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

Chapter 7. THROUGH THE LOOKING GLASS

Though Calabi-Yau manifolds entered physics with a bang, these beguiling shapes were quickly at risk of becoming a whimper, and for reasons that had nothing to do with string theory’s embarrassment of riches (in the form of the multiplicity of theories that Edward Witten ultimately helped sort out). The attraction of these geometric shapes was obvious: As Duke physicist Ronen Plesser describes it, “the hope was that we could classify these spaces, figure out the kind of physics they give rise to, exclude some of them, and then conclude that our universe is described by space number 476, and everything we might want to know could be derived from that.”1

That simple vision has yet to be realized, even today, and as progress stalled and enthusiasm waned more than two decades ago, doubts inevitably crept in. By the late 1980s, many physicists felt that the role of Calabi-Yau manifolds in physics was doomed. Physicists like Paul Aspinwall (now at Duke), who was then finishing his Ph.D. work at Oxford, found it hard to get jobs and secure grants to pursue their research into Calabi-Yau manifolds and string theory. Disenchanted string theory students, including two of Brian Greene’s former class-mates and coauthors at Oxford, started leaving the field to become financiers. Those who remained, such as Greene, found themselves having to fend off charges of “pursuing calculations for their own sake—of learning math and regurgitating it as physics.”2

Perhaps that was true. But given that Greene and Plesser would soon make critical contributions to an idea called mirror symmetry, which would revitalize Calabi-Yau manifolds and rejuvenate a somnolent branch of geometry, I must admit that I’m pleased they settled on that particular line of research rather than, say, stock futures. Before this renaissance came, however, these manifolds would hit a low point that, for a while at least, looked as if it might prove fatal.

Signs of trouble began to emerge through string theory investigations of a concept known as conformal invariance. A string moving through spacetime sweeps out a surface of two real dimensions (one of space and one of time) and one complex dimension called a world sheet. If the string is a loop, the world sheet takes the form of a continuously extending, multidimensional tube (or more precisely, a complex Riemann surface without a boundary), whereas a strand of open string traces an ever-elongating strip (that is, a complex Riemann surface with a boundary). In string theory, we study all possible vibrations of strings, and those vibrations are governed by a physical principle (or action principle ) that depends on the conformal structure of the world sheet, which is itself an intrinsic feature of Riemann surfaces. In this way, string theory has conformal invariance automatically built into it. The theory is also scale invariant, meaning that multiplying distances by some constant shouldn’t change the relationships between points. Thus, you should be able to change the surface—like putting air in a balloon or shrinking the surface by letting air out, or expanding it in other ways that change the distance between two points or change the shape—without changing anything important in string theory itself.

The problems arise when you insist on conformal invariance in a quantum setting. Much as a classical particle follows a geodesic, a path that minimizes the distance traveled (as described by the principle of least action we talked about in Chapter 3), a classical string follows a path of minimal distance. As a result, the world sheet associated with that moving string is a minimal-area surface of a special kind. The area of this two-dimensional world sheet can be described by a set of equations, a two-dimensional field theory, that tells us exactly how a string can move. In a field theory, all forces are described as fields that permeate spacetime. The movement of a string and its behavior in general are a result of the forces exerted on it, and the string moves in such a way as to minimize the area of the world sheet. Among the vast number of possible world sheets, encompassing the vast number of possible ways the string can move, this field theory picks out the one with minimal area.

The quantum version of this field theory must capture not only the gross features of a string moving in spacetime—and the surface that such an object traces out—but also some of the finer details that stem from the fact that the string oscillates as it moves. The world sheet, as a result, will have small-scale features that reflect those oscillations. In quantum mechanics, a particle or string moving in spacetime will take all possible paths. Rather than just selecting the one world sheet possessing minimal area, a quantum field theory takes a weighted average of all possible world sheet configurations and assigns more importance in its equations to surfaces with lower area.

But after all this averaging has been done—by integrating over all possible world sheet geometries—will the two-dimensional quantum field theory that emerges still retain scale invariance and other aspects of conformality? That depends on the metric of the space in which the world sheets exist; some metrics allow for a conformal field theory, and others do not.

To see whether scale invariance is maintained in a specific metric, we calculate something known as the beta function, which measures deviations from conformality. If the value of the beta function is zero, nothing changes when we deform the world sheet by blowing it up, stretching it, or shrinking it, which is good if you like your theories conformal. It had been assumed that the beta function would automatically vanish (or drop to zero) in a Ricci-flat metric such as those we can find in Calabi-Yau manifolds. Unfortunately, as with many of the complicated equations we’ve been discussing, the beta function cannot be calculated explicitly. Instead, we approximate it by an infinite sum known as a power series. The more terms we use from the series, the better our approximation.

You can get a sense of how we do this by imagining that you’re trying to measure the surface area of a sphere by wrapping it in a wire mesh. If the wire consists of a single loop that can assume a single position around the sphere, you can’t get a good area estimate. If you instead had four triangular loops, configured as a tetrahedron surrounding the sphere, you could get a much better area approximation. Increasing the number to twelve loops (shaped like pentagons in the case of a dodecahedron) and to twenty loops (shaped like triangles in the case of an icosahedron) would yield even more refined approximations. Much as in our example, the terms in the beta function’s power series are known as loops. Working with just one term in the series gives us the one-loop beta function; working with the first two terms gives us the two-loop beta function, and so forth.

The problem with adding more loops to the wire mesh is that these beta function calculations, which are extraordinarily difficult to start with, get harder—and more computationally intensive—as the number of loops goes up. Calculations showed that the first three terms of the power series were zero, just as predicted, which was reassuring to physicists. But in a 1986 paper, Marcus Grisaru (a physicist now at McGill) and two colleagues, Anton van de Ven and Daniela Zanon, showed that the four-loop beta function does not vanish. A subsequent calculation by Grisaru and others showed that the five-loop beta function does not vanish, either. This seemed like a major blow to the position of Calabi-Yau manifolds in physics, because it suggested that their metric did not allow for conformal invariance to hold.

“As a believer in both string theory and supersymmetry, our findings were of some concern,” Grisaru says. “We were happy that the result gave us much notoriety, but you don’t always want the notoriety that comes from demolishing a beautiful building. Still, my attitude about science is that you have to accept whatever you get.”3

But perhaps all was not lost. A separate 1986 paper by David Gross and Witten, both of whom were then at Princeton, argued that even though having an exactly Ricci-flat metric on a Calabi-Yau manifold does not work, the metric can be changed slightly so that the beta function will vanish, as required. Tweaking the metric in this way didn’t mean making a single adjustment; instead it meant making an infinite number of adjustments, or quantum corrections. But in situations like this, when an infinite series of corrections are required, there’s always the question as to whether that series will eventually converge on a solution. “Could it be that when you start adding in all the corrections, there’d be no solution at all?” Plesser asks.

