From spinors to horospheres: a geometric tour

Abstract

This article is an exposition and elaboration of recent work of the first author on spinors and horospheres. It presents the main results in detail, and includes numerous subsidiary observations and calculations. It is intended to be accessible to graduate and advanced undergraduate students with some background in hyperbolic geometry. The main result is the spinor--horosphere correspondence, which is a smooth, $SL(2,\mathbb{C})$-equivariant bijection between two-component complex spin vectors and spin-decorated horospheres in three-dimensional hyperbolic space. The correspondence includes constructions of Penrose--Rindler and Penner, which respectively associate null flags in Minkowski spacetime to spinors, and associate horospheres to points on the future light cone. The construction is presented step by step, proceeding from spin vectors, through spaces of Hermitian matrices and Minkowski space, to various models of 3-dimensional hyperbolic geometry. Under this correspondence, we show that the natural inner product on spinors corresponds to a 3-dimensional, complex version of lambda lengths, describing a distance between horospheres and their decorations. We also discuss various applications of these results. An ideal hyperbolic tetrahedron with spin-decorations at its vertices obeys a Ptolemy equation, generalising the Ptolemy equation obeyed by 2-dimensional ideal quadrilaterals. More generally we discuss how real spinors describe 2-dimensional hyperbolic geometry. We also discuss the relationships between spinors, horospheres, and various sets of matrices.