Last week Princeton hosted a workshop on Physics at LHC: From Experiment to Theory. Many of the slides of the talks are available here. The talks covered a wide range of ground, from theory not obviously relevant to the LHC to reports on the status of experiments. The talk by Michael Tuts of ATLAS reports that they may be roughly 5 weeks behind schedule, with nominal schedule setting the start of beam commissioning at 450 GeV late in November. 7 TeV beams won’t be available until summer 2008, with first physics run perhaps in July, and about 10-100 pb^{-1} during 2008, giving similar statistics to what the Tevatron has today for some processes. It looks like it may be the 2009 run that will most likely produce data that will go beyond the Tevatron results, although if the Higgs mass is about 150 GeV, there may be evidence in the 2008 data. Plans for a luminosity upgrade in 2015 are already proceeding.

There was also a plan for a meeting of the String Vacuum Project at the workshop, with a discussion paper available here (via their Wiki).

This week, also down in Princeton, but at the IAS, there will be a Workshop on Homological Mirror Symmetry and Applications, mainly devoted to recent work on geometric Langlands and its relation to mirror symmetry. Since I had to teach I couldn’t make it down there today, so unfortunately missed the first talks by Kapustin and Witten, as well as that of Edward Frenkel, but I’m looking forward to going down tomorrow and maybe later in the week to hear some of the talks.

So, what happened?

Nothing too exciting to report I fear. I attended the talks by Witten, Kapustin and Gukov, which were just about the Witten-Kapustin and Witten-Gukov papers, not something newer. The Witten and Gukov talks were exceptionally clear. In both the Kapustin and Gukov talks, Witten was in the audience and often ended up being the one to respond to questions from other audience members.

Witten was mainly talking about ‘t Hooft loop operators. He tried to jazz things up a bit by telling the mathematicians that studying moduli spaces of bundles in 2d using a 4d QFT was a way of dealing with a problem that they knew, that you have to think about not just the space of bundles on a Riemann surface, but the stack (geometric Hecke operators need to be defined using the stack). He also referred to the construction of the ‘t Hooft loops as a “categorification”.

I noticed few physicists there; the physicists in Princeton don’t seem too interested in following Witten into this area. IAS math will have a whole program next year which will include more activity in this area.

As always, lunch at the IAS was excellent, and many interesting people there to talk to.

Peter,

I’ll be very interested to hear what comes of all this by the end of the week. As I mentioned previously, I’m no expert in Geometric Langlands, but I’m not aware of any spectacular results that have come out of this project to date. Of course, there are lots of indications (even to a non-specialist such as myself) that the project has lots of promise and talented researchers.

But my impression is that the GL project was mostly interesting for its intriguing promise, not its existing results. More specifically, here’s an outline of my understanding:

Many attempts have been made towards a proof (or good understanding) of original Langlands conjecture, which of course applies to number fields. Those attempts have for the most part met with very limited success (one might say, that they’ve mostly “hit the wall”). My impression is that most of the researchers focusing on the original Langlands conjecture have come to believe that we’re laking some essential geometric idea. So far, some new geometric ideas have emerged from the Geometric Langlands project – but these are ideas that pertain with some success to the function field case. However, I am not aware of a real, complete proof even in that case. Nevertheless, the ideas seem real, new and potentilly powerful, which is what I have understood to have generated high expectations for Geometric Langlands – expectations which may or may not be fully warranted.

All that being said, I am not aware of any significant role played by Homological Mirror Symmetry in any of these new and promising geometric ideas – at least the ones that I have heard about.

Am I wrong about the existing state of GL? Is some exciting new idea in Geometric Langlands predicated on Homological Mirror Symmetry expected to be revealed in this workshop?

Just to be clear:

My comments above contemplate Gaitsgory’s fine work as well as the suggestive papers of Kapustin and Witten (Electric-Magnetic Duality and the Geometric Langlands Program) and Hausel and Thaddeus (Mirror symmetry, Langlands duality, and the Hitchin system). Thanks.

