Moduli Spaces in CFT: Large Charge Operators

TL;DR

This paper establishes a necessary condition for conformal symmetry breaking in CFTs with global symmetries, linking large-charge operator spectra to moduli space properties and illustrating it with 3D supersymmetric examples.

Contribution

It proves a general condition connecting large-charge operator spectra to symmetry breaking, extending known BPS results to non-supersymmetric theories.

Findings

A tower of charged operators with linear scaling dimensions indicates symmetry breaking.

In supersymmetric theories, the linear spectrum follows directly from BPS conditions.

Large charge spectra relate to massive particle spectra on the moduli space.

Abstract

Using the large-charge expansion, we prove a necessary condition for a CFT to exhibit conformal symmetry breaking, under the assumption that a continuous global symmetry is ${\it also}$ broken on the moduli space: there must be a tower of charged local operators whose scaling dimensions are asymptotically linear in the charge. In supersymmetric theories with a continuous R-symmetry and a holomorphic moduli space, the existence of such a tower of operators follows trivially from a BPS condition: their scaling dimensions are then exactly linear in the R-charge. We illustrate the more general statement in several examples of three-dimensional ${\cal N}=1$ CFTs, where the leading linear behavior receives nontrivial corrections. By considering a suitable scaling limit, we also relate the spectrum of states with large charge on the cylinder (isomorphic to local operators) to the spectrum of massive particles on the moduli space.