In the best of circumstances, changing the metric by a little bit will yield a solution that itself changes by a little bit. We know, for example, how to solve the equation 2x = 0, the answer being x = 0. “If I try to solve 2x = 0.100, the answer changes just slightly (x = 0.050), which is the situation we like,” Plesser explains. We can also solve the equation x2 = 0 without much difficulty (again, x = 0). “But if you ask me to solve x2 = -0.100, my solution (at least in real numbers) just vanished,” he says. “So we see that a small modification can take you to a solution that is just a little bit different or to a solution that [for real numbers] doesn’t even exist.”4

In this case of the amended Calabi-Yau, Gross and Witten determined that the series would converge. They showed that you can correct the Calabi-Yau metric, term by term, and at the end of that procedure, you’ll be left with a very complicated equation that can be solved nevertheless. In the process, all the loops in the beta function will go to zero.

As a result, explains Shamit Kachru of Stanford, “people didn’t have to throw out Calabi-Yau’s; they just had to modify them slightly. And since you couldn’t write down the Calabi-Yau metric to begin with, the fact that the metric had to be modified slightly wasn’t such a big deal.”5

Further insights on how the Calabi-Yau metric had to be changed came in that same year from the work of Dennis Nemeschansky and Ashoke Sen, who were then based at Stanford. The resultant manifold is still a Calabi-Yau topologically, and its metric is almost Ricci flat but not quite. Nemeschansky and Sen offered an explicit construction, a precise formula, showing how the modified metric differs from Ricci flatness. Their work, along with that of Gross and Witten, “helped save Calabi-Yau’s for physics, for without it, people would have had to abandon the whole approach,” asserts Sen. What’s more, Sen argues, we could have never gotten to the solution without the first assumption that the Calabi-Yau manifold used in string theory was Ricci flat. “For if we didn’t start from a Ricci-flat metric, it’s hard to imagine any procedure we could have used to take us to the corrected metric.”6

I agree with Sen, but that doesn’t mean the Ricci-flat assumption has been rendered useless. One way to think about it is that the Calabi-Yau with a Ricci-flat metric is like the solution to the equation x2 = 2. What you really want, however, is the solution to the equation x2 = 2.0000000001, because the manifold is almost Ricci-flat but not exactly. The only way to get to the modified, corrected metric is to start with the solution to x2 = 2 and move from there to the one you want. For most purposes, however, the solution to x2 = 2 is good enough. As a general matter, the Ricci-flat metric is also the easiest to use and captures most of the phenomena you’re after anyway.

The next big steps in the resurrection of Calabi-Yau manifolds were made by Doron Gepner (then a Princeton physics postdoc) over a several-year period starting in 1986. Gepner devised conformal field theories, each of which bore striking similarities—in terms of the associated physics—to a single Calabi-Yau manifold of a distinct size and shape. Initially Gepner found that the physics related to his field theory—involving certain symmetries, fields, and particles—looked the same as that for a string propagating on a particular Calabi-Yau manifold. That got people’s attention, as there seemed to be a mysterious link between two things, a conformal field theory and a Calabi-Yau, which ostensibly had nothing to do with each other.

One of the people whose interest was piqued was Brian Greene—my Harvard postdoc at the time, who had an expertise in the mathematical underpinnings of Calabi-Yau manifolds, having completed his doctoral dissertation on the subject, and who also had a strong grounding in conformal field theory. He started talking to people in the physics department who also worked on conformal theories, including two graduate students, Ronen Plesser and Jacques Distler. Distler and Greene began investigating correlation functions associated with the field theory and the corresponding Calabi-Yau. The correlation functions in this case involved “Yukawa couplings” that dictate how particles interact, including interactions that confer mass to a given particle. In a paper submitted in the spring of 1988, Distler and Greene found that the correlation functions (or Yukawa couplings) for the two cases numerically agreed—more evidence that the field theory and Calabi-Yau were somehow related if not the same.7 Gepner arrived at a similar conclusion, regarding the match in Yukawa couplings, in a paper submitted shortly thereafter.8

More specifically, Distler and Greene, and Gepner separately, found that for precise settings of manifold size and shape, they could compute all the correlation functions, which are a set of mathematical functions that, taken in sum, completely characterize the conformal field theory. The result, in other words, spelled things out in starkly explicit terms, identifying an exact conformal field theory with all the correlation functions, as well as showing the exact size and shape of the associated Calabi-Yau. For a restrictive class of the Calabi-Yau manifolds known to date, we have since identified a corresponding Gepner model.

That link, which was firmly established by the late 1980s, helped turn around thinking regarding the usefulness of Calabi-Yau manifolds. As Kachru put it, “you couldn’t doubt the existence of his [Gepner’s] conformal field theories, because they were fully solvable, fully computable. And if you can’t doubt those theories, and they show the same properties as his Calabi-Yau compactifications, then you can’t doubt those compactifications, either.”9

“Gepner’s paper really saved Calabi-Yau’s,” claims Aspinwall, at least insofar as physics and string theory were concerned.10 The connection, moreover, between a representative Gepner model and a specific Calabi-Yau compactification helped set the stage for the discovery of mirror symmetry—a finding compelling enough to remove any lingering doubts as to whether Calabi-Yau manifolds were spaces worth studying.

Some of the earliest hints of mirror symmetry came in 1987, when Stanford physicist Lance Dixon and Gepner observed that different K3 surfaces were linked to the same quantum field theory, thereby implying that these disparate surfaces were related through symmetry. Neither Dixon nor Gepner, however, produced a paper on the subject (though Dixon gave some talks), so the first written statement of mirror symmetry probably came in a 1989 paper by Wolfgang Lerche (of Caltech), Cumrun Vafa, and Nicholas Warner (of MIT); they argued that two topologically distinct Calabi-Yau threefolds—six-dimensional Calabi-Yau manifolds rather than four-dimensional K3’s—could give rise to the same conformal field theory and hence the same physics.11 This was a stronger statement than the Dixon-Gepner one because it linked Calabi-Yau manifolds with different topologies, whereas the earlier finding applied to surfaces with the same topology (though with different geometries), owing to the fact that all K3s are topologically equivalent. The problem was that no one knew how to construct the pairs of Calabi-Yau manifolds that might be tied together in this strange way. Gepner models turned out to be one of the keys to unraveling that puzzle, and those same models also helped bring Brian Greene and Ronen Plesser together for the first time.

In the fall of 1988, Brian Greene learned through a conversation with Vafa (whose offices were on the same “theory” floor of the Harvard physics building) about this suspected link between different Calabi-Yau manifolds. Greene realized, right off, that this idea would be hugely important if it could be proved. He joined forces with Vafa and Warner to gain a better understanding of the link between Calabi-Yau manifolds and Gepner models. Greene, Vafa, and Warner spelled out the steps for going from a Gepner model to a particular Calabi-Yau, explains Greene.12 The researchers provided “an algorithm that showed why they are connected and how they are connected. Give me a Gepner model, and I can show you in a flash which Calabi-Yau it’s connected to.”13 The Greene, Vafa, and Warner paper explained why every Gepner model yields a Calabi-Yau compactification. Their analysis took the guesswork out of matching Gepner models with Calabi-Yau manifolds, as Gepner previously had to look through tables to find a Calabi-Yau that produced the desired physics.

