
with a large number $N_{\rm sp}\gg 1$ of species [16, 17], in which
perturbative as well as non-perturbative arguments indicate that the
gravitational interactions become strong at a scale of order

$\frac{2\pi M_{\text{\tiny P}}}{\sqrt{N_{\rm sp}}}\,.$ (3.24)

In our setup, identifying $N_{\rm sp}$ with the number of (massless and
massive) KK modes of mass smaller than $M_{\text{\tiny QG}}$, one can show
that $M_{\text{\tiny QG}}$ is indeed consistent with (3.24), see for instance
[49] and Appendix A for a systematic discussion. Yet, $M_{\text{\tiny QG}}$
ceases to be a reliable measure of the species scale when $M_{\rm
s}<M_{\text{\tiny QG}}$. In such circumstances black-hole arguments [18, 19]
(see also [23]) lead to the identification $M_{\text{sp}}=M_{\rm s}$ adopted
above.121212Applying (3.24) to string excitation modes, one gets $M_{\rm s}$
up to logarithmic corrections [50, 49] – see also related discussions in [51,
52]. One may adopt that “stringy species scale” as a definition of species
scale, but that would not significantly affect our conclusions.

As the tower scale, also the species scale generically depends on the details
of the EFT UV completion. However, it was recently proposed [53] that
information on the species scale is captured by the coefficient of certain
higher-derivative gravitational interactions – see also [54, 55, 56, 57, 58,
59, 60] for related discussions. By applying the proposal of [53] to our
context one gets a relation of the form $\gamma\sim{\cal O}(1){M_{\text{\tiny
P}}^{2}}/{M_{\text{sp}}^{2}}$ between the species scale and GB coefficient
appearing in (2.8). This relation can be understood as follows.

The four-dimensional gravitational theories we consider in this paper
represent the low-energy description of some UV complete theory. The lowest
threshold that characterizes the latter theory has been denoted by
$M_{\text{sp}}$. It is therefore natural to imagine deriving our EFT by
matching it to its UV completion precisely at that scale. On general grounds,
the resulting QFT will contain operators of arbitrary dimensions with
coefficients set by powers of $M_{\text{sp}}$ times dimensionless coefficients
$c$. Perturbativity at the matching scale demands that $|c|\lesssim 1$. One
would thus be naively tempted to claim that $M_{\text{\tiny UV}}$ should be
identified with $M_{\text{sp}}$. However, a further step is usually needed
when $M_{\text{\tiny KK}}<M_{\text{sp}}$. In order to arrive at our four-
dimensional EFT one should first integrate out the KK excitations within the
intermediate higher-dimensional description. In carrying out this last step
some of the higher-dimensional operators of our four-dimensional EFT will
inevitably receive corrections proportional to powers of
$M_{\text{sp}}/M_{\text{\tiny KK}}$. In particular, any operator of the form
“current squared” can in principle receive tree-level corrections from the
integration of KK resonances of the appropriate spin. This is for instance the
case for $R^{2}$ and $R_{ab}R^{ab}$, unless protected by extended
supersymmetries, which may a priori be mediated by scalar and spin-2 KK modes.
On the other hand, there is no KK excitation with the quantum numbers
appropriate to couple linearly to $R_{abcd}$ in any known low-energy
description of string theory, and so the coefficient of $R_{abcd}R^{abcd}$ is
not expected to be renormalized at tree-level. The same tree-level non-
renormalization property thus extends to the GB operator (2.6). Moreover, by
power-counting the coefficient of the latter operator can only receive
logarithmic corrections and cannot depend on inverse powers of $M_{\text{\tiny
KK}}$. This is particularly clear in the ${\cal N}=1$ context we are
considering, see Section 2.2. Taking into account our normalization
conventions, which are discussed in more detail in Appendix A, it follows that

${\gamma}={4\pi^{2}}\,c_{\text{GB}}\frac{M_{\text{\tiny
P}}^{2}}{M_{\text{sp}}^{2}},$ (3.25)

where $c_{\text{GB}}$ is some dimensionless coefficient that, up to
logarithmic radiative corrections, directly arises from some higher-
dimensional description. The consistency condition $|c_{\text{GB}}|\lesssim 1$
implies an upper bound on the GB coefficient, which combined with (3.21) gives

$\frac{\pi N}{6\alpha}\leq{\gamma}\lesssim{4\pi^{2}}\,\frac{M_{\text{\tiny
P}}^{2}}{M_{\text{sp}}^{2}}.$ (3.26)

A particular implication of this relation is an upper bound on the species
scale, as pointed out in [57]:

$M_{\text{sp}}\lesssim M_{\gamma}\equiv\frac{2\pi M_{\text{\tiny
P}}}{\sqrt{\gamma(s)}}\,.$ (3.27)

We emphasize the moduli-dependence of $M_{\gamma}$, which expectedly vanishes
as $\alpha\to 0$. Note that (3.27) is defined in terms of a Wilson
coefficient, and therefore in general the mass scale $M_{\gamma}$ has no
direct physical interpretation, which may for instance provide some sharper
criterion to fix its normalization. Furthermore, in non-supersymmetric as well
as our ${\cal N}=1$ context, the EFT coefficient $\gamma(s)$ receives scheme-
dependent renormalization corrections (see Section 2.2). It could therefore be
useful to identify an alternative and more physical proxy for the species
scale. This is what we will do next.

We propose that an alternative upper bound on the species scale can be derived
from the physics of EFT strings. Take the set of EFT string charges (3.7). As
emphasized above, if an elementary EFT charge ${\bf e}$ has scaling weight
$w_{{\bf e}}=1$, then there exists an asymptotic regime, defined by the
associated EFT string limit, in which this EFT string uplifts to a critical
superstring at weak string coupling. Hence, according to the above
definitions, in this regime we can make the identifications
$M^{2}_{\text{sp}}=M_{\rm t}^{2}=M^{2}_{\rm s}=2\pi{\cal T}_{\bf e}$. In all
the other cases, in which either $w_{{\bf e}}=1$ but the theory is not in the
corresponding asymptotic regime, or $w_{{\bf e}}\geq 2$, the EFT string does
not uplift to a weakly coupled critical superstring. Hence it should not be
quantizable, and so its would-be excitation masses should be above the species
scale, that is $2\pi{\cal T}_{\bf e}\geq M^{2}_{\text{sp}}$. This is what was
emphasized in [61], which compared the species scale with the EFT string
tensions along the corresponding EFT string limits in F-theory models.

The above considerations motivate us to propose the following general upper
bound

$M^{2}_{\text{sp}}\leq M^{2}_{{\cal T}}\,,$ (3.28)

where we have introduced the dominant EFT string scale

$M^{2}_{{\cal T}}(\ell)\equiv\min\big{\\{}2\pi{\cal T}_{\bf e}(\ell)\,|\,{\bf
e}\in{\cal C}^{\text{\tiny EFT}}_{\rm S}\big{\\}}\,,$ (3.29)

with ${\cal T}_{\bf e}(\ell)=\langle{\bm{\ell}},{\bf e}\rangle
M^{2}_{\text{\tiny P}}$, as in (2.21). The strict equality holds only when
$w_{\bf e}=1$ and the saxions are in the asymptotic regime identified by the
corresponding EFT string flow. We stress that the simple formula (2.21) is
fixed by ${\cal N}=1$ supersymmetry and is thus protected against perturbative
corrections. Hence, as a function of the dual saxions $\ell_{i}$, the dominant
EFT string scale (3.29) gives an explicit and robust upper bound on the
species scale, valid for instance also if classical or quantum perturbative
corrections to (2.5) and (2.18) cannot be neglected anymore (while non-
perturbative corrections continue to be negligible). In other words, it enjoys
a sort of non-renormalization theorem. By expressing (3.29) in terms of the
saxions $s^{i}$ by means of (2.11), we would get a formula that formally
depends on the Kähler potential $K$, which itself is sensitive to perturbative
corrections. Despite that, such corrections get all “resummed” in the
$\ell_{i}$, as manifest in the dual saxionic formulation.

Note that the dominant EFT string scale (3.29) depends on the dual saxions
$\ell_{i}$ in such a way that, under an overall constant rescaling of the
saxions, it behaves precisely as $M^{2}_{\gamma}$ in (3.27). However, (3.29)
is fully determined by data available within the two-derivative EFT, namely
the EFT string tensions ${\cal T}_{\bf e}$. Once the Kähler potential and the
saxionic cone are given, or analogously the dual saxions $\ell_{i}$ and the
set of EFT string charges (3.7), (3.29) is determined at each point of the
perturbative region. It is sufficient to restrict to the elementary generators
of ${\cal C}^{\text{\tiny EFT}}_{\rm S}$, compute the corresponding tensions
and identify the lowest one. Of course, the charge corresponding to the lowest
tension generically changes as we move in the saxionic domain. Hence
$M^{2}_{{\cal T}}$ is a continuous but possibly non-smooth function of the
dual saxions.

In Section 4 we will verify the bound (3.28) in explicit string theory models
and compare it to (3.27). Other checks are provided in Appendix C. In fact,
$M_{{\cal T}}$ turns out to provide a good estimate of the species scale, not
only when $M_{\rm sp}=M_{\rm s}$ (in which case by construction $M_{\rm
sp}=M_{{\cal T}}$), but also more generically, at least for “not-too-large”
saxions $s^{i}$. Instead, a large hierarchy $M_{\text{sp}}\ll M_{{\cal T}}$
can occur in extreme limits in asymptotic field space regions where the
species scale is set by $M_{\text{\tiny QG}}$, a typical example being
realized in the strong string coupling limit of M-theory.

We conclude this section by stressing that one of the basic assumptions that
underlie our analysis is that supersymmetry is exact and in particular that no
perturbative stabilizing mechanism for axions and saxions is present. Clearly,
if supersymmetry gets broken at low energies a potential for the saxions is
generically induced. In that case some of the results obtained using our
formulation would be qualitatively wrong. For example one could not reliably
identify the vacuum configuration of a realistic string theory model using Eq.
(2.3). Nevertheless, the considerations presented in our paper are short-
distance in nature and, therefore, largely insensitive to IR deformations like
supersymmetry breaking. To guarantee this we will restrict our attention to
$\Lambda$’s satisfying

$\Lambda>M_{{\text{\tiny IR}}}$ (3.30)

where $M_{{\text{\tiny IR}}}$ is some physical IR mass scale below which the
long-distance modifications of (2.3) can no longer be ignored.

