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


Although geometry has served us well, there is a problem lurking just beneath the surface, and it may portend trouble for the future. To see this, one need travel no farther than to the nearest lake or pond. (And if there are no ponds in your neck of the woods, a backyard pool or bathtub might do.) A lake’s surface may look perfectly smooth on a calm, windless day, but that is an illusion. When we examine the surface at higher resolution, it appears jagged rather than smooth. We see that it’s actually composed of individual water molecules that are constantly jiggling around, moving within the pond itself and passing freely between the water and air. Viewed in this light, the surface is not a static, well-defined thing at all. In fact, it hardly qualifies as a surface, as we commonly use the term.

Classical geometry is like this, too, according to Harvard physicist Cumrun Vafa, in that it only provides an approximate description of nature rather than an exact, or a fundamental, one. To its credit, this approximate description holds up well and describes our universe almost flawlessly, except on the Planck scale (10-33 centimeters)—a realm at which standard geometry gets swamped by quantum effects, rendering simple measurements impossible.

The chief difficulty in resolving things at very fine scales stems from the Heisenberg uncertainty principle, which makes it impossible to localize a single point or to secure a precise fix on the distance between two points. Instead of standing still, everything at the Planck scale fluctuates—including points, lengths, and curvature. Whereas classical geometry tells us that two planes intersect in a line and three planes intersect in a point, the quantum perspective tells us that we might instead imagine three planes intersecting in a sphere that encompasses a range of probable positions for that point.

To probe the universe at the level of the hidden dimensions or individual strings, we’re going to need a new kind of geometry—sometimes referred to as quantum geometry—that is capable of operating on both the largest and smallest scales imaginable. A geometry of this sort would have to be consistent with general relativity on large scales and quantum mechanics on small scales and consistent with both in places where the two theories converge. For the most part, quantum geometry does not yet exist. It is as speculative as it is important, a hope rather than a reality, a name in search of a well-defined mathematical theory. “We have no idea what such a theory will look like, or what it should be called,” Vafa says. “It’s not obvious to me that it should be called geometry.”1 Regardless of its name, geometry as we know it will undoubtedly come to an end, only to be replaced by something more powerful—geometry as we don’t know it. This is the way of all science, as it should be, since stagnation means death.

“We’re always looking for the places where science breaks down,” explains University of Amsterdam physicist Robbert Dijkgraaf. “Geometry is closely tied to Einstein’s theory, and when Einstein’s theory becomes stressed, geometry is stressed, too. Ultimately, Einstein’s equations must be replaced in the same way that Newton’s equations were replaced, and geometry will go along with it.”2

Not to pass the buck, but the problem has more to do with physics than with math. For one thing, the Planck scale where all this trouble starts is not a mathematical concept at all. It’s a physical scale of length, mass, and time. Even the fact that classical geometry breaks down at the Planck scale doesn’t mean there’s anything wrong with the math per se. The techniques of differential calculus that underlie Riemannian geometry, which in turn provides the basis for general relativity, do not suddenly stop working at a critical length scale. Differential geometry is designed by its very premise to operate on infinitesimally small lengths that can get as close to zero as you want. “There’s no reason that extrapolating general relativity to the smallest distance scales would be a problem from a mathematics standpoint,” says David Morrison, a mathematician at the University of California, Santa Barbara. “There’s no real problem from a physics standpoint, either, except that we know it’s wrong.”3

In general relativity, the metric or length function tells you the curvature at every point. At very small length scales, the metric coefficients fluctuate wildly, which means that lengths and curvature will fluctuate wildly as well. The geometry, in other words, would be undergoing shifts so violent it hardly makes sense to call it geometry. It would be like a rail system where the tracks shrink, lengthen, and curve at will—a system that would never deliver you to the right destination and, even worse, would get you there at the wrong time. That’s no way to run a railroad, as they say, and it’s no way to do geometry, either.

