IFT-UAM/CSIC-23-144

On the moduli space curvature at infinity
Fernando Marchesano,1 Luca Melotti 1,2 and Lorenzo Paoloni 1

1 Instituto de Física Teórica UAM-CSIC, c/ Nicolás Cabrera 13-15, 28049
Madrid, Spain
2 Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049
Madrid, Spain
Abstract

We analyse the scalar curvature of the vector multiplet moduli space ${\cal
M}^{\rm VM}_{X}$ of type IIA string theory compactified on a Calabi–Yau
manifold $X$. While the volume of ${\cal M}^{\rm VM}_{X}$ is known to be
finite, cases have been found where the scalar curvature diverges positively
along trajectories of infinite distance. We classify the asymptotic behaviour
of the scalar curvature for all large volume limits within ${\cal M}^{\rm
VM}_{X}$, for any choice of $X$, and provide the source of the divergence both
in geometric and physical terms. Geometrically, there are effective divisors
whose volumes do not vary along the limit. Physically, the EFT subsector
associated to such divisors is decoupled from gravity along the limit, and
defines a rigid ${\cal N}=2$ field theory with a non-vanishing moduli space
curvature $R_{\rm rigid}$. We propose that the relation between scalar
curvature divergences and field theories that can be decoupled from gravity is
a common trait of moduli spaces compatible with quantum gravity.

###### Contents

  1. 1 Introduction
  2. 2 Type IIA large volume limits
  3. 3 The scalar curvature at infinity
    1. 3.1 M-theory analysis
    2. 3.2 The 4d EFT perspective
    3. 3.3 The curvature criterion
  4. 4 The curvature along different limits
    1. 4.1 M-theory limits
    2. 4.2 F-theory limits
    3. 4.3 Emergent string limits
    4. 4.4 Summary
  5. 5 Examples
  6. 6 Conclusions
  7. A Asymptotic perturbative expansion of the metric
  8. B The scaling weight and degeneracy of EFT string limits

##  1 Introduction

The Swampland Programme [1, 2, 3, 4, 5] aims to describe the set of effective
field theories (EFTs) with a gravitational UV completion. One way to do so is
to establish general constraints on the range value and the field dependence
for the couplings of such EFTs. This was the spirit behind [6], where several
criteria were proposed to constrain the geometry of the moduli space of vacua
$\mathcal{M}$ of any EFT compatible with Quantum Gravity. One of these
proposals conjectured that $\mathcal{M}$ should always have points at infinite
field distance, and another one that the scalar curvature of $\mathcal{M}$
near such points should be non-positive. Part of the intuition came from
compactifications of type II string theory on Calabi–Yau (CY) manifolds, that
lead to 4d ${\cal N}=2$ supergravity EFTs. In particular, the vector multiplet
moduli space of such theories ${\cal M}_{\rm 4d}^{\rm VM}$ is endowed with a
Weil-Petersson (WP) metric that is known to be exact, that always contains
points at infinite distance and that provides ${\cal M}_{\rm 4d}^{\rm VM}$
with a finite volume [7, 8, 9]. Moreover, for cases with a single modulus $T$
the Kähler potential that describes the asymptotic WP metric near the
infinite-distance point $\mbox{Im}\,T=\infty$ reads $K=-3\log\mbox{Im}\,T$ and
leads to an asymptotically constant, negative curvature.

However, it is precisely in the context of type II Calabi–Yau
compactifications where counterexamples to the curvature proposal were found
[10]. The counterexamples involved 4d ${\cal N}=2$ vector multiplet moduli
spaces with three moduli, and the scalar curvature of the WP metric was not
only asymptotically positive, but actually divergent. More recently, these
examples have been reconsidered in [11], where it was pointed out that the
divergence disappears if in ${\cal M}_{\rm 4d}^{\rm VM}$ one replaces the WP
metric by the Hodge metric. It was then proposed that scalar curvature
divergences are related to the lack of completeness of the WP metric, and that
the same physical mechanism should shield the EFT from both of these effects.

In this work we aim to shed some light on the asymptotic behaviour of the
moduli space scalar curvature. For this, we classify its behaviour in a wide
class of infinite distance limits, and identify the physical origin of the
divergences whenever they appear. We focus our analysis on infinite-distance
limits within the 4d ${\cal N}=2$ vector multiplet moduli space obtained from
type IIA string theory compactified on an arbitrary Calabi–Yau three-fold $X$.
These have been recently classified in terms of monodromy techniques [12, 13,
14] and the Emergent String Conjecture [15], with the motivation to understand
how the Distance Conjecture (SDC) [6] is realised along them. More precisely,
we focus on limits where the volume of $X$ tends to infinity, which were also
considered in [16] in the context of the Emergence Proposal [17, 12, 18].
Large-volume limits have the advantage that they can be easily mapped to
trajectories in the 5d vector multiplet moduli space ${\cal M}_{\rm 5d}^{\rm
VM}$ of M-theory compactified on $X$, where they become either of finite or
infinite length. This 5d description is instrumental to understand which
limits give rise to a divergence.

In general, we obtain that divergences for the type IIA scalar curvature
$R_{\rm IIA}$ are related to divisors ${\cal D}\subset X$ whose volume stays
constant along the large-volume limit. From the M-theory perspective it is the
volume of $X$ that stays constant along ${\cal M}_{\rm 5d}^{\rm VM}$, while
${\cal D}$ shrinks to zero size. In particular, if the trajectory becomes of
finite distance in ${\cal M}_{\rm 5d}^{\rm VM}$, then ${\cal D}$ must be a
generalised del Pezzo divisor that shrinks to a point. Shrinkable cycles are
known to give rise to limits in which a non-trivial field theory decouples
from gravity [19, 20]. We argue that this is also what happens along the type
IIA infinite-distance trajectories where $R_{\rm IIA}$ diverges.

The Distance Conjecture predicts that, as we approach an infinite distance
point of ${\cal M}$, the maximal cut-off of an EFT $m_{*}$ decreases to zero
in Planck units. If along such a limit there is a field theory sector that
remains dynamical below $m_{*}$, then it will be gradually decoupled from
gravity, resulting in a rigid field theory that dominates the low-energy
dynamics. In our type IIA setup $m_{*}$ is the D0-brane mass, and the
constant-volume divisor ${\cal D}$ hosts a 4d ${\cal N}=2$ field theory
subsector that remains dynamical below this scale. The moduli space and
kinetic terms of this subsector is described by a rigid special Kähler
geometry [21], that it is recovered from the local special Kähler structure
that describes ${\cal M}_{\rm 4d}^{\rm VM}$, as in other gravity-decoupling
limits [22, 23, 24, 25, 26, 27, 28]. Using the standard expressions for the
scalar curvature in both special Kähler geometries, one can see that a non-
trivial curvature $R_{\rm rigid}$ for the ${\cal N}=2$ rigid field theory
below $m_{*}$ results in a (positive) divergence for the ${\cal N}=2$
supergravity curvature measured in Planck units, that goes like $R_{\rm
IIA}\simeq R_{\rm rigid}\frac{M_{\rm P}^{2}}{m_{*}^{2}}$. Via Conjecture 1 we
propose that this is a general mechanism valid beyond our specific setup, and
that it essentially accounts for all the divergences that a moduli space
scalar curvature can have in theories that are compatible with quantum
gravity.

For those limits along which the curvature does not diverge there is a variety
of behaviours, which we classify for the type IIA EFT string limits of [16].
We argue that this is a representative subset of limits, in the sense that it
is dense in the continuum of type IIA large-volume endpoints, a set that can
be identified with the moduli space ${\cal M}_{\rm 5d}^{\rm VM}$. We find that
non-divergent scalar curvatures are asymptotically negative, unless we
consider a limit that it is close to a divergent one in ${\cal M}_{\rm
5d}^{\rm VM}$. As a result, limits with constant positive curvature display a
hierarchy of gauge couplings, and in particular some gauge sectors that
dominate over the remaining gauge interactions and the gravitational ones,
although not parametrically like in the divergent case. If this was a general
lesson, one would recover the physical intuition behind the scalar curvature
proposal in [6], in the sense that whenever gravity is not substantially
suppressed with respect to other interactions one recovers an asymptotic
scalar curvature that is negative.

The rest of this paper is organised as follows. Section 2 describes the type
IIA infinite-distance limits that we consider and their M-theory description.
Section 3 describes the moduli space scalar curvature along such limits, and
identifies its divergences both from an M-theory and a 4d EFT perspective,
leading to Conjecture 1. Section 4 analyses the asymptotic behaviour of the
curvature along different classes of EFT string limits, which is summarised in
table 1. Section 5 contains some examples that illustrate the general picture,
and in section 6 we discuss some possible implications of our findings. Some
technical details have been relegated to the appendices. Appendix A contains
the perturbative expansion of the asymptotic metric and appendix B discusses
the degeneracy of the different EFT string limits. Both sets of results are
used in the classification of limits performed in section 4.

##  2 Type IIA large volume limits

Let us consider type IIA string theory compactified on a Calabi–Yau three-fold
$X$

$ds^{2}=ds_{\mathbbm{R}^{1,3}}^{2}+ds_{X}^{2}\,,$ (2.1)

whose metric is determined by the periods of its holomorphic three-form
$\Omega$ and its Kähler form $J$. The latter can be expanded as

$J=t^{a}\omega_{a}\,,\qquad a=1,\dots h^{1,1}(X),$ (2.2)

where $\ell_{s}^{-2}\omega_{a}$ is a basis of integral harmonic two-forms
Poincaré dual to a basis of Nef divisors, with
$\ell_{s}=2\pi\sqrt{\alpha^{\prime}}$ the string length. We use the same basis
to reduce the B-field and RR potentials

$B=b^{a}\omega_{a}+\dots\,,\qquad C_{1}=\kappa_{4}A_{1}^{0}\,,\qquad
C_{3}=\kappa_{4}\hat{A}_{1}^{a}\wedge\omega_{a}+\dots\,,$ (2.3)

where the dots stand for terms of the expansion that are irrelevant for the
vector multiplet sector. Here $T^{a}=b^{a}+it^{a}$ and $A_{1}^{0}$,
$\hat{A}_{1}^{a}$ represent 4d complex scalars and vector bosons,
respectively. Upon dimensional reduction of the standard type IIA 10d
supergravity action one recovers the following piece of 4d ${\cal N}=2$
Lagrangian describing the vector multiplet sector [29, 23, 30]

$S_{\rm 4d}^{\rm
VM}=\frac{1}{2\kappa_{4}^{2}}\int_{\mathbbm{R}^{1,3}}R*\mathbbm{1}-2g_{ab}dT^{a}\wedge*d\bar{T}^{\bar{b}}+\frac{1}{2}\int_{\mathbbm{R}^{1,3}}I_{AB}F^{A}\wedge*_{4}F^{B}+R_{AB}F^{A}\wedge
F^{B},$ (2.4)

with $A=(0,a)$. Here $F^{A}$ are integrally-quantised two-form field strengths
associated to the U(1) gauge bosons: $F^{0}=dA_{1}^{0}$ is the graviphoton
field strength, while $F^{a}=dA_{1}^{a}\equiv
d(\hat{A}_{1}^{a}-b^{a}A_{1}^{0})$. The moduli space metric in the large-
volume regime is specified by

$g_{ab}=\frac{3}{2}\left(\frac{3}{2}\frac{{\cal K}_{a}{\cal K}_{b}}{{\cal
K}^{2}}-\frac{{\cal K}_{ab}}{{\cal K}}\right),$ (2.5)

with ${\cal
K}_{abc}=\ell_{s}^{-6}\int\omega_{a}\wedge\omega_{b}\wedge\omega_{c}$ the
triple intersection numbers of $X_{6}$, from where we build ${\cal
K}_{ab}={\cal K}_{abc}t^{c}$, ${\cal K}_{a}={\cal K}_{abc}t^{b}t^{c}$ and
${\cal K}={\cal K}_{abc}t^{a}t^{b}t^{c}=6{\cal V}_{X}$. The gauge couplings
are

$I\,=\,-\frac{{\cal
K}}{6}\left(\begin{array}[]{cc}1+4g_{ab}b^{a}b^{b}&4g_{ab}b^{b}\\\
4g_{ab}b^{b}&4g_{ab}\end{array}\right),\qquad
R\,=\,-\left(\begin{array}[]{cc}\frac{1}{3}{\cal
K}_{abc}b^{a}b^{b}b^{c}&\frac{1}{2}{\cal K}_{abc}b^{b}b^{c}\\\
\frac{1}{2}{\cal K}_{abc}b^{b}b^{c}&{\cal K}_{abc}b^{c}\end{array}\right).$
(2.6)

These expressions are a large-volume approximation that receive curvature and
world-sheet instanton corrections. These can be encoded in terms of the full
prepotential

${\cal
F}=-\frac{1}{6}\mathcal{K}_{abc}T^{a}T^{b}T^{c}+\frac{1}{2}K_{ab}^{(1)}T^{a}T^{b}+K_{a}^{(2)}T^{a}+\frac{i}{2}K^{(3)}+(2\pi
i)^{-3}\sum_{\bm{k}}n_{\bm{k}}^{(0)}{\rm Li}_{3}\left(e^{2\pi
ik_{a}T^{a}}\right),$ (2.7)

where $K^{(3)}=\frac{\zeta(3)}{8\pi^{3}}\chi(X_{6})$, $K_{ab}^{(1)}$ and
$K_{a}^{(2)}$ depend on topological data of $X$, Li3 stands for the 3rd
polylogarithmic function and $n_{\bm{k}}^{(0)}$ for the genus zero Gopakumar-
Vafa (GV) invariant of the curve class $k_{a}{\cal C}^{a}$, with ${\cal
C}^{a}$ the dual of $[\ell_{s}^{-2}\omega_{a}]$. In the large-volume regime
the piece $\mathcal{F}^{\rm cl}=-\frac{1}{6}\mathcal{K}_{abc}T^{a}T^{b}T^{c}$
is the dominant one, and it is from this term that (2.5) and (2.6) are
derived. The effect of the polynomial corrections on these quantities have
been discussed in [16, Appendix B], and they can be neglected as long as all
volumes are large. The same applies to the exponentially-suppressed terms that
correspond to world-sheet instanton corrections, although we will see that,
when discussing the moduli space curvature, in some cases they may play a
role.

In this setup, [16] explored a particular class of infinite distance
trajectories, in which a linear combination of Kähler moduli specified by a
vector $\bm{e}$ grows towards infinite volume

$t^{a}=t^{a}_{0}+e^{a}\phi,\quad\text{with}\quad\phi\to\infty,$ (2.8)

with $t^{a}_{0}$ inside the Kähler cone and $e^{a}\in\mathbbm{N}$. Such limits
are motivated by the backreaction of 4d strings in the theory, built by
wrapping an NS5-brane on the divisor class ${\cal
D}_{\bm{e}}=\ell_{s}^{-2}e^{a}[\omega_{a}]$. Using the terminology of [31, 32,
33], they were dubbed EFT string limits. In order to keep the field trajectory
within the vector multiplet moduli space one must rescale the 10d dilaton
accordingly

$g_{s}(\phi)=g_{s,0}\frac{{\cal V}_{X}^{1/2}(\phi)}{{\cal
V}_{X,0}^{1/2}}\to\infty,$ (2.9)

taking us to a region of 10d strong coupling. Nevertheless, from the 4d EFT
perspective we are led to a weakly coupled region, with a simpler microscopic
description in terms of a dual frame.

Trajectories of the form (2.8) are only a subset of the whole set of infinite
distance limits in the vector multiplet moduli space of type IIA Calabi-Yau
compactifications, which have been recently classified in [14, 15]. In general
one obtains three main types of limits, each with a different dual
description. Applied to EFT string limits such a classification translates
into:

  1. 1.

${\bf k}\equiv{\cal K}_{abc}e^{a}e^{b}e^{c}\neq 0$, dubbed type IVd or $n=3$
limits in [14]. Their dual description is M-theory compactified on
$S^{1}\times X$, where the $S^{1}$ radius tends to infinity.

  2. 2.

${\bf k}=0$, but ${\bf k}_{a}\equiv{\cal K}_{abc}e^{b}e^{c}\neq 0$ for some
$a$, dubbed type IIIc or $n=2$ limits in [14], and J-Class A limits in [15].
Their dual description consist of F-theory compactified on ${\bf T}^{2}\times
X$, with ${\bf T}^{2}$ taken to a decompactification limit.

  3. 3.

${\bf k}_{a}=0$, $\forall a$, dubbed type IIb or $n=1$ limits in [14], and
J-Class B limits in [15]. Their dual description features a weakly-coupled,
asymptotically tensionless critical string [15].

The subindices in IIb, IIIc, and IVd correspond to non-negative integers that
classify subclasses of limits, and which for EFT string limits are determined
by

$r={\rm rank}\,\mathbf{K},\qquad\mathbf{K}_{bc}\equiv e^{a}{\cal K}_{abc}\,.$
(2.10)

In particular, it follows from the analysis in [14] that $b=d=r$, while
$c=r-2$. A physical interpretation for the subindex in IIb was given in [16],
in terms of towers of asymptotically massless states. In the following
sections we will see that this lesson is more general and all the subindices
are crucial to describe the set of towers of asymptotically massless states
for this kind of limits, as well as the asymptotic behaviour of the moduli
space scalar curvature.

Finally, the above classification of limits has a physical interpretation in
terms of 4d EFT string solutions [31, 32, 33]. In this setup, the relevant EFT
strings are made up of NS5-brane wrapping Nef divisors $e^{a}D_{a}$ dual to
$\ell_{s}^{-2}e^{a}[\omega_{a}]$ in $X_{6}$, and whose 4d backreaction
generates the saxion flow (2.8) towards their core. The 4d tension ${\cal T}$
of an EFT string varies as $\phi^{-1}$ along its own flow, and sets a
reference mass scale $\sqrt{\cal T}$ that can be compared with the scale
$m_{*}$ of the leading tower of asymptotically massless states. It was
proposed in [32, 33] that along 4d EFT string limits this relation is
universal and quite simple, namely of the form

$\frac{m_{*}^{2}}{M_{\rm P}^{2}}\sim\left(\frac{\cal T}{M_{\rm
P}^{2}}\right)^{w},\qquad w=1,2,3\,,$ (2.11)

where $w$ is dubbed the scaling index. In the setup at hand, one can easily
check that $m_{*}\simeq m_{\rm D0}$ is specified by the mass of a D0-brane and
that $w=n$ corresponds to the singularity type of each limit, as defined in
[12, 13, 14].

