Fortunately for us, Ben Mann moved from the NSF to DARPA to be the program manager in charge of our project. From our first meetings with him, it was clear that Ben was uniquely qualified for this job. He has the vision and the courage to take on a high-risk/high-reward project, find the right people to implement it, and help them develop their ideas to the fullest. And his infectious enthusiasm energizes everybody around him. We were really lucky to have Ben at the helm. We would not have been able to achieve a fraction of what we did without his guidance and support.
As a first order of business, I sent an e-mail to Edward Witten telling him about our grant and asking him whether he would be interested in joining us. Given Witten’s unique position in physics and math, we had to have him on board. Alas, Witten’s first reaction was non-committal. He congratulated us on receiving the grant, but also made it clear that he had many projects to work on, and we shouldn’t count on his participation.
But in a stroke of luck, Peter Goddard, one of the physicists who discovered the electromagnetic duality in non-abelian gauge theories, was about to become the director of the Institute for Advanced Study in Princeton. His more recent research was on things related to representation theory of Kac–Moody algebras, and because of this, I had met Peter at various conferences.
I remembered one of these meetings particularly well. It was in August 1991, and we were at a big workshop on math and quantum physics at Kyoto University in Japan. In the middle of the workshop, we received the alarming news of the coup d’état in the Soviet Union. It looked like the authoritarian regime was coming back to power, and the limited freedoms of perestroika would soon be scaled back. This would mean that the borders would be sealed again, and so there was a very real possibility that I wouldn’t be able to see my family for years. My parents called me right away to tell me that if this were to happen, I should not worry about them and, in any event, I should not try to come back to Russia. As we said good-bye, we were preparing for the worst. It wasn’t even clear that we would be able to talk to each other on the phone again in the near future.
Those were tumultuous days. One night, my good friend physicist Fedya Smirnov and I were in the lounge of one of the guest houses, watching Japanese TV and trying to figure out what was happening in Moscow. Everyone else in the building seemed to be sound asleep. Suddenly, around 3 am, Peter Goddard walked into the lounge, a bottle of Glenfiddich whiskey in hand. He asked us about the latest news, and we all had a drink. He then went back to sleep, but insisted that we keep the bottle – a nice gesture of support.
The next day the coup d’état was defeated, to our great relief. A picture of me and Borya Feigin (who was also at this conference) smiling and pumping fists in the air ended up on the front page of Yomiuri, one of the leading Japanese newspapers.
In my e-mail to Peter, I reminded him of this episode and told him about our DARPA grant. I suggested that we organize a meeting at the Institute for Advanced Study to bring together both physicists and mathematicians to talk about the Langlands Program and dualities in physics, to try to find common ground, so that we could solve the riddle together.
Peter’s response was the best we could hope for. He offered his full support in organizing the meeting.
The institute was a perfect venue for such a meeting. Created in 1930 as an independent center of research and thinking, it has been home to Albert Einstein (who spent the last twenty years of his life there), André Weil, John von Neumann, Kurt Gödel, and other prominent scientists. The current faculty is equally impressive: it includes Robert Langlands himself, who has been a professor there since 1972 (now emeritus), and Edward Witten. Two other physicists on the faculty, Nathan Seiberg and Juan Maldacena, work in closely related areas of quantum physics, and several mathematicians, such as Pierre Deligne and Robert MacPherson, conduct research on topics linked to the Langlands Program.
My e-mail exchange with Goddard resulted in plans for an exploratory meeting in early December 2003. Ben Mann, Kari Vilonen, and I were coming to Princeton, and Goddard promised to participate. We invited Witten, Seiberg, and MacPherson; another Princeton mathematician, Mark Goresky, who was co-managing the DARPA project with Kari and myself, was to join us as well (we also invited Langlands, Maldacena, and Deligne, but they were traveling and could not attend).
The meeting was set to start at 11 am at the conference room next to the Institute cafeteria. Ben, Kari, and I arrived early, about fifteen minutes before the meeting. There was no one else there. As I was pacing nervously around the room, I couldn’t stop thinking: “Is Witten coming?” He was the only one of the invitees who had not confirmed his participation.
Five minutes before the meeting, the door opened. It was Witten! That was the moment when I knew that something good would come out of all this.
A few minutes later, the other participants arrived. We all sat around a big table. After the usual greetings and small talk, there was silence; all eyes turned to me.
“Thank you all for coming,” I began. “It has been known for some time that the Langlands Program and electromagnetic duality share something in common. But the exact understanding of what’s going on has eluded us, despite numerous attempts. I think time has come to unravel this mystery. And now we have the necessary resources because we have received a generous grant from DARPA to support research in this area.”
People at the table were nodding their heads. Peter Goddard asked, “How do you propose we go about it?”
Prior to the meeting, Kari, Ben, and I played out different scenarios, so I was well prepared.
