Quantum spinning strings in AdS_4 x CP^3: testing the Bethe Ansatz proposal

TL;DR

This work resolves a tension between the holographic one-loop string corrections in AdS$_4\times\mathbb{CP}^3$ and a proposed all-loop Bethe Ansatz for ABJM by showing that the interpolating function $h(\lambda)$ acquires a nonzero one-loop correction with $a_1 = -\ln 2/(2\pi)$. Focusing on a basic circular $(S,J)$ string, the authors compute the leading quantum correction, extract the one-loop term in the magnon dispersion, and demonstrate consistency with the Bethe Ansatz framework, including the nontrivial one-loop phase. They also establish a mapping between AdS$_4$ and AdS$_5$ results at strong coupling, clarifying how the two theories relate within the Bethe Ansatz program while noting remaining ambiguities in analytic $1/J^{2n}$ terms. Overall, the study strengthens the case for integrability-based descriptions of ABJM and guides future refinement of the corresponding Bethe Ansatz proposals.

Abstract

Recently, an asymptotic Bethe Ansatz that is claimed to describe anomalous dimensions of "long" operators in the planar N=6 supersymmetric three-dimensional Chern-Simons-matter theory dual to quantum superstrings in AdS_4 x CP^3 was proposed. It initially passed a few consistency checks but subsequent direct comparison to one-loop string-theory computations created some controversy. Here we suggest a resolution by pointing out that, contrary to the initial assumption based on the algebraic curve considerations, the central interpolating function h(λ) entering the BMN or magnon dispersion relation receives a non-zero one-loop correction in the natural string-theory computational scheme. We consider a basic example which has already played a key role in the AdS_5 x S^5 case: a rigid circular string stretched in both AdS_4 and along an S^1 of CP^3 and carrying two spins. Computing the leading one-loop quantum correction to its energy allows us to fix the constant one-loop term in h(λ) and also to suggest how one may establish a correspondence with the Bethe Ansatz proposal, including the non-trivial one-loop phase factor. We discuss some problems which remain in trying to match a part of world-sheet contributions (sensitive to compactness of the string direction) and their Bethe Ansatz counterparts.