# The Swampland at Large Number of Space-Time Dimensions

Quentin Bonnefoy DESY, Notkestraße 85, 22607 Hamburg, Germany
Luca Ciambelli Université Libre de Bruxelles and International Solvay
Institutes, ULB-Campus Plaine CP231, B-1050 Brussels, Belgium
Dieter Lüst Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-
Maximilians-Universität, 80333 München, Germany
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6,
80805, München, Germany
Severin Lüst Jefferson Physical Laboratory, Harvard University, Cambridge, MA
02138, USA

###### Abstract

We discuss some aspects of swampland constraints - especially the swampland
distance conjecture - in a large number of space-time dimensions $D$. We
analyze Kaluza-Klein (KK) states at large $D$ and find that some KK spectra
possess an interesting dependence on $D$. On the basis of these observations
we propose a new large dimension conjecture. We apply it to KK states of
compactifications to anti-de Sitter backgrounds where it predicts an upper
bound on the dimension of space-time as a function of its characteristic
radius. We also apply our conjecture to black hole spacetimes, whose entropies
have a $D$-dependence very similar to that of the KK spectrum.

††preprint: LMU-ASC 40/20††preprint: MPP-2020-192††preprint: DESY 20-185

##  I Introduction

The so-called swampland program Vafa:2005ui ; Ooguri:2006in ; Palti:2019pca
is addressing the important question about what kind of effective field
theories, albeit being apparently fully consistent from the quantum field
theory point of view, cannot be consistently embedded into quantum gravity.
One aspect of the swampland idea is that directions in field space which are
associated to large field values are rejected in quantum gravity in the sense
that they are accompanied by an infinite tower of almost massless particles.

As it is well-known for many years, string theory as consistent theory of
quantum gravity can only exist in a certain critical space-time dimension
$D_{c}$, namely $D_{c}=10$ for the superstring and the heterotic strings,
$D_{c}=26$ for the bosonic string and $D_{c}=11$ for M-theory. There are also
certain attempts for non-critical string theories with $D\neq D_{c}$, however
their full quantum consistency is still unclear. The idea of this paper is to
consider the limit of large number of space-time dimensions,
$D\rightarrow\infty$, in the context of the swampland distance conjecture
Ooguri:2006in . The large-$D$ limit and a corresponding $1/D$ expansion of
general relativity was already considered before in a series of interesting
papers, starting with the work of Emparan:2013moa  (for a recent review see
Emparan:2020inr ). Here we want to present a first attempt to get a (upper)
bound on $D$ via swampland considerations, i.e., we like to address the
question: is large-$D$ quantum gravity in the swampland?

There are no theorems, or even conjectures, classifying quantum gravity
theories in terms of space-time dimensions that would settle this question. We
do not try to follow this route either. Instead, we study the consistency of
known swampland conjectures when they are extended to large space-time
dimensions. In particular, we utilize the swampland distance conjecture
Ooguri:2006in , which is about large distances $\Delta$ in field space of
quantum gravity, such as string theory, and about related towers of massless
states that appear at large distances. As a generalization of this idea, we
propose to determine the distance between space-times of different space-time
dimensions. This amounts to enlarge the parameter space of space-time
geometries to include, besides the geometric distance parameters (like the
compactification radius, the AdS radius or the black hole mass), also the
number of dimensions $D$ of space-time as an additional swampland distance
parameter. This extension is motivated by the non-trivial behaviour of known
towers when they are considered in the large-$D$ limit, which cannot be
captured by usual distances on field spaces. Concretely, we want to derive or
characterize new distance functions $\Delta(D)$ as a function of $D$, as they
might provide some possible constraints on the number of space-time
dimensions.

Given that we lack a dynamical theory for the dimension of spacetime, we
cannot build a distance for it in the usual way. Instead, we investigate
bounds on $D$ from consistency requirements on the towers of states predicted
by the SDC, most notably the KK states. We propose to build a distance out of
the typical mass of the towers and study its $D$-dependence, using it as a
proxy for $\Delta(D)$.

The paper is organized as follows. In the next section, we analyze the
$D$-dependence of several swampland conjectures. Then we will propose a Large
Dimension Conjecture (LDC) that restricts the number of space-time dimensions
to be smaller or equal to some critical value that depends on the typical
space-time size , or more generally on the value of the moduli. After that, we
focus on towers of KK modes that appear in effective field theories (EFTs) of
anti-de Sitter space (AdS), and we will concretely derive the distance
functions $\Delta(D)$ for $AdS_{d}\times S^{d^{\prime}}$ backgrounds. We
furthermore compute the distance function $\Delta(D)$ for black holes. At the
end of the paper we will eventually argue a possible new D-duality symmetry
between large and small dimension.

##  II Swampland conjectures at large number of dimensions

The swampland program aims at formulating conditions that must hold for any
EFT of a quantum gravity theory. It defines a subset of the seemingly
consistent EFTs, strictly smaller than the full set, but large enough so that
the notion of a landscape remains. Within this subset, one may find certain
large-$D$ theories. Our goal in this paper is to study the intersection
between the landscape and the set of large-$D$ gravity EFTs.

Since classifying all possible quantum theories of gravity is not a viable
option, even for backgrounds of fixed dimensions, swampland conjectures have
been originally derived by extrapolating properties of known string vacua, or
by relying on semi-classical black hole physics. As it is well-known, we
cannot use the former at large $D$. Nevertheless, we can choose to take all
swampland conjectures at face values, and see how they behave at large $D$. If
they become extremely stringent, we would take it as a hint that large-$D$
theories are in the swampland. If on the other hand they stop being
constraining, as seems to be the case, the conclusion is less clear. One could
conclude that there are simply less theories in the swampland at large $D$.
One could also extrapolate the known information that the landscape is much
smaller than the set of seemingly consistent EFTs, and state that it should
hold at large $D$ as well. Then, we would deduce that either we need more
conjectures to restrict the large $D$ landscape within the set of large $D$
EFTs, or that the large-$D$ limit was not allowed in the first place and that
large-$D$ theories are in the swampland.

Below, we discuss swampland conjectures which have a well-defined large-$D$
behaviour.

