The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)

Chapter 9. BACK TO THE REAL WORLD

Upon meeting the Good Witch Glinda in The Wizard of Oz, Dorothy recounted the entire story of “how the cyclone had brought her to the Land of Oz, how she had found her companions, and of the wonderful adventures they had met with. ‘My greatest wish now,’ she added, ‘is to get back to Kansas.’”1

Upon hearing this narrative so far, a tale frequented by visits from the Good Doctor Witten and others, during which we’ve heard of some wonderful adventures in the Land of Calabi-Yau—with its hidden dimensions, mirror partners, supersymmetry, and vanishing first Chern classes—some readers may, like Dorothy, be yearning for more familiar surroundings. The question, as always, is this: Can we get there from here? Can the combination of string theory and Calabi-Yau manifolds reveal secrets of a concealed, higher-dimensional domain that we can imagine but never set foot in—the theoretical equivalent of Oz—while also teaching us something new about the more concrete physical realm that some call Kansas?

“We can write physical theories that are interesting to look at from a mathematics standpoint, but at the end of the day, I want to understand the real world,” says Volker Braun, a physicist at the Dublin Institute for Advanced Studies. 2 In our attempt to relate string theory and Calabi-Yau manifolds to the real world, the obvious point of comparison is particle physics.

The Standard Model, which describes matter particles and the force particles that move between them, is one of the most successful theories of all time, but as a theory of nature, it is lacking in several respects. For one thing, the model has about twenty free parameters—such as the masses of electrons and quarks—that the model does not predict. Those values must be put in by hand, which strikes many theorists as unforgivably ad hoc. We don’t know where those numbers come from, and none of them seems to have a rationale in mathematics. String theorists hold out hope of providing this mathematical rationale, with the only free parameter—besides the string’s tension (or linear energy density)—being the geometry of the compact internal space. Once you pick the geometry, the forces and particles should be completely fixed.

The aforementioned 1985 paper by Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten (see Chapter 6) “showed that you could bring all the key ingredients together to get a world that looked, at least in rough terms, like the Standard Model,” says Candelas. “The fact that you could do this in a theory that included gravity led to a lot of interest in string theory.”3 One success of the Candelas et al. model is that it produced chiral fermions, a feature of the Standard Model whereby every matter particle has a kind of “handedness,” with the left-handed version differing from its right-handed mirror image in important ways. As we saw earlier, their model also divided elementary particles into four families or generations, rather than the three that the Standard Model calls for. Although that number was off by one, Candelas says, “the main point was to show that you could get different generations—that you could get the repeated structure we see in the Standard Model.”4

Strominger was equally sanguine, calling their pioneering Calabi-Yau compactifications “a spectacular jump from the basic principles of string theory to something close to the world we live in. It was like shooting a basketball from the far side of the court and having it sail through the hoop,” he says. “In the space of all the things that could possibly happen in the universe, we came amazingly close. Now we want to do better, to find something that is not more or less right but exactly right.”5

A year or so later, Brian Greene and colleagues produced a model that marked a step forward by getting the three generations that our theories require, chiral fermions, the right amount of supersymmetry (which we denote as N = 1), neutrinos with some mass (which is good) but not too much (which is even better), as well as reproducing the fields associated with the Standard Model forces (strong, weak, and electromagnetic). Perhaps the biggest shortcoming of their model was the presence of some extra (unwanted) particles that were not part of the Standard Model and had to be eliminated by various techniques. On the plus side, I was struck by the simplicity of their approach—that all they had to do, in essence, was pick the Calabi-Yau manifold (in this case opting for one that I had constructed), and with that single choice, they carried us closer toward generating the Standard Model. Although progress has been made in a number of areas in the decades since, string theory and string theorists have yet to realize the Standard Model. From our current vantage point, we don’t even know whether string theory can reproduce the Standard Model.

Hard as this task now appears, proponents hope that string theory will not only take us there but eventually take us well beyond the Standard Model, which is where they believe we ultimately must go. For we already know the Standard Model is not the last word in physics. In the past decade or so, that model has been modified, or extended, a number of times to account for new experimental findings, such as the 1998 discovery that neutrinos, previously assumed to be massless, in fact have some mass. Furthermore, we now believe that 96 percent of the universe consists of dark matter and dark energy—mysterious forms of matter and energy about which the Standard Model says nothing at all. We expect there will be other discoveries to grapple with as well—whether they be the detection of supersymmetric particles (some of which are leading dark-matter candidates) or something completely unexpected—as the Large Hadron Collider goes about its business of smashing protons at high energies and seeing what comes out.

Even though Candelas and company and Greene and company weren’t able to replicate the Standard Model, their compactifications surpassed it in at least one respect, as they made inroads toward achieving the Minimal Supersymmetric Standard Model. The MSSM, as it’s called, is an extension of the conventional model, with supersymmetry added to it, meaning that it includes all the supersymmetric partners that are not included in the Standard Model itself. (Subsequent, string-theory-based efforts to realize the Standard Model, which will be discussed later in this chapter, encompass supersymmetry as well.)

For those who believe supersymmetry should be part of a theory of nature—a list that would probably include (though not be limited to) most string theorists— the Standard Model is certainly lacking in that respect. But there’s another major deficiency, which has come up repeatedly in these pages—namely, that the Standard Model, a theory of particle physics, overlooks gravity altogether, which is why it can never provide the ultimate description of our universe. Gravity is left out of the model for two reasons: For one thing, it is so much weaker than the other forces—strong, weak, and electromagnetic—that it is essentially irrelevant for the study of particle interactions on a small scale. (As a case in point, the gravitational force between two protons is roughly 1035 times weaker than the electromagnetic force they experience. Put in other terms, a magnet the size of a button is able—by virtue of the electromagnetic force—to pick a paper clip up off the ground, thereby overcoming the gravitational pull of the entire planet Earth.) The second reason, as has been amply discussed, is that no one yet knows how to tie gravity (described by general relativity) and the other forces (described by quantum mechanics) into a single, seamless theory. If string theory succeeds in reproducing the Standard Model while also bringing gravity into the fold, we’ll be that much closer to having a complete theory of nature. And if string theory achieves this end, we’ll have not only a Standard Model with gravity but also a supersymmetric Standard Model with gravity.

