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Integrability and the AdS(3)/CFT(2) correspondence

A. Babichenko, B. Stefanski, K. Zarembo

TL;DR

The paper develops an integrable framework for the AdS3/CFT2 correspondence with sixteen supercharges by constructing Green-Schwarz–style actions from Z4-graded cosets based on D(2,1;α)×D(2,1;α). It demonstrates classical integrability via a Lax connection and finite-gap construction, and proposes quantum Bethe equations at special α-values that reproduce BMN-like spectra in the strong coupling limit. A key finding is the appearance of massless modes that are not captured by the finite-gap/Bethe approach, signaling a need to extend the formalism to fully account for the massless sector and target-space Virasoro symmetry. The work also analyzes limiting cases (flat space and AdS3×S3×T^4) and develops a large-charge (Landau-Lifshitz) picture, clarifying the structure of excitations and guiding future efforts toward a complete quantum integrable description of AdS3/CFT2.

Abstract

We investigate the AdS(3)/CFT(2) correspondence for theories with 16 supercharges using the integrability approach. We construct Green-Schwarz actions for Type IIB strings on AdS(3)xS(3)xM(4) where M(4)=T(4) or S(3)xS(1) using the coset approach. These actions are based on a Z(4) automorphism of the super-coset D(2,1;alpha)xD(2,1;alpha)/SO(1,2)xSO(3)xSO(3). The equations of motion admit a representation in terms of a Lax connection, showing that the system is classically integrable. We present the finite gap equations for these actions. When alpha=0,1/2,1 we propose a set of quantum Bethe equations valid at all values of the coupling. The AdS(3)/CFT(2) duals contain novel massless modes whose role remains to be explored.

Integrability and the AdS(3)/CFT(2) correspondence

TL;DR

The paper develops an integrable framework for the AdS3/CFT2 correspondence with sixteen supercharges by constructing Green-Schwarz–style actions from Z4-graded cosets based on D(2,1;α)×D(2,1;α). It demonstrates classical integrability via a Lax connection and finite-gap construction, and proposes quantum Bethe equations at special α-values that reproduce BMN-like spectra in the strong coupling limit. A key finding is the appearance of massless modes that are not captured by the finite-gap/Bethe approach, signaling a need to extend the formalism to fully account for the massless sector and target-space Virasoro symmetry. The work also analyzes limiting cases (flat space and AdS3×S3×T^4) and develops a large-charge (Landau-Lifshitz) picture, clarifying the structure of excitations and guiding future efforts toward a complete quantum integrable description of AdS3/CFT2.

Abstract

We investigate the AdS(3)/CFT(2) correspondence for theories with 16 supercharges using the integrability approach. We construct Green-Schwarz actions for Type IIB strings on AdS(3)xS(3)xM(4) where M(4)=T(4) or S(3)xS(1) using the coset approach. These actions are based on a Z(4) automorphism of the super-coset D(2,1;alpha)xD(2,1;alpha)/SO(1,2)xSO(3)xSO(3). The equations of motion admit a representation in terms of a Lax connection, showing that the system is classically integrable. We present the finite gap equations for these actions. When alpha=0,1/2,1 we propose a set of quantum Bethe equations valid at all values of the coupling. The AdS(3)/CFT(2) duals contain novel massless modes whose role remains to be explored.

Paper Structure

This paper contains 22 sections, 159 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $\mathbbm{Z}_4$ cosets
  3. The action and equations of motion
  4. $AdS_3\times S^3\times S^3\times S^1$ supercoset
  5. Flat space limit
  6. Coset model vs. Green-Schwarz string
  7. The BMN limit
  8. The $AdS_3\times S^3\times T^4$ limit
  9. Integrability
  10. The General Scheme
  11. Classical Bethe equations for $D(2,1;\alpha )\times D(2,1;\alpha )$
  12. BMN limit
  13. Quantum Bethe equations
  14. $AdS_3\times S^3\times T^4$
  15. Outlook: Some comments on the massless modes
  16. ...and 7 more sections

Figures (4)

  • Figure 1: The Dynkin diagram of the classical Bethe equations.
  • Figure 2: The Dynkin diagram of the asymptotic Bethe ansatz.
  • Figure 3: The Dynkin diagram for the $PSU(1,1|2)\times PSU(1,1|2)/SU(1,1)\times SU(2)$ coset.
  • Figure 4: The Dynkin diagram for the quantum Bethe equations.