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December 21, 2006

A Little Bit of Geometric Langlands: Relation to Integrable Systems

Posted by Urs Schreiber

As I mentioned in Navigating Geometric Langlands by Analogies, the mathematicians and the string theorists in Hamburg have a a small series of lectures this term, where we try to explain to each other some tiny fraction of what Langlands duality is about.

Last time we had something on the “classical” number-theoretic aspect. I didn’t even try to report on that.

This time, Jörg Teschner spoke about geometric Langlands duality. After briefly mentioning what the main statement is, he concentrated on understanding one aspect of this statement using the language of integrable systems.

Ever since it was found that certain aspects of N=4N=4 super Yang-Mills theory are governed by what are called quantum integrable systems, and due to the impact this has on understanding and testing the AdS/CFT duality, string theorists have been interested in integrable systems. That’s one reason why Jörg Teschner decided to emphasize this aspect of the talk.

Roughly, the main point he tried to make could maybe be summarized like this:

The geometric Langlands duality relates (conjecturally) two different derived categories by an equivalence.

But on top of that, the conjecture states that both these derived categories have something like a nice “basis of eigenstates of some operator” (compare my previous entry # on what such a statement would really mean) and that under the equivalence a basis vector on one side is sent to a basis vector on the other.

Now, on one of these two sides, those “basis vectors” (the Hecke eigensheaves) can be understood as coming from common eigenstates of the set of commuting Hamiltonians of some integrable system.

From this point of view of integrable systems, geometric Langlands duality seems to be a statement about when and how an integrable system admits a separation of variables.

If you like integrable systems, that should be interesting.

In reading the following, you should be well aware that, while some detailed formulas do appear, chances are high that various imperfections and imprecise statements appear. I am not an expert on the following material. I’ll try to reproduce the transcript of the talk as well as I can, but that’s all.

As usual, some personal comments I include are set in italics.

So, these are the notes I have taken:

This is the first of two talks on aspects of the geometric Langlands duality:

1) Perspective of Integrable Models

2) Representation Theoretic Perspective and connection to CFT

Here is part 1).

The geometric Langlands duality conjecture states (see T. Pantev’s lecture I, II for details on that statement) that there is an equivalence

(1)irreducible LGlocal systems onX𝒟modules onBun G(X). \text{irreducible} {}^L G \;\text{local systems on}\; X \;\;\;\; \leftrightarrow \;\;\;\; \mathcal{D}-\text{modules on}\; \mathrm{Bun}_G(X) \,.

In words, this means, roughly, something like: given a complex curve XX and a suitable Lie group GG, then gadgets on the space of all GG-bundles on XX that differential operators may act on are in correspondence with vector bundles on the space of flat LG{}^L G-bundles with connection on XX, where LG{}^L G is the Langlands dual group to GG.

Assuming the genus of XX is

(2)g2, g \geq 2 \,,

one finds that the dimension of the space Bun G(X)\mathrm{Bun}_G(X) is

(3)d g:=dim (Bun G(X))=dim(G)(g1). d_g := \mathrm{dim}_\mathbb{C}(\mathrm{Bun}_G(X)) = \mathrm{dim}(G)(g-1) \,.

Now, and that’s what this talk will be about, the special 𝒟\mathcal{D}-modules on the right side of the Langlands duality conjecture can be understood in terms of certain eigenstates of a certain integrable system, namely

The Hitchin System

Consider the cotangent space

(4)T *Bun G(X) T^* \mathrm{Bun}_G(X)

of the space of GG-bundles on XX. Being a cotangent bundle, it carries a canonical 1-form. The 2-form differential of that defines a Poisson structure on the space of functions on T *Bun GT^* \mathrm{Bun}_G.

Fact i): this structure is classically integrable.

This means that there exists functions

(5)h 1,h 2,,h d g:T *Bun G(X) h_1, h_2, \cdots, h_{d_g} : T^* \mathrm{Bun}_G(X) \to \mathbb{R}

that pairwise Poisson-commute

(6){h r,h 2}=0r,s{1,2,.,d g}. \{h_r, h_2\} = 0 \;\;\;\; \forall r,s \in \{1,2,.\cdots, d_g\} \,.

