#### The Weak Gravity Conjecture and the axionic black hole paradox

HJE
The Weak Gravity Conjecture and the axionic black
Arthur Hebecker 0 1
Pablo Soler 0 1
0 Philosophenweg 19, Heidelberg , D-69120 Germany
1 Institute for Theoretical Physics, University of Heidelberg
In theories with a perturbatively massless 2-form (dual to an axion), a paradox may arise in the process of black hole evaporation. Schwarzschild black holes can support a non-trivial Wilson-line-type eld, the integral of the 2-form around their horizon. After such an `axionic black hole' evaporates, the Wilson line must be supported by the corresponding 3-form eld strength in the region formerly occupied by the black hole. In the limit of small axion decay-constant f , the energy required for this eld con guration is too large. Thus, energy can not be conserved in the process of black hole evaporation. The natural resolution of this paradox is through the presence of light strings, which allow the black hole to shed" its axionic hair su ciently early. This gives rise to a new WeakGravity-type argument in the 2-form context: small coupling, in this case f , enforces the presence of light strings or a low cuto . We also discuss how this argument may be modi ed in situations where the weak coupling regime is achieved in the low-energy e ective theory through an appropriate gauging of a model with a vector eld and two 2-forms.
Black Holes; Gauge Symmetry; Global Symmetries
1 Introduction
2
nal moments of an axionic black hole
2.1 Immediate breakdown at critical radius
2.2
Slow evaporation and spread ux
3 Infrared divergences and quantum e ects 4 5
1
Conclusions
Introduction
3.1
3.2
Non-perturbative e ects
Quantum vs. classical
Systems with (aligned) multiple axions
Very roughly speaking, the Weak Gravity Conjecture (WGC) says that a U(1) gauge theory
with coupling g
1 can only be consistently coupled to quantum gravity if there are
charged states or even a cuto
at the scale gMP
MP [1]. This is expected to extend
to (p + 1)-form gauge theories with charged p-branes in any number of dimensions.1 The
simplest way to formulate the analogous statement about a low cuto is then to say that
p+1
g in Planck units (see [4{20] for a selection of recent related work).
Much of the recent phenomenological interest in the WGC derives from its potential
power to constrain axion in ation [21{37]: to this end, one views axion models as 0-form
gauge theories in the regime where the axionic coupling 1=f is small (i.e. f
MP ).
However, fundamental justi cations for the WGC both in the 1-form, the 0-form and
various other cases and di erent regimes are di cult to obtain. Thus, we believe that
supplying any arguments for or against it, even if in a slightly unusual setting, is important.
It is our aim to analyse possible arguments in favor of the WGC on the dual side of
the more familar axion = large- eld-in ation case mentioned above. Indeed, we dualise the
0-form ' to a 2-form, dB2 = f 2
d', and consider the action
Z
d4x p
g
1
f 2 jdB2j2 +
Z
worldsheet
B2 :
We want to understand whether we can have small f without light charged objects, i.e.
without light strings. In the extreme, the tension
of the lightest string might be
1Notice that our convention for designating generalized gauge theories di ers from that of, e.g. [2, 3].
MP (see [28] for constraints on such regimes in perturbative string
compacti cations). Is this inconsistent in any tangible way?
The standard magnetic WGC would have to argue about the smallest instanton not
yet being a `black hole' (in this case wormhole or gravitational instanton). This is clearly
questionable. The electric WGC would have to argue about the stability of extremal black
strings. But such macroscopic objects appear rather pathological (having a de cit angle
greater than 2 ) and, morevoer, possess no horizon.2
What is worse, both routes rest simply on assumptions, such as `stable extremal black holes (BHs) must not exist' (see however [18] for recent progress towards justifying this requirement rigorously). Here we suggest a completely di erent approach and, with certain caveats to be ex
the integral is over any sphere homotopic to their horizon. Such a `Wilson-loop variable' b
is locally unobservable and carries no energy (the eld strength H3 = dB2 vanishes), and
so the axionic BH behaves at the semi-classical level exactly as a Schwarzschild one.
A paradox arises in the evaporation process of such a black hole: by causality, the B2
boundary integral far away from the BH can not change. Once the BH is gone, spacetime
has become non-singular and the interior of this boundary must, according to Gauss' law,
contain a minimal H3 eld strength. The energy carried by this eld strength is too large
compared to the mass of the black hole just before the low-energy e ective theory has
R B2, where
broken down. Thus, energy can not be conserved.
In slightly more detail, the BH is expected to shrink via Hawking radiation, maintaining
the value of b, at least up to a radius r of the order of the Planck length. While the later
stages in the life of the BH cannot be properly described without a UV complete theory,
it is clear that the only way in which a complete evaporation can be consistent with a
non-zero b is through a `leftover' eld strength:
Z
V (r)
H3 =
Z
S2(r)
B2
b
O(1) :
(1.2)
e ective theory. Unless there is a remnant, we have a contradiction.3
Here the sphere is the boundary of the volume V (r), the latter being a ball of radius r at
the place where the BH used to be. But in the regime of small f , the prefactor 1=f 2 of
the eld energy stored in H3 is huge. It is in fact much larger than the available energy
O(MP ) of the smallest semiclassical BH which was still controlled in the low-energy
This contradiction is resolved if light strings are present in the spectrum of the theory.
