Searching for self-duality in non-maximally supersymmetric backgrounds

Fermionic T-duality is the generalisation to superspace of bosonic T-duality (i.e. to include fermionic degrees of freedom). Originally, T-duality described the equivalence relation between two physical theories, each living on a different background. However, this thesis is concerned with fermionic T-duality and its role in self-duality. The goal is to determine whether AdS backgrounds with less than maximal supersymmetry are self-dual. A background is said to be self-dual if, after a specific sequence of bosonic and fermionic T-duality transformations, the original background is recovered. Self-dual backgrounds are of great interest due to their link to integrability. Fermionic T-duality has played a pivotal role in proving that the maximally supersymmetric background AdS₅ × S⁵ is self-dual. This background is also known to be integrable, therefore, when it was shown to be self-dual, the hypothesis that self-duality implied integrability, and vice-versa, was born. We investigate how far this hypothesis may be stretched for a number of AdS backgrounds, for which integrability has already been determined. The following backgrounds were considered: AdS₂ × S² × T⁶ and AdSd × Sᵈ XT(¹⁰⁻³ᵈ) (d = 2; 3). This question of self-duality was approached in two ways. In the first approach we show that these less supersymmetric backgrounds are self-dual by working with the supergravity fields and using the fermionic Buscher procedure derived by Berkovits and Maldacena. In the second approach, we verify the self-duality of Green-Schwarz supercoset σ-models on AdSd × Sᵈ (d = 2; 3) backgrounds. Furthermore, we prove the self-duality of AdS₅ × S⁵ without gauge fixing K-symmetry. We show that self-duality is a property which holds for the exceptional backgrounds, where the need to T-dualise along one of the spheres arises, again. Nature is not supersymmetric, therefore learning how to do physics in AdS₅ × S⁵ is not enough. In order to understand theories like Quantum Chromodynamics, we need to systematically break the supersymmetry present in our toy models. In this regard, it is easy to appreciate the significance of studying backgrounds with less than maximal supersymmetry

Gauge-gravity duality at finite N

Recently it has been shown that N = 4 super Yang-Mills theory is integrable in the planar limit. Past arguments suggest the integrability is only present in the planar limit. However, this conclusion was shown to be incorrect. Two speci c classes of operators were studied in the O(N) sector. The rst were labelled by Young diagrams having two long columns. The second were labelled by Young diagrams having two long rows. This result was then generalized to p long rows or columns with p xed to be O(1) as N ! 1. For this case, the non-planar limit was found to be integrable. In this dissertation, we extend this work by considering p to be O(N). We have calculated the dilation operator for the case with two impurities