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

TL;DR

This paper investigates the quantum corrections to spinning strings in AdS_4 x CP^3 and refines the Bethe Ansatz proposal by identifying a non-zero one-loop correction to the interpolating function h(λ), improving the match between string theory and integrability predictions.

Contribution

It demonstrates that the interpolating function h(λ) receives a non-zero one-loop correction, resolving previous discrepancies between string theory calculations and the Bethe Ansatz proposal in AdS_4 x CP^3.

Findings

Computed the one-loop correction to the energy of a circular string in AdS_4 x CP^3.

Fixed the constant one-loop term in the function h(λ).

Suggested how to incorporate the one-loop phase factor into the Bethe Ansatz.

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(\lambda) 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(\lambda) 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.