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


When searching for diamonds, if you’re lucky you may find other precious gems as well. When I announced the proof of the Calabi conjecture in a two-page 1977 paper, which was followed by a seventy-three-page 1978 elaboration, I also announced the proof of five other related theorems. The windfall was, in many ways, a consequence of the unusual circumstances under which I approached the Calabi conjecture—first trying to prove it wrong before shifting gears and trying to prove it right. Luckily, it turns out that none of that effort was wasted: I could use every apparent misstep, every ostensible dead end. For my alleged counter-examples—ideas logically implied by the conjecture that I’d hoped to prove false—turned out to be true, as was the conjecture itself. These failed counter-examples were, in fact, real examples and soon became mathematical theorems in themselves—a few of some note.

The most important of these theorems led to the proof of the Severi conjecture—a complex version of the Poincaré conjecture that had remained unsolved for more than two decades. But before getting to that, I had to solve an important inequality for classifying the topology of surfaces, which I became interested in partly as a result of a talk by Harvard mathematician David Mumford, who was passing through California at that time. The problem, first posed by Antonius van de Ven of Leiden University, concerned an inequality among Chern classes for Kähler manifolds. Van de Ven had initially proved that the second Chern class of a manifold multiplied by 8 must be greater than or equal to the square of the first Chern class of that same manifold. Many people believed this inequality could be made much stronger by inserting the constant 3 into the expression in place of 8. Indeed, 3 was considered the optimal value. The question Mumford posed was whether this more stringent inequality could be proved. (Mumford’s version was more stringent because it claimed that a certain quantity, the second Chern class, was greater than something else—hence the inequality—not only when multiplied by 8 but even when multiplied by the smaller number 3.)

He asked this question in September 1976 during a lecture at the University of California, Irvine; I was attending the lecture not long after finishing my proof of the Calabi conjecture. In the middle of the talk, I became almost certain that I’d encountered this problem before. In a discussion after the lecture, I told Mumford I thought I could prove this more difficult case. When I got home, I checked my calculations and found out that just as I’d suspected, I had tried to use this kind of inequality in 1973 to disprove the Calabi conjecture; now I could use the Calabi-Yau theorem to prove that this inequality held. What’s more, I was able to use a special case of that statement—namely, the equality case (3 times the second Chern class equals the square of the first Chern class)—to solve the Severi conjecture, as well.

These two theorems, which proved the Severi conjecture and the more general inequality (sometimes called the Bogomolov-Miyaoka-Yau inequality to acknowledge two other contributors to this problem), were the first major spin-offs of the Calabi proof, but many others were to follow. Calabi’s conjecture was, in fact, broader than I’ve indicated so far. It applied not only to the case of zero Ricci curvature, but to the cases of constant negative and constant positive Ricci curvature as well. No one has yet proved the positive curvature case in its full generality, where Calabi’s original conjecture is known to be false. (I formulated a new conjecture that specified a condition under which metrics with positive Ricci curvature could exist. Over the past two decades, many mathematicians, including Simon Donaldson, have made serious contributions toward this conjecture, but it has yet to be proven.) Nevertheless, I did prove the negative curvature case as part of my overall argument, and this was independently proved by the French mathematician Thierry Aubin as well. The solution to the negative curvature case established the existence of a broad class of objects called Kähler-Einstein manifolds, thereby establishing a new field of geometry that has turned out to be surprisingly fruitful.

It’s fair to say I had a good time pursuing the immediate applications of the Calabi conjecture, knocking off a half dozen or so proofs in short order. It turns out that once you know a metric exists, there are all kinds of consequences. You can use that knowledge to work backward and deduce things about manifold topology, without knowing the exact metric. We can use those properties, in turn, to identify unique features of a manifold in the same way we can identify a galaxy without knowing everything about it, or much as we don’t need a full deck of cards to be able to deduce a great deal about the deck (including the total number of cards and the markings on each one). To me, that’s kind of magical and says even more about the power of mathematics than a situation in which we have every last detail in front of us.

It was gratifying to reap some of the rewards of my arduous labors and to see others pursue avenues that hadn’t occurred to me. Despite that good fortune, something was gnawing at me. Deep down, I felt certain that this work would have implications for physics, in addition to mathematics, though I did not know exactly how. In a way, this sentiment was rather obvious since the differential equations you are trying to solve in the Calabi conjecture—in the zero Ricci curvature case—are literally Einstein’s equations of empty space, corresponding to a universe with no background energy, or a cosmological constant of no value. (These days, the cosmological constant is generally thought to be positive and synonymous with the dark energy that is pushing the universe to expand). Likewise, Calabi-Yau manifolds are regarded as solutions to Einstein’s differential equations, just as a unit circle is the solution to the equation x2 + y2 = 1.

Of course, you need many more equations to describe Calabi-Yau spaces than it takes to describe a circle, and the equations themselves are much more complicated, but the basic idea is the same. The Calabi-Yau equations not only satisfy the Einstein equations, they do so in a particularly elegant way that I, at least, found arresting. That made me think they had to fit into physics somewhere. I just didn’t know where.

Nor was there much I could do about it other than telling my physics postdocs and physicist friends why I thought the Calabi conjecture, and the so-called Yau’s theorem that emerged from it, might be important for quantum gravity. The main problem was that I didn’t understand quantum gravity well enough then to follow up on my intuition. I brought up the idea from time to time, but mainly had to sit back and see what came of it.

The years passed, and as other mathematicians and I continued to follow up on the Calabi conjecture in the course of pushing through our broader agenda in geometric analysis, there were some stirrings in the physics world going on behind the scenes and, for a little while at least, unbeknownst to me. It started in 1984, which turned out to be a landmark year in which string theory took great strides from being a somewhat general idea toward becoming an actual, flesh-and-blood theory.

Before getting to those exciting developments, let’s say a bit more about this theory that brashly attempts to bridge the gap between general relativity and quantum mechanics. At its core is the notion that the smallest bits of matter and energy are not pointlike particles but are instead tiny, vibrating pieces of string, which assume the form of either loops or open strands. Just as a guitar is capable of playing many different notes, these fundamental strings can vibrate in many different ways. String theory posits that strings vibrating in different ways correspond to the different particles in nature as well as the different forces. And, assuming the theory works—a matter yet to be settled—that’s how unification is achieved: These particles and forces share a common bond because they’re all manifestations, and excitations, of the same basic string. You could say that’s what the universe is made of: When you get down to it, at the most elementary level, it’s all strings.

String theory borrows from Kaluza-Klein the general notion that extra dimensions are required for this grand synthesis to be realized. Part of the argument is the same: There is simply not enough room for all the forces—gravity, electromagnetism, weak, and strong—to fit into a single four-dimensional theory. If one were to follow a Kaluza-Klein approach and ask how many dimensions are needed to combine all four forces within a single framework—with five covering gravity and electromagnetism, a couple more for the weak force, and a few more for the strong—you’d need a minimum of eleven dimensions. But it turns out that this approach doesn’t quite work—one of the many aspects of the theory we’ve learned from the physicist Edward Witten.

