The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)
Chapter 4. TOO GOOD TO BE TRUE
The third major success of our new “hammer”—geometric analysis—relates to a conjecture raised in 1953 by Eugenio Calabi, a mathematician who has been at the University of Pennsylvania since 1964. The conjecture has turned out to be a seminal piece of work for the field, as we shall see, and important for my own career as well. I consider myself fortunate to have stumbled upon Calabi’s ideas or perhaps collided with them head-on. (And in those days, no one wore helmets.) Any mathematician with sufficient talent and training is likely to make some contribution to the field, but it can take luck to find the problem especially suited to your talents and style of thought. Although I’ve gotten lucky a few times in math, coming across Calabi’s conjecture was certainly a high point for me in that regard.
The problem takes the form of a proof that links the topology of complex space (which we’ll talk about shortly) to geometry or curvature. The basic idea is that we start with some raw topological space, which is like a bare patch of land that’s been razed for construction. On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways. The question Calabi asked, though original in its details, is of the form we often ask in geometry—namely, given a general topology, or rough shape, what sort of precise geometric structures are allowed?
4.1—The geometer Eugenio Calabi (Photo by Dirk Ferus)
This may not seem like a statement replete with implications for physics. But let’s frame it in a different way. Calabi’s conjecture is concerned with spaces that have a specific type of curvature known as Ricci curvature (to be expanded upon shortly). It turns out that the Ricci curvature of a space relates to the distribution of matter within that space. A space that we call Ricci flat, meaning it has zero Ricci curvature, corresponds to a space with no matter. Viewed in those terms, the question Calabi asked was intimately tied to Einstein’s theory of general relativity: Could there be gravity in our universe even if space is a vacuum totally devoid of matter? Curvature, if Calabi were right, makes gravity without matter possible. He put it more generally as his question pertained to spaces of any even dimension and was not limited to the four-dimensional spacetime posited in general relativity. Yet for me, that was the most exciting way of framing the conjecture. It resonated strongly with my conviction that the deepest ideas of math, if shown to be true, would almost invariably have consequences for physics and manifest themselves in nature in general.
Calabi maintains that when he first hit upon the idea, “it had nothing to do with physics. It was strictly geometry.”1 And I have no doubt that is true. The conjecture could have been posed in exactly the same way even if Einstein had never come up with the idea of general relativity. And it could have been proved in the way that it was, even if Einstein’s theory did not exist. Yet I still believe that by the time Calabi formulated this problem—almost forty years after Einstein published his revolutionary paper—Einstein’s theory was already “in the air.” No one in mathematics could help thinking about Einsteinian physics, even if it wasn’t deliberate. By that time, Einstein’s equations, forever tying curvature to gravity, had become firmly embedded in mathematics. General relativity, you might say, had become part of the general consciousness. (Or perhaps the “collective unconscious,” as Jung put it.)
Regardless of whether Calabi consciously (or unconsciously) considered physics, the link between his geometric conjecture and gravity was a great motivating factor for me to take up this work. Proving the Calabi conjecture, I sensed, could be an important step toward uncovering some deep secrets.
Questions like Calabi’s are frequently framed in terms of the metric, the geometry of a space—the set of functions that enables us to determine the length of every path—which we first encountered in Chapter 1. A given topological space can have many possible shapes and therefore many possible metrics. Thus the same topological space can accommodate a cube, sphere, pyramid, or tetrahedron, all of which are topologically equivalent. So the question the conjecture raises, regarding what kind of metric a space can “support,” can be rephrased in an equivalent way: For a given topology, what kind of geometry is possible?
Of course, Calabi didn’t put it in precisely those terms. He wanted to know, among other things, whether a certain kind of complex manifold—a space that was compact (or finite in extent) and “Kähler”—that satisfied specific topological conditions (concerning an intrinsic property known as a “vanishing first Chern class”) could also satisfy the geometrical condition of having a Ricci-flat metric. All the terms of this conjecture are, admittedly, rather difficult to grasp, and defining the concepts needed to understand Calabi’s statement—complex manifolds, Kähler geometry (and metric), first Chern class, and Ricci curvature—will take some doing. We’ll work up to an explanation gradually. But the main thrust of the conjecture is that spaces meeting that complicated set of demands are indeed mathematically and geometrically possible.
To me, such spaces are rare like diamonds, and Calabi’s conjecture provided a road map for finding them. If you know how to solve the equation for one manifold and can understand the general structure of that equation, you can use the same idea to solve the equation for all Kähler manifolds meeting the same requirements. The Calabi conjecture offers a general rule for telling us that the “diamonds” are there—for telling us that the special metric we seek does, in fact, exist. Even if we cannot see it in its full glory, we can be confident nevertheless that it’s genuine. Among mathematical theories, therefore, this question stood out as a kind of jewel—or diamond in the rough, you might say.
From this sprang the work I’ve become most famous for. One might say it was my calling. No matter what our station, we’d all like to find our true calling in life—that special thing we were put on this earth to do. For an actor, it might be playing Stanley Kowalski in A Streetcar Named Desire. Or the lead role in Hamlet. For a firefighter, it could mean putting out a ten-alarm blaze. For a crime-fighter, it could mean capturing Public Enemy Number One. And in mathematics, it might come down to finding that one problem you’re destined to work on. Or maybe destiny has nothing to do with it. Maybe it’s just a question of finding a problem you can get lucky with.
To be perfectly honest, I never think about “destiny” when choosing a problem to work on, as I tend to be a bit more pragmatic. I try to seek out a new direction that could bring to light new mathematical problems, some of which might prove interesting in themselves. Or I might pick an existing problem that offers the hope that in the course of trying to understand it better, we will be led to a new horizon.
