example reveal that fundamental instantons actually induce multi-fermion and
higher-derivative F-terms as in [24, 25], as opposed to superpotential terms
like those in (7.13). Whether or not this does indeed occur can only be
determined with some knowledge of the microscopic UV completion of the four-
dimensional model. Let us therefore consider some known examples in string
theory.

In string theory models the fundamental instantons sourcing extremal wormholes
uplift to various types of branes wrapping internal cycles, which support
various types of localized world-volume degrees of freedom – see for instance
[108, 109, 110, 111, 112, 113] and [42] for a review. Extra fermionic zero-
modes are known to appear for example as Goldstinos of an “accidental”
enhanced supersymmetry felt by microscopic brane configurations that are
“strongly” isolated, in the sense that these brane configurations as well as
any multiple thereof are isolated and can probe only some local internal
geometry. However, the string theory realizations described in Section 6.5
clearly indicate that such strongly isolated branes correspond to non-EFT
instanton charges, whereas an EFT instanton charge (or an appropriate multiple
thereof) corresponds to a non-isolated brane that can explore the entire
compactification space and in principle “detect” its global minimally
supersymmetric structure. Thus, our EFT considerations should not be affected
by additional Goldstino-like fermionic zero-modes associated with a local
enhanced supersymmetry.

More concretely, in the F-theory models described in Sections 4.1.1, 4.1.2 and
6.5.1, two simple upliftings of EFT instantons are provided by Euclidean
D3-branes wrapping the hyperplane divisor in $\mathbb{P}^{3}$ and the vertical
divisor in a $\mathbb{P}^{1}$ fibration over $\mathbb{P}^{2}$. Even if at
first sight these branes contain extra fermionic (and bosonic) zero-modes,
those are actually lifted by the inclusion of world-volume fluxes [114] and/or
the interaction with background D3-branes [97]. As a result, in these explicit
examples the localized contribution associated to the EFT instanton is known
not to bring extra fermionic zero-modes. By F-theory/heterotic duality, the
F-theory mechanism must have heterotic counterparts. As discussed in Section
6.5.2, in the large volume perturbative regime part of the EFT instantons
correspond to F1-strings wrapping movable curves, which are either non-
isolated or admit a non-isolated multiple. Our remarks then fit well with the
early observation [109] that non-isolated genus zero world-sheet instantons
satisfy all necessary conditions to contribute to the superpotential.

Note that non-EFT instantons may also generate superpotentials of the form
(7.14). In fact, many string theory examples of brane instanton
superpotentials are generated by strongly isolated branes – see e.g. the
review [42]. However, according to the above observations and to the
discussion of Section 5.4, these non-perturbative effects may be interpreted
as being non-gravitational in nature.

These considerations suggest that extremal wormholes generated by EFT
instantons should dominantly match onto effective superpotentials like (7.14)
at low energies. More precisely, because the cleanest examples of [97]
correspond to extremal wormholes in perturbative sectors of homogeneity degree
$n=3$, this conclusion should at least apply to that subclass of extremal
wormholes. This expectation fits well with the Supersymmetric Genericity
Conjecture of [97], which implies that in an ${\cal N}=1$ four-dimensional
theory of quantum gravity a superpotential term should vanish only if the
theory is related to a higher supersymmetric one. In our context, the $n=3$
EFT instantons appear as the natural responsible for the realization of this
conjecture, at least for $t^{i}$ chiral sectors that cannot get superpotential
terms at tree level, as for instance induced by the three-form potentials
mentioned in Footnote 16. Note that the Supersymmetric Genericity Conjecture
is quantum gravitational in nature, which resonates well with the already
emphasized gravitational nature of the EFT instantons, and the fact that their
existence can be guaranteed by invoking the BPS instanton tower or sublattice
WGCs of Section 5.4. These speculations would also imply that the low energy
theory contains a sum $\sum_{\bf q}W_{\bf q}$ of contributions like (7.13),
for all charges populated by EFT instantons with $n=3$, and that according to
BPS instanton tower/sublattice WGCs this sum is infinite.

In Subsection 7.4, we will see how these considerations can be generalized to,
and further supported by, the marginally degenerate wormholes discussed in
Section 6.4.

###  7.3 Wormhole D-terms

Let us now consider the everywhere non-degenerate wormholes (i.e. those with
$n>3$). They do not preserve any of the four supersymmetries of the asymptotic
Minkowski vacuum, and as such they are expected to have four corresponding
Goldstino zero-modes. In the case of homogeneous wormholes of Section 6.1,
such zero-modes are explicitly constructed in Appendix D.2. A computation like
(7.12) now vanishes identically: no F-term can be generated. Instead, one can
check that it is correlators of the type
$\langle\bar{\chi}\bar{\chi}\chi\chi\rangle$ that can be reproduced at large
distances by a local four-fermion operator, since all zero-modes have
wavefunctions that asymptotically behave as the free Feynman propagator
$S_{\rm F}$ for any $n>3$, as one can check by expanding (D.36) for $r\gg L$.
The case $n=3$, as usual, should be treated separately, and will be considered
in Section 7.4.

The leading terms in the low-energy contributions (7.3) should then organize
themselves into a supergravity D-term providing a correction $\Delta K_{\bf
q}$ of the Kähler potential. Following the same procedure used to derive
(7.12), or equivalently adopting the arguments of [115], one finds that
$\Delta K_{\bf q}$ should be given by the superfield extension of the operator
(7.2). Specifying to the homogeneous solution (6.7) we have

$\Delta K_{\bf q}={\cal A}^{\prime}_{\bf q}\,\left(e^{2\pi{\rm i}\langle{\bf
q},{\bm{t}}\rangle}+e^{-2\pi{\rm i}\langle{\bf
q},\overline{\bm{t}}\rangle}\right)e^{-2\pi f({\rm Im\hskip 0.70004pt}t)}\,,$
(7.15)

where ${\cal A}^{\prime}_{\bf q}$ is another unknown normalization constant
containing also loop effects, $t^{i}=a^{i}+is^{i}$ should of course be
considered as chiral superfields, and from the on-shell action (6.10) we read

$2\pi f({\rm Im\hskip 1.00006pt}t)\equiv S|_{\text{hw}}-2\pi\langle{\bf
q},{\bm{s}}\rangle=\left[\sin\left(\frac{\pi}{2}\sqrt{\frac{3}{n}}\right)-1\right]2\pi\langle{\bf
q},{\rm Im\hskip 1.00006pt}{\bm{t}}\rangle\,.$ (7.16)

This expression is reliable as long as ${\bf q}$ is large, $n>3$, and $s^{i}$
is aligned with ${\cal F}^{i}({\bf q})$. For smaller charges and/or non-
homogeneous saxionic wormholes $f({\rm Im\hskip 1.00006pt}t)$ may of course be
different. Still, according to the general estimates given at the beginning of
this section, any non-perturbative correction like (7.15) should be extremely
tiny within the regime of validity of the EFT, and certainly much smaller than
the expected perturbative corrections to the leading Kähler potential (2.5).
However, even if tiny, (7.16) represents a breaking of the axionic shift
symmetries, which cannot be broken by perturbative corrections.

As already emphasized, the two derivative action (6.10), which is always
accurate in the large ${\bf q}$ limit, is lower than the BPS action of an EFT
instanton, or of a multi BPS instanton configuration carrying the same charge.
We have argued in 5.5 that it is precisely because of a finite size throat
that this happens. We are thus tempted to interpret non-extremal half-
wormholes as non-BPS bound states of fundamental instantons describing throats
that open up around the EFT instantons themselves. In this view, the throat-
opening would effectively integrate out whatever localized internal degrees of
freedom the EFT instantons may have. Furthermore, being non-BPS bound states,
non-extremal wormholes would induce D-terms like (7.15), rather than F-terms
as in Section 7.2. Non-extremal wormholes are thus qualitatively different
than EFT instantons, but may simultaneously be intimately connected. This view
is consistent with the ten-dimensional uplifts of similar wormholes recently
constructed in [105], that were interpreted in terms of intersecting Euclidean
branes. In the following subsection we will further elaborate on the
connection between wormholes and instantons.

###  7.4 Marginally degenerate wormholes and superpotentials

It is now time to discuss the physics associated with the marginally
degenerate wormholes ($n=3$). In Section 6.4 we have seen how, slightly away
from the wormhole’s neck, such configurations have approximately the same form
as extremal BPS wormholes associated with EFT instantons of the same charge.
Moreover, after an appropriate IR regularization their action (6.36) looks
approximately like the instanton local term (5.35) that complements extremal
wormholes. These observations strongly suggest a deep connection between
wormholes with $n=3$ and fundamental EFT instantons. In this section we would
like to elaborate upon this connection. We first motivate the apparently ad
hoc IR regularization introduced in Section 6.4 and subsequently discuss how,
extending to our general framework a mechanism proposed in [29, 30],
marginally degenerate wormholes may be able to generate superpotential terms
as those discussed in Section 7.2.

In order to unclutter the notation, here the IR-regularizing cutoff will be
simply denoted as $\Lambda$, rather than $\Lambda_{\text{\tiny IR}}$ as in
Section 6.4. This choice is also consistent with the fact that in this section
$\Lambda$ plays the role of the Wilsonian cutoff of the IR effective theory
that one obtains by “integrating out” the wormhole contributions, as discussed
at the beginning of this section. One should then keep in mind that in this
section $\Lambda$ is much lower than, and should not be confused with, the UV
cutoff of the initial EFT in which the wormhole solutions are derived.

The regularization prescription adopted in Section 6.4 may be interpreted as
identifying approximate wormhole solutions of the equations of motion,
following [73]. One cuts a ball of radius $r_{\Lambda}\equiv 1/\Lambda$ out of
a flat Euclidean background and dual saxions with constant background, and
replaces it with the IR-regularized half-wormhole, smoothly gluing the fields
at the common boundary three-sphere (see the left of Fig. 14). In our case,
the value of the background dual saxions on the boundary three-sphere is given
by ${\bm{\ell}}_{\text{\tiny IR}}={\bm{\ell}}(r_{\Lambda})$. This cut-and-
paste procedure gives a configuration that is an approximate saddle-point
whose action receives non-negligible contributions only from the region $r\leq
r_{\Lambda}$. The result is essentially the regularized action (6.36) with
$\Lambda_{\text{\tiny IR}}\rightarrow\Lambda$, and the dependence of the
associated effective operators (7.2) on the background fields is thus also
expected to be controlled by the same “on-shell” action [116, 73].353535In
principle this regularization procedure may be applied to the everywhere non-
degenerate $n\geq 4$ wormholes as well. However, in those cases the background
fields quickly reach their asymptotic value, cf. (6.32), and one can hence
identify ${\bm{\ell}}_{\Lambda}$ with ${\bm{\ell}}_{\infty}$, as we implicitly
did in our treatment of Section 7.3 (see also the comments at the end of
Section 5.2).

A natural interpretation of the IR regularization of Section 6.4 is to
formally view it as a prescription to “integrate out” the wormhole down to
energy scales $\Lambda\ll 1/L$ [116, 73]. Suppose we are interested in
computing some correlation function at distances $\gg L$. We can take into
account the effect of the wormhole in two equivalent ways. Either we compute
the full path integral for our initial EFT, including the wormhole and its
fluctuations around it, or we use a low-energy effective description defined
below $\Lambda$ where the geometry is essentially flat and the wormhole is
replaced by a set of effective local operators ${\cal O}_{\text{\tiny IR}}$
like those in (7.2). We may call these the “UV” and “IR” descriptions of our
theory. In a semi-classical expansion, the first, “UV” theory gives a factor
$e^{-S|_{\text{hw}}}$, where $S|_{\text{hw}}$ is precisely the wormhole action
we would compute in our initial EFT (2.3). The second gives ${\cal
O}_{\text{\tiny IR}}e^{-S|_{\text{hw}}^{\rm IR}}$, where $S|_{\text{hw}}^{\rm
IR}$ is the on-shell action for the “IR” theory in the presence of the
operator insertion. Because by definition the low-energy degrees of freedom
are precisely the same, the long-distance $r>1/\Lambda$ contributions to the
actions exactly match; they only differ at distances smaller and of order the
matching scale $1/\Lambda$. Therefore, equating the results of the two
descriptions, we learn the effective operator must depend on the background
fields as follows [116]

${\cal O}_{\text{\tiny IR}}\propto e^{-(S|_{\text{hw}}-S|_{\text{hw}}^{\rm
IR})}=e^{-S|_{\text{hw}}^{\Lambda}}.$ (7.17)

The final exponential contains the expression (6.36). The effect of the
marginally degenerate wormhole, once renormalization-group evolved down to
$\Lambda\ll 1/L$, is thus captured by effective local operators suppressed by
the exponential of our IR-regularized action (6.36). The agreement between
(6.36) and (5.35) is not a mere coincidence, as one may have thought by our
analysis in Section 6.4: up to corrections of order $\Lambda^{4}L^{4}$ the
effect of a marginally degenerate wormhole is _exactly equivalent_ to that of
a fundamental BPS instanton of the same charge.

\begin{overpic}[angle={0},width=230.0pt]{Figures/wh3.jpg}
\put(48.1,55.9){${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}L}$}
\end{overpic}

Figure 14: Pictorial representation of a marginally degenerate half-wormhole
surrounded by a radius at the IR cutoff $1/\Lambda\gg L$ (left panel). Sending
$L\to 0$ one formally obtains a fundamental BPS instanton insertion (right
panel).

If what we are arguing is correct, then the results of Section 7.2 would tell
us that marginally degenerate wormholes can induce superpotentials at low
energies, as opposed to their non-degenerate ($n>3$) siblings. At the very
least this conclusion should hold when $\Lambda^{4}L^{4}\to 0$, where the
solution formally coincides with that of a fundamental instanton.

The claim that wormholes can induce F-terms was first made in [29, 30]. At a
first sight, there is an apparent obstruction: for any $L\neq 0$ the $n=3$
wormholes break all supersymmetries and therefore have additional Goldstino-
like zero-modes in the chiral partners $\chi^{i}$ of $t^{i}$, see Appendix
D.3. This would naively suggest that such configurations should generate
D-terms like those of Section 7.3, rather than superpotential terms.
Interestingly, also this conclusion is too rushed. Let us actually ask if our
$n=3$ wormhole describes four-fermion interactions at low energies $\Lambda\ll
1/L$. As mentioned in Section 7.3, a procedure analogous to the one carried
out in (7.12) can be applied to evaluate four-point functions of the form
$\langle\bar{\chi}\bar{\chi}\chi\chi\rangle$. Yet, here that result cannot be
interpreted as the insertion of a local operator involving four free fermions,
since the contribution of the wavefunctions of the extra zero-modes vanishes
away from the wormhole faster than a free Feynman propagator. The reason is
that the additional Goldstino-like zero-modes present in the marginally
degenerate wormhole are localized around the throat, where supersymmetry is
completely broken. Indeed, we have seen in (6.35) that the throat is the only
region where the $n=3$ configuration departs from the extremal solution by
corrections of order ${\cal O}(L^{4}/r^{4})$, and as a consequence that must
also be the only region where the profiles of the zero-modes of the two
backgrounds can differ. This is indeed the case. In conclusion, the fact that
for $n=3$ the two extra zero-modes are localized at the throat prevents us
from obtaining a four-fermion operator at low energies. Still, their presence
seems to obstruct the generation of a superpotential as well. How can we
reconcile this with our earlier observation that marginally degenerate
wormholes are BPS instantons in disguise?