Various approaches to realizing the Standard Model are being tried—those involving orbifolds (which are like folds in flat space), intersecting branes, stacked branes, and the like—with advances being made on multiple fronts. This discussion, however, will focus on just one of those avenues, that of E8 X E8 heterotic string theory, which is one of the theory’s five varieties. We’ve made this choice not because it’s necessarily the most promising (which I’m in no position to judge), but because efforts along these lines are closely tied to geometry, and it is the branch that arguably has the longest tradition of trying to go from Calabi-Yau geometry to the real world.

I’m not playing up the geometry angle simply because it happens to be the main thrust of this book. It’s also vital to the endeavor about which we speak. For one thing, we cannot describe forces—an essential part of the Standard Model (and of any purported theory of nature)—without geometry. As Cumrun Vafa has said, “the four forces all have geometric underpinnings and three of them—the electromagnetic, weak, and strong—are related to each other by symmetry.”6 The Standard Model weaves together three forces and their associated symmetry (or gauge) groups: special unitary group 3, or SU(3), which corresponds to the strong force; special unitary group 2, or SU(2), which corresponds to the weak force; and the first unitary group, or U(1), which corresponds to the electromagnetic force. A symmetry group consists of the set of all operations, such as rotations, that you can do to an object and keep it unchanged. You take the object, apply the symmetry operation to it once or as many times as you want, and in the end, the object looks the same as when you started. In fact, you could not tell that anything was done to it.

Perhaps the simplest group to describe is U(1), which involves all the rotations you can do to a circle that is sitting in a two-dimensional plane. This is a one-dimensional symmetry group, because the rotations are around a single, one-dimensional axis perpendicular to the circle and lying at its center. SU(2) relates to rotations in three dimensions, and SU(3), which is more abstract, very roughly involves rotations in eight dimensions. (The rule of thumb here is that any group SU[n] has a symmetry of dimension n2 - 1.) The dimensions of the three subgroups are additive, which means that the overall symmetry of the Standard Model is twelve-dimensional (1 + 3 + 8 = 12).

As solutions to the Einstein equation, Calabi-Yau manifolds of a particular geometry can help us account for the gravity part of our model. But can these manifolds also account for the other forces—those found in the Standard Model—and if so, how? In answering that question, I’m afraid we’ll have to take a roundabout path. Particle physics, as presently cast, is a quantum field theory, meaning that all the forces, as well as all the particles, are represented by fields. Once we know these fields, which permeate four-dimensional space, we can derive the associated forces. These forces, in turn, can be described by vectors that have both an orientation and a size, meaning that at every point in space, an object will feel a push or pull in a certain direction and at a certain strength. At an arbitrary spot in our solar system, for instance, the gravitational force exerted on an object such as a planet is likely to point toward the sun, and the magnitude of that force will depend on the distance to the sun. The electromagnetic force exerted on a charged particle occupying a given spot, similarly, will depend on its location with respect to other charged particles.

The Standard Model is not only a field theory, but a special kind of field theory called a gauge theory—an approach that progressed greatly in the 1950s through the work of physicists Chen Ning Yang and Robert Mills (first mentioned in Chapter 3). The basic idea is that the Standard Model takes its various symmetries and combines them into a kind of composite symmetry group, which we denote as SU(3) ╳ SU(2) ╳ U(1). What makes these symmetries special and unlike ordinary symmetries is that they are gauged. That means you can take one of the allowed symmetry operations, such as rotation on a plane, and apply it differently at different points in spacetime—rotating by, say, 45 degrees at one point, 60 degrees at another point, and 90 degrees at another. When you do that, despite the nonuniform application of the symmetry, the “equations of motion” (which control the dynamic evolution of fields) don’t change, and neither does the overall physics. Nothing looks different.

Symmetries don’t normally operate this way unless they are gauge symmetries. In fact, the Standard Model has four “global” symmetries relating to matter particles and charge conservation that are not gauged. (These global symmetries act on the matter fields of the Standard Model that we’ll be talking about later in the chapter.) There’s another global symmetry in the Standard Model, and in field theories in general, that is not gauged, either. Called the Poincaré symmetry, it involves simple translations (such as moving the whole universe one foot to the right, or doing the same experiment in two different laboratories) and rotations, where the final outcome looks the same.

However, if you want some of your symmetries to be gauged, Yang and Mills figured out that you need to put something else, something extra, into your theory. That “something else” happens to be gauge fields. In the Standard Model, the gauge fields correspond to the symmetries that are gauged—SU(3) ╳ SU(2) ╳ U(1)—which means, by association, that the gauge fields also correspond to the three forces that are incorporated in the model: the strong, weak, and electromagnetic. Incidentally, Yang and Mills were not the first to construct a U(1) gauge theory describing electromagnetism; that had been achieved decades before. But they were the first to work out a gauge theory for SU(2)—a feat that showed the way to devise SU(n) theories for any n of two or greater, including SU(3).

The introduction of gauge fields enables you to have a theory with symmetries that are gauged, which in turn allows you to keep physics invariant, even when symmetry operations are differentially applied. Physicists didn’t make the Standard Model that way just because it struck them as more elegant or more appealing. They made it that way because experiments told them that this is how nature works. The Standard Model is, therefore, a gauge theory for empirical reasons rather than for, say, aesthetic ones.

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9.1—Chen Ning Yang and Robert Mills, authors of the Yang-Mills theory (Courtesy of C. N. Yang)

While physicists tend to speak in terms of gauge fields, mathematicians often express the same ideas in terms of bundles, which are a mathematical way of representing the fields that relate to the three forces. String theorists straddle the line between physics and math, and bundles play a key role in the heterotic constructions we’ll be talking about momentarily.

