This week at Harvard’s CMSA there’s a program on Arithmetic Quantum Field Theory that is starting up and will continue through March. There’s a series of introductory talks going on this week, by Minhyong Kim, Brian Williams, and David Ben-Zvi. I believe video and/or notes of the talks will be made available.

At the IHES and the Max Planck Institute, the Clausen-Scholze joint course on analytic stacks has just ended. For an article (in German) about them and the topic of the course, see here. What they’re working on provides some new very foundational ideas about spaces and geometry, in both the arithmetic and conventional real or complex geometry contexts. Many of the course lectures are pretty technical, but I recommend watching the last lecture, where Scholze explains what they hope can be done with these new foundations.

Of the applications, the one that interests me most is the one that was a motivation for Scholze to develop these ideas, the question of how to extend his work with Fargues on local Langlands as geometric Langlands to the case of real Lie groups. He’ll be giving a series of talks about this at the IAS next month.

Something to look forward to in the future is seeing the new Clausen-Scholze ideas about geometry and arithmetic showing up in the sort of relations between QFT, arithmetic and geometry being discussed at the CMSA.

I believe the videos should be available on https://www.youtube.com/@harvardcmsa7486/videos, as they have posted various past conference videos on this site. Interested in it, but not having enough time to attend in person. Hopefully, they will post recorded videos on the same site.

I’ve seen samizdat versions of notes from Scholze’s Munster talk. Has he now proved the conjectures he made?

I’d always idly wondered about the category of globalisations of a given Harish-Chandra module – we know it has canonical `maximum’ and `minimum’ elements, but nothing really about the structure of all such. Do these now form a well defined moduli space , and if so, what does its topology (or Hodge theory!) tell us about the representation, and how can we read these invariants from the HC-module, or its BB description on the flag variety?

Its amusing how this counts as an `easy’ application of this Clausen-Scholze technology. We await the subtle applications…

Presumably it is clear how this CS analytic stuff describes the maximal and minimal locally analytic vectors in a Matsuki dual picture to usual BB, as probably done by Kashiwara directly. But how does it control all the `intermediate’ extensions??? Very intriguing! Can one hope for _interesting_ HC modules, with very special moduli spaces of globalisations, parameterised by extremal points in some Langlands dual picture?

Anyhow, I should watch Scholze talk#24 (and talks#1-talks#23) before commenting further.

Did Clausen/Scholze explain (as an obvious example) representations of a Heisenberg group, and how the Stone-von Neumann theorem fits cleanly into their geometry and rep theory and D-modules, not just `sort of analogously’?

Also I hope to learn from these talks: what more does the Clausen-Scholze stuff applied to the usual Hodge package produce in terms of constraints, when compared to the o-minimal analsysis of Klingler, Tsimerman, et al? Presumably quite a bit more, as one can now handle growth conditions….?

anon,

If you have access to notes of the Munster talk, you know a lot more about this than I do. From Scholze’s comments in the last talk of the lecture series and from second hand reports, all that’s clear is that Harish-Chandra modules are not the right thing here.

My understanding is that this is about real local Langlands which is just for reductive groups. If it gives a proof of Stone-von Neumann that would be fun.

@anon

Thanks for your comment/question! Things resolve themselves differently, and in some sense in a more boring way.

Background: We define a category that is “a category of $G$-representations” for a real Lie group $G$, but thinking of representations in terms of sheaves on the classifying stack $[*/G]$. Here $*$ is really the analytic spectrum of the gaseous real numbers, and $G$ is incarnated as an analytic space via regarding it as a real-analytic manifold (and using the corresponding functor to analytic spaces). Then we simply take the (“quasicoherent”) derived category $D([*/G])$ of $[*/G]$.

Now, somewhat surprisingly, every(?) Harish-Chandra module has a unique(?) corresponding object in $D([*/G])$. (Question marks because I did not yet carefully check in general.) The obvious question is, how can we single out a unique $G$-representation associated to a Harish-Chandra module? Part of the answer is that a little bit of magic seems to happen; another part of the answer is that there are actually two ways to turn an object of $D([*/G])$ into a representation of $G$: One can use $*$- or $!$-pullback along $*\to [*/G]$. One of them ($*$-pullback) gives the minimal globalization, the other ($!$-pullback) gives the maximal globalization.

Here’s a related story. There’s this ring of infinite-order differential operators $\hat{D}$ on a complex manifold. Usually it’s subtle to get a good theory of $\hat{D}$-modules working (for reasons of functional analysis). In our setup it’s straightforward. One can then wonder whether this resulting category still has a Riemann-Hilbert correspondence to a category of sheaves. It turns out that $\hat{D}$-modules are just equivalent to sheaves! (No constructibility conditions or anything — all sheaves.) This is related to some classical stuff about hyperfunctions. [Disclaimer: I’m far behind on learning the classical theory…]

This story of $\hat{D}$-modules actually plays very well with the above story of $G$-representations, and they are easy to combine in the Beilinson-Bernstein localization business, leading indeed to the Matsuki dual picture. Many of the ideas here are really due to Rodriguez Camargo, who has developed this in the $p$-adic context. (This leads to the slightly strange situation that many things are now written over the $p$-adics but not over the reals… not because the reals are harder for these things, but mainly because so far it’s mostly $p$-adic people using this theory.)

Dear Peter S,

Thank you for responding to my question/comment.

