Investigating the off-shell stability of anti-de Sitter space in string theory

We propose an investigation of stability of vacua in string theory by studying their stability with respect to a (suitable) world-sheet renormalization group (RG) flow. We prove the geometric stability of (Euclidean) anti-de Sitter (AdS) space (i.e., H-n) with respect to the simplest RG flow in closed string theory, the Ricci flow. AdS space is not a fixed point of the Ricci flow. We therefore choose an appropriate flow for which it is a fixed point, prove a linear stability result for AdS space with respect to this flow and then show that this implies its geometric stability with respect to the Ricci flow. The techniques used can be generalized to RG flows involving other fields. We also discuss tools from the mathematics of geometric flows that can be used to study stability of string vacua.