Conformal Bootstrap for AdS Loops

This thesis studies loop corrections to four-point functions of identical scalar operators in four-dimensional conformal field theories with weakly coupled AdS duals. The main goal is to determine how much of the quantum dynamics in AdS, at all orders in the large-$N$ perturbation, is encoded in lower-order CFT data. We address this question using analytic bootstrap methods. We first consider a scalar theory in AdS with a quartic interaction and isolate a contribution to the correlator that, at any order in $1/N$, is completely determined by tree-level data. This contribution controls the leading logarithmic behavior of double-trace anomalous dimensions in the large-spin expansion. We resum these effects to all orders in $1/N$ and show that they admit an effective description in AdS. In Mellin space, the flat-space limit of the same contribution reproduces consecutive unitarity cuts of the corresponding bubble diagrams. We then study the appearance of higher-trace operators in theories dual to bulk $\phi^3$ and $\phi^4$ interactions. Infinite spin sums of double-trace data generate contributions which, after crossing, have the structure of higher-trace exchanges. This shows how multi-trace sectors arise as a necessary consequence of crossing symmetry and allows us to determine part of their OPE data. Finally, we introduce a diagrammatic framework to organize the different terms in the conformal block expansion. This framework makes the relations between lower- and higher-trace data more transparent and allows crossing symmetry to be applied to individual diagrammatic topologies.