The all loop AdS4/CFT3 Bethe ansatz

TL;DR

Problem: obtain the all-loop asymptotic spectrum for AdS4/CFT3 in the planar limit. Approach: propose a set of five nested Bethe equations organized by the OSp(2,2|6) symmetry, with momentum-carrying roots u4 and u_bar4 and a single interpolating function h(lambda), together with a Beisert–Staudacher–BES-type dressing factor. Findings: the equations reproduce the 2-loop Minahan–Zarembo results, align with the string algebraic curve at strong coupling, and yield the Hernandez–Lopez semi-classical correction in a natural way. Significance: provides a unified integrable framework for AdS4/CFT3 spectra, clarifies the role of symmetry in fixing the scalar factor, and sets the stage for addressing wrapping effects and determining the interpolating function h(lambda).

Abstract

We propose a set of Bethe equations yielding the full asymptotic spectrum of the AdS4/CFT3 duality proposed in arXiv:0806.1218 to all orders in the t'Hooft coupling. These equations interpolate between the 2-loop Bethe ansatz of Minahan and Zarembo arXiv:0806.3951 and the string algebraic curve of arXiv:0807.0437. The several SU(2|2) symmetries of the theory seem to highly constrain the form of the Bethe equations up to a dressing factor whose form we also conjecture.

Figures (3)

• Figure 1: Structure of the $AdS_4/CFT_3$ Bethe equations. The several Bethe equations are organized according to the $OSp(2,2|6)$ symmetry of the problem. The Dynkin diagram associated with this symmetry group is depicted in the figure. Particularly important subsectors are two $SU(2|2)$ obtained by exciting solely momentum carrying roots of one of the wings ($u_4$ or $u_{\bar{4}}$) plus an arbitrary amount of auxiliary roots in the $SU(2|2)$ tail ($u_1$, $u_2$ and $u_3$). Equally important is the $SU(2)\times SU(2)$ subsector obtained by only exciting the momentum carrying roots ($u_4$ and $u_{\bar{4}}$). When we consider higher orders in perturbation theories a dressing kernel appears introducing extra self-interactions for the momentum carrying roots and also a new interaction between the roots $u_4$ and $u_{\bar{4}}$. Perturbatively, this couples the two $SU(2)$'s in the $SU(2)\times SU(2)$ sector starting at eight loops.
• Figure 2: The several states in the Hilbert space can be constructed in the usual oscillator representation. There is one oscillator per Dynkin node of the $OSp(2,2|6)$ super Dynkin diagram. A light (dark) gray shaded node corresponds to an oscillator excited once (twice). The number of times each oscillator is excited is the same as the number of Bethe roots of the corresponding type.
• Figure 3: Bethe equations for the two choices of the grading $\eta=\pm 1$. The subscripts $\pm 1$ in the dashed lines correspond to the kernels $\sigma_{BES}(u,v)$ for $\eta=1$ and to a kernel $\sigma_{BES}(u,v) \frac{x^-(u) -x^+(v)}{x^+(u)-x^-(v)}$ for $\eta=-1$.