If we think of Bun G\mathrm{Bun}_G as a configuration space of some classical physical system, then T *Bun G(X)T^* \mathrm{Bun}_G(X) would be the corresponding phase space and the h rh_r would be a set of commuting “Hamiltonians” or of one Hamiltonian and a couple of mutually commuting “conserved charges”.

Fact ii): the surface of constant {h r} r\{h_r\}_r is almost always a torus

Next, we consider quantizing this classical setup. This proceeds essentially by geometric quantization.

(The statement was that what we do is not precisely the same as geometric quantization, but I can’t tell the difference right now.)

So, we find the canonical line bundle on T *Bun G(X)T^* \mathrm{Bun}_G(X) given by our symplectic form that defines the Poisson structure and we assume we can form a square root bundle of that. A polarization is chosen and this bundle descends to a bundle on Bun G(X)\mathrm{Bun}_G(X) itself.

Quantization sends the functions {h r} r\{h_r\}_r to operators

(7){H r} \{H_r\}

on the space of (suitably well-behaved) sections of that line bundle on Bun G(X)\mathrm{Bun}_G(X).

The next theorem says that we can find a quantization such that even the quantum system is integrable.

Theorem (Beilinson & Drinfeld) There exist differential operators

(8){H 1,H 2,,H d g} \{H_1, H_2, \cdots, H_{d_g}\}

on sections of that line bundle such that

(9)[h r,h 2]=0r,s{1,2,,d g}. [h_r, h_2] = 0 \;\;\;\; \forall r,s \in \{1,2,\cdots, d_g\} \,.

and such that the symbol of H rH_r is h rh_r.

Given any such quantum integrable system we can pose the following spectral problem:

for which constants

(10)μ=(μ 1,μ 2,,μ d g) \mu = (\mu_1,\mu_2, \cdots, \mu_{d_g})

can we find a section ψ\psi such that

(11)H rψ=μ rψ H_r \psi = \mu_r \psi

for all rr ?

At least morally, all ψ\psi of this form are related to the Hecke eigensheaves appearing in the geometric Langlands duality.

The following theorem gives a necessary and sufficient condition for when such ψ\psi exist. The remainder of the talk will be concerned essentially with understanding what the following statement really means.

Theorem (Beilinson-Drinfeld): there exist solutions to the above spectral problem if and only if the μ r\mu_r are the χ r\chi_r-character of a “globally defined” Lie( LG)\mathrm{Lie}({}^L G)-oper.

Next we define what an “oper” is and what that notion of character might mean.

Let Ω\Omega be the canonical line bundle on our complex curve XX.

Definition: An sl n\mathrm{sl}_n-oper is an nn-th order holomorphic differential operator

(12)δ:Ω (n1)/2Ω (n+1)/2 \delta : \Omega^{(n-1)/2} \to \Omega^{(n+1)/2}

locally of the form

(13) z nq 1(z) z n1q n1(z). \partial_z^n - q_1(z)\partial_z^{n-1} - \cdots - q_{n-1}(z) \,.

It’s called a sl(n)\mathrm{sl}(n)-oper because by linearizing this degree nn-differential operator we get an operator

(14) z= z+A(z) \nabla_z = \partial_z + A(z)

where AA is the n×nn\times n-matrix

(15)A(z)=(0 q 1(z) q 2(z) q n1(z) 1 0 0 0 0 1 0 0 0 0), A(z) = \left( \array{ 0 & q_1(z) & q_2(z) & \cdots & q_{n-1}(z) \\ 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots &&&& \vdots \\ 0 &&&& 0 } \right) \,,

which, due to the tracelessness, we can regard as taking values in sl(n)\mathrm{sl}(n).

