IFT-UAM/CSIC-23-31

Small Black Hole Explosions

Roberta Angius, Jesús Huertas, Angel M. Uranga
Instituto de Física Teórica IFT-UAM/CSIC,
C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

Small black holes are a powerful tool to explore infinite distances in moduli
spaces. However, we show that in 4d theories with a scalar potential growing
fast enough at infinity, it is energetically too costly for scalars to diverge
at the core, and the small black hole puffs up into a regular black hole, or
follows a runaway behaviour.

We derive a critical exponent characterizing the occurrence or not of such
small black hole explosions, both from a 4d perspective, and in the 2d theory
after an ${\bf{S}}^{2}$ truncation. The latter setup allows a unified
discussion of fluxes, domain walls and black holes, solving an apparent puzzle
in the expression of their potentials in the 4d ${\cal{N}}=2$ gauged
supergravity context.

We discuss the realization of these ideas in 4d ${\cal{N}}=2$ gauged
supergravities. Along the way we show that many regular black hole
supergravity solutions in the literature in the latter context are incomplete,
due to Freed-Witten anomalies (or duals thereof), and require the emission of
strings by the black hole.

From the 2d perspective, small black hole solutions correspond to dynamical
cobordisms, with the core describing an end of the world brane. Small black
hole explosions represent obstructions to completing the dynamical cobordism.
We study the implications for the Cobordism Distance Conjecture, which states
that in any theory there should exist dynamical cobordisms accessing all
possible infinite distance limits in scalar field space. The realization of
this principle using small black holes leads to non-trivial constraints on the
4d scalar potential of any consistent theory; in the 4d ${\cal{N}}=2$ context,
they allow to recover from a purely bottom-up perspective, several non-trivial
properties of vector moduli spaces near infinity familiar from CY3
compactifications.

###### Contents

  1. 1 Introduction
  2. 2 Small black hole explosions: the 4d view
    1. 2.1 Small black holes
    2. 2.2 Perturbing the small black hole with a 4d potential
    3. 2.3 Solution for the critical case
  3. 3 Small black hole explosions: The 2d view
    1. 3.1 The 2d reduction
    2. 3.2 Interplay of black holes, fluxes and domain walls
    3. 3.3 The entropy function and the effective potential
    4. 3.4 Small black hole explosions from effective potential
  4. 4 Small black hole explosions: The Dynamical Cobordism view
    1. 4.1 The 1d running system
    2. 4.2 Small black hole solutions Redux
    3. 4.3 Small black holes and 2d Dynamical Cobordism
  5. 5 Explosions from topological obstructions
    1. 5.1 Topological obstructions from gaugings
    2. 5.2 A microscopic realization: The Freed-Witten anomaly
    3. 5.3 4d ${\cal N}=2$ gauged supergravity Black Holes come with strings attached
    4. 5.4 Explosions via string banquets
  6. 6 Interplay with the Cobordism Distance Conjecture
    1. 6.1 Small black holes and the Cobordism Distance Conjecture
    2. 6.2 Adding the potential: the $STU$ model
    3. 6.3 Constraints on potentials from small black holes explosions
  7. 7 Conclusions
  8. A All about $h$
    1. A.1 No 4d potential
    2. A.2 Including 4d potential
  9. B The 4d entropy functional computation of $V_{eff}$
  10. C Alternative derivation of the 1d equations of motion
  11. D 4d ${\cal{N}}=2$ supergravity
    1. D.1 Ungauged 4d ${\cal{N}}=2$ supergravity
    2. D.2 Gauged 4d ${\cal{N}}=2$ supergravity
    3. D.3 Black hole central charge

##  1 Introduction

The exploration of regions at infinite distance in moduli spaces (or in
general, scalar field spaces) is a powerful source of new physics results. One
instance is the celebrated swampland distance conjecture Ooguri:2006in 111See
e.g. Grimm:2018ohb ; Corvilain:2018lgw ; Grimm:2019ixq  for recent
developments in CY moduli spaces., derived from adiabatic motion in exact
moduli spaces. However, a physically more realistic avenue for the exploration
of infinite distances in field space, pioneered in Buratti:2018xjt  (see also
Lust:2019zwm ), is the use of spacetime dependent solutions in effective field
theories, in which certain scalars run along some spacetime coordinate.

In several setups, such solutions involve scalars which run off to infinite
distance at a finite distance in spacetime. The resulting singularities
describe physical objects whose properties relate to the infinite distance in
interesting ways. In particular, the end of the world branes in dynamical
cobordisms Buratti:2021yia ; Buratti:2021fiv ; Angius:2022aeq ;
Blumenhagen:2022mqw ; Blumenhagen:2023abk  (see also McNamara:2019rup  for the
swampland cobordism conjecture, Dudas:2000ff ; Blumenhagen:2000dc ;
Dudas:2002dg ; Dudas:2004nd  for early work and Basile:2018irz ;
Antonelli:2019nar ; Mininno:2020sdb ; Basile:2020xwi ; Basile:2021mkd ;
Mourad:2022loy  for other related recent developments), display interesting
scaling relations among the spacetime curvature and the spacetime and field
theory distances. Other setups are 4d EFT strings Lanza:2020qmt ;
Lanza:2021qsu ; Marchesano:2022avb , and small black holes (see e.g.
Dabholkar:2012zz  for a review, and Hamada:2021yxy  for discussion in the
swampland context). In fact, they can also be regarded as dynamical cobordisms
of the theories upon compactification on ${\bf{S}}^{1}$ for EFT strings and
${\bf{S}}^{2}$ for 4d small black holes.

An important advantage of dynamical explorations of infinity in field space,
as compared with the adiabatic one, is that the latter may be obstructed in
the presence of non-trivial scalar potentials growing at infinity222In fact,
taking fixed vevs in the slope of a non-trivial potential leads to dynamical
tadpoles, which besides being unaesthetical, can also lead to specific
incompatibilities with swampland constraints Mininno:2020sdb ., as emphasized
in Gonzalo:2018guu . In the context of ETW branes, the introduction of non-
trivial potentials growing at infinity was included in the analysis in
Angius:2022aeq . In the present work, we deal with this question for small
black holes. For concreteness, we focus on 4d small black holes, but we expect
similar conclusions to apply in other dimensions.

Small black holes are very robust solutions in theories with $U(1)$’s and
scalar-dependent gauge kinetic functions. Extremal small black holes may be
regarded as the endpoint of the evaporation of large black holes with some
charge. In addition, as emphasized in Hamada:2021yxy , they are interesting
probes of infinity in scalar moduli space333See Delgado:2022dkz  for a
different exploration of infinity, using large black holes.. As you approach
the black hole core, a black hole potential drives the scalars towards
infinite distance in field space, in a most radical realization of the
attractor mechanism. Indeed, the near-core solution is independent of the
value of moduli in the asymptotic vacuum infinitely away from the black hole.

One could even expect a similar phenomenon, even in the presence of a 4d
potential fixing the asymptotic moduli; the 4d scalar potential would be
irrelevant near the black hole core, still allowing scalars to attain infinity
in field space. However, this may not be so if the potential grows fast enough
near infinity in field space, so that it is able to drag the scalars back from
infinity. In such case, the small black hole core explodes444The name evokes
the similar (but converse) effect in Green:2006nv , where a black hole
attractor drags the 4d scalar vevs from an asymptotic vacuum to another. Here
we have a 4d potential dragging the scalar from the infinite values at the
small black hole core to some finite values at a regular horizon. Both setups
deal with the competition of black hole attractor potentials and 4d
potentials., possibly puffing up into a black hole solution of finite size
horizon, or perhaps following a runaway with no static endpoint configuration.

In this paper we provide explicit examples of the different possibilities, and
give precise criteria on the 4d potential for each of them. In particular, we
find that for small black holes exploring an infinite distance limit (with
gauge coupling scaling as a decreasing exponential along it), the small black
hole explosion occurs when the 4d potential grows faster (or equal) than a
critical exponent, related to that of the gauge coupling, but independent of
the black hole charge.

We study the implications of small black hole explosions for the capability of
small black holes to explore infinite distance limits in scalar moduli spaces.
Our findings motivate interesting non-trivial constraints on the growth near
infinity of 4d scalar potentials in any consistent theory of quantum gravity.
In fact, in 4d ${\cal{N}}=2$ theories we obtain constraints on the behaviour
of the Kähler potential near infinity in moduli space, matching, from a fully
bottom-up approach, non-trivial properties of CY3 compactifications.

We carry out the analysis both in the 4d picture, as well as in terms of the
2d truncation on the angular ${\bf{S}}^{2}$. The latter viewpoint is
interesting in itself Angius:2022aeq , since the spacetime dependent solutions
correspond to dynamical cobordisms, i.e. running configurations (for the
scalars and the size of ${\bf{S}}^{2}$), with the small black hole cores
corresponding to the ETW branes, i.e. the cobordism defect required to absorb
the charges. From this perspective, small black holes puffed up into regular
black holes correspond to running solutions, which, instead of hitting an ETW
brane which ends spacetime, relax to a minimum of the effective potential
describing the near horizon AdS2 vacuum. They represent solutions which fail
to realize dynamical cobordisms ending spacetime. In this respect, they
provide a large class of realizations of the idea, introduced in
Angius:2022aeq  (in the context of the D2/M2-brane solution Itzhaki:1998dd ),
that crouching corrections in the near core region of an ETW brane can reveal
hidden vacua coming down from infinity in moduli space.

Along the way in this work, we find several other interesting results. We show
that this 2d perspective is optimal to discuss the interplay of domain walls,
black holes and fluxes recently reviewed in Cribiori:2023swd  in the context
of in 4d ${\cal{N}}=2$ (gauged) supergravities. Moreover, our 2d perspective
explains that the seemingly different expression for the black hole and the 4d
potentials in terms of the covariantly holomorphic quantities, actually agree
upon the inclusion of an extra field associated to the ${\bf{S}}^{2}$ size
(see Cribiori:2023swd  for an alternative proposed explanation). We also use a
2d version the entropy function formalism Sen:2005wa  (see Sen:2007qy  for a
review) to provide a novel derivation of the effective potential in
Bellucci:2008cb , controlling the attractor mechanism in the presence of both
black hole and 4d scalar potentials.

Although our discussion is general, a particular setup in which to study the
competition of black hole attractor potentials and 4d potentials is 4d
${\cal{N}}=2$ gauged supergravity. Indeed, regular black holes which feature a
balance of both kinds of potentials have appeared in the literature, including
BPS solutions Cacciatori:2009iz ; DallAgata:2010ejj ; Hristov:2010ri , and
non-supersymmetric solutions Klemm:2012yg ; Klemm:2012vm ; Gnecchi:2012kb
(see Astesiano:2021hro  and references therein for a recent discussion).
However, we argue that, upon a closer look, there is a seemingly unnoticed
consistency condition in general not satisfied by these solutions. The problem
is related to the non-invariance of the configurations under some symmetries
underlying Abelian gaugings, and in certain microscopic setups shows up as a
Freed-Witten anomaly. The configurations can thus be amended by having the
black hole emit a number of 4d strings (or equivalently having those strings
fill up the near horizon AdS2). For regular black holes with large charges
$Q$, the anomaly is a $1/Q$ effect, meaning that these strings can be
described as probes in the above classical geometries. On the other hand, for
small black holes, the strings have a dramatic effect, and lead to a second
mechanism for small black hole explosion, which is possible even for
subcritical 4d potentials.

The paper is organized as follows. In Section 2 we discuss small black hole
explosions in 4d: In section 2.1 we introduce small black hole solutions with
no 4d potential, in section 2.2 we derive the equations of motion in the
presence of 4d potentials and the critical exponent, and in section 2.3 we
construct the critical small black hole solution. In Section 3 we approach the
systems from the perspective of its truncation to 2d, performed in section
3.1. In section 3.2 we describe the interplay of domain walls, black holes and
fluxes, and explain the relation between the seemingly different expression
for the potentials by using the ${\bf{S}}^{2}$ breathing mode. In section 3.3
we adapt the 4d entropy functional formalism to our 2d setup and derive the
effective potential for the scalars. In section 3.4 we use the effective
potential to discuss small black hole explosions and recover the critical
exponent. In Section 4 we discuss running solutions of the 2d theory: In
section 4.1 we use this perspective to recover the equations of motion derived
from 4d approaches in the literature. In section 4.2 we describe solutions for
small black holes in the subcritical and critical cases. In section 4.3 we
match this picture with the local description of dynamical cobordisms. In
Section 5 we describe the consistency conditions requiring certain black holes
to emit strings, and their implications for small and large black holes. In
section 5.1 we discuss dual descriptions of Abelian gaugings and their
corresponding symmetries, and the string emission effect. In section 5.2 we
review this in the illustrative example of the Freed-Witten anomaly. In
section 5.3 we discuss large black holes in the 4d ${\cal{N}}=2$ gauged
supergravity context, and their emission of strings. In section 5.4 we show
that small black holes emitting strings explode, even if the 4d potential is
subcritical. In Section 6 we discuss the ability of small black holes to
continue exploring infinite distance limits in moduli space, even in the
presence of 4d potentials. After reviewing the situation in exact moduli
spaces in section 6.1, we introduce 4d potentials in section 6.2 in an
illustrative example in the context of the $STU$ model in 4d ${\cal{N}}=2$
gauged supergravity. This motivates the proposal of a general criterion
restricting the properties of general 4d theories in section 6.3. In Section 7
we offer some final thoughts. Appendix A contains some technical aspects
complementing the discussion of Section 2. Appendix B provides a derivation of
the effective potential of Bellucci:2008cb  from the entropy function
formalism in 4d. In Appendix C we provide an alternative rederivation of the
equations of motion for running solutions of the 2d theory. Appendix D
collects some basic ingredients of 4d ${\cal{N}}=2$ (possibly gauged)
supergravity.

Note: As this work was being completed, we noticed vandeHeisteeg:2023ubh
which also explores the use of small black holes to constrain the properties
of 4d theories near infinite distance limits. It would be interesting to
explore the interplay of both approaches to these systems.

##  2 Small black hole explosions: the 4d view

In this section we discuss small black holes and their fate under the
introduction of a 4d potential, in a fairly direct 4d description. We follow a
formulation similar to Denef:1998sv ; Denef:2000nb  (see also Hamada:2021yxy
for a recent discussion).

###  2.1 Small black holes

We start with the structure of small black holes in the absence of 4d
potential. We consider 4d Einstein-Maxwell theory coupled to a real scalar
$\phi$:

$S=\int d^{4}x\sqrt{-g}\left[R-2|d\phi|^{2}+\frac{1}{2g^{2}}|F|^{2}\right]$
(2.1)

where the gauge coupling is a function $g(\phi)$.

We are looking for spherically symmetric static solution with electric charge
$Q$. We take the ansatz

$ds^{2}=-e^{2U(\tau)}dt^{2}+e^{-2U(\tau)}\left(\frac{d\tau^{2}}{\tau^{4}}+\frac{1}{\tau^{2}}d\Omega^{2}_{2}\right)$
(2.2)

$F_{2}=2\sqrt{2}g^{2}Qe^{2U}d\tau\wedge dt$ (2.3)

$\phi=\phi(\tau)$ (2.4)

Here $d\Omega_{2}$ is the volume form on a unit ${\bf{S}}^{1}$, and $\tau$
runs from the small black hole core (located at $\tau\to-\infty$) to
asymptotic Minkowski space (located at $\tau=0$). In general, one can
introduce an additional function $h(\tau)$, but it is possible to set $h=1$,
as discussed in Appendix A.1.

Plugging the ansatz into the action, we get the 1-dimensional action:

$S_{1d}=\int
d\tau\left\\{\dot{U}^{2}+\dot{\phi}^{2}+2g^{2}Q^{2}e^{2U}\right\\}$ (2.5)

This describes the dynamics of one particle in two dimensions under a
potential given by (minus) the last term. The equations of motion read

$\displaystyle\ddot{U}=2Q^{2}e^{2U}g(\phi)^{2}\quad,\quad\ddot{\phi}=Q^{2}e^{2U}\left(g(\phi)^{2}\right)^{\prime}$
(2.6)

where prime denotes derivative with respect to $\phi$. This must be
supplemented with the (‘zero energy’) constraint

$\left[\frac{1}{2}(\dot{U}^{2}+\dot{\phi}^{2})\right]-g^{2}Q^{2}e^{2U}=0$
(2.7)

In this work we are going to focus on the particular class of theories with
gauge coupling which for $\phi\to\infty$ behave as

$\displaystyle g=e^{-\alpha\phi}$ (2.8)

This includes theories arising from 4d ${\cal{N}}=2$ supergravity. Moreover,
the exponential dependence is the behaviour required by the distance
conjecture, see Hamada:2021yxy  for a recent discussion. It would be
straightforward to carry out the analysis of our work for other dependencies,
which are left as an exercise for the interested reader.

