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Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansätze and the R-Matrix Formalism

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Abstract

The one-dimensional Heisenberg XXX spin chain appears in a special limit of the AdS/CFT integrable system. We review various ways of proving its integrability, and discuss the associated methods of solution. In particular, we outline the coordinate and the algebraic Bethe ansatz, giving reference to literature suitable for learning these techniques. Finally, we speculate which of the methods might lift to the exact solution of the AdS/CFT system, and sketch a promising method for constructing the Baxter Q-operator of the XXX chain. It allows to find the spectrum of the model using certain algebraic techniques, while entirely avoiding Bethe’s ansatz.

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  1. Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, Johann von Neumann-Haus, Rudower Chaussee 25, 12489, Berlin, Germany

    Matthias Staudacher

  2. Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476, Potsdam, Germany

    Matthias Staudacher

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  1. Matthias Staudacher
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Correspondence to Matthias Staudacher.

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Staudacher, M. Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansätze and the R-Matrix Formalism. Lett Math Phys 99, 191–208 (2012). https://doi.org/10.1007/s11005-011-0530-9

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  • DOI: https://doi.org/10.1007/s11005-011-0530-9

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