(now I am being lazy and skip reproducing a little example, which, personally, I did not find all that illuminating)

Given any such “oper”, we can think of its linearized version as above, regard that as the covariant derivative of a connection and consider the monodromies of that connection. The following statement relates these monodromies to solutions of our spectral problem:

Proposition (E. Frenkel): a necessary condition for the existence of solutions ψ\psi to

(16)H rψ=μ rψ H_r \psi = \mu_r \psi

for all rr is that the sl(2)\mathrm{sl}(2)-oper of the form

(17) z 2q(z) \partial_z^2 - q(z)

with

(18)q(z)= r=1 N(λ r(λ r+2)(zλ r) 2+μ rzλ r) q(z) = \sum_{r=1}^N \left( \frac{\lambda_r(\lambda_r + 2)}{(z - \lambda_r)^2} + \frac{\mu_r}{z - \lambda_r} \right)

for some set of integers {λ r}\{\lambda_r\} has trivial monodromy.

How do we find sets {λ r}\{\lambda_r\} such that the monodromy of the associated oper vanishes?

Answer (E. Frenkel): the q(z)q(z) with this property can always be brought into the form

(19)q(z)=1z(χ(z)) 212 zχ(z), q(z) = \frac{1}{z}(\chi(z))^2 - \frac{1}{2}\partial_z \chi(z) \,,

where

(20)χ(z)= r=1 Nλ rzλ r s=1 N2zw s, \chi(z) = \sum_{r=1}^N \frac{\lambda_r}{z-\lambda_r} - \sum_{s=1}^N \frac{2}{z- w_s} \,,

and where, finally, the w sw_s are solutions of the famous Bethe ansatz equation

(21) r=1 Nλ rw sλ r ss2w sw s=0. \sum_{r=1}^N \frac{\lambda_r}{w_s - \lambda_r} - \sum_{s' \neq s} \frac{2}{w_{s'}- w_s} = 0 \,.

(There is no guarantee that I correctly reproduced these formulas. But I try my best.)

These Bethe Ansatz equations are close to the heart of those field theorists and string theorists working on integrability of N=4N=4 super Yang-Mills and on AdS/CFT. There it is all about solving these equations.

The idea is that the ideals of sections generated by the kernel of (H rμ r)(H_r - \mu_r), which form a 𝒟\mathcal{D}-module, i.e. a module for the algebra of differential operators on Bun G(X)\mathrm{Bun}_G(X) are closely related to the automorphic 𝒟\mathcal{D}-modules, otherwise known as Hecke eigensheaves.

One question after the talk was: “So, what is that curious character ρχ r(ρ)\rho \mapsto \chi_r(\rho) that appeared in the Beilinson-Drinfeld theorem?” Answer: “that’s implictly given by that proposition by E. Frenkel at the end”.

Posted at December 21, 2006 4:16 PM UTC

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Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

In the geometric quantization (or not quite) that you’re doing here, the symplectic form was exact, so the relevant line bundle was trivial… right?

I’m not sure where the square root came in (maybe because I’m used to Dolbeault rather than Spinc quantization), but anyway I hope that the square root chosen was also trivial!

The equivalence

irreducible LG local systems on X ↔ 𝒟−modules on BunG(X)

is best understood in the case X a punctured formal disc, more precisely Spf C[[z]][z-1], and is called the geometric Satake correspondence. Since I don’t remember having seen it discussed on this blog, I’ll mention the basics of this.

In this case BunG(X) is G((z))/G[[z]], containing (and homotopy equivalent to) the space of smooth based loops into the maximal compact subgroup of G. Bott studied this “loop Grassmannian” 50 years ago, and it’s still giving us great mathematics. The finite-dimensional Morse-Bott strata on this give a set of D-modules that I’m pretty sure are the relevant basis on the right side.

The left side is just the representation category of the Langlands dual group, whose basis is the irreducible representations.

So, for example, you can get each irrep of the Langlands dual as the intersection homology of the closure of the corresponding Morse-Bott stratum.

