IFT-UAM/CSIC-24-40

Yukawa Couplings at Infinite Distance
and Swampland Towers in Chiral Theories
G. F. Casas,♢ L. E. Ibáñez ♣,♢ and F. Marchesano♢
♣ Departamento de Física Teórica
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
♢ Instituto de Física Teórica UAM-CSIC, c/ Nicolas Cabrera 13-15, 28049
Madrid, Spain
Abstract

We study limits of vanishing Yukawa couplings of 4d chiral matter fields in
Quantum Gravity, using as a laboratory type IIA orientifolds with D6-branes.
In these theories chiral fermions arise at brane intersections, where an
infinite tower of charged particles dubbed gonions are localised. We show that
in the limit $Y\rightarrow 0$ some of these towers become asymptotically
massless, while at the same time the kinetic term of some chiral fields
becomes singular and at least two extra dimensions decompactify. For limits
parametrised by a large complex structure saxion $u$, Yukawa couplings have a
behaviour of the form $Y\,\sim\,{1}/{u^{r}}$, with $r$ some positive rational
number. Moreover, in this limit some of the gauge couplings associated to the
Yukawa vanish. The lightest gonion scales are of order $m_{\rm gon}\sim
g^{s}M_{\rm P}$ with $s>1$, verifying the magnetic WGC with room to spare and
with no need of its tower/sublattice versions. We also show how this behaviour
can be understood in the context of the emergence of kinetic terms in Quantum
Gravity. All these results may be very relevant for phenomenology, given the
fact that some of the Yukawa couplings in the Standard Model are very small.

###### Contents

  1. 1 Introduction
  2. 2 Yukawa couplings in 4d ${\cal N}=1$ orientifolds
    1. 2.1 Type IIA orientifolds and intersecting D6-branes
    2. 2.2 Yukawas in toroidal and Calabi–Yau orientifolds
    3. 2.3 Kinetic terms of chiral fields
  3. 3 The $Y\to 0$ limit and infinite distance
    1. 3.1 Small Yukawas and gonion masses
    2. 3.2 Gonions, strings and monopoles
    3. 3.3 How $Y\to 0$ implies $g\to 0$
    4. 3.4 Mass scales in simple limits
  4. 4 Explicit examples
  5. 5 Structure of towers: species scale and emergence
    1. 5.1 Tower densities and the Species Scale
    2. 5.2 Emergence of kinetic terms
    3. 5.3 Singular Yukawa couplings and emergence
    4. 5.4 Chiral towers and the (sub)Lattice/Tower WGC
  6. 6 Final comments and outlook
  7. A Gonion spectrum for D6-branes at angles
  8. B Mirror duals of type I magnetised Calabi–Yau compactifications
  9. C Selection rules for Yukawas

##  1 Introduction

The Swampland Programme [1, 2, 3, 4, 5] attempts to identify model-independent
implications of Quantum Gravity (QG) for EFTs coupled to gravity. One of the
leitmotifs is the presence of light towers of infinite number of states when
one goes to the boundary of the moduli space [6]. This is also related to the
fact that, when vacuum expectation values of fields tend to infinity,
different types of global symmetries appear, which are inconsistent with QG.
The canonical example is the case of a $U(1)$ theory with gauge coupling $g$.
The magnetic Weak Gravity Conjecture (WGC) [7] tell us that as $g\rightarrow
0$, a tower of charged particles with characteristic scale $m_{*}$ appears
such that

$m_{*}\ \lesssim gM_{\rm P}^{D/2-1}\,,$ (1.1)

with $D$ the number of dimensions and $M_{\rm P}$ the Planck mass. One sees
the presence of such a tower as a reflection of the fact that in the strict
$g=0$ limit there would be a global $U(1)$ symmetry which would be forbidden
by general QG arguments. There are many other situations in which towers
appear, signalling either a decompactification (via Kaluza–Klein (KK)-like
towers) or an emergent critical string [8], when one takes infinite distance
limits in the moduli space.

By now many different large moduli limits have been explored and tested
against string theory results in a variety of dimensions and with different
amounts of supersymmetry, but setups with gauge couplings have mostly been
tested in the presence of $\geq 8$ SUSY generators, see [3, 4] for reviews.
Setups with 4d ${\cal N}=1,0$ SUSY and asymptotically vanishing gauge
couplings remain relatively unexplored. However, these are the cases of most
phenomenological interest, particularly due to the observed chirality of the
Standard Model (SM) which requires ${\cal N}\leq 1$.

In the present paper we aim to improve the state-of-the-art by studying string
compactifications leading to 4d ${\cal N}=1$ chiral spectra at infinite
distance limits.111Previous works addressing infinite distance limits in 4d
${\cal N}=1$ chiral constructions include [9, 10]. We are particularly
interested in classes of models which contain the ingredients necessary for a
realistic model with SM gauge group and fermion generations. Such settings
will involve new ingredients like e.g. the existence of perturbative trilinear
Yukawa couplings $Y_{ijk}$ involving three charged chiral multiplets. One can
then pose the following question: Is the limit $Y_{ijk}\to 0$ at infinite
distance? What happens along such a limit? Do towers appear? If they do, what
is their structure? The answer to these questions is not immediately obvious,
since, unlike the case of the $U(1)$ gauge symmetry above, it is not clear
what goes wrong (if at all) when $Y_{ijk}\rightarrow 0$. Some previous studies
of Yukawa couplings in the context of the Swampland Programme include [11, 12,
13, 14].

In addition to its theoretical interest, the study of $Y_{ijk}\rightarrow 0$
limits is also interesting phenomenologically, since most of the Yukawa
couplings in the SM are numerically small. Thus in order to reproduce such
small values this limit is the right one to explore. This is particularly the
case of the Yukawa couplings of neutrinos which require values $Y\lesssim
10^{-12}$ in the case of purely Dirac neutrinos, if one wants to match the
experimental data. We study this particularly interesting case in a separate
companion paper [15], see also the related study in [13].

To study these questions we use as a laboratory ${\cal N}=1$ Calabi–Yau (CY)
orientifold compactifications of type IIA string theory. In those theories
chiral matter is localised at the intersection of D6-branes, which wrap three-
cycles in the CY. Such type of models have been studied in much detail and
specific examples with the SM gauge group (or some extension) with three
chiral families have been constructed (see [16, 17, 18, 19, 20, 21, 22] for
reviews). The properties of these constructions may be considered as
representative of the space of string compactifications with chiral matter,
since different dualities connect them with other constructions like type IIB
CY orientifolds (via mirror symmetry) or heterotic compactifications with
$U(1)$ bundles. Thus we believe that our results go beyond this particular
class of theories. We also believe that, due to chirality, many of our results
remain true in models with SUSY spontaneously broken.

Let us summarise some of our results. We find that, indeed, limits in which a
Yukawa coupling of a 4d ${\cal N}=1$ EFT goes to zero are singular and, in
general, lead to the appearance of different towers of states, depending on
the chosen field direction. In 4d ${\cal N}=1$ non-chiral theories, KK-like
and string towers are known to arise from the gravitational sector of the
compactification [23, 24, 25, 26, 27, 28]. Moreover, if D-branes are present,
KK-towers arising from their worldvolume are also expected. However, as soon
as one considers chiral D-brane models there is a new, more exotic class of
towers that appears. They are made of massive particles transforming under
bifundamental representations of the D-brane gauge group, and which in our
setup are localised at the intersections of pairs of D6-branes
D6${}_{a}\cap$D6b, just where the chiral particles live. They were dubbed
gonions in [29] and arise from open strings stretched between both branes.
Despite coming from string fluctuations, their mass spectrum below the string
scale $M_{s}$ is not Hagedorn-like, but is described by a formula of the type
$m_{n}^{2}\simeq nm_{\rm gon}^{2}$, where $m_{\rm
gon}^{2}\simeq\theta_{ab}M_{s}^{2}$ and $\theta_{ab}$ is an intersection angle
between the two branes.222In type IIB mirror setups that feature D-branes with
magnetic worldvolume fluxes, these gonion states correspond to Landau levels.

Figure 1: Yukawa coupling in type IIA orientifolds with intersecting
D6-branes. The holomorphic piece $W_{ijk}$ depends on the shaded region that
connects the three D6-brane intersections, while $\Theta_{ijk}$ depends on the
local intersection angles.

Perturbative Yukawa couplings come from worldsheet disc diagrams with boundary
on three D6α-branes $\alpha=a,b,c$ and that connect three of their pairwise
intersections [29, 30], see figure 1. Schematically the physical, canonically
normalised Yukawa couplings have an expression of the form

$Y_{ijk}\ =\ \frac{e^{\phi_{4}/2}}{{\rm
Vol}_{X}^{1/2}}\Theta_{ijk}^{1/4}W_{ijk}\ .$ (1.2)

Here $W_{ijk}$ is the holomorphic piece of the Yukawa coupling which, at large
compactification volume ${\rm Vol}_{X}$, can be treated as a constant up to
subleading world-sheet instanton corrections. Additionally, ${\phi_{4}}$
stands for the 4d dilaton and $\Theta_{ijk}$ only depends on the intersection
angles (and hence on the complex structure). Not surprisingly, the limit ${\rm
Vol}_{X}\to\infty$ (at fixed $\phi_{4}$ and complex structure) corresponds to
the non-perturbative limit in which there is a decompactification to M-theory.
More interesting are the large complex structure limits at fixed Kähler
moduli, in which one has $|Y_{ijk}|\sim e^{\phi_{4}/2}\Theta_{ijk}^{1/4}$.
Taking $W_{ijk}\sim{\cal O}(1)$, we find that one can write the Yukawa
couplings in terms of the gonion masses at the three intersections involved in
the Yukawa, in the form

$|Y_{ijk}|\ \simeq\ e^{-\phi_{4}}\left(\frac{m_{\rm gon}^{i}}{M_{\rm
P}}\frac{m_{\rm gon}^{j}}{M_{\rm P}}\frac{m_{\rm gon}^{k}}{M_{\rm
P}}\right)^{1/2}\ ,$ (1.3)

where $m_{\rm gon}^{i}$ is the lightest gonion mass at a given intersection.
This expression is fairly general in CY orientifolds,333Slight modifications
arise in Yukawas that involve what we dub ${\cal N}=2$ sectors, see section 3.
since this piece of the Yukawa essentially only depends on local information
about the intersection points. It shows how, whenever a Yukawa coupling goes
to zero, some light gonion tower arises. Moreover, one can see that this
structure comes from the kinetic terms of the three fields that participate in
the Yukawa, which in string units read $K_{i\bar{i}}\simeq M_{\rm P}/m_{\rm
gon}^{i}$. As a result, the Yukawas become singular because, in the 4d EFT, at
least one of the particles involved in the Yukawa has a singular metric. One
also observes that, when a gonion mass $m_{\rm gon}$ goes to zero, at least
two D-brane and bulk dimensions decompactify.

To compute the Yukawa couplings from the above expression one needs to know
the lightest gonion masses at each of the three intersections, or in other
words the intersection angles. Such angles are a function of the complex
structure saxions of the compactification, but the dictionary between both is
in general very involved. We however propose an estimate of $m_{\rm gon}$ in
terms of the tensions $T_{\rm D4}$ of 4d string made up from D4-branes
wrapping three-cycles of the CY, which couple to space-time filling D6-branes
through their Fayet-Ilioupolous (FI) terms. In particular, a D6a-brane that
intersects its orientifold image will have an FI-term of the form
$\pi\xi_{a}=Q_{a}^{K}T_{{\rm D4},K}$, with $2Q_{a}^{K}\in\mathbbm{Z}$, and $K$
running over an appropriate basis of 4d BPS strings. This FI-term sets the
mass of the lightest scalars at the intersection, and it is the signed sum of
three intersection angles. From here we estimate the lightest gonion scale at
such intersection as $m_{\rm gon}^{2}\simeq g_{a}^{2}T_{{\rm D4},K}$, for some
appropriate choice of $K$. With this prescription one can estimate the
behaviour of the Yukawa couplings at large complex structure in a variety of
settings. We consider STU-like models of the type described in [23] as well as
the EFT string limits of [25, 26] or variants thereof that are compatible with
intersecting D6-branes at supersymmetric angles. This general analysis can be
made explicit in toroidal orientifold models, including an example with a
semi-realistic Pati–Salam gauge group.

The casuistics are wide, but there are several general lessons. For limits
parametrised in terms of a largest complex structure saxion $u$,
asymptotically vanishing Yukawa couplings have a general behaviour of the form

$Y_{*}\,\sim\,\frac{1}{u^{r}}\,,$ (1.4)

with $r$ a rational positive number. In most of the examples analysed one has
multiples of $1/4$, inherited from the exponent in eq.(1.2), and more
precisely $r=1/4,1/2,3/4,1$, although other fractional values for $r$ also
appear. As a consequence, Yukawas between branes $a$, $b$, $c$ are related to
at least one of their gauge couplings $g_{*}$ as

$Y_{*}\,\sim\,g_{*}^{2r}\,.$ (1.5)

This already tells us what goes wrong when a chiral Yukawa coupling goes to
zero: some gauge couplings also go to zero, and global symmetries would arise.

It is also interesting that the charged particles in these settings, the
towers of gonions, do not satisfy the tower/sublattice WGC[31, 32, 33, 34] by
themselves, as all the elements of the tower have the same charge. Moreover,
in general they do not saturate the WGC. From the $U(1)$ WGC one expects
$m_{\rm gon}\leq gM_{\rm P}$, but $m_{\rm gon}$ is oftentimes well below. An
interesting class of limits that we find which illustrate this fact feature
only two decompactifying dimensions and a single dominant gonion tower. We
find the relations

$m_{\rm gon}\,\sim\,g_{*}M_{s}\,\sim\,g_{*}^{2}M_{\rm P}\,,\qquad
Y_{*}\,\sim\,g_{*}\ ,\qquad g_{*}\sim e^{\phi_{4}}\,.$ (1.6)

Note the double suppression by $g_{*}^{2}$ of the gonion masses with respect
to the Planck mass, compared to the standard WGC bound. This implies that,
even though there is no lattice of charged states, the WGC itself is not in
risk upon dimensional reduction.

The study of the structure of gonion towers reveals further interesting
properties with regards to the proposal of Emergence in Quantum Gravity [35,
36, 37]. In that scheme it is proposed that the kinetic terms of particles in
the infrared, arise from loops involving towers of states arising for large
moduli. This idea has been tested and discussed in the contexts of string
theory vacua with $\geq 8$ supercharges, see [35, 36, 37, 38, 39, 40, 41, 12,
13, 42, 43], and it is interesting to see if it is consistent with 4d chiral
theories, as studied in this paper. We show that indeed, in the presence of
gonion towers saturating the species scale, chiral massless fields at a given
intersection get kinetic terms with a behaviour in agreement with the known
tree-level results. For this result to be obtained it is crucial the
particular behaviour of gonion towers, i.e., the form of the spectrum and
crucially the fact that all gonions in a given tower have the same charge. We
also check in simple settings how also the gauge kinetic terms are produced by
loops from gonion towers.

The rest of the paper is organised as follows. In section 2 we present a brief
overview of the construction of type IIA 4d ${\cal N}=1$ CY orientifolds with
chiral matter. We describe the computation of the physical Yukawa couplings
and matter kinetic terms in toroidal settings and obtain explicit formulae
applicable to more general CY examples, by exploiting the local character of
both quantities. In section 3 we study the limit $Y_{ijk}\rightarrow 0$ in
general CY orientifolds and show how in this limit the Yukawas may be
expressed in terms of gonion tower masses, which become light along the limit.
We also study how the FI-terms and gonion masses are related to the tension of
4d strings, something that we use to propose an estimate of the gonion scales
in terms of compactification moduli. This estimate is then used to argue that
in general $Y\to 0$ implies the vanishing of some D6-brane gauge coupling, and
to describe the different mass scales and Yukawa scalings in simple limits. In
order to illustrate the emerging picture, in section 4 we analise examples of
these limits in explicit toroidal orientifolds, including a Pati–Salam
semirealistic model. In section 5 we study the structure of the towers in this
class of orientifolds with reference to the Species Scale [44, 45, 46] in
these constructions, as well as its consistency with the Emergence Proposal.
We find that many qualitative features of the towers, kinetic terms, and
Yukawa couplings in these models may be understood within this hypothesis, in
which it is assumed that kinetic terms of IR particles may be obtained from
loop computations involving the towers. Final conclusions and an outlook are
presented in section 6.

Complementary material is presented in the appendices. Appendix A describes
the gonion spectrum at two D6-brane intersections. Appendix B describes how to
translate type I vacua with $U(N)$ bundles to the language used in the main
text. Appendix C describes the selection rules for perturbative Yukawas in
intersecting D6-brane models.

##  2 Yukawa couplings in 4d ${\cal N}=1$ orientifolds

In this section we review the general description of 4d ${\cal N}=1$ type IIA
orientifold compactifications with intersecting D6-branes. These constructions
feature a massless chiral spectrum that arises from the open string sector,
and a set of Yukawa couplings generated by worldsheet instantons. While the
explicit computation of these couplings has only been performed in toroidal
orbifolds, the localisation properties of D-branes allow us to deduce the
expression (1.2) for a general Calabi–Yau setting. As a byproduct, one can
match the kinetic terms for the chiral fields at intersections with the
results previously obtained in the literature and, eventually, express them in
terms of gonion masses. In section 5 we interpret this relation between gonion
towers and kinetic terms from the viewpoint of the Emergence Proposal.

###  2.1 Type IIA orientifolds and intersecting D6-branes

Let us consider type IIA string theory on a background of the form
$X_{4}\times X_{6}$, where $X_{6}$ is a compact Calabi–Yau three-fold with
Kähler form $J$ and holomorphic three-form $\Omega$, and with volume form
$\frac{1}{6}J^{3}=\frac{1}{4}\mbox{Re}\,\Omega\wedge\mbox{Im}\,\Omega$. To
this background we apply the standard orientifold quotient generated by
$\Omega_{\rm ws}(-1)^{F_{L}}{\cal R}$ [16, 17, 18, 20, 21], where $\Omega_{\rm
ws}$ is the worldsheet parity reversal operator, ${F_{L}}$ is the space-time
fermion number for the left-movers and ${\cal R}$ an anti-holomorphic
involution of $X_{6}$ acting as ${\cal R}(J,\Omega)=(-J,\bar{\Omega})$. The
fixed point locus of ${\cal R}$ specifies one or several special Lagrangian
three-cycles calibrated by $\mbox{Re}\,\Omega$, where the O6-plane are
located, and that we denote by $\Pi_{\rm O6}$. To cancel the induced RR
tadpole, one can add background fluxes threading $X_{6}$ and stacks of
D6-branes wrapping its three-cycles.444In principle, one may also cancel the
tadpole with a more general D-brane sector that includes coisotropic D8-branes
[47]. However, as of today the properties of these objects are not well
understood in smooth Calabi–Yau geometries, and for this reason we will not
consider them in the following. Since our main interest is the chiral sector
of the resulting 4d EFT, let us assume that background fluxes are absent and
that the tadpole is cancelled by the D6-brane content, which satisfies the
following equation

$\sum_{\alpha}N_{\alpha}\left([\Pi_{\alpha}]+[\Pi_{\alpha*}]\right)=4[\Pi_{\rm
O6}]\,.$ (2.1)

Here $\Pi_{\alpha}$ is a three-cycle wrapped by a stack of $N_{\alpha}$
D6-branes, and $\Pi_{\alpha*}={\cal R}(\Pi_{\alpha})$ its orientifold image,
while the brackets denote their homology classes in
$H_{3}(X_{6},\mathbbm{Z})$. To preserve the 4d ${\cal N}=1$ supersymmetry of
the background, such three-cycles must satisfy the special Lagrangian
conditions ${\cal F}+iJ|_{\Pi_{\alpha}}=0$ and
$\mbox{Im}\,\Omega|_{\Pi_{\alpha}}=0$, where ${\cal
F}=B|_{\Pi_{\alpha}}+\frac{\ell_{s}^{2}}{2\pi}F$ is the gauge-invariant
D6-brane worldvolume field strength and $\ell_{s}=2\pi\sqrt{\alpha^{\prime}}$
the fundamental string length.