The Calabi conjecture, having been around a couple of decades, fell into the latter category. I latched on to this problem during my first year of graduate school, though sometimes it seemed as if the problem latched on to me. It caught my interest in a way that no other problem had before or has since, as I sensed that it could open a door to a new branch of mathematics. While the conjecture was vaguely related to Poincaré’s classic problem, it struck me as more general because if Calabi’s hunch were true, it would lead to a large class of mathematical surfaces and spaces that we didn’t know anything about—and perhaps a new understanding of spacetime. For me the conjecture was almost inescapable: Just about every road I pursued in my early investigations of curvature led to it.
Before discussing the proof itself, we first need to go over the aforementioned concepts that underlie it. The Calabi conjecture pertains strictly to complex manifolds. These manifolds, as we’ve said, are surfaces or spaces, but unlike the two-dimensional surfaces we’re familiar with, these surfaces can be of any even dimension and are not confined to the usual two. (The restriction to even dimensions pertains only to complex manifolds but, in general, a manifold can be of any dimension, even or odd.) By definition, manifolds resemble Euclidean space on a small or local scale but can be very different on a large or so-called global scale. A circle, for instance, is a one-dimensional manifold (or one-manifold), and every point you pick on that circle has a “neighborhood” around it that looks like a line segment. But taken as a whole, a circle looks nothing like a straight line. Moving up a dimension, we live on the surface of a sphere, which is a two-manifold. If you pick a small enough spot on the earth’s surface, it will look almost perfectly flat—like a disk or a portion of a plane—even though our planet is curved overall and thus non-Euclidean. If you instead look at a much larger neighborhood around that point, departures from Euclidean behavior will become manifest and we’ll have to correct for the curvature.
One important feature of manifolds is their smoothness. It’s built into the definition, because if every local patch on a surface looks Euclidean, then your surface has to be smooth overall. Geometers will characterize a manifold as smooth even if it has some number of funny points where things are not even locally Euclidean, such as a place where two lines intersect. We call such a point a topological singularity because it can never be smoothed over. No matter how small you draw the neighborhood around that point, the cross made by the intersecting lines will always be there.
This sort of thing happens in Riemannian geometry all the time. We may start out with a smooth object that we know how to handle, but as we move toward a certain limit—say, by making a shape pointier and pointier or an edge ever sharper—it develops a singularity nevertheless. We geometers are so liberal, in fact, that a space can have an infinite number of singularities and still qualify in our eyes as a kind of manifold—what we call a singular space or singular manifold, which lies at the limit of a smooth manifold. Rather than two lines intersecting in a point, think instead of two planes intersecting in a line.
So that’s roughly what we mean by manifolds. Now for the “complex” part. Complex manifolds are surfaces or spaces that are expressed in terms of complex numbers. A complex number takes the form of a + ib, where a and bare real numbers and i is the “imaginary unit,” defined as the square root of negative one. Much as two-dimensional numbers of the form (x, y) can be graphed on two axes named x and y, one-dimensional complex numbers of the form a + ib can be graphed on two axes referred to as the real and imaginary axes.
Complex numbers are useful for several reasons—one being that people would like to be able to take the square root of negative numbers as well as positive numbers. Armed with complex numbers, one can solve quadratic equations of the form ax2 + bx + c = 0, regardless of the values of a, b, and c, using the quadratic formula that many first learned in high school: x = [-b ± √(b2 - 4ac)]/2a. When complex numbers are allowed, you don’t have to throw up your arms in despair when b2 - 4ac is negative; you can still solve the equation.
Complex numbers are important, and sometimes essential, for solving polynomial equations—equations, that is, involving one or more variables and constants. The goal in most cases is to find the roots of the equation, the points where the polynomial equals zero. Without complex numbers, some problems of this sort are insoluble. The simplest example of that is the equation x2 + 1 = 0, which has no real solutions. The only time that statement is true, and the equation equals zero, is when x = i or when x = -i.
Complex numbers are also important for understanding wave behavior, in particular the phase of a wave. Two waves of the same size (amplitude) and frequency can either be in phase, meaning they line up and add to each other constructively, or be out of phase, meaning they partially or completely cancel each other in what’s called destructive interference. When waves are expressed in terms of complex numbers, you can see how the phases and amplitudes add up simply by adding or multiplying those complex numbers. (Doing the calculation without complex numbers is possible but much more difficult, just as it’s possible to compute the motions of the planets in the solar system from the perspective of Earth, though the equations become much simpler and more elegant when you put the sun at the center of your frame of reference.) The value of complex numbers for describing waves makes these numbers crucial to physics. In quantum mechanics, every particle in nature can also be described as a wave, and quantum theory itself will be an essential component of any theory of quantum gravity—any attempt to write a so-called theory of everything. For that task, the ability to describe waves with complex numbers is a considerable advantage.
The first well-known calculation involving complex numbers was contained in a book published in 1545 by the Italian mathematician Girolamo Cardano. But it wasn’t until about three hundred years later that complex geometry was established as a meaningful discipline. The person who really brought complex geometry to the fore was Georg Friedrich Bernhard Riemann—the architect of the first complex manifolds ever seriously examined, so-called Riemann surfaces. (These would become important in string theory more than a century after Riemann died. When a tiny loop of string—the basic unit of string theory—moves through higher-dimensional spacetime, it sweeps out a surface that is none other than a Riemann surface. Such surfaces have proven to be quite useful for doing computations in string theory, making them one of the most studied surfaces in theoretical physics today. And Riemann surface theory itself has benefited from the association as well, with equations drawn from physics helping to reinvigorate the math.)
