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Conformal Scale Factor Inversion for Domain Walls and Holography

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 AdS 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.