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Worldsheet Scattering in AdS_5 x S^5

Thomas Klose, Tristan McLoughlin, Radu Roiban, Konstantin Zarembo

TL;DR

This work computes the leading-order worldsheet S-matrix for the AdS_5 x S^5 sigma-model in a light-cone gauge and reveals that the symmetry generators act nonlocally, promoting PSU(2|2)^2 to a Hopf algebra with a nontrivial coproduct. The authors derive explicit tree-level S-matrix elements, show absence of particle production in the bosonic sector, and demonstrate invariance under the Hopf-algebra action, connecting the worldsheet results to the strong-coupling limit of the gauge-theory spin chain S-matrix. A detailed comparison with Beisert’s SU(2|2) S-matrix clarifies the role of length-changing effects (Z markers) and dressing phases, and provides a framework to extend perturbative calculations toward the full integrable structure of the AdS/CFT correspondence. The study highlights the interplay between gauge choices, nonlocal symmetry actions, and integrability, pointing to future work on loops, crossing symmetry, and the analytic structure of the S-matrix.

Abstract

We calculate the S-matrix in the gauge-fixed sigma-model on AdS_5 x S^5 to the leading order in perturbation theory, and analyze how supersymmetry is realized on the scattering states. A mild nonlocality of the supercharges implies that their action on multi-particle states does not follow the Leibniz rule, which is replaced by a nontrivial coproduct. The plane wave symmetry algebra is thus naturally enhanced to a Hopf algebra. The scattering matrix elements obey the classical Yang-Baxter equation modified by the existence of the coproduct. This structure mirrors that of the large 't Hooft coupling expansion of the S-matrix for the spin chain in the dual super-Yang-Mills theory.

Worldsheet Scattering in AdS_5 x S^5

TL;DR

This work computes the leading-order worldsheet S-matrix for the AdS_5 x S^5 sigma-model in a light-cone gauge and reveals that the symmetry generators act nonlocally, promoting PSU(2|2)^2 to a Hopf algebra with a nontrivial coproduct. The authors derive explicit tree-level S-matrix elements, show absence of particle production in the bosonic sector, and demonstrate invariance under the Hopf-algebra action, connecting the worldsheet results to the strong-coupling limit of the gauge-theory spin chain S-matrix. A detailed comparison with Beisert’s SU(2|2) S-matrix clarifies the role of length-changing effects (Z markers) and dressing phases, and provides a framework to extend perturbative calculations toward the full integrable structure of the AdS/CFT correspondence. The study highlights the interplay between gauge choices, nonlocal symmetry actions, and integrability, pointing to future work on loops, crossing symmetry, and the analytic structure of the S-matrix.

Abstract

We calculate the S-matrix in the gauge-fixed sigma-model on AdS_5 x S^5 to the leading order in perturbation theory, and analyze how supersymmetry is realized on the scattering states. A mild nonlocality of the supercharges implies that their action on multi-particle states does not follow the Leibniz rule, which is replaced by a nontrivial coproduct. The plane wave symmetry algebra is thus naturally enhanced to a Hopf algebra. The scattering matrix elements obey the classical Yang-Baxter equation modified by the existence of the coproduct. This structure mirrors that of the large 't Hooft coupling expansion of the S-matrix for the spin chain in the dual super-Yang-Mills theory.

Paper Structure

This paper contains 26 sections, 163 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of results
  3. The fields.
  4. The S-matrix.
  5. A puzzle.
  6. The tree-level S-matrix.
  7. Hopf algebra
  8. Scattering of bosons
  9. The a-gauge
  10. S-matrix
  11. Absence of particle production
  12. Scattering of fermions
  13. Physical degrees of freedom
  14. Action and quantization
  15. Tree-level S-matrix
  16. ...and 11 more sections

Figures (4)

  • Figure 1: The contour $\gamma_y$ for the action of the global charges $Q_{(1)}$ on a field inserted at the position $y$.
  • Figure 2: Contour manipulations leading to nontrivial braiding in the product of mutually-nonlocal fields.
  • Figure 3: Contour manipulations for the action of a charge on single field.
  • Figure 4: Three of the diagrams contributing to $2\rightarrow4$ scattering.