Strings `lassoing' the black hole do interact with the B2 integral, and indeed generate an
e ective potential for the Wilson loop variable of the form V (b)
e 4 r2 cos b. This is
2One might try to overcome this by demanding instead that two same-charge microscopic strings should
always repel.
3Similar ideas have been used in [39, 40] in a somewhat di erent context to argue that certain types of
global symmetries (carried by skyrmions/baryons) can be preserved in BH evaporation.
{ 2 {
exp( 4
R2) and is negligible. However, once R
analogous to the familiar potential for the Wilson line variable H A1 in a standard (1-form)
gauge theory compacti cation from 5d to 4d. As long as the BH radius R is much larger
than the length scale set by the string tension, R
the e ective potential induced by a virtual string `lassoing' the BH has a suppression
1=p , the strings are irrelevant:
1=p , the e ective potential for
the `Wilson loop variable' b ceases to be exponentially suppressed. Now b is driven to zero
dynamically near the BH, the value of b at larger radii is supported by a non-vanishing
eld strength H3 near the BH, and the BH can eventually disappear without a trace. The
problem with the evaporation of axionic BHs is resolved if light strings exist.4
The rest of this paper is organized as follows. In section 2 we estimate the speci c
parametric bounds on the string tension that result from di erent evaporation rates of
small black holes. We spell out in section 3 several assumptions implicit in our arguments
and the physical e ects that motivate them. Finally, in section 4 we study a setup in which
a massless 2-form
eld with small e ective f arises via alignment, upon Higgsing a linear
combination of two B2 elds, in analogy to [14, 26]. Our arguments suggest the appearance
of light monopoles in such a theory.
The simultaneous work [41] also discusses generalized (global) symmetries and their
possible problems from a somewhat di erent perspective.
2
The
nal moments of an axionic black hole
As we have discussed in the Introduction, a contradiction arises in the process of
evaporation of axionic black holes in the absence of light strings.5 This implies a parametric
upper bound on the tension of strings (for a given gauge coupling f ). In order to derive
the parametric form of the bound, we need to make certain assumptions about the latest
stages of the evaporation of axionic BHs. We consider next two possibilities and derive the
parametric form of the constraints that arise in each case.
2.1
Immediate breakdown at critical radius
The simplest assumption, or at least the assumption leading to the simplest estimate, is
a catastrophic, explosion-like evaporation of the BH at the moment when it reaches the
tension of charged objects, i.e.
4The appearance of a potential due to lassoing strings is not undisputed [44]. Indeed, in a partition
function calculation in the scalar
eld basis, a single lassoing string has in nite action. First, this does
not preclude single-instanton contributions in the B2 basis. Second, even staying in the scalar
eld basis,
nothing speaks against contributions from even numbers of lassoing strings. This is analogous to calculating
the familiar instanton-induced cosine potential in quantum mechanics, but using a basis of xed discrete
momenta, such that only even numbers of instantons occur. The toy model calculation of [34] shows that a
potential can still be derived in this way. Moreover, if it still turned out that a potential in the strict sense
does not arise, tunneling processes corresponding to lassoing strings are certainly possible and can give rise
to a dynamical e ect on b. In any event we can argue, without directly referring to light strings, that the
dynamical activation of b kicks in at a scale R
1 given by the cuto
of the theory. A bound on such a
cuto , analogous to the magnetic rather than electric WGC, would then be obtained. Although we mostly
refer to light strings throughout this work, both points of view are related by associating the cuto to the
5There are several important caveats to this simple reasoning that must be considered to reach our
conclusions. In order to avoid obscuring our results, we postpone their discussion to later sections.
{ 3 {
critical radius Rc
1=p . This is not a totally unnatural expectation: according to
the arguments of [8, 12, 13], not just one but many string states should come in and an
extreme growth of the number of degrees of freedom with energy may indeed lead to an
instantaneous evaporation at a temperature Tc
1=Rc
p .
In this case, there is no time for any e ect arising from the dynamical strings to
propagate. The H3 eld they dynamically induce is limited to a ball of radius Rc, the total
resulting eld energy being
E
RcMp2 if energy is to be conserved in the nal stage
of the evaporation. This leads to a lower bound on the string tension
has an energy that scales with the coupling constant g as
1=g2. When both situations
are related by a dynamical change in the topology, energy conservation imposes a bound
MPd 2 & 1=g2. Of course, one needs to complete this inequality to make it dimensionally
consistent. If one can argue that there is only one other scale
Rc 1 involved in the
process, the WGC is recovered.