Now that the relationship between Gepner models and Calabi-Yau manifolds had been solidified, Greene teamed up with Plesser in 1989 in the hopes of going further. One of the first things they realized, says Greene, “is that we now had a potent tool for analyzing very complicated [Calabi-Yau] geometry using a field theory that we had complete control over and, indeed, a complete understanding of.”14 What would happen, they wondered, if they changed the Gepner model a little bit? They thought the altered model would probably correspond to a somewhat different Calabi-Yau. So they tried it out, by performing a rotational symmetry operation (like rotating a square by 90 degrees) on the Gepner model. It left everything in the field theory unchanged. Performing the same symmetry operation on the Calabi-Yau, however, produced manifolds of different topology as well as geometry.

The symmetry operation, in other words, changed the Calabi-Yau topology while leaving the conformal field theory that goes along with it intact. The result, then, was two Calabi-Yau manifolds with two distinct topologies connected to the same physical theory. “And that, in a nutshell, is mirror symmetry,” Gepner notes.15 It is also, more generally, what we call a duality, in which two objects (in this case, the Calabi-Yau manifolds) that appear unrelated nevertheless give rise to the same physics.

Greene and Plesser’s first paper on the subject of mirror symmetry contained ten so-called mirror partners or mirror manifolds in nontrivial (as in not totally flat) Calabi-Yau manifolds, starting with the simplest Calabi-Yau, the quintic threefold. Along with the other nine examples, their paper gave a formula that showed how to construct mirror pairs from any Gepner model—hundreds, if not thousands, of which are known to date.16

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7.1—Brian Greene (Photo © Andrea Cross)

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7.2—Ronen Plesser (Photo by Duke Photography)

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7.3—The double tetrahedron, which has five vertices and six faces, and the triangular prism, which has six vertices and five faces, are simple examples of mirror manifolds. These rather familiar-looking polyhedra can be used, in turn, to construct a Calabi-Yau manifold and its mirror pair, with the number of vertices and faces of the polyhedra relating to the internal structure of the associated Calabi-Yau. (The details of this “construction” procedure are quite technical, however, going well beyond the scope of this discussion.)

Mirror manifolds have some fascinating attributes that came to light by juxtaposing objects that previously seemed unrelated. Greene and Plesser found, for example, that one Calabi-Yau manifold might have 101 possible shape settings and one possible size setting; the mirror would be just the opposite, having one possible shape setting and 101 size settings. Calabi-Yau manifolds have holes of various dimensions (odd- and even-numbered), but Greene and Plesser found a curious pattern between the pairs: The number of odd-dimensional holes in one manifold equaled the number of even-dimensional holes in its mirror, and vice versa. “This means that the total number of holes . . . in each is the same even though the even-odd interchange means that their shapes and fundamental geometrical structures are quite different,” Greene says.17

That still doesn’t quite explain the “mirror” aspect of this symmetry, which might be easier to picture through topology. It was found, for example, that Calabi-Yau mirror pairs had Euler characteristics of opposite sign, thereby implying markedly different topologies, albeit crudely; the numbers themselves convey only a bit of information about a space, and as we’ve seen, many very different-looking spaces—such as a cube, tetrahedron, and sphere—can all have the same Euler characteristic. This crudeness can be refined, however, by breaking the Euler characteristic into sums and differences of integers called Betti numbers, which contain more detailed information about a space’s inner structure.

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7.4—Surfaces (and again, we mean orientable or two-sided ones) can be distinguished from each other topologically by comparing their Betti numbers. In general, the Betti number tells you the number of ways you can cut all the way around a two-dimensional surface without dividing it into two pieces. There are no ways of making such cuts to a sphere, so its Betti number is 0. A donut, on the other hand, can be cut in two different ways without dividing it in half—as shown—so its Betti number is 2.

Any given object has n + 1 Betti numbers, where n is the number of dimensions the object has. A zero-dimensional point, therefore, has a Betti number of 1; a one-dimensional circle has a Betti number of 2; a two-dimensional surface, such as a sphere, has 3; and so forth. The first Betti number is denoted b1, the second b2, and the last bk, where the kth Betti number is the number of independent, k-dimensional cycles or loops that can be wrapped around (or threaded through) the space or manifold in question. (We’ll say a bit more about cycles later in this chapter.)

In the case of two-dimensional surfaces, the first Betti number describes the number of cuts that can be made to a space without dividing it in two. If you take the surface of a sphere, a two-dimensional space, there’s no way to cut all the way around the surface without dividing it into two pieces. That’s equivalent to saying the first Betti number of a sphere equals 0.

Now we’ll take a hollow donut. If you make an incision all the way around the “equator,” you’ll still end up with a single pastry, albeit an eviscerated one. Similarly, if you slice the donut the other way, the short way through the hole, you’ll be left with a severed donut that’s still just a single pastry (albeit an unsightly one). Since there are two different ways of cutting the donut, neither of which divides the space into two, we say its first Betti number is 2.

Next, let’s consider a pretzel with two holes. We can slice through either hole or slice through from one hole to the other or make a slice all the way around the outside edge and still be left with a single pretzel. That makes four different ways of cutting the double-holed pretzel, none of which divide it into two pieces, so its first Betti number is 4. The same pattern holds for a pretzel with 18 holes; its first Betti number is 36.

We can get an even more refined description of the topology of different manifolds, however. Each of the Betti numbers is itself the sum of finer numbers called Hodge numbers, discovered by the Scottish mathematician W. V. D. Hodge. These numbers afford an even higher-resolution glimpse into a space’s substructure. This information is encapsulated in a so-called Hodge diamond.

Hodge diamonds enable us to visualize the “mirror” in mirror symmetry. A given grid of sixteen numbers corresponds to a particular six-dimensional Calabi-Yau manifold, which we’ll call M. To get the Hodge diamond for the mirror manifold, M′, we draw a line that goes from the middle of the lower left edge of the diamond to the middle of the upper right edge. Then we flip the Hodge numbers over this diagonal line. The modified Hodge diamond, which characterizes the mirror partner, is literally a reflection or mirror image of the original.

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7.5—Detailed topological information about a Calabi-Yau manifold (of three complex dimensions) is contained in a four-by-four array of numbers known as a Hodge diamond. Although a Calabi-Yau manifold may not be uniquely characterized by its Hodge diamond, manifolds with different Hodge diamonds are topologically distinct. The two Hodge diamonds shown are mirror images of each other, corresponding to a Calabi-Yau manifold and its mirror partner.

