More on the Weak Gravity Conjecture via Convexity of Charged Operators

TL;DR

This paper investigates the convexity of the conformal dimension function of charged operators in conformal field theories, testing the convex charge conjecture through semiclassical computations across various models and dimensions.

Contribution

It refines and extends the convex charge conjecture testing by analyzing semiclassical contributions beyond leading order and exploring models with complex conformal dimensions.

Findings

Convexity holds for the real part of complex conformal dimensions.

Semiclassical computations confirm the convex charge conjecture in various models.

Models exhibit rich dynamics with phase transitions affecting conformal dimensions.

Abstract

The Weak Gravity Conjecture has recently been re-formulated in terms of a particle with non-negative self-binding energy. Because of the dual conformal field theory (CFT) formulation in the anti-de Sitter space the conformal dimension $\Delta (Q)$ of the lowest-dimension operator with charge Q under some global U(1) symmetry must be a convex function of Q. This property has been conjectured to hold for any (unitary) conformal field theory and generalized to larger global symmetry groups. Here we refine and further test the convex charge conjecture via semiclassical computations for fixed charge sectors of different theories in different dimensions. We analyze the convexity properties of the leading and next-to-leading order terms stemming from the semiclassical computation, de facto, extending previous tests beyond the leading perturbative contributions and to arbitrary charges. In particular, the leading contribution is sufficient to test convexity in the semiclassical computations. We also consider intriguing cases in which the models feature a transition from real to complex conformal dimensions either as a function of the charge or number of matter fields. As a relevant example of the first kind, we investigate the $O(N)$ model in $4+\epsilon$ dimensions. As an example of the second type we consider the $U(N)\times U(M)$ model in $4-\epsilon$ dimensions. Both models display a rich dynamics where, by changing the number of matter fields and/or charge, one can achieve dramatically different physical regimes. We discover that whenever a complex conformal dimension appears, the real part satisfies the convexity property.