Bethe Ansaetze for GKP strings

TL;DR

This work derives and analyzes the Bethe-Yang equations for low-lying excitations on the GKP string in both AdS5×S5 and AdS4×CP3, revealing a striking alignment between the all-loop asymptotic Bethe equations and the spectral data of two distinct low-energy effective models. At weak coupling, holes and anti-holes define the fundamental excitations whose energies and momenta are encoded by universal densities and a twist-dependent boundary structure, leading to SU(4) symmetry restoration via stacks. At strong coupling, the same equations reduce to the O(6) sigma model for N=4 and the Bykov model for ABJM, with matching S-matrices and a consistent interpretation of transmission through string endpoints. The results provide a non-perturbative cross-check of the dressing phase and demonstrate a deep link between gauge-theory integrability and two-dimensional effective theories arising in the holographic dual, including finite-volume selectivity through a Z2 symmetry and fermionic twists. The findings support a coherent, integrable picture linking AdS4/CFT3 and AdS5/CFT4 across coupling regimes, with explicit mappings between spectral data and worldsheet dynamics.

Abstract

Studying the scattering of excitations around a dynamical background has a long history in the context of integrable models. The Gubser-Klebanov-Polyakov string solution provides such a background for the string/gauge correspondence. Taking the conjectured all-loop asymptotic equations for the AdS_4/CFT_3 correspondence as the starting point, we derive the S-matrix and a set of spectral equations for the lowest-lying excitations. We find that these equations resemble closely the analogous equations for AdS_5/CFT_4, which are also discussed in this paper. At large values of the coupling constant we show that they reproduce the Bethe equations proposed to describe the spectrum of the low-energy limit of the AdS_4xCP^3 sigma model.

Figures (5)

• Figure 1: Dynkin diagram in the non-compact grading. The momentum carriers of the alternating spin chain are the roots associated with the nodes $4$ and $\bar{4}$.
• Figure 2: The $SU(4)$ Dynkin diagram for the stacks. The $SU(2)$ subgroup associated to the middle node is directly inherited from the spin chain. The $SU(2)\times SU(2)$ group associated to the left and right nodes are restored through the formation of stacks, whose contents are depicted on top of them.
• Figure 3: Symmetry enhancement at weak coupling. The three bosonic nodes of the $\mathfrak{su}(4)$ symmetry algebra are enhanced by a fermionic node to leading order at weak coupling. The resulting Dynkin diagram is the one of the $\mathfrak{osp}(1,1|4)$ Lie super-algebra.
• Figure 4: The mirror path at small values of the coupling constant. It entails shifting the rapidity $u$ by the imaginary unit $i$ such that one crosses the cut $\mathcal{C}_{+}$ connecting the square-root branch points $i/2\pm 2g$. This cut closes up at weak coupling, hence performing the mirror transformation is impossible perturbatively.
• Figure 5: The mirror path at large values of the coupling constant. At strong coupling the cut $\mathcal{C}_{+}$ opens up hence revealing the simplicity of the mirror transformation.