Supersymmetry in Lorentzian Curved Spaces and Holography

TL;DR

This paper explores the conditions for supersymmetry in four-dimensional Lorentzian curved spacetimes and their five-dimensional holographic duals, revealing a link between conformal Killing spinors and null conformal Killing vectors.

Contribution

It establishes the equivalence between the existence of charged conformal Killing spinors and null conformal Killing vectors in Lorentzian signature, extending previous Euclidean results.

Findings

Supersymmetry requires a null conformal Killing vector in Lorentzian spaces.

For theories with R-symmetry, this vector is a Killing vector.

Results align with classifications of supersymmetric solutions in five-dimensional supergravity.

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.