Rigid Supersymmetry on 5-dimensional Riemannian Manifolds and Contact Geometry

TL;DR

This paper explores the geometric conditions for rigid supersymmetry on 5-dimensional Riemannian manifolds, linking solutions of Killing spinor equations to contact geometry and special manifold structures.

Contribution

It generalizes methods to 5D manifolds, relating supersymmetry solutions to contact structures and classifying geometries based on the number of solutions.

Findings

Existence of solutions relates to almost contact metric structures.

Special cases include manifolds foliated by Quaternion-Kahler submanifolds.

Geometry is constrained to S3 or T3-fibrations with multiple solutions.

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.