Riemann surfaces, like ordinary two-dimensional manifolds, are smooth, but being complex surfaces (of one complex dimension), they have some added structure built in. One feature that automatically comes with complex surfaces, though not always with real ones, is that all the neighborhoods of the surface are related to each other in specific ways. If you take a small patch of a curved Riemann surface and project it onto a flat surface and do the same with other small patches around it, you’ll end up with a map, similar to what you’d end up with if you tried to represent a three-dimensional globe in a two-dimensional world atlas. When you make a map like this with Riemann surfaces, the distance between things can get distorted but the angles between things are always maintained. The same general idea was used in Mercator projection maps, first introduced in the sixteenth century, which treat Earth’s surface as a cylinder rather than a sphere. This angle-preserving characteristic, known as conformal mapping, was important in navigation centuries ago and helped keep ships on course. Conformal mapping can help simplify calculations involving Riemann surfaces, making it possible to prove things about these surfaces that can’t be proved for noncomplex ones. Finally, Riemann surfaces, unlike ordinary manifolds, must be orientable, meaning that the way we measure direction—the frame of reference that we choose—remains consistent regardless of where we are in space. (This is not the case on a Möbius strip, a classic example of a nonorientable surface, where directions are reversed—down becomes up, left becomes right, clockwise becomes counterclockwise—as you loop around to your original spot.)
4.2—These two-dimensional surfaces—bull, bunny, David, and horse—are all examples of Riemann surfaces, which are of great importance in both mathematics and string theory. One can place a checkerboard pattern onto these surfaces by picking a point on the checkerboard, feeding its coordinates into a mathematical “function,” and generating a point on, say, the bunny instead. But the transposed checkerboard cannot be perfect (unless it is being mapped onto a two-dimensional torus), owing to the presence of singular points—such as might be found on the north and south poles of a sphere—that will inevitably crop up on a surface whose Euler characteristic (described later in this chapter) is nonzero.
Nevertheless, the process is still conformal, meaning that angles—including the right angles of the checkerboard—are always preserved in moving from one surface to the other. Although the size of things, such as the checkerboard squares, may get distorted, the corners of the squares are always 90 degrees. This angle-preserving property is one of the special features of a Riemann surface.
As you move from patch to patch on a Riemann surface, you will invariably change coordinate systems, as only a tiny neighborhood around a given point looks Euclidean. But these patches need to be stitched together in just the right way so that movements between them always preserve angles. That’s what we mean by calling such a movement, or “transformation,” conformal. Complex manifolds, of course, come in higher dimensions as well—Riemann surfaces being just the one-dimensional version of them. Regardless of dimensions, the different regions or patches of a manifold have to be properly attached in order for the manifold to qualify as complex. However, on a higher-dimensional complex manifold, angles are not preserved during movements between one patch of the manifold and another and between one coordinate system and another. These transformations are not conformal, strictly speaking, but are rather a generalization of the one-dimensional case.
The spaces Calabi envisioned not only were complex, but also had a special property called Kähler geometry. Riemann surfaces automatically qualify as Kähler, so the real meaning of the term only becomes apparent for complex manifolds of two (complex) dimensions or higher. In a Kähler manifold, space looks Euclidean at a single point and stays close to being Euclidean when you move away from that point, while deviating in a specific way. To qualify that last statement, I should say that space looks “complex Euclidean,” rather than just plain old Euclidean, meaning that the space is even-dimensional and that some of the coordinates defining a point are expressed in terms of complex numbers. This distinction is important because only complex manifolds can have Kähler geometry. And Kähler geometry, in turn, enables us (among other things) to measure distance using complex numbers. The Kähler condition, named after the German mathematician Erich Kähler, provides an indication of how close a space comes to being Euclidean based on criteria that are not strictly related to curvature.
To quantify the closeness of a given manifold to Euclidean space, we need to know the metric for that manifold. In a flat space, where all the coordinate axes are perpendicular to each other, we can simply use the Pythagorean formula to compute distances. In curved spaces, where coordinate axes are not necessarily perpendicular, things become more complicated, and we have to use a modified version of that formula. Distance calculations then involve metric coefficients—a set of numbers that vary from point to point in space and also depend on how the coordinate axes are oriented. Selecting one orientation for the coordinate axes rather than another will change those numbers. What’s important is not the value of the metric coefficients themselves, which is somewhat arbitrary, but rather how they change from place to place in the manifold. For that information tells you where one point lies in relation to every other, which encapsulates everything you need to know about the geometry of the manifold. As we’ve seen in previous chapters, in four-dimensional space, ten such coefficients are needed. (Actually, there are sixteen numbers in all because the metric tensor in this case is a four-by-four matrix. However, the metric tensor is always symmetrical around a diagonal axis running from the upper left corner of the matrix to the lower right. There are four numbers on the diagonal itself, and two sets of six numbers on either side of the diagonal that are the same. Owing to this symmetry, we often need to concern ourselves with just ten numbers—four along the diagonal and six on either side—rather than all sixteen.)
Still, this doesn’t explain how a metric works. Let’s take a reasonably simple example in one complex or two real dimensions—the Poincaré metric on the unit disk in a plane centered at the origin (0, 0) of the x-y coordinate system. This is just the set of points (x, y) that satisfies the inequality x2 + y2 < 1. (Technically this is referred to as an “open” unit disk because it lacks a boundary, the unit circle itself, defined by x2 + y2 = 1.) Because we are in two dimensions, the metric tensor of the Poincaré metric is a two-by-two matrix. Each spot in the matrix is filled with a coefficient of the form Gij where i refers to the row and j to the column. The array, therefore, looks like this:
Owing to symmetry, as discussed above, G12 has to be the same as G21. For the Poincaré metric, these two “off-diagonal” numbers are defined as zero. The other two numbers, G11 and G22, do not have to be the same, but in the Poincaré metric, they are: Both, by definition, are equal to 4/(1 - x2 - y2)2. For any pair of x and y you choose sitting within this disk, the metric tensor tells you the coefficients. For x = 1/2 and y = 1/2, for instance, G11 and G22 both equal 16, whereas the other two coefficients are 0, as they are everywhere, for any point (x, y) in the unit disk.
