Notes on the Twistor $\mathbf P^1$
Abstract
Remarkably, the twistor $\mathbf P^1$ occurs as a fundamental object in both fourdimensional spacetime geometry and in number theory. In Euclidean signature twistor theory it is how one describes spacetime points. In recent work by Fargues and Scholze on the local Langlands conjecture using geometric Langlands on the FarguesFontaine curve, the twistor $\mathbf P^1$ appears as the analog of this curve at the infinite prime. These notes are purely expository, written with the goal of explaining, in a form accessible to both mathematicians and physicists, various different ways in which the twistor $\mathbf P^1$ makes an appearance, often as a geometric avatar of the quaternions.
 Publication:

arXiv eprints
 Pub Date:
 February 2022
 DOI:
 10.48550/arXiv.2202.02657
 arXiv:
 arXiv:2202.02657
 Bibcode:
 2022arXiv220202657W
 Keywords:

 Mathematical Physics;
 Mathematics  Number Theory