d}a^{i}$ corresponds to the 1-form current associated to the axion shift
symmetry, the quantities $q_{i}$ are to be interpreted as the wormhole charges

$\frac{1}{2\pi}\int_{S^{3}}{\cal H}_{3\,i}=q_{i}\in\mathbb{Z}\,.$ (5.8)

Charge quantization can be either seen as a consequence of the axion
periodicity (2.1) (i.e. the momentum conjugate to a periodic variable is
quantized) or, in the dual language, to the quantization of the ${\cal
H}_{3\,i}$ field-strengths.

The $SO(4)$ symmetry ensures that our ansatz can be regarded as a consistent
truncation of the full theory. Inserting it in Eq. (5.1), the action reduces
to

$\displaystyle\left.S\right|_{\rm ansatz}=$
$\displaystyle\,-6\pi^{2}\int_{I}{\rm d}\rho\left[e^{A+B}+\left(\frac{{\rm
d}B}{{\rm
d}\rho}\right)^{2}e^{3B-A}\right]+6\pi^{2}\left[e^{2B}\right]_{\partial I}$
(5.9) $\displaystyle+\int_{I}{\rm d}\rho\,\left[e^{3B-A}\pi^{2}{\cal
G}^{ij}\,\frac{{\rm d}\ell_{i}}{{\rm d}\rho}\frac{{\rm d}\ell_{j}}{{\rm
d}\rho}+e^{A-3B}{\cal G}^{ij}\,q_{i}q_{j}\right]\,.$

Extremizing (5.9) with respect to $A$ we get the constraint

$e^{6B-2A}\pi^{2}{\cal G}^{ij}\,\frac{{\rm d}\ell_{i}}{{\rm d}\rho}\frac{{\rm
d}\ell_{j}}{{\rm d}\rho}-{\cal G}^{ij}\,q_{i}q_{j}=6\pi^{2}\left(\frac{{\rm
d}B}{{\rm d}\rho}\right)^{2}e^{6B-2A}-6\pi^{2}\,e^{4B}\,.$ (5.10)

The equations of motion for $\ell_{i}$ are derived from the second line of
(5.9) and will be shown shortly. The variation of (5.9) with respect to $B$ is
on the other hand redundant once the former two conditions are satisfied, and
hence will not be discussed.

To write the field equations for $\ell_{i}$ it is convenient to introduce [74]
a ‘proper’ radial coordinate $\tau$ such that

${\rm d}\tau=\pm\frac{1}{\pi}e^{A-3B}{\rm d}\rho\,,$ (5.11)

where the two signs correspond to the two possible relative orientations
between $\tau$ and $\rho$. With such a choice, the second line of (5.9)
resembles the action for a particle with non-canonical kinetic term moving in
a non-trivial potential:

$2\pi\int{\rm d}\tau\left[\frac{1}{2}{\cal
G}^{ij}(\ell)\,\dot{\ell}_{i}\dot{\ell}_{j}-V_{\bf q}({\ell})\right]$ (5.12)

with $\dot{\ell}_{i}\equiv\frac{{\rm d}\ell_{i}}{{\rm d}\tau}$. The potential
is

$V_{\bf q}({\ell})\equiv-\frac{1}{2}{\cal
G}^{ij}({\ell})\,q_{i}q_{j}\equiv-\frac{1}{2}\|{\bf q}\|^{2}\,$ (5.13)

where we have introduced the norm defined by the metric (2.14):

$\|{\bf q}\|^{2}\equiv{\cal G}^{ij}(\ell)\,q_{i}q_{j}\,.$ (5.14)

The (dimensionless) “particle” energy

$E\equiv\frac{1}{2}{\cal G}^{ij}({\ell})\,\dot{\ell}_{i}\dot{\ell}_{j}+V_{\bf
q}({\ell})$ (5.15)

is thus manifestly conserved. From (5.12) – or equivalently (5.9) – we may now
obtain the equations of motion for $\ell_{i}$

${\cal G}^{ij}\frac{D\dot{\ell}_{j}}{{\rm d}\tau}=-\frac{\partial V_{\bf
q}}{\partial\ell_{i}}\quad\Leftrightarrow\quad{\cal
F}^{ij}\ddot{\ell}_{j}=\frac{1}{2}{\cal
F}^{ijk}\left(q_{j}q_{k}-\dot{\ell}_{j}\dot{\ell}_{k}\right)\,,$ (5.16)

where ${D\dot{\ell}_{j}}/{{\rm d}\tau}$ is the Levi-Civita covariant
derivative associated with the metric ${\cal G}^{ij}$; we have used (2.14) and
we have introduced the shorthand notation ${\cal F}^{i}\equiv{\partial{\cal
F}}/{\partial\ell_{i}}$, ${\cal F}^{ij}\equiv{\partial^{2}{\cal
F}}/{\partial\ell_{i}\partial\ell_{j}}$, etc. In addition the dual saxions
$\ell_{i}$ satisfy appropriate boundary conditions, which we assume to be
Dirichlet.181818In presence of low-energy supersymmetry breaking, the
asymptotic values ${\bm{\ell}}_{\infty}$ may ultimately be determined by an
hypothetical stabilizing potential for the saxions. We will come back to this
in Section 7.5.

We are interested in asymptotically flat wormhole configurations manifesting a
Euclidean “time-reversal” symmetry.191919The Euclidean time-reversal symmetry
ensures that by cutting these solutions at the minimal radius one gets half-
wormholes which, once analytically continued back to a Lorentzian spacetime,
describe the nucleation or absorption of a baby universe, with real boundary
values of the fields and of their time derivatives [9]. We can take a
symmetric $\tau$ interval, $\tau\in[-\tau_{\infty},\tau_{\infty}]$, with both
endpoints $\tau=\pm\tau_{\infty}$ corresponding to $r=\infty$. The dual
saxions can then flow from some asymptotic value ${\bm{\ell}}_{\infty}$ at
$\tau=-\tau_{\infty}$ to some value ${\bm{\ell}}_{*}$ at the neck of the
wormhole $\tau=0$, and eventually back to the asymptotic value
${\bm{\ell}}_{\infty}$ at $\tau=\tau_{\infty}$. The analogy with the point
particle makes it clear that the saxionic flow can be interpreted as a
scattering process, in which the particle climbs the potential (5.13) until it
reaches the turning point ${\bm{\ell}}_{*}$, at which it stops and then rolls
down back to its original position. Since $V_{\bf q}<0$, the saxions can
bounce only if $E=-|E|<0$. We will however be a bit more general and consider

$E=-|E|\leq 0\,.$ (5.17)

We will refer to the configurations with $E<0$ and $E=0$ as non-extremal and
extremal wormholes, respectively, even though strictly speaking the extremal
configurations do not describe proper wormholes connecting two asymptotically
flat regions, see Section 5.3 below for more details. (Rather, they can be
formally considered as half-wormholes in the extreme $E\rightarrow 0$ limit.)
In the following we will concentrate on these cases, without considering the
configurations with $E>0$, which have a singular geometry. See for instance
the review papers [10, 83, 84] for additional discussions on these three
cases.

Inserting (5.15) in the constraint (5.10), and writing the result in the gauge
(5.5) one gets

$e^{2A}=\frac{1}{1-\frac{L^{4}}{r^{4}}}$ (5.18)

where

$L^{4}\equiv\frac{|E|}{3\pi^{2}M^{4}_{\text{\tiny P}}}=\frac{\|{\bf
q}\|^{2}_{*}}{6\pi^{2}M^{4}_{\text{\tiny P}}}\,,$ (5.19)

with $\|{\bf q}\|^{2}_{*}\equiv\|{\bf q}\|^{2}({\ell}_{*})$. Hence the metric
takes the form

${\rm d}s^{2}=\frac{1}{1-\frac{L^{4}}{r^{4}}}{\rm d}r^{2}+r^{2}{\rm
d}\Omega_{3}^{2},$ (5.20)

with $r\in[L,\infty)$ and $L$ can be identified with the minimal $S^{3}$
radius $r_{*}=L$. In the second equality in (5.19) we used the fact that, if
$E<0$, at the turning point of the wormhole throat the particle stops
($\dot{\ell}_{i}|_{\tau=0}=0$) and $E$ reduces to $V_{\bf q}(\ell_{*})$. We
see that the minimal $S^{3}$ radius of non-extremal wormholes is controlled by
the charge norm $\|{\bf q}\|$ at such radius. If instead $E=0$, then $L=0$ and
the metric becomes flat.

By using (5.18) in (5.11) (in the gauge (5.5)) one gets a differential
relation between $r$ and $\tau$, which can be easily integrated. For non-
extremal ($E<0$) wormholes, the integration constant can be fixed by imposing
that

$\tau=0\quad~{}~{}~{}~{}\Leftrightarrow\quad~{}~{}~{}~{}r=L\,.$ (5.21)

Then (5.11), with $+/-$ sign for positive/negative $\tau$, integrates to

$\tau=\pm\frac{1}{2\pi M^{2}_{\text{\tiny
P}}L^{2}}\left[\frac{\pi}{2}-\arcsin\left(\frac{L^{2}}{r^{2}}\right)\right]$
(5.22)

or similarly ${L^{2}}/{r^{2}}=\cos\left(2\pi M^{2}_{\text{\tiny
P}}L^{2}\tau\right)$. This in particular implies that $e^{-2A}$ as a function
of $\tau$ is given by $e^{-2A}={\sin^{2}\left(2\pi M^{2}_{\text{\tiny
P}}L^{2}\tau\right)}$. The maximal extension of the $\tau$ interval is
$|\tau|\leq\tau_{\infty}$, with

$\tau_{\infty}\equiv\frac{1}{4M^{2}_{\text{\tiny
P}}L^{2}}\quad~{}~{}~{}~{}\Leftrightarrow\quad~{}~{}~{}~{}r_{\infty}=\infty\,,$
(5.23)

where $\cos\left(2\pi M^{2}_{\text{\tiny P}}L^{2}\tau_{\infty}\right)=0$. In
particular, the region $0\leq\tau\leq\tau_{\infty}$ parametrizes the first
half-wormhole, while the interval $-\tau_{\infty}\leq\tau\leq 0$ parametrizes
the second half-wormhole. In other words, the particle takes a proper time
$\tau_{\infty}$ to make half of its journey.

The particle interpretation also suggests a simple characterization of regular
wormholes. The point particle is at rest at $\tau=0$ and starts rolling down
the potential $V_{\bf q}$. If it reaches $\tau_{\infty}$ remaining inside the
perturbative dual saxionic domain, then the corresponding trajectory describes
an everywhere regular on-shell wormhole. Whenever along its path the particle
exits the allowed domain, the solution develops a singularity and should
either be somehow regularized or discarded.

Finally, the on-shell wormhole action can be obtained by plugging (5.10) into
(5.9), using the gauge (5.5) and the explicit solution (5.18). Alternatively,
we can use the traced Einstein equation

$R\,*1={\cal G}^{ij}\left({\rm d}\ell_{i}\wedge*{\rm
d}\ell_{j}-\frac{1}{M^{2}_{\text{\tiny P}}}{\cal H}_{3\,i}\wedge*{\cal
H}_{3\,j}\right)\,$ (5.24)

in (5.1), observing that the asymptotic Gibbons-Hawking boundary term vanishes
for the metric (5.20). In either case, one obtains the wormhole action

$\left.S\right|_{\rm w}=2\pi\int{\rm d}\tau\,\|{\bf
q}\|^{2}=\frac{1}{M_{\text{\tiny P}}^{2}}\int{\cal G}^{ij}{\cal
H}_{3,i}\wedge*{\cal H}_{3,j}\,.$ (5.25)

As argued in [9] (see also Section 7), the physically relevant quantity is
$1/2$ of $\left.S\right|_{\rm w}$, namely the on-shell action of half-
wormhole:

$S|_{\text{hw}}=2\pi\int_{0}^{\tau_{\infty}}{\rm d}\tau\,\|{\bf q}\|^{2}\,.$
(5.26)

The latter can also be written using (5.13), (5.19), and (5.23) in an
alternative form:

$\displaystyle S|_{\text{hw}}$
$\displaystyle=-4\pi\int_{0}^{\tau_{\infty}}{\rm d}\tau\,V_{\bf q}$ (5.27)
$\displaystyle=3\pi^{3}M^{2}_{\text{\tiny
P}}L^{2}+4\pi\int_{0}^{\tau_{\infty}}{\rm d}\tau\,(E-V_{\bf q})$
$\displaystyle\geq 3\pi^{3}M^{2}_{\text{\tiny P}}L^{2}\,.$

The first term in the second line coincides with what one would find for a
purely axionic theory, with dual saxions “frozen” at their throat values
$\ell_{i,*}$.202020Up to a different convention for the Planck scale, the
semi-wormhole action (5.27) agrees with the one first derived in [9]. It also
agrees with [85] provided no Gibbons-Hawking term is added at the inner
boundary of the half-wormhole. The second contribution can be identified with
the integrated “kinetic” energy of the dual saxions, see (5.15), and is hence
always positive.

The on-shell action (5.27) does not include the imaginary contribution of the
boundary term (2.16). These are generically non-vanishing and will be taken
into account in Section 7.

We remind the reader that the results obtained in this section rely on the
assumption that a truncation of the action at the 2-derivative level is
justified. As emphasized in Section 3.4 this requires $L>1/\Lambda$, where
$\Lambda$ is an appropriate UV cutoff. A reasonable bound would be
$\Lambda<M_{\rm t}$, since the tower scale $M_{\rm t}$ represents the highest
possible energy at which our four-dimensional EFT makes sense. However, in
principle a wormhole with $L\lesssim M^{-1}_{\rm t}$ could still make sense in
a weakly coupled higher-dimensional EFT. This is certainly not possible if
$L^{-1}$ exceeds the species scale $M_{\text{sp}}$. So, we will impose the
less conservative bound

$L^{2}>M^{-2}_{\text{sp}}$ (5.28)

as the most fundamental consistency condition. Note that this requirement can
always be satisfied by rescaling ${\bf q}$, so to increase $\|{\bf q}\|$.

The analysis of this section may also require the introduction of an IR
cutoff, as discussed in general at the end of Section 3.4. Suppose for example
that a more complete description includes a small saxion potential $V_{\rm
s}(s)$, see for instance [86, 87] for recent discussions and more references,
as well as Section 7.5. This would contribute to the equations of motion and
to the on-shell action. A sufficient condition for these corrections to be
negligible and not to significantly affect our results is that $r^{6}V_{\rm
s}\ll M_{\text{\tiny P}}^{2}L^{4}$. If the overall size of the potential is
$V_{\rm s}\ll M_{\text{\tiny P}}^{2}/L^{2}$, such condition is certainly
satisfied sufficiently close to the wormhole throat. In the same regime our
equations of motion are reliable as well. Yet, for $r>M^{-1}_{\text{\tiny
IR}}\sim(M_{\text{\tiny P}}^{2}L^{4}/V_{\rm s})^{1/6}$ both the classical
configuration and its action can significantly depart from what we have found.
We should thus either stay away from the asymptotic region, or properly take
into account the non-trivial potential.

###  5.3 Extremal wormholes and fundamental BPS instantons

So far in this section we have not really exploited supersymmetry, and what we
have found would hold even if the kinetic matrices of $\ell_{i}$ and ${\cal
H}_{3,i}$ were different from each other (see for instance the recent work
[88]). On the other hand supersymmetry forces such matrices to be identical,
see (5.1), and this allows for a particularly simple class of BPS extremal
$E=0$ configurations. These solutions can be considered a generalization of
the extremal wormholes found in [89] – see also [90] for their ten-dimensional
counterpart.

Given the form of the potential (5.13), we see that taking
$\dot{\ell}_{i}=-q_{i}$ obviously satisfies (5.15) with $E=0$ (another
possible choice is $\dot{\ell}_{i}=q_{i}$ and can be viewed as being
associated to conjugate charges). The equations of motion (5.16) are
manifestly solved as well. The condition can be easily integrated to
${\bm{\ell}}(\tau)={\bm{\ell}}(\tau_{\infty})+{\bf q}(\tau_{\infty}-\tau)$.
The wormhole metric (5.6) is flat in the extremal case $E=0$, and in the gauge
(5.5) our solution has $e^{2A}=1$ with the coordinate $r\geq 0$ spanning the
entire space. In flat space (5.7) implies that ${\cal
H}_{3,i}=-M^{2}_{\text{\tiny P}}*{\rm d}\ell_{i}$ and ${\rm d}{\cal
H}_{3,i}=2\pi\delta_{4}(0)$. As discussed in Appendix D.1, these relations can
be generalized to BPS multi-centered extremal wormhole solutions.

When considering extremal wormholes it is more convenient introduce a new
proper radial coordinate

$\hat{\tau}\equiv\tau_{\infty}-\tau\,,$ (5.29)

which has opposite orientation with respect to $\tau$ and is such that
$r=\infty$ corresponds to $\hat{\tau}_{\infty}=0$, while $r=0$ corresponds to
$\hat{\tau}_{*}=\tau_{\infty}$. This is motivated by the fact that in the
extremal solution $r=0$ will soon be shown to correspond to a singular point,
which is then moved to infinite $\hat{\tau}$-distance, and this simplifies our
presentation. By solving (5.11) one obtains

$\hat{\tau}=\frac{1}{2\pi M^{2}_{\text{\tiny P}}r^{2}}\,$ (5.30)

and

${\bm{\ell}}(r)={\bm{\ell}}_{\infty}+{\bf
q}\hat{\tau}={\bm{\ell}}_{\infty}+\frac{{\bf q}}{2\pi M^{2}_{\text{\tiny
P}}r^{2}}\,.$ (5.31)

As promised, the solution is singular at $r=0$. The nature of the singularity
will be explored shortly. Similarly, the conditions on ${\bm{\ell}}_{\infty}$
and ${\bf q}$ necessary for (5.31) to lie within the allowed saxion domain
identified in Section 3.1 will be analyzed in detail below.

