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All-loop Bethe ansatz equations for AdS3/CFT2

Riccardo Borsato, Olof Ohlsson Sax, Alessandro Sfondrini

TL;DR

This work provides a comprehensive construction of all-loop Bethe ansatz equations for the AdS3/CFT2 system described by the d(2,1;α)^2 symmetry, introducing a novel mixed grading and four scalar dressing factors constrained by crossing symmetry. Through a detailed nesting procedure, the authors derive the Bethe equations and explore their semiclassical limit, where dressing phases generalize the AFS/BES structure and nontrivially couple nodes of different masses. A crucial part of the analysis compares the semiclassical BA to finite-gap equations and near-BMN results, uncovering a notable mismatch in level-matching and length identifications that challenges a unified all-loop description and signals unresolved aspects of massless modes. The discussion outlines potential resolutions, including a full crossing solution, massless sector integration, and extensions to related AdS3 backgrounds, highlighting the work as a significant step toward a complete integrable framework for AdS3/CFT2.

Abstract

Using the S-matrix for the d(2,1;alpha)^2 symmetric spin-chain of AdS3/CFT2, we propose a new set of all-loop Bethe equations for the system. These equations differ from the ones previously found in the literature by the choice of relative grading between the two copies of the d(2,1;alpha) superalgebra, and involve four undetermined scalar factors that play the role of dressing phases. Imposing crossing symmetry and comparing with the near-BMN form of the S-matrix found in the literature, we find several novel features. In particular, the scalar factors must differ from the Beisert-Eden-Staudacher phase, and should couple nodes of different masses to each other. In the semiclassical limit the phases are given by a suitable generalization of Arutyunov-Frolov-Staudacher phase.

All-loop Bethe ansatz equations for AdS3/CFT2

TL;DR

This work provides a comprehensive construction of all-loop Bethe ansatz equations for the AdS3/CFT2 system described by the d(2,1;α)^2 symmetry, introducing a novel mixed grading and four scalar dressing factors constrained by crossing symmetry. Through a detailed nesting procedure, the authors derive the Bethe equations and explore their semiclassical limit, where dressing phases generalize the AFS/BES structure and nontrivially couple nodes of different masses. A crucial part of the analysis compares the semiclassical BA to finite-gap equations and near-BMN results, uncovering a notable mismatch in level-matching and length identifications that challenges a unified all-loop description and signals unresolved aspects of massless modes. The discussion outlines potential resolutions, including a full crossing solution, massless sector integration, and extensions to related AdS3 backgrounds, highlighting the work as a significant step toward a complete integrable framework for AdS3/CFT2.

Abstract

Using the S-matrix for the d(2,1;alpha)^2 symmetric spin-chain of AdS3/CFT2, we propose a new set of all-loop Bethe equations for the system. These equations differ from the ones previously found in the literature by the choice of relative grading between the two copies of the d(2,1;alpha) superalgebra, and involve four undetermined scalar factors that play the role of dressing phases. Imposing crossing symmetry and comparing with the near-BMN form of the S-matrix found in the literature, we find several novel features. In particular, the scalar factors must differ from the Beisert-Eden-Staudacher phase, and should couple nodes of different masses to each other. In the semiclassical limit the phases are given by a suitable generalization of Arutyunov-Frolov-Staudacher phase.

Paper Structure

This paper contains 22 sections, 97 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The S-matrix
  3. The 11, 1b1b, 33 and 3b3b sectors
  4. The 11b, 1b1, 33b and 3b3 sectors
  5. The 13, 1b3b, 31 and 3b1b sectors
  6. The 13b, 1b3, 31b and 3b1 sectors
  7. Crossing equations
  8. Diagonalisation of the S-matrix
  9. Level-I.
  10. Level-II vacuum.
  11. Propagation.
  12. Scattering.
  13. Bethe Equations
  14. Fermionic duality
  15. Small h limit and Cartan matrix
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Two of the Dynkin diagrams for $\mathfrak{d}(2,1;\alpha)$. The crossed notes are fermionic and the labels indicate the momentum carrying roots in the Bethe equations. The $\mathfrak{d}(2,1;\alpha)^2$ Cartan matrix \ref{['eq:cartan']} corresponds to using diagram \ref{['fig:dynkin-d21a-orig']} for the left-movers and diagram \ref{['fig:dynkin-d21a-dual']} for the right-movers.
  • Figure 2: The Dynkin diagram for $\mathfrak{d}(2,1;\alpha)^2$ in the mixed grading \ref{['eq:cartan']}, with the various interaction terms in \ref{['eq:bethe-equations']} indicated.