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Magnon dispersion to four loops in the ABJM and ABJ models. (English) Zbl 1193.81089

J. Phys. A, Math. Theor. 43, No. 27, Article ID 275402, 17 p. (2010); corrigendum 44, No. 4, Article ID, 2 p. (2011).
Summary: The ABJM model is a superconformal Chern-Simons theory with \({\mathcal N}= 6\) supersymmetry which is believed to be integrable in the planar limit. However, there is a coupling-dependent function that appears in the magnon dispersion relation and the asymptotic Bethe ansatz that is only known to leading order at strong and weak coupling. We compute this function to four loops in perturbation theory by an explicit Feynman diagram calculation for both the ABJM model and the ABJ extension. The ABJM four-loop correction has mixed transcendentality, while the ABJ extension adds a term to the ABJM correction with highest transcendentality. We then compute the four-loop wrapping correction for a scalar operator in the 20 representation of \(\text{SU}(4)\) and find that it agrees with a recent prediction of the ABJM \(Y\)-system by Gromov, Kazakov and Vieira. We also propose a limit of the ABJ model that might be perturbatively integrable at all loop orders but has a short range Hamiltonian.
From the text of the corrigendum: A sign inconsistency in the definition of the Feynman rules alters the overall signs of the expressions for \(F_{tv3}\), \(F_{tv4}\), \(F_{tv5}\) in (3.13). This sign change modifies the final results and requires a number of replacements (for details see the full text).

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
81T18 Feynman diagrams
81Q40 Bethe-Salpeter and other integral equations arising in quantum theory
81R12 Groups and algebras in quantum theory and relations with integrable systems
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