Josherich's Blog

Edward Frenkel - New Frontiers in the Langlands Program for Riemann Surfaces

very happy that Edward has you. He entertained the general public on Monday. He entertained the high school students yesterday and today finally it’s our turn to be entertained.

Oh, thank you. Yeah. Thank you very much. It’s my pleasure to I’m here to entertain you.

So, today I am going to talk about one of the major topics of my own research, mathematical research that I have been working on for many years, for decades, and that is the Langland program. Specifically, I want to talk about recent developments in the study of a particular flavor of the Langland program.

Langland’s program has many flavors. The flavor that has to do with Riemann surfaces is one of them. So the Langland program for complex algebraic curves or Riemann surfaces are two equivalent ways to think about it. But first of all, I want to give a very brief overview of what this Langland program is about.

Here I want to say let me see I have my notes here. So that’s right, a good framework to talk about the Langland program is provided by André. It’s actually a very good story. André, the great French mathematician, was put in jail by the French authorities in 1940 for refusing to serve in the army. And while he was in jail, I guess he continued to work on mathematics. But also, he wrote a letter to his sister Simone Weil, who is actually quite famous in her own right, famous philosopher and humanist, and tried to explain to her what interested him in his mathematical research in a way that even a philosopher can understand.

I mean, I kid philosophers. He formulated his approach as a kind of Rosetta Stone of mathematics where, as you may remember, a Rosetta Stone has texts written in three different languages, and just assuming that they express the same thing gave researchers a way to learn about these languages by translating from between those languages. And that’s what André talked about, three different languages.

So what are the three languages in the mathematical Rosetta Stone? One has to do with number theory. Sorry about that. Number theory is on one side, and on the other side, you have Riemann surfaces, if you will. Riemann surfaces are complex algebraic curves and they seem to be far apart, but there is an area in between. To see that, we have to recognize that Riemann surfaces are the same as smooth, compact, projective algebraic surfaces.

So anyway, algebraic curves are curves over the field of complex numbers, but you can consider curves over other fields, and in particular, you can consider curves over a finite field. Here, q is a power of a prime. So then there is a unique finite field with q elements if q is a power of a prime. You can consider algebraic equations over this field, and in particular, the algebraic varieties defined by these equations, specifically algebraic varieties of dimension one, that is to say algebraic curves.

Interestingly enough, there should be some parallels between these two stories because, after all, if you have a curve defined by an equation that makes sense over Z, then you can take it mod p. So you’ll have a curve over Fp, right? You can also complexify it and you get a curve over complex numbers. There is like an overlap between the two.

But also, you can do things like consider cohomology, consider sheaves, and so on, and a lot of these things run in parallel. On the other hand, there are also parallels between these two fields. The reason is basically that what do I mean by number theory? I guess there are people in this room who probably know more about it than I do. I started on this side, and in some sense, my life’s work was in particular trying to move this direction.

There are people who start here and then they go this way, and then we meet in the middle. But we start with a field of rational numbers. When I give public talks about this, I say rational not because they are reasonable, but because they are ratios. So irrational doesn’t mean they are unreasonable. It just means that they cannot be written as ratios. But you already know that.

A field of rational numbers is a field of fractions of a ring of integers. Whole numbers and integers behave very similarly as a ring. It’s a ring, right? Very similar to the ring of polynomials in one variable over a finite field. Both are integral domains, and the structure is very similar.

Of course, this is a field of fractions of Z, but here you can have the field of fractions of this, and this will be rational functions. These are of the form p(t)/q(t) where p and q are polynomials that are relatively prime and q is non-zero. This is exactly what I’ll call rational functions, similar to rational fq of p1.

The simplest algebraic curve is a curve of genus zero. It’s called the projective line. You can consider the projective line over fq or any other field and consider the field of rational functions on it. It’s exactly this which is very similar because this is also ratios of two polynomials, let’s not call them p and q, m/n where m and n are relatively prime integers and n is not equal to zero. You can consider the number of extensions of fields which are finite extensions of Q.

Here, you can also consider extensions, but here extensions correspond to covers. For instance, elliptic curves would be a double cover of P1, and so on. These two stories are also parallel. André said what interests me is to see how these different subjects explore the analogies between them.

Of course, one of the things he did, one of the really spectacular things he did, was trying to borrow the idea of the Riemann hypothesis and from here he borrowed the idea of cohomology, the idea of finding invariants of geometric shapes of algebraic varieties and more general topological spaces. He said there must be some homology theory here so that we could also construct based on that some kind of analog of the zeta function, and then we would have an analog of the Riemann hypothesis. Indeed, this was done, and this is known as the Bae conjectures. They were proved.

One of the first major works was Grothendieck. Then there was Dwork, and the final step, the analog of the Riemann hypothesis, was completed by Pierre Deligne. So that’s the general setup.

Now, I want to argue that there is a fourth column here, or the fourth language, and that’s quantum field theory, which somehow goes hand in hand with this. Because of the connections here, these quantum field theories have to do—they’re not necessarily two-dimensional, but there are some theories which can be related to two-dimensional theories.

The two-dimensional theory is defined on a two-dimensional manifold which will be Riemannian manifolds, which will be given surfaces. So quantum field theory is absolutely necessary to specify that this is P1 over Fq or just abstract P1.

Well, there is no such thing as abstract P1. In some sense, it has to be at least over Z. We could say over Z, P1 over Z. It’s fine, because if, say 1/q, maybe it means that t to the power q is zero or something.

Oh no, no, no. So it simply means that the scalars, the field of scalars, is fq. That’s all we need. But let’s just—since I already put this here and I am already talking about rational functions, I am already saying so in some sense P1.

I mean, strictly speaking when we are now—we are under the spell of Grothendieck, right? Grothendieck explained how to treat general algebraic varieties and schemes. The idea being that instead of our old approach of thinking of them as collections of points, which by the way is like an idea that everything is a collection of particles.

If you think about it, it’s a particle approach in physics. That was a traditional approach: everything is a collection of particles. But then we realized that those particles can behave like waves. And so what are they, particles or waves?

So then you say, okay, actually they’re neither—they’re fields. But what’s a field? And they say, okay, well we know what classical fields are, and what’s a quantum field? And then you say, well, I’m sorry, I have dental appointments.

