Spinors and horospheres

TL;DR

This paper establishes a bijective link between spinors and horospheres in 3D hyperbolic space, generalizes Penner's lambda lengths, and explores their applications in hyperbolic geometry, Grassmannians, and cluster algebras.

Contribution

It introduces a new correspondence between spinors and horospheres, extending lambda length concepts to three dimensions and connecting hyperbolic geometry with Grassmannian and cluster algebra structures.

Findings

Complex lambda lengths satisfy Ptolemy equations in tetrahedra.

Correspondences between hyperbolic polygons and Grassmannian spaces are established.

Lambda lengths relate to Plücker coordinates, linking geometry and algebra.

Abstract

We give an explicit bijective correspondence between between nonzero pairs of complex numbers, which we regard as spinors or spin vectors, and horospheres in 3-dimensional hyperbolic space decorated with certain spinorial directions. This correspondence builds upon work of Penrose--Rindler and Penner. We show that the natural bilinear form on spin vectors describes a certain complex-valued distance between spin-decorated horospheres, generalising Penner's lambda lengths to 3 dimensions. From this, we derive several applications. We show that the complex lambda lengths in a hyperbolic ideal tetrahedron satisfy a Ptolemy equation. We also obtain correspondences between certain spaces of hyperbolic ideal polygons and certain Grassmannian spaces, under which lambda lengths correspond to Pl\"{u}cker coordinates, illuminating the connection between Grassmannians, hyperbolic polygons, and type A cluster algebras.