Now that we have these numbers, these coefficients, what do we do with them? And how do they relate to distance? Let’s draw a little curve inside the unit disk, but instead of thinking of a curve as a static item, let’s think about it as the path traced out by a particle moving in time from a point A to point B. What is the length of that path, according to the Poincaré metric?
To answer that question, we take the curve s and break it up into tiny line segments—the tiniest line segments you can possibly imagine—and add them up. We can approximate the length of each of those segments by using the Pythagorean theorem. First, we define the x, y, and s values parametrically; that is, as functions of time. Thus, x = X(t), y = Y(t), and s = S(t). The derivatives of those time functions, X′(t) and Y′(t), we treat as the legs of a triangle; when used in the Pythagorean theorem, √([X′(t)]2 + [Y′(t)]2) gives us the derivative S′(t)—that is, the approximate length of one of those minuscule segments we created. Integrating that derivative over A to B gives us the length of the entire curve. Each line segment, in turn, represents a tangent to the curve, known in this context as a tangent vector. But, because we are on the Poincaré disk, we must integrate the product of the Pythagorean result times the metric—namely, √([X′(t)]2 + [Y′(t)]2) ╳ √(4/[1 - x2 - y2]2)—to correct for the curvature.
To simplify the picture further, let’s set Y(t) to 0 and thereby restrict ourselves to the x-axis. We’ll start at 0 and move at a constant rate along the x-axis toward 1. If time is moving from 0 to 1 as well, then X(t) = t, and if Y(t) = 0 (as we’re assuming in this instance), then X′(t) = 1 because we’re taking the derivative of X with respect to time. But since X in this case equals time, we’re really taking the derivative of X with respect to itself, and that’s always 1. If you think of the derivative in terms of a ratio, this last point becomes fairly obvious: The derivative of X here is the ratio of the change in X to the change in X, and any ratio of that form—with the same quantity in the numerator and denominator—has to be 1.
So the fairly messy product of the above paragraph, which is the thing we have to integrate to get the distance, now reduces to 2/(1-x2). It’s fairly easy to see that as x approaches 1, this factor approaches infinity and its integral approaches infinity as well.
One caveat is that just because the metric coefficients—the G11 and G22 terms here—go to infinity doesn’t necessarily mean the distance to the boundary goes there, too. But this does indeed happen in the case of the Poincaré metric and the unit disk. Let’s take a closer look at what happens to those numbers as we move outward in space and time. Starting at the origin, where x = 0 and y = 0, G11 and G22 equal 4. But when we get near the edge of the disk, where the sum of the squares of x and y is close to 1, the metric coefficients get really big, meaning the tangent vector lengths get big as well. When x = 0.7 and y = 0.7, for example, G11 and G22 equal 10,000. For x = 0.705 and y = 0.705, they’re more than 100,000; and for x = 0.7071 and y = 0.7071, they’re more than 10 billion. Moving closer to the disk boundary, these coefficients not only get bigger, but go to infinity—and so do the distances to the edge. If you’re a bug on this surface crawling toward the edge, the bad news is that you’ll never reach it. The good news is that you’re not missing much, since this surface—taken on its own—doesn’t really have an edge. When you put an open unit disk inside a plane, it has a boundary—in the form of a unit circle—as part of the plane. But as an object unto itself, this surface has no boundary, and any bug seeking to get there will die unfulfilled. This unfamiliar, and perhaps counterintuitive, fact is a result of the negative curvature of a unit disk as defined by the Poincaré metric.
We’ve been spending some time talking about the metric in order to get some sense of what a Kähler metric and a Kähler manifold equipped with such a metric are all about. Whether a particular metric is Kähler is a function of how the metric changes as you move from point to point. Kähler manifolds are a subclass of a set of complex manifolds known as Hermitian manifolds. On a Hermitian manifold, you can put the origin of a complex coordinate system at any point, such that the metric will look like a standard Euclidean metric at that point. But as you move away from that point, the metric becomes increasingly non-Euclidean; more specifically, when you move away from the origin by a distance epsilon, ε, the metric coefficients themselves change by a factor on the order of ε. We characterize such manifolds as Euclidean to first order. So if ε were one-thousandth of an inch, when we move by ε, the coefficients in our Hermitian metric would remain constant to within a thousandth of an inch or thereabouts. Kähler manifolds are Euclidean to second order, meaning that the metric is even more stable; the metric coefficients on a Kähler manifold change by ε2 as you move from the origin. To continue the previous example, where ε = 0.001 inch, the metric would change by 0.000001 inch.
So what prompted Calabi to single out Kähler manifolds as the ones of greatest interest? To answer that, we need to consider the range of choices available. If you want to be really stringent, you could insist, for example, that the manifolds must be totally flat. But the only compact manifolds that are totally flat happen to be donuts or tori or close relatives thereof, and this holds for any dimension of two or higher. As manifolds go, tori are rather simple and can thus be rather limiting. We’d prefer to see more variety—a broader range of possibilities. Hermitian manifolds, on the other hand, don’t constrain things enough: the possibilities are just too great. Kähler manifolds, lying between Hermitian and flat, have the kind of properties that we geometers often seek out. They have enough structure to make them easier to work with, yet not so much structure as to be overly restrictive in the sense that no manifold can match your detailed specifications.
Another reason for focusing on Kähler manifolds is that we can use tools introduced by Riemann—some of which were exploited by Einstein—to study these manifolds. These tools work on Kähler manifolds, which are a restrictive class of Hermitian manifolds, but do not apply to Hermitian manifolds in general. We’d like to be able to use these tools, because they were already quite powerful when Riemann first developed them, and mathematicians have had more than a century in which to enhance them further. That makes Kähler manifolds a particularly appealing choice because we have the technology at our disposal to really probe them.
