JHEP01(2017)088
Published for SISSA by Springer
Received: October 19, 2016
Revised: January 14, 2017
Accepted: January 16, 2017
Published: January 20, 2017
Super-Planckian spatial field variations and quantum
gravity
Daniel Klaewer and Eran Palti
Institut f¨ur Theoretische Physik, Ruprecht-Karls-Universit¨at,
Philosophenweg 19, 69120 Heidelberg, Germany
E-mail: klaewer@thphys.uni-heidelberg.de,
palti@thphys.uni-heidelberg.de
Abstract: We study scenarios where a scalar field has a spatially varying vacuum expec-
tation value such that the total field variation is super-Planckian. We focus on the case
where the scalar field controls the coupling of a U(1) gauge field, which allows us to apply
the Weak Gravity Conjecture to such configurations. We show that this leads to evidence
for a conjectured property of quantum gravity that as a scalar field variation in field space
asymptotes to infinity there must exist an infinite tower of states whose mass decreases as
an exponential function of the scalar field variation. We determine the rate at which the
mass of the states reaches this exponential behaviour showing that it occurs quickly after
the field variation passes the Planck scale.
Keywords: Models of Quantum Gravity, Superstring Vacua, Black Holes, Effective field
theories
ArXiv ePrint: 1610.00010
Open Access,
c
The Authors.
Article funded by SCOAP
3
.
doi:10.1007/JHEP01(2017)088
JHEP01(2017)088
Contents
1 Introduction 1
2 A local weak gravity conjecture 3
3 Super-Planckian variations in weakly-curved backgrounds 7
4 Super-Planckian variations in strongly-curved backgrounds 13
5 Discussion 16
1 Introduction
There are a number of general expectations of quantum gravity. One of the most established
being that quantum gravity does not have global symmetries. Such properties can be
utilised as criteria for when an effective theory can be consistent with quantum gravity.
An effective theory that exhibits the required properties is sometimes termed to be in the
Landscape, while one which does not is termed to be in the Swampland [1]. One way of
approaching the question of whether an effective theory is in the Swampland is by coupling
it to gravity and looking at black hole solutions to this system. Then the consistency of
such solutions with expectations from quantum gravity can lead to constraints regarding
properties of the theory. An example of this methodology is the Weak Gravity Conjecture
(WGC) which states that in a theory with a U(1) gauge symmetry, with coupling constant
g, there must exist a charged particle with charge q and mass m
WGC
such that [2]
qgM
p
≥ m
WGC
. (1.1)
The arguments for the WGC are that if it is not satisfied then certain configurations in the
theory, in particular monopoles and black holes, would behave against expectations from
quantum gravity. The WGC has been the subject of intense studies recently, see [3–19]
for an incomplete list of the most recent work. Henceforth we will drop the charge q when
referring to the WGC, it can be reinstated easily should it be required.
In this paper we adopt a similar approach, we consider configurations in the effective
theory, coupled to gravity, where a scalar field in the theory adopts a spatially varying
vacuum expectation value (vev). We will demand that this spatial configuration be con-
sistent with expectations from quantum gravity and deduce from this constraints on the
theory. In particular, we are interested in the case where the total scalar field vev variation
is larger than the Planck scale.
The results of the work are naturally framed in the context of a conjectured property
of quantum gravity proposed as conjecture 2 in [20]. The conjecture was one of a number
– 1 –
JHEP01(2017)088
relating to the idea of the Swampland, but for ease of notation we will refer to it as the
Swampland Conjecture (SC). In [20] the SC was studied within a string theory context,
and the evidence presented for it was based on string theory. We will consider it as a more
general property of quantum gravity and present evidence for it not based on string theory.
Consider an arbitrary point in field space φ
0
, and displace a proper distance in field
space ∆φ. The SC states that there exists an infinite tower of states, with mass scale m
SC
,
which, compared to the theory at φ
0
, are lighter at φ
0
+ ∆φ by a factor of order e
−α
∆φ
M
p
.
