Before we discuss the nature of this theory in any detail, let’s talk about what we mean by a quantum field theory in general. For example, in electromagnetism, we study electric and magnetic fields in three-dimensional space. Each of them is what mathematicians call a vector field. A useful analogy is the vector field describing a wind pattern: at each point in space, the wind blows in a particular direction and has a particular strength – and this is captured by a pointed line segment attached to this point, which mathematicians call a vector. The collection of these vectors attached to all points in space is a vector field. We have all seen the wind represented as a vector field on weather maps.
Likewise, a given magnetic field has a particular direction and strength at each point in space, as can be seen from the picture above. Hence, it is also a vector field. In other words, we have a rule that assigns a vector to each point of our three-dimensional space. Not surprisingly, mathematicians call such a rule “a map” from our three-dimensional space to the three-dimensional vector space. And if we follow how a given magnetic field changes in time, we obtain a map from the four-dimensional space-time to the three-dimensional vector space. (This is like watching how the weather map is changing over time on TV.) Similarly, any given electric field, changing in time, may also be described as a map from the four-dimensional space-time to the three-dimensional vector space. Electromagnetism is a mathematical theory describing these two maps.
In the classical theory of electromagnetism, the only maps that we are interested in are the maps corresponding to solutions of the Maxwell equations. In contrast, in quantum theory we study all maps. Any calculation in quantum field theory, in fact, involves the summation over all possible maps, but each map is weighted, that is, multiplied by a prescribed factor. These factors are defined in such a way that the maps corresponding to solutions of the Maxwell equations make the dominant contribution, but other maps also contribute.
Maps from space-time to various vector spaces appear in many other quantum field theories (for example, in non-abelian gauge theories). However, not all quantum field theories rely on vectors. There is a class of quantum field theories, called the sigma models, in which we consider maps from the space-time to a curved geometric shape, or manifold. This manifold is called the target manifold. For example, it could be a sphere. Though sigma models were first studied in the case of four-dimensional space-time, such a model also makes sense if we take space-time to be a manifold of any dimension. Thus, there is a sigma model for any choice of the target manifold and any choice of the space-time manifold. For example, we can choose a two-dimensional Riemann surface as our space-time, and the Lie group SO(3) as the target manifold. Then the corresponding sigma model will describe maps from this Riemann surface to SO(3).
The picture below illustrates such a map: on the left-hand side we have a Riemann surface, on the right-hand side we have the target manifold, and the arrow represents a map between them; that is, a rule that assigns a point in the target manifold to each point in the Riemann surface.
In the classical sigma model, we consider maps from the space-time to the target manifold that solve the equations of motion (the analogues of the Maxwell equations of electromagnetism); such maps are called harmonic. In the quantum sigma model, all quantities of interest, such as the so-called correlation functions, are computed by summing up over all possible maps, with each map weighted, that is, multiplied by a prescribed factor.
Let’s return to our question: which two-dimensional quantum field theory describes the dimensional reduction of a four-dimensional supersymmetric gauge theory with the gauge group G on ∑ × X as we rescale X so that its size becomes very small? It turns out that this theory is a supersymmetric extension of the sigma model of maps from ∑ to a specific target manifold M, which is determined by the Riemann surface X and the gauge group G of the original gauge theory. Our notation for it should reflect this, so we will denote it by M(X, G).1
As had earlier turned out to be the case with group theory (see Chapter 2), when physicists stumbled upon these manifolds, they discovered that mathematicians had been there before them. In fact, these manifolds had a name: Hitchin moduli spaces, after British mathematician Nigel Hitchin, professor at Oxford University, who had introduced and studied these spaces in the mid-1980s. Although it is clear why a physicist would be interested in these spaces – they appear when we do the dimensional reduction of a four-dimensional gauge theory – the reasons for a mathematician’s interest in these spaces seem less obvious.
Luckily, Nigel Hitchin has given a detailed account2 of the history of his discovery, and it’s actually a great example of the subtle interplay between math and physics. In the late 1970s, Hitchin, Drinfeld, and two other mathematicians, Michael Atiyah and Yuri Manin, studied the so-called instanton equations, which physicists had come up with while studying gauge theories. These instanton equations were written in a flat four-dimensional space-time. Hitchin subsequently studied differential equations in a flat three-dimensional space, called the monopole equations, obtained by dimensional reduction of the instanton equations from four to three dimensions. Those were interesting from a physical point of view, and they also turned out to have an intriguing mathematical structure.
It was then natural to look at the differential equations obtained by reducing the instanton equations from four to two dimensions. Alas, physicists had observed that these equations did not have any non-trivial solutions in the flat two-dimensional space (that is, on the plane), so they did not pursue these equations further. Hitchin’s insight, however, was that these equations could be written on any curved Riemann surface, such as the surface of a donut or a pretzel, as well. Physicists missed this, because at the time (in the early 1980s) they were not particularly interested in quantum field theories on such curved surfaces. But Hitchin saw that, mathematically, solutions on these surfaces were quite rich. He introduced his moduli space M(X, G) as the space of solutions of these equations on a Riemann surface X (in the case of a gauge group G).* He found that it was a remarkable manifold; in particular, it possessed a “hyper-Kähler metric,” of which very few examples were known at the time. Other mathematicians followed in his footsteps.
