On rigid supersymmetry and notions of holomorphy in five dimensions

TL;DR

This paper explores the geometric structures underlying rigid N=1 supersymmetry in five dimensions, focusing on almost complex and CR structures, integrability conditions, and implications for localization and obstructions.

Contribution

It characterizes the geometric conditions for supersymmetric backgrounds in five dimensions, linking supersymmetry equations to CR and transversally holomorphic foliations.

Findings

Manifolds admit families of almost CR structures compatible with supersymmetry.

Derived integrability conditions for CR and THF structures in supersymmetric contexts.

Discussed potential global obstructions to solutions of supersymmetry equations.

Abstract

We study the equations governing rigid N=1 supersymmetry in five dimensions. If the supersymmetry spinor satisfies a reality condition, these are foliations admitting families of almost complex structures on the leaves. In other words, all these manifolds have families of almost Cauchy-Riemann (CR) structures. After deriving integrability conditions under which circumstances the almost CR structure defines a CR manifold or a transversally holomorphic foliation (THF), we discuss implications on localization. We also discuss potential global obstructions to the existence of solutions.