For a second slogan about quantum mechanics I’ve chosen:
Quantum mechanics is evidence of a grand unification of mathematics and physics.
I’m not sure whether this slogan is likely to annoy physicists or mathematicians more, but in any case Edward Frenkel deserves some of the blame for this, since he describes (see here) the Langlands program as a Grand Unified Theory of mathematics, which further is unified with gauge field theories similar to the Standard Model.
This week I’m in Berkeley and have been attending some talks at an MSRI workshop on New Geometric Methods in Number Theory and Automorphic forms. Number theory is normally thought of as a part of mathematics about as far away from physics as you can get, but I’m struck by the way the same mathematical structures appear in the representation theory point of view on quantum mechanics and in the modern point of view on number theory. For example, the lectures on Shimura varieties have taken as fundamental example the so-called Siegel upper-half space, which is the space Sp(2n,R)/U(n). Exactly the same space occurs in the quantization of the harmonic oscillator (see chapters 21 and 22 of my notes), where it parametrizes possible ground states. Different aspects of the structure play central roles in the math and the physics. In the simplest physics examples one works at a fixed point in this space, with Bogoliubov transformations taking one to other points, something which becomes significant in condensed matter applications. In number theory, one is interested not just in this space, but in the action of certain arithmetic groups on it, with the quotient by the arithmetic group giving the object of fundamental interest in the theory.
The workshop is the kick-off to a semester long program on this topic. It will run simultaneously with another program with deep connections to physics, on the topic of Geometric Representation Theory. This second program will deal with a range of topics relating quantum field theory and representation theory, with the geometric Langlands program a major part of the story, one that provides connections to the number theoretical Langlands program topics of this week’s workshop. I’ve got to be in New York teaching this semester, so I’m jealous of those who will get to participate in the two related MSRI programs here in Berkeley. Few physicists seem to be involved in the programs, but these are topics with deep relations to physics. I do think there is a grand unified theory of some kind going on here, although of course one needs to remember that grand unified theories in physics so far haven’t worked out very well. Maybe the problem is just that one hasn’t been been ambitious enough, that one needs to unify not just the interactions of the standard model, but number theory as well…
What underlies the striking “unification” of concepts of QFT with concepts found in number theory generally and in automorphic representation theory particularly is to a large extent attributable to the function field analogy. This together with the Weil uniformization theorem is what makes one guess geometric Langlands (and hence QFT phenomena) from number theoretic Langlands.
However, to date this is really just that, educated guessing based on analogy. What is missing would be a theory that makes the “unification” here a systematic theorem.
For instance at the Oxford meeting on the Langlands program that you mentioned a while back here, Sergey Oblezin had reported on his work with Gerasimov and Lebedev on topological sigma-model mirror symmetry computations related to geometric Langlands, and when asked afterwards if any of these insights could conceivably be carried over to the number theoretic Langlands correspondence, the answer was something like “We could, if only we really knew what it would mean to systematically consider p-adic sigma-models”.
This is open. It is probably somehow related to the phenomenon, maybe not widely appreciated, that also the construction of the refined Witten genus by Ando-Hopkins-Rezk passes all the way through arithmetic geometry (elliptic curves, hence string worldsheets, defined over Spec(Z); in fact complex worldsheets never even appear explicitly in the computation and only the elliptic curves in positive characteristic (the supersingular ones) contribute at the stringy height 2.)
Last week at CUNY in New York I gave a talk (notes and video are here) on what I would modestly suggest might be a mathematical theory that would allow to turn the analogy here into a systematic theorem unifying automorphic mathematics with QFT.
Urs’ notes look really exciting and suggestive! such unifying geometric structures are definitely needed to get a deeper understanding of these analogies, or to see the arithmetic world directly in the light of QFT.
I would disagree though with the suggestion that the geometric Langlands analogy is only an educated guess to date (if it ever was that – starting from Drinfeld’s original work). In fact one of the main themes of our current MSRI semester is the really exciting interactions going on between geometric Langlands and classical Langlands – for the first time in the last couple of years people (in particular Xinwen Zhu and Zhiwei Yun) are directly taking ideas and constructions from the geometric Langlands program and using them to prove (phenomenal) arithmetic results. (Ngo’s proof of the Fundamental Lemma was arguably the beginning of this trend.) The fundamental structures of the Langlands program already span across fields of definition in a natural way.
