DRINFELD
That’s pretty easy: It is the double cover of SO(3). Have you seen the cup trick?
EDWARD
The cup trick? Oh yes, I remember...
FADE TO:
INT. A HOUSE PARTY OF HARVARD GRADUATE STUDENTS
A dozen or so students, in their early to mid-twenties, are talking, drinking beer and wine. EDWARD is talking to a STUDENT.
STUDENT
Here’s how it works.
The STUDENT takes a plastic cup with wine and puts it on the open palm of her right-hand. She then starts rotating her palm and arm (as shown on the series of photographs below). After she makes a full turn (360 degrees), her arm gets twisted. Keeping the cup upright, she continues rotating, and after another full turn - surprise! - her arm and the cup come back to the initial, untwisted, position.3
ANOTHER STUDENT
I’ve heard Filipinos have a traditional wine dance, in which they do this with both hands.4
He picks up two cups of beer and tries to rotate them both at once, but his hands are unsteady, and he quickly spills beer from both cups. Everybody is laughing.
FADE TO:
INT. BACK TO DRINFELD’S OFFICE
DRINFELD
The trick illustrates the fact that there is a closed path on the group SO(3) which is non-trivial, but if we traverse this path twice, we get a trivial path.5
EDWARD
Oh, I see. The first full turn of the cup twists your arm when you do this, and that’s like a non-trivial path on SO(3).
He picks up a cup of tea from the table and goes through the motion of the first twist.
EDWARD
You would think that making the second turn would twist the arm even more. Instead, the second turn untwists the arm.
EDWARD completes the motion.
DRINFELD
Exactly.6
EDWARD
What does this have to do with the Langlands dual group?
DRINFELD
The Langlands dual group of SO(3) is a double cover of SO(3), so...
EDWARD
So for each element of SO(3), there are two elements of the Langlands dual group.
DRINFELD
Because of this, this new group7 won’t have any non-trivial closed paths.
EDWARD
So passing to the Langlands dual group is a way to get rid of this funny twisting?
DRINFELD
That’s right.8 At first glance, this might seem like a minor difference, but in fact it has major effects, such as the difference in behavior between the building blocks of matter, like electrons and quarks, and the particles that carry interactions between them, like photons. For more general Lie groups, the difference between the group and its Langlands dual group is even more pronounced. In fact, in many cases, there is no apparent link between the two dual groups.
EDWARD
Why does the dual group appear in the Langlands relation? Seems like magic...
DRINFELD
We don’t really know.
FADE TO BLACK
The Langlands duality sets up a pairwise relationship between Lie groups: for every Lie group G, there is a Langlands dual Lie group LG, and the dual of LG is G itself.9 It is surprising enough that the Langlands Program relates two different types of objects (one from number theory and one from harmonic analysis), but the fact that two dual groups, G and LG, appear on the two sides of this relation, as shown on the diagram above, is mind-boggling.
We have talked about the Langlands Program connecting different continents of the world of mathematics. By way of analogy, let’s say those continents were Europe and North America, and we had a way to match every person in Europe with a person in North America, and vice versa. Moreover, suppose that under this relation, various attributes, such as weight, height, and age, matched perfectly, but genders got switched: every man was matched with a woman, and vice versa. Then this would be like the switch between a Lie group and its Langlands dual group under the relation predicted by the Langlands Program.
This switch is in fact the most mysterious aspect of the Langlands Program. We know several mechanisms that describe how the dual group appears, but we still don’t understand why this happens. That ignorance is one of the reasons we try to expand the ideas of the Langlands Program to other fields of mathematics (through Weil’s Rosetta stone) and then to quantum physics, as we will see in the next chapter. We want to find more examples of the appearance of the Langlands dual group and hope that this will give us more clues about why it happens, and what it means.
Let’s focus now on the right column of Weil’s Rosetta stone, which concerns Riemann surfaces. As we established in the previous chapter (see the diagram above), in the version of the Langlands relation that plays out in this column, the cast of characters has “automorphic sheaves” in the role of automorphic functions (or automorphic representations) associated to a Lie group G. It turns out that these automorphic sheaves “live” on a certain space attached to a Riemann surface X and the group G, called the moduli space of G-bundles on X. It’s not important to us at the moment what it is.10 On the other side of the relation, the role of the Galois group is played by the fundamental group of this Riemann surface, as we have seen in Chapter 9. We then find that the geometric Langlands relation (also known as the geometric Langlands correspondence) should schematically look as follows:
This means that to each representation of the fundamental group in LG, we should be able to associate an automorphic sheaf. And Drinfeld had a radically new idea about how to do this.
FADE IN:
INT. DRINFELD’S OFFICE
DRINFELD
So we have to find a systematic way to construct these automorphic sheaves. And I think representations of Kac-Moody algebras can do the trick.
EDWARD
Why is that?
DRINFELD
We are now in the world of Riemann surfaces. Such a surface may have a boundary, which consists of loops.
DRINFELD draws a picture on the blackboard.