Fortunately, string theory doesn’t go about things in such an ad hoc fashion, picking an arbitrary number of dimensions, scaling up the size of the matrix or Riemann metric tensor, and seeing what forces you can or cannot accommodate. Instead, the theory tells you exactly how many dimensions are needed for the job, and that number is ten—the four dimensions of the “conventional” spacetime we probe with our telescopes, plus six extra dimensions.

The reason string theory demands ten dimensions is fairly complex and stems from the need to preserve symmetry—essential for any purported theory of nature—and to make the theory consistent with quantum mechanics, which surely has to be one of the key ingredients. But the argument essentially boils down to this: The more dimensions you have, the more possible ways a string can vibrate. To reproduce the full range of possibilities in our universe, string theory requires not only a very large number of potential vibrations but a specific number that you only get in ten-dimensional spacetime. (Later in this chapter, we’ll discuss a variation, or “generalization,” of string theory called M-theory that involves eleven dimensions, but we’ll leave off that for now.)

A string confined to one dimension can only vibrate longitudinally, by stretching and compressing. In two dimensions, a string can vibrate that way and in the perpendicular, or transverse, direction as well. In dimensions three and higher, the number of independent vibrational patterns continues to grow until you reach dimension ten (nine spatial dimensions and one of time), where the mathematical requirements of string theory are satisfied. That’s why string theory requires at least ten dimensions. The reason string theory requires exactly ten dimensions to be consistent, no more and no less, relates to something called anomaly cancellation, which brings us back to our narrative and the year 1984.

Most string theories developed up to that point had been plagued by anomalies or inconsistencies that rendered any predictions they made nonsensical. The theories, for example, tended to have the wrong kind of left-right symmetry—one that was incompatible with quantum theory. A key breakthrough was made by Michael Green, then at Queen Mary College of London, and John Schwarz of the California Institute of Technology. The main problem Green and Schwarz overcame related to parity violation—the idea that the fundamental laws of nature distinguish between left and right and are not symmetric in this respect. Green and Schwarz found ways of formulating string theory such that parity violation was upheld. The quantum effects that had riddled the theory with inconsistencies miraculously canceled themselves out in ten spacetime dimensions, raising hopes that the theory might actually describe nature. Green and Schwarz’s achievement marked the start of what was called the first string revolution. And with those anomalies suddenly dispensed with, it was time to see whether the theory could lead to some realistic physics.

Part of the challenge is to see whether string theory can explain why the universe looks the way it does. That explanation must account for the fact that we inhabit a spacetime that looks four-dimensional, while the theory insists it’s actually ten-dimensional. The answer to that apparent discrepancy, according to string theory, lies in compactification. The notion is not entirely new, as Kaluza and Klein (and Klein, in particular) had already suggested that the extra dimension in their five-dimensional theory was indeed compactified—shrunk down so small we couldn’t see it. String theorists have made a similar case, only they have six dimensions to dispose of rather than just one.

Actually, this last remark is somewhat misleading, as we’re not really trying to get rid of dimensions. The trick, instead, is to wrap them up in a very exacting way—coiled up within a geometry whose precise contours are critical to the magic act that string theory is trying to pull off. There are many geometries to choose from, each of which leads to a different possible compactification.

The whole idea, according to Harvard physicist Cumrun Vafa, can be summed up in a simple equation that everyone can understand: 4 + 6 = 10.1 That’s all there is to it, though you might want to rephrase it as 10 - 6 = 4, indicating that once the six dimensions are concealed (or subtracted away), what is in actuality a ten-dimensional universe appears to have just four. Compactification can, equivalently, be thought of as a funny kind of multiplication known as a Cartesian product—a product in which the number of dimensions are added together rather than multiplied. The relevant equation that describes the product manifold in which the four and six dimensions combine suggests that our ten-dimensional spacetime has a substructure; it is literally the product of four- and six-dimensional spacetime, just as a plane is the product of two lines, and a cylinder is the product of a line and a circle. Moreover, the cylinder, as we’ve seen, is how the Kaluza-Klein concept is often illustrated. If you start by depicting our four-dimensional spacetime as an infinite line that stretches forever in both directions and then snip the line and magnify one of the ends, you’ll see that the line actually has some breadth and is more accurately described as a cylinder, albeit one of minuscule radius. And it is within this circle of tiny radius that the fifth dimension of Kaluza-Klein theory is hidden. String theory takes that idea several steps further, arguing in effect that when you look at the cross-section of this slender cylinder with an even more powerful microscope, you’ll see six dimensions lurking inside instead of just one. No matter where you are in four-dimensional spacetime, or where you are on the surface of this infinitely long cylinder, attached to each point is a tiny, six-dimensional space. And no matter where you stand in this infinite space, the compact six-dimensional space that’s hiding “next door” is exactly the same.

That, of course, is merely a crude, schematic picture that tells us nothing about the actual geometry of this shrunken, six-dimensional world. Imagine, for example, if we took an ordinary sphere—a two-dimensional surface—and reduced it to a point, a zero-dimensional object. In that way, we’ve compactified two dimensions down to none. We could try to reduce ten to four by shrinking a six-dimensional sphere (a2 + b2 + c2 + d2 + e2 + f 2 = 1), but that would not work as the geometry for the extra dimensions; the equations of string theory demand that the six-dimensional space must have a very particular structure that a simple sphere does not possess.

A more complicated shape was clearly needed, and after Green and Schwarz’s success with parity violation, the task of finding that shape became quite urgent. Once physicists had the proper manifold in which to curl up the extra six dimensions, they could finally try to do some real physics.

As an initial follow-up step in 1984, Green, Schwarz, and Peter West of King’s College decided to look at K3 surfaces—a broad class of complex manifolds that had been studied by mathematicians for more than a century, though K3’s attracted the more recent attention of physicists when my proof of the Calabi conjecture showed that these surfaces could support a Ricci-flat metric. “What I understood was that the compact space had to be Ricci flat to insure that the lower-dimensional space we inhabit would not have a positive cosmological constant, which was considered to be a fact of our universe at the time,” recalls Schwarz.2 (In light of the subsequent discovery of dark energy, implying an extremely small but positive cosmological constant, string theorists have devised more complicated ways of producing a tiny cosmological constant in our four-dimensional world from compact, Ricci-flat spaces—a subject to be addressed in Chapter 10.)

