††institutetext: Department of Physics and Astronomy, Uppsala University, Box
516, SE-75120, Uppsala, Sweden

# Charged Nariai black holes on the dark bubble

Ulf Danielsson & Vincent Van Hemelryck<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we realise the charged Nariai black hole on a braneworld from a
nucleated bubble in AdS5, known as the dark bubble model. Geometrically, the
black hole takes the form of a cylindrical spacetime pulling on the dark
bubble. This is realised by a brane embedding in an AdS5 black string
background. Identifying the brane with a D3-brane in string theory allows us
to determine a relation between the fine structure constant and the string
coupling, $\alpha_{\text{EM}}=\frac{3}{2}g_{s}$, which was previously obtained
for a microscopic black hole. We also speculate on the consequences for the
Festina Lente bound and neutrino masses.

††preprint: UUITP-13/24

###### Contents

  1. 1 Introduction
  2. 2 The dark bubble scenario
  3. 3 Reissner-Nordström-de Sitter black holes and the Nariai limit
  4. 4 The 5d black string
  5. 5 Glueing two black string solutions together
    1. 5.1 Solving the junction conditions
    2. 5.2 Matching with 4d Nariai
  6. 6 Relating fine structure constant to the string coupling
  7. 7 Discussion
  8. A The dilatonic coupling
  9. B Energy-momentum tensor on the brane
  10. C Fundamental string charge

##  1 Introduction

The construction of four-dimensional de Sitter vacua from string
compactifications remains a tremendous challenge in string phenomenology
Danielsson:2018ztv ; Obied:2018sgi ; Ooguri:2018wrx ; Bedroya:2019snp .
Additionally, string models with phases of variable cosmic acceleration have
been under pressure as well, see refs. Hebecker:2019csg ; Cicoli:2021fsd ;
Cicoli:2021skd ; Rudelius:2021azq ; Rudelius:2022gbz ; Andriot:2022xjh ;
Shiu:2023nph ; Shiu:2023fhb ; Hebecker:2023qke ; Freigang:2023ogu , however
see refs. Andriot:2023wvg ; Andriot:2024jsh  for studies with open universes.
Therefore, the fate of cosmic acceleration in string compactifications remains
an important topic of study in the Swampland program. All these investigations
start from string compactifications down to a 4d effective field theory, but
contrarily, in the dark bubble scenario, this starting point is left behind.

According to the dark bubble model, the universe is riding an expanding bubble
in AdS5, with the radial direction along the throat Banerjee:2018qey . The
bubble is formed through a phase transition, where a bubble of true vacuum
nucleates. Its wall consists of a brane with a tension that is slightly less
than the critical value, with the difference setting the value of the positive
cosmological constant. The model has been studied and developed in a series of
papers, see refs. Banerjee:2018qey ; Banerjee:2019fzz ; Banerjee:2020wix ;
Banerjee:2020wov ; Banerjee:2021qei ; Danielsson:2022fhd ; Danielsson:2022lsl
; Danielsson:2023alz ; Basile:2023tvh ; Banerjee:2023uto , with the earlier
results reviewed in refs. Banerjee:2021yrb ; Banerjee:2022ree  and related
studies in Berglund:2022qsb ; Bandos:2023yyo ; Koga:2019yzj ; Koga:2020jok ;
Koga:2022opd ; Basile:2020mpt ; Basile:2021vxh . In Danielsson:2022lsl ;
Danielsson:2023alz  the dark bubble model was embedded into a specific string
theory setting, where the unique hierarchy of scales of the model was
explored. In standard compactifications, as well as in RS-brane worlds, the 5d
Planck length is always larger than the 4d one, provided that the size of the
extra dimension, and the AdS length scale, are larger than the higher
dimensional Planck length. The dark bubble, which has a brane with an inside
and an outside, rather than the two insides of the RS-brane, is fundamentally
different. What matters is the difference in AdS scale between the inside and
the outside, which leads to a smaller 5d Planck length than the 4d one. This
leads to several unique phenomenological features - including no need for
scale separation in the higher dimensions.

As the 4d spacetime is defined on a shell rather than in a vacuum of the
theory, the standard rules of effective field theory do not directly apply to
the 4d setup. For this reason, it is rather non-trivial to study the physics
on the dark bubble, such as particle physics, gravitational waves, gauge
theories or even black holes. Nevertheless, the dark bubble model needs to
incorporate the matter fields we are familiar with in 4d. In
Danielsson:2022fhd  it was shown how 4d gravitational waves are induced from
5d ones. The analysis took the expansion of the universe into account, and
provided a non-trivial test of the model, demonstrating the intricate
interplay between the 5d bulk, the embedding of the brane, and the induced
metric in the 4d spacetime. In Basile:2023tvh , a further step was taken,
where electromagnetism was identified with the gauge field in the DBI-action
of a D3-brane on which the universe is riding. To achieve this, it was crucial
to include the presence of the $H=\mathrm{d}B$ three-form field of string
theory. As was shown in Basile:2023tvh , the electromagnetic field strength in
4d sources an $H$-field in the bulk, which then backreacts on the 5d geometry.
This affects the induced 4d metric such that the 4d gravitational backreaction
is accounted for. The analysis there was made in an expanding universe, and
is, again, a very non-trivial result, confirming the consistency of the dark
bubble model.

In this paper, we consider another case that further tests the dark bubble
model: a charged Nariai black hole as large as the cosmological horizon. Such
black holes have been studied in the context of the Swampland paradigm,
Montero:2019ekk , where it has been argued that their existence leads to new
constraints on particle masses beyond those of the weak gravity conjecture. In
this paper, we show that the dark bubble scenario can successfully incorporate
a Nariai black hole. The analysis provides yet another explicit example of how
the dark bubble reproduces 4d Einstein gravity without localising it on the
brane, a characteristic of the RS model. Instead, the bulk plays an important
role in the construction, and the failure to localise yields well-controlled
corrections that become important only at stringy densities, Basile:2023tvh .

Furthermore, we find a relation between the fine structure constant and the
string coupling, which is in agreement with the results of ref.
Danielsson:2023alz . There, using an explicit string theory embedding of a
small, extremal black hole of unit charge, it was shown that the fine
structure constant $\alpha_{\text{EM}}$ is fixed by the string coupling
$g_{s}$ through

$\alpha_{\text{EM}}=\frac{3}{2}g_{s}\,.$ (1)

Remarkably, the same relation can be derived from the embedding of the
macroscopic Nariai black hole.

The outline of the paper is as follows. In section 2 we review some technical
aspects of the dark bubble model. In section 3 we review the 4d Nariai
solution. In section 4 we study a 5d black string solution, which we use in
section 5 to uplift the Nariai solution. In section 6 we match the fine
structure coupling to the string coupling, and in section 7 we end with some
conclusions.

##  2 The dark bubble scenario

In the dark bubble scenario, a bubble with an AdS5 space inside is joined
together with another AdS5 spacetime on the outside by a brane. The dynamics
of the brane are governed by two sets of key equations, i.e. the Israel
junction conditions and the Gauss-Codazzi equations. The junction conditions
are simply the 5d Einstein equations in the presence of a thin brane, relating
the extrinsic curvatures on the in- and outside with the stress tensor on the
brane. The Gauss-Codazzi equation relates the 5d bulk curvatures and the
extrinsic curvatures of the embedding, to the intrinsic 4d curvatures of the
brane. We discuss the junction conditions at length in section 5.1 and refer
the reader to its discussion there.

However, it is interesting to emphasise that these junction conditions and
Gauss-Codazzi equations can be combined into the 4d Einstein equations, as was
done in Banerjee:2019fzz :

$G_{ab}^{(4)}=\kappa_{4}^{2}\left((-\sigma_{c}h_{ab}-S_{ab})+\frac{1}{2\kappa_{5}^{2}}\left[\left(\frac{\mathcal{J}^{+}_{ab}}{k_{+}}-\frac{\mathcal{J}^{-}_{ab}}{k_{-}}\right)-\frac{1}{2}\left(\frac{\mathcal{J}^{+}}{k_{+}}-\frac{\mathcal{J}^{-}}{k_{-}}\right)h_{ab}\right]\right)=\kappa_{4}^{2}T_{ab},$
(2)

where

$\mathcal{J}_{ab}=R^{(5)}_{\alpha\beta\gamma\delta}e^{\alpha}_{\;c}e^{\beta}_{\;a}e^{\gamma}_{\;d}e^{\delta}_{\;b}h^{cd}\,,$
(3)

and $e^{\alpha}_{\;b}$ are the tangent vectors to the brane for every $b$. The
“$-$” and “$+$” correspond to the contributions from the in-in- and outside
spacetimes respectively. The 4d Newton’s constant is related to the 5d one
through

$\kappa_{4}^{2}=\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\kappa_{5}^{2}\,,$ (4)

where $\kappa_{d}^{2}=8\pi G_{d}=\ell_{\text{Pl},d}^{d-2}$.111Here, we see the
cause of the inverted hierarchy mentioned in the introduction. To get the
result for an RS brane we need to replace $k_{-}-k_{+}\rightarrow
k_{-}+k_{+}$. Usually, the scales are also chosen to be equal, i.e.
$k_{-}=k_{+}=k$. In this way, the RS construction is very similar to a
compactification with an extra dimension of size $R_{\text{AdS}}=1/k$. In the
case of the dark bubble, tuning $k_{+}$ to be just slightly smaller than
$k_{-}$, creates an inverted hierarchy with
$\ell_{\text{Pl},5}\ll\ell_{\text{Pl},4}$.  We have also defined the critical
tension

$\sigma_{c}=\frac{3}{\kappa_{5}^{2}}(k_{-}-k_{+})\,.$ (5)

We see that the effective energy-momentum tensor has two contributions. The
direct contribution from the brane itself, together with any matter that it
carries in the form of $S_{ab}$, and a geometric contribution from the bulk
curvature through the tensor $\mathcal{J}_{ab}$. The expression for the
Einstein equation is not exact but has corrections at second order in $S_{ab}$
(but no higher orders), see ref. Banerjee:2019fzz  for more details. These
corrections are small as long as the net contribution to the 4d Einstein
equations does not reach stringy densities Basile:2023tvh .

