# Langlands Program and Physics

One of my minor hobbies over the years has been trying to understand something about the Langlands conjectures in number theory, partly because some of the mathematics that shows up there looks like it might be somehow related to quantum field theory. A few days ago I was excited to run across a web-page for a workshop held in Princeton earlier this year on the topic of the Langlands Program and Physics. Notes from some of the lectures there are on-line.

Unfortunately, after reading through the notes, I’m afraid there’s relatively little there about the potential intersection of the ideas of the Langlands Program with Physics. From the physics end of things there are some pretty illegible notes of a lecture by Witten about the Langlands Dual Group in Physics. Part of this story involves the Montonen-Olive duality of N=4 supersymmetric Yang-Mills. This duality interchanges the coupling constant with its inverse, whiile taking the gauge group G to the Langlands dual group (group with dual weight lattice). The symmetry that inverts the coupling constant is actually part of a larger SL(2, Z) symmetry.

One possible explanation for this SL(2,Z) symmetry is the conjectured existence of a six-dimensional superconformal QFT with certain properties. Witten explains more about this in his lectures at Graeme Segal’s 60th birthday conference in 2002. His article from the proceedings volume, entitled “Conformal Field Theory in Four and Six Dimensions” doesn’t seem to be available online, but his slides are, and they cover much the same material. There has been a seminar going on at Berkeley this past semester in which Ori Ganor has been giving talks on this topic.

While the occurence of the Langlands dual group and SL(2,Z) symmetry are suggestive, the relation of this to the full Langlands program seems to be a bit tenuous. There is however a much closer relation between 2d conformal field theory and the Langlands program, a relation which is part of the story of what is now known as “Geometric Langlands”.

Some of the other lectures at the Princeton workshop give a good explanation of the standard Langlands duality conjectures, although I’m not convinced that many physicists will find them easy going. These conjectures posit a duality between two very different kinds of group representations associated to a one-dimensional field (a number field or function field of a curve over a finite field). On the one side one has an analytic object, an “automorphic representation” on a space of functions on a group G(A), where G is a group over A, the adeles of the field. On the other side one has an arithmetic object, representations of the absolute Galois group of the field in the Langlands dual group to G. Typically this duality is used to get information about arithmetic objects using the more tractable analytic objects. The most famous example of this is the Taniyama-Shimura-Weil conjecture relating the arithmetic of elliptic curves to modular forms, which Wiles (with Taylor) was able to prove enough of to use it to prove Fermat’s last theorem.

In general the Langland conjectures for the case of number fields remain an open problem, but for the case of function fields of a curve, they have been proven for G=GL(n) by Drinfeld for n=2 and Lafforgue for general n (which got both of them Fields medals). The geometric Langlands program involves reformulating the function field case in such a way that it still makes sense when you replace the curve over a finite field by a curve over the field of complex numbers. This idea goes back to Drinfeld and Laumon in the 1980s, and has evolved into a specific conjecture which was recently proved by Frenkel, Gaitsgory and Vilonen.

I confess to still being pretty mystified by this subject. The analog of the arithmetic side is clear enough, it’s a homomorphism of the fundamental group of the curve into the Langlands dual group, or equivalently a vector bundle with holomorphic flat connection. But I still don’t understand the analog of the analytic side, which is some sort of D-module over the moduli space of bundles over the curve, broken up into “Hecke eigensheaves”. My colleague Michael Thaddeus explained to me today over lunch what a “Hecke eigensheaf” is supposed to be, but there’s a whole web of relations of this to representations of affine Lie algebras, CFT and vertex operator algebras that neither of us understands very well.

While I don’t understand this material, I do hope to find time in the future to try and figure some of it out. Various sources that seem to explain this are the following:

Edward Frenkel’s web-site at Berkeley contains a lot of interesting material. Many of his papers are on this topic, especially relevant is his Bourbaki seminar report on Vertex Algebras and Algebraic Curves.

Another relevant web-site is that of David Ben-Zvi at Texas. Look at his very informal surveys of Langlands theory written in 1995 before he gets too embarassed by the mistakes in them and takes them down. He is joint author with Frenkel of a book Vertex Algebras and Algebraic Curves.

There’s an on-going seminar on geometric Langlands at the University of Chicago which has a web-page.

Kari Vilonen has a web-site devoted to geometric Langlands and its relation to physics.