Well, as emphasized around (6.37), $n=3$ wormholes appear in an entire family
of solutions with arbitrary $L$. The extra zero-modes of such solutions would
certainly be relevant for any $L$ within the calculable regime. But the
profiles of the special wormholes obtained in the formal limit $L\to 0$
disappear behind the singularity. If we could ignore curvature singularities,
the extra zero-modes of such wormholes would in some sense become invisible to
our EFT path integral. The subclass of $n=3$ wormholes with $L\to 0$ would
then have effectively two zero-modes like the BPS solution, and hence the
ability to generate a superpotential as in Section 7.2, as proposed in [29,
30]. The limit $L\to 0$ is visually seen passing from the left to the right of
Fig. 14.

The arguments we just presented are essentially a rephrasing, in our more
general setting, of those of [29, 30]. To see this, take our $n=3$ solution
and, instead of regularizing it with an IR cutoff at $r_{\Lambda}$, adopt a
different regularization (see Appendix D.3 for more details). Namely, multiply
the kinetic potential ${\cal F}(\ell)$ by $(1+\varepsilon)$, where
$\varepsilon>0$ is a very small constant, effectively replacing $n=3$ with
$n=3(1+\varepsilon)>3$. This produces an everywhere non-degenerate wormhole
profile, with a precise relation between its IR value $\tilde{\ell}_{\infty}$
and its UV value $\tilde{\ell}_{*}$ (see (6.32)). Using (6.30), and at leading
non-trivial order in $\varepsilon$, this reads

$\tilde{\ell}_{\infty}=\frac{\pi}{4}\varepsilon\tilde{\ell}_{*}=\frac{\varepsilon}{8M^{2}_{\text{\tiny
P}}L^{2}}\,.$ (7.18)

One finally picks $\varepsilon$ so that ${\bm{\ell}}_{\infty}$ in the
$\varepsilon$-regularized theory coincides with the ${\bm{\ell}}_{\Lambda}$ of
our IR-regularized theory. This way we have found a different regularization
of our profile where, if needed, the limit $r\to\infty$ may be taken. The
papers [29, 30] suggest then that the contribution of the $n=3$ wormholes to
the IR effective theory can be computed by taking the limit
$\varepsilon\rightarrow 0$ of the $\varepsilon$-regularized wormholes while
keeping ${\bm{\ell}}_{\infty}$ fixed. From (7.18) we see that this
prescription is actually equivalent to the limit $L\to 0$ we alluded to
earlier. As in the case discussed above, also the $\varepsilon$-regularized
wormholes have extra Goldstino-like zero-modes $\delta\chi^{i}_{\alpha}$
localized around their throat. Taking $\varepsilon\rightarrow 0$, or
equivalently $L\to 0$, the extra Goldstino-like zero-modes are effectively
pushed beyond the EFT and, as such, they are no more integration variables in
the EFT path integral.

In conclusion, the reasoning of [30] applied to our context implies that
marginally degenerate wormholes generate a superpotential of the form (7.14).
Assuming that the $L\rightarrow 0$ limit sensibly connects these wormholes to
the corresponding fundamental instantons, from this we can also infer that the
$n=3$ EFT instantons do not contain any extra fermionic zero-mode and hence do
generate (7.14), as speculated in Section 7.2. The correspondence between
$n=3$ wormholes and EFT instantons has additional important implications. It
says that the wormhole $\alpha$-parameters, whose presence has been for
simplicity suppressed, must be fully determined by an EFT instanton
calculation in the UV complete theory, which has no free parameters at the
onset. This would represent a concrete manifestation of the Baby Universe
Hypothesis [28]. Moreover, a non-trivial wormhole-induced superpotential tells
us that the associated $\alpha$-parameters are generically non-vanishing,
realizing the Supersymmetric Genericity Conjecture of [97]. (Note that, in
some non-generic cases, the UV-complete computation may still give vanishing
$\alpha$-parameters.)

Another interesting consequence of the wormhole argument is that a
superpotential of the form (7.14) should be generated for any charge ${\bf
q}\in{\cal C}_{\text{\tiny WH}}$ corresponding to an $n=3$ sector. This
conclusion may be extended to ${\bf q}\in{\cal C}_{\rm S}^{\text{\tiny
EFT}}-{\cal C}_{\text{\tiny WH}}$ by considering restricted perturbative
regimes, as described in Footnote 24 and for instance exemplified in Section
6.5.2. The picture that emerges is that fundamental EFT instantons carrying
the same charges ${\bf q}$, or at least a multiple thereof (possibly selected
by some stricter ${\cal H}_{3,i}$ quantization rule), should exist as well,
consistently with the BPS instanton tower/sublattice WGCs of Section 5.4.

Finally, as shown in (6.36) in the above $L\rightarrow 0$ limit the on-shell
action reduces to the BPS one, $S|_{\rm hw}\rightarrow
S_{\text{BPS}}=2\pi\langle{\bf q},{\bm{s}}_{\text{\tiny IR}}\rangle$. Since
${\bf q}\in{\cal C}_{\text{\tiny WH}}$, by definition it satisfies
$\langle{\bf q},{\bf e}\rangle\geq 1$ for any ${\bf e}\in{\cal C}_{\rm
S}^{\text{\tiny EFT}}$, see (6.1). We can now run the same argument used to go
from (3.17) to (3.20) and conclude that $\langle{\bf q},{\bm{s}}_{\text{\tiny
IR}}\rangle\geq N/\alpha$ for ant ${\bm{s}}_{\text{\tiny
IR}}\in\hat{\Delta}_{\alpha}$, and therefore:

$e^{-S_{\text{BPS}}}\leq
e^{-\frac{2\pi}{\alpha}N},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}({\bm{s}}_{\text{\tiny
IR}}\in\hat{\Delta}_{\alpha}~{}~{}~{}\&~{}~{}~{}{\bf q}\in{\cal
C}_{\text{\tiny WH}})\,.$ (7.19)

We see that also the contribution of the marginally degenerate wormholes is
strongly suppressed in perturbative regimes associated with $N\gg 1$
(s)axions.

###  7.5 Implications for purely axionic models

The symmetry-breaking mediated by non-perturbative effects can potentially
induce effective axion potentials $V_{\text{eff}}(a)$ at low energies.
Quantifying the impact of these effects is crucial in models for inflation and
for the QCD axion approach to the strong CP problem, where even tiny
corrections can have dramatic phenomenological consequences.

The physics of axionic wormholes has been largely studied in the literature
from an EFT perspective – see for instance the review papers [10, 117] and
references therein. From an agnostic low-energy point of view, the axions
appearing in such bottom-up constructions can either be a subgroup of our
$a^{i}$’s, whose saxionic partners have been stabilized and acquired a mass,
or can be completely independent degrees of freedom, say arising from the
breaking of accidental compact symmetries linearly realized in the EFT. We
will refer to the former as scenarios of “fundamental axions” and the latter
as scenarios of “QFT axions”. In the following we ask what lessons can be
drawn about symmetry-breaking in such scenarios from our results. Our key
assumption is that any such low-energy axion model admits, at sufficiently
high energies, an intermediate ${{\cal N}=1}$ axiverse description as in
Sections 2 and 3 (plus the necessary additional degrees of freedom).

As a preliminary observation we recall that, in a supersymmetric setup like
the one considered in this paper, an ${\cal O}(e^{-S|_{\text{hw}}})$ violation
of axionic symmetries is encoded in the sum, over all wormhole charges, of
Kähler or superpotential corrections of the form (7.15) and (7.14). In the
supersymmetric regime, then, the effective axion potential at momenta $\ll
1/L$ is controlled at least by a factor of order ${\cal
O}(e^{-2S|_{\text{hw}}})$. If, on the other hand, supersymmetry is softly
broken at a scale $M_{\text{\tiny{SSB}}}\ll M_{\text{sp}}$, then the effective
axion potential may also receive corrections of order
$(M_{\text{\tiny{SSB}}}/M_{\text{sp}})^{p}e^{-S|_{\text{hw}}}$ for some $p>0$.
Hence, the effective axion potential is always more suppressed than just
${\cal O}(e^{-S|_{\text{hw}}})$. Let us next estimate what $S|_{\text{hw}}$
can be in scenarios with fundamental or QFT axions.

Consider first the case of fundamental axions. That is, assume the EFT
contains a number $\leq N$ of approximately massless fundamental axions,
precisely like the ones discussed in our paper, while the saxions are instead
stabilized by a SUSY-breaking potential $V_{\rm s}$. The presence of $V_{\rm
s}$ introduces a correction to (5.9) in the form of a new effective potential
for the (dual) saxions

$V_{\bf q}=-\frac{1}{2}\|{\bf q}\|^{2}-\pi^{2}r^{6}M^{2}_{\text{\tiny
P}}V_{\rm s}.$ (7.20)

The saxion potential $V_{\rm s}$ generates an IR threshold mass
$M_{\text{\tiny IR}}$, as anticipated around (3.30). Approximate, purely
axionic (non-extremal) wormholes may be constructed at distances $r\gg
1/M_{\text{\tiny IR}}$, where the saxions are fixed at their asymptotic values
and the only relevant dynamical degrees of freedom are the axions – see [118,
119] for discussions on multi-axion generalizations of the axionic wormhole of
[9]. Their action $S|_{\text{hw}}=3\pi^{3}M_{\text{\tiny P}}^{2}L^{2}$ is
characterized by curvature lengths $L\geq 1/M_{\text{\tiny IR}}$, and
according to (7.7) the associated wormholes would be very inefficient sources
of symmetry breaking in the controllable regime $M_{\text{\tiny IR}}\ll
M_{\text{sp}}$ because $S|_{\text{hw}}\gg N_{\text{sp}}$. In the more
interesting regime $L\leq 1/M_{\text{\tiny IR}}$, wormholes inevitably excite
the saxionic degrees of freedom via (7.20) – see [86, 87] for explicit
examples in simple dilatonic models – and the results of our paper directly
apply. Hence, in the more interesting cases the effect is estimated as
discussed in Section 7.1.

Axions may also originate as approximate Nambu-Goldstone bosons of some
linearly realized accidental compact symmetry. In that case one can build
wormholes with those “QFT axions”, the associated saxionic radial modes, and
possibly other degrees of freedom as well, whereas the fundamental saxions and
axions we have been considering here remain mere silent spectators. One can
now envision QFT models in which the radial modes shrink as one approaches the
wormhole throat, resulting in small on-shell actions (see e.g. [85]). But
small actions are a clear indication of sizable quantum effects, and that may
signal we are outside the perturbative domain. Fortunately, irrespective of
that we can confidently claim that wormhole effects cannot get larger than
indicated in (7.11). Our bound on the GB coefficient is indeed largely
insensitive to the details of the wormhole configuration and impervious to
those violations of perturbativity. Barring unnatural cancellations due to
other higher derivative terms, then, the suppression (7.11) should apply to
QFT wormholes as well, of course provided the UV completion is ultimately well
described by an ${\cal N}=1$ axiverse.

##  8 Conclusions

In this paper we explored various aspects of ${\cal N}=1$ axiverse models
containing an arbitrary number $N$ of fundamental axions, adopting the general
framework developed in [13, 14, 15] and taking into account various quantum
gravity constraints.

We proposed the upper bound (3.28) on the ultimate (moduli-dependent) UV
cutoff of any EFT description of gravity, the so-called species scale. Our
upper limit is set by the dominant EFT string scale (3.29), which is a
perturbatively exact quantity depending on four-dimensional data already
available at the two-derivative level. Our proposal (3.28) has been checked in
a number of string theory models and compared to another upper bound on the
species scale suggested in [53, 57], finding very good agreement.

The approximate axionic shift symmetries that characterize our EFTs are
explicitly broken by two classes of non-perturbative effects. The first class
consists in short distance effects due to fundamental instantons. The second
class is associated to non-perturbative effects within the EFT, part of which
are of gravitational nature and potentially due to axionic wormholes. A
significant portion of our paper was dedicated to the study of axionic
wormholes in ${\cal N}=1$ axiverse models, and of their relation with
fundamental instantons.

Our axiverse models support a large class of extremal and non-extremal
wormhole solutions, which carry specific sets of axionic charges. The extremal
ones are BPS and singular, but their singularity has a clear interpretation as
due to the insertion of fundamental BPS instantons. Among the fundamental BPS
instantons, a special role is played by the EFT instantons [14]. We argued,
both within the macroscopic EFT description and by discussing microscopic
string theory realizations, that EFT instantons are intrinsically
gravitational in nature. The analogy with five-dimensional back holes suggests
that EFT instantons satisfy an axion version of the BPS tower or sublattice
WGC proposed in [92].

Interestingly, a slightly restricted subset of EFT instanton charges always
support homogeneous non-extremal wormhole solutions as well, which describe
$N$ saxions moving along a specific direction in the saxionic space. For the
same set of charges, more general solutions can also be constructed, both
perturbatively, as small deformations of the homogeneous configurations, and
numerically. Crucial to the existence of our homogeneous wormhole solutions is
the homogeneity degree $n$ of the function $P(s)$ characterizing the saxion
Kähler potential (2.5). With $n>3$ the solutions are regular everywhere and in
principle capable of inducing at low energy exponentially suppressed symmetry-
breaking corrections to the Kähler potential. These wormholes may be
interpreted as non-BPS bound states of EFT instantons and may capture, in the
low-energy theory, some of their physical effects. Wormholes with $n<3$,
instead, degenerate at a small distance from their throat and no relevant
semi-classical effects can be associated to them.

The non-extremal configurations with $n=3$ are degenerate only asymptotically
and deserved a separate discussion. We revisited and extended a proposal of
[30], which implies that these marginally degenerate wormholes should generate
effective symmetry-breaking superpotentials. Moreover, we argued that these
configurations can be more directly interpreted as a low-energy manifestation
of EFT instantons, thus identifying a clearer link between two a priori
independent classes of non-perturbative effects. The combination of these
considerations has far-reaching consequences. On the one hand, it implies that
EFT instantons corresponding to $n=3$ sectors are generically expected to
generate superpotential terms, compatibly with the Supersymmetric Genericity
Conjecture [97]. On the other hand, it indicates that Coleman’s
$\alpha$-parameters of the $n=3$ wormholes should be somehow fixed by the UV
complete description of the corresponding EFT instantons, providing a concrete
realization of the Baby Universe Hypothesis [28]. Actually, according to the
Baby Universe Hypothesis the same “natural selection” of the
$\alpha$-parameters should also take place for the $n>3$ regular wormholes and
the corresponding D-terms. We leave a more in depth exploration of these
interesting speculations for the future.

We showed that the effect of non-extremal wormholes associated to ${\cal N}=1$
axiverses is always very suppressed. For non-degenerate wormholes, the basic
reason is that the species scale is expected to decrease parametrically with
increasing $N$, hence there is a maximal curvature mass scale below which the
EFT predictions are reliable, and this suggests that the wormhole on-shell
action should be at least of order $N$. One way to quantify this suppression
is by taking advantage of a lower bound we established on the coefficient of
the Gauss-Bonnet operator: in the EFT domain of validity, its value is always
positive and growing with at least a power of $N$. For $n=3$ marginally
degenerate wormholes one can equally show that the on-shell action is enhanced
by $N$. As a result, we can robustly conclude that in the axiverse the effect
of non-extremal wormholes is suppressed by powers of $e^{-N}$. Despite their
tiny impact on the axiverse dynamics, non-extremal wormholes remain very
interesting and concrete laboratories for the study of quantum gravity and of
its low-energy consequences.

