Form factors and the dilatation operator in $\mathcal{N}=4$ super Yang-Mills theory and its deformations

In the first part of this thesis, we study form factors of general gauge-invariant local composite operators in $\mathcal{N}=4$ super Yang-Mills theory at various loop orders and for various numbers of external legs. We show how to use on-shell methods for their calculation and in particular extract the dilatation operator from the result. We also investigate the properties of the corresponding remainder functions. Moreover, we extend on-shell diagrams, a Gra{\ss}mannian integral formulation and an integrability-based construction via R-operators to form factors, focussing on the chiral part of the stress-tensor supermultiplet as an example. In the second part, we study the $\beta$- and the $\gamma_i$-deformation, which were respectively shown to be the most general supersymmetric and non-supersymmetric field-theory deformations of $\mathcal{N}=4$ super Yang-Mills theory that are integrable at the level of the asymptotic Bethe ansatz. For these theories, a new kind of finite-size effect occurs, which we call prewrapping and which emerges from double-trace structures that are required in the deformed Lagrangians. While the $\beta$-deformation is conformal when the double-trace couplings are at their non-trivial IR fixed points, the $\gamma_i$-deformation has running double-trace couplings without fixed points, which break conformal invariance even in the planar theory. Nevertheless, the $\gamma_i$-deformation allows for highly non-trivial field-theoretic tests of integrability at arbitrarily high loop orders