Spin actions and Polygon spaces

TL;DR

The paper constructs correspondences between polygon spaces in dimensions $2$, $3$, $5$, and $9$ and quotient spaces arising from $2$-Stiefel manifolds over normed division algebras $\\mathbb{F}$ (real, complex, quaternions, and octonions). It employs Hopf maps on $\\mathbb{F}^2$ and spin representations of $SU(2,\mathbb{F})$ to induce $SO(1+\dim \mathbb{F})$ actions, enabling identifications with (or lifts to) Grassmannians and related homogeneous spaces. For $\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$, the authors obtain explicit homeomorphisms $\widetilde{\mathcal{P}}_k(\mathbb{R}^n)$ with quotients of (orbits on) $2$-Grassmannians, while for $\mathbb{O}$ they extend the construction via $SU(2,\mathbb{O})\simeq Spin(9)$ and a compatible octonionic Hopf framework. The octonionic case illustrates how nonassociativity necessitates modified actions and multiplication rules, yet yields a parallel $\widetilde{\mathcal{P}}_k(\mathbb{R}^9) \simeq SU(2,\mathbb{O})\backslash V_{\mathbb{O}}(2,k)/\mathbb{O}(1)^k$ result, broadening Hausmann–Knutson’s 2- and 3-dimensional picture to higher dimensions with a unified spinorial viewpoint.

Abstract

In this article, we construct correspondences between polygon spaces in Euclidean spaces of dimension $2,3,5,9\ $and the quotient spaces of $2$-Steifel manifolds along the normed division algebra$\ \mathbb{F}$ real $\mathbb{R}$, complex $\mathbb{C}$, quaternions $\mathbb{H}$, octonions $\mathbb{O}$. For the purpose, we introduce Hopf map on $\mathbb{F}^{2}\ $and consider the spin action of $SU\left( 2,\mathbb{F}\right) $ to spinor $\mathbb{F}^{2}\ $and the induced $SO\ $action to the Euclidean space $\mathbb{R\oplus F}$. The correspondences are extension of the work of Hausmann and Knutson for polygon spaces of dimension $2,3\ $and $2$-Grassmannians over real and complex.

Theorems & Definitions (33)

• Remark 1
• Example 3
• Lemma 4
• Lemma 5
• Lemma 6
• Corollary 7
• Proposition 8
• Proposition 9
• Remark 10
• Theorem 11