Given this setup, the massless spectrum of the 4d EFT can be divided into two:
the closed and the open string sectors. The former includes the gravity
multiplet and the bulk moduli of the compactification, which can be subdivided
into Kähler and complex structure moduli. At large compactification volumes
the complexified Kähler moduli can be defined as $T^{a}=b^{a}+it^{a}$, where

$J_{c}\equiv B+iJ=(b^{a}+it^{a})\omega_{a}\,,$ (2.2)

and $[\omega_{a}]$ provide a basis for the cohomology classes
$H^{2}_{-}(X_{6},\mathbbm{R})$ which are ${\cal R}$-odd. Similarly, the
complex structure moduli $U^{K}=\zeta^{K}+iu^{K}$ can be defined by the
expansion

$\Omega_{c}\equiv
C_{3}+ie^{-\phi}\mbox{Re}\,\Omega=(\zeta^{K}+iu^{K})\alpha_{K}\,,$ (2.3)

where $C_{3}$ is the three-form RR potential, $\phi$ is the 10d dilaton and
$[\alpha_{K}]$ a basis for the ${\cal R}$-even cohomology classes
$H^{3}_{+}(X_{6},\mathbbm{R})$. The kinetic terms of all these bulk fields are
encoded in a Kähler potential of the form $K\equiv K_{K}+K_{Q}$, where

$K_{K}\equiv-{\rm log}\left({\rm Vol}_{X_{6}}\right)=-{\rm
log}\left(\frac{i}{48}{\cal
K}_{abc}(T^{a}-\bar{T}^{a})(T^{b}-\bar{T}^{b})(T^{c}-\bar{T}^{c})\right)\,,$
(2.4)

with ${\cal
K}_{abc}\equiv\ell_{s}^{-6}\int_{X_{6}}\omega_{a}\wedge\omega_{b}\wedge\omega_{c}$
the triple intersection numbers of $X_{6}$ and

$K_{Q}\equiv-2\log{\cal
H}=-2\log\left(\frac{i}{8\ell_{s}^{6}}\int_{X_{6}}e^{-2\phi}\Omega\wedge\bar{\Omega}\right)=4\phi_{4}\,,$
(2.5)

with $\phi_{4}=\phi-\frac{1}{2}\log{\rm Vol}_{X_{6}}$ the 4d dilaton.

These expressions give a good approximation of the kinetic terms in regimes
where both worldsheet and D2-brane instanton corrections can be ignored, so
that the 4d EFT displays approximate shift symmetries on the fields $b^{a}$,
$\zeta^{K}$, which are considered axions. Typically, one chooses $\omega_{a}$
and $\alpha_{K}$ as harmonic forms such that $b^{a}$ and $\zeta^{K}$ have unit
periodicity. Then, as discussed in [26, section 6.4], one can identify
$\ell_{s}^{-2}[\omega_{a}]$ as Poincaré duals to ${\cal R}$-even four-cycles
of $X_{6}$ wrapped by NS5-branes, and $\ell_{s}^{-3}[\alpha_{K}]$ as duals to
${\cal R}$-odd three-cycles wrapped by D4-branes. Both objects appear as
strings in the 4d EFT, that couple magnetically to the axions. The tension of
such strings is a function of the saxions $t^{a}$ and $u^{K}$, which are also
the fields that describe the infinite-distance geodesics in the large-volume
large-complex-structure regime of the compactification.

Indeed, let us focus on 4d strings coming from D4-branes. Normalising the
axion $\zeta^{K}$ to have unit period implies that $[\Sigma_{K}^{-}]={\rm
P.D.}[\ell_{s}^{-3}\alpha_{K}]$ is an ${\cal R}$-odd three-cycle of $X_{6}$
that, when wrapped by D4-brane, yields a 4d string that sources this axion. If
this 4d string is BPS, then its tension in units of the reduced Planck mass
$M_{\rm P}$ is given by

$\ell_{K}\equiv-\frac{1}{2}\frac{\partial K}{\partial u^{K}}=-\frac{1}{2{\cal
H}}\int_{\Sigma_{K}^{-}}e^{-\phi}\mbox{Im}\,\Omega\,,$ (2.6)

where we have used results of [48], that imply that ${\cal H}$ is a function
of the $u^{K}$ only. Following [26] we name the $\ell_{K}$ as dual saxions,
and we choose $[\Sigma_{K}^{-}]$ to generate a subcone of 4d BPS strings
dubbed EFT strings, which have the property that their tension only vanishes
at infinite distance boundaries of the moduli space of large complex
structures. Near those loci the string is expected to induce a breakdown of
the 4d EFT, via an infinite tower of oscillation modes similar to those of a
tensionless critical string.

Non-Abelian gauge group | $\prod_{\alpha}SU(N_{\alpha})$
---|---
Massless $U(1)$s |  $\sum_{\alpha}c_{\alpha}U(1)_{\alpha}$ such that $\sum_{\alpha}c_{\alpha}([\Pi_{\alpha}]-[\Pi_{\alpha*}])=0$
Chiral multiplets | $\sum_{\alpha<\beta}\,I_{\alpha\beta}({\bf N}_{\alpha},{\bf\bar{N}}_{\beta})\,+\,I_{\alpha\beta*}({\bf N}_{\alpha},{\bf\bar{N}}_{\beta})$
| $\frac{1}{2}(I_{\alpha\alpha*}-I_{\alpha{\rm O6}}){\bf
S}_{\alpha}\,+\,\frac{1}{2}(I_{\alpha\alpha*}+I_{\alpha{\rm O6}}){\bf
A}_{\alpha}$
Table 1: 4d EFT chiral spectrum, in terms of the intersection number
$I_{\alpha\beta}=[\Pi_{\alpha}]\cdot[\Pi_{\beta}]$. ${\bf S}_{a}$ and ${\bf
A}_{a}$ stand for the symmetric and antisymmetric representations of
$U(N_{a})$. If $\Pi_{\alpha}=\Pi_{\alpha*}$, the D6-brane instead hosts either
an $SO(2N_{\alpha})$ or $USp(N_{\alpha})$ gauge group.

The massless open string states determine the gauge sector of the 4d EFT. More
precisely, the 4d chiral spectrum is specified by the topological data of the
three-cycles $\Pi_{\alpha}$ wrapped by D6-branes, as summarised in table 1. To
describe the 4d gauge couplings it is useful to consider a basis of three-
cycle classes $\\{[\Sigma^{J}_{+}],[\Sigma_{K}^{-}]\\}$, where
$[\Sigma_{K}^{-}]$ are as above and $[\Sigma^{J}_{+}]$ is a basis ${\cal
R}$-even classes such that
$[\Sigma_{K}^{-}]\cdot[\Sigma^{J}_{+}]=2\delta^{J}_{K}$. Expanding the three-
cycles wrapped by the D6-branes on such a basis

$[\Pi_{\alpha}]=P_{\alpha\,J}[\Sigma^{J}_{+}]+Q_{\alpha}^{K}[\Sigma_{K}^{-}]\,,\qquad[\Pi_{\alpha*}]=P_{\alpha\,J}[\Sigma^{J}_{-}]-Q_{\alpha}^{K}[\Sigma_{K}^{-}]\,,$
(2.7)

one obtains that the gauge kinetic function associated to each stack is

$2\pi if_{\alpha\alpha}=P_{\alpha\,K}U^{K}\,,$ (2.8)

so that the gauge coupling is $g_{\alpha}^{-2}=(2\pi)^{-1}P_{\alpha\,K}u^{K}$.
The Abelian sector of the gauge group is particularly interesting, as several
$U(1)$ factors develop a Stuc̈kelberg mass as a byproduct of a generalised
Green–Schwarz mechanism [49, 50]. The piece of the 4d EFT Lagrangian
describing such couplings is

$\frac{1}{2}M_{\rm
P}^{2}\int_{X_{4}}G_{KL}\left(d\zeta^{K}-2Q^{K}_{\alpha}A^{\alpha}\right)\wedge*\left(d\zeta^{L}-2Q^{L}_{\beta}A^{\beta}\right)\,,$
(2.9)

where $G_{KL}=\frac{1}{2}\partial_{u^{K}}\partial_{u^{L}}K_{Q}$ and
$A^{\alpha}$ represents the $U(1)_{\alpha}$ gauge boson. It follows that the
$U(1)$ Stückeberg mass matrix reads [51]

$M_{\alpha\beta}^{2}=4G_{KL}Q^{K}_{\alpha}Q^{L}_{\beta}g_{\alpha}g_{\beta}\,M_{\rm
P}^{2}\,.$ (2.10)

In terms of the Kähler potential, (2.9) is encoded in the replacement
$u^{K}\to u^{K}-Q^{K}_{\alpha}V^{\alpha}$ in (2.5), with $V^{\alpha}$ the
vector multiplet associated to $U(1)_{\alpha}$. From here one can deduce that
the chiral fields at D6-brane intersections enter a D-term potential with
field-dependent Fayet-Iliopoulos terms [52]. In particular one finds

$V_{D}=\frac{1}{2}\sum_{\alpha}g_{\alpha}^{2}\left(\xi_{\alpha}+\sum_{i}q^{i}_{\alpha}K_{i\bar{i}}\Phi_{i}\bar{\Phi}_{\bar{i}}\right)^{2}\,,\qquad\text{with}\quad\pi\xi_{\alpha}=Q^{K}_{\alpha}\ell_{K}M_{\rm
P}^{2}\,.$ (2.11)

Here $\Phi_{i}$ represents the scalar component of the 4d chiral field at two
D6-brane intersections, with kinetic terms $K_{i\bar{i}}$ and charge
$q^{i}_{\alpha}$ under $U(1)_{\alpha}$. The D-term-induced mass for fields at
the intersections $\Pi_{\alpha}\cap\Pi_{\beta}$ or
$\Pi_{\alpha}\cap\Pi_{\beta*}$ is555We have redefined the FI terms compared to
other references like [52, 17]: $[\xi_{\alpha}]_{\rm
here}=[\xi_{\alpha}/g_{\alpha}^{2}]_{\rm there}$.

$q^{i}_{\alpha}g_{\alpha}^{2}\xi_{\alpha}+q^{i}_{\beta}g_{\beta}^{2}\xi_{\beta}\,.$
(2.12)

The special Lagrangian condition $-\mbox{Im}\,(e^{-\pi
i\varphi_{\alpha}}\Omega)|_{\Pi_{\alpha}}=0$ with $\varphi_{\alpha}\in
2\mathbbm{Z}$ implies $\xi_{\alpha}=0$, and so when imposed on all D6-branes
the masses (2.12) vanish. As we move in complex structure moduli space
$\\{u^{K}\\}$, the calibration phase $\varphi_{\alpha}$ will vary and we may
generate either positive or tachyonic masses for the scalar fields at the
intersections, describing walls of marginal stability [53]. Microscopically,
one can understand such walls by zooming in near an intersections of two
three-cycles $\Pi_{\alpha}$ and $\Pi_{\beta}$, where they can be approximated
by two special Lagrangian three-planes in flat space, with calibration phases
$\varphi_{\alpha}$ and $\varphi_{\beta}$. Because of this, their relative
orientation can be described by a $U(3)$ rotation $(e^{\pi
i\theta_{\alpha\beta}^{1}},e^{\pi i\theta_{\alpha\beta}^{2}},e^{\pi
i\theta_{\alpha\beta}^{3}})$ such that
$\theta_{\alpha\beta}^{1}+\theta_{\alpha\beta}^{2}+\theta_{\alpha\beta}^{3}=\varphi_{\beta}-\varphi_{\alpha}\mod\mathbbm{Z}$.
One can then apply the angle criterion of Nance and Lawlor [54, 55] to
determine whether or not it is energetically favourable for these three-cycles
to recombine into one. When it is, there is a tachyonic mass term (2.12) for
the field $\Phi_{n}$ at such intersection, which obtains a vev along the
recombination process. The condition

$\theta_{\alpha\beta}^{1}+\theta_{\alpha\beta}^{2}+\theta_{\alpha\beta}^{3}\in
2\mathbbm{Z}\,,$ (2.13)

describes two three-planes related by an $SU(3)$, supersymmetric rotation
[56]. In this case both special Lagrangians three-cycles $\Pi_{\alpha}$ and
$\Pi_{\beta}$ have the same calibration phase and are mutually supersymmetric.

$m_{\Phi_{\alpha\beta}}=0$$m_{{\rm
gon},\alpha\beta}=|\theta_{\alpha\beta}|^{\frac{1}{2}}\,M_{s}$$M_{s}$$M_{\rm
P}$ Figure 2: The spectrum of string oscillator states close to an
intersection at small angle $\theta_{\alpha\beta}$. There is a tower of
gonions with the same charge as the massless mode $\Phi_{\alpha\beta}$ and
characteristic mass $m_{gon,\alpha\beta}$. The tower runs up to the string
scale $M_{s}$, at which the full Hagedorn string spectrum with integer
oscillators is recovered.

Remarkably, the angles $\theta_{\alpha\beta}^{r}$ not only determine the mass
of the lightest scalar field at an intersection, but also the spectrum of an
infinite tower of massive chiral multiplets that transform in the vector-like
representation $({\bf
N}_{\alpha},{\bf\bar{N}}_{\beta})+({\bf\bar{N}}_{\alpha},{\bf N}_{\beta})$. If
we consider the case $|\theta_{\alpha\beta}^{r}|\ll 1$, $r=1,2,3$, the
lightest scalars of such a tower display the following mass spectrum [56, 49]

$m^{2}_{\alpha\beta}(\text{scalar})_{\vec{k},\pm
r}=2\pi\left[\lambda_{\alpha\beta}^{\vec{k}}\pm|\theta_{\alpha\beta}^{r}|\right]\,e^{2\phi_{4}}M_{\rm
P}^{2}\,,$ (2.14)

which has been depicted in figure 2, and where

$\lambda_{\alpha\beta}^{\vec{k}}=\sum_{r}k_{r}|\theta_{\alpha\beta}^{r}|+\frac{1}{2}\sum_{p}|\theta_{\alpha\beta}^{p}|\,,\qquad
k_{r}\in\mathbbm{Z}_{>0}\,.$ (2.15)

Such charged massive states were dubbed gonions in [29], and they form an
infinite tower of states localised at D6-brane intersections that becomes
particularly relevant in models with very small intersection angles. If the
condition (2.13) is satisfied, then the $\frac{1}{2}$-spin states present the
same mass spectrum, combining with the scalars into a tower of 4d ${\cal N}=1$
chiral multiplets. Finally, besides for scalars and fermions, such a tower
also exists for $W$-bosons:

$m^{2}_{\alpha\beta}(\text{vector})_{\vec{k}}=2\pi\lambda_{\alpha\beta}^{\vec{k}}\,e^{2\phi_{4}}M_{\rm
P}^{2}\,.$ (2.16)

We see that in both cases we find a Kaluza–Klein-like tower of states charged
under $U({\bf N}_{\alpha})\times U({\bf N}_{\beta})$. As we reach the string
scale $M_{s}=\ell_{s}^{-1}$ additional states appear, such that we recover a
Hagedorn-like spectrum typical of a critical string, see figure 2. In fact, at
this scale we also find states with higher 4d spin, also charged under the
gauge group. For more details on the gonion spectrum see Appendix A .

Additional aspects of the 4d EFT are revealed from the dimensional reduction
of the D6-brane action [57, 58]. In particular, a smooth three-cycle
$\Pi_{\alpha}$ has $b_{1}(\Pi_{\alpha})$ special Lagrangian deformations and
Wilson line moduli, that are perceived as adjoint fields.666Adjoint fields
develop a superpotential generated by worldsheet instantons [59, 60], but the
resulting masses are typically negligible at large compactification volumes.
Additional superpotential terms are generated for those elements of
$b_{1}(\Pi_{\alpha})$ dual to non-trivial two-cycles of $X_{6}$ [61]. Such
terms incorporate open and closed string moduli in the flux-induced scalar
potential, as discussed in [62, 63, 64, 65]. At the 4d EFT level the D6-brane
moduli redefine the bulk fields $\\{U^{K}\\}$ [57, 58, 62, 63], such that the
Kähler potential is modified and loses its factorised structure. The same
feature is expected for the chiral fields localised at D6-brane intersections.
Their kinetic terms $K_{i\bar{i}}$ should arise from the bulk Kähler potential
$K=K_{K}+K_{Q}$ once the bulk fields have been properly redefined. Although in
this case obtaining such a redefinition is much more involved, below we will
propose an Ansatz based on the known results for physical Yukawa couplings,
which we now turn to discuss.

###  2.2 Yukawas in toroidal and Calabi–Yau orientifolds

Besides the chiral spectrum and FI-terms, key quantities in any 4d chiral EFT
are the physical Yukawa couplings among chiral fields. In the context of type
IIA CY orientifolds with D6-branes, perturbative couplings are generated by
worldsheet disc instantons connecting three D6-brane intersections. The result
takes the form [29, 30]

$Y_{ijk}=h_{qu}\sum_{\vec{n}}d_{\vec{n}}\,e^{2\pi
i\left[\int_{D_{\vec{n}}}J_{c}-\int_{\partial D_{\vec{n}}}\nu\right]}\,.$
(2.17)

Here $D_{\vec{n}}\subset X_{6}$ is a holomorphic disc with boundary on the
three cycles $\Pi_{a}$, $\Pi_{b}$, $\Pi_{c}$, and going through the
intersections $i\in\Pi_{a}\cap\Pi_{b}$, $j\in\Pi_{b}\cap\Pi_{c}$,
$k\in\Pi_{c}\cap\Pi_{a}$. The vector $\vec{n}$ indicates different elements
$[D_{\vec{n}}]$ of the relative homology class
$H_{2}(X_{6};\Pi_{a}\cup\Pi_{b}\cup\Pi_{c},\mathbbm{Z})$ with such boundary
conditions, and $d_{\vec{n}}$ the multiplicity of each class. Finally, $\nu$
is the Wilson line background at each D6-brane and $h_{qu}$ is a contribution
from fluctuations around the semiclassical worldsheet instanton solution [66,
67]. In practice, one can compute this last quantity via CFT correlation
functions [68, 69, 70] or via dimensional reduction of mirror
compactifications of magnetised D9-branes [71, 72]. The resulting overall
scheme is that

  * -

The sum over worldsheet instantons depends on the Kähler moduli of the
compactification and on the D6-brane adjoint moduli.