MSRI ran a workshop on Geometric Aspects of the Langlands Program in 2002 and the talks are on-line.

As usual, Witten has a hand in all of this, see his remarkable paper “Quantum field theory, Grassmanians and algebraic curves”, Communications in Mathematical Physics, 113 (1988) 529-600, and his contribution to the 1987 conference “The Mathematical Heritage of Hermann Weyl” entitled “Free fermions on an algebraic curve”.

For a different conjectural relation between Langlands and QFT, see:
Mikhail Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, in Functional Analysis on the Eve of the 21st Century, Vol. 1, Birkhauser, Boston, pp. 119-151.

Last Updated on

This entry was posted in Langlands. Bookmark the permalink.

### 31 Responses to Langlands Program and Physics

1. plato says:

If this were a moderated usenet forum you’d be told that your posts violate netiquette

As to reading my stuff:)No!No! think of content first, then who wrote it.:)

One of the things I had found was to respect people thoughts and if such a link of paragraph was to be read of theirs, then it must lead back to site where it was taken.

I wanted to respect the source of information and places, so this compromise was thought of. If you do not like the “source,” then I can not be faulted on its content. Only, that I linked it?

So your rules of netiquette do not apply, hence the moderated usenet forum rules do not apply also. Your software should not support this if this is to be taken as netiquette issue. This is easily solved.

In all cases, I link to the sources as best as possible with secondary links sometimes to source as well.

Interesting site of yours, with some debate about it’s authenticity and law? Copyleft idealism?:)

2. Anonymous says:

‘Plato’, since you haven’t noticed yet: Posting hyperlinks that stretch over several lines is not a good thing to do if you want anyone to read your stuff and to take you seriously. If this were a moderated usenet forum you’d be told that your posts violate netiquette. Here this does not happen as others are violating basic principles of netiquette (and of civilized behaviour) even more.

3. Urs says:

A.J. Tolland just explained us this stuff – he was excited that Peter Woit writes about his math research.

What is this referring to? A talk at Harvard by Tolland? Or anything online?

4. Lubos Motl says:

A.J. Tolland just explained us this stuff – he was excited that Peter Woit writes about his math research.

Peter asks too many questions, many of them have a very well-known answer.

The six-dimensional theories that explain the SL(2,Z) symmetry of N=4 Yang-Mills in d=4 are a completely inevitable part of string/M-theory, and there is extensive literature about them. See the (2,0) theories.

Obviously, Tolland et al. have had many more things to say.

5. D R Lunsford says:

Plato-

You ask where to begin? Where most of the rest of us began – with friction problems, inclined planes, and our own common sense and respect for our scientific forebears. Throw in honesty and hard work and you may get somewhere.

-drl

6. D R Lunsford says:

Plato,

He’s not “just” looking – he’s NOT LOOKING AT ALL. No one in the usual gang of suspects is looking at anything except their own twisted hallucinatory reflections. These people are actively trying to destroy Western science, a collective creation of millenia that is disgraced by their presence.

-drl

7. plato says:

DRL,

Greene is justing pointing, and reveals a deeper concept. Oracle or not, the question is on how we percieve the implication AGN Jets in a consistant mathematical framework?

If we deal with microstates, can we ever associate such math structures to explain this consistancy on a macroscale?

I am in a space(between metric points) where NCG is operating and do not know how to express myself:)The point is where to begin? Yet the dynamical relations of the torus and spin, direct the jet?

So if we accept euclidean postulates to the fifth, GR to gravity, Reimann sphere, why not here in this diagram(?) U(1) and say we see it in the jet?

8. D R Lunsford says:

Dear Plato,

Greene is a carnival-barking buffoon – you might want to consider that before setting him up as your oracle.

Yours,

Ari

9. plato says:

Poster below:)

Without repeating ole cliche’s and sounding like some repetitive impressionist, might we be masking the intent of what one might be saying?:)

I know it is very difficult to find the center, and topologically, impress the idea of “AGN jet” as a continued geometrical expression as well?

As part of, some continued function of the mathematical expressions that you fellows seemed engaged in?

I think Greene was specific on this point?

In fact, in the reciprocal language, these tiny circles are getting ever smaller as time goes by, since as R grows, 1/R shrinks. Now we seem to have really gone off the deep end. How can this possibly be true? How can a six-foot tall human being ‘fit’ inside such an unbelievably microscopic universe? How can a speck of a universe be physically identical to the great expanse we view in the heavens above? (Greene, The Elegant Universe, pages 248-249)

Updating on Blogs are slow so this link should encapsulate more of the idea that I am trying to express.