About ten years later, physicists began to appreciate the importance of these manifolds in quantum physics, although that interest did not really catch on before the work of Witten and his collaborators, which I am presently describing. (It is also interesting to note that the Hitchin moduli spaces, which originally appeared in the right column of André Weil’s Rosetta stone, recently found applications to the Langlands Program in the middle column, in which the role of Riemann surfaces is played by curves over finite fields.3)
The interaction between math and physics is a two-way process, with each of the two subjects drawing from and inspiring the other. At different times, one of them may take the lead in developing a particular idea, only to yield to the other subject as focus shifts. But altogether, the two interact in a virtuous circle of mutual influence.
Now, armed as we are with the insights of both mathematicians and physicists, let’s apply the electromagnetic duality to the four-dimensional gauge theory with the gauge group G. We will then obtain the gauge theory with the gauge group LG, the Langlands dual group of G. (Recall that if we apply this duality twice, we get back the original group G. In other words, the Langlands dual group of LG is the group G itself.) The effective two-dimensional sigma models on ∑, associated to G and LG, will then also be equivalent, or dual, to each other. For the sigma models this kind of duality is called mirror symmetry. In one of the sigma models we consider maps from ∑ to the Hitchin moduli space M(X,G) corresponding to G; in the other we consider maps from ∑ to the Hitchin moduli space M(X,LG) corresponding to LG. The two Hitchin moduli spaces, and their sigma models, have nothing to do with each other a priori, so the mirror symmetry between them is just as surprising as the electromagnetic duality of the original gauge theories in four dimensions.
> Physicists’ interest in two-dimensional sigma models of this type is motivated in part by the important role they play in string theory. As I mentioned in Chapter 10, string theory postulates that fundamental objects of nature are not point-like elementary particles (which have no internal geometry and hence are zero-dimensional), but are one-dimensional extended objects called strings, which can be open or closed. The former have two end-points, and the latter are little loops, much like the ones we encountered in Chapter 10.
The idea of string theory is that vibrations of these tiny strings as they move through space-time create elementary particles and forces between them.
Sigma models enter string theory once we begin to consider how strings move. In standard physics, when a point-like particle moves in space, its trajectory is a one-dimensional path. The positions of the particle at different moments in time are represented by points on this path.
If a closed string is moving, however, then its movement sweeps a two-dimensional surface. Now the position of the string at a particular moment in time is a loop on this surface.
The strings may also interact with each other: a string may “split” into two or more pieces, and those pieces may also come together, as shown on the next picture. This gives us a more general Riemann surface with an arbitrary number of “holes” (and with boundary circles). It is called the worldsheet of the string.
Such a trajectory may be represented by a Riemann surface ∑ embedded into space-time S and hence by a map from ∑ to S. These are precisely the kinds of maps that appear in the sigma model on ∑ with the target manifold S. However, things are now turned upside-down: the space-time S is now the target manifold of this sigma model – that is, the recipient of the maps – not the source of the maps, in contrast to the conventional quantum field theories, such as electromagnetism.
The idea of string theory is that by doing calculations in these sigma models and summing up the results over all possible Riemann surfaces ∑ (that is, over all possible paths of the strings propagating in a fixed space-time S)4 we can reproduce the physics that we observe in space-time S.
Unfortunately, the resulting theory is plagued by some serious problems (in particular, it allows for the existence of “tachyons,” elementary particles moving faster than light, whose existence is prohibited by Einstein’s relativity theory). The situation improves dramatically if we consider a supersymmetric extension of string theory. Then we obtain what’s called superstring theory. But again, there is a problem: superstring theory turns out to be mathematically consistent only if our space-time S has ten dimensions, which is at odds with the world we observe having only four (three dimensions of space and one time dimension).
However, it could be that our world is really a product, in the sense explained above, of the four-dimensional space-time we observe and a tiny six-dimensional manifold M, which is so small that we cannot see it using available tools. If so, then we would be in a situation similar to the dimensional reduction (from four to two dimensions) that we have discussed above: the ten-dimensional theory would give rise to an effective four-dimensional theory. The hope is that this effective theory describes our universe, and in particular, includes the Standard Model as well as a quantum theory of gravity. This promise of potential unification of all known forces of nature is the main reason that superstring theory has been studied so extensively in recent years.5
But we have a problem: which six-dimensional manifold is this M?
