The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)
Chapter 10. BEYOND CALABI-YAU
Crafting a successful theory is like running an obstacle course that’s never been run before. You get past one hurdle—going over it, around it, or maybe even under—knowing there are many more to come. And even though you’ve successfully cleared all the hurdles behind you, you don’t know how many lie ahead, or whether one coming up might stop you for good. Such is the case with string theory and Calabi-Yau manifolds, where we know that at least one hurdle still looming is of sufficient magnitude to potentially topple this whole glorious enterprise. I’m talking about the moduli problem, which has been the subject of many talks and papers, as well as the source of much grief and consternation. As we’ll see, what begins with the relatively simple goal of addressing this problem can take us far afield from where we began, at times leaving us without any goalposts in sight.
The size and the shape of any manifold with holes in it are determined by parameters called moduli. A two-dimensional torus, for example, is in many ways defined by two independent loops or cycles, one going around the hole and another going through. The moduli, by definition, measure the size of the cycles, which themselves govern both the size and shape of the manifold. If the cycle going through the donut hole is the smaller of the two, you’ll have a skinny donut; if it’s the larger, you’ll have a fat donut with a relatively small hole in the middle. A third modulus describes the degree to which the torus is twisted.
So much for the torus. A Calabi-Yau, which, as we’ve noted, can have upwards of five hundred holes, comes with many cycles of various dimensions and hence many more moduli—anywhere from dozens to hundreds. One way to picture them is as a field in four-dimensional spacetime. The field for the size moduli, for instance, assigns a number to every point in ordinary space corresponding to the size (or radius) of the unseen Calabi-Yau. A field of this sort—which is completely characterized by a single number at each point in space, with no direction or vector involved—is called a scalar field. One can imagine all sorts of scalar fields around us, such as those measuring the temperature at every point in space or the humidity, barometric pressure, and so forth.
The catch here is that if nothing constrains the manifold’s size and shape, you’re going to run headlong into the aforementioned moduli problem, which will dash any hopes you might have of eliciting realistic physics from this geometry. We’re faced with this problem when the scalar fields that relate to a manifold’s size and shape are massless, meaning that no energy is required to alter them. They are, in other words, free to change value without impediment. Trying to compute the universe under these ever-shifting circumstances is “like running in a race, and the finish line is always moving an inch away from you,” as University of Wisconsin physicist Gary Shiu puts it.1
There’s an even bigger problem: We know that such fields cannot exist in nature. For if they did, there’d be all kinds of massless moduli particles—associated with the scalar (moduli) fields—flying around at the speed of light. These moduli particles would interact with other particles with roughly the same strength as gravitons (the particles thought to mediate the gravitational force), thereby wreaking havoc with Einstein’s theory of gravity. Because that theory, as described in general relativity, works so well, we know these massless fields and particles simply cannot be there. Not only would their presence conflict with well-established gravitational laws, it would also give rise to a fifth force and perhaps additional forces that have never been seen.
So there’s the rub. Given that much of string theory now hinges on compactifications of Calabi-Yau manifolds, which have these moduli with their associated massless scalar fields and particles that don’t appear to exist, is string theory itself doomed? Not necessarily. There might be a way around this problem, because there are other elements of the theory—things that we already knew about but left out to make our calculations simpler. When included, these elements make the situation look considerably different. These additional ingredients include items called fluxes, which are fields like electric fields and magnetic fields, although the new fields from string theory have nothing to do with electrons or photons.
10.1—Fluxes can be thought of as lines of force that are not unlike the magnetic field lines shown here, although string theory includes fields that are more exotic and that point in the six compact directions we can’t see.
Again, let’s consider a two-dimensional torus, in this case, a particularly malleable donut whose shape is constantly shifting between skinny and fat. We can stabilize this torus into a fixed shape by wrapping wires through and around it. That is essentially the role of flux. Many of us have seen an effect like this when, say, a magnetic field is suddenly switched on, and iron filings, which were easily scattered about before, now assume a fixed pattern. The flux holds the filings in place, which is where they’ll stay unless additional energy is applied to move them. In the same way, the presence of fluxes means that it now takes energy to change the manifold’s shape, as the massless scalar fields have thus become scalar fields with mass.
Six-dimensional Calabi-Yau manifolds are more complex, of course, since they can have many more holes than a donut and the holes themselves can be of higher dimensions (up to six). That means there are more internal directions in which the flux can point, leading to many more possible ways of threading the field lines through those holes. Now that you have all these fluxes running through your manifold, you might want to know how much energy is stored in the accompanying fields. To calculate the energy, explains Stanford’s Shamit Kachru, you need to take an integral of the field strength squared “over the precise shape of the compact dimensions”—or, you might say, over the surface of the Calabi-Yau. So you divide the surface into infinitesimally small patches, determine the square of the field strength at each patch, add up all those contributions, divide by the number of patches, and you’ll have your average value or integral. “Since varying that shape will vary the magnitude of the total field energy,” Kachru says, “the shape the manifold chooses is the one that minimizes the flux energy of this field.”2 And that’s how bringing fluxes into the picture can stabilize the shape moduli and in that way stabilize the shape itself.
10.2—Just as we can fix and stabilize the arrangement of iron filings by applying a magnetic “flux,” so too can we, in principle, stabilize the shape or size of a Calabi-Yau manifold by turning on the various fluxes of string theory. (Image courtesy of TechnoFrolics [www.technofrolics.com])
That’s part of the story, though we’ve neglected an important aspect of this stabilization process. Just as the magnetic or electric fields are quantized, the fluxes of string theory are also quantized, assuming integer values only. You can put in 1 unit of flux or 2 units of flux but not 1.46 units of flux. When we say that fluxes stabilize the moduli, we mean they restrict the moduli to particular values. You can’t set the moduli to any value you choose—only to values that correspond to discrete fluxes. In that way, you’ve restricted the manifold—the Calabi-Yau—to a discrete set of shapes as well.
