On one hand, this index may be related to the Euclidean partition function of the theory on S3×S1 with complex chemical potentials, which maps by the AdS/CFT correspondence to a sum over Euclidean gravity solutions. The superconformal index of the N=4 SU(N) supersymmetric Yang-Mills theory counts the 1/16-BPS states in this theory, and has been used via the AdS/CFT correspondence to count black hole microstates of 1/16-BPS black holes. We do not yet understand from the CFT point of view why this is true. In all examples that we tested we find that the conjecture holds. 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 semiclassical methods. 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 formulation is particularly interesting in anti–de Sitter space, because it has a simple conformal field theory (CFT) dual formulation: let Δ(q) be the dimension of the lowest-dimension operator with charge q under some global U(1) symmetry, then Δ(q) must be a convex function of q. We propose a closely related, but distinct, formulation, which is that it should correspond to a particle with non-negative self-binding energy.
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. The weak gravity conjecture is typically stated as a bound on the mass-to-charge ratio of a particle in the theory. We construct an explicit bulk dual in anti-de Sitter space, with couplings of order 1/N, for the SU(N)-singlet sector of QED in d space-time dimensions (2