For the moment let us point out a suggestive fact. By recalling (2.11), one
can easily derive the identity ${\cal G}^{ij}q_{i}q_{j}={\cal
G}^{ij}q_{i}\frac{{\rm d}\ell_{j}}{{\rm d}\hat{\tau}}=-q_{i}\frac{{\rm
d}s^{i}}{{\rm d}\hat{\tau}}$. Thus, paying attention to the relation (5.29),
the on-shell action (5.26) of our extremal solution reads212121The coordinate
$r$ here describes the whole space. Hence the notation “half extremal
wormhole” may be a bit misleading.

$S|_{\text{hw-extr.}}=2\pi\int_{\hat{\tau}_{\infty}}^{\hat{\tau}_{*}}{\rm
d}\hat{\tau}\,{\cal
G}^{ij}q_{i}q_{j}=-2\pi\int_{\hat{\tau}_{\infty}}^{\hat{\tau}_{*}}{\rm
d}\hat{\tau}\,q_{i}\frac{{\rm d}s^{i}}{{\rm d}\hat{\tau}}=2\pi
q_{i}s^{i}_{\infty}-2\pi q_{i}s^{i}_{*}\,.$ (5.32)

This action reminds us of that of a BPS instanton, which appears in the
exponential of (3.3) (with $s^{i}$ replacing $s^{i}_{\infty}$), but differs
from it because of the term $-2\pi q_{i}s^{i}_{*}$. Yet, inspecting (5.31) we
see that the dual saxions diverge as $r\rightarrow 0$. If the Kähler potential
is as in (2.5), this implies that ${\bm{s}}_{*}\to 0$ in this limit. So,
apparently, the two actions do in fact coincide. Note that this in particular
implies that the charge vector ${\bf q}=\\{q_{i}\\}$ should belong to the set
${\cal C}_{\rm I}$ of BPS instanton charges introduced in (3.5). These
observations suggest that extremal wormholes be somehow related to fundamental
BPS instantons. In order to clarify this relation we need to understand how to
interpret the singularity.

The singularity at $r=0$ indicates that the expression (5.31) is not fully
reliable. From a genuinely EFT perspective, that solution should be
interpreted as viable only in a region $r\geq 1/\Lambda$ away from the origin,
where $\Lambda$ lies in the range (3.30) and a derivative expansion is
meaningful. Below the UV cutoff the singularity should be regularized by some
local counterterm placed around $r=0$. Restricting the integral in (5.32) to
the reliable region $r\geq 1/\Lambda$ we obtain the on-shell contribution

$S^{\Lambda}_{\rm bulk}=2\pi q_{i}(s^{i}_{\infty}-s^{i}_{\Lambda})\,,$ (5.33)

where we defined

$s^{i}_{\Lambda}\equiv s^{i}(r=\Lambda^{-1})\,.$ (5.34)

That action should then be supplemented by a cutoff-dependent localized term
$S_{\rm loc}^{\Lambda}$ accounting for the effect of the unknown physics at
scales above $\Lambda$. Such a localized term cannot be determined by sole
considerations of the low-energy observer. Fortunately we have crucial
information about the UV. The analysis presented in Appendix E of [14] shows
that fundamental BPS instantons localized at $r\sim 1/\Lambda$ are captured
within the EFT precisely by the introduction of a localized term of the form

$S^{\Lambda}_{\rm loc}=2\pi q_{i}s^{i}_{\Lambda}=2\pi\langle{\bf
q},{\bm{s}}_{\Lambda}\rangle\,,$ (5.35)

along with its supersymmetric counterpart $-2\pi{\rm i}\langle{\bf
q},{\bm{a}}_{\Lambda}\rangle$. Hence the real part of the complete on-shell
action for an extremal wormhole, including the localized term, should be given
by

$S_{\text{\tiny BPS}}=S^{\Lambda}_{\rm bulk}+S^{\Lambda}_{\rm loc}=2\pi
q_{i}s^{i}_{\infty}\equiv 2\pi\langle{\bf q},{\bm{s}}_{\infty}\rangle$ (5.36)

and exactly reproduces the expression appearing in (3.3).

The BPS fundamental instanton acts as a magnetic source for the potentials
${\cal B}_{2,i}$, as codified by the modified Bianchi identity ${\rm d}{\cal
H}_{3,i}=2\pi\delta_{4}(0)$. In the dual axionic formulation, this arises from
the localized imaginary term written below (5.35), see Appendix E of [14] for
more details. Keeping track of the boundary term (2.16) arising in the duality
transformation, one find that the BPS action, including its imaginary part,
reads

$S_{\text{\tiny BPS}}-2\pi{\rm i}\langle{\bf
q},{\bm{a}}_{\infty}\rangle=-2\pi{\rm i}\langle{\bf
q},{\bm{t}}_{\infty}\rangle\,,$ (5.37)

which is holomorphic, as expected by supersymmetry. Extremal wormholes are
therefore a low-energy description of BPS fundamental instantons.

Not all BPS fundamental instantons can be reliably described by extremal
wormholes within the EFT, though. Those that admit such a description will be
called EFT instantons, as in [14]. This class is the subject of the next
subsection.

###  5.4 EFT instantons

EFT instantons are BPS instantons whose charges belong to the set

${\cal C}_{\rm I}^{\text{\tiny EFT}}\equiv{\cal P}\cap{\cal C}_{\rm I}\,,$
(5.38)

where ${\cal P}$ is the dual saxionic domain (3.8). A trivial example is
provided by the model of Section 4.1.1, in which ${\cal C}_{\rm
I}^{\text{\tiny EFT}}={\cal C}_{\rm I}$ is generated by the hyperplane divisor
$D$. As a less trivial example, in the model of Section 4.1.2 ${\cal C}_{\rm
I}$ is generated by the effective divisors $E^{1},E^{2}$, while ${\cal C}_{\rm
I}^{\text{\tiny EFT}}\subset{\cal C}_{\rm I}$ is generated by the nef divisors
$D^{1},D^{2}$. In more general F-theory models satisfying the conditions of
footnote 14, the set of EFT instantons corresponds microscopically to
D3-branes wrapping movable divisors. If one instead considers the heterotic
models of Section 4.2, according to (4.49) EFT instantons are represented by
F1-strings wrapping movable curves in ${\cal P}^{\rm het}_{\text{\tiny K}}$,
and to NS5-branes wrapping the entire internal Calabi-Yau and possibly
supporting additional F1-charge.

One reason for the requirement (5.38) is the following. In the perturbative
models in which the Kähler potential takes the form (2.5), ${\cal P}$ has
conical shape. Since (5.31) describes a straight dual saxionic line generated
by ${\bf q}$, any ${\bf q}\in{\cal C}_{\rm I}^{\text{\tiny EFT}}$ identifies
profiles completely contained in ${\cal P}$. In general ${\cal P}$ may be non-
convex, and in that case ${\bf q}\in{\cal C}_{\rm I}^{\text{\tiny EFT}}$ is
not enough to ensure that the trajectory (5.31) is completely contained in
${\cal P}$ for any ${\bm{\ell}}_{\infty}\in{\cal P}$. Yet, if ${\bf q}$ is in
the interior of ${\cal P}$ and ${\bm{\ell}}_{\infty}\propto{\bf q}$ then the
solution certainly exists. By continuity we thus expect the same to be true
for nearby choices of ${\bm{\ell}}_{\infty}$. We conclude that for any ${\bf
q}\in{\cal C}_{\rm I}^{\text{\tiny EFT}}$ one can judiciously adjust
${\bm{\ell}}_{\infty}$ in order to get extremal solutions belonging to ${\cal
P}$.

Yet, the discussion of Section 5.3 seems to suggest that any BPS instanton
charge ${\bf q}\in{\cal C}_{\rm I}$, either EFT or non-EFT, can be associated
with a corresponding extremal wormhole provided $r>1/\Lambda$. As we will now
show, for charges in the domain (5.38) there is no obvious obstruction to
extending our solutions all the way to $r\sim 1/M_{\text{sp}}$. Extremal
wormholes carrying non-EFT charge can instead reach a finite distance boundary
of ${\cal P}$ at some finite radial distance, uncorrelated to the species
scale. Hence, in a sense, non-EFT instantons are associated to strong coupling
effects of non-gravitational nature. This does not happen if ${\bf q}$ belongs
to (5.38). In the following we will offer a few arguments motivating this
claim.

Let us first make some preliminary remark on the tensions of BPS strings.
Recall that tensionless strings signal field space boundaries of ${\cal P}$:
tensionless EFT strings are associated with boundaries at infinite field space
distance, while finite distance boundaries may be detected by tensionless BPS
but non-EFT strings. As one approaches either of these two types of
boundaries, the EFT becomes strongly coupled, but in a different way. To see
this difference, recall that the semiclassical description of BPS strings
roughly requires their tension to be larger than the EFT cutoff $\Lambda$:

$\Lambda^{2}<2\pi{\cal T}_{\bf e}\,.$ (5.39)

The violation of (5.39) by EFT and non-EFT strings has different implications,
though. Indeed, in the former case, the bound (3.28) would imply that

$\Lambda^{2}\geq M^{2}_{{\cal T}}\geq M^{2}_{\rm sp}\,.$ (5.40)

That is, the violation of (5.39) by an EFT string implies that the theory has
entered a phase in which gravity does not admit any semiclassical
gravitational description. On the other hand, the tension of an elementary BPS
but non-EFT string is not directly correlated with the species scale, and can
be much smaller. So, the violation of (5.39) by some elementary BPS but non-
EFT string signals an EFT break-down of non-gravitational nature. Phases of
this type have been for instance studied in [91].

Let us now use this consideration to better appreciate the distinction between
EFT and non-EFT string instanton charges. Take first a BPS instanton charge
${\bf q}$ that is not EFT, i.e. that does not belong to (5.38). Recalling
(3.9), this means that there may exist a BPS but non-EFT string charge ${\bf
e}$ such that $\langle{\bf q},{\bf e}\rangle<0$.222222This certainly happens
only if ${\cal P}={\cal C}_{\rm S}^{\vee}$. More generically ${\cal
P}\subset{\cal C}_{\rm S}^{\vee}$, and there may be BPS instanton charges
${\bf q}$ in ${\cal C}_{\rm S}^{\vee}-{\cal P}$, which would then have
$\langle{\bf q},{\bf e}\rangle\geq 0$ for any ${\bf e}\in{\cal C}_{\rm S}$. In
such a case, the corresponding tension decreases along the flow (5.31), as one
approaches the instanton:

${\cal T}_{\bf e}(r)={\cal T}^{\infty}_{\bf e}-\frac{|\langle{\bf q},{\bf
e}\rangle|}{2\pi r^{2}}\,.$ (5.41)

Hence ${\cal T}_{\bf e}(r)$ violates (5.39) at a critical radius

$r^{2}_{\rm cr}=\frac{|\langle{\bf q},{\bf e}\rangle|}{2\pi{\cal
T}^{\infty}_{\bf e}-\Lambda^{2}}\,.$ (5.42)

Since $2\pi{\cal T}^{\infty}_{\bf e}-\Lambda^{2}$ can be much smaller than
$M^{2}_{\rm sp}$, the theory can enter a non-gravitationally strongly coupled
phase at distances $r_{\rm cr}\gg M^{-1}_{\rm sp}$. If ${\bf q}$ is not EFT
and nevertheless $\langle{\bf q},{\bf e}\rangle\geq 0$ for any ${\bf
e}\in{\cal C}_{\rm S}$, we cannot run this argument, but we still expect the
radial flow to reach some finite distance boundary of ${\cal P}$ at some
finite $r_{\rm cr}$ uncorrelated to $M_{\rm sp}$. It would be very interesting
to more thoroughly investigate the possible non-perturbative effects
associated with the various non-EFT instantons.

Suppose instead that ${\bf q}$ belongs to (5.38). In this case the condition
$\langle{\bf q},{\bf e}\rangle\geq 0$ is guaranteed. Hence, not only can one
always find extremal wormholes that never exit ${\cal P}$, but it is also
guaranteed that any BPS (either EFT or non-EFT) string tension always
increases as one approaches $r=0$,

${\cal T}_{\bf e}(r)={\cal T}^{\infty}_{\bf e}+\frac{\langle{\bf q},{\bf
e}\rangle}{2\pi r^{2}}\geq{\cal T}^{\infty}_{\bf e}\,.$ (5.43)

Therefore, if (5.39) is asymptotically satisfied by ${\cal T}^{\infty}_{\bf
e}$, then it will hold true for any $r$ and the theory will never enter a
phase in which the semiclassical EFT description necessarily breaks down.

While along an EFT instanton flow there do not appear dangerous light string
tensions, the theory formally exits the perturbative domain $\hat{\cal
P}_{\alpha}$ as $r\rightarrow 0$, since the dual saxions (5.31) diverge in the
${\bf q}$-direction. This is signaled also by the divergence of BPS string
tensions, which should instead satisfy the condition (3.10) in order not to
have a too strong backreaction. In particular, (3.10) should be satisfied by
all elementary EFT string charges. (At the same time, (3.10) determines an
upper bound on the string charges which can admit a weakly coupled world-sheet
description.) The formal divergence of the string tensions as $r\rightarrow 0$
is not really a problem, since one should actually restrict the flow to
$r\geq\Lambda^{-1}>M^{-1}_{\rm sp}$. Combined with (5.43), this restriction
implies that

${\cal T}_{\bf e}(r)\leq{\cal T}_{\bf e}^{\infty}+\frac{\langle{\bf q},{\bf
e}\rangle}{2\pi}\Lambda^{2}\,.$ (5.44)

Hence, if (3.10) is asymptotically satisfied, then it is satisfied for any
$r\geq\Lambda^{-1}$ provided

$\langle{\bf q},{\bf e}\rangle<\frac{(2\pi M_{\text{\tiny
P}})^{2}}{\Lambda^{2}}\,.$ (5.45)

Since $\Lambda\ll 2\pi M_{\text{\tiny P}}$, this condition can be easily
satisfied if the charges are not too large. This shows that the condition
(3.10) is certainly satisfied by the elementary EFT instantons.

So far we have thus shown that, approaching an EFT instanton string, tensions
never vanish but rather diverge, and this latter behavior does not signal a
loss of calculability. We next proceed our discussion by comparing the
dominant EFT string scale (3.29) to $r^{-1}$, which represents the energy
scale probed by the solution (5.31). From (5.31) it is easy to see that, as
one approaches $r=0$, $M_{{\cal T}}$ is identified by the tension of an EFT
string charge ${\bf e}$ with minimal possible paring $\langle{\bf q},{\bf
e}\rangle$. We will denote such charge as ${\bf e}_{\bf q}$. If $\langle{\bf
q},{\bf e}_{\bf q}\rangle\geq 1$, then $M_{{\cal T}}$ is always larger than
$r^{-1}$ along the entire flow:

$r^{2}M^{2}_{{\cal T}}=2\pi r^{2}{\cal T}^{\infty}_{{\bf e}_{\bf
q}}+\langle{\bf q},{\bf e}_{\bf q}\rangle>\langle{\bf q},{\bf e}_{\bf
q}\rangle\,.$ (5.46)

(Of course, no weakly coupled description is possible anymore below Planckian
radii.) The situation is different if instead $\langle{\bf q},{\bf e}_{\bf
q}\rangle=0$. In such a case $M_{{\cal T}}\equiv 2\pi{\cal T}^{\infty}_{{\bf
e}_{\bf q}}$ would remain constant and there would exist a minimal radius at
which $r^{-1}$ exceeds $M_{{\cal T}}$. Note that this can happen only if ${\bf
q}$ belongs to some infinite distance boundary of ${\cal P}$, and viceversa.
Indeed, for any ${\bf e}\in{\cal C}_{\rm S}^{\text{\tiny EFT}}$, $\langle{\bf
q},{\bf e}\rangle$ can be identified with an EFT string tension in Planck
units evaluated at ${\bm{\ell}}={\bf q}$. If $\langle{\bf q},{\bf e}\rangle=0$
then this tension vanishes and ${\bm{\ell}}={\bf q}$ is at infinite field
distance. Viceversa, if ${\bm{\ell}}={\bf q}$ is at infinite field distance,
then it should correspond to a tensionless point of some EFT string. We will
see that the same distinction will be relevant when we discuss regular
wormholes in Section 6.

Finally, we emphasize that the existence of fundamental instantons depends on
the UV completion of the theory, while our identification of BPS and EFT
instanton charges just uses EFT data. Hence, a priori, one is not guaranteed
that for any ${\bf q}\in{\cal C}_{\rm I}$ or ${\bf q}\in{\cal C}^{\text{\tiny
EFT}}_{\rm I}$ there actually exists a corresponding fundamental instanton
(rather than a multi-instanton configuration carrying total charge ${\bf q}$).
Regarding this point, we notice the analogy between our distinction between
EFT and non-EFT BPS instantons, and the extremal and non-extremal BPS
particles, respectively, discussed in [92] in the context of five-dimensional
${\cal N}=1$ models. In particular, in our axionic context the counterpart of
the BPS tower and sublattice versions of the weak gravity conjecture (WGC)
[93] proposed in [92] may be formulated as follows:232323The five-dimensional
setting considered in [92] is more directly related to an ${\cal N}=2$ version
of our four-dimensional framework, in which the BPS fundamental instantons may
be regarded as BPS particle world-lines wrapped along a compactified Euclidean
time in five dimensions. We leave a more detailed investigation of this
relation, and of its connection with wormhole effects in ${\cal N}=2$ models,
to future work.

BPS instanton tower WGC. For any ${\bf q}\in{\cal C}^{\text{\tiny EFT}}_{\rm
I}$ there exists an integer $k\geq 1$ such that there is a fundamental EFT
instanton of charge $k{\bf q}$.

BPS instanton sublattice WGC. There exists an integer $k\geq 1$ such that for
any ${\bf q}\in{\cal C}^{\text{\tiny EFT}}_{\rm I}$ there is a fundamental EFT
instanton of charge $k{\bf q}$.

The latter conjecture is clearly stronger than the former, but both imply that
there is an infinite subset of EFT instanton charges (which coincides with the
complete ${\cal C}^{\text{\tiny EFT}}_{\rm I}$ if $k=1$) which is populated by
fundamental instantons. On the other hand, in analogy with the results of
[92], we expect no tower or sublattice WGC to hold for BPS but non-EFT
instanton charges. We will see how the above conjectures are compatible with
some of the physical effects associated with non-extremal wormholes that we
will discuss in Section 7.