  * -

The prefactor $h_{qu}$ depends on the complex structure moduli, and in
particular on the D6-brane intersection angles involved in the Yukawa
coupling.

These statements should be taken with care, because of the field redefinitions
that take place when translating the geometric description of the moduli space
of the compactification into 4d supergravity variables. However, one can still
get a precise picture by considering the explicit expressions for Yukawa
couplings obtained in toroidal compactifications. Indeed, on
$\otimes_{r=1}^{3}({\bf T}^{2})_{r}$ naively one obtains a coupling of the
form [30, 71]:

$Y_{ijk}=e^{\phi_{4}/2}\prod_{r=1}^{3}\left(\mbox{Im}\,T^{r}\right)^{1/4}\left[\Theta^{(r)}\right]^{1/4}e^{\frac{1}{2}H^{(r)}}W_{ijk}^{(r)}\,,$
(2.18)

where

$\Theta^{(r)}=2\pi\frac{\Gamma(1-|\chi_{ab}^{r}|)}{\Gamma(|\chi_{ab}^{r}|)}\frac{\Gamma(1-|\chi_{bc}^{r}|)}{\Gamma(|\chi_{bc}^{r}|)}\frac{\Gamma(1-|\chi_{ca}^{r}|)}{\Gamma(|\chi_{ca}^{r}|)}\,,$
(2.19)

is the piece of $h_{qu}$ that depends on the intersection angles. More
precisely, $\chi_{\alpha\beta}^{r}$ is the internal twist felt in the $r^{\rm
th}$ complex plane by the 4d scalar of the $\alpha\beta$ sector that is
involved in the three-point coupling. For small intersection angles we either
have $|\chi_{\alpha\beta}^{r}|=|\theta_{\alpha\beta}^{r}|$ or
$1-|\theta_{\alpha\beta}^{r}|$, see Appendix A for more details. Additionally

$\displaystyle W_{ijk}^{(r)}$ $\displaystyle=$
$\displaystyle\vartheta\left[\begin{array}[]{c}\delta_{ijk}^{r}\\\
0\end{array}\right]\left(\nu^{r},|I_{ab}^{r}I_{bc}^{r}I_{ca}^{r}|T^{r}\right)\,,$
(2.22) $\displaystyle H^{(r)}$ $\displaystyle=$ $\displaystyle 2\pi
i\frac{\nu^{r}\mbox{Im}\,\nu^{r}}{\mbox{Im}\,T^{r}}|I_{ab}^{r}I_{bc}^{r}I_{ca}^{r}|^{-1}\,,$
(2.23)

encode the worldsheet instanton dependence on the Kähler and D6-brane moduli,
with $\vartheta$ the Jacobi theta function, $I_{\alpha\beta}^{r}$ the
intersection number on $({\bf T}^{2})_{r}$,
$\nu^{r}=I_{ab}^{r}\nu_{c}^{r}+I_{bc}^{r}\nu_{a}^{r}+I_{ca}^{r}\nu_{b}^{r}$ a
combination of open string adjoint moduli and
$\delta_{ijk}^{r}=\frac{i^{r}}{I_{ab}^{r}}+\frac{j^{r}}{I_{bc}^{r}}+\frac{k^{r}}{I_{ca}^{r}}$
scanning over the different intersections. We refer to [30, 71] for more
details on these expressions.

Now, as pointed out in [73], in order for the matter fields at the
intersections to transform properly under the torus modular group, one should
redefine them as follows

$\hat{\Phi}_{\alpha\beta}=\left(\prod_{r=1}^{3}(\mbox{Im}\,T^{r})^{1/4}e^{i\pi\frac{\nu_{\alpha\beta}^{r}{\rm
Im}\nu_{\alpha\beta}^{r}}{I_{\alpha\beta}^{r}{\rm
Im}T^{r}}}\right)\Phi_{\alpha\beta}\,,$ (2.24)

where $\nu^{r}_{\alpha\beta}=\nu^{r}_{\alpha}-\nu^{r}_{\beta}$. This implies
that the physical Yukawa couplings are in fact

$Y_{ijk}=e^{\phi_{4}/2}\prod_{r=1}^{3}\left(\mbox{Im}\,T^{r}\right)^{-1/2}\left[\Theta^{(r)}\right]^{1/4}W_{ijk}^{(r)}\,.$
(2.25)

From here, one can deduce the following structure for the physical Yukawas
couplings in a general Calabi–Yau compactification

$Y_{ijk}=\frac{e^{\phi_{4}/2}}{{\rm
Vol}_{X}^{1/2}}\,\Theta_{ijk}^{1/4}\,W_{ijk}\,,$ (2.26)

where $\Theta_{ijk}$ only depends on the intersection angles, which now can
differ for each triplet $ijk$, and $W_{ijk}$ is a holomorphic function on the
complexified Kähler and D6-brane position moduli. The intersection angles, and
therefore $\Theta_{ijk}$ are invariant under an overall rescaling of all
saxions $u^{K}\to\lambda u^{K},\forall K$, since this amounts to perform a 4d
dilaton rescaling $e^{-\phi_{4}}\to\lambda e^{-\phi_{4}}$, and leave the
periods of $\Omega/[\int\Omega\wedge\bar{\Omega}]^{1/2}$ fixed. This implies
that $\Theta_{ijk}$ is a homogeneous function of degree zero on the saxions
$\\{u^{K}\\}$, and as a result the physical Yukawas are homogeneous functions
of degree $-1/2$ on them.

As we will now discuss, the result (2.26) has a neat interpretation in terms
of the Kähler metric for the matter fields at the intersection.

###  2.3 Kinetic terms of chiral fields

One interesting byproduct of an expression like (2.26) is that it gives
valuable information on the kinetic term for the chiral fields at the D6-brane
intersections. Indeed, assuming no kinetic mixing between such fields, one
should be able to express the physical Yukawa coupling in terms of the usual
4d ${\cal N}=1$ supergravity formula

$Y_{ijk}=e^{K/2}\left(K_{i\bar{i}}K_{j\bar{j}}K_{k\bar{k}}\right)^{-1/2}\,W_{ijk}\,.$
(2.27)

Comparing with (2.26) and using the above expressions for $K=K_{K}+K_{Q}$ one
obtains

$K_{i\bar{i}}K_{j\bar{j}}K_{k\bar{k}}=e^{3\phi_{4}}\Theta_{ijk}^{-1/2}\,.$
(2.28)

That is, we recover a simple dependence on the 4d dilaton plus a function on
the intersection angles. Let us consider the explicit result for
$\Theta_{ijk}$ in toroidal compactifications and use that the matter field
metrics should only depend on the angles at their own intersection. One
arrives at the following result

$K_{i\bar{i}}=e^{\phi_{4}}(2\pi)^{-1/2}\prod_{r=1}^{3}\left(\frac{\Gamma(|\chi_{i}^{r}|)}{\Gamma(1-|\chi_{i}^{r}|)}\right)^{1/2}\,.$
(2.29)

This implies that the Kähler metrics are functions of degree $-1$ on the
saxions $\\{u^{K}\\}$.

To see how this expression connects with previous results in the literature,
let us consider the case where
$\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2}>0$ and
$\theta_{\alpha\beta}^{3}<0$. As it follows from the discussion in Appendix A,
the lightest 4d scalar in this sector corresponds to the twist vector
$\chi_{\alpha\beta}=(\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2},\theta_{\alpha\beta}^{3}+1,0)=(|\theta_{\alpha\beta}^{1}|,|\theta_{\alpha\beta}^{2}|,|1-\theta_{\alpha\beta}^{3}|,0)$.
Hence its kinetic term is

$K_{i\bar{i}}=e^{\phi_{4}}(2\pi)^{-1/2}\left[\frac{\Gamma(|\theta_{\alpha\beta}^{1}|)}{\Gamma(1-|\theta_{\alpha\beta}^{1}|)}\frac{\Gamma(|\theta_{\alpha\beta}^{2}|)}{\Gamma(1-|\theta_{\alpha\beta}^{2}|)}\frac{\Gamma(1-|\theta_{\alpha\beta}^{3}|)}{\Gamma(|\theta_{\alpha\beta}^{3}|)}\right]^{1/2}\stackrel{{\scriptstyle|\theta^{r}_{\alpha\beta}|\ll
1}}{{\simeq}}\left[\frac{e^{2\phi_{4}}|\theta_{\alpha\beta}^{3}|}{2\pi|\theta_{\alpha\beta}^{1}||\theta_{\alpha\beta}^{2}|}\right]^{1/2}\,,$
(2.30)

where in the last equality we have assumed that $|\theta_{\alpha\beta}^{r}|\ll
1$ and
$|\theta^{3}_{\alpha\beta}|=|\theta_{\alpha\beta}^{1}|+|\theta_{\alpha\beta}^{2}|$.777If
the last assumption is not true, one should add a factor
$e^{-\gamma_{E}(|\theta_{\alpha\beta}^{1}|+|\theta_{\alpha\beta}^{2}|-|\theta^{3}_{\alpha\beta}|)}$
in the expression for $K_{i\bar{i}}$, where $\gamma_{E}$ is the Euler-
Mascheroni constant [69, 70]. As a result the rhs of (2.30) remains valid.
This agrees with several results in the literature [69, 74, 70, 71, 75, 72],
up to an explicit dependence on the Kähler moduli which is removed by the
redefinition (2.24).

The presence of the 4d dilaton in (2.29) will play an important role in our
discussion. To motivate this sort of dependence, let us consider two
intersecting D6-branes in a smooth compact Calabi–Yau. As mentioned above, one
expects the fields at D6-brane intersections to redefine the 4d supergravity
variables that come from complex structure moduli of the compactification.
Indeed, giving a vev to these fields amounts to perform a recombination of
three-cycles, whose geometry is specified by the complex structure moduli of
$X_{6}$. It is therefore natural to consider the following Ansatz

$K_{Q}=-2\log{\cal H}=-2\log\left(\hat{\cal
H}-f(\Phi^{i},\bar{\Phi}^{\bar{i}})\right)=-2\log\left(\hat{\cal
H}-\frac{1}{2}k_{i\bar{i}}\Phi^{i}\bar{\Phi}^{\bar{i}}+\dots\right)\,,$ (2.31)

where $\hat{\cal H}$ is the same function as ${\cal H}$, but it depends on the
redefined variables $\hat{u}^{K}$, and in the last step we have performed an
expansion in the intersection fields. If we perform the same expansion in the
logarithm we find

$K=-2\log\hat{\cal H}+\frac{k_{i\bar{i}}}{\hat{\cal
H}}\Phi^{i}\bar{\Phi}^{\bar{i}}+\dots$ (2.32)

For vanishing vevs of the intersection fields we can identify $\hat{\cal H}$
with $e^{-2\phi_{4}}$, and so our result above translates into

$k_{i\bar{i}}=e^{-\phi_{4}}(2\pi)^{-1/2}\prod_{r=1}^{3}\left(\frac{\Gamma(|\chi_{i}^{r}|)}{\Gamma(1-|\chi_{i}^{r}|)}\right)^{1/2}\,.$
(2.33)

Notice that, besides the 4d dilaton dependence, the rest of the Kähler metrics
depend on purely local data, namely the intersection angles. As emphasised
before, local intersection data and the associated spectrum of states are
independent on the global features of the compactification manifold. As such,
we expect the result (2.33) to be also valid in for intersecting D6-branes in
smooth Calabi–Yau manifolds.

In the following sections we will be interested in considering Yukawa
couplings and Kähler metrics along limits of small intersecting angles. Let us
first consider the case where all three intersection angles are small but non-
vanishing. If as in (2.30) we take
$\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2}>0$ and
$\theta_{\alpha\beta}^{3}<0$, then the matter field metric will look like the
rhs of this equation. If we now impose the supersymmetry condition
$|\theta_{\alpha\beta}^{3}|=|\theta_{\alpha\beta}^{1}|+|\theta_{\alpha\beta}^{2}|$
then either all angles will be of the same order of magnitude or one of them
will be much smaller than the other two. In both cases the matter field
metrics will go like $(\theta_{\alpha\beta}^{\rm min})^{-1/2}$ where
$\theta_{\alpha\beta}^{\rm min}\equiv{\rm
min}\\{|\theta_{\alpha\beta}^{r}|\\}$. From looking at the spectrum (2.14) we
realise that this determines the mass scale of the lightest gonion tower at
the intersection, which we denote as $m_{\rm gon}^{i}$. More precisely we find

$k_{i\bar{i}}\simeq\frac{M_{\rm P}}{m_{\rm gon}^{i}}\,.$ (2.34)

One can see that this result is valid for other sign choices of the angles, or
even when not all of them are small. Indeed, the setup where two intersecting
D6-branes have $|\theta_{\alpha\beta}^{r}|\ll 1$ is quite standard in D-brane
compactifications, and it is for instance realised by the mirror of two
magnetised D9-branes at very large volume. One may also consider the mirror of
a magnetised D9-brane and a D5-brane [71], or that of two intersecting
D7-branes with worldvolume fluxes [74, 76]. In both cases in the limit of
dilute fluxes one of the angles goes to zero, say $\theta_{\alpha\beta}^{1}\to
0$, while the remaining two remain finite and become equal in magnitude
$|\theta_{\alpha\beta}^{2}|-|\theta_{\alpha\beta}^{3}|\to 0$. Plugged into
(2.29) and taking the limit, one again recovers (2.34). Finally, when no angle
is small, $m_{\rm gon}^{\rm min}$ is of the order of the string scale, and so
(2.34) is still a reasonable estimate of (2.33). In section 5 we will provide
a derivation of this remarkably simple result from the viewpoint of emergence.

There are however exceptions to this rule, like when two D6-branes $\alpha$
and $\beta$ do not intersect transversely, and one of the angles is
identically zero. For instance, one may have two D6-branes intersecting over a
one-cycle, as in the Pati–Salam example of section 4. In that case there are
only two gonion towers below the string scale, and the kinetic terms for the
fields at the intersection are not determined by such gonion masses. While one
can still recover the correct kinetic terms as a limit of the expression
(2.29), it does not make sense to relate it with a (non-existent) gonion
tower. Instead, it is more appropriate to express such a kinetic terms as

$K_{i\bar{i}}=e^{2\phi_{4}}h_{i\bar{i}}\prod_{r=1}^{2}\left(\frac{\Gamma(|\chi_{i}^{r}|)}{\Gamma(1-|\chi_{i}^{r}|)}\right)^{1/2}\simeq
e^{2\phi_{4}}h_{i\bar{i}}\,,$ (2.35)

where $h_{i\bar{i}}$ is a homogeneous function of degree one on the saxions
$\\{u^{K}\\}$, and in the rhs we have used supersymmetry. In explicit examples
one can see that $h_{i\bar{i}}$ satisfies

$h_{i\bar{i}}\lesssim g^{-1}_{\alpha}g^{-1}_{\beta}\,.$ (2.36)

Further exceptions should arise from setups where the would-be lightest gonion
tower is removed by some effect, like an orbifold projection or world-sheet
instanton induced masses. In the mirror type IIB frame, (2.35) is expected to
apply for magnetised D7-branes wrapping the same four-cycle, like in [77], or
for intersecting D7-branes with no chiral matter at their intersection [74].
Note that in flat space (2.35) occurs when two D-branes preserve a common
${\cal N}=2$ supersymmetry. Abusing language, we will dub all these exceptions
as ${\cal N}=2$ sectors, even when they host chiral matter. Notice that
$K_{i\bar{i}}$ are always homogeneous functions of degree $-1$.

Finally, notice that (2.33) is nothing but the Kähler metric for a chiral
field, measured in units of the string scale $M_{s}$. Whenever $m_{\rm
gon}^{i}/M_{\rm P}\to 0$, this metric blows up, and also does so in units of
the lower, 4d EFT scale $\Lambda_{\rm EFT}$. This indicates that one should
decouple kinematically this chiral field from the 4d EFT, and that we are
approaching a singular limit of the EFT. As we will argue in the next section,
this sort of behaviour occurs whenever a Yukawa coupling (2.26) vanishes
asymptotically for a large complex structure limits. The vanishing Yukawa is
accompanied by a vanishing D-brane gauge coupling, under which chiral matter
at some intersection is charged. This triggers the appearance of a gonion
tower at that intersection, preventing the EFT to reach such a singular,
infinite distance limit.

##  3 The $Y\to 0$ limit and infinite distance

In this section we explore the limits $Y\to 0$ for Yukawas of chiral matter
fields. Along large complex structure limits, one finds simple expression for
the Yukawa asymptotic behaviour in terms of gonion masses. In particular, if
the Yukawa connects D6-brane intersections with three gonion towers each, one
recovers the expression (1.3), which shows that a limit in which $Y\to 0$ is
accompanied by a massless gonion tower. If the Yukawa involves an intersection
with only two gonion towers, dubbed ${\cal N}=2$ sector in section 2, the
expression is slightly different. However, we argue that in all cases at least
one of the gauge couplings related to the Yukawa vanishes asymptotically,
which gives a rationale of why a gonion scale tends to zero. We do so by
estimating the lightest gonion masses in terms of the gauge couplings of the
D6-branes and the tensions of the 4d strings that they couple to via their FI-
terms. We use this same estimate to discuss the different mass scales in
simple complex structure limits.

###  3.1 Small Yukawas and gonion masses

Given the expression (2.26) for the Yukawa couplings in Planck units, one may
analyse those limits in which the different Yukawa couplings go to zero value.
For this, let us distinguish their dependence on three different set of
moduli:

  * -

The Yukawas depend on D-brane moduli in the adjoint representation of the
gauge group, through the holomorphic piece $W_{ijk}$. As it is clear for the
toroidal example (2.22) some Yukawas vanish at special vevs of the
complexified D6-brane position moduli $\nu_{r}$ [30], at finite distance
points in moduli space. The presence of such moduli is however unwanted in
realistic constructions, so in the following we will consider Yukawas arising
from D-branes with no adjoint moduli. This can be achieved by considering
D6-branes that wrap three-cycles $\Pi_{\alpha}$ with $b_{1}(\Pi_{\alpha})=0$,
or whose adjoint moduli are projected out via some orbifold action, as in
[78].

