Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 017, 32 pages      arXiv:2406.18337      https://doi.org/10.3842/SIGMA.2025.017

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

Diego Artacho a and Jordan Hofmann b
a) Imperial College London, London SW7 2AZ, UK
b) King's College London, London WC2R 2LS, UK

Received July 01, 2024, in final form March 01, 2025; Published online March 11, 2025

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.

Key words: special spinors; projective spaces; generalized spin structures; spin$^c$; spin$^h$.

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