2.2
Slow evaporation and spread ux
While the result (2.2) is suggestive, the assumption of immediate BH evaporation at a
critical temperature is likely too naive. We may obtain a much more conservative bound if
we assume that, after the e ective potential is activated at a temperature Tc
p , the H3 ux induced has a time tev to spread out radially at the speed of light before the
1=Rc
BH completely evaporates. Assuming that in its latest stages the BH still radiates energy
at rate
dM=dt
zero scales as6 tev
of radius Rc + tev
accounts for the Wilson loop variable b after the BH disappears has spread out to a ball
Mp4=M 2, the evaporation time from the critical mass Mc
RcMp2 to
Mc3=Mp4
Mp2= 3=2
Rc. In this scenario, the left-over eld H3 that
tev
E
Mp2= 3=2. The energy stored in such a ux scales as
1 Z
f 2
V (tev)
d3x pg jH3j2
b
2
f 2te3v
9=2
f 2Mp6
:
(2.3)
6Given the assumptions we have to make about the latest stage of BH evaporation, our bounds on the
tension of strings can only be considered estimates. In fact, the numerical factors we are neglecting can be
very large. For example, the numerical factor in front of tev is 5120 .
{ 4 {
(2.1)
(2.2)
Arguing as before that this energy should be less than the mass of the axionic BH of radius
Rc, we obtain a bound on the string tension
latest stages of the evaporation, and the H3 ux had more time to spread over a broader
region. In fact, the bound would disappear entirely if the decay never reached an end (in
this case there would be a remnant which could by itself support a non-zero value of b). We
nd it reasonable to assume that the decay will not slow down. By contrast, we think it is
likely that the right bound derived this way will lie somewhere between (2.2) and (2.4).
3
Infrared divergences and quantum e ects
We have so far glossed over some key issues that need to be addressed before accepting the
conclusions of last section.
Non-perturbative e ects 3.1
A
rst question one should ask when considering axionic BHs is whether the variable
b = R B2 is measurable and has a physical meaning at all [43{45]. Since B2 is locally pure
gauge, it does not exist in a local sense. The way to measure it is via an Aharonov-Bohm
(AB) type experiment, where strings lassoing the BH acquire a phase proportional to b
which can then be manifested in an interference pattern.
A problem arises because the energy stored in an in nitely long string is logarithmically
divergent in the IR. Strings are hence con ning, and there is a limit in the maximum radius
Rmax
f 1 exp(MP2 =f 2) that a string loop can reach [2, 46, 47]. In order for the axionic
hair on a BH to be measurable, we have to make sure that the AB experiment can be
performed, i.e. that the BH's size is smaller than Rmax. Given the exponential dependence
of Rmax with respect to f (recall that we are interested in the f
MP limit), this condition
can be easily satis ed.
Another e ect that we have not yet considered is that induced by instantons. When
non-perturbative e ects are taken into account, there is no strictly massless propagating
degree of freedom (other than the graviton) in the theory. The instantons induce a mass
gap, albeit an exponentially suppressed one. Under certain conditions this can be described
in terms of a coupling of B2 to a non-dynamical 3-form (see [48{52]), but it is most easily
seen in the dual description where instantons generate a periodic potential for the axion.
Recall that, upon circumnavigating a string, the axion
eld shifts by a period
+ 2 . In the absence of a potential, this shift will be uniformly distributed around the
string, i.e.
, where
is the angle that parametrizes the winding trajectory. When
a potential V ( ) is present, however,
will tend to remain at its minimum for most of
the trajectory. In this case, the string will be the boundary of a domain wall where the
eld
jumps discretely. As before, this puts a limit in the maximum size Rmax of the
string loops one can consider. Beyond Rmax, strings loops are unstable due to nucleation
!
{ 5 {
HJEP09(217)36
of smaller string loops which eat up the axionic domain wall. Also as before, this e ect is
non-perturbatively suppressed (the axionic potential scales typically as e MP =f ) and can
be safely ignored for su ciently small strings.
One may ask, still, whether axionic BHs can even exist once instantons are taken
into account. One possibility would be that instanton e ects induced a non-perturbative
potiential to the Wilson-line variable b. In this case, a non-zero b could not spread all
the way to in nity, since this would carry an in nite potential energy, independently of
how small the instanton e ects are. However, one may simply consider a pair of axionic
BHs, with opposite value of R B2 so that the system still has a
nite energy. As long
as the non-perturbative potential induced is small enough, it will take an exponentially
long time for the system to relax to its stable con guration (by merging the two axionic
BHs). In this note we implicitly work in a regime of parameters where instanton e ects
are highly suppressed and do not play a role for the relatively small black holes that we
are considering.
It is nally also interesting to describe the B2 Wilson-line in the dual axionic picture.
