Notes on the Twistor P¹

I’ve just finished writing up some notes on what the twistor $P^1$ is and the various ways it shows up in mathematics.  The notes are available here, and may or may not get expanded at some point.  The rest of the blog posting will give some background about this.

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. 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.

The Langlands program comes in global and local versions, with the local versions at each point in principle fitting together in the global version. In the simplest arithmetic context, the points are the prime numbers p, together with an “infinite prime”. 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.

Back in 2015 Laurent Fargues gave a talk at Columbia on “p-adic twistors”. I attended the talk, and wrote about it here, but didn’t understand much of it. 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. 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 relation to the Langlands program was a mystery to me. 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 $P^1$.

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. I’ve also been fascinated by the Fargues-Scholze work, while understanding very little of it. 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. He remarked that it’s rather mysterious why the twistor $P^1$ is what is showing up here as the geometrical object governing what is happening at the infinite prime. I was very forcefully struck by seeing that this object was exactly the same object that describes a space-time point in twistor theory and I mentioned this at the end of my talk in Paris back in late October.

Scholze’s comments inspired me to take a much closer look at the twistor $P^1$, beginning by trying to understand a bunch of things that were somehow related, but that I had never really understood. These ranged from Carlos Simpson’s approach to Hodge theory via the twistor $P^1$ to some basic facts about local class field theory, where one gets a simple analog for each prime p of the twistor $P^1$ and the quaternions. Along the way, I finally much better understood something else in number theory that had always fascinated me, see the story explained very sketchily in section 6.2. That the quantum mechanical formalism for a four-dimensional configuration space beautifully generalizes to all primes, with the global picture including an explanation of quadratic reciprocity is not something I’ve seen elsewhere in attempts to bring p-adic numbers into physics. I’d be very curious to hear if someone else knows of somewhere this has been discussed.

Anyway, these new notes are partly for my own benefit, to put what I’ve understood in one place, but I hope others will find something interesting in them. Now I want to get back to thinking about the open questions raised by the twistor unification ideas that I was working on before the last few months. A big question there is to understand what twistor unification might have to do with Witten’s ideas relating geometric Langlands with 4d QFT. Perhaps something I’ve learned by writing these notes will be helpful in that context.

Update: I’ve posted the notes, with an added abstract and a final section of speculations, to the arXiv, see here.