Perturbative and non-perturbative analysis of defect correlators in AdS/CFT

In this thesis, we consider two approaches to the study of correlation functions in one-dimensional defect Conformal Field Theories (dCFT$_1$), in particular those defined by 1/2-BPS Wilson line defects in the three- and four-dimensional superconformal theories relevant in the AdS/CFT correspondence. In the first approach, we use the analytic conformal bootstrap to evaluate two examples of defect correlators. The four-point correlator of the displacement supermultiplet inserted on the 1/2-BPS Wilson line in ABJM theory is computed to the third order in a strong-coupling expansion and reproduces the explicit first-order Witten diagram calculations. The CFT$_1$ data are then extracted from this correlator, and the operator mixing is solved at first order. Consequently, all-order results are derived for the part of the correlator with the highest logarithm power, uniquely determining the double-scaling limit. Then, the five-point correlator of $1/2$-BPS operators inserted on the 1/2-BPS Wilson line in $\mathcal{N}=4$ super Yang-Mills are studied. The superblocks are derived for all channels of the OPE, and the five-point correlator is bootstrapped to first order in a strong coupling expansion. The CFT$_1$ data are then extracted, confirming that operator mixing does not affect the first-order anomalous dimension. The second approach considers the general structure of correlators in effective theories in AdS$_2$. All scalar $n$-point contact Witten diagrams for external operators of integer conformal weight are computed. Effective theories in AdS$_2$ defined by an interaction Lagrangian with an arbitrary number of derivatives are then considered and solved to first order using a new formalism of Mellin amplitudes for 1d CFTs. Finally, the cusped Wilson line discretised action is presented as an alternative way to obtain non-perturbative data: through Lattice Field Theory.