As with many problems we’ve grappled with in this book, this geometric weirdness springs from the fundamental incompatibility of quantum mechanics and general relativity. Quantum geometry might be thought of as the language of quantum gravity—the mathematical formalism needed to fix the compatibility problem—whatever that theory turns out to be. There’s another way to consider this problem, which also happens to be the way that many physicists think about this: Geometry, as it appears in physics, might be a phenomenon that’s “emergent” rather than fundamental. If this view is correct, it might explain why traditional geometric descriptions of the world appear to falter at the realm of the very small and the very energetic.

Emergence can be seen, for instance, in the lake or pond we discussed earlier in this chapter. If you look at a sizable body of water, it makes sense to think of the water as a fluid that can flow and form waves—as something that has bulk properties like viscosity, temperature, and thermal gradients. But if you were to examine a tiny droplet of water under an extremely powerful microscope, it would not look anything like a fluid. Water, as everyone knows, is made out of molecules that on a small scale behave more like billiard balls than a fluid. “You cannot look at waves on the surface of a lake and, from that, deduce anything about the molecular structure or molecular dynamics of H2O,” explains MIT physicist Allan Adams. “That’s because the fluid description is not the most fundamental way of thinking about water. On the other hand, if you know where all the molecules are and how they’re moving, you can in principle deduce everything about the body of water and its surface features. The microscopic description, in other words, contains the macroscopic information.”4 That’s why we consider the microscopic description to be more fundamental, and the macroscopic properties emerge from it.

So what does this have to do with geometry? We’ve learned through general relativity that gravity is a consequence of the curvature of spacetime, but as we’ve seen, this long-distance (low-energy) description of gravity—what we’re now calling classical geometry—breaks down at the Planck scale. From this, a number of physicists have concluded that our current theory of gravity, Einstein’s theory, is merely a low-energy approximation of what’s really going on. Just as waves on the surface of a lake emerge from underlying molecular processes we cannot see, these scholars believe that gravity and its equivalent formulation as geometry also emerge from underlying, ultramicroscopic processes that we assume must be there, even if we don’t know exactly what they are. That is what people mean when they say gravity or geometry is “emergent” from the sought-after Planck scale description of quantum geometry and quantum gravity.

Vafa’s concerns regarding the possible “end of geometry” are legitimate, but such an outcome may not be a tragedy—Greek or otherwise. The downfall of classical geometry should be celebrated rather than dreaded, assuming we can replace it with something even better. The field of geometry has constantly changed over the millennia. If the ancient Greek mathematicians, including the great Euclid himself, were to sit in on a geometry seminar today, they’d have no idea what we were talking about. And before long, my contemporaries and I will be in the same boat with respect to the geometry of future generations. Although I don’t know what geometry will ultimately look like, I fully believe it will be alive and well and better than ever—more useful in more situations than it presently is.

On this point, Santa Barbara physicist Joe Polchinski appears to agree. He doesn’t think the breakdown of conventional geometry at the Planck length signals “the end of the road” for my favorite discipline. “Usually when we learn something new, the old things that took us there are not thrown out but are instead reinterpreted and enlarged,” Polchinski says. Paraphrasing Mark Twain, he considers reports of the death of geometry to be greatly exaggerated. For a brief period in the late 1980s, he adds, geometry became “old hat” in physics. Passé. “But then it came back stronger than ever. Given that geometry has played such a central role in the discoveries to date, I have to believe it is part of something bigger and better, rather than something that will ultimately be discarded.”5 That’s why I argue that quantum geometry, or whatever you call it, has to be an “enlargement” of geometry, as Polchinski put it, since we need something that can do all the great things geometry already does for us, while also providing reliable physical descriptions on the ultra-tiny scale.

