Love and Math

by Frenkel, Edward

  In my thesis, building on the work I did in Moscow with Borya, I was able to construct representations of the Kac–Moody algebra of G parametrized by the opers corresponding to the Langlands dual group LG. The existence of a link between the two was nearly miraculous: the Kac–Moody algebra associated to G somehow “knew” about the Langlands dual group LG, as Drinfeld had predicted. This made his plan work according to the following scheme:14

  My proof of this result was technically quite involved. I was able to explain how the Langlands dual group appeared, but even now, more than twenty years later, I still find mysterious why it appears. I solved the problem, but it was ultimately unsatisfying to feel that something just appeared out of thin air. My research since then has been motivated in part by trying to find a more complete explanation.

  It often happens like this. One proves a theorem, others verify it, new advances in the field are made based on the new result, but the true understanding of its meaning might take years or decades. I know that even if I don’t find the answer, the torch will be passed to new generations of mathematicians who will eventually figure it out. But of course, I would love to get to the heart of it myself.

  Beilinson and Drinfeld subsequently used the theorem from my thesis in their beautiful construction of the geometric Langlands relation (in the right column of Weil’s Rosetta stone). Their spectacular work was the beginning of a new chapter in the Langlands Program, bringing a host of fresh ideas and insights into the subject and expanding it even further.

  I later summarized the research I have done in this area (some of it in collaboration with Borya, and some with Dennis Gaitsgory) in my book Langlands Correspondence for Loop Groups published by Cambridge University Press.15 It came out in 2007, exactly twenty years after I wrote the first formulas for the free-field realization of the Kac–Moody algebras on a night train ride home from Borya’s dacha, a calculation that – little did I know – began my long journey to the Langlands Program.

  As an epigraph to my book, I chose these lines from a 1931 poem by E.E. Cummings, one of my favorite poets:

  Concentric geometries of transparency slightly

  joggled sink through algebras of proud

  inwardlyness to collide spirally with iron arithmethics...

  To me, it sounds like a poetic metaphor for what we are trying to achieve in the Langlands Program: a unity of geometry, algebra, and arithmetic (that is, number theory). A modern-day alchemy.

  The work of Beilinson and Drinfeld solved some long-standing problems, but it also raised more questions. That’s how it is in mathematics: each new result pushes back the veil covering the unknown, but what then becomes known doesn’t simply encompass answers – it includes questions we didn’t know to ask, directions we didn’t know we could explore. And so each discovery inspires us to make new strides and never leaves us satisfied in our pursuit of knowledge.

  In May 1991, I attended the graduation ceremony at Harvard. It was an even more special moment for me because the commencement speaker was Eduard Shevardnadze, one of the architects of the perestroika in the Soviet Union. He had recently resigned his post as foreign minister in protest against the violence in the Baltic republics, warning against an incipient dictatorship.

  Those were turbulent times. We didn’t know about all the turmoil that was yet to come: the coup d’état in August of that year, the subsequent breakup of the Soviet Union, the immense hardship that most people would have to endure in the course of economic reforms. Nor could we anticipate Shevardnadze’s controversial stint as the head of his native Republic of Georgia. But on that glorious day in the sunlit Harvard Yard, I wanted to say “thank you” to the man who helped to free me, and millions of my compatriots, from the communist regime.

  I came up to him after his speech and told him I had just received my Ph.D. from Harvard, which wouldn’t have been possible without perestroika. He smiled and said, in Russian, with his charming Georgian accent, “I am glad to hear this. I wish you great success in your work.” He paused and added, as a true Georgian: “And happiness in your personal life.”

  The next morning I flew to Italy. Victor Kac invited me to a conference he organized in Pisa with his Italian colleague Corrado De Concini. From Pisa, I went to the island of Corsica to attend another meeting, and then to a conference in Kyoto, Japan. These conferences brought together physicists and mathematicians interested in Kac–Moody algebras and their applications to quantum physics. I lectured about the work I had just completed. This was the first time most of the participants heard about the Langlands Program, and they seemed to be intrigued by it. Thinking back to those days, I am amazed how much things have changed since then. The Langlands Program is now considered a cornerstone of modern math and is widely known across diverse disciplines.

  This was the first time I had the opportunity to travel around the world. I was discovering different cultures and also realizing how mathematics, our common language, brings us closer together. Everything was new and exciting, the world – a kaleidoscope of infinite possibilities.

  Chapter 16

  Quantum Duality

  We have seen the Langlands Program reverberate through chambers of mathematics, from number theory to curves over finite fields to Riemann surfaces. Even representations of Kac–Moody algebras have gotten into the mix. Through the lens of the Langlands Program, we observe the same patterns, the same phenomena in these diverse mathematical fields. They manifest themselves in different ways, but some common features (such as the appearance of the Langlands dual group) can always be recognized. They point to a mysterious underlying structure – we might say, the source code – of all mathematics. It is in this sense that we speak of the Langlands Program as a Grand Unified Theory of mathematics.

