Perturbation theory for string sigma models

TL;DR

This thesis explores quantum properties of superstrings in AdS backgrounds, developing perturbative methods, deriving actions, and testing integrability predictions to support the models' quantum consistency.

Contribution

It provides new derivations of superstring actions in various AdS backgrounds, introduces a unitarity-based method for S-matrix calculations, and compares perturbative results with integrability predictions.

Findings

Derived superstring actions for AdS4 and AdS3 backgrounds.

Developed a unitarity-based method for one-loop S-matrix calculations.

Reproduced two-loop free energy and tested integrability predictions.

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.