But that’s not all. Calabi was interested in these manifolds because of the kinds of symmetry they possess. Kähler manifolds, like all Hermitian manifolds, have a rotational symmetry when vectors on them are multiplied by the imaginary unit i. In one complex dimension, points are described by a pair of numbers (a, b) taken from the expression a + bi. Let’s assume that (a, b) defines a tangent vector sticking out from the origin. When we multiply that vector by i, its length is preserved, although it gets rotated by 90 degrees. To see how the rotation works, let’s start out at the point (a, b), or a + ib. Multiplying by i yields ia-bor, equivalently, -b + ia, corresponding to a new point (-b, a) on the complex plane that defines a vector that’s orthogonal (or perpendicular) to the original vector but of the same length.
You can see that these vectors really are perpendicular by plotting the points (a, b) and (-b, a) on simple Cartesian coordinates and measuring the angle between the vectors going to these points. The operation we’re talking about here—switching the x coordinate to -y and switching the y coordinate to x—is called the J transformation, which happens to be the real-number analogue of multiplying by i. In this case, doing the J operation twice, or J2, is the same as multiplying by -1. It’s best to continue this discussion by talking about J rather than i since any way we might try to picture this—either in our heads or on paper—will likely involve real coordinates rather than the complex plane. Just keep in mind, once again, that the J operation is a way of interpreting complex multiplication by i as a transformation of two-dimensional coordinates.
All Hermitian manifolds have this kind of symmetry: J transformations rotate vectors 90 degrees while keeping their length unchanged. Kähler manifolds (being a more restrictive subset of the Hermitian case) have this symmetry as well. In addition, Kähler manifolds have what’s called internal symmetry—a subtle kind of symmetry that must remain constant as you move between any two points in the space or it’s not Kähler geometry. Many of the symmetries we see in nature relate to rotation groups. A sphere, for example, has global symmetry—so called because it applies to every point on the object at once. One such symmetry is rotational invariance, which means you can turn the sphere every which way and it still looks the same. The symmetry of a Kähler manifold, on the other hand, is more local because it only applies to first derivatives of the metric. However, through the techniques of differential geometry, which involve integrating over the whole manifold, we can see that the Kähler condition and its associated symmetry do imply a specific relation between different points. In this way, the symmetry that we initially described as local has, by virtue of calculus, a broader, more global reach that establishes a link between different points on the manifold.
The crux of this symmetry relates to the rotation caused by the J operation and something called parallel translation or transport. Parallel transport, like the J operation, is a linear transformation: It’s a way of moving vectors along a path on a surface or manifold that keeps the lengths of those vectors constant while keeping the angles between any two vectors constant as well. In instances where parallel transport is not easy to visualize, we can determine the precise way of moving vectors around from the metric by solving differential equations.
On a flat (Euclidean) plane, things are simple: You just keep the direction and length constant for each vector. On curved surfaces and general manifolds, the situation is analogous to maintaining a constant bearing in Euclidean space but much more complicated.
Here’s what’s special about a Kähler manifold: If we take a vector V at point P and parallel-transport it along a prescribed path to point Q, we’ll get a new vector at Q called W1. Then we’ll do the J operation on W1, rotating it 90 degrees, which gives us a new vector, JW1, that’s been transformed by the J operation. Alternatively, we can start with vector V at point P and J transform it (or rotate it) right there, giving us the new vector JV (still affixed to point P). We then parallel-transport vector JV to point Q, where we’ll rename it W2. If the manifold is Kähler, the vectors JW1 and W2 will always be the same, regardless of what path you take in getting from P to Q. One way of putting this is that on a Kähler manifold, the J operation remains invariant under parallel transport. That’s not the case for complex manifolds in general. Another way of putting it is that on a Kähler manifold, parallel-transporting a vector and then transforming it by the J operation is the same as transforming the original vector by J and then parallel-transporting it. These two operations commute—the order in which you do them doesn’t matter. That’s not always true, as Robert Greene memorably describes it, “just as opening the door and then walking out of the house is not the same as walking out of the house (or trying to!) and then opening the door.”2
The basic notion of parallel transport is illustrated in Figure 4.3 for a surface of two real dimensions, or one complex dimension, because that’s what we can draw. But this case is rather trivial, because the possibilities for rotation are so limited. There are only two choices: rotating to the left and to the right.
But in complex dimension two, which is real dimension four, there are an infinite number of vectors of a given length perpendicular to a given vector. This infinitude of vectors defines a tangent space, which you might picture—in two dimensions rather than four—as a giant piece of plywood balanced on top of a basketball. In this setting, knowing that a vector is perpendicular to the one you have in hand does little to narrow down the field of possibilities—unless the manifold happens to be Kähler. In that case, if you know how the J transformation works at one point, you know what vector you’ll get when you do the J transformation at some remote point, because you can carry your result from the first point to the second via parallel transport.
4.3—In the first panel, we parallel-transport vector V from point P to Q, where the vector is given the new name W1. We then perform the J operation on W1, which rotates it by 90 degrees. The reoriented vector is called JW1. In the second panel, we perform the J operation on vector V at point P, which gives us a realigned vector (again rotated 90 degrees), called JV. We then parallel-transport JV to point Q, where it is assigned the new name, W2. In both cases, the resultant vectors—JW1 and W2—are the same. This is one of the hallmarks of a Kähler manifold—namely, that you get the same result, the same vector, regardless of whether you parallel-transport a vector and then transform it by J or transform the vector by J first and then parallel-transport it. The two operations commute, meaning that the order in which you do them does not matter.