I mentioned Sklar before, in a topological expressed post, along side of the Kein bottle. Part and parcel of this continued evolution geometrically?

10. Anonymous says:

Favorrite Cliches:

“..asymptotic background..”
“”…robust,,,xyzxxyyx”
“huge number of possible string theory Hamiltonians…”
” my colleague at Columbia …”
“vertex operator algebras that neither of us understands very well..”
.

.

11. Anonymous says:

Sometimes the poets have it right:

Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world,
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.

12. plato says:

Being a non mathematician I am interested in how these visulaizations are generated, so of course I wonder sometimes at statements Peter is making about CFT. This statement below is taken from thread of that discussion. You have to understand the expectancy has a predictive quality about CFT’s future?:)

I was looking for “some consistent method” that would tie together the vast framework of mathematicals in relation to our understood physics approach?

Is that unrealistic?:)

When looking a Klein ordering of Geometries, I had not see any other method that would point to this consistancy, so I am lost when it comes to all the facets of the maths that could be generated.

Maybe lacking a physics approach, mathematics has become limited(like string theory?:) and is no more then a abstract realm that people like to venture, as would artists who have found relevance to signatory styles of expression? There pictures are very unique sometimes like the move to non-eucidean views of Gauss who kept us in suspence?

You would have to forgive my ignorance here, and I fall back on DRL’s encouragement for attempting to comprehend?:)

The gravitational collapse.

How would you define it, if you had a consistent geometrical method from it’s previous developement??:)

If you look to the cosmo, where else will you find the counterparts of this expression of that same mathematics? I believe some have under estimated the value the comsological palette has in which to test the mathematics it is using.

That is my guess, as suspected, I too have no qualifications to speak on this, although I have a keen eye for artistic styles of expression( Penrose and his tillings? Arthur Miller, Gabriele Veneziano-The Myth of Beginning of Time. Scientific America, The Time before Time, May 2004 )

If you think any of these gentlemen deficient in the use of artistic expression even cubist art in relation to the monte carlo effect, maybe the total value of the new math that must emerge will have to wait for another Reinmann?:)

13. Peter says:

I don’t see how moonshine could be related to the Riemann hypothesis, but who knows. Since it is a story about CFT and an interesting group, it may well somehow be related to the Langlands story.

In any case I think the motivation for studying moonshine is that it is (merely!) interesting mathematics. I wasn’t aware of any evidence that the CFT involved could be used to explain anything about the real world.

14. Anonymous says:

“In that interview, Atiyah made more extensive comments than Singer predicting that new ideas in number theory would come from physics. He explicitly mentioned the Langlands program, so was probably thinking about the same things I’ve written about here. He also mentions the Riemann hypothesis, perhaps thinking of Alain Connes’s work on this subject, which has some physics motivation.

But it sounds like he tried this idea out on Andrew Wiles, who was very skeptical about the whole idea that number theoriists would learn very much from physics.”

Yes, sorry, I meant Atiyah.

Do you have any idea if these things (Langlands, Riemann) with regards to their possible connections to physics have anything to do with the moonshine connections to string theory? Which further begs the question if string theory is wrong then what does that say for moonshine – merely interesting mathematics?

15. Urs says:

Sorry, forgot to insert my name. That last comment was of course from me.

16. Anonymous says:

Actually, B takes values in Lie(H) and A in Lie(G), but otherwise, yes.

Note that for the application to Montonen-Olive duality -> 6D SCFTs -> M2s on M5s the kernel of t is nontrivial (in general) as descibed in hep-th/0409200. (We already noted recently that semisimplicity does not seem to allow the observed n-cube scaling for these systems.)

In fact, the kernel of t in these situations is nothing but the abelian group in which the abelian 2-gerbe associated with the bulk of the membrane is associated. This makes it plausible that the abelian 2-gerbe holonomy over the bulk times the nonabelian 1-gerbe holonomy over the boundary can be given a well-defined meaning.

But I agree without you having to convince me: That constraint is unexpected. If you can find a nonabelian gerbe without that constraint but with self-dual 3-form field strength, please drop me a note! ðŸ™‚

Let me see if I get this straight. B takes values in H, A in G, and the kernel of the homomorphisms t : H -> G is an abelian normal subgroup of H. So if H is semisimple then the closed surface holonomy is indeed zero?