Consider for simplicity a quantum mechanical model, obtained by compactifying a 4d
theory with a massless B2 eld on a T 3, parametrized by coordinates (x; y; z). We can turn on
a Wilson-line along any of the torus two-cycles, say Tx2y. The quantum mechanical
canonical momentum of Bxy is given by @tBxy. By Heisenberg's uncertainty principle, a state
with highly localized value of the Wilson line will correspond to a very broad superposition
of states with
= 4B2, such states correspond to
axion ux turned on along the one-cycle dual to Tx2y, i.e. our original state with Wilson-line
Bxy will be described as a superposition of states with di erent values of @z . The value
of the Wilson line corresponds to the phase, analogous to the -angle, introduced in this
superposition. In the case of an axionic BH, the two-cycle is the (homology class of the)
horizon, and its dual corresponds to the (non-compact) radial direction. Hence, in the
scalar language, a non-trivial value of the axionic hair b corresponds to the phase in the
superposition of states with non-trivial axionic gradient @r . Of course, once instanton
effects are taken into account and a non-perturbative potential for
is generated, the duality
between
and B2 becomes more involved.
3.2
Quantum vs. classical
It is essential to understand under which conditions the Wilson line variable b of a BH can
be thought of as a classical degree of freedom. To this end, consider the action
S
Z
d x
4 p
1
g f 2 jdB2j2
in the BH background
ds2 =
(1
R=r)dt2 + (1
R=r) 1dr2 + r2d 2
:
Assuming that B2 is proportional to the normalized harmonic 2-form on S2, parametrized
by b = b(t; r), gives
S
Z
dt
(1
{ 6 {
(3.1)
(3.2)
(3.3)
We see that the kinetic term diverges near the horizon. This agrees with the intuition that
any dynamics near the horizon should be extremely slow in the time variable suitable for
the observer at in nity. Furthermore, the gradient term goes to zero near the horizon. This
is again intuitive since there should be no suppression of con gurations where the values
of b very near the horizon and further away di er signi cantly. Indeed, the (e ectively
frozen) value of b near the horizon should not be able to in uence an observer at a certain
distance. We thus take it for granted that, in total, the e ect of the near-horizon region in
the above action will be as follows: the BH horizon does not x the value of b, in agreement
with the intuitive expectation from the no-hair theorem.
This can be modelled by excising a sphere of some radius & R (we do not care about
O(1) factors) and considering only the dynamics of b in the outer region. At our qualitative
level of analysis we can then also set (1
R=r) to unity and simply write
R
S
Z
dt
Here we took the lower limit of integration to be R for simplicity, although it should
of course be slightly larger as explained above. Clearly, an action like (3.4) requires a
boundary condition at r = R. We choose Neumann boundary conditions for consistency
with the classical shift symmetry of b and our expectation, argued above, that the near
horizon region does not break this invariance classically.7
Crucially, the r integral now converges, such the quantum mechanical model for the
zero mode (the r-independent mode of b) reads
(3.4)
(3.5)
(3.6)
(3.7)
S
Z
1
dt f 2R
In other words, the dynamics is the same that one would obtain from a compacti cation
to one dimension on a compact 3d space of typical radius R with one non-trivial two-cycle.
We are thus dealing with quantum mechanics of a variable with period 2
and a single
mass scale f 2R introduced through the kinetic term. Adopting textbook knowledge to our
setting, it is clear that this mass scale translates into a time-scale which governs the spread
of an optimally localized gaussian wave packet to the maximal width of 2 . On time scales
shorter than
tqm
1=(f 2R)
we can then think of our e ective potential, induced by lassoing strings, as of a classical
force acting on the classical variable b.
Now, in our `immediate breakdown' scenario, the typical time scale is tc
Rc. It is
simply the time a signal neads to travel across the relevant region of space. For our classical
analysis to be meaningful one should then require that this typical time is small enough:
tc < tqm
7A more careful modelling of the dynamics of b in the near horizon region would be interesting but is
not necessary for the point we want to make in this section.
{ 7 {
This comfortably contains the range of very high string tensions constrained by the WGC
and by our analysis.
For the `slow-evaporation' scenario, the much longer time scale of evaporation of a BH
with radius Rc is the relevant one. As we saw, this is tev
Mp2= 3=2. We now have to
impose that
tev < tqm
We evaluate the implications of this bound for our analysis by considering three possible
regimes: rst, if f 2=5MP8=5 . , our classical analysis is valid and hence the paradox derived
in section 2.2 should be taken seriously. Second, if f MP .
. f 2=5MP8=5, our classical
analysis is valid but no paradox arises according to section 2.2. The existence of this region
exempli es how, in the slow-evaporation scenario, our constraint on
is weaker than that
from the WGC. Third, if
. f MP our classical analysis is not trustworthy. However, we
were anyway not trying to make a new statement about this regime of low string tension,
consistent with the WGC. (It might, independently, be interesting to study axionic black
hole evaporation in this regime. But we do not need this for the present paper.)