“I suggest that we organize a meeting here at the institute. We will invite physicists working in related areas and we will organize lectures by mathematicians to present our current state of knowledge in the Langlands Program. We will then discuss together possible links to quantum physics.”
Now all eyes turned to Witten, the dean of quantum physicists. His reaction was crucial.
Tall and physically imposing, Witten projects great intellectual power, to the point where some feel intimidated by him. When he speaks, his statements are precise and clear to a fault; they seem to be made of unbreakable logic. He never hesitates to take a pause, contemplating his answer. At such times, he often closes his eyes and leans his head forward. That’s what he did at that moment.
All of us were waiting patiently. Less than a minute must have passed by, but to me it felt like eternity. Finally, Witten said, “This sounds like a good idea. What dates do you have in mind for the meeting?”
Ben, Kari, and I couldn’t help but look at each other. Witten was on board, and this was a big victory for us.
After a brief discussion we found the dates that were suitable for everyone: March 8–10, 2004. Then someone asked who would be the participants and the speakers. We mentioned a few names and agreed to finalize the list over e-mail and send the invitations shortly. At this, the meeting adjourned. It took no more than fifteen minutes.
Needless to say, Ben, Kari, and I were very pleased. Witten promised to help organize the meeting (which would of course be a big draw for the invitees) and to actively participate in it as well. We also expected that Langlands would take part, as well as other physicists and mathematicians on the Institute faculty who were interested in the subject. Our first goal was accomplished.
In the course of the next few days we finalized the list of participants, and a week later invitations went out. The letter said:
We are writing to invite you to participate in an informal workshop on the Langlands Program and Physics that will be held at the Institute for Advanced Study from March 8 to March 10, 2004. The goal of this workshop is to introduce physicists to recent developments in the geometric Langlands Program in the interest of exploring potential connections between this subject and Quantum Field Theory. We will plan several introductory lectures by mathematicians, and there will be ample time for informal discussions. This workshop is supported by a grant from DARPA.
Normally, conferences like this have fifty to a hundred participants. What often happens is that speakers give their talks while everyone listens politely. A couple of participants might ask questions at the end of the talk, and a few more may engage the speaker afterward. We envisioned something completely different: a dynamic event that was more a brainstorming session than a typical conference. Therefore we wanted to have a small meeting, about twenty people. We hoped that this format would encourage more interaction and freewheeling conversation between participants.
We already had our first meeting in this format in November 2003, at the University of Chicago. There was a small number of mathematicians invited, including Drinfeld and Beilinson (who had both taken professorships at the University of Chicago a few years earlier). That meeting was a success, and it proved to us that this format was working.
We decided that Kari, Mark Goresky, and I would speak, as well as my former Ph.D. student David Ben-Zvi, who was then a professor at the University of Texas at Austin. We broke down the material into four parts, each to be presented by one of us. In our presentations we had to convey the main ideas of the Langlands Program to the physicists who were not familiar with the subject. This was not an easy task.
Preparing for the conference, I wanted to learn more about the electromagnetic duality. We are all familiar with the electric and magnetic forces. Electric force is what makes electrically charged objects attract or repel each other depending on whether their charges are of the same or opposite signs. For example, an electron has negative electric charge, and a proton has a positive charge (of opposite value). The attractive force between them is what makes the electron spin around the nucleus of the atom. Electric forces create what is called an electric field. We have all seen it in action during a lightning strike, which is caused by the movement of warm wet air through an electric field.
Photo by Shane Lear. NOAA photo library.
Magnetic force has a different origin. It is the force that is created by magnets or by moving electrically charged particles. A magnet has two poles: north and south. When we place two magnets with opposite poles facing each other, they attract, whereas the same poles repel each other. The Earth is a giant magnet, and we take advantage of the magnetic force it exerts when we use a compass. Any magnet creates a magnetic field, as we can see clearly on the picture.
In the 1860s, British physicist James Clerk Maxwell developed an exquisite mathematical theory of electric and magnetic fields. He described them by a system of differential equations that now carry his name. You might expect these equations to be long and complex, but in fact they are quite simple: there are only four of them, and they look surprisingly symmetrical. It turns out that if we consider the theory in the vacuum (that is, without any matter present), and exchange the electric field and magnetic fields, the system of equations will not change.5 In other words, the switching of the two fields is a symmetry of the equations. It is called the electromagnetic duality. This means the relationship between the electric and magnetic fields is symmetrical: each of them affects the other in exactly the same way.
Photo by Dayna Mason.6
Now, Maxwell’s beautiful equations describe classical electromagnetism, in the sense that this theory works well at large distances and low energies. But at small distances and high energies, the behavior of the two fields is described by the quantum theory of electromagnetism. In the quantum theory, these fields are carried by elementary particles, photons, which interact with other particles. This theory goes under the name of quantum field theory.