Although we spent the previous chapter exploring the heterotic version of string theory, it turns out that incorporating fluxes into heterotic models is rather difficult. Happily, the process is better understood in Type II string theory (a category that includes both Types IIA and IIB), which is dual to heterotic theory in some circumstances. I’ll now mention an important 2003 analysis, performed in the Type IIB setting, that stands out in this regard.
So far we’ve only talked about stabilizing the shape moduli of a manifold with fluxes. The paper in question (dubbed KKLT after its authors, Shamit Kachru, Renata Kallosh, and Andrei Linde—all from Stanford—and Sandip Trivedi of the Tata Institute in India) is generally considered the first publication to show a consistent way of stabilizing all the moduli of the Calabi-Yau, both the shape moduli and size moduli. Stabilizing size is crucial for any string theory based on Calabi-Yau manifolds, because otherwise, there’s nothing to keep the six hidden dimensions from unwinding and becoming infinitely large—bringing them to the same infinite size that we assume the other four dimensions have as well. If the small, invisible dimensions suddenly sprang free and expanded, we’d then be living in a spacetime of ten large dimensions, with ten independent directions to move in or to search for our missing keys, and we know our world doesn’t look like that. (Which gives us some hope for finding lost keys.) Something has got to hold those dimensions back, and that something—according to the KKLT authors—turns out to be D-branes.3
Stabilizing the six-dimensional Calabi-Yau with branes is something like constraining an inner tube with a steel-belted radial tire. Just as the tire will hold back the tube as you pump air into it, the branes can curb the tiny manifold’s inclination to expand.
“You say the shape and size are stabilized if you try to squash it and something pushes back, and if you try to expand it and something pushes back,” explains Johns Hopkins physicist Raman Sundrum. “The goal was to make a compact, stable spacetime, and KKLT showed us how to do that—not just one way of doing so but many different ways.”4
Having a stabilized volume and size is essential if we hope to explain phenomena like cosmic inflation—an idea that holds that almost all of the features we see in the universe today are the result of a brief though explosive period of exponential growth at the time of the Big Bang. This growth spurt, according to the theory, is fueled by the presence of a so-called inflaton field that endows the universe with a positive energy that drives expansion. “In string theory, we assume that positive energy must come from some kind of ten-dimensional sources, which have the property that as you make the compact [Calabi-Yau] space bigger, the associated energy gets smaller,” says Cornell physicist Liam McAllister. When given a chance, all fields will try to spread out and get dilute. “What this means is that the system is ‘happier’ when the internal space gets bigger and the energy becomes lower,” he says. “The system can reduce its energy by expanding, and it can reduce it to zero by expanding an infinite amount.”5 If there’s nothing to keep the internal space from expanding, it will expand. When that happens, the energy that would otherwise drive inflation dissipates so quickly that the process would stop before it even got started.
In the KKLT scenario, branes provided a possible mechanism for realizing the universe we see—a universe that’s influenced by inflation to a large degree. The goal of this exercise was not to reproduce the Standard Model or to get into the details of particle physics, but rather to go after some broader, qualitative features of our universe, including aspects of cosmology, which is arguably the broadest discipline of all.
For in the end, we want a theory that works on all scales—a theory that gives us both particle physics and cosmology. In addition to providing hints as to how inflation might work in string theory, the KKLT paper, as well as a 2002 paper by Kachru, Steve Giddings, and Joe Polchinski—the latter two of the University of California, Santa Barbara—showed how string theory might account for the apparent weakness of gravity, which is about a trillion trillion trillion times weaker than the electromagnetic force. Part of the explanation, according to string theory, is that gravity permeates all ten dimensions, which dilutes its strength. But in the Giddings-Kachru-Polchinski (GKP) scenario, the effect gets amplified exponentially by the geometrical concept of warping, which will be discussed later in this chapter. This explanation builds on a warped-geometry model first achieved in field theory—by Lisa Randall of Harvard and Sundrum—and later incorporated into string theory by GKP, as well as in the subsequent KKLT work.
Another milestone achieved by KKLT was providing a string theory description of how our universe might be endowed with a positive vacuum energy—sometimes called dark energy—the existence of which has become evident through measurements since the late 1990s. We won’t provide an elaborate description of that mechanism, which gets rather technical and involves the placement of something called an antibrane (the antimatter counterpart to branes) in a warped region of the Calabi-Yau, such as the tip of a so-called conifold singularity—a noncompact, cone-shaped protrusion extending from the “body” of the manifold. At any rate, the exact details are not all that important here, since their study was never supposed to supply the definitive answer to any of these questions, Kachru explains. “KKLT is really intended to be a toy model—the kind of thing that theorists play with in order to study phenomena—although many other constructions are possible.”6
The point, then, is that if the work on the moduli stabilization front keeps progressing and the work on the particle physics front keeps progressing, there is at least the potential, according to McAllister, “of having it all. If you take a Calabi-Yau manifold and throw in D-branes and fluxes, you may have all the ingredients you need in principle to get the Standard Model, inflation, dark energy, and other things we need to explain our world.”7
The upshot of the KKLT paper was that in showing how the moduli can be stabilized, the authors showed how a Calabi-Yau manifold itself can be restricted to a distinct set of stable, or quasi-stable, shapes. That means you can pick a Calabi-Yau of a specific topological type, figure out the ways you can dress it up with fluxes and branes, and literally count the possible configurations. The trouble is that when you do the counting, some people may be unhappy with the result, because the number of possible configurations appears to be preposterously large, upwards of 10500.
