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Algebraic Curves for Integrable String Backgrounds

K. Zarembo

TL;DR

The paper analyzes integrable string backgrounds within AdS/CFT by developing a unified finite-gap framework for coset sigma-models. It derives classical Bethe equations and associated algebraic curves for a broad set of AdS backgrounds (including AdS_5×S^5, AdS_4×CP^3, and multiple AdS_2/AdS_3 variants) using Z_4 symmetric cosets and their Lax formulations. A key result is the explicit construction of density-based integral equations (and their Dynkin-diagram structure) that encode the semiclassical string spectrum and map onto quantum Bethe equations via thermodynamic limits, with connections to the S-matrix and BES dressing phase. The work extends known finite-gap methods beyond AdS_5×S^5, offering a systematic route to semiclassical spectra in diverse RR backgrounds and highlighting the role of inversion symmetries and central (energy-carrying) nodes in the underlying integrable structures.

Abstract

Many Ramond-Ramond backgrounds which arise in the AdS/CFT correspondence are described by integrable sigma-models. The equations of motion for classical spinning strings in these backgrounds are exactly solvable by finite-gap integration techniques. We review the finite-gap integral equations and algebraic curves for coset sigma-models, and then apply the results to the AdS(d+1) backgrounds with d=4,3,2,1.

Algebraic Curves for Integrable String Backgrounds

TL;DR

The paper analyzes integrable string backgrounds within AdS/CFT by developing a unified finite-gap framework for coset sigma-models. It derives classical Bethe equations and associated algebraic curves for a broad set of AdS backgrounds (including AdS_5×S^5, AdS_4×CP^3, and multiple AdS_2/AdS_3 variants) using Z_4 symmetric cosets and their Lax formulations. A key result is the explicit construction of density-based integral equations (and their Dynkin-diagram structure) that encode the semiclassical string spectrum and map onto quantum Bethe equations via thermodynamic limits, with connections to the S-matrix and BES dressing phase. The work extends known finite-gap methods beyond AdS_5×S^5, offering a systematic route to semiclassical spectra in diverse RR backgrounds and highlighting the role of inversion symmetries and central (energy-carrying) nodes in the underlying integrable structures.

Abstract

Many Ramond-Ramond backgrounds which arise in the AdS/CFT correspondence are described by integrable sigma-models. The equations of motion for classical spinning strings in these backgrounds are exactly solvable by finite-gap integration techniques. We review the finite-gap integral equations and algebraic curves for coset sigma-models, and then apply the results to the AdS(d+1) backgrounds with d=4,3,2,1.

Paper Structure

This paper contains 16 sections, 90 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Strings in AdS and integrability
  3. Coset construction
  4. Integrability
  5. Supersymmetry
  6. Finite-gap integration
  7. Classical Bethe equations for various backgrounds
  8. $O(3)$ sigma-model
  9. Strings on $AdS_5\times S^5$
  10. Strings on $AdS_4\times CP^3$
  11. Strings on $AdS_3\times S^3$
  12. Strings on $AdS_3\times S^3\times S^3$
  13. Strings on $AdS_2\times S^2$
  14. $AdS_2\times S^2\times S^2$
  15. $AdS_2\times S^3$
  16. ...and 1 more sections

Figures (8)

  • Figure 1: The Dynkin diagram of $\mathfrak{psu}(4|4)$ in the preferred basis.
  • Figure 2: The Dykin diagram for classical Bethe equations in $AdS_5\times S^5$.
  • Figure 3: The Dynkin diagram for classical Bethe equations in $AdS_4\times CP^3$.
  • Figure 4: The Dynkin diagram for the $PSU(1,1|2)\times PSU(1,1|2)/SU(1,1)\times SU(2)$ coset.
  • Figure 5: The Dynkin diagram of the classical Bethe equations.
  • ...and 3 more figures