I can talk a little bit about the role of homological mirror symmetry in geometric Langlands (although I’d probably be better off letting DBZ do it :)). In the physics formulation of the conjecture, one ends up with a mirror symmetry between the A- and B-models on certain Hitchin moduli spaces. This is in contrast with the mathematical conjecture which is an equivalence between categories of sheaves on the contangent bundle to Bun_G and D-modulies on Bun_Gdual (or something like that). To get that, Witten and Kapustin give some physics arguments that the A-model on a contangent bundle gives rise to D-modules on the base. I think this is a new idea (obviously, it is presaged somewhat by the work of Hausel and Thaddeus). Along the same lines, I don’t think people had been aware that geometric Langlands has the structure of a 4D TQFT. On the math side, the connection between A-branes and D-modules has been proved in certain cases by Getzler and Nalder and Nadler. In the general situation, however, I think there’s a lot more to be said, however, especially in regards to worldsheet instantons.

Robert,

There are many things to say about your question, but I think it’s

quite fair to say

1. the original Langlands program is doing spectacularly well, and is far from stuck at a wall.

2. there are spectacular applications to representation theory as well as to the classical Langlands program that have come from geometric Langlands ideas.

3. there are strong connections between geometric Langlands and

homological mirror symmetry, at least the hyperkahler version thereof.

I don’t feel qualified to say much about 1, other than

to mention Richard Taylor, Michael Harris and many others working in that area and proving fundamental conjectures in the area since

Wiles’ breakthrough – a recent meeting in Luminy on

Langlands program left the uninformed spectators (eg me) in complete awe at the astonishing rate of progress in the area.

For 2 I would focus for example on the work of Bezrukavnikov and collaborators, and that of Ngo and Laumon. Roman B. (together with Mirkovic, Rumynin, Ginzburg, Arkhipov, Finkelberg etc)

has been proving a host of conjectures in “classical” representation theory, or what you might call Lusztig-world, in particular

Lusztig’s conjectures tying modular representaitons with

representations of quantum groups or affine algebras. Also there’s a profound understanding of (affine) braid group actions on various categories coming from this work of Roman that relates to many geometric problems, and work of Bridgeland and others.

One of the most exciting things is the work of Ngo. Ngo was able to look at one of the fundamental geometric aspects of geometric Langlands, the Hitchin fibration, in a brand new way and use it with Laumon to prove the fundamental lemma for unitary groups (and maybe soon for other groups) — this being one of the central problems in the CLASSICAL Langlands program.

There are a bunch of other concrete consequences of geometric

Langlands ideas outside of its internal problems, but that’s maybe enough for now.

Finally for 3 I would point (besides the papers you mention)

to the work of Donagi-Pantev, in which geometric Langlands

is proved (in a certain degenerate limit, on an open subset)

using T-duality (Fourier-Mukai transforms). I would also point

out the work of Nadler and Nadler-Zaslow connecting Fukaya

categories on cotangents to constructible sheaves,

a key step in bridging the homological mirror symmetry

picture of Kapustin-Witten to geometric Langlands.

(Damn — why do I always confuse Getzler and Zaslow? Apologies.)

Aaron,

What do you have in mind about the worldsheet instantons? I thought they were supposed to be trivial?

While we’re on the subject of vague geometric Langlands speculations, I’ve always wondered what’s supposed to happen to twisted N=4 at large N. Is there any sort of AdS dual theory?

David –

Many thanks for your very able and enlightening thoughts. That’s a lot to think about, which I will try to do. Thanks again.

Aaron –

I also appreciate your thoughtful and reasonable response, which I am also going to try to think about.

Hi AJ —

I’m not sure I’m remembering this correctly, but I think the issue is the composition of cc – cc – A strings. There are no instanton contributions to the cc -cc strings or to the cc – A strings, but there might be to the composition law.

Thanks David!

From many conversations with my colleagues here at Columbia, I can vouch for the fact that experts think that there have been big advances in the classical Langlands area during the last year or two, especially Taylor-Harris on Sato-Tate, and Ngo-Laumon on the fundamental lemma.

It also seems to me that there has been an increase in the interest of the rest of the math community in geometric Langlands ideas, and this is not just due to Witten’s work and the relation to mirror symmetry. The connections to other subjects that David mentions have been part of this, as well as efforts by him, Ed Frenkel and others to make these ideas more accessible. A few years ago the widespread perception was that you had to speak Russian and hang out in Hyde Park to have any hope of understanding anything about geometric Langlands, but that has definitely started to change.

