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.
Highlights
Domain wall solutions ofgravity are important both from the gravitational point of view [1] and because they correspond to the holographic renormalization group flows of dual quantum field theories (QFTs) [2,3,4,5,6,7,8,9,10,11]
The UV is a fixed point corresponding to the boundary of an asymptotically AdSdþ1 geometry, and conformal symmetry of the QFTd is broken by a scalar field φ, that drives the RG flow to the IR
A given pair of modes related by conformal scale factor inversion (CSFI) are superpartners, each can be found from acting with the supersymmetry (SUSY) generators on the other
Summary
Domain wall solutions of (super)gravity are important both from the gravitational point of view [1] and because they correspond to the holographic renormalization group flows of dual quantum field theories (QFTs) [2,3,4,5,6,7,8,9,10,11]. We introduce a map between the UV and IR limits of pairs of domain wall solutions related by inversion of the conformal scale factor aðzÞ. The map, which we call conformal scale factor inversion (CSFI), relates two domain wall geometries, DW and DgW, in Einstein gravity, with scalar fields φ and φand potentials VðφÞ and VðφÞ, aðzÞ. A given pair of modes related by CSFI are superpartners, each can be found from acting with the supersymmetry (SUSY) generators on the other This relates the corresponding “wave functions” in such a way that fixing the boundary conditions of fluctuations in one model fixes them in its CSFI-pair as well, even in cases where the extension of CSFI is only asymptotic, such as in asymptotically Liouville/AdS backgrounds.
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