  The Weak Gravity Conjecture postulates the existence of superextremal charged
particles, i.e. those with mass smaller than or equal to their charge in Planck
units. We present further evidence for our recent observation that in known
examples a much stronger statement is true: an infinite tower of superextremal
particles of different charges exists. We show that effective Kaluza-Klein
field theories and perturbative string vacua respect the Sublattice Weak
Gravity Conjecture, namely that a finite index sublattice of the full charge
lattice exists with a superextremal particle at each site. In perturbative
string theory we show that this follows from modular invariance. However, we
present counterexamples to the stronger possibility that a superextremal
particle exists at every lattice site, including an example in which the
lightest charged particle is subextremal. The Sublattice Weak Gravity
Conjecture has many implications both for abstract theories of quantum gravity
and for real-world physics. For instance, it implies that if a gauge group with
very small coupling $e$ exists, then the fundamental gravitational cutoff
energy of the theory is no higher than $\sim e^{1/3} M_{\rm Pl}$.
