Convexity and Liberation at Large Spin

TL;DR

This paper explores the structure of higher-dimensional conformal field theories, revealing convexity properties, large spin behavior, and additive spectra, with implications for various models including the 3d Ising and theories with gravity duals.

Contribution

It provides an analytical characterization of the large spin spectrum, convexity properties, and the additive nature of twists in CFTs, supported by comparisons with perturbative and holographic results.

Findings

Convexity of operator dimensions in the gapped phase.

Large spin operators become effectively free, with 1/s as a weak coupling parameter.

Spectrum exhibits additivity, with twists approaching sums of lower twists.

Abstract

We consider several aspects of unitary higher-dimensional conformal field theories (CFTs). We first study massive deformations that trigger a flow to a gapped phase. Deep inelastic scattering in the gapped phase leads to a convexity property of dimensions of spinning operators of the original CFT. We further investigate the dimensions of spinning operators via the crossing equations in the light-cone limit. We find that, in a sense, CFTs become free at large spin and 1/s is a weak coupling parameter. The spectrum of CFTs enjoys additivity: if two twists tau_1, tau_2 appear in the spectrum, there are operators whose twists are arbitrarily close to tau_1+tau_2. We characterize how tau_1+tau_2 is approached at large spin by solving the crossing equations analytically. We find the precise form of the leading correction, including the prefactor. We compare with examples where these observables were computed in perturbation theory, or via gauge-gravity duality, and find complete agreement. The crossing equations show that certain operators have a convex spectrum in twist space. We also observe a connection between convexity and the ratio of dimension to charge. Applications include the 3d Ising model, theories with a gravity dual, SCFTs, and patterns of higher spin symmetry breaking.