  * -

The squared Yukawas depend on the Kähler moduli as:

$|Y_{ijk}|^{2}=A\,\frac{|W_{ijk}|^{2}}{{\rm Vol}_{X}}\,,$ (3.1)

where $A>0$ is a function of the complex structure saxions $\\{u^{K}\\}$, that
include the 4d dilaton and that we treat as an ${\cal O}(1)$ constant factor.
To reach the limit $|Y|\to 0$ we must take ${\rm Vol}_{X}\to\infty$ and
decompactify some of the extra dimensions.888One could consider trajectories
in which ${\rm Vol}_{X}$ remains finite and $W_{ijk}\to 0$ due to its
dependence on the Kähler moduli. This would however not guarantee the
vanishing of the Yukawa, because corrections coming from D2-brane instantons
[79, 80] would not necessarily vanish along the limit. All of these limits are
of infinite distance and, as it follows from the analysis in [23, 26] the
lightest tower of massive state predicted by the Distance Conjecture is made
up of D0-branes, that satisfy $m_{\rm D0}^{2}\simeq M_{\rm P}^{2}/{\rm
Vol}_{X}$. As a result:

$m_{\rm D0}\sim|Y|M_{\rm P}$ (3.2)

if we consider the case where $|W|$ is of order one and constant along the
limit. Since this limit takes us to a region of strong coupling, to analyse it
properly one should reconsider it in the context of M-theory.

  * -

The squared Yukawas depend on the complex structure saxions as:

$|Y_{ijk}|^{2}=B\,e^{\phi_{4}}\Theta_{ijk}^{1/2}\,,$ (3.3)

where $B>0$ is function of the Kähler moduli, that we treat as an ${\cal
O}(1)$ numerical factor. If the Yukawa involves an ${\cal N}=2$ sector, then
one of the angles within $\Theta_{ijk}$ must be set to zero via the
appropriate limit. In any case, $|Y_{ijk}|^{2}$ is always a homogeneous
function of degree $-1$ on the saxions $\\{u^{K}\\}$, so taking the trajectory
$u^{K}\to\lambda u^{K},\forall K$ with $\lambda\to\infty$ makes
$|Y_{ijk}|^{2}$ scale like $\lambda^{-1}$. At the same time, due to (2.8) all
D6-brane gauge couplings scale like $g_{\alpha}\sim\lambda^{-1/2}\sim|Y|$.
Therefore, asymptotically one recovers a large global symmetry, in agreement
with the fact that this is an infinite distance limit. From general Swampland
arguments, one expects the appearance of asymptotically massless towers of
particles charged under these gauge symmetries, namely gonion towers.

Indeed, because this trajectory amounts to the weak coupling limit
$e^{\phi_{4}}\to e^{\phi_{4}}/\lambda$ and to leave $\Theta_{ijk}$ constant,
all gonion masses scale like $m_{\rm gon}\sim M_{s}\sim\lambda^{-1}M_{\rm P}$.
Hence the gonion towers essentially act like that of a fundamental string, and
we obtain the relation

$m_{\rm gon}\sim|Y|^{2}M_{\rm P}\,.$ (3.4)

Clearly, $|Y|,g\to 0$ occurs at the same time as $m_{\rm gon}/M_{\rm P}\to 0$.
From the results of [23, 26], one expects the presence of additional towers of
Kaluza–Klein and winding modes with the same asymptotic behaviour as $m_{\rm
gon}$, see section 3.4 for more details. Thus, we find a tower spectrum that
would seem to implement an emergent string limit, with the novelty that these
particles are charged under the gauge group hosted by D6-branes.

Let us now consider more involved trajectories the space of saxions
$\\{u^{K}\\}$, in which not all of them but only a subset is very
large.999Considering small $u^{K}$ takes us away from our regime of control.
In that case D2-brane instanton effects become relevant and the corresponding
axionic shift symmetries of (2.5) are no longer present. In such regions
$e^{\phi_{4}}\ll 1$, and so naively the Yukawa couplings (2.26) are very
suppressed unless this is compensated by $\Theta_{ijk}$. In this sense it is
useful to distinguish different cases:

  * i)

The Yukawa does not involve any ${\cal N}=2$ sector. Then, using the same
reasoning to derive (2.34) we conclude that in these regions we have

$|Y_{ijk}|^{2}\simeq B\,e^{-2\phi_{4}}\frac{m_{\rm gon}^{i}}{M_{\rm
P}}\frac{m_{\rm gon}^{j}}{M_{\rm P}}\frac{m_{\rm gon}^{k}}{M_{\rm P}}\,,$
(3.5)

where $m_{\rm gon}^{i}$ is the mass scale of the lightest gonion tower at the
$i^{\rm th}$ intersection in $\Pi_{a}\cap\Pi_{b}$, etc. Just like (2.26), we
expect this relation to hold in general Calabi–Yau compactifications, as long
as we are in large complex structure regions.

Since $e^{-\phi_{4}}\gg 1$ by assumption, any trajectory along which $|Y|\to
0$ implies that at least one gonion tower becomes massless in Planck units.
Reciprocally, because $e^{-\phi_{4}}m_{\rm gon}/M_{\rm P}<1$, if one gonion
tower goes to zero in Planck units all the Yukawas in which such an
intersection is involved will also vanish asymptotically. In section 3.3 we
argue that then at least one gauge coupling under which this gonion tower is
charged must also vanish asymptotically, signalling that we are approaching an
infinite distance limit.

In general, large complex structure limits of infinite distance imply that
$e^{\phi_{4}}\to 0$ and all Yukawas of the form (3.5) vanish. To see their
relation with the lightest gonion mass, let us consider a trajectory where the
scalings satisfy

$\left(\frac{m_{\rm gon}^{i}}{M_{\rm
P}}\right)^{1/w_{i}}\sim\left(\frac{m_{\rm gon}^{j}}{M_{\rm
P}}\right)^{1/w_{j}}\sim\left(\frac{m_{\rm gon}^{k}}{M_{\rm
P}}\right)^{1/w_{k}}\sim e^{\phi_{4}}\,,\quad w_{i},w_{j},w_{k}\geq 1\,.$
(3.6)

Defining $m_{\rm gon}^{ijk}\equiv{\rm min}(m_{\rm gon}^{i},m_{\rm
gon}^{j},m_{\rm gon}^{k})$ and $w_{ijk}\equiv{\rm max}(w_{i},w_{j},w_{k})$,
one obtains

$m_{\rm gon}^{ijk}\sim|Y_{ijk}|^{\frac{2w_{ijk}}{w_{i}+w_{j}+w_{k}-2}}M_{\rm
P}\,,$ (3.7)

Notice that the maximum exponent in the rhs is $|Y|^{2}$, as in the case
(3.4).

  * ii)

One (possibly chiral) ${\cal N}=2$ sector is involved in the Yukawa, say the
intersections in $i\subset\Pi_{a}\cap\Pi_{b}$. Then the Yukawa asymptotic
behaviour is captured by

$|Y_{ijk}|^{2}\simeq B\,e^{-2\phi_{4}}h_{i\bar{i}}^{-1}\frac{m_{\rm
gon}^{j}}{M_{\rm P}}\frac{m_{\rm gon}^{k}}{M_{\rm
P}}=B\,h_{i\bar{i}}^{-1}\frac{m_{\rm gon}^{j}}{M_{s}}\frac{m_{\rm
gon}^{k}}{M_{s}}\,.$ (3.8)

In this case it is not true that the Yukawa coupling must necessarily vanish
whenever $e^{\phi_{4}}\to 0$. Some of the large complex structure infinite
distance trajectories will not modify neither $h_{i\bar{i}}$ nor the
intersection angles involved in the Yukawa, and as such (3.8) will remain
invariant along the limit.

A trajectory for which $|Y|\to 0$ implies that either $m_{\rm gon}/M_{\rm
P}\to 0$ or $h_{i\bar{i}}\to\infty$. The first case is similar to the one
above, while in the second one we also expect a vanishing gauge coupling due
to (2.36). Moreover, in the examples of section 4 we will identify
$h_{i\bar{i}}^{-1}\simeq m_{\rm w}/M_{\rm P}$, where $m_{\rm w}$ is a winding
scale for bifundamental matter (see ft. 17). This suggests that there is
always a gauge coupling and a tower of states charged under it that asymptotes
to zero mass whenever $|Y|\to 0$.

  * iii)

If more than one ${\cal N}=2$ sector is involved in the Yukawa, one may
proceed as before by performing further replacements $m_{\rm gon}^{j}/M_{\rm
P}\to h_{j\bar{j}}$. However, in concrete setups the corresponding Yukawas do
not seem to be related to a chiral sector of the compactification, unless one
generates chirality via an orbifold projection.101010For instance, it was
shown in [81] that for D7-branes wrapping Hirzebruch or del Pezzo surfaces,
Yukawas involving more than one ${\cal N}=2$ sector vanish identically. For
this reason we will not consider this case in the following.

To sum up, the expressions (3.5) and (3.8) encode the possible relations
between the lightest gonion towers and the Yukawa couplings along different
limits. However, they do not explain how these two quantities depend on the
compactification moduli, nor which other mass scales are relevant in this
setup. One can gain some insight on both of these aspects by relating gonion
masses to 4d string tensions, as we now discuss.

###  3.2 Gonions, strings and monopoles

One difficulty in evaluating expressions for the Yukawa couplings like (2.26)
or (3.5) is that they depend on a very different set of data. On the one hand,
the 4d dilaton ${\phi_{4}}$ is a function of the complex structure moduli
vevs, which are global compactification data. On the other hand, the
intersection angles and the gonion masses depend on local data at each
D6-brane intersection. While gonion masses clearly depend on the complex
structure fields, the dictionary between global and local data is very
involved as soon as we consider non-trivial Calabi–Yau metrics. In spite of
this, in the following we would like to propose an estimate for the lightest
gonion mass at the set of intersections $\Pi_{a}\cap\Pi_{b}$, based on the
properties of 4d strings and monopoles in this class of compactifications.
This estimate will be later on used to discuss the spectrum of energy scales
that appear at regions of small Yukawa couplings.

Let us consider a type IIA orientifold compactification with intersecting
D6-branes, as described in section 2.1. To each three-cycle $\Pi_{\alpha}$
wrapped by a D6-brane one can associate an Abelian gauge factor
$U(1)_{\alpha}$, and some linear combinations of these $U(1)$’s remain
massless after the Stückelberg mechanism has been implemented. As captured in
table 1, the condition for a massless $U(1)$ is that the linear combination
$c_{\alpha}[\Pi_{\alpha}]$ minus its orientifold image is a trivial homology
class of [82]. This implies that such a combination of three-cycles can be
connected by a four-chain $\Sigma_{4}$ wrapped by a D4-brane, which appears as
a 4d monopole of this $U(1)$ factor [83]. The simplest case in which such a
monopole appears is when one considers a stack of two D6-branes wrapping a
three-cycle $\Pi$ with a position modulus $\nu$, which one can use to perform
an adjoint Higgssing $U(2)\to U(1)_{a}\times U(1)_{b}$ by giving a relative
vev $\nu_{a}-\nu_{b}=\zeta\ell_{s}$ to the three-cycles. After the Higgssing,
particles with electric and magnetic charges under the massless combination
$U(1)=\frac{1}{2}[U(1)_{a}-U(1)_{b}]$ appear, and for $\zeta\ll 1$ they
satisfy

$m_{\rm mono}\simeq g^{-2}m_{\rm ele}\,,$ (3.9)

as figure 3 illustrates for the case of flat space.

Figure 3: Two D6-branes after the adjoint Higgssing $U(2)\to U(1)\times U(1)$.
Electrically charged particles arise from open strings stretched between both
D6-branes. Magnetic charges arise from D4-branes on four-chains connecting
both three-cycles.

We now consider the case of a $U(1)$ with a non-vanishing Stückelberg
coupling. Now $[\Pi_{a}]\equiv\sum c_{\alpha}[\Pi_{\alpha}]$ will be a non-
trivial three-cycle class in $H_{3}(X_{6},\mathbbm{Z})$ (even after
subtracting its orientifold image) and so there is no four-chain connecting
the constituent three-cycles. To have the analogue of the previous setup we
must first consider a D4-brane wrapping a three-cycle on the class
$[\Pi_{a}]-[\Pi_{a*}]$, and then a second D4-brane on a four-chain
$\Sigma_{4}$ that connects this D4 with the D6-branes wrapping $\Pi_{a}$ and
$\Pi_{a*}$. This new configuration fits well with our field theory intuition
regarding monopoles of $U(1)$’s with a $B\wedge F$ coupling [84]. The 4d
monopole, which in our setup is represented by the D4-brane in $\Sigma_{4}$,
is not a gauge invariant operator by itself, and to cure this it must have a
4d string ending on it, represented by the D4-brane (or set of D4-branes)
wrapping $[\Pi_{a}]-[\Pi_{a*}]$. See figure 4 for a specific flat-space
representation of this setup.

Figure 4: Configuration of D6-branes intersecting at one angle. Particles with
bifundamental electric charge still arise from open strings stretched between
both branes, including the gonion tower. However, now monopoles (D4-branes on
the shaded green area) need to have 4d strings ending on them (D4-branes on
the orange three-cycle).

One may now compare the mass of electrically and magnetically charged objects.
Since we are in a 4d ${\cal N}=1$ setting the mass of the monopole is not
protected by supersymmetry, but the tension of the 4d string is, and we have
the field theory relation between this tension and the mass of the open
strings at the D6-brane intersection. For simplicity, let us consider a single
D6-brane on $\Pi_{a}$ and its orientifold image on $\Pi_{a*}$. In a
configuration like in figure 4, it is easy to convince oneself of the relation

$T_{\rm string}=2\pi|\xi_{a}|\simeq g_{a}^{-2}|m_{\rm ele}^{2}|\,,$ (3.10)

where $\xi_{a}$ is the FI-term of $U(1)_{a}$. Here we assume a regime such
that $\xi_{a}\ell_{s}^{2}\ll 1$, which corresponds to a small intersection
angle and to the field theory approximation. Besides being the (tachyonic)
mass of the lightest state, $m_{\rm ele}$ corresponds to the scale of the
gonion tower at the D6-brane intersection. We thus see that, in this case, the
gonion scale reduces to $g_{a}T_{\rm string}^{1/2}$, which is a simple
function of the complex structure saxions.

This configuration, however, is too simple to represent realistic D6-branes
models. In particular, the 4d string that couples to the D6-branes corresponds
to an EFT string, in the sense of [25, 26, 27]. As mentioned above, this is a
particular class of 4d BPS strings, whose tension only vanishes at infinite
distance boundaries in moduli space, precisely where an approximate axionic
shift symmetry becomes exact. As a result, the corresponding pair of D6-branes
displays a non-vanishing FI-term at any point in moduli space, and $\Pi_{a}$
and $\Pi_{a*}$ are always non-mutually BPS, as indeed happens in the figure,
where there is a single non-trivial angle and $m_{\rm ele}^{2}$ describes a
tachyonic mode.

In the intersecting D6-brane models of section 2.1, the analogue of relation
(3.10) is given by (2.11) and (2.12). To each D6-brane wrapping a three-cycle
$\Pi_{\alpha}$ we associate the homology class
$[\Pi_{\alpha}]-[\Pi_{\alpha^{*}}]=2Q_{\alpha}^{K}[\Sigma_{K}^{-}]$, which
corresponds to a 4d string charge. The tension $T_{\alpha}=2\pi\xi_{\alpha}$
related to this string charge depends on the complex structure saxions
$\\{u^{K}\\}$, and the functions $g_{\alpha}^{2}T_{\alpha}$ determine the mass
of the lightest open strings at the different D6-brane intersections through
(2.12). The dictionary with gonion towers is however no longer as simple as in
figure 4, because now $g_{\alpha}^{2}T_{\alpha}$ corresponds to a (signed) sum
of EFT string tensions and of gonion scales, and cancellations can occur.

Indeed, in the setup of section 2.1 we have chosen the three-cycles
$[\Sigma_{K}^{-}]$ to generate the cone of EFT strings. The corresponding 4d
strings are dubbed elementary EFT strings, and their tension is given by
$T_{K}=\ell_{K}M_{\rm P}^{2}$. The 4d string tension associated to
$\Pi_{\alpha}$ reads $T_{\alpha}=2Q^{K}_{\alpha}T_{K}$, and it cannot
correspond to an EFT string tension because by assumption it vanishes at some
locus in the interior of the moduli space. Necessarily some $Q^{K}_{\alpha}$
must be negative, and so the D-flatness condition
$T_{\alpha}=2\pi\xi_{\alpha}=0$ for $\Pi_{\alpha}$ translates into a relation
between elementary EFT string tensions. Additionally, the condition
$g_{\alpha}^{2}T_{\alpha}=g_{\beta}^{2}T_{\beta}$ implies the supersymmetry
relation (2.13) between the sum of intersection angles at
$\Pi_{\alpha}\cap\Pi_{\beta}$, while
$g_{\alpha}^{2}T_{\alpha}=-g_{\beta}^{2}T_{\beta}$ does the same for
$\Pi_{\alpha}\cap\Pi_{\beta^{*}}$. This suggests that the gonion masses at
these intersection must depend on the complex structure saxions through the
functions $g_{\alpha}^{2}T_{K_{\alpha}}$, $g_{\beta}^{2}T_{K_{\beta}}$, where
${K_{\alpha}}$ runs over the values such that $Q_{\alpha}^{K_{\alpha}}\neq 0$.
In particular, if all these functions have a certain scaling along an infinite
distance limit, we would expect such gonion masses to scale in the same way.

