The Geometry of Generalised Spin$^r$ Spinors on Projective Spaces

TL;DR

This work develops an exterior-forms realization of the spin representation for generalized spin$^r$ structures on the projective spaces $\mathbb{CP}^n$, $\mathbb{HP}^n$, and $\mathbb{OP}^2$, and identifies invariant spin$^r$ spinors across homogeneous realisations. It determines the minimal $r$ for which invariant spin$^r$ spinors exist and describes these spaces explicitly, including pure and parallel instances on $\mathbb{CP}^n$, a unique invariant spin$^{\mathbb{H}}$ spinor on $\mathbb{HP}^n$ when $r=3$, and a $4$-dimensional space of invariant spin$^9$ spinors on $\mathbb{OP}^2$. It further derives geometric consequences, showing, for instance, that the $\mathbb{CP}^n$ metric is Kähler-Einstein for certain spin$^{\mathbb{C}}$ twists, and that the quaternionic Kähler structure on $\mathbb{HP}^n$ arises from the invariant spin$^{\mathbb{H}}$ spinor, while the octonionic case yields parallel invariant spinors under a rank-3 endomorphism twist. Overall, the work merges differential-forms techniques with spin$^r$ geometry to expose how invariant spinors encode and reflect the underlying symmetric-space geometry.

Abstract

In this paper, we adapt the characterisation of the spin representation via exterior forms to the generalised spin$^r$ context. We find new invariant spin$^r$ spinors on the projective spaces $\mathbb{CP}^n$, $\mathbb{HP}^n$, and the Cayley plane $\mathbb{OP}^2$ for all their homogeneous realisations. Specifically, for each of these realisations, we provide a complete description of the space of invariant spin$^r$ spinors for the minimum value of $r$ for which this space is non-zero. Additionally, we demonstrate some geometric implications of the existence of special spin$^r$ spinors on these spaces.

Theorems & Definitions (50)

• Theorem
• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• Proposition 2.4: gen_spin_AL
• Definition 2.5
• Theorem 2.6: gen_spin_AL
• Definition 2.7
• Definition 2.8
• Remark 2.9