Bulk space-time geometries in AdS/CFT

Abstract

The AdS/CFT correspondence provides a geometric description of certain strongly coupled conformal field theories (CFTs). These CFTs are conjectured to have a dual description involving `bulk' space-time geometries that solve Einstein's equation. It is a holographic correspondence, so the CFTs in some sense lives on the boundary of the bulk. In the regime where this description is applicable, the holographic CFTs are at strong coupling, and can be placed on non-trivial curved space-times. In these contexts, other available tools, such as perturbation theory and lattice techniques, break down.

Under this correspondence, physical quantities in the CFT can be extracted from the bulk geometry. This means that properties of the CFT will be reflected in features of the dual bulks. Using a mix of basic geometry and numerical methods, we explore ways in which the bulk space-time geometries in the AdS/CFT correspondence reflect physical properties of the dual CFTs. We will, for instance, discuss the role of certain features of the bulk geometry in describing a large scale limit of the CFT state. This will motivate us to construct a class of bulk geometries numerically that describe this large scale limit.

We will also find that the geometric tools that come with the bulk description allow us to make powerful statements about the CFTs which would be non-trivial to derive using traditional quantum field theory methods. We will be able to derive bounds on the energy gap and vacuum energy density of these CFTs using basic geometric methods. Finally, we will end with a conjecture of a bound on a finite temperature phase transition in the CFT, which we will present analytical and numerical evidence for using similar bulk geometric arguments.