Love and Math

by Frenkel, Edward

  Later in the introduction, Kapustin and Witten credited our meeting at the Institute for Advanced Study (in particular, the talk given there by my former student David Ben-Zvi) as the starting point for their research.

  In the main body of the paper, Kapustin and Witten developed further the ideas that Witten formulated at our Princeton meeting. In particular, they elucidated the structure of the A-branes and B-branes arising in this picture, the mirror symmetry between them, and the link between the A-branes and the automorphic sheaves.

  To explain their results, let us start with a simpler example of mirror symmetry. In the work by Kapustin and Witten, the mirror symmetry is between two Hitchin moduli spaces and the corresponding sigma models. But now let us replace one of these moduli spaces with a two-dimensional torus.

  Such a torus may be viewed as the product of two circles. Indeed, the mesh on the picture shows clearly that the torus is like a bead necklace:

  The role of the beads is played by the vertical circles in the mesh, and the role of the chainlet of the necklace, on which the beads are strung, is played by a horizontal circle, which we can imagine going through the middle of the torus. A mathematician would say that the necklace is a “fibration,” whose “fibers” are beads and whose “base” is the chainlet. By the same token, the torus is a fibration whose fibers are circles and whose base is also a circle.

  Let us call the radius of the base circle (chainlet) R1 and the radius of the fiber circles (beads) R2. It turns out that the mirror dual manifold will also be a torus. But it will be the product of the circles of radii 1/R1 and R2. This inversion of the radius is similar to the inversion of the electric charge that happens under the electromagnetic duality.

  So now we have two mirror dual tori – one of them, call it T, with the radii R1 and R2, and the other, call it Tv, with the radii 1/R1 and R2. Note that if the base circle in T is big (that is, R1 is large), then the base circle in Tv is small (because then 1/R1 is small), and vice versa. This kind of switch between “big” and “small” is typical of all dualities in quantum physics.

  Let’s study B-branes on T and A-branes on Tv. They are matched under mirror symmetry, and this relation is well-understood (sometimes it is referred to as “T–duality” – T for torus).14

  A typical example of a B-brane on the torus T is a so-called zero-brane, which is concentrated at a point p of T. It turns out that the dual A-brane on Tv, in contrast, will be “smeared” all over the torus Tv. What we mean by “smeared” requires an explanation. Without getting into too much detail, which would take us too far afield, this A-brane on Tv is the torus Tv itself equipped with an additional structure: a representation of its fundamental group in the circle group (similar to those we discussed in Chapter 15). This representation is determined by the position of the original point p in the torus T, so that, in fact, there is a one-to-one correspondence between the zero-branes on T and the A-branes “smeared” on Tv.

  This phenomenon is similar to what happens under the so-called Fourier transform widely used in signal processing. If we apply Fourier transform to a signal concentrated near a particular moment in time, we obtain a signal that looks like a wave. The latter is “smeared” over the line that represents time, as shown on the picture.

  The Fourier transform may also be applied to many other types of signals, and there is an inverse transform, which allows us to recover the original signal. Often, complicated signals are transformed into simple ones, and that’s why Fourier transform is useful in applications. Likewise, under mirror symmetry, complicated branes on one torus correspond to simple ones on the other, and vice versa.

  It turns out that we can use this toric mirror symmetry to describe the mirror symmetry between the branes on the two Hitchin moduli spaces. Here we need to use an important property of these moduli spaces, which was described by Hitchin himself. Namely, the Hitchin moduli space is a fibration. The base of the fibration is a vector space, and the fibers are tori. That is, the whole space is a collection of tori, one for each point of the base. In the simplest case, both the base and the toric fibers are two-dimensional, and the fibration looks like this (note that the fibers at different points of the base may have different sizes):

  Think of the Hitchin fibration as a box of donuts, except that there are donuts attached not only to a grid of points in the base of the carton box, but to all points in the base. So we have infinitely many donuts – Homer Simpson would sure love that!

  It turns out that the mirror dual Hitchin moduli space, the one associated to the Langlands dual group, is also a donut/toric fibration over the same base. (“Donuts. Is there anything they can’t do?”) This means that over each point in this base, we have two toric fibers: one in the Hitchin moduli space on the A-model side, and one in the Hitchin moduli space on the B-model side. Moreover, these two tori are mirror dual to each other, in the sense described above (if one of them has radii R1 and R2, then the other one has radii 1/R1 and R2).

  This observation gives us the opportunity to study mirror symmetry between two dual Hitchin moduli spaces fiber-wise, using the mirror symmetry between the dual toric fibers.

  For example, let p be a point of the Hitchin moduli space M(X,LG). Let’s take the zero-brane concentrated at this point. What will be the mirror dual A-brane on M(X, G)?

  The point p belongs to a torus, which is the fiber of M(X,LG) over a point b in the base (the left torus on the picture below, on the B-model side). Consider the dual torus, which is the fiber of M(X, G) over the same point b (the right torus on the picture, on the A-model side). The dual A-brane on M(X, G) we are looking for will be the A-brane “smeared” over this dual torus. It will be the same dual brane as the one we obtain under the mirror symmetry between these two tori.

