### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 027, 13 pages      arXiv:2201.04717      https://doi.org/10.3842/SIGMA.2022.027
Contribution to the Special Issue on Twistors from Geometry to Physics in honor of Roger Penrose

### Twistor Theory of Dancing Paths

Maciej Dunajski
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

Received January 14, 2022, in final form March 28, 2022; Published online March 31, 2022

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.

Key words: path geometry; twistor theory; causal structures.

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