$\textrm{d}s^{2}=f(r)\textrm{d}\tau^{2}+\frac{\textrm{d}r^{2}}{f(r)}+r^{2}\textrm{d}\vec{x}_{D-2}^{2},$
(6.2)

with

$f(r)\equiv
r^{2}-\frac{2\kappa^{2}\epsilon}{(D-2)r^{D-3}}+\frac{\kappa^{2}\rho^{2}}{(D-2)(D-3)r^{2D-6}}.$
(6.3)

Here $\epsilon$ and $\rho$ are the boundary energy and charge densities
respectively, $r_{+}$ is the largest positive zero of $f(r)$, and we have set
the AdS radius to one. If we work at fixed inverse temperature $\beta$ and
chemical potential $\mu$, then we have

$\displaystyle r_{+}$
$\displaystyle=\frac{2\pi}{(D-1)\beta}\left(1+\sqrt{1+\frac{(D-1)(D-3)^{2}\kappa^{2}\beta^{2}\mu^{2}}{4\pi^{2}(D-2)}}\right)$
$\displaystyle\rho$ $\displaystyle=(D-3)r_{+}^{D-3}\mu$
$\displaystyle\epsilon$
$\displaystyle=\frac{(D-2)r_{+}^{D-3}}{2\kappa^{2}}\left(r_{+}^{2}+\frac{(D-3)\kappa^{2}\mu^{2}}{D-2}\right).$
(6.4)

This black brane approaches extremality when $\kappa\beta\mu\gg 1$, with the
extremal radius being given by

$r_{+}|_{\beta=\infty}\equiv
r_{e}=\frac{(D-3)\kappa|\mu|}{\sqrt{(D-1)(D-2)}}.$ (6.5)

At extremality the function $f(r)$ has a double zero at $r=r_{e}$, so the
radial geodesic distance from $r_{e}$ to any large but finite radius $r_{c}$
is logarithmically divergent.

Figure 8: Exponential decay of spatial correlators for the extremal black hole
back brane.

The main point of [142] is that if we study the Hartle-Hawking state of two
such extremal black branes obtained by slicing the Euclidean path integral,
there is a tension between two facts which apparently follow from the bulk
picture together with the holographic correspondence:111111It has gradually
been understood that the semiclassical picture of the bulk needs to be used
with some care at very low temperatures, as quantum effects eventually become
important [143], [144], [145] , [146], [147], [148]. It would be worthwhile to
revisit this argument from the point of view of the modern understanding of
these quantum effects via a dimensional reduction to Jackiw-Teitelboim
gravity, as they could potentially change the conclusions, but we won’t
attempt it here.

  * •

Exponential correlators: the fixed-time correlators of boundary operators
decay exponentially with distance.

  * •

Volume-law entanglement: the union of a boundary subregion $A_{R}$ in the
right CFT and the same subregion $A_{L}$ in the left CFT has a von Neumann
entropy which grows like the volume of the subregion.

There are various ways to understand why the bulk picture implies these
results. One nice way is based on the idea that black hole horizons are
extremal surface barriers [149]. What this means is that an extremal surface
of any co-dimension greater than or equal to two cannot be smoothly deformed
from a surface which does not cross a horizon to a surface which does. The
reason is simple: if such a deformation were possible, then at some point the
surface would have to be tangent to the horizon, but then the extremality
equations would imply that the surface would be entirely contained in the
horizon. In particular for Euclidean states such as the Hartle-Hawking state,
we can approximate the two-point function of as massive field as

$\langle\mathcal{O}(x)\mathcal{O}(y)\rangle\sim\mathrm{e}^{-m|x-y|},$ (6.6)

where $|x-y|$ is the geodesic distance between $x$ and $y$. As shown in figure
8, the geodesic which is relevant for computing the correlator of two boundary
fields necessarily has a length which grows like the boundary distance between
the fields: the horizon at $r=r_{e}$ is an extremal surface barrier, so the
geodesic has no choice but to involve a large extensive component which lies
just outside the horizon. Moreover any geodesics which cross the horizon must
have infinite length, and therefore there is no correlation between boundary
operators on opposite sides.

Figure 9: Competing minimal surfaces to compute the von Neumann entropy of
$A_{L}\cup A_{R}$ in the thermofield double state. At nonzero temperature the
red “area-law” surface on the right always wins for large enough volume, but
at zero temperature its area is infinite so the blue “volume-law” surface on
the left wins for any region size.

The volume law entanglement can be understood along similar lines: we can
compute the von Neumann entropy of the dual CFT on $A_{L}\cup A_{R}$ using the
Ryu-Takayanagi formula, which tells us that it is given by the area of the
minimal-area surface which is homologous to $A_{L}\cup A_{R}$. There are two
possible candidates for the RT surface (see figure 9), only one of which gives
a volume law, and at finite temperature the “area-law surface” eventually wins
for big enough regions. At zero temperature however the “area-law surface”
always has infinite area due to the infinite distance to the horizon, and so
the “volume-law surface” always wins.

The reason that exponential decay of correlators and volume-law entanglement
are in tension is that the former suggests only short-range entanglement is
present, while the latter requires long-range entanglement. If the entropy of
$A_{R}\cup A_{L}$ is growing like the volume, then since the total state is
pure all this entanglement must be purified by something in the complementary
region. Such a purification is unlikely, given the exponential decay of
correlation with distance. This intuition has been formalized in $1+1$
dimensions into a precise theorem [150], and it quite plausibly holds in
general.

If the tension just described indeed constitutes a contradiction, then the
only way out is for the thermodynamic description of the extremal black brane
to break down. There are two ways this can happen. The first way is that there
could be an exactly BPS particle, which turns out to lead to power-law
correlators and thus removes the tension. This is the situation which is
realized for BPS branes in supersymmetric theories. The second possibility,
which has so far been manifested for all non-supersymmetric extremal branes,
is that there is some kind of matter present which causes the brane to be
unstable. Had we been discussing small black holes, such an instability would
have immediately required the existence of superextremal charged particle and
thus given a derivation of some version of the WGC. For large black holes
however there are more possibilities for the instability. Indeed this topic
has a long history in the literature on applications of AdS/CFT to condensed
matter physics, where the various possibilities go under the names
“holographic superconductor” [151], [152], [153] or “holographic Fermi
surface” [154], [155], [156] depending on whether the particle causing the
instability is a boson or a fermion. The rough idea for the bosonic case is as
follows: in the near-horizon region the gauge kinetic term

$-(\nabla^{\mu}\phi-inA^{\mu}\phi)^{\dagger}(\nabla_{\mu}\phi-
inA_{\mu}\phi)-m^{2}\phi^{\dagger}\phi$ (6.7)

for a boson of charge $ne$ leads to an effective mass

$m^{2}_{\mathrm{eff}}=m^{2}-\frac{n^{2}e^{2}}{\kappa^{2}}$ (6.8)

in the near-horizon $AdS_{2}$ region. This leads to an instability if
$m^{2}_{\mathrm{eff}}$ violates the $AdS_{2}$ Breitenlohner-Freedman bound
$m^{2}>-\frac{(D-1)(D-2)}{4}$, so in other words there is an instability for
masses in the range

$-\frac{(D-1)^{2}}{4}<m^{2}\leq\frac{n^{2}e^{2}}{\kappa^{2}}-\frac{(D-1)(D-2)}{4}.$
(6.9)

The first inequality here is the $D$-dimensional Breitenlohner-Freedman bound,
which is necessary for the vacuum to be stable. When $n\neq 0$ the first term
on the right-hand side of (6.9) gives something like the WGC inequality, as
first noticed in [157], but it is missing the factor of $\gamma$. And moreover
due to the second term it is possible to have an instability even if $n=0$, so
the brane can be unstable even if there are no charged particles at all! Thus
any argument which requires an instability only of large extremal black holes
in AdS is not sufficient to imply the validity of the WGC, although it is
certainly suggestive.

###  6.2 The WGC from thermodynamics

####  6.2.1 The WGC and quasinormal mode frequencies

An interesting argument linking the WGC to a bound on the imaginary part of
the frequencies of black hole quasinormal modes has been given in [158]. The
argument relies on a “Universal Relaxation Bound,” previously proposed in
[159]. To derive this bound, Hod begins with Bekenstein’s bound on information
transfer [160] (which itself derives from Bekenstein’s entropy bound [161])
and places an upper bound on the rate at which an observer can receive
information:

$\dot{I}\leq\frac{\pi E}{\log 2}\,.$ (6.10)

Here $I$ is the information and $E$ is the energy of the package containing
the information. Using the relations

$S=I\log 2\,,~{}~{}~{}\dot{S}=\frac{\Delta
S}{\Delta\tau}\,,~{}~{}~{}T=\frac{\Delta E}{\Delta S}\,$ (6.11)

one may rewrite the bound as

$\Delta\tau\geq\frac{1}{\pi T}\,,$ (6.12)

which Hod interprets as a bound on the time $\Delta\tau$ over which a system
can relax to equilibrium, deemed the “Universal Relaxation Bound.” Finally, he
applies this bound to the quasinormal modes of a black hole by setting $T$ to
be the temperature of the black hole and $\Delta\tau$ to be the inverse of the
smallest imaginary part of a quasinormal mode frequency,
$\Delta\tau=[\mathrm{Min}(\mathrm{Im}(\omega))]^{-1}$, arriving at the bound

$\mathrm{Min}[\mathrm{Im}(\omega)]\leq\pi T\,.$ (6.13)

In response, it should be noted that while the derivation of Hod’s bound
(6.12) follows rather straightforwardly from Bekenstein’s bound (6.10), its
interpretation as a universal bound on relaxation times is more suspect.
Bekenstein derived his bound by imagining a scenario in which one observer
sends a package full of information to another. It is not entirely clear how
this scenario can be translated into the case of interest at hand, in which a
black hole is perturbed and relaxes to equilibrium. A sharper derivation of
the proposed Universal Relaxation Bound, particularly in the context of
quasinormal mode frequencies, is clearly desirable. However, it should be
noted that Hod and others have given both numerical and analytical evidence in
favor of the bound (6.13) [159, 162, 163], and bounds similar to (6.12) but
without the precise $O(1)$ factors have been argued for in other contexts (see
e.g. [164] and references therein).

Assuming (6.13), Hod argued for the WGC as follows. None of the quasinormal
modes arising from gravitational and electromagnetic perturbations of a
nearly-extremal Reissner-Nordström black hole obey the bound. However, if we
assume that the bound merely requires that some mode in the black hole
background obeys (6.13), then the bound could be satisfied by a quasinormal
mode of a matter field. In particular, Hod showed analytically that, to
leading order in $T$ in the extremal limit, a charged scalar field has a
quasinormal mode which obeys the bound (6.13) precisely when the scalar
satisfies the WGC [158]. A similar conclusion was also obtained analytically
in the asymptotically AdS${}_{2}\times S^{2}$ near-extremal, near-horizon
limit [165]. These are intriguing results, which call for further study. Black
holes that are far from extremality have modes that comfortably satisfy the
bound. Near extremality, black hole quasinormal modes split into two families:
damped modes, which have $\mathrm{Im}(\omega)$ of order the inverse black hole
radius, and zero-damped modes (ZDMs), which have $\mathrm{Im}(\omega)\to 0$ as
$T\to 0$ [166]. Clearly, the bound (6.13) can only be obeyed by a ZDM. Much of
the literature on numerical computation of quasinormal modes focuses on the
damped modes, whereas the ZDMs are less well-studied [167, 166, 168, 169].
ZDMs for a Reissner-Nordström black hole have been found in the pure Einstein-
Maxwell theory [170], so the precise numerical coefficient in (6.13) is
important for the link to the WGC. Future work could check whether (6.13) is
equivalent to the WGC away from the $T\to 0$ limit. It would also be of
interest to explore this correspondence for more general black holes (e.g.,
dilatonic black holes). Strong numerical evidence for a WGC/quasinormal mode
connection would provide a motivation for further study of the quasinormal
mode relaxation bound itself.

The inequalities (6.12) and (6.13) bear a superficial resemblance to the well-
studied chaos bound [171], which requires a Lyapunov exponent $\lambda\leq
2\pi T$. However, the relaxation rate and the rate of growth of chaos are not,
in general, the same. For example, the Sachdev-Ye-Kitaev model saturates the
chaos bound, but its thermal 2-point function falls exponentially with
timescale $\tau=q/(2\pi T)$ [172], where $q$ is a positive even integer. This
is consistent with (6.12), but does not saturate the bound, except when
$q=2$.121212BH thanks Zachary Fisher and Ziqi Yan for discussions on this
point.

####  6.2.2 The WGC and entropy

As discussed in Section 2, one argument against continuous global symmetries
is based on the existence of finite-mass black hole states of arbitrarily
large global charge, leading to infinite entropy in a finite-size region, in
violation of entropy bounds in quantum gravity. This has motivated studies
relating the WGC to entropy bounds. For small but nonzero gauge coupling, a
WGC-violating theory can have a very large (but finite) number of stable
extremal black holes in a finite mass range. For example, it was suggested in
[173] that the uncertainty in a measurement of the charge of a black hole is
of order $1/e$, leading to an entropy scaling as $\log(1/e)$ and eventually
violating entropy bounds for sufficiently small $e$. However, it is unclear
why one would not be able to measure charge more precisely than $1/e$ (e.g.,
by measuring the motion of charged particles in the long-range electric field
outside the black hole), or why the Bekenstein-Hawking entropy should be the
relevant bound for an ensemble with such a large range of possible charges.
Furthermore, examples in which exactly stable BPS charged black holes exist
(with moduli spaces such that $e$ can be made arbitrarily small) illustrate
that the existence of many (marginally) stable species is not, in itself, in
contradiction with quantum gravity.

Subsequent studies have examined logarithmic corrections to black hole entropy
in the presence of WGC-violating matter [174, 175], [176]. These corrections
have interesting properties, but their computation has not led to an
undisputed proof of the WGC. In particular, [41] claims that the argument of
[176] relies on applying a formula outside its regime of validity.

###  6.3 The WGC from corrections to large black holes

The extremality bound for black holes is derived from the two-derivative
effective action. From the beginning, it was understood that the WGC could
potentially be satisfied by large black holes when higher-derivative
corrections to the effective action are taken into account [1]. Schematically,
these modify the extremality bound to take the form
$|Q|/M\geq(|Q|/M)|_{\mathrm{ext}}(1-c/Q^{2})$, where $c$ is a linear
combination of Wilson coefficients of four-derivative operators and
$(|Q|/M)|_{\mathrm{ext}}$ is the charge-to-mass ratio of asymptotically large
extremal black holes, which is computed with the two-derivative action. We
continue to define a superextremal state as one for which
$|Q|/M\geq(|Q|/M)|_{\mathrm{ext}}$, so that finite-size black holes are
superextremal when $c\geq 0$. The corrections to the extremality bound from
general four-derivative operators added to Einstein-Maxwell theory were
calculated in detail shortly afterward [177], and it was found that certain
black holes in heterotic string compactifications do, in fact, become
superextremal. Further such checks have been carried out in [178, 179].

Before discussing the technical details, it is useful to describe two quite
different interpretations that one might attach to the observation that small
corrections to large black holes can allow them to become superextremal. The
first is that this trivializes the Weak Gravity Conjecture. The WGC in its
most mild form merely requires that some state in the theory be superextremal.
If this state is a large black hole, then the WGC is simply a statement about
the signs of some higher-dimension operators in the effective action, and does
not imply the existence of any light charged particles below the Planck scale.
If a general positivity proof can be constructed for the linear combination of
operator coefficients appearing in the corrected extremality bound, the WGC
will follow, and as such will be reduced to a statement about gravitational
effective field theory. The second viewpoint is that the evidence that we have
for the WGC, as discussed in earlier sections of this review, favors the much
stronger tower/sublattice WGCs, involving an infinite tower of charged
particles of increasing charge and mass, all of which are superextremal. For
very large values of $|Q|$, the charged “single-particle states” simply are
black holes, and so the tower/sublattice WGCs require that they be
superextremal (as depicted in Figure 5). From this perspective, an EFT
argument could explain superextremality far out in the charge lattice, but the
tower/sublattice WGCs will also imply the existence of superextremal states at
smaller $Q$, where the states are no longer well-described as black holes in
EFT. Arguments in favor of superextremality from higher-derivative corrections
cannot decisively favor the former perspective (that EFT is everything) over
the latter (that Swampland constraints go beyond EFT). However, if we find
consistent theories of quantum gravity (not just EFTs) in which large black
holes are subextremal, this would immediately falsify the tower/sublattice
WGCs.

####  6.3.1 The corrected extremality bound

There are several possible four-derivative operators that may be added to the
Lagrangian of Einstein-Maxwell theory, built out of $R_{\mu\nu\rho\sigma}$ and
$F_{\mu\nu}$. For this discussion, we work in the normalization
$\frac{1}{2\kappa^{2}}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ for the two-
derivative Lagrangian. If we limit our attention to CP-conserving terms, there
are four independent physical four-derivative terms. Their contribution to the
effective action can be parametrized as

$\displaystyle
S_{4\partial}=\int\mathrm{d}^{D}x\sqrt{-g}\Big{(}c_{GB}O_{GB}+c_{RF}R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}+c_{T}T_{\mu\nu}T^{\mu\nu}+c_{F}(F_{\mu\nu}F^{\mu\nu})^{2}\Big{)},$
(6.14)

where $T_{\mu\nu}$ is the two-derivative Maxwell stress tensor
$F_{\mu\rho}F_{\nu}^{~{}\rho}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}$,
and
$O_{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$
is the Gauss-Bonnet term (in $D=4$, this is a topological term that does not
affect the extremality bound). This is the basis favored by the discussion in
[180]. All other four-derivative operators can be related to these four terms
(up to terms that are of higher order in the derivative expansion) via
equations of motion (or, equivalently, field redefinitions). For example,
terms involving the Maxwell stress tensor $T_{\mu\nu}$ can be traded for terms
involving $R_{\mu\nu}$ using the Einstein equations, while terms involving
$\nabla_{\mu}F_{\rho\sigma}$ or $\nabla^{\mu}F_{\mu\nu}$ vanish in the pure
Einstein-Maxwell theory without electric or magnetic sources.

The condition for extremal Reissner-Nordström black holes to become (strictly)
superextremal due to four-derivative terms is [177]

$\displaystyle(D-3)\left[(D-2)(Dc_{T}+16c_{F})+8(D-3)c_{RF}\kappa^{2}\right]-4(D-4)(3D-7)c_{GB}\kappa^{4}>0.$
(6.15)

For $D=4$ this simplifies to $c_{T}+4c_{F}+c_{RF}\kappa^{2}>0$.

As discussed in [181], another convenient basis related to familiar anomalies
is

$\displaystyle{\tilde{S}}_{4\partial}=\int\mathrm{d}^{D}x\sqrt{-g}$
$\displaystyle\Big{(}{\tilde{c}}_{W}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}+{\tilde{c}}_{RF}R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}+{\tilde{c}}_{GB}O_{GB}+{\tilde{c}}_{F}(F_{\mu\nu}F^{\mu\nu})^{2}\Big{)}.$
(6.16)

Here $W_{\mu\nu\rho\sigma}$ is the Weyl tensor. The relationship between the
bases (6.16) and (6.14) is

$\displaystyle{\tilde{c}}_{F}=c_{F}+c_{T}\frac{(D-4)^{2}}{16(D-1)},\quad$
$\displaystyle{\tilde{c}}_{W}=\frac{c_{T}}{\kappa^{4}}\frac{D-2}{4(D-3)},$
$\displaystyle{\tilde{c}}_{GB}=c_{GB}-\frac{c_{T}}{\kappa^{4}}\frac{D-2}{4(D-3)},\quad$
$\displaystyle{\tilde{c}}_{RF}=c_{RF}.$ (6.17)

A more detailed discussion of the field redefinitions that can convert between
operator bases may be found in Appendix B of [182].

Early work on this subject derived the condition (6.15) by directly solving
the modified equations of motion in the presence of higher-derivative
operators and extracting the corrected extremality bound from the perturbed
black hole solution. Recently, calculations have been greatly streamlined by
the discovery of elegant formulas relating the change in the extremality bound
to integrals evaluated on the uncorrected black hole solution. Specifically,
the shift of the charge-to-mass ratio $\zeta=\sqrt{2}M_{\mathrm{Pl}}Q/M$ of an
extremal 4d Reissner-Nordström black hole away from $1$ is given by

$\Delta\zeta=\frac{1}{M}\lim_{\zeta\to
1}\left.\int\textrm{d}^{3}x\,\sqrt{-g}\Delta{\cal L}\right|_{\text{RN}},$
(6.18)

where $\Delta{\cal L}$ consists of the higher-derivative corrections to the
leading order Lagrangian and $|_{\text{RN}}$ signals that it is to be evaluted
on the (two-derivative) Reissner-Nordström solution.

There are two key steps in deriving such an expression. First, the Wald
entropy of a black hole [183] is related, through standard thermodynamic
arguments, to the Euclidean action evaluated on the solution. This receives
corrections only from the corrections to the action evaluated on the
uncorrected solution, since the evaluation of the uncorrected action on
corrections to the solution vanishes at first order due to the extremality of
the uncorrected action at an uncorrected solution [182, 184]. Second, the
correction to the extremal charge-to-mass ratio is itself shown to be related
to the change in the black hole entropy [182]. This has been generalized to
rotating and dyonic black holes [185], dilatonic black holes [186], AdS black
holes [187], and dyonic Kaluza-Klein black holes [81]; see further discussion
in [180]. In fact, the extremality/entropy relationship has been proven by
[188] using very general thermodynamic considerations, which imply that when
there is a minimal mass for a given charge, $M>M_{\text{ext}}({\vec{Q}})$,
sensitive to a parameter $\epsilon$ (like the coefficient of a 4-derivative
operator),

$\frac{\partial
M_{\text{ext}}({\vec{Q}},\epsilon)}{\partial\epsilon}=\lim_{M\to
M_{\text{ext}}}\left[-T\left.\left(\frac{\partial
S(M,{\vec{Q}},\epsilon)}{\partial\epsilon}\right)\right|_{M,{\vec{Q}}}\right],$
(6.19)

even outside the black hole context. Recently, similar results have been
derived using the Iyer-Wald covariant phase space formalism [189] (also see
[80]).

These thermodynamic techniques have been adapted to study not only extremality
but long-range forces, to assess whether the Repulsive Force Conjecture is
satisfied by corrected black holes [190] (see also [191]). The results suggest
that the RFC may not be automatically satisfied by four-derivative
corrections. However, they are obtained in EFT examples, rather than explicit
string theory compactifications, so further work should investigate whether
these examples can be realized in a full quantum gravity setting (and hence
provide a counterexample to the RFC for corrected black holes).

####  6.3.2 Overview of arguments

The thermodynamic arguments sketched above have provided an efficient tool for
computing the correction to the extremal charge-to-mass ratio in a given EFT,
as a function of the Wilson coefficients of higher-dimension operators. Such
calculations lead to superextremality conditions that take the form of
positivity bounds like (6.15). A variety of attempts have been made to prove
such bounds from general principles.

In many cases, the general structure of a theory implies that the $c_{T}$ and
$c_{F}$ terms in (6.14), which involve only photons and not gravitons, give
the dominant corrections to the extremality bound. When these coefficients are
explicitly calculable, they are often positive. Indeed, in quantum field
theory (without gravity), one can prove rigorous positivity bounds on the
coefficients of four-derivative operators involving $F_{\mu\nu}$ [192, 193].
In a regime where the dominant contributions to $c_{T}$ and $c_{F}$ arise from
low-energy QFT effects which would persist in the $M_{\mathrm{Pl}}\to\infty$
limit, this is sufficient to prove (6.15), as already noted in [1].

Of course, we are interested in gravitational theories, where completely
general, rigorous positivity arguments are more elusive. In four dimensions,
loop effects involving gravitons can provide dominant contributions to four-
derivative operators in the IR, so that taking the $M_{\mathrm{Pl}}\to\infty$
limit obscures important physics. In general, when the WGC is not satisfied by
light charged particles, but is parametrically saturated (or even violated) by
all light particles, EFT proofs of (6.15) are more difficult to obtain. Below,
we will summarize three broad categories of arguments for positivity: those
that explicitly compute the coefficients in (6.14) within a given EFT; those
that rely on analyticity, unitarity, and/or causality; and those based on
entropy.

####  6.3.3 Explicit computations within low-energy EFTs

Figure 10: Example of a loop diagram leading to logarithmic running of a four-
derivative operator in 4d. Photons scatter via a loop of gravitons; due to two
couplings each scaling as $1/M_{\mathrm{Pl}}^{2}$, this gives rise to a
contribution schematically behaving as
$\frac{1}{M_{\mathrm{Pl}}^{4}}F_{\mu\nu}^{4}\log(E)$.

In four dimensions, the Wilson coefficients in (6.14) exhibit logarithmic
renormalization group evolution. This follows from dimensional analysis; for
example, $[c_{T}]=[c_{F}]=-D$ and $[\kappa^{4}]=4-2D$, which agree precisely
when $D=4$. An explicit example of a loop diagram contributing to such running
is shown in Figure 10. For exponentially large black holes, we expect the
Wilson coefficients (evaluated at a renormalization scale corresponding to the
black hole’s size) to be dominated by RG running. As a result, the sign of the
correction should be determined by such RG effects, independent of details of
the UV completion and the operator coefficients at the cutoff scale. The
consequences were first explored in [181], and more recently in [180]. The
case of multiple $U(1)$s has also been considered [194].

In the basis (6.16) there are logarithmic corrections to ${\tilde{c}}_{W}$ and
${\tilde{c}}_{GB}$, determined by the well-known Weyl anomaly coefficients $c$
and $a$, respectively [181]. The coefficients ${\tilde{c}}_{RF}$ and
${\tilde{c}}_{F}$ do not run. In the basis (6.16), $c_{T}$ and $c_{GB}$ run
[180]; in either basis, the running of $O_{GB}$ is irrelevant for the
extremality bound. In Einstein-Maxwell theory plus any minimally coupled
matter of spin $<3/2$, $c>0$, ensuring the validity of (6.15). Spin-$3/2$
fields contribute negatively to $c$, but a single spin-$3/2$ field is
insufficient to drive the running negative. However, nonminimal couplings,
such as dipole moments for fermions, also contribute negatively to $c$ for a
small range of Planck-suppressed couplings. In cases with ${\cal N}\geq 2$
supersymmetry where extremal black holes are BPS, these negative contributions
precisely cancel positive ones so that the black hole extremality bound
remains uncorrected. However, there are low-energy (non-supersymmetric)
effective Lagrangians with no obvious pathologies in which multiple fields
with finely-tuned nonminimal couplings could lead to a negative running for
$c$, and hence to large black holes that cannot satisfy the WGC.

If the tower or sublattice WGC is true, then the corrections to large black
holes must allow them to become superextremal. Thus, there must be a bound on
the number of fermionic fields with dipole couplings in the limited range
where the running of $c$ is negative; such theories would lie in the
Swampland. In pure QFT, negative coefficients of the $F^{4}$ operators would
violate causality; however, precisely because the negative contributions arise
from gravitational-strength interactions, there is no violation of causality
in the gravitational context [180].

Moving beyond log running in the deep IR, we can also consider the threshold
corrections induced by integrating out specific massive particles. Neutral
bosons coupling to $F_{\mu\nu}F^{\mu\nu}$ or $F_{\mu\nu}{\tilde{F}}^{\mu\nu}$,
exchanged at tree level, generate positive four-derivative operator
coefficients consistent with (6.15) [195]. Loops of charged particles with
sufficiently large charge-to-mass ratio (obeying the WGC themselves, by a safe
enough margin) also satisfy (6.15) [193]. The challenging case, then, is when
there are no light neutral bosons and all of the charged particles have
$m\gtrsim eM_{\mathrm{Pl}}$; then, gravitational-strength ultraviolet
contributions can be competitive, and the sign is not obviously determined.

####  6.3.4 Arguments from analyticity, unitarity, and/or causality

Figure 11: The contour integral for a dispersive proof of positivity of four-
derivative operators. Here, we illustrate the case of $2\to 2$ scattering of a
particle of mass $m$. The amplitude ${\cal A}(s,t=0)$ has poles at
$s=m^{2},3m^{2}$ and branch cuts at $s\leq 0,s\geq 4m^{2}$. The dashed contour
around the singularity inserted at $s_{0}=2m^{2}$ can be deformed to the solid
contour $\gamma$ surrounding subtractable pole contributions, positive branch
cut contributions, and a negligible contour at infinity.

In EFTs embedded within UV-complete quantum field theories, positivity bounds
on certain combinations of operator coefficients (or, more invariantly, on
derivatives of low-energy scattering amplitudes) may be proven using
analyticity, unitarity, and causality [196, 192, 197, 198, 199, 200]. A
prototypical example is the positivity of the $(\partial\phi)^{4}$ operator
coefficient in the theory of a massive scalar field, derived from a forward
dispersion relation. This term contributes an $s^{2}+t^{2}+u^{2}$ term in the
low-energy amplitude ${\cal A}(s,t)$ for $\phi\phi\to\phi\phi$ scattering. The
coefficient of this term can be read off from a second derivative, and in turn
related to a contour integral in the complexified $s$ plane by Cauchy’s
theorem:

$\displaystyle\frac{1}{2}{\cal A}^{\prime\prime}(s_{0},t=0)=\frac{1}{2\pi
i}\oint_{\gamma}\frac{{\cal
A}(s,0)}{(s-s_{0})^{3}}=\frac{1}{\pi}\int_{\mathrm{cuts}}\mathrm{d}s\,\frac{s\sigma_{\mathrm{tot}}(s)}{(s-s_{0})^{3}}>0.$
(6.20)

In the last step, the contour $\gamma$ around $s_{0}$ has been deformed to
enclose the $s$\- and $u$-channel branch cuts and two large arcs at large $s$,
as illustrated in Figure 11. The integrals along the branch cuts, in the $t\to
0$ limit, are related to positive total cross sections by the optical theorem.
The contour at infinity does not contribute, because the Froissart bound (in
conjunction with a Phragmén-Lindelöf theorem) constrains the large-$s$
amplitude to obey ${\cal A}(s,t=0)<s^{2}\log s$. This argument, given in
[192], can be extended to positive $t$ (below the branch cut) [197, 198]. A
version of the argument can also be derived in AdS using CFT crossing
relations [201]. Causality constraints, arising from superluminal propagation
in nontrivial field backgrounds, lead to similar conclusions [192]. Notice
that the positivity bound on the $(\partial\phi)^{4}$ coefficient is a strict
inequality, provided that $\phi$ is not free.

Corrected black holes satisfy the WGC if the inequality (6.15) holds. This
inequality involves four-derivative operators that contribute to scattering
amplitudes of photons and gravitons, so it is natural to seek a general
argument, similar to that for $(\partial\phi)^{4}$, that implies positivity
independent of the details of the UV completion. For example, in a theory of
only photons, the bounds derived from unitarity of forward scattering of
linearly polarized photons (of all possible polarizations) imply

$c_{T}\geq 0\quad\text{and}\quad Dc_{T}+16c_{F}\geq 0,$ (6.21)

in the notation of (6.14). However, the arguments immediately become more
difficult in a gravitational context. The relationship between superluminality
and causality is more subtle because lightcones are not rigid; cf. discussions
in [193, 202, 203, 204]. Unitarity arguments based on forward dispersion
relations face the difficulty that graviton exchange contributes a term
$\propto-G_{N}s^{2}/t$ to scattering amplitudes, rendering the $t\to 0$ limit
ill-defined. Furthermore, high-energy scattering in gravitational theories can
produce large black holes, so QFT bounds on the asymptotic UV behavior of
amplitudes do not necessarily hold. The fact that the $1/t$ graviton-exchange
pole scales as $s^{2}$ poses a particular problem for bounding four-derivative
operators. For example, as shown in [205], if one carries out a contour
integral to read off $O(s^{4})$ coefficients and then sends $t\to 0$, one
obtains candidate positivity constraints on operators involving four Riemann
tensors that are compatible with known string theory examples. On the other
hand, one cannot isolate the $O(s^{2})$ contributions from local operators
from those of graviton exchange in this way. Furthermore, if one simply
discards the $-G_{N}s^{2}/t$ term and follows the logic of the unitarity
bound, one would conclude that (in $D>4$, where it affects $2\to 2$ graviton
scattering) the coefficient of the Gauss-Bonnet term must be both $\geq 0$ and
$\leq 0$ [205]. Theories are known in which this coefficient is nonzero, so it
is clear that discarding the $t$-channel pole is not a strictly correct
procedure. A plausible interpretation of this result is that the coefficient
of the Gauss-Bonnet term cannot be too large with either sign, as further
argued in [206] on causality grounds.