That figure, far from being exact, is meant to provide a rough indication of the number of arrangements (or shapes) you can get in a Calabi-Yau with many, many holes. Consider again a torus with flux winding through a particular hole to stabilize it. Because the flux is quantized, we’ll further suppose it can take on one of ten integer values from 0 to 9. That’s equivalent to saying there are ten stable shapes for the torus. If we had a torus with two holes instead of one and could run flux through each of them, there’d be 102, or 100, possible stable shapes. A six-dimensional Calabi-Yau can, of course, offer many more options. “The number ten to the five hundredth  was obtained by taking from mathematicians the maximum number of holes a manifold could have—on the order of five hundred—and assuming that through each hole, you could place fields or fluxes that have any of ten possible states,” explains Polchinski, one of the people to whom this number is ascribed. “The counting here is really crude. The number could be much larger or much smaller, but it’s probably not infinite.”8
What does a number like this mean? For starters, it means that owing to the topological complexity of a Calabi-Yau manifold, the equations of string theory have a large number of solutions. Each of these solutions corresponds to a Calabi-Yau with a different geometry that, in turn, implies different particles, different physical constants, and so forth. Moreover, because Calabi-Yau manifolds are, by definition, solutions to the vacuum Einstein equations, each of those solutions, which involve different ways of incorporating fluxes and branes, corresponds to a universe with a different vacuum state and, hence, a different vacuum energy. Now here’s the kicker: A fair number of theorists believe that all these possible universes might actually exist.
There’s a picture that goes with this. Imagine a ball rolling on a vast, smooth, frictionless landscape. There being no preferred position, it can go anywhere without costing any energy. This is like the situation of unstabilized moduli and massless scalar fields. Now let’s imagine that this surface is not entirely smooth but instead has little dips in it, in which the ball can get stuck without the input of some energy to dislodge it. This is the situation you get when the moduli are stabilized; each of the dips on the surface corresponds to a different solution to string theory—a different Calabi-Yau occupying a different vacuum state. Because we have such a large number of possible solutions, this “landscape” of different vacuum states is enormous.
Of course, this whole notion—the so-called landscape of string theory—has become extremely controversial. Some people embrace the picture it implies of multiple universes, some abhor it, and others (myself included) regard it as speculative. There is a question in some people’s minds as to the practical value of a purported theory of nature that offers more solutions than we can ever sort through. One also has to wonder, in view of all the possible universes strewn across this landscape, if there’s any conceivable way of finding ours.
More worrisome to others, the landscape idea has become closely tied to so-called anthropic arguments, some of which go like this: The cosmological constant for our universe, as calculated from recent measurements in astronomy, appears to be very small, a factor of about 10120 lower than that predicted by our best physical theories. No one has been able to explain this discrepancy or this constant’s exceedingly small size. But what if all the 10500or so possible vacua in the landscape are actually realized somewhere—each representing a separate universe or subuniverse with a different internal geometry (or Calabi-Yau) and a different cosmological constant? Among all those choices, at least one of those subuniverses is bound to have an extremely low cosmological constant just like ours. And since we have to live somewhere, maybe, by chance, that’s where we ended up. But it’s not entirely dumb luck. For we couldn’t live in a universe with a large cosmological constant, because the expansion there would have been so fast that stars, planets, and even molecules would never have a chance to form. A universe with a large negative cosmological constant would have quickly shrunk down to nothing—or to some violent singularity that would likely ruin your whole day. In other words, we live in the kind of universe in which we can live.
10.3—The energy of empty space, also called the vacuum energy, can assume a vast number of possible values that represent stable, or semistable, solutions to the equations of string theory. The concept of the “landscape” of string theory was invented, in part, to illustrate the idea that the theory has many possible solutions, corresponding to many possible vacuum states, or vacua—each of which could represent a different universe. The stable vacua in this figure are represented by dips or valleys on a sloping, hilly landscape—places in which balls, for instance, might get stuck as they rolled down a mountainside in different directions. The elevation of these troughs corresponds to the energy the vacuum assumes at that particular spot on the landscape. Some theories suggest there might be on the order of 10500 different solutions, each corresponding to a different Calabi-Yau manifold and hence a different geometry for the compact dimensions. Calabi-Yau spaces are an integral part of this picture because it is thought that the bulk of the vacuum energy is used to keep the six extra dimensions of string theory curled up in such spaces rather than allowing them to expand to infinity. (Calabi-Yau images courtesy of Andrew J.Hanson, Indiana University)
The physicist David Gross has compared anthropic-style reasoning of this sort to a virus that ought to be eradicated. “Once you get the bug, you can’t get rid of it,” he complained at a cosmology conference.9 Stanford physicist Burton Richter claims that landscape enthusiasts such as his Stanford colleague Leonard Susskind have “given up. To them the reductionist voyage that has taken physics so far has come to an end,” Richter wrote in the New York Times. “Since that is what they believe, I can’t understand why they don’t take up something else—macramé, for example.”10 Susskind has not taken statements like that lying down: There’s no way around the multiple solutions of string theory, he contends, so, like it or not, the landscape is here to stay. Since that’s the case, we’d better make peace with it and see if there’s anything useful to be learned. “The field of physics is littered with the corpses of stubborn old men who didn’t know when to give up,” he wrote in his book, The Cosmic Landscape, while admitting that he too may be a “crusty old [curmudgeon], battling to the very end.”11
It’s fair to say that things have gotten a little heated. I haven’t really participated in this debate, which may be one of the luxuries of being a mathematician. I don’t have to get torn up about the stuff that threatens to tear up the physics community. Instead, I get to sit on the sidelines and ask my usual sorts of questions—how can mathematics shed light on this situation?
Some physicists had originally hoped there was only one Calabi-Yau that could uniquely characterize string theory’s hidden dimensions, but it became clear early on that there were a large number of such manifolds, each having a distinct topology. Within each topological class, there is a continuous, infinitely large family of Calabi-Yau manifolds. This is perhaps easiest pictured with the torus. A torus is the topological equivalent of a rectangle. (If you roll a rectangular sheet up into a cylinder and smoothly connect the tips, you’ll have a torus.) A rectangle is defined by its height and width, each of which can assume an infinite number of possible values. All of these rectangles and their associated tori are topologically equivalent. They’re all part of the same family, but there’s an infinite number of them. The same holds for Calabi-Yau manifolds. We can take a manifold, modify its “height,” “width,” and various other parameters, and end up with a continuous family of manifolds, all of the same topological type. So KKLT and the related landscape concept didn’t change that situation at all. At best, imposing the constraints that come from physics—by insisting that the flux be quantized—has led to a very large but finite number of Calabi-Yau shapes rather than an infinite number. I suppose that might be viewed as some progress.