Oops, sorry. I evidently didn’t read you whole post.

But can you have non-zero surface holonomy if the surface has no boundary?

20. Urs says:

Thomas –

B is not auxiliary. The constraint says that its image under the Lie algebra homomorphism dt : \h -> \g (which I chose as dt = ad in my previous comment) can be expressed as the curvature of a \g-valued 1-form. The part of B in the kernel of dt is not restricted. By comparison with abelian gerbes which have B_i but not A_i (not to be confused with a_ij) it follows that if anything deserves to be called auxiliary then it is A_i.

The constraint implies that surface holonomy over closed surfaces takes values in the kernel of the homomorphisms t : H -> G, which is an abelian normal subgroup of H. Nothing implies that this holonomy has to vanish.

But I’d be glad to know if this constraint can be relaxed. However, it follows independently from results on standard path space connections, from categorification of ordinary gauge theory and is also the only known solution to the self-duality constraint for nonabelian gerbes. (The use of weak structure 2-groups instead of strict ones in 2-bundles, or equivalently of ‘dynamical’ group products in nonabelian gerbes relaxes it a little, though.)

I do have a nonabelian surface holonomy for nonabelian 2-bundles and nonabelian gerbes for the case ${ad}(B_i) + F_{A_i} = 0$,

Urs, I really don’t understand the logic behind this. Doesn’t B = F(A) mean that B is an auxiliary field, completely determined by A? So you basically have a gauge theory with a 1-form connection. For such a theory, are not the only gauge-invariant quantities line holonomies?

It seems to me that your surface holonomy should equal the line holonomy of its boundary. In particular, can you associate a non-zero surface holonomy to a closed surface?

22. Peter says:

In that interview, Atiyah made more extensive comments than Singer predicting that new ideas in number theory would come from physics. He explicitly mentioned the Langlands program, so was probably thinking about the same things I’ve written about here. He also mentions the Riemann hypothesis, perhaps thinking of Alain Connes’s work on this subject, which has some physics motivation.

But it sounds like he tried this idea out on Andrew Wiles, who was very skeptical about the whole idea that number theoriists would learn very much from physics.

23. Anonymous says:

Is this the same connection between number theory and physics that Singer talks about in his Abel prize interview?

24. Urs says:

Part of this story involves the Montonen-Olive duality of N=4 supersymmetric Yang-Mills. This duality interchanges the coupling constant with its inverse, whiile taking the gauge group G to the Langlands dual group (group with dual weight lattice). The symmetry that inverts the coupling constant is actually part of a larger SL(2, Z) symmetry.

One possible explanation for this ${SL}(2,\mathbb{Z})$ symmetry is the conjectured existence of a six-dimensional superconformal QFT with certain properties. Witten explains more about this in his lectures at Graeme Segal’s 60th birthday conference in 2002. His article from the proceedings volume, entitled ‘Conformal Field Theory in Four and Six Dimensions’ doesn’t seem to be available online, but his slides are, and they cover much the same material.

The abelian case is well understood. The ${SL}(2,Z)$ symmetry of abelian YM follows (at least classically obviously) from realizing it as a toroidal compactification of the theory of an abelian 2-form with self-dual field strength in six dimensions, where the ${SL}(2,Z)$ is just the modular group of the internal torus.

It is believed that something analogous holds true for nonabelian (super)Yang-Mills (for any A-D-E gauge group), i.e. that its Montone-Olive symmetry comes from a toroidal compactification of some 6-dimensional theory involving a non-abelian 2-form.

In this set of slides, Witten calls this nonabelian 6D theory a nonabelian gerbe theory. But certainly that is just a name, to be filled with content, right?

The most glaring problem with making this concrete seems to be this:

What precisely is the duality condition in the nonabelian case and under which conditions can it be imposed?

When I talked to nonabelian gerbe people about this, one thing they said is that it is not clear that in the nonabelian case the self-duality should still be ordinary Hodge self-duality, but that it might involve in addition to the Hodge star an operation on the Lie algebra factor. But I am not quite sure what that should be.

In lack of a better idea, let me assume in the following that we want ordinary Hodge duality. Now, one sufficient condition fulfilled by an ordinary bundle to admit a self-dual field strength is that the field strength transforms covariantly.

