It has only been a couple weeks since my last posting on this topic, but there’s quite a bit of new news on the geometric Langlands front.

One of the great goals of the subject has always been to bring together the arithmetic Langlands conjectures of number theory with the geometric Langlands conjectures, which involved curves over function fields or over the complex numbers. Fargues and Scholze for quite a few years now have been working on a project that realizes this vision, relating the arithmetic local Langlands conjecture to geometric Langlands on the Fargues-Fontaine curve. Their joint paper on the subject has just appeared [*arXiv version here*]. It weighs in at 348 pages and absorbing its ideas should keep many mathematicians busy for quite a while. There’s an extensive introduction outlining the ideas used in the paper, including a long historical section (chapter I.11) explaining the story of how these ideas came about and how the authors overcame various difficulties in trying to realize them as rigorous mathematics.

In other geometric Langlands news, this weekend there’s an ongoing conference in Korea, videos here and here. The main topic of the conference is ongoing work by Ben-Zvi, Sakellaridis and Venkatesh, which brings together automorphic forms, Hamiltonian spaces (i.e classical phase spaces with a G-action), relative Langlands duality, QFT versions of geometric Langlands, and much more. One can find many talks by the three of them about this over the last year or so, but no paper yet (will it be more or less than 348 pages?). There is a fairly detailed write up by Sakellaridis here, from a talk he gave recently at MIT.

In Austin, Ben-Zvi is giving a course which provides background for this work, bringing number theory and quantum theory together, conceptualizing automorphic forms as quantum mechanics on arithmetic locally symmetric spaces. Luckily for all of the rest of us, he and the students seem to have survived nearly freezing to death and are now back at work, with notes from the course via Arun Debray.

For something much easier to follow, there’s a wonderful essay on non-fundamental physics at Nautilus, The Joy of Condensed Matter. No obvious relation to geometric Langlands, but who knows?

**Update**: Arun Debray reports that there is a second set of notes for the Ben-Zvi course being produced, by Jackson Van Dyke, see here.

**Update**: David Ben-Zvi in the comments points out that a better place for many to learn about his recent work with Sakellaridis and Venkatesh is his MSRI lectures from last year: see here and here, notes from Jackson Van Dyke here.

**Update**: Very nice talk by David Ben-Zvi today (3/22/21) about this, see slides here, video here.

Geometric Langlands is reflected in integer/fractional quantum Hall effect and the so-called Hofstadter’s butterfly, according to recent work of Kazuki Ikeda.

Can anyone provide a (hand-wavy) summary for the educated layman (Physics post-grad) of what a Fargues-Fontaine curve is, and what’s special about, e.g. in the context of the Langlands programme?

Re: DBZ’s course, I am not the only student taking notes. Jackson Van Dyke is also posting his notes online: https://web.ma.utexas.edu/users/vandyke/notes/langlands_sp21/langlands.pdf (Github link: https://github.com/jacksontvd/langlands_sp21). Jackson’s notes go into quite a bit more detail, and he’s gone back and added more references and figures than I have. In the end we’ll hopefully combine our notes into one document. If anyone has any questions, comments, or corrections about either set of notes they’re welcome to get in touch with me or Jackson.

Arun,

Thanks a lot, both for producing the notes, and for letting us know about the other ones.

Peter – thanks for the references!

Maybe let me note the series of talks in Korea (by Sakellaridis, Venkatesh and me) are aimed at a more arithmetic audience, some previous talks (eg at MSRI last March) might be more accessible to the audience here.

Also I might add that the perspective on automorphic forms as quantum mechanics is very old and widely used. A newer perspective I’m trying to advertise in the course and talks is to think of automorphic forms (and the Langlands program) as being really about 4d QFT — an arithmetic elaboration of the Kapustin-Witten picture for geometric Langlands. This accounts for many of the special features of quantum mechanics on arithmetic locally symmetric spaces – e.g., dependence on a number field is the analog of considering states on different 3-manifolds, Hecke operators — a form of quantum integrability– come from ‘t Hooft line operators, the choice of level (congruence subgroup) corresponds to consideration of surface defects, and most importantly the relation with Galois representations (the Langlands program) can be viewed as electric-magnetic duality.

The new feature of the work with Sakellaridis and Venkatesh (the first paper should appear relatively soon..) is that the theories of periods of automorphic forms and L-functions of Galois representations can fruitfully be understood as considering boundary conditions in the two dual TQFT, and that the electric-magnetic duality of boundary conditions (as studied by Gaiotto-Witten) can be used to explain the relation between the two (the theory of integral representations of L-functions).

