Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds

TL;DR

This paper provides a unified spinorial framework for describing $ ext{SU}(3)$- and $G_2$-structures, leading to new embedding theorems and methods for constructing conical manifolds.

Contribution

It introduces a uniform spinorial characterization of special geometric structures and applies it to hypersurface theory and manifold construction.

Findings

Derived new embedding theorems for $ ext{SU}(3)$- and $G_2$-manifolds.

Developed a general recipe for building conical manifolds.

Unified various notions of Killing spinors within a single framework.

Abstract

We present a uniform description of $\mathrm{SU}(3)$-structures in dimension $6$ as well as $G_2$-structures in dimension $7$ in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach also enables one to subsume all variations of the notion of a Killing spinor.