###  II.1 The species bound at large D

Let us start with conjectures that can be motivated independently of string
theory and which feature a dimensional dependence. The first one is the
species bound, namely the fact that EFTs must have a cutoff $\Lambda$ much
lower than the Planck scale when a large number of particles $N$ are present.
In four dimensions this bound comes with a precise scaling, namely that
$\Lambda_{sp}\sim N^{-\frac{1}{2}}M_{P}$. One can derive this bound in several
ways, here we follow an argument associated to black hole (BH) physics
Dvali:2007hz . The argument uses a theory with $N$ fields, each of which is
charged under its own gauged discrete symmetry. The idea is then to form a BH
with the maximal discrete charge, and to demand that it can consistently emit
Hawking radiation.

Let us run the argument at any number of dimensions:111In the following the
number of space-time dimensions in the uncompactified effective theory is
denoted by $d$, the number of compact dimensions are given by $d^{\prime}$ and
the number of total dimensions is called $D=d+d^{\prime}$. we consider a
theory of $N$ fields of mass $m$, each of them charged under a gauged $Z_{2}$
symmetry (so that the symmetry is at least $Z_{2}^{N}$). We can form a black
hole charged under the full $Z_{2}^{N}$ group; its charge is measurable via
Aharonov-Bohm experiments, so it must be radiated away at some point of the BH
evolution. Thus, the BH must at least emit each of the $N$ quanta once. If $m$
or $N$ are very large, this has to happen when the BH is heavy, otherwise the
emission is kinematically forbidden. However, when the BH is heavy, its
Hawking temperature is small, so that the emission is unlikely. Thus, we must
be able to ensure

$T_{BH}\gtrsim m\ ,\quad M_{BH}>Nm\ .$ (1)

To be precise, at large $d$ the temperature $T_{BH}$ is not the relevant
quantity to compare to $m$, because it is parametrically different from the
frequency most radiated by the BH. The emission peaks at the energy
$E_{BH}\sim\frac{d^{2}}{R_{BH}}$, while the temperature only scales as
$T_{BH}\sim\frac{d}{R_{BH}}$ Emparan:2013moa , so we must impose

$E_{BH}\gtrsim m\ ,\quad M_{BH}>Nm\ .$ (2)

The radius of the BH, in Planck units, is given in terms of the mass as
follows

$R_{BH}\sim\sqrt{d}M_{BH}^{\frac{1}{d-3}}\ .$ (3)

Thus,

$E_{BH}\sim d^{3/2}M_{BH}^{-\frac{1}{d-3}}\ .$ (4)

Using (2), at large $d$, we find $m\lesssim N^{-\frac{1}{d-2}}d^{3/2}M_{P}$.
Given that all fields have mass $m$, a bound on the latter gives a higher
bound on the cutoff,222We could also think of a tower of states of masses
lying between $m$ and $Nm=\Lambda_{sp}$. Then, the conditions become
$E_{BH}\gtrsim Nm$, $M_{BH}>N^{2}m$ and the species bound is unchanged. so
that we obtain the species bound at large (but fixed) $d$,

$\frac{\Lambda_{sp}}{M_{P}}\sim N^{-\frac{1}{d-2}}\ .$ (5)

This argument assumes that the charge must be Hawking-radiated away. It could
also be that it is radiated at a stage of the BH evolution that is not anymore
described by Hawking radiation. However, the species bound can also be derived
by studying the perturbative running of $M_{P}$. By doing so, we find the same
$N^{-\frac{1}{d-2}}$ scaling. Thus, at fixed but large $d$, the cutoff
decreases very slowly when $N$ increases, and the species bound simply
disappears at infinite $d$.

###  II.2 The TCC and the distance conjecture at large D

The second conjecture that we consider is the Trans-Planckian conjecture (TCC)
Bedroya:2019snp . It states that sub-Planckian quantum fluctuations should
never expand beyond the Hubble horizon. Applied to scalar potentials $V$ in
asymptotic field ranges, it yields

$\absolutevalue{V^{\prime}}\geq\frac{2}{\sqrt{(d-1)(d-2)}}V\ .$ (6)

This condition becomes trivial at large $d$.

In addition, it has been conjectured in Andriot:2020lea  that the factor
$\frac{1}{\sqrt{(d-1)(d-2)}}$ that appears in (6) is a lower bound for the
mass decay rate that appears in the swampland distance conjecture (SDC). More
precisely, the swampland distance conjecture states that at large distances in
field space, one expects towers of states that become light according to
$m\sim e^{-\alpha\phi}$, with $\phi$ the field that undergoes a large
displacement and $\alpha$ an unspecified, order one number. The claim in
Andriot:2020lea  is that

$\alpha\geq\frac{1}{\sqrt{(d-1)(d-2)}}\ .$ (7)

Although this bound has been conjectured based on compactifications of $10D$
supergravities, we can consider extending it to arbitrary dimensions. For
instance, it holds for the tower of KK modes in the simple compactifications
that we discuss later. At large $d$, it becomes unpredictive.

###  II.3 The WGC at large D

The weak gravity conjecture (WGC) can be expressed in arbitrary dimensions for
arbitrary $p$-form gauge fields (with gauge coupling $e$) by reversing the
extremality bound of charged black branes ArkaniHamed:2006dz ;
Heidenreich:2015nta . The claim is that there should be a state of tension $T$
and quantized charge $Q$ such that

$\frac{p(d-p-2)}{d-2}T^{2}\leq e^{2}Q^{2}M_{P}^{d-2}\ .$ (8)

Whatever $p$ is, this bound never becomes empty at large $d$. For a $1$-form
field, it yields the usual

$m\lesssim eQM_{P}^{\frac{d-2}{2}}\ .$ (9)

The strongest bound comes when considering a $\frac{d}{2}$-form field:

$T\lesssim\frac{eQ}{\sqrt{d}}M_{P}^{\frac{d-2}{2}}\ .$ (10)

When the charged brane is wrapped on the $\frac{d}{2}$ dimensions, it yields a
particle which verifies

$m\lesssim\frac{eQ}{\sqrt{d}}M_{P}^{\frac{\frac{d}{2}-2}{2}}\ ,$ (11)

which verifies again the WGC (note that it looks different than (9), due to
the radion couplings in the compactified theory Heidenreich:2015nta ). Thus,
the WGC is a first example of a conjecture that does not seem to weaken at
large $d$, and that can even become very stringent. However, unlike the
previous conjectures, there are unfixed parameters in formulae like (11),
namely the charge and gauge coupling, so that it is unclear what the
dependence with $d$ of the mass really is. For instance, the original version
of the conjecture does not exclude a state whose charge scales with
$\sqrt{d}$. Perturbative unitarity arguments may question the interpretation
of that state as a particle, however it has been argued that the WGC state
could be a BH, which can have a very large charge.

##  III A Large-D distance Conjecture

We just saw that swampland conjectures having an unambiguous $d$-scaling seem
to weaken at large number of dimensions. The interpretation of this fact is
unclear, although we argued that it could be used to motivate an extension of
the current set of swampland conjectures, or simply to reject the large-$D$
limit altogether in specific contexts. In order to gather more information in
this direction, we focus on the swampland distance conjecture and then also on
the AdS distance conjecture at large number of dimensions.

The swampland distance conjecture (SDC) Ooguri:2006in  states that at large
distances $\Delta$ in the field space of a $d$-dimensional (effective) quantum
gravity theory there must be an infinite tower of states with mass scale $m$
such that

$SDC:\quad m=M_{P}e^{-\Delta}\,.$ (12)

We assume that the spectrum of the tower is equidistant, which means that the
number of states $N$ in the tower below the Planck mass is given as

$N={M_{P}\over m}=e^{\Delta}\,.$ (13)

As shown before in eq.(5) for large number of dimensions, $N$ agrees with the
number of states below the species scale.

The SDC predicts that EFTs which describe the bulk of moduli space break down
at its boundaries. This manifests itself by the appearance of towers of states
whose masses are connected to large field displacements. In quantum gravity or
in string theory, the underlying reason why this happens is that objects
revealing the higher-dimensional nature of the theory, or its stringy nature,
become part of the low-energy modes, as happens for instance in a
decompactification limit or a tensionless limit Lee:2019wij .

Decompactification in string theory must be understood in a broader sense than
the usual field theory one. Indeed, while a large volume limit is well
described in the field theory limit of string theory (which already captures
the presence of KK modes necessary to ”build” the compact geometry), a small
volume limit often displays the stringy nature of the theory. Indeed, winding
modes become light and take over the KK modes, which become heavy, to
reconstruct a (T-)dual geometry.

Therefore, concerning the emergence of winding modes we would like to insist
on the following: we could have either noticed that the small volume limit
corresponds to a large distance in field space in the EFT and used the SDC, or
we could have simply remarked that the KK modes become heavy, which weakens
our geometric picture. Then, we would have anticipated that there is a dual
formulation where a geometrical picture emerges again, or that the small
volume limit is not consistent in the first place. Indeed, an immediate
consequence of the SDC is that, if there happens to be an infinite distance
limit in field space without a corresponding tower of light states satisfying
(12), then it cannot be possible to approach this infinite distance point.

Therefore, let us revert the logic of the SDC and define, for a given tower
tower of states $\left|i\right>$, the quantity $\Delta_{i}$ (the “distance”)
as the negative logarithm of its typical mass scale $m_{i}$, i.e.
$\Delta_{i}\sim-\log m_{i}$. Following (12) we normalize $\Delta_{i}$ such
that $m_{i}=M_{P}$ for $\Delta_{i}=0$. At some infinite distance point in
field space, $\Delta_{i}$ agrees with the distance computed from the
$\sigma$-model metric. Analogously, if there is a dual tower of states
$\left|\tilde{\imath}\right>$ we define the dual distance
$\widetilde{\Delta}_{i}\sim-\log\widetilde{m}_{i}$. Following our definition,
$\Delta_{i}$ and $\widetilde{\Delta}_{i}$ are not the same (even though
typically closely related). This is consistent with the SDC as the two towers
become light at different points in moduli space. For example, for string
theory on a circle with $\left|i\right>$ and $\left|\tilde{\imath}\right>$
given by Kaluza-Klein and winding modes we have $\Delta_{i}\sim\log R$ and
$\widetilde{\Delta}_{i}\sim-\log R$. Similarly for compactification on a
2-torus, $\left|i\right>$ and $\left|\tilde{\imath}\right>$ given by two
different Kaluza-Klein modes, and $\Delta_{i}\sim\log(\imaginary\tau)$ and
$\widetilde{\Delta}_{i}\sim-\log(\imaginary\tau)$, where $\tau$ is the complex
structure modulus of $T^{2}$.

On the basis of the SDC and following these observations, we now demand that,
for a particular tower $\left|i\right>$, an ”infinite distance” limit
$\Delta_{i}\rightarrow-\infty$ is obstructed, i.e. not allowed, unless there
is a dual tower $\left|\tilde{\imath}\right>$, which becomes light in this
limit. It would be tempting to dub this requirement the _Negative Distance
Conjecture (NDC)_. Note that in general there is not necessarily a simple
relation between $\Delta_{i}$ and the geometric distance in scalar field
space, so we might not directly be able to use the SDC to infer anything about
the limit $\Delta_{i}\rightarrow-\infty$. In the case of KK modes, the limit
$\Delta_{KK}(\phi)\rightarrow-\infty$ implies $\phi\rightarrow-\infty$ and the
SDC says that there must be a second tower of states or
$\phi\rightarrow-\infty$ cannot be possible. With the NDC we extrapolate from
the KK modes behaviour a general pattern assumed to be true for any tower of
states.

Let us now go back to our discussion of large-$D$ EFTs. In general, the
quantity $\Delta$ that determines the mass scale of the tower in (12) is a
field dependent quantity, $\Delta=\Delta(\phi)$, where $\phi$ denotes the
scalar fields in the EFT. Typically they correspond to geometric length
parameters like the size of a compact space. However, the mass scale of the
tower can also depend on the number of space-time dimensions in a non-trivial
way. As we will extensively discuss later, this is the case for the KK
spectrum. Therefore, we introduce in addition to $\phi$ the space-time
dimension $D$ as a new swampland variable that determines the spectrum and the
validity regime of the EFT.

In particular, we focus our discussion on the dependence of the masses of
known towers of states on the dimension of space-time. Said differently, we
study the distance $\Delta$ as a function of the number $D$ of space-time
dimensions, i.e. $\Delta=\Delta(\phi,D)$. As a consequence, infinite distances
have to be extended to incorporate the $D$-direction. In particular, even for
single field models, there are now different ways to go to infinite distance,
since $D$ can be taken to be fixed or to vary. The usual towers of states,
such as KK modes, may not have the expected behavior once $D$ varies, even
when the distance restricted to the field space of $\phi$ becomes infinite.

What does this extension of the parameter space lead to? We do not have a
dynamical theory of $D$, i.e. there is no kinetic term for $D$ in an EFT, and
we cannot use the SDC in the standard way. But on the basis of the previous
discussion, namely utilizing the NDC, we are now ready to propose the
following _Large Dimension Conjecture (LDC)_ :

_If there is a tower of states $\left|i\right>$, the corresponding “distance”
$\Delta_{i}(\phi,D)$ must be a positive function of the EFT fields $\phi$ and
the dimension $D$,_

$LDC:\quad\Delta_{i}(\phi,D)\geq 0\,,$ (14)

_unless there is a dual tower of states $\left|\tilde{\imath}\right>$ such
that the dual distance $\widetilde{\Delta}_{i}(\phi,D)$ becomes positive when
$\Delta_{i}(\phi,D)$ changes its sign and takes negative values. _

In simple cases one is dealing with a two-parameter space of backgrounds, the
number $D$ of space-time dimensions and one geometric size parameter of the
background, denoted by $\phi\sim\log R$. In this case we can give a simple
interpretation of the LDC: the geometric interpretation of space-time breaks
down when $\Delta_{KK}(R,D)\rightarrow-\infty$. Therefore, we conclude that
something must happen in this limit. Either we expect a dual geometry with a
dual tower of winding modes to emerge or the limit is not consistent.

In the two parameter case we can look for one-dimensional directions of
constant $\Delta$. These lines will separate the parameter space into a region
in which the tower mass scale is smaller than one, i.e. positive $\Delta$, and
another region in which the tower mass scale is larger than one and $\Delta$
is negative. Forbidding this region then provides a possible bound on a
combination of $D$ and $R$.

More precisely, for fixed $D$, the LDC provides a $D$-dependent bound on $R$,
which in general is stronger than just requiring $R>1$. But in the context of
large-$D$ gravity, we also like to treat $D$ as a free variable and then the
LDC provides, for fixed $R$, a bound on the number of space-time dimensions,
namely a critical dimension $D_{0}$. In fact, treating $D$ as a distance
variable and keeping $R$ fixed, $\Delta(D)$ should measure the distance
between backgrounds of different dimension.333One could try to derive
$\Delta(D)$ as the distance between space-time backgrounds using the geometric
distance formula $\Delta\sim\int\left(\frac{1}{{\cal
V}_{M}}\int_{M}\sqrt{g}\tr[\left(g^{-1}\frac{\partial
g}{\partial\tau}\right)^{2}]\right)^{\frac{1}{2}}\text{d}\tau$, where ${\cal
V}_{M}$ denotes the volume of space-time. The parameter $\tau$ is given by the
number of space-time dimensions $D$, i.e. one should compute the distance for
metric variations with respect to $D$. Alternatively, as in Kehagias:2019akr ,
one can also try to set up geometric flow equations, like the Ricci flow,
where one considers the flow of the background with respect to $D$. This will
be discussed elsewhere workinprogress . As we will see for the KK tower or for
black holes the LDC means that $D_{0}$ is determined by the special dimension
in which the space-time volume becomes basically smaller than one in Planck
units, which should not happen in quantum gravity, unless there is a dual
description of the theory.

Let us see what the LDC means for the canonical volume modulus of
compactification spaces with a large number of dimensions. We consider a
$D$-dimensional space-time, which splits into a non-compact space $M_{d}$ and
a compact space $K_{d^{\prime}}$ of dimensions $d$ and $d^{\prime}$
respectively, like for $AdS_{d}\times S^{d^{\prime}}$, so that
$D=d+d^{\prime}$. The ansatz for the $D$-dimensional metric reads

$\mathrm{d}s^{2}_{D}=e^{2\beta\varphi}ds^{2}_{M_{d}}+e^{2\phi}ds^{2}_{K_{d^{\prime}}}\,,$
(15)

where we split the volume modulus $\phi(x^{\mu})=\phi_{0}+\varphi(x^{\mu})$
into a constant piece $\phi_{0}$ and its dynamical part $\varphi$ which
depends only on the coordinates $x^{\mu}$ of the “external” space, but not on
the coordinates $y^{m}$ of the “internal” $K_{d^{\prime}}$. In addition, we
normalize its background value $\phi_{0}$ and the volume of $K_{d^{\prime}}$
according to

${\cal
V}_{d^{\prime}}\equiv\left(M_{P}^{(D)}\right)^{d^{\prime}}\int_{K_{d^{\prime}}}d^{d^{\prime}}y\sqrt{\det(e^{2\phi_{0}}g_{K_{d^{\prime}}})}=e^{d^{\prime}\phi_{0}}\,.$
(16)

Consequently, ${\cal V}_{d^{\prime}}$ is dimensionless and measured in units
of the $D$-dimensional Planck mass.

After reduction of the $D$-dimensional Einstein-Hilbert term, we reach the
lower-dimensional Einstein frame by fixing

$\beta=-\frac{d^{\prime}}{d-2}\,.$ (17)

The relevant terms in the $d$-dimensional effective action look like

$\mathcal{L}=\frac{\bigl{(}M_{P}^{(d)}\bigr{)}^{{d}-2}}{2}\int
d^{d}x\sqrt{-g_{M_{d}}}\left[\mathcal{R}_{d}-\frac{d^{\prime}(D-2)}{d-2}\left|\partial\phi\right|^{2}\right]\,.$
(18)

The lower dimensional Planck mass is related to the higher dimensional Planck
as

$M_{P}^{(d)}=M_{P}^{(D)}{\cal V}_{d^{\prime}}^{1\over d-2}\ .$ (19)

Let us finally introduce the canonically normalized field

$\Phi=\sqrt{\frac{d^{\prime}(D-2)}{d-2}}\phi=\sqrt{\frac{D-2}{d^{\prime}(d-2)}}\log{\cal
V}_{d^{\prime}}\,.$ (20)

According to the infinite distance conjecture, the normalized field $\Phi$ is
related to a tower of states with masses $m$ given by

$m\sim M^{(d)}_{P}e^{-\alpha(d,d^{\prime})\left|\Phi\right|}\,,$ (21)