###  5.5 A non-standard BPS bound

Given the underlying supersymmetric structure of our EFT, one expects some
kind of BPS bound relating the action of the (non-BPS) wormhole (5.26) and the
one of the BPS instanton (5.36). In the following section we will show that
regular wormholes solutions are generically expected to exist only for charges
belonging to (a subset of) ${\cal C}^{\text{\tiny EFT}}_{\rm I}$. However, the
observations of this subsection do not really rely on that, and we can just
assume there exists a regular wormhole carrying a BPS instanton charge ${\bf
q}\in{\cal C}_{\rm I}$. Recalling (5.15), we observe that the extremality
bound (5.17) is equivalent to $\|{\bf q}\|^{2}\geq\|\dot{\bm{\ell}}\|^{2}$.
Employing (2.14) and (2.15) we then have

$\|{\bf q}\|^{2}\geq\|{\bf q}\|\|\dot{\bm{\ell}}\|\geq|q_{i}{\cal
G}^{ij}\dot{\ell}_{j}|=|q_{i}\dot{s}^{i}|\,,$ (5.47)

which is saturated only in the extremal BPS instanton case:
$\dot{\bm{\ell}}=\pm{\bf q}$. The action (5.26) is therefore subject to the
following bound:

$S|_{\text{hw}}\geq 2\pi\int^{\tau_{\infty}}_{0}{\rm
d}\tau\,\left|q_{i}\dot{s}^{i}\right|\geq
2\pi\Big{|}q_{i}\int^{\tau_{\infty}}_{0}{\rm
d}\tau\,\dot{s}^{i}\Big{|}=\big{|}S_{\text{\tiny BPS}}-2\pi\langle{\bf
q},{\bm{s}}_{*}\rangle\big{|}\,,$ (5.48)

where $S_{\text{\tiny BPS}}$ is as in (5.36) and we have again adopted the
condition (5.21). The bound involves in a crucial way the neck contribution
$\langle{\bf q},{\bm{s}}_{*}\rangle$. In the extremal case, as we have seen in
the previous section, that contribution is actually absent once one takes into
account the counterterm representing the insertion of the fundamental BPS
instanton at the singularity. On the other hand, in the non-extremal wormhole
case there is no singularity and no need for a local counterterm. A large
enough $\langle{\bf q},{\bm{s}}_{*}\rangle$ in that case may indicate that
$S|_{\text{hw}}<S_{\text{\tiny BPS}}$, which would contradict the expectation
that supersymmetric solutions saturate inequalities like (5.48). This puzzling
relation has indeed been observed in some explicit wormhole solutions (see
e.g. [74]) and will be encountered and further discussed in the following
section. A possible correction to that puzzling result is suggested at the end
of Section 6.1.

##  6 Wormholes in the ${\cal N}=1$ axiverse

We are now ready to analyze explicit non-extremal (non-supersymmetric)
wormhole configurations. We will restrict our focus on the large class of
models characterized by a Kähler potential of the form (2.5), or equivalently
by a kinetic potential of the form (2.18). We would like to show that, under
rather general conditions, if the degree $n$ of homogeneity of $P(s)$
satisfies $n\geq 3$ then the subset of EFT instanton charges

${\cal C}_{\text{\tiny WH}}\equiv\\{{\bf q}\in{\cal C}^{\text{\tiny EFT}}_{\rm
I}|\ \langle{\bf q},{\bf e}\rangle\geq 1,\ \forall{\bf e}\in{\cal
C}^{\text{\tiny EFT}}_{\rm S}\\}\ \subset\ {\cal C}^{\text{\tiny EFT}}_{\rm
I}\,,$ (6.1)

admits corresponding non-extremal wormhole configurations. Recalling (5.38),
the condition $\langle{\bf q},{\bf e}\rangle\geq 1$ for any ${\bf e}\in{\cal
C}^{\text{\tiny EFT}}_{\rm S}$ is basically equivalent to the requirement that
either ${\bf q}$ belongs to the interior of ${\cal P}$, or to its finite field
distance boundary $\partial{\cal P}|_{\text{fin.dist.}}$. Indeed, without
imposing this condition we would have $\langle{\bf q},{\bf e}\rangle\geq 0$
anyway. Hence the condition can be violated only if there exists an EFT string
charge ${\bf e}$ such that $\langle{\bf q},{\bf e}\rangle=0$. But, as already
discussed below (5.46), this would mean that ${\bf q}$ should lie on an
infinite field distance boundary of ${\cal P}$.242424 ${\cal C}_{\text{\tiny
WH}}$ depends on the boundary properties of the (dual) saxionic cone. A
restriction of the perturbative domain, corresponding to the removal of some
generator ${\bf e}\in\partial\Delta$ of $\Delta$, may result in the inclusion
in the new ${\cal C}_{\text{\tiny WH}}$ of some ${\bf q}\in{\cal
C}^{\text{\tiny EFT}}_{\rm I}$ that were excluded in the initial perturbative
domain. Note also that the existence of the wormholes does not depend on the
overall sign of ${\bf q}$. Hence, if a wormhole exists for ${\bf q}\in{\cal
C}_{\text{\tiny WH}}$, one with $-{\bf q}\in{\cal C}_{\text{\tiny WH}}$ must
exist as well. We will refer to these latter wormholes as anti-wormhole. (Of
course, the distinction between wormholes and anti-wormholes is only a matter
of convention.)

In the following subsections we will provide evidence that the set (6.1)
identifies the charges of physically acceptable non-extremal wormholes. The
general _homogeneous solution_ is introduced in Section 6.1 and its
reliability is investigated in Section 6.2. Considerations applying to more
general wormhole solutions are presented in Section 6.3. Wormholes arising
from kinetic potentials with degree of homogeneity $n=3$ are discussed
separately in Section 6.4. Explicit realizations in string theory models are
finally presented in Section 6.5.

###  6.1 The homogeneous solution

The sharpest statements can be made for wormhole charges ${\bf q}\in{\cal
C}_{\text{\tiny WH}}$ belonging to the interior of ${\cal P}$. For each of
these charges we will show that one can construct an explicit wormhole
configuration, no matter how complicated $P(s)$ or $\tilde{P}(\ell)$ are, as
long as they are homogeneous functions of degree $n\geq 3$. In a certain
sense, our result can be considered as a four-dimensional generalization of
the string theory solutions found in [29], which also applies to many more
string models. So, we will first focus on charges in the interior of ${\cal
P}$, while those belonging to the finite-distance boundary of ${\cal P}$ will
be discussed afterwards.

Now, since ${\cal P}$ is conical, if ${\bf q}$ is in its interior the whole
“ray” generated by ${\bf q}$ is contained in ${\cal P}$. It is then natural to
consider _homogeneous_ wormhole configurations with

${\bm{\ell}}(\tau)=\tilde{\ell}(\tau)\,{\bf q}\,.$ (6.2)

Note that the condition ${\bm{\ell}}\in{\cal P}$ simply reads
$\tilde{\ell}>0$. Inserting the ansatz (6.2) into (5.12) leads to

$\displaystyle 2\pi\int{\rm
d}\tau\,\Big{[}\,\frac{n}{4\tilde{\ell}^{2}}\Big{(}\frac{{\rm
d}\tilde{\ell}}{{\rm
d}\tau}\Big{)}^{2}+\frac{n}{4{\tilde{\ell}^{2}}}\Big{]}\,,$ (6.3)

where we used the identity ${\cal G}^{ij}({\bf q})q_{i}q_{j}=\frac{n}{2}$,
which follows from degree-$n$ homogeneity of $\tilde{P}$. Furthermore, one can
verify that the homogeneity of $\tilde{P}$ guarantees that the equations of
motion (5.16) for $\ell_{i}$ coincide with those derived from the effective
action in Eq. (6.3). This drastically simplifies the study of homogeneous
multi-saxion wormholes.

We have basically reduced the problem to a simple axio-dilaton model with
kinetic potential

$\tilde{\cal F}=n\log\tilde{\ell}\,$ (6.4)

and (effective) unit charge, as the ones studied in [9, 29]. Similar results
also follow. In particular, since the reduced effective potential is

$\tilde{V}(\tilde{\ell})=-\frac{n}{4\tilde{\ell}^{2}}\,,$ (6.5)

the wormhole radius (5.19) and the maximal proper time (5.23) are given
respectively by

$L^{4}=\frac{n}{12\pi^{2}\tilde{\ell}_{*}^{2}M^{4}_{\text{\tiny
P}}}\quad~{}~{}~{}\Rightarrow\quad~{}~{}~{}\tau_{\infty}=\frac{\pi}{2}\sqrt{\frac{3}{n}}\,\tilde{\ell}_{*}\,.$
(6.6)

We note in particular that $L$ explicitly depends on $\tilde{\ell}_{*}$, but
not on ${\bf q}$. The explicit profile $\tilde{\ell}(\tau)$ can be obtained by
integrating directly (5.15) in our reduced one-saxion model. Imposing the
boundary conditions $\tilde{\ell}|_{\tau=0}=\tilde{\ell}_{*}$ and $\frac{{\rm
d}\tilde{\ell}}{{\rm d}\tau}|_{\tau=0}=0$ fixes completely the solution:

$\tilde{\ell}(\tau)=\tilde{\ell}_{*}\cos\left(\frac{\tau}{\tilde{\ell}_{*}}\right)=\tilde{\ell}_{*}\cos\left(\frac{\pi}{2}\sqrt{\frac{3}{n}}\,\frac{\tau}{\tau_{\infty}}\right)\,,$
(6.7)

where the asymptotic value is determined by

$\tilde{\ell}_{\infty}=\tilde{\ell}_{*}\cos\left(\frac{\pi}{2}\sqrt{\frac{3}{n}}\right)\,.$
(6.8)

Since by definition $\tilde{\ell}$ must be positive, Eq. (6.7) describes a
completely smooth wormhole if and only if

$n\geq 3\,.$ (6.9)

In all string theory models we are aware of, $n$ is an integer taking values
from 1 to 7 – see Sections 4 and 6.5 for some examples. We hence have regular
wormholes only in models with $n=3,\ldots,7$. Note that in the limiting case
$n=3$ wormholes are asymptotically degenerate, since the dual saxions reach
the infinite field distance point ${\bm{\ell}}_{\infty}=0$ at spatial
infinity. They are nevertheless sensible and will be further investigated in
Sections 6.4 and 7.4. In Section 6.4 we will also more precisely see how
wormholes corresponding to $n=1,2$ become singular at a radial distance of
order $L$, and should then be discarded. If instead $n>3$ the homogeneous
wormholes are everywhere regular and asymptotically flow to a finite
$\tilde{\ell}_{\infty}>0$. The condition $n>3$ represents a lower bound on the
coefficient of the scalar kinetic term of the type first identified in [9, 29]
for the case of simple dilatonic models, and subsequently generalized in [74].

We can next present the on-shell action. Observing that $\|{\bf
q}\|^{2}=\frac{n}{2\tilde{\ell}^{2}}$, the integral appearing in (5.26) is
easily computed:

$\displaystyle S|_{\text{hw}}$
$\displaystyle=\frac{n\pi}{\tilde{\ell}_{*}}\tan\left(\frac{\pi}{2}\sqrt{\frac{3}{n}}\right)$
(6.10)
$\displaystyle=2\pi\sin\left(\frac{\pi}{2}\sqrt{\frac{3}{n}}\right)\langle{\bf
q},{\bm{s}}_{\infty}\rangle\,.$

Sticking for now to the everywhere controllable solutions corresponding to
$n>3$, we observe that the last line of (6.10), where the relation (6.13) was
used, implies $S|_{\text{hw}}<S_{\text{\tiny BPS}}$, explicitly realizing the
puzzling possibility mentioned in Subsection 5.5. Perhaps the claim that non-
extremal wormholes have actions smaller than the extremal BPS ones is a bit
rushed, though. Other contributions, beyond the two-derivative approximation,
may not always be negligible, at least for not too large wormhole charges.
Consider for example the GB term (5.2). Treating it as a perturbation and
working at the first non-trivial order, it contributes to the on-shell action
without modifying the two-derivative solution. This suggests that GB may
contribute non-negligibly to the on-shell action of non-extremal wormholes,
while of course it does not in the case of (flat) extremal wormholes.
Interestingly, for sufficiently small charges we have checked that
$S_{\text{\tiny BPS}}<(S+S_{\text{\tiny GB}})|_{\text{hw}}$ in some of the
examples discussed in Section 6.5, despite of $S|_{\text{hw}}<S_{\text{\tiny
BPS}}$. What we can infer from this simple exercise is that higher order terms
can significantly affect the relation between the on-shell action of non-
supersymmetric wormholes and the corresponding BPS actions. Still, we cannot
firmly conclude that the puzzle is solved. As soon as the GB starts competing
with the leading two-derivative contribution, one should in principle also
worry about other higher derivative corrections, which we did not consider
because not under theoretical control. Furthermore, for sufficiently large
charges we can show that in the case of homogeneous solutions $S_{\rm
tot}|_{\text{hw}}\simeq S|_{\text{hw}}<S_{\text{\tiny BPS}}$. In fact, by
rescaling ${\bf q}$ (with fixed ${\bm{\ell}}_{*}$), we can make $L$ and (6.10)
arbitrarily large (for fixed ${\bm{s}}_{\infty}$) and all higher-derivative
corrections, including the GB one, arbitrarily small.252525Note, however, that
large charge non-extremal wormholes may in fact be less relevant, not only
because their contribution $e^{-S}$ is strongly suppressed, but also because
their very existence is put under question once one has included the
corrections due to the smaller-charge ones [94, 95].

Let us finally discuss the case in which ${\bf q}\in{\cal C}_{\text{\tiny
WH}}$ belongs to some finite distance boundary of ${\cal P}$. Our proposal
(6.1) is based on the expectation that, being at finite field distance, this
boundary region can be identified with a restricted perturbative sector ${\cal
P}^{\prime}\subset\partial{\cal P}|_{\text{fin.dist.}}$. Furthermore, it
implicitly assumes that this boundary perturbative sector is sufficiently
decoupled from any strongly coupled sector that may appear in the limit, so
that a description according to our general scheme is reliable. If ${\bf q}$
lies in the interior of ${\cal P}^{\prime}$, or if ${\cal P}^{\prime}$ is one-
dimensional, one can in such a case run the same arguments followed above for
charges in the interior of ${\cal P}$. While we do not have a general proof
that these expectations are certainly realized, these are partially supported
by the observation that the homogeneous function $\tilde{P}(\ell)$ appearing
in (2.18) cannot degenerate, either vanishing or diverging, as one approaches
finite distance boundaries.262626Here we are of course using the kinetic
potential (2.18) to compute distances, neglecting corrections that may a
priori be present. Again, we expect these corrections not to significantly
affect our main points. This can be understood in the following way. Take a
finite distance boundary point ${\bm{\ell}}\in\partial{\cal
P}|_{\text{fin.dist.}}$. By definition, it can be connected to an interior
point ${\bm{\ell}}_{0}\in\text{Int}{\cal P}$ by a path $\gamma$ of finite
length ${\rm d}(\gamma)<\infty$. The variation of $\log P$ along this path is
always finite, since:

$\left|\log\tilde{P}(\ell)-\log\tilde{P}(\ell_{0})\right|=\Big{|}\int_{\gamma}\frac{{\rm
d}\tilde{P}}{\tilde{P}}\Big{|}=2\Big{|}\int_{\gamma}{\cal G}^{ij}\ell_{i}{\rm
d}\ell_{j}\Big{|}\leq n\int_{\gamma}\|{\rm
d}{\bm{\ell}}\|=n\,\text{d}(\gamma)<\infty\,,$ (6.11)

where we have again used the degree-$n$ homogeneity of $\tilde{P}(\ell)$.
Since $\log\tilde{P}(\ell_{0})$ is finite by assumption, $\log\tilde{P}(\ell)$
is necessarily finite too, and $\tilde{P}(\ell)$ can neither diverge nor
vanish. Note that by continuity the restriction of $\tilde{P}(\ell)$ to
$\partial{\cal P}|_{\text{fin.dist.}}$ is still homogeneous of degree $n$.

What said so far holds for charges ${\bf q}$ belonging to (6.1). This does
mean that there cannot exist non-extremal wormhole solutions carrying other
charges ${\bf q}$. In particular, there may exist wormholes carrying an EFT
instanton charge belonging to the infinite distance boundary of ${\cal P}$,
for which there exists an EFT string charge ${\bf e}$ such that $\langle{\bf
q},{\bf e}\rangle=0$. This for instance happens in presence of decoupled
perturbative sectors, which separately admit regular wormholes. However, as we
will discuss in Section 6.2, precisely these wormholes appear not to be
consistent with the energetic bounds imposed by the species scale in the
original perturbative regime. Nevertheless, wormholes carrying these charges
may be compatible with the energetic bounds in appropriate restricted
perturbative regimes, which are sufficiently far from the infinite distance
boundary of the original ${\cal P}$.272727 Consider a toy model with ${\cal
P}=\\{{\bm{\ell}}=(\ell_{1},\ell_{2})|\ell_{1},\ell_{2}\in\mathbb{R}_{\geq
0}\\}$, so that ${\cal C}_{\rm I}^{\text{\tiny EFT}}=\\{{\bf
q}=(q_{1},q_{2})|q_{1},q_{2}\in\mathbb{Z}_{\geq 0}\\}$, and ${\cal
F}=\log\ell_{1}+4\log\ell_{2}$. The charges ${\bf q}=(q_{1},0)$ and ${\bf
q}=(0,q_{2})$ are at infinite field distance in ${\cal P}$ and must be
excluded from (6.1): ${\cal C}_{\text{\tiny WH}}=\\{{\bf
q}=(q_{1},q_{2})|q_{1},q_{2}\in\mathbb{Z}_{>0}\\}$. On the other hand, the
charges ${\bf q}=(0,q_{2})$ with $q_{2}>0$ can still be regarded as elements
of the set (6.1) associated with the restricted perturbative regime obtained
by assuming $\ell_{1}\simeq 1$, which is parametrized only by $\ell_{2}$.

###  6.2 Perturbative conditions and relevant scales

In this subsection we verify that the homogeneous wormholes described in
Section 6.1 are compatible with the perturbative domains defined in Section
3.1 and with the energy bounds associated with the species scale (see Section
3.4).