That’s so free fields we can do okay, but then things become more sophisticated. At least we’re moving away from this particle paradigm, shall we say, the particle point of view, everything is a collection of particles.

In mathematics, I think the mathematician who exemplified this shift from the particle point of view to the field point of view was Alexander Grothendieck, where the idea was that actually, first of all, you don’t want to think in terms of points, but you want to think in terms of functions on the variety. If you know the algebra of functions, you can find points easily because points correspond to an ideal if it’s an affine algebraic variety.

A point corresponds to an ideal in the ring of functions, which are all the functions that vanish at the given point. But the language is much more flexible because you can have a ring of functions over R which doesn’t have any R points, like R[x] divided by the ideal generated by x^2 + 1, so it doesn’t have R points because there are no homomorphisms to R.

To have such a homomorphism is to send x to some number alpha such that alpha squared is negative one, because the ideal has to go to zero. But it has complex points. You can send x to i or negative i, so that’s already an improvement.

But then it was more because he said it’s, in fact, let’s not even think in terms of functions, but let’s talk about sheaves. Let’s talk about the category over the sheaf of functions. And that’s like a field in some way you can think about.

So it’s not localized anymore. There are some sheaves which are localized, like the skyscraper sheaf. It’s very confusing unless you speak in French because in English, “field” means two different things. That’s right. Blackboard has two different fields, on left and right field, because you can have—but is it confusing or is it suggesting something?

Good point. So, but going back to your question, to even talk about this algebraic variety, we should talk, well, in this case, it’s not an affine algebraic variety. We should cover it by two, for example, two open affine pieces to define it. We still have to define a ring of functions on it, and then I would have to define, but that ring will be over some field. Therefore, deep down in the definition of it, there has to be some ground field or ground ring or something.

There’s no such thing as P1 absolute P1 unless you want to think of P1 over Z for the integers.

Okay, so that’s the answer. Anyway, this f, what was that? Finite number fields? Finite extensions. So here you have finite extensions to rational numbers. For example, you join i, and you get this field of Gaussian numbers or the field of fractions of Gaussian integers.

But likewise, here you can do it too. Here we can think about it more geometrically because if you have some covering, maybe with ramification points and so on, it’s a finite extension of the field of rational functions upstairs. It’s a finite extension of the field of functions downstairs.

So that’s all in parallel. That’s what I was talking about. Okay. So the Langland program, I like to think about the Langland program in the context of this Rosetta Stone. Now, it’s more than—now the first language is coming in, but initially it was three.

But the idea of the Langland program was to try to solve various questions in number theory by using analysis, by translating them into the language of analysis. The most famous example of this is the so-called modularity conjecture, also known as the Shimura-Taniyama conjecture, which was proved by Andrew Wiles and Richard Taylor, I think in 1995.

Earlier, my colleague at UC Berkeley, Ken Ribet, showed that it implies Fermat’s theorem. So it’s a famous problem.

What’s happening in that problem? What’s happening is that you have a certain equation over the integers and you want to count the number of solutions when you reduce it mod p for every prime, and that’s a hard problem. But it turns out that these numbers appear as coefficients of a modular form in front of q^p where q is the parameter of modular form, the usual not this q—the q is e^(2πi).

When you have a modular form, you can write it as a power series in q. Think about it as a function in the upper half plane, and it’s a Fourier series expansion. The coefficient in front of q^p, lo and behold, is the number of solutions, more or less.

Maybe you have to make a small change of variables—a small transformation. So that’s the original type of questions that we want to solve. Then came up with this idea that something like this could be done for a very large class of problems in number theory.

But in those days, even in 1967 or 1968, when Langland launched what we now call the Langland program, it was clear that number theorists understood very well that these two stories run in parallel, and one can translate Langland’s conjectures or Langland’s correspondences—whatever you want to call it for number fields—into the language of what is called function fields.

At that time, people didn’t think of functions for complex curves. They thought about function fields for curves over finite fields. It was very easy to translate, and so the conjectures were formulated by Langland in both cases for both cases.

This story, though, is a little bit easier. For one thing, we can think about points. Here you have prime numbers; prime numbers here correspond to completions of Q.

Periodic numbers here and prime numbers are points. Points are unlooks because if you have functions on a given surface, you can consider formal Laurent power series at a given point that are like periodic numbers.

So it’s a completion which is associated with a given point. Points in here are like primes here. But here, there is another completion, which is R, and for general number fields, it could be C as well, or R or C. These are called Archimedean completions.

There are none of them here and they cause trouble here in some sense because they are different from periodic numbers. That’s why, in fact, if you look at the history of the subject, it initially was a bunch of conjectures and the first major result—theorems were proved here, I would say.

Let me give you a very quick sort of sense of what it looks like. So middle column, so function field case Fq. So x is a curve over a finite field. You have a field F, which is the field of rational functions on x, like for example, in the case of P1.

Then you have also its algebraic closure. People say separable closure because it’s a non-separable field. But let’s not worry about this.

Let’s say algebraic closure. Then we have the Galois group. Think about it as a certain gadget. You take your curve and you take its maximal covering, possible covering, and take functions on that. That’s algebraic closure, and take the group of deck transformations of this cover.

You already start seeing, ah, it’s just a fundamental group. Yes, it’s a fundamental group but there could be ramification. So it’s a fundamental group of X; that’s one piece of it.

Then you can consider the fundamental group of X without one point, without two points, and so on. Eventually, you have a fundamental group of X without all points, or kind of inductive limit. That’s what the Galois group really is.

It’s a group of deck transformations of the maximal ramified cover. You think about this geometrically. The idea is to consider n-dimensional representations; this is the holy grail you want to understand what this is.

Likewise, for number fields, it’s hard. The big achievement in the 19th century was abelian class field theory which goes back to the theorem that if you take the maximal abelian quotient of the Galois group, that’s basically just describing it.

It’s the same as describing the biggest, the largest field extension for which the Galois group is abelian. The statement is that it’s what you thought it was, namely the field obtained by joining all roots of unity of all orders. This means that the Galois group is basically Q over Z.

So, but it’s an abelian quotient. An abelian quotient means dimensional representations. Every one-dimensional representation of a group factors through its maximal abelian quotient. Then attempts were made to kind of like extend this to higher dimensional representations.