In this case, the equations of motion read

$\displaystyle\ddot{\phi}$ $\displaystyle=$ $\displaystyle-2\alpha
Q^{2}e^{2U-2\alpha\phi}$ $\displaystyle\ddot{U}$ $\displaystyle=$
$\displaystyle 2Q^{2}e^{2U-2\alpha\phi}$ (2.9)

The combination $\alpha U+\phi$ does not appear in the potential, and
corresponds to free motion. Using the initial conditions $U(\tau=0)=0$ (so
that we recover asymptotic flat space) and $\phi(\tau=0)=\phi_{0}$ (the
asymptotic vev at an arbitrary value in the scalar moduli space), we have

$\displaystyle\alpha U+\phi=\phi_{0}+v\tau$ (2.10)

Let us go for the combination that does appear in the potential. For shorthand
notation, we denote

$\displaystyle\varphi\equiv U-\alpha\phi\quad,\quad
q^{2}=2(1+\alpha^{2})Q^{2}$ (2.11)

The equation $\ddot{\varphi}=q^{2}e^{2\varphi}$ admits the solution

$\displaystyle\varphi=-\log(-q\tau+c)\,,$ (2.12)

where the initial conditions at $\tau=0$ implies that the constant
$c=e^{\alpha\phi_{0}}$.

The solution is thus

$\displaystyle U-\alpha\phi$ $\displaystyle=$
$\displaystyle-\log(-q\tau+e^{\alpha\phi_{0}})$ $\displaystyle\alpha U+\phi$
$\displaystyle=$ $\displaystyle\phi_{0}$ (2.13)

where we have already set $v=0$ in (2.10), as this turns out to be required by
the constraint (2.7). This gives

$\displaystyle e^{-2U}$ $\displaystyle=$ $\displaystyle
e^{-\frac{2\alpha}{1+\alpha^{2}}\phi_{0}}\,(-q\tau+e^{\alpha\phi_{0}})^{\frac{2}{1+\alpha^{2}}}$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{1+\alpha^{2}}\phi_{0}+\frac{\alpha}{1+\alpha^{2}}\log(-q\tau+e^{\alpha\phi_{0}})$
(2.14)

The region $\tau\to-\infty$ (the small black hole core) is a point at finite
spacetime distance, at which the scalar goes off to infinite distance with a
logarithmic profile, and the size of the ${\bf{S}}^{2}$, given by
$e^{-2U}/\tau^{2}$ in (2.2), goes to zero. It is important to notice that this
local behaviour is independent of the asymptotic value of $\phi$. As will be
clear in later sections, this is the small black hole version of the attractor
mechanism.

For completeness we also see that at $\tau\to-\infty$, $g$ tends to zero as

$\displaystyle g\sim\tau^{-\frac{\alpha^{2}}{1+\alpha^{2}}}$ (2.15)

###  2.2 Perturbing the small black hole with a 4d potential

We now consider introducing a 4d potential $V(\phi)$ in the theory. The action
reads

$S=\int d^{4}x\sqrt{-g}\left[R-2|d\phi|^{2}+\frac{1}{2g^{2}}|F|^{2}+2V\right]$
(2.16)

The discussion of $h$ now is more involved, but, as explained in Appendix A.2,
one can show that fairly generically $h\simeq 1$ near $\tau\to-\infty$. Since
we are interested in the fate of the small black hole core, we simplify the
discussion and just proceed with $h=1$. The more careful analysis will be
accounted for in the explicit solutions in the next sections.

The dynamics is described by the following 1-dimensional action:

$S_{1d}=\int
d\tau\left\\{\dot{U}^{2}+\dot{\phi}^{2}+2g^{2}Q^{2}e^{2U}-e^{-2U}\tau^{-4}V\right\\}$
(2.17)

The 4d scalar potential can modify the solution substantially. For instance,
near $\tau=0$, the second term dominates, and the scalar will be fixed at the
minimum of the 4d potential (if it exists555We assume that such minimum
exists, and consider small black holes whose asymptotic vev $\phi_{0}$ is
precisely that minimum. The fate of the small black hole core will
nevertheless be independent of this assumption.). On the other hand, near
$\tau\to-\infty$, the black hole potential term would seem to dominate and
maintain the basic features of the small black hole: a scalar going off to
infinity at the black hole core. This is indeed the case unless the 4d
potential grows too fast near $\phi\to\infty$ (i.e. near infinity in field
space), as we explore now in more detail.

In order to proceed, we need to parametrize the asymptotic behaviour of the 4d
scalar near $\phi\to\infty$. As discussed in Angius:2022aeq , a well-motivated
proposal is an exponential666We note that this $\delta$ differs from that in
Angius:2022aeq  by a factor of $\sqrt{2}$, due to different normalization of
the scalar kinetic term.

$\displaystyle V=V_{0}e^{\delta\phi}\quad,\quad V_{0}>0$ (2.18)

We emphasize that we focus on potentials growing at infinity $\phi\to\infty$,
namely $V_{0}>0$ and $\delta>0$, so that they have a chance of competing with
the black hole and prevent the scalars from reaching off to infinity. As we
will see, (2.18) is also a natural parameterization to make an easy comparison
with the black hole pull on scalars, controlled by the exponential gauge
coupling (2.8).

We can now check when the last term in (2.17) becomes relevant as compared
with that for a small black hole solution, by simply evaluating them on the
small black hole solution (2.14). We have that, as $\tau\to-\infty$,

$\displaystyle g^{2}Q^{2}e^{2U}=Q^{2}e^{2(U-\alpha\phi)}\sim\tau^{-2}$
$\displaystyle\tau^{-4}e^{-2U}V\sim\tau^{-2+\frac{\alpha(\delta-2\alpha)}{1+\alpha^{2}}}$
(2.19)

Hence, for the two terms to be comparable as $\tau\to-\infty$, we get the
criticality condition

$\displaystyle\delta=2\alpha$ (2.20)

which remarkably relates the behaviour of the 4d potential and the gauge
kinetic functions near infinity in field space.

For $\delta<2\alpha$, the 4d potential becomes subdominant near
$\tau\to-\infty$, so the small black hole behaviour prevails, and the small
black hole remains small. On the other hand, for $\delta>2\alpha$, the 4d
potential is too strong near the putative small black hole core, so that it is
energetically too costly to maintain the small black hole behaviour; the small
black hole must explode into a finite size solution, either in a runaway
towards larger and larger sized, or to a puffed-up finite size black hole
whose horizon scalar value is located in a regime where both terms can attain
a balance. Which of these two possibilities is realized depends on details
beyond the simple exponential approximation to the 4d potential and gauge
kinetic function.

We will see examples in later sections.

###  2.3 Solution for the critical case

Let us consider the critical case $\delta=2\alpha$. The equations of motion
are

$\displaystyle
h_{0}^{2}\ddot{U}=2Q^{2}e^{2\varphi}+\frac{V_{0}}{\tau^{4}}e^{-2\varphi}$
$\displaystyle h_{0}^{2}\ddot{\phi}=-2\alpha
Q^{2}e^{2\varphi}-\alpha\frac{V_{0}}{\tau^{4}}e^{-2\varphi}$ (2.21)

where $h_{0}$ is a constant discussed in Appendix A.2, and we have defined

$\displaystyle\varphi=U-\alpha\phi$ (2.22)

There is one freely moving combination

$\displaystyle\alpha\ddot{U}+\ddot{\phi}=0\quad\to\quad\alpha U+\phi=0$ (2.23)

(with hindsight we have chosen integration constants equal to zero, since
$\phi_{0}$ is irrelevant for $\tau\to-\infty$, and the velocity is eventually
set to zero by the hamiltonian constraint).

The orthogonal combination $\varphi$ obeys

$\displaystyle\ddot{\varphi}=q^{2}e^{2\varphi}+v_{0}\tau^{-4}e^{-2\varphi}$
(2.24)

with

$\displaystyle q^{2}=\frac{2Q^{2}(1+\alpha^{2})}{h_{0}^{2}}\quad,\quad
v_{0}=\frac{V_{0}(1+\alpha^{2})}{h_{0}^{2}}$ (2.25)

We now explore the existence of small black hole solutions, by using the
following profile for $\varphi$

$\displaystyle\varphi=-\log(-A\tau)$ (2.26)

Plugging the trial function (2.26) in the eom (2.24), we have

$\displaystyle\frac{1}{\tau^{2}}=\frac{q^{2}}{A^{2}\tau^{2}}+\tau^{-4}v_{0}A^{2}\tau^{2}$
(2.27)

So we get a solution if

$\displaystyle v_{0}A^{4}-A^{2}+q^{2}=0$ (2.28)

This is bi-quadratic. We solve for $A^{2}$ as

$\displaystyle A^{2}=\frac{1-\sqrt{1-4q^{2}v_{0}}}{2v_{0}}$ (2.29)

where we choose the negative signs because it reproduces the usual small black
hole when $v_{0}$ is very small. We then have

$\displaystyle
U=-\frac{1}{1+\alpha^{2}}\log(-A\tau)\quad,\quad\phi=\frac{\alpha}{1+\alpha^{2}}\log(-A\tau)$
(2.30)

Notice that this is compatible with actually having constant $h$ in (A.12),

$\displaystyle h_{0}=1+V_{0}A^{2}$ (2.31)

for a suitable choice of $h_{0}$ (determined by combining the equation above
with (2.29) and using the definitions (2.25)). It is also straightforward to
compute that the constraint in the last equation in (A.10) is solved.

Since we are interested in $v_{0}>0$, we get the constraint $q^{2}v_{0}<1/4$.
This means that even black holes of moderate charge admit no solution, hence
cannot remain small, but rather explode. In later sections we will describe
tools characterizing the ultimate fate of the former small black holes. In
particular, in the critical case it is possible that the black holes are
stabilized at a finite horizon puffed up black holes, as we will see in later
sections777In addition to the 4d ${\cal{N}}=2$ examples discussed in Section
5, see Anabalon:2013qua ; Anabalon:2013sra  for examples of puffed up black
holes in a phenomenological bottom up approach. It is easy to check they
correspond to the critical case..

##  3 Small black hole explosions: The 2d view

In this Section we describe small black holes and their possible explosions
from the perspective of the 2d theory resulting after reduction on the angular
${\bf{S}}^{2}$. Note that, because the compactification scale of the
${\bf{S}}^{2}$ will ultimately be comparable with the energy scales of the 2d
theory this is intended as a consistent truncation rather than a reduction to
a 2d effective field theory.

###  3.1 The 2d reduction

In this section we consider the 2d theory resulting from compactification of
the 4d theory on ${\bf{S}}^{2}$, with gauge fluxes888We focus on solutions
with ${\bf{S}}^{2}$ horizons, although in asymptotically AdS spaces there can
be horizons with zero or negative curvature..

Consider the following 4d action for a set of complex scalars $z^{i}$ coupled
to a set of Abelian electric and magnetic gauge bosons 999As compared with
section 2, we have changed the sign of the gauge kinetic terms to adapt it to
the compactification below, and the sign of the scalar potential to adapt to
complex fields and explore asymptotic regions towards $z^{i}\mapsto-i\infty$.

$S_{4d}=\int
d^{4}x\sqrt{-g_{4}}\left\\{R_{4}-2g_{i\bar{\jmath}}\partial_{\mu}z^{i}\partial^{\mu}\bar{z}^{\bar{\jmath}}-\frac{1}{2}{\rm
Im}\,\mathcal{N}_{\Lambda\Sigma}F^{\Lambda}_{\mu\nu}F^{\Sigma\mu\nu}-\frac{1}{2}{\rm
Im}\,\mathcal{N}^{\Lambda\Sigma}G_{\Lambda\mu\nu}G_{\Sigma}^{\mu\nu}-2V\right\\}.$
(3.1)

Here $\mathcal{N}_{\Lambda\Sigma}$ and $\mathcal{N}^{\Lambda\Sigma}$ are the
(possibly non-diagonal) electric and magnetic couplings, $F^{\Lambda}$ and
$G_{\Lambda}$ are the electric and magnetic field strengths, and $V$ is a
general 4d scalar potential.

This action is general, but we maintain a notation adapted to future use in 4d
${\cal{N}}=2$ examples.

We perform a compactification on ${\bf{S}}^{2}$ with the ansatz:

$\displaystyle ds^{2}_{4}=ds^{2}_{2}+e^{2\sigma}d\Omega^{2}_{2}$
$\displaystyle F^{\Lambda}=\sqrt{2}p^{\Lambda}\sin\theta d\theta\wedge
d\varphi\quad,\quad G_{\Lambda}=\sqrt{2}q_{\Lambda}\sin\theta d\theta\wedge
d\varphi$ (3.2)

where $p^{\Lambda}$ and $q_{\Lambda}$ are respectively the electric and
magnetic charges101010Note that this ansatz is implicitly consistent with
electric/magnetic duality, with the proviso that one introduces additions 2d
spacetime filling components of the fields strengths. Also, the 2d spacetime
dependent warp factor $e^{2\sigma}$ controls the size of the ${\bf{S}}^{2}$.
The reduced 2-dimensional action is (after removing an overall $4\pi$ factor):

$\displaystyle S_{2d}=\int d^{2}x\sqrt{-g_{2}}e^{2\sigma}$
$\displaystyle\left[R_{2}-4\Delta\sigma-6\left(\partial\sigma\right)^{2}-2g_{i\overline{j}}\partial_{\mu}z^{i}\partial^{\mu}\overline{z}^{\overline{j}}\right.$
$\displaystyle\left.+2e^{-2\sigma}-2e^{-4\sigma}V_{BH}-2V\right]$

Where we have introduced the black hole potential

$V_{\rm BH}=-\frac{1}{2}\left(p^{\Lambda}{\rm
Im}\,\mathcal{N}_{\Lambda\Sigma}p^{\Sigma}+q_{\Lambda}{\rm
Im}\,\mathcal{N}^{\Lambda\Sigma}q_{\Sigma}\right)$ (3.3)

Integrating by parts the term $\Delta\sigma(r)$, we get an additional
contribution for its kinetic term, giving:

$S_{2d}=\int
d^{2}x\sqrt{-g_{2}}e^{2\sigma}\left[R_{2}+2\left(\partial\sigma\right)^{2}-2g_{ij}\partial
z^{i}\partial\overline{z}^{\overline{j}}+2e^{-2\sigma}-2e^{-4\sigma}V_{BH}-2V\right]$
(3.4)

###  3.2 Interplay of black holes, fluxes and domain walls

Let us make a small aside in this section, which is independent from the
general discussion.

A good motivation for the 2d perspective is that it puts both the 4d potential
and the black hole potential on a similar footing, namely, they both
contribute to the potential of the 2d theory. A particularly nice realization
of this is in the context of 4d ${\cal{N}}=2$ theories, see Appendix D for
notation and definitions, as we discuss next.

In 4d ${\cal{N}}=2$ gauged supergravity, the scalar potential can be regarded
as arising from (possibly generalized) fluxes in the compactification. These
can be regarded as having been produced by domain wall, carrying (3-form)
charges $g_{\Lambda},g^{\Lambda}$ whose central charge ${\cal L}$ gives the
superpotential (D.12)

$\displaystyle{\cal L}=e^{{\cal K}/2}\langle{\cal G},v\rangle=e^{{\cal
K}/2}(Z^{\Lambda}g_{\Lambda}-\partial_{\Lambda}Fg^{\Lambda})$ (3.5)

leading to a 4d scalar potential (D.13)

$\displaystyle V=g^{i{\bar{\jmath}}}\,{\cal D}_{i}{\cal L}\,{\bar{\cal
D}}_{\bar{\jmath}}{\bar{\cal L}}-3|{\cal L}|^{2}$ (3.6)

where we recall that ${\cal
D}_{i}f=\partial_{i}f+\frac{1}{2}(\partial_{i}{\cal K}\cdot f)$

On the other hand, black holes with charges $q_{\Lambda},q^{\Lambda}$ arise
from 4d particles characterized by a central charge (D.19)

$\displaystyle{\cal Z}=e^{{\cal
K}/2}(Z^{\Lambda}q_{\Lambda}-\partial_{\Lambda}Fq^{\Lambda})$ (3.7)

The black hole potential (D.20) is then given by

$\displaystyle V_{BH}=g^{i{\bar{\jmath}}}\,{\cal D}_{i}{\cal Z}\,{\bar{\cal
D}}_{\bar{\jmath}}{\bar{\cal Z}}+|{\cal Z}|^{2}$ (3.8)

The identical structure of the central charges ${\cal L}$ and ${\cal Z}$
arises basically from the fact that, in the microscopic description, both the
domain walls and the 4d particles are obtained from the same kind of object in
what respects to the internal geometry. For instance, in type IIB
compactifications, domain walls arise from D5-branes wrapped on 3-cycles of
the internal CY, while particles arise from D3-branes wrapped on 3-cycles as
well. Hence, the corresponding central charges are controlled by the same
geometric quantities. Clearly, we have a similar picture for the (mirror) type
IIA compactifications, where domain walls arise from D2-, D4-, D6, D8-branes
wrapped on 0-, 2-, 4-, 6-cycles, while 4d particles arise from D0-, D2-, D4-,
D6-branes on the same kind of even-dimensional cycles.

