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Long-Range PSU(2,2|4) Bethe Ansaetze for Gauge Theory and Strings

Niklas Beisert, Matthias Staudacher

TL;DR

The paper develops a comprehensive framework for long-range, asymptotic Bethe Ansatz solutions with psu(2,2|4) symmetry, unifying gauge-theory and string-theory descriptions of planar N=4 SYM. It builds from rank-one sectors to fully-fledged psu(2,2|4) Bethe equations, introducing all-loop S-matrices with dressing factors, nested Bethe Ansätze, and dynamic length features. The authors perform extensive consistency checks, derive all-loop conjectures for several subsectors (notably su(1|2) and su(1,1|2)), and demonstrate agreement with known three-loop results and with classical string limits in the thermodynamic regime. The work provides a blueprint for comparing gauge and string spectra via integrable spin chains, offering tools to explore AdS/CFT beyond perturbation theory and toward a unified quantum-integrable framework.

Abstract

We generalize various existing higher-loop Bethe ansaetze for simple sectors of the integrable long-range dynamic spin chain describing planar N=4 Super Yang-Mills Theory to the full psu(2,2|4) symmetry and, asymptotically, to arbitrary loop order. We perform a large number of tests of our conjectured equations, such as internal consistency, comparison to direct three-loop diagonalization and expected thermodynamic behavior. In the special case of the su(1|2) subsector, corresponding to a long-range t-J model, we are able to derive, up to three loops, the S-matrix and the associated nested Bethe ansatz from the gauge theory dilatation operator. We conjecture novel all-order S-matrices for the su(1|2) and su(1,1|2) subsectors, and show that they satisfy the Yang-Baxter equation. Throughout the paper, we muse about the idea that quantum string theory on AdS_5xS^5 is also described by a psu(2,2|4) spin chain. We propose asymptotic all-order Bethe equations for this putative "string chain", which differ in a systematic fashion from the gauge theory equations.

Long-Range PSU(2,2|4) Bethe Ansaetze for Gauge Theory and Strings

TL;DR

The paper develops a comprehensive framework for long-range, asymptotic Bethe Ansatz solutions with psu(2,2|4) symmetry, unifying gauge-theory and string-theory descriptions of planar N=4 SYM. It builds from rank-one sectors to fully-fledged psu(2,2|4) Bethe equations, introducing all-loop S-matrices with dressing factors, nested Bethe Ansätze, and dynamic length features. The authors perform extensive consistency checks, derive all-loop conjectures for several subsectors (notably su(1|2) and su(1,1|2)), and demonstrate agreement with known three-loop results and with classical string limits in the thermodynamic regime. The work provides a blueprint for comparing gauge and string spectra via integrable spin chains, offering tools to explore AdS/CFT beyond perturbation theory and toward a unified quantum-integrable framework.

Abstract

We generalize various existing higher-loop Bethe ansaetze for simple sectors of the integrable long-range dynamic spin chain describing planar N=4 Super Yang-Mills Theory to the full psu(2,2|4) symmetry and, asymptotically, to arbitrary loop order. We perform a large number of tests of our conjectured equations, such as internal consistency, comparison to direct three-loop diagonalization and expected thermodynamic behavior. In the special case of the su(1|2) subsector, corresponding to a long-range t-J model, we are able to derive, up to three loops, the S-matrix and the associated nested Bethe ansatz from the gauge theory dilatation operator. We conjecture novel all-order S-matrices for the su(1|2) and su(1,1|2) subsectors, and show that they satisfy the Yang-Baxter equation. Throughout the paper, we muse about the idea that quantum string theory on AdS_5xS^5 is also described by a psu(2,2|4) spin chain. We propose asymptotic all-order Bethe equations for this putative "string chain", which differ in a systematic fashion from the gauge theory equations.

Paper Structure

This paper contains 42 sections, 248 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction and Overview
  2. Rank-One Sectors
  3. Review of Hamiltonians
  4. Spectrum
  5. Review of Two-Component Bethe Ansätze
  6. The Spectral Parameter Plane
  7. Bethe Equations for Spins
  8. Bethe Equations for Strings
  9. Two-Excitation States
  10. Thermodynamic Limit
  11. Strong-Coupling Limit
  12. The Factorized S-Matrix of the $\mathfrak{su}(1|2)$ Sector
  13. One-Loop Scattering in the $\mathfrak{su}(1|2)$ Sector
  14. Three-Loop S-Matrix Extraction for $\mathfrak{su}(1|2)$
  15. All-Loop Factorized S-Matrix for $\mathfrak{su}(1|2)$: A Conjecture
  16. ...and 27 more sections

Figures (1)

  • Figure 1: Dynkin diagrams of $\mathfrak{su}(2,2|4)$ for the gradings ${\eta_1},{\eta_2}=\pm1$. The signs in the white nodes indicate the sign of the diagonal elements of the Cartan matrix.