Yes, first homology. This should be the first homology of what? So this curve—if you—is in the geometric case. Yes, yes, yes, absolutely.

Instead of π1, you take H1 homology. That’s right. That’s right. I thought you were talking about—okay. Yeah, yeah, yeah, no.

In the geometric context, yes. There’s a similar result in the functional field case. Anyway, this suggests that maybe we should study higher dimensional representations. If the answer for one-dimensional representations is so nice because to understand one-dimensional representations is the same as to understand representations which one-dimensional representations factor through the maximal abelian quotient.

So that’s why the question became: what are the n-dimensional representations? Now, with over what field, let’s leave this open for now. N-dimensional representations now, okay?

Since I already said the first thing, you can do is complex n-dimensional representations, but you want to have some continuity. There is this topological group with group topology. If you take complex and take the usual topology, it’s too restrictive.

You only get continuous homomorphisms that are the ones whose image is finite. Right. So instead, the proper is to find a prime number L which is relative to P, from the characteristic of the original field and consider elliptic representations.

There is a topology, so you get the topology and continuous representations. They’re going to have infinite—you can have an infinite image. So that’s a proper way to do it.

But then, of course, there’s kind of—you introduce an extra parameter. The idea is that, in some sense, they don’t depend on this parameter, which is kind of an embarrassing, I would say, somewhat embarrassing aspect of it, which I think people don’t really fully understand what’s going on.

In some sense, at least I don’t. Anyway, so you want to classify these, and the idea is that you can classify them by something that has a totally different flavor.

Maybe I should—you know what? Let me introduce what’s called the ring of adelic numbers. That’s a product of all completions of this field. These completions correspond to what’s called points, or more precisely closed points of X. But you take what’s called a restricted product.

So this Fx is a completion which is isomorphic to formal power series in a certain way. I would like to draw a picture pretending that it’s a complex curve. Soon, it will become clear. Here X is a point, and then here’s a point X, and Tx is a coordinate. You have this little disc—think about it’s a formal disc really—but then take a punctured formal disk, and functions on it are formal power series, although not necessarily in your original Fq because Fq is not algebraically closed.

But some finite extension of Fq in general. Okay, so it’s a product of formal power series, but inside formal power series you have Taylor power series, formal Taylor power series.

That’s called the completion of the ring at this point. Restricted means that you consider collections of elements of F(x) which are in OX for all but finitely many points; that’s what the prime means.

So now inside here you have F itself because each rational function can be expanded in a formal power series in the neighborhood of every point. They’re not finite. Yes, that’s a very good point. So only if you consider Fq points, for example.

Here points are maximal ideals, which are the same as prime ideals because it’s a Dedekind domain. The simplest examples are the ideals generated by the polynomial t - a where a is an element of Q.

These are the finite points that you’re referencing. But in fact, there are many irreducible polynomials of higher degree. Each of them generates—they’re all principal ideals, and each of them generates a maximum ideal.

We should think of it as a point but with values in a quotient of this ring by this maximum ideal, which is going to be a finite extension of Fq. It’s going to have Fq to some power, and that power is called the degree of the point.

In fact, there are irreducible polynomials of every degree, and that’s how you get infinitely many points. It’s a slightly different picture from what one would be used to by thinking about curves over C because C is algebraically closed, so every point—but then C is infinite, so it’s also an infinite number of points.

Okay, so F sits in here, X, which is a product of OX. Here I don’t need to put a prime because it’s automatically restricted. Then I want to take GLn of F divided by GLn of O.

Now, here I am considering what’s called unramified versions. I want to simplify things and restrict myself to homorphisms which are unramified. Let me not focus on defining it. If somebody wants to ask me in the end, it’s a technical issue, but it makes things a little bit nicer.

Then basically, you take this quotient, and it has a very nice measure. This is kind of clear because this is the logical group group, and this group has a hard measure, which is left and right invariant, and this measure descends. to the to the quotient. I will give you an interpretation of or definition of this measure, which is much nicer in some sense, much more geometric, much more explicit. But once you have a measure, actually discrete set, if you will, you can take just L2 functions on it. So you get a hybrid space defined in the usual way for a measure, for a space with a measure.

And then interestingly enough, there are certain operators here which commute with each other. They’re called he operators, and they’re also labeled by points. More precisely, there are actually several of them. There is, like, also I from one to N. And so the statement is that I can function. So for every sigma, call it sigma, there is a unique function. So sigma corresponds to a function f sigma, and the eigenvalues of the he operators on it can be expressed in terms of sigma.

So because you see, we can encode the eigenvalues as elementary symmetric functions of a collection of nonzero numbers. So it’s like eigenvalues of an operator and the coefficients of the characteristic polynomial of an n by n matrix. So we can think about joint values of this commuting operators as what do I call sx as a conjugacy class in jn. So we have the collection of values. I meant no, it’s much simpler.

It’s just to think about, let’s say n equal 2. So hx1 f sigma is equal to alpha 1x f sigma and h2x f sigma is alpha 2x. So you can think about these two numbers as encoding eigenvalues. You can also write alpha 1 as x is equal to s1x * s2x and alpha 2x is s1x + s2x and the other way around. Yes, like this.

That’s right. You just write these as characteristic polynomials. Think of the values of these guys as coefficients of the characteristic polynomial and think of this as the numbers. So that means that you think about the conjugacy class of the element sx1 s x2. So this is an unordered collection of n numbers, and the corresponding characteristic polynomial has coefficients which are going to be this and this.

So this is a technical way to rewrite things. There’s no change. Yes, you got a spectral curve. Yeah, but it’s not really smooth. It’s the s of a finite field. But in some sense, you could say that. Yes. In a way, in a way, it’s a yes. Yes. Yes. Correct.

Are we going to see the definition or leave it at the end? Because I like… Yeah. I have still let me get the main stuff out, and then we will come back, circle back to the more technical questions. Okay. But here you have sigma of Rabinius. So this group, more precisely, it’s unramified quotient through which we are now factoring.

Because I said this unramified representation has the so-called Rabinous conjugate classes very, very fairly easy to describe. So let me just say like this, as if they are here. They’re not actually here. First of all, there are not elements here. There are elements in the quotient of this group. But that’s okay because we are only considering representations which factor through that quotient.