A K3 surface—a name that alludes both to the K2 mountain peak and to three mathematicians who explored the geometry of these spaces, Ernst Kummer, the aforementioned Erich Kähler, and Kunihiko Kodaira—was selected for this preliminary inquiry despite the fact that it’s a manifold of just four real (and two complex) dimensions rather than the requisite six, in part because Green, Schwarz, and West had been told by a colleague that there were no higher-dimensional analogues of that manifold. Nevertheless, says Green, “I am by no means convinced we would have been able to figure things out . . . even if we had been given the correct information [about the existence of six-dimensional analogues of Ricci-flat K3’s] at that time.”3 Going with the tried-and-true K3, Schwarz adds, “wasn’t an attempt to do a realistic compactification. We just wanted to play around, see what we got, and see how that meshed with anomaly cancellation.”4 Since then, K3 surfaces have been invaluable to string theorists in this regard as oft-used “toy models” for compactifications. (They are also essential models for exploring the dualities of string theory, which will be discussed in the next chapter.)

At roughly the same time, also in 1984, the physicist Andrew Strominger, now at Harvard but then at the Institute for Advanced Study (IAS) in Princeton, teamed up with Philip Candelas, a mathematically inclined physicist now at Oxford who was then at the University of Texas, to figure out what class of six-dimensional shapes might meet the exacting conditions set by string theory. They knew that the internal space of those shapes had to be compact (to get down from ten dimensions to four) and that the curvature had to satisfy both the Einstein gravity equations and the symmetry requirements of string theory. Their explorations ultimately took them and two other colleagues—Gary Horowitz of the University of California, Santa Barbara, and Witten—to the spaces whose existence I had confirmed in the Calabi conjecture proof (though Witten had followed his own path to those geometric forms). “One of the beautiful things about developments in modern science is that physicists and mathematicians are often led to the same structures for different reasons,” Strominger observes. “Sometimes, the physicists are ahead of the mathematicians; sometimes, the mathematicians are ahead of the physicists. This is a case where the mathematicians were ahead. They understood the significance before we did.”5

Although what Strominger says is true, to an extent, it’s also true that mathematicians like me originally had no idea how Calabi-Yau spaces tied into physics. I investigated them because I thought they were beautiful; it was because of their great beauty that I felt physicists ought to be able to do something with them—that they harbored some mysteries worth uncovering. Ultimately, it was up to the physicists themselves to make that connection, bridging the gap between geometry and physics and, in so doing, initiating a long and fertile collaboration between the two fields—a collaboration that flourishes to this day.

How that connection was made turns out to be an interesting tale in itself. Strominger sums it up this way: “Supersymmetry was the bridge to holonomy, and holonomy was the bridge to Calabi-Yau.”6

As you may recall, we briefly discussed supersymmetry in Chapter 4, in the context of a kind of limited, internal symmetry—as opposed to the more sweeping, global symmetry of an object like a sphere—that Calabi-Yau manifolds (being a class of Kähler manifolds) must possess. This internal symmetry is part of what we mean by supersymmetry, but before we try to paint a clearer picture of that, let’s say a few words about holonomy first.

Loosely speaking, holonomy is a measure of how tangent vectors on a particular surface get twisted up as you attempt to parallel-transport them on a loop around that surface. Imagine, for example, you’re standing on the north pole, holding a spear that is tangent to the earth’s surface. For starters, you’ll walk directly to the equator, following the direction in which the spear is initially pointing—all the while keeping the spear pointing in the same direction, from your perspective, as you proceed.

When you reach the equator, the spear will be pointing down. Now you’ll follow the equator halfway around the earth, keeping the spear pointing down as you do so. Once you’ve gone halfway around, you’ll head to the north pole, with the spear still pointing in the same direction the whole time. When you arrive at the north pole, you’ll find that your spear has rotated 180 degrees from its initial direction, despite your best efforts to maintain a fixed bearing.

We could repeat this process any number of times, making shorter or longer trips along the equator, only to discover that our spear has rotated by different angles, sometimes less than 180 degrees and sometimes more, depending on the length of our trip along the equator. To determine the holonomy of our planet, a two-dimensional sphere, we need to consider all possible paths—all possible loops—you can make on the surface. It turns out that on the surface of a sphere, you can get any angle of rotation you want, from 0 to 360 degrees, by making your loop larger or smaller. (And you can even go beyond 360 degrees, too, by retracing your steps a second time or more.) We say the two-dimensional sphere belongs to the holonomy group SO(2), or special orthogonal group 2, which is a group that contains all possible angles. (Higher-dimensional spheres belong to SO[n], which are the groups that include all possible orientation-preserving rotations, where n refers to the number of dimensions.)


6.1—One way of classifying a space or surface is through holonomy, which tells you what happens to a tangent vector as you parallel-transport it—that is, try to keep it pointing in the same direction as you move it along a path that may itself be curved or twisty. In this example, we’ll start at the north pole with a tangent vector that’s pointing due west and walk to the equator. When we arrive at the equator, our vector is now pointing due south. We’ll keep it pointing south as we walk along the equator from A to B, which will take us halfway around the globe. Now we’ll walk up to the north pole again, keeping the vector fixed as we do so. When we arrive at the north pole, we’ll find that our vector has rotated 180 degrees, despite our best efforts to keep it pointing the same way at all times.

Depending on what path you take on the surface of a globe or sphere, you can end up with any conceivable rotation angle. Once you know the set of all possible angles, you can classify a surface by its holonomy group—a two-dimensional sphere belonging to the special orthogonal group 2, or SO(2).

A Calabi-Yau manifold, on the other hand, belongs to the much more restrictive SU(n) holonomy group, which stands for the special unitary group of n complex dimensions. The Calabi-Yau manifolds of primary interest to string theory have three complex dimensions, which places them in the SU(3) holonomy group. Calabi-Yau manifolds are a lot more complicated than spheres, and SU(3) holonomy is a lot more complicated than our foregoing example of a vector being rotated, despite our best efforts to keep it pointing in the same direction, as we move it on the surface of a sphere. What’s more, because a Calabi-Yau manifold (unlike a sphere) lacks global symmetry, there is no axis around which we can rotate the manifold and leave it unchanged. But it does have a more limited kind of symmetry, as we’ve discussed, which relates both to holonomy and to supersymmetry. For a manifold to have supersymmetry, it must have what is called a covariantly constant spinor. Spinors, though difficult to describe, are analogues of tangent vectors. On a Kähler manifold, there exists one spinor that remains invariant under parallel transport on any closed loop. On Calabi-Yau manifolds—and in the SU(3) group to which they belong—there is an additional spinor that also remains invariant under parallel transport on any closed loop on the manifold.