An important property, peculiar to the dark bubble, is that $S_{ab}$ seemingly
contributes with the wrong sign to the 4d energy density. This is the minus
sign in front of $S_{ab}$ in (2). In the case of a pure brane, we simply have
$S_{ab}=-\sigma h_{ab}$, which is why a larger value of the brane density
$\sigma$ reduces the effective cosmological constant given by
$\Lambda_{4}=\sigma_{c}-\sigma$.222Note that we are using -+++ signature so
that $h_{tt}<0$ has $S_{tt}>0$ (and $S^{t}_{t}<0$) for a positive energy
density. For the dark bubble to nucleate, its tension needs to be smaller than
the critical value $\sigma_{c}$, guaranteeing a positive cosmological
constant.

To get the full energy-momentum tensor $T_{ab}$ in the presence of matter on
the brane, one must take into account its backreaction on the bulk, captured
by $\mathcal{J}_{ab}$. This gives rise to an effective total energy density in
4d that is equal to the physical one. It is this phenomenon that has been
exhibited in earlier work mentioned in the introduction. The choice of the
bulk solution is part of setting up the appropriate initial conditions for the
physical situation of interest. It is a non-trivial result of ref.
Basile:2023tvh  that this can be done for e.g. electromagnetic waves such that
the subsequent time evolution is that of general relativity coupled to Maxwell
theory. A crucial aspect is how the electromagnetic field on the brane sources
a string $B$-field in the bulk. The backreaction of this field on the bulk
metric is what allows the tensor $\mathcal{J}_{ab}$ to give rise to the
desired energy-momentum tensor $T_{ab}$.

Following ref. Basile:2023tvh , we note that the 4d Einstein tensor, and the
electromagnetic energy-momentum tensor on the brane, are both covariantly
conserved. The first by construction, and the latter through its equations of
motion. This is what is needed for any consistent coupling of matter to
gravity. It then follows, by construction, that the extra piece in the 4d
Einstein equations that changes the sign of the energy-momentum tensor, also
is covariantly conserved. Hence, there is no mixing between the terms, and the
Einstein equations remain true for all times, up to corrections that are
relevant only at stringy energy densities.

The goal of this paper is to incorporate the charged Nariai black hole into
the dark bubble scenario, which we review in the next section.

##  3 Reissner-Nordström-de Sitter black holes and the Nariai limit

In this section, we briefly review Reissner-Nordström-de Sitter (RN-dS) black
holes and the Nariai limit in four dimensions. These have been of attention
recently and led to the Festina Lente bound in Montero:2019ekk . These black
holes are solutions of Einstein-Maxwell theory given by the familiar action

$S=\frac{1}{2\ell_{\text{Pl},4}^{2}}\int\mathrm{d}^{4}x\sqrt{-g_{4}}\left(R_{4}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-2\Lambda_{4}\right)\,,$
(6)

with $\Lambda_{4}=+3k_{4}^{2}$. For finding black hole solutions, we choose
the metric to take the usual expression

$\mathrm{d}s_{4}^{2}=-f(\hat{r})\mathrm{d}\hat{t}^{2}+\frac{\mathrm{d}\hat{r}^{2}}{f(\hat{r})}+\hat{r}^{2}\mathrm{d}\Omega_{2}\,.$
(7)

The Maxwell equations then allow for a non-trivial electric field with the
familiar form:

$F=\frac{2Q}{\hat{r}^{2}}\mathrm{d}\hat{t}\wedge\mathrm{d}\hat{r}\,.$ (8)

Finally, the Einstein equations are solved for the following function $f$:

$\displaystyle f(\hat{r})$
$\displaystyle=1-\frac{2M}{\hat{r}}+\frac{Q^{2}}{\hat{r}^{2}}-k_{4}^{2}\hat{r}^{2}$
(9)
$\displaystyle=-\frac{k_{4}^{2}}{\hat{r}^{2}}(\hat{r}+\hat{r}_{-}+\hat{r}_{+}+\hat{r}_{c})(\hat{r}-\hat{r}_{-})(\hat{r}-\hat{r}_{+})(\hat{r}-\hat{r}_{c}).$
(10)

The first formulation (9) contains 3 parameters, where $k_{4}$ is the inverse
dS2 radius and $M$ and $Q$ are the mass and charge parameters, respectively,
both with dimension length. They are related to the physical mass $m$ and
charge $Q_{\text{EM}}$ as follows:

$M=\frac{\hbar c\ell_{\text{Pl},4}^{2}}{8\pi}m\,,\qquad
Q^{2}=\frac{Q_{\text{EM}}^{2}\ell_{\text{Pl},4}^{2}}{32\pi^{2}\epsilon_{0}\hbar
c}.$ (11)

The second formulation (10) contains 4 parameters, the three radii
$\hat{r}_{+},\hat{r}_{-},\hat{r}_{c}$ and $k_{4}$. Without loss of generality,
we take $\hat{r}_{-}\leq\hat{r}_{+}\leq\hat{r}_{c}$, with $\hat{r}_{-}$
($\hat{r}_{+}$) representing the inner (outer) horizon radius of the black
hole and $\hat{r}_{c}$ the cosmological horizon radius. These radii must
satisfy one constraint, being:

$k_{4}^{2}\left(\hat{r}_{-}^{2}+\hat{r}_{+}^{2}+\hat{r}_{c}^{2}+\hat{r}_{-}\hat{r}_{+}+\hat{r}_{-}\hat{r}_{c}+\hat{r}_{+}\hat{r}_{c}\right)=1,$
(12)

which is solved for

$\hat{r}_{-}=\frac{1}{2}\left(-(\hat{r}_{+}+\hat{r}_{c})+\sqrt{4k_{4}^{-2}-3(\hat{r}_{+}^{2}+\hat{r}_{c}^{2})-2\hat{r}_{+}\hat{r}_{c}}\right).$
(13)

Relating the radii to the charge and mass parameters, we find

$Q^{2}=k_{4}^{2}\>\hat{r}_{-}\hat{r}_{+}\hat{r}_{c}\left(\hat{r}_{-}+\hat{r}_{+}+\hat{r}_{c}\right),\qquad
2M=k_{4}^{2}(\hat{r}_{-}+\hat{r}_{+})(\hat{r}_{-}+\hat{r}_{c})(\hat{r}_{+}+\hat{r}_{c}).$
(14)

The constraint (13) and the fact that
$\hat{r}_{-}\leq\hat{r}_{+}\leq\hat{r}_{c}$, do not allow for every
combination of charge and mass. For black holes in Minkowski space, this leads
to the extremality bound $Q\leq M$. The constraint comes from the fact that
the inner horizon cannot cross the outer horizon, i.e.
$\hat{r}_{-}\ngeq\hat{r}_{+}$, preventing the development of a naked
singularity. For the RN-dS black hole, the same reasoning goes true, although
the extremality bound $Q\leq M$ is slightly altered. Additionally, the outer
horizon of the black hole is not allowed to be bigger than the cosmological
horizon. As the three radii are related through eq. (13), these constraints
can be summarised into one inequality, which in terms of the charge and mass
takes the following form:

$27(k_{4}M)^{4}-(k_{4}M)^{2}\left(36(k_{4}Q)^{2}+1\right)+(k_{4}Q)^{2}\left(4(k_{4}Q)^{2}+1\right)^{2}>0.$
(15)

Both of these constraints can be visualised in the phase diagram of RN-dS
black holes, depicted in figure 1.

$k_{4}Q$$k_{4}M$$\hat{r}_{-}\sim\hat{r}_{+}$$\hat{r}_{+}\sim\hat{r}_{c}$extremal
branchNariai branch Figure 1: Only black holes with charge and mass under the
curve are allowed. The top blue line corresponds to extremal black holes where
inner and outer black hole horizons meet, and is referred to as the ‘extremal
branch’. The vertical blue line represents the black holes in the Nariai
limit, where the outer horizon and cosmological horizon meet, which is
referred to as the ‘Nariai branch’.The dashed line represents $Q=M$, and
illustrates that the extremality bound for RN-dS black holes deviates from the
one for RN black holes in Minkowski space.

In the Nariai limit, one lets the cosmological horizon and outer black hole
horizon come close together such that they become indistinguishable for
observers. However, they cannot become strictly equal, and therefore it is
useful to expand the metric for an observer that feels no force in between
those two horizons, located at $\hat{r}=\hat{r}_{g}$ with
$f^{\prime}(\hat{r}_{g})=0$. The Nariai limit can then be understood by
bringing the two horizons close to this static observer, although not
touching. In principle, one can find the solution to the equation
$f^{\prime}(\hat{r}_{g})=0$ in the form
$\hat{r}_{g}=\hat{r}_{g}(\hat{r}_{+},\hat{r}_{c})$. However, the expression
becomes rather lengthy and it is not necessary for our discussion, so we do
not write it here. So in what follows, we want to consider the limit
$\hat{r}_{+}\rightarrow\hat{r}_{g}\leftarrow\hat{r}_{c}$. It proves to be
useful to perform a coordinate transformation as follows:

$\hat{r}\to\sqrt{|f(\hat{r}_{g})|}\>r+\hat{r}_{g},\qquad\hat{t}\to\sqrt{|f(\hat{r}_{g})|}\>t.$
(16)

With this, the 4d metric becomes

$\mathrm{d}s_{4}^{2}=-\frac{f(r)}{|f(\hat{r}_{g})|}\mathrm{d}t^{2}+\frac{|f(\hat{r}_{g})|}{f(r)}\mathrm{d}r^{2}+\left(\sqrt{|f(\hat{r}_{g})|}\;r+\hat{r}_{g}\right)^{2}\mathrm{d}\Omega_{2}.$
(17)

Now we take the Nariai limit
$\hat{r}_{+}\rightarrow\hat{r}_{g}\leftarrow\hat{r}_{c}$. For that, it is very
useful to expand the functions $f(r)/|f(\hat{r}_{g})|$ in terms of $r$.
However, evaluating everything in terms of
$\hat{r}_{g}(\hat{r}_{+},\hat{r}_{c})$, which we did not write, and then
taking the limit $\hat{r}_{+}\to\hat{r}_{c}$, is rather complicated. Instead,
it turns out to be more useful to introduce a small parameter $\epsilon$ as
follows:

$\hat{r}_{g}=\hat{r}_{c}(1-\epsilon),\qquad\hat{r}_{+}=\hat{r}_{c}(1-2\epsilon+O(\epsilon^{2})).$
(18)

Notice that the parameterisation above respects $f^{\prime}(\hat{r}_{g})=0$ or
equivalently $\hat{r}_{g}=\hat{r}_{g}(\hat{r}_{c},\hat{r}_{+})$, up to linear
order in $\epsilon$. Now, we can perform the whole calculation including up to
first order in $\epsilon$ and then take $\epsilon\to 0$ in the Nariai limit.
Let us now perform a series expansion of the metric in small $r$ and up to
leading order in $\epsilon$. To do that, we note that

$\frac{\mathrm{d}}{\mathrm{d}r}=\sqrt{f(\hat{r}_{g})}\frac{\mathrm{d}}{\mathrm{d}\hat{r}}.$
(19)

The metric functions become:

$\displaystyle\frac{f(r)}{|f(\hat{r}_{g})|}$ $\displaystyle=$
$\displaystyle\>1-\left[\left(6k_{4}^{2}-\frac{1}{\hat{r}_{c}^{2}}\right)-\frac{2\epsilon}{\hat{r}_{c}^{2}}\right]r^{2}-\left[\sqrt{6k_{4}^{2}-\frac{1}{\hat{r}_{c}^{2}}}(1-4k_{4}^{2}\hat{r}_{c}^{2})\frac{2\epsilon}{\hat{r}_{c}^{2}}\right]r^{3}+O(r^{4},\epsilon^{2})$
(20) $\displaystyle\hat{r}^{2}$ $\displaystyle=$
$\displaystyle\>\hat{r}_{c}^{2}\left(1-2\epsilon+\left[2\epsilon\sqrt{6k_{4}^{2}-\frac{1}{\hat{r}_{c}^{2}}}\right]r+O(r^{3},\epsilon^{2})\right)$
(21)

In the Nariai limit, i.e. $\epsilon\to 0$, we recover the dS${}_{2}\times
S^{2}$ metric:

$\mathrm{d}s_{4}^{2}=-\left(1-(k_{2}r)^{2}\right)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{\left(1-(k_{2}r)^{2}\right)}+\hat{r}_{c}^{2}\mathrm{d}\Omega_{2}^{2},$
(22)

with

$k_{2}^{2}=6k_{4}^{2}-\frac{1}{\hat{r}_{c}^{2}}.$ (23)

One can express these quantities in terms of the charge parameter $Q$:

$k_{2}^{2}=\frac{6k_{4}^{2}\sqrt{1-12k_{4}^{2}Q^{2}}}{1+\sqrt{1-12k_{4}^{2}Q^{2}}},\qquad\hat{r}_{c}^{2}=\frac{1+\sqrt{1-12k_{4}^{2}Q^{2}}}{6k_{4}^{2}}.$
(24)

It is important to note that, apart from being a near-horizon limit of the RN-
dS solution, the Nariai metric is also a genuine solution of the Einstein-
Maxwell equations with a constant electric field. Indeed, with the metric (22)
and constant electric field like the one in (8) (with $\hat{r}=\hat{r}_{c}$),
the Einstein equations reduce to

$\displaystyle-\frac{1}{\hat{r}_{c}^{2}}=-\frac{Q^{2}}{\hat{r}_{c}^{4}}-3k_{4}^{2}$
(25) $\displaystyle-k_{2}^{2}=+\frac{Q^{2}}{\hat{r}_{c}^{4}}-3k_{4}^{2}\,,$
(26)

which have the same solutions as in eq. (24). Furthermore, the Nariai
background is also a solution of “Einstein-Born-Infeld” theory Cai:2004eh ,
where one replaces the Maxwell action of electromagnetism with the Born-Infeld
action, with a constant electric field.

As explained in the introduction, the goal of this paper is to embed the
Nariai black hole into the dark bubble. Let us therefore end this section with
some expectations on how the 5d embedding should look like. In the dark bubble
model, any 4d energy density on the brane bends it outwards. The first example
was discussed in Banerjee:2019fzz , where point particles in 4d correspond to
strings that pull on the brane in the extra dimension. Additionally, in ref.
Basile:2023tvh  it was shown that in the presence of an electromagnetic field
on the brane, there also exists a $B$-field that pulls the brane outwards.

It is therefore expected that other 4d energy sources, like black holes, also
bend the brane. Pictorially, one can think of a black hole as a very thick
string pulling on the brane, and we expect a throat-like structure as in
figure 2. For an observer outside the black hole, there are two horizons. The
outer one is the cosmological one, while the inner one is that of the black
hole. The mass of the black hole is so large that the brane starts to bend
upwards already far outside of the cosmological horizon centred on the black
hole. As explained above, in the Nariai regime, the black hole horizon and
cosmological horizon do not coincide but are close together such that they
become indistinguishable. In figure 2, this corresponds to a strongly warped
regime where the segment of the brane in between the two horizons becomes a
cylinder rather than a cone.

In the next section, we explain in detail how a Nariai black hole on the brane
can be understood in a 5d embedding. The upshot is that it can be seen as a
perturbed version of a “black string” background, which was discussed in
Bernamonti:2007bu .

Figure 2: A black hole on top of a dark bubble. In the Nariai limit, the
structure close to the horizons becomes a straight cylinder.

##  4 The 5d black string

In this section, we set up the five-dimensional backgrounds that allow us to
embed a brane with the Nariai geometry. For that, we need a five-dimensional
asymptotically AdS spacetime with the proper topology. It turns out that black
string solutions in AdS5 do precisely that. Black strings in such spacetimes
have been extensively studied, see refs. Klemm:2000nj ; Bernamonti:2007bu ;
Brihaye:2007ju ; Brihaye:2009dm ; Harmark:2007md  for a select overview. The
background that is suitable for our purpose is related to the “magnetically
charged black string” in AdS5 that was described in ref. Bernamonti:2007bu .
In the remainder of this section, we introduce a modified version of this
background, allowing for a dS2 subspace.

In particular, we start from five-dimensional Einstein gravity coupled to a
three-form field strength $H$ with following action:

$S=\frac{1}{2\ell_{\text{Pl},5}^{3}}\int\mathrm{d}^{5}x\sqrt{-g_{5}}\left(R_{5}-3\Lambda_{5}-\frac{1}{2}|H|^{2}\right)\,,$
(27)

where $\Lambda_{5}=-4k^{2}$ is the 5d cosmological constant with $k$ the
inverse AdS5 length, and $H$ a three-form field strength with two-form gauge
potential $B$.333In a string theory context, the cosmological constant arises
from moduli stabilisation. Notice also that we have omitted a dilatonic
coupling to $H$. We justify this in Appendix A. For the metric, we take a
modified black string Ansatz that makes the dS2 structure of the Nariai black
hole that we are aiming for more apparent:

$\mathrm{d}s_{5}^{2}=-f(u)\left(1-(k_{2}r)^{2}\right)\mathrm{d}t^{2}+\frac{\tilde{f}(u)}{\left(1-(k_{2}r)^{2}\right)}\mathrm{d}r^{2}+u^{2}\mathrm{d}\Omega_{2}+\frac{\mathrm{d}u^{2}}{w(u)}\,,$
(28)

where $k_{2}$ represents the inverse dS2 radius. When it is put to zero, the
metric corresponds to the Ansatz of Bernamonti:2007bu ; Mann:2006yi  again. We
also turn on one non-vanishing, electric component of the $H$-field, which we
parametrise by

$H=q\>\frac{\sqrt{f(u)\tilde{f}(u)}}{u^{2}\sqrt{w(u)}}\>\mathrm{d}t\wedge\mathrm{d}r\wedge\mathrm{d}u\,,$
(29)

where $q$ is a constant. With this rather particular choice, the field
strength $H$ satisfies

$\star H=q\>\text{vol}(S^{2}),\qquad\mathrm{d}\star H=0\,,$ (30)

hence it automatically solves the equation of motion for $H$ and $q$
represents an electric charge for the field strength $H$.444In ref.
Bernamonti:2007bu , the authors work with a two-form field strength that we
have Hodge dualised into the three-form field strength $H$. In their
prescription, the constant $q$ is a magnetic charge of their two-form field
strength. As usual, the Einstein equations are

$R_{MN}-\frac{1}{2}Rg_{MN}+\frac{3}{2}\Lambda_{5}=\frac{1}{2}\left(|H|_{MN}^{2}-\frac{1}{2}g_{MN}|H|^{2}\right),$
(31)

This gives five non-independent equations to solve (the $tt$-, $rr$-,
$\theta\theta$-, $uu$\- and $ru$-components). They are the following:

$\displaystyle\left(\frac{\tilde{f}^{\prime}}{4\tilde{f}}+\frac{1}{u}\right)w^{\prime}+w\left(-\frac{\tilde{f}^{\prime
2}}{4\tilde{f}^{2}}+\frac{\frac{\tilde{f}^{\prime\prime}}{2}+\frac{\tilde{f}^{\prime}}{u}}{\tilde{f}}+\frac{1}{u^{2}}\right)-6k^{2}+\frac{q^{2}}{u^{4}}-\frac{1}{u^{2}}=0$
(32)
$\displaystyle\left(\frac{f^{\prime}}{4f}+\frac{1}{u}\right)w^{\prime}+w\left(-\frac{f^{\prime
2}}{4f^{2}}+\frac{\frac{f^{\prime\prime}}{2}+\frac{f^{\prime}}{u}}{f}+\frac{1}{u^{2}}\right)-6k^{2}+\frac{q^{2}}{u^{4}}-\frac{1}{u^{2}}=0$
(33)
$\displaystyle\frac{w}{2}\left(\frac{f^{\prime\prime}}{f}+\frac{\tilde{f}^{\prime\prime}}{\tilde{f}}\right)+\frac{wf^{\prime}\tilde{f}^{\prime}}{4f\tilde{f}}+\left(\frac{w^{\prime}}{4}+\frac{w}{2u}\right)\left(\frac{f^{\prime}}{f}+\frac{\tilde{f}^{\prime}}{\tilde{f}}\right)-\frac{w}{4}\left(\frac{f^{\prime
2}}{f^{2}}+\frac{\tilde{f}^{\prime
2}}{\tilde{f}^{2}}\right)-\frac{k_{2}^{2}}{\tilde{f}}$
$\displaystyle-6k^{2}-\frac{q^{2}}{u^{4}}+\frac{w^{\prime}}{2u}=0$ (34)
$\displaystyle\frac{wf^{\prime}\tilde{f}^{\prime}}{4f\tilde{f}}+\frac{w}{u}\left(\frac{f^{\prime}}{f}+\frac{\tilde{f}^{\prime}}{\tilde{f}}\right)-\frac{k_{2}^{2}}{\tilde{f}}-6k^{2}+\frac{q^{2}}{u^{4}}+\frac{w}{u^{2}}-\frac{1}{u^{2}}=0$
(35) $\displaystyle
k_{2}^{2}\left(\frac{\tilde{f}^{\prime}}{\tilde{f}}-\frac{f^{\prime}}{f}\right)=0\,.$
(36)