10.4—Two sides of the landscape debate: (a) Santa Barbara physicist David Gross and (b) Stanford physicist Leonard Susskind (Photo of Susskind by Anne Warren)
Personally, I never shared the dream that some physicists once harbored of there being a single, god-given Calabi-Yau or even just a few. I always assumed that things were going to be more complicated than that. To me, that’s just common sense. After all, who ever said that getting to the bottom of the universe and charting out its intrinsic geometry was supposed to be easy?
So what can we make of this landscape idea that has proved so unsettling to some? One course, I suppose, is to ignore it, as nothing has been settled, nothing proved. Some physicists consider the concept useful for addressing the cosmological constant problem, while others see no utility in it whatsoever. Since the whole notion of the string theory landscape emerged from looking at numerous vacuum states, many if not all of which relate to Calabi-Yau manifolds (depending on which version of this concept you’re talking about), I suggest that one of the things the landscape might be telling us—if we’re to put any stock in the idea at all—is that we need to understand Calabi-Yau manifolds better.
I realize that this statement may be somewhat naive. There are many possible solutions to string theory, and many possible geometries upon which one might compactify the theory’s extra dimensions, with Calabi-Yau manifolds representing just the tip of the iceberg. I’m well aware of this situation and am even looking into some of these new areas myself. Nevertheless, most of the progress we’ve made so far in string theory, and most of the insights gleaned, have come from using Calabi-Yau manifolds as the test case—the model of choice. What’s more, even some of the alternative geometries now being investigated—such as non-Kähler manifolds, which we’ll get into in a moment—are produced by deforming or warping Calabi-Yau manifolds. There’s no shortcut that can take us directly to non-Kähler geometries, which means that we need to understand Calabi-Yau manifolds before we can have a chance of understanding things like non-Kähler manifolds.
This is a common strategy in all areas of exploration: You establish a base camp, which serves as a familiar point of departure, before venturing into the unknown. Remarkably, despite the amount of study that’s gone into the subject since I proved the existence of these manifolds in 1976, there are many simple questions—shockingly simple, in fact—that we cannot yet answer: How many topologically distinct Calabi-Yau manifolds are there? Is that number finite or infinite? And might all Calabi-Yau manifolds be related in some way?
We’ll start with the first question: How many distinct topological varieties, or families, do Calabi-Yau manifolds come in? The short answer is that we don’t know, though we can do a little better than that. More than 470 million Calabi-Yau threefolds have been created by computer so far. For those, we have constructed more than 30,000 Hodge diamonds, which means there are at least 30,000 distinct topologies. (Hodge diamonds, as you may recall from Chapter 7, are four-by-four arrays that sum up basic topological information about a threefold.) The number could be significantly greater than 30,000, however, as two manifolds can have the same Hodge diamond and still be topologically distinct. “No systematic effort has been made to estimate the number of topological types, mostly because no practically calculable numerical test is known to unambiguously distinguish between such threefolds,” explains Howard University physicist Tristan Hubsch. “We still don’t have an unequivocal ID number for a Calabi-Yau manifold. We know the Hodge diamond is part of it, but that doesn’t uniquely define the manifold. It’s more like a partial registration number of a car.”12
It’s not just that we don’t know whether the number of Calabi-Yau manifolds is a little bit more than 30,000 or a lot more; we don’t even know whether the number is finite or not. In the early 1980s, I conjectured that the number is finite, but University of Warwick mathematician Miles Reid holds the opposite opinion, arguing that the number is infinite. It would be nice to find out which view is right. “I suppose that for physicists hoping there were only a handful of Calabi-Yau manifolds, finding out there’s an infinite number would only make things worse,” says Mark Gross of the University of California, San Diego. “From a mathematical viewpoint, it doesn’t really matter. We just want to know the answer. We want to understand the totality of Calabi-Yau manifolds. The conjecture that there is a finite number—be it right or wrong—serves as a kind of measuring stick for our understanding.”13 And from a purely practical standpoint, if the set of manifolds is finite, no matter how large it is, you can always take an average. But we really don’t know how to take the average of an infinite number of objects, which therefore makes it harder for us to characterize those objects.
There have been no contradictions to my conjecture so far. It appears that all the methods we presently know of for constructing Calabi-Yau manifolds will lead to only a finite number of manifolds. It could be a matter of overlooking some kinds of construction, but after a couple of decades of searching, no one has come up with a new method that would lead to an infinite set.
The closest anyone has come to settling this question is a 1993 proof by Mark Gross. He proved that if you think of a Calabi-Yau, loosely speaking, as a four-dimensional surface with a two-dimensional donut attached to each point, there’s only a finite number of these surfaces. “The vast majority of known Calabi-Yau manifolds fall into this category, which happens to be a finite set,” Gross says. That’s the main reason he supports the “finite” hypothesis. On the other hand, he notes, plenty of manifolds don’t fall into this category and we haven’t had any real success in proving anything about these cases.14 That leaves the matter unresolved.
Which brings us to the second question, originally posed by Reid in 1987, and it is equally unsettled: Might there be a way in which all Calabi-Yau manifolds are related? Or as Reid put it: “There are all these varieties of Calabi-Yau’s with all kinds of topological characteristics. But if you look at it from a wider perspective, these could all be the same thing. Basically it’s a crazy idea—something that can’t possibly be true. Nevertheless . . . ” In fact, Reid considered the idea so outlandish, he never referred to it as a conjecture, preferring to call it a “fantasy” instead. But he still believes that someone might be able to prove it.15
Reid speculated that all Calabi-Yau manifolds might be related through something called a conifold transition. The idea—developed in the 1980s by the mathematicians Herb Clemens (then at the University of Utah) and Robert Friedman of Columbia—involves what happens to Calabi-Yau manifolds when you move them through a special kind of singularity. As always, the concept is much easier to picture on a two-dimensional torus. Remember that a torus can be described as a series of circles arranged around a bigger circle. Now we’ll take one of those smaller circles and shrink it down to a point. That’s a singularity because every other spot on the surface is smooth.