The presence of these spinors helps ensure the supersymmetry of the manifolds in question, and the demand for supersymmetry of the right sort is what pointed Strominger and Candelas to SU(3) holonomy in the first place. SU(3), in turn, is the holonomy group associated with compact, Kähler manifolds with a vanishing first Chern class and zero Ricci curvature. SU(3) holonomy, in other words, implies a Calabi-Yau manifold. Putting this in equivalent terms, if you want to satisfy the Einstein equations as well as the supersymmetry equations—and if you want to keep the extra dimensions hidden, while preserving supersymmetry in the observable world—Calabi-Yau manifolds are the unique solution. They are, as Johns Hopkins physicist Raman Sundrum puts it, “the beautiful mathematical answer.”7

“I didn’t know much mathematics at the time, but I made the connection with Calabi-Yau manifolds through the holonomy group that characterizes manifolds,” Strominger explains. “I found Yau’s paper in the library and couldn’t make much sense of it, but from the little I did understand, I realized these manifolds were just what the doctor ordered.”8 While reading my papers is not always a memorable experience, Strominger did tell the New York Times—almost twenty years after the fact—of the excitement he felt upon first stumbling across the proof of the Calabi conjecture.9 Before getting too carried away, though, he called me up first to make sure he’d understood my paper correctly. I told him he had. And it finally dawned on me that after an eight-year wait, physics had finally found Calabi-Yau’s.

So supersymmetry was what brought physicists to this arcane bit of mathematics, but I still haven’t explained why they considered supersymmetry so all-fired important, apart from the general statement that symmetry is essential for understanding any kind of manifold. As Princeton physicist Juan Maldacena explains, “Supersymmetry not only makes calculations easier, it makes them possible. Why? Because it’s easier to describe the motion of a sphere rolling down a hill than the more complicated motion of a football wobbling down a hill.”10

Symmetry makes all sorts of problems easier to solve. Suppose you want to find all the solutions to the equation xy = 4. It would take a while, as there are an infinite number of solutions. If, however, you impose the symmetry condition, x = y, then there are just two solutions: 2 and -2. Similarly, if you know that the points on an x-y plane define a circle around the origin, instead of having two variables to worry about—the values of x and y—you only need to worry about one variable, the radius of the circle, to have all the information you need to reproduce the curve exactly. Supersymmetry, in the same way, cuts down the number of variables, thereby simplifying most problems you might want to solve, because it imposes a constraint on the geometric form that the internalized six dimensions can take. That requirement, notes University of Texas mathematician Dan Freed, “gives you Calabi-Yau’s.”11

Of course, we can’t insist on the existence of supersymmetry in the universe simply to make our calculations easier. There’s got to be more to it than sheer convenience. And there is. One virtue of supersymmetric theories is that they automatically stabilize the vacuum, the ground state of general relativity, so that our universe will not keep descending to lower and lower energy depths. This idea relates to the positive mass conjecture discussed in Chapter 3. In fact, supersymmetry was one of the tools Edward Witten drew upon in his physics-based proof of that conjecture, though supersymmetry did not come up in the more mathematical (and nonlinear) approach pursued by Richard Schoen and me.

But most physicists are interested in supersymmetry for another reason, which is, in fact, how the whole concept originated. To physicists, the most salient aspect of this idea is that of a symmetry that links elementary matter particles like quarks or electrons, which are called fermions, and force particles like photons or gluons, which are called bosons. Supersymmetry establishes a kinship, a kind of mathematical equivalence, between forces and matter and between these two classes of particles. It claims, in fact, that each fermion has an associated boson partner, called a superpartner, and that the reverse holds for every boson. The theory thus predicts a whole new class of particles—with funny names like squarks, selectrons, photinos, and gluinos—that are heavier than their known companions and whose “spin” differs by half an integer. These superpartners have never been seen before, though investigators are looking for them right now in the world’s highest-energy particle accelerators (see Chapter 12).

The world we live in, which physicists characterize as “low-energy,” is clearly not supersymmetric. Instead, current thinking holds that supersymmetry prevails at higher energies, and in that realm, particles and superpartners would appear to be identical. Supersymmetry “breaks,” however, below a certain energy scale, and we happen to live in the realm of broken supersymmetry, where particles and superpartners are distinct in terms of mass and other properties. (After being broken, a symmetry doesn’t disappear altogether but instead goes into hiding.) One way to understand the mass difference, says Howard University physicist Tristan Hubsch (a former postdoc of mine), is to think of supersymmetry as the rotational symmetry around, say, a plastic pen that’s held upright. Imagine you’ve fixed the ends of the pen, and are pushing it from two different angles perpendicular to the length of the pen. Each of these movements takes the same amount of energy, no matter which angle you push on, as long as it’s perpendicular to the pen, explains Hubsch. “And because these little movements are related by rotational symmetry, you can exchange one for the other.”

We can use these pushes to set the pen vibrating with two rotationally symmetric waves. The vibrations would be equivalent, in turn, to two different particles, and the energy of the vibrations would determine the particles’ masses. Because rotational symmetry (or supersymmetry, in the case of string theory) is upheld, the two particles—the particle and superpartner—would have the same mass and would be otherwise indistinguishable.

We can break the rotational symmetry—our stand-in for supersymmetry—by pressing hard on the ends of the pen until it bows. The harder we press, the more severely it bows and the more badly the symmetry is broken. “After symmetry is broken, there are still two kinds of motions, but they are no longer related to each other by rotational symmetry,” Hubsch says. One kind of motion—pushing the pen along the direction it is bowed—still takes energy, and the greater it’s bowed, the more energy it takes. But if you push on the pen in a direction that is not only perpendicular to the pen itself but also perpendicular to the bowing, the pen will swivel around without requiring any energy (assuming there’s no friction between the swiveling pen and the thing holding it in place). In other words, there’s an energy difference or gap between these two motions—one requiring the input of energy and one not—which corresponds to the energy (or mass) gap between a massless particle and its massive superpartner in the broken supersymmetry case.12 Physicists are trying to find signs of this energy gap and thereby establish the existence of massive supersymmetry partners, in high-energy physics experiments now under way at the Large Hadron Collider.

While supersymmetry, in principle, ties forces and matter together in a mathematically beautiful way, string theorists consider it desirable for another reason that goes well beyond the aesthetically pleasing aspects of symmetry. For without supersymmetry, string theory makes little sense. It predicts impossible particles like tachyons, which travel faster than the speed of light and which have a negative mass-squared—that is, their mass is expressed in terms of the imaginary unit i. Our theories of physics can’t readily accommodate bizarre entities like that. Supersymmetry may not need string theory—though it owes much of its development to that theory—but string theory certainly benefits from having supersymmetry around. And supersymmetry, as stated above, was the very thing that brought physicists to the doorstep of Calabi-Yau.

Once Strominger and Candelas had their hands on the Calabi-Yau manifold, they were eager to take the next step—to see if this really was the right manifold, the one responsible for the physics we see. They came to Santa Barbara in 1984 eager to pursue this project and soon got in touch with Horowitz, who had moved from IAS to Santa Barbara a year earlier. They knew, moreover, that Horowitz had been my postdoc and, as a result of that association, had probably heard more than he cared to about the Calabi conjecture. When Horowitz found out what Strominger and Candelas were up to—trying to determine the mathematical requirements of string theory’s internal space—he too saw that the requisite conditions matched those of Calabi-Yau’s. Having more familiarity with this brand of math than the others did, Horowitz became a welcome addition to the team.

