Francesco Fiamberti

We review the general procedure for the field-theoretical computation of wrapping effects in standard and beta-deformed N=4 super Yang-Mills by means of N=1 superspace techniques. In the undeformed theory, these methods allowed to find explicit results at four and five loops for two-impurity operators. In the deformed case, a general expression for the finite-size correction to the anomalous dimension of single-impurity operators at the critical order was obtained.

Several perturbative computations of finite-size effects, performed on the gauge side of the AdS/CFT correspondence by means of superspace techniques, are presented. First, wrapping effects are analyzed in the standard N = 4 theory, by means of the calculation of the four-loop anomalous dimension of the Konishi operator. Then, a similar computation at five loops is described. Afterwards, finite-size effects are studied in the beta-deformed case, where thanks to the reduced number of supersymmetries the simpler class of single-impurity operators can be considered, so that the leading corrections to the anomalous dimensions at generic order can be reduced to the computation of a class of integrals. Explicit results are given up to eleven loops. A further chapter is dedicated to the computation of the leading finite-size effects on operators dual to open strings. In the end, some comments are made and proposals for future developments are discussed.

We compute the anomalous dimension of a length-five operator at five-loop order in the SU(2) sector of N=4 SYM theory in the planar limit. This is critical wrapping order at five loops. The result is obtained perturbatively by means of N=1 superspace techniques. Our result from perturbation theory confirms explicitly the formula conjectured in arXiv:0901.4864 for the five-loop anomalous dimension of twist-three operators. We also explicitly obtain the same result by employing the recently proposed Y-system.

We study the spectrum of one single magnon in the superconformal beta-deformed N=4 SYM theory in the planar limit. We compute the anomalous dimensions of one-impurity operators O_{1,L}= tr(phi Z^{L-1}), including wrapping contributions at their critical order L.

We present the planar four-loop anomalous dimension of the composite operator tr(phi[Z,phi]Z) in the flavour SU(2) sector of the N=4 SYM theory. At this loop order wrapping interactions are present: they give rise to contributions proportional to zeta(5) increasing the level of transcendentality of the anomalous dimension. In a sequel of this paper all the details of our calculation will be reported.