So at this pinch point—a so-called conifold singularity—it’s as if you had two little cones or party hats coming together on an otherwise unremarkable donut. One thing you can now do is what geometers call surgery, which involves cutting out the offending point and replacing it with two points instead. We can then separate those two points, pulling apart the donut at that spot until it assumes a crescent shape. Next, we reconfigure that crescent into its topological equivalent, a sphere. We don’t have to stop there. Suppose we next stretch out the sphere so that it, again, looks more like a crescent. Then we attach those ends to fashion a torus, only this time we’ve been a bit careless, with an extra fold somehow having been introduced into our shape. This gives us a torus with a different topology and two holes instead of one. If we were to continue this process indefinitely—introducing extra folds, or holes, along the way by virtue of our sloppiness—we’d eventually get to all possible two-dimensional tori. The conifold transition is thus a way of connecting topologically distinct tori by way of an intermediary (in this case, a sphere), and this general procedure works for other (nontrivial) kinds of Calabi-Yau manifolds as well.
10.5—The conifold transition is an example of a topology-changing process. In this greatly simplified case, we start with a donut, which is made up of little circles, and shrink one of those circles down to a point. That point is a kind of singularity where two shapes that resemble cones come together. A cone-shaped singularity of this sort is called a conifold. Through a mathematical version of surgery, we replace that singular point with two points and then pull apart those points, so that the donut becomes more of a croissant. We then inflate the croissant to make it like a sphere. In this way, we’ve gone from a donut to a topologically distinct object, a sphere.
10.6—Here’s another way of picturing the conifold transition. We’ll start with the Calabi-Yau manifold on the left. It’s a six-dimensional object because it has a five-dimensional base—being the “product” of a two-dimensional sphere (S2) and a three-dimensional sphere (S3)—plus an added dimension for the height. This Calabi-Yau surface is nice and smooth because it has a two-dimensional sphere (S2) on top. Shrinking that sphere down to a point brings us to the middle picture, the pyramid. The point on the very tip of the pyramid is a singularity, the conifold. If we smooth out that pointy tip, by blowing up the point into a three-dimensional sphere (S3), rather than the two-dimensional sphere (S2) we started with, we’ll arrive at the third panel, manifold M. So the idea here is that the conifold singularity serves as a kind of bridge from one Calabi-Yau to another. (Adapted, with permission, from a figure by Tristan Hubsch)
Six-dimensional Calabi-Yau manifolds aren’t so simple. In our picture of the conifold transition, as suggested by Clemens, instead of shrinking a circle down to a point, we shrink down to a two-dimensional sphere. We’re assuming here that every compact Kähler manifold, and hence every Calabi-Yau manifold, has at least one two-dimensional sphere of a special sort sitting inside. (The Japanese mathematician Shigefumi Mori proved that Kähler manifolds with positive Ricci curvature have at least one such subsurface, and we expect that this condition applies to the Ricci-flat Calabi-Yau case as well. Every Calabi-Yau manifold we know of has a two-dimensional sphere, so our intuition has held up so far. But we still don’t have a proof for Ricci-flat Calabi-Yau manifolds.) After shrinking our two-dimensional sphere down to a point, we can replace that point with a shrunken three-dimensional sphere that can then be reinflated.
If our previous assumption is correct, after this surgery the manifold is no longer Kähler, since it no longer has a two-dimensional sphere, and therefore cannot be a Calabi-Yau. It’s something else, a non-Kähler manifold. Continuing the conifold transition, we can take this non-Kähler manifold, insert a different two-dimensional sphere (where we had previously inserted the three-dimensional sphere), and end up with a different Calabi-Yau.
Although Reid did not invent the conifold transition, he was the first to see how it might be used to forge a link between all Calabi-Yau manifolds. A critical aspect of the conifold transition is that in getting from one Calabi-Yau surface to a different one, the geometry must pass through an intermediary stage—that being a non-Kähler manifold. But what if all these non-Kähler manifolds are connected in the sense that they can be molded into one another by means of squeezing, stretching, or shrinking? That, indeed, is the crux of Reid’s fantasy.
Imagine a giant chunk of Swiss cheese, filled with numerous tiny holes or bubbles. If you live in one bubble, you can’t go far before you hit a boundary, says Allan Adams. “But if you don’t mind going through the cheese, you can get from one bubble to another. Reid conjectured that a conifold transition can take you through the cheese [the non-Kähler part] and into another bubble,” the Kähler part, which is the Calabi-Yau.16 The analogy works in another sense, too, for in this picture, the bulk of space is basically the cheese, save for the tiny bubbles scattered here and there. Those little bubbles are like little bits of Kähler spaces scattered amid a much larger non-Kähler background. And that’s pretty much how we think of it: There’s a vast number of non-Kähler spaces, with Kähler manifolds constituting just a tiny subset.
The general strategy underlying Reid’s conjecture makes sense to Mark Gross. Since, he says, non-Kähler manifolds represent a much bigger set of objects, he says, “if you want to say things are related, allowing them to be part of this much bigger non-Kähler set certainly makes it easier.”17
The situation is kind of like the Six Degrees of Kevin Bacon game, in which players try to show how everyone in Hollywood is connected to the prolific actor. It’s the same with Calabi-Yau manifolds, Adams says. “Are they all neighbors? Can you smoothly deform one into the other? Definitely not. But Reid’s conjecture says that every Calabi-Yau can be deformed into something else [a non-Kähler manifold] that knows all the other Calabi-Yau manifolds.” Say you have a bunch of people and you’re trying to see if they have something in common, he adds. “We just have to show that they know the same gregarious fellow, in which case they’re all part of the same group—the group of this guy’s acquaintances.”18
How does Reid’s proposition, or fantasy, about the connectedness of Calabi-Yau manifolds square with reality? In 1988, Tristan Hubsch and University of Maryland mathematician Paul Green proved that Reid’s conjecture applied to about 8,000 Calabi-Yau manifolds, which included most of the manifolds known at the time. Subsequent generalizations of this work have shown that more than 470 million constructions of Calabi-Yau manifolds—almost all the known threefolds—are connected in the way that Reid suggested.19
Of course, we won’t know it’s true for all cases until it’s proven. And more than two decades after Reid posed his conjecture, this is turning out to be a difficult fantasy to prove. A big part of the challenge, I believe, is that non-Kähler manifolds are not well understood by mathematical standards. We’ll have a better chance of proving this proposition when we understand these manifolds better. As a general matter, we cannot say for sure that such manifolds—the non-Kähler ones—are even real (or mathematically viable). There is no broad existence proof, such as that pertaining to Calabi-Yau manifolds, and existence, so far, has only been established in a few isolated cases.