On the other hand, there is a seemingly puzzling difference in the expression
of the corresponding potentials $V$ and $V_{BH}$. This was recently considered
in Cribiori:2023swd , but here we propose a more natural explanation. The
expression for the potentials differ because they correspond to potentials in
theories of different dimensionality. However, in the context of our
${\bf{S}}^{2}$ compactification, both 4d domain walls and 4d particles
compactify onto 2d particles (or 2d domain walls, since in 2d both kind of
objects are the same). Hence in this context, the contribution of both
potentials must be on equal footing, as we saw in (3.4), and morally, given by
the 2d formula $V=|{\cal D}{\cal Z}|^{2}+|{\cal Z}|^{2}$, where here ${\cal
Z}$ means the central charge of either kind of object.

The point is that, the different microscopic origin of $V$ and $V_{BH}$
manifests in their different coupling with the new field $\sigma$ arising from
the ${\bf{S}}^{2}$ compactification. The presence of this extra field in the
theory accounts for the factor of $+4|{\cal L}|^{2}$ necessary to turn (3.6)
into the form (3.8). The argument is as follows111111 This is analogous to how
in 4d ${\cal{N}}=1$ ‘no-scale’ models, the scalar potential
$|{\tilde{D}}W|^{2}-3|W|^{2}$ turns into $|DW|^{2}$ (where the $\tilde{D}$ in
the first expression includes the Kähler moduli and in the second does not).

Consider complexifying the ${\bf{S}}^{2}$ breathing modulus (e.g. by including
the NSNS 2-form background in type II models) into a new complex modulus
$z=b+ie^{-\sigma}$, and let us introduce a Kähler potential

$\displaystyle k(z,{\overline{z}})=-4\log(-i(z-{\overline{z}}))$ (3.9)

Note that this reproduces the kinetic term of $\sigma$ in (3.4).

There is thus a total Kähler potential given by

$\displaystyle{\tilde{\cal K}}=k(z,{\overline{z}})+{\cal K}$ (3.10)

where the last piece is the usual 4d Kähler potential for vector multiplet
moduli. In what follows, we use a tilde to denote quantities defined with this
modified Kähler potential. For instance, this also modifies the Kähler
derivatives to

$\displaystyle{\tilde{D}}_{i}f=\partial_{i}f+\frac{1}{2}\partial_{i}{\tilde{\cal
K}}\cdot f=\partial_{i}f+\frac{1}{2}\partial_{i}(k+{\cal K})\cdot f$ (3.11)

Let us now consider the modified version of the superpotential (3.5). We have

$\displaystyle{\tilde{\cal L}}=e^{{\tilde{\cal K}}/2}\langle{\cal
G},v\rangle=e^{k/2}{\cal L}$ (3.12)

The 2d potential will be given by the tilded version of (3.6)

$\displaystyle V=|{\tilde{D}}{\tilde{\cal L}}|^{2}-3|{\tilde{\cal L}}|^{2}$
(3.13)

Computing the derivatives

$\displaystyle{\tilde{D}}_{z}{\tilde{\cal
L}}=\frac{4}{(z-{\overline{z}})^{3}}{\cal
L}\quad,\quad{\tilde{D}}_{i}{\tilde{\cal L}}=e^{k/2}D_{i}{\cal L}$ (3.14)

we easily get

$\displaystyle V=g^{z{\overline{z}}}{\tilde{D}}_{z}{\tilde{\cal
L}}{\overline{\tilde{D}}}_{\overline{z}}{\overline{\tilde{\cal
L}}}+e^{k}g^{i{\overline{j}}}D_{i}{\cal
L}{\overline{D}}_{\overline{j}}{\overline{\cal L}}-3e^{k}|{\cal
L}|^{2}=|D{\tilde{\mathcal{L}}}|^{2}+|{\tilde{\mathcal{L}}}|^{2}$ (3.15)

As promised, we recover an expression identical to the black hole potential
(3.8), by simply replacing the corresponding central charges.

Let us finish by mentioning another argument for the fact that the 4d
superpotential must be appropriately dressed with the ${\bf{S}}^{2}$ breathing
mode to becomes on par with the 2d black hole central charge. From our above
discussion, the natural covariantly holomorphic quantity in the 2d theory is a
combination of ${\cal Z}$ and the dressed superpotential (3.12),

$\displaystyle e^{k/2}{\cal L}=-\frac{1}{(z-{\overline{z}})^{2}}{\cal L}\sim
e^{2\sigma}{\cal L}$ (3.16)

This has precisely the required structure of the holomophic superpotential
(D.23) introduced in DallAgata:2010ejj .

###  3.3 The entropy function and the effective potential

In this section we use the 2d action (3.4) to obtain the conditions holding in
the near horizon limit of a general regular black hole solution. The
computation turns out to be a 2d version of the entropy functional formalism
computation, see Appendix B for the 4d computation, following Sen:2005wa  (see
Sen:2007qy  for a review).

We consider an AdS2 ansatz121212We focus on this, rather than flat of dS2, at
least for the time being. for the metric

$ds_{2}^{2}=v_{1}\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}\right)$ (3.17)

We also take a constant value for the moduli $z$, and denote the size of the
${\bf{S}}^{2}$ by $v_{2}$

$\displaystyle e^{2\sigma}=v_{2}$ (3.18)

Evaluating (3.4) on this solution, we get the quantity

$\displaystyle\mathcal{E}(v_{1},v_{2},p,q)=-4\pi\sqrt{-g_{2}}\mathcal{L}_{2}=8\pi
v_{2}-8\pi v_{1}+8\pi V_{BH}\frac{v_{1}}{v_{2}}+8\pi Vv_{1}v_{2}$ (3.19)

where the factor of $4\pi$ is just to meet the conventions of the entropy
functional formalism. Note that in this formalism, as compared with Appendix
B, it is not necessary to perform the Legendre transform, since the magnetic
charges were included in the original action.

The condition to get AdS2 vacua corresponds to the minimization of (3.19) with
respect to the variables $v_{1}$, $v_{2}$ and the moduli $z$. We have

$\displaystyle\textit{(i)}\quad\quad\frac{\partial\mathcal{E}}{\partial
v_{1}}=0\quad,\quad$ $\displaystyle Vv_{2}^{2}-v_{2}+V_{BH}=0$ (3.20)
$\displaystyle\textit{(ii)}\quad\quad\frac{\partial\mathcal{E}}{\partial
v_{2}}=0\quad,\quad$ $\displaystyle 1-V_{BH}\frac{v_{1}}{v_{2}^{2}}+Vv_{1}=0$
$\displaystyle\textit{(iii)}\quad\frac{\partial\mathcal{E}}{\partial
z^{i}}\Big{|}_{z^{i}_{H}}=0\quad,\quad$
$\displaystyle\frac{v_{1}}{v_{2}}\frac{\partial V_{BH}}{\partial
z^{i}}\Big{|}_{z^{i}_{H}}+v_{1}v_{2}\frac{\partial V}{\partial
z^{i}}\Big{|}_{z^{i}_{H}}=0$

The first two give

$\displaystyle v_{2}=\frac{1-\sqrt{1-4VV_{BH}}}{2V}\quad,\quad
v_{1}=\frac{v_{2}}{\sqrt{1-4V_{BH}V}}$ (3.21)

and replacing into the third, we recover the extremization of the quantity
(which in fact corresponds to the ${\bf{S}}^{2}$ size, hence the horizon area)

$\displaystyle V_{eff}=\frac{1-\sqrt{1-4V_{BH}V}}{2V}$ (3.22)

This turns out to be the effective potential derived in Bellucci:2008cb , in a
different approach related to that in section 4.1. This is just a reflection
of the equivalence of the entropy functional and attractor mechanism.

###  3.4 Small black hole explosions from effective potential

It is now a simple exercise to exploit $V_{eff}$ to determine the fate of
small black holes upon the introduction of a 4d potential. Consider a small
black hole with purely electric charge $Q$. We take the $U(1)$ gauge coupling
$g(\phi)=e^{-\alpha\phi}$, as in previous discussions (generalization to other
possible functional dependencies is straightforward), hence

$\displaystyle V_{BH}=2Q^{2}e^{-2\alpha\phi}$ (3.23)

Since we are interested in the fate of the core of the small black hole, where
$\phi\to\infty$, we parametrize the 4d scalar potential by an exponential as
in (2.18)

$\displaystyle V=V_{0}e^{\delta\phi}\quad V_{0}>0$ (3.24)

The effective potential (3.22) is

$\displaystyle
V_{eff}=\frac{1-\sqrt{1-8V_{0}Q^{2}e^{(\delta-2\alpha)\phi}}}{2V_{0}e^{\delta\phi}}$
(3.25)

We can use it to easily recover the different cases introduced in Section 2:

$\bullet$ If $\delta<2\alpha$, we may expand the square root and get

$\displaystyle V_{eff}\simeq 2Q^{2}e^{-2\alpha\phi}=V_{BH}$ (3.26)

The 4d potential is too weak at the black hole core and the attractor is
unchanged. The scalar is dragged to $\phi\to\infty$, and the small black hole
remains small. Note that this does not rule out the possible existence of
finite size black holes, although this is model dependent as it involves the
behaviour of the gauge coupling and 4d potential at finite values of the
moduli; hence, the model independent results on which our analysis focuses is
that the small black hole persists.

$\bullet$ The critical case $\delta=2\alpha$ leads to

$\displaystyle
V_{eff}=\frac{1-\sqrt{1-8V_{0}Q^{2}}}{2V_{0}}\,e^{-2\alpha\phi}$ (3.27)

For $V_{0}Q^{2}\leq 1/8$, the effective potential is exponentially damped as
in $V_{BH}$, but with different prefactor. The scalar is still dragged to
$\phi\to\infty$ by the black hole, which remains small. On the other hand, for
$V_{0}Q^{2}>1/8$, the effective potential is not well defined, meaning that
there is no horizon, neither at finite values of the moduli (and black hole
size) nor at infinity. Of course this conclusion is reliable only as long as
the approximation of the gauge coupling and 4d potential as simple
exponentials holds. If their behaviour away from infinity in moduli space is
different, the small black hole may stabilize into a puffed-up finite size
black hole.

$\bullet$ When $\delta>2\alpha$, the exponential in the square root blows up
and the effective potential is not well defined. As above, the small black
hole explodes in a runaway fashion, at least in the region where the
exponential approximations to the gauge coupling and 4d potential are
reliable.

##  4 Small black hole explosions: The Dynamical Cobordism view

As explained in the introduction, the ${\bf{S}}^{2}$ reduction in the previous
Section is the natural setup in which small black hole solutions can be
regarded as dynamical cobordisms of the resulting 2d theory, given by the
running of the 4d scalar and the ${\bf{S}}^{2}$ size along the radial
direction. The small black hole core plays the role of the ETW brane in the
sense of Angius:2022aeq , namely the cobordism defect that removes the fluxes
and allows spacetime to end in the radial direction. From this perspective,
small black hole explosions signal a failure of the dynamical cobordism, due
to growing corrections as one approaches the putative ETW brane; in case a
puffed-up black hole solution exists, the running solution describes the
eventual stabilization of the running solution in a (previously hidden) AdS2
vacuum.

###  4.1 The 1d running system

In this section we display black holes as running solutions of the 2d theory
(3.4) derived in section 3.1. For convenience we perform the following
redefinition

$\displaystyle e^{2\sigma}=r^{2}e^{2C}$ (4.1)

The effective action in terms of $C$ reads

$\displaystyle S_{2d}=\int
d^{2}x\sqrt{-g_{2}}r^{2}e^{2C}\left\\{R_{2}-2g_{i\overline{\jmath}}\partial_{\gamma}z^{i}\partial^{\gamma}\overline{z}^{\overline{\jmath}}+2\partial_{\gamma}C\partial^{\gamma}C+\frac{4}{r}C^{\prime}e^{-2B}+\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+\frac{2}{r^{2}}(e^{-2B}+e^{-2C})-2\frac{V_{BH}}{r^{4}}e^{-4C}-2V_{4d}\right\\}$
(4.2)

where the prime denotes the derivative respect to the radial coordinate.

Now we consider the following ansatz for the 2d metric:

$ds^{2}_{2}=-e^{2A}dt^{2}+e^{2B}dr^{2}$ (4.3)

and we assume that the functions appearing in the metric and the 2d scalars
$z^{i}$ and $C$ just depend by the radial coordinate. We also restrict to the
simplest setup of one single real scalar $\phi$ with $g_{\phi\phi}=1$.

Plugging in the ansatze in the 2d action (4.2), we get the following 1d action

$\begin{split}S=\int
dre^{A-B+2C}r^{2}&\left\\{C^{\prime}(C^{\prime}+2A^{\prime})+\frac{2}{r}\left(C^{\prime}+A^{\prime}\right)+\frac{1}{r^{2}}(1+e^{2(B-C)})-\phi^{\prime
2}+\right.\\\
&\left.-\frac{e^{2B-4C}}{r^{4}}V_{BH}-e^{2B}V\right\\}\end{split}$ (4.4)

where we have integrated by parts to remove some total derivatives.

This is an effective action controlling the dynamics of running solutions of
the 2d theory. The equations of motion from variations of $A,B,C$ and $\phi$,
respectively, are

$\displaystyle-e^{-2B}\Big{[}2C^{\prime\prime}+C^{\prime}(3C^{\prime}-2B^{\prime})+\frac{2}{r}(3C^{\prime}-B^{\prime})+\frac{1}{r^{2}}\left(1-e^{2(B-C)}\right)+\phi^{\prime
2}\Big{]}=\frac{e^{-4C}}{r^{4}}V_{BH}+V$
$\displaystyle-e^{-2B}\Big{[}C^{\prime}(C^{\prime}+2A^{\prime})+\frac{2}{r}(C^{\prime}+A^{\prime})+\frac{1}{r^{2}}\left(1-e^{2(B-C)}\right)-\phi^{\prime
2}\Big{]}=\frac{e^{-4C}}{r^{4}}V_{BH}+V$
$\displaystyle-e^{-2B}\Big{[}A^{\prime\prime}+C^{\prime\prime}+A^{\prime}(A^{\prime}-B^{\prime})+C^{\prime}(C^{\prime}-B^{\prime}+A^{\prime})+\frac{1}{r}(A^{\prime}-B^{\prime}+2C^{\prime})+\phi^{\prime
2}\Big{]}=$
$\displaystyle\quad\quad\quad\quad\quad=-\frac{e^{-4C}}{r^{4}}V_{BH}+V$
$\displaystyle-e^{-2B}\Big{[}2\phi^{\prime\prime}+2\phi^{\prime}\left(A^{\prime}-B^{\prime}+2C^{\prime}+\frac{2}{r}\right)\Big{]}=-\frac{e^{-4C}}{r^{4}}\frac{\partial
V_{BH}}{\partial\phi}-\frac{\partial V}{\partial\phi}$ (4.5)

The above ansatz and the resulting equations of motion correspond to those in
Bellucci:2008cb  (for the particular case of a single scalar with canonical
kinetic term). We have rederived them as controlling the running solution of
the 2d theory, triggered by the presence of the non-trivial potential in
(3.4), in the spirit of Buratti:2021yia ; Buratti:2021fiv ; Angius:2022aeq .
In Appendix C we provide an alternative rederivation of these equations of
motion.

It is interesting to mention that, as explained in Bellucci:2008cb , it is
possible to derive the effective potential (3.22) from these equations, by
plugging in an AdS2 ($\times{\bf{S}}^{2}$) ansatz with

$\displaystyle A=-B=\log\frac{r}{r_{A}}\quad,\quad C=\log\frac{r_{H}}{r}$
(4.6)

with $r_{A}$, $r_{H}$ the horizon value of AdS2 and ${\bf{S}}^{2}$ radii,
respectively. We refer the reader to Bellucci:2008cb  for details.

###  4.2 Small black hole solutions Redux

In this language, small black hole solutions correspond to dynamical
cobordisms, in which the 1d running solution hits a singularity, at which the
scalars (including the ${\bf{S}}^{2}$ size) go off to infinite distance in
field space, with the small black hole core playing the role of cobordism
defect. On the other hand, finite size black holes correspond to setups in
which the running halts and the 2d theory reaches an AdS minimum.