Let us first discuss the compatibility with the perturbative regime. Without
loss of generality, we can assume that ${\bf q}\in{\cal C}_{\text{\tiny WH}}$
belongs to the interior of ${\cal P}$, since as discussed in Section 6.1 the
case of charges ${\bf q}$ belonging to a finite distance boundary of ${\cal
P}$ can be treated similarly by restricting the perturbative regime. We note
that, once expressed in terms of saxions $s^{i}$, the profiles (6.2) move
along straight saxionic radial lines as well. Indeed from (2.15) and (2.20),
(6.2) is mapped to

$s^{i}(\tau)=\frac{{\cal F}^{i}({\bf q})}{2\tilde{\ell}(\tau)}\,,$ (6.12)

where ${\cal F}^{i}({\bf q})\equiv{\partial{\cal F}}/{\partial\ell_{i}}({\bf
q})$ is a charge dependent constant. Homogeneity also implies that

$\langle{\bf q},{\bm{s}}\rangle=\frac{n}{2\tilde{\ell}}\ .$ (6.13)

Combining (6.12) and (6.7), we clearly see that ${\bm{s}}$ is rescaled by a
number $>1$ as we move away from the wormhole throat. Hence, if the
${\bm{s}}_{*}$ belongs to the perturbative region $\hat{\Delta}_{\alpha}$,
${\bm{s}}_{*}(\tau)$ will remain so for all other values of $\tau$. Since
${\bm{s}}_{*}$ can be freely chosen to be any point of the ray in ${\cal P}$
generated by ${\bf q}$, we can always choose it such that the entire
trajectory lies in the perturbative region $\hat{\Delta}_{\alpha}$. We are
thus confident that our solution is within the “large saxion”, and “small
dual-saxion”, regions identified in Section 3.1. The smallness of the dual
saxions can be bounded as follows. If ${\bm{s}}_{*}$ belongs to
$\hat{\Delta}_{\alpha}$, by repeating the same argument leading to (3.21) with
${\bf q}$ now playing the role of $\tilde{\bf C}$, we get the lower bound
$\langle{\bf q},{\bm{s}}_{*}\rangle\geq c_{{\bf q}}N/\alpha$, where $c_{{\bf
q}}\geq 1$ is some ${\bf q}$-dependent constant scaling as $c_{k{\bf
q}}=kc_{{\bf q}}$. By using (6.13) and the explicit form of the solution
(6.7), this can be translated into an upper bound on $\tilde{\ell}$ along the
entire wormhole
$\tilde{\ell}(\tau)\leq\tilde{\ell}_{*}\leq\frac{n\alpha}{2c_{\bf q}N}$. Note
that combined with (6.2), this sets an upper bound on the value of the
dominant EFT string scale (3.29) along the wormhole:

$M^{2}_{{\cal T}}=2\pi M^{2}_{\text{\tiny P}}\,\tilde{\ell}(\tau)\min_{{\bf
e}\in{\cal C}_{\rm S}^{\text{\tiny EFT}}}\langle{\bf q},{\bf e}\rangle\,.$
(6.14)

For not loo large charges we therefore see that $M^{2}_{{\cal T}}\lesssim
2\pi\alpha M_{\text{\tiny P}}^{2}/N$. This has the same scaling of the upper
bound on $M_{\gamma}^{2}$ obtained by applying (3.26) to (3.27), indicating
that $M_{{\cal T}}$ and $M_{\gamma}$ again agree parametrically with each
other.

We can also get a lower bound on $M^{2}_{{\cal T}}$. Since we are assuming
that ${\bf q}\in{\cal C}_{\text{\tiny WH}}$, by our definition (6.1) we have

$\langle{\bf q},{\bf e}\rangle\geq 1\quad\forall{\bf e}\in{\cal
C}^{\text{\tiny EFT}}_{\rm S}\,.$ (6.15)

Combined with (6.14), this shows that ${\bf q}\in{\cal C}_{\text{\tiny WH}}$
implies

$M^{2}_{{\cal T}}\geq 2\pi M^{2}_{\text{\tiny P}}\,\tilde{\ell}(\tau)\,,$
(6.16)

which suggests that the species scale increases monotonically as one
approaches the wormhole throat. Furthermore, from (6.6) and (6.16) we get the
identity

$L^{2}M_{{\cal T}}^{2}|_{\tau=0}=\sqrt{\frac{n}{3}}\min_{{\bf e}\in{\cal
C}_{\rm S}^{\text{\tiny EFT}}}\langle{\bf q},{\bf e}\rangle\,.$ (6.17)

Therefore, by (6.15) and the regularity condition $n\geq 3$, we deduce that

$LM_{{\cal T}}|_{\tau=0}\geq 1\quad~{}~{}~{}~{}~{}~{}~{}\text{if}\quad{\bf
q}\in{\cal C}_{\text{\tiny WH}}\,.$ (6.18)

This conclusion provides an important and non-trivial consistency check on the
universal reliability of our homogeneous wormhole solutions, even for small
charges ${\bf q}$. It is interesting to compare (6.17) to the analogous (5.46)
for extremal wormholes. In the latter case the charge ${\bf q}$ is bounded
from _above_ by (5.45), while in the regular non-extremal case the charge can
be arbitrarily large. For them such an upper bound does not exist, and
$LM_{{\cal T}}|_{\tau=0}$ can be made arbitrarily large by increasing ${\bf
q}$ (and consistently rescaling ${\tilde{\ell}}$ to keep ${\bm{\ell}}$ of the
same order). Note that these observations hold not only for $n>3$, but also
for the marginally degenerate case $n=3$ discussed in Section 6.4.

Finally, from (6.7) and (6.6), we can also easily compute the field distance
traveled by the saxions, or equivalently the dual saxions, along the half-
wormhole. In Planck units it is given by

${\rm d}({\bm{\ell}}_{*},{\bm{\ell}}_{\infty})=\int^{\tau_{\infty}}_{0}{\rm
d}\tau\sqrt{{\cal
G}^{ij}(\ell)\dot{\ell}_{i}\dot{\ell}_{j}}=\sqrt{\frac{n}{2}}\,\log\left[\cos\left(\frac{\pi}{2}\sqrt{\frac{3}{n}}\right)\right]^{-1}\,.$
(6.19)

This is clearly well defined only for $n>3$. For instance ${\rm
d}({\bm{\ell}}_{*},{\bm{\ell}}_{\infty})|_{n=4}\simeq 2.2$, while ${\rm
d}({\bm{\ell}}_{*},{\bm{\ell}}_{\infty})|_{n=7}\simeq 1.2$. In those cases the
total displacement is moderately super-Planckian, indicating no severe problem
associated to the swampland Distance Conjecture [32]. This is even more so
once we introduce an IR cutoff. Even if, as we will discuss in more detail in
Section 6.4, such an IR regularization is strictly necessary only if $n=3$, it
is physically motivated also for $n>3$.

###  6.3 Non-homogeneous generalization

As emphasized in Section 6.2, the endpoints ${\bm{s}}_{*}$ and
${\bm{s}}_{\infty}$ of the saxionic trajectory along a homogeneous wormhole
can be freely chosen to be any point of the ray in ${\cal P}$ generated by
${\bf q}$. In fact, this property comes from a more general scaling symmetry
of the wormhole equations. Namely, if ${\bm{\ell}}(\tau)$ is a solution of the
equations of motion then
${\bm{\ell}}^{\prime}(\tau^{\prime})=\lambda{\bm{\ell}}(\frac{\tau^{\prime}}{\lambda})$
($\lambda>0$) is a solution too, which starts from
${\bm{\ell}}^{\prime}_{*}=\lambda{\bm{\ell}}_{*}$ (at $\tau^{\prime}=0$) and
arrives at ${\bm{\ell}}^{\prime}_{\infty}=\lambda{\bm{\ell}}_{\infty}$ (at
$\tau^{\prime}_{\infty}=\lambda\tau_{\infty}$). The corresponding saxionic
flow is
${\bm{s}}^{\prime}(\tau^{\prime})=\frac{1}{\lambda}{\bm{s}}(\frac{\tau^{\prime}}{\lambda})$,
which shows that for any given regular wormhole solution we can always make
such a rescaling in order to ensure that the flow is inside
$\hat{\Delta}_{\alpha}$. We would now like to use this property to identify
more general wormhole solutions corresponding to an EFT instanton charge ${\bf
q}\in{\cal C}_{\text{\tiny WH}}$. In order to understand this possibility, it
is useful to discuss some properties of the metric ${\cal G}_{ij}$ and the
potential $V_{\bf q}$.

First of all, the saxionic metric admits as particular geodesics the radial
directions. This “isotropy” is broken by the presence of the potential $V_{\bf
q}(\ell)$ which in fact identifies ${\bf q}$ as preferred direction. Indeed,
assuming that ${\bf q}$ belongs to the interior of ${\cal P}$, if we restrict
along the ${\bf q}$ direction as in (6.2), we get ${\cal G}_{ij}\frac{\partial
V_{\bf q}}{\partial\ell_{j}}=\frac{q_{i}}{\tilde{\ell}}$. Hence along the
radial direction identified by ${\bf q}$ the “force” associated with $V_{\bf
q}$ is directed along $-{\bf q}$. The sign can be understood by noticing that
$V_{\bf q}(\lambda\ell)=\frac{1}{\lambda^{2}}V_{\bf q}(\ell)$ and then, since
$V_{\bf q}$ is negative definite, it decreases as one moves radially towards
the tip of the dual saxionic cone ${\cal P}$. Correspondingly, $V_{\bf
q}(\lambda s)=\lambda^{2}V_{\bf q}(s)$ and then $V_{\bf q}(s)$ decreases as we
move radially away from the tip of the saxionic cone $\Delta$.

We can say something more about the shape of the potential $V_{\bf q}$. Still
assuming that ${\bf q}$ belongs to the interior of ${\cal P}$, take a
particular point $\hat{\bm{\ell}}$ along the ray in ${\cal P}$ generated by
${\bf q}$ and the corresponding point $\hat{\bm{s}}$ along the ray generated
by ${\bm{s}}|_{{\bm{\ell}}={\bf q}}=\frac{1}{2}\frac{\partial{\cal
F}}{\partial{\bm{\ell}}}({\bf q})$,282828By $\frac{\partial{\cal
F}}{\partial{\bm{\ell}}}({\bf q})$ we mean the saxionic vector with components
$\frac{\partial{\cal F}}{\partial{\ell_{i}}}({\bf q})$. Since ${\bf q}$
belongs to ${\cal P}$, then by definition $\frac{\partial{\cal
F}}{\partial{\bm{\ell}}}({\bf q})$ belongs to $\Delta$. and consider the
saxionic plane ${\cal S}_{\hat{\bm{s}}}$ passing through it and ‘orthogonal’
to ${\bf q}$:

${\cal S}_{\hat{\bm{s}}}=\left\\{{\bm{s}}\in\Delta\Big{|}\langle{\bf
q},{\bm{s}}\rangle=\langle{\bf q},\hat{\bm{s}}\rangle\right\\}\,.$ (6.20)

By using again the homogeneity, one can easily show that $V_{\bf
q}(\hat{s})=-\frac{1}{n}\left|\langle{\bf
q},\hat{\bm{s}}\rangle\right|^{2}=-\frac{1}{n}\left|\langle{\bf
q},{\bm{s}}\rangle\right|^{2}|_{{\cal S}_{\hat{\bm{s}}}}$. On the other hand
$\left|\langle{\bf q},{\bm{s}}\rangle\right|^{2}\leq\|{\bf
q}\|^{2}\|{\bm{\ell}}\|^{2}=-nV_{\bf q}({s})$. We deduce that

$V_{\bf q}(\hat{s})\geq V_{\bf q}({s})|_{{\cal S}_{\hat{\bm{s}}}}$ (6.21)

and that the potential restricted to ${\cal S}_{\hat{\bm{s}}}$ has an absolute
maximum at $\hat{\bm{\ell}}$. Hence $V_{\bf q}({s})$ is negative definite and
has a hill shape with crest descending along ${\bm{s}}|_{{\bm{\ell}}={\bf q}}$
(which belongs to $\Delta$) or, correspondingly, $V_{\bf q}({\ell})$ has a
hill shape with crest descending along $-{\bf q}$, as schematically depicted
in figure 6. An additional confirmation comes by inspecting the Hessian of
$V_{\bf q}({\ell})$,

$\frac{\partial^{2}V_{\bf
q}}{\partial\ell_{i}\partial\ell_{j}}|_{\hat{\ell}}=\frac{1}{4}{\cal
F}^{ijkl}(\hat{\bm{\ell}})q_{k}q_{l}=\frac{1}{4\tilde{\ell}^{4}}{\cal
F}^{ijkl}({\bf q})q_{k}q_{l}=\frac{3}{2\tilde{\ell}^{4}}{\cal F}^{ij}({\bf
q})=-\frac{3}{\tilde{\ell}^{2}}{\cal G}^{ij}(\hat{\bm{\ell}})\,,$ (6.22)

where we have set $\hat{\bm{\ell}}=\tilde{\ell}{\bf q}$. As expected, this is
negative definite and also reveals that as we move towards the origin of the
dual saxionic space the steepness of the hill becomes more and more
accentuated.

\begin{overpic}[width=225.84375pt]{Figures/potential_v8.png}
\put(33.0,62.0){\Large$\displaystyle{\bf q}$}
\put(-14.0,70.0){\Large$\displaystyle V_{\bf q}({\ell})$}
\put(-4.0,39.0){\Large$O$} \put(92.0,75.0){\Large$\displaystyle{\bm{\ell}}$}
\end{overpic} Figure 6: Schematic representation of the potential $V_{\bf
q}(\ell)$. The potential has a hill shape with crest descreasing along the
direction $-{\bf q}$. On the homogeneous solution the profile of the dual
saxions follow the geodesic identified by the crest.

It is now clear why, if we start from a point ${\bm{s}}_{*}$ initially at rest
along the ray identified by ${\bm{s}}|_{{\bm{\ell}}={\bf q}}$, the point is
radially driven away from the tip of the cone ${\cal P}$ along the hill’s
crest down to the point ${\bm{s}}_{\infty}$. If $n\geq 3$ one gets precisely
the saxionic counterpart of the homogeneous solution described in Section 6.1
and, as already discussed, we can rescale it in order to make it lie inside
$\hat{\Delta}_{\alpha}$ (or equivalently $\hat{\cal P}_{\alpha}$).

Restrict now to $n>3$, with ${\bf q}\in{\cal C}_{\text{\tiny WH}}$ still in
the interior of ${\cal P}$, so that the corresponding homogeneous wormhole is
everywhere regular and does not degenerate asymptotically. If we slightly move
${\bm{s}}_{*}$ away from the the ray generated by ${\bm{s}}|_{{\bm{\ell}}={\bf
q}}$, the potential will drive it further away the radial direction, but if
${\bm{s}}_{*}$ is close enough to the initial position one should still get a
sensible wormhole solution, ending at some ${\bm{s}}_{\infty}$ inside
$\Delta$. Combining this observation with the above scaling symmetry, we then
expect to be able to fine-tune the initial value ${\bm{s}}_{*}$ to reach any
final point ${\bm{s}}_{\infty}$ inside $\hat{\Delta}_{\alpha}$. For instance,
this is certainly true in the case of a diagonal metric (which corresponds to
a completely factorized
$P(s)=(s^{1})^{n_{1}}(s^{2})^{n_{2}}(s^{3})^{n_{3}}\ldots$, with
$n_{1}+n_{2}+\ldots>3$), which is a case already discussed in the literature –
see for instance [74]. The above argument suggests that this property holds
also for more general models with non-factorizable $P(s)$. While we do not
have a general proof, this expectation is confirmed by the following
perturbative analysis. Consider a given homogeneous solution
${\bm{\ell}}_{0}(\tau)$ of the form (6.2), with $\tilde{\ell}$ as in (6.7),
and take a small deformation
${\bm{\ell}}(\tau)={\bm{\ell}}_{0}(\tau)+\delta{\bm{\ell}}(\tau)$. By
expanding the action (5.12) up to quadratic order, we get the quadratic
contribution

$\displaystyle\pi\int{\rm d}\tau~{}{\cal G}^{ij}({\bf
q})\left\\{\frac{\delta\dot{\ell}_{i}\delta\dot{\ell}_{j}}{\tilde{\ell}^{2}}+\left[1-\sin^{2}\left(\frac{\tau}{\tilde{\ell}_{*}}\right)\right]\frac{\delta\ell_{i}\delta\ell_{j}}{\tilde{\ell}^{4}}\right\\}\,,$
(6.23)

where we have ignored all boundary terms since we are only interested in the
equations of motion for $\delta\ell_{i}$. By setting
$\delta\ell_{i}=\tilde{\ell}f_{i}\ $, (6.23) can be rewritten as

$\pi\int{\rm d}\tau~{}{\cal G}^{ij}({\bf
q})\left\\{\dot{f}_{i}\dot{f}_{j}+\frac{2}{\tilde{\ell}^{2}}f_{i}f_{j}\right\\}\,.$
(6.24)

from which the equations of motion for $f_{i}$ follow

$\ddot{f}_{i}=\frac{2}{\tilde{\ell}^{2}}f_{i}=\frac{2}{\tilde{\ell}^{2}_{*}\cos^{2}\big{(}\frac{\tau}{\tilde{\ell}_{*}}\big{)}}f_{i}\,.$
(6.25)

The general solution of (6.25) satisfying the initial condition
$\dot{f}_{i}(0)=0$, corresponding to $\delta\dot{\ell}_{i}(0)=0$, is 292929
Without loss of generality we can impose the transversality condition ${\cal
G}^{ij}({\bf q})f_{*i}q_{j}=0$, so that the deformation changes the wormhole
radius (5.19) only to second order in $f_{*i}$:
$L^{4}=L^{4}_{(0)}\left(1+\delta\right)\quad~{}~{}~{}~{}\text{with}\quad~{}~{}\delta=\frac{6}{n}{\cal
G}^{ij}({\bf q})f_{*i}f_{*j}$ (6.26) Recalling (5.23), this implies
$\tau_{\infty}$ is only modified by second order corrections too.