If we are intent on learning all we can about the manifolds that have given rise to the landscape conundrum and its attendant cosmological puzzles, it would be helpful to determine whether all Calabi-Yau manifolds are related. A key to doing this, as we’ve just established, may lie in the new frontier of non-Kähler manifolds. These manifolds are of keen interest to physicists not just for the insight they may give us on Calabi-Yau manifolds, but also because they may offer the compactification geometry needed to compute particle masses—one aspect of the quest to realize the Standard Model that has eluded us so far, while physicists have pursued strategies that rely exclusively on Calabi-Yau manifolds.
My colleague Melanie Becker, a physicist at Texas A&M University, believes that non-Kähler approaches may hold the answer. “The way to get particle content and masses,” Becker says, “just might be through the compactification of non-Kähler manifolds.” It could be the geometry we’re looking for—the one that leads us to the promised land of the Standard Model. The reason she thinks so goes back to our discussion in the beginning of this chapter. String theorists introduced fluxes to get rid of massless scalar fields and thereby stabilize the size and shape of a Calabi-Yau manifold. But turning on these powerful fields or fluxes can have another consequence: It can distort the geometry of the manifold itself, changing the metric so that it’s no longer Kähler. “When you turn on the flux, your manifold becomes non-Kähler, and it’s a whole different ball game,” Becker says. “The challenge is that this is really a whole new topic of mathematics. A lot of the math that applies to Calabi-Yau manifolds does not apply to non-Kähler manifolds.”20
From the standpoint of string theory, one of the chief roles intended for these manifolds, be they Calabi-Yau or non-Kähler, is compactification—reducing the theory’s ten dimensions to the four of our world. The easiest way to partition the space is to cut it cleanly, splitting it into four-dimensional and six-dimensional components. That’s essentially the Calabi-Yau approach. We tend to think of these two components as wholly separate and noninteracting. Ten-dimensional spacetime is thus the Cartesian product of its four- and six-dimensional parts, and as we’ve seen, you can visualize it with the Kaluza-Klein-style model we discussed in Chapter 1: In this picture, our infinite, four-dimensional spacetime is like an infinitely long line, except that this line has some thickness—a tiny circle wherein the extra six dimensions reside. So what we really have is the Cartesian product of a circle and a line—a cylinder, in other words.
In the non-Kähler case, the four- and six-dimensional components are not independent. As a result, the ten-dimensional spacetime is not a Cartesian product but rather a warped product, signifying that these two subspaces do interact. Specifically, distances in the four-dimensional spacetime are influenced—and continually rescaled or warped—by the six-dimensional part. The extent to which the four-dimensional spacetime gets expanded or shrunk depends on a number called the warp factor, and in some models the effect, or warping, can be exponential.
10.7—Taking the so-called Cartesian product of a circle and a line segment is like attaching that same line segment to every single point on the circle. The result is a cylinder. Taking the warped product is different. In this case, the length of the line segment does not have to be constant; it can vary depending on where you are on the circle. So in this case, the result is not an actual cylinder; it’s more of a wavy, irregular cylinder.
This is perhaps easiest to picture in our cylinder example. Again, we’ll represent the six-dimensional space with a circle. The four-dimensional part is a line perpendicular to that circle, and we’ll represent it with a unit line segment (rather than an infinite line) to show how distances are affected. If there were no warping, as you moved the line segment to every point on the circle, you’d trace out a perfect (and solid) cylinder. Because of warping, however, the length of that line segment can vary over the surface of the circle. At one point, it may be 1, at another ½, at another 1½, and so forth. What you’ll end up with is not a perfect cylinder but an uneven, wavy cylinder that’s distorted by the warping.
This can all be expressed in more rigorous terms by a set of equations issued in 1986 by the physicist Andrew Strominger. In the earlier (1985) paper he wrote with Candelas, Horowitz, and Witten, which presented the first serious attempt at a Calabi-Yau compactification, they made the simplifying assumption that the four-dimensional and six-dimensional geometries were autonomous, notes Strominger. “And we found solutions in which they were autonomous, even though string theory didn’t demand that. A year later, I worked out the equations you get when you don’t make those assumptions.” These are the so-called Strominger equations, which concern the situation when the fluxes are turned on and the four- and six-dimensional spaces interact. “The possibility of their not being autonomous is interesting because there are some really good consequences,” Strominger adds. Prominent among these consequences is that warping might help explain important phenomena such as the “hierarchy problem,” which relates to why the Higgs boson is so much lighter than the Planck mass and why gravity is so much weaker than the other forces.
The Strominger equations, which apply to non-Kähler manifolds, describe a bigger class of solutions than the equations in the 1985 paper, which only applied to Calabi-Yau manifolds. “In trying to understand the many ways string theory could be realized in nature, one needs to understand the more general solutions,” Strominger says. “It’s important to understand all the solutions to string theory, and Calabi-Yau space does not contain them all.”21Harvard physicist Li-Sheng Tseng (my current postdoc) compares the Calabi-Yau manifold to a circle, “which, of all the smooth and closed one-dimensional curves you can draw, is the most beautiful and special.” The Strominger equations (sometimes referred to as the Strominger system), he says, “involve a relaxation of the Calabi-Yau condition, which is like relaxing the circle condition to the ellipse condition.” If you have a closed loop of string of fixed length, there is only one circle you can possibly make, whereas you can make an endless variety of ellipses by taking that circle and squashing or elongating it to varying degrees. Of all the curves you can make out of that loop, the circle is the only one that remains invariant to rotations around the center.