Before closing this section we note that one might nd it counterintuitive that a
eldtheoretic classical analysis breaks down at large rather than at short times. This apparent
problem can be understood by thinking of our integrated
eld variable b of eq. (3.5) as
of a non-relativistic quantum mechanical particle. Let us denote the position by x and
the mass by m, with the appropriate translation in b, f and R easily derived. Now, if x
is non-compact and the initial state of the particle has a certain Gaussian spread
x, we
can always think classically by asking long-distance questions, i.e. questions for which the
spread in x is negligible (and the spread in the dual variable p is as usual ascribed to our
imprecise knowledge of the initial velocity). Long time scales are helpful since they allow
us to know the momentum and hence the position in future measurements rather precisely.
However, we can also consider (and in our case must consider) quantum mechanics
with a periodic x, say x
x + L. Now, to use the classical intuition of a particle on a
circle we need
x
L. But the width of an optimally localized Gaussian wave packet
grows according to
x(t)
t=(m
x(0)) at late times t (which of course follows simply
from the necessarily present momentum uncertainty). Requiring both
x(0) and
x(t) to
be smaller than L implies t . mL2, which is the analogue of our eq. (3.6) in this quantum
mechanical model. Thus, intuitively speaking, there does indeed exist a maximal time
scale beyond which it makes no sense to talk about a particle localized even approximately
somewhere on a circle.
4
Systems with (aligned) multiple axions
A recurring question when addressing quantum gravity constraints on axion decay
constants, is how these extend to winding trajectories [53] in the
eld space of axions. A
particularly simple case of such trajectories arises in the presence of two axions upon
Higgsing [48, 54] a linear combination of them [26]. It has been argued more generally in [14]
that such theories, even if they satisfy the WGC in their Coulomb phase, could e ectively
{ 8 {
SA;Ba =
Z
d x
4 p
g
we can de ne the variables
The Lagrangian (4.1) is invariant under the gauge transformations: fB1; B2; dAg !
N d 2g. For a spacetime with a non-trivial two-cycle ,
d 1
a
Under gauge transformations with R d i = 2 ci, these variables shift as fb1; b2; ag !
fb1 + c1; b2 + c2; a
c1
N c2g. These continuous shift symmetries are of course broken to
discrete periodicities by the presence of strings.
As mentioned before, the axionic hair can only be measured if there exist strings in
the spectrum charged under the Bi. We will assume the existence of strings
1 and
2
coupled to B1 and B2 with unit charge, i.e.
Sstr =
Z
Z
1
B1 +
2
B2 :
violate it (at least in some of its forms) in their Higgs phase. We would like to study next
how the physical setups and constraints described above behave under Higgsing in a system
with multiple B2 forms.
Let us consider a particular system of two 2-form gauge elds Bi, with i = 1; 2, coupled
to a single one-form
eld A in a Stueckelberg-like manner
S =
Z
d x
Here N is a (large) integer, Hi = dBi and, for simplicity, we have taken identical axion
decay constants f1 = f2
f . The coupling between A and Bi gives a mass to the linear
combination Ba / B1 + N B2. The orthogonal combination Bb /
(perturbatively) massless. It will feature the desired small coupling fe
N B1 + B2 remains
f =N and can
hence be used to construct our axionic BHs. We can rewrite (4.1) in terms of these elds as
HJEP09(217)36
This spectrum determines the periodicity of the \Wilson line" type variables bi. The path
integral measure includes eiSstr (in turn the phase measured by Aharonov-Bohm interference
experiments) which is only invariant under large gauge transformations with ci 2 Z. Hence,
the continuous shift symmetries are broken to discrete periodicities:
fb1; b2; ag ! fb1 + 1; b2; a
1g ;
and
fb1; b2; ag ! fb1; b2 + 1; a
N g : (4.5)
The normalization of Ba and Bb in (4.2) has been chosen so that strings still carry
integral charges, i.e.
Sstr =
Z
1
(Ba
Z
2
N Bb) +
(N Ba + Bb) :
(4.6)
{ 9 {
(4.1)
(4.2)
(4.3)
(4.4)
where ba and bb are de ned analogously to b1 and b2. We see that bb, the massless eld,
still has unit periodicity.
We have constructed an e ective theory of a single light two-form Bb with small
coupling fe . Following the arguments of previous sections, we would naively conclude that
the strings in this setup should be extremely light, otherwise an inconsistency would arise.
As we discuss next, however, there are extra ingredients in this theory that may allow for
a di erent way out.
Consider a monopole of the theory (4.2), i.e. a particle that sources
1 Z
2
S2
dA = a :
For concreteness, consider an (anti-) monopole of charge a =
1. Because of the Stueck
elberg coupling between the A and Ba, such a monopole carries a non-trivial value of ba.
From (4.2) one can see that minimum action con gurations satisfy dA !
r ! 1. This implies that, asymptotically, the magnetic monopole has
(1 + N 2)Ba as
A convenient basis for the space of large gauge transformations (4.5) is given by
fba; bb; ag ! fba; bb + 1; ag ;
and
fba; bb; ag !
ba
1
N
1 + N 2 ; bb +
1 + N 2 ; a + 1 ;
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
ba =
1
It looks as if the particle sources the `Wilson loop' variable associated with the heavy eld
Ba, but this is actually just a matter of gauge choice. From (4.7), we see that (4.9) can be
equivalently written as
Given that the full periodicity of bb is unity, we see that the anti-monopole carries an
axionic charge of about 1=N (of the maximum, which is of course the same as no axionic
charge). Similar objects have been constructed in [45].
