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.
Highlights
Equations arising from the gravitino and dilatino variations
We study the equations governing rigid N = 1 supersymmetry in five dimensions
This is integrable in a sense that we will discuss shortly and it follows that one can introduce differentials ∂b and ∂ ̄b that correspond to the Dolbeault operators ∂ and ∂ ̄ that are familiar from complex geometry
Summary
The bulk of our analysis is based on the following set of bi-spinors that can be defined for any given ξI :. A further consequence of (A.1) is that s ≥ 0 with equality if and only if ξI = 0 It follows that s > 0 everywhere on M since the gravitino equation is linear and of first order. We will refer to vectors and forms parallel to R and κ respectively as vertical and their orthogonal complement as horizontal. The Hodge dual defines the notion of self-dual and anti self-dual forms on the horizontal subspace. Since the ΘIJ are both horizontal and self-dual, ΘIJ = (ΘIJ )+, they define an isomorphism between su(2)R and the su(2)+ factor in the typical so(4) ∼= su(2)+ × su(2)− decomposition of the Lie algebra of the structure group. Once we IJ impose the reality condition (2.10) for mIJ , det m will be positive semi-definite. Note that Φ is invariant under mIJ → f mIJ for any non-zero function f
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