  This kind of fiber-wise description of mirror symmetry – using dual toric fibrations – had been suggested earlier by Andrew Strominger, Shing-Tung Yau, and Eric Zaslow in a more general situation. It is now referred to as the SYZ conjecture, or SYZ mechanism.15 This is a powerful idea: while mirror symmetry for dual tori is very well understood, mirror symmetry for general manifolds (such as the Hitchin moduli spaces) still appears quite mysterious. Therefore we can get a lot of mileage by reducing it to the toric case. Of course, to be able to implement it, we need to represent two mirror dual manifolds as dual toric fibrations over the same base (these fibrations also have to satisfy certain conditions). Luckily, we do have such fibrations in the case of the Hitchin moduli spaces, so we can put the SYZ mechanism to work. (In general, the dimensions of the toric fibers are greater than two, but the picture is similar.16)

  Now we use this mirror symmetry to construct the Langlands relation. First, it turns out that points of the Hitchin moduli space M(X, LG) are precisely the representations of the fundamental group of the Riemann surface X in LG (see endnote 1 to this chapter). Let’s take the zero-brane concentrated at this point. According to the SYZ mechanism, the dual A-brane will be “smeared” over the dual torus (the fiber in the dual Hitchin moduli space over the same point in the base).

  Kapustin and Witten not only described these A-branes in detail, they also explained how to convert them into automorphic sheaves of the geometric Langlands relation. Therefore the Langlands relation is achieved via this flow chart:

  An essential element of this construction is the appearance of intermediate objects: A-branes. Kapustin and Witten proposed that the Langlands relation can be constructed in two steps: they first construct an A-brane using mirror symmetry. And then they construct an automorphic sheaf from this A-brane.17 Up to now, we have only discussed the first step, the mirror symmetry. But the second step is also very interesting. In fact, the link between A-branes and automorphic sheaves was a groundbreaking insight of Kapustin and Witten; before their work, it was not known that there was such a link. Moreover, Kapustin and Witten suggested that a similar link exists in a far more general situation. This striking idea has already spurred a lot of mathematical research.
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  All this stuff, as my dad put it, is quite heavy: we’ve got Hitchin moduli spaces, mirror symmetry, A-branes, B-branes, automorphic sheaves... One can get a headache just trying to keep track of all of them. Believe me, even among specialists, very few people know the nuts and bolts of all elements of this construction. But my point is not for you to learn them all. Rather, I want to indicate the logical connections between these objects and show the creative process of scientists studying them: what drives them, how they learn from each other, how the knowledge they acquire is used to advance our understanding of the key questions.

  But to lighten our loads somewhat, here is a diagram that illustrates the analogies between the objects we have discussed across the columns of Weil’s Rosetta stone, plus an extra column corresponding to quantum physics. (I have combined the left and the middle columns of Weil’s Rosetta stone because the objects appearing in them are quite similar to each other.)

  Having looked at this diagram, my father asked me: How did Kapustin and Witten advance the Langlands Program? This is of course an important question. First of all, linking the Langlands Program to mirror symmetry and electromagnetic duality allows us to use the powerful arsenal of these areas of quantum physics to make new advances in the Langlands Program. Conversely, the ideas of the Langlands Program, transplanted into physics, motivated physicists to ask some questions about the electromagnetic duality that they had never asked before. This has already led to some fascinating discoveries. Second, the language of A-branes turns out to be well-adapted for the Langlands Program. Many of these A-branes have a much simpler structure than the automorphic sheaves, which are notoriously complicated. Therefore, using the language of A-branes, we can unveil some of the mysteries of the Langlands Program.

  I want to show you a concrete example of how this new language can be applied. So let me tell you about my subsequent work18 with Witten, which we completed in 2007. To explain what we did, I have to tell you about a problem that up to now I have sort of swept under the rug. In the above discussion, I pretended that all the fibers appearing in the two Hitchin moduli spaces are smooth tori that we are used to (like the ones shown on the above pictures – perfectly shaped donuts, if you will). In fact, this is only true for most of the fibers. But there are some special fibers that look different: they are degenerations of the smooth tori. If there were no degenerations, the SYZ mechanism would give us a complete description of the mirror symmetry between the branes on the two Hitchin moduli spaces. But the presence of the degenerate tori dramatically complicates this mirror symmetry. The most interesting and complicated part of the mirror symmetry is, in fact, what happens with the branes “living” on these degenerate tori.