Edward Witten seems to be in accord. “What we now call ‘classical geometry’ is much broader than what geometry was understood to be just a century ago,” he says. “I believe that the phenomenon at the Planck scale very likely involves a new kind of generalization of geometry or a broadening of the concept.”6

Generalizations of this sort—which involve taking a theory that is valid in a certain regime and extending its scope and applicability into an even larger milieu—have been introduced to geometry repeatedly. Consider the invention of non-Euclidean geometry. “If you asked Nikolai Lobachevsky what geometry was when he was young,” early in the nineteenth century, “he probably would have listed the five postulates of Euclid,” says Adams. “If you asked him later in his career, he might have said there were five postulates, but maybe we don’t need them all.”7 In particular, he would have singled out Euclid’s fifth postulate—which holds that parallel lines never intersect—as one we could let go. It was Lobachevsky, after all, who realized that excluding the parallel postulate made a whole new geometry—which we call hyperbolic geometry—possible. For while parallel lines clearly do not intersect on a plane—the domain in which Euclid’s plane geometry operates—this is certainly not the case on the surface of a sphere. We know, for instance, that all longitudinal lines on a globe converge at the north and south poles. Similarly, while the angles of triangles drawn on a plane must always add up to 180 degrees, on the surface of a sphere those angles always add up to more than 180 degrees, and on the surface of a saddle they add up to less than 180 degrees.

Lobachevsky published his controversial ideas on non-Euclidean geometry in 1829, although they were buried in an obscure Russian journal called the Kazan Messenger. A few years later, the Hungarian mathematician János Bolyai published his own treatise on non-Euclidean geometry, though the work, unfortunately, was relegated to the appendix of a book written by his father, the mathematician Farkas (Wolfgang) Bolyai. At roughly the same time, Gauss had been developing similar ideas on curved geometry. He immediately recognized that these new notions of curved spaces and “intrinsic geometry” were intertwined with physics. “Geometry should be ranked not with arithmetic, which is purely a priori, but with mechanics,” Gauss claimed.8 I believe he meant that geometry, unlike arithmetic, must draw on empirical science—namely, physics (which was called mechanics at the time)—for its descriptions to carry weight. Gauss’s intrinsic geometry laid the groundwork for Riemannian geometry, which, in turn, led to Einstein’s dazzling insights on spacetime.

In this way, pioneers like Lobachevsky, Bolyai, and Gauss did not throw out all that came before but merely opened the door to new possibilities. And the breakthrough they helped usher in created a more expansive geometry, as its tenets were no longer confined to a plane and could instead apply to all manner of curved surfaces and spaces. Yet the elements of Euclid were still retained in this enlarged, more general picture. For if you take a small patch of Earth’s surface—say, a several-block area in Manhattan—the streets and avenues really are parallel for all practical purposes. The Euclidean description is valid in that limited domain, where the effects of curvature are negligible, but does not hold when you are looking at the planet as a whole. You might also consider a triangle drawn on a spherical balloon. When the balloon is relatively small, the angles will add up to more than 180 degrees. But if we keep inflating the balloon, the radius of curvature (r) will get bigger and bigger, and the curvature itself (which scales with 1/r2) will get smaller and smaller. If we let r go to infinity, the curvature will go to zero and the angles of the triangle, at this limit, will be exactly 180 degrees. As Adams puts it, “there’s this one situation, on a flat plane, where Euclidean geometry works like a champ. It works pretty well on a slightly curved sphere, but as you inflate the balloon and the sphere gets flatter and flatter, the agreement gets better and better. So we can see that Euclidean geometry is really just a special case of a more general story—the case in which the radius of curvature is infinite, the angles of the triangle add up to 180 degrees, and all the postulates of Euclidean geometry are recovered.”9

Similarly, Newton’s theory of gravity was an extremely practical theory, in the sense that it gave us a simple way of computing the gravitational force exerted on any object in a system. Specifically, it worked well so long as the objects in question were not moving too fast or in situations where the gravitational potential is not too large. Then along came Einstein with his new theory, in which gravity was seen as a consequence of curved spacetime rather than a force propagated between objects, and we realized that Newton’s gravity was just a special case of this broader picture that was only valid in the limit when objects were moving slowly and gravity was weak. In this way, we can see that general relativity, as the name suggests, truly was general: It was not only a generalization of Einstein’s special theory of relativity—by taking a theory that did not have gravity and expanding the setting to include gravity—but also a generalization of Newtonian gravity.