### The M-theory perspective

As stressed in [15], type IIA limits with scaling index $w=3$ have a simple
interpretation in terms of M-theory compactified on the Calabi–Yau $X$ [34,
35, 36], as they correspond to a particular point in the interior of the
resulting 5d vector multiplet moduli space. To see this, let us translate the
type IIA trajectory (2.8) in terms of M-theory compactified on $S^{1}\times
X$, assuming for simplicity a vanishing vev $b^{a}=0$ for the B-field axions.
From a 11d perspective, the flow of moduli reads

$2\pi R_{S^{1}}(\phi)=\frac{{\cal V}_{X}^{1/3}}{{\cal V}_{M}^{1/3}},\qquad
t^{a}_{M}(\phi)=t^{a}\frac{{\cal V}_{M}^{1/3}}{{\cal V}_{X}^{1/3}},$ (2.12)

with $R_{S^{1}}$ the $S^{1}$ radius and $t^{a}_{M}$ the Kähler moduli of $X$,
both measured in 11d Planck units $\ell_{11}$. Here ${\cal
V}_{M}=g_{s}^{-2}{\cal V}_{X}$ stands for the volume of $X$ in units of
$\ell_{11}$, a quantity that remains constant along the trajectory. Then, if
the $S^{1}$ radius is much larger than the internal sizes of $X$, it makes
sense to describe the trajectory from a 5d perspective:

$2\pi R_{5}(\phi)={\cal V}_{X}^{1/3},\qquad M^{a}(\phi)=\frac{t^{a}}{{\cal
V}_{X}^{1/3}},$ (2.13)

where $R_{5}$ is the $S^{1}$ radius measured in 5d Planck units
$\ell_{5}=\ell_{11}{\cal V}_{M}^{-1/3}$. The set $\\{M^{a}\\}$ is nothing but
the Kähler coordinates of $X$ normalised such that ${\cal
K}_{abc}M^{a}M^{b}M^{c}=6$, which is a necessary constraint to stay in the 5d
vector multiplet moduli space ${\cal M}_{\rm 5d}^{\rm VM}$ of M-theory
compactified on $X$. To parametrise ${\cal M}_{\rm 5d}^{\rm VM}$ one may
either drop this normalisation and see ${\cal M}_{\rm 5d}^{\rm VM}$ as a set
of real projective classes
$[\bm{t}_{M}]=\\{\lambda\bm{t}_{M}|\lambda\in\mathbb{R}\\}$, or instead define
a set of affine coordinates $\psi^{i}$, with $i=1,\dots,h^{1,1}(X)-1$. With
this second option, the bosonic action of this sector reads

$\displaystyle S_{\rm 5d}^{\rm VM}$
$\displaystyle=\frac{2\pi}{\ell_{5}^{3}}\int_{\mathbbm{R}^{1,4}}\left(R*\mathbbm{1}-2\mathfrak{g}_{ij}d\psi^{i}\wedge*d\psi^{j}\right)-\frac{1}{4\pi\ell_{5}}\int_{\mathbbm{R}^{1,4}}{\cal
I}_{ab}F^{a}\wedge*_{5}F^{b}$
$\displaystyle-\frac{1}{24\pi^{2}}\int_{\mathbbm{R}^{1,4}}{\cal
K}_{abc}\,A^{a}\wedge F^{b}\wedge F^{c}.$ (2.14)

Of particular interest to us will be the 5d gauge kinetic matrix ${\cal
I}_{ab}$, which has a simple expression in terms of the variables $M^{a}$

$\mathcal{I}_{ab}=\frac{1}{4}{\cal K}_{a}^{M}{\cal K}_{b}^{M}-{\cal
K}_{ab}^{M},$ (2.15)

where ${\cal K}^{M}_{a}={\cal K}_{abc}M^{b}M^{c}$ and ${\cal K}^{M}_{ab}={\cal
K}_{abc}M^{c}$. Notice that $\mathcal{I}_{ab}$ is nothing but the image of the
4d kinetic matrix $I_{ab}$ in (2.6) under the map $t^{a}\mapsto
M^{a}=t^{a}/{\cal V}_{X}^{1/3}$. Finally, the kinetic matrix for the scalars
can be expressed as
$\mathfrak{g}_{ij}=\mathcal{I}_{ab}{\partial}_{i}M^{a}{\partial}_{j}M^{b}$.

Asymptotically, the $S^{1}$ radius behaves as $R_{S^{1}}\sim\phi^{w/3}$, and
so we are naively led to a decompactification limit to 5d. This is clearly the
case for $w=3$ limits, because then ${\bf k}\neq 0$ and the Kähler trajectory
(2.13) in terms of the variables $M^{a}$ reads

$M^{a}(\phi)=\left(\frac{6}{{\bf
k}}\right)^{1/3}\left(e^{a}+\phi^{-1}\left(t_{0}^{a}-e^{a}\frac{{\bf
k}_{b}t_{0}^{b}}{{\bf k}}\right)+{\cal O}(\phi^{-2})\right).$ (2.16)

Thus, as $\phi\to\infty$ we reach a point in 5d vector moduli space that is
proportional to $\bm{e}$. Since all the components of $M^{a}(\phi=\infty)$ are
finite, this translates into a finite-distance point in 5d vector multiplet
moduli space ${\cal M}_{\rm 5d}^{\rm VM}$, as advanced above. If some of the
entries of $\bm{e}$ vanish, one is in particular led to a point in 5d moduli
space where some contractible curves or divisors in $X$ shrink, while the rest
remain of finite volume.111This is for instance the case for the so-called
elementary EFT string limits [32], that generate the Kähler cone of $X$, and
where only one component of $\bm{e}$ is non-vanishing and equal to one. In
this case one lands in a boundary of the Kähler cone ${\cal K}_{X}$ of $X$,
which may either be an actual finite-distance boundary of the vector moduli
space of the 5d EFT, or instead a point where a phase transition occurs [37].
In this second case the moduli space ${\cal M}_{{\rm 5d},X}^{\rm VM}$ is
extended by patching, through a flop transition, the Kähler cone of a Calabi-
Yau manifold $X^{\prime}$ that is birationally equivalent to $X$. The sum of
Kähler cones connected by flop transitions ${{\cal K}}_{\rm ext}=\cup_{X}{\cal
K}_{X}$ is called the extended Kähler cone, and it specifies the actual 5d
vector multiplet moduli space ${\cal M}_{\rm 5d}^{\rm VM}=\cup_{X}{\cal
M}_{{\rm 5d},X}^{\rm VM}$. Since these loci turn out to be crucial for our
analysis, let us stress that the shrinking of cycles only occurs in the
M-theory frame. In type IIA units all curves and divisors have large volumes
as long as the initial trajectory point $t_{0}^{a}$ is in the deep interior of
the Kähler cone of $X$. This also guarantees that $R_{S^{1}}^{2}\gg t_{a}^{M}$
at all points in (2.8), and it will be our working assumption in the
following.

Things are different for $w=2$ and $w=1$, limits, where ${\bf k}=0$. There,
the entries $e^{a}$ that are non-vanishing correspond to Kähler variables
$t^{a}_{M}$ or $M^{a}$ that tend to infinity as $\phi\to\infty$, while the
vanishing ones correspond to shrinking cycles. Following the discussion in
[15] it is easy to see that $w=2$ trajectories translate into J-Class A limits
in the vector moduli space of M-theory, that eventually lead to a further
decompactification to F-theory. Similarly, $w=1$ EFT string trajectories are
J-Class B M-theory limits that result in emergent string limits. Even if in
both cases the simplest microscopic description is not in terms of M-theory,
one can still use the dictionary (2.13) to translate the type IIA trajectory
(2.8) into an excursion in M-theory moduli space. In fact, by describing the
5d vector moduli space as a projective space ${\cal M}_{\rm 5d}^{\rm
VM}=\\{[\bm{t}_{M}]\\}$, one can map each 4d EFT string charge to a point in
${\cal M}_{\rm 5d}^{\rm VM}$ via $\bm{e}\mapsto[\bm{e}]$. In this
correspondence, EFT string charges $\mathbf{e}$ that implement $w=3$ limits
are mapped to finite-distance points in ${\cal M}_{\rm 5d}^{\rm VM}$,
including finite-distance boundaries. Charges that generate $w=2$ or $w=1$
limits are instead mapped to points in those boundaries of ${\cal M}_{\rm
5d}^{\rm VM}$ that lie at infinite distance in the 5d metric.

Incidentally, this M-theory perspective gives a specific realisation of the
Distant Axionic String Conjecture [31, 32, 33]. Using it one may identify
${\cal M}_{{\rm 5d},X}^{\rm VM}$, the 5d vector multiplet moduli space of
M-theory on $X$, with the endpoints of large-volume infinite-distance limits
in ${\cal M}_{{\rm 4d},X}^{\rm VM}$, the 4d vector moduli space of type IIA on
$X$. The 4d EFT string flow endpoints are a dense discrete subset in ${\cal
M}_{{\rm 5d},X}^{\rm VM}$, described by $[\bm{t}_{M}]=[\bm{e}]$. Hence, by
appropriately choosing an EFT string charge one may get arbitrarily close to
any large-volume infinite-distance limit endpoint.

In the next section we discuss how this dictionary is relevant to describe the
asymptotic behaviour of the 4d scalar curvature along type IIA large-volume
limits. As in turns out, in this regime the 4d scalar curvature has a very
simple expression in terms of 5d variables.

##  3 The scalar curvature at infinity

In this section we study the scalar curvature $R_{\rm IIA}$ of the vector
moduli space of the type IIA compactification (2.1), and in particular its
dependence along the limits (2.8). In the large-volume regime the 4d vector
multiplet moduli space metric can be approximated by (2.5), a tensor that also
determines the 5d vector multiplet Lagrangian of M-theory compactified in the
same Calabi–Yau, see (2.14). In fact, at large volumes $R_{\rm IIA}$ has a
natural expression in terms of M-theory quantities, a result that we will
exploit to classify its asymptotic behaviour along different limits.

In complex coordinates $(T^{a},\bar{T}^{\bar{a}})$, the Riemann curvature
associated to the metric (2.5) reads

$R_{a\bar{b}c\bar{d}}^{\rm cl}=g_{ab}g_{cd}+g_{ad}g_{bc}-\frac{9}{16{\cal
K}^{2}}g^{ef}{\cal K}_{ace}{\cal K}_{bdf}\,,$ (3.1)

an expression that follows from the general curvature formula of (local)
special Kähler manifolds [38, 23], applied to a region with axionic shift
symmetries like the one at hand [10, 39]. In particular, we are using the
large-volume approximation to replace
${\partial}_{a}{\partial}_{b}{\partial}_{c}\mathcal{F}\to{\partial}_{a}{\partial}_{b}{\partial}_{c}\mathcal{F}^{\rm
cl}=-\mathcal{K}_{abc}$.222This may lead to wrong curvature estimates when
some triple intersection numbers vanish, see section 4.2. From here one
deduces that the Ricci tensor is given by

$R_{a\bar{b}}^{\rm cl}=-g^{c\bar{d}}R_{a\bar{b}c\bar{d}}^{\rm
cl}=-g_{ab}(n_{V}+1)+\frac{9}{16{\cal K}^{2}}g^{cd}g^{ef}{\cal K}_{ace}{\cal
K}_{bdf}\,,$ (3.2)

and finally the scalar curvature reads

$R_{\rm IIA}^{\rm cl}/4=-n_{V}(n_{V}+1)+{\cal Y}^{2}\,,$ (3.3)

where $n_{V}=h^{1,1}(X)$ is the number of vector multiplets and

${\cal Y}^{2}=\frac{9}{16{\cal K}^{2}}g^{ab}g^{cd}g^{ef}{\cal K}_{ace}{\cal
K}_{bdf}\,.$ (3.4)

That is, we have a negative constant piece plus a moduli-dependent positive
piece ${\cal Y}^{2}$. This implies that in the large-volume region the scalar
curvature is bounded from below, while it does not need to be bounded from
above and it may even diverge positively. This divergent behaviour was indeed
observed in [10] by using a specific expression for the scalar curvature valid
for the case $n_{V}=3$, and then evaluating it in a particular asymptotic
region of Kähler moduli space. Since such divergences are the current source
of counterexamples to Conjecture 3 of [6], in the following we would like to
extend the analysis for general Calabi–Yau manifolds and arbitrary large-
volume limits. Our strategy will be to interpret ${\cal Y}^{2}$ in terms of
physical quantities, in order to identify the physical origin of the
divergences.

In an ${\cal N}=1$ setting one could interpret (3.4) in terms of physical
Yukawa couplings, while the analogous ${\cal N}=2$ quantities are dubbed Pauli
couplings and are related to the anomalous magnetic moment of the gauginos. To
describe their magnitude, it is natural to consider the axion-independent
gauge kinetic terms $\tilde{I}_{AB}\equiv-I_{AB}|_{b^{a}=0}$ considered in
[16], which read

$\tilde{I}_{00}=\frac{1}{6}{\cal K}\,,\qquad\tilde{I}_{ab}=\frac{2}{3}{\cal
K}g_{ab}\,,$ (3.5)

with all the remaining components vanishing. This tensor can either be thought
of as the gauge coupling of rotated gauge field strengths as in [16], or an
approximation of the gauge couplings in the large volume limit, since
$I^{ab}=-\tilde{I}^{ab}-b^{a}b^{b}/{\cal V}_{X}$ and so the axion dependence
disappears at large volume. In any case, in terms of this tensor we find

${\cal Y}^{2}={\cal V}_{X}\tilde{I}^{ab}\tilde{I}^{cd}\tilde{I}^{ef}{\cal
K}_{ace}{\cal K}_{bdf}\,,$ (3.6)

which amounts to the triple intersection numbers contracted with the gauge
kinetic terms, and multiplied by a volume factor. Recall that in the limits
under consideration ${\cal V}_{X}\sim\phi^{w}$ and so the volume factor is
always a source of divergence. However, we are also lead to a weakly-coupled
regime in which the components $\tilde{I}^{aa}\simeq g_{a}^{2}$ typically
asymptote as $\phi^{-1}$ or $\phi^{-2}$ [12, 16]. Therefore, depending on the
limit there may be or not a compensating effect that kills the divergence.

One way to generate a divergence for ${\cal Y}^{2}$ is to consider a setup in
which a certain gauge coupling $g_{{\sigma}}$ remains constant along (2.8),
and at the same time ${\cal K}_{{\sigma}{\sigma}{\sigma}}\neq 0$.
Geometrically, we are asking for the existence of a divisor that remains of
constant volume while ${\cal V}_{X}\to\infty$, or equivalently the existence
of a shrinkable divisor in $X$. Such divisors, more precisely generalised del
Pezzo surfaces, appear in Calabi–Yau compactifications where after a gravity-
decoupling limit one still recovers a non-trivial field theory at low energies
[25]. In 4d ${\cal N}=2$ vector multiplet settings, $M_{\rm P}\to\infty$
limits transform a local special Kähler geometry into a rigid special Kähler
one [23, 21], in which (roughly) the Kähler potential is no longer of the form
$K=-\log{\cal K}$, but instead $K=\frac{2}{3}{\cal K}$, and the scalar
curvature is positive definite and of the form ${\cal Y}^{2}/{\cal V}_{X}$.
All this suggests that, in moduli space regions where $R_{\rm IIA}^{\rm
cl}\simeq 4{\cal Y}^{2}$, there is a sector that can be decoupled from gravity
and that is dominating the contributions to the moduli space curvature.

The remainder of this section aims to make this intuition more precise and to
extract its consequences. As we will see, one can arrive to the same
conclusion by using two alternative pictures: the 5d M-theory viewpoint and
the 4d EFT perspective.

###  3.1 M-theory analysis

In section 4 we analyse the scalar curvature along large-volume limits using
(3.6). However, there is an alternative expression that allows us to build a
simple overall picture for the asymptotic behaviour of $R_{\rm IIA}$. Indeed,
notice that in terms of M-theory variables $M^{a}$ we have that

${\cal Y}^{2}=\mathcal{I}^{ab}\mathcal{I}^{cd}\mathcal{I}^{ef}{\cal
K}_{ace}{\cal K}_{bdf}\,,$ (3.7)

where $\mathcal{I}^{ab}$ is the inverse of (2.15). That is, the 4d type IIA
moduli space curvature at large volume is a simple function of the gauge
kinetic functions of the 5d M-theory compactification on $X$, and more
precisely of the physical 5d Chern-Simons couplings. Moreover, as explained in
the last section there is an injective map between the set of EFT string
limits (2.8) and the 5d vector moduli space seen as a projective space, via
$\bm{e}\mapsto[\bm{e}]\in{\cal M}_{\rm 5d}^{\rm VM}$. This implies that, in
order to find out along which limits the IIA curvature tends to infinity one
only needs to find out those loci in ${\cal M}_{\rm 5d}^{\rm VM}$ where ${\cal
Y}^{2}$ diverges. It is easy to see that this can only occur at loci where the
gauge kinetic matrix $\mathcal{I}_{ab}$ degenerates. Being a continuous
physical quantity that describes 5d gauge couplings, this can only happen at
the boundaries of ${\cal M}_{\rm 5d}^{\rm VM}$.

Indeed, let us consider M-theory compactified on $X$. From here we can
construct a 5d vector multiplet moduli space ${\cal M}_{{\rm 5d},X}^{\rm VM}$
that comes either from i) taking the projective quotient of the Kähler cone
${\cal K}_{X}$ under the identification $\bm{t}_{M}\sim\lambda\bm{t}_{M}$, or
ii) the set of coordinates $\\{M^{a}\\}$ where the projective equivalence is
fixed such that ${\cal K}_{abc}M^{a}M^{b}M^{c}=6$. The boundaries of ${\cal
M}_{{\rm 5d},X}^{\rm VM}$ were classified in [37] in terms of the facets of
${\cal K}_{X}$, and they have been recently revisited in the context of the
Swampland Programme in [40, 41, 42]. One finds four different kinds of
boundaries:

  * (i)

A locus where a curve ${\cal C}_{a}\subset X$ collapses to a point, but no
effective divisor collapses.

This always occurs at finite distance in moduli space, and it is not an actual
boundary of the full vector moduli space ${\cal M}_{{\rm 5d}}^{\rm VM}$.
Instead, one must adjoin a new Kähler cone ${\cal K}_{X^{\prime}}$ of a
birationally equivalent Calabi–Yau $X^{\prime}$ along this facet, via a flop
transition where the triple intersection numbers jump as ${\cal
K}_{abc}^{\prime}={\cal K}_{abc}-n^{0}_{{\cal C}_{a}}{\cal C}_{a}{\cal
C}_{b}{\cal C}_{c}$, with $n^{0}_{{\cal C}_{a}}$ the genus zero Gopakumar–Vafa
invariant of this class. By adjoining the Kähler cones of all birationally
equivalent Calabi–Yau manifolds one obtains the extended Kähler cone ${\cal
K}_{\rm ext}=\cup_{X}{\cal K}_{X}$, and from there the actual vector moduli
space ${\cal M}_{{\rm 5d}}^{\rm VM}=\cup_{X}{\cal M}_{{\rm 5d},X}^{\rm VM}$.

