e^{\phi_{4}}(2\pi)^{-1/2}\left[\frac{\Gamma(\frac{1}{2}-\theta_{2})}{\Gamma(\frac{1}{2}+\theta_{2})}\frac{\Gamma(\theta_{3})}{\Gamma(1-\theta_{3})}\right]^{1/2}\,,$
(4.38) $\displaystyle K_{ab}$ $\displaystyle=$ $\displaystyle
e^{\phi_{4}}(2\pi)^{-1/2}\left[\frac{\Gamma(\theta_{2})}{\Gamma(1-\theta_{2})}\frac{\Gamma(\frac{1}{2}-\theta_{3})}{\Gamma(\frac{1}{2}+\theta_{3})}\right]^{1/2}\,,$
(4.39) $\displaystyle K_{bc}$ $\displaystyle=$ $\displaystyle
e^{\phi_{4}}\sqrt{\tau_{1}/2\pi}=2e^{2\phi_{4}}\sqrt{u^{(2)}u^{(3)}/2\pi}=e^{2\phi_{4}}g_{b}^{-1}g_{c}^{-1}\,.$
(4.40)

The main novelty with respect to the metrics of the toy model is that the
kinetic terms of the sectors $ab$ and $ca$ are insensitive to the geometry of
the first torus, since the corresponding angles satisfy
$\theta_{ab}^{1}=\theta_{ca}^{1}=\frac{1}{2}$. This difference disappears when
one considers the physical Yukawa coupling:

$Y_{abc}=B\,\sqrt{2\pi
g_{b}g_{c}}\left[\frac{\Gamma(1-\theta_{2})}{\Gamma(\theta_{2})}\frac{\Gamma(1-\theta_{3})}{\Gamma(\theta_{3})}\frac{\Gamma(\frac{1}{2}+\theta_{2})}{\Gamma(\frac{1}{2}-\theta_{2})}\frac{\Gamma(\frac{1}{2}+\theta_{3})}{\Gamma(\frac{1}{2}-\theta_{3})}\right]^{1/4}\,,$
(4.41)

which yields an identical expression to (4.25). The behaviour of the Yukawa
couplings concerning the limits considered above is also displayed in table 6.
Notice that one again satisfies the pattern (1.4) and (1.5) with
$r=1/4,1/2,3/4,1$.

##  5 Structure of towers: species scale and emergence

We have seen in the previous sections different examples of asymptotic moduli
directions in which towers of particles, both of KK type and gonion appear. In
the limits in which Yukawa couplings go to zero, towers of gonions appear.
Table 7 together with figures 5 and 6 show the structure of scales of some
examples, sometimes quite intricate. At this point it is worth summarising the
general structure of towers. One can show that the different options may be
neatly classified in terms of the density parameters of the towers, as we now
describe.

$p_{\rm KK}/p_{\rm gon}$ | $p_{\rm gon}=0$ | $p_{\rm gon}=2$ | $p_{\rm gon}=4$ | $p_{\rm gon}=6$
---|---|---|---|---
$p=2$ | EFT IIb), IIc) (PS) | quasi-EFT IIa) (TM) | X | X
$p=4$ |  $\mathcal{N}=2$ sectors | non-EFT (PS) | non-EFT (PS)⋆ | X
$p=6$ |  $\mathcal{N}=2$ sectors | EFT IIa) (PS) | EFT IIa) (PS)⋆ | STU-like II) (TM)

Table 7: For the explicit examples considered in section 4, several limits are
displayed in terms of the density parameter $p_{t}$. PS = Pati–Salam model
(4.28) . TM = toy model (4.11). The ⋆ in PS means that the sector $aa^{\star}$
intersects.

###  5.1 Tower densities and the Species Scale

The density parameters $p_{t}$ are defined in terms of a tower spectrum as

$m_{n}^{(t)}\ =\ n_{t}^{1/p_{t}}m_{0}^{(t)}\,,$ (5.1)

with $n_{t}\in\mathbb{Z}$ and $m_{0}^{(t)}$ the characteristic scale of the
tower. The sum in $n_{t}$ goes up to the fundamental Quantum Gravity scale of
the theory which we will call $\Lambda_{QG}$, so that

$\Lambda_{QG}\ =\ N_{t}^{1/p_{t}}m_{0}^{(t)}\ ,$ (5.2)

with $N_{t}$ the total number of states in the tower which fit below the scale
$\Lambda_{QG}$. Thus, e.g., the case of the decompactification of a single
dimension corresponds to $p_{t}=1$ with $m_{0}=m_{\rm KK}$. The case of $p$
decompactified dimensions with the same scale $m_{0}=m_{\rm KK}$, corresponds
to $N_{t}=N^{p}$, with $N$ the number of states in each of the $p$. We will
consider in what follows the cases with $p=2,4,6$ decompactified large
dimensions, appropriate to handle chiral theories with ${\cal N}=1$
supersymmetry. In particular we are considering theories with $p$ large
dimensions and $(6-p)$ dimensions of order of the fundamental scale (and hence
not associated towers). In the case of gonions one has $p_{\rm gon}=2,4,6$
depending on whether, in the studied local brane intersection, there are
gonion towers in one, two, or three complex planes.181818It is interesting to
remark that the gonion towers are the first examples of genuine particle
towers with $p_{t}>1$. The other examples with $p_{t}>1$ so far in the
literature correspond to a combination of $p_{t}$ KK towers, which may be
described effectively as a single tower with $p_{t}>1$..

The different options with $p,p_{\rm gon}=2,4,6$ are shown in table 7. Note
first that necessarily $p_{\rm gon}\leq p$, since gonion towers are only
generated when the corresponding dimensions are large. This is denoted with a
big $X$ in the table. For the other $p,p_{\rm gon}$ choices we have explicit
examples in the text, except for $p_{\rm gon}=0$ and $p>2$, which corresponds
to configurations with only $\mathcal{N}=2$ sectors.