Peter, Aaron and David –

A first, partial thought:

I don’t mean to be harsh or dismissive, but my impression of the work concerning the relationship of mirror symmetry to geometric Langlands is that these papers strongly resemble the physics literature in this area: Written as fast as possible and largely motivated to stake a claim. Lots of claims to have “thrown up bridges” and the like, without too much in the way of bridge engineering detail. Of course, it is entirely possible that some of these ideas will yield real progress, but (for me, at least) it’s far too soon to tell at this point in time.

I’m still thinking about your responses.

RM,

My colleague Michael Thaddeus (with Tamas Hausel) was investigating this example of mirror symmetry several years ago, motivated I think partly by the hope that the Strominger-Yau-Zaslow picture could be worked out in this case, partly by the connection to geometric Langlands. That work was pretty standard mathematics, no connection to physics.

The more recent Kapustin-Witten and Gukov-Witten papers really are written from a physics point of view, and I think it’s up to mathematicians to ultimately figure out how much insight they provide into the mathematics. This is not easy, given the very different point of view, grounded in difficult aspects of QFT. There have been amazing mathematical successes coming from this kind of activity in the past (Seiberg-Witten, knot invariants, counting curves on threefolds, etc.), we’ll see what this leads to. I agree that it’s still unclear…

I don’t think there’s anything tentative at this point

on the relation between geometric Langlands and T-duality.

Not being a physicist I can’t vouch for the relation of this particular

T-duality with mirror symmetry, but Kapustin-Witten explain

it is precisely a hyperkahler instance of mirror symmetry.

From the math point of view if we take homological mirror

symmetry as our reference point, then geometric

Langlands is very close to fulfilling that requirement:

it relates B-branes on one space to D-modules on the dual.

Now you have to decide how close D-modules are to A-branes,

and that’s a big part of the argument in K-W, and on

the math side is the topic of Nadler&Zaslow’s work.

Unfortunately we don’t yet actually have a definition of the category of A-branes outside of the Lagrangian objects (Fukaya category)

so it’s hard to ask the question, but there’s no doubt

the two are very closely related.. I’m not surewhat else you would

look for as “real progress”..

David and Peter-

I’m not ignoring your most recent points, but one of David’s earlier points is in my mind right now – and there’s not that much room up there. I think it is unquestionable that Taylor-Harris, for example, represents real and tremendous progress in establishing the Langlands conjectures for local fields. But establishing the Langlands conjectures for local fields may (in my opinion as a non-specialist) be best viewed as finally making the global conjectures precise. That formulation is a kind of progress on the global case (“real” number field Langlands). But I think it is highly unlikely that Taylor would say that establishing the global conjectures (as now more precisely defined) has been moved significantly closer by his or anyone else’s work under discussion.

RM,

I believe you’re thinking of earlier work by Taylor-Harris, not the recent Sato-Tate work I was referring to. I’ll leave it to people more knowledgable than me to comment on the significance of their local Langlands stuff, but the recent Sato-Tate results are for number fields, not local fields. In my pitiful understanding of the subject, they generalized Taylor-Wiles to symmetric tensor powers.

Peter –

Well, OK. Sato-Tate pertains to global fields and generally fits into the original Langlands program. But so does the Taniyama-Shimura theorem – and I think it’s going rather far to view Taniyama-Shimura (and Fermat?!) as essentially features of the original Langlands program. The original Langlands program is vast and visionary, bold and sweeping. But – again speaking as a non-specialist – I don’t think it’s best practice to see all progress made in any area of mathematics the Langlands program touches as significant progress in

the establishment of non-abelian global calss field for number fields

.

This all reminds me of a quip recently made by an Anglican bishop following some nasty theological dust-up at a bishops’ conference that

pretty soon even the Ayatollah might wake up and find that he’s an Anglican!

Peter –

I’m sorry if this post is redundant, but I’m concerned that my last one was obscure (in part, but not totally, because of its embarrassing typos, for which I apologise).