DRINFELD
Loops on a Riemann surface give us a link to loop groups and hence to Kac-Moody algebras. Using this link, we can convert representations of a Kac-Moody algebra into sheaves on the moduli space of G-bundles on our Riemann surface. Let’s ignore the details for now. Schematically, I expect it to work like this.
He draws a diagram on the blackboard:
DRINFELD
The second arrow is clear to me. The real question is how to construct the first arrow. Feigin told me about your work on the representations of Kac-Moody algebras. I think you can put it to use here.
EDWARD
But then representation of the Kac-Moody algebra of G should somehow “know” about the Langlands dual group LG.
DRINFELD
That’s right.
EDWARD
How is this possible?
DRINFELD
That is a question for you.
FADE TO BLACK
I guess I felt a little like Neo talking to Morpheus in the film The Matrix. It was exciting and also a little scary. Will I really be able to say something new about this field?
In order to explain how I approached this problem, I need to tell you about an efficient method of constructing representations of the fundamental group of a Riemann surface. We do this by using differential equations.
A differential equation is an equation that relates a function and its derivatives. As an example, let’s look at a car moving on a straight road. The road has one coordinate; let’s denote it by x. The position of the car at the moment t in time is then encoded by a function x(t). For example, it could be that x(t) = t2.
The velocity of the car is the ratio of the distance traveled over a small time period Δt to this time period:
If the car were traveling with a constant velocity, it would not matter which time period Δt we take. But if the car is changing its velocity, then a smaller Δt will give us a more accurate approximation for the ve
locity at the moment t. In order to get the exact, instant value of the velocity at that moment, we have to take the limit of this ratio as Δt goes to 0. This limit is the derivative of x(t). It is denoted by x′(t).
For example, if x(t) = t2, then x′(t) = 2t, and more generally, if x(t) = tn, then x′(t) = ntn−1. It’s not difficult to derive these formulas, but this is not essential to us now.
Many laws of nature may be expressed as differential equations, that is, equations involving functions and their derivatives. For example, Maxwell’s equations describing electromagnetism, which we will talk about in the next chapter, are differential equations, and so are the Einstein equations describing the force of gravity. In fact, the majority of mathematical models (be they in physics, biology, chemistry, or financial markets) involve differential equations. Even the simplest questions one can ask about personal finance, such as how to compute compound interest, quickly lead us to differential equations.
Here is an example of a differential equation:
The function x(t) = t2 is a solution of this equation. Indeed, we have x′(t) = 2t and 2x(t)/t = 2t2/t = 2t, so substituting x(t) = t2 into the left- and right-hand sides we obtain the same expression, 2t. Moreover, it turns out that any solution of this equation has the form x(t) = Ct2, where C is a real number independent from t (C stands for “constant”). For example, x(t) = 5t2 is a solution.
Similarly, the solutions of the differential equation
are given by the formula x(t) = Ctn, where C is an arbitrary real number.
Nothing prevents us from allowing n to be a negative integer here. The equation will still make sense, and the formula x(t) = Ctn will still make sense, except that this function will no longer be defined at t = 0. So let’s exclude t = 0 from consideration. Once we do this, we can also allow n to be an arbitrary rational number, and even an arbitrary real number.
And now we make an additional step: in the original formulation of this differential equation, we treated t as time, so it was assumed to be a real number. But now let’s suppose that t is a complex number, so it has the form , where r and s are real numbers. As we discussed in Chapter 9, complex numbers may be represented as points on the plane with coordinates r and s. Once we make t complex, x(t) effectively becomes a function on the plane. Well, the plane minus one point, that is. Since we decided that x(t) may not be defined at the point t = 0, which is the origin on this plane (with both coordinates, r and s, equal to zero), x(t) is really defined on the plane excluding one point, the origin.
Next, we bring the fundamental group into the game. Elements of the fundamental group, as we discussed in Chapter 9, are closed paths. Let’s consider the fundamental group of the plane with one point removed. Then any closed path has a “winding number”: this is the number of times the path goes around the removed point. If the path goes counterclockwise, we count this number with the positive sign, and if it goes clockwise, with the negative sign.11 The closed paths with the winding numbers +1 and −1 are shown on the picture.
A path that spirals around twice and crosses itself to return to where it began would have a winding number of either +2 or −2, and so on for more complicated paths.
Let’s go back to our differential equation:
where n is an arbitrary real number and t now takes values in complex numbers. This equation has a solution x(t) = tn. However, there is a surprise in store: if n is not an integer, then as we evaluate the solution along a closed path on the plane and come back to the same point, the value of the solution at the end point will not necessarily be the value we started with. It will get multiplied by a complex number. In this situation, we say that the solution has undergone a monodromy along this path.
The statement that something changes when we make a full circle may sound counterintuitive and even self-contradictory at first. But it all depends on what we mean by making a full circle. We may be traversing a closed path and returning to the same point in the sense of a particular attribute, such as our position in space. But other attributes may well undergo changes.