In QFT, we can deform a theory by adding relevant operators, without changing
the UV behavior. This provides a method for addressing problematic IR
divergences. In quantum gravity, we do not have this luxury. Quantum gravity
theories are rigid: we cannot simply add terms to the Lagrangian without
modifying the entire theory. On the other hand, we can study a consistent
theory on different backgrounds. This motivated a novel argument aiming to
eliminate the problematic $s^{2}/t$ pole by compactifying to three spacetime
dimensions, where there is no propagating graviton mode [207]. A subtlety is
that, in resolving the IR problem of gravity, a new UV problem arises: 3d
flat-space gravity does not admit localized states of arbitrarily high mass,
because a massive particle has a deficit angle that eventually eats up the
entire space. In other words, the physics of the 3d theory resembles that of
the 4d theory over a range of high energies, but strongly deviates at truly
asymptotic energies. Thus, the meaning of ${\cal A}(s,t)$ becomes obscure in
high-energy regions, where it seems to be not even well-defined, much less
analytic. This was suggested by [208, 209] as a possible culprit behind their
observation that the $t$-channel subtracted positivity bounds derived from
compactification appear to be overly strong. They require new physics to
appear at prematurely small energies, in contradiction with known consistent
theories. These works, reinforcing similar arguments made earlier in [195]
(see also [210]), suggest that positivity arguments can forbid terms of the
form $-c^{2}\frac{s^{2}}{M^{4}}$ with $c\sim O(1)$ and $M$ held fixed in the
limit $M_{\mathrm{Pl}}\to\infty$, but not terms of the form
$-c^{2}\frac{s^{2}}{M^{2}M_{\mathrm{Pl}}^{2}}$, which tend to zero when
gravity is decoupled.

Given the subtleties associated with making completely general and rigorous
arguments in gravitational theories, much of the work on this subject has
focused attention on identifying a sufficient set of conditions to prove
(6.15). As discussed above, explicit computations show that tree-level
exchange of light bosons interacting with photons and loops of light charged
particles both produce corrections to $c_{T}$ and $c_{F}$ that satisfy (6.15).
In these cases, effects from $c_{RF}$ and $c_{GB}$ are subdominant. This is
often the case, as large contributions to $c_{RF}$ or $c_{GB}$ induce
causality violation, in the absence of a tower of high-spin states [206] (see
also [211, 212] for holographic, CFT-based arguments). The most difficult case
to assess is when all contributions to the four-derivative operators arise
from an ultraviolet scale, like the string scale. In this case, additional
assumptions have been invoked. If Regge states associated with the photon have
effects dominating over those associated with the graviton, (6.15) can again
be derived [195]. Similar arguments have been explored for dilatonic black
holes in [186]. Constraints from duality have also been shown to imply
positivity conditions [213, 214].

Recently, new positivity bounds have been derived [215] that, following [206],
avoid the $t$-channel pole problem by studying scattering at fixed impact
parameter, rather than fixed $t$. It remains to be seen whether such an
approach can offer a new perspective on the WGC. A crucial test of any
completely general future proof of a positivity bound is that it must be
compatible with exactly zero correction in the case of BPS black holes.

####  6.3.5 Arguments from entropy

As discussed in Sec. 6.3.1, recent work has shown that the higher-derivative
correction to the black hole extremality bound is related, in a very general
way, to the shift in the Wald entropy of the black hole due to higher-
derivative terms. This raises the intriguing prospect of proving (6.15) by
proving that such corrections to the entropy must be positive [182]. In
particular, an argument based on the Euclidean path integral for black holes
with positive specific heat (including Reissner-Nordström black holes of
sufficiently large charge) establishes that the correction to the Wald entropy
$\Delta S_{4\partial}$ from four-derivative operators is positive whenever the
correction $\Delta F_{4\partial}$ to the free energy of the black hole at
fixed temperature is negative. The restriction to positive specific heat
allows one to conclude that the classical solution minimizes (not just
extremizes) the Euclidean action. Under these conditions, one can show that
any four-derivative operators generated at tree level lead to $\Delta
S_{4\partial}>0$, which in turn implies that corrected black holes satisfy the
WGC. Extending these considerations to rotating dyonic black holes leads to a
range of inequalities generalizing (6.15) [185].

The assumptions in this argument are not universally valid, even for tree-
level exchange [195]. For example, a massive spin-2 field $h_{\mu\nu}$ with a
coupling $h^{\lambda}_{~{}\lambda}F_{\mu\nu}F^{\mu\nu}$ generates a negative
shift in the entropy. It evades the assumptions because the Euclidean action
is not a local minimum with respect to $h^{\lambda}_{~{}\lambda}$. Although
this example evades the entropy argument, it violates unitarity, and so cannot
be embedded in a consistent quantum gravity theory to provide a counterexample
to (6.15). This connection between unitarity and positive contributions to the
Wald entropy may hold more generally, and hint at an argument that could
extend beyond tree level [182]. Modular invariance is another supplementary
assumption that has been invoked to extend the range of validity of entropy
arguments for positivity [216].

##  7 Implications and Connections

###  7.1 Implications for phenomenology and cosmology

####  7.1.1 Direct application of the WGC

Neither the WGC nor the tower/sublattice WGCs have immediate novel
implications for the Standard Model of particle physics. The electromagnetic
coupling constant at low energies is $e=\sqrt{4\pi\alpha}\approx 0.30$, so the
electron satisifies the WGC by more than 20 orders of magnitude. Furthermore,
because $e$ is an order-one number, the tower of charged particles predicted
by the tower/sublattice WGCs could all have mass near or above the Planck
scale. If one applies the WGC to the nonabelian gauge groups of the Standard
Model (above the QCD scale or the electroweak scale, so that the gauge bosons
appear massive), then the gauge bosons themselves obey the WGC, and again a
WGC tower could consistently lie around the Planck scale because the coupling
constants are order one. Perhaps a more interesting statement is that the WGC
implies that a magnetic monopole should exist with a mass near the Planck
scale or below (assuming that the bound is not obeyed only by monopoles of
very large magnetic charge), but this is not a statement that is readily
falsifiable by any conceivable experiment at this time.

Interesting direct applications of the WGC, then, should be sought in new
gauge interactions beyond the Standard Model. These could be previously
undetected forces through which known particles interact, or hidden sector
interactions among particles that are so far unknown (or perhaps detected only
indirectly through their gravitational effects, in the form of dark matter).

Given the minimal Standard Model matter content (without right-handed
neutrinos), the theory can be extended with a single additional $U(1)$ gauge
interaction, coupling to the one of the differences of lepton numbers for
different generations: $L_{e}-L_{\mu}$, $L_{\mu}-L_{\tau}$, or
$L_{e}-L_{\tau}$. At most one of these symmetries can be consistently gauged,
due to mixed ’t Hooft anomalies [217, 218]. The case of $L_{\mu}-L_{\tau}$ is
of particular interest, as it can explain the nearly maximal mixing of muon
and tau neutrinos [219]. Any of these gauge symmetries must be spontaneously
broken. The regime of greatest phenomenological interest involves relatively
large gauge couplings, where the WGC has little power even assuming it applies
in the higgs phase.

A more compelling example is the Standard Model with Dirac neutrino masses,
which admits a different $U(1)$ extension, gauging the difference $B-L$ of
baryon and lepton number [220, 221]. In this case, the associated gauge field
could be exactly massless without contradicting experimental results, provided
that it is extraordinarily weakly coupled: $e_{B-L}\lesssim 10^{-24}$ [222,
223]. The combination of the Planck constraint on the sum of neutrino masses,
$\sum m_{\nu}<0.12\,\mathrm{eV}$ [224] with the values of the neutrino mass-
squared differences inferred from neutrino oscillations, $\Delta
m_{21}^{2}\ll|\Delta m_{31}^{2}|\approx 2.4\times 10^{-3}\,\mathrm{eV}^{2}$
[225], implies that the lightest neutrino has mass $\lesssim
0.03\,\mathrm{eV}$. Thus, the lightest neutrino will obey the WGC for $B-L$
provided that $e_{B-L}\gtrsim 9\times 10^{-30}$. This provides about five
orders of magnitude in allowed $B-L$ coupling in which the mild form of the
WGC would be satisfied. The tower/sublattice WGCs, however, provide a
significant constraint: an infinite tower of $(B-L)$-charged particles should
exist, beginning at masses of order
$e_{B-L}M_{\mathrm{Pl}}\lesssim\mathrm{keV}$ and extending up indefinitely.
This implies that, if $B-L$ is an unbroken gauge symmetry in our universe,
then billions of undetected particles that interact (albeit very weakly) with
ordinary matter exist below the TeV scale. Although this would be surprising,
it is not obviously ruled out by data; it would have phenomenology akin to the
large extra dimensions scenario [226]. A minimal WGC tower of $B-L$ charged
particles would suggest a breakdown of local quantum field theory at energies
$\lesssim e_{B-L}^{1/3}M_{\mathrm{Pl}}\lesssim 10^{10}\,\mathrm{GeV}$, a
scaling analogous to that of Kaluza-Klein theory. (However, because Standard
Model fermions carry $B-L$ charge and are not accompanied by low-mass
excitations of higher $B-L$ charge, we would not expect the $B-L$ gauge group
to literally arise as a Kaluza-Klein gauge field from a circle
compactification.) The tower/sublattice WGCs put the existence of a massless
$B-L$ gauge field in tension with conventional models of GUTs or of high-scale
inflation, which postulate local new physics at energy scales above
$10^{10}\,\mathrm{GeV}$, but is not ruled out by experimental data.

The WGC might also be applied to possible gauge forces in hidden sectors,
possibly related to the dark matter in our universe. One might expect that
forces weak enough to have significant WGC constraints would also be too weak
to have observable consequences. Somewhat surprisingly, it turns out that very
weak forces between dark matter particles can sometimes have observable
consequences in astrophysics or cosmology. Dark matter charged under a
massless abelian gauge field (or “dark photon”) has been considered as a
simple QFT with rich phenomenology [227, 228, 229]. Constraints on the
strength of such a coupling arise from evidence that dark matter is
approximately collisionless. However, even for very small couplings, there can
be collective dark plasma effects [228, 230]. These lead to density
fluctuations in the plasma on a time scale of order the inverse plasma
frequency, $\omega_{p}^{-1}\sim\frac{m_{d}}{e_{d}}\rho^{-1/2}$, where $m_{d}$
is the mass of an individual dark matter particle, $e_{d}$ is the dark photon
coupling, and $\rho$ is the mass density of dark matter (which is directly
inferred from observations). If we suppose that the dark matter particles
themselves obey the WGC for the dark $U(1)$, this can lead to interesting
consequences, as discussed in [231]. In this case, the dark WGC implies that
$\frac{m_{d}}{e_{d}}\lesssim M_{\mathrm{Pl}}$. Quantitative estimates show
that dark plasma fluctuations can then lead to shock waves developing on the
timescale of a merger of colliding galaxy clusters. Thus, it is conceivable
that observations of cluster mergers could reveal dynamical evidence of very
weak gauge forces between dark matter particles that approximately saturate
the WGC. If the dark matter particles are sufficiently light, then the
tower/sublattice WGCs could, in turn, imply important constraints on the UV
cutoff of physics in our universe. Dedicated work, including numerical
simulations, would be necessary to make more precise statements about
observable dark plasma effects.

The tower/sublattice WGCs can also have interesting implications for
nonabelian gauge groups in the dark sector. For example, dark matter charged
under such a gauge group can have distinctive cosmological signatures even for
quite weak couplings, because the dark gluons constitute a form of interacting
dark radiation [232]. The tower/sublattice WGC cutoff on such theories is at
most $g^{1/2}M_{\mathrm{Pl}}$ [43]. Thus, there can potentially be a tension
between cosmological observables associated with interacting dark radiation,
and theories of high-scale inflation.

Another topic of substantial recent phenomenological interest has been kinetic
mixing between a dark $U(1)$ and ordinary electromagnetism [233]. Such a
mixing can be generated by loops of particles that carry both kinds of $U(1)$
charge. The tower/sublattice WGCs imply the existence of such particles, and
hence suggests a minimum kinetic mixing, at least in the absence of gauged
charge conjugation symmetries that enforce an exact cancellation. The size of
kinetic mixing motivated by such an argument has recently been explored, and
compared to concrete string theory examples, in [234] and [235].

####  7.1.2 Bounding the electroweak hierarchy

A longstanding problem in particle physics is the electroweak hierarchy
problem: why is the electroweak energy scale ($v\approx 246\,\mathrm{GeV}$) so
many orders of magnitude below the Planck scale ($M_{\mathrm{Pl}}\approx
2.4\times 10^{18}\,\mathrm{GeV}$)? An enormous hierarchy between the Planck
scale and the masses of the electron, proton, and neutron is necessary in
order to have stable, large objects like stars and planets. However, in the
Standard Model, the electroweak hierarchy is large only in a tiny subset of
the UV parameter space, unlike the hierarchy between the Planck and QCD
scales, which is naturally exponentially large due to asymptotic freedom. This
has motivated a number of suggested extensions of the Standard Model in which
the electroweak hierarchy can naturally become large, ranging from scenarios
where electroweak breaking is triggered by dynamical supersymmetry breaking to
those where the Higgs boson is a composite particle of a strongly-interacting
sector. Traditionally, these models all share the feature that they relate the
electroweak hierarchy to a scale generated by dimensional transmutation, and
they predict new particles with masses near the electroweak scale. LHC
measurements have informed us that the Higgs boson appears to be approximately
elementary (i.e., it has Standard Model-like interactions with other fields),
and additional electroweak-scale particles have not yet been discovered. This
has motivated theorists to pursue novel explanations of the electroweak
hierarchy problem.

Because the WGC gives rise to an upper bound on particle masses, it is
tantalizing to wonder if it could produce an upper bound on the Higgs scale
$v$, thereby explaining why $v$ is so dramatically small compared to
$M_{\mathrm{Pl}}$. This idea was first discussed by [38], and further explored
by [71, 231]. The simplest, original version of the idea is to suppose that
$B-L$ is gauged and that neutrinos, which acquire mass only from electroweak
symmetry breaking, are the particles responsible for satisfying the WGC. The
(Dirac) neutrino mass must then obey
$m_{\nu}=y_{\nu}v/\sqrt{2}<\sqrt{2}e_{B-L}M_{\mathrm{Pl}}$. If we fix
$y_{\nu}$ to its Standard Model value ($\sim 10^{-12}$), and if we postulate a
$B-L$ gauge coupling $e_{B-L}\sim 10^{-28}$ (consistent with the experimental
limits), this inequality tells us that $v\lesssim 10^{-16}M_{\mathrm{Pl}}$,
and thus requires an electroweak hierarchy of the order that we observe in
nature.