$f_{i}(\tau)=f_{*i}\left[1+\frac{\tau}{\tilde{\ell}_{*}}\tan\Big{(}\frac{\tau}{\tilde{\ell}_{*}}\Big{)}\right]=f_{*i}\left[1+\frac{\pi}{2}\sqrt{\frac{3}{n}}\frac{\tau}{\tau_{\infty}}\tan\left(\frac{\pi}{2}\sqrt{\frac{3}{n}}\frac{\tau}{\tau_{\infty}}\right)\right],$
(6.27)

where $f_{*i}=f_{i}(0)$ is the integration constant representing the value of
$f_{i}(\tau)$ at the throat. For the perturbative expansion to make sense
$|f_{i}(\tau)/q_{i}|$ must remain small all along the flow, and since the
perturbation grows monotonically towards $\tau=\tau_{\infty}$ this requires
$|f_{*i}/q_{i}|\ll 1$.303030For $n=3$ the perturbation (6.27) diverges at
infinite radial distance and thus, for any arbitrarily small initial values
$f_{*i}$, the perturbative expansion breaks down at some finite radius.
Nevertheless, as discussed in Section 6.4, it can still make sense by
introducing an IR cutoff. Even though this implies that we cannot fully trust
the quadratic approximation, the growing behavior of the deformation suggests
that in a complete non-perturbative treatment it may be sufficient to pick a
small $|f_{i}(0)/q_{i}|$ to allow for a much larger
$|f_{i}(\tau_{\infty})/q_{i}|$. This is confirmed by the numerical study
presented in section 6.5.1, in which we inspect non-homogeneous wormhole
solutions in some concrete string theory models.

So far we have mostly assumed that ${\bf q}\in{\cal C}_{\text{\tiny WH}}$
belongs to the interior of ${\cal P}$. However, by reasoning as in Section 6.1
one can extend the above arguments to the cases with ${\bf q}\in{\cal
C}_{\text{\tiny WH}}$ belonging to some finite distance boundary of ${\cal
P}$. Hence, at least for $n>3$, all these considerations support the main
claim of this section: if the perturbative regime is described by a Kähler
potential (2.5) (or a kinetic potential (2.18)) with homogeneous $P(s)$ (or
$\tilde{P}(\ell)$) of degree $n\geq 3$, then for each ${\bf q}\in{\cal
C}_{\text{\tiny WH}}$ there exists a corresponding smooth wormhole solution,
and an anti-wormhole solution carrying charge $-{\bf q}$. The marginally
degenerate case $n=3$ will be discussed in Section 6.4.

Finally, let us revisit the consistency condition $LM_{{\cal T}}|_{\tau=0}\geq
1$, already discussed in Section 6.2 for homogeneous wormholes. In that case
we proved (6.17), which shows that the consistency condition is always
satisfied. Even though we are not able to derive such a sharp result for the
more general wormholes discussed in the present subsection, we still expect no
serious issue to show up for the following reasons. From (5.19) and (3.29) we
get

$L^{2}M_{{\cal T}}^{2}|_{\tau=0}=\sqrt{\frac{2}{3}}\|{\bf q}\|_{*}\min_{{\bf
e}\in{\cal C}_{\rm S}^{\text{\tiny EFT}}}\langle{\bm{\ell}}_{*},{\bf
e}\rangle\,.$ (6.28)

Note that all wormhole solutions related by the scaling symmetry discussed at
the beginning of this subsection have the same $LM_{{\cal T}}|_{\tau=0}$,
since (6.28) is invariant under an overall rescaling of ${\bm{\ell}}_{*}$.
Therefore, in order to investigate its behavior as we move ${\bm{\ell}}_{*}$
away from the radial direction identified by ${\bf q}$, with no loss of
generality we can impose ${\bm{s}}_{*}$ to lie in a saxionic hyperplane ${\cal
S}_{*}$ of the form (6.20). Let $\hat{\bm{s}}_{*}$ be the point of ${\cal
S}_{*}$ corresponding to $\hat{\bm{\ell}}_{*}\propto{\bf q}$. At this point
(6.17) holds. Recalling (5.13), the inequality (6.21) translates into

$\|{\bf q}\|_{*}\geq\|{\bf q}\|_{\hat{\bm{\ell}}_{*}}\,.$ (6.29)

Hence the first factor appearing on the right-hand side of (6.28) increases as
${\bm{\ell}}_{*}$ moves away from $\hat{\bm{\ell}}_{*}$. Of course, this may
be compensated by a faster decrease of $\min_{{\bf e}\in{\cal C}_{\rm
S}^{\text{\tiny EFT}}}\langle{\bm{\ell}}_{*},{\bf e}\rangle$. However, since
$\langle{\bm{\ell}}_{*},{\bf e}\rangle$ represents an EFT string tension in
Planck units, its decrease should correspond to approaching an infinite
distance boundary of ${\cal P}$. Qualitatively, we therefore expect the
wormhole to exist only for ${\bm{\ell}}_{*}$ sufficiently close to
$\hat{\bm{\ell}}_{*}$ and away from the infinite distance boundaries, since
otherwise there would be “no room” for accommodating the corresponding flow.
This suggests that an hypothetical decrease of $\langle{\bm{\ell}}_{*},{\bf
e}\rangle$ should not overwhelm the increase of $\|{\bf q}\|_{*}$, so that
overall we expect no violation of $LM_{{\cal T}}|_{\tau=0}\geq 1$. In Section
6.5 we will provide numerical support to this expectation.

###  6.4 Marginally degenerate case

In this section we will have a closer look at the wormhole solutions
corresponding to Kähler and kinetic potentials (2.5) and (2.18) with $n=3$.
But before discussing that specific case, it is instructive to quantify how
much homogeneous wormholes with $n=1,2,3$ fail to be globally non-degenerate.
This will reveal that the “marginally” degenerate case $n=3$ is in fact
special and, as we will see in Section 7, also physically relevant.

We begin observing that, for general $n$, (6.6) implies

$\tilde{\ell}_{*}=\sqrt{\frac{n}{3}}\frac{1}{2\pi M^{2}_{\text{\tiny
P}}L^{2}}$ (6.30)

whereas (5.22) reads

$\frac{\tau}{\tilde{\ell}_{*}}=\sqrt{\frac{3}{n}}\left[\frac{\pi}{2}-\arcsin\left(\frac{L^{2}}{r^{2}}\right)\right].$
(6.31)

The homogeneous solution (6.7) may thus be rewritten as

$\tilde{\ell}(r)=\tilde{\ell}_{*}\cos\left[\sqrt{\frac{3}{n}}\left(\frac{\pi}{2}-\arcsin\left(\frac{L^{2}}{r^{2}}\right)\right)\right]\,.$
(6.32)

This equation shows that if $n\leq 3$ then $\tilde{\ell}(r)$ vanishes at the
radius

$r_{\rm
deg}=\frac{L}{\sqrt{\sin\left[\frac{\pi}{2}\left(1-\sqrt{\frac{n}{3}}\right)\right]}}\,.$
(6.33)

At the degeneration point the theory reaches the tip ${\bm{\ell}}=0$ of the
dual saxionic cone, which is at infinite field distance. This means that all
BPS string tensions (2.21) and the dominant EFT string scale $M_{{\cal T}}$
defined in (3.29) vanish, and with them $M_{\rm t}$ and $M_{\rm sp}$ vanish as
well. When this happens the solution can no longer be trusted. From (6.33) one
gets $r_{\rm deg}|_{n=1}\simeq 1.27L$ and $r_{\rm deg}|_{n=2}\simeq 1.88L$.
Therefore, for $n=1,2$ already at $r_{\rm deg}\sim L$ the solution degenerates
and does not appear to make any physical sense. So, if the $n=1,2$ the charges
belonging to the set (6.1) can only be regarded as charges of fundamental EFT
instantons of the type discussed in Section 5.3.

The story is very different for the $n=3$ case. First of all, in that case
$r_{\rm deg}|_{n=3}\to\infty$. So the solution is everywhere well defined and
degenerates only asymptotically at spatial infinity. The degeneration
${\bm{\ell}}(r\to\infty)\rightarrow 0$ is of course associated to an infinite
field distance limit, as it was for the $n=1,2$ cases, but here this
degeneracy occurs at infinite radius. This is not a real concern since it is
natural to introduce an IR cutoff which sets a finite maximal radial distance
below which the solution is required to be non-degenerate. The IR cutoff may
correspond to a physical mass scale $M_{{\text{\tiny IR}}}$, as mentioned at
the end of Section 3.4. Alternatively, as we will presently elaborate upon, it
may correspond to a new lower Wilsonian cutoff $\Lambda_{\text{\tiny IR}}\ll
L^{-1}$ associated with an infra-red EFT. In any event, the IR cutoff allows
us to make physical sense of the marginally degenerate solution. On the
contrary, there is no way to interpret the $n<3$ configurations as wormholes
connecting asymptotically flat spaces because $r_{\rm deg}|_{n=1,2}\sim L$
would force us to take $M_{{\text{\tiny IR}}},\Lambda_{\text{\tiny IR}}\gtrsim
1/L$, leaving essentially no room for such an interpretation. For completeness
we emphasize that there is instead no urgent reason to introduce a regulator
for $n>3$, because those solutions are regular everywhere. Nevertheless,
physically we should expect a non-trivial $M_{{\text{\tiny IR}}}$ in those
cases as well, or we may still be interested in introducing a lower Wilsonian
cutoff $\Lambda_{\text{\tiny IR}}$, as we will do below – see also the
comments below (5.28) and (6.19).

There is another peculiarity of the marginally degenerate wormhole. Indeed,
for $n=3$ Eq. (6.32) reduces to

$\tilde{\ell}(r)=\frac{1}{2\pi M^{2}_{\text{\tiny
P}}r^{2}}\quad\Rightarrow\quad{\bm{\ell}}(r)=\frac{{\bf q}}{2\pi
M^{2}_{\text{\tiny P}}r^{2}}\,.$ (6.34)

We then see that, as a function of $r$, the dual saxionic profile is the same
as the extremal BPS profile (5.31) with vanishing (and hence degenerate)
asymptotic saxionic value ${\bm{\ell}}_{\infty}=0$. While this observation is
strictly valid only for homogeneous solutions, it remains approximately true
also for more general wormholes corresponding to $n=3$. To see this we look
for new $n=3$ solutions via a perturbative expansion around the homogeneous
configuration as we did in the previous section. At leading order
$\delta\ell_{i}=\ell_{i}-\ell_{i}^{0}\equiv\tilde{\ell}f_{i}$ with $f_{i}$ as
in (6.27). At the same order, specifying $n=3$ and employing (6.31), the
result is

$\displaystyle{\bm{\ell}}$ $\displaystyle\simeq{\bf
q}\tilde{\ell}_{*}\,\cos\Big{(}\frac{\tau}{\tilde{\ell}_{*}}\Big{)}+{\bm{f}}_{*}\,\tilde{\ell}_{*}\Big{[}\cos\Big{(}\frac{\tau}{\tilde{\ell}_{*}}\Big{)}+\frac{\tau}{\tilde{\ell}_{*}}\sin\Big{(}\frac{\tau}{\tilde{\ell}_{*}}\Big{)}\Big{]}$
(6.35) $\displaystyle=\frac{{\bf q}}{2\pi M^{2}_{\text{\tiny
P}}r^{2}}+\frac{{\bm{f}_{*}}}{4M^{2}_{\text{\tiny P}}L^{2}}\left[1+{\cal
O}(L^{4}/r^{4})\right],$

where $f_{*i}$ are arbitrary constants satisfying $|f_{*i}/q_{i}|\ll 1$ and in
the second line we assumed $r/L\gg 1$. Strictly speaking the perturbative
expansion breaks down at very large distances $r^{2}/L^{2}>|q_{i}/f_{*i}|$, as
shown in section 6.3. Nevertheless, we already emphasized that an IR cutoff is
necessary. We can thus safely conclude that, for $n=3$ and within the regime
of validity of our EFT, there exist more general solutions that have
approximately the same saxionic profile as a BPS extremal wormhole, including
a (small but non-vanishing)
${\bm{\ell}}_{\infty}={{\bm{f}_{*}}}/{(4M^{2}_{\text{\tiny P}}L^{2})}$. In the
following subsection we will provide numerical evidence that the same
conclusion generalizes to non-perturbative deformations of the homogeneous
solution. The only qualitative difference between the $n=3$ non-extremal half-
wormholes and the extremal ones is in the metric (which is flat in the
extremal case) and in the range or the radial coordinate $r$, which is
restricted to $r\in[L,\infty)$ in the half-wormhole case. Yet, the two
solutions are _completely indistinguishable_ to an observer at distances $r\gg
L$ from the wormhole throat, up to corrections of order ${\cal
O}(L^{4}/r^{4})$.

This observation extends to the on-shell action as well. Let us start from the
homogeneous solution first, whose complete on-shell action (6.10) diverges,
since the solution degenerates asymptotically. However, as already emphasized,
it is natural to introduce an IR regulator. To compute the action we hence
introduce an IR cutoff $\Lambda_{\text{\tiny IR}}$ as above. We thus assume
$\Lambda_{\text{\tiny IR}}\ll L^{-1}$ and remove the region
$r>\Lambda_{\text{\tiny IR}}^{-1}$, for some $\Lambda_{\text{\tiny IR}}$
comfortably within the region (3.30), from the $n=3$ version of (5.26). This
gives

$\displaystyle S|_{\text{hw}}^{\Lambda_{\text{\tiny IR}}}=$ $\displaystyle
2\pi\int_{\tau_{*}}^{\tau_{\text{\tiny IR}}}{\rm d}\tau\,\|{\bf
q}\|^{2}=\frac{3\pi}{\tilde{\ell}_{*}}\tan\frac{\tau_{\text{\tiny
IR}}}{\tilde{\ell}_{*}}$ (6.36) $\displaystyle=$
$\displaystyle{2\pi}\langle{\bf q},{\bm{s}}_{\text{\tiny
IR}}\rangle\left[1+{{\cal O}}(\Lambda_{\text{\tiny IR}}^{4}L^{4})\right]$

where $\tau_{\text{\tiny IR}}\equiv\tau(r=\Lambda^{-1}_{\text{\tiny IR}})$ and
${\bm{s}}_{\text{\tiny IR}}\equiv{\bm{s}}(\tau_{\text{\tiny IR}})$. Moreover,
we used (6.31) and (6.13) with $n=3$ and $\tilde{\ell}_{\text{\tiny
IR}}=\tilde{\ell}_{*}\cos(\tau_{\text{\tiny IR}}/\tilde{\ell}_{*})$ – see
(6.7). Up to ${\cal O}(\Lambda_{\text{\tiny IR}}^{4}L^{4})$ corrections, the
cutoff on-shell action (6.36) is therefore equivalent to the localized
operator (5.35) accounting for the insertion of a BPS fundamental instanton in
an EFT with Wilsonian cutoff $\Lambda_{\text{\tiny IR}}$. The cutoff on-shell
action for the more general solutions (6.35) is again given by (6.36) within
the quoted uncertainty. Analogously, one can verify that the Gauss-Bonnet term
evaluated on (6.34) vanishes with $\Lambda_{\text{\tiny IR}}^{2}L^{2}\to 0$.

We also note that for $n=3$ (6.30) reduces to

$\tilde{\ell}_{*}=\frac{1}{2\pi M^{2}_{\text{\tiny P}}L^{2}}\,,$ (6.37)

and that this formula is not affected by the above perturbations up to
corrections of order $|f_{*i}|^{2}$ – see section 6.3 for more details. We
emphasize that one can freely change $\tilde{\ell}_{*}$ and $L$ without
affecting the profile (6.34) and its perturbation (6.35), provided $f_{*}$ is
appropriately chosen. This implies that we can consider $L$ and
${\bm{\ell}}_{\text{\tiny{IR}}}$ as independent quantities. This is in sharp
contrast with what happens in the $n>3$ wormholes, in which ${\bm{\ell}}(r)$
depends explicitly on $L$, which is then fixed by ${\bm{\ell}}_{\infty}$.

We finally show that the $n=3$ IR-regularized version of (6.19) reads

${\rm d}({\bm{\ell}}_{*},{\bm{\ell}}_{\text{\tiny
IR}})=-\sqrt{\frac{3}{2}}\log({\Lambda_{\text{\tiny IR}}^{2}L^{2}})\,,$ (6.38)

which e.g. for $\Lambda_{\text{\tiny IR}}^{2}L^{2}=10^{-1}$ or
$\Lambda_{\text{\tiny IR}}^{2}L^{2}=10^{-2}$ gives the moderately super-
Planckian distances ${\rm d}({\bm{\ell}}_{*},{\bm{\ell}}_{\text{\tiny
IR}})\simeq 2.8$ or ${\rm d}({\bm{\ell}}_{*},{\bm{\ell}}_{\text{\tiny
IR}})\simeq 5.6$, respectively. This suggests that marginally degenerate
wormholes (and their deformations) may have potential control issues related
to the Distance Conjecture [32]. Nevertheless, because of the almost-BPS
nature noticed above, we expect their physical effects to be protected by
supersymmetry and hence to be reliable. This expectation will be strengthened
by the discussion of Section 7.4, where we will clarify the physical meaning
of the IR regularization leading to (6.36) and further elaborate on the
almost-BPS nature of the marginally degenerate wormholes.

###  6.5 Examples in string theory models

Let us now discuss explicit realizations of our wormholes in string theory
models.