And also they are not elements, but they’re conjugate classes, which is fine because here also we have conjugate classes. Because you can, of course, exchange this, and you still get the same, and so the statement is that under this correspondence, this data match. So here you have sigma of Fabinius. You see, Fabinius is conjugate class here. Its image is a conjugate class in GN.

Sigma corresponds to an function. Its values we will encode as conjugate classes in GN as well. And under its correspondence, this goes to this. So that’s the language correspondence for GN, essentially. I have to be careful. What has to be taken care of is actually kind of there. There’s some care needs to be taken, but roughly speaking, that’s what it is.

Of course, there is a small issue that here they’re going to be JN or, more generally, there will be JN of Q, some finite extension of QL, and here we are considering complex valid functions because you want to do L2. So you also have to fix an identification between the algebraic closure of QL and C, which is kind of unpleasant.

But well, they actually, if you take the algebraic closure, maybe closure and completion, it can be identified with C by using SMA or something like this. So there is this unpleasant element which I kind of wanted to sweep under the rug, but because there’s no rug, I mentioned that yes, the sort of takeaway, if you want, if you’re in the first grade, right, yes, cowwalk group on this side, no cowwalk group on this side.

You have something else. What’s on this side? Hilbert space and commuting Hilbert space h, so Hilbert space x and the he operators. Oops, they are labeled by points of the curve co points n number i from one to n is JN.

Oh, so if you were over here in Hilbert space and you saw these he operators, you might think of the number theory, not solve, find their functions and eigenvalues. That’s what solving means. You know, it’s a classic situation of integrable model, quantum integral model. You have a Hilbert space; you have a bunch of self-adjoint operators, whatever you want to call it, operators acting on it, which commute.

So they have a joint spectrum. You want to find… ideally, you want to find functions as well, but at the very least, you want to describe the spectrum. You want to describe the collection of joint values. Those joint values are organized as conjugacy classes in GN of C labeled by… Right? As I explained, why conjugacy? Because there are n operators. We think of them as elementary symmetric functions of values of an n by n matrix.

Its conjugate class is well-defined by those elementary symmetric functions. The statement is that for every again function, its collection of conjugate classes can be written in this way for one and only one sigma being which is a homomorphism from Galois to Jella. This is an absolutely mind-boggling statement, if you think about it.

Like why it should be connected? And honestly, I don’t think anybody knows why, even though this has been proved. This is one of those rare instances where a big conjecture has been proved first by Drinfeld, n equal 2 in the late 80s, and then for n greater than two by Lafork around the year 2000.

Lafork, don’t be confused with his brother Vincent Love, who also proved something, who proved essentially for general groups this correspondence, but in one direction. So both, by the way, got Fields Medals for this, deservedly, in my opinion, because it’s really groundbreaking work.

And n greater than two presents much more difficulty than n equal 2. So Lafork definitely deserves it as well because he had to introduce a lot of ingenious ideas in his group.

Okay, so now that’s here. But now let us try to imagine what this correspondence would look like in the case when x now is not over Q but FQ. I’m kind of like a wiggly line. So wiggly arrows. So replacing what should we replace the left-hand side with? And what should we replace the right-hand side with?

And on the left-hand side, it’s kind of clear from what I already said because, in fact, this unramified quotient of the Galois group is, in fact, essentially the fundamental group of x. Because remember I said the f bar is the field of functions on the maximal ramified cover.

But if you want, and then you take the group of deck transformations, but if you want unramified quotient, it’s actually maximal unramified covering. And then the group of deck transformations is a fundamental group. More precisely, you have to pick a point, of course, but I want to emphasize one thing which I kind of skipped when I put this round, this curly brackets.

If you will, I mean equivalence classes, so a group of automorphism is not really a problem. So therefore it doesn’t really depend that much on the choice of a point. Okay, so here, so therefore here we have an analog of this. This is a tal fundamental group and so on, but same principle. You obtain it by taking the maximum unramified cover of x, be it over finite field or over the complex field and sort of taking the group of deck transformations of the schema of the cover, which are identical on the base.

And that’s how in topology we define the fundamental group, if you will, or one of the ways to define it. I mean, another way would be to consider the loops, homotopic class of loops, but one of the ways is to do it like I said.

In the context of algebraic geometry over a finite field, that’s how it’s defined actually because we don’t want to make loops in that situation. At least, nobody knows how to… at least I don’t know how to phrase it in terms of loops. Okay, so therefore here we have the ordinary fundamental group.

So how about considering equivalence classes? And here we don’t have any issues; just treat it as just an abstract group. Consider all now, so that’s a topologist’s way to do it. But there is another way to reformulate which has its advantages, which is more of a differential geometry way, by using what’s called the Riemann-Hilbert correspondence.

Equivalence classes of representations of the fundamental group are in one-to-one correspondence with equivalence classes of, that’s right, vector bundles with flat connections. So you can write this as e, nabla, where e is a rank n vector bundle on x, and nabla is a flat connection on e. Then when you look at the horizontal sections, they give you a homomorphism from pi1 to JN once you fix a reference point with a flat connection on e.

But remember, so yeah, remember x is actually homomorphic as a complex manifold. So in fact, this nabla has the holomorphic part and anti-holomorphic part, and we can say the anti-holomorphic part defines a complex structure on holomorphic structure because now we know which functions, which sections are holomorphic. They are annihilated by nabla db bar, so to speak.

So there’s nabla, z bar locally if you choose coordinates z and z bar holomorphic, not holomorphic. So this can be rewritten, and I will say it maybe like this. Maybe e-holomorphic and then nabla holomorphic, where this is a holomorphic rank n complex vector bundle on x, and this is a holomorphic flat connection.

But now the word flat becomes redundant because it’s one-dimensional holomorphically. So there is no curvature equation. So just a holomorphic connection. It’s the same as this and also gives you this, right? So that’s a language which nowadays people usually talk about in this case.

So that’s on one side fairly easy. The quality of the last line, the previous line, is that’s right. No, no, that’s just a polynomial; the last… no, the last is easy because the connection has two parts, holomorphic and anti-holomorphic. I’m not sure I understand.

That’s called Hubert. No, no, I think it’s the fact that you can choose the complex structure and then it becomes the holomorphic connection. No, I don’t choose. If I have this data, it gives me this first data.