This can be intuitively understood by looking at the (b1; b2) eld space. The monopole
charge (4.9) can be rewritten as
b1 =
1
gure 1, the point in (b1; b2) eld space given by (4.11) can actually
be viewed as a fractional value of the light Bb `Wilson line'.
Crucially, this implies that, from the perspective of the low energy e ective theory, a
BH with appropriate axion charge (roughly a multiple of 1=N , e.g. K=N with K
N=2),
can in principle decay. It would decay to K anti-monopoles of the 1-form theory which
was used to create the small axion decay constant fe ' f =N in the rst place. However,
this clearly requires the relevant monopoles to be light enough.
HJEP09(217)36
shown. The arrow denotes the point associated with an S2 loop around an anti-monopole.
is, m . MP =
p
N .
Let us consider an axionic BH with bb
1=2. If we denote the mass of the magnetic
monopoles by m, these will start being emitted once the BH reaches a radius Rm
1=m,
which corresponds to a BH of mass Mbh
MP2 =m. Since the BH is required to emit
N=2 of these monopoles before its decay, we should require that Mbh & N m=2, that
will be signi cantly lower than m . MP =
p
N .
This is still a conservative estimate, since in principle both monopoles and
antimonopoles will be radiated. In the absence of axionic hair, the net axionic charge carried
by these would be zero. The study of the dynamical e ect that the axionic hair of the BH
has in the process of (anti-)monopole emission, and how this leads to a discharge of the
axionic BH is a complicated subject that we leave for future work. However, we expect
that, once this e ect is taken into account, the actual upper bound on the monopole masses
To put our result in perspective, recall that in the un-Higgsed theory the existence of
strings with tension 1;2
2 . f MP su ces to satisfy the WGC and to avoid the possible
pathologies discussed in section 2. By contrast, the situation in the Higgs phase appears to
be more involved. Indeed, at low energies one encounters a theory with a single light eld
Bb, with an e ective coupling fe
expected cuto
e . p
fe MP
f =N , while the strings are now much heavier than the
pf MP =N . Our pathology of axionic BH evaporation
appears to occur. At the same time, a solution speci c to this type of construction suggests
p
itself: the possible lightness of monopoles of the gauged 1-form theory cures the problem
if m . MP =
N . Notice in particular that if f
MP , one concludes that these monopoles
have to reside precisely at the expected e ective cuto
scale m .
e
may be tempted to conclude that the mechanisms of alignment and Higgsing are insu cient
to generate an e ective theory the cuto of which evades the WGC: new low-scale physics
p
MP =
N . One
may be forced to enter in a di erent sector.
Clearly, other conclusions are also possible. First of all, it may simply be impossible to
construct the theory with A, B1 and B2 and the required large N in consistent quantum
gravity models, even though the WGC does not directly forbid such a situation.
Furthermore, it is conceivable that such models avoid our `axionic BH evaporation problem' in
some other way, not related to light strings or light monopoles. Nevertheless, the possible
way out through light monopoles looks intriguing.
5
Conclusions
In this note we have presented a novel argument that suggests a parametric bound on the
tension of strings in theories with 2-form
elds at weak coupling. While the exact form of
the bound depends on the precise rate of evaporation of small axionic black holes, under
certain assumptions we recover the constraints expected from the WGC. Our arguments
are not air-tight, and allow for ways out that do not involve light strings. If the evaporation
of black holes slowed down at late stages (with respect to the standard rate of Hawking
radiation), or if for some reason the semiclassical description of the evaporation became
invalid at a stage earlier than expected, our bounds would become weaker.
In this respect, one can ask whether paradigms such as the rewall hypothesis would
drastically a ect our results. While precise statements are hard to make, our preliminary
conclusion is that they would not.
As long as one can construct (young) black holes
with axionic charge at about the critical radius Rc, our results follow exclusively from
the requirement of black hole evaporation in a
nite time. Of course, if rewalls were
to signi cantly extend the axionic black hole lifetime, our results would change. If this
modi ed lifetime were known, our analysis could easily be adapted and yield a new
WGClike bound.
Another escape to our conclusions would be the existence of long-lived remnants, which
would by themselves resolve the apparent paradox we have discussed. The absence of
remnants has been extensively used to motivate the WGC as well as other `folk theorems'
of quantum gravity, such as the absence of global symmetries. The latter conjecture
is supported by convincing arguments against remnants that carry arbitrary conserved
charges [42], but these do not directly apply to remnants that merely support a
Wilsonline b (see nevertheless [41]). In this respect, our no-remnant (or equivalently topology
change) hypothesis is a stronger requirement. This is not too surprising, the WGC is a
much stronger statement than the simple no-global symmetry hypothesis, and is hence
much harder to prove. In fact, we do not know of fully convincing arguments for the WGC
based exclusively on semi-classical analysis (see however [55]).