In the previous section we derived the equations of motion (4.5), which
control this running. Using reparametrization of $r$, one can set $A=-B\equiv
U$. After some algebra, the equations can be recast as

$\begin{array}[]{l}U^{\prime 2}+\phi^{\prime
2}+\frac{1}{2}\left(\psi^{\prime\prime}+e^{-2\psi}\right)=\frac{e^{-4C-2U}}{r^{4}}V_{BH};\\\
\\\ C^{\prime\prime}+C^{\prime 2}+\frac{2}{r}C^{\prime}+\phi^{\prime 2}=0;\\\
\\\
\phi^{\prime\prime}+2\phi^{\prime}\psi^{\prime}=\frac{1}{2}\frac{e^{-4C-2U}}{r^{4}}\frac{\partial
V_{BH}}{\partial\phi}+\frac{1}{2}e^{-2U}\frac{\partial V}{\partial\phi};\\\
\\\ \psi^{\prime\prime}+2\psi^{\prime 2}-e^{-2\psi}=-2e^{-2U}V.\end{array}$
(4.7)

where we have introduced $\psi$, defined as

$\displaystyle C+U\equiv\psi-\log r$ (4.8)

It is worthwhile to point out that the functions $B$ and $\psi$ are related to
$U$ and $h$ in Section 2 by

$\displaystyle B=-U\quad,\quad
h(\tau)=\psi^{\prime}(r)e^{\psi}\quad,\quad\tau=e^{-\psi}$ (4.9)

In particular the last equation in (4.7) becomes equivalent to the hamiltonian
constraint from the vanishing of the integrand in (A.11).

We now turn to describing diverse small black hole solutions.

####  4.2.1 Small black holes with zero or subcritical 4d potential

As we anticipated in section 2, the small black hole solution for subcritical
4d potential is just the small black hole solution in the absence of 4d
potential, $V=0$. One can check that this corresponds to $C=-U$, hence, from
(4.8), we have $\psi=\log r$. The equations of motion (4.7) read:

$\begin{array}[]{l}U^{\prime 2}+\phi^{\prime 2}=\frac{e^{2U}}{r^{4}}V_{BH};\\\
\\\ U^{\prime\prime}+\frac{2}{r}U^{\prime}-U^{\prime 2}-\phi^{\prime 2}=0;\\\
\\\
\phi^{\prime\prime}+\frac{2}{r}\phi^{\prime}=\frac{1}{2}\frac{e^{2U}}{r^{4}}\frac{\partial
V_{BH}}{\partial\phi},\end{array}$ (4.10)

We now focus on the behaviour $V_{BH}=Q^{2}e^{-2\alpha\phi}$ near $r\to 0$,
and perform a change of coordinates $r\to-1/\tau$ to get

$\begin{array}[]{l}\dot{U}^{2}+\dot{\phi}^{2}=Q^{2}e^{2(U-\alpha\phi)};\\\ \\\
\ddot{U}=Q^{2}e^{2(U-\alpha\phi)};\\\ \\\ \ddot{\phi}=-\alpha
Q^{2}e^{2(U-\alpha\phi)},\end{array}$ (4.11)

These are precisely the constraint (2.7) and the equations (2.6). The
resulting solution is therefore

$\displaystyle
U=-\frac{1}{1+\alpha^{2}}\log(-q\tau)\quad,\quad\phi=\frac{\alpha}{1+\alpha^{2}}\log(-q\tau),$
(4.12)

where we have dropped some possible integration constants, irrelevant in the
near-core $\tau\to-\infty$ region. These are just the near core behaviour of
the solution (2.14).

####  4.2.2 Small black holes in the Critical case

The critical case of the small black hole is given when in the near horizon
limit ($r\to 0$), the $V$ term is not negligible with respect to $V_{BH}$ but
comparable. Restoring the $V$ term, the equations of motion (4.10) are:

$\begin{array}[]{l}U^{\prime 2}+\phi^{\prime 2}-\psi^{\prime
2}+e^{-2\psi}=e^{2U-4\psi}V_{BH}+e^{-2U}V;\\\ \\\
U^{\prime\prime}+2U^{\prime}\psi^{\prime}=e^{2U-4\psi}V_{BH}-e^{-2U}V;\\\ \\\
\phi^{\prime\prime}+2\phi^{\prime}\psi^{\prime}=\frac{1}{2}e^{2U-4\psi}\frac{\partial
V_{BH}}{\partial\phi}+\frac{1}{2}e^{-2U}\frac{\partial V}{\partial\phi};\\\
\\\ \psi^{\prime\prime}+2\psi^{\prime 2}-e^{-2\psi}=-2e^{-2U}V.\end{array}$
(4.13)

Already here, we can understand the critical case as having $C+U={\rm
const}\neq 0$. Using the last equation of (4.13), this implies that
$e^{-2U}V=c_{0}e^{-2\psi}$, which implies $\psi=\log
r+\frac{1}{2}\log(1-2c_{0})$, where $c_{0}$ is a constant that satisfies
$c_{0}<\frac{1}{2}$.

Since in the critical case both potentials must give comparable contributions,
we have that $e^{2U-4\psi}V_{BH}\sim e^{-2U}V\propto e^{-2\psi}$, which
implies that $V_{BH}\propto e^{-2U+2\psi}$. Hence $V\sim 1/V_{BH}$, in
agreement with the analysis in terms of $V_{eff}$ in section 3.4. Using now
$V_{BH}=Q^{2}e^{-2\alpha\phi}$, hence $V=V_{0}e^{2\alpha\phi}$, and
introducing the $r=-(\tau\sqrt{1-2v_{0}})^{-1}$ change of coordinates, we have

$\begin{array}[]{l}\dot{U}^{2}+\dot{\phi}^{2}=\frac{1}{1-2c_{0}}\left[Q^{2}e^{2(U-\alpha\phi)}-\frac{V_{0}}{\tau^{4}}e^{-2(U-\alpha\phi)}\right];\\\
\\\
\ddot{U}=\frac{1}{1-2c_{0}}\left[Q^{2}e^{2(U-\alpha\phi)}-\frac{V_{0}}{\tau^{4}}e^{-2(U-\alpha\phi)}\right];\\\
\\\ \ddot{\phi}=\frac{-1}{1-2c_{0}}\left[\alpha
Q^{2}e^{2(U-\alpha\phi)}-\alpha\frac{V_{0}}{\tau^{4}}e^{-2(U-\alpha\phi)}\right].\end{array}$
(4.14)

These are precisely the equations (2.21) and the corresponding constraint c.f.
(A.10). The resulting solution is

$\displaystyle
U=-\frac{1}{1+\alpha^{2}}\log(-A\tau)\quad,\quad\phi=\frac{\alpha}{1+\alpha^{2}}\log(-A\tau),$
(4.15)

with

$\displaystyle A^{2}=\frac{1-\sqrt{1-4q^{2}v_{0}}}{2v_{0}}$ (4.16)

where we have chosen the negative sign, since it reproduces the usual small
black hole solution when $v_{0}$ is very small. Note that we have dropped some
possible integration constants which are irrelevant in the near-core
$\tau\to-\infty$ region. These are just (2.30) with $A$ given by (2.29).

###  4.3 Small black holes and 2d Dynamical Cobordism

In this section we would like to make more precise the statement that small
black hole solutions correspond to a 2d dynamical cobordism. Following
Buratti:2021fiv ; Angius:2022aeq , dynamical cobordisms are spacetime
dependent solutions running along one dimension $y$ which hit a singularity at
finite spacetime distance, at which scalars run off to infinite field space
distance and at which spacetime ends. In Angius:2022aeq  it was shown that,
near this end of the world brane, dynamical cobordism solutions are described
by a simple local model, which implies specific universal scaling relations
among the spacetime distance $\Delta$ to the singularity, the field theory
distance $D$ traversed, the scalar curvature $R$, and the scalar potential. In
particular, in our 2d context

$\displaystyle D\simeq-\frac{2}{\delta_{2d}}\log\Delta\quad,\quad
R\sim\Delta^{-2}\quad,\quad V_{2d}(\phi)\simeq e^{\delta_{2d}D}.$ (4.17)

where we have added a $2d$ subscript to the potential and critical exponent,
to distinguish them from their 4d counterparts.

Let us consider the small black hole in the critical case (the case with
subcritical (aka zero) 4d potential can be worked out similarly, as in fact
done in Angius:2022aeq ), namely the solution (4.15), and the 2d action (4.2).
The metric is given by:

$ds^{2}=-e^{2U}dt^{2}+e^{-2U}dr^{2}$ (4.18)

Using (4.15), we have that $R\propto r^{\frac{-2\alpha^{2}}{1+\alpha^{2}}}$.
For the spacetime distance we get

$\Delta\simeq\int e^{-U}dr\propto r^{\frac{\alpha^{2}}{1+\alpha^{2}}}$ (4.19)

Hence we recover the scaling $R\propto\Delta^{-2}$ in (4.17) near $r=0$.

Consider now the field distance, from the kinetic terms for the original
scalar field and the $S^{2}$ radion. Using the profiles for the scalars, we
obtain $D(r)=\pm\frac{\sqrt{1-\alpha^{2}}}{(1+\alpha^{2})}\log r$, which in
terms of the spacetime distance $\Delta$ becomes131313The result below
corrects a typo in eq.(5.40) in Angius:2022aeq ..

$D\simeq-\frac{\sqrt{1-\alpha^{2}}}{\alpha^{2}}\log\Delta\ ,\ \ \ \ \ $ (4.20)

namely the scaling (4.17) with the critical exponent

$\displaystyle\delta_{2d}=\frac{2\alpha^{2}}{\sqrt{1-\alpha^{2}}}\,.$ (4.21)

Finally, consider the 2d potential in (4.2). It is easy to check that, using
the solution, all terms in the potential scale as $V_{2d}\sim
e^{2\alpha\phi}$. Hence we have

$\displaystyle
V_{2d}\sim\exp\Big{(}\,\frac{2\alpha^{2}}{\sqrt{1-\alpha^{2}}}D\,\Big{)}$
(4.22)

Hence reproducing the scaling (4.17) with the critical exponent (4.21).

In the language of dynamical cobordisms, small black hole explosions can be
seen as a failure of the dynamical cobordism, which is obstructed by the
existence of a too fastly growing 4d potential at infinity in scalar field
space. In cases where the small black holes puffs up into a regular black
hole, instead of an ETW brane, the 2d running configuration ends up in an AdS2
vacuum corresponding to the near horizon geometry of the finite size black
hole. Hence, the failure of the dynamical cobordism can be blamed on the fact
that we do not reach a singularity at a finite distance, but rather is
replaced by the infinite AdS2 throat. We will comment more on the impact of
such missing dynamical cobordisms in section 6.

It would be interesting to extend these ideas to higher-dimensional theories
to build non-trivial AdS vacua, possibly near infinity (hence in weakly
coupled regimes) in moduli space.

##  5 Explosions from topological obstructions

In this Section we present a different mechanism which can lead to explosions
for small black holes, even for subcritical exponential behaviour of the
potential. It is based on a simple and familiar mechanism, which has
nevertheless been gone unnoticed in the literature on (finite size) black
holes in 4d ${\cal{N}}=2$ gauged supergravity.

###  5.1 Topological obstructions from gaugings

We explain the mechanism in a simplified, but illustrative enough, setup. It
amounts to the statement that in Abelian gaugings, when the $U(1)$ gauge
symmetry is broken, its magnetic charges are confined. We follow the
discussion in Banks:2010zn .

Consider a $U(1)$ gauge theory and an axion scalar, which we denote $\varphi$
(we note that the only relation with $\phi$ is that they often appear as
axion-saxion partners in complex scalars, e.g. in supersymmetric theories). We
ignore gravity in what follows, since it can be included straightforwardly.
Sketchily, we start from the action

$\displaystyle S=\int d^{4}x(|d\varphi|^{2}+|F|^{2})$ (5.1)

The theory has a $U(1)$ gauge symmetry, and a global $U(1)$ shift symmetry for
$\varphi$ (i.e. a $(-1)$-form global symmetry). Given that the moduli space
has a $U(1)$ isometry, we can now define new theories in which the $U(1)$
gauge transformation acts as a shift along it. The resulting action is

$\displaystyle\int\,d^{4}x\,(\,|d\varphi-kA_{1}|+|F|^{2}\,).$ (5.2)

where $k$ is an integer characterizing the winding number of the gauge $U(1)$
around the $U(1)$ isometry of the scalar. The gauge transformation is

$\displaystyle A_{1}\to
A_{1}+d\Lambda(x)\quad,\quad\varphi\to\varphi+k\Lambda(x)$ (5.3)

If the axion can be regarded as the phase of a charge $k$ complex field, this
just corresponds to a Higgs mechanism, in which the gauge $U(1)$ is broken
down to ${\bf{Z}}_{k}$.

Upon dualizing $\varphi$ into a 2-form $B_{2}$, the action shows a Stückelberg
BF coupling

$\displaystyle\int\,d^{4}x\,(\,|d\varphi|^{2}+kB_{2}F_{2}|H_{3}|^{2}\,)\,$
(5.4)

where $H_{3}=dB_{2}$. We can regard this action as the gauging of a global
1-form symmetry of electromagnetism, by coupling the current $j=F_{2}$ to the
$B_{2}$ gauge field. This can be made more manifest by dualizing the gauge
potential into its magnetic potential $V_{1}$, the action becomes

$\displaystyle\int\,d^{4}x\,(\,|dV_{1}-kB_{2}|^{2}+|H_{3}|^{2}\,)$ (5.5)

So the gauge symmetry is

$\displaystyle B_{2}\to B_{2}+d\Lambda_{1}(x)\quad,\quad V_{1}\to
V_{1}+k\Lambda_{1}$ (5.6)

The gauging has important implications for the structure of observable
operators. In particular, charged operators under the $(-1)$-form symmetry of
$\varphi$ (instantons), which have the structure $e^{i\varphi(x)}$ in the
absence of gauging, are now no longer gauge-invariant under (5.3). They must
be dressed by the emission of $k$ electrically charged particles, i.e.
operators charged under the $A_{1}$ (equivalently the 1-form global symmetry
whose current is $*F_{2}$) along semi-infinite lines $L$ ending at $x$, as

$\displaystyle\exp\,[\,{i\varphi(x)}+k\int_{L}A_{1}\,]\quad,\quad\partial L=x$
(5.7)

Similarly, the gauge transformation (5.6) implies that magnetic monopoles,
charged under $V_{1}$ (equivalently line operators along $L$ charged under the
1-form symmetry whose current is $F_{2}$), must be dressed by the emission of
$k$ strings charged under $B_{2}$ (equivalently, the 2-form global symmetry
whose current is $H_{3}$), along a surface $\Sigma$ whose boundary is $L$

$\displaystyle\exp\,[\,{i\int_{L}V_{1}}+k\int_{\Sigma}B_{2}\,]\quad,\quad\partial\Sigma=L$
(5.8)

Hence, magnetic monopoles come with strings attached. There is a topological
obstruction to the existence of free magnetic charges.

This gauging procedure is standard in supergravity, where it takes ungauged to
gauged supergravity theories. In this context, the Abelian gauging parameters
($k$ in the above example) are also known as FI terms, since they are related
by supersymmetry to BF couplings. Moreover, in 4d ${\cal{N}}=2$ theories, such
gaugings are related to the apperance of a scalar potential, see Appendix D.
We will see more about it later.

###  5.2 A microscopic realization: The Freed-Witten anomaly

The realization of the above ideas in the context of flux compactifications
was discussed in BerasaluceGonzalez:2012zn , where a prominent role is played
by Freed-Witten type consistency conditions for fluxes and
branes141414Actually Freed:1999vc  considered the case D-branes in the
presence of torsion $H_{3}$. The physical picture for D-branes and general
$H_{3}$ appeared in Maldacena:2001xj , and for D$p$-branes and RR field
strength flux $F_{p}$ appeared in Witten:1998xy . Still, we stick to the
widely used term FW anomaly / consistency condition.. In the following we
describe one such realization for later use.