It’s topological. No, it’s not topological; it’s a connection system. I mean the actual operator. I don’t choose how much structure to… oh, I see. Riemann-Hilbert is more in a holomorphic context. Okay. Okay, you’re right. You’re right. Okay. Sorry, you’re right.

So let’s say this is Riemann-Hilbert. Fine. I think Riemann would agree with you, but Hilbert… I’m not sure. Anyway. All right. So what to do with the other thing? I mean, here we have some function and here there was an insight, and this insight goes back to Andre, the same Andre V, which by the way there’s one more wrinkle to the story which I have to share, which is that Langland, when he came up with these ideas, decided he was in Princeton and thought the closest person to this subject is Andre.

It’s like, I want to explain it to Andre, and he wrote this long handwritten manuscript. It’s actually in the archive of the Institute for Advanced Study. He wrote this manuscript which started out with, “Now, by now, I popularized it in love and math,” and so now everybody talks, quotes this that he said, “Professor V, I wanted to…” He met Andre in a corridor somewhere and tried to explain, and my guess is it’s like, “Okay, okay, yeah, next time we’ll talk,” you know, so kind of run away.

But send me a letter; send me notes. So he sends him the notes, and the cover page says, “After talking to you, I decided to summarize my conjectures. And these are really conjectures, and I’m not sure that they’re true. So I hope you will give them some consideration, and if not, I am sure – and this is a direct quote – you have a waste basket handy.” He literally wrote that, okay?

And it’s, well, you see it in the archives. I think I have used it as in slides of various lectures. Okay, so Andre V did not throw it into the waste basket, but he said through his secretary, he said, “I would love to read it, but your handwriting, you know.”

How interesting, like if you look back to the history of mathematics or physics or science, how many interesting anecdotes, huh? At least he did not use the margins, so he asked him to type it, and he did type it. And this typed manuscript is also available in some archives. I think it might be even available online.

So Andre V somehow is involved in all of this. But one of the ways in which he is involved, which is very important, is realization of what this double quotient is. Now why did they come up with this in the first place, you may ask? Well, for instance, in this setting for n equal two, if you don’t divide by this, it’s more or less the upper half plane for the field Q.

So this type of questions they were understood to be important, so it was not completely kind of out of the blue. But what Andre V explained is that it has an in the middle column and actually also in the right column, in the geometric columns, when we talk about curves, it has a very nice interpretation.

Some have a plane for the Archimedean. No, no, you just real. Yes, you realize it because basically the rational numbers kill the non-Archimedean points. So what’s left is R. So there’s a slight tweaking that needs to be done, but basically what will be left here is a compact group, the maximal compact because this is a maximal compact.

So you actually… so I think I maybe I spoke when I said we don’t need to divide. You divide; you need to. I think you need to divide, but at the committing point you divide by the maximal compact of GL2(R). What’s left is a negligible GL2(Z). That’s right; that’s right, no.

Here will be Q; here will be a delta right. If you do it over a, you get H, and over SL2(Z)… what’s left from this side, that’s right. So this is GL2(Q), which is like GL2(Z) times the difference between the two. That difference disappears, but what’s left is SL2(Z), and what’s left here is the maximum compact of SL2(R). So that’s how you get that.

That’s right; so you get a modular curve. That’s right. Okay, but here’s the thing, which kind of gets everything running in the complex case. It’s the realization that this double quotient is the set of equivalence classes of rank n bundles on x. This is absolutely stunning, right? That all of a sudden, see, already we brought into our story bundles with flat connections on the left-hand side, on the Galois side. But now on the automorphic side, rank n bundles themselves make an appearance because you have this double quotient.

I don’t want to write again. Just this one is the set of equivalence classes of rank n bundles on x, both over FQ and over C. And actually, well, I don’t want to say general fields, but for me at least in these two cases, let’s just say that.

And there’s… this is very easy to see why because, you see, and I will explain it in the case when x is a complex curve, even a surface. So the statement is that here, of course, we have to be very careful what we mean by a bundle because a bundle by definition is another variety which fibers over your variety where fibers are isomorphic to your group, and the group acts. So it’s a torser; so the group acts from the right fiber-wise.

And the crucial point is that for every point in the base, there is a small neighborhood of this point in whatever topology you have chosen. That’s what is important that the restriction is isomorphic to the trivial bundle. Trivial bundle means just the product of the base, or in this case, a small neighborhood of a point in GN. So the bundle means not only that the group acts fiber-wise; that’s the first property, but the most important property is that locally it’s trivial.

But in what topology? So of course, if you want to do it in a Zariski topology, then it’s obvious that it becomes trivial when you remove finitely many points. But it’s a Zariski topology, so another possibility is an analytic topology; that’s much nicer. So then small neighborhoods are small disks or like small open patches; so then you get more interesting. But it turns out that this is all the same.

So the Zariski is the same as analytical; and in algebraic geometry, we also have a tal topology, which means that it becomes locally constant after a tal-based change. So by doing sort of a finite cover, pulling back to the finite cover, and there is also growth.

You can introduce other topologies like finitely fully faithful, finitely flat, and so on. These other abbreviations turn out that they’re all the same. Now, let’s think about the Zariski topology. So the thing will become trivial after we remove points, but also it’s trivial on the disk around each point.

Therefore, you have a bundle for which you know these are holes in the Riemann surface. Maybe you know it. Let me do a nicer… oh wow. Okay. So but actually, you know, it’s not bad. I’m kind of I can wrap this up quickly as soon as I explain what’s going on.

So it’s trivial on the disk around any point. Okay, it could be an analytic disk; it could be a formal disk. And it’s an easy-to-prove fact in algebraic geometry that it is, so for formal disks, and it’s also trivial outside of these points. So you have a check cover; you have a union of these disks, and you have the complement of those points.

What are the overlaps? What are the overlaps, anybody? Punctured? Exactly, punctured disks, but they’re small, so they’re more like punctured analytic disks, if you like analytic topology, or they’re punctured formal disks on which functions are formal around power series. I will take the second view.

So then you need transition functions, but transition functions are going to be, let’s call this x1, x2, x3. Transition functions are going to be gii from, you know, will be over some set in this case of three elements, which will be an element of the group GL(n) of formal power series of some coordinate. You see, but that’s my f(x).