At a flop transition a finite number of hypermultiplets become massless, but
this does not affect the continuity of the 5d gauge kinetic matrix
$\mathcal{I}_{ab}$, which must remain non-degenerate as the flop occurs. Even
if the curvature may suffer a discontinuity due to the jump in triple
intersection numbers, ${\cal Y}^{2}$ must remain finite since
$\mathcal{I}^{ab}$ does not diverge.

  * (ii)

A locus where an effective divisor ${\cal D}_{\sigma}\subset X$ collapses to a
curve.

This is an actual boundary of ${\cal M}_{\rm 5d}^{\rm VM}$, that is also
reached at finite distance and where a gauge enhancement $U(1)\to SU(2)$
occurs in the vector multiplet sector. The physical origin of such an
enhancement is a charged BPS vector multiplet that becomes massless. Again, at
this point of enhancement the gauge kinetic matrix $\mathcal{I}_{ab}$ should
remain non-degenerate, which implies that ${\cal Y}^{2}$ stays finite.

  * (iii)

A locus where an effective divisor ${\cal D}_{\sigma}\subset X$ collapses to a
point.

This is also a finite-distance boundary of ${\cal M}_{\rm 5d}^{\rm VM}$, that
is associated with an infinite tower of BPS particles (M2-branes wrapping
curves of ${\cal D}_{\sigma}$) becoming massless, and a magnetic dual BPS
string (an M5-brane wrapping ${\cal D}_{\sigma}$) becoming tensionless. An
infinite tower of states becoming massless at finite distance in moduli space
is only possible together with a strong coupling regime and a non-Lagrangian
description of the theory, which in a gravity-decoupling regime becomes a non-
trivial 5d SCFT [43, 44, 45, 46, 47, 48].

The surfaces that realise this possibility are non-Nef effective divisors that
are generalised del Pezzo, just as in the setup described above where a
divergence for ${\cal Y}^{2}$ was generated. The only difference is that now
we are describing the system from an M-theory perspective, so the type IIA
finite coupling $g_{\sigma}$ becomes divergent in M-theory units. The
divergent 5d coupling and the non-trivial intersection numbers of ${\cal
D}_{\sigma}$ signal these boundaries of ${\cal M}_{\rm 5d}^{\rm VM}$ as loci
where ${\cal Y}^{2}$ diverges. Close to these loci ${\cal Y}^{2}$ is very
large, and so (3.3) is positive.

  * (iv)

A boundary at infinite distance.

These are the endpoints of infinite distance limits in ${\cal M}_{{\rm
5d}}^{\rm VM}$, that have been classified in[15] and recently reconsidered in
[49] from the viewpoint of Gopakumar–Vafa invariants. These limits also
feature a tower of asymptotically massless particles made up from M2-branes
wrapping shrinking curves, and may even include shrinking divisors. However,
the physics is very different from the previous case. The tower of states is
the one predicted by the SDC, and it is charged under a gauge symmetry with an
asymptotically vanishing coupling. Therefore, a priori there is no particular
reason to expect a divergence of ${\cal Y}^{2}$ at these loci.

From the type IIA viewpoint, these boundaries correspond to either $w=2$ or
$w=1$ limits, dual to 6d F-theory compactifications on $X$ and emergent
critical strings, respectively. As we will argue in section 4, in order for
${\cal Y}^{2}$ to diverge asymptotically one needs a divisor ${\cal
D}_{\sigma}$ that plays a similar role to generalised del Pezzo in $w=3$
limits.

To sum up, we find that divergences of the function ${\cal Y}^{2}$ are related
to a non-Nef divisor ${\cal D}_{\sigma}$ that shrinks compared to the Nef
divisors of the Calabi–Yau. For $w=3$ limits seen from an M-theory perspective
the divisor shrinks to a point and the corresponding gauge coupling
$g_{\sigma}^{M}$ tends to infinity, while an infinite tower of particles
becomes massless and their dual, non-critical string becomes tensionless. In
type IIA string units the divisor ${\cal D}_{\sigma}$ does not shrink, but it
is the Nef divisors that tend to infinite volume. Due to the 10d dilaton co-
scaling, we also have a tower of states that become massless. They come from
D2-branes wrapping curves inside ${\cal D}_{\sigma}$, a D4-brane wrapping
${\cal D}_{\sigma}$ and bound states between them and D0-branes. All these
states have a mass that scales like $m_{*}\simeq m_{\rm D0}$ and therefore,
due to the multiplicity of GV invariants, dominate the spectrum of particles
below the species scale of the 4d EFT, compared to the KK spectrum of
D0-branes. The non-critical string now arises from an NS5-brane wrapping
${\cal D}_{\sigma}$ and, since its tension goes as $\mathcal{T}_{\sigma}\sim
m_{*}^{2}$, its oscillation modes must be considered together with the D-brane
particle spectrum. Notice that in the IIA frame the gauge coupling
$g_{\sigma}$ associated to the shrinkable divisor remains constant along the
limit, and that it can be tuned to be arbitrarily weak by appropriately
choosing the initial point $\bm{t}_{0}$ of the trajectory. Therefore, there
should not be subtleties related to the stability of the tower of massive
charged states.

###  3.2 The 4d EFT perspective

The analysis in terms of M-theory data confirms our expectation that those
limits where the curvature diverges are related to shrinkable divisors of the
Calabi–Yau $X$. Shrinking cycles are also associated to gravity-decoupling
limits where, starting from a string compactification, one recovers a gauge
field theory without gravity [19, 20]. This procedure has been studied in
great detail for the vector multiplet moduli space of type II string theory
compactified on a Calabi–Yau [22, 23, 24, 25, 26, 27, 28], which is precisely
our setup. Essentially, by expanding the Calabi–Yau periods in powers of
$M_{\rm P}^{-1}$ one recovers the prepotential of a 4d ${\cal N}=2$ rigid
supersymmetric gauge theory, whose moduli space is rigid special Kähler [23].
While one could apply this standard procedure to our setup, in the following
we would like to implement a more pedestrian approach, along the lines of
[50], combined with the insight of the Distance Conjecture.

Indeed, let us consider an EFT string limit of the form (2.8), and consider
whether or not there is a non-trivial dynamical EFT at low energies. A general
property of these limits is that, if we restrict the trajectory to the vector
multiplet moduli space and so keep $M_{s}/M_{\rm P}$ constant, the lightest
tower of asymptotically massless states predicted by the SDC is the tower of
D0-branes, which sets the maximal cut-off scale $m_{*}$ of a 4d EFT

$m_{*}=\frac{m_{\rm D0}}{\sqrt{4\pi}}=\frac{\sqrt{\pi}}{\ell_{s}g_{s}}\,.$
(3.8)

where the relative numerical factor is chosen for convenience. If we have a
non-trivial 4d field theory along an infinite distance limit, one should be
able to express its action in units of $m_{*}$. Applied to our large-volume
EFT string limits we find that the corresponding expression for the 4d kinetic
terms of the scalars is

$m_{*}^{2}\int_{\mathbbm{R}^{1,3}}\tilde{I}_{ab}\,dT^{a}\wedge*d\bar{T}^{\bar{b}}\,.$
(3.9)

where the dimensionful prefactor accounts for the fact that the $T^{a}$ are
dimensionless fields. One can absorb this prefactor and the kinetic term into
canonically normalised dimensionful scalar fields. Upon normalisation, the
fact that some entries of $\tilde{I}_{ab}$ blow up along the limit
$\phi\to\infty$ translates into some scalar field whose vev diverge in units
of the cut-off $m_{*}$. Hence, following the discussion in [28], one sees that
they should decouple from the dynamics at energies below the cut-off $m_{*}$,
where a ${\cal N}=2$ rigid field theory is defined. In terms of (3.9), one can
select which terms should be kept in the rigid theory by doing an expansion of
$\tilde{I}_{ab}$ in powers of $\phi$, see eq.(4.4) and Appendix A. One deduces
that the scalar fluctuations that remain dynamical at the scale $m_{*}$ are
those directions $\bm{t}_{{\sigma}}$ that satisfy

$t^{a}_{\sigma}{\bf K}_{ab}=0\,\quad\forall b\,.$ (3.10)

That is, field directions that belong to $\ker{\bf K}$. There is an exception
to this rule, namely the direction $\bm{t}_{\bm{e}}=\bm{e}$ for a $w=1$ limit,
which belongs to $\ker{\bf K}$. Since this field also has a large vev upon
canonical normalisation, we do not include it in the rigid theory. Notice that
(3.9) recovers the relation between scalar kinetic terms and gauge kinetic
functions that is standard from ${\cal N}=2$ rigid supersymmetry. Accordingly,
one can deduce (3.10) in terms of the strength of the gauge interactions.
Gauge couplings that belong to $\ker{\bf K}$ stay constant along
$\phi\to\infty$, while others vanish and the corresponding gauge interactions
become negligible.

Having selected the set of directions that stay dynamical below $m_{*}$, one
can build the following low energy action

$S_{\rm 4d,rigid}^{\rm
VM}=m_{*}^{2}\int_{\mathbbm{R}^{1,3}}\tilde{I}_{{\sigma}\rho}dT^{\sigma}\wedge*d\bar{T}^{\bar{\rho}}+\frac{1}{2}\int_{\mathbbm{R}^{1,3}}I_{{\sigma}\rho}F^{\sigma}\wedge*_{4}F^{\rho}+R_{{\sigma}\rho}F^{\sigma}\wedge
F^{\rho},$ (3.11)

which indeed corresponds to the vector multiplet sector of the ${\cal N}=2$
field theory obtained from taking the rigid limit. The classical prepotential
for this theory is $\mathcal{F}^{\rm cl}_{\rm rigid}=-\frac{1}{6}{\cal
K}_{{\sigma}\rho\tau}T^{\sigma}T^{\rho}T^{\tau}+\dots$, where the dots
correspond to terms at most linear in the $T^{\sigma}$, that also depend on
the vev of the non-dynamical fields $T^{\Sigma}$ but not on $\phi$. The moduli
space scalar curvature is positive and reads

$R_{\rm rigid}^{\rm
cl}=\tilde{I}^{{\sigma}\rho}\tilde{I}^{\tau\eta}\tilde{I}^{\mu\nu}{\cal
K}_{{\sigma}\tau\mu}{\cal K}_{\rho\eta\nu}\,.$ (3.12)

When we look at this curvature in 4d Planck units we obtain

$4{\cal V}_{X}R_{\rm rigid}^{\rm cl}=4{\cal
V}_{X}\tilde{I}^{{\sigma}\rho}\tilde{I}^{\tau\eta}\tilde{I}^{\mu\nu}{\cal
K}_{{\sigma}\tau\mu}{\cal K}_{\rho\eta\nu}\simeq 4{\cal Y}^{2}\,,$ (3.13)

where in the last term we have used that all the remaining contributions to
${\cal Y}^{2}$ are subleading in powers of $\phi$. Using that (3.12) is
$\phi$-independent, we identify the source of curvature divergence along the
infinite distance limit $\phi\to\infty$ as a rigid gauge theory that survives
in the IR, with a non-trivial scalar curvature, see figure 1. One can
generalise this discussion to understand divergences in the Riemann curvature
tensor and sectional curvatures of the moduli space.

Figure 1: Representation of a moduli space curvature divergence. Along an
infinite distance limit there is a dynamical EFT with non-trivial curvature
below the cut-off $m_{*}$ and so, as a consequence of the SDC, it decouples
from gravity. Besides the SDC tower there is a tower charged under the rigid
EFT, with similar scaling.

Notice that in terms of the Kähler potential $K=-\log{\cal K}$ of the
classical metric, condition (3.10) splits the Kähler moduli in three sets:
$(\bm{t}_{0},\bm{t}_{I},\bm{t}_{II})$, where $\bm{t}_{0}=\bm{e}\phi$ is the
growing direction, $\bm{t}_{II}$ satisfy (3.10) and $\bm{t}_{I}$ are the
remaining moduli. This fits well with the general scheme for rigid limits
considered in [27], since one has a Kähler potential of the form

$K=-\log\left({\cal K}_{0}+{\cal K}_{II}+{\cal K}_{R}\right),$ (3.14)

where ${\cal K}_{0}$ contains all the $\phi$-dependent terms, ${\cal K}_{II}$
all the $\bm{t}_{II}$-dependent terms, and ${\cal K}_{R}$ the rest. Since
${\cal K}_{0}\simeq 6{\cal V}_{X}+{\cal O}(\phi^{0})$ dominates over the other
two terms one can perform the expansion

$K=-\log{\cal K}_{0}-\frac{{\cal K}_{II}}{{\cal K}_{0}}+\dots$ (3.15)

In terms of the fields $\bm{t}_{II}$ we obtain a constant term that does not
affect the metric, and the following Kähler potential for the rigid metric

$K_{\rm rigid}=-\frac{1}{6{\cal V}_{X}}{\cal K}_{II}\ \to\ K_{\rm
rigid}=-\frac{2}{3}{\cal K}_{II}\,.$ (3.16)

where in the last step we have performed a rescaling to measure the metric in
units of $m_{*}$. From here (3.12) follows.

To connect these results with our discussion of section 3.1 let us analyse the
geometric meaning behind the condition (3.10). Since one can build a basis for
$\ker{\bf K}$ made with vectors of integer entries, any element
$\bm{u}_{\sigma}\in\ker{\bf K}$ can be interpreted as a divisor class ${\cal
D}_{\sigma}$ of $X$. If ${\cal D}_{\bm{e}}$ is the Nef divisor that correspond
to the EFT string charge $\bm{e}$, then $\bm{u}_{\sigma}\in\ker{\bf K}$ simply
translates into the trivial intersection ${\cal D}_{\sigma}\cdot{\cal
D}_{\bm{e}}=0$. In $w=1$ limits one has one Nef divisor that meets this
condition, namely ${\cal D}_{\bm{e}}$, but for any other case ${\cal
D}_{\sigma}$ is necessarily non-Nef, because any pair of non-proportional Nef
divisors have a non-trivial intersection [15]. Moreover, whenever ${\cal
D}_{\sigma}$ is non-Nef and effective, then it is a shrinkable divisor,
because ${\rm Vol}({\cal D}_{\sigma})={\cal K}_{a}v_{\sigma}^{a}$ is constant
along the limit (2.8), while ${\cal V}_{X}\sim\phi^{w}$. Hence, one can
perform an overall rescaling of all Kähler moduli such that ${\cal V}_{X}$
stays constant and ${\rm Vol}({\cal D}_{\sigma})\to 0$. In fact, this is
precisely what happens when we map $t^{a}\mapsto M^{a}$ and we look at the
trajectory from the M-theory perspective of section 2.

The M-theory picture is particularly useful for $w=3$ limits, because then the
trajectory (2.8) becomes (2.16). In that frame ${\rm Vol}_{M}({\cal
D}_{\sigma})\to\phi^{-2}$ and the gauge kinetic matrix (2.15) becomes
degenerate as $\phi\to\infty$. Both statements indicate that ${\cal D}_{A}$ is
an effective generalised del Pezzo divisor contracting to a point along this
limit, in agreement with the discussion of section 3.1. Indeed, the fact that
$\mathcal{I}_{ab}$ degenerates shows that we are reaching a boundary of ${\cal
M}_{{\rm 5d},X}^{\rm VM}$, but it cannot be a flop transition nor a boundary
of $SU(2)$ enhancement, because the kinetic terms should be well-defined in
there, and since we are in a $w=3$ limit it cannot be an infinite distance
boundary either. So we conclude that, in the M-theory frame, as
$\phi\to\infty$ we reach a boundary ${\cal M}_{{\rm 5d}}^{\rm VM}$ of the type
(iii). This is indeed what happens in all the examples of $w=3$ limits
analysed in section 5, where any ${\cal D}_{\sigma}$ that lies in $\ker{\bf
K}$ is an effective generalised del Pezzo divisor.333It would be interesting
to verify this expectation in full generality, perhaps using the recent
approach in [42] to describe the cone of effective divisors in terms of GV
invariants.  As such, it has non-trivial triple intersection number, implying
a non-trivial scalar curvature (3.12) for the rigid field theory and a
divergence in the scalar curvature $R_{\rm IIA}$ measured in Planck units.

###  3.3 The curvature criterion

Notice that the discussion of section 3.2 is based on two simple ingredients.
The Distance Conjecture and the existence of a non-trivial rigid EFT below the
SDC cut-off $m_{*}$. The EFT is non-trivial in the sense that it has a non-
trivial field space with a non-vanishing curvature $R_{\rm rigid}$. If it
remains so along the infinite distance trajectory, then the curvature
divergence follows. This simple picture motivates the following proposal, that
should apply to moduli spaces of vacua of EFTs compatible with quantum
gravity:

######  Conjecture 1 (Curvature Criterion)

Along a geodesic trajectory of infinite distance, a moduli space scalar
curvature that diverges asymptotically indicates the presence of a field
theory sector that can be decoupled from gravity.

Essentially, we are proposing that the existence of a rigid field theory
subsector with a non-vanishing curvature is the only mechanism to generate a
curvature divergence along infinite distance limits. This condition is
important for the divergence to occur and, at least at the classical level,
not always satisfied.

The proposal is very suggestive from the viewpoint of the original Conjecture
3 of [6], because it preserves the intuition that dominant gravitational
effects should lead to a negative asymptotic curvature. Additionally, it may
lead to an interesting interpretation of the smoothing surgery prescription
proposed in [11], in which the Weil-Petersson metric of the vector multiplet
moduli space ${\cal M}_{\rm 4d}^{\rm VM}$ is replaced by the complete Hodge
metric on the same manifold. Since this replacement should remove the
curvature divergences, if our proposal is correct it should have an
interpretation in terms of any sector of the compactification that leads to a
rigid EFT. It is however not clear whether it should amount to integrate in
all the massive states charged under the rigid field theory sector, as
proposed in [11].

Indeed, the massive spectrum above the SDC cut-off $m_{*}$ that is charged
under the rigid sector is quite remarkable. On the one hand, we have the
towers of states predicted by the dual description of the theory, which scale
like $m_{*}$ along the limit, and that were accounted for in [15, 16]. On the
other hand, there are additional infinite towers of states that also scale
like $m_{*}$, but that are charged under the gauge sector that corresponds to
$\ker{\bf K}$. Such towers were described for $w=3$ limit at the end of
section 3.1, and they have the particularity that they are charged under gauge
couplings that are constant along the infinite distance limit, and therefore
they have a finite physical charge. Because the mass of their states scales
like $m_{*}\sim\phi^{-w/2}$, these towers have an extremality factor that
asymptotes to infinity. Notice that this indicates a regime where gauge
interactions dominate parametrically over gravitational ones, again in
agreement with the presence of the rigid theory below the SDC scale. When
going above $m_{*}$ for $w=3$ limits, we recover the strong coupling fixed
points of the 5d supersymmetric gauge theories with $E_{n}$ global symmetries
found in [43], which can be engineered via collapsing del Pezzo divisors in
M-theory and generalisations [44, 45, 46]. In this sense, the results in [51,
52, 53, 54] should be crucial in checking the proposal of [11] mentioned
above.