Here $\alpha(d,d^{\prime})$ is a dimension dependent constant, which we will
determine in the following. For the infinite distance point
$\Phi\rightarrow\infty$ the relevant mass scale $m$ is just given in terms of
the KK mass scale,

$\displaystyle m_{KK}\approx{\cal V}^{-{1\over
d^{\prime}}}_{d^{\prime}}M_{P}^{(D)}\approx M^{(d)}_{P}{\cal
V}_{d^{\prime}}^{-{D-2\over
d^{\prime}(d-2)}}=M^{(d)}_{P}\exp\left[-\sqrt{\frac{D-2}{d^{\prime}(d-2)}}\Phi\right]\,.$
(22)

It follows that the constant $\alpha(d,d^{\prime})$ is given by

$\alpha(d,d^{\prime})=\sqrt{\frac{D-2}{d^{\prime}(d-2)}}\,.$ (23)

Consistently with the infinite distance conjecture, the KK tower becomes light
in the limit of large $\Phi$. Given our definition of $\Phi$, this limit can
be reached in the usual way, namely as a large-$\Phi$ limit. On the other hand
we point out that, for fixed $\Phi$, the KK masses approach $M_{P}^{(d)}$ at
$d,d^{\prime}\rightarrow\infty$.

In any case, using our definition (12), the distance $\Delta$ and $\Phi$ are
related as $\Delta=\alpha\Phi$, and the LDC requires that
$\Delta(d,d^{\prime})>0$, i.e. $m_{KK}<M_{P}^{(d)}$, which means

$\Phi>0\,,$ (24)

unless one identifies a dual tower of states that becomes light in the limit
under consideration.

##  IV The LDC for AdS at large D

###  IV.1 The AdS distance conjecture and KK spectrum in arbitrary number of
dimensions

Now let us briefly recall the anti-De Sitter distance conjecture (ADC)
Lust:2019zwm , which is obtained by comparing AdS spaces with different
cosmological constants $\Lambda$, i.e. varying the metric with respect to
$\Lambda$. The ADC is a bit different from the SDC in the sense that it deals
with the distance between not continuously connected backgrounds, which
implies that the scalar fields are not anymore massless scalars but possess a
potential. Different discrete vacua are labeled by different values of the
cosmological constant or the AdS radius, which are related to different,
discrete flux quantum numbers.

Then ADC states that the limit of a small AdS cosmological constant,
$\Lambda\rightarrow 0$, is at infinite distance in the space of AdS metrics,
and that it is related to an infinite tower of states with typical masses that
behave as

${\rm ADC}\colon\quad m_{AdS}\sim\left|\Lambda\right|^{a}\,,$ (25)

with $a={\cal O}(1)$. The strong version of the ADC proposes that for
supersymmetric backgrounds $a=1/2$. The corresponding distance is given in
terms of the logarithm of $\Lambda$,

$\Delta_{AdS}=-a\log\left|\Lambda\right|\,.$ (26)

Via (12), the distance $\Delta_{AdS}$ immediately leads to the tower behaviour
(25); as long as $a$ is positive, the distance $\Delta_{AdS}$ is positive in
the limit of small $\Lambda$ and the mass scale $m_{AdS}$ is decreasing in
this limit. So in the limit of vanishing cosmological constant, i.e. large AdS
radius, AdS quantum gravity as an effective field theory can only co-exist
together with an infinite tower of additional massless states. This implies
that pure AdS quantum gravity is in the swampland. The tower of states is
typically provided by the KK modes of an extra compact
$d^{\prime}$-dimensional space, e.g. with backgrounds of the form
$AdS_{d}\times S^{d^{\prime}}$. For the case of the strong ADC with $a=1/2$ it
follows that there is no scale separation between the mass scales of $AdS_{d}$
and $S^{d^{\prime}}$.

In the following we will consider AdS spaces in arbitrary dimensions $d$:

$\text{d}s^{2}=R_{A}^{2}\left(-\left(\cosh\rho\right)^{2}\text{d}t^{2}+\text{d}\rho^{2}+\left(\sinh\rho\right)^{2}\text{d}\Omega_{d-2}^{2}\right)\;.$
(27)

Here $R_{A}$ is the AdS radius which is related to the cosmological constant
$\Lambda_{AdS_{d}}$ as

$\Lambda_{AdS_{d}}=-{(d-1)(d-2)\over 2R_{A}^{2}}\,.$ (28)

Since AdSd alone is not a viable background, we consider a background space of
the form

${\cal M}_{D}=AdS_{d}\times S^{d^{\prime}}\,\,{\rm with}\,\,\ d+d^{\prime}=D.$
(29)

Recall that the Ricci scalars of AdSd and of the compact
$d^{\prime}$-dimensional sphere $S^{d^{\prime}}$ of radius $R_{S}$ are given
by

${\cal R}_{A}=-{d(d-1)\over R_{A}^{2}}\,,\quad{\cal
R}_{S}={d^{\prime}(d^{\prime}-1)\over R_{S}^{2}}\,.$ (30)

This background is obtained from the so-called Freund-Rubin compactifications
from $D$ dimensions, which is a solution of the $D$-dimensional Einstein
equations without a cosmological constant by turning on a non-vanishing
$d$-form gauge field strength in the AdS-space:444By Hodge duality, it can
equally lie on the $S^{d^{\prime}}$ factor, and read
$F^{\mu_{1}\dots\mu_{d^{\prime}}}=\frac{\epsilon^{\mu_{1}\dots\mu_{d^{\prime}}}}{\sqrt{g_{S}}}f\,.$
(31)

$F^{\mu_{1}\dots\mu_{d}}=\frac{\epsilon^{\mu_{1}\dots\mu_{d}}}{\sqrt{-g_{A}}}f\,,$
(32)

such that the Ricci scalars are given by

${\cal R}_{A}=-\frac{d(d^{\prime}-1)}{D-2}f^{2}\ ,\quad{\cal
R}_{S}=\frac{d^{\prime}(d-1)}{D-2}f^{2}\ ,$ (33)

in units of $M_{P}^{(D)}$, the $D$-dimensional Planck mass. These expressions
correlate the radii of the two factors:

$(d-1)R_{S}=(d^{\prime}-1)R_{A}\ .$ (34)

In the next step we can compute the spectrum of KK states in the effective
AdSd theory which originate from the compactification on $S^{d^{\prime}}$.
This spectrum obeys the following equation (note that from now on, we measure
the radii in units of D-dimensional Planck mass $M_{P}^{(D)}$)

$m_{l,KK}^{2}=\Bigl{(}M_{P}^{(D)}\Bigr{)}^{2}~{}{l(l+d^{\prime}-1)\over
R_{S}^{2}}\,,$ (35)

where the integer $l$ labels the KK momentum. Comparing this with the AdS
cosmological constant $\Lambda$, the associated KK mass scale ($l=1$) is then
given by

$m_{KK}^{2}=-2{(d-1)d^{\prime}\over(d-2)(d^{\prime}-1)^{2}}\Lambda_{AdS_{d}}\
,$ (36)

so we see that $\alpha=1/2$ and the strong ADC still holds in arbitrary number
of dimensions.

