  The Weak Gravity Conjecture is typically stated as a bound on the
mass-to-charge ratio of a particle in the theory. Alternatively, it has been
proposed that its natural formulation is in terms of the existence of a
particle which is self-repulsive under all long-range forces. We propose a
closely related, but distinct, formulation, which is that it should correspond
to a particle with non-negative self-binding energy. This formulation is
particularly interesting in anti-de Sitter space, because it has a simple
conformal field theory (CFT) dual formulation: let $\Delta(q)$ be the dimension
of the lowest-dimension operator with charge $q$ under some global $U(1)$
symmetry, then $\Delta(q)$ must be a convex function of $q$. This formulation
avoids any reference to holographic dual forces or even to locality in
spacetime, and so we make a wild leap, and conjecture that such convexity of
the spectrum of charges holds for any (unitary) conformal field theory, not
just those that have weakly coupled and weakly curved duals. This Charge
Convexity Conjecture, and its natural generalization to larger global symmetry
groups, can be tested in various examples where anomalous dimensions can be
computed, by perturbation theory, $1/N$ expansions and semi-classical methods.
In all examples that we tested we find that the conjecture holds. We do not yet
understand from the CFT point of view why this is true.