Shortly thereafter, Strominger visited Witten back in Princeton and filled him in on what they’d learned so far. It turned out that Witten had independently arrived at pretty much the same place, though he’d taken a different route in getting there. Candelas and Strominger started from the notion that there are ten dimensions in string theory and that these ten must be compactified in some six-dimensional manifold. The physicists then tried to figure out what kind of six-dimensional space would yield the right sort of supersymmetry, among other requirements. Witten, on the other hand, came at the problem from the perspective of a closed string propagating in spacetime, sweeping out a surface of one complex dimension and two real dimensions, otherwise known as a Riemann surface. Like an ordinary surface in differential geometry, a Riemann surface comes equipped with a metric that tells you the distance between two points, as well as a mechanism for telling you the angle between two points. What makes Riemann surfaces stand out is that, with relatively few exceptions, one can find a unique metric with negative (-1) curvature everywhere.

Witten’s calculations—in a two-dimensional version of quantum theory called conformal field theory—were quite different from those of his coauthors, as he made fewer assumptions about the background spacetime. Yet he reached the same conclusion as the others, namely, that the geometry of the internal space had to be Calabi-Yau. None other would do. “Approaching this question from two directions strengthened the finding,” Horowitz explains. “Moreover, it suggested this was the most natural way of doing the compactification, because you were led to the same conditions from two different starting points.”13

The foursome finished their work in 1984 and promptly distributed their findings to colleagues via preprints, though the paper did not formally come out until the next year. The paper coined the term Calabi-Yau space, introducing the physics world to this strange, six-dimensional realm.


6.2—A rendering of the two-dimensional “cross-section” of a six-dimensional Calabi-Yau manifold (Andrew J. Hanson, Indiana University)


6.3—The Real Calabi-Yau: Eugenio Calabi (left) and Shing-Tung Yau (Photo of Calabi by permission from E. Calabi; photo of Yau by Susan Towne Gilbert)

Prior to the publication of the 1985 paper, Calabi “hadn’t expected there was any physical meaning to this work. It was purely geometric,” he says. The paper, however, changed that, thrusting this mathematical construct into the heart of theoretical physics. It also brought unanticipated attention to the two mathematicians behind these spaces, Calabi recalls, “putting us on the news map. That sort of thing is always flattering, all the notoriety that came with this talk of Calabi-Yau spaces, but it was really not our doing.”14

Our work, for a while at least, became all the rage in physics, but the “news map” spanned even broader terrain. There was an off-Broadway show (Calabi-Yau ), the title of an electro/synthpop album by the band DopplerEffekt (Calabi Yau Space), the name of a painting (Calabi-Yau Monna Lisa) by the Italian artist Francesco Martin, and the butt of a joke in a New Yorker story by Woody Allen (“‘My pleasure,’ she said, smiling coquettishly and curling up into a Calabi-Yau shape”).15 It was a surprising run for such an abstruse idea, given that manifolds of this sort are hard to describe in words and harder still to visualize. A space with six dimensions, as one physicist commented, has “three more than I can comfortably imagine.” The picture is further complicated by the presence of twisting, multidimensional holes running through the space, of which there could be a small number, or—in the deluxe Swiss cheese version—upward of five hundred.

Perhaps the simplest feature of Calabi-Yau spaces is their compactness. Rather than resembling an endless sheet of paper stretching to infinity in all directions, Calabi-Yau manifolds are more like a crumpled-up piece of paper that folds back on itself, only the crumpling-up has to be done in a meticulous way. A compact space contains no infinitely long or wide regions, and if you have a large enough suitcase, the space will fit inside. Another way of putting it, suggests Cornell’s Liam McAllister, is that a compact space “can be covered with a quilt made out of a finite number of patches”—each patch, of course, being of finite size.16 If you’re standing on the surface of such a space and keep walking in the same direction, it may be possible to end up where you started. And even if you don’t make it back exactly, you’ll never get infinitely far away from the starting point, no matter how far you walk. Calling a Calabi-Yau space compact is by no means an exaggeration. Although the exact size of such a manifold is still in question, it is thought to be exceedingly small, with diameters on the order of 10-30 centimeters (making it more than a quadrillion times smaller than an electron). Denizens of the four-dimensional realm like us can’t ever see this six-dimensional realm, but it’s always there, attached to every point in our space. We’re just too big to go inside and look around.

6.4—If string theory is correct, at any point in four-dimensional spacetime there’s a hidden (six-dimensional) Calabi-Yau manifold. (Calabi-Yau images courtesy of Andrew J. Hanson, Indiana University)


That isn’t to say that we don’t interact with those unseen dimensions. As we walk around or sweep our arm through the air, we pass through the hidden dimensions without even realizing it—the movement, in a sense, cancels itself out. Imagine a herd of caribou, 100,000 strong, all headed in the same direction—going from, say, Alaska’s coastal plane to the Brooks Range, where they’ll find a nice valley to spend the winter in. Each animal, explains Allan Adams of MIT, takes a slightly different path over the 800-mile journey, but when looked at in aggregate, all the individual meanderings offset each other and the herd follows just one general path.17 Our brief forays into Calabi-Yau space similarly cancel themselves out, rendering them inconsequential compared with the longer trajectories we follow in the four-dimensional domain.

Another way to think of it is that we live in a space without end; our horizons are vast even if we only manage to visit a tiny fraction of it. Yet everywhere we go in this big wide world, there’s also this tiny, invisible realm, always within striking distance, that we can never fully access. Let’s imagine an unusual x-y axis in which the x direction represents our infinite, four-dimensional space and the y direction represents the internal Calabi-Yau space. At every point along the x-axis, there is a hidden six-dimensional realm. Conversely, at every point along the y-axis, there is an additional four-dimensional space or direction that we can explore as well.

One of the most amazing things is that the concealed, internalized portion of the universe—a place we can never see, touch, smell, or feel—could have a more profound effect on the physics we experience than the concrete world of bricks and stones, cars and rocket ships, and billions and billions of galaxies. Or so string theory proclaims. “All of the numbers we measure in nature—all of the things we consider fundamental, such as the mass of quarks and electrons—all of these are derived from the geometry of the Calabi-Yau,” explains the physicist Joe Polchinski of the University of California, Santa Barbara. “Once we know the shape, in principle, we’d have everything.”18 Or as Brian Greene has put it, “The code of the cosmos may well be written in the geometry of the Calabi-Yau shape.”19 If Einstein’s relativity is proof that geometry is gravity, string theorists hope to carry that notion a good deal further by proving that geometry, perhaps in the guise of Calabi-Yau manifolds, is not only gravity but physics itself.