###  IV.2 The large D behaviour

With the LDC, one can now ask what happens when the dimension $d$ of AdS
or/and $d^{\prime}$ of $S$ is varied and gets large. As we will see, certain
limits for $d,d^{\prime}$ lead to violations of the LDC if we assume that the
tower predicted by the ADC is again the tower of KK modes of a Freund-Rubin
compactification. From that, we conjecture either a bound relating the
dimension of spacetime and geometrical quantities such as radii, or we predict
the presence of a dual tower that should become light in the same limit.

A relevant quantity for the following discussion is the volume of the
$d^{\prime}$-dimensional unit sphere, given by

$\Omega_{d^{\prime}}={2\pi^{(d^{\prime}+1)/2}\over\Gamma\bigl{(}{d^{\prime}+1\over
2}\bigr{)}}\,.$ (37)

For large $d^{\prime}$ this behaves as

$\Omega_{d^{\prime}}\sim\sqrt{2}e\Biggl{(}\sqrt{{2\pi e\over
d^{\prime}}}\Biggr{)}^{d^{\prime}}\rightarrow 0\,,$ (38)

and it therefore becomes increasingly small for large $d^{\prime}$, i.e.
$\Omega_{d^{\prime}}^{1/d^{\prime}}\rightarrow\frac{1}{\sqrt{d^{\prime}}}$ for
$d^{\prime}\rightarrow\infty$. In fact, the area of the unit sphere becomes of
order one for

$d^{\prime}\sim 2\pi e\sim 17\,.$ (39)

Including also the radial dependence, the volume of the
$d^{\prime}$-dimensional sphere of radius $R_{S}$ for large $d^{\prime}$ is
given by ${\cal V}_{d^{\prime}}=R_{S}^{d^{\prime}}\Omega_{d^{\prime}}$.

####  IV.2.1 Small number of internal dimensions

Let us first consider the case where the number of dimensions $d^{\prime}$ of
the sphere is fixed555As a side remark, it is also interesting to consider the
case where the AdS background is fully fixed, namely the case of fixed $d$,
fixed cosmological constant $\Lambda_{AdS_{d}}$ (or AdS radius $R_{A}$) and
fixed $d$-dimensional Planck mass. We then take the large $D$, i.e. large
$d^{\prime}$, limit (that implies that the $D$-dimensional Planck mass
vanishes asymptotically). In this limit, it is easy to see that the KK mass
scales as  $m_{KK}^{2}\sim\frac{\Lambda_{AdS_{d}}}{D}\ .$ Thus, a tower of
states enters the AdS EFT simply because we considered the large $D$ limit.
This is due to the fact that the canonical $\Phi$ in (20) has to vary for the
AdS background to remain fixed, and this behaviour is captured by the SDC. In
this limit, the KK modes are light and the LDC does not bring new information.
and where we take only $d\sim D\to\infty$. The $d$-dimensional Planck mass
reads

$M_{P}^{(d)}=M_{P}^{(D)}{{\cal V}_{d^{\prime}}}^{\frac{1}{d-2}}\sim
M_{P}^{(D)}\ .$ (40)

Consequently, the KK masses are given by their expression in units of
$M_{P}^{(D)}$, given in eq.(36),

$m^{2}_{KK}\sim-{2\over(d^{\prime}-1)}\Lambda_{AdS_{d}}\sim-\Lambda_{AdS_{d}}\,.$
(41)

The corresponding AdS distance is therefore

$\Delta_{AdS}=-{1\over 2}\log\Lambda\,.$ (42)

In terms of $R_{S}$ these relations look like

$m^{2}_{KK}\sim\Bigl{(}M_{P}^{(D)}\Bigr{)}^{2}~{}{1\over
R_{S}^{2}}\,,\quad\Delta_{AdS}=\log R_{S}\,.$ (43)

We see that $m_{KK}$ and $\Delta_{AdS}$ do not depend on the number of space-
time dimensions. Actually this is the same situation as for the KK spectrum
arising from the compactification on a single circle.

On the other hand, in terms of $R_{A}$, the analogous relations look like

$m^{2}_{KK}\sim\Bigl{(}M_{P}^{(D)}\Bigr{)}^{2}~{}{D^{2}\over
R_{A}^{2}}\,,\quad\Delta_{AdS}=\log\biggl({R_{A}\over D}\biggr{missing})\,.$
(44)

Therefore the KK mass scale becomes heavier than the Planck mass for
$D>R_{A}$. Using (34), the latter condition is equivalent to $R_{S}<1$,
consistently with (43).