To appreciate just how daunting this problem is, let’s suppose for the sake of argument that superstring theory were mathematically consistent in six dimensions rather than ten. Then there would only be two extra dimensions and we would have to find a two-dimensional manifold M. In this case, there wouldn’t be so many choices: M would have to be a Riemann surface, which, as we know, is characterized by its genus, that is, the number of “holes.” Furthermore, it turns out that for the theory to work, this M has to satisfy certain additional properties; for example, it has to be what’s called a Calabi–Yau manifold, in honor of two mathematicians, Eugenio Calabi and Shing-Tung Yau, who were the first to study such spaces mathematically (years before physicists got interested in the subject, I might add).6 The only Riemann surface that has this property is the torus. Hence, if M were two-dimensional, we would be able to pin it down – it would have to be a torus.7 However, as the dimension of M grows, so does the number of possibilities. If M is six-dimensional, then by some estimates there are 10500 choices – an unimaginably large number. Which one of these six-dimensional manifolds is realized in our universe and how can we verify this experimentally? This is one of the key questions of string theory that still remains unanswered.8
In any case, it should be clear from this discussion that sigma models play a crucial role in superstring theory, and in fact their mirror symmetry may be traced to a duality in superstring theory.9 Sigma models also have many applications outside of string theory. Physicists have been studying them in great detail, and not only the sigma models in which the target manifold M is six-dimensional.10
So, when Witten spoke at our conference in 2004, he first applied the technique of dimensional reduction (from four to two dimensions), to reduce the electromagnetic duality of two gauge theories (with gauge groups G and LG) to the mirror symmetry of two sigma models (with the targets being the Hitchin moduli spaces associated to the two Langlands dual groups, G and LG). Then he asked: can we connect this mirror symmetry to the Langlands Program?
The answer he suggested was fascinating. Usually, in quantum field theories, we study something called correlation functions, which describe the interaction of particles; for example, one such function might be used to describe the probability of a certain particle emerging from the collision of two others. But it turns out that the formalism of quantum field theory is much more versatile: in addition to these functions, there are also various more subtle objects present in the theory, which are similar to the “sheaves” that we discussed in Chapter 14 in connection with Grothendieck’s dictionary. These objects go under the name “D-branes,” or simply “branes.”
Branes originated in superstring theory, and their name is a truncation of the word “membrane.” Branes arise naturally when we consider the movement of open strings on a target manifold M. The simplest way to describe the positions of both ends of an open string is to stipulate that one of the end points belongs to a particular subset, B1, of M and the other belongs to another subset, B2. This is shown on the picture below, on which the thin curve represents the open string with two end points, one of which is on B1 and the other on B2.
This way, the subsets (or, more properly, submanifolds) B1 and B2 become players in superstring theory and in the corresponding sigma model. These subsets are the prototypes of the general branes that occur in these theories.11
The mirror symmetry between two sigma models gives rise to a relation between the branes in these two sigma models. The existence of this relation was originally proposed in the mid-1990s by mathematician Maxim Kontsevich under the name “homological mirror symmetry.” It has been extensively studied by both physicists and mathematicians, especially in the last decade.
The main idea of Witten’s talk in Princeton was that it is this homological mirror symmetry that should be equivalent to the Langlands relation.
At this point, it is important to note that sigma models come in two flavors, called the “A-model” and the “B-model.” The two sigma models we are considering are in fact different: if the one with the target manifold Hitchin moduli space M(X, G) is the A-model, then the one with the target manifold M(X, LG) is the B-model. Accordingly, the branes in the two theories are called “A-branes” and “B-branes,” respectively. Under the mirror symmetry, for each A-brane on M(X, G) there should be a B-brane on M(X, LG), and vice versa.12
In order to establish the geometric Langlands relation, we need to associate an automorphic sheaf to each representation of the fundamental group of X in L
G. Here is roughly how Witten proposed it could be constructed using mirror symmetry:
Though many details remained to be ironed out, Witten’s talk was a breakthrough; it showed a clear path to establishing a link between electromagnetic duality and the Langlands Program. On the one hand, it brought into the realm of modern math a host of new ideas that mathematicians had not thought of (certainly not in connection with geometric Langlands): categories of branes, the special role played by the Hitchin moduli spaces in the Langlands Program, and the connection between A-branes and automorphic sheaves. On the other hand, this link also enabled physicists to use mathematical ideas and insights to advance the understanding of quantum physics.
Over the next two years, Witten worked out the details of his proposal, in collaboration with a Russian born physicist from Caltech, Anton Kapustin. Their paper13 on this subject (230 pages long) appeared in April 2006 and made a big splash in both physics and mathematics communities. The opening paragraph from this paper describes many of the concepts that we have discussed in this book:
The Langlands program for number fields unifies many classical and contemporary results in number theory and is a vast area of research. It has an analog for curves over a finite field, which has also been the subject of much celebrated work. In addition, a geometric version of the Langlands program for curves has been much developed, both for curves over a field of characteristic p and for ordinary complex Riemann surfaces.... Our focus in the present paper is on the geometric Langlands program for complex Riemann surfaces. We aim to show how this program can be understood as a chapter in quantum field theory. No prior familiarity with the Langlands program is assumed; instead, we assume a familiarity with subjects such as supersymmetric gauge theories, electric-magnetic duality, sigma models, mirror symmetry, branes, and topological field theory. The theme of the paper is to show that when these familiar physical ingredients are applied to just the right problem, the geometric Langlands program arises naturally.
Love and Math Page 24
