From the point of view of the taxonomy of groups in the Kingdom of Mathematics, the family of Lie groups is subdivided into two genera: that of finite-dimensional Lie groups (such as the circle group and the group SO(3)) and that of infinite-dimensional Lie groups. Note that any finite-dimensional Lie group is already infinite, in the sense that it has infinitely many elements. For example, the circle group has infinitely many elements (these are the points of the circle). But it is one-dimensional because all of its elements may be described by one coordinate (the angle). For an infinite-dimensional Lie group, we need infinitely many coordinates in order to describe its elements. This kind of “double infinity” is really hard to imagine. Yet, such groups do arise in nature, so we need to study them as well. I will now describe an example of an infinite-dimensional Lie group known as a loop group.
To explain what it is, let’s first consider loops in the three-dimensional space. Simply put, a loop is a closed curve, such as the one shown on the left-hand side of the picture below. We have already seen them when we talked about braid groups (we called them “knots”).9 I want to stress that non-closed curves, such as the one shown on the right-hand side of the picture, are not considered as loops.
Similarly, we can also consider loops (that is, closed curves) inside any manifold M. The space of all these loops is called the loop space of M.
As we will discuss in more detail in Chapter 17, these loops play a big role in string theory. In conventional quantum physics, the fundamental objects are elementary particles, such as electrons or quarks. They are point-like objects, with no internal structure; that is, zero-dimensional. In string theory it is postulated that fundamental objects of nature are one-dimensional strings.10 A closed string is nothing but a loop embedded in a manifold M (the space-time). That’s why loop spaces are the bread and butter of string theory.
This is a loop
This is not a loop
Now let’s consider the loop space of the Lie group SO(3). Its elements are loops in SO(3). Let’s look at one of these loops closely. First of all, it is similar to the loop pictured above. Indeed, SO(3) is three-dimensional, so on a small scale it looks like the three-dimensional flat space. Second, each point on this loop is an element of SO(3), that is, a rotation of the sphere. Hence our loop is a sophisticated object: it is a one-parameter collection of rotations of the sphere. Given two such loops, we can produce a third by composing the corresponding rotations of the sphere. Thus, the loop space of SO(3) becomes a group. We call it the loop group of SO(3).11 It’s a good example of an infinite-dimensional Lie group: we really cannot describe its elements by using a finite number of coordinates.12
The loop group of any other Lie group (for example, the group SO(n) of rotations of a hypersphere) is also an infinite-dimensional Lie group. These loop groups arise as symmetry groups in string theory.
The second concept relevant to the paper by Feigin and Fuchs that I was studying was the concept of a Lie algebra. Each Lie algebra is in some sense a simplified version of a given Lie group.
The term “Lie algebra” is bound to create some confusion. When we hear the word “algebra,” we think of the stuff we studied in high school, such as solving quadratic equations. However, now the word “algebra” is used in a different connotation: as part of the indivisible term “Lie algebra” referring to mathematical objects with specific properties. Despite what the name suggests, these objects do not form a family in the class of all algebras, the way Lie groups form a family in the class of all groups. But nevermind, we’ll just have to live with this inconsistency of terminology.
To explain what a Lie algebra is, I first have to tell you about the concept of the tangent space. Don’t worry, we are not going off on a tangent; we follow one of the key ideas of calculus called “linearization,” that is, approximation of curved shapes by linear, or flat, ones.
For example, the tangent space to a circle at a given point is the line that passes through this point and is the line closest to the circle among all lines passing through this point. We have already encountered it above when we talked about the dimension of the circle. The tangent line just touches the circle at this particular point, ever so slightly, whereas all other lines passing through this point cross the circle at another point as well, as shown on the picture.
Likewise, any curve (that is, a one-dimensional manifold) can be approximated near a given point by a tangent line. René Descartes, who described an efficient method for computing these tangent lines in his Géométrie, published in 1637, wrote:13 “I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know.” Similarly, a sphere can be approximated at a given point by a tangent plane. Think of a basketball: when we put it on the floor, it touches the floor at one point, and the floor becomes its tangent plane at that point.14 And an n-dimensional manifold may be approximated at a given point by a flat n-dimensional space.
Now, on any Lie group we have a special point, which is the identity element of this group. We take the tangent space to the Lie group at this point – and voilà, that’s the Lie algebra of this Lie group. So each Lie group has its own Lie algebra, which is like a younger sister of the Lie group.15
For example, the circle group is a Lie group, and the identity element of this group is a particular point on this circle16 corresponding to the angle 0. The tangent line at this point is therefore the Lie algebra of the circle group. Alas, we cannot draw a picture of the group SO(3) and its tangent space because they are both three-dimensional. But the mathematical theory describing tangent spaces is set up in such a way that it works equally well in all dimensions. If we want to imagine how things work, we can model them on one- or two-dimensional examples (like a circle or a sphere). In doing so, we use lower-dimensional manifolds as metaphors for more complicated, higher-dimensional manifolds. But we don’t have to do this; the language of mathematics enables us to transcend our limited visual intuition. Mathematically, the Lie algebra of an n-dimensional Lie group is an n-dimensional flat space, also known as a vector space.17
There is more. The operation of multiplication on a Lie group gives rise to an operation on its Lie algebra: given any two elements of the Lie algebra, we can construct a third. The properties of this operation are more difficult to describe than the properties of multiplication in a Lie group, and they are not essential to us at the moment.18 An example, which would be familiar to those readers who have studied vector calculus, is the operation of cross-product in the three-dimensional space.19 If you have seen this operation, you may have wondered about its weird-looking properties. And guess what, this operation actually makes the three-dimensional space into a Lie algebra!
