PERTURBATIVE APPROACH TO INTEGRABILITY IN THREE-DIMENSIONAL CHERN-SIMONS THEORIES

One of the most fascinating discoveries of contemporary Theoretical Physics is the AdS/CFT correspondence relating gauge theories to gravity theories. Soon after its formulation, tremendous developments allowed to obtain a deep comprehension of the four-dimensional N=4 SYM theory and, in particular, led to the discovery of integrable structures both in the gauge theory itself and in its string counterpart. In the last few years, much attention was devoted to the study of supersymmetric Chern-Simons-matter theories in three dimensions. In this class of theories a distinguished role is played by the N=6 ABJM model which is a U(N) x U(N) superconformal gauge theory. Indeed, in the large N limit, the ABJM theory has been conjectured to be the AdS/CFT dual description of M-theory on an AdS4 x S7/Zk background and, for k << N << k^5, of type IIA string theory on AdS4 × CP3. For this reason, soon after its discovery the ABJM model has quickly become the ideal three-dimensional playground to study AdS/CFT as much as N=4 SYM has been in the four-dimensional case. Quite surprisingly, the ABJM model seems to share a number of notable properties with N=4 SYM theory even though the two theories are a priori different in nature. One of the common features is provided by the fact that also in planar ABJM theory integrable structures naturally show up. In particular, a Bethe Ansatz approach to the computation of anomalous dimensions is possible and a set of all-loop Bethe equations was formulated. These equations are very similar to those of SYM theory, however, starting at eight loops, important new features are believed to appear, in particular in connection with the appearance of the dressing phase. With the aim to check such a picture, we analyze for the first time the form of the eight-loop dilatation operator in the SU(2) x SU(2) sector and, adopting a perturbative approach based on superspace techniques, we directly extract the value of the unknown leading order coefficient of the dressing phase from supergraphs involving maximal interactions.