Some limited references: Ginzburg’s mysteriously unpublished paper, Mirkovic-Vilonen, more recent developments in Anderson and Kamnitzer.

Posted by: Allen Knutson on December 21, 2006 6:43 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

In the geometric quantization (or not quite) that you’re doing here, the symplectic form was exact, so the relevant line bundle was trivial… right?

I guess so. I was wondering about that, too, but forgot to ask.

I’m not sure where the square root came in

Right, me neither. I can try to check again with Jörg.

A well kept secret concerning this lecture series of ours is that none of us is actually working on anything really related, so every lecturer is to a large extent forced to try to reproduce information from reviews he hasn’t fully absorbed yet.

I guess one could answer all these questions by taking the time to actually read and absorb Frenkel’s lecture notes in detail.

Some limited references: […]

Thanks a lot, indeed!

Posted by: urs on December 21, 2006 6:55 PM | Permalink | Reply to this

D-modules

To get a better feeling for these beasts, I would enjoy seeing some extremely elementary examples.

For instance: what are all the 𝒟\mathcal{D}-modules on S 1S^1?

I have Björk, Analytic 𝒟\mathcal{D}-Modules and Applications in front of me, but if anyone likes to reply before I finish reading everything in there, that would be great.

Posted by: urs on December 22, 2006 12:26 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

Urs said:

Ever since it was found that certain aspects of N=4 super Yang-Mills theory are governed by what are called quantum integrable systems, and due to the impact this has on understanding and testing the AdS/CFT duality, string theorists have been interested in integrable systems.

Hi Urs. Actually, it is worth noting that string theorists have been interested in integrable systems for a lot longer than that. Among the many motivations include the non-critical string theory studies tha t resulted in the non-perturbative formulations of string theories I’ve talked about a bit before here, here and here. Integrable systems of KdV type (of a somewhat different sort, I know, and classical) seem to organize the content of the (quantum) string theory -fully non-perturbatively- in a manner that is highly instructive, continues to fascinate, and seems to me to be part of a very important story to uncover further. I often wonder if there is any mileage to be gained in trying to see if there are ways to connect some of the lessons learned in that context to the newly appeared role of integrable systems in AdS/CFT. Or vice-versa. Or perhaps they are just two totally disconnected devices.

Sorry to possibly pollute with the irrelevant, but you did ask yesterday about blog posts of mine containing research discussion. ;-)

Cheers,

-cvj

Posted by: cvj on December 21, 2006 8:36 PM | Permalink | Reply to this

Integrable systems in string theory

string theorists have been interested in integrable systems for a lot longer than that

Right, I should have mentioned that. And not only string theorists. “Integrable systems” (with slightly varying definition of what that actually means) appear all over the place.

Not the least, the worldsheet theory of the bosonic string itself is an integrable system, in that it has infinitely many conserved charges.

Traditionally these are encoded in the DDF operators. But if one applies the Lax pair method of integrable systems to the bosonic string and turns the crank, one finds the so-called Pohlmeyer invariants.

the non-critical string theory studies that resulted in the non-perturbative formulations of string theories I’ve talked about a bit

Very interesting, yes, thanks.

Do you ever invoke Bethe Ansatz and the like in that context?

Sorry to possibly pollute with the irrelevant,

Not at all, thanks for the comment!

but you did ask yesterday about blog posts of mine containing research discussion.

Yes, and I did mean it! :-)

Posted by: urs on December 22, 2006 9:20 AM | Permalink | Reply to this

Re: Integrable systems in string theory

Hi Urs,

Thanks for the information. I like the relation to the bosonic string snd related matters you pointed out, but I find that the integrability in the non-critical (or minimal) string setup is more tantalizing. At least for my particular tastes and goals right now. This is because it is fully non-perturbative in its formulation, and furthermore there are several vitally important features of the string theory that are natural structures of the integrable system itself. D-branes and fluxes are recent examples - in type 0A they are essentially special solitons of the KdV system and are acted on by the classic Backlund transformations of the KdV system! I find this sort of thing quite remarkable (see my earlier posts mentioned in comment above for references to my own work on this with Carlisle and Pennington, and also to work of Seiberg and Shih), and I suspect that we can learn a lot more about string theory - in terms of clues to more non-perturbative phenomena, and maybe clues as to what it really is- from some of this type of formulation. (I’ve done some work in that spirit and described it in this post. More to come too.) The integrability has been key here in allowing the sharp and important structures to float free of the clutter.