The fact that the Hodge numbers for the manifolds are reversed is a consequence rather than an explanation of mirror symmetry, because this can also happen in two manifolds that are not actual mirror pairs. The reflected Hodge numbers seen by Greene and Plesser were merely the first hints, rather than proof, that they’d stumbled upon an interesting new symmetry. Far more compelling, says Plesser, was determining that the physics (or conformal field theories) associated with the mirror pairs was “literally identical.”18

Outside corroboration came a few days after Greene and Plesser had submitted their paper in 1989. Candelas informed Greene that he and two students had found a striking pattern after surveying a large number of Calabi-Yau manifolds they’d created by computer. These manifolds, they noticed, seemed to come in pairs in which the number of even-dimensional holes of one equaled the number of odd-dimensional holes of the other. The observed interchange in number of holes, shape and size settings, and Hodge numbers between the two manifolds was certainly intriguing, but could have been mere mathematical coincidence, says Greene, “perhaps of no more consequence to physics than going to one store, where milk was a dollar and juice two dollars, and then finding another store, where milk was two dollars and juice just one dollar. What clinched the deal was the argument that Plesser and I gave establishing that pairs of different Calabi-Yau’s yielded the same physics. That’s the real definition of mirror symmetry—the one from which all the interesting consequences follow—and it is much more than the simple interchange of two numbers.”19

The two efforts were not only parallel but “complementary,” according to Greene. While he and Plesser had gone deeper by exploring the physical consequences of this correspondence, Candelas and his students—by virtue of their computer program—had found a much larger sample of Calabi-Yau shapes whose Hodge numbers fell into mirror pairs. With these two papers (both published in 1990), Greene proclaimed that the “mirror symmetry of string theory” had finally been established.20

Vafa, for one, was happy to see the conjecture he’d contributed to upheld, though he’d felt all along that mirror symmetry would prove correct. “I sometimes say it was more courageous of us to have formulated it without any known examples,” he jokes.21

Initially, I had been skeptical of Vafa and Greene’s program, as I told them, because almost all of the Calabi-Yau manifolds we’d found up to that point had negative Euler characteristics. If what they were saying was true, and these manifolds really came in pairs of opposite Euler characteristics, then we should have been finding about as many with positive Euler characteristics as those with negative values, given that the Euler characteristic of the manifold and its mirror must have opposite signs. Fortunately, those concerns were not sufficient to make Vafa, Greene, Plesser, and others abandon their investigations into the possibility of this new kind of symmetry. (The moral being it’s often better to go ahead and look, rather than concluding in advance that something cannot be found.) And before long, we started finding more Calabi-Yau manifolds with positive Euler characteristics—enough, in fact, to put my earlier concerns to rest.

I soon arranged for Greene to talk about mirror symmetry to mathematicians, and several big shots—including I. M. Singer from MIT—were planning to be there. A physicist by training, Greene was nervous about speaking before such a crowd. I told him he should try to use the word quantum liberally in his lecture, as mathematicians tended to be impressed with that word. Perhaps, I suggested, he could describe mirror symmetry in terms of “quantum cohomology”—a term I coined at the time.

Cohomology has to do with the cycles, or loops, of the manifold and how they intersect. Cycles, in turn, are linked to subsurfaces within the manifold— also called submanifolds—that have no boundaries. To get a better idea of what we mean by submanifold, imagine a hunk of Swiss cheese cut into the shape of a ball. You could think of that entire Swiss cheese ball as a three-dimensional space that could be wrapped along the outside with a plastic sheet. But inside you might also find hundreds of smaller holes—subsurfaces within the larger surface—that you might be able to wrap or string something like a rubber band through. The submanifold is a geometric object with a precisely defined size and shape. A cycle, to a physicist, is a less sharply defined loop based solely on topology, whereas most geometers don’t see any difference between a cycle and a submanifold. Nevertheless, we tend to use cycles—such as a circle winding through a donut hole—to extract information about a manifold’s topology.

Physicists have a way of associating a quantum field theory with a given manifold. But since that manifold normally has an infinite number of cycles, they use an approximation that drives that number down to a finite, and therefore manageable, value. This process is called quantization—taking something with an infinite number of possible settings (such as frequencies on the FM radio dial) and saying that only certain values are going to be allowed. Doing that involves making quantum corrections to the original equation—an equation that is all about cycles and therefore all about cohomology, too. Hence the name quantum cohomology.

It turns out that there is more than one way to do these quantum corrections. Thanks to mirror symmetry, we can take a Calabi-Yau manifold and produce its physically equivalent mirror partner. The partners are described by two apparently different but fundamentally equivalent versions of string theory, Type IIA and Type IIB, that describe the same quantum field theory. We can do these quantum correction calculations relatively easily for the B model, where the quantum corrections turn out to be zero. The A model calculation, where the quantum corrections are not zero, is practically impossible.

About a year after Greene and Plesser’s paper, a further development in mirror symmetry commanded the attention of the mathematics community. That’s when Candelas, Xenia de la Ossa, Paul Green, and Linda Parks showed how mirror symmetry could help solve mathematical puzzles, particularly in the fields of algebraic and enumerative geometry—some of which had resisted solution for decades or more. The problem Candelas and his colleagues had taken on, the so-called quintic threefold problem, had certainly been around for a while. Also called the Schubert problem, in honor of Hermann Schubert, the nineteenth-century German mathematician who solved the first part of this puzzle, the problem concerns the number of rational curves—that is, curves with genus 0, or no holes, such as a sphere—that can fit on a quintic (six-dimensional) Calabi-Yau.

Counting in this way might sound like an odd pastime, unless you’re an enumerative geometer, in which case your days are filled with activities such as this. The task, however, is rarely as simple as emptying the contents of a jelly-bean bowl onto a table and counting the number of treats. Counting the number of objects on a manifold, and finding the right way to frame the problem so that the number you get is useful, has been a continuing challenge for mathematicians for a century or more. The number we seek at the end of this process must be finite, so we must restrict our search to compact space rather than to, say, an infinite plane. If we want, for example, to count the number of points of intersection between two curves, complications arise when these two curves are tangential or touch each other. Enumerative geometers have developed techniques to handle such situations and to arrive, hopefully, at a discrete number.

One of the earliest problems of this sort was posed around the year 200 B.C. by the Greek mathematician Apollonius, who asked how many circles can be drawn that are tangent to three given circles. The general answer to this question, which happens to be eight, can be obtained with a ruler and compass. But more sophisticated computational techniques are needed for the Schubert problem.

Mathematicians have attacked this problem in phases, taking it one degree at a time. A degree, in this context, refers to the highest exponent of any term in the polynomial equations that describe the term. For example, the degree of the polynomial 4x2 - 5y3 is three, and 6x3y2 + 4x is of degree five (the x3y2 exponents get combined), whereas 2x + 3y - 4 is of degree one and corresponds to a straight line. So the point of this exercise is to pick the manifold—the quintic threefold in this case—and pick the degree of the curves in question and then ask how many of these curves there are.

