Title: Correlators on the Wilson Line Defect CFT

Abstract: Conformal field theory (CFT) plays a key role in modern theoretical physics. Through CFT we describe real physical systems at criticality and fixed points of the renormalization group flow. It is also central in the study of quantum gravity, thanks to the AdS/CFT correspondence. This thesis originates in the context of the N=4 supersymmetric Yang-Mills (SYM) theory, which represents the CFT side of this correspondence. This work mainly revolves around the supersymmetric Wilson line and its interpretation as a conformal defect in N=4 SYM. Particularly, we focus on excitations localized on the defect called insertions, whose correlators are described by a one-dimensional CFT. The first main result of this work is an efficient algorithm for computing multipoint correlation functions of scalar insertions on the Wilson line, consisting of recursion relations up to next-to-leading order at weak coupling. We show various computations of such four-, five- and six-point correlators, and discuss their properties. Moreover, we use the four-point function case to illustrate the power of the Ward identities, which are crucial in deriving a next-to-next-to-leading order result. Thanks to these perturbative results, we find a family of differential operators annihilating our correlation functions, which we conjecture to be a multipoint extension of the Ward identities satisfied by the four-point functions. These non-perturbative constraints are shown to be fundamental ingredients in the bootstrap of a five-point function at strong coupling. To conclude, we define an inherently one-dimensional Mellin amplitude at the non-perturbative level with appropriate subtractions and analytical continuations. The efficiency of the 1d Mellin formalism is manifest at the perturbative level. We find a closed-form expression for the Mellin transform of leading order contact interactions and use it to extract CFT data.