[Paper Review] On Convexity of Charged Operators in CFTs and the Weak Gravity Conjecture

Ofer Aharony, Eran Palti|arXiv (Cornell University)|Aug 10, 2021

Black Holes and Theoretical PhysicsPhysics and Astronomy84 references53 citations

TL;DR

This paper proposes the Charge Convexity Conjecture: in any unitary conformal field theory (CFT) with a global U(1) symmetry, the scaling dimension ∆(q) of the lowest-dimension operator of charge q must be a convex function of q. The conjecture arises from a reformulation of the Weak Gravity Conjecture in terms of non-negative self-binding energy, which dualizes to convexity in the CFT spectrum. The authors test the conjecture across multiple CFTs using perturbation theory, 1/N expansions, and semi-classical methods, finding consistent support across all tested examples.

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.

Motivation & Objective

• To propose a new, universal formulation of the Weak Gravity Conjecture in terms of non-negative self-binding energy for charged particles.
• To translate this gravitational conjecture into a CFT dual statement: the scaling dimension ∆(q) of charged operators must be convex in charge q.
• To test the convexity conjecture across diverse CFTs using perturbative and large-N methods.
• To establish that this convexity property holds in all tested unitary CFTs, even those without weakly-curved gravitational duals.
• To motivate a deeper CFT-based understanding of why such convexity should hold universally.

Proposed method

• Propose the Positive Binding Conjecture: a charged particle must have non-negative self-binding energy, defined as the energy difference between a two-particle state and twice the one-particle energy.
• Map this binding energy condition to the CFT side via holography, leading to the requirement that ∆(q) be convex in q for U(1) global symmetry.
• Generalize the conjecture to non-Abelian global symmetries using symmetric product representations: ∆(Sym^n(r₀)) must be convex in n.
• Test the conjecture in explicit CFTs using perturbation theory (e.g., 4−ϵ, 3−ϵ), 1/N expansions, and semi-classical methods.
• Compute one-loop anomalous dimensions of multi-meson operators and verify that their contributions to the scaling dimension are convex.
• Analyze cancellation patterns in Feynman diagrams (e.g., gluon exchange between mesons) that lead to vanishing or positive net anomalous dimensions.

Experimental results

Research questions

• RQ1 Does the scaling dimension ∆(q) of the lowest-dimension operator with charge q exhibit convexity in q for any unitary CFT with a U(1) global symmetry?
• RQ2 Can the Weak Gravity Conjecture be reformulated in terms of non-negative self-binding energy of charged states, and does this formulation extend beyond AdS and weakly-curved gravity?
• RQ3 Is the convexity of ∆(q) robust across diverse CFTs, including those with non-trivial dynamics like O(N) models and Chern-Simons theories?
• RQ4 Why does the convexity condition hold in all tested unitary CFTs, and is there a fundamental CFT mechanism underlying it?
• RQ5 Does the conjecture fail in non-unitary theories, and can this be used as a diagnostic for unitarity?

Key findings

• The Charge Convexity Conjecture holds in all tested examples of unitary CFTs, including the U(1) model in 4−ϵ dimensions, the quartic O(N) model in 4−ϵ, and the sextic O(N) model in 3−ϵ dimensions.
• In the large-N O(N) model in d=5, the conjecture fails, consistent with the known non-unitarity of the full CFT despite perturbative unitarity.
• One-loop anomalous dimensions for multi-meson operators in gauge theories with scalars show cancellation between diagrams with opposite gluon exchange signs, leading to a net positive contribution from φ⁴ interactions.
• The conjecture is satisfied in supersymmetric theories and in 3d U(Nc) gauge theories with fermions, where mixing between operators of different spin must be carefully accounted for.
• The conjecture is not satisfied in non-unitary theories such as the O(N) model in 4 < d < 6, suggesting convexity may be a signature of unitarity.
• The conjecture is robust under various expansions (perturbative, 1/N, semi-classical), and no counterexamples have been found in unitary CFTs.