####  6.5.1 Wormholes in F-theory models

Consider first the F-theory models discussed in subsection 4.1. In this case
case, we recall that the set ${\cal C}_{\rm I}$ of BPS instanton charges can
be identified with the cone of effective divisors. Microscopically, these
instantons correspond to Euclidean D3-branes wrapping effective divisors. In
order to identify the possible wormhole configurations, we must first identify
the set (5.38) of possible EFT instanton charges. Recalling the discussion
around (4.17), we immediately conclude that ${\bf q}=q_{a}D^{a}\in{\cal
C}^{\text{\tiny EFT}}_{\rm I}$ is a nef divisor in some space $X^{\prime}$
which can be obtained from $X$ through small transitions. In the following we
will for simplicity take ${\bf q}=q_{a}D^{a}$ to be nef already in $X$. Note
that the cone generated by nef divisors is contained in the cone generated by
movable divisors [96]. Any movable effective divisor $D\subset X$ admits a
multiple $mD$, for some integer $m\geq 1$, that can be freely deformed along
the entire $X$. Hence for any EFT instanton of charge ${\bf
q}=q_{a}D^{a}\in{\cal C}^{\text{\tiny EFT}}_{\rm I}$, there exists an EFT
instanton of charge $m{\bf q}$, with $m\geq 1$, corresponding to a non-
isolated Euclidean D3-brane that can freely explore the entire internal space.
Even though if $m>1$ a single Euclidean D3-brane of charge ${\bf q}$ does not
have such a property, $m$ such D3-branes can recombine into a single non-
isolated one that does. This shows the intrinsically gravitational nature of
all these EFT instantons (for both $m=1$ or $m>1$), which would not exist if
we decoupled gravity by decompactifying the internal space. We expect these
qualitative features to characterize the UV completion of EFT instantons also
in other models. We will provide other examples in the following subsections.
Note also that, if correct, the BPS instanton tower and sublattice WGCs of
Section 5.4 imply the existence of an infinite set of Euclidean D3-brane
wrapping irreducible movable divisors populating a (possibly non-strict)
infinite subset of ${\cal C}^{\text{\tiny EFT}}_{\rm I}$.

As already mentioned below (4.18), the $\ell_{a}$-sector alone corresponds to
$n=3$. This means that, in addition to the extremal wormholes corresponding to
EFT instantons, it only allows for the marginally degenerate wormholes of
Section 6.4. As anticipated therein and further elaborated in Section 7, it
makes perfect sense to regularize these wormholes by introducing an IR cutoff.
However, for illustrative purposes, in this section we prefer to first discuss
everywhere regular wormholes, which can be obtained by extending the
$\ell_{a}$-sector in order to get an overall $n>3$. One such extension is
provided by Sen’s weak coupling limit, which adds the IIB axio-dilaton, and
hence one extra dual saxion $\hat{\ell}$ as defined in (4.19), getting an
overall $n=4$.313131This is just one simple possibility. For instance, we may
also turn on charges corresponding to the (s)axionic sector appearing in some
asymptotic region of the complex structure moduli space. In this case we know
that the family of homogeneous wormholes certainly exists for any charge
vector $(\hat{q},{\bf q})$ belonging to the set ${\cal C}_{\text{\tiny WH}}$
defined in (6.1). Since the set of EFT string charges take the form
$(\hat{e},{\bf e})$, where $\hat{e}\geq 0$ and ${\bf e}=e^{a}\Sigma_{a}$ is a
(possibly trivial) movable curve, this means that $\hat{q}>0$ and ${\bf
q}=q_{a}D^{a}$ represents a nef divisor in $X$ with non-vanishing (positive)
intersection with any non-trivial movable curve. In these wormholes the dual
saxions have a profile of the form
$\hat{\ell}(\tau)=\hat{q}\,\tilde{\ell}(\tau)$ and
${\bm{\ell}}(\tau)=\ell_{a}(\tau)D^{a}={\bf q}\,\tilde{\ell}(\tau)$, where
$\tilde{\ell}(\tau)$ is as in (6.7) with $n=4$. Note that, since we will
consider the two sectors as decoupled as in (4.20), by “forgetting” the
$\hat{\ell}$ direction, one gets information on the profile of the marginally
degenerate wormhole of charge ${\bf q}$ as well.

Figure 7: Sample of $\bm{x}(\tau)$ trajectories corresponding to the charge
vector $\mathbf{q}=(1,1,1)$ in the F-theory model 2 of Section 4.1.2 (with
$p=3$), with throat position $\bm{x}_{*}$ displaced from the homogeneous
solution, indicated by a black star in the plot. The gray region is an
interpolation of the red $\bm{x}_{*}$ acceptable domain of figure 8(a). Small
displacements around the star lead to completely different trajectories.

According to the general arguments of Section 6.3, we also expect that there
exist other wormhole solutions whose initial position
$(\hat{\ell}_{*},{\bm{\ell}}_{*})$ is non-aligned but sufficiently closed to
the direction identified by the charge vector $(\hat{q},{\bf q})$. Note that
the scaling symmetry discussed at the beginning of Section 6.3 identifies
equivalence classes of solutions. It is hence convenient to rephrase our
discussion in the projectivized dual saxionic cone, obtained by modding out
their overall rescaling. In the present case, it is conveniently parametrized
by the vector

${\bm{x}}(\tau)\equiv
x_{a}(\tau)D^{a}\equiv\frac{\hat{q}\,{\bm{\ell}}(\tau)}{\hat{\ell}(\tau)}\,,$
(6.39)

which characterizes wormholes that are not equivalent under scaling symmetry.
Note that the homogeneous solution corresponds to the constant profile
${\bm{x}}(\tau)\equiv{\bf q}$ (hence with ${\bm{x}}_{*}={\bm{x}}_{\infty}={\bf
q}$), while other equivalence classes of solutions correspond to any other
profile ${\bm{x}}(\tau)$ completely contained in ${\cal P}_{\text{\tiny K}}$
and with ${\bm{x}}_{*}\neq{\bf q}$. Not only do we expect to find admissible
solutions for any ${\bm{x}}_{*}$ in a sufficiently small neighborhood of ${\bf
q}$, but we also expect that we can tune ${\bm{x}}_{*}$ in this neighborhood
to get any ${\bm{x}}_{\infty}\in{\cal P}_{\text{\tiny K}}$. Note also that by
the scaling symmetry, for any admissible solution ${\bm{x}}(\tau)$, we can
always find a non-empty subset of corresponding flows
$(\hat{\ell}(\tau),{\bm{\ell}}(\tau))$ that lie inside the $\alpha$-saxionic
convex hull.

(a) Scanned throat values ${\bm{x}}_{*}\equiv{\bm{x}}|_{r=L}$.

(b) Corresponding asymptotic values
${\bm{x}}_{\infty}\equiv{\bm{x}}|_{r=\infty}$.

Figure 8: Numerical scan obtained by integrating the equations of motion
(5.16) with charge vector $\mathbf{q}=(1,1,1)$ in the F-theory model of
Section 4.1.2 with $p=3$ and with random initial conditions for the dual
saxions. The first plot contains the allowed throat points ${\bm{x}}_{*}$
identified by the scan: all the points in the box have been randomly scanned,
but we just show the ones that lead to acceptable solutions, i.e. with
$x_{1},x_{2}>0$ all along their trajectory. The shape of the domains of
acceptable solutions are clearly distinguished and visible and have been
indicated with different colors. The black star denotes the point
corresponding to the homogeneous solution and the red dots in its neighborhood
can be identified with the solutions predicted by our general arguments, while
the blue set on the left represents “accidental” solutions, not predicted by
our argument. The second plot shows the asymptotic values ${\bm{x}}_{\infty}$
associated to the acceptable solutions. We can again clearly distinguish two
sets of points, corresponding to the two disjoint sets of the first plot: the
set corresponding to the red neighborhood of the black star in the first plot,
which is spread out all over $\mathbb{R}^{2}_{>0}$, as predicted by our
general argument; the set corresponding to the accidental solutions, which
identifies the blue thin denser cone which is visible in the second plot.

(a) Throat values ${\bm{x}}_{*}\equiv{\bm{x}}|_{r=L}$ of solutions with
boundary charge vector $\mathbf{q}=(1,0,1)\in{\cal C}_{\text{\tiny WH}}$.

(b) Asymptotic values ${\bm{x}}_{\infty}\equiv{\bm{x}}|_{r=\infty}$ of
solutions with boundary charge vector $\mathbf{q}=(1,0,1)\in{\cal
C}_{\text{\tiny WH}}$.

(c) Throat values ${\bm{x}}_{*}\equiv{\bm{x}}|_{r=L}$ of solutions with
boundary charge vector $\mathbf{q}=(1,1,0)\notin{\cal C}_{\text{\tiny WH}}$.

(d) Asymptotic values ${\bm{x}}_{\infty}\equiv{\bm{x}}|_{r=\infty}$ of
solutions with boundary charge vector $\mathbf{q}=(1,1,0)\notin{\cal
C}_{\text{\tiny WH}}$.

Figure 9: Same scan as in Figure 8 but with charges $\mathbf{q}=(1,0,1)$ and
$\mathbf{q}=(1,1,0)$, as indicated in the captions of the plots. While the
solution with vanishing $q_{1}$ charge is expected, the one with vanishing
$q_{2}$ charge is apparently surprising as the points with $\ell_{2}=q_{2}=0$
are at infinite distance. Indeed, a closer look at 9(c) reveals how getting
closer to the point associated to the would-be homogeneous solution (indicated
as star in the plot) the domain shrinks so that the the homogeneous solution
is never actually reached, confirming that $\ell_{2,*}=0$ never delivers a
valid configuration. This is a completely different behavior with respect to
figure 9(a), where the homogeneous solution has a “dense” neighborhood of
valid solutions.

We can more explicitly test our expectation by considering the simple models
described in Section 4.1.2 and in the Appendix C.1, in which ${\bm{x}}(\tau)$
should belong to $\mathbb{R}^{2}_{>0}$ and $\mathbb{R}^{3}_{>0}$ respectively.
Consider first the model of subsection 4.1.2. Ignoring for the moment the
sector associated with the dilaton (4.19), the cone of BPS instanton charges
is generated by the effective divisors $E^{1}$ and $E^{2}$. Hence, in the
basis of nef divisors $D^{1},D^{2}$ the components of a BPS instanton charge
${\bf q}=q_{1}D^{1}+q_{2}D^{2}\in{\cal C}_{\rm I}$ must satisfy
$q_{1}+pq_{2}\geq 0$ and $q_{2}\geq 0$. On the other hand, ${\bf q}$ belongs
to the subset ${\cal C}^{\text{\tiny EFT}}_{\rm I}$ of EFT instanton charges
only if $q_{1}\geq 0$ and $q_{2}\geq 0$. For instance the charge vector ${\bf
q}=E^{2}$, which corresponds to $(q_{1},q_{2})=(-p,1)$, is BPS but not EFT.
Including back the dilaton, according to our general claim there should exist
a large family of wormhole solutions for charges in the set (6.1), in addition
to the corresponding homogeneous wormholes which certainly exist. Recalling
(4.30), we see that ${\cal C}_{\text{\tiny WH}}$ includes charge vectors
$(\hat{q},q_{1},q_{2})$ with $\hat{q},q_{2}>0$ and $q_{1}\geq 0$. Let us first
consider charge vectors which are in the interior of ${\cal P}$, that is with
$\hat{q},q_{1},q_{2}>0$. One can numerically integrate the equations (5.16)
for different choices of the twisting constant $p>0$ (the case $p=0$ reducing
to a trivial factorized model), charges $\hat{q},q_{1},q_{2}>0$ and initial
values $(x_{*1},x_{*2})$. The results confirm our expectations, as exemplified
in Figs. 7 and 8. We stress that the specific choice of the charges used in
these plot has been made for purely visual reasons and we explicitly verified
that our qualitative conclusions hold also for more general charges. The plots
show how, even if the allowed throat values $(x_{1,*},x_{2,*})$ are confined
to a sharp specific sub-region, the corresponding asymptotic values
$(x_{1,\infty},x_{2,\infty})$ spread all over $\mathbb{R}^{2}_{>0}$, as
predicted by our general arguments. The set ${\cal C}_{\text{\tiny WH}}$
includes also charge vectors with $q_{1}=0$ and $\hat{q},q_{2}>0$, which are
on the finite distance boundary $\ell_{1}=0$ of ${\cal P}$, and we thus expect
to find non-homogeneous wormhole solutions around the homogeneous one for
these charges as well. This is indeed what we find, as shown in the plots of
Fig. 9(a) and 9(b). Almost surprisingly, as shown in Fig. 9(c) and 9(d), we
also find solutions for charge vectors with $\hat{q},q_{1}>0$ and $q_{2}=0$,
which are not included in ${\cal C}_{\text{\tiny WH}}$ and for which no
homogeneous solution is possible. A closer inspection of the plots reveals
however how these solutions do not form a dense open set around the would-be
homogeneous solution, with a spread distribution of end points
$(x_{1,\infty},x_{2,\infty})$, but rather look like the class of “accidental”
solutions of Fig. 8.

It is also interesting to explicitly check the correspondence between
fundamental EFT instantons and $n=3$ wormholes of section 6.4 beyond the
homogeneous and perturbative regime. We can do this by keeping only the
$\ell_{a}$ sector, which identifies an $n=3$ model, and performing the
numerical scan. Applying the IR regularization outlined in Section 6.4, which
introduces an IR cutoff radius $r_{\text{\tiny IR}}=\Lambda^{-1}_{\text{\tiny
IR}}$, we can test the correspondence by comparing the complete numerical
action of the IR regulated half-wormhole solutions to the associated BPS value
(5.35). The result of the scan is reported in the plots of Fig. 10, where we
show both the usual spread in the asymptotic values associated to deformations
around the homogeneous solutions and the comparison between the complete and
the BPS action. If we introduce the expansion parameter $\epsilon_{\text{\tiny
IR}}\equiv\Lambda^{2}_{\text{\tiny IR}}L^{2}\ll 1$, in the neighborhood of the
homogeneous solutions the actions coincide up to
$\mathcal{O}(\epsilon_{\text{\tiny IR}}^{2})$, exactly as predicted by (6.36).
Interestingly, a new region around $\ell_{1}=0$ also appears. This is not
surprising as it closely resemble the situation of Figs. 8 and 9. This time
the agreement between the complete and BPS action is roughly of
$\mathcal{O}(\epsilon_{\text{\tiny IR}})$.

(a) Scanned throat and asymptotic values of $x=\ell_{1}/\ell_{2}$.

(b) Ratio between the complete and the BPS action as a function of $x_{*}$.

Figure 10: Numerical scan obtained by integrating the equations of motion
(5.16) with charge vector $\mathbf{q}=(1,1)$ in the F-theory model of Section
4.1.2 without the dilaton and with random initial conditions for the dual
saxions. The left panel compares the allowed throat values of
$x_{*}=\ell_{1,*}/\ell_{2,*}$ to the asymptotic values $x_{\text{\tiny
IR}}=\ell_{1,\text{\tiny IR}}/\ell_{2,\text{\tiny IR}}$ found by the scan,
where acceptable solutions have been identified as the ones in which the dual
saxions degenerate at a distance greater than $r_{\text{\tiny IR}}=10L$ from
the throat. The black star denotes the homogeneous solution and the red dots
in its neighborhood can be identified with the solutions predicted by the
general perturbative argument, while the blue set on the left represents the
usual accidental solutions. We note that, as expected, a tiny $x_{*}$ interval
of $1$ is mapped to a much larger $x_{\text{\tiny IR}}$ interval. The second
plot shows the ratio $S_{\text{hw}}/S_{\text{BPS}}$ associated to the
acceptable solutions, where $S_{\text{hw}}$ is obtained by cutting off the
integration up to $r_{\text{\tiny IR}}$ as outlined in the main text and
$S_{\text{BPS}}$ is given by (5.35) . Around the homogeneous solution the two
actions are compatible up to $\mathcal{O}(\epsilon_{\text{\tiny IR}}^{2})$
terms, with $\epsilon_{\text{\tiny IR}}\equiv\Lambda_{\text{\tiny
IR}}^{2}L^{2}$, as predicted by (6.36), while in the accidental region the
deviation is slightly bigger (but still below
$\mathcal{O}(\epsilon_{\text{\tiny IR}})$).

A last crucial check regards the compatibility of our solutions with the
energy bounds imposed by the dominant EFT string scale (3.29). In section 6.2
we showed how on the homogeneous solution the condition $L^{2}M_{{\cal
T}}^{2}\geq 1$ is always guaranteed if $\mathbf{q}\in\mathcal{C}_{\text{WH}}$,
and argued that this should be the case also on more general solutions that
deviate from the homogeneous one. We quantitatively verified this claim on a
set of numerically obtained non-homogeneous solutions with charges
$q=(1,0,1)$, $q=(1,1,1)$ and $q=(3,1,2)$. The results are reported in figure
11, where we show the density histogram of $L^{2}M^{2}_{{\cal T}}$ normalized
to the one of the associated homogeneous solution. It is clearly visible how
the ratio between the dominant EFT string mass squared ($M^{2}_{{\cal T}}$)
and the wormhole scale squared ($1/L^{2}$) does not deviate much from one,
corresponding of the homogeneous solutions, guaranteeing the stability of
these numerical solution and supporting our previous claim.

(a) $(L^{2}M^{2}_{\mathcal{T}})_{\text{hom}}=4/\sqrt{3}$.

(b) $(L^{2}M^{2}_{{\cal T}})_{\text{hom}}=4/\sqrt{3}$.

(c) $(L^{2}M^{2}_{{\cal T}})_{\text{hom}}$ $=8/\sqrt{3}$.

Figure 11: Density histograms of $L^{2}M_{{\cal T}}^{2}$ evaluated at the
throat of a set of non-homogeneous numerical wormhole solutions for the
F-theory model of Section 4.1.2. The values of $L^{2}M_{{\cal T}}^{2}$ have
been normalized to the values of the associated homogeneous solution. The
results clearly show how the values do not deviate substantially from the ones
of the homogeneous solutions, guaranteeing $L^{2}M_{{\cal T}}^{2}\gtrsim 1$
for all these solutions, as required by compatibility with the species scale.

