Love and Math

by Frenkel, Edward

  As Niels Bohr, one of the creators of quantum mechanics, famously said to Wolfgang Pauli, “We are all agreed that your theory is crazy. The question that divides us is whether it is crazy enough to have a chance of being correct.”

  In the case of supersymmetry, we still don’t know whether it is realized in nature, but the idea has become popular. The reason is that many of the issues that plague conventional quantum field theories are eliminated when supersymmetry is introduced. Supersymmetric theories are generally more elegant and easier to analyze.

  Quantum electromagnetism is not supersymmetric, but it has supersymmetric extensions. We throw in more particles, both bosons and fermions, so that the resulting theory exhibits supersymmetry.

  In particular, physicists have studied the extension of the electromagnetism with the maximal possible amount of supersymmetry. And they showed that in this extended theory the electromagnetic duality is indeed realized.

  To summarize, we don’t know whether a form of quantum electromagnetic duality exists in the real world. But we do know that in an idealized, supersymmetric, extension of the theory, the electromagnetic duality is manifest.

  There is one important aspect of this duality that we haven’t yet discussed. The quantum field theory of electromagnetism has a parameter: the electric charge of the electron. It is negative, so we write it as −e, where e = 1.602 · 10−19 Coulombs. It is very small. The maximal supersymmetric extension of electromagnetism has a similar parameter, which we will also denote by e. If we perform the electromagnetic duality and exchange all things electric by all things magnetic, we will get a theory in which the charge of the electron will be not e, but its inverse, 1/e.

  If e is small, then 1/e is large. So if we start with the theory with a small charge of the electron (as is the case in our world), then the dual theory will have a large charge of the electron.

  This is hugely surprising! In terms of our soup analogy, imagine that e is the soup temperature. Then the duality would mean that switching the ingredients such as carrots and beets would suddenly convert a cold borscht into a hot one.

  This inversion of e is in fact a key aspect of the electromagnetic duality, which has far-reaching consequences. The way quantum field theory is set up, we have a good handle on the theory only for small values of the parameter such as e. We don’t even know a priori that the theory makes sense at large values of the parameter. Electromagnetic duality tells us not only that it makes sense, but that it is in fact equivalent to the theory with small values of the parameter. This means that we have a chance to describe the theory for all values of the parameter. That’s why this kind of duality is considered as a Holy Grail of quantum physics.

  Our next question is whether electromagnetic duality exists for quantum field theories other than the electromagnetism and its supersymmetric extension.

  Apart from the electric and magnetic forces, there are also three other known forces of nature: gravity, which we all know and appreciate, and the two nuclear forces with rather mundane names: strong and weak. The strong nuclear force is keeping quarks inside elementary particles such as protons and neutrons. The weak nuclear force is responsible for various processes transforming atoms and elementary particles, such as the so-called beta-decay of atoms (emission of electrons or neutrinos) and hydrogen fusion, which powers stars.

  These forces seem to be quite distinct. It turns out, however, that the theories of electromagnetic, weak, and strong forces have something in common: they are what we call gauge theories, or Yang–Mills theories, in honor of the physicists Chen Ning Yang and Robert Mills, who wrote a groundbreaking paper about them in 1954. As I mentioned at the beginning of this chapter, gauge theory has a symmetry group, called gauge group. It is a Lie group, a concept we talked about in Chapter 10. The gauge group of the theory of electromagnetism is the group that I introduced at the very beginning of this book, the circle group (also known as SO(2) or U(1)). It is the simplest Lie group, and it is abelian. We already know that many Lie groups are non-abelian, such as the group SO(3) of rotations of a sphere. The idea of Yang and Mills was to construct a generalization of the electromagnetism in which the circle group would be replaced by a non-abelian group. It turned out that gauge theories with non-abelian gauge groups accurately describe the weak and strong nuclear forces.

