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

TL;DR

The paper proposes a new convexity-based formulation of the Weak Gravity Conjecture in CFTs, suggesting that the spectrum of operator dimensions with charge should be convex, a property verified in various examples.

Contribution

It introduces the Charge Convexity Conjecture, asserting convexity of operator dimensions with charge in all unitary CFTs, extending beyond holographic duals.

Findings

Conjecture holds in tested examples via perturbation, $1/N$, and semi-classical methods.

Convexity of $\Delta(q)$ observed in all examined cases.

Proposes a universal property of CFT spectra related to 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 $\Delta(q)$ 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.