The model described in the appendix C.1 can be analyzed in the same way. In
this case, in addition to the dilaton (4.19), we have a three-dimensional
vector ${\bm{\ell}}=\ell_{1}D^{1}+\ell_{2}D^{2}+\ell_{3}D^{3}$ that takes
values in the Kähler cone. So, with focus on this sector, any ${\bf q}\in{\cal
C}^{\text{\tiny EFT}}_{\rm I}$ can be identified with the generic nef divisor
${\bf q}=q_{1}D^{1}+q_{2}D^{2}+q_{3}D^{3}$, with $q_{1},q_{2},q_{3}\geq 0$.
Again, the set ${\cal C}_{\rm I}$ of BPS instanton charges is larger, and is
generated by the effective divisors $E^{1,2,3}$ defined in (C.1). In
particular, if $p,h>0$, the generators $E^{1}$ and $E^{3}$ identify BPS
instanton charges $(q_{1},q_{2},q_{3})=(1,-p,0)$ and
$(q_{1},q_{2},q_{3})=(-h,0,1)$, respectively, which do not belong to ${\cal
C}^{\text{\tiny EFT}}_{\rm I}$. Including the dilaton as in the previous
model, the set (6.1) includes charge vectors with $\hat{q},q_{3}>0$ and
$q_{1},q_{2}\geq 0$. Again, we expect that for any choice of charges in ${\cal
C}_{\text{\tiny WH}}$, there should exist a large family of wormhole solutions
reaching all possible asymptotic values
$(x_{1,\infty},x_{2,\infty},x_{3,\infty})\in\mathbb{R}^{3}_{>0}$. By
numerically integrating (5.16) with initial conditions
$(x_{1,*},x_{2,*},x_{3,*})\in\mathbb{R}^{3}_{>0}$ one gets another clear
confirmation of our expectation, as evident from the plots in Fig. 12.

(a) Scanned throat values ${\bm{x}}_{*}\equiv{\bm{x}}|_{r=L}$.

(b) Corresponding asymptotic values
${\bm{x}}_{\infty}\equiv{\bm{x}}|_{r=\infty}$.

Figure 12: Numerical scan obtained by integrating the equations of motion
(5.16) with charge vector $\mathbf{q}=(1,1,1,1)$ in the $\mathbb{P}^{1}$ over
$\mathbb{F}_{p}$ F-theory model of App. C.1, with $p=h=1$ and random initial
conditions for the dual saxions. As in figure 8, the first plot contains the
acceptable throat points ${\bm{x}}_{*}$ identified by the scan. The shape of
the domain of acceptable solutions is clearly visible and generates a dense
set around the homogeneous solution, indicated as a black ball connected by
dashed gray lines for clarity. The second plot shows the asymptotic values
${\bm{x}}_{\infty}$ associated to the acceptable solutions, which spreads all
over $\mathbb{R}^{3}_{>0}$, in agreement with our general argument.

####  6.5.2 Wormholes in heterotic models

EFT instantons and wormholes in the heterotic models of Section 4.2 can be
discussed in a similar way. Hence we will be briefer, outlining only some
relevant points.

The set of BPS instanton charges $(q_{0},{\bf q})\in{\cal C}_{\rm I}$ is
realized by F1-strings wrapping effective curves ${\bf q}=q_{a}\Sigma^{a}$ and
$q_{0}$ NS5-branes wrapping the internal space – see [37] for more details.
The subset of EFT instanton charges ${\cal C}_{\rm I}^{\text{\tiny EFT}}$ is
obtained by imposing that the charges $(q_{0},{\bf q})$ satisfy the same
constraints (4.49) that must be obeyed by $(\ell_{0},{\bm{\ell}})$. Let us
first set $q_{0}=0$, so that (4.49) reduce to ${\bf q}=q_{a}\Sigma^{a}\in{\cal
P}^{\text{\tiny het}}_{\text{\tiny K}}$. The BPS instanton tower or sublattice
WGCs of Section 5.4 translate into geometrical statements on the existence of
irreducible effective curves that populate an infinite subset of ${\cal
C}_{\rm I}^{\text{\tiny EFT}}$ – see [92] for the corresponding statements in
M-theory compactification to five dimensions. Such EFT instantons correspond
to F1-strings wrapping movable curves. As for the movable divisors encountered
in Section 6.5.1, this implies that for some $m\in\mathbb{Z}_{\geq 1}$ the
charge $m{\bf q}$ can be represented by a curve that can freely move along the
internal space. Hence, as already observed in F-theory models, EFT instantons
correspond to microscopic configurations which can probe the entire
compactification space. These EFT instanton charges do not belong to the set
${\cal C}_{\text{\tiny WH}}$ associated with the complete perturbative regime
described by $(\ell_{0},{\bm{\ell}})$, since they have vanishing pairing with
EFT string charges $(e^{0},{\bf 0})$, $e^{0}>0$. Nevertheless, they
characterize the restricted perturbative regime obtained by removing the EFT
string charge $(1,{\bf 0})$ from the generators of $\Delta$, realizing an
example of the mechanism outlined in Footnote 24. In this regime, one can
“ignore” the saxionic combination $\hat{s}^{0}\equiv
s^{a}-\frac{1}{2}p_{a}s^{s}$ and the corresponding dual saxion $\ell_{0}$, and
only the second term of (4.46) remains relevant, whose homogeneity degree is
$n=3$. Hence, the corresponding EFT instanton charges can be at best supported
by marginally degenerate wormholes of the type discussed in Section 6.4. Since
EFT strings are realized by NS5-branes wrapping nef divisors [14, 37], the set
(6.1) corresponding to this restricted theory identifies movable curves ${\bf
q}=q_{a}\Sigma^{a}$ which have non-vanishing (positive) intersection with any
nef divisor. Our general arguments imply that marginally degenerate wormholes
should exist for any such curve. This certainly happens when ${\bf q}$ lies in
the interior of ${\cal P}^{\text{\tiny het}}_{\text{\tiny K}}$, and in these
cases at least the homogeneous wormholes certainly exist. Note that these
homogeneous wormholes coincide with the string theory wormholes first
constructed and studied in [29, 30]. We emphasize that this reasoning is based
on an EFT which, as discussed in Section 4.2, is fully reliable only if
$s^{0}\gg p_{a}s^{a}$. In this regime ${\cal T}_{*}=M^{2}_{\text{\tiny
P}}\ell_{0}$ determines a constant upper bound on the species scale, so that
(6.18) does not hold for such wormholes. This is expected, since as observed
above these charges do not belong to the set ${\cal C}_{\text{\tiny WH}}$
associated with the full perturbative regime, which includes $\ell_{0}$. In
order to make these wormholes fully reliable, we should go beyond the
$s^{0}\gg p_{a}s^{a}$ regime, as in the toy model of Footnote 27, but
unfortunately this is presently out of our reach. On the other hand, the
quasi-BPSness mentioned in Section 6.4, and further discussed in 7.4 below,
strongly suggests that these wormholes should survive away the $s^{0}\gg
p_{a}s^{a}$ regime.

We can then turn on a charge $q_{0}>0$, so that the EFT instanton uplifts to a
configuration involving Euclidean NS5-branes wrapping the entire internal
space. For generic choices of ${\bf q}=q_{a}\Sigma^{a}$ in the interior of
${\cal P}^{\text{\tiny het}}_{\text{\tiny K}}$, the entire Kähler and kinetic
potentials (4.46) and (4.48) are relevant, which have homogeneity $n=4$. Hence
there should exist corresponding homogeneous as well as non-homogeneous
solutions, which are fully non-degenerate. Finally, one could also pick
charges so that the relevant EFT has again $n=3$. Explicit examples could be
obtained through heterotic/F-theory duality, which maps NS5 instantons to some
D3-brane instantons in F-theory – see for instance the detailed discussion of
[97] and Appendix B. As an example, a D3-brane wrapping the divisor $D^{2}$ in
the F-theory model of Section 4.1.2 corresponds to an NS5-brane instanton in
the dual heterotic description. By duality the relevant EFT instanton has
homogeneity $n=3$ and admits marginally degenerate wormholes. Other explicit
examples could be obtained from the heterotic/F-theory dictionary discussed in
Appendix B.

Finally we notice that we can adapt (part of) of the above discussion to
wormholes associated with the Kähler moduli sector of IIA compactifications,
either with orientifolds or not (and, in the latter case, with enhanced ${\cal
N}=2$ supersymmetry), and with the complex structure sector of IIB models by
mirror symmetry. More general wormholes of type IIA orientifold
compactifications can be obtained by taking particular limits of the wormholes
discussed in the next section.

####  6.5.3 Wormholes in M-theory models

Finally, we would like to briefly discuss how our general considerations apply
to M-theory models compactified on $G_{2}$ manifolds – see Section 6.5 of [14]
and Section 8 of [37] for a summary of the relevant ingredients in the present
context. The effective theory of these models does not generically admit an
explicit description, basically because of the absence of a complex and Kähler
structure on the internal space. One can nevertheless say something about the
existence of corresponding wormholes.

The information on the $G_{2}$-holonomy metric of the internal seven-
dimensional space $Y$ is completely encoded in the associative three-form
$\Phi\in H^{3}(Y,\mathbb{R})$ [98]. The saxions are then obtained by expanding
$\Phi=s^{i}\omega_{i}$ in an appropriate cohomology basis $\omega_{i}\in
H^{3}(Y,\mathbb{Z})$. Alternatively, the same information is encoded in the
coassociative four-form $*_{Y}\Phi$, and its rescaled counterpart:

$\hat{\Phi}\equiv\frac{1}{2V_{Y}}*_{Y}\Phi\,,$ (6.40)

where $V_{Y}$ is the volume of $Y$. The dual saxions $\ell_{i}$ are then
obtained from the expansion

$\hat{\Phi}=\ell_{i}\hat{\omega}^{i}\,,$ (6.41)

where $\hat{\omega}^{i}\in H^{4}(Y,\mathbb{Z})$ is the dual cohomology basis,
such that $\int_{Y}\omega_{i}\wedge\hat{\omega}^{j}=\delta_{i}^{j}$. Both
$\Phi$ and $\hat{\Phi}$ parametrize corresponding cones, which can be
identified with the saxionic and the dual saxionic cone, respectively. BPS
instantons uplift to M2-branes wrapping internal submanifolds $\Sigma_{\bf q}$
calibrated by $\Phi$. The corresponding charge vectors can be identified with
the Poincaré dual four-forms ${\bf q}=[\Sigma]=q_{i}\hat{\omega}^{i}\in
H^{4}(Y,\mathbb{Z})$. Hence, according to our general definition the
cohomology class ${\bf q}$ identifies an EFT instanton if it admits, among its
representative closed four-forms, a coassociative four-form $\hat{\Phi}_{\bf
q}$. Note that the existence of calibrated submanifolds $\Sigma_{k{\bf q}}$
for any such ${\bf q}$ and some integer $k\geq 1$ is not obvious at all, and
would be necessary in order to realize the BPS instanton tower or sublattice
WGCs of Section 5.4.

The Kähler potential is given by the geometrical formula
$K=-3\log\int_{Y}\Phi\wedge*_{\Phi}\Phi$ [99], and then by (6.40) the kinetic
potential for the dual saxions takes the form (2.18) with

$\tilde{P}(\ell)=\left(\int_{Y}\hat{\Phi}\wedge*_{\hat{\Phi}}\hat{\Phi}\right)^{3}\,.$
(6.42)

In these geometric formulas the Hodge-operators $*_{\Phi}$ and
$*_{\hat{\Phi}}$ are implicitly defined by $\Phi$ and $\hat{\Phi}$,
respectively. Even if it is generically hard to make explicit the dependence
of (6.42) on the dual saxions, it is easy to see that it is homogeneous of
degree $n=7$. Indeed, a rescaling $\ell_{i}\rightarrow\lambda\ell_{i}$
corresponds to $\hat{\Phi}\rightarrow\lambda\hat{\Phi}$. This induces a
rescaling ${\rm d}s^{2}_{Y}\rightarrow\lambda^{-\frac{2}{3}}{\rm d}s^{2}_{Y}$
of the corresponding metric, and hence
$*_{\hat{\Phi}}\hat{\Phi}\rightarrow\lambda^{\frac{4}{3}}*_{\hat{\Phi}}\hat{\Phi}$.

From our general discussion we conclude that whenever we pick ${\bf q}$ inside
the dual saxionic cone ${\cal P}$, we certainly have ${\bf q}\in{\cal
C}_{\text{\tiny WH}}$ and there exists a corresponding homogeneous wormhole
with $n=7$, as well as its non-homogeneous variations. We expect similar
conclusions if ${\bf q}$ belongs to some finite field distance boundary of
${\cal P}$. Charges ${\bf q}$ on some infinite distance boundary of ${\cal P}$
can still correspond to sensible wormholes in restricted perturbative regimes,
possibly with $n<7$ – see the discussion at the end of Section 6.1.
Furthermore, by considering $G_{2}$ compactifications admitting a weakly
coupled IIA description [100], one can immediately adapt these conclusions to
type IIA compactifications on Calabi-Yau spaces with O6-planes.

##  7 Low-energy implications

In this section we would like to discuss some of the physical implications of
wormholes. We mostly follow the standard literature [9, 26, 101], see also
[10] for a review and a more complete list of references.

Consider a full (two-sided) non-extremal wormhole solution of the type
discussed in the previous sections, illustrated in Fig. 13(a). One can view it
as a conduit between two distinct asymptotically flat universes. Yet, the same
solution also provides an accurate description of a short cut connecting two
regions of the same background geometry separated by a distance much larger
than the wormhole “thickness” $L$, as shown in figure 13(b). In either case,
at energy scales much smaller than $1/L$, wormholes can be “integrated out”
and their effect codified in the appearance of appropriate interactions in an
EFT defined at a new lower Wilsonian cutoff $\Lambda\ll 1/L$. These
interactions are intrinsically non-local, and the leading order effects are
captured by bi-local operators of the form [102, 101]

$\int{\rm d}^{4}x\sqrt{|g(x)|}\int{\rm
d}^{4}y\sqrt{|g(y)|}~{}C^{IJ}\,\overline{\cal O}_{I}(x){\cal O}_{J}(y)\,,$
(7.1)

with ${\cal O}_{I}$ local gauge-invariant operators and
$C^{IJ}=\overline{C^{JI}}$ denoting a dimensionless non-degenerate constant
matrix. In our setting, some properties of the operators ${\cal O}_{I}(x)$ can
be understood from general principles. Given a wormhole charge ${\bf
q}\in{\cal C}_{\text{\tiny WH}}$, one can pick a basis in which ${\cal
O}_{I}(y)$ represents the insertion of the corresponding half-wormhole, while
$\overline{\cal O}_{I}(x)$ represents the insertion of the anti-half-wormhole
of charge $-{\bf q}$. Based on symmetry considerations alone, each ${\cal
O}_{I}(x)$ should be proportional to $e^{2\pi{\rm i}q_{i}a^{i}(x)}$ and each
$\overline{\cal O}_{I}$ to $e^{-2\pi{\rm i}q_{i}a^{i}}$. These phases
precisely match the contribution of the boundary term (2.16) evaluated on the
asymptotic boundary of each half-wormhole. Moreover, the entire contribution
in (7.1) should be suppressed by $e^{-S_{\text{tot}}|_{\rm w}}$, where
$S_{\text{tot}}|_{\rm w}$ denotes the _real_ part of the complete double-sided
wormhole on-shell action (including possible higher-derivative terms). These
two effects combine into the following universal dependence of the effective
operators on the background fields

${\cal O}_{I}\ \propto\ e^{-S_{\text{tot}}|_{\rm hw}}e^{2\pi{\rm
i}q_{i}a^{i}}\,,$ (7.2)

where the action of half a wormhole $S_{\text{tot}}|_{\rm
hw}\equiv\frac{1}{2}S_{\text{tot}}|_{\rm w}$ has made its appearance, as we
anticipated above (5.26).

(a) Wormhole connecting two different asymptotic spaces.

\begin{overpic}[angle={0},width=230.0pt]{Figures/wh1prova.jpg}
\put(49.0,53.0){$L$} \end{overpic} (b) Wormhole connecting two regions of the
same asymptotic space.

Figure 13: Cartoons of a wormhole connecting two different asymptotic spaces
(left panel) and two regions of the same asymptotic space separated by a
distance much greater than the wormhole neck radius $L$ (right panel).

The bi-local contributions (7.1) can be rewritten in a manifestly local form
by introducing the so-called Coleman’s $\alpha$-parameters [26]. Explicitly,
(7.1) is equivalent to the insertion of the local terms

$S_{\rm eff}=\int{\rm d}^{4}x\sqrt{|g|}\,\alpha^{I}{\cal
O}_{I}(x)+\text{c.c.}\,,$ (7.3)

provided one includes the Gaussian integration

$\int{\rm d}\alpha\,{\rm
d}\bar{\alpha}\,e^{-\alpha^{I}C^{-1}_{IJ}\bar{\alpha}^{J}}$ (7.4)

in the path integral. Corrections to (7.1), as well as a non-trivial
integration over otherwise disconnected geometries, affect the Gaussian weight
but do not change the qualitative conclusion [102, 101]: wormholes can be
viewed at low-energies as insertions of local operators provided an
integration over seemingly innocuous “parameters” $\alpha^{I}$ is performed.

Wormholes are equivalently understood as non-perturbative tunneling processes
in which “baby universes” are created or annihilated with amplitude $\sim
e^{-S_{\text{tot}}|_{\rm hw}}$. The terms in (7.3) describe the creation and
absorption of any number of baby universes by the perturbative vacuum state
$|0\rangle$ we started from. The $\alpha^{I}$-parameters are expectation
values of the creation-annihilation operators $\hat{\alpha}^{I}$ of such baby
universes. One can then go to the so-called $\alpha$-vacua $|\alpha\rangle$,
in which $\hat{\alpha}^{I}|\alpha\rangle=\alpha^{I}|\alpha\rangle$. The
$\alpha$-vacua decohere and a given universe can be thought of as being in a
superselection sector labeled by a specific set of $\alpha^{I}$-parameters,
which appear in the low-energy effective term (7.3).

The presence of an Hilbert space of baby universe is one of the most striking,
but also subtle, outcomes of Coleman’s viewpoint on wormholes – see also [27]
for an interesting recent revisitation of this interpretation. The associated
integration over $\alpha^{I}$-parameters makes it impossible for the cluster
decomposition principle to hold. This in general implies the non-factorization
of correlation functions [74]. In the context of the present paper, the
fundamental quantum indeterminacy of the $\alpha$-parameters also appears in
tension with experience from string theory, in which all couplings are
dynamical and no free parameters exist, as also expected for more general
quantum gravity models [28].