In theories of QG in which a large number of species are present, a prominent
role is played by the species scale $\Lambda_{QG}$ [44, 45, 46]. It may be
defined in different ways, presumably equivalent, and corresponds to the scale
$\Lambda_{\rm QG}$ at which quantum gravity effects become strong and can no
longer be ignored. This scale is well below the Planck scale in the presence
of a large number $N$ of species and in $D$ dimensions is of order

$\Lambda_{QG}\ \lesssim\frac{M_{\rm P}^{(D)}}{N^{1/(D-2)}}\ ,$ (5.3)

where $M_{\rm P}^{(D)}$ is the Planck scale in $D$ dimensions. When we have a
tower whose total number of states is $N_{t}=N$, one says that this tower
saturates the species bound. When a tower saturates the species scale then,
combining eqs.(5.1) and (5.3) one obtains (in Planck units)

$\Lambda_{QG}\ \simeq\ \left(m_{0}^{(t)}\right)^{p_{t}/(D-2+p_{t})}$ (5.4)

In our case, we have three types of towers: 1) bulk KK+winding (multiplicity
$p$), 2) KK+winding towers on the branes, 3) gonion (multiplicity $p_{\rm
gon}$) and 3) 4d strings (either fundamental or dual). In all the limits
considered in section 3 that are not emergent string limits, and in particular
in all the limits of section 4, the Species Scale is the string scale,
$\Lambda_{\rm QG}\simeq M_{s}$ and the geometric KK towers saturate the
species scale. Thus when we have $p=6$ the above formula yields $M_{s}\simeq
m_{\rm KK}^{3/4}$ which is consistent with the example e.g. in figure 5-II) in
which six dimensions are decompactified. When $p=2$, as in figure 6-IIa) with
two dimensions large, one rather has $M_{s}\simeq m_{\rm KK}^{1/2}$.

Gonion towers may or may not saturate the species bound. When $p_{\rm gon}=p$
there is saturation but there is not for $p_{\rm gon}<p$. This depends on
whether the total number of gonion states $N_{g}$ equals the KK total
$N=(M_{s}/M_{\rm P})^{2}$. In general the number of gonions along the $i^{\rm
th}$ complex dimension is $\sim(M_{s}/m_{\rm gon}^{i})^{2}$ so that

$N_{g}\ \simeq\ \prod_{i=1}^{p_{\rm gon}/2}\left(\frac{M_{s}}{m_{\rm
gon}^{i}}\right)^{2}\,.$ (5.5)

If the gonion tower saturates the species scale (which only happens for
$p_{\rm gon}=p=2,4,6$), one has $N_{g}=N=(M_{\rm P}^{2}/M_{s}^{2})$ and one
can write (in Planck units)

$M_{s}^{(p_{\rm gon}+2)}\ \simeq\ \prod_{i=1}^{p_{\rm gon}/2}(m_{\rm
gon}^{i})^{2}\ .$ (5.6)

Threfore, one has $m_{\rm gon}\simeq M_{s}^{1+2/p}/M_{\rm
P}^{2/p}=e^{2\phi_{4}/p}M_{s}$. Note that this agrees with the results
described in the examples of section 4 and figure 6-IIa) above. Note also that
in the cases where the gonion tower does not saturate the bound (5.6) does not
apply.

Let us consider for example the case of the toy model (4.11) and the gonion
towers living at the intersection $aa^{*}$. Let us in particular consider
$u\simeq u^{(i)}$ and the limit $s\sim u^{3}\sim\lambda$. We have gonions at
the three complex planes and 6 decompactifying dimensions, so that $p_{\rm
gon}=p=6$. The gonion masses are all of order $m^{i}_{\rm gon}\simeq 1/u^{2}$.
From table 5 one can compute the total number of gonion states $N_{aa^{*}}$ in
the sector $aa^{*}$

$N_{aa^{*}}\ \simeq\prod_{i}\left(\frac{M_{s}}{m^{i}_{\rm gon}}\right)^{2}\
\simeq\ k^{3}s^{2}e^{2\phi_{4}}\ \simeq\ k^{3}s\,.$ (5.7)

On the other hand one can compute the total number of species from the
geometric KK states using e.g. the species scale expression $N=M_{\rm
P}^{2}/\Lambda_{QG}^{2}\simeq u^{3}\sim s$. Hence $N_{g}=N$ and the gonion
tower at $aa^{*}$ also saturates the species scale. On the other hand at the
intersection $ab$ there is only a tower in the second complex plane so that
$N_{ab}\simeq(M_{s}/m_{{\rm gon},2})^{2}\simeq ku\simeq kN^{1/3}$ and the
tower does not saturate the species scale.

###  5.2 Emergence of kinetic terms

We have shown in previous sections how in the limit in which Yukawa couplings
vanish, towers of charged states (gonions) become light, and have also
discussed how in that limit the metric of the chiral fields localised at brane
intersections becomes singular in string units. It has been recently argued
that all kinetic terms of massless fields in QG may vanish in the UV and that
they are generated in the IR by summing over loop contributions over towers of
states. This is the so called Emergence proposal [35, 36, 37] (see also [38,
39, 40, 41, 12, 13, 42, 43]) which in its stronger version may be stated as
follows:

In a theory of Quantum Gravity all light particles in a perturbative regime
have no kinetic terms in the UV. The required kinetic terms appear as an IR
effect due to loop corrections involving the sum over a tower of
asymptotically massless states.

A milder version of this proposal states that for any singularity at infinite
distance in the moduli space, there is an infinite tower of states becoming
massless which induces quantum corrections to the metrics matching the tree
level behaviour.

It is interesting to check whether our findings in the previous sections are
consistent with this hypothesis or not. Specifically let us consider the
subsector of the theory in which two D6-branes, D6α and D6β branes intersect
at small angles. There will be towers of gonion states localised at their
intersection. We would like to check whether the sum over one-loop diagrams
involving the gonions are able to reproduce the form of the kinetic terms of
the chiral massless states and the $U(1)$ gauge boson under which the tower is
charged.

$\psi_{n}$$\psi_{n}$$q$$q-p$$V$$V$$p$$p$

$\tilde{\psi}_{n}$$\tilde{\psi}_{n}$$q$$q-p$$V$$V$$p$$p$

Figure 7: One-loop contributions to the gauge kinetic functions from a tower
of gonions.