Consider this example. Rick met Ilsa at a dinner party on March 14, 2010, and instantly fell in love. Ilsa didn’t think much of Rick at first but agreed to go on a date with him anyway. And then on another date. And another. Ilsa started to like Rick; why, he is funny and smart, taking good care of her. Next thing you know, Ilsa was also in love; she even changed her Facebook status to “in a relationship,” and so did Rick. Time flew fast, and soon it was March 14 again, the one-year anniversary of the day they first met. From the point of view of the calendar – if we only pay attention to the month and day and ignore the year – Rick and Ilsa had made full circle. But things changed. On the day they met, Rick was in love, and Ilsa wasn’t. But a year later, that is no longer the case; in fact, they could be equally in love with each other, or perhaps, Ilsa, head over heels, and Rick just so-so. It is even possible that Rick would have fallen out of love with Ilsa and started secretly seeing someone else. We don’t know. What is important to us is that even though they had come back to the same calendar date, March 14, their love for each other may well have changed.
Now, my father tells me that this example is confusing because it seems to suggest that Rick and Ilsa came back to the same point in time, which is impossible. But what I am focusing on are particular attributes: specifically, month and day. In that respect, going from March 14, 2010, to March 14, 2011, is really making a circle.
But maybe it’s better to consider instead a closed path in space. So suppose that while they were together, Rick and Ilsa went on a trip around the world. As they were traveling, their relationship was evolving, so when they came back to the same point in space – their hometown – their love for each other may have changed.
In the first case, we have a closed path in time (more precisely, in the month-and-date calendar), and in the second case, a closed path in space. But the conclusions are similar: a relationship may change along a closed path. Both scenarios illustrate a phenomenon that we could call the monodromy of love.
Mathematically, we can represent Rick’s love for Ilsa by a number x, and Ilsa’s love for Rick by a number y. Then the state of their relationship at each moment would be represented by a point on the plane with coordinates (x, y). For example, in the first scenario, on the day they first met, this was the point (1,0). But then, as they were moving along a closed path (in time or in space), the position of the point changed. Hence the evolution of their relationship is represented by a trajectory on the xy-plane. The monodromy is simply the difference between the initial point and the end point of this trajectory.
Here is a less romantic example. Suppose you climb a spiral staircase and make a full turn. As far as the projection of your position on the floor is concerned, you have made a full circle. But another attribute – your altitude – has changed: you have moved on to the next level. That’s also a monodromy. We can tie this with our first example because the calendar is like a spiral: 365 days of the year is like a circle on the floor, and the year is like the altitude. Moving from a given date, such as March 14, 2010, to the same date a year later is therefore similar to climbing a staircase.
Let’s go back to the solution of our differential equation. A closed path on the plane is like the closed path of your projection on the floor. The value of the solution is like the altitude of your position on the staircase. From this point of view, it should not come as a surprise that the value of the solution as we make a full turn would be different from the initial value.
Taking the ratio of these two values, we obtain the monodromy of the solution along this path. It turns out that we can interpret this monodromy as an element of the circle group.12 To illustrate this, imagine that you could bend a candy cane so that it would take the shape of a donut. Then follow the red swirl. Moving along the cane is like following a closed path on our plane, the swirl being our solution. When we make a full circle on the cane, the swirl will in general come back to a different poin
t than where it started. That difference is like the monodromy of our solution. It corresponds to a rotation of the cane by some angle.
The computation presented in endnote 12 shows that the monodromy along a closed path with the winding number +1 is the element of the circle group corresponding to the rotation by 360n degrees. (For example, if n is 1/6, then to this path we assign the rotation by 360/6 = 60 degrees.) Likewise, the monodromy along the path with the winding number w is the rotation by 360wn degrees.
The upshot of this discussion is that the monodromies along different paths on the plane without a point give rise to a representation of its fundamental group in the circle group.13 More generally, we can construct representations of the fundamental group of any Riemann surface (possibly, with some points removed, as in our case) by evaluating the monodromy of differential equations defined on this surface. These equations are going to be more complicated, but locally, within a small neighborhood of a point on the surface, they all look similar to the one above. Using the monodromy of solutions of even more sophisticated equations, we can construct in a similar way representations of the fundamental group of a given Riemann surface in Lie groups other than the circle group. For example, we can construct representations of the fundamental group in the group SO(3).
Let’s return to the problem I was facing: we start with a Lie group G and take the corresponding Kac–Moody algebra. Drinfeld’s conjecture required finding a link between representations of this Kac–Moody algebra and representations of the fundamental group in the Langlands dual group LG.
The first step is to replace representations of the fundamental group by suitable differential equations whose monodromy takes values in LG. This makes the question more algebraic and hence closer to the world of Kac–Moody algebras. The kinds of differential equations that are relevant here were introduced earlier (essentially, in the case of a plane without a point, as above) by Drinfeld and Sokolov during the time when Drinfeld was “exiled” to Ufa. Beilinson and Drinfeld subsequently generalized this work to arbitrary Riemann surfaces and called the resulting differential equations “opers.” The word “oper” is derived from “operator,” but it was also partly a joke, because in Russian it is a slang word for a police officer, like “cop.”
Love and Math Page 20