##  4 String theory models

In order to make the general discussion of Sections 2 and 3 more concrete, we
now describe two broad classes of string theory models, namely the F-theory
and heterotic models in the large volume regime. These have the advantage that
can be described quite easily in our general framework, and will allow us to
provide a few explicit examples thereof to better illustrate and check our
main points. As in [14, 37], our general claims also apply to other string
theory models or perturbative regimes – e.g. type I, type IIA and M-theory –
which are however either very similar/dual to the heterotic and F-theory
cases, or admit a less explicit EFT description. Hence, for concreteness and
clarity, in Section 4.1 we focus on the F-theory and in Section 4.2 on
heterotic models. We will encounter M-theory models on $G_{2}$ manifolds in
Section 6.5.3.

For clarity, we collect here the conventions we adopt on the relevant scales
in string/M-theory. The ten-dimensional Ricci scalar for type I, IIA, IIB
appears in the Einstein and string frame actions as

$\displaystyle\frac{2\pi}{l_{(10)}^{8}}\int
R_{(10)}=\frac{2\pi}{l_{(10)}^{8}}\int e^{-2\phi}R^{(\rm s)}_{(10)}\,,$ (4.1)

where $l_{(10)}=2\pi\sqrt{\alpha^{\prime}}$ represents the ten-dimensional
Planck length in the Einstein frame, and the string length in the string
frame. The string and Einstein frame metrics are related by ${\rm
d}s^{2}_{\rm(s)}=e^{\frac{\phi}{2}}{\rm d}s^{2}_{\text{\tiny(E)}}$. Concerning
M-theory, the eleven-dimensional Einstein-Hilbert term is

$\displaystyle\frac{2\pi}{l_{(11)}^{9}}\int R_{(11)},$ (4.2)

where $l_{(11)}$ is the Planck length in M-theory. Choosing

${\rm d}s_{11}^{2}=e^{-\frac{1}{6}\phi}{\rm
d}s_{10}^{2}+l_{(11)}^{2}e^{\frac{4}{3}\phi}{\rm d}y^{2}\,,$ (4.3)

and compactifying on the interval $y\in[0,1]$, (4.2) reduces to the Einstein
frame action in (4.1) with $l_{(10)}=l_{(11)}$.

In both cases, the four-dimensional metric is embedded in the higher-
dimensional one according to the ansatz

${\rm d}s^{2}_{d}=e^{2A}{\rm d}s^{2}_{4}+{\rm d}s^{2}_{X}.$ (4.4)

The Weyl rescaling factor $e^{2A}=M^{2}_{\text{\tiny P}}l_{(d)}^{2}/(4\pi
V_{X})$, where $V_{X}$ denotes the volume of the $(d-4)$-dimensional
compactified space in $l_{(d)}$ units, is necessary to identify ${\rm
d}s^{2}_{4}$ with the four-dimensional Einstein frame metric. Explicit
expressions of this quantity for our models will be provided below. Note that
the appropriate dimension $d$ generically depends on the saxions.

According to Appendix A, the strong coupling scales for ten-dimensional string
theory and M-theory are, respectively,

$\widehat{M}_{(10)}=\frac{(2\pi)^{\frac{3}{4}}}{l_{(10)}},\quad\quad\widehat{M}_{(11)}=\frac{(2\pi)^{\frac{2}{3}}}{l_{(11)}}\,.$
(4.5)

The quantum gravity scale introduced in Section 3.4 is then given by
$M_{\text{\tiny{QG}}}=e^{A}\widehat{M}_{(d)}$.

###  4.1 F-theory/type IIB orientifold models

An important large class of examples is provided by the F-theory
compactifications – see e.g. [62, 63] for reviews. An F-theory model
corresponds to a type IIB compactification on a Kähler space $X$ in presence
of 7-branes. The space $X$ can be regarded as the base of an elliptically
fibered Calabi-Yau four-fold, whose fiber’s complex structure can be
identified with the type IIB axio-dilaton. In particular, this requires the
base $X$ to have an effective anti-canonical divisor
$\overline{K}_{X}$.131313We adopt the quite common usage of denoting
holomorphic line bundles and corresponding divisors by the same symbol. In the
following we will for simplicity assume that the elliptically fibered Calabi-
Yau four-fold has vanishing third Betti number, so to avoid technical
complications associated with moduli of the M-theory gauge three-form.

These models admit a natural perturbative regime corresponding to the large
volume limit. Let us pick a basis of divisors $D^{a}\in H_{4}(X,\mathbb{Z})$
of $X$ and a dual basis of two-cycles $\Sigma_{a}\in H_{2}(X,\mathbb{Z})$,
such that $D^{a}\cdot\Sigma_{b}=\delta^{a}_{b}$, $a=1,\ldots,b_{2}(X)$. The
Kähler moduli $v_{a}$ are obtained by expanding the (Einstein frame) Kähler
form $J$ of $X$ in the Poincaré dual basis $[D^{a}]\in H^{2}(X,\mathbb{Z})$.
Keeping Poincaré duality implicit, we can write

$J=v_{a}D^{a}\quad~{}~{}~{}~{}\Leftrightarrow\quad~{}~{}~{}~{}v_{a}=\int_{\Sigma_{a}}J=\Sigma_{a}\cdot
J\,.$ (4.6)

The corresponding saxions $s^{a}$ are then defined as follows:

$s^{a}=\frac{1}{2}\kappa^{abc}v_{b}v_{c}\,,$ (4.7)

where we have introduced the triple intersection numbers
$\kappa^{abc}=D^{a}\cdot D^{b}\cdot D^{c}$. Hence in this case $N=b_{2}(X)$.

As discussed in [14], one can identify the saxionic cone with the cone ${\rm
Mov}_{1}(X)$ generated by movable curves (see e.g. [64]). We can then write

${\bm{s}}=s^{a}\Sigma_{a}\in\Delta_{\text{\tiny K}}\simeq{\rm Mov}_{1}(X)\,.$
(4.8)

Note that the string charge vectors ${\bf e}$ can be identified with effective
curves $\Sigma_{\bf e}=e^{a}\Sigma_{a}$ and, in particular, the EFT string
charges correspond to movable curves

${\cal C}^{\text{\tiny EFT}}_{\rm S}\simeq{\rm Mov}_{1}(X)_{\mathbb{Z}}\,.$
(4.9)

Physically, EFT strings are realized by D3-branes wrapping movable curves. The
corresponding BPS instantons are instead realized by Euclidean branes wrapping
effective divisors $D\in{\rm Eff}^{1}(X)_{\mathbb{Z}}$, so that we can make
the identification ${\cal C}_{\rm I}\simeq{\rm Eff}^{1}(X)_{\mathbb{Z}}$.

The constants $\tilde{C}_{a}$ defining $\gamma(s)$ as in (2.9) admit a nice
geometrical interpretation [37]:

$\tilde{C}_{a}=6\overline{K}_{X}\cdot\Sigma_{a}\,.$ (4.10)

In particular, the pairing appearing in (3.16) corresponds to the intersection
number

$\langle\tilde{\bf C},{\bf e}\rangle=6\overline{K}_{X}\cdot\Sigma_{\bf e}\,.$
(4.11)

Recalling that $\overline{K}_{X}$ is an effective divisor, the bound (3.16) is
always satisfied, since movable curves can be precisely characterized as those
curves that have non-negative intersection with all effective divisors [65].
In order to test the bound (3.21), which is expected to hold up to subleading
corrections in $1/N\ll 1$, let us focus on the large class of models with
toric $X$ – see for instance [66]. Then the anti-canonical divisor is given by

$\overline{K}_{X}=\sum_{I\in\text{toric div.}}{\cal D}_{I}\,,$ (4.12)

where the sum is over the set of prime toric divisors ${\cal D}_{I}$,
$I=1,\ldots,N+3$. All these divisors are effective, and in fact generate the
whole cone of effective divisors. So any movable curve $\Sigma_{\bf e}$ has
strictly positive intersection number with at least one toric divisor ${\cal
D}_{I}$, and then

$\overline{K}_{X}\cdot\Sigma_{\bf e}\geq 1\,.$ (4.13)

Combined with (4.12), this implies that

$\langle\tilde{\bf C},{\bf e}\rangle\geq
6\,,\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall{\bf e}\in{\cal
C}^{\text{\tiny EFT}}_{\rm S}\,.$ (4.14)

We then see that, in this large class of models, the condition (3.18) is
indeed satisfied, strengthened by a factor of 6, and then the bound (3.21) is
realized in the stronger form:

$\gamma(s)\geq\frac{N\pi}{\alpha}\,.$ (4.15)

The effective theory is more easily described in the dual saxionic formulation

$\ell_{a}=\frac{3v_{a}}{\kappa(J,J,J)}\,,$ (4.16)

where $\kappa(J,J,J)\equiv J\cdot J\cdot J=\kappa^{abc}v_{a}v_{b}v_{c}$. The
dual saxionic cone ${\cal P}_{\text{\tiny K}}$ can be identified with an
“extended” Kähler cone ${\cal K}(X)_{\rm ext}$ obtained by gluing different
spaces connected by flop transitions, in which curves collapse or blow-up:

${\cal P}_{\text{\tiny K}}={\cal K}_{\rm ext}(X)=\bigcup_{X^{\prime}\sim
X}\overline{{\cal K}(X^{\prime})}\,.$ (4.17)

Here $X^{\prime}\sim X$ means that $X^{\prime}$ can be obtained from $X$ by a
chain of flops (which may also be trivial, corresponding to $X^{\prime}=X$).
Hence ${\bm{\ell}}\in{\cal P}_{\text{\tiny K}}$ if there exists one chamber of
${\cal K}_{\rm ext}(X)$, associated with a compactification space
$X^{\prime}\sim X$, in which ${\bm{\ell}}=\ell_{a}\,D^{a}$ is a nef
$\mathbb{R}$-divisor, that is ${\bm{\ell}}\in\overline{{\cal
K}(X^{\prime})}$.141414 In the following we will often focus on spaces $X$
which are toric or orientifold quotients of Calabi-Yau three-folds. In these
cases ${\cal K}(X)_{\rm ext}$ can be identified with the space of the so-
called movable divisors. Hence we can write
${\bm{\ell}}=\ell_{a}\,D^{a}\in{\cal P}_{\text{\tiny K}}\simeq{\rm
Mov}^{1}(X)$. This identification can actually hold more generically – see
[14] for more details.