In the same way, quantum mechanics is a generalization of Newtonian mechanics, but we don’t need to invoke quantum mechanics to play baseball or tiddlywinks. Newton’s laws of motion work fine for large objects like baseballs (and even for smaller objects like winks), where the corrections imposed by quantum theory would be immeasurably small and can thus be safely overlooked. But the macroscopic domain in which baseballs and rockets fly about, and Newton’s laws prevail, is just a special case of the broader, more general domain of quantum theory, which also holds for objects that are considerably smaller. Using quantum mechanics, we can accurately predict the trajectories of relativistic electrons at a high-energy collider, whereas in that domain, Newtonian mechanics will let us down.

Now we’re approaching this very same juncture with respect to geometry. Classical Riemannian geometry is not capable of describing physics at the quantum level. Instead we seek a new geometry, a more general description, that applies equally well to a Rubik’s Cube as it does to a Planck-length string. The question is how to proceed. To some extent, we’re groping in the dark, as Isaac Newton might have been when trying to write his own theory of gravity.

Newton had to invent new techniques to achieve that end, and out of that, calculus was born. Just as Newton’s mathematics was motivated by physics, so must ours be today. We can’t create quantum geometry without some input from physics. While we can always conjure up some new interpretation of geometry, if it is to be truly fruitful, geometry must describe nature at some basic level. And for that, as Gauss wisely acknowledged, we need some guidance from the outside.

The relevant physics gives us the technical demands that our math must satisfy. If classical geometry is used, physics at the Planck scale would appear to involve discrete changes and sudden discontinuities. The hope is that quantum geometry will eliminate those discontinuities, creating a smoother picture that’s simpler to grasp and easier to work with.


14.1—The physicist John Wheeler’s concept of quantum foam. The top panel looks completely smooth. But if that surface is blown up by twenty orders of magnitude (middle panel), irregularities become evident. When the surface is blown up another thousandfold, all the little bumps become mountains, and the surface is now the antithesis of smooth.

String theory, almost by definition, is supposed to deal with problems of that sort. Because “the fundamental building block of . . . string theory is not a point but rather a one-dimensional loop, it is natural to suspect that classical geometry may not be the correct language for describing string physics,” Brian Greene explains. “The power of geometry, however, is not lost. Rather, string theory appears to be described by a modified form of classical geometry . . . with the modifications disappearing as the typical size in a given system becomes large relative to the string scale—a length scale which is expected to be within a few orders of magnitude of the Planck scale.”10

Previous theories of fundamental physics regarded their basic building block, the particle, as an infinitely small, zero-dimensional point—an object that the mathematics of the time was ill-equipped to handle (and that the mathematics of today is still ill-equipped to handle). Strings are not infinitely small particles, so the quantum fluctuations, which had been so troublesome for classical geometry at ultramicroscopic scales, are spread out over a substantially larger area, thereby diminishing their strength and making the fluctuations more manageable. In this way, the vexing problem of singularities in physics, where the curvature and density of spacetime blow up to infinity, is deftly bypassed. “You never get to the point where the disasters happen,” says Nathan Seiberg of the Institute of Advanced Study. “String theory prevents it.”11


14.2a—This photograph is called “The Blue Marble” to suggest that when viewed from a distance, Earth’s surface looks as smooth and unblemished as a marble. (NASA Goddard Space Flight Center)


14.2b—A close-up photograph of Santa Fe, New Mexico (which lies near the center of the Blue Marble image), as taken by the Landsat 7 Earth-observing satellite, showing that the surface is anything but smooth. Together, these two photographs illustrate the notion of quantum foam—the idea that what might appear to be a smooth, featureless object from a distance can look extremely irregular from up close. (Visualization created by Jesse Allen, Earth Observatory; data obtained, coregistered, and color balanced by Laura Rocchio, Landsat Project Science Office)

Even if outright catastrophe is averted, it is still instructive to look at the close calls and near misses. “If you want to study situations where geometry breaks down, you want to pick cases where it only breaks down a little bit,” says Andrew Strominger. “One of the best ways of doing that is studying how Calabi-Yau spaces break down, because in those spaces, we can isolate regions where spacetime breaks down, while the rest of the region stays nice.”12