Schubert solved the problem for the first degree, showing that the number of lines on the quintic is exactly 2,875. In 1986, roughly a century later, Sheldon Katz (currently at the University of Illinois) demonstrated that the number of curves of degree two (such as circles) is 609,250. Candelas, de la Ossa, Green, and Parks tackled the degree-three case, which loosely translates to the number of spheres that can fit inside a particular Calabi-Yau space. But they employed a trick based on mirror symmetry. Whereas solving the problem on the quintic itself was exceedingly difficult, the quintic’s mirror manifold—which Greene and Plesser had constructed—offered a much easier setting in which to address the question.

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7.6—In a celebrated result of nineteenth-century geometry, the mathematicians Arthur Cayley and George Salmon proved that there are exactly 27 lines on a so-called cubic surface, as illustrated. Hermann Schubert later generalized this result, which is known as the Cayley-Salmon theorem. (Image courtesy of the 3D-XplorMath Consortium)

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7.7—A common problem in algebraic and enumerative geometry is to count the number of lines or curves on a surface. To gain a sense of what it means to have lines on the surface, one can see that the doubly ruled hyperboloid shown is a surface made up entirely of lines. It’s called doubly ruled because there are two distinct lines going through every point. Such a surface is not a good candidate for enumerative geometry, however, since an infinite number of lines can be drawn. (Photo by Karen Schaffner, Department of Mathematics, University of Arizona)

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7.8—The Apollonius problem, one of the most famous problems in all of geometry, asks how many circles can be drawn that are tangent to three circles in a plane. The framing of this problem, as well as its original solution, has been attributed to the Greek geometer Apollonius of Perga (circa 200 B.C.). The eight solutions to this problem—eight different tangent circles—are shown. The mathematician Hermann Schubert extended that result two thousand years later, showing that there are sixteen spheres tangent to four given spheres.

In fact, Greene and Plesser’s original paper on mirror symmetry had identified this basic approach, showing that a physical quantity, the Yukawa couplings, could be represented by two very different mathematical formulas—one corresponding to the original manifold and the other corresponding to its mirror pair. One formula, which involved the number of rational curves of various degrees that could be found on the manifold, was absolutely “horrendous” to deal with, according to Greene. The other formula, which depended on the shape of the manifold in a more general way, was much simpler to work with. But since both formulas described the same physical object, they had to be the same—just as the words cat and gato look different yet also describe the same furry creature. Greene and Plesser’s paper had an equation that explicitly stated that these two different-looking formulas were equal. “You can have an equation that you know is abstractly correct, but it can nevertheless be a major challenge to evaluate that equation with adequate precision to extract numbers from it,” says Greene. “We had the equation but we didn’t have the tools to leverage it into the determination of numbers. Candelas and his collaborators developed the tools to do that, which was a huge accomplishment” that has had a major impact on geometry. 22

This illustrates the potential power of mirror symmetry. You may not have to bother counting the number of curves in a Calabi-Yau space, when an entirely different calculation—which seems to have nothing to do with the original counting chore—yields the same answer. And when Candelas and company applied this approach to the number of degree-three curves on a quintic threefold, they got 317,206,375.

Our interest, however, is not strictly in the number of rational curves but extends to the manifold itself. For in the course of this curve counting, we’re essentially moving the curves around, using well-established techniques, until we’ve covered the whole space. In the course of doing that, we’ve defined that space—be it a quintic threefold or some other manifold—in terms of those curves.

The effect of all of this was to reinvigorate an area in geometry that had been largely moribund. The idea of using mirror symmetry to solve problems in enumerative geometry, which was pioneered by Candelas and his collaborators, led to the rebirth of that whole field, according to Mark Gross, a mathematician at the University of California, San Diego. “At the time, the field was more or less dead,” Gross says. “As the old problems had been solved, people went back to check Schubert’s numbers with modern techniques, but that was getting pretty stale.” Then he adds, out of the blue, “Candelas brought in some new methods that went far beyond anything Schubert had envisioned.”23 The physicists had been eagerly borrowing from mathematics, but before mathematicians would borrow from physics, they were demanding more proof of the rigor of Candelas’s method.

Fortuitously, at about this time—May 1991, to be exact—I had organized a conference at the Mathematical Sciences Research Institute (MSRI) at Berkeley for mathematicians and physicists to talk about mirror symmetry. I. M. Singer, one of the founders of MSRI, had originally selected a different topic for the conference, but I mentioned to him that some new developments in mirror symmetry were particularly exciting. Having attended Brian Greene’s talk somewhat earlier, Singer agreed with me and asked me to chair the weeklong event.

My hope for the proceedings was to overcome barriers stemming from differences in language and assumed knowledge between the respective fields. Candelas discussed his result for the Schubert problem there, but his number disagreed with the number obtained through ostensibly more rigorous techniques by the Norwegian mathematicians Geir Ellingsrud and Stein Arild Strømme, who had arrived at 2,682,549,425. The algebraic geometers in the crowd tended to be arrogant, assuming the physicists must have made the mistake. For one thing, explains University of Kaiserslautern mathematician Andreas Gathmann, “mathematicians didn’t understand what the physicists were doing, because they [the physicists] were using completely different methods—methods that didn’t exist in math and didn’t always appear to be justified.”24

Candelas and Greene worried that they might have made an error, but they couldn’t see where they had gone wrong. I spoke with both of them at the time, especially Greene, wondering whether something might have gone awry during the process of integrating over an infinite-dimensional space that then has to be reduced to finite dimensions. Choices have to be made in the course of doing that, and none of the options are perfect. While that made Candelas and Greene somewhat uneasy, we couldn’t find any flaws in their reasoning, which was based on physics rather than an outright mathematical proof. Moreover, they remained confident about mirror symmetry in general, despite the mathematicians’ criticism.

This all became clear a month or so later, when Ellingsrud and Strømme found an error in their computer program. Upon rectifying it, they got the same answer that Candelas et al. had. The Norwegian mathematicians showed great integrity in rerunning their program, checking their results, and publicizing their mistake. Many people in the same position try to conceal a mistake for as long as possible, but Ellingsrud and Strømme did the opposite, promptly informing the community of their error and subsequent correction.

This proved to be a big moment for mirror symmetry. The announcement of Ellingsrud and Strømme not only advanced the science of mirror symmetry, but also helped change attitudes toward the subject. Whereas many mathematicians had previously considered mirror symmetry rubbish, they now came to realize there was something to be learned from the physicists after all. As a case in point, the mathematician David Morrison (then at Duke University) had been one of the most vocal critics at the Berkeley meeting. But his thinking turned around, and before long, he would make many important contributions to mirror symmetry, string theory, and topology-changing transitions of Calabi-Yau manifolds.