While this offers an interesting perspective on how quantum gravity might
affect low-energy particle physics in surprising ways, several elements of
this argument are unsatisfactory. One is that it seeks to explain the origin
of a mysterious factor of order $10^{-16}$ in terms of another small number of
order $10^{-28}$, which is unexplained. This is viewed as progress because the
electroweak hierarchy is not robust against quantum corrections (the Higgs
mass acquires additive corrections of order $\frac{h^{2}}{16\pi^{2}}M^{2}$
when coupled to heavy fields of mass $M$ via interactions of size $h$, which
must be “tuned away” through cancellations against other contributions),
whereas the smallness of the gauge coupling $e_{B-L}$ is “technically natural”
(its corrections are all proportional to $e_{B-L}$ itself). Nonetheless, if
our goal is to understand the origin of small numbers in our theory of nature,
this at best shifts the problem to explaining the origin of the small number
$e_{B-L}$. One might hope that such a problem has a solution, for instance, in
terms of a dynamical mechanism of moduli stabilization. Nonetheless, this
shift of the hierarchy problem toward a problem of explaining an exponentially
tiny $e_{B-L}$ is in some tension with the spirit of the WGC itself. A very
small value of $e_{B-L}$ restores a global symmetry of the theory, and so
quantum gravity should resist attempts to generate exponentially tiny gauge
couplings. This suggests that perspectives rooted in a very literal
interpretation of technical naturalness may encounter obstacles in a quantum
gravity setting. A sharper version of this concern is that the magnetic WGC
tells us that $e_{B-L}M_{\mathrm{Pl}}$ serves as an ultraviolet cutoff on our
EFT. This is particularly problematic from the viewpoint of the
tower/sublattice WGCs, which postulate a tower of $(B-L)$-charged particles
appearing at this mass scale. If infinitely many particles in such a tower
obey the WGC, then it was unnecessary to require that the neutrinos obey the
WGC, destroying the link between a small $e_{B-L}$ and the Higgs scale $v$.

One refinement of the argument [71, 231] draws on the Repulsive Force
Conjecture in the presence of scalar fields [70], arguing that the bound
assumes the schematic form $m\leq\sqrt{g^{2}-\mu^{2}}M_{\mathrm{Pl}}$ where
$g$ is a gauge coupling and $\mu$ is a coupling to scalars. In cases where
$g^{2}\approx\mu^{2}$, this can be a much stronger bound than simply $m\leq
gM_{\mathrm{Pl}}$. This offers the opportunity to push the magnetic WGC scale
$gM_{\mathrm{Pl}}$ up to higher energies, where it has less effect on the
argument. On the other hand, it introduces yet another small number, the ratio
$\sqrt{g^{2}-\mu^{2}}/g$, which requires explanation. One must postulate a
specific form of the scalar couplings of the light, WGC-obeying matter fields
in order to make this argument. It should be different from the scalar
couplings of black holes; otherwise, the tower/sublattice WGC tower would
begin at the same scale, $\sqrt{g^{2}-\mu^{2}}M_{\mathrm{Pl}}$, rather than
$gM_{\mathrm{Pl}}$. This application of the RFC requires that the scalar
providing the additional force remain light, which itself requires explanation
and can lead to additional naturalness constraints on the EFT [231].

The most recent variations on the argument [231] explore new forces under
which no Standard Model particle is charged. One could, for example, consider
a scalar field $\Phi$ charged under a new $U(1)_{X}$ gauge interaction, which
we suppose should satisfy the WGC for $U(1)_{X}$. If we further posit that
$\Phi$ couples to the Higgs boson through a quartic coupling
$\kappa|\Phi|^{2}|h|^{2}$, then the additive shift of the $\Phi$ mass-squared
by $\frac{1}{2}\kappa v^{2}$ could cause $\Phi$ to fail to obey the WGC if $v$
is too large. Similar models can be constructed with fermionic fields. These
models make distinctive phenomenological predictions, relative to the original
$B-L$ model, and could have implications for dark matter dynamics.

These attempts to bound the electroweak scale $v$ using Weak Gravity arguments
all invoke a similar set of assumptions. We must assume the existence of very
small (but technically natural) couplings. We must also assume that specific
particles in the theory, which happen to interact with the Higgs boson, are
the ones that satisfy the WGC. If the WGC were satisfied by an independent set
of particles, not interacting with the Higgs, then the link to the electroweak
hierarchy would be severed. Finally, we must suppose that this restricted set
of theories is relevant for the world that we live in. If the landscape of
quantum gravity contains many universes resembling our own that do not contain
the postulated $U(1)_{B-L}$ or $U(1)_{X}$ force and the specific connections
assumed between these forces and the electroweak scale, then there is no
reason why we would expect our universe to obey the assumptions. The argument
that the WGC constrains the electroweak scale would only be plausible if such
vacua are overwhelmingly more common than others, or overwhelmingly more
likely to be populated by cosmology. In recent years, there has been a
proliferation of models that link cosmology to particle physics by postulating
the existence of a landscape that takes a very specific form, where for
instance certain couplings are assumed to exist and take on fixed values in
all vacua, and only a limited set of parameters “scan” from one vacuum to
another. These have been referred to as “artificial landscapes” [236, 237],
and in the absence of evidence that they resemble the true landscape of
quantum gravity, it is unclear what lessons one can draw from them.

Finally, it may be worth emphasizing that the examples in which we have checks
of the WGC are cases where we compute the mass at leading order in a
perturbative expansion, or where the mass is protected by supersymmetry. As a
result, we have no explicit examples in which the WGC is satisfied by a state
whose mass is fine-tuned to be light due to a cancellation. Indeed, such
examples would be extremely difficult to generate. If one could find such
examples in the string theory landscape, they would at least serve as an
interesting proof of principle that the WGC could require a fine-tuning that
would appear accidental from the viewpoint of low-energy effective field
theory.

####  7.1.3 Other applications to the hierarchy problem

In [238], a dynamical mechanism known as “cosmological relaxation” was
proposed as a solution to the hierarchy problem. In this scenario, the Higgs
field $h$ is coupled to a real scalar field $\phi$ through a potential of the
form

$V=(-M^{2}+g\phi)|h|^{2}+\left(gM^{2}\phi+g^{2}\phi^{2}+\cdots\right)+\Lambda^{4}\cos(\phi/f)\,,$
(7.1)

where $M$ is the cutoff of the effective field theory and $\Lambda$ depends on
the vev of $h$. Initially, the dynamics of $\phi$ are dominated by the
polynomial terms, and the cosine term is negligible. When $\phi\sim M^{2}/g$,
however, the Higgs field acquires a vev, and the scale $\Lambda$ for the
cosine terms grows, creating a barrier which stabilizes the axion and leaves
the Higgs with a mass well below the EFT cutoff $M$. In order for this
mechanism to work, however, the cosine terms must eventually be able to
compete with the $gM^{2}\phi$ term. In typical relaxation scenarios, this
requires $g$ to be roughly of order $10^{-34}$. Furthermore, inflation must
last long enough for $\phi$ to scan the entire range of the Higgs mass. This
places an additional bound on the cutoff given by $M\lesssim(\Lambda
M_{\textrm{Pl}})^{1/2}$, which yields $M\lesssim 10^{9}$ GeV for
$\Lambda=\Lambda_{\text{QCD}}$.

The tiny coupling $g\sim 10^{-34}$ is “technically natural,” but this does not
necessarily mean that the model can be UV-completed. In particular as we take
$g\to 0$ the theory (7.1) has an exact global symmetry
$\phi^{\prime}=\phi+2\pi f$, and if the arguments against exact global
symmetries have any robustness then they should also rule out sufficiently
small values of $g$. One possible approach to avoiding this problem is to view
$\phi^{\prime}=\phi+2\pi f$ as a gauge symmetry, or in other words to turn
$\phi$ into an axion. This however forbids most of the terms in (7.1) (a small
explicit violation of a gauge symmetry is just as bad as a large one), and
thus kills the feasibility of the model.

So far the most promising proposal for obtaining a large scalar field
excursion that is consistent with all versions of the WGC is the “axion
monodromy” proposal of [103, 104].131313This was originally proposed as a
model of inflation, as we will discuss later in this section, but it can also
be used as a mechanism to implement cosmological relaxation as we discuss
here. The most basic version of this proposal [239] uses an axion coupled to a
3-form gauge field $A_{3}$ via the Lagrangian (which we here write as a
$4$-form)

$L=-\frac{1}{2}\textrm{d}\phi\wedge\star\textrm{d}\phi-\frac{1}{2e_{3}^{2}}F_{4}\wedge\star
F_{4}+\frac{g}{e_{3}}\phi F_{4}\,.$ (7.2)

with $F_{4}=\textrm{d}A_{3}$. Naively one might think that the coupling $g$
should be zero to respect the axion periodicity $\phi\sim\phi+2\pi f$, but as
is usual for Chern-Simons type interactions the fact that the integral of
$F_{4}$ obeys the quantization

$\int F_{4}=2\pi m,\qquad m\in\mathbb{Z}$ (7.3)

means that it is enough that we have

$g=\frac{ke_{3}}{2\pi f},\qquad k\in\mathbb{Z}.$ (7.4)

We have normalized $g$ here so that it matches the $g$ in (7.1), so we now
have two ways to get a small $g$: either we can take $f$ large in Planck units
or we can take $e_{3}$ small. To avoid trouble with the axion WGC (3.24) we do
not want to take $f$ large in Planck units (we’ll discuss this more in Section
7.1.5), so our task is to understand how constrained we are by the WGC for the
three-form gauge field $A_{3}$. Before discussing that however it is perhaps
worth explaining in more detail how the theory (7.2) allows for a super-
Planckian field excursion. The equations of motion following from (7.2) are

$\displaystyle\star\textrm{d}\star\textrm{d}\phi+\frac{g}{e_{3}}\star
F_{4}=0\,,~{}~{}~{}~{}~{}\textrm{d}(ge_{3}\phi-\star F_{4})=0,$ (7.5)

so the quantity

$\widetilde{F}_{0}\equiv\star F_{4}-ge_{3}\phi$ (7.6)

is constant. In fact it is quantized: the integral of $A_{3}$ over space is a
periodic variable, and $\widetilde{F}_{0}$ is proportional to its canonical
conjugate. Working this out gives the quantization

$\frac{1}{e_{3}^{2}}\widetilde{F}_{0}=n,$ (7.7)

with $n\in\mathbb{Z}$. Substituting this back into the first equation of
motion we find

$\star\textrm{d}\star\textrm{d}\phi+g(e_{3}n+g\phi)=0,$ (7.8)

which is the equation of motion for a scalar field with a potential that is a
second order polynomial just as in (7.1). This equation may appear not to be
gauge-invariant, but it actually is since $n$ is a dynamical variable and the
relevant gauge transformation is

$\displaystyle\phi^{\prime}=\phi+2\pi f\,,~{}~{}~{}~{}~{}n^{\prime}=n-k.$
(7.9)

Here $k$ is the integer defined by (7.4). One way to think about the apparent
non-periodicity of the potential is to observe that although $\phi$ is
periodic, $\star F_{4}$, which is not periodic, is also rolling in order to
ensure that $\widetilde{F}_{0}$ is constant. Indeed one can say that $\star
F_{4}$ is really the gauge-invariant scalar which is rolling in axion
monodromy.

We now turn to applying the WGC for 3-form gauge fields to axion monodromy
[240]. The objects to which it applies are domain walls of tension $T_{3}$,
across which $\frac{1}{e_{3}^{2}}\widetilde{F}_{0}$ changes by an integer $Q$,
and the WGC says there should be such domain walls with

$T_{3}\leq\frac{4\pi fgM_{\mathrm{Pl}}Q}{k}\,.$ (7.10)

The danger here is that an upper bound on the domain wall tension also likely
gives some sort of lower bound on the rate for bubbles bounded by the domain
wall to nucleate, and if this happens too often it destroys the relaxation
mechanism. The domain walls separate regions whose potential energy differs by

$\Delta V\sim ge_{3}\phi\sim fg^{2}\phi/k\,.$ (7.11)

The bounce action computed by [240] is not accurately described by the thin-
wall approximation, but involves important gravitational backreaction [241].
The result is a bubble nucleation probability

$P\sim\exp(-B)\,,~{}~{}~{}~{}B\approx w(b)\frac{2\pi^{2}T_{3}}{H^{3}}\,,$
(7.12)

where $H$ is the Hubble scale during inflation. The parameter $b$ is defined
as

$b=\frac{\Delta V}{HT_{3}},$ (7.13)

and it turns out that in the parameter range of interest, $w(b)\sim O(1)$ and
$b\lesssim 1$. Using $b$, we can rewrite the bounce action estimate as:

$B\sim 2\pi^{2}\frac{T_{3}^{4}b^{3}}{(\Delta V)^{3}}.$ (7.14)

The WGC provides a constraint, following [240], because we require $B\gg 1$
for an exponentially suppressed tunneling probability, but the WGC implies
that $B<B_{\mathrm{max}}$ where $B_{\mathrm{max}}$ is obtained when $T_{3}$
saturates (7.10). These can only be consistent when $B_{\mathrm{max}}\gg 1$.
Together with the estimates $\phi\sim M^{2}/g$ and $gM^{2}\sim\Lambda^{4}/f$
required for consistent relaxion phenomenology, this inequality translates
into a bound on the EFT scale:

$M\lesssim\left(4\pi^{2}b^{3}\right)^{1/8}\sqrt{\Lambda M_{\text{Pl}}}\,.$
(7.15)

For $\Lambda=\Lambda_{\text{QCD}}$, this bound becomes

$M\lesssim b^{3/8}\times 2.5\times 10^{9}\text{ GeV},$ (7.16)

which for $b\sim 1$ rivals the bound for consistency of the relaxion model
discussed previously.

Thus, the three-form WGC provides an interesting constraint on cosmological
relaxation implemented via axion monodromy. On the other hand a similar
constraint may also be derived independently of the WGC, and in the original
paper it was shown that this constraint could be satisfied without spoiling
the model. Therefore neither the axion WGC nor the three-form WGC seem to pose
a fatal challenge to the axion monodromy version of the cosmological
relaxation model. It is worth mentioning, however, that embedding the model
into string theory nonetheless seems to be very challenging, if not
impossible. In particular, [242] argued that within a string compactification,
the huge winding number of the relaxion corresponds to a huge charge carried
by branes or fluxes (this is already apparent in the model (7.2) since $\star
F_{4}$ is rolling). This charge backreacts on the compactification geometry
and eventually spoils the relaxation mechanism. The relaxion scenario may lie
in the Swampland, but if so then the most stringent top-down constraints do
not come from the WGC.

####  7.1.4 Mass of the photon or dark photons

Conventionally, we assume that the photon is a massless gauge field. However,
theories of massive, abelian spin-1 particles are perfectly consistent, either
with a simple mass term and no gauge invariance at all [243] or with a real
scalar field added to provide the longitudinal mode, together with a gauge
invariance to eliminate the redundant degree of freedom [244]. If the photon
has a very small mass, the longitudinal mode is extremely weakly coupled, so
it is difficult to experimentally distinguish from a massless photon despite
the change from two to three independent propagating polarization states
[245]. Of course, the photon in our universe must be extremely light. There is
a large literature on experimental constraints on the mass, to which a few
interesting entry points are [246, 247, 248].

In four dimensions a massive photon in the Stueckelberg regime can be
described by BF theory: we have a 1-form gauge field $A$ with field strength
$F=\textrm{d}A$ and a 2-form gauge field $B$ with field strength
$H=\textrm{d}B$, interacting via the Lagrangian

$S=\int\left(-\frac{1}{2f^{2}}H\wedge\star H-\frac{1}{2e^{2}}F\wedge\star
F+\frac{k}{2\pi}B\wedge F\right),$ (7.17)

where $k\in\mathbb{Z}$ just as in the axion monodromy discussion of the
previous subsection. The gauge coupling $f$ of the 2-form field has mass
dimension 1. This theory describes a massive gauge field with mass

$m=\frac{k}{2\pi}ef.$ (7.18)

For $k\neq 0$, taking the gauge field mass to zero requires either $e\to 0$ or
$f\to 0$. In either case, we are taking a gauge coupling to zero, and so the
WGC imposes some constraint. In particular, if we send $e\to 0$, the magnetic
WGC tells us that there is a UV cutoff on the theory at the scale
$eM_{\mathrm{Pl}}$, and the tower/sublattice WGCs suggest that there effective
field theory breaks down irrevocably at some scale, possibly a higher one like
$e^{1/3}M_{\mathrm{Pl}}$ (as in Kaluza-Klein theory). If we send $f\to 0$, the
mild WGC for the 2-form gauge field $B$ implies that strings charged under $B$
should exist with tension $T\lesssim fM_{\mathrm{Pl}}$.

