Back in October Peter Scholze wrote to me to tell me he had taken a look at my Brown lecture and was interested in twistors, due to the fact that the twistor $P^1$ was the infinite prime analog of the Fargues-Fontaine curve. I’ve also been fascinated by the Fargues-Scholze work, while understanding very little of it. Over the past couple years I’ve gotten much more deeply involved in twistor theory, working on some ideas about how to get unification out of the Euclidean version of it. Some years later I did talk about this a little with David Ben-Zvi, who explained to me that his work with David Nadler (see for instance here) relating geometric Langlands with the representation theory of real Lie groups involved a similar relation between local Langlands at the infinite prime and geometric Langlands on the twistor $P^1$. The relation to the Langlands program was a mystery to me. What I did get from the Fargues talk was that the analog at the infinite prime of the Fargues-Fontaine curve (which I couldn’t understand) was something called the twistor $P^1$, which I could understand.
The appearance of “twistors” was intriguing, although they didn’t seem to have much to do with Penrose’s twistor geometry that had always fascinated me. I attended the talk, and wrote about it here, but didn’t understand much of it. A major development of the past few years has been the recent proof by Fargues and Scholze that the arithmetic local Langlands conjecture at a point can be formulated in terms of the geometric Langlands conjecture on the Fargues-Fontaine curve.īack in 2015 Laurent Fargues gave a talk at Columbia on “p-adic twistors”. In the simplest arithmetic context, the points are the prime numbers p, together with an “infinite prime”. The Langlands program comes in global and local versions, with the local versions at each point in principle fitting together in the global version. I’ve always been fascinated by the relations between these subjects and fundamental physics, with quantum theory closely related to representation theory, and gauge theory based on the geometry of bundles and connections that also features prominently in this story.
One of the major themes of modern mathematics has been the bringing together of geometry and number theory as arithmetic geometry, together with further unification with representation theory in the Langlands program. The rest of the blog posting will give some background about this. The notes are available here, and may or may not get expanded at some point. I’ve just finished writing up some notes on what the twistor $P^1$ is and the various ways it shows up in mathematics.
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