Beyond solving the degree-three Schubert problem, Candelas and company had used their mirror symmetry method to calculate the solutions for degrees one through ten, while producing a general formula for solving every order of the quintic threefold problem and for predicting the number of rational curves for any and all degrees. In so doing, they’d taken major strides toward answering a centuries-old challenge cited in 1900 by the German mathematician David Hilbert as one of the twenty-three biggest problems in all of mathematics—namely, trying to establish a “rigorous foundation of Schubert’s enumerative calculus” such that “the degree of the final equations and the multiplicity of their solutions may be foreseen.”25 Candelas’s formula took many of us by surprise. The numerical solutions to the Schubert problem appeared to be just a string of numbers with no pattern and no apparent relationships between them. The work of Candelas and colleagues showed that rather than being random digits, these numbers were actually part of an exquisite structure.

The structure Candelas and his colleagues observed gave rise to a formula for doing the work, which was tested via mathematically intensive computations for polynomials of degrees one through four. The first three problems have already been discussed, whereas the fourth-degree problem was solved in 1995 by the mathematician Maxim Kontsevich (currently at the Institut des Hautes Études Scientifiques, IHES), who obtained the number 242,467,530,000. Although the Candelas group’s formula agreed with all known data points, one had to wonder whether this proposition could actually be proved. Many mathematicians, including Kontsevich, took additional steps to help put the equations into the form of a full-fledged conjecture, mainly by supplying definitions for some terms in that equation. The resulting statement, which came to be known as the mirror conjecture, could then be put to the ultimate test: a mathematical proof. Proving the mirror conjecture would provide mathematical validation for the notion of mirror symmetry itself.

Here again we venture into one of those areas of controversy that pop up from time to time in mathematics. These things happen, I suppose, because we live in an imperfect world populated by imperfect beings, and because mathematics—despite the popular image—is not a strictly intellectual pursuit, done in total isolation, divorced from politics, ambition, competition, and emotion. It often seems that in matters like this, the smaller the stakes, the bigger the controversy.

My colleagues and I had been studying the mirror conjecture, and its generalizations, since 1991—pretty much as soon as the Candelas results were announced. In a paper posted on the math archives in March 1996, Alexander Givental of the University of California, Berkeley, claimed to have proved the mirror conjecture. We scrutinized his paper very carefully and were not alone in finding it hard to follow. That year, I personally invited an MIT colleague who is an expert on the subject (and who wishes to remain anonymous) to lecture on Givental’s proof in my seminar. He politely declined, citing his serious misgivings about the arguments presented in that paper. My colleagues and I also failed in our attempt to reconstruct Givental’s entire argument, despite our attempts to contact him and piece together the steps we found most puzzling. We consequently abandoned that effort and pursued our own proof of the mirror conjecture, which we published a year later.

Some observers, including Gathmann, called our paper the “first complete rigorous proof ” of the conjecture, arguing that Givental’s “proof was hard to understand and at some points incomplete.”26 Amherst mathematician David Cox, coauthor (with Katz) of Mirror Symmetry and Algebraic Geometry, similarly concluded that we had provided the conjecture’s “first complete proof.”27 On the other hand, some people take a different view, maintaining that Givental’s proof, published a year before ours, was complete, containing no significant gaps. While people are free to debate the matter further, I believe the best thing to say at this point is that collectively the two papers constitute a proof of the mirror conjecture and to leave it at that. Continuing to quarrel over credit makes little sense, especially when there are so many unsolved problems in math that we might instead devote our energy to.

Controversy aside, what did those two papers actually prove? For starters, the mirror conjecture proof showed that Candelas had the right formula for predicting the number of curves of a given degree. But our proof was a good deal broader. The Candelas formula applied only to counting curves on quintic threefolds, whereas our results applied to a much broader class of Calabi-Yau manifolds, including those of interest to physicists, and to other objects, such as vector bundles, about which we’ll be saying more in Chapter 9. Moreover, in our generalization, the mirror conjecture involved not just counting curves but counting other geometric characteristics as well.

As I see it, proving this conjecture offered a consistency check on some of the ideas of string theory—a check that’s grounded in rigorous mathematics, thereby putting the theory on firmer mathematical footing. But string theory has more than returned the favor, as mirror symmetry has helped create a new industry in algebraic geometry—enumerative geometry, a branch thereof, being one of the main beneficiaries—while contributing to the solution of long-standing problems in those fields. Indeed, many colleagues in algebraic geometry have told me they haven’t done anything interesting in the past fifteen years except for the work inspired by mirror symmetry. The great windfall from string theory for mathematics suggests to me that the intuition of physicists must carry some weight. It means there must be some truth to string theory, even if nature doesn’t work exactly the way the theory posits, because we have used it to solve classic problems that mathematicians had been unable to solve on their own. Even now, years later, it’s hard to imagine an independent way of getting a formula like that of Candelas et al. were it not for string theory.

Ironically, one thing the proof of the mirror conjecture did not do was explain mirror symmetry itself. In many ways the phenomenon that physicists hit upon, and mathematicians have subsequently exploited, remains a mystery, although two main avenues of inquiry now under way—one called homological mirror symmetry, the other going by the acronym SYZ—are actively seeking such an explanation. The SYZ work attempts to provide a geometric interpretation of mirror symmetry, whereas homological mirror symmetry takes a more algebraic approach.

We’ll start with the one I’m more invested in, the SYZ conjecture, perhaps because the acronym is short for the three authors of the original 1996 paper: Andrew Strominger is the S, Eric Zaslow of Northwestern University is the Z, and I’m the Y. Collaborations of this sort rarely have a formal starting point, and this one began in some sense with casual conversations I had with Strominger at a 1995 conference in Trieste, Italy. Strominger discussed a paper he’d recently written with Katrin Becker and Melanie Becker, physicist sisters now at Texas A&M. As D-branes were then entering string theory in a big way, almost to the point of taking over the field, their paper looked at how these branes fit in with Calabi-Yau geometry. The idea was that branes can wrap around submanifolds that sit inside a Calabi-Yau space. Becker, Becker, and Strominger were studying a class of submanifolds that preserve supersymmetry and that, as a result, have very desirable properties. Strominger and I were curious about the role these submanifolds might play in mirror symmetry.

I returned to Harvard tantalized by this possibility, which I immediately took up with Zaslow, a physicist-turned-mathematician who was my postdoc at the time. Before long, Strominger came from Santa Barbara to visit Harvard, which was actively recruiting him (though it took more than a year before he switched coasts). The three of us met then, which finally put S, Y, and Z in the same room at the same time and eventually on the same page, as we submitted our paper for publication in June 1996.