####  IV.2.2 Large number of internal dimensions

Next let us assume that the radii of the two subspaces are of the same order,
i.e. $R_{S}=R_{A}=R$. It then follows that also the respective numbers of
dimensions must be the same: $d=d^{\prime}=D/2$. It also follows that in terms
of the flux number $f$ we have that $R\simeq Df^{2}$. In the limit of large
dimensions, using eq.(36), the KK mass scale behaves as

$m_{KK}^{2}\sim{|\Lambda_{AdS_{d}}|\over D}\,.$ (45)

Instead of expressing the KK scale in terms of $\Lambda_{AdS_{d}}$, we can
also express it in terms of the radius $R$:

$m_{KK}^{2}\sim\Bigl{(}M_{P}^{(D)}\Bigr{)}^{2}~{}\biggl{(}{1\over{\cal
V}_{d^{\prime}}}\biggr{)}^{2/d^{\prime}}\sim\Bigl{(}M_{P}^{(D)}\Bigr{)}^{2}~{}{D\over
R^{2}}\,,$ (46)

In this relation $m_{KK}$ is measured in units of the higher dimensional
Planck mass $M_{P}^{(D)}$. In our case, the Planck mass in the effective AdS-
theory behaves as

$M_{P}^{(d)}\sim M_{P}^{(D)}\biggl{(}{R\over\sqrt{D}}\biggr{)}\,.$ (47)

Therefore in terms of $M_{P}^{(d)}$, the KK masses become

$m_{KK}^{2}\sim\Bigl{(}M_{P}^{(d)}\Bigr{)}^{2}~{}\biggl{(}{D\over
R^{2}}\biggr{)}^{2}\,,$ (48)

where $R$ is measured in units of $M_{P}^{(D)}$. Comparing with the
$D/2$-dimensional effective action, the normalized field $\Phi$ depends on $R$
and $D$ as

$\Phi(R,D)\simeq\sqrt{D}\log\biggl({R\over\sqrt{D}}\biggr{missing})\,,$ (49)

and becomes therefore negative for $D>R^{2}$.

So we see that, at fixed radius, all KK masses become super Planckian above a
certain critical dimension. The corresponding AdS distance is given as

$\Delta_{AdS}=2\log\biggl({R\over\sqrt{D}}\biggr{missing})\,,$ (50)

and it is positive for $D$ only smaller than a certain critical dimension
$D_{0}$, namely

$D\leq D_{0}=R^{2}\,,$ (51)

where all KK tower starts below the Planck scale. This condition is nothing
but the application of (24) to the case of $AdS_{D/2}\times S^{D/2}$. It gives
us a non-trivial constraint on $R$ and $D$ because the volume of the sphere
has a non-trivial dimension dependence.

On the other hand, if this condition is violated, all KK masses are super
Planckian and then the notion of space-time is in general lost. Actually the
KK modes are part of the space-time geometry - they build geometry - and
certain KK towers must start below the Planck mass, if one wants to have a
geometrical interpretation of space-time. Therefore, a KK tower that starts
above the Planck mass appears pathological in quantum gravity and one can
conjecture that this regime of the parameter space is in the swampland (unless
there is a dual tower). This observation is the reasoning for the LDC, which
leads to the upper bound (51) on $D$ for $AdS_{d}\times S^{d^{\prime}}$
geometries.

However note that for AdS space-times of radius $R$ (see (27) for the metric)
many of the curvature invariants do not vanish, for instance:

$\displaystyle{\cal R}_{\mu\nu\sigma\rho}{\cal
R}^{\mu\nu\sigma\rho}=\frac{2D(D-1)}{R^{4}}\ ,\quad{\cal R}_{\mu\nu}{\cal
R}^{\mu\nu}$ $\displaystyle=\frac{D(D-1)^{2}}{R^{4}}\ ,$ (52)
$\displaystyle{\cal R}^{2}=\frac{D^{2}(D-1)^{2}}{R^{4}}\ .$

So demanding that the curvature stays less than the Planck mass, i.e. ${\cal
R}^{2}\lesssim M_{P}^{4}$, implies $R\gtrsim D$.

In summary, for the case with both $AdS_{d}$ and $S^{d^{\prime}}$ having large
dimensions there are three regions,666For the case with only AdSd having a
large dimension, but the sphere having a finite dimension there are two
regimes: $D<R:$ small curvature and good notion of space-time geometry. $R<D$:
large curvature and no good notion of space-time geometry, the KK tower is
above the Planck scale. when comparing $R$ and $D$:

A: $D<R$: small curvature, light KK tower and good notion of space-time
geometry.

B: $R<D<R^{2}$: large curvature but still good notion of space-time geometry,
i.e. the KK tower is still below the Planck mass.

C: $R^{2}<D$: large curvature and no notion of space-time geometry, as the KK
tower is above the Planck mass.

Given that the LDC only brings something non-trivial in regimes of large
curvature, we cannot exclude that our bounds are too naive and are affected by
strong gravitational corrections. It would be interesting to investigate the
constraints imposed by the LDC in geometries where the curvature is
controllable, to firmly establish the bounds on $R$ and $D$, the typical size
of the geometry and its dimension workinprogress .

We would like to conclude with a remark about IIB string theory on
$AdS_{5}\times S^{5}$ or M-theory on $AdS_{4}\times S^{7}$ or $AdS_{7}\times
S^{4}$ backgrounds. There are no winding modes on the sphere, however there
exists the classical limit of small sphere radius where the KK modes decouple.
There, $\Delta_{KK}<0$ without any dual tower, which is excluded by the LDC,
and indeed flux quantization on the sphere forbids that we take the small
radius limit or the large dimension limit, respectively. The flux quantization
condition emerges from the existence of objects with Planck-scale charge,
namely $D3$, $M2$ or $M5$ branes. Equivalently, these backgrounds can also be
obtained as the near-horizon geometry of these branes, whose number cannot be
smaller than one. It is thus conceivable that the LDC is connected to the
existence of such objects in large $D$, as also required by the completeness
conjecture Polchinski:2003bq .

##  V The black hole entropy distance at large $D$

As it was recently shown in Bonnefoy:2019nzv , the limit of large entropy,
${\cal S}\rightarrow\infty$, is at infinite distance in the space of black
hole metrics. Therefore, similarly to the ADC, it was argued Bonnefoy:2019nzv
that there exists a black hole entropy distance conjecture (BHEDC) such that

$\Delta_{bh}=\alpha\log{\cal S}\,,$ (53)

stating that the large entropy limit of black holes is also followed by a
tower of light modes:

$m_{bh}\sim{1\over{\cal S}^{\alpha}}\,.$ (54)

A possible interpretation of this tower of states is closely related to the
black hole micro states.