$V_{n}$$\psi_{n}$$q$$q-p$$\chi$$\chi$$p$$p$

$\tilde{V}_{n}$$\tilde{\psi}_{n}$$q$$q-p$$\chi$$\chi$$p$$p$

Figure 8: One-loop contributions to a chiral fermion kinetic metric from a
tower of gonions and KK gauge tower.

Let us start with $g_{\alpha\beta}^{-2}$, the gauge kinetic function of the
gauge group generated by $Q_{\alpha\beta}\equiv Q_{\alpha}-Q_{\beta}$, which
is the one felt by the towers at the intersection. All the tower of gonions
has the same charge  which we denote by $q_{\alpha\beta}$. Gonions will
include vector-like ${\cal N}=1$ chiral multiplets which will contribute to
the renormalisation of the gauge kinetic function at one loop. Loops like
those in figure 7 contribute to the gauge kinetic function of the gauge group
$Q_{\alpha\beta}$ (see [12] for details of the computation)191919 To match the
results known in explicit string models it is crucial that all the states in
the tower have the same charge. This is unlike e.g. $U(1)$ BPS towers in which
charges grow with the level.

$\delta(g_{\alpha\beta}^{-2})\,\simeq\,q_{\alpha\beta}^{2}\sum_{k=1}^{N_{g}}\log(\Lambda_{QG}^{2}/m_{k}^{2})\,\simeq\,q_{\alpha\beta}^{2}\
N_{g}=q_{\alpha\beta}^{2}\prod_{i=1}^{p_{\rm gon}/2}\left(\frac{M_{s}}{m_{\rm
gon}^{i}}\right)^{2}\,.$ (5.8)

In terms of gonion angles, i.e. $(m_{\rm gon}^{i}/M_{s})^{2}\simeq\theta_{i}$,
one has

$\delta(g_{\alpha\beta}^{-2})\,\simeq\,q_{\alpha\beta}^{2}\
\prod_{i=1}^{p_{\rm gon}/2}\frac{1}{\theta_{i}}\,.$ (5.9)

So we learn that as the gonions become light, the gauge coupling becomes
small. The idea of emergence implies that if $N_{g}=N$ and the tower saturates
the species scale, the full gauge kinetic function will be given by

$g_{\alpha\beta}^{-2}\,\simeq\,q_{\alpha\beta}^{2}N\,\simeq\
q_{\alpha\beta}^{2}\left(\frac{M_{\rm
P}}{M_{s}}\right)^{2}\,=\,q_{\alpha\beta}^{2}\ e^{-2\phi_{4}}\,,$ (5.10)

where $e^{\phi_{4}}$ is the 4d dilaton. Note that this is independent of the
tower structure and only rests on the equality $N_{g}=N$.

Using the above expressions one can also write an expression in which $M_{s}$
does not appear explicitly

$g_{\alpha\beta}^{-2}\ \simeq\ \prod_{i=1}^{p_{\rm gon}/2}\left(\frac{M_{\rm
P}}{m_{\rm gon}^{i}}\right)^{4/(2+p_{\rm gon})}\,.$ (5.11)

On the other hand, if the tower of gonions does not saturate the species
scale, then there is a contribution to the $g_{\alpha\beta}^{-2}$ gauge
kinetic function as in eq.(5.8) but the above expression does not apply. Note
that $g_{\alpha\beta}^{-2}$ is just the contribution of the intersection
$\alpha\beta$ to the gauge group with generator $Q_{a}-Q_{\beta}$, On the
other hand there may be in general other contributions to the gauge kinetic
terms of $U(1)_{\alpha}$ and $U(1)_{\beta}$ from other intersections. Let us
again consider the toy example (4.11). The contribution from the intersection
of the $aa^{*}$ to the gauge coupling $g_{a}$ will be

$\delta(g_{a}^{-2})\,\simeq\,(2)^{2}N_{g}\,\simeq\,4k^{3}s\,.$ (5.12)

Interestingly, this matches with the tree level value for the coupling
constant $g_{a}$ in eq.(4.15), as expected from emergence in the limit $s\sim
u^{3}\simeq\lambda$ in which the $u$-dependent pieces are subleading. In fact
one can see that such subleading terms are induced by the contribution to
$g_{aa}^{-2}$ from the intersections $ab$ and $ac$:

$\delta(g_{b}^{-2})\,\simeq\,\left(\frac{M_{s}}{m_{{\rm
gon},2}}\right)^{2}\,\simeq\,ku\,,\qquad\delta(g_{c}^{-2})\,\simeq\,\left(\frac{M_{s}}{m_{{\rm
gon},3}}\right)^{2}\,\simeq\,ku\,.$ (5.13)

Let us consider now the metric of the chiral fields at the D6-brane
intersection $\alpha\beta$. In this case in the loop circulates a tower of
gonions, with the same charge as the massless field, along with a tower of KK
states living in the worldvolume of the intersecting branes. The trilinear
couplings involving the massless chiral field, a gonion and a KK state was
described in e.g. [99, 100]. The multiplicities of both towers are the same
and the one loop diagrams as in figure 8 have analogous overall behaviour than
the previous computation. One thus has again

$K_{\alpha\beta}\,\simeq\,\prod_{i=1}^{p_{\rm gon}/2}\left(\frac{M_{s}}{m_{\rm
gon}^{i}}\right)^{2}\,.$ (5.14)

Again, if the tower of gonions saturates the spacies scale one gets

$K_{\alpha\beta}\,\simeq\,\prod_{i=1}^{p_{\rm gon}/2}\left(\frac{M_{\rm
P}}{m_{\rm gon}^{i}}\right)^{4/(2+p_{\rm gon})}\,.$ (5.15)

Note that this result matches with that in eq.(2.34), assuming the gonion
tower saturates the species scale. Note also that in that saturation case one
has

$K_{\alpha\beta}\,\simeq\,g_{\alpha\beta}^{-2}\,\simeq\,N_{g}\,\simeq
N\,\simeq\,\left(\frac{M_{\rm P}}{M_{s}}\right)^{2}\,=\,e^{-2\phi_{4}}\,,$
(5.16)

which relates the metric at the singular intersection and the gauge kinetic
function of the gauge boson coupling to $Q_{\alpha}-Q_{\beta}$. This means
that when the metric at the intersection becomes singular, these gauge
couplings go to zero. This is in agreement with what we have seen in previous
sections.

