# Analyticity in Spin in Conformal Theories

## TL;DR

This paper introduces an analytic formula for extracting conformal theory data as a function of spin, revealing insights into the spectrum and interactions, especially in large-N and sparse spectrum theories.

## Contribution

It presents a novel spin-analyticity formula for conformal correlators, connecting Lorentzian continuation, Regge bounds, and large-spin expansions, with implications for holography.

## Key findings

- The formula converges thanks to bounds on the high-energy Regge limit.
- In large-N theories, the imaginary part is dominated by single-trace operators.
- Sparse spectra show suppression of bulk higher-derivative interactions.

## Abstract

Conformal theory correlators are characterized by the spectrum and three- point functions of local operators. We present a formula which extracts this data as an analytic function of spin. In analogy with a classic formula due to Froissart and Gribov, it is sensitive only to an "imaginary part" which appears after analytic continuation to Lorentzian signature, and it converges thanks to recent bounds on the high-energy Regge limit. At large spin, substituting in cross-channel data, the formula yields 1/J expansions with controlled errors. In large-N theories, the imaginary part is saturated by single-trace operators. For a sparse spectrum, it manifests the suppression of bulk higher-derivative interactions that constitutes the signature of a local gravity dual in Anti-de-Sitter space.