Twistor Theory of Dancing Paths

TL;DR

This paper develops a twistor-theoretic framework linking path geometries on surfaces to causal structures on higher-dimensional manifolds, with specific examples involving projective planes, lines, ellipses, and conics.

Contribution

It introduces a novel construction of causal structures from path geometries and characterizes when these structures are conformal, including explicit examples and extensions to conic configurations.

Findings

Causal structures correspond to path geometries on the projective plane.

Explicit example with a symmetric sextic and SL(2,R)-invariant structure.

Extension to causal structures involving conics on a seven-dimensional manifold.

Abstract

Given a path geometry on a surface $\mathcal{U}$, we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on $\mathcal{U}$. This causal structure corresponds to a conformal structure if and only if $\mathcal{U}$ is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds on an ${\rm SL}(2,{\mathbb R})$-invariant projective structure where the paths are ellipses of area $\pi$ centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane.