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Scattering Equations in AdS: Scalar Correlators in Arbitrary Dimensions

Lorenz Eberhardt, Shota Komatsu, Sebastian Mizera

Abstract

We introduce a bosonic ambitwistor string theory in AdS space. Even though the theory is anomalous at the quantum level, one can nevertheless use it in the classical limit to derive a novel formula for correlation functions of boundary CFT operators in arbitrary space-time dimensions. The resulting construction can be treated as a natural extension of the CHY formalism for the flat-space S-matrix, as it similarly expresses tree-level amplitudes in AdS as integrals over the moduli space of Riemann spheres with punctures. These integrals localize on an operator-valued version of scattering equations, which we derive directly from the ambitwistor string action on a coset manifold. As a testing ground for this formalism we focus on the simplest case of ambitwistor string coupled to two current algebras, which gives bi-adjoint scalar correlators in AdS. In order to evaluate them directly, we make use of a series of contour deformations on the moduli space of punctured Riemann spheres and check that the result agrees with tree level Witten diagram computations to all multiplicity. We also initiate the study of eigenfunctions of scattering equations in AdS, which interpolate between conformal partial waves in different OPE channels, and point out a connection to an elliptic deformation of the Calogero-Sutherland model.

Scattering Equations in AdS: Scalar Correlators in Arbitrary Dimensions

Abstract

We introduce a bosonic ambitwistor string theory in AdS space. Even though the theory is anomalous at the quantum level, one can nevertheless use it in the classical limit to derive a novel formula for correlation functions of boundary CFT operators in arbitrary space-time dimensions. The resulting construction can be treated as a natural extension of the CHY formalism for the flat-space S-matrix, as it similarly expresses tree-level amplitudes in AdS as integrals over the moduli space of Riemann spheres with punctures. These integrals localize on an operator-valued version of scattering equations, which we derive directly from the ambitwistor string action on a coset manifold. As a testing ground for this formalism we focus on the simplest case of ambitwistor string coupled to two current algebras, which gives bi-adjoint scalar correlators in AdS. In order to evaluate them directly, we make use of a series of contour deformations on the moduli space of punctured Riemann spheres and check that the result agrees with tree level Witten diagram computations to all multiplicity. We also initiate the study of eigenfunctions of scattering equations in AdS, which interpolate between conformal partial waves in different OPE channels, and point out a connection to an elliptic deformation of the Calogero-Sutherland model.

Paper Structure

This paper contains 52 sections, 195 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Note added.
  4. Bosonic ambitwistor string on a coset manifold
  5. Flat space
  6. Reparametrization invariance.
  7. Ambitwistor symmetry.
  8. Group manifold
  9. Classical theory.
  10. Quantum theory.
  11. Coset manifold
  12. Classical theory.
  13. Quantum theory.
  14. Anomalies.
  15. Ghosts.
  16. ...and 37 more sections

Figures (2)

  • Figure 1: Diagrammatic depiction of the recursion relation from \ref{['recursion-relation']}. The new external states (bold) have conformal generators determined by the Ward identities $\sum_i D_i = 0$ for each term on the right-hand side individually.
  • Figure 2: Two examples of Witten diagrams corresponding to the expressions \ref{['eq:simple4pt']} and \ref{['eq:simple5pt']} respectively.