Perturbation theory for string sigma models

Abstract

In this thesis we investigate quantum aspects of the Green-Schwarz superstring in various AdS backgrounds relevant for the AdS/CFT correspondence, providing several examples of perturbative computations in the corresponding integrable sigma-models. We start by reviewing in details the supercoset construction of the superstring action in $AdS_5 \times S^5$, pointing out the limits of this procedure for $AdS_4$ and $AdS_3$ backgrounds. For the $AdS_4 \times CP^3$ case we give a thorough derivation of an alternative action, based on the double-dimensional reduction of eleven-dimensional super-membranes. We then consider the expansion about the BMN vacuum and the S-matrix for the scattering of worldsheet excitations in the decompactification limit. To evaluate its elements efficiently we describe a unitarity-based method resulting in a very compact formula yielding the cut-constructible part of any one-loop two-dimensional S-matrix. In the second part of this review we analyze the superstring action on $AdS_4 \times CP^3$ expanded around the null cusp vacuum. The free energy of this model, whose computation we reproduce up to two-loops at strong coupling, is related to the cusp anomalous dimension of the ABJM theory and, indirectly, to a non-trivial effective coupling $h(\lambda)$ featuring all integrability-based calculations in $AdS_4/CFT_3$. Finally, we extensively discuss the comparison of the perturbative results and the integrability predictions for the one-loop dispersion relation of GKP excitations. Our results provide valuable data in support of the quantum consistency of the string actions - often debated due to possible issues with cancellation of UV divergences and the lack of manifest power-counting renormalizability - and furnish non-trivial stringent tests for the quantum integrability of the analyzed models.