It is important to note that all the derivatives appearing above are with
respect to the $u$-coordinate and that the differential equations involve no
$r$-dependence. As we mentioned before, when $k_{2}$ vanishes, one recovers
the Einstein equations for the charged black string in AdS5 Bernamonti:2007bu
. Fully analytic solutions have only been found for special values of the
charge $q$, see e.g. Bernamonti:2007bu ; Chamseddine:1999xk ; Klemm:2000nj ,
and when $k^{2}q^{2}=1/12$ the solution is even supersymmetric
Chamseddine:1999xk ; Klemm:2000nj . Nevertheless, the Einstein equations can
be solved both for small and large values of $ku$ for any charge $q$. We are
interested in the latter case, where the functions $f$, $\tilde{f}$ and $w$
allow Fefferman-Graham expansions of the following form Bernamonti:2007bu ;
Mann:2006yi 555As explained in appendix A, if a dilatonic coupling to the
$H$-field were to be included, its equation of motion would impose its non-
constant part to be much more subleading in the large $(ku)$-limit than the
order we are considering. Therefore, it is justified to have omitted it.:

$\displaystyle
f^{(0)}(u)=(ku)^{2}+\frac{1}{2}+\frac{1}{12}(1-12(kq)^{2})\frac{\log(ku)}{(ku)^{2}}+\frac{c_{t}}{(ku)^{2}}+O\left(\frac{\log(ku)}{(ku)^{4}}\right)$
(37)
$\displaystyle\tilde{f}^{(0)}(u)=(ku)^{2}+\frac{1}{2}+\frac{1}{12}(1-12(kq)^{2})\frac{\log(ku)}{(ku)^{2}}+\frac{c_{r}}{(ku)^{2}}+O\left(\frac{\log(ku)}{(ku)^{4}}\right)$
(38) $\displaystyle
w^{(0)}(u)=(ku)^{2}+\frac{2}{3}+\frac{1}{6}(1-12(kq)^{2})\frac{\log(ku)}{(ku)^{2}}+\frac{c_{t}+c_{r}+(kq)^{2}/3}{(k_{\pm}u)^{2}}+O\left(\frac{\log(ku)}{(ku)^{4}}\right)$
(39)

The constants $c_{t}$ and $c_{r}$ determine the mass and tension of the black
string and can be determined in terms of the horizon radius by matching these
expressions to the near-horizon expansions of the black string solutions, as
was done in ref. Bernamonti:2007bu .

When $k_{2}$ is turned on, the Einstein equations receive additional terms and
hence the solutions change. First, eq. (36) informs us that $f=\tilde{f}$ and
we can take $c^{\prime}=1$ as it can be absorbed by redefinition of the time
coordinate $t$. This implies that we need to take $c_{t}=c_{r}$, and such 5d
solutions turn out to be extremal at zeroth order in $k_{2}$. We then solve
the system of differential equations perturbatively when $k_{2}^{2}/k^{2}$
becomes small. We can write the expansions

$\displaystyle
f(u)=f^{(0)}(u)+\frac{k_{2}^{2}}{k^{2}}f^{(1)}(u)+O(k_{2}^{4}/k^{4})$ (40)
$\displaystyle
w(u)=w^{(0)}(u)+\frac{k_{2}^{2}}{k^{2}}w^{(1)}(u)+O(k_{2}^{4}/k^{4})\,.$ (41)

When plugging these into the Einstein equations, we find the following
Fefferman-Graham expressions of the first order corrections:

$\displaystyle
f^{(1)}(u)=a_{1}(ku)^{2}+\frac{1}{2}(a_{1}-1)+\frac{1}{12}a_{1}(1-12(kq)^{2})\frac{\log(ku)}{(ku)^{2}}+\frac{c_{1}}{(ku)^{2}}+O\left(\frac{\log(ku)}{(ku)^{4}}\right)$
(42) $\displaystyle
w^{(1)}(u)=-\frac{1}{3}+\frac{-2(-c_{1}+c_{r}a_{1})}{(ku)^{2}}+O\left(\frac{\log(ku)}{(ku)^{4}}\right)\,.$
(43)

This introduces two new parameters, $a_{1}$ and $c_{1}$, which are not fixed
by the Einstein equations. One could try to see whether these parameters are
constrained when matching these expansions with small $(ku)$-expansions as in
Bernamonti:2007bu  and how the mass and tension of the black string background
are affected, but we postpone this for the future.

##  5 Glueing two black string solutions together

###  5.1 Solving the junction conditions

At this stage, we are ready to glue two different black string spacetimes
together. They are joined together by sharing a boundary, which corresponds to
a (dynamical) shell. In our setting, the shell is provided by a three-brane
that is embedded at a fixed location in the $u$-direction, i.e. at $u=a$ with
$a$ constant. The prescription for gluing spacetimes was originally developed
by Israel in the form of junction conditions Israel:1966rt . The first
condition requires the induced metric on the shell (three-brane) to be
continuous:

$h_{mn}^{-}=h_{mn}^{+}\,$ (44)

where $h_{mn}$ is the induced metric at $u=a$ and the “$-$” and “$+$” refer to
the spacetime on the inside and outside respectively. The second junction
condition requires that the extrinsic curvature must be discontinuous across
the shell. The jump in the extrinsic curvature is provided by the stress
tensor of the shell, formulated in the second junction condition:

$\Delta K_{m}^{n}-\Delta Kh_{m}^{n}=-\ell_{\text{Pl},5}^{3}S_{m}^{n},$ (45)

where $S_{mn}$ is the stress tensor on the brane and the $\Delta$ refers to
the difference of the quantities between inside and outside spacetimes, i.e.

$\Delta K_{m}^{n}=(K_{-})_{m}^{n}-(K_{+})_{m}^{n},\qquad\Delta K=K_{-}-K_{+}.$
(46)

Before discussing the extrinsic curvature contributions, let us first take a
look at the stress tensor on the brane. We use a three-brane with a world-
volume field strength $\mathcal{F}$, described by a DBI action of the
following form:

$S_{\text{DBI}}=-\sigma\int\mathrm{d}^{4}x\sqrt{-\det({h_{ab}}+\tau\mathcal{F}_{ab})}\,.$
(47)

The field strength $\mathcal{F}$ is a combination of a worldvolume U(1) gauge
field $A$ with field strength $F$ and the pull-back of the $B$-field on the
brane:

$\tau\mathcal{F}=\tau F_{ab}+B_{ab}\,,$ (48)

where $H=\mathrm{d}B$ locally. With a stringy embedding in mind, we think of
the brane as a D3-brane and we have $\tau=2\pi\alpha^{\prime}$, while the
tension of the D3-brane is $\sigma=\frac{1}{(2\pi)^{3}\alpha^{\prime
2}g_{s}}-\delta\sigma$. The correction $\delta\sigma$ reduces the tension so
that $\sigma<\sigma_{c}$, and we get a positive cosmological constant. Its
presence was argued by referring to the weak gravity conjecture in
Danielsson:2022lsl , and it was explicitly calculated in Danielsson:2023alz
using stringy $R^{2}$-corrections to the brane action. We must also allow for
a constant electric field on the brane. Due to the coupling to the $B$-field,
such an electric field sources a jump in the $H$-field across the brane. As we
have already imposed the expression of the $H$-field in the previous section,
we can use the $H$-equation of motion with the three-brane source to evaluate
the electric field and hence the stress tensor in terms of a jump in the
$H$-field. The details are worked out in appendix B, and when we evaluate the
stress tensor in the large $u=a$ limit, we find

$\displaystyle
S_{t}^{t}=S_{r}^{r}=-\sigma-\frac{1}{2\ell_{\text{Pl},4}^{2}}\frac{E^{2}}{2}=-\sigma-\frac{1}{8}\frac{(\Delta
q)^{2}}{\kappa_{5}^{4}}\frac{1}{\sigma}\frac{1}{a^{4}}$ (49) $\displaystyle
S_{\theta}^{\theta}=S_{\phi}^{\phi}=-\sigma+\frac{1}{2\ell_{\text{Pl},4}^{2}}\frac{E^{2}}{2}=-\sigma+\frac{1}{8}\frac{(\Delta
q)^{2}}{\kappa_{5}^{4}}\frac{1}{\sigma}\frac{1}{a^{4}}.$ (50)

Here $\Delta q=q_{+}-q_{-}$ corresponds to the difference in 5d charges on the
in- (“$-$”) and outside (“$+$”) as defined in eq. (29). Now we turn back to
the geometrical part. We write the metric on the out- and inside as follows:

$\mathrm{d}s_{5}^{2}=-f_{\pm}(u)\left(1-(k_{2\pm}r_{\pm})^{2}\right)\mathrm{d}t_{\pm}^{2}+\frac{f_{\pm}(u)}{\left(1-(k_{2\pm}r)^{2}\right)}\mathrm{d}r_{\pm}^{2}+u^{2}\mathrm{d}\Omega_{2}+\frac{\mathrm{d}u^{2}}{w_{\pm}(u)},$
(51)

Note that we allowed for different time and spatial coordinates on the in- and
outside with $t_{\pm}$ and $r_{\pm}$, and for different inverse de Sitter
radii $k_{2\pm}$. Additionally, the functions $f_{\pm}(u)$ and $w_{\pm}(u)$
have the same functional form as in eqns. (37)-(39) and (42)-(43), but we
allow all the parameters to be different. This amounts to replacing $k\to
k_{\pm}$, $c_{r}\to c_{r\pm}$, $q\to q_{\pm}$, $a_{1}\to a_{1\pm}$ and
$c_{1}\to c_{1\pm}$.

