Spinors and horospheres

TL;DR

The paper develops a spinorial framework unifying 3D hyperbolic geometry with higher-rold Teichmüller theory and Grassmannians. It constructs an explicit SL(2,$\mathbb{C}$)-equivariant bijection between nonzero spinors and spin-decorated horospheres in $\mathbb{H}^3$, encodes complex distances and lambda lengths, and proves a 3D Ptolemy equation for ideal tetrahedra. It further establishes deep connections to Grassmannians via Plücker coordinates, identifying decorated Teichmüller spaces with Grassmannian cones and embedding cluster algebra structures of type A into this geometric framework. The results generalize Penner’s 2D theory to 3D, providing a cohesive bridge between hyperbolic geometry, spinor calculus, and algebraic combinatorics with potential applications to hyperbolic 3-manifolds and higher Teichmüller theory.

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 Plücker coordinates, illuminating the connection between Grassmannians, hyperbolic polygons, and type A cluster algebras.

Figures (5)

• Figure 1: Left: null flag corresponding to a spinor. Right: point on light cone (red), with corresponding horosphere $H$ in $\mathbb{H}^2$ (blue).
• Figure 2: Left: Null flag corresponding to $(\xi, \eta)$, and corresponding horosphere. Right: decorated horospheres as they appear in the upper half space model $\mathbb{U}$.
• Figure 3: Complex distance between horospheres.
• Figure 4: Decorated horospheres and complex lambda lengths along the edges of an ideal tetrahedron.
• Figure 5: Tetrahedron with vertices labeled $0$, $1$, $2$, $3$.

Theorems & Definitions (54)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 2.1: Penrose_Rindler84
• Definition 2.3
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Lemma 2.8