At large volume, the kinetic potential ${\cal F}(\ell)$ takes the form (2.18):

${\cal F}_{\text{\tiny
K}}(\ell)=\log\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})\,.$ (4.18)

Hence $\tilde{P}(\ell)=\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})$, which is
clearly homogeneous as in (2.20), with $n=3$.

If one can take Sen’s orientifold limit, the space $X$ can be regarded as the
$\mathbb{Z}_{2}$-orientifold quotient of a Calabi-Yau three-fold $\hat{X}$. A
new saxion $\hat{s}\equiv e^{-\phi}$ appears, detected by $D(-1)$-instantons,
where $\phi$ is the standard type IIB dilaton, so that we now have
$N=b_{2}(X)+1$. The corresponding dual saxion is

$\hat{\ell}=\frac{1}{2}e^{\phi}\,.$ (4.19)

In the perturbative regime described by the dual saxions
$\ell_{i}=(\hat{\ell},\ell_{a})$, the leading contribution to the Kähler
potential is given by

${\cal F}=\log\hat{\ell}+{\cal F}_{\text{\tiny
K}}(\ell)=\log\hat{\ell}+\log\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})$
(4.20)

and can then be written as in (2.18) with
$\tilde{P}=\hat{\ell}\,\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})$, which has
homogeneity $n=4$.151515In fact, the saxionic and dual saxionic cones are
expected to receive corrections coming from higher derivative terms. This type
of effect has been discussed in some detail for heterotic models in [37] and
we will encounter it in subsection 4.2 – see e.g. (4.38). In particular, if we
choose $\ell_{0}$ so that $\ell_{0}M^{2}_{\rm P}$ gives the tension of the
lightest D7-string, we generically have $\ell_{0}=\hat{\ell}+c^{a}\ell_{a}$,
where $c^{a}\in\mathbb{Q}$ accounts for possible world-volume curvature/bundle
corrections. This means that in (4.20) we should set
$\hat{\ell}=\ell_{0}-c^{a}\ell_{a}$, which induces also a shift
$s^{a}\rightarrow\hat{s}^{a}=s^{a}+c^{a}s^{0}$ in the Kähler potential. These
subtleties will be studied in more detail elsewhere, but since they are not
crucial to our purposes for simplicity we will just ignore them, tacitly
keeping them in mind.

The conversion factor from ten- to four-dimensional scales, appearing in
(4.4), is given by

$e^{2A}=\frac{M^{2}_{\text{\tiny P}}l^{2}_{(10)}}{4\pi
V(X)}=\frac{M^{2}_{\text{\tiny
P}}l^{2}_{(10)}}{2\pi}\sqrt{\frac{\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}{3}}\,,$
(4.21)

where $V(X)=\kappa(J,J,J)/6$ is the compactification volume in
$l_{(10)}$-units. By applying this conversion factor to $\widehat{M}_{(10)}$
in (4.5) we get

$M^{2}_{\text{\tiny
QG}}=e^{2A}\widehat{M}_{(10)}^{2}=\sqrt{\frac{2\pi\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}{3}}\,M^{2}_{\text{\tiny
P}}\,.$ (4.22)

It is interesting to explicitly check how the powers of $2\pi$’s precisely
combine so that $M^{2}_{\text{\tiny QG}}$ is the square of $2\pi
M_{\text{\tiny P}}$ — the scale a naive low-energy observer would identify as
the quantum gravity scale — times a non-trivial suppression
$\sim(\ell/2\pi)^{3/2}$ controlled by the “loop parameter” (3.2).

####  4.1.1 Model 1: ${\mathbb{P}}^{3}$

For illustrative purposes, it is useful to describe a couple of simple
explicit models (though with a small number of (s)axions) and their relevant
energy scales. The first and easiest example is obtained by choosing
$X=\mathbb{P}^{3}$.

In this case, the set of effective divisors
$\operatorname{Eff}^{1}(X)_{\mathbb{Z}}$ is spanned by a single element, the
hyperplane divisor $D\equiv H$, which has triple self-intersection
$\kappa(D,D,D)=1$. Hence, in the large volume perturbative regime,
${\bm{\ell}}=\ell D$, the dual saxionic cone is just given by $\ell\geq 0$
(including the degenerate boundaries) and (4.18) reduces to

${\cal F}_{\text{\tiny K}}=3\log\ell\,.$ (4.23)

The corresponding saxionic cone is spanned by the curve $\Sigma\equiv D\cdot
D$. The only saxion $s$ of this model encodes the volume of the hyperplane
divisor, has Kähler potential $K=-3\operatorname{log}s$ and is related to the
dual saxion by $\ell=\frac{3}{2s}$, see (2.11). The saxionic cone is simply
given by $s\geq 0$. Moreover the $\alpha$-saxionic convex hull is just
$\hat{\Delta}_{\alpha}=\\{s\geq 1/\alpha\\}$, and then $\hat{\cal
P}_{\alpha}=\\{\ell\leq\frac{3}{2}\alpha\\}$.

The hyperplane divisor $D$ generates the set of BPS instanton charges ${\cal
C}_{\rm I}=\\{{\bf q}=qD|q\in\mathbb{Z}_{\geq 0}\\}$, while the curve $\Sigma$
defines the elementary EFT string charge, which generates the set of EFT
string charges ${\cal C}^{\text{\tiny EFT}}_{\rm S}=\\{{\bf
e}=e\Sigma|e\in\mathbb{Z}_{\geq 0}\\}$. All these EFT string charges have
scaling weight $w=2$ [14], and the tension of the elementary string is given
by ${\cal T}=M^{2}_{\text{\tiny P}}\,\ell$. The anti-canonical divisor in this
setting is just $\overline{K}_{X}=4H$ and, by using (2.9) and (4.10), it
yields

$\gamma(s)=4\pi s\,.$ (4.24)

This is manifestly positive in the saxionic cone and we have also
$\gamma(s)|_{\hat{\Delta}_{\alpha}}\geq\frac{4\pi}{\alpha}$, stricter than
(3.21) with $N=1$ by a factor of 24.

Let us now discuss the species scale. The upper bounds given by (3.27) and
(3.29) become

$M^{2}_{\gamma}=\frac{\pi M^{2}_{\text{\tiny P}}}{s}\,,\quad\quad M^{2}_{{\cal
T}}=2\pi M^{2}_{\text{\tiny P}}\ell=\frac{3\pi M^{2}_{\text{\tiny P}}}{s}\,.$
(4.25)

We see that $M_{\gamma}\sim M_{{\cal T}}$ in the entire perturbative domain.
Because the dominant EFT string scale $M_{{\cal T}}$ is associated to a $w=2$
string and there are no $w=1$ EFT strings (and, correspondingly, no weak
string coupling), we expect $M_{\rm sp}=M_{\text{\tiny QG}}<M_{{\cal T}}$,
realizing the bound (3.28) in its strict form. We can verify this by using the
quantum gravity scale (4.22), which reads

$M^{2}_{\text{\tiny QG}}=\sqrt{\frac{2\pi\ell^{3}}{3}}\,M^{2}_{\text{\tiny
P}}=\frac{3}{2}\sqrt{\frac{\pi}{s^{3}}}\,M^{2}_{\text{\tiny P}}\,.$ (4.26)

It follows that $M_{{\cal T}}/M_{\rm sp}=(4\pi
s)^{\frac{1}{4}}\geq(4\pi/\alpha)^{\frac{1}{4}}$, and the strict form of the
bound (3.28) is always satisfied in the perturbative regime $\alpha/2\pi\ll
1$. Nevertheless, $M_{{\cal T}}$ provides a good proxy for $M_{\rm sp}$, say
up to an ${\cal O}(10)$ factor, for $s\lesssim 10^{3}$.

####  4.1.2 Model 2: $\mathbb{P}^{1}$ fibration over $\mathbb{P}^{2}$

In our context this model has already been discussed in [14], which we can
then follow. The internal space $X$ is a $\mathbb{P}^{1}$ fibration over
$\mathbb{P}^{2}$, and the fibration is specified by the integer $p\geq 0$.

The cone $\operatorname{Eff}^{1}(X)_{\mathbb{Z}}$ of effective divisors, which
can be identified with the cone of BPS instanton charges ${\cal C}_{\rm I}$,
is simplicial and is generated by two effective divisors $E^{1},E^{2}$:
$E^{1}$ is the divisor obtained by restricting the $\mathbb{P}^{1}$ fibration
over $\mathbb{P}^{1}\subset\mathbb{P}^{2}$, while $E^{2}$ corresponds to a
global section of the $\mathbb{P}^{1}$ fibration. One can then identify a
basis of nef divisors $D^{1}=E^{1}$ and $D^{2}=E^{2}+pE^{1}$, which generate
the Kähler cone: $J=v_{1}D^{1}+v_{2}D^{2}$, with $v_{1,2}>0$. The triple
intersection numbers are given by the coefficients of the formal object ${\cal
I}(X)=(D^{1})^{2}D^{2}+pD^{1}(D^{2})^{2}+p^{2}(D^{2})^{3}$. Hence, by using
the expansion ${\bm{\ell}}=\ell_{a}D^{a}$ the kinetic potential (4.18) becomes

${\cal F}_{\text{\tiny
K}}=\log\kappa\left({\bm{\ell}},{\bm{\ell}},{\bm{\ell}}\right)=\log\left(3\ell_{1}^{2}\ell_{2}+3p\ell_{1}\ell_{2}^{2}+p^{2}\ell_{2}^{3}\right)$
(4.27)

In this model the dual saxionic cone coincides with the closure of the Kähler
cone: ${\cal P}_{\text{\tiny
K}}=\\{{\bm{\ell}}=\ell_{1}D^{1}+\ell_{2}D^{2}|\ell_{1}\geq 0,\ell_{2}\geq
0\\}$. From (2.15) one can obtain the corresponding saxions:

$s^{1}=\frac{6\ell_{1}+3p\ell_{2}}{6\ell_{1}^{2}+6p\ell_{1}\ell_{2}+2p^{2}\ell_{2}^{2}}\quad,\quad
s^{2}=\frac{3(\ell_{1}+p\ell_{2})^{2}}{6\ell_{1}^{2}\ell_{2}+6p\ell_{1}\ell_{2}^{2}+2p^{2}\ell_{2}^{3}}\,.$
(4.28)