The most exciting thing around these parts though is the shockingly futuristic sounding course announcement (http://math.berkeley.edu/courses/fall-2014-math-274-001-lec) by Peter Scholze, who seems to be able to work with number fields as if they were function fields:
Syllabus: Originally defined by Drinfel’d, and then used extensively by L. Lafforgue and V. Lafforgue, moduli spaces of shtukas have proved to be a powerful tool in the study of the Langlands correspondence over function fields. It is an important question whether something similar could work over number fields.
In this course, we want to sketch a strategy to define moduli spaces of local shtukas over mixed-characteristic fields such as Qp (leaving open the problem of assembling these spaces for varying primes p). These spaces should generalize Rapoport-Zink spaces, as well as the conjectural theory of local Shimura varieties that has recently been suggested by Rapoport-Viehmann. Notably, the spaces of local shtukas should overcome the ‘minuscule’ condition inherent in the theory of Shimura varieties, so that they can be seen as very general p-adic period domains (which exist even in situations where Griffiths transversality is a nontrivial condition). We wll start by reviewing the theory of (local and global) shtukas over function fields. Next, we will define shtukas in mixed characteristic in an absolute setup; this is closely related to the notion of Breuil-Kisin modules, and has been the subject of recent investigations of Fargues, some of whose results we will recall.
To set things into perspective, we will spend some time detailing the case of p-divisible groups, explaining the equivalence between p-divisible groups and certain shtukas (due to Breuil, Kisin, and Fargues). We will then use the joint work with Weinstein to identify Rapoport-Zink spaces with moduli spaces of shtukas.
We will then be able to define general spaces of shtukas. They will turn out to be somewhat esoteric objects, called ‘diamonds’, so quite a bit of time will be spent on explaining the definition of diamonds. In particular, we will explain how they overcome the problem that a general space of shtukas should live over ‘a product of several copies of Spa Qp’ (taken over some absolute base ‘F1’), by giving a highly nontrivial definition of ‘a product of several copies of Spa Qp’ in a way that makes Drinfel’d’s lemma true.
If time permits, we will try to (conjecturally) understand the étale cohomology of these spaces, and their relation to the local Langlands correspondence.defined by Drinfel’d, and then used extensively by L. Lafforgue and V. Lafforgue, moduli spaces of shtukas have proved to be a powerful tool in the study of the Langlands correspondence over function fields. It is an important question whether something similar could work over number fields.
In this course, we want to sketch a strategy to define moduli spaces of local shtukas over mixed-characteristic fields such as Qp (leaving open the problem of assembling these spaces for varying primes p).
I wonder if philosophy might be somewhat to blame. Or is this to take on to much of a role for my discipline? Perhaps if we had at least pressed mathematicians over the past, say, 60 years to try make clearer sense to us their conceptions of space, it might have stimulated some thinking along these lines.
It’s notable in this respect that Weyl’s gauge theory grew out of a deep reading of Husserl. And if Urs’s approach were to flourish, a philosophically-oriented mathematician, William Lawvere, will be in line for the credit of devising the notion of ‘cohesion’.
I used to just believe the advertisement that the statement of geometric Langlands duality follows, at the moment , by more than educated guesswork. It was an eye-opener when Robert Langlands himself publically expressed doubts recently. In this vein it pays to look at page 4 of
* Dima Arinkin, Dennis Gaitsgory, “Singular support of coherent sheaves, and the geometric Langlands conjecture” (arXiv:1201.6343)
The usual statement of geometric Langlands duality, as found in the standard reviews, they call a “naive guess”, pointing out that V. Lafforgue had explicitly shown it to be wrong in general. Then they discuss the “heuristic reason” for this failure. Check out some sample text from this page 4 to see the educated guesswork in action:
Needless to say that I don’t find fault with educated guesswork, and that I find the work by all people involved here admirable. But where it involves educated guesswork one should call it such, for the sake of the subject. Speaking to actual number theorists reveals that a derivation of geometric Langlands and its QFT incarnation not involving handwaving would be much appreciated.