  The gauge group of the theory of the weak force is the group called SU(2). It is the Langlands dual group to SO(3) and is twice as big (we talked about it in Chapter 15). The gauge group of strong nuclear force is called SU(3).7

  So gauge theories provide a universal formalism describing three out of four fundamental forces of nature (we count electric and magnetic forces as parts of one force of electromagnetism). Moreover, in subsequent years, it was realized that these are not just three separate theories but parts of a whole: there is one theory, commonly referred to as the Standard Model, which includes the three forces as different pieces. It is therefore what we could call a “unified theory” – something Einstein sought unsuccessfully during the last thirty years of his life (though at that time only two forces were known: electromagnetism and gravity).

  We have already talked at length about the importance of the idea of unification in mathematics. For example, the Langlands Program is a unified theory in the sense that it describes a wide range of phenomena in different mathematical disciplines in similar terms. The idea of building a unified theory from as few first principles as possible is especially appealing in physics, and it’s clear why. We would like to reach the most complete understanding of the inner workings of the universe, and we hope that the ultimate theory – if it exists – is simple and elegant.

  Simple and elegant do not mean easy. For example, Maxwell’s equations are deep, and it takes one some effort to understand what they mean. But they are simple in the sense that they are most economical in expressing the truth about the electric and magnetic forces. They are also elegant. So are Einstein’s equations of gravity and the equations of non-abelian gauge theory found by Yang and Mills. A unified theory should combine all of them, like a symphony weaving together the voices of different instruments.

  The Standard Model is a step in this direction, and its experimental confirmation (including the recent discovery of the Higgs boson) was a triumph. However, it is not the ultimate theory of the universe: for one thing, it does not include the force of gravity, which has proved to be the most elusive one. Einstein’s general relativity theory gives us a good understanding of gravity classically, that is, at large distances, but we still don’t have an experimentally testable quantum theory that would describe the force of gravity at very short distances. Even if we focus on the three other forces of nature, the Standard Model still leaves too many questions unanswered and does not account for a big chunk of matter observed by astronomers (called “dark matter”). So the Standard Model is but a partial draft of the ultimate symphony.

  One thing is clear: the final score of the ultimate symphony will be written in the language of mathematics. In fact, after Yang and Mills wrote their celebrated paper introducing non-abelian gauge theories, physicists realized, to their astonishment, that the mathematical formalism needed for these theories was developed by mathematicians decades earlier, without any reference to physics. Yang, who went on to win a Nobel Prize, described his awe in these words:8

  [I]t was not just joy. There was something more, something deeper: After all, what could be more mysterious, what could be more awe-inspiring, than to find that the structure of the physical world is intimately tied to the deep mathematical concepts, concepts which were developed out of considerations rooted only in logic and the beauty of form?

  The same sense of wonder was expressed by Albert Einstein when he asked,9 “How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?” The concepts that Yang and Mills used to describe forces of nature appeared in mathematics earlier because th
ey were natural also within the paradigm of geometry that mathematicians were developing following the inner logic of that subject. This is a great example of what another Nobel Prize–winner, physicist Eugene Wigner, called the “unreasonable effectiveness of mathematics in the natural sciences.”10 Though scientists have been exploiting this “effectiveness” for centuries, its roots are still poorly understood. Mathematical truths seem to exist objectively and independently of both the physical world and the human brain. There is no doubt that the links between the world of mathematical ideas, physical reality, and consciousness are profound and need to be further explored. (We will talk more about this in Chapter 18.)

  We also need new ideas in order to go beyond the Standard Model. One such idea is supersymmetry. Whether it is present in our universe is the subject of a big debate. So far, no traces of it have been discovered. The experiment is the ultimate judge of a theory, so until proved experimentally, supersymmetry remains a theoretical construct, no matter how beautiful and alluring the idea may be. But even if it turns out that supersymmetry is not realized in the real world, it provides a convenient mathematical apparatus that we can use for building new models of quantum physics. These models are not that far away from the models governing the physics of the real world but are often much easier to analyze because of the greater degree of symmetry that they exhibit. We hope that what we learn about these theories will have a bearing on the realistic theories of our universe, regardless of whether supersymmetry exists in it or not.

  Just like the theory of electromagnetism has a maximal supersymmetric extension, so do non-abelian gauge theories. These supersymmetric theories are obtained by throwing in more particles into the mix, both bosons and fermions, to reach the most perfect possible balance between them. It is then natural to ask: do these theories possess an analogue of the electromagnetic duality?

  Physicists Claus Montonen and David Olive tackled this question11 in the late 1970s. Building on an earlier work12 by Peter Goddard (the future director of the Institute for Advanced Study), Jean Nuyts, and David Olive, they came up with a startling conclusion: yes, there is an electromagnetic duality in the supersymmetric non-abelian gauge theories, but these theories are not self-dual in general, the way electromagnetism is. As we discussed above, if we replace all things electric by all things magnetic and vice versa in electromagnetism, we will get back the same theory, with the inverted charge of electron. But it turns out that if we do the same in a general supersymmetric gauge theory with a gauge group G, we will obtain a different theory. It will still be a gauge theory, but with a different gauge group (and also with the inverted parameter, which is the analogue of the charge of the electron).

  And what will the gauge group be in the dual theory? It turns out to be LG, the Langlands dual group of the group G.

  Goddard, Nuyts, and Olive discovered it by performing a detailed analysis of the electric and magnetic charges in the gauge theory with a gauge group G. In electromagnetism, which is the gauge theory with the gauge group being the circle group, the values of both charges are integers. When we exchange them, one set of integers gets exchanged with another set of integers. Hence the theory stays the same. But they showed that, in a general gauge theory, the electric and magnetic charges take values in two different sets. Let’s call them Se and Sm. They can be expressed mathematically in terms of the gauge group G (it is not important at the moment how, exactly).13

  It turns out that under the electromagnetic duality, Se becomes Sm and Sm becomes Se. So the question is whether there is another group G′, for which Se is what was Sm for G, and Sm is what was Se for G (this should also be compatible with some additional data determined by G and G′). It is not obvious whether such a group G′ exists, but they showed that it does and gave a construction. They didn’t know at the time that G′ had already been constructed by Langlands a decade earlier in much the same way, even though Langlands’ motivation was entirely different. This group G′ is nothing but the Langlands dual group LG.

  Why the electromagnetic duality leads to the same Langlands dual group that mathematicians discovered in a totally different context was the big question we were going to tackle at the meeting in Princeton.

  Chapter 17

  Uncovering Hidden Connections

  Just about an hour by train from New York City, Princeton looks like a typical Northeastern suburban town. The Institute for Advanced Study, known in the scientific community simply as “the institute,” is on the outskirts of Princeton, literally in the woods. The area around it is quiet and picturesque: ducks swimming in small ponds, trees reflected in still water. The institute, a cluster of two- and three-story brick buildings with the feel of the 1950s, radiates intellectual power. One can’t help but savor its rich history wandering in the hushed corridors and the main library, which was used by Einstein and other giants.

  This is where we had our meeting in March 2004. Despite the short notice, the response to the invitations we sent in December was overwhelmingly positive. There were about twenty participants – so when I opened the meeting, I asked those present to take turns and introduce themselves. I felt like pinching myself: Witten and Langlands were there, sitting close by, as was Peter Goddard – and several of their colleagues from both School of Mathematics and School of Natural Sciences. David Olive, of the Montonen–Olive and Goddard–Nuyts–Olive papers, was also present. And of course Ben Mann was with us as well.

  Everything went according to plan. We were essentially recounting the story that you have been reading in this book: the origins of the Langlands Program in number theory and harmonic analysis, the passage to curves over finite fields and then to the Riemann surfaces. We also spent quite a bit of time explaining the Beilinson–Drinfeld construction and my work with Feigin on Kac–Moody algebras, as well as its links to the two-dimensional quantum field theories.