To see that a circle is just a special case of an ellipse, we need look no further than the equation that defines an ellipse on the Cartesian (x-y) plane: x2/a2 + y2/b2 = 1, where a and b are positive, real numbers. A curve so described is never a circle except when a = b. Moreover, it takes two parameters, a and b, to define an ellipse, and just one (since a = b) to define a circle. That makes an ellipse a somewhat more complicated system than a circle, just as the Strominger (non-Kähler) system is more complicated to deal with than a Calabi-Yau, which can be described with fewer parameters.
10.8—If you have a loop of fixed length, you can make an infinite number of ellipses—some pointier, some rounder—but you can only make one circle of that circumference. In other words, by relaxing some of the properties that make a circle so special, you can end up with any number of ellipses. Similarly, a Calabi-Yau manifold, which has Kähler symmetry by definition, is (like the circle) much more special than a non-Kähler manifold, which satisfies less rigorous conditions and encompasses a much broader class of objects.
Even though going from a circle to an ellipse, or from a Calabi-Yau to a non-Kähler manifold, may represent a step down in symmetry and beauty, Tseng notes, “it’s clear that nature doesn’t always choose the most symmetric configuration. Think, for example, of the elliptical orbits of the planets. So it’s also possible that the internal six-dimensional geometry that describes our natural universe may not be quite so symmetric as the Calabi-Yau, but just slightly less so, as in the Strominger system.”22
The system that Strominger proposed is no picnic to deal with, because it consists of four differential equations that have to be solved simultaneously—any one of which can be nightmarish to solve. There are two Hermitian Yang-Mills equations, which have to do with the gauge fields (see Chapter 9). Another equation ensures that the whole geometry is supersymmetric, while the last is designed to make the anomalies cancel, which is essential for the consistency of string theory.
As if this weren’t challenging enough, each of these four equations is really a system of equations rather than a single equation. Each can be written as a single matrix (or tensor) equation, but since the matrix itself has many variables, you can split that single equation into separate component equations. For this same reason, the famous Einstein equation that encapsulates the theory of general relativity is really a set of ten field equations that describe gravity as the curvature of spacetime caused by the presence of matter and energy, even though it can be written as a single tensor equation. In the proof of the Calabi conjecture, solving the Einstein equation in the vacuum condition reduced to a single equation, albeit a rather imposing one. Non-Kähler manifolds are harder to work with than Calabi-Yau manifolds, because the situation has less symmetry and therefore more variables—all of which results in more equations to be solved. Furthermore, we don’t really have the mathematical tools to understand this problem well at the moment. In the Calabi case, we drew on algebraic geometry, which has developed tools over the previous two centuries for dealing with Kähler manifolds but not their non-Kähler counterparts.
Still, I don’t believe these two classes of manifolds are that different, from a mathematical standpoint. We’ve used geometric analysis to build Calabi-Yau manifolds, and I’m confident these techniques can help us build non-Kähler manifolds as well, assuming that first we can either solve the Strominger equations or at least prove that solutions exist. Physicists need to know whether non-Kähler manifolds can exist and whether it’s possible to satisfy all four equations at once, for if it’s not possible, people working on them may be wasting their time. I looked at the problem for nearly twenty years after Strominger proposed it and couldn’t find a solution. Or, I should say, a smooth solution, a solution without singularities, as Strominger did find some solutions with singularities (but those are messy and extremely difficult to work with). After a while, people began to believe that a smooth solution did not exist.
Then a minor breakthrough occurred. Some colleagues and I found smooth solutions in a couple of special cases. In the first paper, which I completed in 2004 with Stanford mathematician Jun Li (a former graduate student of mine), we proved that a class of non-Kähler manifolds was mathematically possible. In fact, for each known Calabi-Yau manifold, we proved the existence of a whole family of non-Kähler manifolds that are similar enough in structure to be from the same neighborhood. This was the first time the existence of these manifolds was ever mathematically confirmed.
Although solving the Strominger equations is extraordinarily difficult, Li and I did about the easiest thing you could do in this area. We proved that those equations could be solved in the limited case, where the non-Kähler manifold is very close to the Calabi-Yau. In fact, we started with a Calabi-Yau manifold and showed how it could be deformed until its geometry, or metric, was no longer Kähler. Although the manifold could still support a Calabi-Yau metric, its metric was now non-Kähler, hence offering solutions to the Strominger system.
Probably more significantly, Li and I generalized the DUY theorem (as mentioned in Chapter 9, the acronym for the theorem’s authors, Donaldson, Uhlenbeck, and me) to cover basically all non-Kähler manifolds. Having DUY in hand was of great practical value because it automatically took care of two of the four Strominger equations—those relating to Hermitian Yang-Mills theory—leaving the supersymmetry and anomaly cancellation equations to be solved. Given that DUY has been instrumental for Calabi-Yau compactifications (in terms of reproducing the gauge fields), we’re hoping it will do the same for non-Kähler compactifications as well.
One promising avenue for generating non-Kähler manifolds, as suggested by Reid’s conjecture, is to start with an already-known Calabi-Yau manifold and take it through a conifold transition. I recently looked into this with Jun Li and Ji-Xian Fu, a former Harvard postdoc of mine now based at Fudan University in Shanghai. The basic manifold we started with came from Herb Clemens, one of the architects of the conifold transition, but he provided just a general topology—a manifold without a metric and hence no geometry. Fu, Li, and I tried to give this manifold some shape by showing the existence of a metric that would satisfy the Strominger equations.