One may now consider infinite distance limits where the scalar curvature
becomes positive asymptotically, although this time bounded from above. In our
type IIA setup, these are for instance $w=3$ limits where $[\bm{e}]\in{\cal
M}_{\rm 5d}^{\rm VM}$ lies outside of a boundary of the type (iii), but close
to it. In this case all gauge couplings are asymptotically vanishing at the
same rate, but there is still a hierarchy between gauge sectors. The coupling
$g_{\sigma}$ that corresponds to a small del Pezzo divisor ${\cal D}_{\sigma}$
will be much stronger than a coupling related to any Nef divisor, and the
tower of states made of D2-branes wrapping curves of ${\cal D}_{\sigma}$ will
be much lighter than other D2-brane towers, dominating the massive spectrum
below the species scale. This again implies a gauge subsector that dominates
over the remaining gauge interactions and the gravitational ones, although not
parametrically like when $[\bm{e}]$ lies in the boundary of ${\cal M}_{\rm
5d}^{\rm VM}$. From the viewpoint of the 5d gauge theories of [43] the limit
specifies a point in the Coulomb branch away from the fixed point.

Interestingly, our results from sections 4 and 5 suggest that the only way to
obtain an asymptotically positive scalar curvature is to consider a limit that
is close to a divergent curvature limit, in agreement with the spirit of
Conjecture 1.

##  4 The curvature along different limits

In the following we analyse in more detail the dependence of the scalar
curvature (3.3) along the different large-volume EFT string limits (2.8), in
order to substantiate the picture presented in the last section. A first
question is whether or not the saxion-dependent quantity (3.6) tends to
infinity along a given limit. This can be addressed by diagonalising the gauge
kinetic matrix $\tilde{I}_{ab}$, and finding how each of its eigenvalues
depend on the parameter $\phi$. While in general
$\tilde{I}_{ab}=\frac{2}{3}{\cal K}g_{ab}$ leads to moduli-dependent
eigenvectors and eigenvalues, one can focus on their leading behaviour with
respect to $\phi$, which turns out to be specified by topological data. One
then obtains an approximate eigenvector expression

$u^{a}_{M}(\phi)=v^{a}_{M}+{\cal O}(\phi^{-1})\,,$ (4.1)

where $\bm{v}_{M}$ is $\phi$-independent and has integer entries, so it can be
identified with a divisor class ${\cal D}_{M}$ of $X$. When contracted with
the kinetic matrix, one can extract its asymptotic behaviour along this
particular direction

$\tilde{I}_{MM}\equiv\tilde{I}_{ab}v_{M}^{a}v_{M}^{b}\sim\phi^{w_{M}}\,,\qquad
w_{M}\in\mathbb{Z}\,,$ (4.2)

which captures the leading asymptotic behaviour of the corresponding
eigenvalue. Let us call $w_{M}$ the weight of the vector $v_{M}$. Plugged back
into (3.6) one finds that

${\cal Y}^{2}\sim\sum_{M,N,L}\left(v_{M}^{a}v_{N}^{b}v_{L}^{c}{\cal
K}_{abc}\right)^{2}\phi^{w-w_{M}-w_{N}-w_{L}}\,,$ (4.3)

from where one can determine if the curvature diverges or not. The details of
this computation are described in Appendix A, and it turns out that the result
essentially depends on the vector $\bm{e}$ specifying the limit and its
contractions with ${\cal K}_{abc}$. Therefore the calculation varies for each
class $w=1,2,3$ of limits, as we discuss below.

Since $\tilde{I}_{ab}$ is a Hodge star metric, one can give a more accurate
description of its asymptotic diagonalisation by using the same techniques as
in [55, 56]. This method is more effective the less degenerate the limit is,
in the sense of [13] and [32]. As discussed in Appendix B, limits with scaling
weight $w=3$ are maximally degenerate, those with $w=2$ are partially
degenerate and $w=1$ corresponds to non-degenerate limits. We thus apply the
mixed Hodge structure approach of [55, 56] to $w=2$ and $w=1$ limits.
Remarkably, we find that in a large fraction of limits the (classical) scalar
curvature is asymptotically negative and has a universal behaviour.

###  4.1 M-theory limits

We first analyse type IIA limits with scaling weight $w=3$, which we also dub
as M-theory limits. As explained around (2.16), they correspond to a
decompactification limit to M-theory on $X$, and their endpoints have a
particularly simple description in terms of M-theory variables.

Following the strategy described above, we consider the diagonalisation of the
kinetic matrix $\tilde{I}_{ab}=\frac{2}{3}{\cal K}g_{ab}$ along a $w=3$ limit
(2.8). For this we expand the elements that define the metric in powers of
$\phi$

$\begin{split}&{\cal K}={\bf k}\phi^{3}+3{\bf k}_{a}t^{a}_{0}\phi^{2}+3{\bf
K}_{ab}t^{a}_{0}t^{b}_{0}\phi+{\cal K}_{abc}t^{a}_{0}t^{b}_{0}t^{c}_{0}\,,\\\
&{\cal K}_{a}={\bf k}_{a}\phi^{2}+2{\bf K}_{ab}t^{b}_{0}\phi+{\cal
K}_{abc}t^{b}_{0}t^{c}_{0}\,,\\\ &{\cal K}_{ab}={\bf K}_{ab}\phi+{\cal
K}_{abc}t^{c}_{0}\,,\end{split}$ (4.4)

where ${\bf k}\equiv{\cal K}_{abc}e^{a}e^{b}e^{c}$, ${\bf k}_{a}\equiv{\cal
K}_{abc}e^{b}e^{c}$, and ${\bf K}_{ab}={\cal K}_{abc}e^{c}$ have been defined
as in section 2. From the results of Appendix A one can see that there are
three classes of leading vectors in the expansion (4.1), which can be defined
as

$\bm{v}_{0}\in\ker{\bf K}\,,\qquad v_{I}^{a}{\bf k}_{a}=0\text{ but
}\bm{v}_{I}\notin\ker{\bf K}\,,\qquad\bm{v}_{e}=\bm{e}\,,$ (4.5)

and that the weight (4.2) for each kind of eigenvector is

$w_{0}=0,\qquad w_{I}=1,\qquad w_{e}=1.$ (4.6)

Notice that a vector of the form $\bm{v}_{I}$ can be shifted by one in the
class $\bm{v}_{0}$ without spoiling its definition, but as long as
$w_{0}<w_{I}$ the weights are well-defined. Also, the the weight $w_{e}=w-2$
follows from the general properties of EFT string, that constrain its tension
${\cal T}_{e}=3e^{a}{\cal K}_{a}/2{\cal K}$ to asymptote like $\phi^{-1}$
along its own EFT string flow [32]. It follows from (4.6) that we can only
have a divergent term in (4.3) whenever a vector of the form $\bm{v}_{0}$ is
involved, which implies that the matrix ${\bf K}$ defined in (2.10) must be
singular. In the language of [14], this means that along limits of the form
IV${}_{n_{V}}$ the curvature does not diverge, but along those of the type
IV${}_{d<n_{V}}$ it will, provided that the elements within $\ker{\bf K}$ have
a non-vanishing triple intersection. This is the case for generalised del
Pezzo divisors, which at the end of section 3.1 were identified with the
condition $\bm{v}_{0}\in\ker{\bf K}$. Such divisors define a rigid field
theory with classical prepotential

$\mathcal{F}_{\rm rigid}^{\rm
cl}=-\frac{1}{6}\mathcal{K}_{\alpha\beta{\gamma}}T^{\alpha}T^{\beta}T^{\gamma}-\frac{1}{6}\mathcal{K}_{\alpha\beta\Gamma}T^{\alpha}T^{\beta}T_{0}^{\Gamma}-\frac{1}{6}\mathcal{K}_{\alpha\Sigma\Gamma}T^{\alpha}T_{0}^{\Sigma}T_{0}^{\Gamma}-\frac{1}{6}\mathcal{K}_{\Lambda\Sigma\Gamma}T_{0}^{\Lambda}T_{0}^{\Sigma}T_{0}^{\Gamma}\,,$
(4.7)

where ${\cal D}_{\alpha}$ form an integer basis of $\ker{\bf K}$, and ${\cal
D}_{\Gamma}$ form a basis for the vectors of the form $\bm{v}_{I}$. Some
examples of this rigid prepotential are worked out in section 5 and in [50,
section 4].

One can also see in a more direct manner that the curvature does not diverge
when $\ker{\bf K}$ is trivial. Indeed, let us consider the explicit expression
for the inverse of $\tilde{I}_{ab}$

$\tilde{I}^{ab}=3\frac{t^{a}t^{b}}{\mathcal{K}}-\mathcal{K}^{ab}\,,$ (4.8)

where ${\cal K}^{ab}$ is the inverse of ${\cal K}_{ab}$. Plugging this
expression into (3.6) we obtain

${\cal Y}^{2}=\frac{3}{2}n_{V}-{\cal V}_{X}{\cal K}^{ab}{\cal K}^{cd}{\cal
K}^{ef}{\cal K}_{ace}{\cal K}_{bdf}\,.$ (4.9)

If ${\bf K}_{ab}$ is invertible with inverse ${\bf K}^{ab}$, then we have that

${\cal K}^{ab}={\bf K}^{ab}\phi^{-1}+{\cal O}(\phi^{-2})\,,$ (4.10)

and so

${\cal Y}^{2}=\frac{3}{2}n_{V}-{\cal V}_{X}\phi^{-3}{\bf K}^{ab}{\bf
K}^{cd}{\bf K}^{ef}{\cal K}_{ace}{\cal K}_{bdf}+{\cal O}(\phi^{w-4})\,.$
(4.11)

The second term scales like $\phi^{w-3}$ and so it will only asymptote to a
non-vanishing constant for $w=3$ limits. Because ${\bf K}$ is the intersection
matrix of curves within the divisor ${\cal D}_{\bm{e}}$, it has signature
$(1,r-1)$, so most of its eigenvalues are negative while its trace is a
positive integer. Therefore, whenever ${\bf K}$ has a very small eigenvalue
$\lambda_{\sigma}$ it must be negative, and one can estimate the second term
in (4.11) by $\frac{\bf k}{6}|\lambda_{\sigma}|^{-3}$. One expects that ${\cal
Y}^{2}$ is asymptotically finite but large when the vector $\bm{e}$ is close
to a vector for which ${\bf K}$ is singular. To make this intuition more
precise let us rewrite (4.11) as

${\cal Y}^{2}=\frac{3}{2}n_{V}-{\bf K}_{M}^{ab}{\bf K}_{M}^{cd}{\bf
K}_{M}^{ef}{\cal K}_{ace}{\cal K}_{bdf}+{\cal O}(\phi^{w-4})\,,$ (4.12)

where ${\bf K}_{M}^{ab}$ is the inverse of ${\bf K}^{M}_{ab}\equiv(6/{\bf
k})^{1/3}{\cal K}_{abc}e^{c}$. The eigenvalues of ${\bf K}^{M}_{ab}$ depend
continuously on the M-theory parametrisation (2.16), and so ${\cal Y}^{2}$
will be large when $[\bm{e}]\in{\cal M}_{\rm 5d}^{\rm VM}$ is close in the
M-theory moduli space metric to a singularity in ${\bf K}_{M}^{ab}$. This
locus coincides with a singularity of ${\cal I}_{ab}$, or in other words with
a boundary of type (iii), as expected from our previous discussion.

###  4.2 F-theory limits

Let us consider $w=2$ limits, that correspond to decompactification limits to
F-theory on $X$, focusing first on the case of a smooth elliptic fibration.
Following the convention in [14, section 3.4] and [16, section 4], we take a
Kähler cone basis $\omega_{a}=\\{\omega_{E},\omega_{\alpha}\\}$ where
$\omega_{\alpha}=\pi^{*}\omega^{\prime}_{\alpha}$ is the pull-back of a
simplicial Kähler cone basis of $B_{2}$. Their triple intersection numbers are

${\cal K}_{EEE}=\eta_{\alpha\beta}c_{1}^{\alpha}c_{1}^{\beta},\qquad{\cal
K}_{EE\alpha}=\eta_{\alpha\beta}c_{1}^{\beta},\qquad{\cal
K}_{E\alpha\beta}=\eta_{\alpha\beta},\qquad{\cal K}_{\alpha\beta{\gamma}}=0,$
(4.13)

where $c_{1}(B_{2})=c_{1}^{\alpha}\omega^{\prime}_{\alpha}$ and
$\eta_{\alpha\beta}$ is a symmetric matrix with signature
$(1,h^{1,1}(B_{2})-1)$. We now obtain a $w=2$ limit by taking a vector
$\bm{e}$ of the form $(0,e^{\alpha})$, $e^{\alpha}\in\mathbb{N}$. Then, one
can again consider the expansion (4.4), but now since ${\bf k}=0$ and

${\bf k}_{a}=\eta_{\alpha\beta}e^{\alpha}e^{\beta}\,\delta_{aE},\qquad{\bf
K}=\begin{pmatrix}\eta_{\alpha\beta}e^{\alpha}c_{1}^{\beta}&\eta_{\alpha\beta}e^{\beta}\\\
\eta_{\alpha\beta}e^{\beta}&0\end{pmatrix},$ (4.14)

the classes of leading eigenvectors in (4.1) read

$\bm{v}_{0}\in\ker{\bf
K}=\\{(0,f^{\beta})\,|\,\eta_{\alpha\beta}e^{\alpha}f^{\beta}=0\\}\,,\qquad\bm{v}_{e}=\bm{e}=(0,e^{\alpha})\,,\qquad\bm{v}_{E}=(1,\vec{0})\,,$
(4.15)

see Appendix A. Working out the metric expansion one also deduces that

$w_{0}=0\,,\qquad w_{e}=0\,,\qquad w_{E}=2\,,$ (4.16)

so naively one could have divergent terms in the sum (4.3), if only vectors of
the form $\bm{v}_{0}$ and $\bm{v}_{e}$ are involved. However, because all non-
vanishing triple intersection numbers involve at least one vector in
$\bm{v}_{E}$, one cannot build a divergent term, and so ${\cal Y}^{2}$ must be
asymptotically constant.

It turns out that for smooth elliptic fibrations one cannot only show that
${\cal Y}^{2}$ does not diverge along $w=2$ limits, but also that (3.3)
asymptotes to a specific negative value. To see this, let us apply the
approach used in [55, 56] to the present setup. Given the triple intersection
numbers (4.13), we have that the Kähler potential $K=-\log(\frac{4}{3}{\cal
K})$ is determined by

${\cal
K}=\eta_{\alpha\beta}c_{1}^{\alpha}c_{1}^{\beta}t_{E}^{3}+3\eta_{\alpha\beta}c_{1}^{\alpha}t^{\beta}t_{E}^{2}+3\eta_{\alpha\beta}t^{\alpha}t^{\beta}t^{E}.$
(4.17)

Following the philosophy of [55, 56], we consider the following limit

$t^{\alpha}=t^{\alpha}_{0}+\phi^{\delta}+e^{\alpha}\phi,\qquad
t^{E}=t^{E}_{0},\qquad{\delta}\in(0,1).$ (4.18)

This represents a growth sector in which the Kähler direction along $\bm{e}$
grows much faster than any other direction of the base, which in turn grow
faster than the Kähler coordinate $t^{E}$ that measures the area of the
elliptic fibre and that it pairs up with $\bm{v}_{E}$ in the Kähler form. The
mixed Hodge structure approach used in [55, 56] allow us to keep an
approximate form of the Kähler potential to describe the metric along this
growth sector, namely

$K=-\log\left(4\eta_{\alpha\beta}t^{\alpha}t^{\beta}t^{E}+\dots\right)\,,$
(4.19)

where we have neglected subleading terms in $\phi$. The metric that one
derives from this approximate expression reads

$g=\begin{pmatrix}\frac{1}{4(t^{E})^{2}}\\\
&\frac{\eta_{\alpha}\eta_{\beta}}{\eta^{2}}-\frac{\eta_{\alpha\beta}}{2\eta}\end{pmatrix}\,,$
(4.20)

where $\eta_{\alpha}=\eta_{\alpha\beta}t^{\beta}$ and
$\eta=\eta_{\alpha\beta}t^{\alpha}t^{\beta}$. We therefore find a gauge
kinetic matrix of the form

$\tilde{I}=\begin{pmatrix}\frac{\eta}{2t^{E}}\\\
&t^{E}\left(2\frac{\eta_{\alpha}\eta_{\beta}}{\eta}-\eta_{\alpha\beta}\right)\end{pmatrix},$
(4.21)

up to subleading terms. Remarkably, the result is independent of the value of
$\delta$ and, because $\eta_{\alpha\beta}$ is non-degenerate, reproduces the
weights (4.16). The inverse matrix at leading order reads

$\tilde{I}^{-1}=\begin{pmatrix}\frac{2t^{E}}{\eta}\\\
&\frac{1}{t^{E}}\left(2\frac{t^{\alpha}t^{\beta}}{\eta}-\eta^{\alpha\beta}\right)\end{pmatrix},$
(4.22)

matching the results of Appendix A. From here we obtain

${\cal Y}^{2}=3(n_{V}-1)+{\cal O}(\phi^{-1})\,.$ (4.23)

As a result, we find a scalar curvature with the negative asymptotic value

$R_{\rm IIA}^{\rm cl}/4\ =-n_{V}^{2}+2n_{V}-3+{\cal O}(\phi^{-1})\,.$ (4.24)

Now, recall that (4.24) is the curvature associated to the classical
prepotential $\mathcal{F}^{\rm cl}$. The fact that it is negative corresponds
to a rigid theory with no curvature at the classical level. Indeed, in the
case at hand, the rigid ${\cal N}=2$ theory has as dynamical fields the Kähler
variables $\tilde{T}^{\beta}$ that correspond to the vectors in $\bm{v}_{0}$.
By construction, these are divisors which are pullbacks from curves of the
base with negative definite self-intersection matrix
$\tilde{\eta}_{\alpha\beta}$. The classical prepotential of the rigid theory
reads

$\mathcal{F}_{\rm rigid}^{\rm
cl}=\frac{1}{2}\tilde{\eta}_{\alpha\beta}\tilde{T}^{\alpha}\tilde{T}^{\beta}T_{0}^{E}+\dots$
(4.25)

where the dots contain at most linear terms on the dynamical fields. Since
there is no cubic term, from (3.12) we have that $R_{\rm rigid}^{\rm cl}=0$,
in agreement with the fact that (4.25) corresponds to a flat classical metric
$g_{\alpha\beta}^{\rm cl}=T_{0}^{E}\tilde{\eta}_{\alpha\beta}$. Polynomial
curvature corrections in (2.7) will not change this result, but exponential
world-sheet corrections could do so. Indeed, whenever $\ker{\bf K}$ we have a
set of vertical divisors ${\cal D}_{\beta}$ that do not intersect ${\cal
D}_{e}$. Projected to the base $B_{2}$, this lead to a set of curves ${\cal
C}_{\beta}=\pi({\cal D}_{\beta})$ that do not intersect $\pi({\cal D}_{e})$ in
the base. As such they must have negative self-intersection and a finite
number of nonzero genus zero GV invariants that will contribute to (2.7).
Unless all GV invariants vanish, this will lead to terms that are independent
of $\phi$ if we take $\delta=0$ in (4.18). Even if exponentially suppressed
for large values of $t^{\beta}_{0}$, they will give a non-vanishing,
$\phi$-independent contribution to
${\partial}_{\beta}{\partial}_{\gamma}{\partial}_{\delta}\mathcal{F}$, where
all indices belong to $\ker{\bf K}$. Therefore, when $\ker{\bf K}$ is non-
trivial and its projection to the base leads to curves ${\cal C}_{\beta}$ with
non-vanishing GV invariants, we expect to have a curvature divergence along
$w=2$ EFT string limits (2.8), sourced by quantum corrections in the rigid
field theory.444It may seem surprising that world-sheet instanton corrections
can modify the large-volume asymptotic curvature, since under the assumption
$t_{0}^{a}\gg 1$ one can neglect them to compute the metric. Note however that
the curvature also depends on derivatives of the metric, and that (2.5) has
plenty of axionic isometries. Only world-sheet instanton terms break such
isometries, so they are indeed the ones that are able to correct the
curvature. It would be important to test this expectation explicitly and, if
true, to understand it from an F-theory viewpoint.

