This dissertation will focus on exploring the AdS/CFT correspondence, both as a tool to probe the behavior of strongly coupled conformal field theories (CFTs) and as a fundamental duality to help us understand the holographic connection between quantum gravity and gauge theories.

We will begin with an overview of this (two-part) dissertation, followed by an introduction to gravity in asymptotically locally AdS spacetimes.

In the first part of the dissertation, we will then discuss the use of AdS/CFT as a tool to probe the dynamics of heat transport in strongly coupled CFTs. We will begin with a simple case in three bulk spacetime dimensions, and then construct a four-dimensional black hole solution which is dual to heat flow in a three-dimensional CFT. This black hole solution is stationary, but its horizon is not a Killing horizon, making it an interesting gravitational solution in its own right. We will then construct the gravitational dual to a CFT on a rotating black hole, and find that the CFT does not carry heat away from the black hole, but rather is confined to a halo around it. This is an artefact of strong coupling, and we comment on possible connections to soft condensed matter phenomena.

In the second part of this dissertation, we probe the AdS/CFT dictionary via entanglement entropy. Specifically, we show that the Hubeny-Rangamani-Takayanagi (HRT) prescription, which is a prescription for computing CFT entanglement entropy holographically, requires modification. We comment on possible modifications, and explore in depth the possibility of using complexified surfaces in the HRT prescription. Finally, we will approach the issue of bulk reconstruction via hole-ography, which attempts to reconstruct the bulk geometry from the entanglement entropy of regions of the CFT. We put some constraints on when this approach can succeed, and comment on why it might fail when it does.