It turns out that this is in fact the Lie algebra of the Lie group SO(3). So the esoteric-looking operation of cross-product is inherited from the rule of composition of rotations of the sphere.
You may be wondering why we care about Lie algebras if the operation on them is so weird-looking. Why not stick with Lie groups? The main reason is that, unlike a Lie group, which is usually curved (like a circle), a Lie algebra is a flat space (like a line, a plane, and so on). This makes the study of Lie algebras much simpler than the study of Lie groups.
For example, we can talk about the Lie algebras of loop groups.20 These Lie algebras, which we should think of as simplified versions of the loop groups, are called Kac–Moody algebras, after two mathematicians: Victor Kac (Russian-born, emigrated to the U.S., now Professor at MIT) and Robert Moody (British-born, emigrated to Canada, now Professor at University of Alberta). They started investigating these Lie algebras independently in 1968. Since then, the theory of Kac–Moody algebras has been one of the hottest and fastest growing areas of mathematics.21
It was these Kac–Moody algebras that Fuchs had suggested as the topic of my next research project. When I started learning all this, I saw that I had to do a lot of studying before I could get to the point where I coul
d do something on my own. But I was fascinated with the subject.
Fuchs lived in the northeastern part of Moscow, not far from a train station where I could catch a train to my hometown. I used to go home every Friday for the weekend, so Fuchs suggested that I come to his place every Friday at 5 pm and then take the train home after our meeting. I would usually work with him for about three hours (during which he would also feed me dinner), and then I would catch the last train, arriving home around midnight. Those meetings played a big role in my mathematical education. We had them week after week, the entire fall semester of 1986, and then the spring semester of 1987 as well.
It wasn’t until January 1987 that I finished reading the long paper by Feigin and Fuchs and felt that I could start working on my research project. By that time, I was able to get a pass to the Moscow Science Library, a huge repository of books and journals, not only in Russian (many of which Kerosinka’s library also had), but in other languages as well. I started going there regularly to pore over dozens of math journals, looking for articles about Kac–Moody algebras and related subjects.
I was also eager to learn about their applications to quantum physics, which was of course a huge draw for me. As I mentioned above, Kac–Moody algebras play an important role in string theory, but they also appear as symmetries of models of two-dimensional quantum physics. We live in a three-dimensional space, so realistic models describing our world should be three-dimensional. If we include time, we get four dimensions. But mathematically, nothing precludes us from building and analyzing models describing worlds of other dimensions. The models in dimensions less than three are simpler, and we have a better chance of solving them. We can then use what we learn to tackle the more sophisticated three- and four-dimensional models.
This is in fact one of the main ideas of the subject called “mathematical physics” – study models of different dimensions that may not be directly applicable to our physical world, but share some of the salient features of the realistic models.
Some of these low-dimensional models also have real-world applications. For example, a very thin metal layer may be viewed as a two-dimensional system and hence may be effectively described by a two-dimensional model. A famous example is the so-called Ising model of interacting particles occupying the nodes of a two-dimensional lattice. The exact solution of the Ising model by Lars Onsager provided valuable insights into the phenomenon of spontaneous magnetization, or ferromagnetism. At the core of Onsager’s calculation was a hidden symmetry of this model, underscoring once again symmetry’s paramount role in understanding physical systems. It was subsequently understood that this symmetry is described by the so-called Virasoro algebra, a close cousin of Kac–Moody algebras.22 (In fact, it was the Virasoro algebra that was the main subject of the paper by Feigin and Fuchs which I was studying.) There is also a large class of models of this type in which symmetries are described by the Kac–Moody algebras proper. The mathematical theory of Kac–Moody algebras is essential for understanding these models.23
Kerosinka’s library subscribed to a publication called Referativny Zhurnal, the Journal of References. This journal, published monthly, had short reviews of all new articles, in all languages, organized by subject, with a short summary of each. I started reading it regularly, and what a valuable source it turned out to be! Every month a new volume about math papers would come, and I would fish through the relevant sections trying to find something of interest. If I found something that sounded exciting, I would write down the reference and get it on my next visit to the Moscow Science Library. This way, I discovered a lot of interesting stuff.
One day, while turning the pages of the Referativny Zhurnal, I stumbled upon a review of a paper by a Japanese mathematician Minoru Wakimoto, which was published in one of the journals I was paying close attention to, Communications in Mathematical Physics. The review did not say much, but the title referred to the Kac–Moody algebra associated to the group of rotations of the sphere, SO(3), so I took down the reference and on my next visit to the Science Library I read the article.