Let us now consider non-smooth elliptic fibrations, whose triple intersection
numbers do not correspond to (4.13). In the case that $r=$ rank ${\bf
K}=n_{V}$, we can use (4.11) to deduce that

$R_{\rm IIA}^{\rm cl}/4\ =-n_{V}^{2}+\frac{1}{2}n_{V}+{\cal O}(\phi^{-1})\,.$
(4.26)

The remaining cases are $w=2$ limits which, in the nomenclature used in [14],
correspond to type IIIc limits with $0<c<n_{V}-2$, and so ${\bf K}$ has a rank
larger than 2. The classes of leading eigenvectors (4.1) that generalise
(4.14) are

$\bm{v}_{0}\in\ker{\bf K}\,,\qquad\bm{v}_{e}=\bm{e}\,,\qquad{v}_{I}^{a}{\bf
k}_{a}=0\ \text{but}\ \bm{v}_{I}\notin\ker{\bf
K}\oplus\langle\bm{e}\rangle\,,\qquad{v}_{E}^{a}{\bf k}_{a}\neq 0\,.$ (4.27)

As in (4.5), the split between these classes of vectors is not unique, but the
weights are well defined as long as $w_{0},w_{e}<w_{I}<w_{E}$, which should be
the case. More precisely we expect

$w_{0}=0\,,\qquad w_{e}=0\,,\qquad w_{I}=1\,,\qquad w_{E}=2\,.$ (4.28)

To see this notice that the vectors $\bm{v}_{0}$, $\bm{v}_{e}$ represent
vertical divisors as in the smooth fibration case, and $w_{0}=w_{e}=0$ follows
from general arguments. The vector $\bm{v}_{E}$ again represents the dual
divisor to the generic fibre, and so again $w_{E}=2$ is expected. Finally, the
vectors of the form $\bm{v}_{I}$ represent the divisors that arise from fibral
degenerations, and their weight is motivated by examples that we analyse in
section 5, although it would be nice to confirm its value in general.

With this set of weights one cannot have asymptotic divergences for ${\cal
Y}^{2}$, even for this more general class of $w=2$ limits. Indeed, the
intersection of two vertical divisors is always proportional to the generic
fibre, and as a result in the sum (4.3) two vectors of the form $\bm{v}_{0}$
or $\bm{v}_{e}$ should always combine with a vector $\bm{v}_{E}$, as in the
smooth case. Additionally, we can have one vector $\bm{v}_{0}$ or $\bm{v}_{e}$
coupling to two vectors $\bm{v}_{I}$, but this combination also kills the
potential divergence. Therefore, we expect an asymptotic behaviour for ${\cal
Y}^{2}$ that is analogous to the smooth fibration and maximal rank cases. A
natural guess for this value is

${\cal Y}^{2}=3(n_{V}-r/2)+{\cal O}(\phi^{-1})\,.$ (4.29)

It would be interesting to test this guess with explicit examples. Again,
whenever $r<n_{V}$ it could be that world-sheet instanton corrections generate
a non trivial $R_{\rm rigid}$, and therefore a divergence for the asymptotic
scalar curvature, if the appropriate GV invariants are non-trivial.

Finally, let us consider the tower of states that are charged under the gauge
group of the rigid theory. In $w=2$ EFT string limits we have two universal
towers of asymptotically massless states that combine into a lattice. These
are D0-branes, D2-branes wrapping the generic fibre, and bound states of them.
The gauge kinetic matrix of this lattice blows up along the limit, signalling
a decompactification limit to F-theory. However, as pointed out in [16,
section 4] there can be a larger lattice of light states that tend to zero
mass as fast as the D0-brane tower. To see this, let us consider an effective
vertical divisor ${\cal D}_{\sigma}$ of $X$ that corresponds to a vector of
the form $\bm{v}_{0}$ in (4.15). If we set $\delta=0$ in (4.18), the volume of
this divisor will remain constant as $\phi\to\infty$, and so the DBI action of
a D4-brane wrapped around such divisor only depends on $\phi$ through the 10d
dilaton dependence, just like D0-branes. The same is true if we add an Abelian
worldvolume flux ${\cal F}$ threading the internal D4-brane worldvolume.
Scanning over the worldvolume flux quanta, we generate an infinite lattice of
D4/D2/D0 BPS bound states. Such a lattice is charged under $U(1)$’s with
constant gauge couplings, and so we again have a tower of particles with an
extremality factor ${\gamma}$ that asymptotes to infinity. This infinite
lattice disappears once that we analyse the system from the viewpoint of
F-theory compactified on $X$, in contrast with the towers with
${\gamma}\to\infty$ of $w=3$ limits in the M-theory picture.

###  4.3 Emergent string limits

Let us finally consider $w=1$ limits, that should correspond to emergent
string limits. In this case we have that ${\bf k}={\bf k}_{a}=0$ and

${\bf K}=\begin{pmatrix}0&0\\\ 0&Q_{pq}\end{pmatrix},$ (4.30)

where the first entry corresponds to $\bm{e}$. Geometrically, $\bm{e}$
represents a Nef divisor class ${\cal D}_{e}$ whose topology is either that of
a $K3$ or a $T^{4}$, and $Q$ determines the lattice dual to
$\Lambda^{\vee}=\iota^{*}H^{2}(X_{6},\mathbbm{Z})$, where $\iota$ is the
embedding map of such surface into $X_{6}$ [15]. Using that all non-
proportional Nef divisors should intersect, one sees that the quadratic form
${\bf Q}$ is such that all Kähler moduli besides $t^{a}\propto e^{a}$ appear
in $Q_{pq}t^{p}t^{q}$, although this does not mean that $r={\rm rank}\,{\bf
Q}$ has the maximal value $n_{V}-1$. In the classification of [14] these are
limits of type IIr.

To describe the asymptotic behaviour of the metric it is useful to consider
the same classes of eigenvectors as in [16, eq.(5.14)]:

$\bm{v}_{e}=\bm{e}\,,\qquad\bm{v}_{0}\in\ker{\bf
K}\,,\qquad\bm{v}_{I}\notin\ker{\bf K}\,,$ (4.31)

where the $\bm{v}_{0}$’s are linearly independent from $\bm{e}$. Then from
expanding the gauge kinetic matrix in powers of $\phi$ one obtains that

$w_{e}=-1,\qquad w_{0}=0,\qquad w_{I}=1,$ (4.32)

in agreement with the results of [16]. Again, the value of $w_{e}$ is fixed by
EFT string asymptotic behaviour ${\cal T}_{e}\sim\phi^{-1}$ along its own
limit. While this negative power could be a naive source of divergences for
${\cal Y}^{2}$, the structure of triple intersection numbers implies that each
non-vanishing element of the sum (4.3) that contains $\bm{v}_{e}$ must also
contain two vectors $\bm{v}_{I}$, and so such a combination cannot be a source
of divergence. One may worry that quantum corrections can modify this result,
but quantum-corrected cubic terms that involve the vector $\bm{e}$ have an
exponential suppression of the form $\exp(-2\pi k\phi)$, and so they become
irrelevant at $\phi\to\infty$. A linear divergence for the scalar curvature
only arises if we have vectors or the form $\bm{v}_{0}$ with non-trivial
triple intersection numbers, which requires to consider limits with
$r<n_{V}-1$.

If $\ker{\bf K}=\langle\bm{e}\rangle$ (i.e., $r=n_{V}-1$) one can get a
precise expression for the asymptotic value of the scalar curvature. As shown
in [32] and Appendix B, $w=1$ limits are non-degenerate, which makes them
particularly suitable for applying the mixed Hodge structure techniques that
we applied to $w=2$ limits. In this case the limit (2.8) corresponds to the
growth sector $t\gg\tilde{t}^{p}$, where $t$ corresponds to the Kähler modulus
along $\bm{e}$ and $\tilde{t}^{p}$ to the remaining ones, among which no
growth hierarchy is needed to pursue the analysis. The approximate Kähler
potential reads

$K=-\log\left(4tQ_{pq}\tilde{t}^{p}\tilde{t}^{q}+\dots\right)\,,$ (4.33)

where the terms that we are neglecting are $\phi$-independent. Due to the
similarity with (4.19) everything works as in the smooth-fibration $w=2$
limit, except that now it could be that ${\bf Q}$ has a non-trivial kernel. We
find that

$\tilde{I}=\begin{pmatrix}\frac{Q}{2t}\\\
&t\left(2\frac{Q_{p}Q_{q}}{Q}-Q_{pq}\right)\end{pmatrix}\sim\begin{pmatrix}\phi^{-1}\\\
&\phi\\\ &&\ddots\end{pmatrix},$ (4.34)

where $Q_{p}=Q_{pq}\tilde{t}^{q}$ and $Q=Q_{pq}\tilde{t}^{p}\tilde{t}^{q}$.
The dots represent saxion directions in the kernel of ${\bf Q}$, which do not
scale with $\phi$. We therefore reproduce the results in (4.32), and find
again that the elements of $\ker{\bf Q}$ are the only possible sources of
divergence for ${\cal Y}^{2}$. Notice moreover that when $\ker{\bf Q}$ is
trivial the leading kinetic matrix $\tilde{I}$ is completely analogous to that
in (4.21), in the sense that its inverse looks like (4.22), with the
replacement $\bm{\eta}\to{\bf Q}$. The computation of ${\cal Y}^{2}$ works
exactly the same as in that case, and at the end one obtains an asymptotically
constant and negative curvature of the form (4.24) for all these type of
limits.

###  4.4 Summary

Let us summarise the results of this section, which are gathered in table 1.

Case | $R_{\rm IIA}^{\rm cl}(\phi\to\infty)$ | $R_{\rm IIA}(\phi\to\infty)$
---|---|---
$w$=3 | $r=n_{V}$ | $-4n_{V}^{2}+2n_{V}+\mathfrak{C}$ | $-4n_{V}^{2}+2n_{V}+\mathfrak{C}+\mathcal{O}(e^{-t_{0}^{\Sigma}})$
$r<n_{V}$ | $R_{\rm rigid}^{\rm cl}\frac{2}{3}\bf{k}\phi^{3}$ | $\left(R_{\rm rigid}^{\rm cl}+\mathcal{O}(e^{-t_{0}^{\sigma}})\right)\frac{2}{3}\bf{k}\phi^{3}$
$w$=2 | $r=n_{V}$ | $-4n_{V}^{2}+2n_{V}$ | $-4n_{V}^{2}+2n_{V}+\mathcal{O}(e^{-t_{0}^{\Sigma}})$
$r<n_{V}$, smooth | $-4(n_{V}^{2}-2n_{V}+3)$ | $\left[\mathcal{O}(e^{-t_{0}^{\sigma}}){\bf k}_{\Sigma}t^{\Sigma}_{0}\phi^{2}\right]^{*}$
$r<n_{V}$, non-smooth | $\left[-4n_{V}^{2}+8n_{V}-6r\right]^{*}$ | $\left[\mathcal{O}(e^{-t_{0}^{\sigma}}){\bf k}_{\Sigma}t^{\Sigma}_{0}\phi^{2}\right]^{*}$
$w$=1 | $r=n_{V}-1$ | $-4(n_{V}^{2}-2n_{V}+3)$ | $-4(n_{V}^{2}-2n_{V}+3)+\mathcal{O}(e^{-t_{0}^{\Sigma}})$
$r<n_{V}-1$ | $R_{\rm rigid}^{\rm cl}2{\bf K}_{\Sigma\Lambda}t^{\Sigma}_{0}t^{\Lambda}_{0}\phi$ | $\left(R_{\rm rigid}^{\rm cl}+\mathcal{O}(e^{-t_{0}^{\sigma}})\right)2{\bf K}_{\Sigma\Lambda}t^{\Sigma}_{0}t^{\Lambda}_{0}\phi$
Table 1: Leading asymptotic behaviour of the moduli space scalar curvature
along different type IIA vector multiplet limits (2.8). Here $R_{\rm IIA}^{\rm
cl}$ corresponds to (3.3), the curvature of the large-volume approximate
metric (2.5), $R_{\rm IIA}$ to its quantum-corrected version and $R^{\rm
cl}_{\rm rigid}$ to (3.12). The rest of the quantities are topological data
defined as $n_{V}=h^{1,1}(X)$, ${\bf k}\equiv{\cal K}_{abc}e^{a}e^{b}e^{c}$,
${\bf k}_{a}\equiv{\cal K}_{abc}e^{b}e^{c}$, ${\bf K}_{ab}={\cal
K}_{abc}e^{c}$ and $r={\rm rank}\,{\bf K}$. The $t^{\sigma}$ are dynamical
saxions in the rigid theory while $t^{\Sigma}$ are frozen. Finally, the
entries marked as $[\dots]^{*}$ are the result of guesswork.

The general pattern is that the scalar curvature does not diverge along the
EFT string limits (2.8) when the rank of the matrix ${\bf K}_{ab}={\cal
K}_{abc}e^{c}$ is maximal, while it does so as ${\cal V}_{X}\sim\phi^{w}$ for
all the other cases. The physical source of divergence is the curvature
$R_{\rm rigid}$ of the rigid field theory below the SDC scale $m_{*}\simeq
m_{\rm D0}$. Such a curvature vanishes at the classical level for $w=2$ limits
but, barring the absence of GV invariants in the rigid sector, it is expected
to be present when world-sheet instanton corrections are taken into account.

If the scalar curvature does not diverge along the limit it is asymptotically
negative, with the exception of $w=3$ limits with $r=n_{V}$, where the moduli-
independent quantity

$\mathfrak{C}=-\frac{\bf k}{6}{\bf K}^{ab}{\bf K}^{cd}{\bf K}^{ef}{\cal
K}_{ace}{\cal K}_{bdf}$ (4.35)

can be arbitrarily large and positive. As explained in section 3.1 this
happens precisely when the vector $\bm{e}$, interpreted as a point in the
M-theory moduli space ${\cal M}_{\rm 5d}^{\rm VM}$, is close to a boundary
where the gauge kinetic function $\mathcal{I}_{ab}$ degenerates and the
curvature $R_{\rm IIA}$ diverges. In practice, this large value for (4.35) is
attained when ${\bf K}$ has a hierarchy of eigenvalues. Physically, this
corresponds to a hierarchy between gauge couplings in the vector multiplet
sector, and in particular some couplings that are much larger than the one of
the graviphoton. The quotient of couplings does however not grow as
$\phi\to\infty$. That would correspond to the case where we recover a rigid
${\cal N}=2$ theory below $m_{*}$ for any value of $\phi$, and the curvature
blows up.

Let us stress that the content of table 1 is only valid for limits of the form
(2.8) under the assumption $t_{0}^{a}\gg 1$. For other infinite distance
limits there may be other ways different from a non-trivial $\ker{\bf K}$ in
which one can decouple a subset of the moduli from the trajectory of infinite
distance, and one should define the rigid field theory accordingly.555In other
words, since we are close to an M-theory regime, table 1 reflects the
available mechanisms for M-theory compactified on $X$ to engineer rigid field
theories. Other limits will correspond to additional type II engineerings. A
clear example are $w=1$ limits that lead to Seiberg-Witten points, which have
been recently studied in [15, 57]. As pointed out in [57], in the region in
which the limit is defined one typically has to add some world-sheet instanton
corrections in order for the metric not to be degenerate, and this
automatically implements a hierarchy between different moduli. It would be
very interesting to check if Conjecture 1 is valid along these and other
limits of the vector multiplet moduli space.

Finally, note that a positive scalar curvature does not tell us anything about
the sign of the sectional curvatures. Indeed, it was shown in [32] that along
EFT string limits at least one holomorphic sectional curvature must be
negative asymptotically. In principle one can study the asymptotic behaviour
of sectional curvatures and the Ricci tensor by considering (3.1) and (3.2)
and applying similar techniques to the ones used in this section. It would be
interesting to see which pattern of asymptotic sectional curvatures allows to
have a divergent scalar curvature and at the same time a moduli space of
finite volume, as it is the case for the type IIA vector moduli space with
Weil-Petersson metric [7, 8, 9]. We leave these questions for future work.

##  5 Examples

In this section we work out some examples that illustrate our discussion and
results of the previous sections. On each example we describe the asymptotic
behaviour of the type IIA scalar curvature $R_{\rm IIA}$ in terms of the
associated M-theory vector moduli space. For those limits where the curvature
blows up, we reproduce the divergence from the curvature of the associated
rigid theory. Our examples correspond to Calabi–Yau manifolds with two or
three moduli, but in some cases the Kähler cone needs to be extended to
construct the full set of limits. As discussed in section 3, curvature
divergences can appear for vectors $\bm{e}$ that lie in the boundary of the
(extended) Kähler cone. In these examples with few moduli, they oftentimes
corresponds to generators of the cone, and more precisely to elementary
limits, in the sense of footnote 1.