Jim Eadon,

let me try to answer. This paper is about the (local) Langlands correspondence over the $p$-adic numbers $\mathbb Q_p$. Recall that $p$-adic numbers can be thought of as power series $a_{-n}p^{-n} + \ldots + a_0 + a_1 p + a_2p^2 + \ldots$ in the “variable” $p$ — they arise by completing the rational numbers $\mathbb Q$ with respect to a distance where $p$ is small. They are often thought of as analogous to the ring of meromorphic functions on a punctured disc $\mathbb D^*$ over the complex numbers, which admit Laurent series expansions $a_{-n} t^{-n} + \ldots + a_0 + a_1 t + a_2t^2 + \ldots$. More precisely, there is this “Rosetta stone” going back to Weil between meromorphic functions over $\mathbb C$, their version $\mathbb F_p((t))$ over a finite field $\mathbb F_p$, and $\mathbb Q_p$.

However, there is an important difference: $t$ is an actual variable, while $p$ is just a completely fixed number — how should $p=2$ ever vary? In geometric Langlands over $\mathbb C$, it is critical to take several points in the punctured disc $\mathbb D^\ast$ and let them move, and collide, etc. What should the analogue be over $\mathbb Q_p$, where there seems to be no variable that can vary?

In one word, what the Fargues–Fontaine curve is about is to build an actual curve in which $p$ is the variable, so “turn $\mathbb Q_p$ into the functions on an actual curve”. It then even becomes possible to take two independent points on the curve, and let them move, and collide. With this, it becomes possible to adapt all (well, at least a whole lot of) the techniques of geometric Langlands to this setup.

This idea of “turning $p$ into a variable and allowing several independent points” is something that number theorists have long been aiming for, and is basically the idea behind the hypothetical “field with one element”. I would however argue that our paper is the first paper to really make profitable use of this idea.

David,

Thanks! I’ve added some links to your MSRI talks, which do look like a better place for people to start.

I’ve always been fascinated by analogies between number theory and QM/QFT, the new angle on this you’re pointing out is really remarkable, looks like a significant deep link between the subjects.

Could you (or Peter Scholze!) comment on any relation of this to the other topic of the posting (local arithmetic Langlands as geometric Langlands on the Fargues-Fontaine curve)?

Professor Scholze,

You give a sense of how exciting it is, to, literally? connect the dots between different fields. I’m glad I studied pure mathematics as a hobby enough to get the gist of your explanation. I will re-read your reply a few times, as it’s deep, and connects several fascinating objects and techniques.

I really appreciate you taking the time to engage, it means a lot. And thanks too, to Professor Woit, for bringing such mathematics to my (and others) attention, I enjoy the blog.

Peter – the way I see it (somewhat metaphorically) is as follows. In extended 4d topological field theory we seek to attach vector spaces to 3-manifolds, categories to 2-manifolds etc. The Langlands program fits beautifully into this if you accept the “arithmetic topology” analogy: besides ordinary 3-manifolds we consider (Spec of ring of integers of) global fields (number fields and function fields over finite fields) as “3-manifolds”. Besides surfaces we also admit local fields (such as p-adics or Laurent series over finite fields) and curves over the algebraic closure of finite fields as “2-manifolds” (this is the theme I’d like to get to in my course, though still a way to go).

If you accept this ansatz, there’s no “geometric Langlands” and “arithmetic Langlands”, we’re just considering different kinds of “manifolds” as inputs. For example geometric Langlands on surfaces and local (arithmetic) Langlands both concern equivalences of categories (in the latter case, one seeks descriptions of categories of reps of reductive groups over local fields are described in terms of spaces of Galois representations).

The Fargues-Scholze work is (among many other things!) a spectacular realization of this kind of idea. They show that the local Langlands program can be (and arguably is best) considered as geometric Langlands on an actual curve attached to the local field (the Fargues-Fontaine curve). Moreover the most crucial structure here, the Hecke operators, are miraculously described in a geometric way (factorization — the colliding points in Peter’s response) that descends directly (via Beilinson-Drinfeld) from the structure of operator products in 2d QFT.

The wonderful recent work of Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum and Varshavsky that Will Sawin mention in a recent comment also fits into this general paradigm, in that they show that [unramified] arithmetic and geometric Langlands in the function field setting are precisely related by “dimensional reduction” – you pass from “2-manifolds” (curves over alg closure of finite fields) to “3-manifolds” (curves over the finite fields — which you should think of as mapping tori of the Frobenius map, so 3-manifolds fibering over the circle) by taking trace of Frobenius, just as TFT would tell you.