So if $U = \lbrace U_i\rbrace_{i\in I}$ is a good covering of the base space with open sets and $F_{A_i}$ is the field strength on $U_i$, then on double overlaps

$F_i = g_ij F_j g_ij^{-1}\,,$

obviously.

Since the covariant transformation respects Hodge self-duality, it is consistent to impose Hodge self-duality in overlapping patches $U_i$.

It is not clear at all that this remains true in general for nonabelian gerbes!

For nonabelian gerbes the general transition law for the nonabelian 3-form field strenth $H_i$ has a covariant part

$H_i = g_ij(H_j) + …$

plus a mess of noncovariant terms

$\cdots + \mathbf{d} d_{ij} + [a_{ij},d_{ij}] – A_i(d_{ij}) + \cdots }] and in particular involving this term \[ \cdots + (F_{A_i} + {ad}(B_i))(a_{ij}) \,. }] (The notation here is taken from equation (55) in hep-th/0409200.) Suppose we want H to be Hodge self-dual and hence H_i to be Hodge-self-dual on each U_i. This implies that on every double overlap all these additional terms in the above transition law have to be self-dual by themselves! So self-duality on H implies further self-duality conditions on the fields A_i, B_i, a_{ij}, d_{ij} (which are the connection 1-form, it’s 2-form cousin and two ‘transition forms’ that measure the failure of A_i and B_i to transform as usual.) But these fields don’t transform covariantly themselves. So the self-duality condition on them involves still more conditions, now on triple overlaps. And so on. It is a huge mess of ever more complicated conditions that arise this way. (Unless there is some simplifying principle hidden in them, which I currently cannot see.) It will be hard to find solutions to these conditions. One solution, though, is easy to see. Obviously, for H to be self-dual it is sufficient that \[ d_{ij} = 0$

(actually this seems to be easy to weaken somewhat)

and

${ad}(B_i) + F_{A_i} = 0 \,.$

The big question is: Are there any further restrictions on the cocycle data of a nonabelian gerbe that would allow Hodge-self-dual H? In particular, are there any with ${ad}(B_i) + F_{A_i} \neq 0$?

The above choice is curious, since it implies that, while $A_i$ and $B_i$ are nonabelian, $H_i$ takes value in an abelian subalgebra of the full nonabelian Lie algebra.

It is also the only case so far in which we know (so far) how to associate a nonabelian 2-holonomy with the nonabelian gerbe. (A paper on that is due out by end of the year. Really, I should not be blogging but be working on that…)

The existence of that nonabelian 2-holonomy seems to be, apart from the self-duality of $H$, a further important condition on whatever Witten may mean by nonabelian gerbe field theory:

We known that when lifted to M-theory these nonabelian 6-D theories come from stacks of coinciding M5s with M2s ending in them. The action of these M2s should involve the abelian volume holonomy of an abelian 2-gerbe characterized by the 4-form $dC_3$, where $C_3$ is the supergravity 3-form potential, over the world-volume of the membrane, call that suggestively but by abuse of the integral notation $\exp(i \int_V C_3)$, times a nonabelian surface holonomy of the nonabelian 2-form living on the M5s over the worldsheet of the boundary of the M2, call that ${Tr}{hol}_{\partial V}(B)$.

Due to global issues (completely analogous to how the coupling of the string to an abelian 2-form involves abelian gerbe holonomy) the product

$\exp(i \int_V (C_r)) {Tr}{hol}_{\partial V}(B)$

has a couple of subtleties. (For the case of 1-dimensional lower these, and their solution, are nicely discussed in the above mentioned paper by Aschieri& Jurčo).

Therefore, in order to understand nonabelian theories in 6D (and, incidentally, the general configuration of the fundamental objects of M-theory) it would be very helpful to have a notion of nonabelian surface holonomy ${hol}_{\partial V}(B)$ that makes the above expression well-defined.

I do have a nonabelian surface holonomy for nonabelian 2-bundles and nonabelian gerbes for the case ${ad}(B_i) + F_{A_i} = 0$, i.e. for the only known case in which the existence of a self-dual 3-form field strength is known. But I have not yet checked if it makes the above action for the M2 brane globally well defined.

(also posted to the String Coffee Table)

25. Aaron says:

For the physics side of geometric langlands (although they might not have exactly known it at the time), you can see

hep-th/9501022
hep-th/9501096

26. Pingback: The String Coffee Table

27. Pingback: The String Coffee Table