There is yet another implication of the above logic that may appear puzzling –
see also the discussions in [103, 104]. It may seem to imply that no effective
potential for our axions can be induced. In fact, according to (7.1), complete
wormholes subtract charge from $x$ and place it in another point $y$. An
observer around $x$ would thus experience a _local_ violation of charge
conservation, signaled by an anomaly in the Noether current:

$\nabla_{\mu}J^{\mu}=\sum_{IJ}q_{J}C^{IJ}\left[\left(\int\overline{\cal
O}_{I}\right){\cal O}_{J}-\overline{\cal O}_{I}\left(\int{\cal
O}_{J}\right)\right].$ (7.5)

Yet, _globally_ charge remains conserved.323232This continues to hold when
corrections to the dilute instanton gas approximation, in the form of wormhole
and instanton interactions as described in [101], are included. In this
situation, no effective potential for the axion can be generated. In order to
induce such effect, the anomaly in (7.5) should have an overlap with the
vacuum and a state of axions at rest, and this does not seem to happen here
because the right-hand side of (7.5) is completely neutral. The only logical
way to obtain an axion potential is if for some mysterious reason part of the
wormholes did not bring back the charge to our Universe. In that case the
symmetry between outgoing and incoming charge would be broken, and the vacuum
would spontaneously break the shift symmetry. In the language of the
$\alpha^{I}$-parameters, that peculiar condition is realized when some of the
$\alpha^{I}$’s acquire a non-trivial vacuum expectation value, in conflict
with what (7.4) seems to indicate.

More generally, a natural way to avoid all the problems mentioned above is the
existence of a huge gauge redundancy that identifies different $\alpha$
states, implying that the Hilbert space of baby universes is in fact one-
dimensional [27, 28]. In this case all $\alpha$ parameters would have unique
values, controlled by some moduli fields. They would hence reduce to ordinary
EFT couplings and no violation of factorization would take place. The Baby
Universe Hypothesis [28] proposes that this is indeed what happens in any
consistent $d>3$-dimensional quantum gravity model. If correct, the Baby
Universe Hypothesis would remove any arbitrariness in the
$\alpha^{I}$-parameters appearing in (7.3), allowing wormholes to break
axionic symmetries, at least in principle.

In the following we assume that this “natural selection” of the
$\alpha^{I}$-parameters actually takes place, but also provide arguments
indicating that such selection must actually occur, at least in the case of
$n=3$ non-extremal wormholes.

We begin with a discussion of the impact of the Gauss-Bonnet term on the low-
energy theory (Section 7.1). Subsequently we analyze more explicitly the
structure of the low-energy EFT with $\Lambda\ll 1/L$. Specifically, the
underlying supersymmetry imposes important restrictions on the wormhole-
induced effective action: the local operators ${\cal O}_{I}(x)$ should
organize themselves in supersymmetric multiplets, and enter the low-energy
effective action either through D-terms or F-terms. The a priori infinite
series appearing in (7.3) is expected to be dominated by the lowest derivative
terms. We see in Section 7.2 that F-terms can be generated by EFT instantons
(extremal wormholes), whereas regular wormholes can only generate D-term
corrections to the Kähler potential at leading order (see Section 7.3). As we
will see in Section 7.4, the particular case of marginally degenerate
wormholes ($n=3$) is associated with possible non-perturbatively generated
superpotentials. Some phenomenological consequences for axion physics are
discussed in Section 7.5.

###  7.1 Large $N$ suppressions and Gauss-Bonnet

The two-derivative contribution to the half-wormhole action is bounded from
below by (5.27), which codifies the combined gravitational-axionic
contribution of the wormhole neck. Combining this observation with (5.28),
which implies that any non-extremal wormhole solution can make sense only if
$L^{-1}$ is smaller than the species scale evaluated at the wormhole’s neck,
and the consistency condition $M_{\text{sp}}\leq M_{\text{\tiny QG}}$ (see
Section 3.1 and (3.24)), we obtain

$S|_{\rm hw}\geq 3\pi^{3}\frac{M^{2}_{\text{\tiny
P}}}{M_{\text{sp}}^{2}(s_{*})}\geq\frac{3\pi}{16}N_{\text{sp}}\,$ (7.6)

where $N_{\text{sp}}>N$ by construction. At the leading two-derivative level
the low-energy operators (7.2) are therefore suppressed by

$e^{-S|_{\rm hw}}\leq e^{-\frac{3\pi}{16}N_{\text{sp}}}\,.$ (7.7)

Even before asking whether F-terms or D-terms are actually induced or not (for
that we will have to wait Sections 7.2, 7.3, 7.4), we therefore see that
wormhole effects must be very small. Clearly, the larger the number of degrees
of freedom accessible to the QFT description, the smaller the UV cutoff, and
the less relevant wormholes are.

Strictly speaking, (7.7) is an accurate estimate of the size of wormholes
insertions only if a perturbative four-dimensional description can be extended
all the way to the species scale. Presumably, this holds when the tower scale
$M_{\text{t}}$ of Section 3.4 corresponds to the string mass scale. In that
case, generic higher dimensional operators are not expected to affect our
estimate qualitatively because suppressed by powers of $1/(M_{\text{sp}}L)<1$.
On the other hand, if $M_{\text{t}}$ denotes a KK mass, generically our four-
dimensional description would be accurate only for
$L>1/M_{\text{\tiny{KK}}}>1/M_{\text{sp}}$. Hence the purely four-dimensional
contribution to the wormhole action would be $S|_{\rm hw}\geq
3\pi^{3}M^{2}_{\text{\tiny P}}/M_{\text{\tiny KK}}^{2}$ and the suppression of
a wormhole insertion much more significant than shown in (7.7). Of course,
more interesting effects may potentially arise integrating in the KK
resonances. For $L\ll 1/M_{\text{\tiny KK}}$ one may find higher-dimensional
uplifts of our wormholes – see [105] for explicit examples of uplifted
wormholes in string theory – that potentially have smaller actions than in a
strict four-dimensional regime with $L\gtrsim 1/M_{\text{\tiny KK}}$. A
careful investigation of $d$-dimensional uplift of our wormholes would be
necessary to confirm that, but this is beyond the scope of the present paper.
In the following we will thus proceed using only results that are under our
control. In that view (7.7) provides a conservative measure of the strength of
non-perturbative gravitational effects in the axiverse.

While the previous discussion indicates that higher-derivative operators are
generically not expected to change our conclusions qualitatively, as we
emphasized in Section 2.2 this may not be the case for (semi-)topological
terms. Let us thus discuss them. When considering wormhole physics, it turns
out that the Pontryagin term is completely irrelevant. As discussed below
(5.7), the Pontryagin operator itself exactly vanishes by spherical symmetry
on the wormhole solution (5.20). More generally, the topological structure of
a general non-spherically-symmetric wormhole configuration is such that the
Pontryagin index (supplemented with the appropriate boundary term) vanishes.
We can therefore safely ignore this term and focus on GB.

The potential relevance of the GB term for wormhole physics was emphasized in
[9, 85]. We find that indeed GB gives a _positive_ contribution to the on-
shell action, at least if we are allowed to treat it as a perturbation. In
such a case, at leading non-trivial order its effect on the equations of
motion can be neglected when evaluating the on-shell action. Within this
approximation $(S+S_{\text{\tiny GB}})|_{\text{hw}}$ is simply obtained by
plugging in the leading order solution. Furthermore, in evaluating (5.2) on
our solution we can practically ignore the asymptotic boundary term, since the
wormhole metric (5.20) quickly becomes flat and so $Q\rightarrow Q_{0}$ as
$r\rightarrow\infty$. Hence, the on-shell action (5.2) for the wormhole
reduces to the purely bulk term

$\displaystyle S_{\text{\tiny GB}}=-\int_{\cal
M}\sqrt{g}\,\gamma(\ell)\,E_{\text{\tiny GB}}\,.$ (7.8)

Now, the GB density (2.6) of a wormhole metric (5.20) is negative definite
since $E_{\text{\tiny GB}}|_{\text{wh}}=-{3L^{8}}/({2\pi^{2}r^{12}})$. So,
denoting by $\gamma_{\rm min}$ the minimum value attained by $\gamma(\ell)$
along the dual saxion flow, we conclude that the on-shell Gauss-Bonnet term
(7.8) of a semi-wormhole satisfies a lower bound

$S_{\text{\tiny GB}}|_{\text{hw}}=-\int_{{\rm
hw}}\sqrt{g}\,\gamma(\ell)\,E_{\text{\tiny GB}}\geq-\gamma_{\rm min}\int_{\rm
hw}\sqrt{g}\,E_{\text{\tiny GB}}=\gamma_{\rm min}\,,$ (7.9)

where in the last step we used the fact that $\int_{\rm
hw}\sqrt{g}\,E_{\text{\tiny GB}}=-1$, which we obtained by dividing by two the
integral of $E_{\text{\tiny GB}}$ over the full wormholes of Fig.
13.333333Recalling (5.3), for asymptotically flat spaces with no finite
distance boundaries we have $\int_{{\cal M}}\sqrt{g}\,E_{\text{\tiny
GB}}=\chi({\cal M})-\chi({\cal M}_{0})$, where ${\cal M}$ denotes a general
wormhole configuration with $S^{3}$ boundaries at asymptotically flat
infinities and ${\cal M}_{0}$ the corresponding (possibly reducible) flat
space with the same asymptotic behavior, since the corresponding boundary
terms in (5.3) cancel each other. If ${\cal M}$ is as in Fig. 13(a), then
$\chi({\cal M})=0$ and $\chi({\cal
M}_{0})=\chi(\mathbb{E}_{4}\cup\mathbb{E}_{4})=2$, which gives $\int_{{\cal
M}}\sqrt{g}\,E_{\text{\tiny GB}}=-2$. Consistently, repeating the calculation
for ${\cal M}$ as in Fig. 13(b) one gets the same result.

The r.h.s. of (7.9) reproduces the topological contribution considered in [9],
where a constant $\gamma$ was assumed. Restricting our considerations to the
controllable domain $\hat{\Delta}_{\alpha}$ and combining (7.9) and (3.21), we
get

$S_{\text{\tiny GB}}|_{\text{hw}}\geq\gamma_{\rm
min}\geq\frac{N\pi}{6\alpha}\,,$ (7.10)

indicating that (7.2) contains, on top of (7.7), a further suppression of
order

$e^{-S_{\text{\tiny GB}}|_{\text{hw}}}\leq e^{-\frac{N\pi}{6\alpha}}.$ (7.11)

The condition (3.27) is equivalent to $S|_{\rm hw}\gtrsim S_{\text{\tiny
GB}}|_{\rm hw}$, as required in a self-consistent derivative expansion. Hence,
(7.11) only provides a correction to the dominant effect shown in
(7.7).343434It is interesting to observe that the large suppression of
$e^{-\gamma}$ also guarantees the weak-coupling regime of a possible third-
quantization of the type proposed in [106].

In conclusion, it is not possible to precisely quantify the suppression of
wormholes insertions, as the dominant contribution (7.7) depends on unknown
details of the UV completion. Yet, the relation $S|_{\rm hw}\gtrsim
S_{\text{\tiny GB}}|_{\rm hw}$ reminds us that a rough lower bound may be
provided by the GB contribution alone. Interestingly, this itself is bounded
from below by (7.10), via a bound expressed solely in terms of quantities
accessible to the low-energy observer. We can thus confidently claim that
wormholes are substantially suppressed in the $N\gg 1$ limit. Even for
moderately small couplings $\alpha=\pi/6\sim 0.5$ and just $N=100$ axions –
which is a quite natural number in stringy axiverse models – the upper bound
in (7.11) is of order $\sim 10^{-44}$. The actual suppression could be much
stronger than (7.11), though, since (3.21) is generically quite conservative,
as remarked below that equation. For instance, in the F-theory models of
Section 4.1 the argument in the exponential of (7.11) must be corrected at
least by a factor of six – see (4.15). Furthermore, in the heterotic models of
Section (4.2), the lower bound on the GB coefficient is even stronger. The
numerical investigation summarized in Fig. 2 confirms our analytic
expectations and also suggests that the GB coefficient $\gamma(s)$ can in fact
scale with higher powers of $N$ as soon as $N>100$. Wormhole effects are
evidently very suppressed in the axiverse.

###  7.2 Extremal wormholes and F-terms

We now turn to a more explicit discussion of the leading low-energy terms that
can be induced by wormholes. We begin with the BPS extremal wormholes
introduced in Section 5.3. Conceptually, those objects appear to be
qualitatively different from non-extremal wormholes, since they are more
naturally interpreted as configurations sourced by localized EFT instantons
and do not entail the introduction of $\alpha^{I}$-parameters nor baby
universes. Nevertheless, the present discussion will serve as a reference for
our subsequent investigation of non-extremal wormholes.

Our main concern here is to understand whether extremal wormholes are
described at low energies via superpotential or Kähler terms. A necessary
condition for a superpotential term to be generated is that the extremal
wormhole configuration admits exactly two fermionic zero-modes. These would
correspond to ’t Hooft vertices with two fermion insertions, which is
something that only a superpotential term can provide. As shown in Appendix
D.1, extremal wormholes break two of the four supercharges preserved by the
underlying flat spacetime. The corresponding two Golstino-like zero-modes
(D.13) are contained in the chiralinos $\bar{\chi}^{i\,\dot{\alpha}}$, the
partners of $\bar{t}^{i}$. One would therefore expect extremal wormholes to be
able to induce an effective superpotential at low energies. Let us elaborate
on this possibility.

Because of the two zero-modes, a single extremal wormhole can only generate
non-vanishing contributions to correlation functions involving at most two
$\bar{\chi}^{i\,\dot{\alpha}}$ in excess over $\chi^{i}_{\alpha}$. We thus
focus on the two-point function
$\langle\bar{\chi}^{i\dot{\alpha}}(x)\bar{\chi}^{j\dot{\beta}}(y)\rangle$. We
estimate it in a UV theory sourced by (5.35) ignoring for the moment possible
subtleties related to the insertion of the fundamental instanton (we will come
back to this shortly). Explicitating only the integration over the zero-modes
– the massive modes can only affect the overall factor – we schematically get

$\displaystyle\langle\bar{\chi}^{i\,\dot{\alpha}}(x)\bar{\chi}^{j\,\dot{\beta}}(y)\rangle\propto$
$\displaystyle\ e^{-S_{\rm BPS}+2\pi{\rm i}\langle{\bf
q},{\bm{a}}_{\infty}\rangle}\int{\rm d}^{4}x_{0}\int{\rm
d}^{2}\theta\,[\bar{\chi}_{(0)}^{i\,\alpha}(x-x_{0})]^{\dot{\alpha}}\theta_{\alpha}\,[\bar{\chi}_{(0)}^{j\,\beta}(y-x_{0})]^{\dot{\beta}}\theta_{\beta}$
(7.12) $\displaystyle\sim$ $\displaystyle\
\,e^{\frac{1}{2}K_{\infty}}e^{2\pi{\rm i}\langle{\bf
q},{\bm{t}}_{\infty}\rangle}{{\cal G}}^{im}_{\infty}q_{m}{{\cal
G}}^{jn}_{\infty}q_{n}\int{\rm d}^{4}x_{0}\,[S_{\rm
F}(x-x_{0})]^{\dot{\alpha}}_{\alpha}[S_{\rm
F}(y-x_{0})]^{\dot{\beta}}_{\beta}\varepsilon^{\alpha\beta}$

with the complete complex BPS on-shell action (5.37) appearing as an overall
factor. In deriving (7.12) we replaced
$\bar{\chi}^{i\,\dot{\alpha}}(x)=[\bar{\chi}_{(0)}^{i\,\beta}(x-x_{0})]^{\dot{\alpha}}\theta_{\beta}+\cdots$
(the dots refer to non-zero-modes), where the zero-mode content of the field
is parametrized by the wavefunction defined in (D.13), with $x_{0}$ and
$\theta_{\alpha}$ denoting the associated bosonic and Grassmanian parameters.
In the final step we used the fact that in the limit
$|x-x_{0}|,|y-x_{0}|\to\infty$ the asymptotic dependence of the Goldstino
wavefunction is precisely the same as that of the Feynman’s propagator $S_{\rm
F}$, see (D.15), whereas the overall factor of $e^{K_{\infty}/2}$ is due to
its normalization.

It is now easy to show that the above correlator can be reproduced, at
distances sufficiently far from the singularity, by a supergravity F-term (we
adopt the conventions of [38])

$\int{\rm d}^{2}\theta\,2{\cal E}\,W_{\bf q}\,$ (7.13)

where

$W_{\bf q}={\cal A}_{\bf q}\,M^{3}_{\text{\tiny P}}\,e^{2\pi{\rm i}\langle{\bf
q},{\bm{t}}\rangle}\,,$ (7.14)

and ${\cal A}_{\bf q}$ includes the contribution of the non-zero-modes. The
superpotential (7.14) induces an effective fermionic vertex
$e^{K/2}(\partial_{i}\partial_{j}W_{\bf
q})\chi^{i}_{\alpha}\chi^{j}_{\beta}\varepsilon^{\alpha\beta}$, and observing
that
$\langle\chi^{i}_{\alpha}(x_{0})\bar{\chi}^{j\dot{\alpha}}(x)\rangle\propto{\cal
G}^{ij}[S_{\rm F}(x-x_{0})]^{\dot{\alpha}}_{\alpha}$, we see that a single
insertion of (7.14) would precisely reproduce the structure in (7.12),
provided ${\bm{t}}_{\infty}$ is identified with the classical background in
the effective field theory. In analogy to [107] we can thus conclude that at
low energies the BPS extremal wormhole of charge ${\bf q}\in{\cal
C}^{\text{\tiny EFT}}_{\rm I}$ generates the superpotential (7.14). BPS
extremal anti-wormholes of charge $-{\bf q}\in{\cal C}^{\text{\tiny EFT}}_{\rm
I}$ would instead generate the complex conjugate of (7.13). See [89] for a
discussion along similar lines.

Before jumping to the conclusion that extremal wormholes can generate
superpotentials at low energies, we should however address the subtlety we
alluded to earlier. Recall that extremal wormholes are singular and that a
localized UV-sensitive contribution (5.35) had to be included in our argument.
Additional contributions may be present, though. Indeed, fundamental
instantons generically carry additional internal degrees of freedom, which can
in principle contribute both zero and non-zero fermionic modes. The presence
of additional fermionic zero-modes may invalidate our calculation and for