Sato-Tate deals with points on elliptic curves reduced mod p for all p – and definitely pertains to global number fields. Moreover, Langlands personally has had interesting things to say about Sato-Tate. But while Sato-Tate generally fits into the Langlands program and pertains to global number fields, it does not seem best practice (to me, as a non-specialist) to view Sato-Tate as a creature of the Langlands program. After all, Taniyama-Shimura (and Fermat!) also generally fit into the Langlands program and pertain to global number fields. In my opinion, Sato-Tate and Taniyama-Shimura can well be taken as evidence that the original Langlands program is on target. But I don’t think any of these things significantly advances us towards a proof of a global nonabelian class field theory for number fields.

So, in summary, I don’t think Sato-Tate is a significant advance of the classical Langlands program, even though Sato-Tate is a very significant advance!

Disclaimer: I am, at best, an interested outside to number theory and might suffer from severe delusions about the mathematical content of what I say.

Robert,

I think it’s incorrect to claim that Shimura-Taniyama does not represent a significant advance towards the global Langlands program, and to claim that it’s “rather far” to view the former as being a “feature” of the former. Here’s why:

The Langlands program relates automorphic representations and Galois representations over global fields (or, more philosophically, automorphic objects and geometric ones). Given class field theory, the first non-trivial case is the two dimensional one. In this case, there’s a standard method to produce tons of interesting examples on both sides. For the automorphic side, one takes (weight 2) modular forms, and for the Galois side one takes the Tate modules (homology) of elliptic curves defined over such fields. Given this, the natural question to ask is if these two are Langlands-related (i.e: have the same L-function)? In this sense, Eichler-Shimura and Wiles’ theorem provide the first non-trivial instance of the Langlands program — the former says that such forms have associated geometric objects, and the latter produces an inverse association.

Of course, I realise that this doesn’t quite prove it (even for n=2, where it’s still actively being researched by Kisin and company!), but it still seems like a fairly significant advance towards Langlands’ conjecture to be able verify it in the first non-trivial case one can write down!

bb –

As one non-specialist to another, I think I mostly agree with you. “Eichler-Shimura and Wiles’ theorem provide the first non-trivial instance of the Langlands program?” Of course, although I might quibble about the “first” qualifier, since I think lots of earlier examples by Langlands and others were “non-trivial.” Maybe Fermat is the first “spectacular” example. Sato-Tate? Ditto. These results are all – to my limited understanding – very good examples of the kinds of things Langlands and his school explicitly wanted to be able to get at with the classical Langlands program. That’s why I think that “Sato-Tate and Taniyama-Shimura can well be taken as evidence that the original Langlands program is on target.”

So, yes, seeing that various “predicted” by the classical Langlands program are correct and can be proved using techniques of the type Langlands and his school advanced is definitely progress. And in that sense I see David and Aaron as being definitely and completely correct in pointing out that classical Langlands techniques have NOT “hit a wall” in reaching such spectacular and wonderful examples.

But, as preliminary and possiblly epistomological quibble, Fermat rather predates Langlands, and lots of people made lots of progress on Fermat before Wiles – and not just Faltings. Sato-Tate is from 1960, Taniyama-Shimura from 1955, etc. Each of these conjectures have many lives and characteristic techniques that are independent of the Langlands project, even though the Langlands project had (has) something important to contribute to their resolutions. Wiles’ critical use of Galois deformation in proving Fermat is not really part of the Langlands project, either – a fact that even the Langlands-imperialists (that a joke) at Columbia Math have admited in public writing! So I don’t think these conjectures are best thought of as mostly creatures of the Langlands project. Rather, the Langlands project exists in large measure (but not solely) to contribute some techniques that address such conjectures!

But my second, more basic, point is that it is my impression (as an outsider) that the leading people in this field believe that (1) the classical Langlands program is a huge mass of conjectures, of which we are only beginning to scratch the surface, which tells us what the structure is, and (2) some powerful new geometric ideas are needed before real progress towards Langlands’ goal of a class field theory for nonabelian global number fields. My further impression is that Geometric Langlands is thought to be a good place to look for such ideas.

None of that is intended as criticism. No one could reasonably critize Taylor or Harris or Shahidi or anyone else just because they don’t yet have the tools to complete the clasical Langlands program and are looking at GL (among other places) for new ideas!

Anyway, I go on far too long and I don’t mean to dominate Peter’s blog. So I’ll shut up now.

And, Peter, what’s wrong with Hyde Park? I love Hyde Park!