The (Mori) cone of effective curves is generated by $\Sigma_{1}=E^{1}\cdot
E^{2}$ and $\Sigma_{2}=(E^{1})^{2}$, which are dual to the nef divisors
$D^{a}$: $D^{a}\cdot\Sigma_{b}=\delta^{a}_{b}$. The cone of movable curves is
instead generated by $\hat{\Sigma}_{1}=D^{1}\cdot
D^{2}=\Sigma_{1}+p\Sigma_{2}$ and $\hat{\Sigma}_{2}=(D^{1})^{2}=\Sigma_{2}$,
which are dual to the effective divisors $E^{a}$:
$E^{a}\cdot\hat{\Sigma}_{b}=\delta^{a}_{b}$. Hence
${\bm{s}}=s^{1}\Sigma_{1}+s^{2}\Sigma_{2}=s^{1}\hat{\Sigma}_{1}+(s^{2}-ps^{1})\hat{\Sigma}_{2}$,
and the saxionic cone is $\Delta=\\{{\bm{s}}=s^{a}\Sigma_{a}|s^{1}\geq 0\,,\
s^{2}\geq ps^{1}\\}$. One can also invert the relation between saxions and
dual saxions:

$\ell_{1}=\frac{3p\sqrt{s^{2}-ps^{1}}}{2\left[(s^{2})^{\frac{3}{2}}-(s^{2}-ps^{1})^{\frac{3}{2}}\right]}\quad,\quad\ell_{2}=\frac{3\left(\sqrt{s^{2}}-\sqrt{s^{2}-ps^{1}}\right)}{2\left[(s^{2})^{\frac{3}{2}}-(s^{2}-ps^{1})^{\frac{3}{2}}\right]}$
(4.29)

As in our general discussion, we can characterize the boundaries of ${\cal
P}_{\text{\tiny K}}$ in terms of tensionless strings. The set of EFT string
charges

${\cal C}^{\text{\tiny EFT}}_{\rm S}=\\{{\bf
e}=e^{1}\Sigma_{1}+e^{2}\Sigma_{2}|(e^{1},e^{2})\in\mathbb{Z}^{2}\,,\
e^{1}\geq 0\,,\ e^{2}\geq pe^{1}\\}\,$ (4.30)

is generated by ${\bf e}_{(1)}=\hat{\Sigma}_{1}=\Sigma_{1}+p\Sigma_{2}$ and
${\bf e}_{(2)}=\hat{\Sigma}_{2}=\Sigma_{2}$, which have tensions ${\cal
T}_{(1)}=M^{2}_{\text{\tiny P}}(\ell_{1}+p\ell_{2})$ and ${\cal
T}_{(2)}=M^{2}_{\text{\tiny P}}\ell_{2}$. We notice that ${\cal T}_{(2)}$
vanishes at $\ell_{2}=0$, while ${\cal T}_{(1)}$ vanishes at the tip
$\ell_{1}=\ell_{2}=0$. These are infinite distance boundary components of
${\cal P}_{\text{\tiny K}}$. On the other hand, on the boundary component
$\ell_{1}=0$ no EFT string tension vanishes. This is instead characterized by
the vanishing of the tension $M^{2}_{\text{\tiny P}}\ell_{1}$ associated with
the non-EFT string charge $\Sigma_{1}$, which together with $\Sigma_{2}$
generates the set of BPS charges ${\cal C}_{\rm S}$. This implies that, even
if the saxionic convex hull is simply given by
$\hat{\Delta}_{\alpha}=\\{s^{1}\geq\frac{1}{\alpha},s^{2}-ps^{1}\geq\frac{1}{\alpha}\\}$,
the corresponding dual saxionic image $\hat{\cal P}_{\alpha}$ is more
complicated – see figure 1.

(a) Saxionic convex hull $\hat{\Delta}_{\alpha}$.

(b) Dual saxionic convex hull $\hat{\mathcal{P}}_{\alpha}$.

Figure 1: Saxionic convex hull $\hat{\Delta}_{\alpha}$ and dual saxionic
convex hull $\hat{\mathcal{P}}_{\alpha}$ for the F-theory model 2. The plot
has been drawn with the reference values $\alpha=1/10$ and $p=3$. The hatched
area in figure 1(a) is outside the saxionic cone $\Delta$, which comprise the
gray and white regions.

Remember that the GB coefficient is determined by the anti-canonical divisor.
In the present examples, the latter is given by

$\displaystyle\overline{K}_{X}$
$\displaystyle=(3-p)D^{1}+2D^{2}=(3+p)E^{1}+2E^{2}\,.$ (4.31)

From (2.9) and (4.10) we then get

$\gamma(s)=\pi\left[(3-p)s^{1}+2s^{2}\right]\,,$ (4.32)

which is positive since $s^{2}\geq ps^{1}$. Furthermore, we see that
$\gamma(s)|_{\hat{\Delta}_{\alpha}}\geq\frac{(5+p)\pi}{\alpha}$, which is
stronger than (3.21) with $N=2$. It is for instance sufficient to take
$\alpha\leq\frac{1}{10}$ and $p\geq 1$ to get
$\gamma(s)|_{\hat{\Delta}_{\alpha}}>188$.

Now let us turn our attention to the relevant energy scales at play. It is
easy to check that $K\simeq-\log\sigma$ asymptotically along the EFT string
flow associated with ${\bf e}_{(2)}$, while $K\simeq-3\log\sigma$ along the
EFT string flow associated with ${\bf e}_{(1)}$. This is consistent with the
fact that only ${\bf e}_{(2)}$ has scaling weight $w=1$, while ${\bf e}_{(1)}$
has $w=2$ [14]. Indeed, the string obtained by wrapping a D3 on
$\hat{\Sigma}_{2}$ is dual to a fundamental heterotic superstring via
F-theory/heterotic duality – see Appendix B for a more general discussion. We
can then distinguish different regimes set by the two elementary EFT string
tensions ${\cal T}_{{(1)}}=M^{2}_{\text{\tiny P}}(\ell_{1}+p\ell_{2})$ and
${\cal T}_{(2)}=M^{2}_{\text{\tiny P}}\ell_{2}$. Let us start assuming that
$p>0$, the particular case $p=0$ will be discussed at the end.

For $p>0$ we have ${\cal T}_{(2)}\leq{\cal T}_{{(1)}}$ and for any value of
the saxions the dominant EFT string scale (3.29) is given by

$M^{2}_{{\cal T}}=2\pi M^{2}_{\text{\tiny P}}\ell_{2}\,.$ (4.33)

Furthermore, the condition (3.10) requires that $\ell_{2}<2\pi/p$, and so
$M_{{\cal T}}<2\pi M_{\text{\tiny P}}/\sqrt{p}$. We can distinguish two
regimes, namely ${\cal T}_{(2)}\ll{\cal T}_{{(1)}}$ or ${\cal
T}_{(2)}\simeq{\cal T}_{{(1)}}$. If $\ell_{2}\ll\ell_{1}$ we are in the first
regime, where ${\cal T}_{(2)}$ corresponds to the tension of a dual weakly
coupled critical string. Here (3.28) is actually saturated. The second regime
is defined by ${\cal T}_{(2)}\simeq{\cal T}_{(1)}$. Since ${\cal
T}_{(1)}/{\cal T}_{(2)}=p+\ell_{1}/\ell_{2}>p$, this can be reached only if
$p\sim{\cal O}(1)$ and $\ell_{1}/\ell_{2}\lesssim 1$. In this second regime
there should not exist a controlled dual weakly coupled string theory
description. The species scale should then be identified with the quantum
gravity scale (4.22). By combining (4.22) and (4.27) we get

$M^{2}_{\text{\tiny
QG}}=\sqrt{2\pi\left(\ell_{1}^{2}\ell_{2}+p\ell_{1}\ell_{2}^{2}+\frac{1}{3}p^{2}\ell_{2}^{3}\right)}\,M^{2}_{\text{\tiny
P}}\lesssim\sqrt{2\pi\ell_{2}^{3}}\,M^{2}_{\text{\tiny
P}}=\sqrt{\frac{\ell_{2}}{2\pi}}\,M^{2}_{{\cal T}}\,,$ (4.34)

where in the second step we have used
$\ell_{1}\lesssim\ell_{2}<\frac{2\pi}{p}$ and $p\sim{\cal O}(1)$, and we have
neglected an ${\cal O}(1)$ overall constant. Consistently with the bound
(3.28) we find that $M_{\text{\tiny QG}}\lesssim M_{{\cal T}}$ whereas
$M_{\text{\tiny QG}}\ll M_{{\cal T}}$ for $\ell_{2}\ll 2\pi$, that is, far
away from the tip of the saxionic domain. Other regimes can be better studied
through the dual heterotic M-theory description, which will be discussed in
subsection 4.2.1 and will confirm that $M^{2}_{\text{sp}}$ is still bounded by
(4.33).

It is instructive to also discuss the bound (3.27) for this model. Recalling
(4.32) and (4.28) we get

$M^{2}_{\gamma}=\frac{4\pi M^{2}_{\text{\tiny
P}}}{(3-p)s^{1}+2s^{2}}=\frac{8\pi\ell_{2}(3\ell_{1}^{2}+3p\ell_{1}\ell_{2}+p^{2}\ell_{2}^{2})}{6\ell_{1}^{2}+(3+p)(6\ell_{1}\ell_{2}+3p\ell_{2}^{2})}M^{2}_{\text{\tiny
P}}.$ (4.35)

A comparison between the two mass scales (4.33) and (4.35) gives:

$\frac{M^{2}_{{\cal
T}}}{M^{2}_{\gamma}}=\frac{1}{2}\left[1+\frac{18\ell_{1}\ell_{2}+(9p+p^{2})\ell_{2}^{2}}{4(3\ell_{1}^{2}+3p\ell_{1}\ell_{2}+p^{2}\ell_{2}^{2})}\right]\,.$
(4.36)

Since we are assuming $p>0$, we have

$\frac{1}{2}\leq\ \frac{M^{2}_{{\cal
T}}}{M^{2}_{\gamma}}\leq\frac{5}{8}+\frac{9}{8p}\leq\frac{7}{4}\,,$ (4.37)

where the two extrema correspond to $\ell_{2}=0$ and $\ell_{1}=0$,
respectively. This shows that $M_{{\cal T}}$ and $M_{\gamma}$ are always of
the same order, and then the upper bound (3.27) is satisfied too.