Emergence ideas may also give information about the kinetic term of the moduli
involved in the problem. Thus e.g., consider the asymptotic EFT string limits.
There the masses of the KK states behave like

$m_{\rm KK}(\lambda)\,\simeq\,\frac{1}{\lambda^{w/2}}\,.$ (5.17)

One can then compute the metric of the $\lambda$ field by computing the one-
loop corrections involving the full tower of KK states up to the species
scale. This is by now a well known computation which yields for the metric
(see [36, 37, 12]):

$K_{\lambda}\,\simeq\,\frac{1}{m_{\rm KK}^{2}}\left(\partial_{\lambda}m_{\rm
KK}\right)^{2}\,\simeq\,\frac{1}{\lambda^{2}}\,,$ (5.18)

and hence a leading term for the $\lambda$-dependent Kähler potential

$K_{Q}(\lambda)\,\sim\,-\log\lambda\ ,$ (5.19)

as expected.

###  5.3 Singular Yukawa couplings and emergence

In [12, 13] it was already stated, using emergence arguments, that the (non-
vanishing) Yukawa couplings of 3-particles with associated towers of states
with characteristic scales $m_{a}$ takes the qualitative form

$Y_{ijk}\simeq\left(\frac{m_{i}}{M_{s}}\right)^{\gamma_{i}}\left(\frac{m_{j}}{M_{s}}\right)^{\gamma_{j}}\left(\frac{m_{k}}{M_{s}}\right)^{\gamma_{k}}\,,$
(5.20)

with the three factor arising from wave function renormalisation of the three
legs, and $\gamma_{a,b,c}$ being positive rational numbers depending on the
structure of the towers. In the present paper we confirm that expected
structure but within the realm of specific string models, which also feature
specific charged towers: the gonion particles. We also work within ${\cal
N}=1$ supergravity (not only rigid SUSY as in [13]). Thus in section 3 we
obtained the expression

$Y_{ijk}\simeq e^{\phi_{4}/2}\left(\frac{m^{i}_{\rm
gon}}{M_{s}}\right)^{1/2}\left(\frac{m^{j}_{\rm
gon}}{M_{s}}\right)^{1/2}\left(\frac{m^{k}_{\rm gon}}{M_{s}}\right)^{1/2}\,,$
(5.21)

where $m^{i}_{\rm gon}$ is the lightest of the gonion towers at the
intersection corresponding to the particle $i$, and the same for $j,k$. In the
interesting case in which only one of the intersections (say that associated
to particle $a$) is singular, there are no towers associated to the other two
particles in the coupling and one has

$Y_{ijk}\,\simeq\,e^{\phi_{4}/2}\left(\frac{m^{i}_{\rm
gon}}{M_{s}}\right)^{1/2}\,.$ (5.22)

If in addition the gonion towers associated to particle $i$ saturate the
species scale so that $N_{g}=N$ one can write

$Y_{ijk}\,\simeq\,e^{\phi_{4}/2}N^{-1/2}\,\simeq\,e^{\phi_{4}}\,\simeq\,g_{\alpha\beta}\,,$
(5.23)

where $g_{\alpha\beta}$ is the gauge coupling of the gauge boson coupling to
$Q_{\alpha}-Q_{\beta}$. An example of this setup is the toy model with the
limit (4.20) where $s,u^{(2)}\sim\lambda$ and $u^{(1)}$, $u^{(3)}$ are
bounded. This yields a spectrum of the EFT string II-a) type, as in figure 6.
In this case we have $p=p_{\rm gon}=2$ and a tower of gonions in the
intersection $ab$ in the second complex plane. Since $p=2$ one has
$M_{s}^{2}=m_{\rm KK}M_{\rm P}\sim 1/u$, so that $N=u$. On the other hand the
number of gonions is $N_{g}=(M_{s}/m_{{\rm gon},2})^{2}=u$, so the gonion
tower saturates. So one concludes that the Yukawa coupling should go like
$Y\sim g_{a}\sim 1/\lambda^{1/2}\simeq 1/u^{1/2}$, which agrees with what is
found in the previous chapters.

###  5.4 Chiral towers and the (sub)Lattice/Tower WGC

As we have seen, the towers of gonions at an intersection of two branes $a,b$
all have the same bifundamental charge. They are in fact charged under the
linear combination of $U(1)$’s proportional to $Q_{a}-Q_{b}$, but not the
orthogonal one. This $U(1)$ gets a Stückelberg mass, but this mass is in
general much lighter than the gonion masses, so that there is typically a
large range of scales at which we have a tower of states, with the same
charge, coupling to an approximately massless $U(1)$. As we mentioned in the
introduction, this violates the (sub)Lattice and Tower WGC, since at least at
the particle level, there is no trace of a tower containing a full spectrum of
$U(1)$ charges. In fact this is not totally surprising since examples of
violations have appeared already in the literature in models with $\geq 8$
supercharges [101, 28, 102, 103] . The inspiration to put forward those
extensions came from demanding the consistency of the WGC under dimensional
reduction [104, 31, 33, 32]. In particular, starting with an extremal particle
under a $U(1)$ and compactifying on a circle, an extra $U(1)_{\rm KK}$ gauge
boson appears. Studying the convex hull for these two $U(1)$’s one can easily
see that the WGC is violated. This problem can be cured if there is a full
tower of states with all the charges, as in those mentioned extensions.
However in our case we have seen that the charged gonions are not extremal but
rather have a further power of $g$ suppression, with $m_{\rm gon}\simeq
g^{2}M_{\rm P}$. Then one can see that a compactification down to three
dimensions cannot lead to any violation of the WGC (see figure 9). As far as
we are aware, this way of avoiding WGC inconsistencies under dimensional
reduction is novel, it would be interesting to explore possible connections
with other exceptions to the (sub)Lattice/Tower WGC discussed in the
literature.

$Z_{U(1)}$$Z_{KK}$$e^{-\phi_{4}}$$1$$1$ Figure 9: The ‘convex hull’ (see [104]
for notation) of gonions and a KK tower in circle compactification, here just
the vertical dot line, always contains the extremal circle, if we remain in a
perturbative region $e^{-\phi_{4}}=g^{-1}\geq 1$.