I’m looking forward to reading the definition of your category of G(R)-modules — it sounds wonderful. (I’m *very* curious how it encodes the basic finiteness theorems and theorems about density of analytic vectors that drive the usual story of this, so that such modules have underlying HC modules, what is the notion of equivalence of modules, and the way that bad Banach space examples get excluded).

Also: Do you have a range of extension functors interpreting between * and ! pushforward, analogous to Beilinson’s extension functors? (that have to do with properties – growth ? – of the function f^s? )

But I’m reeling from what you write about infinite order differential operators — that solves a problem that plagued the Kyoto school for many decades (with their various microlocal differential operator rings E, variants all defined with different growth conditions). What an extraordinary RH theorem you have – just the right analytic notion to capture *all* sheaves. That is extraordinary. Where can I read it!?!!! (And are there notes from your course, or should one just watch it all?) I’m happily imagining all of the questions one can now answer with these tools… wow.

(Trivia: I didnt quite understand from the above if the classification of your G(R)-modules is `automatic’ from the definition – is it straightforward from the definition that they are global sections of sheaves on your gaseous classifying space, and those with trivial central character give good sheaves on your gaseous quotient flag variety/G(R)? What is your analogue of the BB-theorem that the flag variety is D-affine, or its mixed version, the Grauert Riemanschneider vanishing theorem. Or are you assuming a BB classification of HC-modules? Given a HC-module with generalised central character, don’t we already know how to get the globalisations from the Matsuki dual sheaf to the BB-localisation, I think more or less functorially? If this is how it works, it feels a bit like cheating, or at least criminally underutilising your rather major breakthroughs. )

Anyhow, thank you. Sorry for burbling – I’m still trying to process your response, even without the supporting definitions.

Apologies, I see I read too fast – * and ! are for pullbacks to a point, not pushforward of an open subvariety, hence are different notions of `underlying vector (Banach?) space’ for your equivariant sheaf? I retract my question about analogues of Beilinson’s functors, which is nonsense in this context. Indeed ‘things resolve themselves differently’.

Please ignore any further nonsense in my previous question — I won’t understand what is actually going on until I work through the definitions and compute what happens for SL_2

Peter W — thank you for allowing these questions on your blog. And for your links, which are very thought provoking

anon,

“I won’t understand what is actually going on until I work through the definitions and compute what happens for SL_2”

You’re not the only one…

@anon

Thanks for your excitement ;-). Just some brief answers to a few points:

Our representations usually are on some (dual) nuclear Frechet space. Banach (or even Hilbert) spaces do not play a prominent role in our work. More broadly, norms/metrics in the classical sense are less relevant for us. (They are however key to many results — like the Hodge decomposition — but we hope that we can get such results using a very different incarnation of metric information, related to the “extended Berkovich line”.)

About the Riemann-Hilbert theorem: We hoped that we would find time to discuss it in the course, but in the end there was no time. But it’s almost a tautology for us! Namely, we realize $\hat{D}$-modules again as quasicoherent sheaves on some stack, the “analytic de Rham stack” $X_{\mathrm{dR}}$. This is the quotient of the complex manifold $X$ by the “overconvergent neighborhood of the diagonal”. In the $p$-adic context, see the recent work of Rodriguez Camargo (on arXiv). On the other hand, usual (in arithmetic geometry-speak, “Betti”) sheaves can be realized as quasicoherent sheaves on some other stack, $X_{\mathrm{Betti}}$. This is a priori built in a slightly crazy way, covering $X$ by totally disconnected sets; it was briefly discussed in the course, in my lecture giving the definition of analytic stacks.

Now our Riemann-Hilbert statement follows from a more primitive geometric statement, namely that the two stacks are isomorphic, $X_{\mathrm{dR}}\cong X_{\mathrm{Betti}}$! And once one understands what’s going on, this is really a triviality. Namely, it’s clear that there is a surjective map $X\to X_{\mathrm{Betti}}$ and that the induced equivalence relation on $X$ is precisely the overconvergent neighborhood of the diagonal.

(Much of the hard work in the course was to get such a flexible notion of analytic stacks where in particular these things turn out to be isomorphic even though they are presented in very different ways. The key is the $!$-topology whose discussion was probably the most technical part of the course.)

PS: In the Liquid Tensor Experiment, I said that the theorem on the existence of the liquid analytic ring structure on the reals may be my most important theorem. Everybody seemed very puzzled about this judgment. But it is the key to get machinery like the above.

(Now we also have the gaseous analytic ring structure on the reals, which is easier to construct and also works for most applications, in particular the stuff above. But gaseous vector spaces are a much more radical departure from the usual setting of complete locally convex vector spaces, so for some purposes it is still very useful to have the liquid theory as well. I discussed one instance in the last lecture. In case someone wonders whether the techniques of our course help in the liquid case: So far no — we now have rather nice arguments for the existence of the solid and the gaseous analytic ring structure, but the existence of the liquid one is still a very difficult theorem.)

[My apologies to all physicists reading this for our abuse of physical terminology. Explanation: There are “condensed” vector spaces (=a variant of topological vector spaces), and within those various kinds of more “complete” ones. The weakest notion is “gaseous”, next comes “liquid” (at various “temperatures” = degree of convexity), and in the non-archimedean case things can even get “solid” (at “temperature 0”).]

I am quite bedazed by all of this, but if nomenclature requests are being entertained, I hope that in applications to number theory, we will soon have room-temperature superconductors?