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Integrability, spin-chains and the AdS3/CFT2 correspondence

O. Ohlsson Sax, B. Stefanski

TL;DR

The paper extends integrability methods to the AdS3/CFT2 setting by formulating an all-loop Bethe Ansatz for strings on $AdS_3\times S^3\times S^3\times S^1$ with symmetry $D(2,1;\alpha)^2$, using an $\alpha$-dependent generalisation of the Zhukovsky map to capture massive and massless modes. It derives the weak-coupling spin-chain limits, identifies the alternating spin-chain representations $(-\frac{\alpha}{2};\frac{1}{2};0)$ and $(-\frac{1-\alpha}{2};0;\frac{1}{2})$, and constructs the corresponding integrable Hamiltonians from the $R$-matrix, providing a concrete lattice realisation of the BA. In the $\alpha\to1$ limit, the algebra reduces to $psu(1,1|2)$ and the authors show the integrable structure remains well-behaved even though certain Hamiltonians diverge, suggesting the appearance of the missing massless modes and motivating a generalized BA to fully capture the massless sector. The work also clarifies the connection to ABJM-like structures at special points (e.g., $\alpha=1/2$) and lays groundwork for identifying the elusive CFT2 dual to the AdS3 backgrounds by linking weak-coupling spin-chains to the BA spectra. Overall, the results demonstrate that integrability provides a coherent framework to incorporate massless modes and to probe the AdS3/CFT2 correspondence across the full $\alpha$-family.

Abstract

Building on arXiv:0912.1723, in this paper we investigate the AdS3/CFT2 correspondence using integrability techniques. We present an all-loop Bethe Ansatz (BA) for strings on AdS_3 x S^3 x S^3 x S^1, with symmetry D(2,1;alpha)^2, valid for all values of alpha. This construction relies on a novel, alpha-dependent generalisation of the Zhukovsky map. We investigate the weakly-coupled limit of this BA and of the all-loop BA for strings on AdS_3 x S^3 x T^4. We construct integrable short-range spin-chains and Hamiltonians that correspond to these weakly-coupled BAs. The spin-chains are alternating and homogenous, respectively. The alternating spin-chain can be regarded as giving some of the first hints about the unknown CFT2 dual to string theory on AdS_3 x S^3 x S^3 x S^1. We show that, in the alpha to 1 limit, the integrable structure of the D(2,1;alpha) model is non-singular and keeps track of not just massive but also massless modes. This provides a way of incorporating massless modes into the integrability machinery of the AdS3/CFT2 correspondence.

Integrability, spin-chains and the AdS3/CFT2 correspondence

TL;DR

The paper extends integrability methods to the AdS3/CFT2 setting by formulating an all-loop Bethe Ansatz for strings on with symmetry , using an -dependent generalisation of the Zhukovsky map to capture massive and massless modes. It derives the weak-coupling spin-chain limits, identifies the alternating spin-chain representations and , and constructs the corresponding integrable Hamiltonians from the -matrix, providing a concrete lattice realisation of the BA. In the limit, the algebra reduces to and the authors show the integrable structure remains well-behaved even though certain Hamiltonians diverge, suggesting the appearance of the missing massless modes and motivating a generalized BA to fully capture the massless sector. The work also clarifies the connection to ABJM-like structures at special points (e.g., ) and lays groundwork for identifying the elusive CFT2 dual to the AdS3 backgrounds by linking weak-coupling spin-chains to the BA spectra. Overall, the results demonstrate that integrability provides a coherent framework to incorporate massless modes and to probe the AdS3/CFT2 correspondence across the full -family.

Abstract

Building on arXiv:0912.1723, in this paper we investigate the AdS3/CFT2 correspondence using integrability techniques. We present an all-loop Bethe Ansatz (BA) for strings on AdS_3 x S^3 x S^3 x S^1, with symmetry D(2,1;alpha)^2, valid for all values of alpha. This construction relies on a novel, alpha-dependent generalisation of the Zhukovsky map. We investigate the weakly-coupled limit of this BA and of the all-loop BA for strings on AdS_3 x S^3 x T^4. We construct integrable short-range spin-chains and Hamiltonians that correspond to these weakly-coupled BAs. The spin-chains are alternating and homogenous, respectively. The alternating spin-chain can be regarded as giving some of the first hints about the unknown CFT2 dual to string theory on AdS_3 x S^3 x S^3 x S^1. We show that, in the alpha to 1 limit, the integrable structure of the D(2,1;alpha) model is non-singular and keeps track of not just massive but also massless modes. This provides a way of incorporating massless modes into the integrability machinery of the AdS3/CFT2 correspondence.