In the case where the photon mass arises from the Higgs mechanism, there is no
fundamental obstruction to sending $f\to 0$: this corresponds to turning off
the Higgs vev, which can be accomplished just by giving the Higgs a positive
mass term around the origin. In this case, the $B$-field may be thought of as
an emergent gauge field in the IR below the scale of the Higgs vev, and the
charged strings predicted by the 2-form WGC are simply ANO strings [249, 250].
In the core of an ANO string, the Higgs vev is zero; in the limit that the
Higgs vev is taken to zero, an ANO string simply becomes more and more diffuse
and fades away. The WGC, then, is compatible with small masses arising from
the Higgs mechanism.

By contrast, there are massive gauge theories which are fundamentally of
Stueckelberg type. In this case, the strings charged under the $B$ field are
fundamental (e.g., the F-strings or D-strings of string theory). The core of
the string is not well-described by effective field theory, and there is no
finite-distance point in field space at which the gauge boson mass can be sent
to zero. In this case, the limit $f\to 0$ corresponds to a theory of
fundamental, tensionless strings, signaling a complete breakdown of local
effective field theory. In such a case, the fundamental quantum gravity cutoff
energy is bounded, $\Lambda_{\mathrm{QG}}\lesssim\sqrt{2\pi
fM_{\mathrm{Pl}}}$.

The WGC, then, imposes an ultraviolet cutoff on theories of a massive gauge
boson with mass arising from a fundamental Stueckelberg term [52]. This can
also be understood as a consequence of the Swampland Distance Conjecture: for
mass terms of fundamental Stueckelberg type, the $m\to 0$ limit is an infinite
distance limit, and so an infinite tower of light states appears when one
approaches this limit.

This constraint on massive, abelian gauge bosons has potentially important
implications for the Standard Model photon and for potential dark photons
[52]. First, consider the Standard Model photon. A conservative bound,
obtained from the arrival time of different frequencies from Fast Radio
Bursts, is that $m_{\gamma}\lesssim 10^{-14}~{}\mathrm{eV}$ [248]. Stronger
bounds exist, but involve more assumptions, so we will work with this very
simple kinematic bound; our conclusions can be readily adapted to other
constraints. If we assume that electromagnetic charge is quantized in the
usual way, the only way to obtain a small Standard Model photon mass of
fundamental Stueckelberg type is by taking $f$ to be very small:
$ef/(2\pi)\lesssim 10^{-14}~{}\mathrm{eV}$ requires $f\lesssim 10^{-22}~{}{\rm
GeV}$. But then the Weak Gravity Conjecture would require fundamental strings
with tension $\lesssim fM_{\mathrm{Pl}}\lesssim(20~{}\mathrm{MeV})^{2}$.
However, we know that gravity does not become strongly coupled near the MeV
scale, so we cannot have a fundamental string with its associated tower of
high-spin modes at such a low scale. This strongly suggests that the only way
for the Standard Model photon to be massive is if it is Higgsed.

Could the Standard Model photon be Higgsed? The short answer is “probably
not,” and it becomes “no” provided that we assume that the ratios among
charges of light particles in the theory are at most $O(1)$ (such an
assumption is common in discussions of phenomenological implications of the
WGC). With this assumption, if the Standard Model photon were Higgsed then we
would already have discovered the corresponding Higgs boson. The only way to
avoid this is for the Standard Model photon to obtain a mass from a Higgs
field with a charge that is a tiny fraction of the electron’s charge, in which
case the associated Higgs boson could remain hidden from experiments. For
example, suppose that the fundamental unit of electric charge is not $e$ but
some $e_{0}=e/N$ where $N$ is a very large integer. Then our calculation
becomes quite different: we have $m_{\gamma}=e_{0}f/(2\pi)$ and, rather than
small $f$, we are free to take very small $e_{0}$. Furthermore, if $f$ is a
Higgs vev, rather than a Stueckelberg scale, then there are no associated
fundamental strings providing a UV cutoff. As an example, the choice
$f\sim\mathrm{eV}$ and $e_{0}\sim 10^{-14}$ could be consistent with
experimental bounds on millicharged particles. It implies a WGC tower of
states with tiny electric charge beginning at the scale
$e_{0}M_{\mathrm{Pl}}\sim 10~{}\mathrm{TeV}$, which is allowed by data. Apart
from a new hierarchy puzzle associated with the small mass of the new Higgs
field, the cost of considering such a theory is the introduction of the
enormous integer $N\sim 10^{14}$. The Standard Model fermions would carry
electric charge on this order, in units of the fundamental charge. There are
no known consistent theories of quantum gravity that can produce such large
ratios of charges among light particles. On the other hand, there are
phenomenological models in which such a large integer could be obtained as a
product of smaller integers, as in the clockwork scenario [251], [252, 253],
adapted to this context by [254]. It remains to be seen if such scenarios can
be found in the Landscape.

The WGC can constrain not only the possibility that the Standard Model photon
is massive, but also the possibility that a dark photon $A^{\prime}$ has a
fundamental Stueckelberg mass. One application is to dark photon dark matter,
which is now the target of many dedicated experiments. Dark photon dark matter
can arise from the primordial fluctuations of a massive vector field
$A^{\prime}$ during inflation, which can account for the observed dark matter
relic abundance if $m_{A^{\prime}}\gtrsim 10^{-5}\,\mathrm{eV}$ [255].
However, for such a light dark photon, if the mass is of fundamental
Stueckelberg type then the WGC implies a cutoff lower than the inflationary
Hubble scale assumed in the calculation of the dark matter relic abundance.
This constraint excludes a substantial part of the parameter space of such
models [52].

####  7.1.5 Axion inflation

The spectra of temperature and polarization fluctuations in the Cosmic
Microwave Background radiation (CMB) strongly suggest that the universe
experienced an early period of accelerated expansion known as inflation [256],
[257]. This idea [258, 259] is most easily realized by a scalar field rolling
slowly down a potential [260]. The action describing this scenario is

$S=\int\textrm{d}^{4}x\sqrt{-g}\left[\frac{M_{\mathrm{Pl}}^{2}}{2}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]\,,$
(7.19)

where the scalar $\phi$ is called the inflaton. At leading order the metric
$g_{\mu\nu}$ is taken to be the FRW metric,

$\textrm{d}s^{2}=-\textrm{d}t^{2}+a^{2}(t)\textrm{d}\vec{x}^{2}\,,$ (7.20)

and the scalar field $\phi$ is taken to be homogenous in space,
$\phi(t,\vec{x})=\phi(t)$. The equations of motion are

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0,~{}~{}~{}~{}~{}\frac{1}{3M_{\mathrm{Pl}}^{2}}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right)=\left(\frac{\dot{a}}{a}\right)^{2},$
(7.21)

and inflation happens when $H\equiv\frac{\dot{a}}{a}$ is approximately
constant. Requiring $|\dot{H}|\ll H^{2}$ implies that $\dot{\phi}^{2}\ll V$,
and it is usually also assumed that the acceleration of $\phi$ is small,
$|\ddot{\phi}|\ll\dot{\phi}H$ (see, e.g., [261]). Together these requirements
are equivalent to the “slow roll conditions”

$\displaystyle\frac{|V^{\prime}|}{V}M_{\mathrm{Pl}}$ $\displaystyle\ll 1$
$\displaystyle\frac{|V^{\prime\prime}|}{V}M_{\mathrm{Pl}}^{2}$
$\displaystyle\ll 1.$ (7.22)

Inflation ends when these conditions are violated, after which the field is
usually expected to oscillate about its current minimum and in some manner
(called reheating) decay into the dense gas of hot particles we usually call
the big bang.

CMB observables give us data about the inflaton potential $V$. Especially
noteworthy for our purposes are the primordial scalar and tensor power spectra

$\displaystyle k^{3}P_{s}(k)$
$\displaystyle=\frac{H^{4}}{2\dot{\phi}^{2}}\approx\frac{V^{3}}{6M_{\mathrm{Pl}}^{6}V^{\prime
2}}\,,$ $\displaystyle k^{3}P_{t}(k)$
$\displaystyle=\frac{4H^{2}}{M_{\mathrm{Pl}}^{2}}\approx\frac{4V}{3M_{\mathrm{Pl}}^{4}}\,,$
(7.23)

where quantities on the right are evaluated at the value of $\phi$
corresponding to the time when modes of wave number $k$ were exiting the
inflationary horizon (see, e.g., [262]). The scalar power spectrum has been
well-measured by the temperature anisotropy of the CMB, so it is the tensor
spectrum, which causes anisotropy in the B-mode polarization of the CMB, which
is of most interest in learning more about the physics of inflation. In
particular from (7.23) we see that a measurement of tensor modes would give us
direct information about the overall scale of the inflationary potential. The
tensor amplitude is usually expressed via the tensor-to-scalar ratio

$r\equiv\frac{P_{t}(k_{*})}{P_{s}(k_{*})}=\frac{8\dot{\phi}^{2}}{H^{2}M_{\mathrm{Pl}}^{2}}\approx\frac{8M_{\mathrm{Pl}}^{2}V^{\prime
2}}{V^{2}},$ (7.24)

where $k_{*}$ is a typical wave number of large scale structure, say
$.05\,\mathrm{Mpc}^{-1}$. For high-scale inflation such as might lead to
observable tensor modes, this corresponds to the value of the inflaton about
$60$ e-foldings before the end of inflation (i.e., at the time $t_{*}$ when
$\log\left(a(t_{\text{end}})/a(t_{*})\right)\approx 60$).

From (7.24), we see that the tensor-to-scalar ratio measures how quickly the
inflaton is rolling. We also know how long inflation lasts (60 e-folds for
high-scale inflation). Putting these together, and integrating $\dot{\phi}$
over time, we obtain a rough estimate on the distance traveled by the field
$\phi$ during the course of inflation, which is known as the Lyth bound [263]:

$\Delta\phi\approx\mathcal{O}(1)\left(\frac{r}{0.01}\right)^{1/2}M_{\mathrm{Pl}}\,.$
(7.25)

The Lyth bound tells us that if tensor modes can be detected in the CMB in the
near future (which would require $r\gtrsim 0.001-0.01$), then the inflaton
must traverse a distance of order 1 in Planck units.

This “super-Planckian” field range is important because such large field
excursions can run into trouble with effective field theory: quantum gravity
sets an EFT cutoff no larger than the Planck scale, so a perturbative
expansion of $V(\phi)$ in powers of $\phi$ suppressed by $M_{\mathrm{Pl}}$
will run into trouble when $\Delta\phi$ is $O(M_{\rm Pl})$ or larger. If
tensor modes are observed in the CMB, therefore, we will need a model of
large-field inflation that is not destroyed by large corrections arising from
quantum gravity.

Historically, axions have been considered the most promising route for
circumventing this issue. An axion has a discrete gauge symmetry,
$\phi\rightarrow\phi+2\pi$. This shift symmetry protects the axion potential
from Planck-suppressed operators $\phi^{n}/M_{\textrm{Pl}}^{n-4}$. The
dynamics are instead controlled by a periodic potential, which is often
assumed to be dominated by instanton effects, leading to an action of the form

$S=\int\textrm{d}^{4}x\left[-\frac{1}{2}f^{2}(\partial_{\mu}\phi)^{2}-V(\phi)\right]\,,$
(7.26)

where

$V(\phi)=\Lambda_{\textrm{UV}}^{4}\mathrm{e}^{-S_{\textrm{inst}}}\left(1-\cos\left(\phi\right)\right)+O(\mathrm{e}^{-2S_{\textrm{inst}}})\,.$
(7.27)

Here $f$ is the “axion decay constant,” $S_{\textrm{inst}}$ is the instanton
action, and higher harmonics of the potential are suppressed by additional
powers of $\mathrm{e}^{-S_{\textrm{inst}}}$. The resulting model of inflation
is called “natural inflation” [264], and it yields phenomenologically viable
models of large-field inflation with detectable tensor modes ($r>0.01$) for
$f\gtrsim 10M_{\textrm{Pl}}$.

Although natural inflation is an appealing model, it has proven difficult to
implement in string theory when $f\gtrsim M_{\mathrm{Pl}}$ and
$S_{\mathrm{inst}}\gg 1$, with the basic problem being that such large
excursions in scalar field space often lead to additional light degrees of
freedom appearing which spoil inflation [90]. Indeed this difficulty is part
of what led AMNV to propose the WGC in 2007: the axion WGC (3.24) gives an
upper bound on $f$, which in $D=4$ and assuming the instantons satisfying the
WGC have small instanton number tells us that

$fS_{\mathrm{inst}}\lesssim M_{\mathrm{Pl}}.$ (7.28)

Thus any natural inflation model with observable tensor modes and a computable
potential (meaning $f\gtrsim 10M_{\mathrm{Pl}}$ and $S_{\mathrm{inst}}\gg 1$)
is in strong tension with the axion WGC.

Figure 12: In N-flation, a large number $N$ of axions with individually sub-
Planckian decay constants give rise to an effective decay constant
$f_{\textrm{eff}}$ that can be arbitrarily large, realized by traveling along
the space diagonal of the $N$-dimensional hypercube.

Already by the time the WGC was introduced, various works had considered
possible ways to get around the above difficulties and realize a model of
large-field natural inflation consistent with quantum gravity. One such
proposal is N-flation [265], [132]. As its name suggested, N-flation invokes
not just 1, but $N$ axion fields. If each field has a decay constant $f$, then
by traveling along the diagonal in field space, one sees an effective decay
constant of $f_{\textrm{eff}}=\sqrt{N}f$ (using the simple fact that an
$N$-dimensional hypercube of side length $f$ has a diagonal of length
$\sqrt{N}f$; see figure 12). A related idea is decay constant alignment [266]:
here, only two axions are needed, but their decay constants are “aligned” so
that the fundamental axion domain is not a square, but rather an elongated
parallelogram, as shown in figure 13. Even though each individual axion may
have a sub-Planckian decay constant, the diagonal direction in field space may
be much larger than $M_{\rm Pl}$, thereby generating a model of natural
inflation with a super-Planckian effective decay constant $f_{\textrm{eff}}\gg
M_{\rm Pl}$.

Figure 13: In decay constant alignment, two axions with individually sub-
Planckian decay constants are aligned so that their diagonal can be
arbitrarily large.

However, in their simplest incarnations, neither N-flation nor decay constant
alignment evade the axion WGC. The individual instantons involved may be
superextremal, but together they do not satisfy the convex hull condition, as
shown in figure 14. Said differently, there are no superextremal instantons
associated with the diagonal directions of field space.

We can make a more general argument. Suppose our theory features $n$
instantons, with action

$S=\int\textrm{d}^{4}x\left[-\frac{1}{2}\partial_{\mu}\vec{\phi}\cdot\mathbf{K}\cdot\partial^{\mu}\vec{\phi}-V(\vec{\phi})\right]\,,$
(7.29)

where $\mathbf{K}$ is the kinetic matrix for the axions. We further suppose
that instantons generate a leading-order potential of the form

$V(\vec{\phi})=\sum_{k}\Lambda_{\textrm{UV}}^{4}\mathrm{e}^{-S_{k}}\left(1-\cos\left(\vec{Q}_{k}\cdot\vec{\phi}\right)\right)\,,$
(7.30)

Next, suppose we want to inflate in the $\vec{e}$ direction of field space, so
that the inflaton starts at the point $\vec{\phi}=\phi_{0}\hat{e}$ and rolls
to the minimum at the origin in approximately a straight line in field space.
We assume that the largest value of $\phi_{0}$ allowed satisfies

$\phi_{0}\left(\vec{Q}_{k}\cdot\hat{e}\right)\leq\pi~{}~{}\text{for all
$k$}\,,$ (7.31)

since otherwise the inflaton sits in a cosine well of the $k$th potential term
which does not contain the origin, which will presumably lead the inflaton to
roll into a neighboring vacuum rather than the vacuum at $\vec{\phi}=0$.

