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Counting Conformal Correlators

Petr Kravchuk, David Simmons-Duffin

TL;DR

This work develops a cohesive, group-theoretic framework for classifying conformally invariant tensor structures in CFT$_d$ by gauge-fixing to a conformal frame and counting invariants under stabilizer groups. It provides explicit counting rules linking conformal correlators to invariant subspaces of restricted representations, extends the formalism to conserved currents, and systematically addresses parity and permutation symmetries. The authors supply concrete 3d results, detailed analyses of three- and four-point functions, and a clear mapping to flat-space scattering amplitudes in $d{+}1$ dimensions, establishing a robust toolbox for bootstrap applications. The approach yields both structural classifications and explicit bases, enabling practical construction of tensor structures and streamlined crossing equations, with illustrative examples including four Majorana fermions. Overall, the paper offers a unifying representation-theoretic method to organize conformal tensor structures across dimensions and signatures, with direct implications for bootstrap program efficiency and amplitude correspondence.

Abstract

We introduce simple group-theoretic techniques for classifying conformally-invariant tensor-structures. With them, we classify tensor structures of general n-point functions of non-conserved operators, and $n\geq 4$-point functions of general conserved currents, with or without permutation symmetries, and in any spacetime dimension d. (The case n=3 for conserved operators will appear in subsequent work.) Our techniques are useful for bootstrap applications. The rules we derive simultaneously count tensor structures for flat-space scattering amplitudes in d+1 dimensions.

Counting Conformal Correlators

TL;DR

This work develops a cohesive, group-theoretic framework for classifying conformally invariant tensor structures in CFT by gauge-fixing to a conformal frame and counting invariants under stabilizer groups. It provides explicit counting rules linking conformal correlators to invariant subspaces of restricted representations, extends the formalism to conserved currents, and systematically addresses parity and permutation symmetries. The authors supply concrete 3d results, detailed analyses of three- and four-point functions, and a clear mapping to flat-space scattering amplitudes in dimensions, establishing a robust toolbox for bootstrap applications. The approach yields both structural classifications and explicit bases, enabling practical construction of tensor structures and streamlined crossing equations, with illustrative examples including four Majorana fermions. Overall, the paper offers a unifying representation-theoretic method to organize conformal tensor structures across dimensions and signatures, with direct implications for bootstrap program efficiency and amplitude correspondence.

Abstract

We introduce simple group-theoretic techniques for classifying conformally-invariant tensor-structures. With them, we classify tensor structures of general n-point functions of non-conserved operators, and -point functions of general conserved currents, with or without permutation symmetries, and in any spacetime dimension d. (The case n=3 for conserved operators will appear in subsequent work.) Our techniques are useful for bootstrap applications. The rules we derive simultaneously count tensor structures for flat-space scattering amplitudes in d+1 dimensions.

Paper Structure

This paper contains 33 sections, 4 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Conformal correlators of long multiplets
  3. Conformal invariance
  4. Conformal frame
  5. $n$-point functions
  6. Parity
  7. Permutation symmetry
  8. Three-point structures
  9. Four-point structures
  10. Summary: tensor structures of long-multiplets
  11. Conservation conditions
  12. Multiple conserved operators and permutation symmetries
  13. Correlation functions in 3d
  14. Conventions for $SO(2,1)$
  15. Three-point structures
  16. ...and 18 more sections

Key Result

Proposition 1

$S_2$-symmetric tensor structures are in one-to-one correspondence with even-spin traceless symmetric tensors in $\widehat{\mathrm{S}}^2\rho_1 \otimes \rho_3$ plus odd-spin traceless-symmetric tensors in $\widehat{\wedge}^2\rho_1 \otimes \rho_3$. Here, $\widehat{\mathrm{S}}^2$ denotes the symmetric

Theorems & Definitions (4)

  • Proposition 1: $S_2$
  • Proposition 2: $S_3$
  • Proposition 3: $\mathbb{Z}_2$ and $\mathbb{Z}_2^2$
  • Theorem 1