Goldstone Bosons and Convexity

TL;DR

This paper investigates the spectrum of charged operators in conformal field theories with a $U(1)$ symmetry, demonstrating that the presence of a Goldstone boson in a certain regime enforces convexity in the operator spectrum.

Contribution

It establishes a link between Goldstone boson physics and the convexity of the operator spectrum in CFTs with a $U(1)$ symmetry, a novel theoretical insight.

Findings

Goldstone bosons emerge in the large $f$ regime of CFTs with $U(1)$ symmetry.

Operator spectra must be convex in charge when Goldstone boson physics is consistent.

Convexity is derived from the requirement of consistent Goldstone boson realization.

Abstract

We study the spectrum of scalar charged operators in Conformal Field Theories (CFTs) with a $U(1)$ global symmetry. The charged operators are dual, by the state-operator correspondence, to homogenous charged states on the sphere. Such states can break the $U(1)$ symmetry, and we define what we call the large $f$ regime in the CFT as one where the symmetry breaking scale is much higher than the scale of the CFT sphere. In such a regime, there is (an approximate) Goldstone boson associated to the breaking. We show that consistency of the Goldstone boson physics implies that the spectrum of states, and therefore of operators, must be convex in charge. More precisely, we show that any family of operators of different charges, which are lowest dimension of their charge, and which additionally share the same realisation of the Goldstone boson in terms of the degrees of freedom of the CFT, must be convex.