Now we have everything to solve the junction conditions, for two spacetimes
joint together by a three-brane at $u=a$. The first junction condition
requires us to rescale the time and radial coordinates as follows, defining
new coordinates $t$ and $r$:

$\sqrt{f_{+}(a)}\>t_{+}=\sqrt{f_{-}(a)}\>t_{-}=t,\qquad\sqrt{f_{+}(a)}\>r_{+}=\sqrt{f_{-}(a)}\>r_{-}=r$
(52)

Additionally, as the induced metric has to be continuous, we have to impose
that

$k_{2}\equiv\frac{k_{2+}}{\sqrt{f_{+}(a)}}=\frac{k_{2-}}{\sqrt{f_{-}(a)}}\,,$
(53)

where the new $k_{2}$ is now the physical inverse dS2 radius that an observer
on the brane would measure. With all this, induced metric becomes indeed of
the dS${}_{2}\times S^{2}$ type:

$h_{mn}^{\pm}\mathrm{d}x^{m}\mathrm{d}x^{n}=-\left(1-(k_{2}r)^{2}\right)\mathrm{d}t^{2}+\frac{1}{\left(1-(k_{2}r)^{2}\right)}\mathrm{d}r^{2}+a^{2}\mathrm{d}\Omega_{2}$
(54)

Because the solutions are constructed in the large $(k_{\pm}a)$\- and small
$(k_{2\pm}/k_{\pm})$-limit, we infer from eq. (37) and (53) that $k_{2}$ must
satisfy

$k_{2}(k_{\pm}a)\sim
k_{2\pm}\Rightarrow\frac{k_{2}}{k_{\pm}}\sim\frac{k_{2\pm}}{k_{\pm}}\frac{1}{k_{\pm}a}\ll\frac{1}{k_{\pm}a}\ll
1\,.$ (55)

Next, we compute the extrinsic curvature combinations appearing in the
junction condition (45). The $tt$\- and $rr$-components are the same, the
$\theta\theta$\- and $\phi\phi$-component likewise, which results in only two
independent components, which are

$\displaystyle\Delta K_{t}^{t}-\Delta
Kh_{t}^{t}=\frac{2}{a}\left(\sqrt{w_{-}(a)}-\sqrt{w_{+}(a)}\right)+\frac{1}{2}\left(\frac{\sqrt{w_{-}(a)}\>f_{-}^{\prime}(a)}{f_{-}(a)}-\frac{\sqrt{w_{+}(a)}\>f_{+}^{\prime}(a)}{f_{+}(a)}\right)$
(56) $\displaystyle\Delta K_{\theta}^{\theta}-\Delta
Kh_{\theta}^{\theta}=\frac{1}{a}\left(\sqrt{w_{-}(a)}-\sqrt{w_{+}(a)}\right)+\left(\frac{\sqrt{w_{-}(a)}\>f_{-}^{\prime}(a)}{f_{-}(a)}-\frac{\sqrt{w_{+}(a)}\>f_{+}^{\prime}(a)}{f_{+}(a)}\right)$
(57)

These expressions are valid for any functions $f_{\pm}(a)$ and $w_{\pm}(a)$.

We are now ready to glue two particular geometries together. For the inside,
we choose a geometry described by (37)-(39) and (42)-(43), with appropriate
constants $c_{r}$ etc. such that the 5d geometry in the $k_{2}\to 0$ limit is
a 5d black string solution as in ref. Bernamonti:2007bu .

On the outside, we allow us a bit more freedom. We still choose a similar
black string solution, but we allow ourselves to choose the constant $c_{r}$
as we please. We can do so for the following reason: the values of $c_{r}$
that were calculated in ref. Bernamonti:2007bu  were obtained by requiring
that the large $(ku)$-expansion of the functions can be glued onto the
expressions of the near-horizon expansion, where the horizon was located at
$ku_{h}\lesssim O(1)$. However, in our setup on the outside, we do not need
such requirements. What happens for small $ku$, all the way down to the would-
be horizon, is taken care of by the geometry on the inside of the bubble. This
allows for more freedom in choosing the parameters. A similar situation
occurred in ref. Danielsson:2022odq , where over-extremal Reisner-Nordström
geometries were constructed, also by glueing two AdS spacetimes together with
a spherical shell. A naked singularity did not occur, as the inside of the
shell did carry no charge.

With these comments, we are now ready to compute the extrinsic curvature
contributions to the junction conditions. As we discussed before, the
functions $f_{\pm}$ and $w_{\pm}$ are defined up to first order in
$k_{2\pm}^{2}/k^{2}$ as in (42)-(43). These should now be expressed in terms
of the physical $k_{2}$ through the first junction condition (53). Since we
have constructed the solutions perturbatively in $k_{2\pm}$, we can identify
the physical $k_{2}$ in a similar manner, by which we mean that

$k_{2\pm}^{2}=k_{2}^{2}f_{\pm}(a)=k_{2}^{2}\left(f_{\pm}^{(0)}(a)+\frac{k_{2\pm}^{2}}{k_{\pm}^{2}}f_{\pm}^{(1)}(a)+\cdots\right)$
(58)

and hence our perturbative expansion allows us to drop the second term and
write

$k_{2\pm}^{2}=k_{2}^{2}f_{\pm}^{(0)}(a).$ (59)

Additionally, we note that the functions $w_{\pm}$ and $f_{\pm}$ are defined
up to order $(k_{\pm}a)^{-2}$, but we expand the extrinsic curvature
contributions (56)-(57) up to order $(k_{\pm}a)^{-4}$. We should do this
because the functions $w_{\pm}$ and $f_{\pm}$ are leading with
$(k_{\pm}a)^{2}$ and approximated up to $(k_{\pm}a)^{-2}$, but by taking the
root in (56)-(57) lowers that with one power, and dividing by $a$ or taking a
derivative lowers it with an additional one. Hence the combinations are
leading by $(k_{\pm}a)^{0}$ and are truncated after $(k_{\pm}a)^{-4}$.
Finally, we also allow us to throw away terms of the order
$(k_{2}^{2}/k_{\pm}^{2})/(k_{\pm}a)^{2}$ because these are much smaller than
$1/(k_{\pm}a)^{4}$ by eq. (55). With all these truncations, the extrinsic
curvature contributions become

$\displaystyle\Delta K_{t}^{t}-\Delta Kh_{t}^{t}=$ $\displaystyle
3(k_{-}-k_{+})-\frac{k_{-}-k_{+}}{2k_{-}k_{+}}\frac{1}{a^{2}}+\left(\frac{c_{r-}-\frac{1}{24}}{k_{-}^{3}}-\frac{c_{r+}-\frac{1}{24}}{k_{+}^{3}}\right)\frac{1}{a^{4}}$
$\displaystyle+\left(\frac{(1-12k_{-}^{2}q_{-}^{2})\log(k_{-}a)}{12k_{-}^{3}}-\frac{(1-12k_{+}^{2}q_{+}^{2})\log(k_{+}a)}{12k_{+}^{3}}\right)\frac{1}{a^{4}}$
(60)
$\displaystyle+k_{2}^{2}\Biggl{[}(a_{1-}k_{-}-a_{1+}k_{+})a^{2}+\left(\frac{3-2a_{1+}}{6k_{+}}-\frac{3-2a_{1-}}{6k_{-}}\right)\Biggr{]}$
$\displaystyle\Delta K_{\theta}^{\theta}-\Delta Kh_{\theta}^{\theta}=$
$\displaystyle
3(k_{-}-k_{+})-\left(\frac{c_{r-}-\frac{1}{24}-\frac{1-12k_{-}^{2}q_{-}^{2}}{24}}{k_{-}^{3}}-\frac{c_{r+}-\frac{1}{24}-\frac{1-12k_{+}^{2}q_{+}^{2}}{24}}{k_{+}^{3}}\right)\frac{1}{a^{4}}$
$\displaystyle-\left(\frac{(1-12k_{-}^{2}q_{-}^{2})\log(k_{-}a)}{12k_{-}^{3}}-\frac{(1-12k_{+}^{2}q_{+}^{2})\log(k_{+}a)}{12k_{+}^{3}}\right)\frac{1}{a^{4}}$
$\displaystyle+k_{2}^{2}\Biggl{[}2(a_{1-}k_{-}-a_{1+}k_{+})a^{2}+\left(\frac{3-4a_{1+}}{6k_{+}}-\frac{3-4a_{1-}}{6k_{-}}\right)\Biggr{]}$
(61)

We notice that the logarithmic dependence on $a$ in both of the extrinsic
curvature contributions are the same, and that the junction conditions become
insensitive to the constants $c_{1\pm}$ due to our approximations. Equating
these to the stress tensor contributions (49)-(50) and rearranging the terms,
we find that the junction conditions become

$\displaystyle-\frac{1}{a^{2}}+k_{2}^{2}\left(A(k_{\pm},a_{1\pm})\;a^{2}+B(k_{\pm},a_{1\pm})\right)=-\frac{Q_{t}^{2}}{a^{4}}-\frac{\Xi(a)}{a}-3k_{4}^{2}$
(62) $\displaystyle-
k_{2}^{2}\left(2A(k_{\pm},a_{1\pm})\;a^{2}+2B(k_{\pm},a_{1\pm})+1\right)=+\frac{Q_{\theta}^{2}}{a^{4}}+\frac{\Xi(a)}{a^{4}}-3k_{4}^{2},$
(63)

where we have introduced the following definitions:

$\displaystyle
A(k_{\pm},a_{1\pm})=2\frac{k_{-}k_{+}}{k_{-}-k_{+}}(a_{1-}k_{-}-a_{1+}k_{+})$
(64) $\displaystyle
B(k_{\pm},a_{1\pm})=2\frac{k_{-}k_{+}}{k_{-}-k_{+}}\left(\frac{3-2a_{1+}}{6k_{+}}-\frac{3-2a_{1-}}{6k_{-}}\right)$
(65)
$\displaystyle\Xi(a)=\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\left(\frac{(1-12k_{-}^{2}q_{-}^{2})\log(k_{-}a)}{12k_{-}^{3}}-\frac{(1-12k_{+}^{2}q_{+}^{2})\log(k_{+}a)}{12k_{+}^{3}}\right)$
(66) $\displaystyle
Q_{t}^{2}=\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\left(-\frac{1}{8}\frac{(\Delta
q)^{2}}{\ell_{\text{Pl},5}^{3}}\frac{1}{\sigma}+\frac{c_{r-}-\frac{1}{24}}{k_{-}^{3}}-\frac{c_{r+}-\frac{1}{24}}{k_{+}^{3}}\right)$
(67) $\displaystyle
Q_{\theta}^{2}=Q_{t}^{2}+\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\left(\frac{1-12k_{+}^{2}q_{+}^{2}}{24k_{+}^{2}}-\frac{1-12k_{-}^{2}q_{-}^{2}}{24k_{-}^{2}}\right)$
(68) $\displaystyle
3k_{4}^{2}=-\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\left(\ell_{\text{Pl},5}^{3}\sigma-3(k_{-}-k_{+})\right)\,.$
(69)

###  5.2 Matching with 4d Nariai

Ideally, we want the junction conditions (62)-(63) to take the form of the 4d
Einstein equations in the usual 4d Nariai setting:

$\displaystyle-\frac{1}{a^{2}}$
$\displaystyle=-\frac{Q^{2}}{a^{4}}-3k_{4}^{2}$ (70) $\displaystyle-k_{2}^{2}$
$\displaystyle=+\frac{Q^{2}}{a^{4}}-3k_{4}^{2}.$ (71)

This is not quite the case yet. For instance, we want no term proportional to
$k_{2}^{2}$ in (62), meaning that $A(k_{\pm},a_{1\pm})=0$ and
$B(k_{\pm},a_{1\pm})=0$. Additionally, we want no term proportional to
$k_{2}^{2}a^{2}$ in (63), the only term involving $k_{2}^{2}$ being exactly
proportional to $-1$, which is accounted for by the same requirements. These
are solved by

$a_{1\pm}=\frac{3k_{\mp}}{2(k_{-}+k_{+})}.$ (72)

With these choices, there are a few other terms that we do not want to be
there, e.g. the ones with a logarithmic dependence in $a$. There are a few
ways to tackle this problem.

#### A special case

If one tunes the 5d charges so they obey

$k_{+}^{2}q_{+}^{2}=\frac{k_{-}^{3}-k_{+}^{3}(1-12k_{-}^{2}q_{-}^{2})}{12k_{-}^{3}}\,,$
(73)

we find that the function $\Xi(a)$ becomes constant and that the two charge
parameters are the same:

$\Xi(a)=\Xi=\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\frac{(1-12k_{-}^{2}q_{-}^{2})\log\left(\frac{k_{-}}{k_{+}}\right)}{12k_{-}^{3}},\qquad
Q_{t}^{2}=Q_{\theta}^{2}\,.$ (74)

Now the two junction conditions (62)-(63) nicely become the 4d Einstein
equations of the Nariai spacetime with the 4d charge satisfying:

$Q^{2}=Q_{t}^{2}+\Xi=\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\left[-\frac{1}{8}\frac{(\Delta
q)^{2}}{\ell_{\text{Pl},5}^{3}}\frac{1}{\sigma}+\frac{c_{r-}-\frac{1}{24}}{k_{-}^{3}}-\frac{c_{r+}-\frac{1}{24}}{k_{+}^{3}}+\frac{(1-12k_{-}^{2}q_{-}^{2})\log\left(\frac{k_{-}}{k_{+}}\right)}{12k_{-}^{3}}\right]$
(75)

In these expressions it is useful to identify the 4d Planck scale using the
standard dark bubble relation

$\ell_{\text{Pl},4}^{2}=\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\ell_{\text{Pl},5}^{3}.$
(76)

Per usual, the Nariai system is solved for $a$ and $k_{2}$ in terms of $Q$ and
$k_{4}$, but here we would like to solve for $a$ and $\sigma$ (or $k_{4}$)
instead. We find that

$a^{2}=\frac{1-\sqrt{1-8k_{2}^{2}Q^{2}}}{2k_{2}^{2}},\qquad
k_{4}^{2}=\frac{1+4k_{2}^{2}Q^{2}+\sqrt{1-8k_{2}^{2}Q^{2}}}{24Q^{2}}$ (77)

and hence

$\ell_{\text{Pl},5}^{3}\sigma=3(k_{-}-k_{+})-\frac{3}{2}\left(\frac{1}{k_{+}}-\frac{1}{k_{-}}\right)\frac{1+4k_{2}^{2}Q^{2}+\sqrt{1-8k_{2}^{2}Q^{2}}}{24Q^{2}}$
(78)

We need to check as a consistency requirement that $(k_{2}a)^{2}\ll 1$ and
$\sigma>0$, and preferably that
$\ell_{\text{Pl},5}^{3}\sigma-3(k_{-}-k_{+})\ll 1$. This is accounted for when
$k_{2}^{2}Q^{2}\ll 1$, which can be arranged for with an appropriate choice of
$c_{r+}$. We want to stress again that, although $q_{+}$ and $c_{r+}$ are not
completely independent in the ordinary black string setup, we can take them
independent here as we do not need to meet the requirements that the solution
on the outside extrapolates to a solution for small values of $a$ without
naked singularities.

#### Unconstrained charges

If the extra term on the right-hand side of eq. (68), as well as the term
$\Xi$ in eq. (62), were subleading, we would not need to impose any further
constraints on the charges. This can be achieved provided that $\Delta
k=k_{-}-k_{+}\ll k_{-}$, and by taking $q_{-}=0$, $k_{+}q_{+}\ll 1$. Having
$k_{-}\sim k_{+}$ is natural in the stringy embeddings considered in
Danielsson:2022lsl ; Danielsson:2023alz . As in the examples studied in
Basile:2023tvh , it is also natural to have $q_{-}=0$ and the exterior charged
$q_{+}$ fully sourced by the presence of an electromagnetic field on the
brane. As we verify later, we also find that, even though the 4d Nariai is
near-extremal, the 5d uplift has $k_{+}q_{+}\ll 1$, which means that it is far
from supersymmetric.

##  6 Relating fine structure constant to the string coupling

In this section, we derive a relation between the gauge coupling of the
electric field, i.e. the fine structure constant, and the string coupling.
Before doing so, let us remind ourselves of some qualitative features of our
setup. As we explained at the end of section 3, it was shown in
Banerjee:2019fzz  that fundamental strings pulling on the brane provide very
small black holes on the brane, and we argued that larger black holes can be
seen as thick strings. Those can be seen as multiple strings puffed up into a
cylinder-like structure, where the strings dissolve into electric flux on the
brane. Let us tie this idea with the properties of the mass-charge diagram or
sharkfin of 4d black holes in figure 1. A black hole arising from one
fundamental string pulling on the brane, which has unit values of charge and
mass, sits in the lower-left corner of the diagram and corresponds to the
smallest possible, extremal black hole. If several such strings are added on
top of each other, one moves up along the extremal branch, where the inner and
the outer horizons of the black hole coincide. This can be done up to the
maximal Nariai value in the upper right corner of the diagram, where both
black hole horizons coincide with the cosmological horizon.

There now exists a very non-trivial way to test the dark bubble model of the
Nariai black hole. In ref. Banerjee:2019fzz , it was shown that a string
pulling on the brane corresponds to a particle of mass $\tau_{\text{ps}}L$,
where $\tau_{\text{ps}}$ is the effective tension of the string and $L=1/k$,
the AdS radius. The effective tension can be obtained by integrating the
energy density over the $S^{2}$. The difference in tension (including the
energy density of the electric flux) compared to the critical tension
contributes to the energy density as the cosmological constant.

According to the picture painted above (i.e. seeing the extremal Nariai
solution as a polarised collection of fundamental hanging strings), this
should be true also for the thick string representing the extremal Nariai
solution, implying that

$\tau_{\rm Nariai}L=m\,.$ (79)

In the following, we show that this indeed holds, provided that the fine
structure constant of electromagnetism is related to the string coupling,
precisely as in ref. Danielsson:2023alz .

To test this condition, let us first compare our charge $Q$ with its usual
definition in Maxwell theory in terms of the fine structure constant and
fundamental charge. We make use of the assumptions at the end of the previous
section and find using eq. (67)

$Q^{2}=\ell_{\text{Pl},4}^{2}\left(-\frac{1}{8}\frac{q_{+}^{2}}{\ell_{\text{Pl},5}^{6}}\frac{1}{\sigma}+\frac{c_{M}}{\ell_{\text{Pl},5}^{3}k^{3}}\right)\,,$
(80)

where

$c_{M}=c_{r-}-c_{r+}\,.$ (81)

Here we only include leading contributions in $k_{-}-k_{+}$ and have put
$k_{-}\sim k_{+}\sim k$. We double-check our approximations at the end. Note
that $c_{M}>0$ and $c_{r+}<0$. We first need to properly quantise the charges.
To do this, we recall eq. (11) and reinstate the speed of light and reduced
Planck’s constant:

$M=\frac{\hbar c\ell_{\text{Pl},4}^{2}}{8\pi}m\,,\qquad
Q^{2}=\frac{Q_{\text{EM}}^{2}\ell_{\text{Pl},4}^{2}}{32\pi^{2}\hbar\epsilon_{0}c}=N^{2}\alpha_{\text{EM}}\frac{\ell_{\text{Pl},4}^{2}}{8\pi}.$
(82)

We then take $Q_{\text{EM}}=Ne$, where $e$ is the elementary charge, and use
$\alpha_{\text{EM}}=\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}$, the fine structure
constant. As explained above, we can think of this 4d charge being
supplemented by $N$ fundamental strings pulling on the brane. These strings
also source the $H$-field on the outside, providing the 5d charge $q_{+}$.
Considering how a fundamental string sources the $H$-field (see Appendix C),
we find

