Quantum spinning strings in AdS ₄ × ℂ ℙ ³: Testing the Bethe Ansatz proposal

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.