Pure spinors, intrinsic torsion and curvature in odd dimensions

TL;DR

This paper investigates the geometric and algebraic properties of odd-dimensional complex manifolds with pure spinors, focusing on intrinsic torsion, curvature, and their implications for differential equations and integrability conditions.

Contribution

It develops a spinor calculus framework to analyze intrinsic torsion and curvature in odd dimensions, linking geometric structures to spinorial differential equations and integrability conditions.

Findings

Characterization of intrinsic torsion via spinor calculus

Conditions for integrability of null structures

Relation between curvature properties and spinor solutions

Abstract

We study the geometric properties of a $(2m+1)$-dimensional complex manifold $\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset \mathrm{Spin}(2m+1,\mathbb{C})$, 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, i.e.\ 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 $\mathcal{N}_\xi$ and of its rank-$(m+1)$ orthogonal complement $\mathcal{N}_\xi^\perp$ 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. Notably, we relate the integrability properties of $\mathcal{N}_\xi$ and $\mathcal{N}_\xi^\perp$ to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we 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.