In the following we would like to translate this intuition into a proposal for
estimating the lightest gonion scale coming from the intersections
$\Pi_{\alpha}\cap\Pi_{\beta}$, at asymptotic regions in moduli space. Let us
first consider the case where $\Pi_{\alpha}\cap\Pi_{\beta}$ corresponds to an
${\cal N}=2$ sector, so there are only two intersection angles
$\theta_{\alpha\beta}^{1}$ and $\theta_{\alpha\beta}^{2}$. For the three-cycle
$\Pi_{\alpha}$ we split the subset of indices $K_{\alpha}\in K$ as
$\\{K_{\alpha}\\}=\\{K_{\alpha}^{+},K_{\alpha}^{-}\\}$ such that
$Q^{K_{\alpha}^{+}}_{\alpha}>0$ and $Q^{K_{\alpha}^{-}}_{\alpha}<0$, and we do
the same for $\Pi_{\beta}$. We now define

$m_{\alpha\beta,+}^{2}=g_{\alpha}^{2}T_{\alpha}^{+}-g_{\beta}^{2}T_{\beta}^{-}\,,\qquad
m_{\alpha\beta,-}^{2}=-g_{\alpha}^{2}T_{\alpha}^{-}+g_{\beta}^{2}T_{\beta}^{+}\,,$
(3.11)

where
$T_{\alpha}^{+}=\sum_{K_{\alpha}^{+}}Q^{K_{\alpha}^{+}}_{\alpha}T_{K_{\alpha}^{+}}$
and
$T_{\alpha}^{-}=\sum_{K_{\alpha}^{-}}Q^{K_{\alpha}^{-}}_{\beta}T_{K_{\alpha}^{-}}$,
and similarly for $T_{\beta}^{\pm}$.111111An alternative definition would
involve the decomposition
$[\Pi_{\alpha}]-[\Pi_{\alpha^{*}}]=2\tilde{Q}^{K}_{\alpha}[\tilde{\Sigma}_{K}^{-}]$,
where $[\tilde{\Sigma}_{K}^{-}]$ is a basis of odd three-cycles hosting 4d BPS
strings, in the particular region under consideration. From here one can
define analogous quantities to $T_{\alpha}^{\pm}$, $m_{\alpha\beta,\pm}^{2}$
and $m_{\alpha\beta,{\rm min}}^{2}$. Because the cone of 4d BPS strings is
larger than that of 4d EFT strings (see [26] for examples), the former
involves smaller tensions that vanish at finite-distance boundaries of the
moduli space. It would be interesting to see whether this would give a more
accurate gonion mass estimate. Notice that $m_{\alpha\beta,\pm}^{2}>0$ at any
point in moduli space, and that at the supersymmetric locus we have that
$m_{\alpha\beta,+}^{2}=m_{\alpha\beta,-}^{2}$, which implies
$|\theta_{\alpha\beta}^{1}|=|\theta_{\alpha\beta}^{2}|$. If the intersection
angles are small, it is natural to identify $m_{\alpha\beta,\pm}$ with the
gonion scale at this intersection.

The case of three intersection angles is more involved. Let us for instance
consider the case where $|\theta_{\alpha\beta}^{r}|\ll 1$ and
$|\theta^{3}_{\alpha\beta}|=|\theta_{\alpha\beta}^{1}|+|\theta_{\alpha\beta}^{2}|$.
Then one can identify $m_{\alpha\beta,\pm}$ with the largest gonion scale of
the three and, if they are all of the same order, it provides a good estimate
of the lightest one as well. If, on the contrary, there is a hierarchy between
min$(|\theta_{\alpha\beta}^{1}|,|\theta_{\alpha\beta}^{2}|)$ and
$|\theta^{3}_{\alpha\beta}|$ then $m_{\alpha\beta,\pm}$ only serves as an
upper bound for the lightest gonion scale $m_{\rm gon}^{\rm min}$. One expects
that such a scale is approximated by one of the summands that appear in either
$m_{\alpha\beta,+}^{2}$ or $m_{\alpha\beta,-}^{2}$, although a priori it is
not clear which one. However, one may set a lower bound on $m_{\rm gon}^{\rm
min}$ by defining

$m^{2}_{\alpha\beta,{\rm min}}={\rm
min}_{K}\,\left(\left|g_{\alpha}^{2}Q_{\alpha}^{K}-g_{\beta}^{2}Q_{\beta}^{K}\right|T_{K}\right),$
(3.12)

with no summation over indices involved. In fact, because gonions are located
at D6-brane intersections, one can propose a more precise prescription, in
which the indices $K$ that appear in (3.12) contribute to the intersection
number
$I_{\alpha\beta}=2(Q_{\alpha}^{K}P_{\beta\,K}-Q_{\beta}^{K}P_{\alpha\,K})$. In
practice, this amounts to require that either $P_{\alpha\,K}$ or
$P_{\beta\,K}$ is non-vanishing for a given value of $K$, or in 4d EFT terms
that

$\nabla g_{\alpha}^{-2}\cdot\nabla T_{K}\neq
0\,,\qquad\text{and/or}\qquad\nabla g_{\beta}^{-2}\cdot\nabla T_{K}\neq 0\,.$
(3.13)

To sum up, when at least one intersection angle is small, we estimate that the
scale for the lightest gonion tower at the intersections
$\Pi_{\alpha}\cap\Pi_{\beta}$ lies in the range

$m_{\alpha\beta,{\rm min}}\lesssim m_{\rm gon}^{\rm min}\lesssim
m_{\alpha\beta,\pm}\,.$ (3.14)

Notice that both $m_{\alpha\beta,{\rm min}}/M_{\rm P}$ and
$m_{\alpha\beta,\pm}/M_{\rm P}$ are homogeneous functions of degree $-1$ on
the complex structure saxions $\\{u^{K}\\}$, just like $m_{\rm gon}^{\rm
min}/M_{\rm P}$ should be.

In specific setups, one may give an even more precise prescription for this
estimate. Let us for instance consider a region in which a single field
$u^{I}\in\\{u^{K}\\}_{K}$ is much larger than the rest. Then by the results of
[25, 26, 27] we have that $T_{I}\sim M_{\rm P}^{2}/u_{I}$ in this region. If
this EFT string appears in the decomposition $T_{\alpha}=Q_{\alpha}^{K}T_{I}$
and $g_{\alpha}^{2}$ depends on $u^{I}$, then we have that the definition
(3.12) gives $m_{\alpha\beta,{\rm min}}/M_{\rm P}\sim 1/u^{I}$, which is the
smallest possible value given the degree of this function. A typical setup
where this occurs is in (the mirror of) type I compactifications with
magnetised D9-branes, with $u^{I}=e^{-\phi}{\rm Vol}_{X}$ and
$Q_{\alpha}^{I}=\frac{1}{6}\int_{X}F_{\alpha}\wedge F_{\alpha}\wedge
F_{\alpha}$ [85, 71, 86]. In this case it is known that the role of $T_{I}$ in
$T_{\alpha}$ is to account for the $\alpha^{\prime}$ corrections to the FI-
term $\xi_{\alpha}$ [87], and as such its presence is irrelevant for
estimating the gonion masses. Therefore, in this kind of regimes $T_{I}$
should be excluded when computing the lower bound for the gonion scale, and
the rhs of (3.12) should be modified as ${\rm
min}_{K}\left(\dots\right)\to{\rm min}_{K\neq I}\left(\dots\right)$.

###  3.3 How $Y\to 0$ implies $g\to 0$

In the following we would like to argue that, in the setup at hand, a large
complex structure limit in which $Y_{ijk}\to 0$ implies the vanishing of a
gauge coupling under which the chiral fields $i$, $j$ or $k$ are charged. Our
reasoning will rely on the gonion mass estimate (3.14) and the analysis made
in [26] for type II orientifolds limits of the kind that we are considering in
this work.

We begin by considering the asymptotic expressions for the Yukawa couplings
(3.5) and (3.8). We have that $Y\to 0$ implies either $h_{i\bar{i}}\to\infty$
or $m_{\rm gon}/M_{\rm P}\to 0$. As mentioned above, in the first case a gauge
coupling should vanish due to the inequality (2.36), which we will assume to
hold for any ${\cal N}=2$ sector. One may then turn to the second case, and in
particular assume an asymptotically massless gonion tower that transforms in
the bifundamental representation of $U(1)_{\alpha}\times U(1)_{\beta}$. Then,
using the estimate (3.14) one finds two non-exclusive options:

  * a)

At least one of the two gauge couplings $g_{\alpha}^{2}$ or $g_{\beta}^{2}$
goes to zero.

  * b)

At least one of the string tensions $T_{K}$ in (3.12) goes to zero in Planck
units.

In case a) we are done. To analyse case b) we will assume that a) does not
happen, which implies $m_{\rm gon}\gtrsim T_{\rm D4}^{1/2}$. In the following
we will argue that this is not a reasonable scale for a gonion tower with a
vanishing Yukawa.

Since by construction $T_{\rm D4}$ is the tension of a 4d EFT string, the
framework considered in [26] is particularly useful to evaluate this scale. In
there, one considers one or several field directions
$u^{I}\subset\\{u^{K}\\}$, growing at the same rate

$u^{I}\sim\lambda\to\infty\,,$ (3.15)

while the remaining ones stay constant. These were dubbed EFT string limits in
[25, 26, 27] and they will be reviewed in more detail in the next subsection.
In fact, the following analysis also applies to a more general class of
limits, where those saxions that do not grow linearly with $\lambda$ are not
constant, but instead bounded from above. We dub such limits as quasi-EFT
string limits, see section 4 for an explicit example.

The homogeneity of ${\cal H}$ in the complex structure saxions $\\{u^{K}\\}$
implies the following equality between them an their duals (2.6)

$u^{K}\ell_{K}=2\,.$ (3.16)

As a result, when we grow a field direction as in (3.15) necessarily
$\ell_{I}\sim 1/\lambda$, and so the corresponding EFT string tension scales
as $T_{{\rm D4},I}=\ell_{I}M_{\rm P}^{2}\sim M_{\rm P}^{2}/\lambda$. The
reverse is not true, because there are many $\ell_{K}$ that vanish
asymptotically when $u^{I}\to\infty$. In particular, in [26] the number $p$ of
dual saxions $\ell_{K}$ that asymptote to zero were called the degeneracy of
the limit, $p=1$ corresponding to non-degenerate limits in which only
$\ell_{I}\sim 1/\lambda$ and the rest of the dual saxions were asymptotically
constant.

By assumption, both gauge couplings $g_{\alpha}$ and $g_{\beta}$ are
independent of the growing direction (3.15). If $u^{K}_{\alpha,\beta}$ are the
saxions on which these couplings depend, then due to the condition (3.13), the
lightest gonion mass is determined by the duals $\ell_{K}^{\alpha,\beta}$ of
these saxions. As a result, in order to achieve a limit in which $m_{\rm
gon}/M_{\rm P}\to 0$ with constant gauge couplings, we necessarily need a
degenerate limit, because $\ell_{I}\notin\ell_{K}^{\alpha,\beta}$. We will
assume that the tensions of the EFT strings degenerate with $T_{{\rm D4},I}$
also scale like $T_{\rm D4}\sim M_{\rm P}^{2}/\lambda$, although for milder
scalings our reasoning works as well.

As mentioned before, along large complex structure limits the 4d dilaton
always vanishes asymptotically. Hence one may use its asymptotic behaviour to
classify the different limits. This is related to the asymptotic behaviour of
the Kähler potential, which in terms of $\lambda$ reads

$K_{Q}=-n\log\lambda+\dots$ (3.17)

Here $n$ is typically an integer, which in the language of [36] corresponds to
the singularity type. For us, due to the relation $K_{Q}=4\phi_{4}$, it
describes the asymptotic behaviour of the fundamental string scale:

$M_{s}\sim M_{\rm P}\lambda^{-n/4}\,.$ (3.18)

In the following we analyse the gonion scale for different values of $n$:

  * -

$n=1$

In this case we have that $T_{{\rm D4},I}^{1/2}\sim M_{s}/\lambda^{1/4}\ll
M_{s}$, and it is this EFT string that sets the species scale along the limit.
Given this feature and the low singularity type, the orientifold limits
analysed in [26] suggest that this is an emergent string limit, and so by the
Emergent String Conjecture it should be non-degenerate. As mentioned above,
this contradicts our assumption that $m_{\rm gon}/M_{\rm P}\to 0$.

  * -

$n=2$

In this case $T_{I}\sim M_{s}^{2}$. If the gauge couplings are constant and
the limit is degenerate, we can have all gonions towers at the scale $M_{s}$.
While this is a plausible scenario, it implies that the whole D-brane sector
related to the Yukawa remains invariant along the limit, when measuring things
in string units. In other words, the D-brane subsector $a$, $b$, $c$ are
isolated from the rest of the compactification, where the infinite
distance/gravity-decoupling trajectory happens. As a result, one should
recover a 4d EFT with non-vanishing gauge couplings and Yukawas that decouples
from gravity along the limit, as in the setups of [88] and [89]. Then,
applying an argument similar to that in [88], it would not make sense that the
Yukawa is described by (3.5), because one would then have $Y\sim
e^{\phi_{4}}\to 0$ instead of a constant physical Yukawa. The only reasonable
option seems that the Yukawa involves an ${\cal N}=2$ sector, and so the
expression to consider is (3.8). Then $Y\to 0$ implies that we should either
have $h_{i\bar{i}}\to\infty$, which was already excluded, or $m_{\rm
gon}/M_{s}\to 0$, which contradicts our assumptions.

  * -

$n>2$

In this case we find that $T_{{\rm D4},I}\sim\lambda^{n/2-1}M_{s}^{2}$, and so
$M_{s}\ll T_{I}^{1/2}$. In this case applying (3.14) with constant gauge
couplings would give us $m_{\rm gon}\sim\lambda^{\frac{n-2}{4}}M_{s}$, which
does not make sense because $m_{\rm gon}\leq M_{s}$.

Finally, as we discuss in section 5, that a light gonion mass implies the
vanishing of a gauge coupling can be motivated from the viewpoint of the
Emergence Proposal. Indeed, in the presence of a gonion tower, the gauge
couplings of the gauge bosons that couple to them receive large contributions
at one loop from the graphs in figures 7 and 8. The associated gauge kinetic
functions grow to infinity as the gonion masses go to zero. In any event, it
would be interesting to see if some of the working assumptions that we have
used could be circumvented in some setup, and in particular if they are
modified when one considers more involved limits for the growing fields, like
the growth sectors of [90]. In the following we will aim to get more intuition
on these and other limits of light gonion towers, by analysing the scale
spectrum that emerges along them.

###  3.4 Mass scales in simple limits

To gain a better physical picture of the limits under discussion, in the
following we discuss the relevant mass scales in a couple of simple setups.
First, we consider an STU-like model that arises from freezing complex
structure moduli with the vanishing FI-term condition (2.13), which we analyse
along the lines of [23]. Second, we comment on the mass scales that appear in
the type IIA EFT string limits of [26].

#### STU-like model

To build a simple STU-like model one may consider a type IIA orientifold
compactification with several complex structure moduli, and then impose that
the FI-terms $\xi_{\alpha}$ for some D6-branes wrapping three-cycles
$\Pi_{\alpha}$ vanish.121212This requirement is imposed by hand, in order to
maintain a non-trivial chiral spectrum along the trajectories that we take.
One then obtains a slice of the saxionic moduli space of much lower dimension,
which we will assume to be parametrised by only two moduli: $s\equiv u^{0}$
and $u$.

To engineer such a setup and at the same time have some control over the
compactification scale, we consider the type IIA dual of the type I and
$SO(32)$-heterotic Calabi–Yau compactifications considered in [86, 91], which
in Appendix B we translate to the language of section 2. In particular we
consider a compactification with only two complex structure moduli, so that we
have

$e^{-\phi}\mbox{Re}\,\Omega=s\alpha_{0}+u^{1}\alpha_{1}+u^{2}\alpha_{2}\,,$
(3.19)

and a Kähler potential that reads

$K_{Q}=-\log s-\log\kappa(u^{1},u^{2})\,,$ (3.20)

where

$\kappa(u^{1},u^{2})\equiv\frac{1}{6}\left[\kappa_{111}(u^{1})^{3}+3\kappa_{112}(u^{1})^{2}u^{2}+3\kappa_{122}u^{1}(u^{2})^{2}+\kappa_{222}(u^{2})^{3}\right]\,,$
(3.21)

with $\kappa_{ijk}\geq 0$. The dual saxions then read

$\ell_{0}=\frac{1}{2s}\,,\qquad\ell_{1}=\frac{\partial_{u^{1}}\kappa(u^{1},u^{2})}{2\kappa(u^{1},u^{2})}\,,\qquad\ell_{2}=\frac{\partial_{u^{2}}\kappa(u^{1},u^{2})}{2\kappa(u^{1},u^{2})}\,.$
(3.22)

We now introduce a single D6-brane wrapping a three-cycle $\Pi_{\alpha}$ with
homology class

$[\Pi_{\alpha}]=\frac{1}{2}[A]+\frac{1}{2}f_{\alpha}^{i}[D_{i}]-\left(\frac{1}{2}\kappa_{ijk}f_{\alpha}^{j}f_{\alpha}^{k}+\frac{1}{24}c_{2\,i}\right)[C^{i}]+\left(\frac{1}{6}\kappa_{ijk}f_{\alpha}^{i}f_{\alpha}^{j}f_{\alpha}^{k}+\frac{1}{24}c_{2\,i}f_{\alpha}^{i}\right)[B]\,,$
(3.23)

where $[B]={\rm P.D.}[\alpha_{0}]$, $[D_{i}]=2{\rm P.D.}[\alpha_{i}]$,
$i=1,2$, and $[A]\cdot[B]=2$, $[C^{i}]\cdot[D_{j}]=2\delta^{i}_{j}$. Here
$f_{\alpha}^{i}\in\mathbbm{Z}$ specify the D6-brane wrapping numbers and are
mirror to the magnetic worldvolume flux quanta of a D9-brane, while $c_{2\,i}$
are fixed by the topology of the mirror manifold $Y$. The orientifold image of
this D6-brane wraps a three-cycle whose homology class $[\Pi_{\alpha^{*}}]$ is
obtained by performing the sign flip $f_{\alpha}^{i}\to-f_{\alpha}^{i}$ in
(3.23). The gauge coupling of this D6-brane is given by

$\frac{2\pi}{g_{\alpha}^{2}}=s-\left(\frac{1}{2}\kappa_{ijk}f_{\alpha}^{j}f_{\alpha}^{k}+\frac{1}{24}c_{2\,i}\right)u^{i}\,,$
(3.24)

while the condition for a vanishing Fayet-Iliopoulos term is

$\ell_{i}f^{i}_{\alpha}=\frac{1}{2s}\left(\frac{1}{6}\kappa_{ijk}f_{\alpha}^{i}f_{\alpha}^{j}f_{\alpha}^{k}+\frac{1}{24}c_{2\,i}f^{i}_{\alpha}\right)\,.$
(3.25)

Clearly, imposing this condition removes one of the three saxion moduli
$\\{s,u^{1},u^{2}\\}$ of the model. In the regime of control of the mirror
type I setup we have the hierarchy $s\gg u^{i}\gg f^{i}_{\alpha}$, which we
will also assume in our type IIA setting. From there one can see that
neglecting the rhs of (3.25) is a good approximation of the vanishing FI-term
condition. From there one sees that $f^{1}_{\alpha}$ and $f^{2}_{\alpha}$ must
be of opposite sign and that, unless there is a hierarchy between them or the
$\kappa_{ijk}$ we have that $u^{1}\simeq u^{2}\simeq u$. Moreover, given a
solution $(\bar{u}^{1},\bar{u}^{2})$ of the approximate D-flatness equation,
the rescaling $\bar{u}^{i}\mapsto\lambda\bar{u}^{i}$ provides a new one. When
we restore the rhs of (3.25) this scaling symmetry is no longer there, but
given the hierarchy $s\gg u^{i}\gg f^{i}_{\alpha}$ the difference is going to
be subleading. Hence, for our purposes, we can assume that after imposing
(3.25) we have two saxions $s$ and $u$, whose only constraint is $s\gg u$.

