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Spinning Conformal Correlators

Miguel S. Costa, Joao Penedones, David Poland, Slava Rychkov

TL;DR

Spinning Conformal Correlators develops an embedding-space, index-free framework to handle symmetric traceless tensor operators in conformal field theories. The approach encodes tensors as polynomials in polarization vectors and uses embedding-space building blocks to construct all conformally invariant tensor structures for n-point functions, including conservation constraints. A key result is a precise counting correspondence: the number of three-point tensor structures matches the number of on-shell spinning-particle S-matrix structures in one higher dimension, and a practical counting formula for three-point structures is derived, with special cases in 3D and parity-odd sectors. The paper also connects these tensor-structure counts to AdS/CFT via bulk vertices and discusses implications for the conformal bootstrap and tensor operator analyses.

Abstract

We develop the embedding formalism for conformal field theories, aimed at doing computations with symmetric traceless operators of arbitrary spin. We use an index-free notation where tensors are encoded by polynomials in auxiliary polarization vectors. The efficiency of the formalism is demonstrated by computing the tensor structures allowed in n-point conformal correlation functions of tensors operators. Constraints due to tensor conservation also take a simple form in this formalism. Finally, we obtain a perfect match between the number of independent tensor structures of conformal correlators in d dimensions and the number of independent structures in scattering amplitudes of spinning particles in (d+1)-dimensional Minkowski space.

Spinning Conformal Correlators

TL;DR

Spinning Conformal Correlators develops an embedding-space, index-free framework to handle symmetric traceless tensor operators in conformal field theories. The approach encodes tensors as polynomials in polarization vectors and uses embedding-space building blocks to construct all conformally invariant tensor structures for n-point functions, including conservation constraints. A key result is a precise counting correspondence: the number of three-point tensor structures matches the number of on-shell spinning-particle S-matrix structures in one higher dimension, and a practical counting formula for three-point structures is derived, with special cases in 3D and parity-odd sectors. The paper also connects these tensor-structure counts to AdS/CFT via bulk vertices and discusses implications for the conformal bootstrap and tensor operator analyses.

Abstract

We develop the embedding formalism for conformal field theories, aimed at doing computations with symmetric traceless operators of arbitrary spin. We use an index-free notation where tensors are encoded by polynomials in auxiliary polarization vectors. The efficiency of the formalism is demonstrated by computing the tensor structures allowed in n-point conformal correlation functions of tensors operators. Constraints due to tensor conservation also take a simple form in this formalism. Finally, we obtain a perfect match between the number of independent tensor structures of conformal correlators in d dimensions and the number of independent structures in scattering amplitudes of spinning particles in (d+1)-dimensional Minkowski space.

Paper Structure

This paper contains 28 sections, 201 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Embedding Formalism
  3. Correlators: Simplest Examples
  4. Encoding Tensors by Polynomials
  5. Tensors in the Physical Space
  6. Tensors in the Embedding Space
  7. Example
  8. Correlation Functions of Spin $l$ Primaries
  9. Two-Point Functions
  10. Three-Point Functions
  11. Scalar-Scalar-Spin $l$
  12. General Spins $l_1,l_2$ and $l_3$
  13. Parity Odd Three-Point Functions
  14. Relation to Leading OPE Coefficient
  15. Three Dimensions
  16. ...and 13 more sections

Figures (3)

  • Figure 1: Light cone in the embedding space; light rays are in one-to-one correspondence with physical space points. The Poincaré section of the cone is also shown.
  • Figure 2: Schematic representation of one of the tensor structures appearing in the (spin 5)-(spin 3)-(spin 7) three-point function. $V_i$'s are represented as disconnected dots at the vertices and $H_{ij}$'s as lines joining the vertices.
  • Figure 3: Same as Fig. \ref{['Diagram3pt']} but for a five-point function. The isolated dots representing $V$'s appear in several colors because for an $n$-point function there are several possible $V$'s per vertex.