Acknowledgments

We thank M. Bianchi, V. Cagioni, D. Cassani, L. McAllister, F. Marchesano, J.
Moritz, G. Shiu, J. Stout, A. Strominger, C. Vafa, I. Valenzuela, D. van de
Heisteeg, T. Van Riet, T. Weigand, and M. Wiesner for useful discussions. LM
thanks the Jefferson Physical Laboratory of Harvard University, and the
members of the Harvard Swampland Initiative for hospitality during the final
stage of this work. This work was supported in part by the Italian MUR
Departments of Excellence grant 2023-2027 “Quantum Frontiers”, by the MUR-PRIN
contract 2022YZ5BA2 - Effective Quantum Gravity, and by the MIUR-PRIN contract
2017CC72MK003. AV acknowledges the Swiss National Science Foundation (SNF) for
funding through the Eccellenza Professorial Fellowship “Flavor Physics at the
High Energy Frontier” project number 186866. The work of LV was partly
supported by the Italian MIUR under contract 202289JEW4 (Flavors: dark and
intense), the Iniziativa Specifica “Physics at the Energy, Intensity, and
Astroparticle Frontiers” (APINE) of Istituto Nazionale di Fisica Nucleare
(INFN), and the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No 860881-HIDDeN.

Appendix

##  Appendix A Naive Dimensional Analysis

In this appendix we review the concept of Naive Dimensional Analysis (NDA) to
explain the structure of the derivative expansion in our EFT, justify the
definitions (4.5) of the strong coupling scales in string and M-theory, and
derive (3.24) and (3.25). In the process we will clarify the origin of the
“geometric” $2\pi$-factors that appear throughout the paper; all other factors
of order unity cannot be detected by NDA arguments and will be ignored. NDA
was first used in the context of QCD by S. Weinberg in [120] and later
formalized by [121] (see also [122] and references therein and, e.g. [123] for
a $d$-dimensional analysis). Its application to string theory models is not as
popular.

For clarity we anticipate here the proxy for the strong coupling scale of a
$d$-dimensional gravitational theory that we found to apply to all scenarios
of interest to us:

$\widehat{M}_{(d)}^{d-2}\equiv(2\pi)^{\lfloor\frac{d}{2}\rfloor}{M}_{(d)}^{d-2}.$
(A.1)

Here $M_{(d)}$ denotes the $d$-dimensional Planck scale which, to avoid
unnecessary $1/2$ factors, in this appendix is defined via 363636$M_{(4)}$ is
thus related to the $M_{\text{\tiny P}}$ used in the rest of the paper via
$M^{2}_{\text{\tiny P}}=2M^{2}_{(4)}$.

${\cal L}_{(d)}\supset\sqrt{-g_{(d)}}M_{(d)}^{d-2}R_{(d)}.$ (A.2)

The quantity $\widehat{M}_{(d)}$ identifies the highest possible scale at
which a $d$-dimensional gravitational EFT can be extrapolated. If a
parametrically large number $N_{(d)}$ of $d$-dimensional degrees of freedom is
present, (A.1) must be further reduced by a factor of $1/N_{(d)}$, so (A.1) is
to be interpreted as an upper bound. Eq. (4.5) shows the ten- and eleven-
dimensional incarnations of (A.1), respectively.

### The strong coupling scale

Imagine we match a $d$-dimensional EFT to its UV completion at the lowest
energy threshold ${\mathtt{M}}_{(d)}$ of the UV theory. Using the terminology
of Section (3.4), ${\mathtt{M}}_{(d)}$ denotes the $d$-dimensional version of
the species scale. A simple $\hbar$ power counting reveals that the matching
procedure must result in the following effective Lagrangian (we focus on
gravity, but our arguments are completely general)

${\cal
L}_{(d)}=\frac{{\mathtt{M}}_{(d)}^{d}}{{\mathtt{g}}_{(d)}^{2}}\sqrt{-g_{(d)}}\left\\{\frac{R_{(d)}}{{\mathtt{M}}_{(d)}^{2}}+c_{2}\frac{R_{(d)}^{2}}{{\mathtt{M}}_{(d)}^{4}}+c_{3}\frac{R_{(d)}^{3}}{{\mathtt{M}}_{(d)}^{6}}+\cdots\right\\}$
(A.3)

where $c_{2,3,\cdots}$ are pure numbers at most of order unity that depend on
ratios of mass scales and/or ratios of couplings. The strength of the
Einstein-Hilbert term at momenta $p\sim{\mathtt{M}}_{(d)}$ is measured by the
EFT coupling ${\mathtt{g}}_{(d)}$, which has units of
$[{\mathtt{g}}_{(d)}]=[1/\sqrt{\hbar}]$. Denoting by $N_{(d)}$ the number of
$d$-dimensional degrees of freedom below ${\mathtt{M}}_{(d)}$, a typical loop
integral with momenta $p\sim{\mathtt{M}}_{(d)}$ is estimated to be of order

${\mathtt{g}}_{(d)}^{2}N_{(d)}\int\frac{{\rm
d}\Omega_{d}}{(2\pi)^{d}}=\frac{2\,{\mathtt{g}}_{(d)}^{2}N_{(d)}}{(4\pi)^{\frac{d}{2}}\Gamma\left(d/2\right)}\sim\frac{{\mathtt{g}}_{(d)}^{2}}{(2\pi)^{\lfloor\frac{d+1}{2}\rfloor}}$
(A.4)

with $\lfloor\cdots\rfloor$ the floor function. In the last step of (A.4) we
kept track of the powers of $2\pi$ but neglected a factor $N_{(d)}/(d-2)!!$
for $d={\text{even}}$ or $2N_{(d)}/(d-2)!!$ for $d={\text{odd}}$. It turns out
that in the higher-dimensional scenarios we consider such factors are of order
one. They will hence be neglected along with all other factors of order unity
because NDA cannot keep track of them. The symbol $\sim$ will be used when our
NDA prescription of ignoring numbers of order unity is employed.

Conventionally, a coupling is called maximally strong when loop effects are of
order unity [122, 123]. According to (A.4), in our scenarios this occurs when
${\mathtt{g}}_{(d)}^{2}\sim{\mathtt{g}}_{(d)}^{2}|_{\text{max}}\equiv(2\pi)^{\lfloor\frac{d+1}{2}\rfloor}$.
Now, from (A.3) it follows that the $d$-dimensional Planck scale can be
expressed in terms of the matching scale and the EFT coupling. Recalling our
convention (A.2) we have
$M_{(d)}^{d-2}={{\mathtt{M}}_{(d)}^{d-2}}/{{\mathtt{g}}_{(d)}^{2}}$. This says
that, for fixed $M_{(d)}$, the higher the matching scale the stronger the
coupling. The maximally allowed value ${\mathtt{M}}_{(d)}^{2}|_{\text{max}}$
is obtained with the maximally strong coupling
${\mathtt{g}}_{(d)}^{2}|_{\text{max}}$ identified above:

$\left.{\mathtt{M}}_{(d)}^{d-2}\right|_{\text{max}}\equiv(2\pi)^{\lfloor\frac{d+1}{2}\rfloor}M_{(d)}^{d-2}.$
(A.5)

This expression coincides with (A.1) for any even dimensionality, but is
slightly larger when $d$ is odd. We will see that (A.5) does indeed represent
a good proxy for strong coupling in ten-dimensional string theories, but can
be slightly refined for eleven-dimensional M-theory. A perturbative
$d$-dimensional theory must always satisfy
${\mathtt{g}}_{(d)}^{2}\ll{\mathtt{g}}_{(d)}^{2}|_{\text{max}}$ so as to
ensure that the description remains perturbative all the way up to the
matching scale. For the particular case $d=4$ this logic explains the $2\pi$’s
in Section 3. In general the ratio
${\mathtt{M}}_{(d)}^{d-2}/{\mathtt{M}}_{(d)}^{d-2}|_{\text{max}}$ parametrizes
the expansion parameter on which the coefficients $c_{2,4,\cdots}$ depend.

To check the consistency of these considerations let us first consider type
IIA string theory at not-too-large dilaton. At weak string coupling the
matching scale ${\mathtt{M}}_{(10)}$ in (A.3) corresponds to the string mass
scale in the Einstein frame:

$\displaystyle{\mathtt{M}}_{(10)}^{2}=2\pi T_{\rm
F1}=\frac{(2\pi)^{2}}{l_{(10)}^{2}}e^{\phi/2}.$ (A.6)

As the string coupling $g_{\rm s}=e^{\phi}$ increases, ${\mathtt{M}}_{(10)}$
grows and eventually is expected to reach the value
${\mathtt{M}}_{(10)}|_{\text{max}}$ at which both the UV theory and the EFT
are strongly coupled. Recalling (4.1) we have
$M_{(10)}^{8}=2\pi/l_{(10)}^{8}$, which combined with (A.5) gives

$\displaystyle{\mathtt{M}}_{(10)}^{2}|_{\text{max}}=\,$
$\displaystyle(2\pi)^{\frac{5}{4}}M_{(10)}^{2}=\frac{(2\pi)^{\frac{3}{2}}}{l_{(10)}^{2}}.$
(A.7)

To verify that ${\mathtt{M}}_{(10)}|_{\text{max}}$ gives also a good measure
of the scale at which the string coupling gets strong we inspect the
coefficients of the $R_{(10)}^{4}$ operators in the EFT. The latter can be
arranged as shown in (A.3) with coefficients (see for instance [124], and
references therein)

$\displaystyle c_{4}|_{\text{Type IIA}}=\,$
$\displaystyle\zeta(3)\,T_{1}+\frac{\pi^{2}e^{2\phi}}{3}T_{2}=\zeta(3)T_{1}+\frac{1}{12}\frac{{\mathtt{M}}_{(10)}^{8}}{{\mathtt{M}}_{(10)}^{8}|_{\text{max}}}T_{2},$
(A.8)

where $T_{1}$ and $T_{2}$ are appropriate tensor structures contracting the
indices of the Riemann tensors. Consistently with our expectation, the loop
corrections become of the same order as the tree-level terms when $(2\pi
e^{\phi})^{2}={\mathtt{M}}_{(10)}^{8}/{\mathtt{M}}_{(10)}^{8}|_{\text{max}}$
gets of order one. We can therefore conclude that
${\mathtt{M}}_{(10)}|_{\text{max}}$ is a physically reasonable estimate of the
strong coupling scale in Type IIA string theory, at least for $2\pi
e^{\phi}={\mathtt{M}}_{(10)}^{4}/{\mathtt{M}}_{(10)}^{4}|_{\text{max}}<1$, and
that $2\pi e^{\phi}$ is an appropriate expansion parameter. For ten-
dimensional string theory the strong coupling scale introduced in Section 3.4
can thus be taken to be
$\widehat{M}_{(10)}={\mathtt{M}}_{(10)}|_{\text{max}}$, as anticipated in
(4.5).

In the eleven-dimensional description of M-theory the approximation in (A.4)
is again justified. The matching scale (or, equivalently, spieces scale)
${\mathtt{M}}_{(11)}$ should correspond to a strong coupling scale, but the
story here is a little more subtle than before because of the presence of
branes. We will argue that it is more appropriate to identify the strong
coupling scale of M-theory with (A.1) rather than (A.5). While in Type IIA the
strongest coupling at the matching scale is set by gravity, from which (A.5)
follows, in M-theory it is not a priori evident which coupling is the
strongest. Its action contains the standard Einstein-Hilbert term as well as
M2-branes and M5-branes. The coefficients of these terms are completely fixed
by a number of independent arguments. The purely gravitational contributions
are given by

$\frac{2\pi}{l_{(11)}^{9}}\int
R_{(11)}+\frac{2\pi}{l_{(11)}^{3}}\int_{\text{M2}}{\text{vol}}_{\text{M2}}+\frac{2\pi}{l_{(11)}^{6}}\int_{\text{M5}}{\text{vol}}_{\text{M5}}\,.$
(A.9)

Let us estimate the maximal strong coupling scale for each term in turn. The
one for the Einstein-Hilbert interaction can be obtained via the logic leading
to (A.5). Combining with our definition (4.2) one gets
${\mathtt{M}}_{(11)}^{9}|_{\text{max,EH}}=(2\pi)^{7}/l_{(11)}^{9}$. The world-
volume theory supported by a $p$-brane of tension $T_{p}$ is effectively
described by a $p+1$-dimensional action with coupling
${\mathtt{g}}_{(p+1)}^{2}={\mathtt{M}}_{(11)}^{p+1}|_{\text{Mp}}/T_{p}$. The
corresponding maximal coupling scale can be determined in a way completely
analogous to what done for the bulk gravitational theory in (A.4), though now
the dimensionality is not that of spacetime. Ignoring again factors of order
one coming from the number of degrees of freedom living on the branes, for the
M5 brane that logic gives
${\mathtt{M}}_{(11)}^{6}|_{\text{max,M5}}=(2\pi)^{4}/l_{(11)}^{6}$ and for the
M2 brane we have
${\mathtt{M}}_{(11)}^{3}|_{\text{max,M2}}=(2\pi)^{3}/l_{(11)}^{3}$. The actual
regime of strong coupling of M-theory is presumably controlled by the smallest
of these scales. According to our NDA estimate, that is the one associated to
the M5 brane. It is hence natural to take

${\mathtt{M}}_{(11)}=\widehat{M}_{(11)}=\frac{(2\pi)^{2/3}}{l_{(11)}}.$ (A.10)

This is smaller than ${\mathtt{M}}_{(11)}|_{\text{max}}$ given in (A.5) by
just a factor $(2\pi)^{1/9}$. While such difference is very small, we are
tempted to take our estimate seriously because NDA can only identify the
geometric powers of $2\pi$, and according to a $2\pi$ counting (A.10) should
be our candidate scale.

Our choice (A.10) can also be justified by an alternative criterion, which is
essentially semiclassical and may be formulated as follows: a $d$-dimensional
theory is said to be maximally strong at $p\sim{\mathtt{M}}_{(d)}$ if its
action is one (in $\hbar=1$ units) when evaluated on spherically symmetric
configurations controlled by distance and curvature scales $\sim
1/{\mathtt{M}}_{(d)}$. With this new prescription the matching scale of a
strongly-coupled theory would always be set by (A.1) rather than (A.5), so the
new criterion would agree with the one discussed earlier only for even
dimensionalities. Interestingly, when we apply the new criterion to M-theory
one finds that all three terms in (A.9) become strong at the very same scale
(A.10). This coincidence seems to support our identification.

An empirical non-trivial check that the powers of $2\pi$ suggested by (A.10)
have some truth in them is obtained noting that according to (A.3) we should
expect the coefficients of the $R_{(11)}^{4}$ terms in M-theory to be
proportional to

$\frac{M_{(11)}^{9}}{{\widehat{M}}_{(11)}^{6}}=\frac{M_{(11)}^{3}}{(2\pi)^{10/3}}$
(A.11)

up to a pure number. Amusingly, the powers of $2\pi$ obtained from that
relation match exactly those obtained in explicit calculations, as for
instance collected in [124] and [57].

### The four-dimensional theory

Finally, we would like to see how the Lagrangian (A.3) appears to a four-
dimensional observer at energy scales $<M_{\text{\tiny KK}}$ below the Kaluza-
Klein mass scale. We look for gravitational solutions with line element as in
(4.4) and proceed as follows. We first derive the dimensionally reduced four-
dimensional version of (A.3) with all the KK resonances integrated in.
Subsequently we integrate out the KK resonances to obtain an EFT at the scale
$M_{\text{\tiny KK}}$.

