Geometry and entanglement in AdS/CFT and beyond

Holographic duality is the most well-understood non-perturbative framework of quantum gravity that we have. In particular, it has revealed a deep connection between quantum entanglement and the dual gravitational geometry. A semiclassical analysis on geometries has shed insights on the non-perturbative aspects of quantum field theories, some of which does not seem to require an asymptotic anti-de Sitter boundary. This raises the question, “What does the semiclassical gravity actually know, and how?” This thesis aims to approach a very narrow aspect of this question by, first, summarizing developments in the leading order perturbation theory. Then, we make some small advances on generalizing the perturbative framework to geometries beyond AdS/CFT. On the field theory we side, we discuss perturbation theory with irrelevant deformations; on the gravity side, we study cutoff AdS and D3 brane geometry. The result is not conclusive, but we believe that the framework we set up, along with knowledge in the first few chapters, will lead us to a better understanding of what the perturbation theory of semiclassical gravity is actually capable of.S.B