Here α is a positive constant which is fixed by the choice of direction of displacement in
field space. As ∆φ → ∞ the tower of states becomes massless. We can write this as
m
SC
(φ
0
+ ∆φ) = m
SC
(φ
0
) Γ (φ
0
, ∆φ) e
−α
∆φ
M
p
. (1.2)
The function Γ (φ
0
, ∆φ) accounts for the statement that the mass variation is of order
the exponential. Our interpretation is that the SC is a statement about the asymptotic
behaviour of field space, while Γ accounts for the relatively unconstrained local structure
of field space. The conjecture that the tower of states becomes massless implies that as
∆φ → ∞ the magnitude of Γ should be sub-dominant to the exponential factor. The
quantitative behaviour of Γ for finite ∆φ is less clear, especially for arbitrary α.
To make the finite ∆φ behaviour more quantitative, let us denote as the Refined
Swampland Conjecture the statement that the mass of the tower of states quickly flows
to exponential behaviour for any ∆φ > M
p
. More precisely, that Γ (φ
0
, ∆φ) e
−α
∆φ
M
p
< 1,
and continues to decrease monotonically with an approximate minimal rate of e
−α
∆φ
M
p
, for
∆φ > O(1) M
p
. We will make the O(1) factor more precise, but the important point is that
the refined SC refers to the idea that the SC behaviour is tied to Planck scale field variations.
So not only a statement about asymptotic behaviour but also neither a statement about
sub-Planckian ∆φ. This is supported by evidence from string theory. A non-trivial example
was studied in [21] where variations in field space of so-called monodromy axions was
studied. It was found that the exponential behaviour of the SC did manifest, and was
reached quite rapidly for ∆φ > M
p
.
The refined SC and the WGC can be naturally related in a number of ways. The
WGC can also be written as a statement about axions and instantons [2, 5, 10, 13]. It
implies Sf ≤ qM
p
where here S is the action of instantons coupling to the axion, q is the
instanton number and f is the axion decay constant. Requiring the instanton expansion
to be under control, S > 1, leads to the statement that the axion period, and therefore
maximal variation, should be sub-Planckian. This result ties naturally to the refined SC
since the exponential in (1.2) is incompatible with the periodicity of an axion. Therefore
the field space can not change towards it for ∆φ > M
p
and the only way to respect it is
for the axion variation to be sub-Planckian. Another relation, pointed out in [21], is to
consider a supersymmetric setting with a saxion field u. Then the axion decay constant
maps to the field space metric for u, while u also controls the instanton action, so that we
have
√
g
uu
u ≤ M
p
. Therefore for u ≥ M
p
the proper (canonically normalised) field space
variation grows at best logarithmically with u. Schematically ∆φ ≤ ln u for u ≥ M
p
. This
– 2 –
JHEP01(2017)088
logarithmic behaviour is tied to the exponential behaviour of the SC as long as there is a
tower of states whose mass is controlled by u.
A particularly important relation for the purposes of this paper follows from the results
in [10, 16] which lead to a statement that the particle of the WGC is the first in an infinite
tower of states all of which satisfy the WGC. This was termed the (sub-)Lattice Weak
Gravity Conjecture and is a natural sharpening of the Completeness Conjecture [22, 23].
The statement that the gauge coupling g measures a mass scale of a tower of states rather
than the mass of a single state also matches its interpretation as a cut-off scale of the
theory.
1
It is natural to identify the tower of states of the Lattice WGC with that of the
SC. If we do this we can formulate a similar statement to the SC as a statement about the
field dependence of a U(1) gauge coupling. Specifically the coupling should have a field
dependence g (φ) such that
g (φ
0
+ ∆φ) = g (φ
0
) Γ (φ
0
, ∆φ) e
−α
∆φ
M
p
. (1.3)
The statement (1.3) will be the relevant one for the analysis in this paper and we will
therefore refer to it as the SC with the assumption of the relation to the WGC remai-
ning implicit.
