Supersymmetry in Lorentzian Curved Spaces and Holography

TL;DR

The paper analyzes how supersymmetry can be preserved for four-dimensional Lorentzian curved spaces and their five-dimensional holographic duals. It shows that a charged conformal Killing spinor on the boundary is equivalent to the existence of a null conformal Killing vector, and that with an R-symmetry this becomes a Killing vector, linking boundary SUSY to bulk minimal gauged supergravity. The work derives precise geometric data—spinor bilinears, intrinsic torsions, and CKY forms—and provides explicit gauge-field expressions and coordinate forms, establishing a consistent boundary-bulk dictionary in Lorentzian signature. It also connects to the new minimal supergravity framework and analyzes time-like and null bulk solutions, including explicit examples and Fefferman–Graham expansions, highlighting the distinct features from the Euclidean case. Overall, the results unify boundary supersymmetric backgrounds with holographic bulk solutions and set the stage for constructing regular bulk geometries with given Lorentzian boundary data.

Abstract

We consider superconformal and supersymmetric field theories on four-dimensional Lorentzian curved space-times, and their five-dimensional holographic duals. As in the Euclidean signature case, preserved supersymmetry for a superconformal theory is equivalent to the existence of a charged conformal Killing spinor. Differently from the Euclidean case, we show that the existence of such spinors is equivalent to the existence of a null conformal Killing vector. For a supersymmetric field theory with an R-symmetry, this vector field is further restricted to be Killing. We demonstrate how these results agree with the existing classification of supersymmetric solutions of minimal gauged supergravity in five dimensions.