The case $p=0$, which we ignored so far, is characterized by $M^{2}_{{\cal
T}}=2\pi M^{2}_{\text{\tiny P}}\min\\{\ell_{1},\ell_{2}\\}$ and
$M^{2}_{\gamma}=4\pi\frac{\ell_{1}\ell_{2}}{\ell_{1}+3\ell_{2}}M^{2}_{\text{\tiny
P}}$, so that $\frac{1}{2}\leq M^{2}_{{\cal
T}}/M^{2}_{\gamma}\leq\frac{3}{2}$. Again, the two upper bounds on the species
scale parametrically agree. Invoking (3.10) we require $\ell_{1},\ell_{2}\leq
2\pi$ and obtain the inequality $M^{2}_{{\cal T}}\leq M^{2}_{\text{\tiny QG}}$
for $\ell_{2}\leq\ell^{2}_{1}/2\pi$, consistently with the identification
$M^{2}_{\text{sp}}=M^{2}_{{\cal T}}=M^{2}_{\text{\tiny P}}\ell_{2}$, while
$M^{2}_{\text{sp}}=M^{2}_{\text{\tiny QG}}\leq M^{2}_{{\cal T}}$ for
$\ell_{2}\geq\ell^{2}_{1}/2\pi$.

A similar discussion can be carried out for the models where $X$ is
$\mathbb{P}^{1}$ fibration over an Hirzebruch surface $\mathbb{F}_{p}$, and is
presented in Appendix C.1.

###  4.2 Heterotic models

Our second class of models is given by $E_{8}\times E_{8}$ heterotic
compactifications on Calabi-Yau spaces, and their M-theory counterpart, at
large volume. (The $SO(32)$ case is completely analogous.) As discussed in
[37], the relevant saxionic cone is affected by ten- and eleven-dimensional
higher derivative terms. Here we summarize only the necessary information.

We consider a perturbative regime associated to $N=b_{2}(X)+1$ saxions
$s^{i}=(s^{0},s^{a})$, which include the Kähler moduli $s^{a}$ of the Calabi-
Yau compactification space $X$. These are obtained by expanding the string
frame Kähler form $J=s^{a}D_{a}$ in (a basis Poincaré dual to) a basis of
divisors $D_{a}$, $a=1,\ldots,b_{2}(X)$. The remaining saxion $s^{0}$ combines
the dilaton and the Kähler moduli:

$s^{0}=\frac{1}{6}e^{-2\phi}\kappa_{abc}s^{a}s^{b}s^{c}+\frac{1}{2}p_{a}s^{a}\,.$
(4.38)

where $\kappa_{abc}\equiv D_{a}\cdot D_{b}\cdot D_{c}$ and

$p_{a}\equiv-\int_{D_{a}}\left[\lambda(E_{2})-\frac{1}{2}c_{2}(X)\right]\in\mathbb{Z}\,,$
(4.39)

where $E_{1}$ and $E_{2}$ denote the two $E_{8}$ internal bundles, and

$\lambda(E)\equiv-\frac{1}{16\pi^{2}}\mathop{\rm tr}(F\wedge F)\,.$ (4.40)

The tadpole cancellation condition imposes the topological constraint
$\lambda(E_{1})+\lambda(E_{2})=c_{2}(X)$. Let us also introduce the integer

$n_{a}\equiv\frac{1}{2}\int_{D_{a}}c_{2}(X)\,.$ (4.41)

Notice that $n_{a}s^{a}=\frac{1}{2}\int_{X}c_{2}(X)\wedge J\geq 0$ [67] and
that, by supersymmetry, the internal $E_{8}$ bundles must satisfy
$\int_{X}\lambda(E_{1,2})\wedge J\geq 0$. Combining these positivity
conditions with (4.39) and the tadpole condition, one gets

$\left|p_{a}s^{a}\right|\leq n_{a}s^{a}\,.$ (4.42)

One could also include NS5/M5-branes wrapping internal curves (see [37]), but
for simplicity here we will not do that.

The saxionic cone is given by

$\Delta=\left\\{(s^{0},s^{a})\ \big{|}\ {\bm{s}}=s^{a}D_{a}\in\overline{{\cal
K}(X)}\,,\ s^{0}\geq 0\,,\ s^{0}\geq p_{a}s^{a}\right\\}\,,$ (4.43)

and the GB coupling (2.9) takes the form

$\gamma(s)=\pi\left(2s^{0}-p_{a}s^{a}+\frac{1}{6}n_{a}s^{a}\right)\,.$ (4.44)

Let us also recall that in the M-theory realization [68, 69], the Calabi-Yau
$X$ three-fold is fibered over an interval, representing the 11-th M-theory
direction. Then $s^{0}$ and $s^{0}-p_{a}s^{a}$ can be interpreted as the
volume of $X$ at the two endpoints of this interval [37]. For simplicity we
will henceforth assume that $p_{a}s^{a}\geq 0$. (By (4.42), we can actually
have $0\leq p_{a}s^{a}\leq n_{a}s^{a}$.) In this case the saxionic cone (4.43)
reduces to

$\Delta=\left\\{(s^{0},{\bm{s}})\ \big{|}\
{\bm{s}}=s^{a}D_{a}\in\overline{{\cal K}(X)}\,,\ s^{0}\geq
p_{a}s^{a}\right\\}\,.$ (4.45)

The Kähler potential can be in principle obtained by dimensionally reducing
the ten- and eleven-dimensional heterotic (M-)theory. One must take into
account the corrections discussed in [70] (see also [71]). These affect the
choice of saxionic variables and induce the tilting of the saxionic cone
(4.45) due to the constants $p_{a}$, which encode the effect of ten- and
eleven-dimensional higher derivative terms. Moreover, these terms may induce
additional corrections to the Kähler potential. Fully determining these
corrections is beyond the scope of the present paper, and so we will content
ourselves with considering contributions coming from the leading heterotic
M-theory terms [69], while taking into account the tilting of the saxionic
cone (4.45) and possible additional information coming from heterotic/F-theory
duality.

Under these working assumptions and using the above saxionic parametrization,
the Kähler potential takes the form

$K=-\log(s^{0}-\frac{1}{2}p_{a}s^{a})-\log\kappa({\bm{s}},{\bm{s}},{\bm{s}})\,,$
(4.46)

where $\kappa({\bm{s}},{\bm{s}},{\bm{s}})\equiv\kappa_{abc}s^{a}s^{b}s^{c}$,
and we ignore irrelevant additional constants. By (2.11), the corresponding
dual saxions are then given by

$\ell_{0}=\frac{1}{2(s^{0}-\frac{1}{2}p_{b}s^{b})}\quad,\quad\ell_{a}=\frac{3\kappa_{abc}s^{b}s^{c}}{2\kappa({\bm{s}},{\bm{s}},{\bm{s}})}-\frac{p_{a}}{4(s^{0}-\frac{1}{2}p_{b}s^{b})}\equiv\hat{\ell}_{a}-\frac{1}{2}p_{a}\ell_{0}\,,$
(4.47)

and their kinetic potential takes the form

${\cal F}=\log\ell_{0}+\log\hat{P}({\hat{\ell}})\,.$ (4.48)

Here $\hat{\ell}_{a}\equiv\ell_{a}+\frac{1}{2}p_{a}\ell_{0}$ and
$\log\hat{P}({\hat{\ell}})$ are the dual saxions and the kinetic potential
that one would obtain by ignoring the $s^{0}$ saxion and starting from a
Kähler potential $\hat{K}=-\log\kappa({\bm{s}},{\bm{s}},{\bm{s}})$.

Note that the Calabi-Yau volume changes along the M-theory interval [70] and
that $s^{0}-p_{a}s^{a}$ represents its smallest value in $l_{(11)}$ units.
Hence the assumed validity of the geometric heterotic M-theory regime of [69]
requires that $s^{0}-p_{a}s^{a}\gg 1$, which in turn implies that the
$p_{a}s^{a}$ contribution in (4.46) may be considered as a subleading
contribution, potentially of the same order of other neglected corrections
coming from higher-derivative M-theory terms. More information can be obtained
by looking at the models that admit a dual F-theory description, whose
perturbative regime described in Section 4.1 should correspond to freezing one
of the heterotic saxions $s^{a}$. In Appendix B we show how, in the regime
$\ell_{0}\ll|\ell_{a}|$, the corresponding restriction of (4.47) matches the
F-theory Kähler potential (4.18), up to $\ell_{0}^{2}$ corrections. We then
expect the F-theory Kähler potential (4.18) to capture possible corrections to
the corresponding restricted version of (4.47).

These uncertainties clearly affect the identification of the dual saxionic
cone. First focus on the modified dual saxionic vector
$\hat{\bm{\ell}}={\bm{\ell}}+\frac{1}{2}\ell_{0}{\bf
p}=\hat{\ell}_{a}\Sigma^{a}$, where $\Sigma^{a}$ is the basis of curves dual
to $D_{a}$ ($D_{a}\cdot\Sigma^{b}=\delta_{a}^{b}$). The second relation in
(4.47) implies that $\hat{\bm{\ell}}\in{\cal P}^{\text{\tiny
het}}_{\text{\tiny K}}$, where ${\cal P}^{\text{\tiny het}}_{\text{\tiny
K}}\subset H_{2}(X,\mathbb{R})$ is the Poincaré dual of the closure of the
image of ${\cal K}(X)$ under the map $J\rightarrow J\wedge J$. Note that
${\cal P}^{\text{\tiny het}}_{\text{\tiny K}}$ is a subcone of the cone
Mov${}_{1}(X)$ introduced in (4.8). However, this condition may not precisely
represent the dual saxionic domain, as we may be missing modifications of the
dual saxionic cone which are negligible only if $\ell_{0}\ll|\ell_{a}|$.
Indeed, as discussed in more detail in Appendix B, in models admitting a dual
F-theory description ${\bm{\ell}}$, rather than $\hat{\bm{\ell}}$, should
belong to ${\cal P}^{\text{\tiny het}}_{\text{\tiny K}}$. This suggests the
following possible refinement of the dual saxionic cone ${\cal P}$:

${\cal P}\simeq\\{(\ell_{0},{\bm{\ell}})\ |\ \ell_{0}\geq 0\,,\
{\bm{\ell}}\in{\cal P}^{\text{\tiny het}}_{\text{\tiny K}}\\}\,.$ (4.49)

While ${\cal P}$ may receive further corrections and it would certainly be
more satisfying to have a more precise derivation thereof, for concreteness we
will henceforth assume (4.49), keeping in mind that its reliability is more
robust for models admitting an F-theory dual in the regime
$\ell_{0}\ll|\ell_{a}|$.