My modest point is that if one had a systematic “inter-geometric” theory which would work both for complex analytic as well as for arithmetic gemetry, then this kind of problem of wrong guesses might not occur.
For instance one central part of the analogy is that automorphic representations on the arithmetic side are supposed to be analogous to Hitchin connections on bundles of conformal blocks over a moduli stack of fields of a gauged WZW model.
Now if one had a truly inter-geometric theory, then it might be possible to axiomatize what the Hitchin connection on the bundle of conformal blocks over the moduli of fields of the WZW model would be (indeed, that is what the “cohesive” axioms are aiming at, we are almost there), and then interpret these axioms systematically in arithmetic geometry, such that one would obtain systematically the desired number-theoretic analog, together with the guarantee that it has all the relevant properties.
In other words, with a genuine inter-geometric theory one might be able to systematically say what Yang-Mills/Chern-Simons/Wess-Zumino-Witten QFT is in arithmetic geometry (something like — but much more comprehensive than — what is currently found under the headline “p-adic string theory”) and then systematically work out what the bundle of conformal blocks is when interpreted over some global field. If the inter-geometric theory works well, then it would guarantee that the arithmetic-geometry incarnation of this quantum theory would satisfy the same kind of general properties (e.g. the relevant arithmetic version of S-duality) that its complex-analytic cousin does.
Such a theory would be an actual “unification of interactions in physics and number theory” as in the thread above. Whether or not my humble suggestion for how to go about this works out, I believe such a theory would be as desireable as it is clearly missing at present, all the excitement about undeniable progress nonwithstanding.
… the way the same mathematical structures appear in the representation theory point of view on quantum mechanics and in the modern point of view on number theory …
Is there any way an interested outsider can get a feel for what you’re doing here? It sounds fascinating and beautiful, but also frustratingly unreachable. As an engineer I have some grasp of mathematics but this is no mathematics “as we know it”.
As I understand them, the issues that Arinkin-Gaitsgory discuss don’t in any way reveal a shakiness of the analogy or an indication of guesswork (I haven’t understood Langlands’ notes so can’t comment). Part of the problem is that the words “geometric Langlands” stand for several layers of statements. The traditional one, formulated by Drinfeld, has to do with individual cuspidal Hecke eigensheaves corresponding to irreducible local systems. This statement is the one that’s easiest to see in analogy with number theory (matching Hecke eigenforms with Galois representations) though even here one has to take account of a basic failure of naive dictionaries: the geometric analog of a number field or function field in finite characteristic should not be a Riemann surface, but roughly a surface bundle over the circle. This explains the “categorification” (need for a function-sheaf dictionary, which is the weak part of the analogy) that takes place in passing from classical to geometric Langlands — if you study the corresponding QFT on such three-manifolds, you get structures much closer to those of the classical Langlands correspondence.
In any case, this form of the geometric Langlands correspondence is now a theorem for GL_n. The categorical form that people now mean by GLC is an attempt to understand something more subtle, namely the functional analysis aspect of Langlands: we want to decompose the geometric analog of L^2 of the adelic double quotient. So it is very natural to expect that functional analytic issues emerge, i.e., you have to be very careful about setting up the correct function space. This has been clear to experts from the start, with the stated versions of the categorical correspondence being a guideline rather than a statement to be taken literally. But remarkably, despite the extra categorifications, the functional analytic issues are precisely analogous to those that arise in number theory (in the Langlands decomposition and Arthur-Selberg trace formula), and that goes for Arinkin-Gaitsgory’s work as well. In fact their work is for me further justification of the robust nature of the analogy between number theory and QFT since even at this greater resolution both sides give rise to the same structures.
If anything it seems that the two sides keep getting closer – for example Xinwen Zhu’s striking recent version of geometric Satake in mixed characteristic. His story doesn’t have a direct number field analogue of factorization, the main geometric structure to come out of the geometric Langlands program, but rumor has it Scholze does! Likewise classical Langlands doesn’t seem to have much use for many of the categorical structures of geometric Langlands, but apparently the thriving p-adic Langlands program is finding many of the same issues and structures as well. None of which should take anything away from the inter-geometric theory Urs is developing, which sounds very exciting, just to say the analogies and bridges between the areas are stronger than ever (and to my mind were never questionable, just only recently starting to really live up to their potential).