Extended Supersymmetry on Curved Spaces

TL;DR

This paper characterizes the conditions for preserving N=2 supersymmetry on four-dimensional curved manifolds, providing explicit solutions for auxiliary fields and showing the independence of supersymmetry conditions from spacetime signature.

Contribution

It derives and solves differential constraints for N=2 superconformal theories on curved spaces, offering explicit auxiliary field expressions and analyzing signature independence.

Findings

Supersymmetry conditions depend mainly on the existence of a conformal Killing vector.

Explicit auxiliary field solutions are provided for general backgrounds.

Supersymmetry constraints are largely signature-independent, except in special cases.

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.