Scale | value in $M_{\rm P}^{2}$ units
---|---
$M_{s}^{2}$ | $1/\sqrt{su^{3}}$
$T_{\rm D4}$ |  $1/s$, $1/u$
$m_{\rm gon}^{2}$ | $1/su$
$m_{\rm KK}^{2}$ or $m_{\rm w}^{2}$ | $1/su$
$m_{\rm Stu}^{2}$ | $1/su^{2}$
Table 2: Scales of the STU-like model in Planck units.

The different mass scales of this model have been collected in table 2, in
terms of $s$ and $u$. The string scale $M_{s}=e^{\phi_{4}}M_{\rm P}$ can be
obtained from $K_{Q}=4\phi_{4}$ and (3.20). The tension of 4d strings made up
of D4-branes wrapping odd three-cycles can be computed from
$T_{K}=\ell_{K}M_{\rm P}^{2}$ and (3.22). The gonion mass scale $m_{\rm gon}$
refers to the intersections of $\Pi_{\alpha}$ with its orientifold image
$\Pi_{\alpha^{*}}$, and can be estimated via (3.12). Since $s\gg u$ by
assumption, one must not include $K=0$ in (3.12), and so the two bounding
scales $M_{\alpha\alpha^{*}}$ in (3.14) coincide, with
$g_{\alpha}^{2}=g_{\alpha^{*}}^{2}\simeq 1/s$ and $|T_{\alpha}^{\pm}|\sim
T_{u}\sim M_{\rm P}^{2}/u$. Alternatively, one may compute the gonion scale by
going to the mirror type I frame. There, in the approximation of slowly
varying field strengths the gonion masses can be estimated by
$f_{a}^{i}/A_{i}$, where $A_{i}$ is some area of the mirror manifold $Y$.
Therefore, they coincide with a mirror Kaluza–Klein scale $m_{\rm KK}(Y)$,
which given the isotropy of areas induced by (the mirror of) the FI-term
condition (3.25) can be estimated as $m_{\rm KK}^{2}(Y)\sim{\rm
vol}_{Y}^{-1/3}=\sqrt{u/s}M_{s}^{2}=M_{\rm P}^{2}/su$, where we have applied
the mirror map to express the result in type IIA variables. In the type IIA
frame $m_{\rm KK}(Y)$ translates into a KK or winding scale, also displayed in
table 2. Finally, the Stückelberg mass can be easily estimated from (2.10),
and using that $\xi_{\alpha}=0$:

$M_{\alpha\alpha}^{2}\sim
g_{\alpha}^{2}Q_{\alpha}^{K}Q_{\alpha}^{L}\frac{\partial_{K}\partial_{L}(s\kappa)}{s\kappa}\,M_{\rm
P}^{2}\sim\frac{M_{\rm P}^{2}}{su^{2}}\,.$ (3.26)

and it is the lowest of all scales.

Additionally, one may study the dependence of Yukawa couplings on complex
structure saxions. With only two D6-branes there are no Yukawas couplings, but
one may add an additional D6-brane of the form (3.23) (mirror to a magnetised
D9-brane) or a D6-brane on the class $[C^{i}]$, mirror to a D5-brane. In both
cases there are no ${\cal N}=2$ sectors, and so one may estimate the Yukawa
coupling either using (3.5) or by performing a computation along the lines of
[71, 72] in the mirror frame. One then obtains

$|Y|^{2}\sim 1/s\,.$ (3.27)

Notice that, because the gauge couplings of D9-branes are given by (3.24), one
has the relation $|Y|\sim g_{\alpha}$, which corresponds to $r=1/2$ in (1.4)
and (1.5).

Let us now consider a couple of scaling limits which preserve the hierarchy
$s\gg u$:

$\text{I)}\quad s\sim u\sim\lambda\,,\qquad\text{II)}\quad s\sim
u^{3}\sim\lambda\,.$ (3.28)

The first limit corresponds to scale the 4d dilaton as $e^{\phi_{4}}\to
e^{\phi_{4}}/\lambda$ without changing the periods of $\Omega$, and the second
to the scaling of the mirror volume ${\rm vol}_{Y}\to\lambda\,{\rm vol}_{Y}$
without rescaling the 4d dilaton. We see that in the first limit the scaling
relations are:

$\text{I)}\quad\frac{m_{\rm Stu}}{M_{\rm P}}\sim\left(\frac{m_{\rm KK}}{M_{\rm
P}}\right)^{3/2}\sim\left(\frac{m_{\rm gon}}{M_{\rm
P}}\right)^{3/2}\sim\left(\frac{M_{s}}{M_{\rm
P}}\right)^{3/2}\sim\left(\frac{T_{\rm D4}^{1/2}}{M_{\rm P}}\right)^{3},\quad
m_{\rm gon}\sim|Y|^{2}M_{\rm P}\,,$ (3.29)

while in the second one we have instead

$\text{II)}\ \frac{m_{\rm Stu}}{M_{\rm P}}\sim\left(\frac{m_{\rm KK}}{M_{\rm
P}}\right)^{5/4}\sim\left(\frac{m_{\rm gon}}{M_{\rm
P}}\right)^{5/4}\sim\left(\frac{T_{\rm D4}^{1/2}}{M_{\rm
P}}\right)^{5/3}\sim\left(\frac{M_{s}}{M_{\rm P}}\right)^{5/3},\quad m_{\rm
gon}\sim|Y|^{4/3}M_{\rm P}\,.$ (3.30)

In both cases we have taken $T_{\rm D4}\sim M_{\rm P}^{2}/s$ since it
corresponds to the lightest 4d string scale. Notice that the Yukawa relation
in (3.29) coincides with (3.4), as expected, while the one in (3.30) coincides
with (3.7) when one set $w_{i}=w_{i}=w_{k}=4/3$. The generated hierarchy of
scales in both kinds of limits has been depicted in figure 5. Notice that,
even when they scale similarly, $m_{\rm KK}\leq m_{\rm gon}$ because
$|f_{\alpha}^{1}|^{1/2}\geq 1$ contributes to the gonion mass, and $m_{\rm
gon}<M_{s}$ because of the hierarchy $s\gg u$. Notice also that in both cases
the Stückelberg-induced mass is parametrically below the Kaluza–Klein
scale.131313A similar hierarchy between KK and Stückelberg scales was observed
in [9]. Above this scale we should still have a 4d EFT in which the massive
$U(1)$’s become dynamical gauge fields. Despite the 4d chiral anomalies
associated to some of them, we still have a consistent EFT thanks to the
Green–Schwarz mechanism.

Figure 5: Scales in both limits of the STU-like model. Here $m_{\rm KK}$
stands for either a Kaluza–Klein or a winding scale.

#### EFT string limits

Another interesting set of limits are the EFT string limits of [25, 26, 27].
Applied to the context of complex structure saxions of type IIA orientifolds,
they arise from the backreaction of 4d string coming from certain D4-branes,
and they essentially correspond to take the trajectory
$u^{K}=u^{K}_{0}+q^{K}\lambda$, with $q^{K}\in\mathbbm{N}$ and
$\lambda\to\infty$, in the moduli space of complex structure saxions, see [26,
section 6.4] for more details. Since the Kähler potential takes the form (2.5)
where ${\cal H}$ is a homogeneous polynomial of degree two on the saxions
$\\{u^{K}\\}$, the value of any dual saxion $\ell_{K}$ for which $q^{K}\neq 0$
scales as $\lambda^{-1}$ along the limit. In particular, the tension of the 4d
string sourcing this limit is given by $T_{\rm D4}=q^{K}\ell_{K}M_{\rm
P}^{2}$, and along the limit it decreases as $T_{\rm D4}\sim M_{\rm
P}^{2}/\lambda$. A remarkable feature of these limits is that they seem to
satisfy the Integral Weight Conjecture, which states that along EFT limits the
scale $m_{*}$ of lightest tower of states scales as [26, 27]

$\frac{m_{*}^{2}}{M_{\rm P}^{2}}\sim\left(\frac{T_{\rm D4}}{M_{\rm
P}^{2}}\right)^{w},$ (3.31)

where $w=1,2,3$ is called the scaling weight. In [26] several instances of
complex structure limits in type IIA orientifolds and of mirror Kähler limits
in type I and type IIB orientifolds were analysed, finding that only the
weights $w=1$ and $w=2$ were realised, with $m_{*}$ corresponding to either a
KK or a winding scale. In the following we will analyse these two
possibilities.

An important restriction to keep in mind is that, given a pair of D6-branes
wrapping $\Pi_{\alpha}$ and $\Pi_{\beta}$, we are interested in those limits
such that the FI-terms $\xi_{\alpha}$ and $\xi_{\beta}$ remain zero along
them, or else remain small enough such that one can attain a supersymmetric
locus by adjusting the non-growing saxions saxions of the compactification.
When applied to EFT string limits, this favours linear combinations of dual
saxions $Q_{\alpha}^{K}\ell_{K}$ and $Q_{\beta}^{K}\ell_{K}$ that are
essentially independent of $\lambda$. The simplest way in which this can
happen is if all the dual saxions in such linear combination scale the same
along the limit, or in other words if the tensions of all 4d strings that
couple to $\Pi_{\alpha}$ and $\Pi_{\beta}$ scale similarly along
$\lambda\to\infty$. In the following we will assume that this is the case.
Notice that, as a result, the upper and lower bounds in (3.14) coincide, and
we have a much sharper estimate for the gonion masses.

Let us first consider $w=1$ limits, where we have:

$\frac{m_{\rm KK/w}}{M_{\rm P}}\sim\frac{T_{\rm D4}^{1/2}}{M_{\rm P}}\sim
1/\sqrt{\lambda}\,.$ (3.32)

Based on the cases analysed in [26], one expects these limits to be non-
degenerate, in the sense that along the limit $T_{\rm D4}$ is the smallest 4d
string tension, and that there is no other EFT string with similar or faster
scaling. This non-degeneracy is also motivated by the Emergent String
Conjecture [8], since $w=1$ limits are expected to correspond to emergent
string limits. One also expects them to be elementary [89], meaning that
$q^{K}$ has a single non-vanishing entry and there is only one dual saxion
that scales as fast as $1/\lambda$, say $\ell_{I}$. Then, because there is
always at least two dual saxions involved in the FI-terms, our working
assumption to keep $\pi\xi_{\alpha/\beta}=Q^{K}_{\alpha/\beta}\ell_{K}$
vanishing along the limit implies that $Q_{\alpha/\beta}^{I}=0$ and so the
dual saxions in the sums $Q_{\alpha/\beta}^{K}\ell_{K}$ scale slower than
$1/\lambda$. Based on the examples of [26], we in particular expect them to be
asymptotically constant. This implies that the string tensions in (3.11) and
(3.12) scale like $M_{\rm P}^{2}$, and so the gonion masses are determined by
the scaling of gauge couplings along the limit.

In the following we would like to argue that either $g_{\alpha}^{2}\sim
1/\lambda$ or $Q_{\alpha}^{K}=0$, $\forall K$, so that $\xi_{\alpha}\equiv 0$
identically, and the same for $g_{\beta}^{2}$, $\xi_{\beta}$. The reasoning
again stems from the Emergent String Conjecture, which suggests that the
D4-brane sourcing the limit should be dual to a fundamental heterotic string
on a smooth Calabi–Yau compactification, and the growing saxion dual to the
heterotic 4d dilaton. In this frame, the D6-branes wrapping $\Pi_{\alpha}$ and
$\Pi_{\beta}$ should become $U(n)$ bundles, as in [86, 91], or alternatively
one of them may become an space-time filling NS5-brane.141414Due to the
assumption of chirality, it cannot be that both $\Pi_{\alpha}$ and
$\Pi_{\beta}$ are dual to NS5-branes. In the former case the 4d dilaton
dependence of the heterotic bulk gauge group gives $g\sim 1/\sqrt{\lambda}$,
and in the latter $g$ is independent of the 4d dilaton but there is no FI-
term. When plugged into (3.11) or (3.12) all this results into

$\frac{m_{\rm gon}^{\rm min}}{M_{\rm P}}\sim 1/\sqrt{\lambda}\,,$ (3.33)

which coincides with (3.32). More precisely, one would expect that $m_{\rm
KK/w}\leq m_{\rm gon}^{\rm min}\leq T_{\rm D4}^{1/2}$, as it happens in the
STU-like model.

Let us now turn to $w=2$ limits, where

$\frac{m_{\rm KK/w}}{M_{\rm P}}\sim\frac{T_{\rm D4}}{M_{\rm P}^{2}}\sim
1/\lambda\,.$ (3.34)

Typically, these limits are partially degenerate (see e.g. [89, Appendix B]).
As a result one may consider two simple options to build FI-terms independent
of $\lambda$. Either they are made of 4d EFT strings whose tension scale i)
like $T_{\rm D4}\sim M_{\rm P}^{2}/\lambda$ or ii) slower. In the second case
we assume that their tension is asymptotically constant in $M_{\rm P}^{2}$
units.

In case i) we have two possibilities: either $g_{\alpha}^{2}\sim 1/\lambda$ or
it is independent of this EFT string limit. In the first case the contribution
$g_{\alpha}^{2}T_{\alpha}^{+}$ in (3.11) to the squared gonion mass goes like
$M_{\rm P}^{2}/\lambda^{2}$, and in the second it goes like $M_{\rm
P}^{2}/\lambda$. In case ii) we expect things to work like for $w=1$ limits
and the gauge coupling to scale like $g_{\alpha}^{2}\sim 1/\lambda$, since
otherwise we would obtain a region in which $m_{\rm gon}^{\rm min}\gg M_{s}$,
which would be inconsistent. Therefore in this second case the contribution
$g_{\alpha}^{2}T_{\alpha}^{+}$ to the squared gonion mass also goes like
$M_{\rm P}^{2}/\lambda$. Unless the FI-term for the D6-brane wrapping
$\Pi_{\beta}$ is trivial, the same analysis applies to the contribution
$-g_{\beta}^{2}T_{\beta}^{-}$ to the squared gonion mass, and the final
scaling will be the larger of the two. We thus recover two generic scalings:

$m_{\rm gon}^{\rm min}\sim M_{\rm P}/\lambda\,,\qquad\text{or}\qquad m_{\rm
gon}^{\rm min}\sim M_{\rm P}/1\sqrt{\lambda}\,.$ (3.35)

In the first case the gonion tower scales like the lightest KK or winding
modes, and in the second it scales like the lightest 4d EFT string. Note that
this result is again compatible with the ordering $m_{\rm KK/w}\leq m_{\rm
gon}^{\rm min}\leq T_{\rm D4}^{1/2}$ of the STU-like model.

Figure 6: Generic asymptotic scales in different EFT string limits. I)
represents $w=1$ limits and II$\alpha$) $w=2$ limits of type ($\alpha$), with
$\alpha=$ a, b, c in (3.37). To compare with figure 5 we have identified the
EFT string flow parameter $\lambda$ with the growing field direction $u$, and
$m_{\rm KK}$ with the lowest Kaluza–Klein or winding scale. The string scale
$M_{s}$ is missing as it cannot be inferred from the general considerations in
the main text.

Let us discuss the scaling of the Stückelberg mass terms (2.10) in all these
cases. Given the structure (2.5) and that the FI-terms
$2\pi\xi_{\alpha}=Q_{\alpha}^{K}\ell_{K}M_{\rm P}$ vanish, we have

$M_{\alpha\beta}^{2}=g_{\alpha}g_{\beta}Q_{\alpha}^{K}Q_{\beta}^{L}\frac{\partial_{K}\partial_{L}{\cal
H}}{{\cal H}}\,M_{\rm P}^{2}\,.$ (3.36)

For $w=1$ limits, the non-degeneracy and our assumption that
$Q^{I}_{\alpha/\beta}=0$ imply that $M_{\alpha\beta}^{2}\sim
g_{\alpha}g_{\beta}M_{\rm P}^{2}$. Combined with our previous claim on
$g_{\alpha}$ for this case, we obtain that $m_{\rm Stu}\sim M_{\rm
P}/\lambda^{1/2}$. For $w=2$ limits, we assume that ${\cal
H}^{-1}\partial_{K}\partial_{L}{\cal H}$ has the same scaling as
$\ell_{K}\ell_{L}$ in terms of $\lambda$, which translates into the estimate
$M_{\alpha\beta}^{2}\sim g_{\alpha}g_{\beta}T_{\alpha}^{+}T_{\beta}^{+}M_{\rm
P}^{-2}$. Applied to the three setups considered for this case one finds three
different behaviours:

(a) $\displaystyle\frac{m_{\rm gon}^{\rm min}}{M_{\rm
P}}\sim\left(\frac{m_{\rm Stu}}{M_{\rm P}}\right)^{2/3}\sim 1/\lambda\,,$
(3.37a) (b) $\displaystyle\frac{m_{\rm gon}^{\rm min}}{M_{\rm
P}}\sim\left(\frac{m_{\rm Stu}}{M_{\rm P}}\right)^{1/2}\sim
1/\sqrt{\lambda}\,,$ (3.37b) (c) $\displaystyle\frac{m_{\rm gon}^{\rm
min}}{M_{\rm P}}\sim\frac{m_{\rm Stu}}{M_{\rm P}}\sim 1/\sqrt{\lambda}\,.$
(3.37c)

All these scenarios are illustrated in figure 6.

#### Back to the Yukawas

Let us now discuss what the above scenarios based on EFT string limits imply
for the Yukawa couplings. Although from this generic viewpoint we have to rely
on previous assumptions and cannot be exhaustive, the results will give as an
idea on what to expect for the asymptotic behaviour of Yukawas in the type II
orientifold limits under discussion. In the next section we will confirm our
expectations with explicit constructions.

We first consider Yukawas along limits with scaling weight $w=1$ and no ${\cal
N}=2$ subsector, so that the asymptotic Yukawa behaviour is dictated by (3.5).
To evaluate this expression, one needs the behaviour of the 4d dilaton along
the limit, which as discussed previously should be of the form (3.18). More
precisely, since this should be an emergent string limit, a typical value that
one obtains in orientifold setups is $n=1$, from where one obtains a scaling
of the form $|Y|^{2}\sim 1/\lambda$. Notice that this coincides with (3.27) if
we identify $\lambda$ with the field direction $s$ in the STU-like model,
which is consistent with the fact that such a field is S-dual to the heterotic
4d dilaton.