  Kapustin and Witten only considered in their paper the mirror symmetry restricted to the smooth tori. This left the question of the degenerate tori open. In our paper, Witten and I explained what happens in the case of the simplest degenerate tori, those with the so-called “orbifold singularities,” such as this pinched torus:

  This is in fact the picture of a degenerate fiber arising in the case that our Riemann surface X is itself a torus, and the group LG is SO(3) (it is taken directly from my paper with Witten). The base of the Hitchin fibration is a plane in this case. For every point on this plane, except for three special points, the fibers are the usual smooth tori. So outside these three points the Hitchin fibration is just a family of smooth tori/donuts. But in the neighborhood of each of the three special points, the “neck” of the toric fiber/donut collapses, as shown on the next picture, on which we keep track of the fibers over points within a given path in the base.

  It’s as if Homer Simpson got so excited about having a box with infinitely many donuts that he accidentally stepped on it, crushing some of the donuts (but don’t worry about Homer; there would still be infinitely many perfectly shaped donuts left).

  As we approach the marked point in the base (which is one of the three special points in the base), the neck of the torus in the fiber becomes thinner and thinner, until it collapses at the marked point. The fiber at the marked point is shown under a different angle on the picture above. It is not a torus anymore; it is what we could call a “degenerate” torus.

  The question we need to answer is what happens when the zero-brane on the Hitchin moduli space is concentrated at the special point of the degenerate torus like the one marked on the above picture, at which the neck collapses. Mathematicians call it an orbifold singularity.

  It turns out that this point has an additional symmetry group. In the example shown above, it is the same as the group of symmetries of a butterfly. In other words, it consists of the identity element and another element, corresponding to flipping the wings of the butterfly. This implies that there is not one, but two different zero-branes concentrated at this point. The question then is: what will be the corresponding two A-branes on the mirror dual Hitchin moduli space? (Note that in this case, G will be the group SU(2), which is the Langlands dual group of SO(3).)

  As Witten and I explain in our paper, at each of the three special points in the base of the Hitchin fibration, the degenerate torus on the mirror dual side will look like this (the picture is taken from our paper):

  It appears in the Hitchin fibration in a similar fashion to what was shown on the previous picture, except that now, as we approach one of the special points in the base, the neck of the torus in the fiber becomes thinner and thinner at two places, and the neck collapses at both of them when we reach the special point on the base.

  The corresponding degenerate fiber is quite different from the previous one because now the torus collapses at two points instead of one. Hence this degenerate torus has two pieces. Mathematicians call them components. Now we can answer our question: the two A-branes we are looking for (mirror dual to the two zero-branes concentrated at the singular point of the first degenerate torus) will be the A-branes “smeared” on each of the two components of the dual degenerate torus.

  This is a prototype for what happens in general. When we look at the two Hitchin moduli spaces as fibrations over the same base, there will be degenerate fibers on both sides. But the mechanisms of degeneration will be different: if on the B-model side there is an orbifold singularity with an inner symmetry group (like the butterfly group in the above example), then the fiber on the A-model side will consist of several components, like the two components on the above picture. It turns out that there will be as many of these components as the number of elements in the symmetry group on the B-model side. This ensures that the zero-branes concentrated at the singular points precisely match the A-branes “smeared” on those different components.

  In my paper with Witten, we analyzed this phenomenon in detail. Somewhat surprisingly, this led us to some new insights not only into the geometric Langlands Program for Riemann surfaces, but also into the middle column of Weil’s Rosetta stone, which is about curves over finite fields. This is a good example of how ideas and insights in one area (quantum physics) could propagate back all the way to the roots of the Langlands Program.

  Hereby lies the power of these connections. We now have not three, but four columns in Weil’s Rosetta stone: the fourth one is for quantum physics. When we discover something new in this fourth column, we look at what the analogous results should be in the other three columns, and this may well become a source of new ideas and insights.

  Witten and I started working on this project in April 2007 when I was visiting the Institute in Princeton, and the paper was finished on Halloween, October 31. (I remember the date well, because after posting it online I went to a Halloween party to celebrate.) During these seven months, I came to the Institute three times, each time for about a week. Every day we would work together at Witten’s comfortable office. The rest of the time we were at different locations. I split my time between Berkeley and Paris in those days, and I also spent a couple of weeks visiting a math institute in Rio de Janeiro. But my whereabouts didn’t matter. As long as I had a working
Internet connection, we could collaborate effectively. During the most intense periods, we would exchange a dozen e-mails a day, ponder questions, send each other drafts of the paper, etc. Since we share the same first name, there was a kind of mirror symmetry between our e-mails: each would start with “Dear Edward” and end with “Best, Edward.”

  This collaboration gave me the opportunity to observe Witten up close. I was amazed by both his intellectual power and work ethics. I sensed that he gives a lot of thought to the choice of a problem to work on. I have talked about this earlier in the book: there are problems that may take 350 years to solve, so it is important to estimate the ratio of importance of a given problem to the probability of success within a reasonable period of time. I think Witten has a great intuition for this, as well as great taste. And once he chooses the problem, he is relentless in pursuing it, like Tom Cruise’s character in the film Collateral. His approach is thorough, methodical, with no stone left unturned. Like everyone else, he gets perplexed and confused from time to time. But he always finds his way. Working with him was inspiring and enriching on many levels.