On Convexity of Charged Operators in CFTs and the Weak Gravity Conjecture

Abstract

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 ∆(q) be the dimension of the lowest-dimension operator with charge q under some global U (1) symmetry, then ∆(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.

Corrections to Extremal Black Holes from Iyer-Wald Formalism

“…However, because they appear as boundary terms they have support on the bifurcation surface of the black hole and therefore do contribute to the Wald entropy. This is consistent with (33) which only involves the Bekenstein-Hawking entropy, but seemingly in conflict with entropy/extremality relationships involving the Wald entropy [22]. The resolution lies in the fact that in a microcanonical ensemble, higher-derivative corrections only shift the temperature of a black hole when its area is corrected.…”

“…Similar relationships between entropy and extremality corrections have appeared before in the literature in various guises and have been derived by studying explicit corrections to black hole solutions as well as by using a thermodynamic approach [5-8, 14, 18, 19, 22] Let us now comment upon the validity and use of (33). First, it should be stressed that the order of limits in the entropy correction is important.…”

“…In particular, the WGC does not follow from requiring entropy corrections to be positive in the presence of topological terms. Nonetheless, we still believe (33) to be interesting because in many cases of interest the correction to the temperature of an extremal black hole in a microcanonical ensemble scales as T ∼ O( √ α i ), where α i denotes a Wilson coefficient. In that case, the shift in the area also scales with a square root leading to δS BH ∼ O( √ α i ) such that δS = δS BH + O(α i ).…”

“…From this perspective, it might be more appropriate to refer to (33) as an area/extremality relation.…”

“…However, for black holes that are asymptotically AdS the correct formulation of the WGC is less clear although proposals exist, such as [31,32]. Recently, a version of the WGC in AdS has been formulated [33] that is inspired by the Repulsive Force Conjecture (RFC) [34,35]. The RFC states that for two identical charged particles, their total long-range force should be repulsive rather than attractive (or vanish).…”