For black holes at large space-time dimensions we can use the result for the
black hole entropy at large $D$, which is given by the following expression
Emparan:2013moa :

${\cal S}\sim\biggl{(}{R_{h}\over\sqrt{D}}\Biggr{)}^{D-2}\,.$ (55)

Here $R_{h}$ (in Planck units) plays the role of the radius of the black hole
horizon. For $D\rightarrow\infty$ the entropy becomes very small. Note that
the $1/\sqrt{D}$ in the denominator again originates from the area of the unit
sphere in large dimensions in the same way as for the KK mass scale in the
previous section. So comparing the large-$D$ entropy with the distance
$\Delta_{bh}$ we obtain

$\Delta_{bh}=\alpha\log{\cal
S}=\alpha(D-2)\log\biggl({R_{h}\over\sqrt{D}}\Biggr{missing})\,.$ (56)

We see that the black hole distance has the property that it is only positive
in a finite interval of $D$,

$\Delta(D)_{bh}\geq 0\quad{\rm for}\quad 2\leq D\leq R_{h}^{2}\,,$ (57)

with a maximum of the distance somewhere between $D=2$ and $D=R_{h}^{2}$.
Therefore the LDC again implies the following critical dimension:

$D_{0}\sim R_{h}^{2}\,.$ (58)

Respecting the bound means that the black hole entropy should be larger than
one. Now the LDC is based on the argument that black holes with entropy
smaller than one are not consistent objects in quantum gravity. Note that the
case $D=2$ is special: the distance is zero and the black hole tower contains
only one state below the Planck mass.

##  VI Discussion

In this paper we have considered some swampland aspects for gravity theory in
an arbitrary number and in particular in a large number of space-time
dimensions. The discussion was mainly focussed on the large distance
conjecture with KK modes, whose masses exhibit a non-trivial dependence on the
number of space-time dimensions. In addition we also looked at the black hole
distance at large $D$. Based on these results we formulated a new large-$D$
conjecture stating that certain distance functionals of the swampland distance
conjecture must be positive as a function of the number of space-time
dimensions. Being certainly satisfied in critical string theory, the LDC is
supposed to provide a general swampland condition for quantum gravity in
arbitrary dimensions. Applying it to AdS spaces and to black hole geometries,
it then leads to an upper critical dimensions from the requirement that the
volume of the compact space must be larger than one or, in the context of
black holes, that the black hole entropy should be larger than one at large D.
Since $D$ is a discrete number, the existence of a critical dimension would
strictly spoken also require that the AdS sizes and the black hole entropies
that saturate the LDC bound are also quantized. In fact the possibility of
discrete and quantized black hole entropies was considered in the context of
quantum black holes in Bekenstein:1995ju ; Mukhanov:2018tzo .

As we have seen from the masses of the KK modes the critical dimension depends
on the size of the compact space, e.g. $D_{0}=R^{2}$ in the case of large
dimensions for the internal and AdS spaces. In particular, for large AdS
spaces (or black holes), i.e. $R\gg 1$ (or $R_{h}\gg 1$), the critical
dimension $D_{0}$ is very large. But what about the possible existence of a
critical dimension, which is independent from the length scale $R$? For that
let us impose the much stronger requirement that all possible backgrounds such
as all possible AdS spaces or all possible black holes must satisfy the LDC.
Clearly, the strongest bound arises for Planck size compact spaces $R\sim{\cal
O}(1)$ or Planck size black holes $R_{h}\sim{\cal O}(1)$. Then one gets an
absolute bound on $D_{0}$ of the order ${\cal O}(2-20)$. Another speculative
way to obtain a critical dimension is via the exact formula of volume of the
unit sphere $S^{D/2}$ in eq.(37): $\Omega_{D/2}$ is an increasing function
with $D$ for small $D$ and then decreases for large $D$ with a maximum volume
at $D/2=6$. So the $AdS_{5}\times S^{5}$ background with $D/2=5$ in
superstring theory is tantalizingly close to the unit sphere of maximal
volume.

We also like to recall that for the case of the curved $AdS_{d}\times
S^{d^{\prime}}$ backgrounds the KK masses in AdS space start to become heavier
than the Planck mass in the regime where the space-time curvature is already
large. One might therefore wonder if our ”naive” estimates for the KK masses
are still valid, or if there are non-negligible gravitational contributions to
the KK masses that keep them always below the Planck mass. On the other hand
for the flat background, there are no curvature corrections and hence the KK
spectrum at large dimensions would be under control. So it would be very
interesting to see how the large $D$ spectrum for flat backgrounds depends on
the number of space-time dimensions, like for tori or Ricci flat spaces at
high number of dimensions, or also for other high-dimensional spaces with non-
vanishing torsion classes workinprogress .

As discussed before, the distance conjecture often contains the expectation
that at large distances in field space one ends up with a dual description,
where states getting light at large distances are accompanied by a dual tower
of heavy states for large field values. Moreover often there is a dual limit
in field space, with a dual distance $\tilde{\Delta}$, where the situation is
reversed. Here we have seen that states get heavy at large $D$. Therefore,
non-perturbative branes wrapped around certain high-dimensional cycles might
become light in the large-$D$ limit workinprogress . In view of the possible
presence of these non-perturbative states, it would be interesting to
investigate the existence of a novel non-perturbative duality symmetry in
quantum gravity, denoted by D-duality, which could act in following way on $R$
and $D$:

$(\sqrt{D}/R)\,\longleftrightarrow\,(R/\sqrt{D})\,.$ (59)

One may then consider two limiting cases with either the dimension or the size
fixed. The former could be realized by known dualities: for superstring theory
in the critical dimension $D_{0}=10$, D-duality would act on the internal
radius, e.g. in the four-dimensional heterotic strings on $T^{6}$ it may act
like S-duality Font:1990gx  exchanging strings with wrapped NS-5 branes. The
latter much more intriguing but also exotic case would exchange quantum
gravity theories in different dimensions.

Finally, how the LDC is precisely related to the critical dimension in string
theory is not obvious at the moment. One possible way to relate the swampland
bounds on the number of space-time dimensions to the critical dimension in
string theory might be via recent arguments about entropy bounds and the
unitarity of scattering amplitudes Dvali:2020wqi .

Acknowledgements

We thank Gia Dvali, Eran Palti, Cumrun Vafa and Timo Weigand for very useful
discussions. Q.B. is supported by the Deutsche Forschungsgemeinschaft under
Germany’s Excellence Strategy EXC 2121 “Quantum Universe” - 390833306. L.C. is
supported by the ERC Advanced Grant High-Spin-Grav. The work of D.L. is
supported by the Origins Excellence Cluster. The work of S.L. is supported by
the ANR grant Black-dS-String ANR-16-CE31-0004-01, the ERC grant 772408
“String landscape” and the US National Science Foundation grant PHY-1915071.

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