There’s another way of showing what this simple operation, the J transformation, has to do with symmetry. It’s what we call a fourfold symmetry because each time you operate by J, you rotate the vector 90 degrees. When you do it four times, or 360 degrees, you’ve come full circle and are back where you started. Another way to think of it is that transforming by J twice is the same as multiplying by -1. Transforming by J four times is -1 ╳ -1 = 1. Back to identity, as we say.
Given that this fourfold symmetry applies only to the tangent space of a single point, in order for it to be useful, it has to behave consistently as you move around in the space. This consistency is an important feature of internal symmetry. You might compare it to a compass that has twofold symmetry in the sense that it can only point in two directions: north and south. If you moved this compass around in space and it randomly pointed north or south, with no rhyme or reason to it, you might conclude that your space was not particularly symmetrical, nor did it have a discernible magnetic field (or that it was time to buy a new compass). Similarly, if your J operation gave you different results depending on where and how you moved in your manifold, it would lack the order and predictability that symmetry normally confers. Not only that, you’d know that your manifold was not Kähler.
4.4—A demonstration of the simple and rather obvious fact that a square has fourfold symmetry about its center, which is another way of saying that if you rotate it by 90 degrees, four times consecutively, you’ll end up where you started. Since the J transformation also involves a 90-degree rotation, this operation has fourfold symmetry as well, with four rotations bringing you back to the starting point. (Technically speaking, the J operation acts on tangent vectors, so it’s only roughly analogous to the rotation of an object like a square.) The J transformation, as discussed in the text, is the real-number analogue of the i multiplication. Multiplying by i four times is the same as multiplying by 1, which—like doing the J operation four times—will inevitably bring you back to where you started.
This internal symmetry, which in many ways defines Kähler manifolds, is restricted to the space tangent to the manifolds. That can be advantageous because, on a tangent space, the results of an operation do not depend on your choice of coordinates. In general, that’s what we look for in both geometry and physics—that the results of an operation are independent of the coordinate system that was selected. Simply put, we don’t want the answers we get to be a consequence of an arbitrary decision regarding how the coordinate axes were oriented or which point was picked as the origin.
This symmetry imposes another set of constraints on the mathematical world Calabi was imagining, drastically simplifying it and making the problem of its proof seem potentially soluble. But it had other consequences that he did not foresee; indeed, this internal symmetry that he was proposing is a special kind of supersymmetry—an idea that would prove to be of vital importance to string theory.
The last two pieces in our puzzle—Chern class and Ricci curvature—are related and grow out of attempts made by geometers to generalize Riemann surfaces from one dimension into many, and then to characterize the differences between those generalizations mathematically. This leads us to an important theorem that applies to compact Riemann surfaces—as well as to any other compact surface without a boundary. (The definition of a boundary in topology is rather intuitive: A disk has a boundary or clearly defined edge, whereas a sphere has none. You can move in any direction you want on the surface of a sphere, and go as far as you want, without ever reaching, or approaching, an edge.) The theorem was formulated in the nineteenth century by Carl Friedrich Gauss and the French mathematician Pierre Bonnet, connecting the geometry of a surface to its topology.
The Gauss-Bonnet formula holds that the total Gauss curvature of such surfaces equals 2π times the Euler characteristic of that surface. The Euler characteristic—known as χ (“chi”)—in turn, is equal to 2 - 2g, where g is the genus or number of “holes” or “handles” that a surface has. The Euler characteristic of a two-dimensional sphere, for example, which has zero holes, is 2. Euler had previously devised a separate formula for finding the Euler characteristic of any polyhedron: χ = V - E + F, where V is the number of vertices, E the number of edges, and F the number of faces. For a tetrahedron, χ = 4 - 6 + 4 = 2, the same value we derived for the sphere. For a cube with 8 vertices, 12 edges, and 6 faces, χ = 8 - 12 + 6 = 2, again the same as the sphere. It makes sense for those topologically identical (though geometrically distinct) objects to have the same value of χ, given that the Euler characteristic relates to an object’s topology rather than its geometry. The Euler characteristic, χ, was the first major topological invariant of a space: a property that remains constant, or invariant, for spaces that may look dramatically different—like a sphere, tetrahedron, and cube—yet are topologically equivalent.
Going back to the Gauss-Bonnet formula, the total Gauss curvature of a two-dimensional sphere is thus 2π times 2, or 4π. For a torus, it’s 0, as χ for a two-dimensional torus with one hole is 0 (2 - 2g = 2-2 = 0). Attempting to generalize the Gauss-Bonnet principle to higher dimensions leads us to Chern classes.
4.5—An orientable (or two-sided) surface is described topologically by its Euler characteristic, or Euler number. There’s a simple formula for computing the Euler characteristic of a polyhedron, which, loosely speaking, is a geometric object with flat faces and straight edges. The Euler characteristic, represented by the Greek letter χ (chi), equals the number of vertices minus the number of edges plus the number of faces. For the rectangular prism (or “box”) in this example, that number turns out to be 2. It’s also 2 for a tetrahedron (4 - 6 + 4) and for a square pyramid (5 - 8 + 5). The fact that these spaces have the same Euler characteristic (2) isn’t surprising, since all these objects are topologically equivalent.
Chern classes were developed by my mentor and adviser S. S. Chern as a crude method for mathematically characterizing the differences between two manifolds. Simply put, if two manifolds have different Chern classes, they cannot be the same, though the converse is not always true: Two manifolds can have the same Chern classes and still be different.
For Riemann surfaces of one complex dimension, there’s just one Chern class, the first Chern class, which in this case equals the Euler characteristic. The number of Chern classes that a manifold can be assigned depends on the number of dimensions. A manifold of two complex dimensions, for instance, has a first and a second Chern class. For manifolds of keen interest in string theory—those with three complex dimensions (or six real ones)—there are three Chern classes. The first Chern class assigns integer coefficients to subspaces, or submanifolds, of two real dimensions sitting inside the six-dimensional space (much as two-dimensional surfaces such as paper can be fit into your three-dimensional office). The second Chern class assigns numbers to submanifolds of four real dimensions inside the six-dimensional space. The third Chern class assigns a single number—χ , the Euler characteristic—to the manifold itself, which has three complex dimensions and six real dimensions. For a manifold of n complex dimensions, the last Chern class—or nth Chern class—is always equal to the Euler characteristic.