The sort of picture that is emerging is very nice, and that’s why I wonder whether contact can be made to some of the recent work going on elsewhere in using integrability in strings (AdS/CFT, etc) …. but it might be that this is silly. It might be that it is like trying to fundamentally connect together any two areas of science that are both using differntial calculus!


As to the Bethe ansatz in this context…. I don’t think it has yet played a central role, to my knowledge.

Best,

-cvj

Posted by: cvj on December 23, 2006 6:47 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

Urs wrote:

Last time we had something on the “classical” number-theoretic aspect. I didn’t even try to report on that.

Too bad! I’ve been spending the last few years studying number theory with James Dolan. First we learned about class field theory and Artin reciprocity (the abelian case of the Langlands correspondence, worked out much earlier). More recently we’ve been tackling the full-fledged Langlands correspondence, with mixed success. We started this course of study before Witten did his thing, and we haven’t tried at all to understand what Witten did — but I think we do understand the analogy between number fields and function fields which underlies the analogy between ‘classical’ Langlands theory and the ‘geometric’ Langlands theory. It’s quite old, and it goes like this:

NUMBER THEORY                 COMPLEX GEOMETRY                                 
Integers                      Polynomial functions on the complex plane
Rational numbers              Rational functions on the complex plane
Prime numbers                 Points in the complex plane            
Integers mod p^n              (n-1)st-order Taylor series
p-adic integers               Taylor series
p-adic numbers                Laurent series
Adeles for the rationals      Adeles for the rational functions
Fields                        One-point spaces
Homomorphisms to fields       Maps from one-point spaces
Algebraic number fields       Branched covering spaces of the complex plane

Physicists can imagine the complex plane implicit in the right-hand column as a patch of a string world-sheet.

I can certainly understand why most physicists don’t want to think about the ‘classical’ Langlands correspondence, since it involves weird stuff like finite fields and p-adic numbers. But, eventually someone will take the new string theory ideas and push them into the classical Langlands theory using the analogy between number fields and function fields… and we’ll see how number theory is inherently ‘stringy’!

(Maybe someone is already doing it — I try not to keep up with the latest trends, and right now geometric Langlands has become painfully trendy.)

(Nonetheless, Urs, the math you’re descibing is beautiful stuff!)

Posted by: John Baez on December 22, 2006 5:47 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

Or how physics is inherently arithmetical…

Posted by: James on December 22, 2006 7:50 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

Too bad!

I agree, it is too bad that I did not feel like I had absorbed the idea of the classical Langlands program well enough to be able to say anything sensible about it.

For the “geometric” version, while I would have to struggle to get in full control of all the technical details, I do understand the general statement: given a complex curve we can consider two different moduli spaces of structures on that curve and two different derived categories of certain sheaves on these moduli spaces. And the statement is that these derived categories are equivalent.

What would be the analog of that equivalence of derived categories in the classical number theoretic version of Langlands?

the analogy between number fields and function fields which underlies the analogy between ‘classical’ Langlands theory and the ‘geometric’ Langlands theory

This analogy is essentially the idea behing algebraic geometry. Depending on our taste, it allows us to think of geometry in terms of rings or of number theory as an instance of “exotic” geometry.

I am hoping that when I understand “geometric Langlands duality” I can think of “classical Langlands duality” as geometric Langlands applied to “exotic geometries”.

Is this hope justified?

Posted by: urs on December 27, 2006 10:37 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

Urs wrote:

I am hoping that when I understand “geometric Langlands duality” I can think of “classical Langlands duality” as geometric Langlands applied to “exotic geometries”.

