Notes on the Twistor $\mathbf P^1$

TL;DR

This paper explores the fundamental role of the twistor  in both space-time geometry and number theory, highlighting its appearances as a geometric avatar of quaternions in various contexts.

Contribution

It provides an accessible exposition of the twistor 's appearances in physics and mathematics, connecting twistor theory with recent developments in number theory and geometric Langlands.

Findings

Twistor  describes space-time points in Euclidean signature.

It appears as an analog of the Fargues-Fontaine curve at the infinite prime.

The notes clarify the geometric and algebraic significance of twistor .

Abstract

Remarkably, the twistor $\mathbf P^1$ occurs as a fundamental object in both four-dimensional space-time geometry and in number theory. In Euclidean signature twistor theory it is how one describes space-time points. In recent work by Fargues and Scholze on the local Langlands conjecture using geometric Langlands on the Fargues-Fontaine 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.