  We have also seen some of the most common and intuitive notions of mathematics that we study at school: numbers, functions, equations – twisted, warped, sometimes even shattered. Many have proved to be nowhere near as fundamental as they had seemed. In modern math there are concepts and ideas that are deeper and more versatile: vector spaces, symmetry groups, arithmetic modulo prime numbers, sheaves. So mathematics has a lot more than meets the eye, and it is the Langlands Program that lets us begin to see what we could not see before. So far, we have only been able to catch glimpses of the hidden reality. And now, like archaeologists faced with a fractured mosaic, we try to piece together the evidence we were able to collect. Every new piece of the puzzle gives us new insights, new tools to unravel the mystery. And each time, we are dazzled by the seemingly inexhaustible richness of the emerging picture.

  I found my own entry point into this magical world when Drinfeld connected my work on Kac–Moody algebras to the Langlands Program. This vast subject and its omnipresence in mathematics have fascinated me ever since. I became compelled to learn more and more about various tracks of the Program that are discussed in this book, and most of my research ever since has been either on the Langlands Program or was inspired by it in one way or another. This has forced me to travel across mathematical continents, learning different cultures and languages.

  Like any traveler, I was bound to be surprised by what I saw. And now we come to one of the biggest surprises: it turns out that the Langlands Program is also inextricably linked to quantum physics. The key is duality, in physics as in math.

  It might seem strange to look for a duality in physics, but in a sense this is a concept we are all already familiar with. Take electricity and magnetism. Even though these two forces seem to be quite different, they are actually described by a single mathematical theory, called electromagnetism. This theory possesses a hidden duality that exchanges electric and magnetic forces. (We will discuss it in detail below.) In the 1970s, physicists tried to generalize this duality to the so-called non-abelian gauge theories. These are the theories that describe nuclear forces: the “strong” force, which keeps quarks inside protons, neutrons, and other elementary particles; and the “weak” force, responsible for
things like radioactive decay.

  At the core of every gauge theory is a Lie group, which is called the gauge group. Electromagnetism is in a sense the simplest of gauge theories, and the gauge group is in this case our old friend, the circle group (the group of rotations of any round object). This group is abelian; that is, the multiplication of any two elements does not depend on the order in which the it is taken: a · b = b · a. But for the theories of strong and weak interactions, the corresponding gauge groups are non-abelian, that is, a · b ≠ b · a in the gauge group. And so we call them non-abelian gauge theories.

  Now, in the 1970s, physicists found that there was an analogue of electromagnetic duality in the non-abelian gauge theories, but with a surprising twist. It turned out that if we start with the gauge theory whose gauge group is G, then the dual theory will be the gauge theory with another gauge group. And lo and behold, that group turned out to be nothing but the Langlands dual group LG, which is a key ingredient of the Langlands Program!

  Think about it this way: mathematics and physics are like two different planets; say, Earth and Mars. On Earth, we discover a relation between different continents. Under this relation, every person in Europe gets matched with one in North America; their heights, weights, and ages are the same. But they have opposite genders (this is like switching a Lie group and its Langlands dual Lie group). Then one day we receive a visitor from Mars who tells us that on Mars they have also discovered a relation between their continents. Turns out every Martian on one of their continents can be matched with a Martian on another continent, so that their heights, weights, and ages are the same, but... they have opposite genders (who knew Martians had two genders, just like us?). We can’t believe what we are hearing: it appears that the relation we have on Earth and the relation they have on Mars are somehow connected. But why?

  Likewise, because the Langlands dual group appears in both math and physics, it is natural to assume that there must be a connection between the Langlands Program in mathematics and electromagnetic duality in physics. But for almost thirty years, no one could find it.

  I discussed this question on several occasions over the years with Edward Witten. Professor at the Institute for Advanced Study in Princeton, he is considered as one of the greatest living theoretical physicists. One of his amazing qualities is the ability to use the most sophisticated apparatus of quantum physics to make astonishing discoveries and conjectures in pure mathematics. His work has inspired several generations of mathematicians, and he became the first physicist to earn the Fields Medal, one of the most prestigious prizes in mathematics.

  Curious about a possible link between the quantum dualities and the Langlands Program, Witten would ask me about it from time to time. We would discuss it at my office at Harvard when he came to visit Harvard or MIT, or at his office in Princeton when I was there. The discussions were always stimulating, but we never got very far. It was clear that some essential elements were missing, still waiting to be discovered.

  We got help from an unexpected source.

  At a conference in Rome in May 2003,1 I receive an e-mail from my old friend and colleague Kari Vilonen. Originally from Finland, Kari is one of the most gregarious mathematicians I know. When I first came to Harvard, he and his future wife Martina took me to a sports bar in Boston to watch a playoff baseball game in which the Red Sox were playing. Alas, the Sox lost, but what a memorable experience it was. We have been friends ever since, and years later we co-authored several papers on the Langlands Program (together with another mathematician, Dennis Gaitsgory). In particular, we proved together an important case of the Langlands relation.

  In his e-mail, Kari (who was then a professor at Northwestern University) wrote that he had been contacted by people at DARPA who wanted to give us a grant to support research on the Langlands Program.