Before getting to that, we need to explain how Calabi-Yau manifolds relate to these gauge fields (which mathematicians refer to as bundles). The fields that we see—four-dimensional gravity and the SU(3) ╳ SU(2) ╳ U(1) gauge fields associated with the other three forces—unquestionably exist in the four-dimensional realm we inhabit, in keeping with our observations. Yet the gauge fields actually exist in all ten dimensions that string theory describes. The component that lies in the six compactified dimensions of the Calabi-Yau gives rise to the four-dimensional gauge fields of our world and hence to the strong, weak, and electromagnetic forces. In fact, it’s fair to say that those forces are generated by the internal structure of the Calabi-Yau, or so string theory maintains.

We’ve said a bit about symmetry so far without mentioning the challenge that model builders face in what’s called symmetry breaking. In the heterotic version of string theory we’ve been discussing, the ten-dimensional spacetime from which we start is endowed with what’s called E8 X E8 symmetry. E8 is a 248-dimensional symmetry group that can be thought of, in turn, as a gauge field with 248 components (much as a vector pointing in some arbitrary direction in three dimensions has three components—described as the xy, and z components). E8 X E8 is an even bigger symmetry group of 496 (248 + 248) dimensions, but for practical purposes, we can ignore the second E8. Of course, even the 248 symmetry dimensions pose a problem for re-creating the Standard Model, which has only twelve symmetry dimensions. Somehow, we’ve got to “break” the 248-dimensional symmetry of E8 down to the twelve we want.

Let’s go back to our example of a two-dimensional sphere or globe, which has rotational symmetry in three dimensions and belongs to the symmetry group SO(3). (The term SO here is short for special orthogonal group, because it describes rotations around an orthogonal axis.) You can take that sphere and spin it around any of three axes—xy, and z—and it will still look the same. But we can break that symmetry in three dimensions by insisting that one point must always stay fixed. On our planet, we could single out the north pole as that point. Now only one set of rotations, those that happen around the equator (on an axis that runs between the two poles), will keep that point (the north pole) fixed and unwavering. In this way, the threefold symmetry of the sphere has been broken and reduced to a one-dimensional symmetry, U(1).7

In order to get down to four dimensions and the Standard Model with its twelve-dimensional symmetry group, we have to find some way of breaking the symmetry of the E8 gauge group. In the E8 case, we can break symmetry by choosing a particular configuration in which some of the 248 components of the big gauge field are turned on or turned off. In particular, we’ll find a way to leave twelve of those little fields turned off, which is kind of like insisting that one spot on the sphere, the north pole, is not going to move. But they can’t be just any twelve fields; they have to be the right ones to fit into the SU(3) ╳ SU(2) ╳ U(1) symmetry groups. In other words, when you’re done breaking down the massive E8 group, what you’ll have left in four dimensions are just the gauge fields of the Standard Model. The other fields, which correspond to the broken symmetries, don’t disappear entirely. By virtue of being turned on, they’ll reside at a high-energy regime that puts them far beyond our reach, totally inaccessible to us. You might say the extra symmetries of E8 are hidden away in the Calabi-Yau.

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9.2—Owing to its great symmetry, a sphere remains unchanged under rotations under any axis running through the center. We can “break symmetry,” however, by insisting that the north pole must remain fixed during rotation. Now rotation is only permitted on one axis (running between the north and south poles) rather than on any axis. Adherence to this condition breaks, or constrains, the unlimited rotational symmetry of the sphere.

Nevertheless, a Calabi-Yau manifold can’t reproduce the Standard Model on its own. This is where bundles, which are literally extensions of the manifold, come in. Bundles are defined as groups of vectors attached to every point on the manifold. The simplest type of bundle is known as the tangent bundle. Every Calabi-Yau has one, but—as the tangent bundle of a Calabi-Yau is even harder to picture than the manifold itself—let’s instead consider the tangent bundle of an ordinary two-dimensional sphere. If you pick a point on the surface of that sphere and draw two vectors tangent to that point, those vectors will define a plane (or a disk within a plane if you limit the vectors to an arbitrary length). If we do the same thing at every point on the surface and put all those planes (or disks) together, that collective entity will be the bundle.

Note that the bundle necessarily includes the manifold itself because the bundle contains, by definition, every single point on the manifold’s surface. For that reason, the tangent bundle of a two-dimensional sphere is actually a four-dimensional space, because the tangent space has two degrees of freedom—or two independent directions in which to move—and the sphere itself, being part of the bundle, has another two degrees of freedom that are themselves independent of the tangent space. The tangent bundle of a six-dimensional Calabi-Yau, similarly, is a twelve-dimensional space, with six degrees of freedom in the tangent space and another six degrees of freedom in the manifold itself.

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9.3—At each point on the surface of a sphere, there is a tangent plane intersecting the sphere at just that point and nowhere else. The tangent bundle for the sphere consists of all the planes tangent to every point on the sphere. Since the tangent bundle, by definition, includes every point on the sphere, it must also include the sphere itself. Although it would be impossible to draw a tangent bundle (with its infinite number of intersecting tangent planes), instead we show a sphere with patches of tangent planes at a few representative points.

Bundles are critical to string theorists’ attempts to re-create particle physics in terms of Yang-Mills theory, under which gauge fields are described by a set of differential equations, unsurprisingly called the Yang-Mills equations. What we’d like to do here, specifically, is to find solutions to the equations for the gauge fields that live on the Calabi-Yau threefold. Since the principal reason Calabi-Yau manifolds entered string theory in the first place was to satisfy the requirement for supersymmetry, the gauge fields must also obey supersymmetry. That means we have to solve a special version of the Yang-Mills equations, the supersymmetric version, which are called the Hermitian Yang-Mills equations. In fact, these equations yield the least amount of supersymmetry you can have (what’s known as N = 1 supersymmetry), which is the only amount of supersymmetry consistent with present-day particle physics.

