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Asymptotic Bethe ansatz \(S\)-matrix and Landau-Lifshitz-type effective 2D actions. (English) Zbl 1109.81071

Summary: Motivated by the desire to relate Bethe ansatz equations for anomalous dimensions found on the gauge-theory side of the AdS/CFT correspondence to superstring theory on AdS\(_{5} \times S^{5}\) we explore a connection between the asymptotic S-matrix that enters the Bethe ansatz and an effective two-dimensional quantum field theory. The latter generalizes the standard ‘non-relativistic’ Landau-Lifshitz (LL) model describing low-energy modes of ferromagnetic Heisenberg spin chain and should be related to a limit of superstring effective action. We find the exact form of the quartic interaction terms in the generalized LL-type action whose quantum S-matrix matches the low-energy limit of the asymptotic S-matrix of the spin chain of Beisert, Dippel and Staudacher (BDS). This generalizes to all orders in the ’t Hooft coupling \(\lambda\) an earlier computation of Klose and Zarembo of the S-matrix of the standard LL model. We also consider a generalization to the case when the spin-chain S-matrix contains an extra ’string’ phase and determine the exact form of the LL 4-vertex corresponding to the low-energy limit of the ansatz of Arutyunov, Frolov and Staudacher (AFS). We explain the relation between the resulting ‘non-relativistic’ non-local action and the second-derivative string sigma model. We comment on modifications introduced by strong-coupling corrections to the AFS phase. We mostly discuss the SU(2) sector but also present generalizations to the SL(2) and \(\text{SU}(1|1)\) sectors, confirming universality of the dressing phase contribution by matching the low-energy limit of the AFS-type spin-chain S-matrix with tree-level string-theory S-matrix.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81U20 \(S\)-matrix theory, etc. in quantum theory
81Q40 Bethe-Salpeter and other integral equations arising in quantum theory
81T60 Supersymmetric field theories in quantum mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D40 Statistical mechanics of magnetic materials
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