Conformal scale factor inversion for domain walls and holography

Abstract

We describe a correspondence between domain wall solutions of Einstein gravity with a single scalar field and self-interaction potential. The correspondence we call conformal scale factor inversion (CSFI) is a map comprising the inversion of the scale factor in conformal coordinates, and a transformation of the field and its potential which preserves the form of the Einstein equations for static and isotropic domain walls. By construction, CSFI maps the asymptotic anti-de Sitter boundary to the vicinity of a naked singularity in a theory with a special Liouville (exponential) scalar field potential; it is also a map in the parameter space of exponential potentials. The correspondence can be extended to linear fluctuations, being akin to an S-duality, and can be interpreted in terms of "SUSY quantum mechanics" for the fluctuation modes. The holographic implementation of CSFI relates the UV and IR regimes of a pair of holographic renormalization group flows; in particular, it is a symmetry of the GPPZ flow.