“Before string theory forced us to get fancy, most physicists didn’t think much about geometry and topology,” says University of Pennsylvania physicist Burt Ovrut. “We just wrote down equations like the Yang-Mills equations and tried to solve them.” The only catch is that the Hermitian Yang-Mills equations are highly nonlinear differential equations that nobody knows how to solve. “To this day,” Ovrut says, “there is not a single known [explicit] solution to the Hermitian Yang-Mills equations on a six-dimensional Calabi-Yau manifold. So you’d have to stop there, were it not for the work of some geometers who showed us another way to proceed.”8

Bundles give us a way around this nonlinear differential logjam, because we can think of the bundle attached to the Calabi-Yau as an alternate description of the gauge fields that the Yang-Mills equations define. Exactly how that can be done is described by the DUY theorem, whose name is an acronym for Simon Donaldson (now at Imperial College), Karen Uhlenbeck at the University of Texas, and my name.

The idea behind the theorem is that the Hermitian Yang-Mills equations define a field that can be represented by a vector bundle. We proved that if you can construct a bundle on the Calabi-Yau that satisfies a specific topological condition—namely, that it is stable (or, to put it more technically, slope-stable) —then that bundle will admit a unique gauge field that automatically satisfies those equations. “This wouldn’t have helped if you had traded in one infinitely difficult problem for another infinitely difficult problem,” notes Ovrut. “But the second problem of constructing a stable bundle is much easier, and as a result, you don’t have to solve those horrible differential equations at all.”9

In other words, we found a geometric solution to a problem that we’d been unable to solve by other means. We showed that you don’t have to worry about fields or differential equations. All you have to worry about is constructing a stable bundle. So what does it mean for a bundle to be slope-stable? The slope of a curve, as we’ve discussed, is a number that relates to curvature, and slope stability in this case relates to the curvature of the bundle. Putting it in loose terms, “the slope expresses a sense of balance,” explains University of Pennsylvania mathematician Ron Donagi. “It says that the curvature in one direction can’t be much bigger than the curvature in another direction. No matter which way you face, no direction can be too extreme in relation to the others.”10 Any bundle can be divided up into smaller pieces, or sub-bundles, and the stability condition means that the slope of any of these sub-bundles cannot be bigger than the slope of the bundle as a whole. If that condition is met, the bundle is slope-stable and the gauge fields will satisfy the Hermitian Yang-Mills equations. Supersymmetry will be satisfied as a result.

In some ways, the idea of slope stability, which is central to the DUY theorem, is a consequence of the Calabi-Yau theorem, because the theorem imposed specific curvature requirements on a Calabi-Yau manifold guaranteeing that its tangent bundle must be slope-stable. And the fact that the Calabi-Yau equations and Hermitian Yang-Mills equations are the same for the tangent bundle when the background metric is Calabi-Yau—another consequence of the Calabi conjecture proof—inspired me to think about the relation between slope stability and the Hermitian Yang-Mills equations. The idea that emerged was that a bundle will satisfy those equations if and only if it is stable.

That, in fact, is what Donaldson proved in his part of DUY—which he published in 1985, specifically regarding the special case of two complex dimensions. Uhlenbeck and I worked independently of Donaldson, and in a paper that came out a year later, we proved that the same result applied to any complex dimension and, consequently, to any space with an even number of real dimensions. I still consider this one of the most difficult theorems I’ve ever proved—or, in this case, co-proved. Our work, in combination with Donaldson’s, is now collectively referred to as DUY.

The theorem is very much analogous to the proof of the Calabi conjecture, as both reduce a problem involving some nasty nonlinear system of equations that we don’t know how to deal with to a problem of geometry, where we might have an inkling of how to proceed. In the Calabi case, I never solved the relevant differential equations explicitly. I just showed that if a manifold satisfies certain conditions (compact, Kähler, vanishing first Chern class) that can be checked through standard procedures of algebraic geometry, then a solution to those equations (in the form of a Ricci-flat metric) must exist. DUY works the same way, specifying the conditions on the bundle (namely, slope stability) such that a solution to the Hermitian Yang-Mills equations always exists. Algebraic geometers have also developed methods to assess a bundle’s stability, although this turns out to be more complicated to check than whether a manifold’s first Chern class is vanishing or not.

Some people, including physicists outside this particular area of mathematics, find DUY amazing because, on the face of it, the bundle conditions appear to have nothing to do with the differential equations you’re hoping to solve. But it wasn’t amazing to me, and if anything, it seemed like a natural extension of the Calabi conjecture. The Calabi proof is all about the manifold, the Calabi-Yau, whereas the DUY theorem is all about the bundle. You’re looking for the metric of the bundle, but the metric for the manifold itself is given to you as part of the starting information. You can pick any “background” metric you want, including the Calabi-Yau metric.

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9.4—Karen Uhlenbeck (Image courtesy of the University of Texas at Austin)

The point of intersection between the Calabi conjecture and the DUY theorem is the tangent bundle. And here’s why: Once you’ve proved the existence of Calabi-Yau manifolds, you have not only those manifolds but their tangent bundles as well, because every manifold has one. Since the tangent bundle is defined by the Calabi-Yau manifold, it inherits its metric from the parent manifold (in this case, the Calabi-Yau). The metric for the tangent bundle, in other words, must satisfy the Calabi-Yau equations. It turns out, however, that for the tangent bundle, the Hermitian Yang-Mills equations are the same as the Calabi-Yau equations, provided the background metric you’ve selected is the Calabi-Yau. Consequently, the tangent bundle, by virtue of satisfying the Calabi-Yau equations, automatically satisfies the Hermitian Yang-Mills equations, too.

The upshot of this is that the tangent bundle is really the first special case of the DUY theorem—the theorem’s first solution—although that fact came out of the proof of the Calabi conjecture ten years before DUY was conceived. That’s not the most interesting thing about DUY, however. The true power of DUY lies in prescribing the conditions (again regarding stability) that other bundles (not just the tangent bundle) must satisfy in order to ensure that a solution to the Hermitian Yang-Mills equations exists.