Paper Structure

This paper contains 28 sections, 116 equations, 4 figures.

Table of Contents

  1. Introduction
  2. All-loop Bethe equations for $d(2,1;\alpha)^2$
  3. Classical Bethe equations for $d(2,1;\alpha)^2$
  4. Weak-coupling limit
  5. An integrable $d(2,1;\alpha)^2$ spin-chain
  6. The alternating $d(2,1;\alpha)$ spin-chain
  7. The integrable $d(2,1;\alpha)$ hamiltonian
  8. The $sl(2|1)$ subsector
  9. The lift to $d(2,1;\alpha)$
  10. The $su(2) \times su(2)$ sector
  11. The full $d(2,1;\alpha)^2$ spin-chain
  12. Fermionic duality
  13. Dualization of the full Bethe equations?
  14. The $\alpha\rightarrow 1$ limit and $AdS_3 \times S^3 \times T^4$
  15. The weakly-coupled $\alpha=1$ Bethe equations
  16. ...and 13 more sections

Figures (4)

  • Figure 1: An illustration of the short $\left(-\tfrac{\alpha}{2};\tfrac{1}{2},0\right)$$d(2,1;\alpha)$ module. For clarity, in the figure we have neglected to include the exact coefficients of the linear maps for the representation. These can be read-off from equations \ref{['eq:d21a-chiral-rep-LR']}-\ref{['eq:d21a-chiral-rep-Q']} in Appendix \ref{['sec:d21a-algebra']}. To emphasise this we use the notation $\Bigl<\dots\Bigr>$ to denote the span of $\dots$. The states in the left-hand side column have $(L_5,R_8)=(\pm\tfrac{1}{2},0)$ and those in the right-hand side column have $(L_5,R_8)=(0,\pm\tfrac{1}{2})$. Starting at the top, the rows in the diagram have $S_0$ eigenvalues equal to $-\tfrac{\alpha}{2}\,,\,-\tfrac{\alpha+1}{2} ; -\tfrac{\alpha+2}{2}\,,\dots$
  • Figure 2: An illustration of the short $\left(\tfrac{\alpha}{2},-\tfrac{\alpha}{2}\right)$$sl(2|1)$ module. For clarity, in the figure we have neglected to include the exact coefficients of the linear maps for the representation. These can be read-off from equations \ref{['eq:d21a-chiral-rep-LR']}-\ref{['eq:d21a-chiral-rep-Q']} in Appendix \ref{['sec:d21a-algebra']} using the embedding \ref{['eq:sl21embedding']}. To emphasise this we use the notation $\Bigl<\dots\Bigr>$ to denote the span of $\dots$. The states in the left-hand side column have $B=-\tfrac{\alpha}{2}$ and those in the right-hand side column have $B=\tfrac{1-\alpha}{2}$. Starting at the top, the rows in the diagram have $J_0$ eigenvalues equal to $\tfrac{\alpha}{2}\,,\,\tfrac{\alpha+1}{2}\,,\,\tfrac{\alpha+2}{2}\,,\dots$
  • Figure 3: Two of the Dynkin diagrams for $d(2,1;\alpha)$. The crossed notes are fermionic and the labels indicate the momentum carrying roots in the Bethe equations. The original equations \ref{['eq:BE-d21a-one-loop-orig']} corresponds to the diagram \ref{['fig:dynkin-d21a-orig']}, while the dualized equations \ref{['eq:BE-d21a-one-loop-dual']} corresponds to \ref{['fig:dynkin-d21a-dual']}.
  • Figure 4: Three Dynkin diagrams for $psu(1,1|2)$.