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Higher Spin ANEC and the Space of CFTs

David Meltzer

TL;DR

The paper develops and exploits higher spin ANEC as a robust set of positivity constraints on the leading Regge trajectory in unitary CFTs, deriving new bounds on spinning, charged operators and showing that saturation of Hofman-Maldacena bounds implies higher-spin symmetry. By combining light-cone OPE, Rindler positivity, and the analytic bootstrap, it connects HS ANEC to OPE data, including three-point couplings, and tests these constraints in Ising-like 3d CFTs and holographic CFTs with large spin, clarifying when HS ANEC holds at large and finite spin. It also reveals that HS ANEC constrains the interplay between TTT and higher-spin correlators, and demonstrates that ANEC saturation generally forbids nontrivial spin-4 couplings unless higher-spin symmetry is present. Overall, the results illuminate how causality and positivity encode deep structural features of CFTs and their AdS duals, and establish a program to probe the space of consistent theories using HS ANEC and analytic bootstrap.

Abstract

We study the positivity properties of the leading Regge trajectory in higher-dimensional, unitary, conformal field theories (CFTs). These conditions correspond to higher spin generalizations of the averaged null energy condition (ANEC). By studying higher spin ANEC, we will derive new bounds on the dimensions of charged, spinning operators and prove that if the Hofman-Maldacena bounds are saturated, then the theory has a higher spin symmetry. We also derive new, general bounds on CFTs, with an emphasis on theories whose spectrum is close to that of a generalized free field theory. As an example, we consider the Ising CFT and show how the OPE structure of the leading Regge trajectory is constrained by causality. Finally, we use the analytic bootstrap to perform additional checks, in a large class of CFTs, that higher spin ANEC is obeyed at large and finite spin. In the process, we calculate corrections to large spin OPE coefficients to one-loop and higher in holographic CFTs.

Higher Spin ANEC and the Space of CFTs

TL;DR

The paper develops and exploits higher spin ANEC as a robust set of positivity constraints on the leading Regge trajectory in unitary CFTs, deriving new bounds on spinning, charged operators and showing that saturation of Hofman-Maldacena bounds implies higher-spin symmetry. By combining light-cone OPE, Rindler positivity, and the analytic bootstrap, it connects HS ANEC to OPE data, including three-point couplings, and tests these constraints in Ising-like 3d CFTs and holographic CFTs with large spin, clarifying when HS ANEC holds at large and finite spin. It also reveals that HS ANEC constrains the interplay between TTT and higher-spin correlators, and demonstrates that ANEC saturation generally forbids nontrivial spin-4 couplings unless higher-spin symmetry is present. Overall, the results illuminate how causality and positivity encode deep structural features of CFTs and their AdS duals, and establish a program to probe the space of consistent theories using HS ANEC and analytic bootstrap.

Abstract

We study the positivity properties of the leading Regge trajectory in higher-dimensional, unitary, conformal field theories (CFTs). These conditions correspond to higher spin generalizations of the averaged null energy condition (ANEC). By studying higher spin ANEC, we will derive new bounds on the dimensions of charged, spinning operators and prove that if the Hofman-Maldacena bounds are saturated, then the theory has a higher spin symmetry. We also derive new, general bounds on CFTs, with an emphasis on theories whose spectrum is close to that of a generalized free field theory. As an example, we consider the Ising CFT and show how the OPE structure of the leading Regge trajectory is constrained by causality. Finally, we use the analytic bootstrap to perform additional checks, in a large class of CFTs, that higher spin ANEC is obeyed at large and finite spin. In the process, we calculate corrections to large spin OPE coefficients to one-loop and higher in holographic CFTs.

Paper Structure

This paper contains 21 sections, 117 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Summary
  3. Review of the Lightcone OPE and HS ANEC
  4. Bounds on Higher Spin Couplings
  5. Bounds for Charged Operators
  6. HS ANEC Examples
  7. Spin-2/Spin-4 Mixed Systems
  8. Saturation of ANEC
  9. Application to the 3d Ising CFT
  10. Comparison to Analytic Bootstrap
  11. $\mathrm{SL}(2,\mathbb{R})$ Expansion
  12. Isolated Operator Exchange
  13. Double-twist Exchange
  14. Conclusion
  15. Integrals of Three-Point Functions
  16. ...and 6 more sections

Figures (5)

  • Figure 1: $s=t$ crossing equation for $c_{\phi\phi[\phi\phi]_{0,s}}c_{\psi\psi[\phi\phi]_{0,s}}$ with isolated operator in $t$-channel.
  • Figure 2: Crossing equation with double-twist dominance in $t$-channel
  • Figure 3: Correction to $[\phi\psi]$ due to $T$ exchange
  • Figure 4: A universal AdS loop diagram for theories with gravity. We label AdS fields by their dual CFT operators.
  • Figure 5: Ladder diagram for graviton exchange in AdS.