Before our paper on the subject came out in 1986, I told Edward Witten that Yang-Mills theory seemed to fit quite naturally with Calabi-Yau manifolds and therefore ought to be important for physics. Witten didn’t see the relevance at first, but within about a year, he took my suggestion even farther, showing how this approach could be used in Calabi-Yau compactifications. Once Witten’s paper came out, given his stature in the field, other people became interested in applying DUY to string theory. So this is another example of geometry’s taking the lead even though it doesn’t always go that way.

Now let’s see how we can put some of this geometry (and topology) to use in order to generate particle physics from string theory. The first step is to pick a Calabi-Yau manifold, but not just any manifold will do. If we want to utilize certain approaches that have proved effective in the past, we need to pick a non-simply connected manifold—that is, a manifold with a nontrivial fundamental group. This, as I hope you recall, means you can find in that space a loop that cannot be shrunk to a point. The manifold, in other words, has to be like a torus rather than a sphere and have at least one hole. The presence of such a hole, cycle, or loop inevitably affects the geometry and topology of the bundle itself, which in turn affects the physics.

The second step is to construct a bundle that not only gives you the gauge fields of the Standard Model but also cancels the anomalies—the negative probabilities, unwanted infinities, and other irksome features—that had beset some of the earliest versions of string theory. When Michael Green and John Schwarz showed how to cancel anomalies in their momentous 1984 paper, their argument was framed in terms of gauge fields. Expressing this same idea in geometric and topological terms, one could say that a bundle will satisfy the anomaly cancellation requirement if its second Chern class equals the second Chern class of the tangent bundle.

We have already discussed the notion of Chern class, a technique for classifying topological spaces and measuring crude differences between them (see Chapter 4). The first Chern class, as noted, is vanishing (or zero) if you can orient all the tangent vectors on a manifold in the same direction. In some sense, this is like being able to comb a head of hair without getting a cowlick somewhere. This is impossible on a two-dimensional sphere, but you can avoid cowlicks on the surface of a two-dimensional torus. Thus we say the torus has a vanishing first Chern class, whereas the sphere’s first Chern class is nonvanishing.

The second Chern class can be described in roughly the same way except that we need to consider two vector fields on some manifold rather than just one. (The vector fields we’re talking about here are complex, meaning that the individual vector coordinates are described by complex numbers.) Assuming these two vectors fields are independent, at most points on the manifold they are likely to be pointing in different directions. But at certain points, a vector from each field may point in the same complex direction, or both vectors may go to zero. In fact, there may be a whole set of points where this is true. This set of points forms a closed, two-dimensional surface within our six-dimensional manifold, and collectively, these points represent the second Chern class.

So how does this tie in with anomaly cancellation? Green and Schwarz showed that no matter how bad the anomalies may be, if you can get them to cancel each other out and thereby disappear, you might have a viable theory after all. One way of getting rid of these bothersome anomalies is to make sure that the bundle you pick has the same second Chern class as that of the tangent bundle.

As for why that might work, we must remember that the bundles we’re talking about here are, in a sense, stand-ins for the background fields, the gravitational and gauge fields, from which the forces of nature can be derived. The tangent bundle of the Calabi-Yau, for example, is a good facsimile of the gravitational field because the Calabi-Yau, as defined by a special metric, solves Einstein’s gravity equations. Gravity, in other words, is literally encoded in that metric. But the metric also ties in with the tangent bundle, and here’s why: The metric, as stated before, provides a function for computing the distance between any two points, A and B, on the manifold. We take all possible paths between A and B and break up each path into a set of tiny vectors, which are, in fact, tangent vectors; taken together, these vectors form a tangent bundle. That’s why in our attempts at anomaly cancellation, we can use the tangent bundle of a Calabi-Yau to cover the gravity end of things.

We’ll then choose an additional vector bundle for the purpose of reproducing the gauge fields of the Standard Model. So now we have our two bundles, one giving us the gravitational field and the other the gauge fields. Unfortunately, each field will inevitably have anomalies in it—there’s no way to keep them out—but Green and Schwarz showed us there’s no need for despair. They demonstrated, Donagi explains, “that the anomaly coming from gravity has the opposite sign as the anomaly coming from the gauge field. So if you can engineer things so that they have the same magnitude, they will cancel each other out.”11

To find out if this works, we take the second Chern class of both the tangent bundle of the Calabi-Yau and the gauge field bundle. The answer we get for each bundle depends on those out-of-the-ordinary points where the vector fields align or vanish. However, you can’t just count the number of such points, because there’s actually an infinite number of them. What we can do, instead, is to compare the curves (of one complex dimension) that those points trace out. The curves corresponding to each of those bundles do not have to be identical for the second Chern classes to match, but they do have to be homologous.

Homology is a subtle concept, perhaps best defined through example, and I’m going to try to pick the simplest example possible—a donut with two holes. Each hole is cut out by a circle—a one-dimensional object—but each circle bounds a hole that is two-dimensional. And that’s what we mean by homologous, the two circles of our double-donut being an example. Stating it in broader terms, we call two curves or cycles homologous if they are of the same dimension and bound a surface or manifold of one dimension higher. We use the term Chern class to indicate that there is a whole class of curves that are linked in this way through homology. The reason we brought this concept up in the first place is that if the curves for our two bundles are homologous—the tangent one representing gravity and the other one representing the gauge fields—then the second Chern class of these bundles will match. And as a result, the anomalies of string theory will magically cancel out, which is what we were after all along.

