A counterexample to the CFT convexity conjecture

TL;DR

This paper constructs a specific weakly-coupled CFT model that serves as a counterexample to the conjecture that the minimal operator dimension at fixed charge is convex, challenging a previous theoretical assumption.

Contribution

It provides the first explicit counterexample to the CFT convexity conjecture, using a clockwork-like model with modifications to ensure weak coupling.

Findings

Counterexample disproves the convexity conjecture

Model demonstrates non-convex behavior of operator dimensions

Discussion of alternative conjectures and applications

Abstract

Motivated by the weak gravity conjecture, arXiv:2108.04594 conjectured that in any CFT, the minimal operator dimension at fixed charge is a convex function of the charge. In this letter we construct a counterexample to this convexity conjecture, which is a clockwork-like model with some modifications to make it a weakly-coupled CFT. We also discuss further possible applications of this model and some modified versions of the conjecture which are not ruled out by the counterexample.