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The Analytic Bootstrap and AdS Superhorizon Locality

A. Liam Fitzpatrick, Jared Kaplan, David Poland, David Simmons-Duffin

Abstract

We take an analytic approach to the CFT bootstrap, studying the 4-pt correlators of d > 2 dimensional CFTs in an Eikonal-type limit, where the conformal cross ratios satisfy |u| << |v| < 1. We prove that every CFT with a scalar operator φmust contain infinite sequences of operators O_{τ,l} with twist approaching τ-> 2Δ_φ+ 2n for each integer n as l -> infinity. We show how the rate of approach is controlled by the twist and OPE coefficient of the leading twist operator in the φx φOPE, and we discuss SCFTs and the 3d Ising Model as examples. Additionally, we show that the OPE coefficients of other large spin operators appearing in the OPE are bounded as l -> infinity. We interpret these results as a statement about superhorizon locality in AdS for general CFTs.

The Analytic Bootstrap and AdS Superhorizon Locality

Abstract

We take an analytic approach to the CFT bootstrap, studying the 4-pt correlators of d > 2 dimensional CFTs in an Eikonal-type limit, where the conformal cross ratios satisfy |u| << |v| < 1. We prove that every CFT with a scalar operator φmust contain infinite sequences of operators O_{τ,l} with twist approaching τ-> 2Δ_φ+ 2n for each integer n as l -> infinity. We show how the rate of approach is controlled by the twist and OPE coefficient of the leading twist operator in the φx φOPE, and we discuss SCFTs and the 3d Ising Model as examples. Additionally, we show that the OPE coefficients of other large spin operators appearing in the OPE are bounded as l -> infinity. We interpret these results as a statement about superhorizon locality in AdS for general CFTs.

Paper Structure

This paper contains 26 sections, 125 equations.

Table of Contents

  1. Introduction and Review
  2. Lightning Bootstrap Review
  3. The Bootstrap and Large $\ell$ Operators
  4. An Elementary Illustration from Mean Field Theory
  5. An Elementary Illustration from Mean Field Theory
  6. Existence of Twist $2 \Delta_{\phi} + 2n + \gamma(n, \ell)$ Operators at Large $\ell$
  7. Relation to Numerical Results and the 3d Ising Model
  8. Properties of Isolated Towers of Operators
  9. Implications for SCFTs in 4d
  10. Bounding Contributions from Operators with General Twists
  11. Failure in Two Dimensions
  12. AdS Interpretation
  13. Discussion
  14. Properties of Conformal Blocks
  15. Factorization at Large $\ell$ and Small $u$
  16. ...and 11 more sections