Is this hope justified?

I don’t understand the technical details well enough to answer your question from a position of real knowledge… but from everything I’ve read, I know the answer is yes.

For example, here is something Kevin Buzzard wrote in reply to week218 (see the Addenda). Kevin is a number theorist who is much more familiar with ‘classical’ Langlands duality:

Actually I think there is even a kind of Langlands philosophy for (z)\mathbb{C}(z) and its finite extensions nowadays worked out recently by Beilinson and Drinfeld. I saw Beilinson give several lectures on it, more than once, and still didn’t really get it, I am too number-theoretic.

I replied:

Is this the “geometric Langlands program” stuff? Physicists are getting interested in that…

He replied:

Yes.

A much more detailed story can be found here:

He explains several versions of the Langlands program: first one for number fields like (5)\mathbb{Q}(\sqrt{5}), then one for function fields of curves over finite fields, then one for function fields of curves over \mathbb{C}.

The last-named entities are secretly just Riemann surfaces seen through the lens of algebraic geometry. And Riemann surfaces, of course, are secretly just string worldsheets.

So, in the previous two cases, we are just generalizing ideas from the geometric Langlands program (or string theory, if you like) to ‘exotic geometries’ - the geometries number theorists like to study.

But of course, historically, the number theorists got there first!

Posted by: John Baez on December 28, 2006 6:59 AM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

If you had one sentences and one equation and had to say what the gist of the number theoretic Langlands duality conjecture were. What would you say? Something about a correspondence between reps of Galois groups and automorphic forms – but what exactly? What is the big picture?

Is there a single crisp statement as for geometric Langlands? Some equivalence of categories, maybe?

Posted by: urs on December 28, 2006 4:56 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

Urs wrote:

If you had one sentence and one equation and had to say what the gist of the number theoretic Langlands duality conjecture was, what would you say?

I’d say: “Wait until I understand this better.”

Is there a single crisp statement as for geometric Langlands? Some equivalence of categories, maybe?

I suspect the ‘classical’ and ‘geometric’ cases are are EXACTLY analogous, once you find the right framework. The only serious difference should be that one uses a number field while the other uses the field of functions on a Riemann surface.

But right now, it seems easier to find clear statements of Langlands duality as an equivalence of categories in the ‘geometric’ case. In the ‘classical’ case one tends to read statements of Langlands duality that sound like decategorified consequences of some equivalence of categories: for example, an equation between L-functions.

But, I think this is just a matter of history: the classical case came first, and people usually start by understanding things in a decategorified way!

I began my struggles to understand this stuff by learning how zeta functions are obtained by decategorification. I described this in week216. L-functions are just a slight generalization of zeta functions, so next I should tackle these.

However, now I think there’s a much faster way for me to learn about the classical Langlands program: simply take what people say about the geometric case and replace the function fields by number fields!

For example, you and Allen have emphasized the important role of this category:

[vector bundles on the moduli space of flat HH-bundles over a complex curve XX]

where HH is some reductive Lie group — the Langlands dual of the reductive Lie group you’re calling GG.

Actually, Allen emphasized the case where XX is not a complex curve, but the punctured complex plane. This is the simplest case of all! But never mind: we’re talking grand strategy here, not tactical details.

We can translate the above category into the language of algebraic geometry, describing it in terms of the field of functions on XX, instead of XX itself.

Then, we can replace this function field by a number field. We should get something familiar from the ‘classical’ Langlands program. In fact I can almost see how!

One key step is to realize that the moduli space of flat HH-bundles over XX is the same as

hom(π 1(X),H)/H.hom(\pi_1(X),H)/H .

That is: the space of homomorphisms from the fundamental group π 1(X)\pi_1(X) to HH, modulo conjugation by elements of HH.