Figure 14: In their simplest incarnations, both N-flation and decay constant
alignment violate the axion WGC: the charge-to-action vectors
$\vec{z}_{k}=\vec{Q}_{k}/S_{k}$ of the instantons are superextremal, but their
convex hull does not contain the unit ball.

Next, we assume the axion WGC, which implies that for the given direction
$\hat{e}$, there exists a superextremal instanton satisfying

$\frac{\vec{Q}_{k}\cdot{\hat{e}}}{fS_{k}}\gtrsim\frac{1}{M_{\rm Pl}}\,,$
(7.32)

where $f:=\sqrt{\hat{e}\cdot\mathbf{K}\cdot\hat{e}}$ is the axion decay
constant for the direction $\hat{e}$. Finally, we assume $S_{k}>1$ for
perturbative control of the instanton expansion. Together with (7.31), this
gives a bound on the physical displacement of the field,

$||\vec{\phi}||:=\sqrt{\vec{\phi}\cdot\mathbf{K}\cdot\vec{\phi}}=f\phi_{0}\lesssim\pi
M_{\rm Pl}\,.$ (7.33)

Thus, the axion WGC constrains the maximum axion field range to be $O(\pi
M_{\rm Pl})$: too small to generate a successful model of natural inflation
with observable tensor modes [92], [93], [94], [79], [96]. The generality of
this observation re-invigorated hopes that the consistency of quantum gravity
might lead to testable predictions for cosmology, and generated renewed
interest in the WGC and the Swampland program more generally.

There are several caveats to the above argument, however, which need to be
discussed. First of all, it assumes the axion WGC, which as we have seen is on
somewhat shakier footing than the higher-form versions of the WGC. In
particular, it is not immediately related to black hole evaporation. Relatedly
the $O(1)$ coefficient in the axion WGC is so far not decisively fixed, hence
our argument produced only a “squiggly” $\lesssim$ statement rather a sharper
$\leq$ statement. Various possibilities for the precise $O(1)$ coefficient in
the axion WGC bound have been suggested in [59, 213]. Moreover, the bound
(7.28) relies on assuming that the instantons obeying the axion WGC have
instanton number which is $O(1)$: this is natural from the point of view of
the tower/sublattice WGCs, and also from the point of view of the idea that
there should be objects obeying the WGC which are not black holes (or in the
axion case which do not have large gravitational backreaction), but it only
follows from tower/sublattice WGCs if we assume the relevant tower/sublattice
is not too sparse.

Another related issue is that the bound (7.31) assumes that every instanton
whose charge-to-action vector $\vec{Q}_{k}/S_{k}$ contributes to the convex
hull also contributes significantly to the axion potential. However, it is
conceivable that the dominant contributions to the axion potential could come
from instantons which violate the axion WGC, whereas the instantons that
satisfy the WGC give only subleading, unimportant contributions to the
potential. In this case, the inflationary dynamics are unconstrained by the
axion WGC. This “extra instanton loophole” has driven a lot of interest in
strong forms of the WGC [267, 95]. However, even the lattice WGC is not quite
sufficient to close this loophole [268]. On the other hand, threading the
extra instanton loophole seems to require a fair bit of tuning [268], and so
far super-Planckian axion decay constants have yet to be realized in string
theory [100].

It is also possible to try to relax the assumption that $S_{k}\gg 1$, which we
suggested is required for perturbative control of the instanton expansion. In
string compactifications, $S_{k}$ is typically the size of some cycle of the
Calabi-Yau in string units, so the $\alpha^{\prime}$ expansion of string
theory is not valid unless $S_{k}\gg 1$. However, in “extranatural inflation”
[269], instantons in 4d come from the particles in 5d wrapping the
compactification $S^{1}$. The instanton action for a particle of mass $m$
wrapping a circle of radius $R$ is given by $S_{\textrm{inst}}=2\pi mR$, and
the contribution to the axion potential is given by:

$\displaystyle V(\phi)$ $\displaystyle=\frac{3(-)^{S}}{4\pi^{2}}\frac{1}{(2\pi
R)^{4}}\sum_{n\in\mathbb{Z}}c_{n}\mathrm{e}^{-2\pi
nRm_{5}}\mathrm{e}^{in\phi}\,,$ $\displaystyle c_{n}$
$\displaystyle=\frac{(2\pi Rm_{5})^{2}}{3n^{3}}+\frac{2\pi
Rm_{5}}{n^{4}}+\frac{1}{n^{5}}\,.$ (7.34)

In this context, there is no problem with taking $S_{\textrm{inst}}\ll 1$:
this simply corresponds to a light particle with $m\ll 1/R$. Likewise, there
is no problem with perturbative control of the potential: the $1/n^{5}$ term
in $c_{n}$ suppresses higher harmonics even for $S_{\textrm{inst}}\ll 1$ [36].
This significantly weakens the WGC bound (7.28) on the axion decay constant.
By imposing the convex hull condition on both the $U(1)$ of the parent 5d
theory and the Kaluza-Klein $U(1)$, one can strengthen the bound to $f\lesssim
M_{\textrm{Pl}}/S_{\textrm{inst}}^{1/2}$, but this is still insufficient to
close this “small action loophole” in the context of extranatural inflation
[270].

Finally, and perhaps most importantly, the above arguments only apply to
theories where the only fields relevant during inflation are axions and the
metric. Including non-periodic scalars would only re-introduce the UV-
sensitivity we avoided with axions, but we saw above in our discussion of the
axion monodromy model (7.2) that the inclusion of a three-form gauge field
$A_{3}$ coupled to an axion by a Chern-Simons term offers a simple mechanism
whereby an axion with $f\lesssim M_{\mathrm{Pl}}$ can nonetheless lead to a
scalar $\star F_{4}$ which rolls down a potential for many Planck distances.
Moreover we saw that the three-form WGC applied to $A_{3}$ leads to only weak
constraints on the range of this rolling. We presented this model in the
context of cosmological relaxation, but its most compelling application is
really inflation, which indeed is what it was originally proposed for [103],
[104]. Moreover proposals have been given for embedding this model into a
consistent string compactification, although there are still certainly details
remaining to be worked out and so far no detailed model has been given where a
large field range is realized [271], [242], [272].

It is sometimes suggested that string theory does not allow for models with
observable tensor modes, and the axion WGC was proposed in part to give an
explanation for this claim, but axion monodromy casts serious doubt on it.
Natural inflation with observable tensor modes may well be in the Swampland,
but the right lesson from this may just be that we should see what kind of
predictions follow from the more general axion models which do seem to work.
In particular, given the ever-improving observational upper bounds on $r$, it
is natural to ask whether axion monodromy models exist which aren’t excluded
but nonetheless predict observable tensor modes. The simplest potential, a
quadratic one (see equation (7.8)), is now already excluded by the Planck
satellite and various ground-based experiments, but other powers are possible
[273]. Perhaps the detailed issues remaining to be resolved in realizing axion
monodromy in a genuine string compactification may yet lead to distinctive
predictions. If so, then the various forms of the WGC will likely be important
tools in guiding us towards models that work. Either way, it is remarkable
that ongoing observations are teaching us concrete things about physics near
the Planck scale.

###  7.2 Implications for mathematics

The WGC is a statement about the charges and masses of particles in effective
field theory. In string/M-theory, supersymmetric effective field theories
arise from compactifying on Calabi-Yau manifolds. Charged particles arise from
$p$-branes wrapping $p$-cycles of the Calabi-Yau manifold. The charge of such
a particle is determined by the homology class $\Sigma$ wrapped by the brane,
and the mass of the particle is determined by the volume of the wrapped cycle.
Thus, the WGC translates into geometric statements about the volumes of
representatives of various cycles in a Calabi-Yau manifold.

For concreteness, let us consider the case of M-theory on a Calabi-Yau
threefold $X$. This produces a 5d supergravity theory, and charged particles
arise from M2-branes wrapping 2-cycles of $X$. The charge lattice of the
theory is identified with the homology lattice $H_{2}(X,\mathbb{Z})$.

The resulting supergravity theory has a BPS bound: the mass of a particle of
charge $q_{I}$ is constrained to satisfy

$m\geq\left(\frac{(2\pi)^{2}}{2\kappa_{5}}\right)^{1/3}|\zeta_{q}|\,,$ (7.35)

where $\zeta_{q}$ is the “central charge,” a quantity that depends linearly on
$q_{I}$. It is sometimes remarked that the BPS bound is a sort of converse to
the WGC bound, and there is a precise sense in which this is true: if there
exist BPS black holes in a given direction $\hat{q}$ in the charge lattice,
then the BPS bounds and extremality bounds coincide in this direction in the
lattice. The only way a particle of charge $q_{I}\propto\hat{q}$ can satisfy
both the WGC bound and the BPS bound is if it saturates both bounds. A
particle that saturates the BPS bound is called a BPS particle: therefore, the
tower/sublattice WGCs require an infinite tower of BPS particles of increasing
mass/charge in every direction in the charge lattice for which the BPS bound
coincides with the extremality bound.

Geometrically, BPS particles arise from M2-branes wrapping “holomorphic”
curves of $X$. Here, a curve $\Sigma$ is “holomorphic” if its volume is given
by integrating the Kähler form $J$ over it, $V_{\Sigma}=\int_{\Sigma}J$.
Equivalently, we say that the curve is “calibrated” by the Kähler form.

The upshot of this is that the tower/sublattice WGCs imply the existence of an
infinite tower of holomorphic curves in any direction $\hat{q}$ of the
homology lattice $H_{2}(X,\mathbb{Z})$ for which the BPS bound coincides with
the extremality bound. In fact, the condition that the BPS bound and
extremality bounds coincide in the direction $\hat{q}$ can also be given a
geometric interpretation: these bounds necessarily coincide for any $\hat{q}$
that resides in the so-called “cone of moving curves”
$\mathcal{K}^{\vee}\subset H_{2}(X,\mathbb{R})$, which is equal to the cone
dual of the “cone of effective divisors” [46].141414If the theory in question
allows certain $U(1)$ gauge fields to be enhanced to a larger nonabelian
group, the effective cone of divisors may change under a phase transformation
representing a Weyl reflection of the nonabelian gauge group. In this case,
the BPS and extremality bounds coincide in an even larger cone in the charge
lattice, which is geometrically the cone dual to the intersection of the
effective cone of divisors over all phases of the theory. [274] Thus, the
tower/sublattice WGCs translate to the nontrivial geometric statement that
there must exist an infinite tower of holomorphic curves for any rational
direction $\hat{q}$ within the cone of moving curves $\mathcal{K}^{\vee}$.

This statement is powerful in that (a) it is a purely geometric statement,
with no reference to physics, and (b) it can actually be verified in examples.
Gopakumar-Vafa (GV) invariants [275, 276, 277] count the number of BPS
particles (i.e., holomorphic curves) of a given charge $q_{I}$ in the
lattice,151515More accurately, GV invariants compute an index of BPS particles
of a given charge, meaning that a nonzero GV invariant implies a nonzero
number of BPS particles, whereas a vanishing GV invariant could result from an
equal number of BPS hypermultiplets and BPS vector multiplets of a given
charge.  and these can be computed for many Calabi-Yau hypersurfaces [278] and
complete intersection Calabi-Yau manifolds (CICYs) [279]. At the same time,
the cone of moving curves $\mathcal{K}^{\vee}$ can often be computed (with
some input from the 5d supergravity theory) using the methods of [46].
Together with the automated computation of GV invariants introduced in [280],
this has enabled thorough checks of this geometric version of the
tower/sublattice WGCs in over 1400 Calabi-Yau manifolds [274].

So far, we have focused our attention on BPS particles, which are required by
the tower/sublattice WGCs in certain directions in the charge lattice, where
the BPS bound and extremality bound coincide. In a theory with 8 supercharges,
however, there will generically be some directions for which the BPS bound and
the extremality bound do not coincide. The tower/sublattice WGCs still require
the existence of superextremal particles, which means they still impose
constraints on the volumes of cycles of the Calabi-Yau manifold. To be more
precise, the charge-to-mass vector of a particle associated with a $p$-cycle
$\Sigma$ of a Calabi-Yau manifold is given by [281],

$\vec{z}=\frac{V_{X}^{1/2}\vec{q}_{\Sigma}}{V_{\Sigma}}\,,$ (7.36)

where $V_{X}$ is the volume of the Calabi-Yau $X$, $V_{\Sigma}$ is the volume
of $\Sigma$, and $\vec{q}_{\Sigma}$ labels the charge vector associated with
the homology class. The norm $||\vec{q}_{\Sigma}||$ is the norm of the
harmonic form related to $\Sigma$ using the metric on $X$.

The particle is superextremal when $||\vec{z}||\geq\gamma_{4}^{1/2}$. In
general, $\gamma_{4}$ depends on the direction $\hat{q}$ as well as the
massless scalar fields in the theory, but it necessarily satisfies
$\gamma_{4}\geq\frac{1}{2}$, which means that a superextremal particle has

$\frac{V_{X}^{1/2}||\vec{q}_{\Sigma}||}{V_{\Sigma}}\geq\frac{1}{\sqrt{2}}\,.$
(7.37)

The tower WGC therefore implies that for a given cycle $[\Sigma]$, there
exists an integer $n$ and a representative of $n[\Sigma]$ satisfying (7.37).
The sublattice WGC further implies that there exists a universal $n$, which is
independent of $[\Sigma]$.

In some cases, this bound leads to surprising, nontrivial mathematical
results. In particular, any 4-cycle $[\Sigma]$ in a Calabi-Yau threefold $X$
with $h^{2,0}(X)=0$, can be represented as a union of holomorphic and
antiholomorphic representatives; upon wrapping D4-branes on these
representatives, these correspond to BPS and anti-BPS particles, respectively.
The minimal volume representative of $[\Sigma]$ therefore has a volume no
larger than the sum of the volumes of these holomorphic and antiholomorphic
representatives, which we denote $V(\Sigma_{\cup})$. But satisfying (7.37) may
require this union of holomorphic and antiholomorphic representatives to
recombine into a new representative $\Sigma_{\text{min}}$ whose volume is
significantly smaller than $V(\Sigma_{\cup})$. More precisely, the
“recombination fraction”

$\tau_{\Sigma}:=\frac{V(\Sigma_{\cup})-V(\Sigma_{\text{min}})}{V(\Sigma_{\text{min}})}$
(7.38)

may be much larger than 1 [282]. Physically, this recombination corresponds to
D4-branes wrapping these representatives recombining and fusing, and particles
in 4d binding to form bound states of significantly smaller energy.
Mathematically, the existence of representatives $\Sigma_{\text{min}}$ with
large recombination fraction $\tau_{\Sigma}\gg 1$ is a nontrivial consequence
of the WGC, which has been verified in some examples [283].

Finally, let us remark that within the context of 5d M-theory
compactifications, there are interesting connections between the tower WGC,
the WGC for strings, and the Swampland Distance Conjecture and various
mathematical conjectures about Calabi-Yau manifolds known as “cone
conjectures” [284, 285]. More on these connections can be found in [286].

###  7.3 Implications for general relativity

The (weak) cosmic censorship hypothesis holds that for generic initial data,
the maximal Cauchy development possesses a complete future null infinity
[287]. Colloquially, there can be no naked singularities visible at future
null infinity: any such singularity must be hidden behind a horizon.

There is strong numerical evidence that cosmic censorship is violated in more
than four spacetime dimensions [288], as black strings may pinch off and
develop singularities due to the Gregory-Laflamme instability [289]. In four
dimensions, however, such instabilities do not exist, and violation of cosmic
censorship is much less certain.161616Even in higher dimensions, known
violations of cosmic censorship have zero mass and occur in Planck-sized
regions where quantum gravitational effects become important. It has been
argued that such quantum effects restore some notion of cosmic censorship, so
that these counterexamples to cosmic censorship are relatively benign [290].