#### Two-moduli example

Let us first consider a two-modulus example based on the Calabi–Yau manifold
$X=\mathbb{P}^{(1,1,1,6,9)}[18]$, which was studied in [58] and can either be
regarded as a weighted projective space or as an smooth elliptic fibration
over $\mathbb{P}^{2}$. This manifold has two Kähler moduli and a simplicial
Kähler cone spanned by two Nef divisors ${\cal D}_{1}$ and ${\cal D}_{2}$. In
this basis the Kähler form reads

$J=t^{1}\omega_{1}+t^{2}\omega_{2}\,,$ (5.1)

where $[\omega_{i}]$ is the Poincaré dual class to ${\cal D}_{i}$ and
$t^{i}\geq 0$. The triple intersection numbers are

$\mathcal{K}_{111}=9\,,\qquad\mathcal{K}_{112}=3\,,\qquad\mathcal{K}_{122}=1\,,\qquad\mathcal{K}_{222}=0\,.$
(5.2)

The different discrete cones relevant to infinite distance large-volume limits
have been discussed in [31, section 4.2]. In particular the cone of effective
divisors is given by

$\mathcal{E}=\\{(e^{1},e^{2})\in\mathbb{Z}^{2}|e^{1}\geq 0,3e^{1}+e^{2}\geq
0\\}\,,$ (5.3)

generated by the classes ${\cal D}_{2}$ and ${\cal D}_{B}={\cal D}_{1}-3{\cal
D}_{2}$, where the latter corresponds to the ${\mathbb{P}^{2}}$ base.

The asymptotic classical scalar curvature $R^{\rm cl}_{\rm IIA}$ along the
M-theory moduli space

$\displaystyle\mathcal{M}_{{\rm 5d},X}^{\rm VM}$
$\displaystyle=\left\\{(M^{1},M^{2})\ |\ M^{i}\geq 0\quad\text{and}\quad
9(M^{1})^{3}+9(M^{1})^{2}M^{2}+3M^{1}(M^{2})^{2}=6\right\\}$ (5.4)

is depicted in figure 2. One observes a divergence at
$(M^{1},M^{2})=(\left(\frac{2}{3}\right)^{1/3},0)$, which corresponds to the
elementary limit $\bm{e}=(1,0)$ and an asymptotic negative value at
$(M^{1},M^{2})=(0,\infty)$, which corresponds to the elementary limit
$\bm{e}=(0,1)$. Let us analyse both of these limits separately.

Figure 2:

  * •

Limit $t^{1}\to\infty$ or $\bm{e}=(1,0)$: this is a $w=3$ limit with ${\bf
k}=9$. The matrix

${\bf K}=\begin{pmatrix}9&3\\\ 3&1\end{pmatrix}$ (5.5)

has rank one and a kernel generated by $\bm{v}_{0}=(1,-3)$, that corresponds
to the effective divisor class of the base ${\cal D}_{B}$. In the basis
$\\{{\cal D}_{e},-{\cal D}_{B}\\}$ the triple intersection numbers read666The
sign flip in the effective non-Nef divisor corresponds to the fact that, from
the viewpoint of the local Calabi–Yau geometry, $-{\cal D}_{B}$ becomes Nef.
See [59] for a discussion in the context of M-theory CY compactifications.

$\mathcal{K}_{eee}=9\,,\qquad\mathcal{K}_{BBB}=-9\,,$ (5.6)

and the Kähler form is $J=t^{e}\omega_{e}+t^{B}\omega_{B}$, where the rotated
moduli are $(t^{e},t^{B})=(t^{1}+\frac{1}{3}t^{2},\frac{1}{3}t^{2})$. One can
also compute the rigid prepotential (4.7)

$\displaystyle\mathcal{F}_{\rm rigid}^{\rm
cl}=\frac{3}{2}\left(T^{B}\right)^{3}\,,$ (5.7)

and see that $T^{B}=\frac{1}{3}T^{2}$ is the only field in the rigid field
theory limit. The metric of this theory reads $g_{B\bar{B}}=9t^{B}=3t^{2}$,
which leads to the curvature

$R_{\rm rigid}=\frac{3}{(t^{2})^{3}}\,.$ (5.8)

Using that ${\cal V}_{X}\sim\frac{3}{2}\phi^{3}$ along this limit it is easy
to see that (3.13) reproduces the asymptotic value of the 4d supergravity
moduli space classical scalar curvature

$R_{\rm IIA}^{\rm
cl}=\frac{18}{\left(t_{0}^{2}\right)^{3}}\phi^{3}+\mathcal{O}(\phi^{2})\,.$
(5.9)

  * •

Limit $t^{2}\to\infty$ or $\bm{e}=(0,1)$: this is a $w=2$ limit where the
matrix

${\bf K}=\begin{pmatrix}3&1\\\ 1&0\end{pmatrix},$ (5.10)

has maximal rank. As such, there is no rigid theory in the IR and no scalar
curvature divergence. The scalar curvature of the 4d supergravity moduli space
asymptotes to a negative constant, and more precisely to

$R_{\rm IIA}^{\rm cl}=-12+\mathcal{O}(\phi^{-2})\,,$ (5.11)

in agreement with the results of section 4.

#### Two moduli with a flop

We now consider a Calabi–Yau three-fold $X$ with two Kähler moduli, that was
also studied in [42]. $X$ can be described as a hypersurface embedded in a
toric fourfold constructed from a reflexive polytope and it admits a flop
transition to a birationally equivalent Calabi–Yau $X^{\prime}$. Given a basis
of divisors $\\{\mathcal{D}_{1},\mathcal{D}_{2}\\}$, we can expand the Kähler
form as

$J=t^{1}\omega_{1}+t^{2}\omega_{2}\,,$ (5.12)

and the (simplicial) extended Kähler cone, given by the union of the Kähler
cones of the two phases, reads ${\cal K}_{\rm ext}=\\{(t^{1},t^{2})\ |\
5t^{1}+t^{2}\geq 0,t^{2}\geq 0\\}$. In the first phase $X$ the given divisor
basis is Nef and the triple intersection numbers read

$\mathcal{K}_{111}=5\,,\qquad\mathcal{K}_{112}=5\,,\qquad\mathcal{K}_{122}=5\,,\qquad\mathcal{K}_{222}=3\,,$
(5.13)

while in the phase $X^{\prime}$ they become

$\mathcal{K}_{111}=-15\,,\qquad\mathcal{K}_{112}=5\,,\qquad\mathcal{K}_{122}=5\,,\qquad\mathcal{K}_{222}=3\,.$
(5.14)

One can perform the change of coordinates

$y^{1}=-t^{1}\,,\qquad y^{2}=5t^{1}+t^{2}\,,$ (5.15)

which amounts to take as basis of divisors
$\mathcal{D}^{\prime}_{1}=-\mathcal{D}_{1}+5\mathcal{D}_{2}$ and
$\mathcal{D}^{\prime}_{2}=\mathcal{D}_{2}$, and get the following triple
intersection numbers

$\mathcal{K}^{\prime}_{111}=90\,,\qquad\mathcal{K}^{\prime}_{112}=30\,,\qquad\mathcal{K}^{\prime}_{122}=10\,,\qquad\mathcal{K}^{\prime}_{222}=3\,.$
(5.16)

The cone of effective divisors is given by

$\mathcal{E}=\\{(e^{1},e^{2})\in\mathbb{Z}^{2}\ |\ e^{1}+e^{2}\geq
0,2e^{1}+e^{2}\geq 0\\}\,,$ (5.17)

generated by $\mathcal{D}_{A}=\mathcal{D}_{1}-\mathcal{D}_{2}$ and
$\mathcal{D}_{B}=-\mathcal{D}_{1}+2\mathcal{D}_{2}$. The M-theory moduli space
is described by

$\begin{split}\mathcal{M}_{{\rm 5d},X}^{\rm VM}=\Bigl{\\{}(M^{1},M^{2})\ |\
M^{i}\geq 0\quad\text{and}\quad
5(M^{1})^{3}&+15(M^{1})^{2}M^{2}+15M^{1}(M^{2})^{2}+3(M^{2})^{3}=6\Bigr{\\}}\\\
&\cup\\\ \Bigl{\\{}(M^{1},M^{2})\ |\ M^{1}\leq
0,5M^{1}+M^{2}\geq&\,\,0\quad\text{and}\\\
-15(M^{1})^{3}+&15(M^{1})^{2}M^{2}+15M^{1}(M^{2})^{2}+3(M^{2})^{3}=6\Bigr{\\}}\end{split}$
(5.18)

and the classical scalar curvature $R_{\rm IIA}^{\rm cl}$ along it is shown in
figure 3.

Figure 3:

One can detect divergences at
$(M^{1},M^{2})=\left(\left(\frac{6}{5}\right)^{\frac{1}{3}},0\right)$ and
$(M^{1},M^{2})=\left(\frac{6}{110}\right)^{\frac{1}{3}}\left(-1,5\right)$,
which respectively correspond to the limits $\bm{e}=(1,0)$ in the original
variables $t^{a}$ and $\bm{e}^{\prime}=(1,0)$ in the new variables $y^{a}$. On
the other hand, the curvature reaches a negative constant value along the
limits $\bm{e}=(0,1)$ and $\bm{e}^{\prime}=(0,1)$, that correspond to
approaching the facet where the flop transition occurs from the two sides.
When going from one side to the other, the metric is continuous, but there is
a finite jump in the scalar curvature, due to the change in the triple
intersection numbers (5.14). Let us analyse each elementary limit.

  * •

Limit $t^{1}\rightarrow\infty$ or $\bm{e}=(1,0)$: this is a $w=3$ limit with
${\bf k}=5$, the matrix

${\bf K}=\begin{pmatrix}5&5\\\ 5&5\end{pmatrix}$ (5.19)

has rank one and a kernel generated by $\bm{v}_{0}=(1,-1)$, that corresponds
to the divisor class $\mathcal{D}_{A}$. Going to the new basis
$\\{\mathcal{D}_{e},-\mathcal{D}_{A}\\}$ we get

$\mathcal{K}_{eee}=5\,,\qquad\mathcal{K}_{AAA}=-2\,,$ (5.20)

with the rotated moduli $(t^{e},t^{A})=(t^{1}+t^{2},t^{2})$. In this limit the
rigid prepotential reads

$\mathcal{F}_{\rm rigid}^{\rm cl}=\frac{1}{3}\left(T^{A}\right)^{3}\,,$ (5.21)

where $T^{A}=T^{2}$ is the only dynamical field in the rigid theory. From this
one can compute the metric $g_{A\bar{A}}=2t^{2}$ and the curvature

$R_{\rm rigid}=\frac{1}{2(t^{2})^{3}}\,,$ (5.22)

that multiplied by $4\mathcal{V}_{X}\sim\frac{10}{3}\phi^{3}$ reproduces the
leading term of the type IIA classical moduli space curvature

$R_{\rm IIA}^{\rm
cl}=\frac{5}{3(t_{0}^{2})^{3}}\phi^{3}+\mathcal{O}(\phi^{2})\,.$ (5.23)

  * •

Limit $t^{2}\rightarrow\infty$ or $\bm{e}=(0,1)$: this is a $w=3$ limit where

${\bf K}=\begin{pmatrix}5&5\\\ 5&3\end{pmatrix},$ (5.24)

has maximal rank. As a consequence, there is no rigid theory in the IR and the
scalar curvature stays finite and it asymptotically tends to a negative
constant, namely

$R_{\rm IIA}^{\rm cl}=-\frac{57}{5}+\mathcal{O}(\phi^{-1})\,,$ (5.25)

in agreement with the results of section 4. However, as previously commented,
this is not a real boundary of the M-theory moduli space, as crossing it gives
rise to a flop transition and takes us to the second phase $X^{\prime}$.

  * •

Limit $y^{1}\rightarrow\infty$ or $\bm{e}^{\prime}=(1,0)$: this is a $w=3$
limit with ${\bf k}^{\prime}=90$, the matrix

${\bf K}^{\prime}=\begin{pmatrix}90&30\\\ 30&10\end{pmatrix},$ (5.26)

has rank one and a kernel generated by $\bm{v}^{\prime}_{0}=(1,-3)$ – or
$\bm{v}_{0}=(-1,2)$ in the original coordinates – that corresponds to the
divisor class $\mathcal{D}_{B}$. In the new basis
$\\{\mathcal{D}_{e},-\mathcal{D}_{B}\\}$, we get

$\mathcal{K}_{eee}=90\,,\qquad\mathcal{K}_{BBB}=-9\,,$ (5.27)

with the rotated moduli
$(t^{e},t^{B})=(y^{1}+\frac{1}{3}y^{2},\frac{1}{3}y^{2})$. One can then
compute the rigid prepotential

$\mathcal{F}_{\rm rigid}^{\rm cl}=\frac{3}{2}\left(T^{B}\right)^{3}\,,$ (5.28)

the metric $g_{B\bar{B}}=3y^{2}$ and finally the rigid scalar curvature

$R_{\rm rigid}=\frac{3}{(y^{2})^{3}}\,.$ (5.29)

From this, taking into account that $\mathcal{V}_{X}\sim 15\phi^{3}$, one can
reproduce the scaling of

$R_{\rm IIA}^{\rm
cl}=\frac{180}{(y_{0}^{2})^{3}}\phi^{3}+\mathcal{O}(\phi^{2})\,.$ (5.30)

  * •

Limit $y^{2}\rightarrow\infty$ or $\bm{e}^{\prime}=(0,1)$: this is a $w=3$
limit where

${\bf K}=\begin{pmatrix}30&10\\\ 10&3\end{pmatrix},$ (5.31)

has maximal rank. It follows that there is no rigid theory in the IR and the
scalar curvature tends to a negative constant, more precisely

$R_{\rm IIA}^{\rm cl}=-\frac{33}{5}+\mathcal{O}(\phi^{-1})\,.$ (5.32)

As one can see, approaching the flop transition in the $X^{\prime}$ phase
gives a scalar curvature that differs by a finite constant from the curvature
in the $X$ phase (5.25).

### Three-moduli example

We consider the CY three-fold $\mathbb{P}^{5}_{1,1,2,8,12}[24]$, also studied
in [20, 15], which can be seen as a K3 fibration over $\mathbb{P}^{1}_{b}$,
where the fibre K3 is itself an elliptic fibration over $\mathbb{P}^{1}_{f}$.
$X$ has three Kähler moduli and a simplicial Kähler cone generated by a basis
of Nef divisors $\mathcal{D}_{1},\mathcal{D}_{2}$ and $\mathcal{D}_{3}$. The
Kähler form can be expanded as

$J=t^{1}\omega_{1}+t^{2}\omega_{2}+t^{3}\omega_{3}\,,$ (5.33)

with $t^{a}\geq 0$, and the triple intersection numbers are

$\mathcal{K}_{122}=2\,,\qquad\mathcal{K}_{123}=1\,,\qquad\mathcal{K}_{222}=8\,,\qquad\mathcal{K}_{223}=4\,,\qquad\mathcal{K}_{233}=2\,.$
(5.34)

The cone of effective divisors is given by

$\mathcal{E}=\left\\{(e^{1},e^{2},e^{3})\in\mathbb{Z}^{3}\ |\ e^{2}\geq
0,2e^{2}+e^{3}\geq 0,e^{1}+4e^{2}+2e^{3}\geq 0\right\\}\,,$ (5.35)

and is generated by the divisor classes $\mathcal{D}_{A}=\mathcal{D}_{1}$,
$\mathcal{D}_{B}=\mathcal{D}_{2}-2\mathcal{D}_{3}$ and
$\mathcal{D}_{C}=-2\mathcal{D}_{1}+\mathcal{D}_{3}$, where geometrically
$\mathcal{D}_{A}$ and $\mathcal{D}_{B}$ correspond to the K3 fibre and the
Hirzebruch surface $\mathbb{F}_{2}$ that is the base of $X$ seen as an
elliptic fibration, respectively. Here the 5d M-theory moduli space

$\begin{split}\mathcal{M}_{{\rm 5d},X}^{\rm
VM}=\Bigl{\\{}(&M^{1},M^{2},M^{3})\ |\ M^{i}\geq 0\quad\text{and}\\\
&6M^{1}(M^{2})^{2}+6M^{1}M^{2}M^{3}+8(M^{2})^{3}+12(M^{2})^{2}M^{3}+6M^{2}(M^{3})^{2}=6\Bigr{\\}}\end{split}$
(5.36)

is two-dimensional. By looking at the classical scalar curvature along this
moduli space, we find that it asymptotes to a negative constant at the points
$(M^{1},M^{2},M^{3})=(\infty,0,0)$ and $(M^{1},M^{2},M^{3})=(0,0,\infty)$,
corresponding respectively to the limits $\bm{e}=(1,0,0)$ and
$\bm{e}=(0,0,1)$, and the same happens for any positive linear combination of
these vectors of charges. On the other hand, it diverges positively at the
point $(M^{1},M^{2},M^{3})=(0,\left(\frac{3}{4}\right)^{\frac{1}{3}},0)$,
corresponding to the limit $\bm{e}=(0,1,0)$. Let us see more in detail the
elementary limits in this case.