So you get an element. So together, they live in the product of I from one to I, but that is inside the Adeles because the Adeles ring of Adeles is a product. Sorry, Jillian. A product of the Adeles… an Adele is a collection of elements of this formal power series for every point.

So here we take almost everywhere one identity element, and at our points x1, x2, x3, say we take the transition functions. Do you see that? So transition functions give you an Adele G, an element in Gnels. And what about this? So of course, the point is that it’s not uniquely determined because I can change the realizations.

I can change the realization on each disk. That’s why I have to divide by this. I have to change the relation by outside of the points. That’s why I divide by this. So that’s how you see it. That’s a crucial element.

So then next what do you do? You could say, “Oh, let’s consider some functions here,” and let’s introduce analogs of he operators, and that would be a natural course to take. But that’s not what the subject… not the course that the subject has taken, because people who worked on this initially, at least, and well up to now, anyway, came from algebraic geometry, and for an algebraic geometry, they don’t like analysis, if you know it might be a strong statement.

There’s also a joke: Gelfand at his seminar he repeatedly told this joke about Nymark, who was his co-author who was also a great mathematician. And they both wrote books and made discoveries: Galois and Nymark theorem. So Nymark, when he was in the army, his lieutenant, his commander, was making fun of him because he knew that he worked, he was a mathematician.

He would say, “This is not algebra; here you have to think.” And in Russia, they say algebra, algebra, it’s not algebra; you have to think. And Gant would always say, “That’s what analysis is; it’s not algebra; here you really have to think.” You know, so you see, and I think the people who worked, including myself in this subject, we wanted to do everything algebraically, which here means holomorphically. But then you don’t have holomorphic functions, for instance, if it is GL1, if n equal 1.

What you have is essentially the Picard variety. It’s a compact algebraic variety; there are only constant functions on it. So that’s why people had to become more creative and they said, “Actually, we have to modify the whole thing, and instead of considering functions on it, we’ll consider sheaves; instead of he operators, we will consider Hecke functors.

And instead of decomposing honest-to-goodness Hilbert space, let’s look at the category of sheaves of a certain kind, called D-modules, or more precisely, the derived category of D.” Modules and try to decompose it with respect in terms of shifts of the hecky functors. So that’s how the lang geometric language correspondence appeared. It’s a categorical statement where everything is elevated to the level of categories.

Now I promised to give you a description of the measure of the measure here and the measure is the following that each bundle has a group of automorphisms. So here is kind of like a remark if you have P in and this is called bun, dream introduces very nice notation bun so bun g, but here is bun gn in general. By the way, of course, the story is a generalization to arbitrary reductive algebraic groups and so on, but I am sticking for now to JN.

To simplify, bun, so you have bun, so it’s a rank and bundle, but in the case when x is over fq, then you have the group of automorphisms. So, the measure associated to P and this again is a discrete set in this case. To describe the measure you have to assign a number to each point and of course, you can you already guessed the number is one divided by the order of a group of automorphisms of P, which is well defined; it’s because it’s finite group.

So you see the difficulty. It’s not even clear what measure to take when we are over C because this is going to be some reductive groups over C. You cannot count the number of points at infinity, so that that ship kind of sailed and it became this big project of trying to first formulate the notion of heagen sheaf, construct heagen sheaf, and then eventually prove the equivalence of categories.

So you have a category of shaves now, but then you have a category of sheaf and here you view this as points of a moduli space of another moduli space of flat columoric bundles. So here you consider sheaves which are D-modules; here, sheaves which are coherent sheaves or quasi-coherent sheaves or more precisely they’re called incoherent sheaves. Okay, so don’t ask. So it becomes more sophisticated, and in that formulation, it was proved recently by a team led by Dennis Gaitsgory and Sam Raskin, and Dennis got a breakthrough prize for this work recently, fully deservedly in my opinion. This is really a major breakthrough.

Now, however, what emerged also so maybe I should mention one thing that balance on D-field was fundamental. Since there are people here who are interested in conformal field theory, I want to mention that the work on the construction of balance and dreamfield, which is incorporated in the work of Gaitsgory, Raskin, and others, relies on a certain construction which comes from number one representations of affine casimor algebras, number two conformable blocks but at the weird value of the level called the Kriscoll level.

So, in fact, this is how I got involved in the subject and you can read about this in my book “Love and Math,” how it happened. One of my early works was with Boris Fagan where we were able to describe the center of the developing algebra of affine cuspidal algebra in terms of the dual group. The dual group also appears if we were to replace G_ln here by G here; it will be homomorphism not to G_N but to the dual group. So what we got is the description of the center in terms of the dual group, the so-called opers.

So, conformal field theory kind of is lurking in the background and that’s one of the reasons why I said quantum field theory is a kind of a separate language here. There is another, also interesting way in which it appears through supersymmetric Yang-Mills theory, which was started in the work of Kapustin and Witten, but I also should mention Nikita Nikasov; he wrote a paper with Witten on the subject and also connecting it to his work on instant partition functions and so on.

However, this is now already be interesting and that’s how I would end my talk six years ago. However, there is also, so this is called geometric languageless correspondence, language correspondent or categorical. However, you can also do functions if only you don’t mind to bring together left and right hands. If you don’t insist on doing everything with the right hand, holomorphic and anti-holomorphic.

This is a work by maybe here I should write balance, Gaitsgory, and so on, and this is this new version which we call analytic lang correspondence. It appeared in a series of papers I wrote recently with David Kazhdan, which you can find on archive. There are four papers; if you just search under our names, you’ll find there a series of four papers.

Now my time is up, but I want to explain one thing: where is the measure? The point is you don’t need a measure. You simply take half densities. You take half forms. So, on a manifold, if you want to integrate things on functions on a manifold, you have to pick a measure of integration. But if you consider the line bundle of volume forms and extract its square root and consider sections of that, then there is a— you get a mission inner product for free because if you multiply two half forms, so to speak, half densities, you get sections of the square root of the bundle of top forms, you get a top form, and the top form you can integrate without any choices.

So that’s what you do, and that, in fact, corresponds to the critical level, which was already built in anyway, but you need to take, of course, not holomorphic functions but C-infinity functions, and then take the Hilbert space completion. That somehow defeats the purpose because if I think C-infinity language, it’s the same thing. I mean open subset of your G-space that’s right connections.