Going back to the saxionic coordinates, the saxionic convex hull is now given
by

$\hat{\Delta}_{\alpha}=\left\\{(s^{0},{\bm{s}})\ \big{|}\
{\bm{s}}=s^{a}D_{a}\in\hat{\cal K}_{\alpha}(X)\,,\
s^{0}\geq\frac{1}{\alpha}+p_{a}s^{a}\right\\}\,.$ (4.50)

where $\hat{\cal K}_{\alpha}(X)$ is the Kähler convex hull defined as in the
generic saxionic case, which is contained in the stretched Kähler cone
introduced in [11]. If one restricts to $\hat{\Delta}_{\alpha}$, (4.44)
satisfies

$\gamma(s)|_{\hat{\Delta}_{\alpha}}\geq\pi\left[\frac{2}{\alpha}+\big{(}p_{a}+\frac{1}{6}n_{a}\big{)}s^{a}|_{\hat{\cal
K}_{\alpha}(X)}\right]\,.$ (4.51)

For concreteness, consider for instance the models with $\lambda(E_{2})=0$. In
this case $p_{a}=n_{a}$ and by (4.42) $p_{a}s^{a}$ takes its highest possible
value, i.e. $n_{a}s^{a}\geq 0$. (Smaller values of $p_{a}s^{a}\geq 0$ lead to
similar conclusions.) By applying the same arguments that led us to (3.21), we
expect $\big{(}n_{a}s^{a}\big{)}|_{\hat{\cal K}_{\alpha}(X)}\geq
b_{2}(X)/\alpha$, which implies that

$\gamma(s)|_{\hat{\Delta}_{\alpha}}\geq\pi\left(\frac{2}{\alpha}+\frac{7}{6}n_{a}s^{a}|_{\hat{\cal
K}_{\alpha}(X)}\right)\geq\frac{(5+7N)\pi}{6\alpha}\,.$ (4.52)

Note that this lower bound is stronger than (3.21). We numerically tested this
bound in a set of explicit Calabi-Yau compactifications with CYtools [72]. The
result is reported in figure 2, in which we plot the value of
$\alpha\gamma(s)$ evaluated at the tip of stretched Kähler cone against
$b_{2}(X)=N-1$. Because the saxionic convex hull $\hat{\Delta}_{\alpha}$ is
contained in the stretched Kähler cone, the numerical analysis provides an
important non-trivial check of our general bound (4.52), as well as of (3.21),
which relied on certain non-trivial quantum gravity constraints.

Figure 2: Value of $\alpha\gamma(s)$ at the tip of the $\alpha$-stretched
saxionic cone as a function of $b_{2}(X)=N-1$. This quantity, which is
$\alpha$-invariant, has been computed in an explicit set of Calabi-Yau
compactifications analyzed with CYtools [72]. The set consists of 71908
Calabi-Yau manifolds with $b_{2}(X)$ ranging from 2 to 491, with up to 100
polytopes per fixed $b_{2}(X)$ and with 25 triangulations per polytope
obtained with the random_triangulation_fast method. The quantity $n_{a}s^{a}$
of equation (4.44) has been obtained with the second_chern_class method and
evaluated at the tip of the $\alpha$-stretched Kähler cone. The black solid
line refers to the lower bound of equation (4.52). The best fit of our
numerical result, obtained assuming a power-law scaling, is
$\alpha\gamma\simeq 10^{0.9}b_{2}^{1.7}$, and is shown by the black dashed
line.

####  4.2.1 Energy scales in heterotic models

Finally, let us consider the scales characterizing the heterotic models
discussed in this section. By discretizing (4.45) one gets ${\cal C}_{\rm
S}^{\text{\tiny EFT}}$, which is generated by $(1,{\bf 0})$ and vectors of the
form $(e^{a}p_{a},{\bf e})$, where ${\bf e}\equiv e^{a}D_{a}$ are generators
of the cone of nef divisors. These EFT strings have tensions

$\displaystyle{\cal T}_{*}$ $\displaystyle=M^{2}_{\text{\tiny
P}}\ell_{0}=\frac{M^{2}_{\text{\tiny
P}}}{2(s^{0}-\frac{1}{2}p_{a}s^{a})}=\frac{3M^{2}_{\text{\tiny
P}}e^{2\phi}}{\kappa({\bm{s}},{\bm{s}},{\bm{s}})}\,,$ (4.53a)
$\displaystyle{\cal T}_{\bf e}$ $\displaystyle=M^{2}_{\text{\tiny
P}}(e^{a}\ell_{a}+e^{a}p_{a}\ell_{0})=\left(\frac{3\kappa({\bf
e},{\bm{s}},{\bm{s}})}{2\kappa({\bm{s}},{\bm{s}},{\bm{s}})}+\frac{e^{a}p_{a}}{2(s^{0}-\frac{1}{2}p_{b}s^{b})}\right)M^{2}_{\text{\tiny
P}}\,,$ (4.53b)

where we have used (4.47). By looking at the behavior of (4.46) under the
corresponding EFT string flows we can check that $K\sim-\log\sigma$ under the
$(1,{\bf 0})$ flow, and $w=1$, consistently with the fact that $(1,{\bf 0})$
indeed represents a critical heterotic string. On the other hand, under the
flow of the EFT string charges $(e^{a}p_{a},{\bf e})$ we have
$K\sim-n\log\sigma$, with integer $n$ determined by the self-intersections of
${\bf e}$. If $e^{a}p_{a}\geq 1$, one has $n=2,3,4$, and consistently these
strings have scaling weights $w=2$ or $w=3$ [37]. If instead $e^{a}p_{a}=0$,
one can also have $n=1$ and $w=1$ [14].

It is now interesting to compare the dominant EFT string scale $M_{{\cal T}}$
introduced in (3.29) with $M_{\gamma}$, see (3.27), and the species scale
$M_{\text{sp}}$. We will approach this task first analytically, in certain
controllable limits, and then numerically in more general setups.

Consider first the asymptotic regime identified by the EFT string flow
generated by $(1,{\bf 0})$, that is, $s^{0}\gg 1$ with $s^{a}$ fixed within
(4.50). Inspecting (4.38) and (4.53), one finds that this limit corresponds to
the weak string coupling limit $e^{\phi}\ll 1$, and that in this limit ${\cal
T}_{*}\ll{\cal T}_{\bf e}$, for any nef divisor ${\bf e}$. Hence in this
regime $M^{2}_{{\cal T}}=2\pi{\cal T}_{*}$ and the bound (3.28) is saturated,
since ${\cal T}_{*}$ corresponds to the critical string tension: $M_{\rm
sp}=M_{\rm s}=M_{{\cal T}}$. In order to move away from this specific regime,
we have to distinguish whether there exists or there does not exist a nef
divisor ${\bf e}=e^{a}D_{a}$ such that $e^{a}p_{a}=0$.

Assume first that $e^{a}p_{a}\geq 1$ for any nef divisor ${\bf e}$, which also
implies the strict positivity $p_{a}s^{a}>0$. In this case, the equations
(4.53) clearly show that ${\cal T}_{*}\leq{\cal T}_{\bf e}$, and therefore
that $M^{2}_{{\cal T}}=2\pi{\cal T}_{*}$, at any point of (4.50). We can now
compare compare $M^{2}_{{\cal T}}$ to $M^{2}_{\gamma}$:

$\frac{M^{2}_{{\cal
T}}}{M^{2}_{\gamma}}=\frac{1}{2}\left[1+\frac{n_{a}s^{a}}{12(s^{0}-\frac{1}{2}p_{a}s^{a})}\right]\,,$
(4.54)

where $M^{2}_{\gamma}$ can be obtained by using (4.44) in the definition
(3.27). Since $s^{0}\geq p_{a}s^{a}$ and $n_{a}s^{a}\geq 0$, we can then
conclude that

$\frac{1}{2}\leq\frac{M^{2}_{{\cal
T}}}{M^{2}_{\gamma}}\leq\frac{1}{2}\left(1+\frac{n_{a}s^{a}}{6p_{a}s^{a}}\right)\,.$
(4.55)

By (4.42) the smallest possible value of the upper bound in (4.55) is
$\frac{7}{12}$ and is obtained by picking a trivial $E_{2}$ bundle, that is
$p_{a}=n_{a}$. Therefore, in this case the two scales clearly agree within a
factor of order one. Even though for more general $p_{a}$ the upper bound
appearing in (4.55) is a priori larger than $\frac{7}{12}$, by our assumption
that $p_{a}e^{a}\geq 1$ for any nef divisor ${\bf e}$ we expect $p_{a}s^{a}$
not to be much smaller than $n_{a}s^{a}$ in our perturbative domain. Verifying
this expectation would require a thorough investigation of the internal bundle
structure, which enters the definition (4.39), but this is beyond the scope of
the present paper. We can however get some qualitative information by
rewriting (4.54) in the form

$\frac{M^{2}_{{\cal
T}}}{M^{2}_{\gamma}}=\frac{1}{2}\left[1+\frac{1}{2}e^{2\phi}\frac{n_{a}s^{a}}{\kappa({\bm{s}},{\bm{s}},{\bm{s}})}\right]\,.$
(4.56)

By using CYtools [72] we have evaluated numerically the ratio
$n_{a}s^{a}/\kappa({\bm{s}},{\bm{s}},{\bm{s}})$ at the tip of the
$\alpha$-stretched Kähler cone of a large number of models. The result is
shown in Fig. 3, which presents the value of the $\alpha$-invariant
combination $n_{a}s^{a}/[\alpha^{2}\kappa({\bm{s}},{\bm{s}},{\bm{s}})]$. If
for instance $\alpha=0.1$, $n_{a}s^{a}/\kappa({\bm{s}},{\bm{s}},{\bm{s}})$ is
roughly given by $N^{-3}$ for large $N$. Moving away from the tip of the
stretched Kähler cone we expect an even larger suppression. This suggests that
$M_{{\cal T}}$ and $M_{\gamma}$ basically agree also in more general $N\gg 1$
models, at least if $e^{\phi}$ is not unnaturally large.

