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.

I really enjoy how unabashed you are at acknowledging when you do not understand something. I have come across way too many in the scientific environment who seem to be unable to do that, even when it is fairly obvious that such is the case.

FWIW the table of comparisons at the very end of the notes looks wonderful to me.

About Section 6.2, see

https://mathscinet.ams.org/mathscinet-getitem?mr=1474565

and

https://mathscinet.ams.org/mathscinet-getitem?mr=1896980

noneone,

Thanks. I have seen those. The first doesn’t really say anything about QM, does give a more abstract approach to the reciprocity law than Weil’s. The second does do a bit with QM and this kind of idea, there are some other things at Varadarjan’s site

https://www.math.ucla.edu/~vsv/

The most detailed relevant things to read that I know about are Michael Berg’s expository book about the Weil paper

“The Fourier Analytic Proof of Quadratic Reciprocity”

and a long article relating some of this to path integrals, see

https://www.sciencedirect.com/science/article/pii/S0723086915000857

What I haven’t seen is anything using the specific choice of orthogonal space V to be the quaternions. This case plays a big role in mathematics (Jacquet-Langlands), but I haven’t seen anything trying to relate it to physics, where it might give a relativistic particle theory that generalizes to all p at once.

Years back, you posted your position as radical platonism and explained that to mean that at a fundamental level, physical objects are actually mathematical objects. Wouldn’t it t be fair to say that these developments (and Langlands trends) continue to point in this direction?

MT,

Yes. This is to my mind very much evidence for what I was calling “radical Platonism”. That the description of a space-time point (a fundamental construction of physics) is the precise analog at the infinite prime of the Fargues-Fontaine curve (a fundamental construction of number theory) at the finite primes indicates a very close and unexpected relationship between physical reality and pure mathematics.

Happy to see you’re allowed to post to the arXiv again after all the drama of years past…

Sebastian Thaler,

I should make clear that I’ve never had a problem posting preprints to the arXiv. The reason there aren’t many there from me in past years is because I haven’t written much that I was happy with, not because moderators rejected submissions. So you should attribute the lack of arXiv articles to my laziness or to my high standards, the blame there does not lie with the arXiv. That I’ve posted two things recently reflects the fact that I’m very excited about what I think are some genuinely new and important ideas (or, in the twistor P1 case, new connections between ideas), that keep looking very interesting to me the more I think about them.

The trackback thing is a different story. There’s a blanket ban at the arXiv on trackback links to my blog. For instance, when someone at Quanta mentions an arXiv preprint in an article, a link appears on the arXiv at that article. For an example, see

https://arxiv.org/abs/1804.10624

which has a link to the recent Quanta article

https://www.quantamagazine.org/a-correction-to-einstein-hints-at-evidence-for-string-theory-20220121/

that I wrote about. When I similarly link to an arXiv paper in a blog discussion of the paper, no link to it appears, even though my blog software sends a trackback request. This is a blanket ban, links to my commentary on my own papers are even banned.

Way back when, there was a public statement from the arXiv that trackbacks to my blog were banned since I was not an “active researcher”. This no longer seems to be the criterion they use (the author of the Quanta piece is not a researcher but a journalist). I just made another attempt recently to find out what their policy is and why trackbacks to my blog are banned, but the response I got indicated that this is something I’ll likely never find out.