Consider type IIB compactified on a CY threefold ${\bf{X}}_{6}$ and introduce
RR 3-form fluxes on a symplectic basis $\\{a_{\Lambda},b^{\Lambda}\\}$ of
3-cycles

$\displaystyle\int_{a_{\Lambda}}F_{3}=g^{\Lambda}\quad,\quad\int_{b^{\Lambda}}F_{3}=-g_{\Lambda}\quad,\quad
g^{\Lambda},g_{\Lambda}\in{\bf{Z}}$ (5.9)

We ignore other possible fluxes (such as NSNS 3-form fluxes), and focus on
these. Consider the electric and magnetic gauge potentials arising from the RR
4-form

$\displaystyle A^{\Lambda}=\int_{a_{\Lambda}}C_{4}\quad,\quad
A_{\Lambda}=\int_{b^{\Lambda}}C_{4}$ (5.10)

The reduction of the 10d Chern-Simons coupling $B_{2}F_{3}F_{5}$ leads to
Stückelberg couplings for their field strengths $F^{\Lambda}$, $F_{\Lambda}$

$\displaystyle\int_{4d}\,\int_{{\bf{X}}_{6}}\,B_{2}F_{3}F_{5};\to\;\int_{4d}\,B_{2}\,(\,g^{\Lambda}F_{\Lambda}+g_{\Lambda}F^{\Lambda}\,).$
(5.11)

This implies the gauging of a particular linear combination of $U(1)$’s.
Equivalently, the different (electric and magnetic) $U(1)$’s act on the same
scalar $\varphi$, the dual of $B_{2}$. We can introduce the vector of electric
and magnetic gauging

$\displaystyle{\cal G}=(g^{\Lambda};g_{\Lambda})$ (5.12)

Consider now a particle carrying charges $q_{\Lambda}$, $q^{\Lambda}$, under
$A^{\Lambda}$, $A_{\Lambda}$, respectively. We introduce the charge vector

$\displaystyle Q=(q^{\Lambda};q_{\Lambda})$ (5.13)

This corresponds to D3-branes wrapped on the 3-cycle

$\displaystyle\Pi_{\rm
D3}=\sum_{\Lambda}(q_{\Lambda}a_{\Lambda}+q^{\Lambda}b^{\Lambda})$ (5.14)

As is familiar, a D$p$-brane wrapped on a $p$-cycle with $k$ units of RR
$F_{p}$ field strength flux is not consistent by itself, due to a worldvolume
tadpole Witten:1998xy , and must emit $k$ fundamental strings. In our example

$\displaystyle\int_{\Pi_{\rm
D3}}F_{3}=q_{\Lambda}g^{\Lambda}-q^{\Lambda}g_{\Lambda}\,\equiv\,-\langle{\cal
G},Q\rangle$ (5.15)

where $\langle\cdot,\cdot\rangle$ denotes the symplectic pairing. Hence, when
$k\equiv-\langle{\cal G},Q\rangle\neq 0$, there must be $k$ fundamental
strings emitted by the D3-brane.

There are other interesting related ways to reach this conclusion, as follows.
We can regard the $F_{3}$ flux as having been created by a domain wall given
by a D5-brane on the dual 3-cycle

$\displaystyle\Pi_{\rm
D5}=\sum_{i}(g^{\Lambda}b^{\Lambda}+g_{\Lambda}a_{\Lambda})\,.$ (5.16)

Now, consider the D3-brane wrapped on (5.14) on the flux-less side of the
domain wall, hence with no strings attached. When the D3-branes is moved
across the D5-brane domain wall to the fluxed side, there is a string creation
effect Bachas:1997ui ; Danielsson:1997wq ; Bergman:1997gf  (dual to the
Hanany-Witten effect Hanany1997 ) at each intersection of the corresponding
3-cycles. Hence, the D3-brane ends up with $k$ strings attached, with

$\displaystyle k=\Pi_{\rm D3}\cdot\Pi_{\rm
D5}=q_{\Lambda}g^{\Lambda}-q^{\Lambda}g_{\Lambda}=-\langle{\cal G},Q\rangle$
(5.17)

In this picture, the strings attached to the D3-branes can be derived from the
FW anomaly due to the $F_{3}$ flux created by the D5-branes (as discussed
above), and the strings attached to the D5-brane arise from the FW anomaly
created by the $F_{5}$ flux created by the D3-brane.

It is straightforward to derive a similar picture for other string setups, for
instance type IIA CY3 compactifications with $F_{0}$, $F_{2}$, $F_{4}$ and
$F_{6}$ flux. In fact, we would like to emphasize that the effect is present
independently of the particular microscopic realization, since it follows from
the structure of topological operators in the presence of gauging c.f. section
5.1.

###  5.3 4d ${\cal N}=2$ gauged supergravity Black Holes come with strings
attached

There is a substantial literature on black holes in 4d ${\cal{N}}=2$ gauge
supergravity, including BPS solutions Cacciatori:2009iz ; DallAgata:2010ejj ;
Hristov:2010ri , and non-supersymmetric solutions Klemm:2012yg ; Klemm:2012vm
; Gnecchi:2012kb  (see Astesiano:2021hro  and references therein for a recent
discussion)151515As in previous sections, we focus on black holes with
spherical horizons. In asymptotic AdS vacua, it is possible to also have black
holes with zero or negative curvature. Although we do not discuss them, we
note that the negative curvature cases also experience the phenomenon
explained in this section..

Interestingly, there are many examples of regular black holes based on
(sometimes BPS) solutions whose existence depends on having a non-zero value
of the symplectic product

$\displaystyle\langle{\cal G},Q\rangle=-k\in{\bf{Z}}$ (5.18)

From our above discussion, these black holes solutions are not totally
complete, due to a subtle lack of gauge invariance which requires the
introduction of $k$ strings sticking out of the black hole (equivalently,
filling out the AdS2 2d geometry in the near horizon region).

These strings are not included in the discussions in the supergravity
literature, and the detailed study of their properties and backreaction is
beyond the scope of this work. Let us however point out that their effect is
very different (and potentially much milder) from the one in small black
holes. Indeed, for regular black holes, the value of $V_{BH}$ at the horizon
is a finite quantity, which moreover is controlled by $Q^{2}$, whereas the
number $k$ of attached strings does not scale with $Q$, and can in fact remain
small (typical values being of order 1). Hence for large black holes with
large charges, the introduction of the strings corresponds to a $1/Q^{2}$
modification of $V_{BH}$, and hence to subleading corrections to the
properties of the black hole.

It is interesting that the topological obstruction explained in previous
sections provides a complementary explanation for the quantization condition
(5.18); it simply relates to the integer number of strings emitted by the
black hole. Also, in the next section we will argue that the fact that
regularity of the black hole solutions is naturally related to a non-zero
value of $\langle{\cal G},Q\rangle$, is related to a novel mechanism for small
black hole explosions.

In section 6.3 we will see another interesting avatar of the constraint in
$\langle{\cal G},Q\rangle$, related to the capability of small black holes to
explore all possible infinite distance limits in moduli space in the presence
of 4d ${\cal{N}}=2$ gauging potentials.

###  5.4 Explosions via string banquets

As we have discussed in section 3.2, in certain theories (such as 4d
${\cal{N}}=2$ supergravity, in previous section) the presence of gauging also
implies the appearance of a 4d potential $V$ (3.6). From the above discussion,
it is clear that in the presence of a potential arising from a gauging vector
${\cal G}$, a black hole with charge vector $Q$ will emit a number of strings
(magnetically charged under the axion being gauged) given by the symplectic
pairing $\langle{\cal G},Q\rangle$. This leads to an interesting mechanism for
the explosion of small black holes161616Strictly speaking, such small black
holes are not gauge invariant as isolated objects, hence we regard them as
defined as endpoint configurations of the corresponding strings. It is in this
sense that one may ask whether it is dynamically preferred for these
configurations to remain small (and hence, continue exploring infinite
distance in field space) or puff up into a finite size configuration (or
possibly a runaway version thereof).

We can use the effective potential approach to address this question. Indeed,
if one regards a possibly puffed up finite size version of the charged black
hole, its near horizon limit corresponds to an AdS${}_{2}\times{\bf{S}}^{2}$
configuration, with the black hole charges realized as fluxes piercing the
${\bf{S}}^{2}$. Due to the latter, the $BF$ couplings (5.4) implied by the
gauging lead to a tadpole for $B_{2}$ in the 2d non-compact dimensions.
Cancellation of this tadpole is achieved by the introduction of strings
(charged under $B_{2}$) spanning the AdS2 factor, and located at a point on
${\bf{S}}^{2}$. This is just the near horizon version of the emitted strings
mentioned above.

It is easy to derive this tadpole in certain microscopic realizations. For
instance, in the type IIB setup in section 5.2, it arises from the reduction
of the 10d Chern-Simons term

$\displaystyle\int_{2d}\,\int_{{\bf{X}}_{6}\times{\bf{S}}^{2}}\,B_{2}F_{3}F_{5};\to\;\int_{2d}\,B_{2}\,(\,g^{i}q_{i}-g_{i}q^{i}\,)\,=\,\int_{2d}\,B_{2}\langle{\cal
G},Q\rangle.$ (5.19)

Hence we need to add $-\langle{\cal G},Q\rangle=-k$ strings filling AdS2.

The presence of these strings modifies the system. A full discussion of this
backreaction is beyond the scope of this work, but we can describe the key
feature by modifying the structure of the 2d action, by introducing an extra
source of tension, localized on the ${\bf{S}}^{2}$ and filling the AdS2,
namely.

$\displaystyle 2kT_{0}\int d^{2}x\sqrt{-g_{2}}$ (5.20)

with the tension $T_{0}>0$ and where the factor of $2$ is for convenience. As
such, they behave as an extra term $2kT_{0}e^{-2\sigma}$ inside the bracket of
the 2d action (3.4), with $T_{0}$ taken constant for simplicity171717Let us
note that by constant, we mean that it does not depend on vector multiplet
moduli. This statement however may be model-dependent and depend on the
particular setup or microscopic embedding in string theory.. Note that a
positive tension $T_{0}$ contributes with same sign as the black hole or 4d
potentials, and opposite to the contribution of the ${\bf{S}}^{2}$ curvature.

Let us now consider the fate of the core of former small black holes in the
presence of the gauging potential, and thus, of the extra strings. Consider
the simplified case of a single scalar. Following the steps in section 3.3,
the structure of the effective potential (3.22) is

$\displaystyle V_{eff}=\frac{(1-kT_{0})-\sqrt{(1-kT_{0})^{2}-4V_{BH}V}}{2V}$
(5.21)

Note that the string contributes as the tadpole generated by the
${\bf{S}}^{2}$ compactification, and they contribute with opposite sign.
Moreover, in microscopic realizations, the strings are locally BPS objects
(such as the fundamental strings in Freed-Witten anoamlies), and by charge
quantization we expect that $T_{0}=1$. Hence, in the presence of such strings,
$V_{eff}$ goes negative and according to the solutions in section 4.2, there
is no small hole solution. Remarkably, this conclusion is independent of the
value of $\delta$ and how it compares with $\alpha$. Namely, in the presence
of topological obstructions which force the introduction of strings sticking
out of the small black hole, the small black holes explodes even for
subcritical values of the exponent in the 4d potential.

An intuitive interpretation is that small black holes with strings attached
tend to feast on them and grow. As usual, our exponential approximations near
infinity in field space do not suffice to assess or not the existence of an
endpoint for this process, a topic to which we turn in the next section.

Incidentally, we note that certain regular black holes, such us the example in
the STU model in DallAgata:2010ejj , do not correspond to small black holes
even in the absence of 4d potential. In this sense, they cannot be regarded as
a puffed up version of small black holes.

##  6 Interplay with the Cobordism Distance Conjecture

###  6.1 Small black holes and the Cobordism Distance Conjecture

The Cobordism Distance Conjecture proposed in Buratti:2021fiv  states that in
a consistent theory of quantum gravity, every infinite field distance limit
can be realized as a dynamical cobordism running into an end of the world
brane (possibly in a suitable compactification of the theory). In fact, for 4d
theories and upon compactification on ${\bf{S}}^{1}$ this encompasses the
Distant Axionic String Conjecture in Lanza:2021qsu . Similarly, this suggests
that small black holes, which are dynamical cobordisms of the 4d theory
compactified on ${\bf{S}}^{2}$, are a powerful tool to explore infinite
distance limits of the theory. In this section we explore to what extent the
set of small black holes is rich enough to actually explore all those possible
limits, and to what extent the obstructions due to small black hole explosions
imply some limitations in this exploration.

A clear restriction in this regard is that small black holes allow to explore
infinite distance limits for scalars controlling gauge couplings of the
theory. This is implicit in the discussion below. In particular, we will frame
the discussion in the context of 4d ${\cal{N}}=2$ supergravity, which provides
a rich arena to carry out the discussion in explicit examples.

We start with the theory in the absence of 4d potentials (see Gendler:2020dfp
for a related discussion). In this context, the structure of infinite distance
limits in the vector multiplet moduli space has been studied in Grimm:2018ohb
; Grimm:2018cpv ; Corvilain:2018lgw . In particular, every infinite distance
limit corresponds to a regime in which some gauge coupling goes to zero, hence
allowing the construction of small black hole solution exploring that infinite
distance limit Hamada:2021yxy . Note that in general an infinite distance
limit can involve combinations of different moduli to go off to infinity (as
classified by the corresponding growth sector), hence the corresponding small
black hole must carry the appropriate combination of charges to couple to the
gauge coupling becoming small. Hence, this implicitly profits from the
Completeness Conjecture, which ensures that for any possible charge vector one
may need, there are appropriate charged states in the theory (and which for
large charges constitute a small black hole, rather than a fundamental
particle).

###  6.2 Adding the potential: the $STU$ model

####  6.2.1 Generalities

Let us now consider the introduction of a 4d potential. In order to be
explicit, we focus on those arising from Abelian gaugings of 4d ${\cal{N}}=2$
theories, as in the previous section, see Appendix D for background
information. In particular, we can focus on the illustrative example of the
$STU$ model. We will subsequently study the extension to more general 4d
${\cal{N}}=2$ models, in particular covering the moduli space of any CY3
compactification.

We follow closely the conventions in DallAgata:2010ejj , and refer the reader
to Appendix D for details.

The model (in a specific symplectic frame) is defined by the prepotential

$F=\frac{X^{1}X^{2}X^{3}}{X^{0}}\,.$ (6.1)

We introduce the coordinates $S=X^{1}/X^{0}$, $T=X^{2}/X^{0}$ and
$U=X^{3}/X^{0}$, in terms of which the Kähler potential is

$K=-\log[-i(S-\bar{S})(T-\bar{T})(U-\bar{U})],$ (6.2)

and ${\cal V}$ is given by

${\cal V}=e^{K/2}\,(1,S,T,U,-STU,TU,SU,ST)^{T}.$ (6.3)

We consider a general set of charges/gaugings, given by

$Q=\left(p^{0},p^{1},p^{2},p^{3},q_{0},q_{1},q_{2},q_{3}\right)^{T}\quad\quad\quad\mathcal{G}=\left(g^{0},g^{1},g^{2},g^{3},g_{0},g_{1},g_{2},g_{3}\right)^{T}$
(6.4)

A rich class of models was introduced in DallAgata:2010ejj  by taking gauging
charges of the form ${\cal G}=(0,g^{1},g^{2},g^{3},g_{0},0,0,0)^{T}$ and black
hole charges $Q=(p^{0},0,0,0,0,q_{1},q_{2},q_{3})$. For the following
discussion, we prefer to allow for general choices (in principle constrained
only by the Freed-Witten like consistency condition discussed in the previous
section).

The black hole central charge (D.19) and 4d superpotential (D.12) are thus

$\displaystyle{\cal
Z}=e^{K/2}(p^{0}STU-p^{1}TU-p^{2}SU-p^{3}ST+q_{0}+q_{1}S+q_{2}T+q_{3}U)$ (6.5)
$\displaystyle{\cal
L}=e^{K/2}(g_{0}+g^{0}STU-g^{1}TU+g_{1}S-g^{2}SU+g_{2}T-g^{3}ST+g_{3}U).$
(6.6)

Since we are interested in exploring infinite distance limits in moduli space,
the asymptotic behaviours are controlled by the imaginary parts of the moduli,
hence we replace $S\mapsto is$, $T\mapsto it$ and $U\mapsto iu$, with
$s,t,u<0$.

From the expressions for the black hole potential (3.8), namely (D.20), and
the 4d scalar potentials (3.6), namely (D.13), we have

$\displaystyle
V_{BH}=\frac{q_{0}^{2}+q_{1}^{2}s^{2}+q_{2}^{2}t^{2}+q_{3}^{2}u^{2}+(p^{1})^{2}t^{2}u^{2}+(p^{2})^{2}s^{2}u^{2}+(p^{3})^{2}s^{2}t^{2}+(p^{0})^{2}s^{2}t^{2}u^{2}}{-2stu}$
$\displaystyle V=\frac{-g_{0}(g^{1}tu+g^{2}su+g^{3}st)-g_{1}g_{2}st-
g_{2}g_{3}tu-g_{1}g_{3}su}{-stu}+$
$\displaystyle\quad\quad\quad\quad+\frac{(g^{0}g_{1}-g^{2}g^{3})s^{2}tu+(g^{0}g_{2}-g^{1}g^{3})st^{2}u+(g^{0}g_{3}-g^{1}g^{2})stu^{2}}{-stu}$
(6.7)

It is now a simple exercise to pick sets of gauging/charges which illustrate
different behaviours for small black holes. We refrain from the discussion of
random examples, but rather focus on a more fundamental question in the next
section.

####  6.2.2 Small black holes: exploration or explosion

As explained, a natural question to what extent small black hole explosions
may hamper their ability to explore infinite distance limits in a given
theory. In this section, we carry out this discussion in the $STU$ model, we
may consider the simplified setup in which all moduli are identified i.e.
scale to the limit in the same way. More general situations will be discussed
later on.