Figure 3: Scaling of the ratio
$n_{a}s^{a}/\alpha^{2}\kappa({\bm{s}},{\bm{s}},{\bm{s}})$, appearing in
(4.56), as a function of $b_{2}(X)=N-1$. The ratio has been evaluated at the
tip of the $\alpha$-stretched Kähler cone in the same set of Calabi-Yau
compactifications analyzed in figure 2. The best fit for the approximate
scaling with $N$, again assuming a power-law behavior, is given by
$n_{a}s^{a}/\alpha^{2}\kappa({\bm{s}},{\bm{s}},{\bm{s}})\simeq
10^{2.1}\,b_{2}^{-3.3}$ and is qualitatively compatible with the scaling of
$n_{a}s^{a}$ from Fig. 2 and with that of $\kappa({\bm{s}},{\bm{s}},{\bm{s}})$
from [11] (modulo a small difference due to the algorithms used to generate
Calabi-Yau manifold, which are different in the latter reference and lead to
manifolds with qualitatively different properties).

Let us now allow for the existence of one or more nef divisors $\hat{\bf
e}=\hat{e}^{a}D_{a}$ such that $p_{a}\hat{e}^{a}=0$. We will restrict to the
elementary charges $\hat{\bf e}$ of this type. The corresponding tensions
(4.53b) reduce to

${\cal T}_{\hat{\bf e}}=M^{2}_{\text{\tiny
P}}\hat{e}^{a}\ell_{a}=M^{2}_{\text{\tiny P}}\frac{3\kappa(\hat{\bf
e},{\bm{s}},{\bm{s}})}{2\kappa({\bm{s}},{\bm{s}},{\bm{s}})}\,,$ (4.57)

and control $M^{2}_{{\cal T}}$ in the region of the saxionic domain such that
$e^{2\phi}\geq\frac{1}{2}\kappa(\hat{\bf e},{\bm{s}},{\bm{s}})$ for some of
these $\hat{\bf e}$. By recalling (4.38), this condition corresponds to
$s^{0}\leq\frac{M^{2}_{\text{\tiny P}}}{2{\cal T}_{\hat{\bf
e}}}+\frac{1}{2}p_{a}s^{a}$. Unfortunately, in this case it is not easy to
draw general conclusions about the ratio $M^{2}_{{\cal T}}/M^{2}_{\gamma}$. In
order to get some quantitative understanding, we pragmatically assume that
$p_{a}\equiv 0$, and we again use CYtools [72] to numerically investigate
$M^{2}_{{\cal T}}/M^{2}_{\gamma}$. The result is shown in Fig. 4. Again we see
that, up to irrelevant numerical factors of order one, the two scales agree at
the tips of the stretched Kähler cones of the entire set of geometries that we
explored. Note that these results hold also for the more general case with
non-vanishing $p_{a}$, as long as we go far enough along the EFT string flows
${\bm{s}}={\bm{s}}_{0}+\sigma\hat{\bf e}$, $\sigma\gg 1$. Indeed, precisely in
this limit we can neglect the contribution $p_{a}s^{a}\simeq p_{a}s^{a}_{0}$,
which does not scale with $\sigma$, and ${\cal T}_{\hat{\bf e}}$ is expected
to identify the lightest EFT string tension, and then to determine $M_{{\cal
T}}$. Furthermore, the same conclusions immediately apply to ${\cal N}=2$ type
IIA models. This clearly indicates that $M^{2}_{{\cal T}}$ agrees (in the
geometric regime) with the estimate of the species scale proposed for these
models in [53].

Figure 4: Sample of values of the ratio between $M^{2}_{{\cal T}}$ and
$M^{2}_{\gamma}$ in a large class of heterotic models, assuming $p_{a}\equiv
0$. This ratio, which is $\alpha$-invariant, is evaluated at the tip of the
$\alpha$-stretched saxionic cone, that is, at the tip of of the
$\alpha$-stretched Kähler cone and with $s^{0}=1/\alpha$, in an explicit set
of 11962 distinct Calabi-Yau compactifications analyzed with CYtools [72]. The
manifolds have been obtained with the random_triangulation_fast method with up
to 100 polytopes per fixed $b_{2}(X)$ and 25 triangulations per polytope.
$M^{2}_{\gamma}$ can be obtained by using (4.44) in the definition (3.27),
with the numerical values of $n_{a}$ obtained with the second_chern_class
function. $M^{2}_{{\cal T}}$ has been estimated using (4.53b), in which we
employed the minimum volume among the nef divisors that we found in each
compactification. These have been individuated requiring $e^{a}M_{\alpha
a}\geq 0\,\forall\,\alpha$, where $M_{\alpha a}$ is the matrix of Mori cone
generators associated to the basis of the inherited $b_{2}(X)+4$ prime toric
divisors obtained using the toric_mori_cone function. Their volumes have been
then computed at the tip of the Kähler cone with compute_divisor_volumes. As
typically only a small fraction of the basis divisors are also nef divisors,
the available statistic is reduced compared to the analysis of figure 2.

Let us now turn to the verification of the bound (3.28) beyond the weak string
coupling limit, that is, in the regime in which $M_{\rm sp}$ is given by the
quantum gravity scale $M_{\text{\tiny QG}}$, rather than the string scale
$M_{\rm s}$. Let us again first assume that $e^{a}p_{a}\geq 1$ for any nef
divisor ${\bf e}=e^{a}D_{a}$, so that $M^{2}_{{\cal T}}=2\pi{\cal T}_{*}$ as
already pointed out. We can check (3.28) analytically in the M-theory
supergravity regime. This requires $e^{\phi}$ and
$e^{-\frac{2\phi}{3}}{\bm{s}}$ to be large enough, since the internal six-
dimensional M-theory and heterotic string frame metrics are related by ${\rm
d}s^{2}_{\text{\tiny M}}(X)=e^{-\frac{2\phi}{3}}{\rm d}s^{2}_{\text{st}}(X)$.
For instance we may require that $e^{-\frac{2\phi}{3}}{\bm{s}}\in\hat{\cal
K}_{\bf\alpha}$. Recalling the ansatz (4.3) (which implies
$l_{(11)}=l_{(10)}$), the scaling factor appearing in (4.4) is given by

$e^{2A}=\frac{M^{2}_{\text{\tiny P}}l^{2}_{(11)}}{4\pi
V(X)}e^{-\frac{2}{3}\phi}\,,$ (4.58)

where $V(X)=\frac{1}{6}e^{-2\phi}\kappa({\bm{s}},{\bm{s}},{\bm{s}})$ is the
average value of the Calabi-Yau volume, as measured in M-theory and in
$l_{(11)}$ units, along the M-theory interval. The quantum gravity scale is
then given by

$M^{2}_{\text{\tiny
QG}}=e^{2A}\widehat{M}^{2}_{(11)}=\frac{1}{2}\left[\frac{12\pi}{(s^{0}-\frac{1}{2}p_{a}s^{a})^{2}\kappa({\bm{s}},{\bm{s}},{\bm{s}})}\right]^{\frac{1}{3}}\,M^{2}_{\text{\tiny
P}}\,,$ (4.59)

where $\widehat{M}^{2}_{(11)}$ is as in (4.5), and we have used (4.38). Since
$M^{2}_{{\cal T}}=2\pi{\cal T}_{*}$, with ${\cal T}_{*}$ as in (4.53a), we
conclude that

$\frac{M^{2}_{{\cal T}}}{M^{2}_{\text{\tiny
QG}}}=\left[\frac{(2\pi)^{2}\kappa({\bm{s}},{\bm{s}},{\bm{s}})}{6(s^{0}-\frac{1}{2}p_{a}s^{a})}\right]^{\frac{1}{3}}=\left(2\pi
e^{\phi}\right)^{\frac{2}{3}}\quad~{}~{}~{}~{}~{}\text{(for $2\pi
e^{\phi}>1$)}\,.$ (4.60)

(This formula could have been derived more directly by identifying
$M^{2}_{{\cal T}}/2\pi={\cal T}_{*}$ with the tension of a string
corresponding to an open M2-brane stretching along the M-theory interval,
which has length $l_{(11)}e^{2\phi/3}$.)

Note that the ratio (4.60) is controlled by the expansion parameter $2\pi
e^{\phi}$, which more precisely distinguishes the ten- and eleven-dimensional
regimes, as also discussed in Appendix A. The bound (3.28) is satisfied almost
tautologically, since the M-theory regime we are working in requires $2\pi
e^{\phi}>1$. If instead $2\pi e^{\phi}<1$, the M-theory description is not
trustable anymore, and one should rather compute $M^{2}_{\text{\tiny QG}}$
starting from $\widehat{M}_{(10)}$ in (4.5). Since ${\cal T}_{*}$ corresponds
to an F1-string tension, the ratio (4.60) is more easily obtained by comparing
$2\pi{\cal T}_{\rm F1}=(2\pi)^{2}l^{-2}_{(10)}e^{\frac{1}{2}\phi}$ and
$\widehat{M}^{2}_{(10)}$ directly in ten dimensions. This gives (see also
Appendix A)

$\frac{M^{2}_{{\cal T}}}{M^{2}_{\text{\tiny QG}}}=\frac{2\pi{\cal T}_{\rm
F1}}{\widehat{M}^{2}_{(10)}}=\left(2\pi
e^{\phi}\right)^{\frac{1}{2}}\quad~{}~{}~{}~{}~{}\text{(for $2\pi
e^{\phi}<1$)}\,.$ (4.61)

Clearly $M_{{\cal T}}<M_{\text{\tiny QG}}$ if $2\pi e^{\phi}<1$, and hence the
bound (3.28) is saturated: $M_{\rm sp}=M_{\rm s}=M_{{\cal T}}$.