$q_{\text{F1}}=\frac{\ell_{\text{Pl},5}^{3}}{4\pi^{2}\alpha^{\prime}}\,,$ (83)

and hence we can take $q_{+}=Nq_{\text{F1}}$ (and $q_{-}=0$). All this implies
that

$N^{2}\alpha_{\text{EM}}\frac{\ell_{\text{Pl},4}^{2}}{8\pi}=-\frac{N^{2}g_{s}\ell_{\text{Pl},4}^{2}}{16\pi}+\frac{\ell_{\text{Pl},4}^{2}c_{M}}{\ell_{\text{Pl},5}^{3}k^{3}}\,,$
(84)

and we conclude that we must choose

$\frac{\ell_{\text{Pl},4}^{2}c_{M}}{\ell_{\text{Pl},5}^{3}k^{3}}=\frac{N^{2}\ell_{\text{Pl},4}^{2}}{8\pi}\left(\alpha_{\text{EM}}+\frac{g_{s}}{2}\right)\,.$
(85)

To further test eq. (79), let us consider an extremal Nariai black hole. Such
black holes sit in the upper right corner of the charge-mass diagram in figure
1, have a Mink${}_{2}\times S^{2}$ near-horizon geometry and correspond to
taking $k_{2}=0$. We now want to view the extremal Nariai black hole as a
thick string pulling on the dark bubble. To do this we must first identify the
effective tension. This is obtained by integrating the correct energy density
over the $S^{2}$. The total energy density as measured in 4d is given by
$\frac{1}{\ell_{\text{Pl},4}^{2}a^{2}}$. This is not the relevant quantity as
observed from 5d. Several quantities contribute to the total energy density
(or tension) and they can be read from how the 5d spacetime is curved. The
strategy to find this energy density is to view one of the junction conditions
as the energy conservation equation. Following Brown:1988kg , we multiply the
junction condition in (45) by the factor
$\left(\sqrt{w_{-}(a)}+\sqrt{w_{+}(a)}\right)/2$, evaluate the extrinsic
curvature contributions and rearrange to get (to leading orders)

$-S^{t}_{t}\frac{\sqrt{w_{-}(a)}+\sqrt{w_{+}(a)}}{2}-\frac{3(k_{-}^{2}-k_{+}^{2})a}{2\ell_{\text{Pl},5}^{3}}=\frac{c_{M}}{\ell_{\text{Pl},5}^{3}k^{2}a^{3}}$
(86)

Integrating over $S^{2}$ to get a tension, multiplies the expression with
$4\pi a^{2}$. The first (positive) term on the left-hand side is the total
energy density of the brane (expressed in 5d units), including the
electromagnetic contribution, while the second (negative) term is due to the
lower energy inside the bubble. The right-hand side is the total effective
tension, $\tau_{\rm Nariai}$, as viewed from afar. We find

$m=\tau_{\rm Nariai}L=\frac{4\pi
Lc_{M}}{\ell_{\text{Pl},5}^{3}k^{2}a}=N^{2}\left(\alpha_{\text{EM}}+\frac{g_{s}}{2}\right)\frac{1}{2a}$
(87)

We find the same formula by looking at the masses of the black string
spacetimes, which were computed in ref. Bernamonti:2007bu . The effective
energy or mass of the string should be equal to the difference in masses of
the out- and inside spacetimes as energy is conserved in the bubble nucleation
process. From ref. Bernamonti:2007bu , we see that the mass, adapted to our
case, is

$m_{\pm}=\frac{4\pi
L}{\ell_{\text{Pl},5}^{3}k_{\pm}\sqrt{f_{\pm}(a)}}\left(\frac{1}{24}-c_{r\pm}\right).$
(88)

We have an additional factor $\sqrt{f_{\pm}(a)}$ due to the coordinate
rescalings that we have introduced in eq. (52). The effective mass of the
brane to leading order in our approximations is

$m=m_{+}-m_{-}=\frac{4\pi Lc_{M}}{\ell_{\text{Pl},5}^{3}k^{2}a},$ (89)

which agrees perfectly with (87).

To proceed, we look at the extremal Nariai limit, for which

$a=\frac{1}{\sqrt{6}k_{4}}\qquad
Q=\frac{1}{\sqrt{12}k_{4}}\qquad\frac{\ell_{\text{Pl},4}^{2}}{8\pi}m=\frac{2\sqrt{2}}{3}Q<Q\,,$
(90)

where the third relation is obtained by putting
$\hat{r}_{-}=\hat{r}_{+}=\hat{r}_{c}$ in eq. (14). This leads to

$m=\tau_{\rm Nariai}L=\frac{8\pi
Q^{2}}{\ell_{\text{Pl},4}^{2}\alpha_{\text{EM}}}\left(\alpha_{\text{EM}}+\frac{g_{s}}{2}\right)\frac{1}{\sqrt{2}Q}=\frac{3m}{4\alpha_{\text{EM}}}\left(\alpha_{\text{EM}}+\frac{g_{s}}{2}\right)$
(91)

This equation is only consistent when we require that
$\alpha_{\text{EM}}=\frac{3g_{s}}{2}$. What we see is that our identification
of the 4d gauge field with the electromagnetic Maxwell field forces the string
coupling $g_{s}$ to be directly related to the fine structure constant
$\alpha_{\text{EM}}$ in 4d.

This is exactly the same value obtained through the very different calculation
of Danielsson:2023alz  using unit charged extremal black hole as a fundamental
string pulling on the dark bubble. The calculation does not make use of any
resolution of the string into a 4d structure as in the case of the
cosmological Nariai black hole. Instead, it relies on the detailed microscopic
stringy embedding into higher dimensions, in particular on an AdS${}_{5}\times
S^{5}$ background Danielsson:2022lsl . Let us review the calculation and
compare.

The 4d extremal RN black hole with unit charge has the mass

$m_{\text{RN}}=\sqrt{\frac{8\pi\alpha_{\text{EM}}}{\ell_{\text{Pl},4}^{2}}}\,,$
(92)

while the effective 4d mass of the pulling string is given by

$m_{\text{ps}}=\tau_{\text{ps}}L=\frac{L}{2\pi\alpha^{\prime}}=\sqrt{\frac{12\pi
g_{s}}{\ell_{\text{Pl},4}^{2}}}\,.$ (93)

Here we have used the relation

$\frac{L^{2}\ell_{\text{Pl},4}^{2}}{\alpha^{\prime 2}}=48\pi^{3}g_{s}\,.$ (94)

This can be obtained by combining different equations, such as the relation
between the 10d and 5d Planck scale, as well as the expressions of the AdS5
scale and $S^{5}$ radius in terms of the $F_{5}$-flux quantum $N_{c}$, which
are

$\ell_{\text{Pl},5}^{3}=\frac{\kappa_{10}^{2}}{{\rm
Vol}(S^{5})}=2^{6}\pi^{4}g_{s}^{2}\frac{\alpha^{\prime 4}}{L^{5}}\,,\qquad
L^{4}=4\pi g_{s}N_{c}\alpha^{\prime 2}\,.$ (95)

The final piece of information needed for eq. (94) is the junction condition
(76), which by eq. (95), is

$\ell_{\text{Pl},4}^{2}=\frac{3N_{c}}{L}\ell_{\text{Pl},5}^{3}\,.$ (96)

The relation eq. (94) is also what guarantees that the tension of a pure
D3-brane (ignoring the subleading correction that gives the positive
cosmological constant) is exactly equal to the critical one $\sigma_{c}$. That
is,

$\frac{1}{(2\pi)^{3}\alpha^{\prime 2}g_{s}}=\sigma_{c}=\frac{3\Delta
k}{\ell_{\text{Pl},5}^{3}}=\frac{6}{L^{2}\ell_{\text{Pl},4}^{2}}\,.$ (97)

Demanding $m_{\text{RN}}=m_{\text{ps}}$, we again find
$\alpha_{\text{EM}}=\frac{3}{2}g_{s}$.

What is remarkable for the extremal Nariai black hole, is that the derivation
of the relation between the string coupling and the fine structure constant is
done without needing details of a stringy embedding. Nevertheless, this was
necessary in the analysis in ref. Danielsson:2023alz , for a single string as
reviewed above. This indicates that the dark bubble is a very robust
construction. In fact, the logic can be turned around by using the dark bubble
to derive the necessary properties of the fundamental objects representing,
e.g., unit charged extremal black holes. What we find is that strings in a
background such as AdS${}_{5}\times S^{5}$ have exactly those properties.

##  7 Discussion

In this paper, we embedded the extremal Nariai black hole into the dark bubble
scenario. We did so by joining two (modified) AdS5 black string backgrounds
together, accommodating the proper isometries and asymptotics of the Nariai
solution. We interpreted this as the black hole representing a thick string
pulling on the brane. Nevertheless, what happens for more general black holes
remains an interesting question. Such black holes are expected to warp the
brane more, and it would be interesting to see how the junction conditions can
be solved for such backgrounds. See ref. Banerjee:2021qei , for some related
results.

Next, one could wonder about Gregory-Laflamme instabilities Gregory:1993vy
for the setup we consider. According to the Gubser-Mitra conjecture
Gubser:2000ec ; Gubser:2000mm , a classical Gregory-Laflamme instability
should correspond to a thermodynamic instability, i.e. when the specific heat
becomes negative. In Bernamonti:2007bu , a critical value of the 5d charge was
identified beyond which the specific heat is always positive. Below this
value, three different branches open up of which two are stable. It turns out
that the extremal black string solution we consider on the outside, i.e. with
$c_{r}=c_{t}$, has zero Hawking temperature and belongs to a stable branch.
For the inside, where we are also interested in extremal background with
vanishing 5d charge, it was shown in ref. Brihaye:2007ju  that the classical
instability does arise for this regular background. A possible solution to
this problem is to consider a non-vanishing 5d charge on the inside as well.
Notice that all these observations are only studied for the case $k_{2}=0$. It
would be interesting for the future to see how the situation is affected for
non-vanishing $k_{2}$. Additionally, one should remember that these
instabilities have been investigated for static backgrounds. In the full dark
bubble setup that we have in mind, the black hole is more than the cylinder,
as depicted in figure 2. The black hole is riding a brane that is expanding.
It would be interesting to see if there is a connection between this and
classical instability.