But what does the Chern class really tell us? In other words, what’s the purpose of assigning all those numbers? It turns out the numbers don’t tell us anything of great value about the submanifolds themselves, but they can tell us a lot about the bigger manifold. (That’s a fairly common practice in topology: We try to learn about the structure of complex, higher-dimensional objects by looking at the number and types of subobjects they can hold.)
Suppose, for example, you’ve assigned a different number to every person in the United States. The number given to an individual doesn’t really tell you anything about him or her. But all those numbers taken together can tell you something interesting about the bigger “object”—the United States itself—such as the size of its population, or the rate of population growth.
Here’s another example that can provide a picture to go along with this rather abstract concept. As we often do, we’ll start by looking at a fairly simple object, a sphere—a surface of one complex dimension or two real dimensions. A sphere has only one Chern class, which in this case equals the Euler characteristic. In Chapter 2, as you may recall, we discussed some of the meteorological (and fluid dynamical) implications of living on a spherically shaped planet. Let’s suppose the wind blows from west to east at every spot on Earth’s surface. Well, at almost every spot. You can imagine the wind blowing in the easterly direction on the equator and at every possible latitudinal line north or south of the equator. At two points, however, lying at the absolute centers of the north and south poles (which can be regarded as singularities), the wind does not blow at all—an inevitable consequence of spherical geometry. For a surface like that, which has those special points that stick out like a sore thumb, the first Chern class is not equal to zero. In other words, it is not vanishing.
Now let’s consider the donut. Winds could blow on the surface in any direction you want—in long rings around the hole, in short rings through the hole, or in more complicated spiral patterns—without ever hitting a singularity, a place where the flow stops. You can keep going around and around and never hit a snag. For a surface like that, the first Chern class is equal to zero or vanishing, as we say.
4.6—The first Chern class (which has the same value as the Euler characteristic for two-dimensional surfaces such as these) relates to places where the flow in a vector field totally shuts down. One can see two such spots on the surface of a sphere such as the familiar globe. If, for example, everything flows from the north pole to the south pole (as in the left-hand sphere), there will be zero net flow at each of the poles, because all the vectors representing flows will cancel each other out. Similarly, if everything flows from west to east (as in the right-hand sphere), there will be exactly two dead spots—again, one on the north pole and one on the south pole—where nothing flows at all, because at these spots, there is no east or west.
This is not the case on the surface of a donut, however, where things can flow vertically (as in the left-hand donut) or horizontally (as in the right-hand donut) without ever hitting a dead spot. That’s why the first Chern class is zero for a donut, which lacks these singular spots, but not for a sphere.
4.7—Determining the first Chern class of an object comes down to finding places where flows in a vector field drop to zero. Places like that can be found in the center of a vortex, such as the eye of a hurricane—a circular region of calm weather, anywhere from 2 to 200 miles in diameter, surrounded by some of the stormiest conditions found on Earth. This photo was taken of Hurricane Fran in 1996, just before the storm ravaged the eastern United States, causing billions of dollars’ worth of damage. (Photo by Hasler, Chesters, Griswold, Pierce, Palaniappan, Manyin, Summey, Starr, Kenitzer, and de La Beaujardière, Laboratory for Atmospheres, NASA Goddard Space Flight Center)
To pick another example, so-called K3 surfaces of complex dimension two and real dimension four have a first Chern class of zero (Chapter 6 discusses K3 surfaces in more detail). According to Calabi’s conjecture, that would enable them to support a Ricci-flat metric, much as a torus could. But, unlike a two-dimensional torus (whose Euler characteristic is zero), the value of χ for a K3 surface is 24. The point is that the Euler characteristic and first Chern class can be quite different in higher dimensions, even though they are the same in the case of one complex dimension.
The next item on our list is Ricci curvature, which is a concept essential to understanding what the Calabi conjecture is all about. Ricci curvature is a kind of average of a more detailed type of curvature known as sectional curvature. To see how it works, let’s start with a simple picture: a sphere and the space (a plane) tangent to its north pole. The plane—which is perpendicular to the line containing the sphere’s center and the selected point on the sphere—contains all the tangent vectors to the sphere at that point. (Similarly, a three-dimensional surface has a three-dimensional tangent space consisting of all vectors tangent to the point, and so on for higher dimensions.) Every vector in that tangent plane is also tangent to a great circle on the globe that runs through the north and south poles. If we take all the great circles that are tangent to vectors in the plane and put those circles together, we can assemble a new two-dimensional surface. (In this case, that surface will just be the original sphere, but in higher dimensions, the surface so constructed will be a two-dimensional submanifold sitting within a larger space.) The sectional curvature of the tangent plane is simply the Gauss curvature of the newly formed surface associated with it.
To find the Ricci curvature, pick a point on the manifold and find a vector tangent to that point. You then look at all the two-dimensional tangent planes that contain that vector; each of those planes has a sectional curvature attached to it (which, as we’ve just said, is the Gauss curvature of the surface associated with that plane). The Ricci curvature is the average of those sectional curvatures. A Ricci-flat manifold means that for each vector you pick, the average sectional curvature of all the tangent planes containing that vector equals zero, even though the sectional curvature of an individual plane may not be zero.