We have further applied our arguments to a setup in which a light 2-form
eld with
small e ective decay constant fe arises upon Higgsing an `aligned' linear combination of
two 2-forms. Interestingly, rather than constraining the string spectrum, the evaporation of
axionic black holes suggests the appearance of light states in a di erent sector, namely light
monopoles which carry fractional amounts of axion hair. Given the prominent role played
by similar models in the paradigm of large eld (`natural') in ation, it would be interesting
to understand whether our conclusions can be con rmed by alternative considerations
and/or for objects of di erent dimensionality.
Concerning objects of di erent dimensionality, one encounters some obvious obstacles.
First, in four or less at space-time dimensions black holes are the only objects with horizon
and a corresponding Hawking evaporation process. So our constraint on 2-forms does not
generalize in any obvious way. One might be tempted to consider BTZ black holes in
d = 3 AdS and try to constrain the corresponding Wilson line and hence the 1-form gauge
theory, but also here our logic does not go through straightforwardly. Crucially, we can not
follow the evaporation of a macroscopic black hole all the way to empty space. Turning to
space-time dimensions above four, we can in principle consider black branes in
at space,
but then we would face the Gregory-La amme instability. Other higher dimensional black
objects such as black rings, whose horizons have richer topologies (S1
Sd 3), will also
be interesting to consider. We leave the study of how to adopt our logic to these cases to
future work.
Possibly, the right way to extend and generalize our argument is not via other black
objects but via space-time topology changes. Indeed, the main idea we have used is the
disappearance of an (e ectively) non-trivial 2-cycle of our space time - the 2-cycle around
the black hole horizon. Now, one might instead consider a geometry where two copies of
Rd are linked by a throat with cross-section Sp (p
d
2), i.e., a wormhole, possibly
bered over an appropriate space of dimension d
p
2. Dynamically, this wormhole
should collapse in pure gravity. But now, one may consider putting a non-zero p-form
`Wilson-line' on its non-trivial cycle and demand a consistent dynamical transition to the
trivial topology of two separate Rd spaces. It is conceivable that, following the logic of
section 2.1, one can recover the WGC bound. We leave a more detailed study of such
possibilities to future work.
Acknowledgments
We would like to thank William Cottrell and Miguel Montero for useful discussions. P.S.
would like to thank Gary Shiu and the Hong Kong Institute for Advanced Study for
hospitality during the completion of this work. This work was supported by the DFG
Transregional Collaborative Research Centre TRR 33 \The Dark Universe".
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
[2] T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev.
D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
[3] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP
02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
09 (2006) 049 [hep-th/0606277] [INSPIRE].
[4] T. Banks, M. Johnson and A. Shomer, A Note on Gauge Theories Coupled to Gravity, JHEP
[5] C. Cheung and G.N. Remmen, Naturalness and the Weak Gravity Conjecture, Phys. Rev.
Lett. 113 (2014) 051601 [arXiv:1402.2287] [INSPIRE].
[6] B. Bellazzini, C. Cheung and G.N. Remmen, Quantum Gravity Constraints from Unitarity
and Analyticity, Phys. Rev. D 93 (2016) 064076 [arXiv:1509.00851] [INSPIRE].
[7] Y. Nakayama and Y. Nomura, Weak gravity conjecture in the AdS/CFT correspondence,
Phys. Rev. D 92 (2015) 126006 [arXiv:1509.01647] [INSPIRE].
[8] B. Heidenreich, M. Reece and T. Rudelius, Sharpening the Weak Gravity Conjecture with
Dimensional Reduction, JHEP 02 (2016) 140 [arXiv:1509.06374] [INSPIRE].
[9] D. Harlow, Wormholes, Emergent Gauge Fields and the Weak Gravity Conjecture, JHEP 01
(2016) 122 [arXiv:1510.07911] [INSPIRE].
[10] L.E. Iban~ez, M. Montero, A. Uranga and I. Valenzuela, Relaxion Monodromy and the Weak
Gravity Conjecture, JHEP 04 (2016) 020 [arXiv:1512.00025] [INSPIRE].
[11] A. Hebecker, F. Rompineve and A. Westphal, Axion Monodromy and the Weak Gravity
Conjecture, JHEP 04 (2016) 157 [arXiv:1512.03768] [INSPIRE].
[12] B. Heidenreich, M. Reece and T. Rudelius, Evidence for a sublattice weak gravity conjecture,
JHEP 08 (2017) 025 [arXiv:1606.08437] [INSPIRE].
[13] M. Montero, G. Shiu and P. Soler, The Weak Gravity Conjecture in three dimensions, JHEP
10 (2016) 159 [arXiv:1606.08438] [INSPIRE].
025013 [arXiv:1608.06951] [INSPIRE].
