Holography, Probe Branes and Isoperimetric Inequalities

TL;DR

This paper demonstrates that a key relation in holography between probe brane actions and bulk gravitational actions depends on an isoperimetric inequality in Poincaré-Einstein spaces, ensuring consistency of the holographic correspondence.

Contribution

It establishes a connection between holographic relations and isoperimetric inequalities, deriving the latter from mathematical theorems by Lee and Wang.

Findings

The relation between probe brane and bulk actions is validated.

The isoperimetric inequality is shown to hold in a broad class of spaces.

Mathematical theorems underpin the physical consistency of holography.

Abstract

In many instances of holographic correspondences between a d dimensional boundary theory and a d+1 dimensional bulk, a direct argument in the boundary theory implies that there must exist a simple and precise relation between the Euclidean on-shell action of a (d-1)-brane probing the bulk geometry and the Euclidean gravitational bulk action. This relation is crucial for the consistency of holography, yet it is non-trivial from the bulk perspective. In particular, we show that it relies on a nice isoperimetric inequality that must be satisfied in a large class of Poincar\'e-Einstein spaces. Remarkably, this inequality follows from theorems by Lee and Wang.