A promising class of counterexamples to cosmic censorship in four dimensions
were proposed in [291], and strong numerical evidence for these
counterexamples was subsequently provided in [292]. These examples involve a
$U(1)$ gauge field coupled to gravity in asymptotically AdS space, with action
(as usual we set the AdS radius to one)

$S=\frac{1}{2\kappa^{2}}\int\textrm{d}^{4}x\sqrt{-g}\left(R+6-F^{\mu\nu}F_{\mu\nu}\right)\,,$
(7.39)

with $F=\textrm{d}A$. The boundary metric is chosen to be flat,

$\textrm{d}s_{\partial}^{2}=-\textrm{d}t^{2}+\textrm{d}r^{2}+r^{2}\textrm{d}\varphi^{2}\,,$
(7.40)

and the only nonzero component of the potential at the boundary is the time
component:

$A_{\partial}=+\frac{a(t)\textrm{d}t}{(1+r^{2})^{n/2}}\,$ (7.41)

where $n$ is an integer controlling the fall-off of the field at large $r$.

Apparent violations of cosmic censorship arise when $a(t)$ is chosen to vanish
at $t=0$ but increases to a constant value larger than some critical value
$a_{\text{max}}$. In this case, there is no smooth static endpoint of the
evolution, so one expects the curvature $F^{2}$ will grow indefinitely at late
times. Numerical simulations confirm this expectation for $n=1$ [292]. Note
that the curvature does not diverge in finite time, so this example does not
quite violate the letter of cosmic censorship, though it does violate the
spirit of it.

This class of counterexamples disappears, however, in the presence of a
superextremal scalar field [291, 293]. In particular, suppose we add a charged
scalar $\Phi$ to the action:

$S_{\Phi}=-\frac{1}{4\pi
G}\int\textrm{d}^{4}x\sqrt{-g}\left[(D_{\mu}\Phi)(D^{\mu}\Phi)^{\dagger}+m^{2}\Phi\Phi^{\dagger}\right]\,,$
(7.42)

with $D_{\mu}=\nabla_{\mu}-i\widetilde{q}A_{\mu}$ (here
$\widetilde{q}=\frac{\sqrt{2}e}{\kappa}q$, where $q$ is the integral charge
we’ve been using throughout).171717We observed below equation (3.6) that the
WGC bound (3.4) does not involve any powers of $\hbar$, and thus potentially
has classical consequences, but those remarks applied for the case of a
classical particle. A similar statement applies for a charged classical field
such as $\Phi$, but we need to be a bit more careful since if we restore $c$
and $\hbar$ then (in Heaviside-Lorentz units) the “mass” $m$ and “charge”
$\widetilde{q}$ appearing in (7.42) both have units of inverse length. The
true charge and mass of a particle appearing after this field is quantized are
related to these by powers of $\hbar$ and $c$, but in writing the WGC
inequality the powers of $\hbar$ and $c$ drop out since $m$ and
$\widetilde{q}$ have the same units and $\kappa$ has already been absorbed
into $\widetilde{q}$. The proposed WGC bound in AdS (3.20) then becomes

$\widetilde{q}\geq\Delta=\frac{3}{2}+\sqrt{\frac{9}{4}+m^{2}}\,.$ (7.43)

When this bound is satisfied, for all choices of $n$ perturbations of $\Phi$
become unstable before $a$ grows to the critical value $a_{\text{max}}$, and
cosmic censorship is restored. When this bound is violated, the solution with
the scalar field is still singular, and it is once again likely that cosmic
censorship is violated [291]. Thus, there is evidently a one-to-one
correspondence between satisfying the WGC and obeying cosmic censorship in
this setup.

Already, this result is quite suggestive. But the connection between the WGC
and cosmic censorship becomes even more impressive when including dilatonic
couplings and multiple scalar fields, as was done in [294]. In the dilatonic
case, one begins with the action

$S=\frac{1}{2\kappa^{2}}\int\textrm{d}^{4}x\sqrt{-g}\left(R+6-\mathrm{e}^{-2\alpha\phi}F^{\mu\nu}F_{\mu\nu}-2\nabla_{\mu}\phi\nabla^{\mu}\phi\right)\,,$
(7.44)

in place of (7.39). Here, $\phi$ is a massless, uncharged scalar field which
we will refer to as the dilaton, not to be confused with the massive, charged
scalar field $\Phi$. In the presence of this dilatonic coupling, the WGC bound
for $\Phi$ is modified to

$\widetilde{q}\geq\widetilde{q}^{W}\equiv\Delta(1+\alpha^{2})^{1/2}\,,$ (7.45)

where $\Delta$ is given by (7.43). Notably, the minimal charge-to-mass ratio
$\widetilde{q}/\Delta$ varies continuously with the parameter $\alpha$.

Figure 15: For fixed $n=4$, $\Delta=2$, and dilatonic coupling $\alpha=0.9$,
the condition to preserve cosmic censorship is precisely the WGC bound (7.45).
Here, blue dots indicate the onset of solutions with $\Phi\neq 0$, and red
dots indicate the approximate location of singular solutions. Figure adapted
from [294].

[294] constructed numerical solutions to the equations of motion in the
presence of the dilaton, with a boundary vector potential given by

$A_{\partial}=\frac{a\textrm{d}t}{(1+r^{2})^{n}}\,,$ (7.46)

focusing in particular on the case $n=4$. They found that for $\alpha<1$,
cosmic censorship is again preserved precisely when the WGC bound (7.45) is
satisfied, as shown in figure 15. This is remarkable, in that it establishes a
WGC-cosmic censorship connection over a one-parameter family of theories,
indexed by $\alpha$. For $\alpha>1$, numerical solutions suggest that it may
be possible to preserve cosmic censorship even when
$\widetilde{q}/\widetilde{q}^{W}$ (or equivalently $q/q^{W}$) is slightly
smaller than 1, as shown in figure 16. It is possible that this conclusion
could be modified at large values of $a$, and the one-to-one correspondence
between the WGC and cosmic censorship could be restored.

Figure 16: For fixed $n=4$, $\Delta=2$, and dilatonic coupling
$\alpha=\sqrt{3}$, tcondition to preserve cosmic censorship does not quite
seem to match with the WGC bound (7.45), though it is possible that
modifications at large $a$ could restore the correspondence. Here, blue dots
indicate the onset of solutions with $\Phi\neq 0$, and red dots indicate the
approximate location of singular solutions. Figure adapted from [294].

Finally, [294] also considered the relationship between cosmic censorship and
the WGC in theories with two gauge fields. In their analysis, the asymptotic
profile of the gauge fields is taken to be

$A_{I\,\partial}=\frac{a_{I}}{(1+r^{2})^{n}}\textrm{d}t\,,$ (7.47)

focusing again on the case of $n=4$, with $a_{1}=\lambda a_{2}$. There are now
two massive scalar fields, $\Phi_{1}$ and $\Phi_{2}$; the former has charge
$\widetilde{q}_{1}$ under the first gauge field and is uncharged under the
second gauge field, whereas the latter carries charge $\widetilde{q}_{2}$
under the second gauge field and is uncharged under the first gauge field.
Since there are multiple gauge fields, the WGC bound is equivalent to the
convex hull condition (see Section 3.3), which is given by

$\frac{1}{z_{1}^{2}}+\frac{1}{z_{2}^{2}}\leq 1\,,$ (7.48)

with $z_{I}=\widetilde{q}_{I}/\Delta_{I}$. By constructing numerical solutions
to the equations of motion for various choices of $\lambda$,
$\widetilde{q}_{I}$, and $\Delta_{I}$, [294] provided strong evidence that
cosmic censorship is preserved precisely when the convex hull condition is
satisfied. Figure 17 depicts this correspondence for one particular choice of
$\lambda$, $\widetilde{q}_{2}$, $\Delta_{1}$ $\Delta_{2}$, varying
$\widetilde{q}_{1}$.

Figure 17: For fixed $n=4$, $a_{1}=3a_{2}$, $q_{2}=4$,
$\Delta_{1}=\Delta_{2}=2$, the condition to preserve cosmic censorship is
precisely the convex hull condition (7.48). Here, blue dots indicate the onset
of solutions with $\Phi_{1}\neq 0$, and red dots indicate the approximate
location of singular solutions. Figure adapted from [294].

The work of [291, 292, 294] thus reveals a remarkable and surprising
correspondence between the WGC and the weak cosmic censorship conjecture. A
couple of aspects of this correspondence are worth further attention. First of
all, note that the mildest form of the WGC, which simply requires the
existence of a superextremal state, is not sufficient to preserve cosmic
censorship. The mild WGC could be satisfied in principle by a finite-sized
black hole state due to subleading corrections that slightly increase $Q/M$
relative to $Q/M$ for an infinitely large, extremal black hole. In the present
scenario, however, preserving cosmic censorship requires a superextremal,
field $\Phi$ (and thus quantum mechanically a superextremal particle) and not
merely a superextremal black hole.

Second, we observe that in all examples so far the superextremal field which
saves cosmic censorship is a scalar. The usual formulations of the WGC do not
put any restrictions on the spins of the superextremal particles which are
required, but it is interesting to consider if there might be some such
restriction. One natural question for future study is whether or not
superextremal bosonic fields of nonzero spin similarly prevent violations of
cosmic censorship. More generally one can also consider the question of
whether or not fermions can do the same once quantum effects are included. So
far all string compactifications we are aware of in fact do have a
superextremal scalar, but this may be an accident of supersymmetry so there is
no strong evidence so far for an interplay between the WGC and spin.

Finally, let us mention an interesting connection between the WGC and a
different kind of gravitational censorship, namely of super-Planckian spatial
field variations. Simple dimensional analysis suggests such a field
configuration could collapse into a black hole. Indeed, this is often true
[295]. However, one case in which a classically stable field configuration
with an arbitrarily large scalar field variation, not screened by a horizon,
can be constructed is a charged Kaluza-Klein bubble stabilized by flux [296].
The scalar field is the radion, which traverses an infinite distance in field
space to the bubble wall where $R\to 0$. Thus, arbitrarily large field values
are not classically censored. In the quantum theory, the solution becomes
unstable if charged matter satisfying the WGC exists, due to Schwinger pair
production, which dynamically censors the large field excursion [297].

The general picture painted by these examples is that the WGC can play an
important role in ensuring the validity of effective field theory. It prevents
a low-energy observer from accessing arbitrarily high energy scales (in the
case of cosmic censorship) or field values (in the case of super-Planckian
censorship).

##  8 Outlook

In this review, we have seen that the Weak Gravity Conjecture potentially
offers a deep organizing principle for unlocking the puzzle of quantum
gravity. In particular the Landscape of string vacua is very large, yet as far
as we can tell, the WGC is obeyed in all of them. Suitably strong forms of the
WGC place meaningful constraints on particle physics and cosmology, and have
further consequences for black holes, pure mathematics, conformal field
theories, and more. Thus, whereas most Swampland conjectures fall either into
the “rigorous but uninteresting” category or the “interesting but not
rigorous” category, the WGC has a claim at both rigor and importance.

Nonetheless, despite all that we have learned about it, the WGC remains
shrouded in mystery. We saw in Section 5 of this review that several arguments
point qualitatively to the validity of the WGC: it is quite plausible that the
WGC is satisfied up to $O(1)$ coefficients. However, examples in string theory
suggest something stronger: in all such examples, the WGC is satisfied with
the precise $O(1)$ coefficient determined by the black hole extremality bound.
This suggests that the WGC may be required for consistency of black hole
physics, but it is not yet clear what goes wrong if the WGC is violated.

The evidence for the WGC coming from string theory is strong, but there is
still a chance that the WGC suffers from a lamppost effect: the known string
examples necessarily involve either (a) weak gauge coupling or (b) BPS
particles—the only particles we currently know how to track into a strongly
coupled regime. An example in which the WGC is satisfied by non-supersymmetric
states in a regime of strong coupling is highly desirable, though perhaps
unfeasible. Without this, a more compelling black hole argument is likely
required to rule out the possibility of a lamppost effect.

Another interesting direction for future research involves the classification
of weak coupling limits, as initiated in e.g. [44, 110, 112, 89, 115, 113,
114]. The Emergent String Conjecture of [112], in particular, suggests that
any infinite distance/weak coupling limit must be either a decompactification
limit or an emergent string limit, in which a fundamental string becomes
tensionless. In the former case, a tower of light, superextremal Kaluza-Klein
modes emerge.181818In the supersymmetric case, these KK modes are BPS and
saturate the WGC bound. In the non-supersymmetric case, these KK modes satisfy
the WGC bound with room to spare after the radion is stabilized. See Section
4.1 above for more details. In the latter case, the modular invariance
argument of Section 4.4 ensures that the particles are superextremal. Thus, at
weak coupling, the tower/sublattice WGCs follows from the Emergent String
Conjecture (which could have stronger phenomenological implications).

In Section 4, we encountered examples in which the sublattice WGC is satisfied
by a sublattice of superextremal particles of coarseness $n>1$ (and index
greater than 1). This raises the question of how large the coarseness may
become, or equivalently, how sparse the sublattice is allowed to be. This
question is important, because the consequences of the sublattice WGC for low-
energy physics can be arbitrarily weak if the coarseness is allowed to be
arbitrarily large. Fortunately, the maximum coarseness we encountered was
$n=3$: the sublattice of superextremal particles is not very sparse, and it is
plausible that the sublattice WGC will always be satisfied with a coarseness
of $O(1)$.

Before we may attempt to place any sort of universal upper bound on the
coarseness/index of the sublattice, however, we must first understand how this
sublattice shows up in the low-energy data of the theory. All of the examples
with coarseness $n>1$ constructed so far are orbifold models, and $n$ divides
the order of the orbifold group. However, we do not yet have a clear
understanding of how this UV relationship manifests in the IR. One possibility
is that the coarseness is related to the global structure of the low-energy
gauge group, but more work is needed to clarify this picture.

As a final direction for future research, let us remark that the statement of
the WGC in the presence of Chern-Simons terms is, at present, not well
understood. The recent work [137] presented an example of WGC mixing, in which
the WGC for different $p$-form gauge fields are mixed up in the presence of
Chern-Simons terms (see also [64, 298, 299, 141]). These Chern-Simons terms
imply that the gauge symmetry acquires a higher-group structure [300, 301],
and it seems likely that the full statement of the WGC is modified in the
presence of such higher-group symmetries, reminiscent of how the WGC is
modified to the Convex Hull Condition in theories with multiple $U(1)$s.
Understanding this better might have interesting implications for axion
monodromy.

The Weak Gravity Conjecture has produced no shortage of surprises over the
course of its fifteen-year existence, providing us with new insights into
quantum gravity and unexpected connections between disparate areas of
theoretical physics. Yet many of the most important questions remain open: is
the Weak Gravity Conjecture true? If so, why? And which version(s) of the
conjecture are the right ones? The answers to these questions may well lead us
to even greater surprises than the ones we have already met.

##  9 Acknowledgments

We are grateful to many physicists for conversations that have shaped our
understanding of the Weak Gravity Conjecture over the years. We would
particularly like to thank Murad Alim, Nima Arkani-Hamed, Tom Banks, Clifford
Cheung, Clay Córdova, Thomas Dumitrescu, Netta Engelhardt, Muldrow Etheredge,
Gary Horowitz, Manki Kim, Cody Long, Matteo Lotito, Juan Maldacena, Liam
McAllister, Jacob McNamara, Miguel Montero, Hirosi Ooguri, Kantaro Ohmori,
Grant Remmen, Steve Shenker, Gary Shiu, John Stout, Leonard Susskind, Cumrun
Vafa, and Irene Valenzuela for collaboration or discussions, but this is by no
means an exhaustive list of those to whom we owe thanks.