To avoid confusion, I want to stress that the term “quantum field theory” has two different connotations: in a broad sense, it means the general mathematical language that is used to describe the behavior and interaction of elementary particles; but it may also refer to a particular model of such behavior – for example, quantum electromagnetism is a quantum field theory in this sense. I will mostly use the term in the latter sense.
In any such theory (or model), some particles (like electrons and quarks) are the building blocks of matter, and some (like photons) are the conduits of forces. Each particle has various characteristics: some familiar ones, like mass and electric charge, and some less familiar, like “spin.” A particular quantum field theory is then a recipe to combine them together.
Actually, the word “recipe” points us toward a useful analogy: think of a quantum field theory as a culinary recipe. Then the ingredients of the dish we are cooking are the analogues of particles, and the way we mix them together is like the interaction between the particles.
For example, let’s look at this recipe of the Russian soup borscht, a perennial favorite in my home country. My mom makes the best one (of course!). Here’s what it looks like (the picture was taken by my dad):
Obviously, I have to keep my mom’s recipe secret. But here’s a recipe I found online:
8 cups of broth (beef or vegetable)
1 pound slice of bone-in beef shank
1 large onion
4 large beets, peeled
4 carrots, peeled
1 large russet potato, peeled
2 cups of sliced cabbage
3/4 cup of chopped fresh dill
3 table spoon of red wine vinegar
1 cup sour cream
Salt
Pepper
Think of this as the “particle content” of our quantum field theory. What would the duality mean in this context? It would simply mean exchanging some of the ingredients (“particles”) with others, so that the total content stays the same.
Here is how such a duality could work:
beet → carrot
carrot → beet
onion → potato
potato → onion
salt → pepper
pepper → salt
All other ingredients stay put under the duality; that is,
broth → broth
beef shank → beef shank
and so on.
Since the amounts of the ingredients we exchange are the same, the result will be the same recipe! This is the meaning of duality.
If, on the other hand, we exchanged beets for potatoes, we would get a different recipe: one that would have four potatoes and only one beet. I haven’t tried it, but I am guessing it would taste awful.
It should be clear from this example that a symmetry of a recipe is a rare property, from which we can learn something about the dish. The fact that we can switch beets with carrots without affecting the outcome means that our borscht is well-balanced between them.
Let’s go back to quantum electromagnetism. Saying that there is a duality in this theory means that there is a way to exchange the particles so that we end up with the same theory. Under the electromagnetic duality we want all “things electric” to become “things magnetic,” and vice versa. So, for instance, an electron (an analogue of a beet in our soup) carries an electric charge, so it should be exchanged with a particle that carries a magnetic charge (an analogue of a carrot).
The existence of such a particle contradicts our everyday experience: a magnet always has two poles, and they cannot be separated! If we break a magnet in two pieces, each of them will also have two poles.
Nonetheless, the existence of a magnetically charged elementary particle, called magnetic monopole, has been theorized by physicists; the first was one of the founders of quantum physics, Paul Dirac, in 1931. He showed that if we allow something funny to happen to the magnetic field at the position of the monopole (this is what a mathematician would call a “singularity” of the magnetic field), then it will carry magnetic charge.
Alas, magnetic monopoles have not been discovered experimentally, so we don’t know yet whether they exist in nature. If they don’t exist, then an exact electromagnetic duality does not exist in nature at the quantum level.
The jury is still out on whether this is the case or not. Regardless, we can try to build a quantum field theory that is close enough to nature and exhibits the
electromagnetic duality. Going back to our kitchen analogy, we can try to “cook up” new theories that possess dualities. We can change the ingredients and their quantities in recipes we know, get rid of some of them, throw in something extra, and so on. This kind of “experimental cuisine” may lead to variable results. We may not necessarily want to “eat” these imagined dishes. But edible or not, it may be worthwhile to study their properties in our dreamed-up kitchen – they may give us some clues about the dishes that are edible (that is to say, the models that could describe our universe).
This trial-and-error “model building” is a path along which progress has been made in quantum physics for decades (just as it was in the culinary art). And symmetry is a powerful guiding principle that has been used in creating these models. The more symmetrical a model is, the easier it is to analyze.
At this point, it is important to note that there are two kinds of elementary particles: fermions and bosons. The former are the building blocks of matter (electrons, quarks, etc.), and the latter are the particles that carry forces (such as photons). The elusive Higgs particle, discovered recently at the Large Hadron Collider under Geneva, is also a boson.
There is a fundamental difference between the two types of particles: two fermions cannot be in the same “state” simultaneously, whereas any number of bosons can. Because their behavior is so radically different, for a long time physicists assumed that any symmetry of a quantum field theory had to preserve a distinction between the fermionic and bosonic sectors – that nature forbids them to be mixed together. But in the mid-1970s several physicists suggested what looked like a crazy idea: that a new type of symmetry was possible that would exchange bosons with fermions. It was christened supersymmetry.
Love and Math Page 22
