Einstein gravity 3-point functions from conformal field theory

TL;DR

<3-5 sentence high-level summary> In four-dimensional, large-N conformal field theories with a sparse spectrum, the paper investigates why stress-tensor three-point functions must adopt an Einstein-gravity form to preserve causality. Using conformal block analysis, Regge and bulk-point kinematics, wavepacket smearing, and the chaos bound, the authors show that stress-tensor exchange alone would violate causality unless the TT structure matches Einstein gravity, fixing a = c. They analyze possible cancellations by other operators and conclude that either an infinite tower of higher-spin states above the gap must exist (suppressing deviations by Δ_gap^{-2}) or the TT couplings are exactly Einstein-like. The work thus connects bulk locality, causality constraints, and holographic universality in large-N CFTs, providing a CFT-based rationale for Einstein-gravity behavior in holographic duals.

Abstract

We study stress tensor correlation functions in four-dimensional conformal field theories with large $N$ and a sparse spectrum. Theories in this class are expected to have local holographic duals, so effective field theory in anti-de Sitter suggests that the stress tensor sector should exhibit universal, gravity-like behavior. At the linearized level, the hallmark of locality in the emergent geometry is that stress tensor three-point functions $\langle TTT\rangle$, normally specified by three constants, should approach a universal structure controlled by a single parameter as the gap to higher spin operators is increased. We demonstrate this phenomenon by a direct CFT calculation. Stress tensor exchange, by itself, violates causality and unitarity unless the three-point functions are carefully tuned, and the unique consistent choice exactly matches the prediction of Einstein gravity. Under some assumptions about the other potential contributions, we conclude that this structure is universal, and in particular, that the anomaly coefficients satisfy $a\approx c$ as conjectured by Camanho et al. The argument is based on causality of a four-point function, with kinematics designed to probe bulk locality, and invokes the chaos bound of Maldacena, Shenker, and Stanford.

Figures (1)

• Figure 1: The lightcone limit is $v \to 0$ with $u$ fixed. The Regge limit is $v \to 0$ with $uv$ fixed. In general, these limits do not commute.