The dimensionally-reduced four-dimensional EFT for the graviton zero-mode and
KK resonances is, taking care of the appropriate factors of $e^{2A}$, a
Lagrangian completely analogous to (A.3):

$\left.{\cal
L}\right|_{\Lambda<M_{\text{sp}}}=\frac{M_{\text{sp}}^{4}}{{\mathtt{g}}^{2}}\sqrt{-g}\left\\{\frac{R}{M_{\text{sp}}^{2}}+c_{2}\frac{R^{2}}{M_{\text{sp}}^{4}}+c_{3}\frac{R^{3}}{M_{\text{sp}}^{6}}+\cdots+{\text{KK-
resonances}}\right\\}$ (A.12)

where

$M_{\text{sp}}^{2}=e^{2A}{\mathtt{M}}_{(d)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\mathtt{g}}^{2}=\frac{{\mathtt{g}}_{(d)}^{2}}{{\mathtt{V}}_{(d-4)}},$
(A.13)

and ${\mathtt{V}}_{(d-4)}={\mathtt{M}}_{(d)}^{d-4}\int{\rm
d}^{d-4}y~{}\sqrt{g_{(d-4)}}$. The matching scale, converted to the four-
dimensional Einstein frame, is what we called species scale in Section 3.4.
The quantity ${\mathtt{V}}_{(d-4)}$ is the volume normalized in units of the
fundamental scale ${\mathtt{M}}^{-1}_{(d)}$ in $d$-dimensions. It is related
to the number of four-dimensional KK degrees of freedom below
${\mathtt{M}}_{(d)}$ by Weyl’s asymptotic formula (see for instance [125])

$N_{\text{\tiny
KK}}\simeq\frac{N_{(d)}\Omega_{(d-4)}{\mathtt{V}}_{d-4}}{(d-4)(2\pi)^{d-4}}=\frac{(d-2)\,N_{(d)}{\mathtt{V}}_{d-4}}{2(4\pi)^{\frac{d}{2}-2}\Gamma\left(d/2\right)}\,.$
(A.14)

If we identify $N_{\text{\tiny KK}}$ with the number of four-dimensional
degrees of freedom, then the ’t Hooft expansion parameter in the reduced four-
dimensional theory is

$\frac{{\mathtt{g}}^{2}N_{\text{\tiny
KK}}}{(2\pi)^{2}}\simeq(d-2)\,\frac{2\,{\mathtt{g}}_{(d)}^{2}\,N_{(d)}}{(4\pi)^{\frac{d}{2}}\Gamma\left(d/2\right)}\,.$
(A.15)

Up to to the $(d-2)$ factor, this is precisely the $d$-dimensional loop
estimate appearing in (A.4). This simply reminds us that a weakly-coupled
$d$-dimensional description necessarily leads to a weakly-coupled four-
dimensional reduction. Now, following the convention in (A.2), (A.12) leads to
the identification $M_{\text{sp}}={\mathtt{g}}M_{(4)}$ which, combined with
(A.15) and adopting the same approximation as in (A.4), reads

$M_{\text{sp}}^{2}\sim\frac{{\mathtt{g}}_{(d)}^{2}}{(2\pi)^{\lfloor\frac{d+1}{2}\rfloor}}\frac{(2\pi
M_{(4)})^{2}}{N_{\text{\tiny KK}}}\,.$ (A.16)

This relation says that the species scale seen by the four-dimensional
observer satisfies an upper bound $M_{\text{sp}}\lesssim 2\pi
M_{(4)}/\sqrt{N_{\text{\tiny KK}}}\sim 2\pi M_{\text{\tiny
P}}/\sqrt{N_{\text{\tiny KK}}}$, which we recognize to be (3.24) whenever the
large number of species can be identified with the number of KK modes. This is
just the statement that the species scale and the maximally strong coupling
scale (A.5) coincide only when the expansion parameter appearing on the r.h.s.
of (A.4) is one.

The next step consists in integrating out the KK modes to obtain a four-
dimensional EFT at scales $\lesssim M_{\text{\tiny KK}}$. In carrying out this
final step we note that the tree-level exchange of a KK resonance $\Phi$ can
only be relevant when linear couplings of the form $\Phi J$ are present in
(A.12). According to the discussion in Section 3.4, neither Einstein-Hilbert
nor Gauss-Bonnet are corrected by tree-level KK exchange in string theory
compactifications. Moreover, loops cannot introduce additional powers of
$M_{\text{sp}}^{2}/M_{\text{\tiny KK}}^{2}$ because controlled by irrelevant
couplings. Hence we deduce that neither the Einstein-Hilbert term nor the
Gauss-Bonnet term are qualitatively affected by KK exchange. 373737On the
other hand, Ricci squared interactions (as well as higher powers of the
curvature) may receive corrections already at tree-level. Ignoring numbers of
order unity we thus conclude that below the KK scale our four-dimensional EFT
formally reads

$\left.{\cal L}\right|_{\Lambda<M_{\text{\tiny
KK}}}=M_{(4)}^{2}M_{\text{sp}}^{2}\sqrt{-g}\left\\{\frac{R}{M_{\text{sp}}^{2}}+c^{\prime}_{2}\frac{R^{2}}{M_{\text{sp}}^{4}}+c^{\prime}_{3}\frac{R^{3}}{M_{\text{sp}}^{6}}+\cdots\right\\}.$
(A.17)

Let us consider the Riemann squared operator, which is particularly relevant
to us because related to the GB term. Exploiting its symmetry properties we
observe that $R_{abcd}R^{abcd}=4\sum_{a<b,c<d}R_{abcd}R^{abcd}$. Therefore it
is reasonable to expect a combinatorial factor of $1/4$ in front of the Wilson
coefficient. We thus take the $c_{2}$ associated to the Riemann squared
operator in (A.3) to be $c_{2}=c_{\text{GB}}/4$ with $c_{\text{GB}}$ of order
one. In addition, we just argued that $c^{\prime}_{\text{GB}}\sim
c_{\text{GB}}$ (whereas, typically, $c^{\prime}_{R^{2}}\sim
c_{R^{2}}M_{\text{sp}}^{2}/M_{\text{\tiny KK}}^{2}$). Using now our
conventions in (2.6) and (2.8), the coefficient of the four-dimensional Gauss-
Bonnet term in (A.17) reads as shown in (3.25).

##  Appendix B Matching in heterotic/F-theory dual models

In this section we more explicitly show how the Kähler/kinetic potentials
(4.46) and (4.18) of heterotic/F-theory dual models match in an appropriate
limit. F-theory compactifications on a $\mathbb{P}^{1}$ fibered space $X$ is
dual to a heterotic compactification on an elliptically fibered Calabi-Yau
three-fold $\hat{X}$ over the same base two-fold $B$. We assume $B$ to be
weak-Fano, and then to support smooth elliptic fibrations [126]. By
definition, such a base has a nef and big anti-canonical divisor
$\overline{K}_{B}$. Let us denote it as

$c_{B}\equiv\overline{K}_{B}\in{\rm Nef}^{1}(B)\,.$ (B.1)

The bigness condition is equivalent to the self-intersection positivity
$c_{B}\cdot c_{B}>0$. Let us also introduce two dual bases of 2-cycles
$c_{\alpha},\tilde{c}^{\alpha}\in H_{2}(B,\mathbb{Z})$, such that
$c_{a}\cdot\tilde{c}^{\beta}=\delta_{\alpha}^{\beta}$.

###  B.1 F-theory side

The space $X$ can be described as a projectivized bundle [127]

$X_{\text{F}}=\mathbb{P}({\cal O}_{B}\oplus{\cal L})\,,$ (B.2)

where the line bundle ${\cal L}$ on the base $B$ determines the twist of the
$\mathbb{P}^{1}$-fibration. We denote by $\pi:X\rightarrow P$ the
corresponding projection map. Introducing fibral projective coordinate
$[x:y]$, we can identify the global section $\sigma:B\rightarrow X$, defined
by $x=0$, and a corresponding effective divisor $S=\sigma_{*}B\subset X$. The
cone of effective divisors ${\rm Eff}^{1}(X)\simeq{\cal C}_{\rm I}$ is
generated by $S$ and the divisors of the form $E=\pi^{*}(e)$, where $e\in{\rm
Eff}_{1}(B)$, i.e. $e$ is an effective curve in $B$. We will also assume that
the line bundle ${\cal L}$ is positive, that is

$c_{\cal L}\equiv c_{1}({\cal L})\equiv p_{\alpha}\tilde{c}^{\alpha}\in{\rm
Eff}_{1}(B)\,,$ (B.3)

where again Poincaré duality is implicit. The cone ${\rm Nef}^{1}(X)$ of nef
divisors is generated by the divisor

$H=S+\pi^{*}c_{\cal L}$ (B.4)

and the divisors of the form $\pi^{*}c$, where $c\in{\rm Nef}^{1}(B)\simeq{\rm
Mov}_{1}(B)$.

We can expand the dual saxion vector ${\bm{\ell}}$ in the basis of four-cycles
provided by $H$ and the four-cycles $\pi^{*}\tilde{c}^{\alpha}$:

${\bm{\ell}}=\ell_{0}H+\ell_{\alpha}\pi^{*}\tilde{c}^{\alpha}\,.$ (B.5)

Then, according to our general discussion of Section 4.1, the dual saxionic
domain is defined by $\ell_{0}\geq 0$ and
$\ell_{\alpha}\tilde{c}^{\alpha}\in{\rm Nef}^{1}(B)_{\mathbb{R}}$. A simple
example is provided by the model of Section 4.1.2, which corresponds to
$B=\mathbb{P}^{2}$, $(D^{1},H)=(D^{1},D^{2})$ and the redefinition
$(\ell_{1},\ell_{0})\rightarrow(\ell_{1},\ell_{2})$.

The basis $(H,\pi^{*}\tilde{c}^{\alpha})\in H_{4}(X,\mathbb{Z})$ is dual to
the basis $(C_{\rm F},C_{\alpha})$, where $C_{\rm F}$ is the
$\mathbb{P}^{1}$-fiber $C_{\rm F}$ and $C_{\alpha}\equiv\sigma_{*}c_{\alpha}$.
The saxions $s^{i}=(s^{0},s^{\alpha})$ can be obtained by expanding
${\bm{s}}\in{\rm Mov}_{1}(X)$ in this basis:

${\bm{s}}=s^{0}C_{\rm
F}+s^{\alpha}C_{\alpha}=(s^{0}-p_{\alpha}s^{\alpha})C_{\rm
F}+s^{\alpha}H\cdot\pi^{*}c_{\alpha}\,,$ (B.6)

where we have used
$C_{\alpha}=S\cdot\pi^{*}c_{\alpha}=H\cdot\pi^{*}c_{\alpha}-p_{\alpha}C_{\rm
F}$. The cone of movable curves is generated by $C_{\rm F}$ and curves of the
form $H\cdot\pi^{*}(c)$, with $c\in{\rm Nef}^{1}(B)$. Hence, the saxionic cone
is defined by the conditions

$s^{0}\geq p_{\alpha}s^{\alpha}\quad,\quad s^{\alpha}c_{\alpha}\in{\rm
Nef}^{1}(B)\,.$ (B.7)

The kinetic potential (4.18) takes the form

${\cal F}=\log\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})=\log\left[{\cal
I}^{\alpha\beta}p_{\alpha}p_{\beta}\ell_{0}^{3}+3{\cal
I}^{\alpha\beta}p_{\alpha}\ell_{\beta}\ell_{0}^{2}+3{\cal
I}^{\alpha\beta}\ell_{\alpha}\ell_{\beta}\ell_{0}\right]\,,$ (B.8)

where ${\cal I}^{\alpha\beta}\equiv\tilde{c}^{\alpha}\cdot\tilde{c}^{\beta}$.
Applying (2.15), we get

$s^{0}=\frac{3({\cal I}^{\alpha\beta}p_{\alpha}p_{\beta}\,\ell_{0}^{2}+2{\cal
I}^{\alpha\beta}p_{\alpha}\ell_{\beta}\ell_{0}+{\cal
I}^{\alpha\beta}\ell_{\alpha}\ell_{\beta})}{2\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}\quad,\quad
s^{\alpha}=\frac{3({\cal I}^{\alpha\beta}p_{\beta}\ell_{0}^{2}+2{\cal
I}^{\alpha\beta}\ell_{\beta}\ell_{0})}{2\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}\,.$
(B.9)

We now take the limit $\ell_{0}\ll|\ell_{\alpha}|$ limit, in which

${\cal F}\simeq\log\ell_{0}+\log\left[{\cal
I}^{\alpha\beta}(\ell_{\alpha}+\frac{1}{2}p_{\alpha}\ell_{0})(\ell_{\beta}+\frac{1}{2}p_{\beta}\ell_{0})\right]\,,$
(B.10)

up to a term of order ${\cal O}(\ell^{2}_{0}/|\ell_{\alpha}|^{2})$, and an
irrelevant additive constant. In this limit we have

$s^{0}-\frac{1}{2}p_{\alpha}s^{\alpha}\simeq\frac{1}{2\ell_{0}}\gg
1\quad,\quad s^{\alpha}\simeq\frac{{\cal
I}^{\alpha\beta}(\ell_{\beta}+\frac{1}{2}p_{\beta}\ell_{0})}{{\cal
I}^{\gamma\delta}(\ell_{\gamma}+\frac{1}{2}p_{\gamma}\ell_{0})(\ell_{\delta}+\frac{1}{2}p_{\delta}\ell_{0})}\,.$
(B.11)

and the Kähler potential is then given by

$K\simeq-\log\left(s^{0}-\frac{1}{2}p_{\alpha}s^{\alpha}\right)-\log\left({\cal
I}_{\alpha\beta}s^{\alpha}s^{\beta}\right)\,,$ (B.12)

where ${\cal I}_{\alpha\beta}=c_{\alpha}\cdot c_{\beta}$ is the inverse of
${\cal I}^{\alpha\beta}$.

###  B.2 Heterotic side

The elliptically fibered Calabi-Yau $\hat{\pi}:\hat{X}\rightarrow B$ admits
the global section $\hat{\sigma}:B\rightarrow\hat{X}$. One can then identify
the nef divisor $\hat{H}=\sigma_{*}B+\pi^{*}c_{B}$. The Kähler saxionic vector
${\bm{s}}$ appearing in (4.45) can then be expanded as follows

${\bm{s}}=\hat{s}\hat{H}+s^{\alpha}\pi^{*}c_{\alpha}\,,$ (B.13)

so that the saxionic cone condition ${\bm{s}}\in\overline{{\cal K}(\hat{X})}$
reads $\hat{s}\geq 0$ and $s^{\alpha}c_{\alpha}\in{\rm Nef}^{1}(B)$. The
corresponding contribution to the Kähler potential (4.46) is

$-\log\kappa({\bm{s}},{\bm{s}},{\bm{s}})=-\log\left[(c_{B}\cdot
c_{B})\hat{s}^{3}+3(c_{B}\cdot
c_{\alpha})s^{\alpha}\hat{s}^{2}+3(c_{\alpha}\cdot
c_{\beta})s^{\alpha}s^{\beta}\hat{s}\right]$ (B.14)

The saxion $\hat{s}$ measures the volume of the elliptic fiber. Let us apply
these results to the models of Section 4.2, taking the limit in which the
fiber is of stringy size, and hence much smaller than the base. This describes
a restricted perturbative regime parametrized only by the saxion $s^{0}$ and
the base Kähler saxions $s^{\alpha}$. In this limit
$p_{a}s^{a}=\hat{p}\hat{s}+p_{\alpha}s^{\alpha}\simeq p_{\alpha}s^{\alpha}$,
and the Kähler potential (4.46) becomes

$-\log(s^{0}-\frac{1}{2}p_{a}s^{a})-\log\kappa({\bm{s}},{\bm{s}},{\bm{s}})\simeq-\log(s^{0}-\frac{1}{2}p_{\alpha}s^{\alpha})-\log\left({\cal
I}_{\alpha\beta}s^{\alpha}s^{\beta}\right)+\ldots\,,$ (B.15)

which indeed matches (B.12), provided we identify the constants $p_{\alpha}$
appearing in these two different settings. This identification was already
proposed in [37], and is also consistent with the identification of the
saxionic cone conditions (B.7) and (4.45).