I’m certainly not going to be the one to cast doubt on these sweeping assertions. But a reasonable person might wonder, just the same, that if the Calabi conjecture was considered too good to be true, what do we call this? And how on earth can we explain it? I’m afraid the real explanation may strike some as unsatisfying and perhaps even circular—that being that Calabi-Yau manifolds can pull off this miraculous feat simply because that ability is, from the very start, built into string theory’s basic machinery. Even so, it still may be possible to provide some inkling as to how that “machine”—which takes in ten-dimensional manifolds and spits out four-dimensional physics—actually works. Here’s an admittedly simplistic picture of how the particles and their masses can be derived from a given Calabi-Yau manifold if we assume that the manifold in question is what is known as non-simply connected. A non-simply connected manifold is like a torus with one or more holes, where some loops sitting on the surface cannot be shrunk down to points. (This is in contrast to a sphere, a simply connected surface, where every loop can be shrunk to a point like an extremely taut rubber band sliding off the equator and wrapping around the north pole.) Starting with a complicated, six-dimensional manifold with some number of holes, you can figure out all the different ways in which strings can wrap around the manifold, going through various holes one or more times. It’s a complicated problem, as there are many ways you can do the wrappings, and each loop or cycle you make can have different sizes that depend, in turn, on the size of the holes. From all these possibilities, you draw up a list of potential particles. The masses can then be determined by multiplying the length of the string by its tension, which is equivalent to its linear energy density, throwing in the kinetic energy of motion as well. The objects you fashion in this way can have any number of dimensions between zero and six. Some objects are allowed; some are not. If you take all the allowed objects and all the allowed motions, you end up with a list of particles and masses.

Another way to view this is that in the reigning picture of quantum physics, owing to the central tenet known as wave-particle duality, every particle can be thought of as a wave and every wave can be thought of as a particle. Particles in string theory, as has been stated before, correspond to a specific vibrational pattern of a string, and a string vibrating in a characteristic, well-prescribed way is also very much like a wave. The question then is to discern how the geometry of this space influences the waves that can form.

Let’s suppose the space in question is the Pacific Ocean, and we’re floating in the middle of it, thousands of miles from the nearest continent and far above the ocean floor. One can imagine that in general, waves forming on the surface here are relatively unaffected by the shape or topography of the seafloor many miles below. But the situation is entirely different in a confined space, such as a shallow and narrow cove—where a mild bump far out at sea becomes a relative tsunami in the shallow waters near shore or, to take a less extreme example, where the reefs and rocks beneath the surface have a major bearing on how and where waves form and break. In this example, the open ocean is like a noncompact (or extended) space, whereas the shallow coastal waters are more like the small, compact dimensions of string theory, where the geometry dictates the kind of waves that can form and so, by extension, the kind of particles that can form.

A musical instrument, such as a violin, is another example of a compact space that produces characteristic vibrational patterns or waves, corresponding to notes rather than particles. The sounds produced by plucking a string depend not only on the length and thickness of the string but also on the shape of the instrument’s interior—the acoustic chamber—where waves of certain frequencies resonate at maximum amplitude. The strings are named for their fundamental frequencies, which on most violins happen to be G, D, A, and E. Physicists—like instrument makers seeking the right shape for the sounds they hope to produce—are hunting for the Calabi-Yau manifold with the proper geometry to give rise to the waves and particles that we see in nature.

The way physicists normally attack a problem of this sort is to find solutions to the wave equation, more formally known as the Dirac equation. The solutions to the wave equation are, not surprisingly, waves and their corresponding particles. But this is a very difficult equation to solve, and we normally cannot solve it for all the possible particles to be found in nature—only for the so-called massless ones that correspond to the lowest or fundamental frequencies of a particular string. The massless particles include all the particles we see, or intuit, in the world around us, including those spotted within our high-energy physics accelerators. Some of these particles, like electrons, muons, and neutrinos, do actually have mass in spite of the terminology “massless.” But they acquire their mass through a mechanism totally different from how the so-called massive particles, which are expected to form at the much higher-energy “string scale,” acquire their mass. The mass of ordinary particles like electrons, moreover, is so much lighter than that of these heavier particles—by a factor of roughly a quadrillion or more—that it is considered a fair approximation to call these ordinary particles massless.

While limiting ourselves to massless particles makes solving the Dirac equation somewhat more manageable, it is still not easy. Fortunately, Calabi-Yau manifolds have certain attributes that help. The first is supersymmetry, which reduces the number of variables, converting a second-order differential equation (in which some derivatives are taken twice) into a first-order differential equation (in which derivatives are taken just once). Another way supersymmetry helps is that it pairs each fermion with its own special boson. If you know all the fermions, then you automatically know all the bosons and vice versa. So you just have to work things out for one class of particles or the other, and you get to pick the one whose equations are easier to solve.

Another special feature of Calabi-Yau manifolds and their geometry in particular is that the solutions to the Dirac equation—in this case, the massless particles—are the same as the solutions to another mathematical formulation known as the Laplace equation, which is considerably easier to work with. The biggest advantage stems from the fact that we can get the answers to the Laplace equation—and identify the massless particles—without having to solve any differential equations at all. Nor do we need to know the exact geometry of the Calabi-Yau or its metric. Instead, we can get all that we need from topological “data” about the Calabi-Yau, all of which is contained in a 4-by-4 matrix called a Hodge diamond. (We’ll be discussing Hodge diamonds in the next chapter, so I won’t be saying more about this now, other than that this topological sleight of hand enables us to round up the massless particles rather effortlessly.)

Getting the particles, however, is just the beginning. Physics, after all, is more than just a bunch of particles. It also includes the interactions or forces between them. In string theory, loops of string moving through space may either join together or split apart, and their inclination to do one or the other depends on the string coupling, which is a measure of the force between strings.

Calculating the strength of particle interactions is a painstaking task—requiring almost the full arsenal of string theory tools—that can, in current practice, take a year to work out for a single model. Supersymmetry, again, makes the computations somewhat less taxing. Mathematics can help, too, since this kind of problem has long been familiar to geometers and, as a result, we have many tools to bring to bear on it. If we take a loop and let it move and vibrate in Calabi-Yau space, it can become a figure eight and then split into two separate circles or loops. Conversely, two circles can come together to form a figure eight. The passage of these loops through spacetime sweeps out a Riemann surface, and that is precisely the picture we have of string interactions, although mathematicians had not connected it to physics until string theory arrived on the scene.

Given those tools, how close can physicists come to matching their predictions to the world we see? This topic will be the focus of Chapter 9, but let’s consider the 1985 paper of Candelas, Horowitz, Strominger, and Witten, which represented the first serious attempt to match string theory—by way of Calabi-Yau compactifications—to the real world.20 Even then, the physicists were able to get a lot of things right. Their model produced, for example, the desired amount of supersymmetry in four dimensions (denoted as N = 1, meaning the space remains invariant under four symmetry operations, which can be thought of as four different kinds of rotations). That, in itself, is a great success. For if they’d instead come up with maximal supersymmetry (N = 8, implying the more demanding situation of invariance under thirty-two different symmetry operations), that would have constrained physics so much that our universe would have to be just flat space without any of the curvature that we believe to exist or any of the complexities, like black holes, that make life so interesting—at least for the theorists. If Candelas and his colleagues had failed on this front, and one could prove that this six-dimensional space could never furnish the appropriate supersymmetry, then compactification in string theory, at least in this instance, would have failed.