Next, we managed to relate the electromagnetic fine structure constant to the
string coupling, in agreement with the calculation for an extremal black hole
of unit charge, as in Danielsson:2023alz . Remarkably, the exact same relation
is obtained even though our calculation for the cosmological Nariai black hole
does not make use of the microscopic relations between, e.g., the string scale
and the higher dimensional scale $L$. This suggests an internal consistency of
the dark bubble that is replicated by string theory.

Finally, as we constructed Nariai geometries, it is only natural to consider
the Festina Lente bound, discussed in refs. Montero:2019ekk ; Montero:2021otb
, which states that any charged particle in a (quasi-) de Sitter background
must obey

$m^{4}\geq m_{4}^{2}H^{2}\sim 1/(N_{c}^{2}\ell_{\text{Pl},4}^{4})\sim
1/L^{4},$ (98)

where we have expressed the bound in quantities relevant to the dark bubble.
The reason for this bound becomes apparent when the charge-mass diagram of
figure 1 is studied. The weak gravity conjecture ensures that a charged black
hole can be discharged by emitting particles of mass $Q>M$. However, when the
mass of the particle becomes too small, the charged black holes close to the
Nariai branch in figure 1, could exit the allowed region, and collapse into a
naked singularity. The Festina Lente bound prevents this from happening.

We believe that our embedding of the Nariai black hole into the dark bubble
can be used to further examine this bound. As we see from (98), the bound
exactly coincides with the size of the AdS5 scale and the size of the extra
dimension. According to Festina Lente, all charged particles must have a mass
at least set by this scale. This is particularly true for all fermions. As
already speculated in Basile:2023tvh , we expect that the embedding of the
standard model into the dark bubble involves sets of branes, possibly
separated on the internal five-dimensional manifold. Most of the fermions in
the Standard Model, such as the electrons, should have masses which are given
by short strings connecting separated branes. This is equivalent to a coupling
to the Higgs field. The neutrinos, which happen to have masses of the order
$1/L$, are a possible exception. Here, the masses could be due to a coupling
between massless modes on the brane (the flavour eigenstates) to massive modes
probing the bulk. It could even be that all fermions are of this form, with a
minimum mass set by $1/L$. It is just that most fermions have further
contributions to the mass from the coupling to the Higgs fields. This, then,
would suggest an intriguing connection between the Festina Lente bound and the
neutrino masses.

## Acknowledgements

We want to thank Daniel Panizo and Thomas Van Riet for interesting
discussions. VVH is supported by Olle Engkvists Stifelse.

##  Appendix A The dilatonic coupling

In this section we write down the conventions for the scales and couplings we
use, argue that the dilaton can be taken as a constant, and derive the energy-
momentum tensor of the brane, which was used in section 5. To connect the
black string backgrounds with string theory, we start from the 10d NSNS action
in Einstein frame for the dilaton:

$S_{\text{NSNS}}=\frac{1}{2\kappa_{10}^{2}g_{s}^{2}}\int\mathrm{d}x^{10}\sqrt{-g_{10}}\left(R_{10}-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}\mathrm{e}^{-(\phi-\phi_{0})}|H|^{2}\right)\,,$
(99)

with $H=\mathrm{d}B$ and where $\mathrm{e}^{\phi_{0}}=g_{s}$. Notice that this
has been obtained by Weyl rescaling the original string frame action with the
dynamical part of the 10d dilaton only. When the action is reduced to 5d,
where we allow only for an $H$-field along the 5d external spacetime, the
reduced NSNS action retains the same form

$S_{\text{NSNS}}=\frac{1}{2\kappa_{5}^{2}}\int\mathrm{d}x^{5}\sqrt{-g_{5}}\left(R_{5}-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}\mathrm{e}^{-(\phi-\phi_{0})}|H|^{2}\right)\,,$
(100)

where the 5d Planck constant satisfies

$\kappa_{5}^{2}=\text{Vol}_{5}\;g_{s}^{2}\;\kappa_{10}^{2}$ (101)

Notice that a 5d cosmological constant can arise from moduli stabilisation
from internal fluxes (RR and NSNS). We also remark that there is a dilatonic
coupling to the $H$-field. We have not included it in the main text, as we
assumed the dilaton to be constant, such that the dilatonic coupling
disappears. We justify this assumption here below. With the presence of a
dynamical dilaton, we infer that its equation of motion is

$\nabla^{2}\phi+\frac{1}{2}\mathrm{e}^{-(\phi-\phi_{0})}|H|^{2}=\frac{1}{\sqrt{-g_{5}}}\partial_{\mu}\left(g^{\mu\nu}\sqrt{-g_{5}}\;\partial_{\nu}\phi\right)+\frac{1}{2}\mathrm{e}^{-(\phi-\phi_{0})}|H|^{2}=0\,.$
(102)

We make the dilaton dependent on the $u$-coordinate, such that with the metric
(28) and $H$-field (29), this equation of motion can be written as

$\frac{\sqrt{w}}{fu^{2}}\>\left(f\sqrt{w}\;u^{2}\>\frac{g^{\prime}}{g}\right)^{\prime}+\frac{1}{2}\frac{g_{s}}{g}u^{-4}q^{2}=0$
(103)

where $\mathrm{e}^{\phi(u)}=g(u)$. With the large $ku$-expansions (37)-(39)
for $f$ and $w$, one can solve this for $g(u)$ and finds that the
$ku$-dependence is subleading:

$g(u)=g_{s}+O\left(\frac{\log(ku)}{(ku)^{4}}\right)\,,$ (104)

which justifies taking the dilaton to be constant.

##  Appendix B Energy-momentum tensor on the brane

In this appendix, we derive the stress tensor on the brane. We do so by
looking at the DBI action of a D3-brane in Einstein frame for the dilaton. It
is given by

$S_{\text{DBI}}=-\sigma\int\mathrm{d}x^{4}\sqrt{-\det\left(h_{ab}+\tau\mathcal{F}_{ab}\mathrm{e}^{-(\phi-\phi_{0})/2}\right)}\,,$
(105)

with $\sigma=\mu_{3}/g_{s}=1/((2\pi)^{3}\alpha^{\prime 2}g_{s})$,
$\tau=2\pi\alpha^{\prime}$ and

$\tau\mathcal{F}_{ab}=\tau F_{ab}+B_{ab}\,.$ (106)

When using a constant dilaton, we have

$S_{\text{DBI}}=-\sigma\int\mathrm{d}x^{4}\sqrt{-\det\left(h_{ab}+\tau\mathcal{F}_{ab}\right)}\,.$
(107)

Now we discuss how this brane sources the $B$-field. Let us expand the square
root such that the DBI action becomes:

$S_{\text{DBI}}=\int\mathrm{d}x^{4}\sqrt{-h_{4}}\left(-\sigma-\frac{\sigma\tau^{2}}{2}|\mathcal{F}|^{2}\right)$
(108)

This reminds of the Maxwell action where $\sigma\tau^{2}$ is the inverse gauge
coupling squared. This field strength sources the $H$-field. Indeed, we find
that

$\frac{\delta S_{\text{DBI}}}{\delta
B}=-\sigma\tau\star_{4}\mathcal{F}\wedge\delta(u-a)\mathrm{d}u,$ (109)

On the other hand, we find from (100) that

$\frac{\delta S_{\text{NSNS}}}{\delta
B}=\frac{1}{2\kappa_{5}^{2}}\mathrm{d}\star_{5}H\,,$ (110)

so the equation of motion for $H$ is

$\frac{1}{2\kappa_{5}^{2}}\mathrm{d}\star_{5}H=\sigma\tau\star_{4}\mathcal{F}\wedge\delta(u-a)\mathrm{d}u\,.$
(111)

With an electric Ansatz $\mathcal{F}=E\mathrm{d}t\wedge\mathrm{d}r$ and using
our expression for $H$, eq. (29) in terms of the 5d magnetic charges, we find
that this equation is solved when the electric field satisfies:

$E=\frac{\Delta q}{2\kappa_{5}^{2}}\frac{1}{\sigma\tau}\frac{1}{a^{2}}.$ (112)

At last, we compute the stress tensor on the brane, which is given by

$\displaystyle
S_{t}^{t}=S_{r}^{r}=-\sigma-\sigma\tau^{2}\frac{E^{2}}{2}=-\sigma-\frac{1}{2}\frac{(\Delta
q)^{2}}{(2\kappa_{5}^{2})^{2}}\frac{1}{\sigma}\frac{1}{a^{4}}$ (113)
$\displaystyle
S_{\theta}^{\theta}=S_{\phi}^{\phi}=-\sigma+\sigma\tau^{2}\frac{E^{2}}{2}=-\sigma+\frac{1}{2}\frac{(\Delta
q)^{2}}{(2\kappa_{5}^{2})^{2}}\frac{1}{\sigma}\frac{1}{a^{4}}\,.$ (114)

Notice that instead of expanding the square root in the DBI action, one could
also choose not to do so. The $H$ equation of motion then imposes that the
electric field $E$ must take the standard Born-Infeld form. Fortunately,
expanding that expression for large $a$ yields the same electric field at
leading order and hence the same stress tensor as was found above.

##  Appendix C Fundamental string charge

Finally, we briefly comment on how a fundamental string sources the $B$-field,
to derive the fundamental charge in 5d. A fundamental string couples to the
$B$-field as follows:

$S_{\text{WZ, F1}}=-\frac{1}{2\pi\alpha^{\prime}}\int B\,.$ (115)

It therefore sources the $B$-field in its equation of motion by

$\frac{1}{2\kappa_{5}^{2}}\mathrm{d}\star_{5}H-\frac{1}{2\pi\alpha^{\prime}}\delta_{\text{F1}}=0.$
(116)

Integrating this and using Gauss’ law, we obtain the fundamental charge as

$q_{\text{F1}}=\frac{\kappa_{5}^{2}}{4\pi^{2}\alpha^{\prime}}\,.$ (117)

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