Supersymmetry on Curved Spaces and Holography

TL;DR

The work establishes a concrete link between holography and supersymmetry on curved spaces by deriving and analyzing charged conformal Killing spinor equations on the boundary of asymptotically locally AdS spaces. It shows that in four dimensions a CK spinor exists on any complex manifold when a (generally complex) R-symmetry background is allowed, implying at least one supercharge for any superconformal (and more generally any R-symmetric) theory on such spaces. The authors connect CK spinors to new minimal supergravity, provide intrinsic-torsion formalisms in 4D and 3D, and give explicit geometric classes (Kähler, Sasaki, Sasaki-Einstein, etc.) that support supersymmetry, including dimensional reduction to 3D with concrete round and squashed sphere and Sasaki-manifold examples. These results offer practical templates for constructing holographic CFTs on curved manifolds and for extracting boundary Lagrangians from bulk AdS solutions.

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.