The paper was a great leap forward and is now considered part of the first string revolution, but it did miss the mark in other areas, such as the number of families of particles. In the Standard Model of particle physics—a model that has ruled particle physics over recent decades and incorporates the electromagnetic, weak, and strong forces—all the elementary particles from which matter is composed are divided into three families or generations. Each family consists of two quarks, an electron or one of its relatives (a muon or tau), and a neutrino, of which there are three varieties: the electron, muon, or tau neutrino. The particles in family 1 are the most familiar ones in our world, being the most stable and the least massive. Those in family 3 are the least stable and the most massive, whereas the members of family 2 lie somewhere in between. Unfortunately for Candelas and company, the Calabi-Yau manifold they worked with yielded four families of particles. They were off by just one, but in this case the difference between three and four was huge.

In 1984, Strominger and Witten started inquiring about the number of families, and I was eventually asked if I could come up with a Calabi-Yau that yielded three families. Horowitz had stressed the importance of this to me as well. What was needed was a manifold with an Euler characteristic of 6 or -6, as Witten had shown a few years before that for a certain class of Calabi-Yau manifolds (those with a nontrivial fundamental group—or noncontractible loop—among other features) the number of families equals the absolute value of the Euler characteristic divided by two. A version of this formula even appeared in the quartet’s widely cited 1985 paper.

I found some time to work on the problem later that year while flying from San Diego to Chicago en route to Argonne National Laboratory, which was hosting one of the first major string theory conferences ever. I was giving a talk there and had set aside the time on the plane to prepare my remarks. It occurred to me that since my physics friends considered the three-family business so important, maybe I should give that matter some thought. And luckily, I did come up with a solution during the flight—a Calabi-Yau manifold whose Euler number was -6, making it the first manifold ever constructed that would yield the three families of particles the standard model demanded. While this wasn’t a great leap forward, it was a “small breakthrough,” as Witten put it.21

The method I used for constructing that manifold was rather technical, but later proved quite useful. For starters, I took the Cartesian product of two cubic hypersurfaces. A hypersurface is a submanifold—that is, a surface of one dimension lower than the ambient space in which it sits, like a disk tucked inside a sphere or a line segment tucked inside a disk. The hypersurface of a cubic surface of three complex dimensions has two complex dimensions overall. The product of two of these hypersurfaces has 2 ╳ 2 = 4 complex dimensions. That’s one dimension too many, and I cut that down to three complex dimensions (or six real dimensions), which is what you want in string theory, by taking a cross-section, or slice, of the four-dimensional product manifold.

Unfortunately, that procedure doesn’t quite give us the manifold we wanted, because that manifold would yield nine families of particles rather than the desired three. But the manifold had threefold symmetry, which enabled me to create a so-called quotient manifold in which every point corresponds to three points in the original. Taking the quotient, in this case, was like dividing the original manifold into three equivalent pieces. In this way, the number of points was decreased by a factor of three, as was the number of families.

To my knowledge, this quotient manifold was the first Calabi-Yau ever constructed—and the only one available for a long time—that had an Euler characteristic of 6 or -6, which could be exploited to generate three families of particles. In fact, I hadn’t heard of anything like it until late 2009, when Candelas and two colleagues—Volker Braun of the Dublin Institute for Advanced Studies and Rhys Davies of Oxford—did something similar, constructing a Calabi-Yau manifold with an Euler characteristic of -72 and a quotient manifold with an Euler characteristic of -6. Ironically, in the late 1980s, Candelas had constructed the original (or “parent”) Calabi-Yau—one of eight thousand manifolds so created at the time—with two colleagues, but did not recognize its potential applicability until more than twenty years later.22


6.5—In geometry we can reduce the dimensions of an object by taking a slice of it and looking only at the revealed cross-section. For example, if you slice through a three-dimensional apple, you’ll expose a two-dimensional surface—one of many such surfaces you could expose depending on where and how you choose to slice. An additional cut into that surface would pick out a one-dimensional line sitting on the surface. Cutting that line, in turn, would select a single (zero-dimensional) point. Therefore, each successive slice (up to that final point) reduces by one the dimensionality of the object at hand.

I bring this up because back in 1986, when Brian Greene began trying to get realistic physics out of a Calabi-Yau manifold, there weren’t a lot of manifolds around to choose from. To get the number of families right, he went with the Calabi-Yau that I had fabricated in 1984 en route to Argonne. Working on this problem first as an Oxford graduate student and then as my postdoc at Harvard, Greene—along with Kelley Kirklin, Paul Miron, and Greene’s former Oxford adviser Graham Ross—came even closer to the Standard Model than Candelas, Horowitz, Strominger, and Witten had a year or so earlier. Greene’s model provided more details—a step-by-step prescription—for generating physics from a Calabi-Yau manifold. He and his colleagues got supersymmetry right, as well as the number of families right, massive neutrinos (with extremely small mass), and pretty much everything else you wanted, except for a few extra particles that weren’t supposed to be there. So this Calabi-Yau manifold came close—indeed closer than we’d ever been before—but didn’t quite do the job, either. This shouldn’t be taken as criticism of their work, since almost a quarter of a century later, no one has finished “the job” yet.

In those early days, physicists had hoped there was just one Calabi-Yau manifold to worry about—a unique solution from which we could calculate everything—or just a handful, in which case they could quickly weed out the less worthy examples and select the right one. When Strominger and Witten first started asking me about the number of known (and already constructed) Calabi-Yau manifolds, I only knew of two examples for certain. One, the quintic threefold , is arguably the simplest Calabi-Yau you can have. (It’s called a quintic because it’s described by a fifth-degree polynomial equation that takes the general form z15 + z25z35 + z45 + z55 = z1 x z2 x z3 X z4z5. It’s called a threefold because it’s a manifold with three complex dimensions.) The second Calabi-Yau manifold was made by combining (or taking the “product” of) three complex one-dimensional tori and then modifying the resultant manifold.

Around this time, Strominger asked me about the total number of Calabi-Yau manifolds possible. I told him there were probably tens of thousands—each representing a different topology and a different solution to the equations of string theory. Within each of those topologically distinct families, moreover, lay an infinite variety of possible shapes. I delivered the same news to a larger crowd of physicists during my 1984 Argonne lecture, and many of them were dismayed when I tossed out the round figure of ten thousand or so—an estimate that has held up pretty well to this day.