DH is supported by the Simons Foundation as a member of the “It from Qubit”
collaboration, the Sloan Foundation as a Sloan Fellow, the Packard Foundation
as a Packard Fellow, and the Air Force Office of Scientific Research under the
award number FA9550-19-1-0360, and the US Department of Energy under grants
DE-SC0012567 and DE-SC0020360. The work of BH is supported by NSF grant
PHY-1914934. The work of MR is supported in part by the NASA Grant
80NSSC20K0506 and the DOE Grant DE-SC0013607. The work of TR is supported by
NSF grant PHY-1820912, the Simons Foundation, and the Berkeley Center for
Theoretical Physics.

##  Appendix A The black hole extremality bound

In a gravitational theory with no naked singularities, a black hole of non-
zero charge $Q$ cannot be arbitrarily light:

$M_{\text{BH}}\geqslant M_{\text{ext}}(Q)>0\,,$ (A.1)

where the _extremal mass_ $M_{\text{ext}}(Q)$ is defined as the infinimum of
the set of possible masses for black holes of charge $Q$. This bound arises
because the energy stored in the electromagnetic field is a positive-energy
source for the gravitational field, and in a theory with matter obeying
reasonable energy conditions this gravitational flux cannot be cancelled
without introducing a naked singularity as a negative-energy source. A black
hole saturating this bound is _extremal_ , whereas all others are
_subextremal_.

To determine the extremal mass $M_{\text{ext}}(Q)$ and thereby the extremality
bound (A.1) it would obviously suffice to find an extremal black hole solution
of charge $Q$ and read off its mass. However, an extremal black hole solution
of a given charge does not always exist because taking the extremal limit
sometimes generates a singularity at the event horizon. Furthermore,
identifying whether a given solution is extremal is not straightforward.
Experience with Reissner-Nordström black holes suggests that vanishing surface
gravity is closely connected to extremality, but as we’ll see in §A.4, not
every black hole solution with vanishing surface gravity is extremal!

To solve the first problem, we expand our field of interest to include charged
solutions that are merely _limits_ of black holes, not necessarily black holes
themselves. In a convenient abuse of terminology, we call such limiting cases
“singular black holes.” Familiar examples of singular black holes include,
e.g., the background generated by $N\gg 1$ D0 branes in type IIA string
theory. Understanding what happens near a singular black hole usually requires
UV information which goes beyond effective field theory, for example near a
stack of D0 branes the string coupling gets large and there is a dual M-theory
description.

To solve the second problem, note that Hawking radiation must shut off in the
extremal limit to satisfy cosmic censorship, so either the surface gravity
$g_{h}$ (Hawking temperature) or horizon area $A_{h}$ (Bekenstein-Hawking
entropy) must go to zero in this limit. We refer to black holes with either of
these two properties as _quasiextremal_ , where those with $A_{h}\to 0$ are
singular. Cosmic censorship requires extremal black holes to be quasiextremal,
but as noted above quasiextremal black holes are not always extremal; see §A.4
for examples. We call black holes that are neither extremal nor quasiextremal
_nonextremal_.

In this appendix, we review general techniques for determining the extremality
bound in the large $Q$ limit (where derivative corrections can be ignored) for
theories with multiple $U(1)$ gauge fields $A^{A}$, massless scalars
$\phi^{i}$, and vanishing cosmological constant. For simplicity, we assume
that the lightest black holes of a given charge $Q$ are spherically
symmetric.191919Without this assumption, the problem is unsolved in general,
except in special cases where a BPS-like bound can be derived using spinor
methods as in, e.g., [302]. Temporarily ignoring the possibility of magnetic
charge, the relevant terms in the low-energy Einstein-frame effective action
at two-derivative order are

$\displaystyle
S_{0}=\int\mathrm{d}^{d}x\sqrt{-g}\biggl{[}\frac{1}{2\kappa^{2}}R-\frac{1}{2}t_{AB}(\phi)F^{A}_{2}\cdot
F^{B}_{2}-\frac{1}{2}G_{ij}(\phi)\nabla\phi^{i}\cdot\nabla\phi^{j}\biggr{]},$
(A.2)

where $F_{2}^{A}=\mathrm{d}A_{1}^{A}$. The types of two-derivative terms
omitted above do not affect spherically symmetric black holes with purely
electric charge [73].

###  A.1 Black hole solutions

In a convenient gauge, a spherically symmetric metric ansatz takes the form:

$\mathrm{d}s^{2}=-\mathrm{e}^{2\psi(r)}f(r)\mathrm{d}t^{2}+\mathrm{e}^{-\frac{2\psi(r)}{d-3}}\left[\frac{\mathrm{d}r^{2}}{f(r)}+r^{2}\mathrm{d}\Omega^{2}_{d-2}\right]$
(A.3)

for functions $\psi(r),f(r)$ to be determined, where
$\mathrm{d}\Omega^{2}_{d-2}$ is the round metric of unit radius on $S^{d-2}$.
The electric charge of the solution is

$Q_{A}=\oint_{S^{d-2}}t_{AB}(\phi)\star F^{B}_{2},$ (A.4)

where the integral is taken over a sphere enclosing the horizon. Spherical
symmetry fixes the electric field to be

$F^{A}_{2}=-\frac{t^{AB}(\phi)Q_{B}}{V_{d-2}}\frac{\mathrm{e}^{2\psi}\mathrm{d}t\wedge\mathrm{d}r}{r^{d-2}},$
(A.5)

where $t^{AB}(\phi)$ is the inverse of $t_{AB}(\phi)$ and
$V_{d-2}=2\pi^{\frac{d-1}{2}}/\Gamma\left(\frac{d-1}{2}\right)$ is the volume
of $S^{d-2}$.

One component of Einstein’s equations is now

$f^{\prime\prime}(r)+\frac{3d-8}{r}f^{\prime}(r)+2\frac{(d-3)^{2}}{r^{2}}(f(r)-1)=0\,,$
(A.6)

with the solution $f(r)=1+A/r^{d-3}+B/r^{2(d-3)}$. To interpret $A$ and $B$,
we switch to infalling coordinates:

$\mathrm{d}s^{2}=-\frac{F(\rho)\,\mathrm{d}v^{2}}{R^{2(d-3)}(\rho)}+\frac{2\,\mathrm{d}v\,\mathrm{d}\rho}{(d-3)R^{d-4}(\rho)}+R^{2}(\rho)\mathrm{d}\Omega_{d-2}^{2},$
(A.7)

where $\rho=r^{d-3}$, $R(\rho)=r\mathrm{e}^{-\frac{\psi}{d-3}}$ and
$F(\rho)=r^{2(d-3)}f(r)$. A smooth event horizon occurs when $F\rightarrow 0$
with $R$ finite. If one exists, $F(\rho)=\rho^{2}+A\rho+B$ can be factored

$F(\rho)=(\rho-\rho_{+})(\rho-\rho_{-}),\quad\rho_{+}\geq\rho_{-},$ (A.8)

leading to an outer (inner) horizon at $\rho=\rho_{+}$ ($\rho=\rho_{-}$). We
set $\rho_{-}=0$ using the residual gauge symmetry
$\rho\rightarrow\rho+\text{constant}$ with $F(\rho)$ and $R(\rho)$ held fixed,
so that

$F(\rho)=\rho(\rho-\rho_{h})\qquad\Leftrightarrow\qquad
f(r)=1-\frac{r_{h}^{d-3}}{r^{d-3}}$ (A.9)

with $\rho_{h}=r_{h}^{d-3}$.

In terms of $z\mathrel{:=}\frac{1}{(d-3)V_{d-2}r^{d-3}}$,202020In comparison
with [73], $z^{(\text{here})}=z^{(\text{there})}/V_{d-2}$. $f(z)=1-z/z_{h}$
and the remaining equations of motion are:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}[f\dot{\phi}^{j}]+\Gamma^{i}_{\;jk}\dot{\phi}^{j}\dot{\phi}^{k}$
$\displaystyle=\frac{1}{2}G^{ij}Q^{2}_{,j}(\phi)\mathrm{e}^{2\psi},$ (A.10)
$\displaystyle k_{N}^{-1}\frac{\mathrm{d}}{\mathrm{d}z}[f\dot{\psi}]$
$\displaystyle=\mathrm{e}^{2\psi}Q^{2}(\phi),$ (A.11) $\displaystyle
k_{N}^{-1}\dot{\psi}(f\dot{\psi}+\dot{f})+fG_{ij}(\phi)\dot{\phi}^{i}\dot{\phi}^{j}$
$\displaystyle=\mathrm{e}^{2\psi}Q^{2}(\phi),$ (A.12)

where $\dot{~}{}{}=\frac{\mathrm{d}}{\mathrm{d}z}$, $G^{ij}(\phi)$ is the
inverse of $G_{ij}(\phi)$,
$\Gamma^{i}_{\;jk}=\frac{1}{2}G^{il}(G_{lj,k}+G_{lk,j}-G_{jk,l})$ are the
associated Christoffel symbols,

$Q^{2}(\phi)\mathrel{:=}t^{AB}(\phi)Q_{A}Q_{B},$ (A.13)

and

$k_{N}\mathrel{:=}\frac{d-3}{d-2}\kappa^{2}$ (A.14)

is the rationalized Newton force constant (such that
$F_{\text{grav}}=-\frac{k_{N}}{V_{d-2}}\frac{mm^{\prime}}{r^{d-2}}$).

Note that (A.10–A.12) are $z$-translation invariant,212121To preserve the
boundary condition $\psi_{\infty}=\psi(z=0)=0$ we then shift $\psi$ and
rescale $z$ to compensate. so for any solution passing through a point
$\phi^{i}_{0}=\phi^{i}(z_{0})$ there is corresponding solution
${\phi^{i}}^{\prime}(z)=\phi^{i}(z+z_{0})$ with
${\phi^{i}_{\infty}}^{\prime}=\phi^{i}_{0}$. Moreover, (A.12) is a consistent
constraint in that the derivative of $f$ times (A.12) is a linear combination
of (A.10, A.11).

When $r_{h}>0$ ($z_{h}$ is finite), a smooth horizon requires
$\dot{\psi}(z_{h})=-k_{N}z_{h}\mathrm{e}^{2\psi_{h}}Q^{2}(\phi_{h})\leq 0$ by
evaluating (A.11) at $z=z_{h}$ where $\psi_{h}=\psi(z_{h})$. Likewise, when
$r_{h}=0$ ($z_{h}=\infty$), a smooth horizon requires
$z\mathrm{e}^{\psi}\propto R^{-(d-3)}$ to approach a non-zero constant as
$z\to\infty$, hence $\psi\to-\log z+\text{constant}$, implying that
$\dot{\psi}\to-\frac{1}{z}<0$. Combining (A.11, A.12),

$\ddot{\psi}=\dot{\psi}^{2}+k_{N}G_{ij}(\phi)\dot{\phi}^{i}\dot{\phi}^{j},$
(A.15)

hence $\ddot{\psi}\geq 0$ and we conclude that

$\dot{\psi}\leq 0\quad\text{for all}\quad z\leq z_{h}$ (A.16)

is required for a smooth horizon. Conversely, (A.16) together with
$\ddot{\psi}\geq 0$ and $\psi_{\infty}=\psi(z=0)=0$ gives
$z\dot{\psi}_{\infty}\leq\psi(z)\leq 0$ for $0\leq z\leq z_{h}$, so when
$r_{h}>0$ ($z_{h}$ is finite), $\psi_{h}$ is finite and the horizon is smooth.

Because a condition of the form (A.16) is preserved under limits, singular
black holes must also satisfy (A.16). Likewise, because any $r_{h}>0$ solution
satisfying (A.16) is smooth, $r_{h}=0$ solutions satisfying (A.16) are limits
of smooth solutions, and so solutions to (A.10)–(A.12) are (possibly singular)
black holes if and only if (A.16) holds.

Such black hole solutions have ADM mass

$M=k_{N}^{-1}\bigl{[}-\dot{\psi}_{\infty}+\frac{1}{2z_{h}}\bigr{]}$ (A.17)

(positive by (A.16)) and surface gravity and horizon area

$\displaystyle
g_{h}=\frac{d-3}{2r_{h}}\mathrm{e}^{\frac{d-2}{d-3}\psi_{h}},\;\;\;A_{h}=V_{d-2}r_{h}^{d-2}\mathrm{e}^{-\frac{d-2}{d-3}\psi_{h}}\;\;\;\Longrightarrow\;\;\;g_{h}A_{h}=(d-3)V_{d-2}r_{h}^{d-3}=\frac{1}{2z_{h}}\,.$
(A.18)

Therefore $r_{h}=0$ ($z_{h}=\infty$) is the quasiextremal case, with
coincident (possibly singular) inner and outer horizons, and $r_{h}>0$
($z_{h}$ finite) is the nonextremal (invariably smooth) case.

####  A.1.1 Magnetic charge

In 4d, spherical symmetry allows black holes to carry both electric and
magnetic charge. The theta term—which had no effect on purely electrically
charged black holes—then becomes important

$S=S_{0}-\frac{1}{8\pi^{2}}\int\theta_{AB}(\phi)F_{2}^{A}\wedge F_{2}^{B}.$
(A.19)

The electric and magnetic charges are defined by

$\displaystyle Q_{A}=\oint\left[t_{AB}\star
F^{B}+\frac{\theta_{AB}}{4\pi^{2}}F^{B}\right],~{}~{}~{}~{}~{}~{}\tilde{Q}^{A}$
$\displaystyle=\frac{1}{2\pi}\oint F^{A}.$ (A.20)

The black hole equations (A.10–A.12) take the same form as before (see, e.g.,
[73]) but with

$\displaystyle
Q^{2}(\phi)=t^{AB}(\phi)\biggl{[}Q_{A}-\frac{\theta_{AC}(\phi)}{2\pi}\tilde{Q}^{C}\biggr{]}\biggl{[}Q_{B}-\frac{\theta_{BD}(\phi)}{2\pi}\tilde{Q}^{D}\biggr{]}+4\pi^{2}t_{AB}(\phi)\tilde{Q}^{A}\tilde{Q}^{B}.$
(A.21)

####  A.1.2 Black branes

Generalizing to homogenous, isotropic, and spherically symmetric black
$(p-1)$-branes, the relevant effective action is

$\displaystyle
S=\int\textrm{d}^{d}x\sqrt{-g}\biggl{[}\frac{1}{2\kappa^{2}}R-\frac{1}{2}t_{AB}(\phi)F^{A}_{p+1}\cdot
F^{B}_{p+1}-\frac{1}{2}G_{ij}(\phi)\nabla\phi^{i}\cdot\nabla\phi^{j}\biggr{]}.$
(A.22)

With the appropriate ansatz (see [73] with
$z^{(\text{here})}=z^{(\text{there})}/V_{d-p-1}$) the black hole equations
again take the form (A.10–A.12) with $k_{N}$ replaced by the rationalized
gravitational force constant for $(p-1)$-branes
$k_{(p)}=\frac{p(d-p-2)}{d-2}\kappa^{2}$. Magnetic charge can be added
consistent with spherical symmetry when $d=2p+2$; see [73] for details.

###  A.2 Quasiextremal black holes

The quasiextremal case, in which $f(r)=1$, has several interesting properties
that play an important role in determining the extremality bound.

####  A.2.1 Vanishing self-force

Evaluating (A.12) at $z=0$ ($r=\infty$), we obtain

$k_{N}M^{2}+G^{ij}_{\infty}\mu_{i}\mu_{j}=t^{AB}_{\infty}Q_{A}Q_{B}$ (A.23)

where $\mu_{i}$ is the scalar charge appearing in, e.g.,
$\phi^{i}(z)=\phi^{i}_{\infty}-G^{ij}_{\infty}\mu_{j}z+O(z^{2})$. Thus, the
long-range self-force between an identical pair of quasiextremal black holes
vanishes [73]. As a corollary,
$M=\sqrt{\frac{1}{k_{N}}(t^{AB}_{\infty}Q_{A}Q_{B}-G^{ij}_{\infty}\mu_{i}\mu_{j})}\leqslant\sqrt{\frac{1}{k_{N}}t^{AB}_{\infty}Q_{A}Q_{B}}$,