For a single modulus case $s=t=u\equiv z$, we have

$\displaystyle
V_{BH}=\frac{q_{0}^{2}+(q_{1}^{2}+q_{2}^{2}+q_{3}^{2})z^{2}+\,[\,(p^{1})^{2}+(p^{2})^{2}+(p^{3})^{2}\,]\,z^{4}+(p^{0})^{2}z^{6}}{-2z^{3}}$
(6.8) $\displaystyle
V=\frac{(-g_{0}(g^{1}+g^{2}+g^{3})-g_{1}g_{2}-g_{2}g_{3}-g_{1}g_{3})z^{2}}{-z^{3}}+$
$\displaystyle\quad\quad\quad\quad+\frac{(g^{0}g_{1}+g^{0}g_{2}+g^{0}g_{3}-g^{1}g^{2}-g^{2}g^{3}-g^{1}g^{3})z^{4}}{-z^{3}}$
(6.9)

As explained in section 6.1, we are interested in testing possible limitations
in the ability of small black holes to explore infinite distance limits, due
to small black hole explosions.

Hence, we focus on 4d potentials with the highest possible value of $\delta$
in the corresponding limit181818Note that we are also not insisting on the 4d
theory to admit an asymptotic AdS4 vacuum, since the discussion of the small
black hole cores does not require information about the asymptotics, which may
well correspond to a 4d running solution.. Considering the limit of large
moduli191919We Note that there is a similar limit, corresponding to $z\to 0$,
which can be analyzed similarly, with identical results up to reshuffling of
terms. This is just a reflection of the duality of the $STU$ model., the most
dangerous term is the second line in (6.9), which scales as $V\sim z$. There
are basically two possibilities (or simple combinations thereof):

$\bullet$ We may choose $g^{0}\neq 0$, and get $V\sim z$ by choosing some of
the $g_{i}\neq 0$. In order to explore this infinite distance limit, one may
be tempted to just take a black hole with $q_{0}\neq 0$ as only charge, so
that $V_{BH}\sim z^{-3}$, for which the 4d potential is subcritical. However,
this is not possible, since it would lead to Freed-Witten-like anomalies.
However, it is possible to choose black holes with some $q_{0}=0$, $q_{i}\neq
0$ for $i=1,2,3$, which avoid the Freed-Witten-like anomalies, and lead to
$V_{BH}\sim z^{-1}$, for which the 4d potential is critical.

$\bullet$ We may get $V\sim z$ by keeping $g^{0}=0$ and turning on two
different $g^{i}$’s, for instance $g^{1}$ and $g^{2}$. In such case, we can
take a black hole with $q_{1},q_{2}\neq 0$, with lead to $V_{BH}\sim z^{-1}$,
hence a critical case, and which easily avoid the Freed-Witten-like anomalies
by demanding that $g^{1}q_{1}+g^{2}q_{2}=0$ (e.g. $q_{1}=g^{2}n$ ,
$q_{2}=-g^{1}n$, with arbitrary $n\in{\bf{Z}}$).

We have thus checked that, even for the most dangerous 4d potential, there are
specific small black holes for which the 4d potential is critical; hence they
in principle have a chance to remain small and be able to explore this
infinite distance limit. We thus conclude that, also in the presence of 4d
potentials, the Completeness Conjecture is enough to guarantee that, for any
gauging potential and any infinite distance limit, there is always some small
black hole in the theory which remains small and still allows to explore the
infinite distance limit.

We finish by noting that we have not discussed other possible sources of 4d
potentials, which in principle could lead to a modification of the above
discussion. On the other hand, turning things around, the above discussion
motivates a proposal restricting the form of the possible 4d potentials, as we
discuss in the next section.

###  6.3 Constraints on potentials from small black holes explosions

In this section we propose the use of our ideas above to constrain the
possible 4d scalar potentials, from their properties regarding small black
holes and their explosions. We show that this can lead to powerful
restrictions, not only on 4d potentials but (in theories with extended
supersymmetry) also on the geometry of the moduli space near infinity.

####  6.3.1 The Small Black Hole Criterion

In the example of the $STU$ model in the previous section, we showed the
interesting fact that the available 4d potentials from gaugings are such that
there still remain enough small black holes (i.e. those possibly not exploding
under the effect of the potential) to still allow the exploration of all
infinite distance limits for the scalars.

It is natural to propose promoting this idea to a general principle, as
follows:

Small Black Hole Criterion: In a consistent theory of quantum gravity with
scalar-dependent Abelian gauge couplings, the allowed 4d potentials are
constrained by the requirement that they allow for the existence of a set of
small black holes whose scalars are still able to explore any infinite
distance limit of the theory (i.e. remain subcritical or critical).

In a sense, this idea is not a new conjecture, but rather a mere reformulation
of the Cobordism Distance Conjecture in Buratti:2021fiv  for the 2d theories
resulting from the 4d theories upon ${\bf{S}}^{2}$ compactification. On the
other hand, its explicit formulation facilitates its use to put constraints on
the 4d theories, as we discuss in the next section.

####  6.3.2 Recovering the CY vector moduli space asymptotics

In this section we put the above proposal to work and show that it leads to
remarkably strong results. In particular, we will argue it implies that, for
any 4d ${\cal{N}}=2$ theory, the Kähler potential for the vector multiplets
satisfies, in the large moduli limit, the equality

$\displaystyle g^{i{\bar{\jmath}}}\partial_{i}{\cal
K}\partial_{\bar{\jmath}}{\cal K}=3$ (6.10)

This is precisely the asymptotic identity obeyed by vector moduli in a CY
threefold compactification of type II string theory (and via duality, of K3
compactifications of heterotic, or other dual realizations of 4d
${\cal{N}}=2$). We are however about to derive it from a purely 4d bottom-up
perspective.

The argument is as follows. Consider a general 4d ${\cal{N}}=2$ with
prepotential $F(X)$. Using homogeneity of the prepotential, the covariantly
holomorphic vector ${\cal V}$ always contains an entry equal to $e^{{\cal
K}/2}$, its symplectic dual contains and entry $e^{{\cal K}/2}F(z)$, where $z$
are the moduli (i.e. affine coordinates). Let us focus on the limit of all
moduli becoming large, in which this entry dominates the dynamics. For
instance, from (D.6) the Kähler potential scales as

$\displaystyle{\cal K}\sim-\log[i(F-{\bar{F}})]$ (6.11)

Consider now a possible gauging of the theory, analogous to $g^{0}$ in the
$STU$ model in section 6.2, namely

$\displaystyle{\cal L}\sim g^{0}F(z)+\ldots$ (6.12)

The corresponding scalar potential (3.6), namely (D.13), is

$\displaystyle
V=\frac{1}{i(F-{\bar{F}})}\,\big{[}\,(\,g^{i{\bar{\jmath}}}\partial_{i}{\cal
K}\partial_{\bar{\jmath}}{\cal K}-3\,)\,(g^{0})^{2}\,F{\bar{F}}\big{]}$ (6.13)

which scales as $V\sim F$ in the large moduli limit unless (6.10). We now show
that this would be in contradiction with our Small Black Hole Criterion, hence
(6.10) must be satisfied.

Consider the above potential and demand that there must exist some black holes
which is critical with respect to that term in the potential, namely
$V_{BH}\sim F^{-1}$. Clearly, this must correspond to a black hole whose only
charge, $q_{0}$, is associated to the entry $e^{{\cal K}/2}$ in ${\cal V}$,
namely the symplectic dual to $g^{0}$. This would superficially allow to
satisfy the Small Black Hole Criterion. However, we should realize that
precisely this required charge vector suffers the problem pointed out in
Section 5, namely

$\displaystyle\langle{\cal G},Q\rangle=g^{0}q_{0}\neq 0$ (6.14)

Namely, it is not an allowed state in the theory, as it is forbidden by gauge
invariance. Alternatively, even if the black hole is made consistent by
allowing it to emit the required number of strings, it does not remain small,
but rather explodes by the string banquet mechanism in section 5.4, and does
not explore the infinite distance limit. Hence, as announced, the Small Black
Hole Criterion implies the property (6.10) in the infinite moduli limit of the
${\cal{N}}=2$ vector moduli space of any consistent theory.

The above argument is a general description of how the $(g^{0})^{2}$ term does
not arise in the potential (6.7) of the $STU$ model in section 6.2. In fact
this model is one of the simplest toy models to characterize the large moduli
behaviour of any CY compactification

$\displaystyle F=-D_{IJK}X^{I}X^{J}X^{k}/X^{0}$ (6.15)

with $6D_{IJK}$ are the familiar (integer) triple intersection numbers of CY
compactifications.

It is an amusing fact that we recover the very familiar property (6.10) of CY3
compactifications in a 4d theory which has a priori no information about CY’s
or compactifications. In this sense, the result resonates with the
reconstruction of compactification geometries from cobordism considerations in
Hamada:2021bbz .

##  7 Conclusions

In this work we have explored the ability of 4d small black holes to explore
infinite distance limits in scalar field space in the presence of scalar
potentials growing near infinity. We have derived the critical relation
distinguishing when the small black hole remains small from when it explodes
by puffing up into a regular black holes or following a runaway behaviour.
Although the discussion is general we have studied the particular case of 4d
${\cal{N}}=2$ gauged supergravity, which allows a rich arena for this
exploration. In fact, in that context we have uncovered a constraint,
previously unnoticed in the literature on regular black holes in this setup,
which requires the black holes to emit strings.

We have used these findings to motivate a general constraint on scalar
potentials in 4d theories, based on the principle that the set of small black
holes should still be rich enough to allow for exploration of all possible
infinite distance limits. We have shown that this leads to non-trivial
constraints, and that in 4d ${\cal{N}}=2$ theories it allows to reproduce some
of the properties of vector moduli spaces of CY3 compactifications.

Our discussion has emphasized the use of a 2d description, after truncation on
${\bf{S}}^{2}$. This allowed us to discuss 4d potentials and black hole
potentials on an equal footing, and, in the 4d ${\cal{N}}=2$ context, to
satisfactorily explain the difference between their expressions in terms of
the underlying covariantly holomorphic quantities. More importantly, the 2d
perspective permits the translation of the result to the language of dynamical
cobordisms, where the small black hole core corresponds to the end of the
world brane, and small black hole explosions signal the failure of completing
a cobordism ending spacetime. In this picture, puffed up black holes
correspond to running solutions of the 2d theory which end up relaxing onto a
newly appeared AdS2 vacua.

There are many possible generalizations of our work and other related ideas.
For instance:

$\bullet$ We expect that further results about the structure of infinite
distance regimes in moduli space can be derived for further exploration of
large and small black holes, in particular considerations about their entropy
and its microscopic explanation, possibly along the lines of
vandeHeisteeg:2023ubh .

$\bullet$ Generalization to black strings and other extended objects, in
general dimensions, and their possible explosions upon the introduction of
scalar potentials. A particular instance of this is the fate of the 4d EFT
strings in Lanza:2020qmt ; Lanza:2021qsu ; Marchesano:2022avb  in the presence
of potentials growing fast at the string core.

$\bullet$ The failure of 2d running solutions to complete dynamical cobordisms
and rather end up relaxing onto AdS2 vacua is a particularly rich instance of
a phenomenon which also exists in higher dimensions Angius:2022aeq . It would
be interesting to explore its possible use as a technique to build or detect
AdS vacua hidden near infinity in moduli space.

$\bullet$ In the supercritical case, the small black hole explosion is
expected to lead to time dependent runaway configurations. The construction of
such full solution is presumably difficult, due to non-trivial dependence in
two coordinates. However, its asymptotic behaviour at late times/distances may
admit a simpler description, depending only on a single light-cone coordinate,
in analogy with expanding domain walls in e.g. Hellerman:2004qa ;
Hellerman:2006nx ; Hellerman:2006hf ; Hellerman:2006ff ; Hellerman:2007fc ;
Angius:2022mgh .

$\bullet$ The Freed-Witten-like constraint in 4d ${\cal{N}}=2$ gauged
supergravity models forces the introduction of strings, which in the near
horizon regime of regular black holes are spacetime filling in the 2d AdS2. In
cases where this extra ingredient breaks supersymmetry i.e. is an anti-string
(see Sugimoto:1999tx ; Antoniadis:1999xk ; Aldazabal:1999jr , also
Angelantonj:1999ms ; Angelantonj:2000xf ; Rabadan:2000ma  for similar
phenomena in higher dimensions), this is reminiscent of the antibrane uplift
in Kachru:2003aw . It would be interesting to explore actual quantitative
connections between the two phenomena.

We hope to come back to these and other interesting topics in future work.

## Acknowledgments

We are pleased to thank Gianguido Dall’Agata, Iñaki Garcia-Etxebarria, Luis
Ibáñez, Fernando Marchesano, Miguel Montero, Tomás Ortín, Irene Valenzuela for
useful discussions, and José Calderón-Infante and Matilda Delgado for
collaboration on related topics. This work is supported through the grants
CEX2020-001007-S and PID2021-123017NB-I00, funded by
MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe. The work by
R.A. is supported by the grant BESST-VACUA of CSIC. The work by J. H. is
supported by the FPU grant FPU20/01495 from the Spanish Ministry of Education.

##  Appendix A All about $h$

###  A.1 No 4d potential

The most general ansatz for an spherically symmetric static solution with
electric charge $Q$ in the theory described by (2.1) is actually

$ds^{2}=-e^{2U(\tau)}dt^{2}+e^{-2U(\tau)}\left(\frac{d\tau^{2}}{\tau^{4}h(\tau)^{2}}+\frac{1}{\tau^{2}}d\Omega^{2}_{2}\right)\quad,\quad
F_{2}=2\sqrt{2}Qg^{2}e^{2U}d\tau\wedge dt$ (A.1)

Plugging the ansatz in the 4d action (2.1) leads to the 1d action

$S_{1d}=\int
d\tau\left\\{\frac{h}{2}\left(\dot{U}^{2}+\dot{\phi}^{2}\right)+\frac{1}{h}g^{2}Q^{2}e^{2U}-\frac{1}{\tau^{2}}(\frac{1}{h}-h+2\tau\dot{h})\right\\}$
(A.2)

The equations of motion read

$\displaystyle\frac{d}{d\tau}\left(h\dot{U}\right)=\frac{2}{h}Q^{2}e^{2U}g(\phi)^{2}$
(A.3)
$\displaystyle\frac{d}{d\tau}\left(h\dot{\phi}\right)=\frac{1}{h}Q^{2}e^{2U}\left(g(\phi)^{2}\right)^{\prime}$
(A.4)
$\displaystyle\left[\frac{1}{2}(\dot{U}^{2}+\dot{\phi}^{2})-\frac{1}{\tau^{2}}\right]h^{2}-g^{2}Q^{2}e^{2U}+\frac{1}{\tau^{2}}=0\,.$
(A.5)

As mentioned in Denef:1998sv , the action (A.2) implies that $h$ is not an
actual dynamical variable, but rather imposes a constraint, associated to
reparametrizations of $\tau$. To make this explicit, consider the variation of
the action under changes of $U$, $\phi$ of the form

$\displaystyle
U(\tau+\delta\tau)=U(\tau)+\dot{U}(\tau)\delta\tau\quad,\quad\phi(\tau+\delta\tau)=\phi(\tau)+\dot{\phi}(\tau)\delta\tau\,.$
(A.6)

Using (A.3), (A.4) we get:

$\delta S_{1d}=\int
d\tau\left\\{\frac{d}{d\tau}\left(h\dot{U}^{2}\delta\tau\right)+\frac{d}{d\tau}\left(h\dot{\phi}\delta\tau\right)+\frac{2}{\tau^{3}}\left(\frac{1}{h}-h\right)\delta\tau\right\\},$
(A.7)

namely, up to total derivatives, demanding $\delta S_{1d}=0$ requires

$\displaystyle\int
d\tau\frac{2}{\tau^{3}}\left(\frac{1}{h}-h\right)\delta\tau=const$ (A.8)

which can be solved by setting $h=1$. Then (A.3), (A.4) correspond to the
equations of motion (2.6), and (A.5) leads to the constraint (2.7).