Let us finally comment on an important difference between the $U(1)$’s we are
referring to here and those studied in non-chiral ${\cal N}\geq 2$ theories.
To start with they are not strictly massless, but get in general a Stückelberg
mass. Furthermore, the $U(1)$ coupling to the massless chiral fields at the
intersection leads to anomalies. This anomaly will be generically cancelled by
a 4d Green–Schwarz mechanism, as is well known happens in this class of
theories.

##  6 Final comments and outlook

In recent years much has been learned about the behaviour of EFT theories from
QG and string theory at infinite distance limits in moduli space. However most
of the setups explored so far display $\geq 8$ supercharges. In the present
paper we have concentrated instead on the case of 4d ${\cal N}=1$ chiral
theories, which are more relevant for possible phenomenological applications.
In fact, independently of any possible phenomenological application, this
class of theories present novel features like chiral fermions and Yukawa
couplings, which we have found give rise to new, not yet well explored
phenomena. We have used as a laboratory type IIA CY orientifolds with chiral
charged multiplets localised at $D6$-brane intersections. We believe though
that, thanks to the different dualities to type IIB, F-theory and heterotic
strings, the qualitative results are expected to apply in more general
settings.

One first question we have asked is whether, in analogy to gauge couplings,
the limit $Y\to 0$ is at infinite distance. We have found that indeed this is
the case, both at large Kähler and complex structure moduli. In the former
case as the Yukawas vanish a tower of D0’s appears. The Yukawas also vanish
along trajectories of large complex structure, a type of limits that turns out
to be much richer. Indeed, in addition to the expected KK and string towers,
we have found that in that limit a slightly more exotic type of towers appear.
They correspond to towers of gonions: charged chiral multiplets in massive
vector-like representations of the D6-brane gauge group, all having the same
bifundamental charge as the zero modes and also living at the intersections of
D6-branes. Although in type IIA these states come from open string oscillators
stretched from one intersecting brane to another, the spectrum is not
Hagedorn-like but rather has the form $m_{gon}^{2}=\theta_{ab}M_{s}^{2}$. In
fact one can write the Yukawa couplings in these limits in terms of the gonion
masses, as in (1.3), showing explicitly how a vanishing Yukawa coupling
implies in these limits the presence of a light tower of charged gonions.

We find that, in limits parametrised by a single complex structure field one
gets a general behaviour of the form $Y_{*}\simeq 1/u^{r}$, with $r>0$ a
rational number, which take values $r=1/4,1/3,1/2,3/4,1$ in most of the
examples explored. Interestingly, this type of behaviour is consistent with
recent results [92, 93] in heterotic string theory, and it would be
interesting to explore possible connections. One also finds that, as the
gonion towers become light, some of the kinetic terms of chiral fermions
involved in the Yukawa coupling diverge, which is at the root of the vanishing
of the Yukawa, once one goes to canonical kinetic terms. At the same time,
some D6-brane gauge group factors coupling to both chiral matter and their
gonions go also to zero as the Yukawa does, like $Y_{*}\sim g_{*}^{2r}$. This
also explains why the limit $Y_{*}\rightarrow 0$ must be at infinite distance,
since such a limit would imply the presence of global symmetries.

The towers of charged gonions are quite interesting theoretically, since they
appear to provide examples in which a (sub)Lattice/Tower WGC conjecture does
not hold. In particular, the particles in the gonion towers have all the same
charge as the massless chiral fermions and there is no trace of particles with
multiple charges. Still, this does no lead to any contradiction upon
dimensional reduction since gonion masses are not extremal particles and have
a further $g_{*}$ suppression factor in their mass. A different direction that
we study is whether our findings concerning the kinetic terms of chiral fields
and gauge bosons as well as the Yukawa couplings are consistent with the
Emergence hypothesis. We find that indeed that is the case, and the particular
form of the gonion towers is crucial to reproduce what is known from explicit
examples. It would be interesting to make a more thorough analysis in a
variety of explicit models to make further tests of this hypothesis, as this
class of ${\cal N}=1$ 4d models offers very rich possibilities.

We think we have only scratched the surface of ${\cal N}=1$ 4d models in what
concerns infinite distance properties and Swampland constraints, and we
believe that it may give us many surprises. Furthermore, the comparison of our
findings with alternative computations of Yukawa couplings in other type of
compactifications [92, 93] would be very interesting. Last but not least, the
limit of small Yukawas may be of much phenomenological relevance since, as we
said, most of the Yukawa couplings of the SM are small. We will present
results in this direction, particularly in the context of neutrino masses, in
a companion paper [15].

Acknowledgments

We would like to thank Alberto Castellano, José Luis Hernando and Ignacio Ruiz
for discussions. This work is supported through the grants CEX2020-001007-S
and PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by ERDF
A way of making Europe. G.F.C. is supported by the grant PRE2021-097279 funded
by MCIN/AEI/ 10.13039/501100011033 and by ESF+.

##  Appendix A Gonion spectrum for D6-branes at angles

In this appendix we describe the mass spectrum of open string states between
two intersecting D6-branes, following [49, 105]. For examples of such spectra
in models with realistic gauge sectors see for instance [50].

Two static D6-branes can intersect transversely in at most three angles. At
such intersection, one can model the open-string spectrum by considering the
flat-space setup of [56, 106], and then applying the bosonisation language
that is standard from orbifold twisted sectors [107, 108]. In this language
the intersection angles are encoded in the four-dimensional twist vector
$v_{\theta}=(\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2},\theta_{\alpha\beta}^{3},0)$
of the $\alpha\beta$ sector, where $-1<\theta_{\alpha\beta}^{r}\leq 1$
measures the three intersection angles in units of $\pi$ and the last entry is
identified with the complex dimension filled by the D6-branes besides the
light-cone directions. Oscillation states are described by the sum
$\chi=r+v_{\theta}$, with $r\in\mathbbm{Z}^{4}$ in the Neveu–Schwarz and
$r\in(\mathbbm{Z}+\frac{1}{2})^{4}$ in the Ramond sector, and where the GSO
projection is implemented via $\sum_{i}r^{i}=$ odd. Finally, if one of the
angles vanishes then the D6-branes are parallel in one dimension and can be
separated by a distance $L$. In terms of these quantities the mass of an open
string state in the $\alpha\beta$ sector is given by