[18] W. Cottrell, G. Shiu and P. Soler, Weak Gravity Conjecture and Extremal Black Holes,
arXiv:1611.06270 [INSPIRE].
[19] T. Banks, Note on a Paper by Ooguri and Vafa, arXiv:1611.08953 [INSPIRE].
[20] A. Hebecker, P. Henkenjohann and L.T. Witkowski, What is the Magnetic Weak Gravity
Conjecture for Axions?, Fortsch. Phys. 65 (2017) 1700011 [arXiv:1701.06553] [INSPIRE].
[21] A. de la Fuente, P. Saraswat and R. Sundrum, Natural In ation and Quantum Gravity,
Phys. Rev. Lett. 114 (2015) 151303 [arXiv:1412.3457] [INSPIRE].
[22] T. Rudelius, Constraints on Axion In ation from the Weak Gravity Conjecture, JCAP 09
(2015) 020 [arXiv:1503.00795] [INSPIRE].
[arXiv:1503.03886] [INSPIRE].
[23] M. Montero, A.M. Uranga and I. Valenzuela, Transplanckian axions!?, JHEP 08 (2015) 032
[24] J. Brown, W. Cottrell, G. Shiu and P. Soler, Fencing in the Swampland: Quantum Gravity
Constraints on Large Field In ation, JHEP 10 (2015) 023 [arXiv:1503.04783] [INSPIRE].
[25] T.C. Bachlechner, C. Long and L. McAllister, Planckian Axions and the Weak Gravity
Conjecture, JHEP 01 (2016) 091 [arXiv:1503.07853] [INSPIRE].
[26] A. Hebecker, P. Mangat, F. Rompineve and L.T. Witkowski, Winding out of the Swamp:
Evading the Weak Gravity Conjecture with F-term Winding In ation?, Phys. Lett. B 748
(2015) 455 [arXiv:1503.07912] [INSPIRE].
JHEP 02 (2016) 128 [arXiv:1504.03566] [INSPIRE].
[31] N. Kaloper, M. Kleban, A. Lawrence and M.S. Sloth, Large Field In ation and Gravitational
Entropy, Phys. Rev. D 93 (2016) 043510 [arXiv:1511.05119] [INSPIRE].
(2016) 653 [arXiv:1511.05560] [INSPIRE].
520 [arXiv:1511.07201] [INSPIRE].
JHEP 01 (2017) 088 [arXiv:1610.00010] [INSPIRE].
[36] A. Herraez and L.E. Iban~ez, An Axion-induced SM/MSSM Higgs Landscape and the Weak
Gravity Conjecture, JHEP 02 (2017) 109 [arXiv:1610.08836] [INSPIRE].
[37] M.J. Dolan, P. Draper, J. Kozaczuk and H. Patel, Transplanckian Censorship and Global
Cosmic Strings, JHEP 04 (2017) 133 [arXiv:1701.05572] [INSPIRE].
[38] M.J. Bowick, S.B. Giddings, J.A. Harvey, G.T. Horowitz and A. Strominger, Axionic Black
Holes and a Bohm-Aharonov E ect for Strings, Phys. Rev. Lett. 61 (1988) 2823 [INSPIRE].
[39] G. Dvali and A. Gu mann, Skyrmion Black Hole Hair: Conservation of Baryon Number by
Black Holes and Observable Manifestations, Nucl. Phys. B 913 (2016) 1001
[arXiv:1605.00543] [INSPIRE].
[40] G. Dvali and A. Gu mann, Aharonov-Bohm protection of black hole's baryon/skyrmion hair,
Phys. Lett. B 768 (2017) 274 [arXiv:1611.09370] [INSPIRE].
123 [arXiv:1702.06147] [INSPIRE].
[42] L. Susskind, Trouble for remnants, hep-th/9501106 [INSPIRE].
[43] J. Preskill and L.M. Krauss, Local Discrete Symmetry and Quantum Mechanical Hair, Nucl.
Phys. B 341 (1990) 50 [INSPIRE].
(1992) 175 [hep-th/9201059] [INSPIRE].
B 254 (1991) 355 [INSPIRE].
215 (1988) 67 [INSPIRE].
[44] S.R. Coleman, J. Preskill and F. Wilczek, Quantum hair on black holes, Nucl. Phys. B 378
[45] L.M. Krauss and S.-J. Rey, Duality, axion charge and quantum mechanical hair, Phys. Lett.
[46] A.G. Cohen and D.B. Kaplan, The Exact Metric About Global Cosmic Strings, Phys. Lett. B
[47] J. Polchinski, Open heterotic strings, JHEP 09 (2006) 082 [hep-th/0510033] [INSPIRE].
(2016) 113002 [arXiv:1602.03191] [INSPIRE].
HJEP09(217)36
[hep-ph/0409138] [INSPIRE].
(2009) 121301 [arXiv:0811.1989] [INSPIRE].
[arXiv:1705.06287] [INSPIRE].