###  A.2 Including 4d potential

Let us now introduce a 4d potential term in the action c.f. (2.16)

$S=\int d^{4}x\sqrt{-g}\left[R-2(d\phi)^{2}+\frac{1}{2g^{2}}F^{2}+2V\right].$
(A.9)

Using the ansatz (A.1), we get the 1d action

$\displaystyle S_{1d}=\int
d\tau\left\\{\frac{h}{2}\left(\dot{U}^{2}+\dot{\phi}^{2}\right)+\frac{1}{h}Q^{2}g(\phi)^{2}e^{2U}-\frac{1}{h}\frac{V(\phi)}{2\tau^{4}}e^{-2U}-\frac{1}{2\tau^{2}}\left(\frac{1}{h}-h+2\tau\dot{h}\right)\right\\}.$

The equations of motion are

$\displaystyle\frac{d}{d\tau}\left(h\dot{U}\right)=\frac{2}{h}Q^{2}g(\phi)^{2}e^{2U}+\frac{1}{h}\frac{V(\phi)}{\tau^{4}}e^{-2U}$
$\displaystyle\frac{d}{d\tau}\left(h\dot{\phi}\right)=\frac{1}{h}Q^{2}\left(g(\phi)^{2}\right)^{\prime}e^{2U}-\frac{1}{h}\frac{V^{\prime}(\phi)}{2\tau^{4}}e^{-2U},$
$\displaystyle\left[\frac{1}{2}\left(\dot{U}^{2}+\dot{\phi}^{2}\right)-\frac{1}{2\tau^{2}}\right]h^{2}-g(\phi)^{2}Q^{2}e^{2U}+\frac{V}{2\tau^{4}}e^{-2U}+\frac{1}{2\tau^{2}}=0$
(A.10)

The change of the action under variations (A.6) requires

$\delta S_{1d}=\int
d\tau\frac{1}{\tau^{3}}\left(\frac{1}{h}-h+2\tau\dot{h}+\frac{1}{h}\frac{2V(\phi)}{\tau^{2}}e^{-2U}\right)\delta\tau=0.$
(A.11)

which can be solve by choosing

$\dot{h}\simeq 0\quad\quad\quad h^{2}=1+\frac{2V(\phi)}{\tau^{2}}e^{-2U}$
(A.12)

Although this is more involved, it simplifies in particular cases. For
instance, if $Ve^{-2U}$ grows slower that $\tau^{2}$ we recover $h^{2}=1$ near
$\tau\to-\infty$, just like in the case with no 4d potential; this corresponds
to the subcritical case in the main text. On the other hand, the critical case
solution in section 2.3 corresponds to a constant $h\equiv h_{0}\neq 1$.

##  Appendix B The 4d entropy functional computation of $V_{eff}$

For completeness we include a quick derivation of $V_{eff}$ from the entropy
functional from the perspective of the 4d solutions. The 2d computation of
$V_{eff}$ in section 3.3 is a version of this.

We start with the following 4d action202020This action is equivalent to the 4d
action (3.1) used at the beginning of section 3.1, where we made explicit the
democratic treatment of the electric and magnetic gauge fields. Here it is the
Legendre transformation (B.4) with the condition (B.5) that will restore the
not-manifest electric/magnetic duality.

$S_{4d}=\int
d^{4}x\sqrt{-g_{4}}\left\\{R_{4}-2g_{i\overline{j}}\partial_{\mu}z^{i}\partial^{\mu}\overline{z}^{\overline{j}}+Im\mathcal{N}_{\Lambda\Sigma}F^{\Lambda}_{\mu\nu}F^{\Sigma\mu\nu}-2V\right\\}$
(B.1)

Recall that, although it has the structure of the bosonic sector of
$\mathcal{N}=2$ (possibly gauged) supergravity action, but is intended as a
general theory.

We are interested in the near horizon limit of extremal black hole solutions
with spherical horizon, which corresponds to AdS${}_{2}\times{\bf{S}}^{2}$
with a 2-form field strength background, of the form:

$ds_{4}^{2}=v_{1}\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}\right)+v_{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)$ (B.2)

$F^{\Lambda}_{rt}=\frac{e^{\Lambda}}{8\pi}\quad\quad\quad
F^{\Lambda}_{\theta\phi}=p^{\Lambda}\sin\theta$ (B.3)

where $v_{1},v_{2},e$ and $p$ are constants. In addition we require that the
scalars be regular when we approach the horizon: $\lim_{r\mapsto
r_{H}}z^{i}=z^{i}_{H}$.
Following Sen:2005wa  (see Sen:2007qy  for a review), we introduce the entropy
function, defined by

$\mathcal{E}(v_{1},v_{2},z^{i},p,q)=2\pi\left(q_{\Lambda}e^{\Lambda}-f(v_{1},v_{2},z^{i},p,e)\right)$
(B.4)

where $q$ is the conjugate quantity to $e$:

$\frac{\partial\mathcal{E}}{\partial e}=0$ (B.5)

and $f$ is the action evaluation on the near horizon ansatz solution

$f(v_{1},v_{2},z^{i},p,e)=\int d\theta d\varphi\sqrt{-g_{4}}\mathcal{L}_{4}.$
(B.6)

The parameters for the actual solution, including the attractor values for the
scalars, are obtained by extremizing the entropy function

$\displaystyle\textit{(i)}\quad\frac{\partial\mathcal{E}}{\partial
v_{1}}=0\quad,\quad\textit{(ii)}\quad\frac{\partial\mathcal{E}}{\partial
v_{2}}=0\quad,\quad\textit{(iii)}\quad\frac{\partial\mathcal{E}}{\partial
z^{i}}\Big{|}_{z^{i}_{H}}=0$ (B.7)

Moreover, the value of $\mathcal{E}$ at this extremum gives the black hole
entropy.

Using the ansatzs (B.2) and (B.3) the entropy functional (B.4) reads:

$\begin{split}&\mathcal{E}(v_{1},v_{2},z^{i}_{H},p,q)=2\pi\left\\{q_{\Lambda}e^{\Lambda}+8\pi
v_{2}-8\pi v_{1}+\right.\\\ &\left.-4\pi
Im\mathcal{N}_{\Lambda\Sigma}p^{\Lambda}p^{\Sigma}\frac{v_{1}}{v_{2}}+\frac{1}{16\pi}Im\mathcal{N}_{\Lambda\Sigma}e^{\Lambda}e^{\Sigma}\frac{v_{2}}{v_{1}}+8\pi
Vv_{1}v_{2}\right\\}\\\ \end{split}$ (B.8)

Here the conjugate variable $q$ is defined in terms of the other parameters by
the (B.5):

$e^{\Lambda}=-8\pi\frac{v_{1}}{v_{2}}Im\mathcal{N}^{\Lambda\Sigma}q_{\Sigma}.$
(B.9)

where $Im\mathcal{N}^{\Lambda\Sigma}$ denotes the inverse matrix of
$Im\mathcal{N}_{\Lambda\Sigma}$.
Using this last equation, (B.8) becomes:

$\displaystyle\mathcal{E}=2\pi\left\\{8\pi v_{2}-8\pi v_{1}-4\pi
Im\mathcal{N}_{\Lambda\Sigma}p^{\Lambda}p^{\Sigma}\frac{v_{1}}{v_{2}}-4\pi
Im\mathcal{N}^{\Lambda\Sigma}q_{\Lambda}q_{\Sigma}\frac{v_{1}}{v_{2}}+8\pi
Vv_{1}v_{2}\right\\}$ (B.10)

Note that the electric and magnetic charge contributions couple to $v_{1}$ and
$v_{2}$ in the same way. This motivates introducing the black hole potential:

$V_{BH}=-\frac{1}{2}\left(p^{\Lambda}Im\mathcal{N}_{\Lambda\Sigma}p^{\Sigma}+q_{\Lambda}Im\mathcal{N}^{\Lambda\Sigma}q_{\Sigma}\right)$
(B.11)

Now (B.10) takes the form:

$\mathcal{E}(v_{1},v_{2},p,q)=2\pi\left[8\pi v_{2}-8\pi v_{1}+8\pi
V_{BH}\frac{v_{1}}{v_{2}}+8\pi Vv_{1}v_{2}\right]$ (B.12)

The extremization conditions (B.7) give:

$\begin{split}&\textit{(i)}\quad
Vv_{2}^{2}-v_{2}+V_{BH}=0\quad\quad\quad\longmapsto\quad\quad\quad
v_{2}=\frac{1\pm\sqrt{1-4VV_{BH}}}{2V}\\\ &\textit{(ii)}\quad
1-V_{BH}\frac{v_{1}}{v_{2}^{2}}+Vv_{1}=0\quad\quad\longmapsto\quad\quad\quad
v_{1}=\frac{v_{2}}{\sqrt{1-4V_{BH}V}}\\\
&\textit{(iii)}\quad\frac{v_{1}}{v_{2}}\frac{\partial V_{BH}}{\partial
z^{i}}\Big{|}_{z^{i}_{H}}+v_{1}v_{2}\frac{\partial V}{\partial
z^{i}}\Big{|}_{z^{i}_{H}}=0\\\ \end{split}$ (B.13)

These conditions exactly reproduce those in Bellucci:2008cb , derived from the
attractor mechanism.
Replacing the values of $v_{1}$ and $v_{2}$, one can define the effective
potential

$\displaystyle V_{eff}=\frac{1-\sqrt{1-4V_{BH}V}}{2V}$ (B.14)

introduced in (3.22) in the main text.

##  Appendix C Alternative derivation of the 1d equations of motion

In this Appendix we provide an alternative derivation of the equations of
motion in section 4.1.

We slightly generalize the ansatz (3.2) for the reduction of the 4d metric to
2d:

$ds^{2}_{4}=e^{\alpha\sigma}ds^{2}_{2}+e^{\beta\sigma}d\Omega^{2}_{2}$ (C.1)

The resulting 2d action, after integrating by parts some terms, is:

$\begin{split}S_{2d}=\int
d^{2}x\sqrt{-g_{2}}e^{\beta\sigma(r)}&\left\\{R_{2}-\beta\left(\alpha+\frac{\beta}{2}\right)\left(\partial\sigma\right)^{2}-2g_{i\overline{j}}\partial_{\mu}z^{i}\partial^{\mu}\overline{z}^{\overline{j}}\right.\\\
&\left.+2e^{(\alpha-\beta)\sigma(r)}-2V_{BH}e^{(\alpha-2\beta)\sigma(r)}-2V_{4d}e^{\alpha\sigma(r)}\right\\}\\\
\end{split}$ (C.2)

One can then obtain the equation of motion for this action upon variations of
the 2d metric, $\alpha$, $\beta$, $\sigma$ and the moduli $z^{i}$.

One can then use the following 2d ansatz

$ds^{2}_{2}=-dt^{2}+e^{2b(r)}dr^{2}.$ (C.3)

The relation with the quantities $A,B,C$ in the main text is

$e^{\alpha\sigma(r)}=e^{2A(r)}\quad\quad\quad
e^{\alpha\sigma(r)+2b(r)}=e^{2B(r)}\quad\quad\quad
e^{\beta\sigma(r)}=r^{2}e^{2C(r)}.$ (C.4)

Upon setting, without loss of generality, $\beta=2$, we recover the equation
of motion (4.5).

##  Appendix D 4d ${\cal{N}}=2$ supergravity

A general setup in which 4d potentials are naturally included is gauged 4d
${\cal{N}}=2$ supergravity. Moreover this provides a template for flux
compactifications. Here we review its basic ingredients, see Lauria:2020rhc
for a thorough discussion.

###  D.1 Ungauged 4d ${\cal{N}}=2$ supergravity

We start with the ungauged 4d ${\cal{N}}=2$ coupled to $n_{V}$ abelian vector
multiplets, and ignore any hypermultiplets in the discussion. There are
$n_{V}$ complex moduli $z^{i}$ labelled by $i=1,\ldots,n_{V}$. Including the
graviphoton, there are $n_{V}+1$ gauge bosons, labelled by
$\Lambda=0,\ldots,n_{V}$. The structure of the bosonic lagrangian is

$\displaystyle{\mathscr{L}}=R-2g_{i\bar{\jmath}}\,\partial_{\mu}z^{i}\partial^{\mu}\bar{z}^{\bar{\jmath}}+\hbox{Im}{\cal
N}_{\Lambda\Sigma}\,F^{\Lambda}_{\mu\nu}\,F^{\Sigma\,\mu\nu}+\hbox{Re}{\cal
N}_{\Lambda\Sigma}\,F^{\Lambda}_{\mu\nu}\,\frac{\epsilon^{\mu\nu\rho\sigma}}{2\sqrt{-g}}F^{\Sigma}_{\rho\sigma}$
(D.1)

where the different quantities are defined using special geometry. The scalars
parametrize a special Kähler moduli space, i.e. is the base of a symplectic
bundle, with covariantly holomorphic sections

$\displaystyle{\mathcal{V}}=\begin{pmatrix}X^{\Lambda}\\\
F_{\Lambda}\end{pmatrix}\quad,\quad{\cal D}_{\bar{\imath}}{\cal
V}=\partial_{\bar{\imath}}{\cal
V}+\frac{1}{2}\partial_{\bar{\imath}}{\cal{K}}\,{\cal V}=0,$ (D.2)

(where ${\cal K}$ is the Kähler potential) and obeying the symplectic
constraint

$\displaystyle i\langle{\cal V},{\bar{\cal
V}}\rangle=F_{\Lambda}{\bar{X}}^{\Lambda}-X^{\Lambda}{\bar{F}_{\Lambda}}=1$
(D.3)

It is useful to write

$\displaystyle{\cal V}=e^{{\cal K}(z,{\bar{z}})/2}v$ (D.4)

The holomorphic symplectic vector

$\displaystyle v(z)=\begin{pmatrix}Z^{\Lambda}\\\
\partial_{\Lambda}F\end{pmatrix}\quad,\quad\partial_{\bar{i}}v=0$ (D.5)

is defined using the prepotential212121Although there are 4d ${\cal{N}}=2$
theories with no prepotential, we focus on the usual case in which it exists.
$F(X)$, a holomorphic function of degree 2. Then, (D.3) becomes

$\displaystyle e^{-{\cal K}}=i\langle v,{\bar{v}}\rangle$ (D.6)

The matrix ${\cal N}_{\Lambda\Sigma}$ determining the coupling between the
scalars and the vectors is defined by the relations

$\displaystyle F_{\Lambda}={\cal N}_{\Lambda\Sigma}X^{\Sigma}\quad,\quad{\cal
D}_{\bar{\imath}}{\bar{F}}_{\Lambda}={\cal N}_{\Lambda\Sigma}{\cal
D}_{\bar{\imath}}{\bar{X}}^{\Sigma}$ (D.7)

For completeness, we present the microscopic realization of this in type II
compactification on a Calabi-Yau threefold ${\bf{X}}_{6}$. Start with type
IIB, where the vector moduli space corresponds to the complex structure
moduli. Introduce a symplectic basis of 3-cycles $A_{\Lambda}$, $B^{\Sigma}$
(namely we have intersection numbers $A_{\Lambda}\cdot
A_{\Sigma}=B^{\Lambda}\cdot B^{\Sigma}=0,A_{\Lambda}\cdot
B^{\Sigma}=\delta_{\Lambda}^{\Sigma}$). The holomorphic vector $v$ is given by

$\displaystyle
Z^{\Lambda}=\int_{A_{\Lambda}}\Omega\quad,\quad\partial_{\Lambda}F=\int_{B^{\Lambda}}\Omega$
(D.8)

and

$\displaystyle
K=-\log\big{(}\,i\int_{{\bf{X}}_{6}}\Omega\wedge{\bar{\Omega}}\big{)}$ (D.9)

The electric and magnetic gauge potentials arise from the redution of the 10d
RR 4-form

$\displaystyle A_{1}^{\Lambda}=\int_{A_{\Lambda}}C_{4}\quad,\quad
A_{1\,\Lambda}=\int_{B^{\Lambda}}C_{4}$ (D.10)

For type IIA one has a similar story with even-dimensional cycles replacing
the 3-cycles and $e^{J}$ playing the role of $\Omega$.

###  D.2 Gauged 4d ${\cal{N}}=2$ supergravity

In this section we review some useful formulas for Abelian gaugings in
supergravity. Gaugings are defined by introducing a vector of parameters (FI
terms)

$\displaystyle{\cal G}=\begin{pmatrix}g^{\Lambda}\cr
g_{\Lambda}\end{pmatrix},$ (D.11)

corresponding to electric or magnetic gaugings, respectively. The physical
interpretation of the gauging is reviewed in section 5.1 in the main text.

The gauging introduces a 4d superpotential

$\displaystyle{\cal L}=\langle{\cal G},{\cal V}\rangle=e^{{\cal
K}/2}(Z^{\Lambda}g_{\Lambda}-\partial_{\Lambda}Fg^{\Lambda})$ (D.12)

This leads to a 4d scalar potential222222Notice that this reproduces the usual
formula of 4d ${\cal{N}}=1$ with the $e^{\cal K}$ prefactor arising from
squaring the $e^{{\cal K}/2}$ in the defining of ${\cal V}$, manifest in the
last expression in (D.12).

$\displaystyle V=g^{i{\bar{\jmath}}}\,{\cal D}_{i}{\cal L}\,{\bar{\cal
D}}_{\bar{\jmath}}{\bar{\cal L}}-3|{\cal L}|^{2}$ (D.13)

The intuition is very simple. The gaugings correspond to the introduction of
fluxes in the compactification (including field strength fluxes, but also
geometric or generalized fluxes). This can be described in terms of a domain
wall connecting the vacuum of the ungauged theory (i.e. no fluxes) and the
vacuum of the theory with gauging ${\cal G}$. The domain wall is a codimension
1 object in 4d, and its internal structure is such that its tension is given
by (D.12). This is the standard argument for the superpotential from $p$-form
field strength fluxes in M-theory or type II compactifications Gukov:1999ya ,
which for completeness we repeat for type IIB 3-form fluxes Taylor:1999ii .