$\alpha^{\prime}m_{\alpha\beta}^{2}=\frac{L^{2}}{4\pi\alpha^{\prime}}+N_{bos}(\theta_{\alpha\beta}^{r})+\frac{(r+v_{\theta})^{2}}{2}-\frac{1}{2}+E_{\alpha\beta},$
(A.1)

where $E_{\alpha\beta}$ is the vacuum energy

$E_{\alpha\beta}=\sum_{r=1}^{3}\frac{1}{2}|\theta_{\alpha\beta}^{r}|\left(1-|\theta_{\alpha\beta}^{r}|\right)\,.$
(A.2)

Finally, $N_{bos}(\theta_{\alpha\beta}^{r})$ stands for the contribution
coming from worldsheet bosonic oscillations on top of either either of the
above R or NS states. The creator operators that implement twisted
oscillations are of the form $\alpha_{-|\theta^{r}|}$ or
$\alpha_{|\theta^{r}|-1}$ and increase the mass of the state by
$|\theta_{\alpha\beta}^{r}|/\alpha^{\prime}$ and
$(1-|\theta_{\alpha\beta}^{r}|)/\alpha^{\prime}$, respectively.

Given these formulae it is easy to construct the spectrum at each D6-brane
intersection and see its explicit dependence on the intersection angles.
Notice that the fourth entry of the vector $r+v_{\theta}$, which will be
either an integer (NS sector) or a half integer (R sector), provides the 4d
Lorentz quantum number of the open string state. Let us for instance consider
an intersection such that
$\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2}>0$,
$\theta_{\alpha\beta}^{3}<0$. and $|\theta_{\alpha\beta}^{r}|\ll 1$. Then it
is easy to check that we will have a unique massless state coming from the R
sector:

$\chi=r+v_{\vartheta}=(\theta_{\alpha\beta}^{1}-\frac{1}{2},\theta_{\alpha\beta}^{2}-\frac{1}{2},\theta_{\alpha\beta}^{3}+\frac{1}{2},-\frac{1}{2})\,,$
(A.3)

which, in terms of four-dimensional physics, corresponds to a Weyl fermion of
negative chirality. On the other hand, the lightest excitations in the NS
sector are given by:

$\begin{array}[]{ccc}{\bf State}&&{\bf Mass}^{2}\\\
\chi=(\theta_{\alpha\beta}^{1}-1,\theta_{\alpha\beta}^{2},\theta_{\alpha\beta}^{3},0)&&\alpha^{\prime}m_{\alpha\beta}^{2}=\frac{1}{2}\left(-|\theta^{1}_{\alpha\beta}|+|\theta_{\alpha\beta}^{2}|+|\theta_{\alpha\beta}^{3}|\right)\\\
\chi=(\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2}-1,\theta_{\alpha\beta}^{3},0)&&\alpha^{\prime}m_{\alpha\beta}^{2}=\frac{1}{2}\left(|\theta^{1}_{\alpha\beta}|-|\theta_{\alpha\beta}^{2}|+|\theta_{\alpha\beta}^{3}|\right)\\\
\chi=(\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2},\theta_{\alpha\beta}^{3}+1,0)&&\alpha^{\prime}m_{\alpha\beta}^{2}=\frac{1}{2}\left(|\theta^{1}_{\alpha\beta}|+|\theta_{\alpha\beta}^{2}|-|\theta_{\alpha\beta}^{3}|\right)\\\
\chi=(\theta_{\alpha\beta}^{1}+1,\theta_{\alpha\beta}^{2},\theta_{\alpha\beta}^{3},0)&&\alpha^{\prime}m_{\alpha\beta}^{2}=\frac{1}{2}\left(3|\theta^{1}_{\alpha\beta}|+|\theta_{\alpha\beta}^{2}|+|\theta_{\alpha\beta}^{3}|\right)\\\
\chi=(\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2}+1,\theta_{\alpha\beta}^{3},0)&&\alpha^{\prime}m_{\alpha\beta}^{2}=\frac{1}{2}\left(|\theta^{1}_{\alpha\beta}|+3|\theta_{\alpha\beta}^{2}|+|\theta_{\alpha\beta}^{3}|\right)\\\
\chi=(\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2},\theta_{\alpha\beta}^{3}-1,0)&&\alpha^{\prime}m_{\alpha\beta}^{2}=\frac{1}{2}\left(|\theta^{1}_{\alpha\beta}|+|\theta_{\alpha\beta}^{2}|+3|\theta_{\alpha\beta}^{3}|\right)\end{array}$
(A.4)

These states will all correspond to $4d$ scalars, as its associate helicity
shows. To each of these states we can apply the bosonic creation operators

$(\alpha_{-|\theta_{\alpha\beta}^{1}|})^{k_{1}}\,(\alpha_{-|\theta_{\alpha\beta}^{2}|})^{k_{2}}\,(\alpha_{-|\theta_{\alpha\beta}^{2}|})^{k_{3}}\,,$
(A.5)

from where the spectrum (2.14) is obtained. Finally, the lightest $W$-boson
state is

$\begin{array}[]{ccc}\chi=(\theta_{\alpha\beta}^{1},\theta_{\alpha\beta}^{2},\theta_{\alpha\beta}^{3},\pm
1)&&\alpha^{\prime}m_{\alpha\beta}^{2}=\frac{1}{2}\left(|\theta^{1}_{\alpha\beta}|+|\theta_{\alpha\beta}^{2}|+|\theta_{\alpha\beta}^{3}|\right)\end{array}$
(A.6)

and so from applying (A.5) we recover the tower of states (2.16). As we reach
the string scale, further states will appear that come from either applying
bosonic oscillators of the form $\alpha_{|\theta^{r}|-1}$ or considering more
general vectors $r$. The latter incorporates 4d helicities of higher spin, and
so at this scale we recover the higher-spin, Hagedorn-like spectrum of a
critical string.

##  Appendix B Mirror duals of type I magnetised Calabi–Yau compactifications

In this appendix we translate the model building setup of [86, 91], which
features type I compactifications on smooth Calabi–Yau with $U(n)$ bundles, to
the type IIA setting that we use along the main text. This framework is
particularly useful for the construction of STU-like models in section 3.