##  Appendix C Other tests of the species scale bound

In this appendix we discuss other non-trivial examples in string theory
compactifications where we can test explicitly our proposal on the relevant
energy scales formulated in Section 3.4. The reader can see this as a natural
addition to Section 4. In particular, we are going to explore the case of a
$\mathbb{P}^{1}$ fibration over a Hirzebruch surface $\mathbb{F}_{p}$ in
F-theory, and delve a bit deeper in the various Kähler moduli limits of
heterotic/IIA models.

###  C.1 Another F-theory model: $\mathbb{P}^{1}$ fibration over
$\mathbb{F}_{p}$

Another class of simple F-theory models where we can check our proposal is
obtained by choosing $X$ to be a $\mathbb{P}^{1}$ fibration over the
Hirzebruch surface $\mathbb{F}_{p}$, which in turn can be described as a
$\mathbb{P}^{1}$ fibration over $\mathbb{P}^{1}$, specified by the integer
$p\geq 0$. This model has been recently discussed in a closely related
framework by [61], to which we refer for more details. For our purposes it is
sufficient to restrict to a $\mathbb{P}^{1}$ fibration over $\mathbb{F}_{p}$
specified a single non-negative integers $h\in\mathbb{Z}_{\geq 0}$,383838
Following [127], the $\mathbb{P}^{1}$ fibration could be defined in terms of a
line bundle ${\cal L}=hd_{1}+td_{2}$, where $d_{1},d_{2}$ are elementary nef
divisors over $\mathbb{F}_{p}$. In particular, their intersection numbers are
given by the coefficients of ${\cal I}(\mathbb{F}_{p})=pd_{1}^{2}+d_{1}d_{2}$.
In the notation of [61], our integers $(h,t)$ correspond to $(s,t)$. Here we
are restricting to fibrations with $t=0$. and to identify the relevant cones
of divisors and curves and their intersection numbers.

The cone of effective divisors is simplicial and is generated by three
effective divisors $E^{a}$, $a=1,2,3$. These three effective divisors can be
roughly regarded as twisted products of the possible pairs of the three
$\mathbb{P}^{1}$’s involved in the geometry. We can also introduce a basis of
nef divisors $D^{a}$, with respect to which

$E^{1}=D^{1}-pD^{2}\,,\quad E^{2}=D^{2}\,,\quad E^{3}=D^{3}-hD^{1}\,.$ (C.1)

The divisors $D^{a}$ generate all the other nef divisors, as well as the
Kähler cone. Hence, by using these divisors in the expansion (4.6) the Kähler
cone corresponds to $v_{1},v_{2},v_{3}>0$. The triple intersections are given
by the coefficients of the formal object

$\displaystyle{\cal I}(X)$
$\displaystyle=D^{1}D^{2}D^{3}+p(D^{1})^{2}D^{3}+hpD^{1}(D^{3})^{2}+hD^{2}(D^{3})^{2}+h^{2}p(D^{3})^{3}\,.$
(C.2)

By using the expansion ${\bm{\ell}}=\ell_{a}D^{a}$, the kinetic potential
(4.18) becomes

${\cal F}_{\text{\tiny
K}}=\log\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})=\log\left(6\ell_{1}\ell_{2}\ell_{3}+3p\ell_{1}^{2}\ell_{3}+3hp\ell_{1}\ell_{3}^{2}+3h\ell_{2}\ell_{3}^{2}+h^{2}p\ell_{3}^{3}\right)\,.$
(C.3)

The condition that $J$ belongs to the Kähler cone is equivalent to
$\ell_{a}>0$. In order to understand the complete (dual) saxionic domain, we
have to consider the saxions
$s^{a}=\frac{3\kappa^{abc}\ell_{b}\ell_{c}}{2\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}$,
which identify the $\mathbb{R}$-effective curves:

${\bm{s}}=s^{a}\Sigma_{a}=\frac{3\,{\bm{\ell}}\cdot{\bm{\ell}}}{2\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}\,,$
(C.4)

where $\Sigma_{a}$ are effective curves

$\Sigma_{1}=E^{2}\cdot E^{3}\quad,\quad\Sigma_{2}=E^{1}\cdot
E^{3}\quad,\quad\Sigma_{3}=E^{1}\cdot E^{2}\,,$ (C.5)

which generate the whole cone of effective curves ${\rm Eff}_{1}(X)$ and are
dual to the nef divisors $D^{1}$: $D^{a}\cdot\Sigma_{b}=\delta^{a}_{b}$. On
the other hand, the saxionic cone can be identified with the cone of movable
curves – see (4.8) – which is generated by the (effective) movable curves

$\hat{\Sigma}_{1}=D^{2}\cdot
D^{3}=\Sigma_{1}+h\Sigma_{3}\,,\quad\hat{\Sigma}_{2}=D^{1}\cdot
D^{3}=p\Sigma_{1}+\Sigma_{2}+hp\Sigma_{3}\,,\quad\hat{\Sigma}_{3}=D^{1}\cdot
D^{2}=\Sigma_{3}\,,$ (C.6)

which are dual to the effective divisors $E^{a}$: $\hat{\Sigma}_{a}\cdot
E^{b}=\delta_{a}^{b}$. Hence, if we use the expansion
${\bm{s}}=\hat{s}^{a}\hat{\Sigma}_{a}$, the saxionic cone is defined by the
positivity conditions $\hat{s}^{a}>0$. By using (C.1) and (C.4) we can compute
the components
$\hat{s}^{a}=E^{a}\cdot{\bm{s}}=\frac{3E^{a}\cdot{\bm{\ell}}\cdot{\bm{\ell}}}{2\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}$,
getting

$\hat{s}^{1}=\frac{6\ell_{2}\ell_{3}}{2\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}\quad,\quad\hat{s}^{2}=\frac{3(2\ell_{1}\ell_{3}+h\,\ell_{3}^{2})}{2\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}\quad,\quad\hat{s}^{3}=\frac{3(2\ell_{1}\ell_{2}+p\,\ell_{1}^{2})}{2\kappa({\bm{\ell}},{\bm{\ell}},{\bm{\ell}})}\,.$
(C.7)

The saxionic cone condition $\hat{s}^{a}>0$ is clearly satisfied if
$\ell_{a}>0$.393939Note that, if we consider fibrations of the kind described
in footnote 38 with $t>0$, in general the image of the cone $\\{\ell_{a}>0\\}$
under the map (C.7) does not cover the entire saxionic cone $\Delta$ – see
[61] – as discussed in general in Section 4.1. This issue is absent if we set
$t=0$.

The ${\cal P}_{\text{\tiny K}}$ boundaries can again be characterized in terms
of tensionless string limits. The set ${\cal C}^{\text{\tiny EFT}}_{\rm S}$ of
EFT string charges is generated by the movable curves (C.6) and hence, in the
basis $\Sigma_{a}$, we identify the following corresponding tensions:

${\cal T}_{\hat{\Sigma}_{1}}=M^{2}_{\text{\tiny
P}}(\ell_{1}+h\ell_{3})\,,\quad{\cal T}_{\hat{\Sigma}_{2}}=M^{2}_{\text{\tiny
P}}(\ell_{2}+p\ell_{1}+hp\ell_{3})\,,\quad{\cal
T}_{\hat{\Sigma}_{3}}=M^{2}_{\text{\tiny P}}\ell_{3}\,.$ (C.8)

We then see that, assuming $p,h>0$, ${\cal T}_{\hat{\Sigma}_{3}}=0$ on the
two-dimensional boundary component $\\{\ell_{3}=0\\}$, ${\cal
T}_{\hat{\Sigma}_{1}}=0$ on the one dimensional boundary component
$\\{\ell_{1}=\ell_{3}=0\\}$, while ${\cal T}_{\hat{\Sigma}_{2}}=0$ at the tip
$\\{\ell_{1}=\ell_{2}=\ell_{3}=0\\}$. These boundary components are at
infinite field distance. On the other hand, the BPS but non-EFT strings of
charges $\Sigma_{1}$ and $\Sigma_{2}$ have tensions ${\cal
T}_{\Sigma_{1}}=M^{2}_{\text{\tiny P}}\ell_{1}$ and ${\cal
T}_{\Sigma_{2}}=M^{2}_{\text{\tiny P}}\ell_{2}$, which vanish on the
boundaries $\ell_{1}=0$ and $\ell_{2}=0$, respectively, which are at finite
field distance (if $\ell_{3}>0$). More precisely, they correspond to the
finite distance $\Delta$ boundaries $\hat{s}^{3}=0$ and $\hat{s}^{1}=0$,
respectively, while $\ell_{3}\rightarrow 0$ corresponds to the infinite
distance limit $\hat{s}^{3}\rightarrow\infty$. Viceversa, a limit
$\hat{s}^{2}\rightarrow 0$ (with fixed $\hat{s}^{1},\hat{s}^{3}$) corresponds
to a limit $\ell_{2}\rightarrow\infty$. So, as in subsection 4.1.2, while the
saxionic convex hull is simply given by
$\hat{\Delta}_{\alpha}=\\{\hat{s}^{a}\geq\frac{1}{\alpha}\\}$, its dual
saxionic counterpart $\hat{\cal P}_{\alpha}$ is more complicated – see figure
15.

Figure 15: Dual saxionic convex hull $\hat{\mathcal{P}}_{\alpha}$ for the
F-theory model $\mathbb{P}^{1}$ over $\mathbb{F}_{p}$. The plot has been drawn
with the reference value $\alpha=1/10$. $\hat{\mathcal{P}}_{\alpha}$
corresponds to the red region.

The anti-canonical divisor is

$\displaystyle\overline{K}_{X}$
$\displaystyle=2E^{3}+(2+h)E^{1}+(2+p+ph)E^{2}$ (C.9)
$\displaystyle=2D^{3}+(2-h)D^{1}+(2-p)D^{2}\,,$

which, combined with (2.9) and (4.10), gives

$\gamma(s)=\pi\left[2\hat{s}^{3}+(2+h)\hat{s}^{1}+(2+p+ph)\hat{s}^{2}\right]\,.$
(C.10)

Hence

$\gamma(s)|_{\hat{\Delta}_{\alpha}}\geq\frac{\pi(6+p+h+ph)}{\alpha}\,,$ (C.11)

which is again stronger than (3.21) with $N=3$. If for instance $p,h\geq 1$
and $\alpha\leq\frac{1}{10}$, we get the lower bound
$\gamma(s)|_{\hat{\Delta}_{\alpha}}>282$.

###  C.2 Energy scales in the $\mathbb{P}^{1}$ over $\mathbb{F}_{p}$ model

The tensions (C.8) are associated to the generators of the cone of EFT string
charges. From (C.8), and recalling that $\ell_{a}>0$, is clear that ${\cal
T}_{\hat{\Sigma}_{3}}$ is always the lowest of the three when $p,h>0$, and can
thus be identified with the dominant EFT string scale (3.29)

$M^{2}_{{\cal T}}=2\pi{\cal T}_{\hat{\Sigma}_{3}}=2\pi M^{2}_{\text{\tiny
P}}\ell_{3}\,.$ (C.12)

Noticing that

${\cal T}_{\hat{\Sigma}_{3}}\leq{\cal T}_{\hat{\Sigma}_{1}}\leq{\cal
T}_{\hat{\Sigma}_{2}}\,,$ (C.13)

the limit ${\cal T}_{\hat{\Sigma}_{3}}\ll{\cal T}_{\hat{\Sigma}_{1}}$, or
equivalently $\ell_{3}\ll\ell_{1}$, corresponds to the weak string coupling
limit of a dual heterotic model, and ${\cal T}_{\hat{\Sigma}_{3}}$ represents
the corresponding critical string tension. In this regime $(2\pi{\cal
T}_{\hat{\Sigma}_{3}})^{1/2}$ can be identified with the species scale, and
(3.28) is saturated.

If instead ${\cal T}_{\hat{\Sigma}_{3}}\simeq{\cal T}_{\hat{\Sigma}_{1}}$ the
species scale may be given by the quantum gravity scale (4.22), which for the
present model reads

$\displaystyle M^{2}_{\text{\tiny QG}}$
$\displaystyle=\sqrt{2\pi\left(2\ell_{1}\ell_{2}\ell_{3}+p\,\ell_{1}^{2}\ell_{3}+hp\ell_{1}\ell_{3}^{2}+h\ell_{2}\ell_{3}^{2}+\frac{1}{3}h^{2}p\ell_{3}^{3}\right)}\,M^{2}_{\text{\tiny
P}}$ (C.14)
$\displaystyle=\sqrt{\frac{1}{2\pi}\left[\frac{\ell_{1}}{\ell_{3}}\left(p\,\ell_{1}+2\ell_{2}\right)+h\left(p\ell_{1}+\ell_{2}+\frac{1}{3}hp\ell_{3}\right)\right]}\,M^{2}_{{\cal
T}}\,.$

Imposing the constraint (3.10) on ${\cal T}_{\hat{\Sigma}_{2}}$ gives
$p\ell_{1}+\ell_{2}+hp\ell_{3}<2\pi$, which in turn implies
$p\ell_{1}+\ell_{2}<2\pi$. Applying these two inequalities we find

$M^{2}_{\text{\tiny
QG}}<\sqrt{\frac{\ell_{1}}{\ell_{3}}\left(2-p\frac{\ell_{1}}{2\pi}\right)+h\left(1-hp\frac{\ell_{3}}{3\pi}\right)}\,M^{2}_{{\cal
T}}\,.$ (C.15)

Since

$\frac{{\cal T}_{\hat{\Sigma}_{1}}}{{\cal
T}_{\hat{\Sigma}_{3}}}=\frac{\ell_{1}}{\ell_{3}}+h>h\,,$ (C.16)

the assumption ${\cal T}_{\hat{\Sigma}_{3}}\simeq{\cal T}_{\hat{\Sigma}_{1}}$
requires that $h\sim{\cal O}(1)$ and $\ell_{1}/\ell_{3}\lesssim 1$. Hence
(C.15) implies that $M^{2}_{\text{\tiny QG}}\lesssim M^{2}_{{\cal T}}$, up to
a factor of order one, compatibly with (3.28). The precise order-one factor
depends on the dual saxions, and may be both smaller as well as (slightly)
greater than one. One may then reverse the logic, and use (3.28) as concrete
criterion to more precisely identify the species scale, as the smallest one
between (C.12) and (C.14).