Those equations seemed appropriate here, because they apply not only to non-Kähler manifolds but also to Calabi-Yau manifolds, which are a special case. And Reid’s conjecture, too, involves a procedure that takes you from Calabi-Yau manifolds to non-Kähler ones and back. So if you want a set of equations that can cover both these geometries, Strominger’s formulations might be just the thing. So far, my colleagues and I have proved that Clemens’s manifold does indeed satisfy three of the four Strominger equations, though we haven’t found a solution to the most difficult one, the anomaly cancellation equation. I’m still pretty confident that this manifold exists. After all, in most human endeavors, three out of four is pretty good. But until we solve that last equation, we won’t have proved a thing.
Fu and I went farther with another example, showing how to construct a topologically diverse class of non-Kähler manifolds that satisfies the Strominger equations. Built from scratch, rather than constructed by modifying known Calabi-Yau manifolds, these manifolds are intrinsically non-Kähler. They consist of K3 surfaces (four-dimensional Calabi-Yau manifolds) with a two-dimensional torus attached at every point. Solving the Strominger equation, in this case, involved solving a Monge-Ampère equation (a class of nonlinear differential equations discussed in Chapter 5) that was more complicated than the one I had to solve for the Calabi proof. Fortunately, Fu and I were able to build from the earlier argument. Our method, as with the Calabi proof, involved making a priori estimates, which means that we had to make guesses regarding the approximate values of various parameters.
Fu and I found a special method that enabled us to solve not just one equation but all four. Whereas in the case of the Calabi conjecture, I was able to obtain all possible solutions to the Monge-Ampère equation, this time Fu and I obtained just a subset of possible solutions. Unfortunately, we don’t understand the system well enough to know how large or small that subset is.
At least we’ve taken some preliminary steps. Most physicists who have begun working on non-Kähler compactifications are assuming that the Strominger equations can be solved without bothering to prove it. Li, Fu, and I have shown that these equations can be solved in the isolated approaches we’ve identified so far, which is another way of saying that these particular manifolds—a fraction of all non-Kähler manifolds—really do exist. This is just the starting point for the bigger problem I want to tackle: finding a metric that satisfies the Strominger system, and all its equations, in the most general terms. While no one has come close to accomplishing this yet—and all signs suggest that a proof will not come easily—my colleagues and I, with our more modest steps, have at least raised the possibility.
Becker tells me that if I succeed in this venture, it will be even more important than the proof of the Calabi conjecture. She could be right, but it’s hard to tell. Before I solved the Calabi conjecture, I didn’t know its full significance. And even after I solved it, physicists did not recognize the importance of the proof and accompanying theorem until eight years after the fact. But I continued to explore Calabi-Yau spaces because, to me, they looked pretty. And these spaces characterized by the Strominger system have a certain allure as well. Now we’ll have to see how things pan out.
In the meantime, Fu and I have offered the manifolds we’ve produced so far to fellow physicists through a collaboration with Melanie Becker, Katrin Becker, Tseng, and others—maybe even Strominger if we can bring him into the fold. Since then, this group has constructed more examples of the original Fu-Yau model, while exploring the physics in a preliminary way. Unlike some of the heterotic string theory compactifications described in the last chapter, this team has not been able to get the right “particle content” or the three generations of particles we see in the Standard Model. “What we do have,” says Melanie Becker, “are stabilized moduli, which is a prerequisite to everything else, as well as an actual way to compute masses.”23
At this stage, it’s hard to know what exactly will come from the efforts of physicists currently toying with non-Kähler compactifications and the many other alternatives to Calabi-Yau manifolds (including an area called nongeometric compactifications) that are currently being investigated. It’s fair to ask whether Calabi-Yau compactifications are the right description of our universe or merely the simplest example from which we’ve learned—a fantastic experiment for letting us discover how string theory works and how we can have supersymmetry, all the forces, and other things we want in an “ultimate” theory. In the end, though, this exercise may yet lead us to a different kind of geometry altogether.
For now, we’re simply trying to explore some of the many possibilities lying before us on the string theory landscape. But even amid all those possibilities, we still live in just one universe, and that universe could still be defined by Calabi-Yau geometry. I personally think Calabi-Yau manifolds are the most elegant formulation, as well as the most beautiful manifolds constructed so far among all the string vacua. But if the science leads us to some other kind of geometry, I’ll willingly follow.
“In the past twenty years, we’ve uncovered many more solutions to string theory, including non-Kähler ones,” says Joe Polchinski. “But the first and simplest solutions—Calabi-Yau manifolds—still look the closest to nature.”24
I’m inclined to agree, though there are plenty of top-notch researchers who have a different opinion. Melanie Becker, for one, is a champion of the non-Kähler approach. Strominger, who has made major contributions in both the Calabi-Yau and non-Kähler realms, doesn’t think Calabi-Yau spaces will ever become obsolete. “But we want to use everything we encounter as stepping-stones to the next level of understanding,” he says, “and Calabi-Yau manifolds have been stepping-stones in many directions.”25
Before long, hopefully, we’ll have a better sense of where they might lead us. Despite my affection for Calabi-Yau manifolds—a fondness that has not diminished over the past thirty-some years—I’m trying to maintain an open mind on the subject, keeping to the spirit of Mark Gross’s earlier remark: “We just want to know the answer.” If it turns out that non-Kähler manifolds are ultimately of greater value to string theory than Calabi-Yau manifolds, I’m OK with that. For these less-studied manifolds hold peculiar charms of their own. And I expect that upon further digging, I’ll come to appreciate them even more.
University of Pennsylvania physicist Burt Ovrut, who’s trying to realize the Standard Model through Calabi-Yau compactifcations, has said he’s not ready to take the “radical step” of working on non-Kähler manifolds, about which our mathematical knowledge is presently quite thin: “That will entail a gigantic leap into the unknown, because we don’t understand what these alternative configurations really are.”26
While I agree with Ovrut’s statement, I’m always up for a new challenge and I don’t mind the occasional plunge into uncharted waters. But since we’re often told not to swim alone, I’m not averse to dragging a few colleagues along with me.