Supersymmetry on Three-dimensional Lorentzian Curved Spaces and Black Hole Holography

TL;DR

We develop a framework for N≤2 superconformal theories on Lorentzian 3-manifolds by analyzing charged conformal Killing spinors (CKS) and their relation to conformal Killing vectors (CKV). We show that a charged CKS exists iff the boundary has a CKV that is null or timelike, reproducing the bulk dichotomy of AdS4 BPS black holes via boundary data, with explicit boundary flux patterns such as $F o0$ for 1/2 BPS and $F o- frac12\mathrm{vol}_2$ for 1/4 BPS cases. We provide coordinate-realizations for null and timelike CKVs, relate boundary backgrounds to AlAdS bulk geometries, and discuss holographic RG flows from a 3D SCFT on $\mathbb{R}\times\Sigma_2$ to a 1D SCQM on the horizon, including magnetic, toroidal, and higher-genus horizons. This work clarifies how CKV/CKS data constrain holographic black hole microphysics and lays a path to derive horizon SCQM algebras such as ${\rm SU}(1,1|1)$ and ${\rm U}(1|1)$ from boundary theory.

Abstract

We study N <= 2 superconformal and supersymmetric theories on Lorentzian threemanifolds with a view toward holographic applications, in particular to BPS black hole solutions. As in the Euclidean case, preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a (charged) "conformal Killing spinor" on the boundary. We find that such spinors exist whenever there is a conformal Killing vector which is null or timelike. We match these results with expectations from supersymmetric four-dimensional asymptotically AdS black holes. In particular, BPS bulk solutions in global AdS are known to fall in two classes, depending on their graviphoton magnetic charge, and we reproduce this dichotomy from the boundary perspective. We finish by sketching a proposal to find the dual superconformal quantum mechanics on the horizon of the magnetic black holes.