Bethe Ansatz in Stringy Sigma Models

TL;DR

The paper demonstrates that exact two-body S-matrices and Bethe-Ansatz solutions can be obtained for three AdS5×S5 subsector sigma-models—landau-Lifshitz on S^2, the AAF su(1|1) fermionic model, and the FR model on S^3×R—by exploiting world-sheet perturbation theory and the assumption of integrability with factorized scattering. It provides explicit S-matrices and Bethe equations for each model, highlighting how finite-volume spectra can be accessed through two-body data and how vacuum structure and non-perturbative states emerge in these integrable string-theoretic settings. The results connect to BMN and Heisenberg-spin-chain limits, offering concrete, solvable laboratories to probe the AdS/CFT spectral problem and to test Bethe-ansatz techniques in both non-relativistic and fermionic 2D quantum field theories. Overall, the work showcases a practical route to deriving and validating Bethe equations directly from world-sheet S-matrices for string-theoretic subsectors, informing our understanding of the AdS5×S5 integrable structure and its finite-volume implications.

Abstract

We compute the exact S-matrix and give the Bethe ansatz solution for three sigma-models which arise as subsectors of string theory in AdS(5)xS(5): Landau-Lifshitz model (non-relativistic sigma-model on S(2)), Alday-Arutyunov-Frolov model (fermionic sigma-model with su(1|1) symmetry), and Faddeev-Reshetikhin model (string sigma-model on S(3)xR).

Figures (11)

• Figure 1: Pole prescription in Landau-Lifshitz model.
• Figure 2: Non-renormalization theorem. Any closed loop of likewise oriented propagators $D_{mn} \equiv D(t_m-t_n,x_m-x_n)$ vanishes.
• Figure 3: Poles for a closed loop. All propagators in a closed have their poles in the lower half plane. Hence the integral (\ref{['eqn:energy-integral ']}) over the energy flowing around the loop vanishes.
• Figure 4: Generic loop diagram for the two-body S-matrix.
• Figure 5: Cutting a generic diagram. Due to charge conservation the number of future directed propagators minus the number of past directed propagators has to be the same at any moment in time. Since any past directed propagator is identically equal to zero, the number of propagators at any cut is the same and is equal to the number of external incoming/outgoing legs.