I’m late but I’m going to say a few words to complement Peter’s comments. Since this is a Physics blog I’m going to give a few key words that may speak to physicists. A lot of things work by analogies in this work, trying to put together some ideas from arithmetic and geometry together, make some mental jumps and trying to fill the gaps.

When Peter is saying “turning into a variable and allowing several independent points” this is analog to the fusion rules in conformal fields theory. There is the possibility in the world of diamonds to take different copies of the prime number p and fuse them into one copy. Here there reference, if I dig in my mind the first time I heard about this, is the work of Beilinson Drinfeld on factorization sheaves in terms of D-modules. You will find this fusion process in the Verlinde formula too in a coherent sheaves setting for compact Riemann surfaces in the work of Beauville “Conformal blocks, fusion rules and the Verlinde formula” for example where you fuse different points on a Riemann surface. I typically remember a talk by Kapranov about ‘The formalism of factorizability’ and did not get why the Russian peoples, who are known to have a huge background in physics, were such obsessed with this. No doubt this is linked to vertex algebras, where there are fusion rules, too and plenty of things of interest for physicists. For arithmeticians the declic, I remember saying to myself “at least I understand why peoples are obsessed with those factorization stuff”, came from Vincent Lafforgue who remarked that if you work in an étale setting instead of a D-module setting, the moduli spaces of Shtukas (a vast generalization of modular curves for functions fields over a finite field) admits a factorization structure and this factorization structure gives you the Langlands parameter for global Langlands over a function field over a finite field.

For the curve here is what I can say. If you take an hyperbolic Riemann surface it is uniformized by the half plane on which you have a complex coordinate z. You can imagine the same type of things for the curve where the variable is the prime number p.

By the way there is an object in the article that may speak to physicists : Bun_G, the moduli of principal G-bundles on the curve. The analog for physicists would be the moduli of principal G-bundles on a compact Riemann surface, where now G is a compact Lie group, that shows up in the work of Atiyah and Bott. Still by the way, one of the origin of my geometrization conjecture is trying to understand the reduction theory “à la Atiyah Bott” for principal G-bundles on the curve (the analog for any G of the work on the indian school (Narasimhan-Seshadri) + Harder for GL_n i.e. usual vector bundles).

There are other objects that may speak, by analogies, to physicists in this paper. Typically the so called local Shtuka moduli spaces “with one paw” (i.e. only one copy of the prime number p). The archimedian analogs are hermitian symmetric spaces. Realizing local Langlands in the cohomology of those local Shtuka moduli spaces has an archimedian analog : Schmid realization of Harisch-Chandra discrete series in the L^2 cohomology of symmetric spaces. Schmid uses the Atiyah-Singer index formula to obtain his result, a tool well know to Physicists. Hermitian symmetric spaces are moduli of Hodge structures and this has been a great thing to realize that p-adic Hodge structures à la Fontaine are the same as “geometric Hodge structures” linked to the curve.

I could speak about this during hours and do some name-dropping that speaks to physicists but one thing is sure: there is no link with the multiverse, this I’m sure. Anyway, I have no idea how the curve looks like in other universes of the multiverse.

Thanks Laurent!

The Atiyah-Bott story (involving gauge fields + the Yang-Mills equations) and the Atiyah-Schmid story (involving, for SL(2,R), the Dirac operator on a 2d space) are two of my favorite topics. They’re essentially the two main components of the Standard model (the Dirac equation for matter fields, the Yang-Mills equation for gauge fields). Only difference is that they’re in 2d rather than 4d….

I hope you’re still planning to come to Columbia for fall 2022, look forward to seeing you then!

My Eillenberg lectures are reported to 2023 sadly, because of the virus. I’ll try not to enter into the technical details and give some general picture of the objects showing up in this work.

By the way, the cancelled program “The Arithmetic of the Langlands program” jointly organized with Calegari, Caraiani and Scholze is officially reported to 2023, same period of the year as before.

A very nice popular sketch of the idea of “the curve” by Matthew Morrow:

https://webusers.imj-prg.fr/~matthew.morrow/Morrow,%20Raconte-moi%20la%20Courbe.pdf

& more:

https://webusers.imj-prg.fr/~matthew.morrow/Exp-1150-Morrow,%20La%20Courbe%20de%20FF.pdf

@Thomas thanks for the link.