Gravity on the Bruhat-Tits Tree

Abstract

It has been noted that the set of p-adic numbers and the Bruhat-Tits tree serve as a useful pairing of boundary and bulk on which a theory of AdS/CFT can be formulated. In this thesis, we assume there are two fields that live on the tree: a scalar matter field and a graviton field. We compute bulk-to-boundary propagators for both fields as a first step towards calculating how their behavior in the bulk corresponds with the CFT side of the duality. We also extend a formulation of the theory of gravity on this tree developed by Gubser et al. to include these scalars, and we explore perturbative solutions to the resulting Einstein equation on the tree. We find that our solutions depend on the geometry of our fields, but for fields with reasonable symmetry, we find solutions that are consistent with expectations about how the fields should act at the boundary. This is true even when we introduce the notion of a Euclidean black hole on the tree, which is represented as a loop in the tree. Though these solutions are arrived at via pertubative methods, in some circumstances they are valid and bounded throughout the entirety of the tree. Furthermore, we find these solutions correspond to adding relevant operators to our conformal field theory, and because the renormalization group flow in our CFT has a low-energy cutoff in some of our geometries, set by the existence of a "center'' of our tree, this shows that we will not flow out of the perturbative regime and can take our perturbative solutions as reliable throughout the tree.