Pure spinors, intrinsic torsion and curvature in even dimensions

TL;DR

This paper explores the geometric and curvature properties of complex manifolds with pure spinor fields, developing a spinor calculus to analyze intrinsic torsion and curvature, with applications to differential equations and real manifolds.

Contribution

It introduces a spinor calculus framework to relate intrinsic torsion and curvature of pure spinor structures in complex manifolds, and proposes a Goldberg-Sachs-type conjecture.

Findings

Derived conditions for integrability of null structures from pure conformal Killing spinors.

Developed a spinor calculus encoding geometric properties of null distributions.

Proposed a conjecture linking curvature to the existence of special null structures.

Abstract

We study the geometric properties of a $2m$-dimensional complex manifold $\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset \mathrm{Spin}(2m,\mathbb{C})$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $\mathcal{M}$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $\xi$ up to scale. The defining property of $\xi$ is that it determines an almost null structure, ie an $m$-plane distribution $\mathcal{N}_\xi$ along which $g$ is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of $\mathcal{N}_\xi$ corresponding to the algebraic properties of the intrinsic torsion of the $P$-structure. This is the failure of the Levi-Civita connection $\nabla$ of $g$ to be compatible with the $P$-structure. In a similar way, we examine the algebraic properties of the curvature of $\nabla$. Applications to spinorial differential equations are given. In particular, we give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. We also conjecture a Goldberg-Sachs-type theorem on the existence of a certain class of almost null structures when $(\mathcal{M},g)$ has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds.