Exciting the GKP string at any coupling

TL;DR

This work provides a comprehensive all-loop characterization of the spectrum around the GKP long string in N=4 SYM by embedding excitations into the Beisert-Staudacher Bethe Ansatz and expressing their dispersion relations parametrically via the BES equation. It unifies weak- and strong-coupling analyses, showing precise matches to known string results (e.g., Frolov-Tseytlin fluctuations) and clarifying how SU(6) symmetry is dynamically restored on the GKP background. The approach yields explicit weak-coupling expansions (up to three loops for some sectors) and a detailed strong-coupling picture with multiple regimes (perturbative, near-flat space, giant hole) and bound-state interpretations. The results illuminate the integrable structure of the gauge/string system, offer a path toward connecting to scattering amplitudes, and suggest extensions to related theories like ABJM via analogous Bethe-Ansatz/Y-system formulations.

Abstract

We analyze the spectrum of excitations around the Gubser-Klebanov-Polyakov (GKP) rotating string in the long string limit and construct a parametric representation for their dispersion relations at any value of the string tension. On the gauge theory side of the AdS/CFT correspondence, i.e., in the planar $\mathcal{N}=4$ Super-Yang-Mills theory, the problem is equivalent to finding the spectrum of scaling dimensions of large spin, single-trace operators. Their scaling dimensions are obtained from the analysis of the Beisert-Staudacher asymptotic Bethe ansatz equations, which are believed to solve the spectral problem of the planar gauge theory. We examine the resulting dispersion relations in various kinematical regimes, both at weak and strong coupling, and detail the matching with the Frolov-Tseytlin spectrum of transverse fluctuations of the long GKP string. At a more dynamical level, we identify the mechanism for the restoration of the SO(6) symmetry, initially broken by the choice of the Berenstein-Maldacena-Nastase vacuum in the Bethe ansatz solution to the mixing problem.

Figures (6)

• Figure 1: Spectrum of masses for gauge and string elementary excitations. The question mark is for the missing mass $2$ boson predicted by the perturbative analysis in the string theory FT02AM07. Its mass should not receive corrections AM07. There are candidates for it but their stability at non-zero momentum is not guaranteed, since decay channels into two fermions are allowed.
• Figure 2: The two fermion branches.
• Figure 3: Two realizations for the contour of integration in the integral representation (\ref{['CIRa']}), associated to a rapidity $u$ with $u^2 \gtrless (2g)^2$, respectively. Here $k = \ell$ fixes the size of the bound state, i.e., the end points $u \pm ik/2 = u\pm i\ell/2$ of the contour of integration. The latter is chosen such as to avoid the cuts in the function $\Lambda$, running from $\pm 2g$ to $\pm \infty$ and depicted by the two wavy lines.
• Figure 4: Sketch of the complex $u$-planes for energy and momentum of a large (lower sheet) and small (upper sheet) fermion. Crosses indicate square-root branch points. Except for the central cut, the upper sheet is free from singularities. The lower sheet contains an infinite series of cuts which extends along the imaginary axis, with a spacing between two consecutive cuts equal to $\pm i$. The dashed line indicates the path followed during the analytic continuation from the small to large fermion domain.
• Figure 5: Complex $u$-plane for energy and momentum of scalar, gauge field, and bound states. Crosses indicate square-root branch points. Wavy lines indicate a choice of cuts that is natural from the weak coupling perspective. The dashed lines represent the real and imaginary $u$-axis, respectively. The integer $k$ depends on the excitation. It is equal to $1$ for a scalar and to $\ell$ for a bound state of $\ell$ gauge fields. For a large fermion, the picture with $k=2$ can be applied if completed with an extra cut along the real $u$-axis. For a small fermion, we would only keep the latter cut and remove the two semi-infinite towers in the upper- and lower-half plane. For all excitations, within the weak coupling expansion, the sequence of cuts degenerates into a sequence of poles along the imaginaris $u$-axis.