The hope to which my colleague speaks is that we might be able to gain some insights on quantum geometry, and what it entails, by doing string theory in the controlled setting of Calabi-Yau spaces—a theme, of course, that runs throughout this book. One promising avenue has been to look for situations in string theory where geometry behaves differently than it does in classical geometry. A prime example of this is a topology-changing transition that can sometimes proceed smoothly in string theory but not in conventional physics. “If you’re restricted to standard geometric techniques—and by that I mean always keeping a Riemannian metric in place—the topology cannot change,” says Morrison. 13 The reason topological change is considered a big deal is that you can’t transform one space into another without ripping it in some way—just as you can’t scramble eggs without breaking some shells. Or turn a sphere into a donut without making a hole.

But piercing a hole in an otherwise smooth space creates a singularity. That, in turn, poses problems for general relativists, who now have to contend with infinite curvature coefficients and the like. String theory, however, might be able to sidestep this problem. In 1987, for example, Gang Tian (my graduate student at the time) and I demonstrated a technique known as a flop transition, which yielded many examples of closely related but topologically distinct Calabi-Yau manifolds.

Conifold transitions, which were discussed in Chapter 10, are another, even more dramatic example of topological change involving Calabi-Yau spaces. Think of a two-dimensional surface such as a football standing on end inside a Calabi-Yau space, as illustrated in Figure 14.3. We can shrink the football down to an increasingly narrow strand that eventually disappears, leaving behind a tear—a vertical slit—in the fabric of spacetime. Then we tilt the slit by pushing the “fabric” above it and that below it toward each other. In this way, the vertical slit is gradually transformed into a horizontal slit, into which we can insert and then reinflate another football. The football has now “flopped” over from its original configuration. If you do this procedure in a mathematically precise way—tearing space at a certain point, opening it up, reorienting the tear, and inserting a new two-dimensional surface with a shifted orientation back into the six-dimensional space—you can produce a Calabi-Yau space that is topologically distinct, and thus has a different overall shape, from the one you started with.

The flop transition was of interest mathematically because it showed how you could start with one Calabi-Yau, whose topology was already familiar, and end up with other Calabi-Yau spaces we’d never seen before. As a result, we mathematicians could use this approach to generate more Calabi-Yau spaces to study or otherwise play around with. But I also had a hunch that the flop transition might have some physical significance as well. Looking back with the benefit of hindsight, one might think I was particularly prescient, though that’s not necessarily the case. I feel that any general mathematical operation we can do to a Calabi-Yau should have an application in physics, too. I encouraged Brian Greene, who was my postdoc at the time, to look into this, as well as mentioning the idea to a few other physicists who I thought might be receptive. Greene ignored my advice for several years, but finally started working on the problem in 1992 in concert with Paul Aspinwall and Morrison. In light of what they came up with, it was well worth the wait.


14.3—One way to think about the flop transition is to make a vertical slit in a two-dimensional fabric. Now push on the fabric from both the top and the bottom so that the vertical slit becomes wider and wider and eventually becomes a horizontal slit. So in this sense, the slit or tear, which was once standing upright, has now “flopped” over onto its side. Calabi-Yau manifolds can undergo flop transitions, too, when internal structures are toppled over in similar fashion (often after an initial tear), resulting in a manifold that’s topologically distinct from the one you started with. What makes the flop transition especially interesting is that the four-dimensional physics associated with these manifolds is identical, despite the difference in topology.

Aspinwall, Greene, and Morrison wanted to know whether something like a flop transition could occur in nature and whether space itself would rip apart, despite general relativity’s picture of a smoothly curved spacetime that is not prone to rupture. Not only did the trio want to determine whether this type of transition could occur in nature, they also wanted to know whether it could occur in string theory.