No, but we don’t want unitary because here on this side is a complex. Just no, but these are complex. These are not unitary; these are complex. These are holomorphic double quotient, yes, yes, yes, they have that right sort of, but you could take stable. You can think of them if you like.

Of course, there is the soal non-abelian what’s it called. So this is by the way, sure, there is a measure. You can also do it this way; sure, for sure. But I feel it’s much more direct to say half densities because you actually have to choose a line bundle. It becomes meaningful if you care about factorization because then it’s yes, yes, yes, yes, yes, for sure.

Because the point is, what kind of operators? What kind of operators are we considering as a clamor? What Nikita is pointing out is a C-infinity manifold. This has different interpretation or as unit modular space of flat unit connections on say models.

But here we really want to think about it. When I say C-infinity, I don’t mean forgetting about complex structure. I don’t mean that. I’m just saying bring together complex and holomorphic and anti-holomorphic structures. But you need functions on this space, not functions but densities, half densities, and half densities are essential anyway because it is only then that we have computing differential operators, for example, acting on simple forms, too, which you can use simplex form to have a volume form as well.

This is an equivalent language. Yeah, yeah, yeah, all right, you need gold simple automatic. Let’s wrap this up, and then we’ll have a discussion because I feel like I’m going over time. It’s on my, it’s on my. I don’t want it to be on my time.

Okay, I wish I had a few more hours to tell you more about this because there are many interesting applications, for example to quantum interval systems and so on. But one last thing I’ll say that so you have ban G. So in general, you have this bun G; you have a certain line bundle on it, which is a square root of the line bundle of volume forms, the maximum exterior power of the tangent bundle.

So in fact, you know, we can take just open 10 subset, which is stable bundles essentially or regularly stable in general, so we don’t want to have automorphisms. But it’s a technical thing; this RS, which is actually bunji itself is a stack, and one of the difficulties of the categorical geometric respondents is that it’s not an algebraic variety if the group is non-abelian. For GL1, you can think about the picard. So things simplify, and by the way, for those of you who are familiar with the famokai transform, the GL1 case is closely connected to the famokai transform.

In a sense, what we are doing if we’re doing at the level of sheaves, but at the level of functions it’s actually non-abelian fy transform because for GL1, in this case, it’s really just writing harmonics interpreting harmonics, um fiat harmonics on torus in terms of some fundamental representation of the fundamental group.

Okay, so there is this line bundle and then you take the Hilbert space, which is L2 of this L2 completion of the smooth complex supported sections of this function. Because of the simplex function, the same you could also think of them as functions.

But the question, so you may I finish this? Okay, so then what are the operators? So, it turns out that those different global differential operators, which are holomorphic, there are global differential operators which are anti-holomorphic, and these guys, they are called quantum hitchin Hamiltonians.

So, here’s another term. There’s another connection to an important branch of mathematics. In this case, it’s the work of hitchin about completely integrable systems algebra in the algebra geometricalization. These are quantizations of hitchin Hamiltonians whose existence follows from the work of failing and myself that I mentioned. They are used very heavily in the geometric/categorical construction. But until our work, at least I’m not aware of anybody also saying that we can also take the anti-holomorphic conjugates, and they commute with each other because holomorphic and anti-holomorphic differential commute, and both of them will act on this line bundle because this line bundle is essentially the product of the square root of the canonical holomorphic one and anti and this complex conjugate.

So these guys act here, and these guys act on this factor. That’s the first. But in addition, there are also analogs of he operators. So in general, for GLN, there’s an additional label I which corresponds to the fundamental representation and the determinant representation, the last one.

But in general, they will correspond to reducible representations of the langlands dual group. They also exist. So in our first work with Pavl Eting and Dave Kazhdan, we’ll focus just on the differential operators and try to set up the spectral theory. But differential operators are not unbounded, and we need it’s very difficult to find what’s called the essentially self-adjoint extension of these operators.

However, in our second and subsequent works, we also introduce the hick operators. These are integral operators. They’re very similar to the original he operators, and our conjecture is that they’re actually compact, and see I learned something about functional analysis in the process.

So they’re actually compact, and therefore they have very nice spectral theory. The Hilbert space breaks down into finitely many, sorry, infinitely many but into finite dimensional spaces. But because they commute with the differential operators, we can then be sure that the differential operators act on each of those.

Because we know that the spectrum of these operators has to do with opers, we can describe the values in terms of opers, and the answer is that the spectrum of these guys, the joint spectrum of these guys, is in one-to-one correspondence with so-called LG opers with real monodromy on X.

And this is very interesting because, for example, if this group is PGL2R, one example of this is a uniformization. So these are fs and differential operators, and we want fs and differential operators with real monodromy, and uniformization, for example, gives us one of those.

So here I have two experts on this, Leon and Nikit Rasv, and many others I suppose, but they have done actually some original work on this. So those opers appear here as describing the spectrum. But see how nice; here the spectrum was described by homomorphism from the Galois group to GLN, or in general, LG dual group.

If here I consider G. But now this Galois data are replaced by this real opers with real monodromy. And this creates a connection between this theory and things like. So there is this description of these real opers in terms of grafting contours. It goes back to Thurston and to Goldman and others.

So it’s some very beautiful sort of differential geometry that somehow now becomes part of the analytic language correspondence. So thank you very much.

Very good. Any questions?

I was going to ask a question. Yes.

Okay. Yes. Excuse me. Okay. That’s how we do it. All right. Could you explain how the Langlands duality groups and the Langlands dual groups fit into this in a short way?

Sure. Okay. Of course. Yeah.

So the idea is the following. Suppose you have.

Uh, this suppose you have, oh yeah, and I didn’t have time to talk about quantum fields theory but okay maybe just two words but reductive algebraic group like GLN or orthogonal symplectic E8 and so on. Let’s say over algebraically closed field to simplify over C. Then it has a maximum torus which is isomorphic to a power of C star, right? At least a set of C context points is isomorphic to and this comes with two lattices.

You have characters. Well, let’s pretend I know what the duality between groups and the Langlands dual is. How does it fit into this story?

Oh, you already know it. I thought you asked me to explain what it is.

Oh, I want to know how it—oh, but I already explained.

So going back to this scenario when we are over finite field, I talked about GLM. Suppose I want.