  * •

Limit $t^{1}\rightarrow\infty$ or $\bm{e}=(1,0,0)$: this is a $w=1$ limit, the
matrix

${\bf K}=\begin{pmatrix}0&0&0\\\ 0&2&1\\\ 0&1&0\end{pmatrix}$ (5.37)

has rank two and a kernel generated by the EFT string limit direction
$\bm{e}$. As a consequence, there is no rigid IR theory and the type IIA
classical scalar curvature goes like

$R_{\rm IIA}^{\rm cl}=-24+\mathcal{O}(\phi^{-1})\,.$ (5.38)

  * •

Limit $t^{2}\rightarrow\infty$ or $\bm{e}=(0,1,0)$: this is a $w=3$ limit with
${\bf k}=8$, the matrix

${\bf K}=\begin{pmatrix}0&2&1\\\ 2&8&4\\\ 1&4&2\end{pmatrix},$ (5.39)

has rank two and its kernel is generated by $\bm{v}_{0}=(0,1,-2)$, which
corresponds to the divisor class $\mathcal{D}_{B}=\mathbb{F}_{2}$. To
construct a basis, we follow (4.5) and we pick the vector
$\bm{v}_{I}=(-2,0,1)$, corresponding to the effective divisor class
$\mathcal{D}_{C}$. In the new basis
$\\{\mathcal{D}_{e},\mathcal{D}_{C},-\mathcal{D}_{B}\\}$ the triple
intersection numbers become

$\mathcal{K}_{eee}=8\,,\qquad\mathcal{K}_{eCC}=-2\,,\qquad\mathcal{K}_{CCB}=2\,,\qquad\mathcal{K}_{BBB}=-8\,,$
(5.40)

and we have the rigid prepotential

$\mathcal{F}_{\rm rigid}^{\rm
cl}=\frac{4}{3}\left(T^{B}\right)^{3}-(T_{0}^{C})^{2}T^{B}\,,$ (5.41)

where $T^{B}=\frac{1}{4}(T^{1}+2T^{3})$ is the only dynamical field in the
rigid theory. From here we compute the metric $g_{B\bar{B}}=8t^{B}$ and the
curvature

$R_{\rm rigid}=\frac{8}{(t^{1}+2t^{3})^{3}}\,,$ (5.42)

that multiplied by $4\mathcal{V}_{X}\sim\frac{16}{3}\phi^{3}$ reproduces

$R_{\rm IIA}^{\rm
cl}=\frac{128}{3\left(t_{0}^{1}+2t_{0}^{3}\right)^{3}}\phi^{3}+\mathcal{O}(\phi^{2})\,.$
(5.43)

  * •

Limit $t^{3}\rightarrow\infty$ or $\bm{e}=(0,0,1)$: this is a $w=2$ limit, the
matrix

${\bf K}=\begin{pmatrix}0&1&0\\\ 1&4&2\\\ 0&2&0\end{pmatrix},$ (5.44)

has rank two and the triple intersection numbers respect the structure of a
smooth elliptic fibration (4.13). The kernel is generated by
$\bm{v}_{0}=(-2,0,1)$, namely the divisor class $\mathcal{D}_{C}$. To complete
our new basis, see (4.14), we take the vector $\bm{v}_{E}=(0,1,0)$ that
corresponds to ${\cal D}_{E}={\cal D}_{2}$. In the basis
$\\{\mathcal{D}_{2},\mathcal{D}_{e},-\mathcal{D}_{C}\\}$ the triple
intersection numbers are

$\mathcal{K}_{222}=8\,,\qquad\mathcal{K}_{22e}=4\,,\qquad\mathcal{K}_{2ee}=2\,,\qquad\mathcal{K}_{2CC}=-2\,,$
(5.45)

and the rigid prepotential

$\mathcal{F}_{\rm rigid}^{\rm cl}=T_{0}^{2}\left(T^{C}\right)^{2}\,,$ (5.46)

from which we get $g_{C\bar{C}}=2t_{0}^{2}$ and $R_{\rm rigid}=0$, in
agreement with an asymptotically constant classical curvature in the 4d EFT,
see (4.24),

$R_{\rm IIA}^{\rm cl}=-24+\mathcal{O}(\phi^{-2})\,.$ (5.47)

### Three moduli with divergent emergent string limit

Then we consider a CY that is a non-smooth elliptic fibration over a
$\mathbb{P}^{2}$ base, which was studied in [60]. This CY has three Kähler
moduli and a simplicial Kähler cone. In a basis of Nef divisor we have the
triple intersection numbers

$\begin{split}&\mathcal{K}_{122}=18\,,\qquad\mathcal{K}_{123}=6\,,\qquad\mathcal{K}_{133}=2\,,\\\
&\mathcal{K}_{222}=63\,,\qquad\mathcal{K}_{223}=15\,,\qquad\mathcal{K}_{233}=3\,.\end{split}$
(5.48)

By looking at the behaviour of the classical moduli space scalar curvature
along the 5d M-theory moduli space we observe the following. There is a cubic
divergence at the point
$(M^{1},M^{2},M^{3})=(0,\left(\frac{2}{21}\right)^{\frac{1}{3}},0)$,
corresponding to the limit $\bm{e}=(0,1,0)$, and a linear divergence at
$(M^{1},M^{2},M^{3})=(\infty,0,0)$, corresponding to $\bm{e}=(1,0,0)$. For
positive linear combinations of these two vectors we get a cubic divergence as
well. Finally, at $(M^{1},M^{2},M^{3})=(0,0,\infty)$, namely $\bm{e}=(0,0,1)$,
the curvature is a negative constant. Let us analyse these three elementary
limits.

  * •

Limit $t^{1}\rightarrow\infty$ or $\bm{e}=(1,0,0)$: this is a $w=1$ limit, the
matrix

${\bf K}=\begin{pmatrix}0&0&0\\\ 0&18&6\\\ 0&6&2\end{pmatrix},$ (5.49)

has rank one and a null eigenvector besides $\bm{e}$, namely
$\bm{v}_{0}=(0,1,-3)$, which corresponds to the divisor class
$\mathcal{D}_{A}=\mathcal{D}_{2}-3\mathcal{D}_{3}$. To construct a basis we
pick the vector $\bm{v}_{I}=(0,1,0)$, that represents the divisor class
$\mathcal{D}_{2}$, see (4.31). In the new basis
$\\{\mathcal{D}_{e},\mathcal{D}_{2},-\mathcal{D}_{A}\\}$ the triple
intersection numbers are

$\mathcal{K}_{e22}=18\,,\qquad\mathcal{K}_{222}=63\,,\qquad\mathcal{K}_{22A}=-18\,,\qquad\mathcal{K}_{AAA}=-9\,,$
(5.50)

and the rigid prepotential

$\displaystyle\mathcal{F}_{\rm rigid}^{\rm
cl}=\frac{3}{2}\left(T^{A}\right)^{3}+9\left(T^{2}_{0}\right)^{2}T^{A}\,,$
(5.51)

where $T^{A}=\frac{1}{3}T^{3}$ is the only dynamical field in the rigid
theory. From here we compute the metric $g_{A\bar{A}}=9t^{A}$ and the
curvature

$R_{\rm rigid}=\frac{3}{(t^{3})^{3}}\,,$ (5.52)

that multiplied by $4\mathcal{V}_{X}\sim 4(3t^{2}_{0}+t^{3}_{0})^{2}\phi$
reproduces

$R_{\rm IIA}^{\rm
cl}=\frac{12(3t^{2}_{0}+t^{3}_{0})^{2}}{(t_{0}^{3})^{3}}\phi+\mathcal{O}(\text{const})\,.$
(5.53)

  * •

Limit $t^{2}\rightarrow\infty$ or $\bm{e}=(0,1,0)$: this is a $w=3$ limit with
${\bf k}=63$. The matrix

${\bf K}=\begin{pmatrix}0&18&6\\\ 18&63&15\\\ 6&15&3\end{pmatrix},$ (5.54)

has rank two and null eigenvector $\bm{v}_{0}=(-1,1,-3)$, which corresponds to
the divisor class
$\mathcal{D}_{B}=-\mathcal{D}_{1}+\mathcal{D}_{2}-3\mathcal{D}_{3}$. To
construct a basis, we pick the vector $\bm{v}_{I}=(5,0,-6)$, representing the
divisor class $\mathcal{D}_{C}=5\mathcal{D}_{1}-6\mathcal{D}_{3}$. In the new
basis $\\{\mathcal{D}_{e},\mathcal{D}_{C},-\mathcal{D}_{B}\\}$ the triple
intersection numbers are

$\begin{split}\mathcal{K}_{eee}=63\,,\qquad\mathcal{K}_{eCC}=-252\,,\qquad\mathcal{K}_{CCC}=1080\,,\\\
\mathcal{K}_{CCB}=-36\,,\qquad\mathcal{K}_{CBB}=18\,,\qquad\mathcal{K}_{BBB}=-9\,,\end{split}$
(5.55)

and the rigid prepotential

$\mathcal{F}_{\rm rigid}^{\rm
cl}=\frac{3}{2}\left(T^{B}\right)^{3}-9T^{C}_{0}\left(T^{B}\right)^{2}+18\left(T^{C}_{0}\right)^{2}T^{B}\,,$
(5.56)

where $T^{B}=\frac{1}{21}(6T^{1}+5T^{3})$ is the only dynamical field in the
rigid theory. From here we compute the metric
$g_{B\bar{B}}=9t^{B}-18t_{0}^{C}$ and the curvature

$R_{\rm rigid}=\frac{3}{(t^{3})^{3}}\,,$ (5.57)

that multiplied by $4\mathcal{V}_{X}\sim 42\phi^{3}$ reproduces

$R_{\rm IIA}^{\rm
cl}=\frac{126}{(t_{0}^{3})^{3}}\phi^{3}+\mathcal{O}(\phi^{2})\,.$ (5.58)

  * •

Limit $t^{3}\rightarrow\infty$ or $\bm{e}=(0,0,1)$: this is a $w=2$ limit
where the matrix

${\bf K}=\begin{pmatrix}0&6&2\\\ 6&15&3\\\ 2&3&0\end{pmatrix},$ (5.59)

has maximal rank. As a consequence, there is no rigid IR theory and the type
IIA classical scalar curvature goes like

$R_{\rm IIA}^{\rm cl}=-30+\mathcal{O}(\phi^{-1})\,.$ (5.60)

Following (4.27) we take as a new basis the following vectors:
$\bm{v}_{E}=(-1,1,0)$, that corresponds to the divisor class
$\mathcal{D}_{E}=-\mathcal{D}_{1}+\mathcal{D}_{2}$ dual to the generic
elliptic fiber, $\bm{v}_{e}=(0,0,1)$ and $\bm{v}_{I}=(-3,2,0)$, namely the
divisor class $\mathcal{D}_{D}=-3\mathcal{D}_{1}+2\mathcal{D}_{2}$. In this
basis, the inverse gauge kinetic function has the leading behaviour

$\tilde{I}^{ab}=\begin{pmatrix}2(2t^{1}+3t^{2})\phi^{-2}&-\frac{3t^{2}(4t^{1}+5t^{2})}{2t^{1}+3t^{2}}\phi^{-2}&-2(t^{1}+t^{2})\phi^{-2}\\\
-\frac{3t^{2}(4t^{1}+5t^{2})}{2t^{1}+3t^{2}}\phi^{-2}&\frac{1}{2t^{1}+3t^{2}}&-\frac{t^{1}+t^{2}}{2t^{1}+3t^{2}}\phi^{-1}\\\
-2(t^{1}+t^{2})\phi^{-2}&-\frac{t^{1}+t^{2}}{2t^{1}+3t^{2}}\phi^{-1}&\frac{1}{12}\phi^{-1}\\\
\end{pmatrix}\quad+\text{ subleading terms}\,.$ (5.61)

As one can clearly see, the weights for these three vectors are respectively,
$w_{E}=2$, $w_{e}=0$ and $w_{I}=1$, in agreement with our guess (4.28).

### Three-moduli non-smooth fibration

Finally, we consider another non-smooth elliptic fibration over a
$\mathbb{P}^{2}$ base, also studied in [60]. It also has three Kähler moduli,
a simplicial Kähler cone and in a Nef divisors basis we have

$\begin{split}\mathcal{K}_{111}=50\,,\qquad\mathcal{K}_{112}=10\,,\qquad\mathcal{K}_{122}=2\,,\qquad\mathcal{K}_{113}=80\,,\qquad\mathcal{K}_{123}=16\,,\\\
\mathcal{K}_{223}=3\,,\qquad\mathcal{K}_{133}=128\,,\qquad\mathcal{K}_{233}=25\,,\qquad\mathcal{K}_{333}=203\,.\end{split}$
(5.62)

In the M-theory moduli space the classical scalar curvature diverges
positively at
$(M^{1},M^{2},M^{3})=(\left(\frac{3}{25}\right)^{\frac{1}{3}},0,0)$,
corresponding to the limit $\bm{e}=(0,1,0)$, and at
$(M^{1},M^{2},M^{3})=(0,0,\left(\frac{6}{203}\right)^{\frac{1}{3}})$,
corresponding to $\bm{e}=(0,0,1)$. Any positive linear combinations of these
two vectors also gives a divergence. At the point
$(M^{1},M^{2},M^{3})=(0,\infty,0)$, reached along the limit $\bm{e}=(0,1,0)$,
the curvature tends to a negative constant. Let us study the three elementary
limits more in detail.

  * •

Limit $t^{1}\rightarrow\infty$ or $\bm{e}=(1,0,0)$: this is a $w=3$ limit with
${\bf k}=50$. The matrix

${\bf K}=\begin{pmatrix}50&10&80\\\ 10&2&16\\\ 80&16&128\end{pmatrix}$ (5.63)

has rank one and a kernel generated by the null eigenvectors
$\bm{v}_{0,1}=(1,-5,0)$ and $\bm{v}_{0,2}=(8,0,-5)$, corresponding to the
divisor classes $\mathcal{D}_{A}=\mathcal{D}_{1}-5\mathcal{D}_{1}$ and
$\mathcal{D}_{B}=8\mathcal{D}_{1}-5\mathcal{D}_{3}$. In the basis
$\\{\mathcal{D}_{e},-\mathcal{D}_{A},-\mathcal{D}_{B}\\}$ the triple
intersection numbers become

$\mathcal{K}_{eee}=50\,,\quad\mathcal{K}_{AAA}=-50\,,\quad\mathcal{K}_{AAB}=-25\,,\quad\mathcal{K}_{ABB}=-75\,,\quad\mathcal{K}_{BBB}=-225\,,$
(5.64)

and the rigid prepotential is

$\mathcal{F}_{\rm rigid}^{\rm
cl}=25\left[\frac{1}{3}(T^{A})^{3}+\frac{1}{2}(T^{A})^{2}T^{B}+\frac{3}{2}T^{A}(T^{B})^{2}+\frac{3}{2}(T^{B})^{3}\right]\,,$
(5.65)

where this time we have two dynamical fields $T^{A}=\frac{1}{5}T^{2}$ and
$T^{B}=\frac{1}{5}T^{3}$. From the last expression one can compute the metric

$g_{i\bar{j}}=25\begin{pmatrix}2t^{A}+t^{B}&t^{A}+3t^{B}\\\
t^{A}+3t^{B}&3t^{A}+9t^{B}\end{pmatrix},$ (5.66)

and then the rigid scalar curvature

$R_{\rm
rigid}=3\left[\frac{1}{(t_{0}^{2})^{3}}+\frac{5}{(t_{0}^{2}+3t_{0}^{3})^{3}}\right]\,,$
(5.67)

and, taking into account that $\mathcal{V}_{X}\sim\frac{25}{3}\phi^{3}$, one
can reproduce the 4d classical moduli space scalar curvature

$R_{\rm IIA}^{\rm
cl}=100\left[\frac{1}{(t_{0}^{2})^{3}}+\frac{5}{(t_{0}^{2}+3t_{0}^{3})^{3}}\right]\phi^{3}+\mathcal{O}(\phi^{2})\,.$
(5.68)

  * •

Limit $t^{2}\rightarrow\infty$ or $\bm{e}=(0,1,0)$: this is a $w=2$ limit
where the matrix

${\bf K}=\begin{pmatrix}10&2&16\\\ 2&0&3\\\ 16&3&25\end{pmatrix}$ (5.69)

has maximal rank. As a consequence, there is no rigid IR theory and the
classical type IIA scalar curvature is

$R_{\rm IIA}^{\rm cl}=-30+\mathcal{O}(\phi^{-1})\,.$ (5.70)

We take as new basis the following vectors: $\bm{v}_{E}=(-1,0,1)$, that
corresponds to the divisor class
$\mathcal{D}_{E}=-\mathcal{D}_{1}+\mathcal{D}_{3}$ dual to the generic
elliptic fiber, $\bm{v}_{e}=(0,1,0)$ and $\bm{v}_{I}=(3,0,-2)$, namely the
divisor class $\mathcal{D}_{C}=3\mathcal{D}_{1}-2\mathcal{D}_{3}$. In this
basis, the inverse gauge kinetic function has the leading behaviour

$\tilde{I}^{ab}=\begin{pmatrix}2(2t^{1}+3t^{3})\phi^{-2}&-\frac{10(t^{1})^{2}+32t^{1}t^{3}+25(t^{3})^{2}}{2t^{1}+3t^{3}}\phi^{-2}&-\frac{2(t^{1}+t^{3})}{(t^{2})^{2}}\phi^{-2}\\\
-\frac{10(t^{1})^{2}+32t^{1}t^{3}+25(t^{3})^{2}}{2t^{1}+3t^{3}}\phi^{-2}&\frac{1}{2t^{1}+3t^{3}}&-\frac{t^{1}+t^{3}}{2t^{1}+3t^{3}}\phi^{-1}\\\
-\frac{2(t^{1}+t^{3})}{(t^{2})^{2}}\phi^{-2}&-\frac{t^{1}+t^{3}}{2t^{1}+3t^{3}}\phi^{-1}&\frac{1}{2}\phi^{-1}\\\
\end{pmatrix}\,,$ (5.71)

up to subleading terms. As one can clearly see, the weights for these three
vectors are, respectively, $w_{E}=2$, $w_{e}=0$ and $w_{I}=1$, again matching
(4.28).

  * •

Limit $t^{3}\rightarrow\infty$ or $\bm{e}=(0,0,1)$: this is a $w=3$ limit with
${\bf k}=203$. The matrix

${\bf K}=\begin{pmatrix}80&16&128\\\ 16&3&25\\\ 128&25&203\end{pmatrix}$
(5.72)

has rank 2 and its kernel is generated by $\bm{v}_{0}=(-1,-3,1)$, which
corresponds to the divisor class
$\mathcal{D}_{D}=-\mathcal{D}_{1}-3\mathcal{D}_{2}+\mathcal{D}_{3}$. To
construct a basis, we pick the vector $\bm{v}_{I}=(25,-128,0)$, representing
the divisor class $\mathcal{D}_{F}=25\mathcal{D}_{1}-128\mathcal{D}_{2}$. In
the new basis $\\{\mathcal{D}_{e},\mathcal{D}_{F},-\mathcal{D}_{D}\\}$, the
triple intersection numbers are

$\begin{split}\mathcal{K}_{eee}=203\,,\qquad\mathcal{K}_{eFF}=-3248\,,\qquad\mathcal{K}_{FFF}=838850\,,\\\
\mathcal{K}_{FFD}=-16384\,,\qquad\mathcal{K}_{FDD}=384\,,\qquad\mathcal{K}_{DDD}=-9\,.\end{split}$
(5.73)

The rigid prepotential is

$\mathcal{F}_{\rm rigid}^{\rm
cl}=\frac{3}{2}(T^{D})^{3}-192(T^{D})^{2}T_{0}^{D}+8192T^{D}(T_{0}^{F})^{2}\,,$
(5.74)

where the only dynamical field is $T^{D}=\frac{1}{203}(128T^{1}+25T^{2})$. The
rigid moduli space metric reads $g_{D\bar{D}}=9t^{D}-384t^{F}$ and the rigid
scalar curvature

$R_{\rm rigid}=\frac{3}{(t_{0}^{2})^{3}}\,,$ (5.75)

which using $\mathcal{V}_{X}\sim\frac{203}{6}\phi^{3}$ correctly reproduces

$R_{\rm IIA}^{\rm
cl}=\frac{406}{(t_{0}^{2})^{3}}\phi^{3}+\mathcal{O}(\phi^{2})\,.$ (5.76)

##  6 Conclusions

In this work we have analysed the asymptotic behaviour of the moduli space
scalar curvature in a large class of trajectories of infinite distance. More
precisely, we have focused on the vector multiplet sector of 4d ${\cal N}=2$
supergravity theories obtained from compactifying type IIA on a Calabi–Yau
three-fold $X$. Precisely in this setup, large-volume trajectories of infinite
distance are known to provide counterexamples [10] to the expectation that the
curvature $R_{\rm IIA}$ should be asymptotically negative [6]. We have
performed a general analysis of the asymptotic behaviour of $R_{\rm IIA}$, but
instead of simply describing when Conjecture 3 of [6] fails and when it does
not, we have tried to understand the physics behind the different asymptotic
behaviours of the curvature. Our results have led us to Conjecture 1 which, if
true, could lead us to a powerful criterion to describe the physics of
infinite distance limits in terms of the scalar curvature in field space.