To find out, they took a Calabi-Yau manifold with a sphere (rather than football) sitting inside it, put it through a flop transition, and used the resultant (topologically altered) manifold to compactify six of spacetime’s ten dimensions to see what kind of four-dimensional physics it yielded. In particular, they wanted to predict the mass of a certain particle, which, in fact, they were able to compute. They then repeated the same process, this time using the mirror partner of the original Calabi-Yau space. In the mirror case, however, the sphere did not shrink down to zero volume as it went through the flop transition. In other words, there was no tearing of space, no singularity; the string physics, as Greene puts it, was “perfectly well behaved.”14 Next, they computed the mass of that same particle—this time associated with the mirror manifold—and compared the results. If the predictions agreed, that would mean the tearing of space and the singularity that came up in the first case were not a problem; string theory and the geometry on which it rests could handle the situation seamlessly. According to their calculations, the numbers matched almost perfectly, meaning that tears of this sort could arise in string theory without dire consequences.

One question left unanswered by their analysis was how this could be true—how, for instance, could a sphere shrink down to zero volume (the size of a point in traditional geometry) when the smallest allowable size was that of an individual string? Possible answers were contained in a Witten paper that came out at the same time. Witten showed how a loop of string could somehow encircle the spatial tear, thereby protecting the universe from the calamitous effects that might otherwise ensue.

“What we learned is that when the classical geometry of the Calabi-Yau appears to be singular, the four-dimensional physics looks smooth,” explains Aspinwall. “The masses of particles do not go to infinity, and nothing bad happens.” So the quantum geometry of string theory must somehow have a “smoothing effect,” taking something that classically looks singular and making it nonsingular.15

The flop transition can shed light on what quantum geometry might look like, by showing us those situations that classical geometry cannot handle. Classical geometry can describe the situation at the beginning and end of the flop with no trouble, but fails at the center, where the width of the football (or basketball) shrinks to zero. By seeing exactly what string theory does differently, in this case and in as many other cases as we can find, we can figure out how classical geometry has to be modified—what kind of quantum corrections are needed, in other words.

The next question to ask, says Morrison, “is whether the quantum modifications we have to make to geometry are sufficiently geometric-looking to still be called geometry or whether they are so radically different that we would have to give up the notion of geometry altogether.” The quantum corrections we know about so far, through examples like the flop transition, “can still be described geometrically, even though they may not be easily calculated,” he says. But we still don’t know whether this is generally true.16

I personally am betting that geometry will prevail in the end. And I believe the term geometry will remain in currency, not simply for the sake of nostalgia but rather because the field itself will continue to provide useful descriptions of the universe, as it has always done in the past.

Looking to the future, it’s clear that creating a theory of quantum geometry, or whatever you care to call it, surely stands as one of the greatest challenges facing the field of geometry, if not all of mathematics. It’s likely to be a decades-long ordeal that will require close collaboration between physicists and mathematicians. While the task certainly demands the mathematical rigor we always try to apply, it may benefit just as much from the intuition of physicists that never ceases to amaze us mathematicians.

At this stage of my career, having already been in the game for about forty years, I certainly have no illusions of solving this problem on my own. In contrast to a more narrowly defined proof that one person might be able to solve single-handedly, this is going to take a multidisciplinary effort that goes beyond the labors of a lone practitioner. But given that Calabi-Yau spaces have been central in some of our early attempts to gain footholds on quantum geometry, I’m hoping to make some contributions to this grand endeavor as part of my long-standing quest to divine the shape of inner space.

Ronnie Chan, a businessman who generously supported the Morningside Institute of Mathematics in Beijing (one of four math institutes I’ve helped start in China, Hong Kong, and Taiwan), once said, “I’ve never seen any person as persistent on one subject as Yau. He only cares about math.” Chan is right about my persistence and devotion to math, though I’m sure that if he were to look hard enough, he’d find plenty of people as persistent and devoted to their own pursuits as I am to mine. On the other hand, the question I have thrown myself at—trying to understand the geometry of the universe’s internal dimensions—is an undeniably big one, even though the dimensions themselves may be small. Without some degree of persistence and patience, my colleagues and I would never have gotten as far as we have. That said, we still have a long way to go.

I read somewhere, perhaps in a fortune cookie, that life is in the getting there—the time spent negotiating a path between point A and point B. That is also true of math, especially geometry, which is all about getting from A to B. As for the journey so far, all I can say is that it’s been quite a ride.