Supersymmetry on Curved Spaces and Holography

TL;DR

This paper explores the geometric conditions for supersymmetry on curved spaces in three and four dimensions, with implications for holography and superconformal theories, showing that complex and Sasaki manifolds admit supersymmetric structures.

Contribution

It demonstrates that superconformal theories can preserve supersymmetry on any complex four-manifold and provides examples of supersymmetric theories on Sasaki three-manifolds, expanding the understanding of supersymmetry in curved spaces.

Findings

Existence of conformal Killing spinors on complex four-manifolds.

Superconformal theories admit at least one supercharge on these manifolds.

Examples of supersymmetric theories on Sasaki spaces provided.

Abstract

We study superconformal and supersymmetric theories on Euclidean four- and three-manifolds with a view toward holographic applications. Preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a (charged) "conformal Killing spinor" on the boundary. We study the geometry behind the existence of such spinors. We show in particular that, in dimension four, they exist on any complex manifold. This implies that a superconformal theory has at least one supercharge on any such space, if we allow for a background field (in general complex) for the R-symmetry. We also show that this is actually true for any supersymmetric theory with an R-symmetry. We also analyze the three-dimensional case and provide examples of supersymmetric theories on Sasaki spaces.