So, our category is really

[vector bundles on hom(π 1(X),H)/Hhom(\pi_1(X),H)/H ]

The next step is to translate this into the language of algebraic geometry. For this, we should realize that π 1(X)\pi_1(X) is secretly a Galois group, following the usual yoga relating Galois groups to fundamental groups, as explained in week205.

Morally speaking, π 1(X)\pi_1(X) is the Galois group Gal(F¯/F)Gal(\overline{F}/F), where FF is the field of functions on XX, and F¯\overline{F} is its algebraic closure — roughly the field of functions on the universal cover of XX.

So, in the ‘classical’ case, I think we should talk about

[vector bundles on hom(Gal(F¯/F),H)/Hhom(Gal(\overline{F}/F),H)/H ]

where now FF is a number field!

This seems about right to me. For example, if H=GL(n)H = GL(n), then

hom(Gal(F¯/F),H)/Hhom(Gal(\overline{F}/F),H)/H

is the moduli space of nn-dimensional representations of the Galois group of FF — which indeed is something people always talk about in the classical Langlands program!

There are a vast number of technical details I dimly know I’m ignoring here. For example, we may need to use the moduli stack

hom(π 1(X),H)//Hhom(\pi_1(X),H)//H

given by the weak quotient of hom(π 1(X),H)hom(\pi_1(X),H) by HH. And, in the ‘classical’ case, we should let HH be an algebraic group over the adeles, instead of a Lie group.

But, the big picture should ultimately be very nice.

Posted by: John Baez on December 28, 2006 8:18 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

π 1(X)\pi_1(X) is secretly a Galois group

Ah, great. That’s an important puzzle piece.

For example, we may need to use the moduli stack

Yes, that’s what Tony Pantev very much emphasized in his talk #.

It turns out that this stack is in fact “transitive” and “locally non-empty” - hence a gerbe. This is supposed to be important in the end. But maybe not at the point at which we are at the moment.

Posted by: urs on December 28, 2006 9:34 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

This seems about right to me. For example, if H=GL(n)H=GL(n), then

(1)hom(Gal(F¯/F),H)/H \mathrm{hom}(Gal(\bar F/F),H)/H

is the moduli space of nn-dimensional representations of the Galois group of FF — which indeed is something people always talk about in the classical Langlands program!

That looks good. This is exactly the kind of translation I was looking for.

Now, curiously, the geometric duality tells us that we should be interested in “vector bundles” (really: complexes of coherent sheaves) on this space of nn-dimensional representations.

Maybe this is best thought of as an iteration of the procedure of replacing spaces by function fields. Now we are replacing 2-spaces by 2-functions. (Handle this statement with care. :-) As Bruce Bartlett emphasized #, passing to vector bundles over a space is like passing to reps of the discrete category over that space, which is like applying the categorified Gelfand-Naimark duality.

(But please intervene if you think I am getting astray here…)

Posted by: urs on December 28, 2006 9:48 PM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

Here some remarks that might be helpful.

The ‘geometric’ in ‘geometric Langlands’ usually refers to categorification, not to the passage from number fields to functions fields. Consider these three cases:

(A) Number fields: Q, Q(i),… (Langlands, Wiles, …)
(B) Function fields over a finite field F: F(t), F(t)[y]/(y^2-t^3-at-b), … (Laumon, Lafforgue,…)
(C) Function fields over the complex field C: C(t),… (Beilinson-Drinfeld, …)

Usual Langlands makes sense only in cases A and B, whereas (to the best of my knowledge) geometric Langlands makes sense only in cases B and C. The relation between the two is in case B. This is usually called Grothendieck’s functions-sheaves correspondence.

Let me first explain a topological analogue of it. The point is that from the point of view of the fundamental group, a finite field is analogous to a little circle. (The absolute Galois group of any finite field is the free (pro-finite) group generated by the Frobenius element. This is just the standard fact from a first course in Galois theory that the Galois group of any extension of finite fields is generated by the Frobenius element.) Thus an algebraic curve over a finite field should, from the point of view of fundamental groups, be thought of as an algebraic curve over a little circle, or a three-manifold X fibered in Riemann surfaces over a circle S with a fixed generator of its fundamental group.