So then Galois appears on both sides and Galois is self-dual so it’s kind of deceiving. Let’s do the following: on this side, take the adels, take a general reductive algebraic group and take adels over it and take this double quotient. Then they will still be he operators; they will be acting on the Hilbert space on this side.

So the question is what will replace these objects, and this is where GLN will have to be replaced by the Langlands D group.

Okay. Now in the complex case, you consider bundles, more precisely the locus of varieties of regularly stable bundles, which is a smooth non-compact complex algebraic variety. There is a line bundle, C-infinity line bundle of half forms; half I call them half densities. There is a Hilbert space associated to it, and on it act some unbounded differential operators, holomorphic, non-holomorphic, and some bounded operators, at least conjecturally initially. They define, actually, they’re bounded, but we believe that they are bounded in general.

They’re conjecturally that they are bounded and compact, and they commute with each other, and we want to describe the joint spectrum of these operators, kind of like here we have heck operators we want to describe joint spectrum.

And the idea is that it can be parameterized by such homomorphisms where for general G it will be LG replacing GLN, so that the data of the Fabinius conjugate classes or rather their images under sigma would match the values of the he operators.

And in general, for a general G, it turns out that the values of HE operators can be packaged as conjugate classes in the Langlands dual group. This is called Satake; it comes from what’s called Satake correspondences. There’s a certain combinatorics here which I’m not explaining which replaces this idea of substituting conjugate class for the elementary symmetric functions of the values.

Okay, but here we expect something of Galois type for the dual group.

And here is what it is. It is what’s called LG opers. So these are certain differential operators you could say on the curve which actually correspond to, you can also think of them as certain LG flat bundles.

OPER means flat LG bundle with Borel reduction to LD. I call it Borel Borel subgroup. So like upper triangular matrices for GN satisfying a certain transversality condition.

Okay, so that’s one way to think about it. But at the very—it’s important to point out is that an oper is, first of all, a flat LG bundle. Therefore, it gives rise to homomorphisms from the fundamental group to LG, and here what appears as the data parameterizing the joint spectrum are the opers with a special property that they have real monodromy.

This is a discrete set. Well, conjecture for actually proving that for GLN, for GL2, for PGL2, it’s a discrete set, but we believe that it is a discrete set and actually, interestingly enough, this does not have any continuous spectrum, which is weird because in the here there is actually continuous spectrum. The Einstein series give you continuous spectrum in the complex curve case, but we believe that there is no continuous spectrum.

So the joint spectrum is in one-to-one correspondence with objects of this kind.

Okay. Any more questions or comments?

Is the absence of the spectrum has to do with the fact that the volume of that space is finite, but the volume of that space is very—it has to, I think it has to do with the following: there is such a thing which is called funality in language. Most people know about language correspondence, but there is also funality, and funality has to do with the fact that how these correspondences behave with respect to homomorphisms between different groups.

And the point is that if here you have another group LH, then you have this way to assign to lang to Galois data for H; it maps to Galois data for G. Therefore, there has to be a way to go from automorphic forms for H to automorphisms for G.

And in fact, Einstein series come this way in the case when LH is a torus, a maximal torus because here any homomorphism essentially is allowed. But here only the ones which are operomorphisms under which the first of all the group has to have a SL2 triple has to be non-abelian and there has to be homomorphisms of groups for which the SL2 triple of LH goes to SL2 triple of LG, and there are very few such homomorphisms.

So this is one indication that you should not have Einstein series, and if you don’t have Einstein series, it’s kind of clear that it shouldn’t be any continuous spectrum. There could be other explanations, but that’s sort of one that I consider kind of convincing.

But do we have LLAS operator? Because in number field, you have we have many LLAS operators, original like regular representation, but then so these are this—well, yes, this holomorphic differential operators called quantum Hamiltonians. They come from the center, but for the enveloping of Casimird algebra, a Casimir algebra.

So you could bar because there are like equals to rank like for original lang. Yes, one kmir. That’s right, and that’s quadratic. So for GL2, there will be quadratic differential operators, holomorphic differential operators. There are three G minus three of them. More precisely, the algebra of global holomorphic differential operators acting on the square root of the canonical line bundle is a polynomial algebra and 3 G minus 3 generators.

If the group is SL3, there will also be cubic operators and so on. So they correspond to Casimir but the Casimirs in the final algebra, they’re called higher sugar, single sugar operators.

So I remember when the physicist discovered and pointed out that Langlands duality in groups was a generalization of electromagnetic duality. So does that suggest there should be a physical connection?

Yes, and so that’s what happened. But I mentioned briefly the work of Kapustin and Witten in 2006. I should probably erase this because it’s confusing because in physics alpha is one over 37.

Kapustin and Witten in 2006 gave an explanation of the link between electromagnetic duality, which was theorized in the 70s to exist in NQL 4 maximum supersymmetric pure rank meals, and so this was the work of Montene and Olive, Knights, and Godd, Knights, and Olive in the late 70s.

What they did is the following: they said if you have this equal four—well, they didn’t use the term I think equal four; but essentially considered that a coupling constant, if the group is U1, we know that there’s electromagnetic duality with a theory which has basically the inverse coupling constant.

So they asked what if we take the group to be a general compactly group; will there be some theory which is equivalent? And then they saw an interesting aspect of it, that you have the lattices of electric charges and magnetic charges, and they exactly correspond to electric and respond to characters and magnetic respond to co-characters.

But then in the dual theory, you have to switch electric and magnetic; that’s why it’s called electromagnetic duality. So that means you need to have another group in which the two things correspond to the opposite of the original, and that’s the Langlands dual group.

That’s how it appears in the electromagnetic duality. How they connected it to Langlands is more complicated. They connected it through the so-called sigma models with target space being the so-called hitchin moduli space.

And then basically, there is an equivalence of categories of A-brains and B-brains that they related to the categorical geometric correspondence. But after Kazhdan and I came up with this analytic language correspondence, Witten also wrote a paper with Gaitsgory—so this is 2021, I believe—in which they also can interpret the analytic correspondence in terms of this, sometimes also called soromagnetic duality or S-duality for this, for the super symmetric Young-Mills.

Thank you. Okay, well, I think that with this, we’ll end the lecture. Thank you very much.