We can also compare (C.12) to (3.27). By (C.10), in the present model we have

$M^{2}_{\gamma}=\frac{4\pi M^{2}_{\text{\tiny
P}}}{2\hat{s}^{3}+(2+h)\hat{s}^{1}+(2+p+ph)\hat{s}^{2}}\,.$ (C.17)

Recalling (C.7) we then obtain

$\frac{M^{2}_{{\cal
T}}}{M^{2}_{\gamma}}=\frac{1}{4}\left[2+\frac{12\ell_{2}\ell_{3}+6(2+p)\ell_{1}\ell_{3}+(6h+3ph+ph^{2})\ell_{3}^{2}}{6\ell_{1}\ell_{2}+3p\ell_{1}^{2}+3hp\ell_{1}\ell_{3}+3h\ell_{2}\ell_{3}+ph^{2}\ell_{3}^{2}}\right]\,.$
(C.18)

We can see this ratio is bounded from below and from above such that it is
always an ${\cal O}(1)$ quantity

$\frac{1}{2}\leq\frac{M^{2}_{{\cal
T}}}{M^{2}_{\gamma}}\leq\frac{1}{2}+\operatorname{max}\left\\{\frac{1}{h},\frac{1}{4}+\frac{3(2+p)}{4ph}\right\\}\leq\frac{3}{4}+\frac{3(2+p)}{4ph}\leq
3\,.$ (C.19)

This confirms that these two upper bounds on the species scale are of the same
order.

###  C.3 Asymptotic tests of our species scale bound

In this appendix we would like to provide further evidence for the validity of
the bound (3.28). We will study the heterotic models discussed in Section 4.2
in various EFT string limits, assuming that all EFT string charges satisfy
$e^{a}p_{a}\geq 0$ as in Section 4.2. Notice that the results in the case
$p_{a}=0$ apply also to ${\cal N}=2$ models obtained from type IIA Calabi-Yau
compactifications, or from the mirror type IIB models.

The infinite distance limit associated with an EFT string charge vector
$(e^{0},{\bf e})$ is given by the $\sigma\rightarrow\infty$ limit of the
saxionic flow

$s^{0}=s^{0}_{0}+e^{0}\,\sigma\quad,\quad{\bm{s}}(\sigma)={\bm{s}}_{0}+{\bf
e}\,\sigma\,.$ (C.20)

Let us assume that ${\bf e}$ is an elementary charge, and that $e^{0}$ is the
minimal one compatible with the saxionic cone (4.45): $e^{0}=p_{a}e^{a}$.
According to the discussion of Section 3.4, along the limit (C.20) the species
scale is either given by $\sqrt{2\pi{\cal T}_{\bf e}}$, if $w_{\bf e}=1$, or
by $M_{\text{\tiny QG}}$, if $w_{\bf e}\geq 2$. Some of these limits will exit
the M-theory/heterotic (or IIA) geometric regime and would require a
dualization to an alternative description. While (3.28) is automatically
satisfied and saturated if $w_{\bf e}=1$, in the $w_{\bf e}\geq 2$ case the
identification of the appropriate decompactification frame, and of the
corresponding $M_{\text{\tiny QG}}$, may not be obvious. In such cases, we
will rather apply (3.24) as a short-cut to estimate $M_{\text{\tiny QG}}$, and
verify (3.28). We will hence need to identify the relevant light masses that
can appear in these limits.

Consider first the KK scale along the M-theory interval/circle. Taking into
account the ansatz (4.3) and Einstein frame rescaling (4.58), the
corresponding KK mass is given by

$M_{\text{\tiny
KK}}^{2}=\frac{(2\pi)^{2}e^{2A}}{l_{(11)}^{2}e^{\frac{4}{3}\phi}}=\frac{6\pi
M^{2}_{\text{\tiny P}}}{\kappa({\bm{s}},{\bm{s}},{\bm{s}})}\,.$ (C.21)

We will also encounter towers of light states corresponding to M2-branes
multiply wrapped around curves ${\cal C}$. Using again (4.58), and the
relation ${\rm d}s^{2}_{\text{\tiny M}}(X)=e^{-\frac{2}{3}\phi}{\rm
d}s^{2}_{\text{st}}(X)$ between the M-theory and the string frame Calabi-Yau
metric, their minimal mass is given by:

$M_{\text{\tiny M2}}^{2}=\frac{(2\pi)^{2}e^{2A}V({{\cal
C}})^{2}}{l_{(11)}^{2}}=\frac{6\pi M^{2}_{\text{\tiny P}}V_{\rm st}({\cal
C})^{2}}{\kappa({\bm{s}},{\bm{s}},{\bm{s}})}\,,$ (C.22)

where $V_{\rm st}({\cal C})={\bm{s}}\cdot{\cal C}=s^{a}D_{a}\cdot{\cal C}$ is
the string frame volume of ${\cal C}$. As we will see, there will also appear
additional light towers of KK-states associated to the internal Calabi-Yau.

Which towers of states should be considered in computing the quantum gravity
scale along these EFT string limits depends on whether $p_{a}e^{a}=0$ or
$p_{a}e^{a}>0$, and on the intersection properties of the nef divisor ${\bf
e}=e^{a}D_{a}$. As remarked in [14] and [37], following [48], the latter can
grouped into three cases. We will then separately discuss these three
possibilities, for both $p_{a}e^{a}=0$ and $p_{a}e^{a}>0$, in turn.

####  C.3.1 EFT string limits with $p_{a}e^{a}=0$

Let us first assume that $e^{0}=p_{a}e^{a}=0$. The saxion $s^{0}$, the average
M-theory Calabi-Yau volume $V(X)=s^{0}-\frac{1}{2}p_{a}s^{a}$, and the tension
${\cal T}_{*}$ defined in (4.53a) remain constant along the flow (C.20). On
the other hand, (4.38) implies that

$e^{2\phi}=\frac{\kappa({\bm{s}},{\bm{s}},{\bm{s}})}{6(s^{0}-\frac{1}{2}p_{a}s^{a})}\,,$
(C.23)

which, as we will see, diverges as some positive power of $\sigma$.
Furthermore (4.53b) reduces to the first contribution appearing on its r.h.s.
and determines, in the limit $\sigma\rightarrow\infty$, the dominant EFT
string scale $M_{\cal T}$ defined in (3.29):

$M^{2}_{\cal T}=\frac{3\pi\kappa({\bf
e},{\bm{s}},{\bm{s}})}{\kappa({\bm{s}},{\bm{s}},{\bm{s}})}\,M^{2}_{\text{\tiny
P}}\,.$ (C.24)

Following [14], we can then distinguish three cases. Note that, while it is
not a priori obvious how to concretely realize the condition $p_{a}e^{a}=0$ in
heterotic compactifications, if we set $p_{a}\equiv 0$ the following results
apply anyway to ${\cal N}=2$ type IIA compactifications over $X$.

Case 1: $\kappa({\bf e},{\bf e},{\bf e})>0$

In this case, evaluating (C.21) and (C.24) along (C.20) and taking the limit
$\sigma\rightarrow\infty$, one gets $M^{2}_{\cal T}\simeq\frac{3\pi
M^{2}_{\text{\tiny P}}}{\sigma}$ and $M_{\text{\tiny KK}}^{2}\simeq\frac{6\pi
M^{2}_{\text{\tiny P}}}{\kappa({\bf e},{\bf e},{\bf e})\sigma^{3}}$. Moreover,
from (C.23) we see that $e^{2\phi}$ diverges as $\sigma^{3}$, and then the
M-theory description is more appropriate. This is a $w=3$ EFT string limit
[14]. The Calabi-Yau KK mass-squared scales as $\sigma^{-1}$ and is
parametrically higher than $M_{\text{\tiny KK}}$. This limit corresponds to a
partial M-theory decompactification, and the species scale should then be
given by the quantum gravity scale (4.59), which asymptotically behaves as

$M^{2}_{\text{\tiny
QG}}\simeq\left[\frac{12\pi}{(s^{0}_{0}-\frac{1}{2}p_{a}s^{a}_{0})^{2}\kappa({\bf
e},{\bf e},{\bf e})}\right]^{\frac{1}{3}}\frac{M^{2}_{\text{\tiny
P}}}{2\sigma}\simeq\frac{M^{2}_{\cal
T}}{\left[18\pi^{2}(s^{0}_{0}-\frac{1}{2}p_{a}s^{a}_{0})^{2}\kappa({\bf
e},{\bf e},{\bf e})\right]^{\frac{1}{3}}}\,.$ (C.25)

Our saxionic convex hull conditions (4.50) requires
$s^{0}_{0}\geq\frac{1}{\alpha}+p_{a}s_{0}^{a}$, with $\alpha\leq 2\pi$. Hence
(C.25) implies that $M_{\text{\tiny QG}}$ and $M_{\cal T}$ scale
asymptotically in the same way, but with $M_{\text{\tiny QG}}<M_{\cal T}$,
hence realizing (3.28) with $M_{\rm sp}=M_{\text{\tiny QG}}$.

Case 2: $\kappa({\bf e},{\bf e},{\bf e})=0$ but $\kappa({\bf e},{\bf e},{\bf
e}^{\prime})>0$ for some ${\bf e}^{\prime}\in{\rm Nef}_{\mathbb{Z}}(X)$

In this limit, the Calabi-Yau base $X$ can be seen as a $T^{2}$ fibration over
a base two-fold $B$, with the non-trivial curve ${\cal C}={\bf e}\cdot{\bf e}$
being a multiple of the $T^{2}$ fiber. From (C.24) and (C.21) get $M^{2}_{\cal
T}\simeq 2\pi M^{2}_{\text{\tiny P}}/\sigma$ and $M_{\text{\tiny
KK}}^{2}\simeq 2\pi M^{2}_{\text{\tiny P}}/[\kappa({\bf e},{\bf
e},{\bm{s}}_{0})\sigma^{2}]$, realizing a $w=2$ EFT string limit. The M-theory
volume $V({\cal C})=e^{-2\phi/3}V_{\rm st}({\cal C})=e^{-2\phi/3}\kappa({\bf
e},{\bf e},{\bm{s}})$ of ${\cal C}$ vanishes as $\sigma^{-2/3}$, while the
M-theory volume of base two-fold $B$ grows as $\sigma^{2/3}$. Hence the base
KK mass-squared scales as $1/\sigma$, and is again much heavier than
$M_{\text{\tiny KK}}$. On the other hand, since we can identify ${\cal C}$
with an integral $k$-multiple of the $T^{2}$ fiber, the mass (C.22) of an
M2-brane on the $T^{2}$ fiber is asymptotically given by $M_{\text{\tiny
M2}}^{2}\simeq 2\pi M^{2}_{\text{\tiny P}}\kappa({\bf e},{\bf
e},{\bm{s}}_{0})/(k^{2}\sigma^{2})$, and hence scales with the same rate of
$M^{2}_{\text{\tiny KK}}$. According to the ESC [48], there should exist some
dual description in which this limit corresponds to a decompactification
limit, which then realizes the species scale as a corresponding quantum
gravity scale. Indeed, as in the F-theory limit of elliptically fibered
M-theory compactifications, while one looses the two directions of the
shrinking $T^{2}$ fiber, the above light M2 states should correspond to KK
modes of a new emergent compact direction. Therefore, the dual
compactification space should roughly be given by a (possibly twisted) product
of the M-theory interval/circle, the new emergent circle, and the base $B$.
The corresponding KK species can then be approximately considered as
multiplicative – see for instance [128] for a discussion on multiplicative
species. More precisely, we estimate the species scale by applying (3.24) with
404040According to Weyl’s rule, which can be extracted from Eq. (A.14), the
number of KK resonances associated with an $n$-dimensional space with mass
below the species scale is well approximated by the formula
$N_{(n)}\sim{\mathtt{V}}_{n}/(2\pi)^{\lfloor(n+1)/2\rfloor}$, up to numerical
factors which will not alter our main conclusion. On the other hand, in the
present setting, we have ${\mathtt{V}}_{6}\simeq{\mathtt{V}}_{1,{\text{\tiny
KK}}}{\mathtt{V}}_{1,{\text{\tiny M2}}}{\mathtt{V}}_{4,B}$ and $N_{\rm
sp}\simeq N_{6}N_{(10)}$. Combining these estimates we obtain (C.26).

$N_{\rm sp}\simeq 2\pi N_{\text{\tiny KK}}N_{\text{\tiny
M2}}N_{B}N_{(10)}=\frac{2\pi M^{2}_{\text{sp}}}{M_{\text{\tiny
KK}}M_{\text{\tiny M2}}}\,N_{B}N_{(10)}$ (C.26)

where $N_{\text{\tiny KK}}$, $N_{\text{\tiny M2}}$, $N_{B}$ denote the numbers
of KK modes with mass below the species scale, corresponding to each geometric
(local) factor, and $N_{(10)}$ denotes the number of light degrees of freedom
in the dual ten-dimensional theory. Using (C.26) in (3.24), with the above
asymptotic values of $M_{\text{\tiny KK}}$ and $M_{\text{\tiny M2}}$, we get

$M^{2}_{\rm sp}\simeq\frac{1}{k\sqrt{N_{B}N_{(10)}}}\,\frac{2\pi
M^{2}_{\text{\tiny P}}}{\sigma}\simeq\frac{\,M^{2}_{\cal
T}}{k\sqrt{N_{B}N_{(10)}}}\,.$ (C.27)

Since in the limit the base volume grows, we expect $N_{B}\gg 1$. Moreover, in
string theory $N_{(10)}$ is significantly larger than one too. Hence
$M^{2}_{\rm sp}<M^{2}_{\cal T}$, in agreement with (3.28).

Case 3: $\kappa({\bf e},{\bf e},{\bf e}^{\prime})=0$ for any ${\bf
e}^{\prime}\in{\rm Nef}_{\mathbb{Z}}(X)$

In this case we have ${\bf e}\cdot{\bf e}=0$ and in this limit the Calabi-Yau
$X$ can be seen as a K3 or $T^{4}$ fibration over a $\mathbb{P}^{1}$, where
the fiber can be identified with the divisor ${\bf e}=e^{a}D_{a}$. Along this
EFT string limit, which has scaling weight $w=1$, the M-theory volume of the
base $\mathbb{P}^{1}$ grows as $\sigma^{\frac{2}{3}}$, while the M-theory
volume of K3 or $T^{4}$ decreases as $\sigma^{-\frac{2}{3}}$. By the ESC [48]
there should exist a dual description in which this EFT string limit is a
tensionless critical string limit. Hence in this limit $M_{\rm sp}=M_{\cal T}$
and (3.28) is satisfied and saturated.

####  C.3.2 EFT string limits with $p_{a}e^{a}\geq 1$

Looking at the asymptotic behavior of (4.53) along (C.20), it is clear that if
$p_{a}e^{a}\geq 1$ the dominant EFT string scale is always given by the EFT
string tension (4.53a):

$M_{\cal T}^{2}=2\pi{\cal T}_{*}=\frac{\pi M^{2}_{\text{\tiny
P}}}{(s^{0}-\frac{1}{2}p_{a}s^{a})}\simeq\frac{2\pi M^{2}_{\text{\tiny
P}}}{p_{a}e^{a}}\,\frac{1}{\sigma}\,.$ (C.28)

Differently from what happened in Section C.3.1, now the average M-theory
Calabi-Yau volume diverges,
${V}(X)=s^{0}-\frac{1}{2}p_{a}s^{a}\simeq\frac{1}{2}p_{a}e^{a}\sigma$, and
this also affects the asymptotic behavior of the dilaton (C.23).

We can then proceed discussing in turn the three cases already considered as
in in Section C.3.1, following [37].

Case 1: $\kappa({\bf e},{\bf e},{\bf e})>0$

In this case both the M-theory interval and the Calabi-Yau volume diverge.
Hence the species scale $M_{\rm sp}$ is given by (4.59). Evaluating (4.60)
along the flow (C.20), in the limit $\sigma\rightarrow\infty$ we get
$M^{2}_{\cal T}/M^{2}_{\text{\tiny QG}}\propto\sigma^{2/3}$, and then (3.28)
is certainly satisfied.

