Rigid Supersymmetry on 5-dimensional Riemannian Manifolds and Contact Geometry

TL;DR

The paper investigates rigid supersymmetry on 5D Riemannian manifolds by deriving and analyzing the Killing spinor equation from 5D minimal off-shell supergravity. It shows that the existence and number of spinor solutions impose geometric structures ranging from almost contact metric structures to Quaternion-Kähler-like foliations and constrained isometry algebras, including S^3- and T^3-fibrations. The authors partially solve auxiliary fields in terms of spinor bilinears, develop a compatible connection framework, and discuss dimensional reductions to 4D theories. They also propose a supersymmetric vector multiplet theory on K-contact backgrounds and illustrate the framework with explicit examples such as M = S^1 × S^4 and M = S^2 × S^3. Overall, the work clarifies how 5D geometry dictates supersymmetry and opens routes to constructing 5D supersymmetric theories on nontrivial backgrounds.

Abstract

In this note we generalize the methods of [1][2][3] to 5-dimensional Riemannian manifolds M. We study the relations between the geometry of M and the number of solutions to a generalized Killing spinor equation obtained from a 5-dimensional supergravity. The existence of 1 pair of solutions is related to almost contact metric structures. We also discuss special cases related to $M = S1 \times M4$, which leads to M being foliated by submanifolds with special properties, such as Quaternion-Kahler. When there are 2 pairs of solutions, the closure of the isometry sub-algebra generated by the solutions requires M to be S3 or T3-fibration over a Riemann surface. 4 pairs of solutions pin down the geometry of M to very few possibilities. Finally, we propose a new supersymmetric theory for N = 1 vector multiplet on K-contact manifold admitting solutions to the Killing spinor equation.