At first, the physicists weren’t in a position to construct Calabi-Yau manifolds on their own, as the mathematics was still too unfamiliar to them, which meant they were dependent on people like me to tell them about such objects. But once they became conversant in the literature, the physicists moved fast and constructed many examples themselves, working independently of mathematicians. Shortly after my Argonne lecture, Candelas and his students took the same general approach I’d used in constructing the first Calabi-Yau that gave rise to three families of particles, computerized the technique, and in this way generated many thousands of Calabi-Yau manifolds (as mentioned a moment ago). I’d only worked out a few of them myself and was never very good at computer calculations. But in light of Candelas’s achievement and the output of his computer, the notion of a large number of manifolds was no longer abstract or merely an estimate made by a partisan mathematician. It was a fact, and if you had any doubts on the subject, you needed to look no further than Candelas’s published database.

The upshot of this was that string theory was starting to look far more complicated than initially envisioned. It was no longer a matter of taking the Calabi-Yau manifold and wringing every last drop of physics out of it you could. Before doing that, you first had to answer the question “Which Calabi-Yau?” And, as will be seen in Chapter 10, the problem of too many Calabi-Yau manifolds has gotten worse over the years rather than better. Yet this problem had even become apparent in 1984, when, says Strominger, “the uniqueness of string theory was already in question.”23

As if that weren’t bad enough, there was another numbers problem afflicting string theory in the early days, and this one had to do with the number of string theories itself. There wasn’t just one string theory. There were five separate theories—with the names Type I, Type IIA , Type IIB, Heterotic SO(32), and Heterotic E8 X E8—that differed, for example, as to whether the strings must be closed loops or could be open strands as well. Each theory belonged to a different symmetry group, and each contained a unique set of assumptions about things like the chirality (or handedness) of fermions and so forth. A competition of sorts arose as to which of these five candidate theories might ultimately prevail and become the true Theory of Everything. In the meantime it was paradoxical—not to mention just plain embarrassing—to have five “unique” theories of nature.

In 1995, in an intellectual tour de force, Witten showed that all five string theories represented different corners of the same overarching theory, which he called M-theory. Witten never said what the M stood for, but speculation has included terms like master, magical, majestic, mysterious, mother, matrix, and membrane.


6.6—At first the five different string theories were regarded as competing theories that were studied separately and considered distinct from each other. Edward Witten and other architects of the “second string revolution” showed that the five theories are all related—tied together through a common framework called M-theory (though, apparently, no one knows what the M stands for).

The last word in that series is of special significance because the fundamental ingredients of M-theory are no longer just strings. They’re more general objects, called membranes or branes, that could have anywhere from zero to nine dimensions. The one-dimensional version, a “one-brane,” is the same as a familiar string, whereas the two-brane is more like what we think of as a membrane, and a three-brane is like three-dimensional space. These multidimensional branes are called p-branes, and a category of these objects called D-branes are subsurfaces within a higher-dimensional space to which open strings (as opposed to closed loops) attach. The addition of branes has made string theory richer and better equipped to take on a broader range of phenomena (as will be explored in later chapters). Furthermore, demonstrating that the five string theories were all connected to each other in fundamental ways meant that one could choose to work with whatever version made solving a particular problem easiest.

M-theory has another important feature that distinguishes it from string theory. The theory exists in eleven dimensions rather than ten. “Physicists claim to have a beautiful and consistent theory of quantum gravity, yet they can’t agree on the number of dimensions,” Maldacena notes. “Some say ten, and some say eleven. Actually, our universe may have both—ten and eleven dimensions.”24

Strominger agrees that the “notion of dimension is not absolute.” He compares string theory and M-theory to different phases of water. “If you put it below freezing it’s ice; above freezing it’s liquid; above boiling it’s steam,” he says. “So it looks very different depending on what phase it’s in. But we don’t know which of these phases we’re actually living in.”25


6.7—Edward Witten at the Institute for Advanced Study (Photo by Cliff Moore)

Even the principal author of M-theory, Witten himself, concedes that the ten- and eleven-dimensional descriptions of the universe “can both be correct. I don’t consider one more fundamental than the other,” he says, “but at least for some purposes, one might be more useful than the other.”26

As a practical matter, physicists have had better success so far in explaining the physics of our four-dimensional world when starting from a ten- rather than eleven-dimensional perspective. Researchers have tried to go from eleven dimensions directly to four by compactifying on a seven-dimensional so-called G2 manifold—the first compact version of which was constructed in 1994 by Dominic Joyce, a mathematician currently at Oxford. So one might think that much of what we’ve talked about so far—such as deriving four-dimensional physics from a ten-dimensional universe by way of six-dimensional Calabi-Yau manifolds (4 + 6 = 10)—could have been suddenly rendered obsolete by Witten’s eureka moment. Fortunately, at least for the purposes of the current discussion, that is not the case.

One drawback of the G2 approach, as explained by Berkeley physicist Petr Horava, a Witten collaborator and a key contributor to M-theory, is that we can’t recover four-dimensional physics by compactifying on a “smooth” seven-dimensional manifold. Another problem is that a seven-dimensional manifold, unlike Calabi-Yau manifolds, cannot be complex, because complex manifolds must have an even number of dimensions. That’s probably the most important difference, Horava adds, “because complex manifolds are much better behaved, much easier to understand, and much easier to work with.”27

Furthermore, there’s still much to be learned about the existence, uniqueness, and other mathematical properties of seven-dimensional G2 manifolds. Nor do we have a systematic way of searching for these manifolds or a general set of rules to draw on, as we do for Calabi-Yau manifolds. Both Witten and I have looked into developing something similar to the Calabi conjecture for G2 manifolds, but so far neither he nor I nor anybody else has gotten very far. That’s one reason M-theory is not as developed as string theory, because the mathematics is much more complicated and not nearly as well established.

Because of the difficulties with the G2 manifolds, most efforts in M-theory have focused on indirect routes of compactifying from eleven dimensions to four. First, eleven-dimensional spacetime is treated as the product of ten-dimensional spacetime and a one-dimensional circle. We compactify the circle, shrinking it down to a minuscule radius, which leaves us with ten dimensions. We then take those ten dimensions and compactify on a Calabi-Yau manifold, as usual, to get down to the four dimensions of our world. “So even in M-theory, Calabi-Yau manifolds are still in the center of things,” says Horava.28 This approach—pioneered by Witten, Horava, Burt Ovrut, and others—is called heterotic M-theory. It has been influential in introducing the concept of brane universes, in which our observable universe is thought to live on a brane, and has also spawned alternative theories of the early universe.

So for now, at least, it appears that all roads lead through Calabi-Yau. To get realistic physics out of string theory and M-theory, and some cosmology as well, this is still the geometry that holds the “cosmic code”—the space wherein the master plan resides. Which is why Stanford physicist Leonard Susskind, one of the founders of string theory, claims that Calabi-Yau manifolds are more than just the theory’s supporting structure or scaffolding. “They are the DNA of string theory,” he says.29