Now suppose we have a “local system” on X, ie a representation of the fundamental group of X. Then for any circle S’ contained in X which sits as a covering space over the base circle S, we have a representation of its fundamental group. But this fundamental group has a canonical generator (‘Frobenius’), and so we can take its trace. Therefore, given any local system on X, we get a function on the set of circles in X of the type above.

Now go back to case B. Then S is the point defined over a finite field F, and X is an algebraic curve over S. The role of S’ is played by a point of X defined over a finite extension of F. As above, any representation of the fundemental group of X gives a function on the set of S’ – by functoriality we have a representation of the fundamental group of S’ and it has a canonical generator. But now the set of such S’ are precisely the same as the set of closed points of X in the sense of scheme theory. Thus, from a local system on X, we get a canonical function on X (or rather the set of closed points of X, if its viewed as a scheme). This is the decategorification, the sheaves-to-functions direction.
You can’t do this for algebraic curves defined over the field of complex numbers because there is no Frobenius automorphism. However, you could do it over any “quasi-finite” field, one whose absolute Galois group is pro-finitely free on one generator – for example C((t)).

On the other hand, coming up with a geometric (=categorified) version of Langlands in case A would require, I think, some new ideas. There is no base field/point over which everything lives, and this has many consequences. For example, in geometric Langlands, one looks at moduli spaces of G-bundles. When G=GL(1), these are just Jacobian varieties. While Jacobians do exist in cases B and C, in case A they are just finite sets (ie class groups). (In case B, they are also finite sets in a certain sense, but since we have a base field, we can enrich them to be varieties defined over our finite field, which are much richer than plain old finite sets.) So, in cases B and C, the Jacobian has interesting geometry, but in case A it does not.

Of course, case A is the most interesting one. It’s like with the Riemann hypothesis: in case C there’s nothing, in case B it was worth a Fields’ medal or two, and in case A it’s the most important problem in mathematics.

Anyway, I hope some people find that helpful. I’m certainly not an expert, but I did pick up a few things a while back.

Posted by: James on December 29, 2006 1:10 AM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

James wrote:

Here some remarks that might be helpful.

The ‘geometric’ in ‘geometric Langlands’ usually refers to categorification, not to the passage from number fields to functions fields.

Wow, that’s really helpful! And embarrassing that I hadn’t caught on. I guess it’s because, as you say, nobody knows how to do the ‘geometric’ approach for number fields, and physicists always use ‘geometric’ when discussing complex function fields. Function fields over finite fields is the case I’ve read least about! So, I guessed ‘geometric’ meant ‘function fields’.

Everything else you write is very helpful too. Thanks a million!

Posted by: John Baez on December 29, 2006 4:54 AM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

I’m sure I’ve seen this reference posted here before, but the details of how one goes from Langlands to geometric Langlands is explained extensively by Ed Frenkel here.

One of the things I find rather cool about the physics story for all of this is that the D-modules arise from strings between A-branes, ie, objects in the Fukaya category (and more). In particular, there’s a correspondence between objects in the Fukaya category of a cotangent bundle and holonomic D-modules on the base. There’s a mathematical version of this in Nadler and Zaslow and Nadler which relates the derived category or constructible sheaves on the base with a version of the Fukaya category on the cotangent bundle.

Needless to say, I understand very little of this, but perhaps if I mumble the words “Riemann-Hilbert correspondence” the relation between constructible sheaves and D-modules will become clear to someone.

Posted by: Aaron Bergman on December 29, 2006 5:27 AM | Permalink | Reply to this

Re: A Little Bit of Geometric Langlands: Relation to Integrable Systems

About the relative difficulty in James’ situations A-B-C, the last one (zetas of Riemann surfaces) was also worth a fields medal or so, as it was the origin of Selberg’s work on discontinuous groups.

Posted by: complexification on January 14, 2007 6:29 AM | Permalink | Reply to this

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