Let us consider type I string theory compactified on a smooth Calabi–Yau $Y$,
with a basis of Nef divisors $\ell_{s}^{-2}\omega_{i}$ and triple intersection
numbers
$\kappa_{ijk}=\ell_{s}^{-6}\int_{Y}\omega_{i}\wedge\omega_{j}\wedge\omega_{k}$.
The Kähler potential for the Kähler sector reads

$K_{Q}=-\log s-\log\frac{1}{6}\kappa_{ijk}u^{i}u^{j}u^{k}\,,$ (B.1)

with

$s=e^{-\phi}V_{Y}\,,\qquad u^{i}=e^{-\phi}t^{i}\,,$ (B.2)

$J=t^{i}\omega_{i}$ the string frame Kähler form of $Y$ and
$V_{Y}=\frac{1}{6}\kappa_{ijk}t^{i}t^{j}t^{k}$ its volume. A single D9-brane
$\alpha$ with a worldvolume flux $f_{\alpha}^{i}\omega_{i}$ can be represented
as

$[\Pi_{\alpha}]=\frac{1}{2}[A]+\frac{1}{2}f_{\alpha}^{i}[D_{i}]-\left(\frac{1}{2}\kappa_{ijk}f_{\alpha}^{j}f_{\alpha}^{k}+\frac{1}{24}c_{2\,i}\right)[C^{i}]+\left(\frac{1}{6}\kappa_{ijk}f_{\alpha}^{i}f_{\alpha}^{j}f_{\alpha}^{k}+\frac{1}{24}c_{2\,i}f_{\alpha}^{i}\right)[B]\,,$
(B.3)

with $[D_{i}]=2\ell_{s}^{-2}\omega_{i}$, and
$[C^{i}]=\ell_{s}^{-4}\tilde{\omega}^{i}$ a four-forms basis such that
$\ell_{s}^{-6}\int_{Y}\omega_{i}\wedge\tilde{\omega}^{j}=\delta_{i}^{j}$. In
addition $[B]=d{\rm vol}_{Y}/V_{Y}$ is a normalised volume form and $[A]=2$.
Finally, $c_{2}(Y)=c_{2\,i}[C^{i}]$ is the decomposition of the second Chern
class of $Y$. The orientifold image of D9-brane is obtained by performing the
sing flip $f_{\alpha}^{i}\to-f_{\alpha}^{i}$. The chiral index between two
D9-branes reads

$[\Pi_{\alpha}]\cdot[\Pi_{\beta}]=\frac{1}{6}\kappa_{ijk}(f_{\beta}^{i}-f_{\alpha}^{i})(f_{\beta}^{j}-f_{\alpha}^{j})(f_{\beta}^{k}-f_{\alpha}^{k})+\frac{1}{12}c_{2\,i}(f_{\beta}^{i}-f_{\alpha}^{i})\,,$
(B.4)

where we have used the intersection rules $[A]\cdot[B]=2$,
$[C^{i}]\cdot[D_{j}]=2\delta^{i}_{j}$. This choice of normalisation reflects
that $[A]$ is Poincaré dual to a pair of Euclidean D5-branes wrapping $Y$ and
with gauge group USp$(2)$, while $[C^{i}]$ are instead duals to $O(1)$
instantons made up from Euclidean D1-branes wrapping two-cycles of $Y$.
Correspondingly, the class $[C^{i}]$ also represents space-time filling
D5-branes wrapped on such two-cycles. Finally, the orientifold plane content
amounts to the charge $[\Pi_{\rm O}]=-16[A]+2c_{2\,i}[C^{i}]$.

From here one can derive the basic model-building rules used in [86, 91], and
one can see that the gauge couplings and FI-terms of these references map
under mirror symmetry to the ones used in the main text. The mirror map to
type IIA translates the classes $[A]$ and $[C^{i}]$ into odd three-forms,
while $[B]$ and $[D_{i}]$ are mapped to even three-forms of the mirror
Calabi–Yau $X$.

##  Appendix C Selection rules for Yukawas

In this appendix we describe the selection rules for the Yukawa couplings. In
doing so, we will get a rich picture of the constraints this imposes in terms
of angle selection and gonion mass dependencies, which are important for the
results of the main text.

Yukawa selection rules have been studied extensively in the context of
heterotic orbifold theory [67, 66, 109, 110, 111]. They are derived from the
structure of the vertex operators and the orbifold symmetries. In particular,
twisted vertex operators in the right-moving sector of the theory carry the
so-called H-momentum, that is, the momentum along the bosonised fermions. Such
momentum conservation is responsible for providing non-trivial selection rules
that indeed forbid many couplings for both twisted and untwisted fields.

This rationale is also applicable to the intersecting branes picture. Indeed,
by dimensional reduction from 10d theory, one should be able to analogously
translate the above H-momentum conservation condition into constraints on the
twisted vector $r$ defined in (A.3). These constraints read

$r^{f}_{i}+r^{f}_{j}+r^{b}_{k}=\left(0,0,0,\pm 1\right)$ (C.1)

or equivalently,

$\sum_{i}r_{i}^{b}=\left(1,1,1,0\right),$ (C.2)

where $r^{f}$ and $r^{b}$ denote the fermionic and bosonic twisted vectors,
respectively. One way to motivate these rules is from the viewpoint of a
Higgssed 4d ${\cal N}=4$ gauge theory. Indeed, let us consider a stack of
three coincident D6-branes in flat space, with a $U(3)$ gauge theory in their
worldvolume, and with an ${\cal N}=4$ superpotential
$W=W_{ijk}\mbox{Tr}\\{\Phi^{i}[\Phi^{j},\Phi^{k}]\\}$ for its adjoint fields.
We now give a linear vev to the off-diagonal entries of its adjoint fields
such that it implements the gauge breaking $U(3)\rightarrow U(1)_{a}\times
U(1)_{b}\times U(1)_{c}$.202020Geometrically, we are describing the
intersection of three D6-branes. See e.g. [112] for a similar analysis in the
context of intersecting D7-branes. From the initial ${\cal N}=4$
superpotential one obtains a Yukawa coupling of the form $Y_{ijk}\sim
Y\varepsilon_{ijk}\psi^{i}\psi^{j}\phi^{k}$, in agreement with the selection
rules given by (C.2). In a mirror picture of magnetised D9-branes, the same
kind of rules would arise from the dimensional reduction procedure described
in [71]. For further selection rules on type II Yukawas see e.g. [113, 114,
115].