Case 2: $\kappa({\bf e},{\bf e},{\bf e})=0$ but $\kappa({\bf e},{\bf e},{\bf
e}^{\prime})>0$ for some ${\bf e}^{\prime}\in{\rm Nef}_{\mathbb{Z}}(X)$

This limit has scaling weight $w=2$ [37]. As already discussed in Section
C.3.1, the Calabi-Yau $X$ can be regarded as a $T^{2}$ fibration over a base
two-fold $B$. In M-theory, the $T^{2}$ fiber has volume $V(T^{2})=V({\cal
C})/k=\frac{1}{k}e^{-2\phi/3}\kappa({\bf e},{\bf
e},{\bm{s}})\propto\sigma^{-1/3}$, where $k\geq 1$ is the number of $T^{2}$
fibers contained in ${\cal C}$, and hence the base $B$ volume grows as
$\sigma^{4/3}$. From (C.21) and (C.22) we get the same asymptotic behaviors
found in the $p_{a}e^{a}=0$ case: $M_{\text{\tiny KK}}^{2}\simeq 2\pi
M^{2}_{\text{\tiny P}}/[\kappa({\bf e},{\bf e},{\bf s}_{0})\sigma^{2}]$ and
$M_{\text{\tiny M2}}^{2}\simeq 2\pi M^{2}_{\text{\tiny P}}\kappa({\bf e},{\bf
e},{\bm{s}}_{0})/(k^{2}\sigma^{2})$. Now we cannot use (4.59) and (4.60), but
we can proceed as in the discussion of Case 2 in Section C.3.1, and use (3.24)
as estimate of the species scale, with $N_{\rm sp}$ as in (C.26). The first
estimate in (C.27) still holds. Recalling (C.28), it can be rewritten as

$M^{2}_{\rm sp}=\frac{p_{a}e^{a}}{k\sqrt{N_{B}N_{(10)}}}\,M^{2}_{\cal T}\,.$
(C.29)

Recalling that the base $B$ decompactifies faster than in the $p_{a}e^{a}=0$
case, we expect an even larger $N_{B}\gg 1$, implying that (3.28) is again
satisfied.

Case 3: $\kappa({\bf e},{\bf e},{\bf e}^{\prime})=0$ for any ${\bf
e}^{\prime}\in{\rm Nef}_{\mathbb{Z}}(X)$

This is a $w=2$ limit [37]. As discussed in Section C.3.1, $X$ is a K3/$T^{4}$
fibration over $\mathbb{P}^{1}$, and we can identify the fiber with the nef
divisor ${\bf e}=e^{a}D_{a}$. Eq. (C.23) implies that the dilaton tends to the
constant value $e^{2\phi}\simeq\kappa({\bf
e},{\bm{s}}_{0},{\bm{s}}_{0})/(p_{a}e^{a})$. Similarly, the M-theory volume of
the K3/$T^{4}$ fiber is asymptotically constant too: $V_{\bf
e}=\frac{1}{2}e^{-\frac{4}{3}\phi}\kappa({\bf
e},{\bm{s}},{\bm{s}})\simeq\frac{1}{2}(p_{a}e^{a})^{\frac{2}{3}}\kappa({\bf
e},{\bm{s}}_{0},{\bm{s}}_{0})^{\frac{1}{3}}$. Hence the $\mathbb{P}^{1}$ base
diverges as $\sigma$, the heterotic/M-theory EFT description remains valid,
and the discussion of Section 4.2.1 applies. In particular, if the asymptotic
value of $2\pi e^{\phi}$ is smaller than one, then $M_{\rm sp}$ asymptotically
coincides with (C.28), and saturates the bound (3.28). If instead it is larger
than one, then $M_{\rm sp}$ is given by (4.59) and (4.60) implies that the
bound (3.28) is satisfied.

##  Appendix D Wormholes and supersymmetry

In this paper we have focused on asymptotically flat wormholes, regarded as
quantum excitations of a supersymmetric Minkowski vacuum preserving ${\cal
N}=1$ supersymmetry. In general a wormhole breaks, at least partially, the
corresponding four supercharges. In this appendix we discuss at a more
quantitative level this breaking and the presence of corresponding Goldstino-
like fermionic zero-modes.

We will use the four- and two-component spinor notations. In four-component
notation the Majorana supersymmetry generators split into
$\epsilon=\epsilon_{\text{\tiny L}}+\epsilon_{\text{\tiny R}}$, with
$\gamma_{5}\epsilon_{\text{\tiny L}}=\epsilon_{\text{\tiny L}}$ and
$\gamma_{5}\epsilon_{\text{\tiny R}}=-\epsilon_{\text{\tiny R}}$. The relation
with the two-component notation [38] is given by

$\gamma^{a}={\rm i}\left(\begin{array}[]{cc}0&\sigma^{a}\\\
\bar{\sigma}^{a}&0\end{array}\right)\,,\quad\gamma_{5}=-{\rm
i}\gamma^{\underline{0123}}=\left(\begin{array}[]{cc}-\mathds{1}&0\\\
0&\mathds{1}\end{array}\right)\,,\quad\epsilon=\left(\begin{array}[]{c}\epsilon_{\text{\tiny
R}}\\\ \epsilon_{\text{\tiny
L}}\end{array}\right)=\left(\begin{array}[]{c}\epsilon_{\alpha}\\\
\bar{\epsilon}^{\dot{\alpha}}\end{array}\right)\,.$ (D.1)

where $(\sigma^{a})_{\alpha\dot{\beta}}=(-\mathds{1},\vec{\sigma})$ and
$(\bar{\sigma}^{a})^{\dot{\alpha}\beta}=(-\mathds{1},-\vec{\sigma})$. We use
Latin letters from the middle of the alphabet ($m,n,\ldots$) to denote curved
indices, and from the beginning of the alphabet ($a,b,\ldots$) to denote flat
indices. If necessary, to avoid further ambiguities, we underline flat
indices. In two-component notation the Majorana condition relates left- and
right-moving components:

$\bar{\epsilon}_{\dot{\alpha}}\equiv\varepsilon_{\dot{\alpha}\dot{\beta}}\bar{\epsilon}^{\dot{\beta}}=(\epsilon_{\alpha})^{*}\,.$
(D.2)

In four-component notation the supersymmetry transformations of gravitino and
chiralinos (the supersymmetric partners of $t^{i}$) are [38]

$\displaystyle\delta\psi_{m}$ $\displaystyle=\Big{[}\nabla_{m}-\frac{{\rm
i}}{2}{\rm Im\hskip
1.00006pt}\left(\partial_{m}t^{i}\partial_{i}K\right)\gamma_{5}\Big{]}\epsilon\,,$
(D.3a) $\displaystyle\delta\chi^{i}_{\text{\tiny R}}$
$\displaystyle=\sqrt{2}\,\partial_{m}t^{i}\gamma^{m}\epsilon_{\text{\tiny
L}}\,,\quad\delta\chi^{i}_{\text{\tiny
L}}=\sqrt{2}\,\partial_{m}\bar{t}^{i}\gamma^{m}\epsilon_{\text{\tiny R}}\,.$
(D.3b)

In this paper we restrict to Kähler potential that depend on $t^{i}=a^{i}+{\rm
i}s^{i}$ only through the saxions $s^{i}={\rm Im\hskip 1.00006pt}t^{i}$.
Recalling (2.11), (D.3a) can be rewritten as

$\delta\psi_{m}=\Big{(}\nabla_{m}-\frac{{\rm
i}}{2}\,\ell_{i}\partial_{m}a^{i}\,\gamma_{5}\Big{)}\epsilon\,.$ (D.4)

In the Wick-rotated Euclidean formulation
$\gamma^{\underline{4}}=\gamma_{\underline{4}}={\rm i}\gamma^{\underline{0}}$
and, correspondingly, $\sigma^{\underline{4}}=\sigma_{\underline{4}}={\rm
i}\sigma^{\underline{0}}=-{\rm i}\mathds{1}$ and
$\bar{\sigma}^{\underline{4}}=\bar{\sigma}_{\underline{4}}={\rm
i}\bar{\sigma}^{\underline{0}}=-{\rm i}\mathds{1}$ whereas $\vec{\sigma}$
remain unchanged, so that the representation (D.1) formally still holds.
Moreover, the left- and right-moving components of the Majorana spinors become
independent fields and are not related by complex conjugation anymore, but
must be regarded as analytic continuation of the Lorentzian ones. In
particular, the Majorana condition (D.2) must be relaxed.

###  D.1 Extremal BPS wormholes

We first focus on the extremal wormholes of Section 5.3. These are
characterized by the following BPS-like equations [14]:

${\cal H}_{3,i}=-M_{\text{\tiny P}}^{2}*{\rm d}\ell_{i}\,.$ (D.5)

The metric is flat and, in the case of the extremal wormhole sourced by a
fundamental instanton centered at $x_{0}$, the Bianchi identity (2.17) must be
corrected by a localized term: ${\rm d}{\cal H}_{3,i}=2\pi\delta_{4}(x_{0})$.
Hence, the dual saxions profiles must satisfy ${\rm d}*{\rm d}\ell_{i}=-2\pi
M_{\text{\tiny P}}^{-2}\delta_{4}(x_{0})$, which is solved by

$\ell_{i}=\ell_{i}^{\infty}+\frac{q_{i}}{2\pi M^{2}_{\text{\tiny
P}}\,|x-x_{0}|^{2}}\,,$ (D.6)

where $|x|^{2}\equiv\delta_{mn}x^{m}x^{n}$. The solution (D.6) can be
immediately generalized to multi-centered ones, sourced by multiple
fundamental instantons. Given the Euclidean axion/two-form duality relation

${\rm d}a^{i}=\frac{{\rm i}}{M_{\text{\tiny P}}^{2}}{\cal G}^{ij}*{\cal
H}_{3,j}\,,$ (D.7)

in terms of the axions $a^{i}$ the BPS conditions (D.5) reads414141Recall that
in Euclidean space the axions dual to a 2-form become purely imaginary fields.

${\rm d}a^{i}={\rm i}{\cal G}^{ij}{\rm d}\ell_{j}=-{\rm i}{\rm d}s^{i}\,.$
(D.8)

By using this and (2.11), and the fact that the metric is flat, the gravitino
transformation (D.4) on the multicenter extremal wormhole solution becomes

$\delta\psi_{m}=(\partial_{m}+\frac{1}{4}\partial_{m}K\gamma_{5})\epsilon=e^{-\frac{1}{4}K\gamma_{5}}\partial_{m}\left(e^{\frac{1}{4}K\gamma_{5}}\epsilon\right)\,.$
(D.9)

We then see that the gravitino remains invariant, $\delta\psi_{m}=0$, under
the supersymmetry transformations

$\epsilon=e^{-\frac{1}{4}(K-K_{\infty})\gamma_{5}}\eta\quad\Leftrightarrow\quad\epsilon_{\alpha}=e^{\frac{1}{4}(K-K_{\infty})}\eta_{\alpha}\quad,\quad\bar{\epsilon}^{\dot{\alpha}}=e^{-\frac{1}{4}(K-K_{\infty})}\bar{\eta}^{\dot{\alpha}}\,,$
(D.10)

with $\eta_{\alpha}$ and $\bar{\eta}^{\dot{\alpha}}$ independent constant
anticommuting spinors. We have chosen the integration constant
$e^{\frac{1}{4}K_{\infty}\gamma_{5}}$, where $K_{\infty}\equiv K(s_{\infty})$,
so that $\epsilon\rightarrow\eta$ at infinite spatial distance. The four
constant components $(\eta_{\alpha},\bar{\eta}^{\dot{\alpha}})$ parametrize
four independent global supersymmetry transformations which leave the
gravitino invariant – see also [129].

Let us now turn to the chiralino transformations (D.3b). We first notice that
the BPS condition (D.8) is equivalent to ${\rm d}t^{i}=0$, while ${\rm
d}\bar{t}^{i}=-2{\rm i}{\rm d}s^{i}$. This implies that

$\delta\chi^{i}_{\alpha}=0\,,\quad\delta\bar{\chi}^{\dot{\alpha}\,i}=2\sqrt{2}e^{\frac{1}{4}(K-K_{\infty})}\partial_{m}s^{i}(\bar{\sigma}^{m})^{\dot{\alpha}\beta}\eta_{\beta}\,,$
(D.11)

under the transformations generated by (D.10). We see that the extremal
wormhole solutions preserve the two supersymmetries generated by
$\bar{\eta}^{\dot{\alpha}}$, while break the ones generated by
$\eta_{\alpha}$. In particular, because $\eta_{\alpha}$ acts non-trivially on
the background, the configuration

$\bar{\chi}^{\dot{\alpha}\,i}(x)\equiv[\bar{\chi}^{i\,\beta}_{(0)}(x-x_{0})]^{\dot{\alpha}}\eta_{\beta}\,,$
(D.12)

with ($\beta=1,2$)

$\displaystyle[\bar{\chi}^{i\,\beta}_{(0)}(x-x_{0})]^{\dot{\alpha}}={\cal
N}\,e^{\frac{1}{4}(K-K_{\infty})}(\bar{\sigma}^{m})^{\dot{\alpha}\beta}\partial_{m}s^{i}\,,$
(D.13)

corresponds to zero-modes of the linearized equations of motion around the
background. Note that the location of the fundamental instanton $x_{0}$ is a
free parameter. The normalization constant $\mathcal{N}$ is fixed by

$\delta^{\beta\beta^{\prime}}=\int{\rm d}^{4}x\,{\cal
G}_{ij}\left\\{[\bar{\chi}^{i\,\beta}_{(0)}]^{\dot{\alpha}}\right\\}^{\dagger}[\bar{\chi}^{j\,\beta^{\prime}}_{(0)}]^{\dot{\alpha}}.$
(D.14)

Since the integral should be restricted to the controllable regime
$|x-x_{0}|^{2}\geq\Lambda^{-2}$, the normalization constant ${\cal N}$ in
general satisfies ${\cal N}\propto\Lambda\,e^{K_{\infty}/4}$ up to a
dimensionless function of $M_{\text{\tiny P}}^{2}/{\Lambda^{2}}$. Furthermore,
as $\partial_{m}s^{i}\sim M_{\text{\tiny P}}^{-2}{\cal
G}^{ij}_{\infty}q_{j}\,\partial_{m}|x-x_{0}|^{-2}$ for large $|x-x_{0}|^{2}$,
the zero-modes (D.12) behave as the components of Feynman’s fermionic
propagator in the asymptotic vacuum:

$\displaystyle[\bar{\chi}^{i\,\beta}_{(0)}]^{\dot{\alpha}}\propto{\cal
N}\,e^{\frac{1}{4}(K-K_{\infty})}\frac{{\cal G}^{ij}q_{j}}{2\pi M_{\text{\tiny
P}}^{2}}\,(\bar{\sigma}^{m})^{\dot{\alpha}\beta}\partial_{m}\frac{1}{|x-x_{0}|^{2}}\propto\varepsilon^{\beta\alpha}q_{j}{\cal
G}^{ij}[S_{\rm F}(x-x_{0})]_{\alpha}^{\dot{\alpha}}\,.$ (D.15)

###  D.2 Non-extremal wormholes

In order to study the fermionic zero-modes of regular non-extremal wormholes
it is convenient, following [115], to introduce a new dimensionless radial
coordinate

$y\equiv 2\pi M^{2}_{\text{\tiny
P}}L^{2}\tau\quad~{}~{}~{}\Leftrightarrow\quad~{}~{}~{}\cos
y=\frac{L^{2}}{r^{2}}\,.$ (D.16)