As you may have surmised, this means our example of a two-dimensional sphere, where one might take a vector tangent to the north pole, is rather uninteresting because there’s only one tangent plane containing that vector. In this case, the Ricci curvature is just the sectional curvature of that plane, which, in turn, is just the Gauss curvature of a sphere (which is 1 for a sphere of unit radius). But when you go to higher dimensions—anything beyond one complex or two real dimensions—there are lots of tangent planes to choose from, and as a consequence, a manifold can be Ricci flat without being flat overall—that is, without having zero sectional curvature and zero Gauss curvature.
The sectional curvature completely determines the Riemann curvature, which in turn encodes all the curvature information you could possibly want about a surface. In four dimensions, this takes twenty numbers (and more for higher dimensions). The Riemann curvature tensor can itself be split into two terms, the Ricci tensor and something called the Weyl tensor, which we won’t go into here. The main point is that of the twenty numbers or components needed to describe four-dimensional Riemann curvature, ten describe Ricci curvature while ten describe Weyl curvature.
The Ricci curvature tensor, a key term in the famous Einstein equation, shows how matter and energy affect the geometry of spacetime. In fact, the left-hand side of this equation consists of what’s called the modified Ricci tensor, whereas the right-hand side of the equation consists of the stress energy tensor, which describes the density and flow of matter in spacetime. Einstein’s formulation, in other words, equates the flow of matter density and momentum at a given point in spacetime to the Ricci tensor. Since the Ricci tensor is just part of the total curvature tensor, as discussed above, we cannot rely on it alone to determine the curvature completely. But if we draw on our knowledge of the global topology, we may have hopes of deducing the curvature of spacetime.
In the special case where the mass and energy are zero, the equation reduces to this: The modified Ricci tensor = 0. That’s the vacuum Einstein equation, and although it may look simple, you have to remember it’s a nonlinear partial differential equation, which is almost never easy to solve. Moreover, the vacuum Einstein equation is actually ten different nonlinear partial differential equations bundled together, because the tensor itself consists of ten independent terms. This equation is similar, in fact, to the Calabi conjecture, which sets the Ricci curvature to zero. It’s not too surprising that one can find a trivial solution to the vacuum Einstein equation—a trivial solution being the least interesting one in which you have a spacetime where nothing happens: no matter, no gravity, and nothing much going on. But there’s a more intriguing possibility, and this is precisely what the Calabi conjecture gets at: Can the vacuum Einstein equation have a nontrivial solution, too? The answer to that question is yes, as we shall see in due time.
Soon after Chern came up with the idea of Chern classes in the mid-1940s, he showed that if you have a manifold with zero Ricci curvature—with a certain geometry, that is—then its first Chern class must also be zero. Calabi flipped that over, asking whether certain topological conditions are sufficient in themselves to dictate the geometry—or, more precisely, to allow that particular geometry to be dictated. Reversals of this sort are not always true. For example, we know that a smooth surface (without edges) whose Gauss curvature is greater than one must be bounded or compact. It cannot wander off to infinity. But as a general matter, compact, smooth surfaces need not have a metric with a Gauss curvature greater than one. A donut, for instance, is perfectly smooth and compact, yet its Gauss curvature cannot always be positive, let alone greater than one. Indeed, a metric of zero Gauss curvature, as discussed previously, is entirely possible, whereas a metric with positive curvature everywhere is not.
So Calabi’s conjecture faced two big challenges: Just because it was the converse of a well-established proposition was not sufficient to make it true. And even if it were true, proving the existence of a metric that meets the desired requirements would be extremely difficult. Like the Poincaré conjecture that came before it, Calabi’s conjecture—or, I should say, an important case of this conjecture—can be summed up in a single sentence: A compact Kähler manifold with a vanishing first Chern class will admit a metric that is Ricci flat. Yet it took more than two decades to prove the contents of that sentence. And several decades after the proof, we’re still exploring the full range of its implications.
As Calabi recalls, “I was studying Kähler geometry and realized that a space that admits a Kähler metric admits other Kähler metrics as well. If you can see one, you can easily get all the others. I was trying to find out if there is one metric that is better than the others—a ‘rounder’ one, you might say—one that gives you the most information and smoothes out the wrinkles.” The Calabi conjecture, he says, is about trying to find the “best” metric.3
Or as Robert Greene puts it, “you’re trying to find the one metric given to you by God.”4
The best metric for the purposes of geometry sometimes means “homogenous,” which implies a certain uniformity. When you know one part of a surface, you pretty much know it all. Owing to its constant curvature—and by that we also mean constant sectional curvature—a sphere is like that. The pinnacle of regularity, it looks the same everywhere (unlike, say, a football, whose sharp ends stand out, literally and physically, from other parts of the surface). While spheres are attractive for this reason, a Calabi-Yau manifold of complex dimension greater than one cannot have constant sectional curvature, unless it is completely flat (in which case its sectional curvature is zero everywhere). If this property is ruled out in the manifolds we seek, which are not totally flat and uninteresting, “the next best thing is to try to make the curvature as nearly constant as possible,” Calabi says.5 And the next best thing happens to be constant Ricci curvature—or, more specifically, zero Ricci curvature.
The full Calabi conjecture is more general than just setting the Ricci curvature to zero. The constant Ricci curvature case is also very important, especially the negative curvature case, which I eventually drew on to solve some noted problems in algebraic geometry (as will be discussed in Chapter 6). Nevertheless, the zero Ricci curvature case is special because we’re saying that the curvature is not only constant, but it’s actually zero. And this posed a special challenge: finding the metric for a manifold, or class of manifolds, that comes close to perfection without being totally boring.
There was a catch, however. Two decades after Calabi issued his proposition, very few mathematicians—save for the author of the conjecture himself—believed it to be true. In fact, most people thought it was too good to be true. I was among them, but was not content to stay on the sidelines, quietly harboring my doubts. On the contrary, I was dead-set on proving this conjecture wrong.