Consider type IIB on a CY3 ${\bf{X}}_{6}$ as described around (D.8) and
introduce RR 3-form fluxes

$\displaystyle\int_{A_{\Lambda}}F_{3}=g^{\Lambda}\quad,\quad\int_{B^{\Lambda}}F_{3}=g_{\Lambda}$
(D.14)

For clarity here and in what follows we ignore constant factors.

Denoting the basis of 3-forms $\alpha_{\Lambda}$, $\beta^{\Sigma}$ Poincaré
dual to $A_{\Lambda}$, $B^{\Sigma}$, the flux quanta (D.14) give the 3-form
flux cohomology class

$\displaystyle[F_{3}]=g^{\Lambda}\alpha_{\Lambda}+g_{\Lambda}\beta^{\Lambda}$
(D.15)

This flux can be regarded as being created by a domain wall, given by a D5
wrapped on the 3-cycle $\Pi_{\rm D5}$ whose Poincaré dual is

$\displaystyle\delta(\Pi_{\rm
D5})=F_{3}=g^{\Lambda}\alpha_{\Lambda}+g_{\Lambda}\beta^{\Lambda}$ (D.16)

The tension of this domain wall can be obtained as

$\displaystyle e^{\cal K}\int_{\Pi_{\rm D5}}\Omega=e^{\cal
K}\int_{{\bf{X}}_{6}}\Omega\wedge\delta(\Pi_{\rm D5})=e^{\cal
K}(Z^{\Lambda}g_{\Lambda}-\partial_{\Lambda}Fg^{\Lambda})=e^{{\cal
K}/2}\langle{\cal G},v\rangle={\cal L}$ (D.17)

###  D.3 Black hole central charge

In 4d ${\cal{N}}=2$ theories there are BPS particle states, which can
correspond to charged black holes. We introduce a vector of gauge charges

$\displaystyle Q=\begin{pmatrix}p^{\Lambda}\cr q_{\Lambda}\end{pmatrix},$
(D.18)

They have an associated central charge

$\displaystyle{\cal Z}=\langle Q,{\cal V}\rangle=e^{{\cal
K}/2}(Z^{\Lambda}q_{\Lambda}-\partial_{\Lambda}Fp^{\Lambda})$ (D.19)

The black hole potential which controls the radial flow of scalars in the
attractor mechanism is given by

$\displaystyle V_{BH}=g^{i{\bar{\jmath}}}\,{\cal D}_{i}{\cal Z}\,{\bar{\cal
D}}_{\bar{\jmath}}{\bar{\cal Z}}+|{\cal Z}|^{2}$ (D.20)

The intuition is very simple. For instance in the type IIB setup, BPS
particles arise from D3-branes wrapped on a (special lagrangian) 3-cycle
$\Pi_{\rm D3}$ whose Poincaré dual is

$\displaystyle\delta(\Pi_{\rm
D3})=F_{3}=p^{\Lambda}\alpha_{\Lambda}+q_{\Lambda}\beta^{\Lambda}$ (D.21)

in the symplectic basis introduced in the previous section. The BPS mass can
be obtained as

$\displaystyle e^{\cal K}\int_{\Pi_{\rm D3}}\Omega=e^{\cal
K}\int_{{\bf{X}}_{6}}\Omega\wedge\delta(\Pi_{\rm D3})=e^{\cal
K}(Z^{\Lambda}q_{\Lambda}-\partial_{\Lambda}Fp^{\Lambda})=e^{{\cal
K}/2}\langle Q,v\rangle={\cal Z}$ (D.22)

In the presence of a gauging introducing a potential, the complete attractor
flow is determined by a combination of ${\cal Z}$ and ${\cal L}$. In
particular, in DallAgata:2010ejj  it was shown that for BPS solutions the
attractor flow is controlled by the superpotential

${\cal W}=e^{U}|{\cal Z}+ie^{2\sigma}{\cal L}|$ (D.23)

where $U$ and $\sigma$ are functions of the radial coordinate controlling the
non-compact 2d geometry and the ${\bf{S}}^{2}$ size, respectively

## References

  * (1) H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B766 (2007) 21–33, [hep-th/0605264].
  * (2) T. W. Grimm, E. Palti, and I. Valenzuela, Infinite Distances in Field Space and Massless Towers of States, JHEP 08 (2018) 143, [arXiv:1802.08264].
  * (3) P. Corvilain, T. W. Grimm, and I. Valenzuela, The Swampland Distance Conjecture for Kähler moduli, JHEP 08 (2019) 075, [arXiv:1812.07548].
  * (4) T. W. Grimm, C. Li, and I. Valenzuela, Asymptotic Flux Compactifications and the Swampland, arXiv:1910.09549.
  * (5) G. Buratti, J. Calderón, and A. M. Uranga, Transplanckian axion monodromy!?, JHEP 05 (2019) 176, [arXiv:1812.05016].
  * (6) D. Lüst, E. Palti, and C. Vafa, AdS and the Swampland, arXiv:1906.05225.
  * (7) G. Buratti, M. Delgado, and A. M. Uranga, Dynamical tadpoles, stringy cobordism, and the SM from spontaneous compactification, JHEP 06 (2021) 170, [arXiv:2104.02091].
  * (8) G. Buratti, J. Calderón-Infante, M. Delgado, and A. M. Uranga, Dynamical Cobordism and Swampland Distance Conjectures, JHEP 10 (2021) 037, [arXiv:2107.09098].
  * (9) R. Angius, J. Calderón-Infante, M. Delgado, J. Huertas, and A. M. Uranga, At the end of the world: Local Dynamical Cobordism, JHEP 06 (2022) 142, [arXiv:2203.11240].
  * (10) R. Blumenhagen, N. Cribiori, C. Kneissl, and A. Makridou, Dynamical Cobordism of a Domain Wall and its Companion Defect 7-brane, arXiv:2205.09782.
  * (11) R. Blumenhagen, C. Kneissl, and C. Wang, Dynamical Cobordism Conjecture: Solutions for End-of-the-World Branes, arXiv:2303.03423.
  * (12) J. McNamara and C. Vafa, Cobordism Classes and the Swampland, arXiv:1909.10355.
  * (13) E. Dudas and J. Mourad, Brane solutions in strings with broken supersymmetry and dilaton tadpoles, Phys. Lett. B 486 (2000) 172–178, [hep-th/0004165].
  * (14) R. Blumenhagen and A. Font, Dilaton tadpoles, warped geometries and large extra dimensions for nonsupersymmetric strings, Nucl. Phys. B 599 (2001) 241–254, [hep-th/0011269].
  * (15) E. Dudas, J. Mourad, and C. Timirgaziu, Time and space dependent backgrounds from nonsupersymmetric strings, Nucl. Phys. B 660 (2003) 3–24, [hep-th/0209176].
  * (16) E. Dudas, G. Pradisi, M. Nicolosi, and A. Sagnotti, On tadpoles and vacuum redefinitions in string theory, Nucl. Phys. B 708 (2005) 3–44, [hep-th/0410101].
  * (17) I. Basile, J. Mourad, and A. Sagnotti, On Classical Stability with Broken Supersymmetry, JHEP 01 (2019) 174, [arXiv:1811.11448].
  * (18) R. Antonelli and I. Basile, Brane annihilation in non-supersymmetric strings, JHEP 11 (2019) 021, [arXiv:1908.04352].
  * (19) A. Mininno and A. M. Uranga, Dynamical Tadpoles and Weak Gravity Constraints, arXiv:2011.00051.
  * (20) I. Basile, On String Vacua without Supersymmetry: brane dynamics, bubbles and holography.  PhD thesis, Scuola normale superiore di Pisa, Pisa, Scuola Normale Superiore, 2020.  arXiv:2010.00628.
  * (21) I. Basile, Supersymmetry breaking, brane dynamics and Swampland conjectures, JHEP 10 (2021) 080, [arXiv:2106.04574].
  * (22) J. Mourad and A. Sagnotti, A 4D IIB Flux Vacuum and Supersymmetry Breaking. I. Fermionic Spectrum, arXiv:2206.03340.
  * (23) S. Lanza, F. Marchesano, L. Martucci, and I. Valenzuela, Swampland Conjectures for Strings and Membranes, arXiv:2006.15154.
  * (24) S. Lanza, F. Marchesano, L. Martucci, and I. Valenzuela, The EFT stringy viewpoint on large distances, arXiv:2104.05726.
  * (25) F. Marchesano and M. Wiesner, 4d strings at strong coupling, arXiv:2202.10466.
  * (26) A. Dabholkar and S. Nampuri, Quantum black holes, Lect. Notes Phys. 851 (2012) 165–232, [arXiv:1208.4814].
  * (27) Y. Hamada, M. Montero, C. Vafa, and I. Valenzuela, Finiteness and the Swampland, arXiv:2111.00015.
  * (28) E. Gonzalo, L. E. Ibáñez, and A. M. Uranga, Modular Symmetries and the Swampland Conjectures, JHEP 05 (2019) 105, [arXiv:1812.06520].
  * (29) M. Delgado, M. Montero, and C. Vafa, Black Holes as Probes of Moduli Space Geometry, arXiv:2212.08676.
  * (30) D. R. Green, E. Silverstein, and D. Starr, Attractor explosions and catalyzed vacuum decay, Phys. Rev. D 74 (2006) 024004, [hep-th/0605047].
  * (31) N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, Supergravity and the large N limit of theories with sixteen supercharges, Phys. Rev. D 58 (1998) 046004, [hep-th/9802042].
  * (32) N. Cribiori, A. Gnecchi, D. Lüst, and M. Scalisi, On the correspondence between black holes, domain walls and fluxes, arXiv:2302.03054.
  * (33) A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038, [hep-th/0506177].
  * (34) A. Sen, Black Hole Entropy Function, Attractors and Precision Counting of Microstates, Gen. Rel. Grav. 40 (2008) 2249–2431, [arXiv:0708.1270].
  * (35) S. Bellucci, S. Ferrara, A. Marrani, and A. Yeranyan, d=4 Black Hole Attractors in N=2 Supergravity with Fayet-Iliopoulos Terms, Phys. Rev. D 77 (2008) 085027, [arXiv:0802.0141].
  * (36) S. L. Cacciatori and D. Klemm, Supersymmetric AdS(4) black holes and attractors, JHEP 01 (2010) 085, [arXiv:0911.4926].
  * (37) G. Dall’Agata and A. Gnecchi, Flow equations and attractors for black holes in N = 2 U(1) gauged supergravity, JHEP 03 (2011) 037, [arXiv:1012.3756].
  * (38) K. Hristov and S. Vandoren, Static supersymmetric black holes in AdS4 with spherical symmetry, JHEP 04 (2011) 047, [arXiv:1012.4314].
  * (39) D. Klemm and O. Vaughan, Nonextremal black holes in gauged supergravity and the real formulation of special geometry, JHEP 01 (2013) 053, [arXiv:1207.2679].
  * (40) D. Klemm and O. Vaughan, Nonextremal black holes in gauged supergravity and the real formulation of special geometry II, Class. Quant. Grav. 30 (2013) 065003, [arXiv:1211.1618].
  * (41) A. Gnecchi and C. Toldo, On the non-BPS first order flow in N=2 U(1)-gauged Supergravity, JHEP 03 (2013) 088, [arXiv:1211.1966].
  * (42) D. Astesiano, S. L. Cacciatori, and A. Marrani, Black hole attractors and U(1) Fayet-Iliopoulos gaugings: analysis and classification, JHEP 04 (2022) 099, [arXiv:2112.04962].
  * (43) D. van de Heisteeg, C. Vafa, and M. Wiesner, Bounds on Species Scale and the Distance Conjecture, arXiv:2303.13580.
  * (44) F. Denef, Attractors at weak gravity, Nucl. Phys. B 547 (1999) 201–220, [hep-th/9812049].
  * (45) F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050, [hep-th/0005049].
  * (46) A. Anabalon, D. Astefanesei, and R. Mann, Exact asymptotically flat charged hairy black holes with a dilaton potential, JHEP 10 (2013) 184, [arXiv:1308.1693].
  * (47) A. Anabalón and D. Astefanesei, On attractor mechanism of $AdS_{4}$ black holes, Phys. Lett. B 727 (2013) 568–572, [arXiv:1309.5863].
  * (48) T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D83 (2011) 084019, [arXiv:1011.5120].
  * (49) M. Berasaluce-Gonzalez, P. G. Camara, F. Marchesano, and A. M. Uranga, Zp charged branes in flux compactifications, JHEP 04 (2013) 138, [arXiv:1211.5317].
  * (50) D. S. Freed and E. Witten, Anomalies in string theory with D-branes, Asian J. Math. 3 (1999) 819, [hep-th/9907189].
  * (51) J. M. Maldacena, G. W. Moore, and N. Seiberg, D-brane instantons and K theory charges, JHEP 11 (2001) 062, [hep-th/0108100].
  * (52) E. Witten, Baryons and branes in anti-de Sitter space, JHEP 07 (1998) 006, [hep-th/9805112].
  * (53) C. P. Bachas, M. R. Douglas, and M. B. Green, Anomalous creation of branes, JHEP 07 (1997) 002, [hep-th/9705074].
  * (54) U. Danielsson, G. Ferretti, and I. R. Klebanov, Creation of fundamental strings by crossing D-branes, Phys. Rev. Lett. 79 (1997) 1984–1987, [hep-th/9705084].
  * (55) O. Bergman, M. R. Gaberdiel, and G. Lifschytz, Branes, orientifolds and the creation of elementary strings, Nucl. Phys. B 509 (1998) 194–215, [hep-th/9705130].
  * (56) A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B492 (1997) 152–190, [hep-th/9611230].
  * (57) N. Gendler and I. Valenzuela, Merging the Weak Gravity and Distance Conjectures Using BPS Extremal Black Holes, arXiv:2004.10768.
  * (58) T. W. Grimm, C. Li, and E. Palti, Infinite Distance Networks in Field Space and Charge Orbits, JHEP 03 (2019) 016, [arXiv:1811.02571].
  * (59) Y. Hamada and C. Vafa, 8d supergravity, reconstruction of internal geometry and the Swampland, JHEP 06 (2021) 178, [arXiv:2104.05724].
  * (60) S. Hellerman and X. Liu, Dynamical dimension change in supercritical string theory, hep-th/0409071.
  * (61) S. Hellerman and I. Swanson, Cosmological solutions of supercritical string theory, Phys. Rev. D 77 (2008) 126011, [hep-th/0611317].
  * (62) S. Hellerman and I. Swanson, Cosmological unification of string theories, JHEP 07 (2008) 022, [hep-th/0612116].
  * (63) S. Hellerman and I. Swanson, Dimension-changing exact solutions of string theory, JHEP 09 (2007) 096, [hep-th/0612051].
  * (64) S. Hellerman and I. Swanson, Charting the landscape of supercritical string theory, Phys. Rev. Lett. 99 (2007) 171601, [arXiv:0705.0980].
  * (65) R. Angius, M. Delgado, and A. M. Uranga, Dynamical Cobordism and the beginning of time: supercritical strings and tachyon condensation, JHEP 08 (2022) 285, [arXiv:2207.13108].
  * (66) S. Sugimoto, Anomaly cancellations in type I D-9 - anti-D-9 system and the USp(32) string theory, Prog. Theor. Phys. 102 (1999) 685–699, [hep-th/9905159].
  * (67) I. Antoniadis, E. Dudas, and A. Sagnotti, Brane supersymmetry breaking, Phys. Lett. B464 (1999) 38–45, [hep-th/9908023].
  * (68) G. Aldazabal and A. M. Uranga, Tachyon free nonsupersymmetric type IIB orientifolds via Brane - anti-brane systems, JHEP 10 (1999) 024, [hep-th/9908072].
  * (69) C. Angelantonj, I. Antoniadis, G. D’Appollonio, E. Dudas, and A. Sagnotti, Type I vacua with brane supersymmetry breaking, Nucl. Phys. B 572 (2000) 36–70, [hep-th/9911081].
  * (70) C. Angelantonj, R. Blumenhagen, and M. R. Gaberdiel, Asymmetric orientifolds, brane supersymmetry breaking and nonBPS branes, Nucl. Phys. B589 (2000) 545–576, [hep-th/0006033].
  * (71) R. Rabadan and A. M. Uranga, Type IIB orientifolds without untwisted tadpoles, and nonBPS D-branes, JHEP 01 (2001) 029, [hep-th/0009135].
  * (72) S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D68 (2003) 046005, [hep-th/0301240].
  * (73) E. Lauria and A. Van Proeyen, ${\cal N}=2$ Supergravity in $D=4,5,6$ Dimensions, vol. 966.  3, 2020.
  * (74) S. Gukov, C. Vafa, and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69–108, [hep-th/9906070]. [Erratum: Nucl.Phys.B 608, 477–478 (2001)].
  * (75) T. R. Taylor and C. Vafa, R R flux on Calabi-Yau and partial supersymmetry breaking, Phys. Lett. B 474 (2000) 130–137, [hep-th/9912152].