The case in which there is some nef divisor $\hat{\bf e}=\hat{e}^{a}D_{a}$
such that $p_{a}\hat{e}^{a}=0$ again requires a separate discussion. In this
case, the corresponding lightest possible tension (4.57) determines
$M^{2}_{{\cal T}}$ if $(2\pi e^{\phi})^{2}\geq\frac{1}{2}\kappa(\hat{\bf
e},2\pi{\bm{s}},2\pi{\bm{s}})$. Clearly, in our perturbative saxionic regime
this can happen only if $2\pi e^{\phi}>1$, namely in the M-theory regime,
since ${\bm{s}}\in\hat{\cal K}_{\alpha}$ and then we certainly have
$\frac{1}{2}\kappa(\hat{\bf e},2\pi{\bm{s}},2\pi{\bm{s}})>1$. As discussed
above, while we will not attempt to make exact statements, we can nevertheless
get non-trivial information assuming that we can set $p_{a}\equiv 0$, either
exactly or approximately. Hence, picking the charge $\hat{\bf e}$ with
lightest tension, which thus determines $M^{2}_{{\cal T}}$, combining (4.57)
and (4.59) we get

$\frac{M^{2}_{{\cal T}}}{M^{2}_{\text{\tiny QG}}}=\frac{\kappa(\hat{\bf
e},2\pi{\bm{s}},2\pi{\bm{s}})}{2(2\pi e^{\phi})^{\frac{4}{3}}}\,.$ (4.62)

Applying the perturbative condition (4.50) to (4.38) with $p_{a}=0$ we get

$e^{2\phi}\leq\frac{\alpha}{6}\kappa({\bm{s}},{\bm{s}},{\bm{s}})$ (4.63)

and hence

$\frac{M^{2}_{{\cal T}}}{M^{2}_{\text{\tiny
QG}}}\geq\left(\frac{2\pi}{\alpha}\right)^{\frac{2}{3}}\frac{V(\hat{\bf
e})}{V(X)^{\frac{2}{3}}}=\frac{1}{2}\left(\frac{12\pi}{\alpha}\right)^{\frac{2}{3}}\frac{\kappa(\hat{\bf
e},{\bm{s}},{\bm{s}})}{\kappa({\bm{s}},{\bm{s}},{\bm{s}})^{\frac{2}{3}}}\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{(for
$p_{a}=0$)}\,.$ (4.64)

We have numerically evaluated $\kappa(\hat{\bf
e},{\bm{s}},{\bm{s}})/\kappa({\bm{s}},{\bm{s}},{\bm{s}})^{\frac{2}{3}}$ at the
tip of the stretched Kähler cone of a large sample of models using again
CYtools [72]. The result is shown in Fig. 5 and, since $2\pi/\alpha\gg 1$ and
the combination $\kappa(\hat{\bf
e},{\bm{s}},{\bm{s}})/\kappa({\bm{s}},{\bm{s}},{\bm{s}})^{\frac{2}{3}}$ is
invariant under an overall saxionic rescaling, it clearly indicates that
(4.64) should always be greater than one, compatibly with the bound (3.28)
with $M_{\rm sp}=M_{\text{\tiny QG}}$.

One can further check the bound (3.28) in other limits. These are discussed in
Appendix C.3, which partly apply also to ${\cal N}=2$ models.

Figure 5: Plot of the lower bound (4.64) evaluated evaluated at the tip of the
$\alpha$-stretched saxionic cone with $\alpha=1$ in the same set of Calabi-Yau
manifolds of figure 4. Since the r.h.s. of (4.64) is invariant under an
overall rescaling of the saxions $s^{i}$, the result for more general $\alpha$
can be obtained through a rescaling by $\alpha^{-\frac{2}{3}}$.

##  5 $SO(4)$-symmetric wormhole configurations

In this section we show that the broad class of models described by the two-
derivative action (2.3), or equivalently (2.13), admits non-extremal and
extremal wormhole configurations with $SO(4)$ symmetry, after continuation to
Euclidean space. These solutions can be considered as generalizations of the
ones provided in the seminal paper [9] and encompass several other
generalizations already appeared in the literature – see for instance [10] for
a review. In Sections 5.1 and 5.2 we present the Euclidean formulation of our
models, derive the equations of motion associated to the $SO(4)$-symmetric
ansatz, and finally calculate the on-shell action for non-extremal wormholes.
This part of our work is not new but serves to set the stage for the original
results presented in the subsequent sections. Extremal wormholes are discussed
in Section 5.3 and their relation to fundamental instantons is clarified. The
concept of EFT instantons is introduced in Section 5.4. A puzzling BPS bound
on the on-shell action of non-extremal wormholes is briefly commented upon in
Section 5.5, a possible interpretation of which is suggested in later
sections.

###  5.1 Euclidean action

When discussing non-perturbative effects such as wormholes we have to pass to
the Euclidean formulation. The way in which one identifies charged saddles in
the axionic Euclidean action has been clarified in [73] (see also [74]) and
involves a continuation to imaginary axion fields as well as the introduction
of boundary terms. No such subtleties appear if one works with the dual
formulation, which is what we do. One just has to Wick rotate (2.13), giving

$\displaystyle S=$ $\displaystyle\,-\frac{1}{2}M_{\text{\tiny
P}}^{2}\int_{\cal M}\sqrt{g}\,R-\frac{1}{2}M_{\text{\tiny
P}}^{2}\int_{\partial\cal M}\sqrt{h}(K-K_{0})$ (5.1)
$\displaystyle\,+\frac{1}{2}M_{\text{\tiny P}}^{2}\int_{\cal M}{\cal
G}^{ij}{\rm d}\ell_{i}\wedge*{\rm d}\ell_{j}+\frac{1}{2M_{\text{\tiny
P}}^{2}}\int_{\cal M}{\cal G}^{ij}{\cal H}_{3,i}\wedge*{\cal H}_{3,j}\,,$

where the Gibbons-Hawking term [75] for asymptotically flat vacua has been
included. In this dual formulation ${\cal H}_{3,i}$ must satisfy (2.17), up to
possible localized terms corresponding to fundamental instantons.161616Apart
from the additional contribution to ${\rm d}{\cal H}_{3}$ of standard gauge
fields (see for instance [37]), which we set to zero, ${\rm d}{\cal H}_{3}$
can also get a contribution from the field-strengths of three-forms
potentials. As discussed in [76, 77, 41], in ${\cal N}=1$ supersymmetry this
effect is dual to the presence of special multi-branched superpotentials which
are often realized in flux compactifications [78]. In this paper we assume
that such superpotential terms are not present or do not affect the (s)axionic
sector under discussion. The GB term (2.8) will play an important role.
Rotated to Euclidean signature, this reads

$S_{\text{\tiny GB}}\equiv-\int_{\cal M}\sqrt{g}\,\gamma(\ell)\,E_{\text{\tiny
GB}}-\int_{\partial{\cal M}}\sqrt{h}\,\gamma(\ell)\,(Q-Q_{0})\,,$ (5.2)

where in
$\gamma(\ell)=\frac{\pi}{6}\tilde{C}_{i}s^{i}\equiv\frac{\pi}{6}\langle\tilde{\bf
C},{\bm{s}}\rangle$ (see (2.9)) we regard $s^{i}$ as functions of the dual
saxions $\ell_{i}$ as defined by (2.15), and we included the appropriate
boundary term [79] necessary to make the variational principle well defined.
This term is analogous to the Gibbons-Hawking term in (5.1), and we will not
need its precise form. Suffice it to say that it allows one to make the
identification [80] (see also [81, 82])

$\int_{\cal M}\sqrt{g}\,\,E_{\text{\tiny GB}}+\int_{\partial{\cal
M}}\sqrt{h}\,Q=\chi({\cal M})\,,$ (5.3)

where $\chi({\cal M})=\sum_{k}(-)^{k}b_{k}({\cal M})$ is the Euler
characteristic of ${\cal M}$. On the other hand, the counterterm
$\int_{\partial{\cal M}}\sqrt{h}\,\gamma(\ell)Q_{0}$ in (5.2) subtracts the
contribution of a flat vacuum configuration so as to have a vanishing action
in flat space, analogously to what done with the Gibbons-Hawking term.

###  5.2 Effective one-dimensional action and equations of motion

In deriving the relevant equations of motion we will follow the approach of
[74], but will work with the formulation in terms of gauge two-forms ${\cal
B}_{2\,i}$ and dual linear multiplets.

We are looking for Euclidean wormholes preserving $SO(4)$ rotational symmetry.
This means that we can restrict the metric to take the form

${\rm d}s^{2}=\frac{1}{M^{2}_{\text{\tiny P}}}\left[e^{2A(\rho)}{\rm
d}\rho^{2}+e^{2B(\rho)}{\rm d}\Omega^{2}\right]\,.$ (5.4)

Here $\rho\in I$ is an arbitrary dimensionless radial coordinate taking values
in some interval $I\subset\mathbb{R}$, to be defined below, and ${\rm
d}\Omega^{2}$ is the line element of a three-sphere of unit radius, which has
volume $2\pi^{2}$. One can of course remove the arbitrariness of $\rho$ by
gauge-fixing $e^{A(\rho)}$ or $e^{B(\rho)}$. A particularly convenient choice
is given by

$e^{B}=\rho=M_{\text{\tiny P}}r$ (5.5)

where $r$ represents the radius of the three-sphere. This gauge fixing
corresponds to the following line element

${\rm d}s^{2}=e^{2A(r)}{\rm d}r^{2}+r^{2}{\rm d}\Omega^{2}.$ (5.6)

Note that, however, in this case $r$ can smoothly parametrize only half
wormhole.171717This issue is avoided if instead we use the ‘geodesic’ radial
coordinate $\chi$ defined by $M_{\text{\tiny P}}{\rm d}\chi=e^{A}{\rm d}\rho$.
With this coordinate the metric takes the form ${\rm d}s^{2}={\rm
d}\chi^{2}+M_{\text{\tiny P}}^{-2}e^{2B(\chi)}{\rm d}\Omega^{2}$,
corresponding to the gauge-fixing $e^{A}=1$ and the identification $\rho=\chi
M_{\text{\tiny P}}$. Another useful parametrization is discussed below, but
for the moment we keep $\rho$ arbitrary.

In addition to the metric, we will allow the fields $\ell_{i},{\cal H}_{3,i}$
to have non-trivial profiles. By $SO(4)$ symmetry, they can only depend on the
radial coordinate: $\ell_{i}=\ell_{i}(\rho)$, or equivalently
$s^{i}=s^{i}(\rho)$. Instead the field-strengths ${\cal H}_{3\,i}$ must
necessarily take the form

${\cal H}_{3\,i}=\frac{1}{\pi}q_{i}\,{\rm vol}_{S^{3}}\,,$ (5.7)

with $q_{i}$ constants. Notice that such a choice satisfies ${\rm d}{\cal
H}_{3,i}\equiv 0$, and this is consistent with (2.17) since the $SO(4)$
symmetry implies that $\mathop{\rm tr}({\cal R}\wedge{\cal R})\equiv 0$.
Observing that $*{\cal H}_{3\,i}=M_{\text{\tiny P}}^{2}{\cal G}_{ij}{\rm