When people first began testing out these ideas—as Candelas, Horowitz, Strominger, and Witten did in their 1985 paper—they almost always used the tangent bundle, which was the only bundle that was well-known at the time. If you use the tangent bundle, then the second Chern class of your bundle cannot help but match the second Chern class of the tangent bundle. So you’re covered on that score, but the tangent bundle will also satisfy the stability condition (which, as mentioned earlier, is a direct consequence of the Calabi conjecture proof). But investigators felt, nevertheless, that if other bundles met the above requirements—including stability—that could allow for more flexible options in terms of physics. Candelas says that even back in 1985, “we realized there were more general ways of doing things, bundles other than the tangent bundle that we might use. Although we knew it could be done, we didn’t know then how to do it in a hands-on way.”12

In the meantime, since the advent of the “second string revolution” in the mid-1990s, researchers saw that it was possible to loosen the restrictions on bundles further, thereby opening up many new possibilities. In M-theory, there is an extra dimension, and that gives you the freedom—and more elbow room—to accommodate extra fields that correspond to branes, the essential new ingredients ushered in by M-theory. With this extra ingredient, the brane now in the picture, the second Chern class of the gauge bundle no longer had to be equal to the second Chern class of the tangent bundle; it could instead be less than or equal. That’s because the brane itself—or the curve upon which it’s wrapped—has its own second Chern class, which can be added to the second Chern class of the gauge bundle to match that of the tangent bundle and thereby ensure anomaly cancellation. As a result, physicists now had a broader variety of gauge bundles to work with.

“Every time you weaken a condition—in this case, changing an equality into an inequality—you have more examples to draw on,” Donagi explains.13 Going back to our earlier example of a sphere or a beach ball, instead of attaching a tangent plane (or piece thereof) to every point on the ball’s surface, we could attach a “normal” bundle with vectors pointing out from the surface. There are many other bundles one might construct by attaching a particular vector space to different points on the manifold and then gluing all these vector spaces together to make a bundle.

While this new freedom from M-theory has enabled researchers to explore a wider range of bundles, so far they haven’t come up with many more examples that actually work. But at least the possibility now exists. The first step, again, is selecting a bundle that is stable and gets rid of the anomalies. From the DUY theorem, we know that such a bundle can give you the gauge fields, or forces, of the Standard Model.

Of course, the Standard Model is not just about forces. It’s a theory of particle physics, so it’s got to say something about particles, too. The question then is whether or how the particles of nature might be tied up with Calabi-Yau manifolds. There are two kinds of particles to talk about—matter particles, which are the things we can touch, and force-mediating particles, which include the photons that deliver light along with other particles we can’t see, like weak bosons and gluons.

The force particles are in some sense easier to derive because if you got all the gauge fields right in the previous step, with all the right symmetry groups, then you already have these particles. They are literally part of the force fields, and the number of symmetry dimensions in each gauge field corresponds to the number of particles that communicate the force. Thus the strong force, which is endowed with eight-dimensional SU(3) symmetry, is mediated by eight gluons; the field of the weak force, which is endowed with three-dimensional SU(2) symmetry, is mediated by three particles, the W+, W-, and Z bosons; and the electromagnetic field, which is endowed with one-dimensional U(1) symmetry, is mediated by a single particle, the photon.

We can picture these particles in action fairly easily. Suppose two guys are roller-skating in parallel, and one throws a volleyball to the other. The guy who throws the ball will veer off in the direction opposite to that in which the ball travels, while the guy who catches the ball will veer off in the same direction as the ball is thrown. If you viewed this interaction from an airplane flying high enough above that you couldn’t see the ball, it would appear as if there were a repulsive force pushing them apart. But if you looked extremely closely at that repulsive force and essentially “quantized” it, you’d see that the movements of the skaters were being caused by a discrete object, a volleyball, rather than by some invisible field. Quantizing the fields, either the matter fields or the gauge (force) fields, means that among all possible fluctuations or vibrations, you will only allow certain ones. Each specially selected fluctuation corresponds to a wave at a specific energy level and hence to a specific particle.

“That’s what happens in the Standard Model,” Ovrut says. “The matter particles are like the guys on roller skates, and the force particles are the volleyballs— the photons, gluons, and the W+, W-, and Z bosons—that are exchanged between them.”14

The ordinary matter particles will, however, take a bit more explaining. All normal matter particles, such as electrons and quarks, have spin-½, spin being an intrinsic, quantum mechanical property of all elementary particles that relates to a particle’s internal angular momentum. These spin-½ particles are solutions to the Dirac equation, which was discussed in Chapter 6. In string theory, you have to solve this equation in ten dimensions. But when you fix the background geometry by selecting a Calabi-Yau manifold, the Dirac equation can be divided into a six-dimensional and a four-dimensional component. Solutions to the six-dimensional Dirac equation fall into two categories: heavy particles—many trillions of times heavier than anything observed in high-energy-accelerator experiments—and the particles of everyday life, whose mass is so small we can call it zero.

Regardless of the particle mass, finding the solutions to those component equations is quite difficult. Fortunately, geometry and topology again can save us from having to solve nearly impossible differential equations. In this case, we have to figure out the cohomology of the gauge bundle, as researchers at the University of Pennsylvania—including Braun (formerly at Penn), Donagi, Ovrut, and Tony Pantev—have shown. Cohomology is closely related to homology and, like homology, is concerned with whether two objects can be deformed into each other. The two concepts, as Donagi puts it, represent distinct ways of keeping track of the same properties.15 Once you determine the cohomology class of a bundle, you can use it to find solutions to the Dirac equation, and generate the matter particles. “It’s a beautiful mathematical approach,” Ovrut claims.16

Employing these techniques and others, Vincent Bouchard of the University of Alberta and Donagi, as well as Ovrut and his colleagues, have produced models that appear to get a lot of things right. Both groups claim to get the right gauge symmetry group, the right supersymmetry, chiral fermions, and the right particle spectrum—three generations of quarks and leptons, plus a single Higgs particle, and no exotic particles, such as extra quarks or leptons that are not in the Standard Model.

But there is considerable debate regarding how close these groups have actually come to the Standard Model. Some questions have been raised, for instance, about methodologies and phenomenological details, such as the presence of “moduli particles,” which will be discussed in the next chapter. Physicists I’ve heard from are of mixed opinion on this subject, and I’m not yet sold on this work or, frankly, on any of the attempts to realize the Standard Model to date. Shamit Kachru of Stanford considers the recent efforts another step forward, following on the advances of Candelas and Greene and their colleagues. “But no one,” Kachru says, “ has yet produced a model that hits it on the nose.”17 Michael Douglas of the Simons Center for Geometry and Physics at Stony Brook University agrees: “All of these models are kind of first cuts; no one has yet satisfied all the consistency checks of the real world. But even though both models are incomplete, we are still learning from all of this work.”18 Candelas credits the Bouchard-Donagi and Ovrut et al. models for showing us how to use bundles other than the tangent bundle. He believes this work will eventually point the way toward other models, noting that “it’s likely there are other possibilities out there. But until you do it, you don’t know what works.”19 Or if it works at all.