  DARPA is the acronym for the Defense Advanced Research Projects Agency, the research arm of the U.S. Department of Defense. It was created in 1958 in the wake of the Sputnik launch with the mission to advance science and technology in the United States and to prevent the kind of technological surprise that Sputnik represented.

  I read the following paragraph on the DARPA website:2

  To fulfill its mission, the Agency relies on diverse performers to apply multi-disciplinary approaches to both advance knowledge through basic research and create innovative technologies that address current practical problems through applied research. DARPA’s scientific investigations span the gamut from laboratory efforts to the creation of full-scale technology demonstrations.... As the DoD’s primary innovation engine, DARPA undertakes projects that are finite in duration but that create lasting revolutionary change.

  Over the years, DARPA funded numerous projects in applied math and computer science; for example, it was responsible for the creation of ARPANET, the progenitor of the Internet. But as far as I knew, they had not supported projects in pure math. Why would they want to support research in the Langlands Program?

  This area appeared to be pure and abstract, without immediate applications. But we have to realize that fundamental scientific research forms the basis of all technological progress. Often, what looked like the most abstract and abstruse discoveries in math and physics subsequently led to innovations that we now use in our everyday life. Think of the arithmetic modulo primes, for example. When we see it for the first time, it looks so abstract that it seems impossible something like this could have any real world applications. In fact, English mathematician G.H. Hardy famously argued that “great bulk of higher mathematics is useless.”3 But the joke was on him: many apparently esoteric results in number theory (his field of expertise) are now ubiquitous in, say, online banking. When we make online purchases, arithmetic modulo N springs into action (see the description of the RSA encryption algorithm in endnote 7 to Chapter 14). We should never try to prejudge the potential of a mathematical formula or idea for practical applications.

  History shows that all spectacular technological breakthroughs were preceded, often decades earlier, by advances in pure research. Therefore, if we limit support for basic science, we limit our progress and power.

  There is also another aspect of this: as a society, we are defined to a large extent by our scientific research and innovation. It is an important part of our culture and well-being. Robert Wilson, the first director of the Fermi National Laboratory, in which the largest particle accelerator of its era was created, put it this way in his testimony to the Congressional Joint Committee on Atomic Energy in 1969. Asked whether this multi-million dollar machine could help the country’s security, he said:4

  Only from a long-range point of view, of a developing technology. Otherwise, it has to do with: Are we good painters, good sculptors, great poets? I mean all the things that we really venerate and honor in our country and are patriotic about. In that sense, this new knowledge has all to do with honor and country but it has nothing to do directly with defending our country except to make it worth defending.

  Anthony Tether, who served as the director of DARPA from 2001 to 2009, recognized the importance of fundamental research. He challenged his program managers to find a good project in pure mathematics. One of the managers, Doug Cochran, took this call seriously. He had a friend at the National Science Foundation (NSF) by the name of Ben Mann. A specialist in the field of topology, Ben had left his academic position and come to Washington to serve as a program director at the Division of Mathematical Sciences of the NSF.

  When Doug asked him to suggest a worthy project in pure mathematics, Ben thought of the Langlands Program. Even though this was not his area of expertise, he saw its importance from the grant proposals in this area submitted to the NSF. The quality of the projects and the fact that the same ideas propagated through diverse mathematical disciplines made a great impression on him.

  So Ben suggested to Doug that DARPA support research in the Langlands Program, and that’s why Kari, I, and two other mathematicians were contacted and asked t
o write a proposal that Doug would present to the director of DARPA. The expectation was that if the director approved it, we would receive a multi-million grant to direct research in this area.

  In all honesty, we hesitated at first. This was an uncharted territory: no mathematicians we knew had ever received grants of this magnitude before. Normally, mathematicians receive relatively small individual grants from the NSF (a little travel money, support for a graduate student, and maybe some summer support). Here we would have to coordinate the work of dozens of mathematicians with the goal of making a concerted effort in a vast area of research. Because the grant would be so large, we would be subjected to much greater public scrutiny, and probably some measure of suspicion and jealousy from our colleagues. We recognized that without significant progress coming out of this project, we would be ridiculed, and that such a failure might close the door to funding other worthy projects in pure math by DARPA.

  Despite our trepidation, we wanted to make an impact in the Langlands Program. And the idea of replacing the traditional, conservative scheme of funding mathematical research by a large injection of funds into a promising area sounded appealing and exciting. We simply couldn’t say no.

  The next question was what we should focus on in our project. The Langlands Program, as we have seen, is multi-faceted and relevant to many fields of mathematics. It would be easy to write half a dozen proposals on this general topic. We had to make a choice, and we decided to focus on what we thought was the biggest mystery: the potential link between the Langlands Program and dualities in quantum physics.

  A week later, Doug made a presentation of our proposal to the DARPA director, and by all accounts, it was a success. The director approved a multi-million dollar funding for this project for three years. This was, as far as we could tell, the biggest grant awarded to research in pure mathematics to date. Expectations, obviously, were high. It was a moment of great excitement, but also some anxiety.