Quantum spinning strings in AdS₄× ℂℙ³: testing the Bethe Ansatz proposal

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 × 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 × 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 oneloop 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. * Also at Lebedev Institute Moscow

Y-system, TBA and Quasi-Classical Strings in AdS 4 × CP3

“…There is yet another way to compute the one-loop shift directly from the world-sheet action which is similar to the algebraic curve computation. Whereas for the folded string both computations give the same excitation frequencies 15 , for the circular string a negative result was obtained in [109]. Recently it was shown that one should be more carefull with the periodicity of the fermionic fields in the world-sheet approach and the corrected derivation leads to agreement with the algebraic curve frequencies [110] so we assume all approaches to be consistent with each other.…”

“…At strong coupling the situation is less clear: in [75] and [78] the coefficient h 0 was argued to be 0 whereas in [109] some evidence was given in favor of a different value − log 2 2π (see also [74]). Hopefully this issue could be analyzed from world-sheet sigma model first principles calculation like in [79,80].…”

Dualities for loop amplitudes of N = 6 Chern-Simons matter theory

“…We note that there already exists evidence that the quantum corrections of the two theories are closely connected. 9 Indeed it was conjectured in [40], and tested for in [41,42,43,44], that the anomalous dimensions of twist operators of ABJM theory can be obtained from N = 4 super Yang-Mills, up to normalization factors which can be easily determined, simply by substituting,…”

BPS Wilson loops and Bremsstrahlung function in ABJ(M): a two loop analysis

“…However, in the three dimensional case there is still one player missing in the integrability game: the infamous function h(λ), mastering the dispersion relation of a single magnon moving on the spin chain [72][73][74]. Weak [75,76] and strong [77] coupling expressions have been obtained at leading and subleading orders, but no systematic method for computing h(λ) exists yet. On the other hand, it should be possible to find a three dimensional analogue of the set of TBA integral equations, used in [51,62], and apply it to the actual computation of B(λ), as done in [78,79].…”