Extended Supersymmetry on Curved Spaces

TL;DR

Extends the study of SUSY on curved spaces by solving the generalized $CKS$ equation for ${\cal N}=2$ theories on Lorentzian and Euclidean four-manifolds. The main finding is that the existence of a conformal Killing vector $z$ suffices to preserve some supersymmetry, with all other background fields fixed by the geometry; explicit expressions for the background gauge fields, tensor $T^+$, and scalar $d$ are given, including plus-or-minus signs in different cases. The authors illustrate the framework with examples such as round and squashed spheres, topological twists, and the $\Omega$-background, and discuss holographic and localization applications. The results are largely signature-independent and extend previous ${\cal N}=1$ analyses, offering a unified approach to rigid ${\cal N}=2$ SUSY on curved spaces and potential higher-dimensional extensions.

Abstract

We study N=2 superconformal theories on Euclidean and Lorentzian four-manifolds with a view toward applications to holography and localization. The conditions for supersymmetry are equivalent to a set of differential constraints including a "generalised" conformal Killing spinor equation depending on various background fields. We solve these equations in the general case and give very explicit expressions for the auxiliary fields that we need to turn on to preserve some supersymmetry. As opposed to what has been observed for the N=1 case, the conditions for unbroken supersymmetry turn out to be almost independent of the signature of spacetime, with the exception of few degenerate cases including the topological twist. Generically, the only geometrical constraint coming from supersymmetry is the existence of a conformal Killing vector on the manifold, all other constraints determine the background auxiliary fields.