If SYZ is correct, it would offer a deeper insight into the geometry of Calabi-Yau spaces, while validating the existence of a Calabi-Yau substructure. It argues that a Calabi-Yau can essentially be divided into two three-dimensional spaces that are highly entangled. One of these spaces is a three-dimensional torus. If you separate the torus from the other part, “invert” it (by switching its radius from r to 1/r), and reassemble the pieces, you’ll have the mirror manifold of the original Calabi-Yau. SYZ, asserts Strominger, “provides a simple physical and geometrical picture of what mirror symmetry corresponds to.”28

The key to understanding mirror symmetry, according to the SYZ conjecture, resides within the submanifolds of a Calabi-Yau and the way in which they’re organized. You might recall our discussion earlier in this chapter of a Swiss cheese chunk as a surface containing many subsurfaces, or submanifolds, within. The submanifolds in this case are not any old random patch of surface but rather a discrete piece (of lower dimension than the manifold itself), defined by a single hole in the “cheese,” that could be wrapped around or wound through individually. Similarly, the submanifolds in a Calabi-Yau of concern to the SYZ conjecture are wrapped by D-branes. (Not to confuse matters further, but one can also think of those D-branes as being the submanifolds rather than just wrapping them. Physicists tend to think in terms of branes, whereas mathematicians are more comfortable with their own terminology.) Subspaces of this sort that can satisfy supersymmetry are called special Lagrangian submanifolds, and as their name implies, they have special features: They have half the dimension of the space within which they sit, and they have the additional attribute of being length-, area-, or volume-minimizing, among other properties.

Let’s consider as an example the simplest possible Calabi-Yau space, a two-dimensional torus or donut. The special Lagrangian submanifold in this case will be a one-dimensional space or object consisting of a loop through the hole of the donut. And since that loop must have the minimum length, it must be a circle, the smallest possible circle going through that hole, rather than some arbitrary, squiggly, or otherwise meandering loop. “The entire Calabi-Yau, in this case, is just a union of circles,” explains Mark Gross, who has probably done more to follow up on the SYZ conjecture since it was first posed than anyone else. “There is an auxiliary space, call it B, that tells you about the set of all those circles, and it is a circle itself.”29 B is said to parameterize that set of circles, meaning that every point on B corresponds to a different circle, just as every circle looping through the donut hole corresponds to a different point on B. Another way to put it is that B, which is called the moduli space, contains an index of every subspace in the bigger manifold. But B is more than just a glorified list, as it shows how all these subspaces are arranged. B may, in fact, be the lynchpin to the whole SYZ conjecture, claims Gross. Accordingly, we’re going to spend a bit of time trying to understand this auxiliary space.

If we go up by one complex dimension from two real dimensions to four, the Calabi-Yau becomes a K3 surface. Instead of being circles, the submanifolds in this case are two-dimensional tori—a whole bunch of them fitting together. “I can’t draw this four-dimensional space,” says Gross, “but I can describe B, which tells us the setting in which all these subobjects, these donuts, sit.”30 In this case, B is just a two-dimensional sphere. Every point on this sphere B corresponds to a different donut, except for twenty-four bad points corresponding to “pinched donuts” that have singularities—the significance of which shall be explained shortly.

Now we’ll go up one more complex dimension, so that the manifold in question becomes a Calabi-Yau threefold. B now becomes a 3-sphere—a sphere, that is, with a three-dimensional surface (which is not one we can readily picture)—and the subspaces become three-dimensional donuts. In this case, the set of “bad” points, corresponding to singular donuts, fall on line segments that connect to each other in a netlike pattern. “Every point on this line segment is a ‘bad’ [or singular] point, but the vertices of the net, where the three line segments come together, represent the worst points of all,” says Gross. They correspond, in turn, to the most scrunched-up donuts.31

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7.9—The SYZ conjecture—named after its inventors, Andrew Strominger, the author (Shing-Tung Yau), and Eric Zaslow—offers a way of breaking up a complicated space such as a Calabi-Yau manifold into its constituent parts, or submanifolds. Although we cannot draw a six-dimensional Calabi-Yau manifold, we can draw the only two- (real) dimensional Calabi-Yau, a donut (with a flat metric). The submanifolds that make up the donut are circles, and all these circles are arranged by a so-called auxiliary space B, which is itself a circle. Each point on B corresponds to a different, smaller circle, and the entire manifold—or donut—consists of the union of those circles.

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7.10—The SYZ conjecture offers a new way of thinking about K3 surfaces, which constitute a class of four-dimensional Calabi-Yau manifolds. We can construct a K3 surface, according to SYZ, by taking a two-dimensional sphere (the auxiliary space, B, in this example) and attaching a two-dimensional donut to every point on that sphere.

This is where mirror symmetry comes in. Working with the initial idea of SYZ, the Oxford geometer Nigel Hitchin, Mark Gross, and some of my former students (Naichung Leung, Wei-Dong Ruan, and others) helped build the following picture: You have a manifold X made up of all kinds of submanifolds that are catalogued by the moduli space B. Next you take all those submanifolds, which have a radius of r, and invert them to a radius of 1/r. (One of the strange though nice features of string theory—something that doesn’t happen in classical mechanics—is that you can make a switch like this, flipping the radius of, say, a cylinder or a sphere, without affecting the physics at all.) If you have a point particle on a circle of radius r, its motion can be described strictly in terms of its momentum, which is quantized, meaning it can just assume certain (integer) values. A string, similarly, has momentum as it moves around a circle, but it can also wrap around the circle one or more times. The number of times it wraps around is called the winding number. So the motion of the string, unlike that of the particle, is described by two numbers: its momentum and its winding number, both of which are quantized. Let’s suppose that on one hand we have a string of winding 2 and momentum 0 on a circle of radius r, and on the other hand we have a string of winding 0 and momentum 2 on a circle of radius 1/r. Although the two situations sound different and conjure up quite different pictures, mathematically they are identical and so is the associated physics. This is known as T duality. “This equivalence extends beyond circles to products of circles or tori,” says Zaslow.32 In fact, the T in T duality stands for tori. Strominger, Zaslow, and I considered this duality so instrumental to mirror symmetry that we titled our original SYZ paper “Mirror Symmetry Is T Duality.”33

Here’s a simple way of seeing how T duality and mirror symmetry go hand in hand. Suppose manifold M is a torus—the product of two circles of radius r. Its mirror, M′, is also a torus—the product of two circles of radius 1/r. Let’s further suppose that r is extremely small. Because M is so tiny, if you want to understand the physics associated with it, quantum effects must be taken into account. So your calculations necessarily become much more difficult. Getting the physics from the mirror manifold, M′, is much easier because if r is very small, then 1/r is very big, and quantum effects can, therefore, be safely ignored. So mirror symmetry, under the guise of T duality, can simplify your calculations (as well as your life) tremendously.