The next steps involve not only getting the right particles but also trying to compute their masses, without which we cannot make meaningful comparisons between a given model and the Standard Model. Before we can compute the mass, we need to determine the value of something called the Yukawa coupling constant, which describes the strength of interactions between particles—the interactions between the matter particles of the Standard Model and the Higgs field, and its associated particle (the Higgs boson), being of greatest relevance here. The stronger the coupling, the greater the mass of the particle.

Let’s take one particle to start with, say, the down quark. As with other matter particles, the field description of the down quark has two components—one corresponding to the right-handed form of this particle and one corresponding to the left-handed form. Because mass, in quantum field theory, comes from interactions with the Higgs field, we multiply the two fields for the down quark—the right- and left-handed versions—by the Higgs field itself. The multiplication in this case actually corresponds to that interaction, with the size of that product, or triple product, telling you how strongly or weakly the down quark and Higgs field interact.

But that’s just the first part of this complex procedure. One complication arises because the size of the triple product can vary as you move around on the “surface” of the Calabi-Yau. The Yukawa coupling constant, on the other hand, is not a variable quantity that depends on your location on the manifold. It is a global measure, a single number, and the way to compute that number is by integrating the size of the product of the down quark and Higgs fields over the entire manifold.

Remember that integration is really a process of averaging. You have some function (in this case, the product of three fields) that assumes different values at different spots on the manifold, and you want to get its average value. You have to do that because the Yukawa coupling is, in reality, a number rather than a function, just as the mass of a particle is a number as well. So what do you do? You chop the manifold up into tiny patches and determine the value of the function at each patch. Then you add up all those values and divide by the number of patches, and you’ll get the average. But while that approach can carry you pretty far, it won’t give you exactly the right answer. The problem is that the space we’re working on here, the Calabi-Yau manifold, is really curvy, and if you were to take a tiny “rectangular” patch (assuming, for the moment, that we’re in two dimensions) of size dx by dy, the size of that patch will vary depending on how flat or curved it is. So instead, you take the value of the function at a point in a particular patch and then weight that number by the size of the patch. In other words, you need a way of measuring how big the patch is. And for that we need the metric, which tells us the manifold’s geometry in exacting detail. The only catch here, as we’ve said many times before, is that no one has yet figured out a way to calculate the Calabi-Yau metric explicitly, which is to say, exactly.

That could be the end of the line: Without the metric, we can never get the mass and thus will never know whether the model we have is anywhere near the Standard Model. But there are some mathematical methods—numerical (as in computer-based) techniques—we can use to approximate the metric. Then it becomes a question of whether your approximation is good enough to yield a reasonable answer.

Two general approaches are presently being tried out, and both rely on the Calabi conjecture in some way. That conjecture, as noted many times before, says that if a manifold satisfies certain topological conditions, a Ricci-flat metric exists. Without producing the metric itself, I was able to prove that such a metric exists. The proof employed a so-called deformation argument, which basically involves showing that if you start with something—let’s say, some sort of metric—and keep deforming it in a certain way, that process will eventually converge on the metric you want. Once you can prove that this deformation process converges on the desired solution, there is a good chance that you can find a numerical scheme that converges as well.

Recently, two physicists—Matt Headrick of Brandeis University and Toby Wiseman of Imperial College—have performed numerical computations along these lines, working out an approximate metric for a K3 surface, the four-dimensional Calabi-Yau we’ve touched on often. The general strategy they used, called discretization, takes an object with an infinite number of points, such as the points tracing out a continuous curve, and represents it by a finite (discrete) number of points, with the hope being that this process will eventually converge on the curve itself. Headrick and Wiseman believe their process does converge, and while their results look good, they have not yet proved this convergence. One drawback of this approach has nothing to do with their analysis per se, but instead relates to the limitations of present-day technology: Available computers simply do not have the capacity to produce a detailed metric for six-dimensional Calabi-Yau manifolds. It all boils down to the fact that the calculation in six dimensions is much bigger, with many more numbers to be crunched, than the four-dimensional problem.

Computers will, no doubt, continue to improve and may eventually become powerful enough to bring the six-dimensional calculation within reach. In the meantime, there’s another way to proceed that faces fewer computational constraints. The approach dates back at least to the 1980s, when I proposed that a Ricci-flat metric can always be approximated by placing (or, more technically, “embedding”) a Calabi-Yau manifold in a very high-dimensional background space. This background space is called projective space, which is like a complex version of flat Euclidean space except that it’s compact. When you put something like a manifold in a bigger space, the subspace automatically inherits a metric (what we call an induced metric) from the background space. A similar thing happens when you put a sphere in ordinary Euclidean space; the sphere adopts the metric of the background space. Drawing on a familiar analogy, we can also think of a hole in Swiss cheese as being embedded in the larger space. And assuming we know how to measure distances in that larger space—the “Big Cheese,” as it were—then we know how to measure the size of that hole as well. In that sense, the embedded space, or hole, inherits a metric from the cheesy background space in which it sits.

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9.5—Through the process of discretization, you can approximate a one-dimensional curve and a two-dimensional surface with a finite number of points. The approximation, naturally enough, gets better and better as you increase the number of points.