Note that $y\in(-\frac{\pi}{2},\frac{\pi}{2})$. We recall that $\tau$ is
related to the three-sphere radius $r$ by (5.22). Then $e^{-2A}=\sin^{2}y$ and
the metric (5.20) becomes

${\rm d}s^{2}=L^{2}\left(\frac{{\rm d}y^{2}}{4\cos^{3}y}+\frac{{\rm
d}\Omega^{2}_{3}}{\cos y}\right)\,.$ (D.17)

Let us pick the vielbein $e^{a}=(e^{\underline{4}},e^{\underline{i}})$, with

$e^{\underline{4}}=\frac{L}{2(\cos y)^{\frac{3}{2}}}{\rm d}y\quad,\quad
e^{\underline{i}}=\frac{L}{\sqrt{\cos y}}\,\hat{e}^{\underline{i}}\,,$ (D.18)

where $e^{\underline{i}}$ is a vielbein for a three-sphere of radius $r=1$.
The components of the spin connection $\omega^{a}{}_{b}=\omega^{a}{}_{bm}{\rm
d}y^{m}$ are

$\omega^{\underline{i}}{}_{\underline{j}}=\hat{\omega}^{\underline{i}}{}_{\underline{j}}\quad,\quad\omega^{\underline{i}}{}_{\underline{4}}=\sin
y\,e^{\underline{i}}\,.$ (D.19)

By combing (D.7) and (5.7) we have

${\rm d}a^{i}=-\frac{{\rm i}{\cal G}^{ij}q_{j}}{2\pi M^{2}_{\text{\tiny
P}}L^{2}}{\rm d}y\,.$ (D.20)

Plugging (D.20) inside (D.4) we get

$\delta\psi_{m}=\left(\nabla_{m}-\frac{\langle{\bf q},{\bm{s}}\rangle}{4\pi
M^{2}_{\text{\tiny P}}L^{2}}\delta^{y}_{m}\gamma_{5}\right)\epsilon\,.$ (D.21)

Let us now gauge-fix the local supersymmetry by imposing the transverse
gravitino gauge

$\gamma^{m}\psi_{m}=0\,,$ (D.22)

where $\gamma_{m}=e_{m}^{a}\gamma_{a}$. Imposing that (D.22) is preserved
under (D.21) and using (D.19), we get the following condition on $\epsilon$:

$2\cos
y\,\partial_{y}\epsilon+\gamma^{\underline{4}}\hat{\gamma}^{i}\hat{\nabla}_{i}\epsilon+\frac{3}{2}\sin
y\,\epsilon=\frac{\langle{\bf q},{\bm{s}}\rangle}{2\pi M^{2}_{\text{\tiny
P}}L^{2}}\cos y\,\gamma_{5}\epsilon\,.$ (D.23)

We would like to determine the form of $\epsilon$ imposing that it becomes
asymptotically covariantly constant on one of the two sides of the wormhole:
$\nabla_{m}\epsilon|_{y=\pm\frac{\pi}{2}}=0$. Since

$\nabla_{i}\epsilon=\hat{\nabla}_{i}\epsilon+\frac{1}{2}\sin
y\,\hat{\gamma}_{i}\gamma_{\underline{4}}\epsilon\,,$ (D.24)

we get the asymptotic conditions

$\hat{\nabla}_{i}\epsilon|_{\pm\frac{\pi}{2}}=\mp\frac{1}{2}\hat{\gamma}_{i}\gamma_{\underline{4}}\epsilon|_{y=\pm\frac{\pi}{2}}\,.$
(D.25)

Given the $O(4)$ symmetry, these conditions should in fact be satisfied by
$\epsilon$ at any $y\in(-\frac{\pi}{2},\frac{\pi}{2})$. We must then consider
two possibilities

$\hat{\nabla}_{i}\epsilon_{\pm}=\mp\frac{1}{2}\hat{\gamma}_{i}\gamma_{\underline{4}}\epsilon_{\pm}\,.$
(D.26)

So $\epsilon_{+}$ can asymptotically tend to a non-vanishing covariantly
constant spinor on the first half-wormhole, $y\in[0,\frac{\pi}{2})$, while it
should quickly vanish in the second half, for $y\rightarrow-\frac{\pi}{2}$.
Viceversa, $\epsilon_{-}$ tends to a covariantly constant spinor on the second
half-wormhole, $y\in(-\frac{\pi}{2},0]$ and must be quickly vanishing for
$y\rightarrow\frac{\pi}{2}$. In the following we will focus on $\epsilon_{+}$,
omitting the subscript for simplicity. (The corresponding results for
$\epsilon_{-}$ can be immediately obtained by inverting the role of the two
half-wormholes.)

So, taking into account (D.26), (D.23) becomes

$2\cos y\,\partial_{y}\epsilon+\frac{3}{2}(\sin
y-1)\,\epsilon=\frac{\langle{\bf q},{\bm{s}}\rangle}{2\pi M^{2}_{\text{\tiny
P}}L^{2}}\cos y\,\gamma_{5}\epsilon\,.$ (D.27)

The solution to this equation can be written in the form

$\epsilon=\mathbf{M}(y)\,\eta\quad~{}~{}~{}~{}\text{with}\quad~{}~{}~{}~{}~{}\mathbf{M}(y)=M_{1}(y)+M_{2}(y)\gamma_{5}\,,$
(D.28)

where $\eta$ is an $y$-independent spinor satisfying the same equation (D.26)
of $\epsilon_{+}$:
$\hat{\nabla}_{i}\eta=-\frac{1}{2}\hat{\gamma}_{i}\gamma_{\underline{4}}\eta$.
We will impose that ${\bf M}(y=\frac{\pi}{2})=\mathds{1}$, so that $\eta$
represents the asymptotic covariantly constant spinor. The condition (D.27)
translates into the following equation for the matrix $\mathbf{M}(y)$:

$\partial_{y}\log\mathbf{M}=-\frac{3}{4}\tan\left(\frac{y}{2}-\frac{\pi}{4}\right)+\frac{\langle{\bf
q},{\bm{s}}\rangle}{4\pi M^{2}_{\text{\tiny P}}L^{2}}\,\gamma_{5}\,.$ (D.29)

This can be integrated into

$\mathbf{M}(y)=\left[\cos\left(\frac{y}{2}-\frac{\pi}{4}\right)\right]^{\frac{3}{2}}\exp\left[-\frac{\gamma_{5}}{4\pi
M^{2}_{\text{\tiny P}}L^{2}}\int^{\frac{\pi}{2}}_{y}{\rm
d}\tilde{y}\,\langle{\bf q},{\bm{s}}(\tilde{y})\rangle\right]\,,$ (D.30)

where we have fixed the integration constants by imposing ${\bf
M}(y)|_{y=\frac{\pi}{2}}=\mathds{1}$. The right- and left-handed components
$(\epsilon_{\alpha},\bar{\epsilon}^{\dot{\alpha}})$ are then given by

$\epsilon_{\alpha}=\eta_{\alpha}\left[\cos\left(\frac{y}{2}-\frac{\pi}{4}\right)\right]^{\frac{3}{2}}e^{f(y)}\,,\quad\bar{\epsilon}^{\dot{\alpha}}=\bar{\eta}^{\dot{\alpha}}\left[\cos\left(\frac{y}{2}-\frac{\pi}{4}\right)\right]^{\frac{3}{2}}e^{-f(y)}\,,$
(D.31)

with

$f(y)\equiv\frac{1}{4\pi M^{2}_{\text{\tiny
P}}L^{2}}\int^{\frac{\pi}{2}}_{y}{\rm d}\tilde{y}\,\langle{\bf
q},{\bm{s}}(\tilde{y})\rangle\,.$ (D.32)

The common cosine factor implies that all components of $\epsilon$ quickly
vanish as one approaches $y=-\frac{\pi}{2}$, as expected. The exponential
factors instead give an enhancement for $\epsilon_{\alpha}$ and a further
suppression for $\bar{\epsilon}^{\dot{\alpha}}$. One can get an idea of the
possible form of $e^{f(y)}$ by restricting to the homogeneous wormholes of
Section 6.1. In terms of the $y$ radial coordinate, (6.7) becomes

$\tilde{\ell}(y)=\frac{1}{2\pi M^{2}_{\text{\tiny
P}}L^{2}}\sqrt{\frac{n}{3}}\cos\left(\sqrt{\frac{3}{n}}\,y\right)\,.$ (D.33)

From (6.13), (6.8) and (6.6), we get

$e^{f(y)}=\left[\frac{\tan\left(\frac{\pi}{4}-\frac{1}{2}\sqrt{\frac{3}{n}}\,y\right)}{\tan\left(\frac{\pi}{4}-\frac{\pi}{4}\sqrt{\frac{3}{n}}\right)}\right]^{\frac{n}{4}}\,.$
(D.34)

Using (D.31) inside (D.3) one gets the corresponding (unnormalized) Goldstino
zero-modes,

$\displaystyle\begin{aligned} \delta\chi^{i}_{\text{\tiny
R}}&=\frac{2\sqrt{2}}{L}\,\partial_{y}t^{i}\left(\cos
y\right)^{3/2}\sigma^{4}\epsilon_{\text{\tiny L}}\,,\\\
\delta\chi^{i}_{\text{\tiny
L}}&=\frac{2\sqrt{2}}{L}\,\partial_{y}\bar{t}^{i}\left(\cos
y\right)^{3/2}\bar{\sigma}^{4}\epsilon_{\text{\tiny R}}\,.\end{aligned}$
(D.35)

These explicitly show that, in general, no supersymmetry is preserved by non-
extremal wormhole configurations. We can further verify this by inserting the
profile of $t^{i}$ on the homogeneous solution into (D.35), obtaining

$\displaystyle\begin{aligned} \delta\chi^{i}_{\text{\tiny
R}}&=-\text{i}\frac{6\sqrt{2}\pi M^{2}_{\text{\tiny
P}}L}{n}\left[1-\sin\left(\sqrt{\frac{3}{n}}y\right)\right]\frac{\left[\cos
y\cos\left(\frac{y}{2}-\frac{\pi}{4}\right)\right]^{3/2}}{\left[\cos\left(\sqrt{\frac{3}{n}}y\right)\right]^{2}}e^{-f(y)}\,\sigma^{4}\eta_{\text{\tiny
L}}\,,\\\ \delta\chi^{i}_{\text{\tiny L}}&=-\text{i}\frac{6\sqrt{2}\pi
M^{2}_{\text{\tiny
P}}L}{n}\left[1+\sin\left(\sqrt{\frac{3}{n}}y\right)\right]\frac{\left[\cos
y\cos\left(\frac{y}{2}-\frac{\pi}{4}\right)\right]^{3/2}}{\left[\cos\left(\sqrt{\frac{3}{n}}y\right)\right]^{2}}e^{+f(y)}\,\bar{\sigma}^{4}\eta_{\text{\tiny
R}}\,,\\\ \end{aligned}$ (D.36)

with $e^{f(y)}$ as in (D.34). For $n>3$ these expressions are regular and
behave as $\sim r^{-3}$ as we approach the asymptotic flat space, like a free
Feynman propagator analogously to (D.15). The case $n=3$ corresponds to the
marginally degenerated case and will be discussed below.

###  D.3 Marginally degenerate wormholes

In Section (6.4) we have regularized the $n=3$ marginally degenerate wormholes
by introducing an IR cutoff $r_{\text{\tiny IR}}$. As discussed in Section
7.4, this prescription receives a natural justification if one aims at finding
the effective operator encoding the wormhole effects at distances larger than
$\Lambda^{-1}_{\text{\tiny IR}}\gg L$. In Section 6.4 we have also observed
how the marginally degenerate wormholes look very similar to extremal BPS
wormholes carrying the same charge. The deviation from the extremal case is
concentrated around the wormhole’s neck. It is then clear that, at radii
slightly larger than $L$, the marginally degenerate wormhole should admit, to
very good approximation, two Killing spinors $\bar{\epsilon}^{\dot{\alpha}}$
of the form (D.10). This approximation of course breaks down as we approach
the throat. However, the corresponding zero-modes $\delta\chi^{i}_{\text{\tiny
R}}$ are necessarily localized around the wormhole throat and quickly vanish
as we approach $r_{\text{\tiny IR}}$.

In order to more explicitly check this qualitative expectation, it is
convenient to use the alternative (smooth) regularization ${\cal
F}=\log\tilde{P}\rightarrow(1+\varepsilon)\log\tilde{P}$, with $\varepsilon\ll
1$, which amounts to setting

$n=3\left(1+\varepsilon\right)\,.$ (D.37)

In order to illustrate the form of the zero-modes, let us focus for simplicity
on the homogeneous case. The regularization implies a finite

$\tilde{\ell}_{\infty}=\frac{\pi}{4}\varepsilon\tilde{\ell}_{*}\,.$ (D.38)

Recalling the discussion in section 6.4 we may pick
$\varepsilon=\frac{4}{\pi}\epsilon_{\Lambda}=\frac{4}{\pi}\Lambda_{\text{\tiny
IR}}^{2}L^{2}$ so that we can identify $\tilde{\ell}_{\infty}$ with
$\tilde{\ell}_{\Lambda}$, formally identifying a correspondence between the
$\Lambda_{\text{\tiny IR}}$ and $\varepsilon$ regularization procedures. In
this way, Eq. (D.34) becomes

$e^{-f(y,\varepsilon)}\simeq\left[\frac{\pi\varepsilon}{8}\tan\left(\frac{\pi}{4}+\frac{y}{2\sqrt{1+\varepsilon}}\right)+{\cal
O}(\varepsilon^{2})\right]^{\frac{3}{4}}\,.$ (D.39)

so that $\delta\chi_{\text{\tiny R}},\delta\chi_{\text{\tiny L}}$ of (D.36)
are now well defined in the whole domain of $y$. The quantitative expectation
that the modes associated to $\delta\chi_{\text{\tiny R}}$ become unobservable
far from the wormhole throat can then be verified by inspecting the functional
dependence on $y$ of the regularized wavefunctions associated to the zero-
modes. In particular, their ratio is given by

$\displaystyle\left[\frac{1-\sin\left(\frac{y}{\sqrt{1+\varepsilon}}\right)}{1+\sin\left(\frac{y}{\sqrt{1+\varepsilon}}\right)}\right]e^{-2f(y,\varepsilon)}.$
(D.40)

Taking $y=\pi/2$, the above expression becomes

$\displaystyle\left[\frac{1-\sin\left(\frac{\pi}{2\sqrt{1+\varepsilon}}\right)}{1+\sin\left(\frac{\pi}{2\sqrt{1+\varepsilon}}\right)}\right]e^{-2f(\pi/2,\varepsilon)}\approx\frac{\pi^{2}\varepsilon^{2}}{64}=\frac{\Lambda_{\text{\tiny
IR}}^{4}L^{4}}{4}\,,$ (D.41)

where in the last step we exploited the correspondence between the two
regularization procedures. Already at distances $1/\Lambda_{\text{\tiny
IR}}=5L$ this gives a relative ${\cal O}(10^{-4})$ relative suppression. Hence
from a IR viewpoint only the zero-mode $\delta\chi_{\text{\tiny L}}$ is
observable. At $y=-\pi/2$ the situation is reversed, with the zero-mode
$\delta\chi_{\text{\tiny L}}$ being suppressed in the IR. This is consistent
with the identification of the $y=-\pi/2$ side as an anti-extremal BPS
wormhole conserving the opposite supercharges.

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