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Relativistic limit on the $AdS_3 \times S^3 \times S^3 \times S^1$ string worldsheet

Emanuele Maria Cattaneo, Davide Polvara, Alessandro Sfondrini

TL;DR

The paper derives and tests the relativistic limit of the worldsheet S-matrix for AdS$_3\times$S$^3\times$S$^3\times$S$^1$ strings with mixed RR/NSNS flux, completing the S-matrix bootstrap with dressing factors for both fundamental and bound states. It identifies two families of massive bound states arising from each sphere, computes their S-matrices in the relativistic limit, and demonstrates consistency with the two-sphere fusion structure and with the relativistic limit of the Frolov–Sfondrini dressing factors. A competing construction based on a fictitious minimal-mass particle is analyzed and found not to reconcile with the Frolov–Syndrini proposal, highlighting the physical relevance of the two-sphere picture. The results provide a concrete relativistic benchmark for dressing factors in this AdS$_3$ background and clarify how bound-state fusion operates in a mixed-flux, two-sphere environment with nontrivial automorphisms and central extensions. This advances understanding of integrable structure, bound-state spectra, and potential connections to QSC frameworks for this class of theories.

Abstract

We study the relativistic limit of the worldsheet S matrix of $AdS_3 \times S^3 \times S^3 \times S^1$ strings in the presence of mixed Ramond-Ramond (RR) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) flux, and complete the S-matrix bootstrap including the dressing factors for fundamental particles and bound states. Our results agree with the relativistic limit of the recent proposal arXiv:2512.07721.

Relativistic limit on the $AdS_3 \times S^3 \times S^3 \times S^1$ string worldsheet

TL;DR

The paper derives and tests the relativistic limit of the worldsheet S-matrix for AdSSSS strings with mixed RR/NSNS flux, completing the S-matrix bootstrap with dressing factors for both fundamental and bound states. It identifies two families of massive bound states arising from each sphere, computes their S-matrices in the relativistic limit, and demonstrates consistency with the two-sphere fusion structure and with the relativistic limit of the Frolov–Sfondrini dressing factors. A competing construction based on a fictitious minimal-mass particle is analyzed and found not to reconcile with the Frolov–Syndrini proposal, highlighting the physical relevance of the two-sphere picture. The results provide a concrete relativistic benchmark for dressing factors in this AdS background and clarify how bound-state fusion operates in a mixed-flux, two-sphere environment with nontrivial automorphisms and central extensions. This advances understanding of integrable structure, bound-state spectra, and potential connections to QSC frameworks for this class of theories.

Abstract

We study the relativistic limit of the worldsheet S matrix of strings in the presence of mixed Ramond-Ramond (RR) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) flux, and complete the S-matrix bootstrap including the dressing factors for fundamental particles and bound states. Our results agree with the relativistic limit of the recent proposal arXiv:2512.07721.
Paper Structure (44 sections, 116 equations)

This paper contains 44 sections, 116 equations.

Table of Contents

  1. Introduction
  2. String on the mixed-flux $\mathbf{AdS}_{\mathbf{3}}\boldsymbol{\times}\mathbf{S}^{\mathbf{3}}\boldsymbol{\times}\mathbf{S}^{\mathbf{3}}\boldsymbol{\times}\mathbf{S}^{\mathbf{1}}$ background
  3. Symmetries in ligthcone gauge
  4. Representations and particles.
  5. Representations with bosonic highest-weight state, $\rho^{\text{B}}_\star (m,p)$.
  6. Representations with bosonic highest-weight state, $\rho^{\text{F}}_\star (m,p)$.
  7. Sphere-1 bosons and their superpartners.
  8. Sphere-2 bosons and their superpartners.
  9. AdS-bosons and their superpatrners.
  10. "Massless" particles.
  11. Isomorphisms between left and right representations.
  12. Expected fusion in the string and mirror models
  13. Fusion in the string model.
  14. Fusion in the mirror model.
  15. Relativistic limit
  16. ...and 29 more sections