Let us now turn to $w=2$ limits. Here the different gonion scalings (3.37) can
combine into a single Yukawa, as long as they all correspond to the same
asymptotic behaviour of $M_{s}$. Since we are no longer in an emergent string
limit, we will assume that $M_{s}$ plays the role of the species scale and
therefore lies in the range $m_{\rm gon}\leq M_{s}\leq T^{1/2}_{\rm D4}$.
Then, a Yukawa made of three intersections with gonion scaling (3.37a) are
compatible with the values $n=2,3,4$, from where one obtains a Yukawa scaling
of the form $|Y|^{2}\sim 1/\lambda^{3-n/2}$. For $n=2$ one can actually
consider any combination of the three scenarios in (3.37), obtaining the
scalings $|Y|^{2}\sim 1/\lambda^{q}$ with $q=2,3/2,1,1/2$.

To sum up, for Yukawas that satisfy (3.5) the above scenarios based on EFT
string limits give us the asymptotic behaviour

$|Y|\sim\frac{1}{u^{r}}\,,\qquad\text{with}\quad
r=\frac{1}{4},\frac{1}{2},\frac{3}{4},1\,,$ (3.38)

where we have again identified $\lambda$ with the field direction $u$.
Remarkably, a behaviour of the type $Y\sim 1/u^{r}$ seems consistent with
recent results obtained in the context of heterotic compactifications in
smooth Calabi–Yau manifolds [92, 93], where numeric techniques were employed
to compute physical Yukawas. Thus e.g. a behaviour like $Y\sim 0.025/u^{1/2}$
could match the slope of the Yukawas in fig.3 of [93], if one identifies
$u=\psi$. In any event, it would be interesting to further explore this
possible connection.

Finally, one could consider Yukawas with ${\cal N}=2$ sectors, whose
asymptotic behaviour is dictated by (3.8). While in general one expects a
milder scaling as compared to (3.5), for these cases the limits of vanishing
Yukawas are more involved to discuss in general, unless one makes some
assumption on the factors $h_{i\bar{i}}$. For this reason we will address
their computation in explicit examples, which we now turn to discuss. As we
will see, in simple limits these Yukawas are also described by the rule
(3.38).

##  4 Explicit examples

The scenarios obtained at the end of the last section depend on a number of
assumptions and simplifications which may be more or less generic depending on
the specific setup. Moreover, from figures 5 and 6 there is no precise way to
determine neither the string scale nor the density of states in the KK and
gonion towers. In this section we will recover most of these scenarios via an
explicit class of intersecting D6-brane models, namely type IIA compactified
on toroidal orientifolds, and more precisely on the orbifold first analysed in
[94, 95]. These constructions have the advantage that both the Kaluza–Klein,
winding and gonion scales can be computed explicitly, as well as the density
of states for each tower. The same applies to other toroidal orbifold
backgrounds, see e.g. [16, 17, 18] for reviews, to which our discussion can be
easily generalised.

To describe these models we follow the conventions of [26, section 6.4], where
the scales of the closed string sector were analysed for different EFT string
limits. We start from the orbifold background $X_{6}=(({\bf
T}^{2})_{1}\times({\bf T}^{2})_{2}\times({\bf T}^{2})_{3})/\Gamma$, with
$\Gamma=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ has the same action and choice of
discrete torsion as in [95]. Each $({\bf T}^{2})_{i}$ is parametrised by a
complex coordinate $z_{i}=y_{2i-1}+i\tau_{i}y_{2i}$, where the $y_{j}$ have
unit period and the complex structure is $\tau_{i}={\rm
i}\frac{R_{2i}}{R_{2i-1}}$, with $R_{i}$ the compactification radii in units
of $\ell_{s}$. In terms of these coordinates the orientifold anti-holomorphic
involution reads ${\cal
R}:(z_{1},z_{2},z_{3})\mapsto(\bar{z}_{1},\bar{z}_{2},\bar{z}_{3})$, and the
holomorphic $(3,0)$-form reads

$\displaystyle\Omega$ $\displaystyle=\ell_{s}^{3}R_{1}R_{3}R_{5}\,dz_{1}\wedge
dz_{2}\wedge dz_{3}$ (4.1) $\displaystyle=\ell_{s}^{3}(R_{1}dy_{1}+{\rm
i}R_{2}dy_{2})\wedge(R_{3}dy_{3}+{\rm i}R_{4}dy_{4})\wedge(R_{5}dy_{5}+{\rm
i}R_{6}dy_{6})\,.$

A harmonic basis for even three-form classes $[\alpha_{K}]=\ell_{s}^{3}{\rm
P.D.}[\Sigma_{K}^{-}]$ is given by

$\displaystyle\alpha_{0}=4dy_{1}\wedge dy_{3}\wedge dy_{5}\,\ell_{s}^{3}\,,$
$\displaystyle\quad\alpha_{1}=-4dy_{1}\wedge dy_{4}\wedge
dy_{6}\,\ell_{s}^{3}\,,$ (4.2) $\displaystyle\alpha_{2}=-4dy_{2}\wedge
dy_{3}\wedge dy_{6}\,\ell_{s}^{3}\,,$
$\displaystyle\quad\alpha_{3}=-4dy_{2}\wedge dy_{4}\wedge
dy_{5}\,\ell_{s}^{3}\,,$

while one for odd three-form classes $[\beta^{J}]=\ell_{s}^{3}{\rm
P.D.}[{\Sigma}_{+}^{J}]$ is

$\displaystyle\beta^{0}=-2dy_{2}\wedge dy_{4}\wedge dy_{6}\,\ell_{s}^{3}\,,$
$\displaystyle\quad\beta^{1}=2dy_{2}\wedge dy_{3}\wedge
dy_{5}\,\ell_{s}^{3}\,,$ (4.3) $\displaystyle\beta^{2}=2dy_{1}\wedge
dy_{4}\wedge dy_{5}\,\ell_{s}^{3}\,,$
$\displaystyle\quad\beta^{3}=2dy_{1}\wedge dy_{3}\wedge
dy_{6}\,\ell_{s}^{3}\,.$

Notice that their Poincaré duals satisfy
$\Sigma_{K}^{-}\cdot\Sigma^{L}_{+}=2\delta_{K}^{L}$ when one takes into
account the orbifold quotient. The relative factor of 2 in between these two
basis takes into account that D2-branes wrapping the even three-cycles
${\Sigma}_{+}^{L}$ dual to (4.3) yield $O(1)$ instantons. Plugging the
$\alpha_{K}$ into (2.3) one identifies
$\zeta^{K}=\frac{1}{\ell_{s}^{3}}\int_{\Sigma_{+}^{K}}C_{3}$ and

$\displaystyle s\equiv u^{(0)}=\frac{1}{4}e^{-\phi}R_{1}R_{3}R_{5}\,,$
$\displaystyle\quad u^{(1)}=\frac{1}{4}e^{-\phi}R_{1}R_{4}R_{6}\,,$ (4.4)
$\displaystyle u^{(2)}=\frac{1}{4}e^{-\phi}R_{2}R_{3}R_{6}\,,$
$\displaystyle\quad u^{(3)}=\frac{1}{4}e^{-\phi}R_{2}R_{4}R_{5}\,.$

The saxions $u^{K}$ measure the internal volume of $O(1)$ instantons wrapping
calibrated three-cycles, while $2u^{K}$ measure the volume of $U(1)$
instantons. Similarly, $2\zeta^{K}$ would be the axion with unit periodicity
in the unorientifolded theory, while $\mbox{Re}\,U^{K}=\zeta^{K}$ is the
actual unit periodicity axion upon orientifolding. In terms of these fields we
have

${\cal H}=\frac{{\rm i}}{32\ell_{s}^{6}}\int_{{\bf
T}^{6}}e^{-2\phi}\Omega\wedge\bar{\Omega}=4\sqrt{u^{(0)}u^{(1)}u^{(2)}u^{(3)}}\,,$
(4.5)

and so the Kähler potential reads $K_{Q}=-\sum^{3}_{K=0}\log
2\mbox{Im}\,U^{K}$. The dual saxions are

${\ell}_{K}=\frac{1}{2u^{K}}\,.$ (4.6)

One may describe several scales of the compactification in terms of these bulk
moduli. For instance, in the regime $\tau_{i}\ll 1$, the lightest KK and
winding scales are

$m_{{\rm KK},i}=\frac{2M_{s}}{R_{2i-1}}=\frac{M_{\rm
P}}{A_{i}^{1/2}2\sqrt{su^{(i)}}}\,,\qquad m_{{\rm
w},i}=\frac{1}{2}R_{2i}M_{s}=\frac{A_{i}^{1/2}M_{\rm P}}{2\sqrt{su^{(i)}}}\,,$
(4.7)

where $A_{i}=\frac{1}{4}R_{2i}R_{2i-1}$, and we have used that the fundamental
string scale is given by

$M_{s}^{2}=\frac{M_{\rm P}^{2}}{4\sqrt{su^{(1)}u^{(2)}u^{(3)}}}\,.$ (4.8)

To build our models we consider fractional factorisable three-cycles classes
of the form

$\displaystyle[\Pi_{\alpha}]$ $\displaystyle=$ $\displaystyle
2[(n_{\alpha}^{1},m_{\alpha}^{1})(n_{\alpha}^{2},m_{\alpha}^{2})(n_{\alpha}^{3},m_{\alpha}^{3})]\,,\qquad
n_{\alpha}^{i},m_{\alpha}^{i}\in\mathbb{Z}$ $\displaystyle=$ $\displaystyle
n_{\alpha}^{1}n_{\alpha}^{2}n_{\alpha}^{3}\Sigma^{0}_{+}-n_{\alpha}^{1}m_{\alpha}^{2}m_{\alpha}^{3}\Sigma^{1}_{+}-m_{\alpha}^{1}n_{\alpha}^{2}m_{\alpha}^{3}\Sigma^{2}_{+}-m_{\alpha}^{1}m_{\alpha}^{2}n_{\alpha}^{3}\Sigma^{3}_{+}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(m_{\alpha}^{1}m_{\alpha}^{2}m_{\alpha}^{3}\Sigma_{0}^{-}-m_{\alpha}^{1}n_{\alpha}^{2}n_{\alpha}^{3}\Sigma_{1}^{-}-n_{\alpha}^{1}m_{\alpha}^{2}n_{\alpha}^{3}\Sigma_{2}^{-}-n_{\alpha}^{1}n_{\alpha}^{2}m_{\alpha}^{3}\Sigma_{3}^{-}\right)\,,$

where we follow the notation of [95]. That is, a three-cycle
$(n^{1},m^{1})(n^{2},m^{2})(n^{3},m^{3})$ corresponds, in the covering space
${\bf T}^{6}$, to the class
$[\Pi]=\left(n^{1}a_{1}+m^{1}a_{2}\right)\times\left(n^{2}a_{3}+m^{2}a_{4}\right)\times\left(n^{3}a_{5}+m^{3}a_{6}\right)$,
where $a_{i}$ is the one-cycle class of ${\bf T}^{6}$ along $y_{i}$. These
fractional three-cycles have a reduced moduli space, and for simplicity we
will assume them to cross the same fixed point of the orbifold action, say the
origin $z_{i}=0$. Moreover, if they are not invariant under the orientifold
projection one needs to add the orientifold image, which lies at

$[\Pi_{\alpha^{*}}]=2[(n_{\alpha}^{1},-m_{\alpha}^{1})(n_{\alpha}^{2},-m_{\alpha}^{2})(n_{\alpha}^{3},-m_{\alpha}^{3})]\,.$
(4.10)

#### A toy model

Let us first consider a set of three D6-brane stacks $a$, $b$, $c$, wrapping
the three-cycles

$\displaystyle\Pi_{a}$ $\displaystyle=2(k,1)(k,1)(k,-1)\,,$
$\displaystyle\Pi_{b}$ $\displaystyle=2(0,1)(1,0)(0,-1)\,,$ (4.11)
$\displaystyle\Pi_{c}$ $\displaystyle=2(0,1)(0,-1)(1,0)\,\,$

with $k\in\mathbbm{N}$. From here one obtains a gauge group
$U(1)_{A}\times{\rm USp}(2)_{b}\times{\rm USp}(2)_{c}$, up to additional
D6-branes needed to cancel tadpoles. The D6-branes $b$ and $c$ are invariant
under both the orbifold and orientifold action, and as a result each of them
hosts a USp$(2)$ gauge group and has a vanishing FI-term. The D6-brane $a$ is
not invariant under ${\cal R}$ and has an orientifold image $\Pi_{a^{*}}$ and
an FI-term of the form

$\frac{\pi\xi_{a}}{M_{\rm
P}^{2}}=-\frac{1}{4s}-\frac{k^{2}}{4u^{(1)}}-\frac{k^{2}}{4u^{(2)}}+\frac{k^{2}}{4u^{(3)}}\,,$
(4.12)

where we have identified $s\equiv u^{(0)}$. Using the dictionary

$2s=e^{-\phi_{4}}\frac{1}{\sqrt{\tau_{1}\tau_{2}\tau_{3}}}\,,\quad
2u^{(i)}=e^{-\phi_{4}}\sqrt{\frac{\tau_{j}\tau_{k}}{\tau_{i}}}\,,$ (4.13)

one can see that the condition $\xi_{a}=0$ is equivalent to impose

$\theta_{1}+\theta_{2}=\theta_{3}\,,\qquad\text{where}\quad\pi\theta_{i}=\tan^{-1}\frac{\tau_{i}}{k}\,.$
(4.14)

where we have used the identity
$\tan(\vartheta+\varphi)=\frac{\tan\vartheta+\tan\varphi}{1-\tan\vartheta\tan\varphi}$.
One also finds the following gauge couplings for the different D6-branes

$\frac{2\pi}{g_{a}^{2}}=2(k^{3}s+ku^{(1)}+ku^{(2)}-ku^{(3)})\,,\qquad\frac{2\pi}{g_{b}^{2}}=2u^{(2)}\,,\qquad\frac{2\pi}{g_{c}^{2}}=2u^{(3)}\,,$
(4.15)

from where one deduces the following D-term induced (squared) mass (2.12)

$q_{a}g_{a}^{2}\xi_{a}=q_{a}\tan\left[\pi(\theta_{3}-\theta_{2}-\theta_{1})\right]M_{s}^{2}\,,$
(4.16)

with $q_{a}$ the charge of the open string state under $U(1)_{a}$. Finally,
out of the three D6-branes above, the Stückelberg mass is only generated for
the $U(1)_{a}$ boson and reads

$M_{aa}^{2}=\frac{1}{2}g_{a}^{2}\left(\frac{1}{s^{2}}+\frac{k^{4}}{(u^{(1)})^{2}}+\frac{k^{4}}{(u^{(2)})^{2}}+\frac{k^{4}}{(u^{(3)})^{2}}\right)M_{\rm
P}^{2}\,.$ (4.17)

What is particularly interesting for us is that we have precise expression for
scales of the gonion towers. For instance in the $aa^{*}$ sector we have that
$m_{{\rm gon},i}^{2}=2\theta_{i}M_{s}^{2}$. In general these scales are an
involved function of the complex structure saxions, but it simplifies in
certain limits, like for instance $\tau_{i}/k\ll 1$. In this regime, which
realises the STU-like hierarchy $s\gg u^{(i)}$, we have

$\frac{m_{{\rm gon},i}}{M_{\rm P}}\simeq\frac{1}{2\sqrt{ksu^{(i)}}}\,.$ (4.18)

This expression describes all those gonion scales that are small in string
units, while the rest are of the order of $M_{s}$. We have summarised in table
3 the different gonion scales for each open string sector.

Sector | $m_{{\rm gon},1}/M_{\rm P}$ | $m_{{\rm gon},2}/M_{\rm P}$ | $m_{{\rm gon},3}/M_{\rm P}$
---|---|---|---
$aa^{*}$ | $1/\sqrt{ksu^{(1)}}$ | $1/\sqrt{ksu^{(2)}}$ | $1/\sqrt{ksu^{(3)}}$
$ab$ | $e^{\phi_{4}}$ | $1/\sqrt{ksu^{(2)}}$ | $e^{\phi_{4}}$
$bc$ | $e^{\phi_{4}}$ | $e^{\phi_{4}}$ | $e^{\phi_{4}}$
$ca$ | $e^{\phi_{4}}$ | $e^{\phi_{4}}$ | $1/\sqrt{ksu^{(3)}}$
Table 3: Gonion scales for different sectors of the toy model, up to numeric
factors.

Notice that, by increasing the value of $n$ and keeping the Kähler moduli
fixed, one may achieve the ordering $m_{{\rm gon},i}<m_{{\rm KK},i}<m_{{\rm
w},i}$, cf. (4.7). In this case, the gonion scale $m_{{\rm gon},i}$ is below
any tower from the bulk, even if it cannot be made parametrically lighter
because it shares the same dependence on complex structure saxions with the
lightest of them.151515Even then, the scale at which the gonion tower displays
its Hagedorn spectrum, namely $M_{s}$, is always above either $m_{{\rm KK},i}$
or $m_{{\rm w},i}$. It is however interesting to notice that, even in this
case, there is still a Kaluza–Klein tower which is lighter than any gonion
tower. Indeed, on each ${\bf T}^{2}$ we have the KK and winding modes of each
D6-brane, which for the D6-brane $a$ read

$m_{{\rm
KKD6}_{a},i}=\frac{2M_{s}}{R_{2i-1}\sqrt{k^{2}+\tau_{i}^{2}}}\simeq\frac{M_{\rm
P}}{A_{i}^{1/2}2k\sqrt{su^{(i)}}}\,,\quad m_{{\rm
wD6}_{a},i}=\frac{R_{2i}}{2\sqrt{k^{2}+\tau_{i}^{2}}}M_{s}\simeq\frac{A_{i}^{1/2}M_{\rm
P}}{2k\sqrt{su^{(i)}}}\,.$ (4.19)

In particular we have that $m_{{\rm KKD6}_{a},i}<m_{{\rm gon},i}$ for any
value of $k$. Again, the dependence on the complex structure saxions is
similar for all these towers, so taking limits on the moduli space of complex
structures they cannot be separated parametrically from each other. Hence,
when we take different infinite distance limits we find several towers of
different nature that become light at the same time.

Indeed, let us for instance consider the limit $\tau_{i}=$ const. and
$e^{\phi_{4}}\sim\lambda^{-1}\to 0$ which realises the STU-like limit I), or
the limit $e^{\phi_{4}}\sim\tau_{i}^{3/2}\sim\lambda^{-3/2}\to 0$ that
implements the STU-like limit II). In both cases it is easy to check that the
content of table 2 is realised with $u\simeq u^{(i)}$. In the second case, the
lightest towers scale as $m_{\rm tower}^{\rm min}\sim u^{-2}M_{\rm P}$, as in
figure 5. More precisely we find the following towers with this scaling:

  * -

Three towers of bulk KK modes $m_{{\rm KK},i}$ and three towers of bulk
winding modes $m_{{\rm w},i}$, $i=1,2,3$.

  * -

In the $aa$ sector, three towers of KK modes $m_{{\rm KKD6}_{a},i}$, and three
towers of winding modes $m_{{\rm wD6}_{a},i}$, $i=1,2,3$.