Indeed, in the overall picture that we have obtained, an important role is
played by those limits in which $R_{\rm IIA}$ diverges positively along the
limit. In our setup, this only happens if one can see the infinite distance
trajectory as a gravity decoupling limit in which a non-trivial field theory
survives below the maximal cut-off $m_{*}$ set by the SDC leading tower. For
the divergence to occur, the rigid field theory must non-trivial in the sense
that its curvature $R_{\rm rigid}$ does not vanish. Remarkably, we have
obtained the same result for all the three classes of large-volume limits,
although in a particular class ($w=2$ limits) world-sheet instanton
corrections would be needed to generate $R_{\rm rigid}$. The different cases
have been summarised in table 1, and the essential condition that determines
if we have a rigid theory below the SDC scale is the rank $r$ of the matrix K
in (2.10). More precisely, the number of fields in the rigid theory is given
by $n_{V}-r$, where $n_{V}=h^{1,1}(X)$ is the number of vector multiplets.
Geometrically, this is the number of independent shrinkable, non-Nef divisors
whose volume stays constant along the limit (2.8), and that host the rigid
${\cal N}=2$ field theory that survives below the SDC scale.

The existence of a rigid field theory below the SDC scale at any point along
an infinite distance limit is interesting by itself, and it opens a new
direction in the classification of infinite distance limits. Notice that the
quantity that tells us whether such a rigid field theory should exist or not
is the rank $r$ in (2.10) which, as explained in section 2, is part of the
classification of limits performed in [12, 13, 14]. The details of the rigid
theory, and in particular its curvature, seems to be additional information,
which nevertheless is directly related to the asymptotic behaviour of the
moduli space metric. It would then be very interesting to see if our findings
can be generalised to the whole set of type II CY vector multiplet limits
using the techniques of [12, 13, 14], and in particular to verify the content
of Conjecture 1. In addition, it would be interesting to extend this analysis
to the hypermultiplet moduli space of type II CY compactifications, although
it could be that in this case the obstructions observed in [61, 62] prevent
divergences of the moduli space curvature at infinity. We hope to report on
these topics in the future.

Another result that is worth mentioning is the massive spectrum of states that
comes with the rigid field theory. In our setup we have found that rigid field
theories below the SDC scale come with their own infinite tower of charged
states. These towers are different from the SDC leading tower and, even if
they lie above the SDC cut-off $m_{*}$, they scale just like $m_{*}$ along the
limit and they display an extremality factor that tends to infinity. This is
particularly dramatic in $w=3$ limits where, due to the multiplicity of GV
invariants within curves in a del Pezzo surface, their density is much larger
that that of a Kaluza-Klein spectrum describing the tower of D0-branes. In our
setup, the interpretation of $w=3$ in terms of M-theory compactified on $X$
relates this new tower of charged states with a decompactification to 5d with
a non-Lagrangian SCFT at strong coupling [43, 44, 45, 46]. It would be
interesting to see if this is a general lesson beyond our setup, and if there
is a whole new casuistic to be explored. Additionally, it would be important
to understand how these new towers influence the recent computation of the
species scale [63, 64, 65, 66, 67] and more precisely the role that it plays
in recent tests of the Emergence Proposal [16, 68, 69, 70].

From a more phenomenological perspective, it would be interesting to see if
infinite distance limits with rigid field theories describe realistic corners
of the string Landscape [71]. Indeed, several string theory scenarios use
localisation properties of the compactification to engineer a field theory
that can be parametrically decoupled from gravity. In this way one can easily
reproduce the disparity of strengths between the gravitational and gauge
interactions observed empirically. Infinite distance limits with a rigid field
theory below the SDC scale (or even limits that are close to them)
automatically implement this feature. Following the reasoning of section 3 and
assuming supersymmetry not far from the 4d cut-off scale this would seem to
indicate that, for EFTs that reproduce the observed gauge-gravitational
hierarchy, the field space curvature in Planck units should be positive.
Reversing the logic, if the Curvature Criterion is correct, one could use the
field space scalar curvature to scan over regions of the string Landscape, in
order to select those corners that are more likely to reproduce our Universe.
It would be quite remarkable if something as simple as the field space scalar
curvature could describe one of the most mysterious hierarchies observed in
Nature!

Acknowledgments

We would like to thank Alberto Castellano, Luis E. Ibáñez, Luca Martucci,
Miguel Montero, Tom Rudelius, Ángel M. Uranga, Timo Weigand and Max Wiesner
for discussions. This work is supported through the grants CEX2020-001007-S
and PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by ERDF
A way of making Europe. LM is supported by the fellowship LCF/BQ/DI21/11860035
and LP by the fellowship LCF/BQ/DI22/11940039 from “La Caixa” Foundation (ID
100010434).

##  Appendix A Asymptotic perturbative expansion of the metric

In this section we analyse the asymptotic scaling of the gauge kinetic
function

$\tilde{I}_{ab}=\frac{2}{3}\mathcal{K}g_{ab}=\frac{3}{2\mathcal{K}}\mathcal{K}_{a}\mathcal{K}_{b}-\mathcal{K}_{ab}\,,$
(A.1)

along each class of limits. According to the type of limit, it might be
convenient to rotate to a new basis.

####  $w=3$ limits

We start by considering an EFT string limit with scaling index $w=3$. We can
use the expansions (4.4) to compute (A.1) and then go to the new basis
$\\{\bm{v}_{e},\bm{v}_{I,p},\bm{v}_{0,\alpha}\\}$, defined in (4.5). Here and
in the following the indices $p,q$ run over the basis elements of type
$\bm{v}_{I}$, while $\alpha,\beta$ are used for vectors of type $\bm{v}_{0}$.
Taking into account that ${\bf K}_{ab}v_{0,\alpha}^{b}=0$, the rotated gauge
kinetic function at leading order reads

$\tilde{I}_{MN}=\tilde{I}_{ab}v^{a}_{M}v^{b}_{N}=\begin{pmatrix}\frac{1}{2}{\bf
k}\phi&2{\bf k}_{q}^{0}&\frac{3}{2}{\bf k}_{\beta}^{00}\phi^{-1}\\\ 2{\bf
k}_{p}^{0}&-{\bf K}_{pq}\phi&-{\bf K}_{\beta p}^{0}\\\ \frac{3}{2}{\bf
k}_{\alpha}^{00}\phi^{-1}&-{\bf K}_{\alpha q}^{0}&-{\bf
K}_{\alpha\beta}^{0}\end{pmatrix}+\text{subleading terms}\,,$ (A.2)

where we have defined

$\displaystyle{\bf k}^{0}_{p}\equiv{\bf K}_{ab}v_{I,p}^{a}t^{b}_{0}\,$
$\displaystyle,\quad{\bf
k}^{00}_{\alpha}=\mathcal{K}_{abc}v_{0,\alpha}^{a}t^{b}_{0}t^{c}_{0}\,,$ (A.3)
$\displaystyle{\bf
K}^{0}_{\alpha\beta}=\mathcal{K}_{abc}v_{0,\alpha}^{a}v_{0,\beta}^{b}t^{c}_{0}\,,\quad{\bf
K}_{\alpha p}^{0}$
$\displaystyle\equiv\mathcal{K}_{abc}v_{I,p}^{a}v_{0,\alpha}^{b}t^{c}_{0}\,,\quad{\bf
K}_{pq}\equiv{\bf K}_{ab}v_{I,p}^{a}v_{I,q}^{b}$

Notice that in (A.2) the off-diagonal components might be further suppressed,
even though we expect this to happen just for particular initial
configurations of the saxions. In any case, this result is enough to compute
the leading term of the inverse kinetic matrix

$\tilde{I}^{MN}=\begin{pmatrix}\frac{2}{\bf
k}\phi^{-1}&\mathcal{O}(\phi^{-2})&\mathcal{O}(\phi^{-2})\\\
\mathcal{O}(\phi^{-2})&-A^{pq}\phi^{-1}&\mathcal{O}(\phi^{-1})\\\
\mathcal{O}(\phi^{-2})&\mathcal{O}(\phi^{-1})&-B^{\alpha\beta}\\\
\end{pmatrix}+\text{subleading terms}\,,$ (A.4)

where $A^{pq}$ and $B^{\alpha\beta}$ are the inverses of ${\bf K}_{pq}$ and
${\bf K}_{\alpha\beta}^{0}$, respectively. From here one can check that the
weights of the vectors of the new basis indeed match with (4.6).

####  $w=2$ limits, smooth elliptic fibration

We now consider a $w=2$ limit, restricting to the case of a smooth elliptic
fibration, where the triple intersection numbers take the form (4.13). In this
case we are already using a good basis, as one can see in (4.14), and actually
there is no need to distinguish between $\bm{v}_{0}$ and $\bm{v}_{e}$ when
computing the leading term of $\tilde{I}_{ab}$, as it is sufficient to split
the coordinates into $\\{t^{E},t^{\alpha}\\}$ to get

$\tilde{I}_{ab}=\begin{pmatrix}\frac{\eta}{2t^{E}}&\frac{1}{2}C_{\alpha\beta}c_{1}^{\alpha}t^{E}\\\
\frac{1}{2}C_{\alpha\beta}c_{1}^{\beta}t^{E}&C_{\alpha\beta}t^{E}\end{pmatrix}+\text{subleading
terms}\,,$ (A.5)

where we defined

$C_{\alpha\beta}\equiv\left[2\frac{\eta_{\alpha}\eta_{\beta}}{\eta}-\eta_{\alpha\beta}\right]\,,$
(A.6)

and $\eta_{\alpha}=\eta_{\alpha\beta}t^{\beta}$ and
$\eta=\eta_{\alpha\beta}t^{\alpha}t^{\beta}$. From (A.5) one can compute the
inverse gauge kinetic matrix

$\tilde{I}^{ab}=\begin{pmatrix}\frac{2t^{E}}{(\eta
ee)}\phi^{-2}&\mathcal{O}(\phi^{-2})\\\
\mathcal{O}(\phi^{-2})&C^{\alpha\beta}\frac{1}{t^{E}}\end{pmatrix}+\text{subleading
terms}\,,$ (A.7)

where $C^{\alpha\beta}$ is the inverse of $C_{\alpha\beta}$. From these
expressions one can extract the weights (4.16).

####  $w=1$ limits

Finally, we consider a $w=1$ limit. We can use the expansions (4.4) to compute
(A.1) and then go to the new basis
$\\{\bm{v}_{e},\bm{v}_{I,p},\bm{v}_{0,\alpha}\\}$, defined in (4.31). As
before, the indices $p,q$ run over the basis elements of type $\bm{v}_{I}$,
while $\alpha,\beta$ are used for vectors of type $\bm{v}_{0}$. Taking into
account that ${\bf K}_{ab}v_{0,\alpha}^{b}=0$, the rotated gauge kinetic
function at leading order reads

$\tilde{I}_{MN}=\begin{pmatrix}\frac{1}{2}{\bf
k}^{00}\phi^{-1}&\left(\frac{1}{2}{\bf k}^{00}_{q}-\frac{1}{3}\frac{{\bf
k}^{000}}{{\bf k}^{00}}{\bf k}^{0}_{q}\right)\phi^{-1}&\frac{1}{2}{\bf
k}_{\beta}^{00}\phi^{-1}\\\ \left(\frac{1}{2}{\bf
k}^{00}_{p}-\frac{1}{3}\frac{{\bf k}^{000}}{{\bf k}^{00}}{\bf
k}^{0}_{p}\right)\phi^{-1}&D_{pq}\phi&\frac{{\bf k}^{00}_{\beta}{\bf
k}^{0}_{p}}{{\bf k}^{00}}-{\bf K}^{0}_{\beta p}\\\ \frac{1}{2}{\bf
k}_{\alpha}^{00}\phi^{-1}&\frac{{\bf k}^{00}_{\alpha}{\bf k}^{0}_{q}}{{\bf
k}^{00}}-{\bf K}^{0}_{\alpha q}&-{\bf K}_{\alpha\beta}^{0}\end{pmatrix}$ (A.8)

up to subleading terms, where we have defined

$\displaystyle{\bf k}^{00}\equiv{\bf K}_{ab}t^{a}_{0}t^{b}_{0}\,,\quad{\bf
k}^{000}\equiv\mathcal{K}_{abc}t^{a}_{0}t^{b}_{0}t^{c}_{0}\,,\quad
D_{pq}\equiv 2\frac{{\bf k}^{0}_{p}{\bf k}^{0}_{q}}{{\bf k}^{00}}-{\bf
K}_{pq}\,,\quad{\bf
k}^{00}_{p}\equiv\mathcal{K}_{abc}v_{I,p}^{a}t^{b}_{0}t^{c}_{0}\,.$ (A.9)

The leading term of the inverse kinetic matrix

$\tilde{I}^{MN}=\begin{pmatrix}\frac{2}{{\bf
k}^{00}}\phi&\mathcal{O}(\phi^{-1})&\mathcal{O}(\phi^{-1})\\\
\mathcal{O}(\phi^{-1})&D^{pq}\phi^{-1}&\mathcal{O}(\phi^{-1})\\\
\mathcal{O}(\phi^{-1})&\mathcal{O}(\phi^{-1})&-B^{\alpha\beta}\\\
\end{pmatrix}+\text{subleading terms}\,,$ (A.10)

where $D^{pq}$ and $B^{\alpha\beta}$ are the inverses of $D_{pq}$ and ${\bf
K}_{\alpha\beta}^{0}$, respectively. From these expressions, one can check
that the weights of the vectors of the new basis indeed match with (4.32).

##  Appendix B The scaling weight and degeneracy of EFT string limits

In this section we explain the relations between the degeneracy of EFT string
limits and their scaling weight $w$. From the viewpoint of [32] an EFT string
limit (2.8) is said to be non-degenerate if the EFT string with charge
$\bm{e}$ defining the limit is the only one becoming tensionless along the
limit. If there are more EFT strings becoming tensionless, the limit is said
to be degenerate and the order of degeneracy $p$ is the number of
asymptotically tensionless independent (or equivalently elementary) EFT
strings. In the following we will show that

  1. 1.

$w=3\qquad\qquad\qquad\qquad\quad\,\,\,\Rightarrow\quad p=n_{V}$, the limit is
maximally degenerate.

  2. 2.

$w=2$ (smooth fibration) $\quad\Rightarrow\quad p=n_{V}-1$, the limit is non-
maximally degenerate.

  3. 3.

$w=1\qquad\qquad\qquad\qquad\quad\,\,\,\Rightarrow\quad p=1$, the limit is
non-degenerate.

Notice that for $w=2$ limits we are only considering smooth $T^{2}$ fibrations
with the structure (4.13). In the non-smooth case intermediate values of $p$
could be possible.

  1. 1.

Consider a $w=3$ limit. Then ${\bf k}\neq 0$ and one has

$\mathcal{K}={\bf
k}\phi^{3}+\mathcal{O}(\phi^{2})\,,\qquad\mathcal{K}_{a}={\bf
k}_{a}\phi^{2}+\mathcal{O}(\phi)\,,$ (B.1)

which imply, denoting by $\mathcal{T}_{a}/M_{P}^{2}=\ell_{a}$ the tension of
the $a^{\rm th}$ elementary EFT string,

$\frac{\mathcal{T}_{a}}{M_{P}^{2}}\sim\frac{3}{2}\frac{{\bf k}_{a}}{\bf
k}\phi^{-1}\sim\phi^{-1}\,,\,\forall a\,.$ (B.2)

Then, all the EFT strings become tensionless and we have a maximally
degenerate limit, with $p=n_{V}$. Notice that in a $w=3$ limit, ${\bf
k}_{a}={\cal D}_{a}\cdot{\cal D}_{e}\cdot{\cal D}_{e}\neq 0,\,\forall a$,
where ${\cal D}_{e}=e^{a}{\cal D}_{a}$ is the Nef divisor wrapped by the NS5
that defines the limit we are considering. One can show this by using the fact
that two non-proportional Nef divisors always have non-trivial intersection
and applying Lemma 5 of Appendix B of [15].

  2. 2.

Consider a $w=2$ that is a smooth $T^{2}$ fibration. Then one can split the
charges as $(e^{E},e^{\alpha})$ and the limit in consideration is given by
$\bm{e}=(0,e^{\alpha})$. In this case one has ${\bf k}=0$, ${\bf
k}_{a}=(\eta_{\alpha\beta}e^{\alpha}e^{\beta},0)$ and

$\begin{split}&\mathcal{K}=3\eta_{\alpha\beta}e^{\alpha}e^{\beta}t^{E}_{0}\phi^{2}+\mathcal{O}(\phi)\,,\\\
\mathcal{K}_{E}=\eta_{\alpha\beta}e^{\alpha}e^{\beta}&\phi^{2}+\mathcal{O}(\phi)\,,\qquad\mathcal{K}_{\alpha}=2\eta_{\alpha\beta}e^{\beta}t^{E}_{0}\phi+\mathcal{O}(\text{const})\,.\end{split}$
(B.3)

Notice that $\eta_{\alpha\beta}e^{\beta}={\cal D}_{\alpha}\cdot{\cal
D}_{e}\neq 0,\,\forall\alpha$, where ${\cal D}_{e}=e^{\beta}{\cal D}_{\beta}$,
because any two non-proportional Nef divisors have non-trivial intersection.
This implies

$\frac{\mathcal{T}_{a}}{M_{P}^{2}}\sim\begin{cases}\phi^{-1}\quad a=\alpha\\\
\text{const}\quad a=E\end{cases}\,,$ (B.4)

and then the limit is degenerate with order $p=n_{V}-1$.

  3. 3.

Consider a $w=1$ limit. Then ${\bf k}={\bf k}_{a}=0$ and one has

$\mathcal{K}=3{\bf
K}_{ab}t_{0}^{a}t_{0}^{b}\phi+\mathcal{O}(\text{const})\,,\qquad\mathcal{K}_{a}=2{\bf
K}_{ab}t_{0}^{b}\phi+\mathcal{O}(\text{const})\,.$ (B.5)

Notice that the fact that it is $w=1$ also implies that it is an elementary
limit in the sense of footnote 1, namely with just one non-vanishing charge.
As a consequence, ${\bf K}_{ab}t_{0}^{b}\neq 0$ if and only if $e^{a}=0$ and
then

$\frac{\mathcal{T}_{a}}{M_{P}^{2}}\sim\begin{cases}\phi^{-1}\quad e^{a}\neq
0\\\ \text{const}\quad e^{a}=0\end{cases}\,.$ (B.6)

So the only EFT string becoming tensionless is the one with charge $e^{a}$ and
the limit is non-degenerate.

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