The Lipschitz Spinor-Higher Horosphere Correspondence

TL;DR

This paper generalizes the correspondence between spinors and horospheres from low dimensions to higher-dimensional hyperbolic spaces using Lipschitz spinors and Clifford algebras.

Contribution

It introduces an equivariant correspondence linking Lipschitz spinors, null multiflags, and higher-dimensional horospheres with extended spin decorations.

Findings

Established a generalized spinor-horosphere correspondence in higher dimensions.

Extended Mathews' isomorphism to Lipschitz spinors in Clifford algebra contexts.

Enabled application of spinors to hyperbolic spaces of any dimension.

Abstract

In a paper of Mathews, an isomorphism is constructed between two-component complex spinors and horospheres in H^3 carrying `spin decorations'. A recent arXiv preprint of Mathews and Varsha arXiv:2412.06572 extends this result to the case of `quaternionic spinors' and spin decorated horospheres in H^4. The following work generalises these results to an equivariant correspondence between two-component `Lipschitz spinors' with entries drawn from the Lipschitz group of a Clifford algebra, null multiflags in generalised Minkowski space, and higher-dimensional horospheres that carry an extension of the Mathews spin decoration. This correspondence allows spinors to be applied to horospheres in any dimension of hyperbolic space.