  * -

In the $bb$ sector, one tower of KK modes $m_{{\rm KKD6}_{b},2}$ and one tower
of winding modes $m_{{\rm wD6}_{c},2}$.

  * -

In the $cc$ sector, one tower of KK modes $m_{{\rm KKD6}_{c},3}$ and one tower
of winding modes $m_{{\rm wD6}_{c},3}$.

  * -

In the $aa^{*}$ sector, three towers of gonion modes $m_{{\rm gon},i}$
$i=1,2,3$.

  * -

In the $ab$ sector, one tower of gonion modes $m_{{\rm gon},2}$.

  * -

In the $ca$ sector, one tower of gonion modes $m_{{\rm gon},3}$.

The towers on each item combine multiplicatively among themselves into a
lattice, while those in different items only combine additively. In this
sense, the towers that have higher density dominate the light spectrum. Notice
that due to the spectrum (2.14) one gonion tower has comparable density as the
combination of one Kaluza–Klein and one winding tower. Therefore, in this toy
model the towers that dominate the lightest spectrum correspond to the bulk KK
and winding, $aa$ sector KK and winding, and $aa^{*}$ gonion towers. In
section 5 we will further analyse this and other light tower spectra from the
viewpoint of the species scale and the Emergence Proposal.

One may also consider a different regime for this toy model, in which only a
single complex structure modulus is small compared to the other two. Let us
for instance consider the case where
$e^{2\phi_{4}}\sim\tau_{2}/k\sim\lambda^{-1}\ll 1$ while
$\tau_{1}/k,\tau_{3}/k\sim{\cal O}(1)$. In terms of complex structure saxions,
this corresponds to the limit

$s,u^{(2)}\sim\lambda\,,\qquad u^{(1)},u^{(3)}\ \text{bounded}\,.$ (4.20)

The saxions $u^{(1)},u^{(3)}$ cannot be constant along the limit, because one
needs to adjust them as $s$ and $u^{(2)}$ grow, in order to satisfy the
supersymmetry condition $\xi_{a}=0$, cf. (4.12). However, they will be
functions of $\lambda$ bounded from above.161616For instance, one could set
$u^{(1)}=u_{0}$ and
$\frac{k^{2}}{u^{(3)}}=\frac{1+k^{2}}{\lambda}+\frac{k^{2}}{u_{0}}$, i.e.
$u^{(3)}$ bounded from above. In this sense, this limit is an example of a
quasi-EFT string limit, in the sense defined in section 3.3. For this case in
particular, because we have two growing saxions, we would expect the limit to
be quasi-EFT with $w=2$. Indeed, we will see that its tower spectrum resembles
to great extent the scenario IIa) in figure 6. First, the scales corresponding
to 4d strings coming from D4-branes and the fundamental string scale read

$T_{{\rm D4},0}^{1/2},T_{{\rm D4},2}^{1/2}\sim\lambda^{-1/2}M_{\rm P}\,,\qquad
M_{s}\sim\lambda^{-1/2}M_{\rm P}\,.$ (4.21)

Second, we find the lightest set of towers at the scale $m_{\rm tower}^{\rm
min}\sim\lambda^{-1}M_{\rm P}$. These are:

  * -

One tower of bulk KK modes $m_{{\rm KK},2}$ and one of bulk winding modes
$m_{{\rm w},2}$.

  * -

In the $aa$ sector, one KK mode $m_{{\rm KKD6}_{a},2}$, and one winding mode
tower $m_{{\rm wD6}_{a},2}$.

  * -

In the $aa^{*}$ sector, one tower of gonion modes $m_{{\rm gon},2}$.

  * -

In the $ab$ sector, one tower of gonion modes $m_{{\rm gon},2}$.

This tower spectrum is much simpler than in the previous limit, and now each
item contains a similar density of states. We therefore find agreement with
the tower spectrum of the scenario IIA in figure 6 if we identify $\lambda\sim
u$, and we include the string scale at $M_{s}=u^{-1/2}M_{\rm P}$. The estimate
for the Stückelberg scale, however, reads in case $m_{\rm
Stu}\sim\lambda^{-1/2}M_{\rm P}$, which is in disagreement with the strict EFT
string limit scenario IIa) in the figure. It thus seems that the estimate for
this scale is not particularly robust when generalising an EFT string limit to
a quasi-EFT string one. Finally, one could also consider a third limit in
which the role of $u^{(1)}$ and $u^{(2)}$ is interchanged in (4.20). The
description above would work the same up to redefinitions, except for the
light gonion tower in the $ab$ sector, which would not be present.

This relatively small difference turns out to be quite relevant when we look
at the Yukawa couplings between massless chiral fields. First, the selection
rules of Appendix C imply that there is a single non-vanishing Yukawa in the
massless sector, namely the one involving the sectors $ab$, $bc$ and $ca$. The
kinetic terms for each of these sectors are:

$\displaystyle K_{ca}$ $\displaystyle=$ $\displaystyle
e^{\phi_{4}}(2\pi)^{-1/2}\left[\frac{\Gamma(\frac{1}{2}-\theta_{1})}{\Gamma(\frac{1}{2}+\theta_{1})}\frac{\Gamma(\frac{1}{2}-\theta_{2})}{\Gamma(\frac{1}{2}+\theta_{2})}\frac{\Gamma(\theta_{3})}{\Gamma(1-\theta_{3})}\right]^{1/2}\,,$
(4.22) $\displaystyle K_{ab}$ $\displaystyle=$ $\displaystyle
e^{\phi_{4}}(2\pi)^{-1/2}\left[\frac{\Gamma(\frac{1}{2}+\theta_{1})}{\Gamma(\frac{1}{2}-\theta_{1})}\frac{\Gamma(\theta_{2})}{\Gamma(1-\theta_{2})}\frac{\Gamma(\frac{1}{2}-\theta_{3})}{\Gamma(\frac{1}{2}+\theta_{3})}\right]^{1/2}\,,$
(4.23) $\displaystyle K_{bc}$ $\displaystyle=$ $\displaystyle
e^{\phi_{4}}\sqrt{\tau_{1}/2\pi}=2e^{2\phi_{4}}\sqrt{u^{(2)}u^{(3)}/2\pi}\,.$
(4.24)

The metric for the first two sectors is obtained by direct application of
(2.29), while the third one corresponds to an ${\cal N}=2$ sector and it is
obtained as a limit of the general expression [74]. Notice that from (4.23) we
obtain $h_{bc}=g_{b}^{-1}g_{c}^{-1}$ in (2.35), which saturates the inequality
(2.36). When plugged into the supergravity formula (2.27) one obtains the
following Yukawa coupling

$Y_{abc}=B\,\sqrt{2\pi
g_{b}g_{c}}\left[\frac{\Gamma(1-\theta_{2})}{\Gamma(\theta_{2})}\frac{\Gamma(1-\theta_{3})}{\Gamma(\theta_{3})}\frac{\Gamma(\frac{1}{2}+\theta_{2})}{\Gamma(\frac{1}{2}-\theta_{2})}\frac{\Gamma(\frac{1}{2}+\theta_{3})}{\Gamma(\frac{1}{2}-\theta_{3})}\right]^{1/4}\,,$
(4.25)

where $B$ is a function of the Kähler moduli that we treat as a constant.

Let us now see how the Yukawa behaves under the different limits considered
above. For the limit $e^{\phi_{4}}\sim\tau_{i}^{3/2}\sim\lambda^{-1/2}\to 0$
that implements the STU-like limit II), we find

$Y_{abc}\sim\left(\frac{\theta_{2}\theta_{3}}{u^{(2)}u^{(3)}}\right)^{1/4}\sim\frac{e^{-\phi_{4}}}{\sqrt{su^{(2)}u^{(3)}}}\sim\lambda^{-1/3}\,.$
(4.26)

This result can also be recovered from (3.8), with
$h_{bc}=g_{b}^{-1}g_{c}^{-1}$ . The difference between this scaling and the
generic one in (3.27), which would correspond to $Y\sim\lambda^{-1/2}$ is due
to the presence of an ${\cal N}=2$ sector in the Yukawa, which typically
softens the suppression.

If instead we consider the quasi-EFT string limit (4.20) we obtain

$Y_{abc}\sim\left(\frac{\theta_{2}}{u^{(2)}}\right)^{1/4}\sim\frac{e^{-\phi_{4}/2}}{\sqrt{s^{1/2}u^{(2)}}}\sim\lambda^{-1/2}\,.$
(4.27)

Finally, if one considers the limit which interchanges the roles of $u^{(2)}$
and $u^{(1)}$ in (4.20), one obtains that the Yukawa coupling remains
constant, due to cancellations involving the ${\cal N}=2$ sector. We have
summarised these results in table 4, together with the asymptotic behaviour
for each gauge coupling involved in the Yukawa. The pattern that we find is
$Y_{abc}\sim g_{b}g_{c}$, and we also see that eqs.(1.4) and (1.5) are
satisfied, with $g_{*}$ the largest of the three gauge couplings and with
$r=\frac{1}{3},\frac{1}{2}$. Finally, notice that the limit (4.20) realises
the scenario (1.6) highlighted in the introduction.

Limit | Scenario | $g_{a}$ | $g_{b}$ | $g_{c}$ | $Y_{abc}$
---|---|---|---|---|---
$s\sim(u^{(i)})^{3}\sim\lambda$ | STU-like II) | $\lambda^{-1/2}$ | $\lambda^{-1/6}$ | $\lambda^{-1/6}$ | $\lambda^{-1/3}$
(4.20) | quasi-EFT IIa) | $\lambda^{-1/2}$ | $\lambda^{-1/2}$ | const. | $\lambda^{-1/2}$
(4.20)′ | quasi-EFT IIa) | $\lambda^{-1/2}$ | const. | const. | const.
Table 4: Different limits discussed in the text, together with the asymptotic
behaviour of the gauge and Yukawa couplings. The limit (4.20)′ corresponds to
(4.20) with the behaviour for $u^{(1)}$ and $u^{(2)}$ interchanged.

#### A Pati–Salam model

Let us now consider a more realistic example, which realises a Left-Right
model first considered in [96, 30] or its Pati–Salam extension
$SU(4)_{a}\times SU(2)_{b}\times SU(2)_{c}$. The embedding of such model into
the $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orientifold background that we are
considering was realised in [97, 98], including the additional D6-branes
needed to cancel all tadpoles. For simplicity, we will not include the latter
into this discussion.

The three stacks of D6-branes relevant for our discussion are

$\displaystyle\Pi_{a}$ $\displaystyle=8(1,0)(k,1)(k,-1)\,,$
$\displaystyle\Pi_{b}$ $\displaystyle=2(0,1)(1,0)(0,-1)\,,$ (4.28)
$\displaystyle\Pi_{c}$ $\displaystyle=2(0,1)(0,-1)(1,0)\,\,,$

where $k\in\mathbbm{N}$ determines the number of chiral families. Notice that
the D6-branes $b$ and $c$ are similar to the ones in the previous toy model,
while the D6-brane $a$ is different. In particular, it does not intersect its
orientifold image, and as a result the lightest gonion in the $aa^{*}$ sector
can be made of the order of the fundamental string scale $M_{s}$ by separating
$a$ from $a^{*}$ in $({\bf T}^{2})_{1}$. The gauge coupling associated to
$U(1)_{a}\subset U(4)_{a}$ is given by

$\frac{2\pi}{g_{a}^{2}}=2(k^{2}s+u^{(1)})\,,$ (4.29)

while its FI-term reads

$\frac{\pi\xi_{a}}{M_{\rm
P}^{2}}=\frac{k}{4}\left(-\frac{1}{u^{(2)}}+\frac{1}{u^{(3)}}\right)\,,$
(4.30)

such that the supersymmetry condition $\xi_{a}=0$ is equivalent to
$u^{(2)}=u^{(3)}$ and to $\theta_{2}=\theta_{3}$, with
$\pi\theta_{j}=\tan^{-1}(\tau_{j}/k)$, $j=2,3$. More precisely, we have that

$q_{a}g_{a}^{2}\xi_{a}=q_{a}\tan\left[\pi(\theta_{3}-\theta_{2})\right]M_{s}^{2}\,,$
(4.31)

with $q_{a}$ the charge of the open string state under $U(1)_{a}$. The
Stückelberg mass term is

$M_{aa}^{2}=\frac{1}{2}g_{a}^{2}\left(\frac{k^{2}}{(u^{(2)})^{2}}+\frac{k^{2}}{(u^{(3)})^{2}}\right)M_{\rm
P}^{2}\,.$ (4.32)

Sector | $m_{{\rm gon},1}/M_{\rm P}$ | $m_{{\rm gon},2}/M_{\rm P}$ | $m_{{\rm gon},3}/M_{\rm P}$
---|---|---|---
$aa^{*}$ | $e^{\phi_{4}}$ | $e^{\phi_{4}}\ \text{or}\ 1/\sqrt{ksu^{(2)}}$ | $e^{\phi_{4}}\ \text{or}\ 1/\sqrt{ksu^{(3)}}$
$ab$ | $e^{\phi_{4}}$ | $1/\sqrt{ksu^{(2)}}$ | $e^{\phi_{4}}$
$bc$ | $e^{\phi_{4}}$ | $e^{\phi_{4}}$ | $e^{\phi_{4}}$
$ca$ | $e^{\phi_{4}}$ | $e^{\phi_{4}}$ | $1/\sqrt{ksu^{(3)}}$
Table 5: Gonion scales for different sectors of the PS model, up to numeric
factors.

As in the toy model, one may now consider different infinite distance limits
within the regime $\tau_{i}\ll 1$. The gonion scales are quite similar to
those of the toy model, and have been summarised in table 5. The main
difference is that now all towers in the $aa^{*}$ sector can be made of the
order of the fundamental string scale by separating the $\Pi_{a}$ from
$\Pi_{a^{*}}$ in the first two-torus.

Another important difference is that $\tau_{1}$ does not influence neither the
supersymmetry condition $\xi_{a}=0$ nor the gonion tower scales. As a result
one may conceive different strict EFT string limits that are compatible with
supersymmetry. A few of them are:

$\displaystyle s\sim\lambda\,,\quad u^{(i)}=\,\text{const.}$
$\displaystyle\implies e^{\phi_{4}}\sim\lambda^{-1/4}\,,$ (4.33)
$\displaystyle s,u^{(2)}=u^{(3)}\sim\lambda\,,\quad u^{(1)}=\,\text{const.}$
$\displaystyle\implies e^{\phi_{4}}\sim\lambda^{-3/4}\,,$ (4.34)
$\displaystyle u^{(2)}=u^{(3)}\sim\lambda\,,\quad s,u^{(1)}=\,\text{const.}$
$\displaystyle\implies e^{\phi_{4}}\sim\lambda^{-1/2}\,,$ (4.35)
$\displaystyle s,u^{(1)}\sim\lambda\,,\quad u^{(2)}=u^{(3)}=\,\text{const.}$
$\displaystyle\implies e^{\phi_{4}}\sim\lambda^{-1/2}\,,$ (4.36)

with $\lambda\to\infty$. In addition we will consider the limit

$s\sim\lambda\,,\quad u^{(2)}=u^{(3)}\sim\lambda^{1/2}\,,\quad
u^{(1)}=\,\text{const.}\,\implies e^{\phi_{4}}\sim\lambda^{-1/2}\,.$ (4.37)

Limit | Scenario | $\\#T_{\rm D4}^{1/2}$ | $\\#m_{{\rm KK}}^{\rm bulk}$ | $\\#m_{{\rm KK}}^{\rm brane}$ | $\\#m_{{\rm gon}}$ | $g_{a}$ | $g_{b}$ | $g_{c}$ | $Y_{abc}$
---|---|---|---|---|---|---|---|---|---
(4.33) | EFT I) | 1 | 6 | 6 | 2 or 4 | $\lambda^{-1/2}$ | const. | const. | $\lambda^{-1/4}$
(4.34) | EFT IIa) | - | 6 | 4 | 2 or 4 | $\lambda^{-1/2}$ | $\lambda^{-1/2}$ | $\lambda^{-1/2}$ | $\lambda^{-3/4}$
(4.35) | EFT IIb) | - | 2 | 2 | - | const. | $\lambda^{-1/2}$ | $\lambda^{-1/2}$ | $\lambda^{-1/2}$
(4.36) | EFT IIc) | - | 2 | 2 | - | $\lambda^{-1/2}$ | const. | const. | const.
(4.37) | non-EFT | - | 4 | 4 | 2 or 4 | $\lambda^{-1/2}$ | $\lambda^{-1/4}$ | $\lambda^{-1/4}$ | $\lambda^{-1/2}$
Table 6: Different limits discussed for the Pati–Salam model, together with
their corresponding scenario in figure 6, the dimension of the lattice for
each kind of leading tower, and the asymptotic behaviour of the gauge and
Yukawa couplings. $\\#m_{\rm KK}$ stands for both KK and winding towers.

For each of these limits one can perform a similar analysis to the one made in
the toy model, regarding the spectrum of light towers and the Yukawa
couplings. We have summarised the results regarding the tower spectra in table
6. One finds that each of the limits (4.33)-(4.36) falls into a different
scenario of figure 6, and that the behaviour for the scales on each scenario
is exactly reproduced in this toroidal setting, upon the identification
$\lambda=u$. What remains to specify for each of these limits that is
particularly relevant for the low energy physics, is the degeneracy of each of
the leading towers, which increases when towers add multiplicatively. In table
6 we have encoded this information by specifying the dimension of the lattice
of states for each leading tower, when present in a given limit. Within the
same limits, the degeneracy of the largest gonion tower (which is twice
compared to that of a KK tower) changes depending on whether or not we
separate the branes $a$ and $a^{*}$ in the first two-torus. The number of
leading gonion towers that are replicas of chiral fields involved in the non-
vanishing Yukawa $Y_{abc}$ is however not affected by this choice.

Notice that the limits (4.34) and (4.35) are special, in the sense that
$\tau_{1}\sim\lambda^{1/2}$ and we are taken away from the regime $\tau_{i}\ll
1$, $\forall i$. This implies that the lightest KK and winding scales coming
from the first torus are not captured by (4.7).171717Instead, we have that the
lightest KK mode is $m_{{\rm KK},1}=\frac{M_{s}}{R_{2}}\simeq\frac{M_{\rm
P}}{2A_{1}^{1/2}\sqrt{u^{(2)}u^{(3)}}}=\frac{M_{\rm P}}{2\pi
A_{1}^{1/2}h_{i\bar{i}}}$.  The limit (4.37) does not fall into the category
of EFT string limits, and the scaling for each of its towers goes as $m_{\rm
tower}^{\rm min}\sim\lambda^{-3/4}$.

The selection rules for Yukawa couplings described in Appendix C imply that
the only non-vanishing Yukawa formed by zero mode couplings is the one from
the sector $abc$, as in the previous example. The kinetic terms for each
sector have the form:

$\displaystyle K_{ca}$ $\displaystyle=$ $\displaystyle