Now let’s see how all these notions fit together, starting with our two-dimensional example above. When you invert the radii of all the submanifolds (circles), the big manifold you’re left with—which is made up of all these circles—has a different radius from what you started with, but it’s still a torus. As a result, we call this example trivial, because the manifold and its mirror are topologically identical. Our four-dimensional example with K3 surfaces is also trivial in the same respect because all K3 surfaces are topologically equivalent. The six-dimensional example involving Calabi-Yau threefolds is more interesting. The components of this manifold include three-dimensional tori. Applying T duality will invert the radii of those tori. For a nonsingular torus, this radius change will not change the topology. However, Gross explains, “even if all the original submanifolds were of the ‘good’ [nonsingular] variety, changing the radius can still change the topology of the big manifolds because the pieces . . . can be put together in a non-trivial way.”34

This might be best understood through analogy: You could take a stack of line segments—or, say, toothpicks—and use them to make a cylinder by sticking the toothpicks into a corkboard in a circular pattern. Instead of making a cylinder (which has two sides), you could also take those same toothpicks, introduce a twist, and assemble them into a single-sided Möbius strip. So you’re using the same pieces, the same submanifolds, but the objects you’re making have a very different topology.35

The point is that after the T duality shift and the different ways of assembling the submanifolds, we’re left with two topologically distinct manifolds that are indistinguishable from a physics standpoint. That’s part of what we mean by mirror symmetry but not quite the whole story, as another interesting feature of this duality is that mirror pairs should have opposite Euler numbers or characteristics. But all the submanifolds we’re talking about here—the special Lagrangians—have an Euler characteristic of 0, which does not change when the radii are inverted.

Although what I just said is true with respect to the “good” (nonsingular) submanifolds, it does not hold for the “bad” (singular) ones. T duality interchanges the Euler characteristic of those submanifolds from +1 to -1, or vice versa. Suppose your original manifold includes thirty-five bad submanifolds, twenty-five of which have an Euler characteristic of +1, and ten with an Euler characteristic of -1. As Gross has shown, if you add all those numbers, you get the Euler characteristic of the manifold as a whole, which turns out to be +15 in this case. In the mirror manifold, everything is reversed: twenty-five submanifolds with a -1 Euler characteristic and ten with a +1 Euler characteristic, for a total of -15, which is the opposite of what we started with and which is just what we wanted.

These bad submanifolds, as discussed earlier, correspond to bad points on the moduli space B. “Everything interesting in mirror symmetry, all the topological changes, are happening at the vertices of B,” Gross explains. So the emerging picture puts this space B at the center of mirror symmetry. Beforehand, the whole phenomenon had a mysterious air. “We had these two manifolds, X and X′, that were somehow related, but it was hard to see what they had in common,” Gross adds. What they have in common is this other object, B, which nobody knew about at first.36

Gross thinks of B as a kind of blueprint. If you look at the blueprint from one perspective, you’ll build one structure (or manifold), and if you look at it another way, you’ll build a different structure. And those differences stem from the funny (singular) points on B where T duality doesn’t work so well, and things change as a result.

That’s pretty much where our current picture of mirror symmetry, as seen through the lens of SYZ, stands. One of the chief virtues of SYZ, claims Strominger, is that “mirror symmetry was demystified a little bit. Mathematicians liked it because it provided a geometric picture of where mirror symmetry comes from, and they can use that picture without reference to string theory.”37 Besides offering a geometric explanation of mirror symmetry, Zaslow says, “it also offers a process for constructing mirror pairs.”38

It’s important to bear in mind that SYZ is still a conjecture that has only been proved in a few select cases but not in a general way. Although the conjecture may not be provable as originally stated, it has been modified in light of new insights to its present incarnation, which, according to Gross, “is slowly absorbing all of mirror symmetry.”39

Some would consider this last point debatable—and perhaps an overstatement. But SYZ has already been used—by Kontsevich and Yan Soibelman of Kansas State University, for instance—to prove a specific example of homological mirror symmetry, the other leading attempt at a fundamental mathematical description of mirror symmetry.

Homological mirror symmetry was first unveiled by Kontsevich in 1993 and has been developing ever since, sparking considerable activity in both physics and math. Mirror symmetry, as originally formulated, didn’t make much sense to mathematicians, as it concerned two different manifolds giving rise to the same physics. But as Soibelman explains, “in mathematics, there really is no concept of a physical theory associated with manifolds X and X′. So Kontsevich has tried to make that statement mathematically precise,” expressing it in a manner that is not tied to physics.40

Perhaps the simplest way of describing homological mirror symmetry is in terms of D-branes, even though Kontsevich’s idea predated the discovery of D-branes by a year or two. Physicists think of D-branes as subsurfaces upon which the endpoints of open strings must attach. Homological mirror symmetry anticipated D-branes, while providing a more refined description of these entities that became among the most basic ingredients of string theory, or M-theory, after the second string revolution. It’s the familiar story in which physics (through the discovery of mirror symmetry) gave rise to mathematics, and mathematics repaid its debt generously.

One of the main ideas behind homological mirror symmetry is that there are two kinds of D-branes involved in this phenomenon, A-branes and B-branes (which are terms that Witten introduced). If you have a mirror pair of Calabi-Yau manifolds, X and X′, A-branes on X are the same as B-branes on X′. This concise formulation, according to Aspinwall, “made it possible for mathematicians to clearly state what mirror symmetry is. And from that statement, you can produce everything else.”41

It’s as if we had two boxes of building blocks with different shapes, suggests Stony Brook University physicist Michael Douglas. “Yet when you stack them up, you can make the exact same set of structures.”42 This is similar to the correspondence between A-branes and B-branes posited by homological mirror symmetry.

A-branes are objects defined by what’s called symplectic geometry, whereas B-branes are objects of algebraic geometry. We’ve already touched on algebraic geometry to some extent, which describes geometric curves in algebraic terms and solves geometric problems with algebraic equations. Symplectic geometry includes the notion of Kähler geometry that is central to Calabi-Yau manifolds, but it is more general than that. Whereas spaces in differential geometry are typically described by a metric that is symmetric across a diagonal line, the metric in symplectic geometry is antisymmetric along that same line, meaning that the signs change across the diagonal.

“These two branches of geometry used to be considered totally separate, so it was a big shock when someone came along and said the algebraic geometry of one space is equivalent to the symplectic geometry of another space,” says Aspinwall. “Bringing two disparate fields together, finding that they are related in some sense through mirror symmetry, is one of the best things you can do in mathematics because you can then apply methods from one field to the other. Normally that really opens the floodgates, and many a Fields Medal has come from it.”43

In the meantime, homological mirror symmetry is reaching out to other branches of mathematics, while also reaching out to SYZ. As of yet, however, there is “no strict mathematical equivalence between the two, [but] they each support the other,” Gross claims. “And if both are correct, we should eventually find that on some level they are equivalent.”44

It’s an unfolding story. We’re still figuring out what mirror symmetry means through our investigations of SYZ, homological mirror symmetry, and other avenues. The phenomenon has branched off, leading to new directions in mathematics that no longer involve mirror symmetry at all, and no one knows how far these explorations will take us, or where they will end up. But we do know where it started: with the discovery of an unusual property of compact Kähler manifolds bearing the name Calabi-Yau—spaces that were almost given up for dead more than two decades ago.