The relation between the WGC and SC also extends naturally to the methodology of
this paper of studying spatial field variations. The WGC was motivated by considering
charged black holes and monopoles [2], but if the gauge coupling is scalar field dependent
then these objects induce a flow of the scalar field from a free value at infinity towards
the black hole horizon or monopole centre. Importantly such flows can range over super-
Planckian distances and therefore form testing grounds for the SC. Indeed, the connection
between the SC and spatial field flows is also naturally related in the context of the At-
tractor Mechanism of Black Holes (see [24] for a review). For extremal black holes the
proper distance to the horizon is infinite and this means that scalar fields flow to universal
behaviour near the horizon, independent of their values at spatial infinity. This is tied
to entropy properties of black holes. The horizon area depends on the scalar field values
and the attractor mechanism ensures that they are fixed solely in terms of the quantised
black hole charges. This is similar to the behaviour of the SC where a long flow in field
space leads to universal behaviour, independent of the initial point, which is tied to quan-
tum gravity physics. This relation between spatial distance and field distance will play an
important role in our analysis.
2 A local weak gravity conjecture
We are interested in spatial field variations and would like to utilise the WGC. The first
question is therefore how the WGC generalises for a spatially varying gauge coupling. For
our purposes it is useful to consider the so-called magnetic formulation of the WGC [2].
The conjecture follows from the statement that the minimally charged monopole should
1
While gM
p
sets the typical mass splitting of the tower, the lightest state can be of course much lighter
in the non-BPS case.
– 3 –
JHEP01(2017)088
not be a black hole. Consider a point monopole solution and associate to it a UV cut-off
radius r
Λ
. The monopole mass behaves as m
Mon
'
1
r
λ
g
2
, and therefore for the monopole
not to be a black hole we require
1
r
Λ
< gM
p
. We can rewrite this in terms of the energy
density ρ in the gauge field at r
Λ
as
gM
2
p
> ρ (r
Λ
)
1
2
. (2.1)
So the magnetic WGC argument states that at energy densities above gM
2
p
some QG
physics becomes relevant. This can be naturally interpreted in terms of the electric WGC.
The condition for a state of mass m
WGC
, which interacts only gravitationally, to be con-
sistently decoupled from an effective field theory is that the Hubble scale H of the the-
ory satisfies HM
p
' ρ
1
2
< m
WGC
M
p
< gM
2
p
.
2
This also fits naturally with the Lattice
WGC [10, 16] since the effective theory can not include an infinite tower of states. Note
that the interpretation leads to a stronger condition than just the electric and magnetic
WGC combined, since it imposes a constraint on the relative magnitude of the inequalities
in the two statements.
We would like to generalise the WGC to the case of spatially varying gauge cou-
pling. The expressions we will use are the natural local generalisations of the electric and
magnetic WGCs
g (r) M
p
≥ m
LWGC
(r) , (2.2)
g (r) M
2
p
> ρ (r)
1
2
. (2.3)
We term these the electric and magnetic Local Weak Gravity Conjectures (LWGC). Here we
restrict for simplicity to a spherically symmetric spatial configuration in which r denotes
the radial co-ordinate. m
LWGC
(r) denotes the energy scale associated to the (possible
tower of) states of the WGC evaluated at r. Perhaps a helpful way to think about the
statement (2.2) is to consider the tower of states as KK modes of an extra circle dimension
whose radius L (r) varies over space and is given by L (r) =
1
m
LWGC
(r)
.
The first motivation for the LWGC comes from thinking about the realisation of the
WGC in string theory. There the WGC amounts to an inequality between the magnitude
of four-dimensional fields (see, for example, [4, 5, 13, 15, 16, 25] for work on this). Typically
these parametrise the extra-dimensional geometry. Since the fields can vary over space, the
string theory setting naturally extends to a local statement. This is certainly manifest for
the supersymmetric case where the WGC inequality is saturated. For example, so-called
STU Black Hole solutions can be realised in string theory by considering type IIA string
theory on a six-torus. The spatially varying fields are moduli parametrising the size of the
tori. Dimensionally reducing the DBI action for wrapped branes on supersymmetric cycles
matches the expression, in terms of the four-dimensional fields, of the associated closed-
string gauge field coupling thereby realising the WGC equality locally. Once the local
2
The states of the WGC are also charged under the U(1) and so should interact not only gravitationally.
It would be interesting to better understand why this is still consistent with m
WGC
M
p
> ρ
1
2
rather than
m
WGC
> ρ
1
4
. One possiblity is that the WGC captures the constraint from gravitational physics only, while
there could also be other constraints from non-gravitational interactions.
– 4 –