### Abstract

Recently, an asymptotic Bethe Ansatz that is claimed to describe anomalous dimensions of ''long'' operators in the planar = 6 supersymmetric three-dimensional Chern-Simons-matter theory dual to quantum superstrings in AdS _{4} × ℂ ℙ ^{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} × S ^{5} case: a rigid circular string stretched in both AdS _{4} and along an S ^{1} of ℂ ℙ ^{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 worldsheet space-like direction) and their Bethe Ansatz counterparts.

Original language | English (US) |
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Article number | 069 |

Journal | Journal of High Energy Physics |

Volume | 2008 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2008 |

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

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## Cite this

_{4}× ℂ ℙ

^{3}: Testing the Bethe Ansatz proposal.

*Journal of High Energy Physics*,

*2008*(11), [069]. https://doi.org/10.1088/1126-6708/2008/11/069