In it, the author constructed novel realizations of the Kac–Moody algebra associated to SO(3). To give the gist of what they are, I will use the language of quantum physics (which is relevant here because Kac–Moody algebras describe symmetries of models of quantum physics). Realistic quantum models, like those describing the interaction of elementary particles, are quite complicated. But we can construct much simpler, idealized, “free-field models,” in which there is no or almost no interaction. The quantum fields in these models are “free” from each other, hence the name.24 It is often possible to realize a complicated, and hence more interesting, quantum model inside one of these free-field models. This allows us to dissect and deconstruct the complicated models, and perform computations that are not accessible otherwise. Such realizations are very useful as the result. However, for quantum models with Kac–Moody algebras as symmetries, the known examples of such free-field realizations had been rather narrow in scope.
As I was reading Wakimoto’s paper, I saw right away that the result could be interpreted as giving the broadest possible free-field realization in the case of the simplest Kac–Moody algebra, the one associated to SO(3). I understood the importance of this result, and it made me wonder: where did this realization come from? Is there a way to generalize it to other Kac–Moody algebras? I felt that I was ready to tackle these questions.
How to describe the excitement I felt when I saw this beautiful work and realized its potential? I guess it’s like when, after a long journey, suddenly a mountain peak comes in full view. You catch your breath, take in its majestic beauty, and all you can say is “Wow!” It’s the moment of revelation. You have not yet reached the summit, you don’t even know yet what obstacles lie ahead, but its allure is irresistible, and you already imagine yourself at the top. It’s yours to conquer now. But do you have the strength and stamina to do it?
Chapter 11
Conquering the Summit
By summer, I was prepared to share my findings with Fuchs. I knew he would be as excited about Wakimoto’s paper as I was. I went to see Fuchs at his dacha, but when I arrived, he told me that there was a slight problem: he had made appointments with me and with his collaborator and former student Boris Feigin on the same day – inadvertently, he said, although I didn’t believe him (much later Fuchs confirmed that this was indeed intentional).
Fuchs had introduced me to Feigin a few months earlier. It was before one of Gelfand’s seminars, soon after I finished my paper on braid groups and was starting to read the article by Feigin and Fuchs. Prompted by Fuchs, I asked Feigin for suggestions as to what else I should be reading. Boris Lvovich, as I addressed him, was then thirty-three years old but already considered one of the biggest stars in the Moscow mathematical community. Wearing a pair of jeans and well-worn sneakers, he appeared to be quite shy. Large thick glasses covered his eyes, and for the most part during our conversation he looked down to avoid making eye contact. Needless to say, I was also shy and not too sure of myself: I was just a beginning student, and he was already a famous mathematician. So this wasn’t the most engaging of encounters. But every once in a while he would raise his eyes and look at me with a big disarming smile, and this broke the ice. I could sense his genuine kindness.
However, Feigin’s initial suggestion startled me: he told me that I should read the book Statistical Mechanics by Landau and Lifshitz, a prospect that I found absolutely dreadful at the time, partly because of the resemblance, in size and weight, between that thick volume and the textbook on the history of the Communist Party that we all had to study in school.
In Feigin’s defense, this was a solid advice – this book is indeed important, and eventually my research turned precisely in that direction (even though I have to admit, to my shame, that I still haven’t read the book). But at the time this idea did not resonate with me at all, and perhaps it was partly for this reason that our initial conversation did
n’t go anywhere. In fact, I never spoke to Feigin again, besides saying “Hello” when I saw him at Gelfand’s seminar, until that day at Fuchs’ dacha.
Soon after my arrival, I saw Feigin through the window dismounting his bike. After greetings and some small talk, we all sat down at a round table in the kitchen and Fuchs asked me, “So, what’s new?”
“Well... I have found this interesting paper by a Japanese mathematician, Wakimoto.”
“Hmmm...” Fuchs turned to Feigin: “Do you know about this?”
Feigin shook his head no, and Fuchs said to me, “He always knows everything... But it’s good that he hasn’t seen this paper – then it will be interesting for him to hear you as well.”
I set out to describe Wakimoto’s work to both of them. As expected, they were both very interested. This was the first time that I had a chance to discuss mathematical concepts in-depth with Feigin, and instantly I had a feeling that we clicked. He was listening attentively and asking all the right questions. He clearly understood the importance of this stuff, and even though his demeanor remained relaxed and casual, he seemed to be excited about it. Fuchs was mostly looking on, and I am sure he was happy that his secret plan to get Feigin and me more closely acquainted worked so well. It really was an amazing conversation. I felt like I was on the verge of something important.
Fuchs seemed to feel the same way. As I was leaving, he told me, “Well done. I wish this were your paper. But I think you are now ready to take it to the next level.”
I went back home and continued to study the questions raised by Wakimoto’s paper. Wakimoto didn’t give any explanations for his formulas. I was doing what amounted to forensic work – trying to find the traces of a big picture behind his formulas.
Love and Math Page 14