ANOTHER DAY, ANOTHER DONUT - The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis

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


Recently, one of us—the less mathematically inclined of the two—stood in the halls of the Jefferson Laboratory’s theory group at Harvard, waiting to speak with Andrew Strominger, who was ensconced in an animated conversation with a colleague. Some minutes later, Cumrun Vafa burst out of the office, and Strominger apologized for the delay, saying that “Cumrun had a new idea related to Calabi-Yau spaces that couldn’t wait.” After a brief pause, he added, “It seems I hear a new idea about Calabi-Yau’s just about every day.”1

Upon further reflection, Strominger downgraded that statement to “about every week.” Over the past several years, consistent with Strominger’s remark, scholarly articles with the term “Calabi-Yau” in the title are being published at the rate of more than one per week—in the English language alone. These manifolds are not just relics of the first string revolution or mathematical curiosities of only historical import. They are alive and well and, if not living in Paris, are at least still prominent fixtures on the mathematics and theoretical physics archives.

That’s not bad considering that in the late 1980s, many physicists thought that Calabi-Yau spaces were dinosaurs and that the whole subject was doomed. Calabi-Yau enthusiasts like myself—the more mathematically inclined of our duo—have often been told that we were talking nonsense. In that era, Philip Candelas, among others, got a bad grant review, which reduced his funding drastically. The cutbacks came for the simple reason that he was still investigating Calabi-Yau spaces, which were considered “the language of the past.” A physicist who was then teaching at Harvard criticized the whole approach in even stronger terms: “Why are you idiots still working on this stupid theory?” Although I was taken aback by the question at the time, I have, after two decades of sustained cogitation, formulated what seems to be an appropriate response: “Well, maybe it’s not so stupid after all.”

Strominger, for one, doesn’t think so, but then again, Calabi-Yau spaces have figured prominently in his career. In fact, it’s possible he’s done more than just about any other person to establish the importance of this class of spaces in physics. “It’s surprising after all this time how central the role of Calabi-Yau’s has remained,” he says. “They keep popping up again and again—the black hole story being one example.”2 Another example involves a new strategy for realizing the Standard Model in eight-dimensional (rather than six-dimensional) Calabi-Yau manifolds—where the shape of the two extra dimensions is determined by string coupling constants—as discussed in recent papers by Vafa, Chris Beasley, and Jonathan Heckman and separately by Ron Donagi and Martijn Wijnholt.3

“It’s not often an idea holds center stage for so long,” adds Strominger, referring to the enduring reign of Calabi-Yau manifolds in string theory. “And it’s not as if the idea has just stuck around for old-time’s sake. It’s not just a bunch of fogies from the eighties remembering the good old days. The idea has continued to branch off and spawn new buds.”4

Vafa, his collaborator from across the hall, concurs: “If you’re interested in four-dimensional gauge theories, you might think they have nothing to do with Calabi-Yau manifolds. Not only do they have something to do with Calabi-Yau manifolds, they have something to do with Calabi-Yau threefolds, which are of greatest interest to string theory. Similarly, you might think the theory of Riemann surfaces has nothing to do with Calabi-Yau threefolds, but studying them in the context of these threefolds turns out to be the key to understanding them.”5

And then there’s Edward Witten, the physicist sometimes called Einstein’s successor (and if string theory is ever proven right, that might be a fair comparison). Witten has had what might reasonably be called an intimate relationship with Calabi-Yau manifolds, as well as with string theory as a whole, where he’s contributed mightily to the first two string “revolutions.” And he is likely to have a hand (or foot) in the third, if and when it ever comes to pass. For as Brian Greene once said, “Everything I’ve ever worked on, if I trace its intellectual roots, I find they end at Witten’s feet.”6

During a meeting with Strominger at Princeton, Witten recently mused, “Who would have thought, twenty-some years ago, that doing string theory on Calabi-Yau manifolds would turn out to be so interesting?” He went on to say: “The deeper we dig, the more we learn because Calabi-Yau’s are such a rich and central construction.” Almost every time we learn a new way of looking at string theory, he adds, these manifolds have helped us by providing basic examples.7

Indeed, almost all of the major calculations in string theory have been done in a Calabi-Yau setting simply because that is the space where we know how to do the calculations. Thanks to the “Calabi-Yau theorem” that emerged from the Calabi conjecture proof, says mathematician David Morrison of the University of California, Santa Barbara, “we have these techniques from algebraic geometry that enable us, in principle, to study and analyze all Calabi-Yau manifolds. We don’t have techniques of similar strength for dealing with non-Kähler manifolds or the seven-dimensional G2 manifolds that are important in M-theory. As a result, a lot of the progress that’s been made has come from Calabi-Yau manifolds because we have the tools for studying them we don’t have for other kinds of solutions.”7 In that sense, Calabi-Yau manifolds have provided a kind of laboratory for experiments, or at least thought experiments, that can teach us about string theory and hopefully about our universe as well.

“It’s a testament to the human mind that we began thinking about Calabi-Yau’s strictly as mathematical objects, before there was any obvious role for them in physics,” notes Stanford mathematician Ravi Vakil. “We’re not forcing Calabi-Yau’s on nature, but nature seems to be forcing them upon us.”89

That does not mean, however, that Calabi-Yau spaces are necessarily the last word or that we even live in such a space. The study of these manifolds has enabled physicists and mathematicians to learn many interesting and unexpected things, but these spaces can’t explain everything, nor can they take us everywhere we might conceivably want to go. Although Calabi-Yau spaces may not be the ultimate destination, they may well be “stepping-stones to the next level of understanding,” Strominger says.10

Speaking as a mathematician, which I suppose is the only way I can speak (with any authority, that is), I can say that a complete understanding of Calabi-Yau spaces is not there yet. And I have my doubts as to whether we can ever know everything there is to know about such spaces. One reason I’m skeptical stems from the fact that the one-dimensional Calabi-Yau is called an elliptic curve, and these curves—solutions to cubic equations in which at least some of the terms are taken to the third power—are enigmatic objects in mathematics. Cubic equations have fascinated mathematicians for centuries. Although the equations assume a simple form (such as y2 = x3 + ax + b), which might even look familiar to someone in a high school algebra class, their solutions hold many deep mysteries that can transport practitioners to the farthest reaches of mathematics. Andrew Wiles’s famous proof of Fermat’s Last Theorem, for example, revolved around understanding elliptic curves. Yet despite Wiles’s brilliant work, there are many unsolved problems associated with such curves—and, equivalently, with one-dimensional Calabi-Yau manifolds—for which there appears to be no resolution in sight.

There is reason to believe that generalizations of elliptic curves to higher dimensions, of which Calabi-Yau threefolds offer one possibility, might be used to address similarly deep puzzles in mathematics, for we often learn something new by taking a special case—such as the elliptic curve—into a more general and higher-dimensional (or arbitrary dimensional) setting. On this front, the study of Calabi-Yau spaces of two complex dimensions, K3 surfaces, has already helped answer some questions in number theory.

But this work is just beginning, and we have no idea where it might take us. At this stage, it’s fair to say that we’ve barely scratched the surface—regardless of whether that surface is the K3 or another variety of Calabi-Yau. Which is why I believe a thorough understanding of these spaces may not be possible until we understand a large chunk of mathematics that cuts across geometry, number theory, and analysis.

Some might take that as bad news, but I see it as a good thing. It means that Calabi-Yau manifolds, like math itself, are developing stories on a road that undoubtedly holds many twists and turns. It means there’s always more to be learned, always more to be done. And for those of us who worry about keeping employed, keeping engaged, and keeping amused, it means there should be plenty of challenges, as well as fun, in the years ahead.