It should be emphasised that the above equations place strong constraints on
the allowed Yukawa coupling, which are directly linked to the patterns and
findings observed along this paper. To illustrate this with an example, let us
consider three stacks of D6-branes with the following intersecting angles

$\displaystyle\theta_{ab}$ $\displaystyle=$
$\displaystyle\left(\theta_{ab}^{1},\theta_{ab}^{2},\theta_{ab}^{3},0\right),$
(C.3) $\displaystyle\theta_{bc}$ $\displaystyle=$
$\displaystyle\left(\theta_{bc}^{1},\theta_{bc}^{2},\theta_{bc}^{3},0\right),$
(C.4) $\displaystyle\theta_{ca}$ $\displaystyle=$
$\displaystyle\left(\theta_{ca}^{1},\theta_{ca}^{2},\theta_{ca}^{3},0\right),$
(C.5)

and impose the supersymmetry condition
$\sum_{i}\theta_{\alpha\beta}^{i}=0$.212121In general, supersymmetry requires
$\sum_{i}\theta_{\alpha\beta}^{i}=0$ mod $2\pi$. We will not consider
$\sum_{i}\theta_{\alpha\beta}^{i}\neq 0$ for the present example.  Moreover,
by the definition of $\theta_{\alpha\beta}^{i}$, the sum
$\theta_{ab}^{i}+\theta_{bc}^{i}+\theta_{ca}^{i}$ also cancels. This greatly
restricts the number of possible configurations of $\theta_{\alpha\beta}^{i}$.
The parameter space is given by four variables, namely given
$\\{\theta^{1}_{ab},\theta^{2}_{ab},\theta^{1}_{bc},\theta^{2}_{bc}\\}$ the
full configuration is determined. Let us in particular take the following
configuration

$\displaystyle\theta_{ab}$ $\displaystyle=$
$\displaystyle\left(|\theta_{ab}^{1}|,|\theta_{ab}^{2}|,-|\theta_{ab}^{3}|,0\right),$
(C.6) $\displaystyle\theta_{bc}$ $\displaystyle=$
$\displaystyle\left(|\theta_{bc}^{1}|,-|\theta_{bc}^{2}|,|\theta_{bc}^{3}|,0\right),$
(C.7) $\displaystyle\theta_{ca}$ $\displaystyle=$
$\displaystyle\left(-|\theta_{ca}^{1}|,|\theta_{ca}^{2}|,|\theta_{ca}^{3}|,0\right),$
(C.8)

and assume that none of the angles vanish. The lightest boson in each
intersection is given by the following twisted vectors

$r_{ab}^{b}=\left(0,0,1,0\right),\quad r_{bc}^{b}=\left(0,1,0,0\right),\quad
r_{ca}^{b}=\left(1,0,0,0\right),\ $ (C.9)

or equivalently, in terms of their supersymmetric fermionic partners,

$r_{ab}^{f}=\frac{1}{2}\left(-,-,+,-\right),\quad
r_{bc}^{f}=\frac{1}{2}\left(-,+,-,-\right),\quad
r_{ca}^{f}=\frac{1}{2}\left(+,-,-,-\right).$ (C.10)

As can be seen, the selection rules for the lightest fields in each
intersection are fulfilled. Notice that this will not hold for all Yukawas,
and in the model of section 4 some Yukawas will be forbidden at tree level. In
that case, Yukawa couplings arise between zero modes and heavier modes.

Applying equations (2.29) and (C.9), the Kähler metrics entering the Yukawa
are

$\displaystyle K_{ab}$ $\displaystyle=$ $\displaystyle
e^{\phi_{4}}(2\pi)^{-1/2}\left(\frac{\Gamma(|\theta^{1}_{ab}|)\,\Gamma(|\theta^{2}_{ab}|)\,\Gamma(1-|\theta^{3}_{ab}|)}{\Gamma(1-|\theta^{1}_{ab}|)\,\Gamma(1-|\theta^{2}_{ab}|)\,\Gamma(|\theta^{3}_{ab}|)}\right)^{1/2}$
(C.11) $\displaystyle K_{bc}$ $\displaystyle=$ $\displaystyle
e^{\phi_{4}}(2\pi)^{-1/2}\left(\frac{\Gamma(|\theta^{1}_{bc}|)\,\Gamma(1-|\theta^{2}_{bc}|)\,\Gamma(|\theta^{3}_{bc}|)}{\Gamma(1-|\theta^{1}_{bc}|)\,\Gamma(|\theta^{2}_{bc}|)\,\Gamma(1-|\theta^{3}_{bc}|)}\right)^{1/2}$
(C.12) $\displaystyle K_{ca}$ $\displaystyle=$ $\displaystyle
e^{\phi_{4}}(2\pi)^{-1/2}\left(\frac{\Gamma(1-|\theta^{1}_{ca}|)\,\Gamma(|\theta^{2}_{ca}|)\,\Gamma(|\theta^{3}_{ca}|)}{\Gamma(|\theta^{1}_{ca}|)\,\Gamma(1-|\theta^{2}_{ca}|)\,\Gamma(1-|\theta^{3}_{ca}|)}\right)^{1/2}$
(C.13)

which, at small angles (or diluted fluxes from the mirror IIB perspective),
reduces to

$K_{\alpha\beta}\simeq\left[\frac{e^{2\phi_{4}}|\theta_{\alpha\beta}^{i}|}{2\pi|\theta_{\alpha\beta}^{j}||\theta_{\alpha\beta}^{k}|}\right]^{1/2}$
(C.14)

as it is well known in the literature [71, 72]. Finally, from (C.14), the
Yukawa coupling takes the form already presented in (3.5), in terms of gonion
masses

$K_{\alpha\beta}\simeq e^{2\phi_{4}}\frac{M_{\rm P}}{m_{\rm gon}}\,.$ (C.15)

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