In the 1950s, John Nash had proved that if you put a Riemannian manifold in a space of high enough dimensions, you can get any induced metric that you want. The Nash embedding theorem, which is one of this illustrious mathematician’s greatest works (among a long and diverse list, I might add), only applies to real manifolds sitting in real space. In general, the complex version of Nash’s theorem is not true. But I suggested that a complex version of the theorem might be true under certain circumstances. I argued, for example, that a large class of Kähler manifolds can be embedded into a higher-dimensional projective space in a manner such that the induced metric is arbitrarily close to the original metric, provided the induced metric is suitably scaled or “normalized”—meaning that all its vectors are multiplied by a constant. Being a special case of Kähler manifolds, Calabi-Yau manifolds with a Ricci-flat metric satisfy this topological condition. That means the Ricci-flat metric can always be induced, and can always be approximated, by embedding the manifold in a background or projective space of sufficiently high dimension. My graduate student at the time, Gang Tian, proved this in a 1990 paper, which was in fact his thesis work. Several important refinements of my original statement have been made since. This includes the thesis of another of my graduate students, Wei-Dong Ruan, who proved that an even better, more precise approximation of a Ricci-flat metric was possible.

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9.6—In geometry, we often talk about “embedding” an object or space in a higher-dimensional “background space.” Here we embed a square—a one-dimensional object, as it consists of a line segment bent several times—in a two-dimensional background space, a sphere.

The main refinement has to do with how you embed the Calabi-Yau in the background space. You can’t just plop it in some haphazard way. The idea is to pick a proper embedding so that the induced metric will be arbitrarily close to the Ricci-flat metric. To do this, you put the Calabi-Yau in the best place possible, the so-called balanced position, which is the one position, among all possibilities, where the inherited metric comes closest to Ricci flat. The notion of a balanced position was introduced in 1982 by Peter Li and me for the case of submanifolds (or subsurfaces) in a sphere sitting in real space. We later extended that result to the more general case of submanifolds in a complex, high-dimensional background (or projective) space. Jean-Pierre Bourguignon, the current director of IHES, then joined our collaboration, coauthoring a 1994 paper with us on the subject.

I had previously conjectured—at a geometry conference at UCLA—that every Kähler manifold that admits a Ricci-flat metric (including a Calabi-Yau) is stable, but the word stable is hard to define. In subsequent geometry seminars, I continued to stress the relevance of the Bourguignon-Li-Yau work, as it’s now called, to the notion of stability. Finally, some years later, my graduate student Wei Luo (who was based at MIT) made the connection between the stability of a Calabi-Yau and the balance condition. With the link provided by Luo, I was able to recast my conjecture to say that if you embed a Calabi-Yau in a very high-dimensional space, you can always find a position where it is balanced.

Simon Donaldson proved that conjecture to be true. His proof also validated the main thrust of this new approximation scheme—namely, that if you embed the Calabi-Yau in a background space of higher and higher dimension and satisfy the balance condition, the metric will get closer and closer to Ricci flat. Donaldson proved this by showing that the induced metrics form a sequence, in background spaces of increasing dimension, and that the sequence converges, approaching perfect Ricci flatness at infinity. The statement only holds, however, because the Calabi conjecture is true: When Donaldson demonstrated that the metric converges to the Ricci-flat metric, his conclusion hinged on the existence of a Ricci-flat metric.

Donaldson’s proof had practical ramifications as well, because he showed that there was a best way of doing the embedding—the balanced way. Framing the problem in this way gives you a means of attacking it and a possible computational strategy. Donaldson utilized this approach in 2005, numerically computing the metric for a K3 surface, and he showed there were no fundamental barriers to extending the technique to higher dimensions.20 In a 2008 paper, Michael Douglas and colleagues built on Donaldson’s result, deriving a numerical metric for a family of six-dimensional Calabi-Yau manifolds, the aforementioned quintic. Douglas is now collaborating with Braun and Ovrut on a numerical metric for the Calabi-Yau manifold in their model.

So far, no one has been able to work out the coupling constants or mass. But Ovrut is excited by the mere prospect that particle masses might be computed. “There’s no way to derive those numbers from the Standard Model itself,” he says, “but string theory at least offers the possibility, which is something we’ve never had before.” Not every physicist considers that goal achievable, and Ovrut admits that “the devil is in the details. We still have to compute the Yukawa couplings and the masses, and that could turn out completely wrong.”21

It’s doubtful, says Candelas, that the models we have in hand now will turn out to be the ultimate model of the universe. In trying to construct such a theory, he says, you have “an awful lot to get right. As you dig deeper into these models, sooner or later we’re likely to come to things that don’t work.”22 So rather than regarding the current models as the last word on the subject, we should view these efforts as part of a general learning process during which critical tools are being developed. Similar caveats apply to parallel efforts to realize the Standard Model that involve branes, orbifolds, or tori, none of which has yet reached that end, either.

There has been progress, says Strominger. “People have found more and more models, and some of the models are getting closer to what we observe around us. But there hasn’t been a repetition of making that basketball shot from clear across the court. And that’s what we’re still waiting for.”23

Invoking another sports analogy, Strominger compared the original Calabi-Yau compactification paper of 1985, which he coauthored with Candelas, Horowitz, and Witten, to hitting a golf ball from two hundred yards away and coming close to the hole. “There was a feeling that it was going to take only one more shot to get it in,” he recalls. But a couple of decades have passed, he says, and “physicists have yet to pick up that gimme.”24

“Twenty-five years is a long time in theoretical physics, and it’s only now that groups are making substantial headway,” says Candelas. “We’re finally reaching a stage where people can do something practical with these new ideas.”25

While acknowledging that researchers have made noteworthy strides, MIT’s Allan Adams argues that “it’s not correct to assume that the nearness to the Standard Model means we’re almost done.” On the contrary, he says, it’s hard to know how far we still have to go. Although we may appear to be close to our goal, there’s still a “great gulf” between the Standard Model and where we stand today.26

At the end of Dorothy’s adventures in the Land of Oz, she learned that she had the powers to get back home all along. After some decades of exploring the Land of Calabi-Yau, string theorists and their math colleagues (even those equipped with the penetrating powers of geometric analysis) are finding it hard to get back home—to the realm of everyday physics (aka the Standard Model)—and, from there, to the physics that we know must lie beyond. If only it were as easy as closing our eyes, tapping our heels together, and saying “There’s no place like home.” But then we’d miss out on all the fun.