  Unlike a typical conference, there was a lot of give and take between the speakers and the audience. It was an intense meeting, with discussions continuing from seminar room to cafeteria and back.

  Throughout, Witten was in high gear. He was sitting in the front row, listening intently and asking questions, constantly engaging the speakers. On the morning of the third day, he said to me, “I’d like to speak in the afternoon; I think I have an idea what’s going on.”

  After lunch, he gave an outline of a possible connection between the two subjects. This was the beginning of a new theory bridging math and physics, which he and his collaborators, and then many others, have been pursuing ever since.

  As we discussed, in the third column of André Weil’s Rosetta stone, the geometric version of the Langlands Program revolves around Riemann surfaces. All of these surfaces are two-dimensional. For example, as we discussed in Chapter 10, the sphere – the simplest Riemann surface – has two coordinates: latitude and longitude. That’s why it is two-dimensional. All other Riemann surfaces are two-dimensional as well because a small neighborhood of each point looks like a piece of a two-dimensional plane, so it may be described by two independent coordinates.

  On the other hand, gauge theories, in which the electromagnetic duality is observed, are defined in the four-dimensional space-time. In order to bridge the two, Witten started by applying a “dimensional reduction” of a four-dimensional gauge theory from four to two dimensions.

  Dimensional reduction is actually a standard tool in physics: we approximate a given physical model by focusing on some degrees of freedom while ignoring others. For example, suppose you are flying on a plane, and a flight attendant, standing in the aisle, gives you a glass of water. Assume for simplicity that the motion of the flight attendant’s hand is perpendicular to the motion of the plane. The velocity of the glass has two components: the first is the velocity of the plane, and the second is the velocity of the flight attendant’s hand, passing you the glass. But the former is much larger than the latter, so if we were to describe the motion of the glass in the air from the point of view of a stat
ic observer on the ground, we could safely ignore the second component of velocity and simply say that the glass is moving with the same velocity as the plane. Therefore, we can reduce a two-dimensional problem involving two components of velocity to a one-dimensional problem involving the component that dominates.

  In our context, the dimensional reduction is realized as follows: we consider a geometric shape (or manifold), which is the product of two Riemann surfaces. Here, “product” means that we consider a new geometric shape whose coordinates are the coordinates of each of these surfaces, taken together.

  As a simpler example, let’s consider the product of two lines. Each line has one coordinate, so the product will have two independent coordinates. Hence it will be a plane: each point on the plane is represented by a pair of coordinates. These are the coordinates of the two lines, taken together.

  Likewise, the product of a line and a circle is a cylinder. It also has two coordinates; one circular and one linear.

  When we take the product, the dimensions add up. In the examples we have just considered, each of the initial two objects is one-dimensional, and their product is two-dimensional. Here is another example: the product of a line and a plane is the three-dimensional space. Its dimension is 3 = 1 + 2.

  Likewise, the dimension of the product of two Riemann surfaces is the sum of their dimensions, that is 2 + 2, which is 4. We can draw a picture of a Riemann surface (we have seen some of them earlier), but we can’t draw a four-dimensional manifold, so we will just have to study it mathematically, using the same methods that we use for the lower-dimensional shapes, which we can imagine more easily. Our ability to do so is a good illustration of the power of mathematical abstraction, as we have already discussed in Chapter 10.

  Now suppose that the size of one of the two Riemann surfaces – call it X – is much smaller than the size of the other one, which we will call ∑. Then the effective degrees of freedom will be concentrated on ∑ and we will be able to approximately describe the four-dimensional theory on the product of the two surfaces by a theory on ∑, which physicists refer to as an “effective theory.” This theory will then be two-dimensional. This approximation will become better and better as we rescale X to make its size smaller and smaller, while preserving its shape (note that this effective theory will still depend on the shape of X). Thus, we pass from the four-dimensional supersymmetric gauge theory on the product of X and ∑ to a two-dimensional theory defined on ∑.