After the Eyjafjallajökull volcano erupted in Iceland in 2010, flight cancellations left Miranda Cheng stranded in Paris. While waiting for the ash to clear, Cheng, then a postdoctoral researcher at Harvard University studying string theory, got to thinking about a paper that had recently been posted online. Its three coauthors had pointed out a numerical coincidence connecting far-flung mathematical objects. “That smells like another moonshine,” Cheng recalled thinking. “Could it be another moonshine?”

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Original story reprinted with permission from Quanta Magazine, an editorially independent division of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences


She happened to have read a book about the “monstrous moonshine,” a mathematical structure that unfolded out of a similar bit of numerology: In the late 1970s, the mathematician John McKay noticed that 196,884, the first important coefficient of an object called the j-function, was the sum of one and 196,883, the first two dimensions in which a giant collection of symmetries called the monster group could be represented. By 1992, researchers had traced this farfetched (hence “moonshine”) correspondence to its unlikely source: string theory, a candidate for the fundamental theory of physics that casts elementary particles as tiny oscillating strings. The j-function describes the strings’ oscillations in a particular string theory model, and the monster group captures the symmetries of the space-time fabric that these strings inhabit.

By the time of Eyjafjallajökull’s eruption, “this was ancient stuff,” Cheng said—a mathematical volcano that, as far as physicists were concerned, had gone dormant. The string theory model underlying monstrous moonshine was nothing like the particles or space-time geometry of the real world. But Cheng sensed that the new moonshine, if it was one, might be different. It involved K3 surfaces—the geometric objects that she and many other string theorists study as possible toy models of real space-time.

By the time she flew home from Paris, Cheng had uncovered more evidence that the new moonshine existed. She and collaborators John Duncan and Jeff Harvey gradually teased out evidence of not one but 23 new moonshines: mathematical structures that connect symmetry groups on the one hand and fundamental objects in number theory called mock modular forms (a class that includes the j-function) on the other. The existence of these 23 moonshines, posited in their Umbral Moonshine Conjecture in 2012, was proved by Duncan and coworkers late last year.

Meanwhile, Cheng, 37, is on the trail of the K3 string theory underlying the 23 moonshines—a particular version of the theory in which space-time has the geometry of a K3 surface. She and other string theorists hope to be able to use the mathematical ideas of umbral moonshine to study the properties of the K3 model in detail. This in turn could be a powerful means for understanding the physics of the real world where it can’t be probed directly—such as inside black holes. An assistant professor at the University of Amsterdam on leave from France’s National Center for Scientific Research, Cheng spoke with Quanta Magazine about the mysteries of moonshines, her hopes for string theory, and her improbable path from punk-rock high school dropout to a researcher who explores some of the most abstruse ideas in math and physics. An edited and condensed version of the conversation follows.

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QUANTA MAGAZINE: You do string theory on so-called K3 surfaces. What are they, and why are they important?

MIRANDA CHENG: String theory says there are 10 space-time dimensions. Since we only perceive four, the other six must be curled up or “compactified” too small to see, like the circumference of a very thin wire. There’s a plethora of possibilities—something like 10500—for how the extra dimensions might be compactified, and it’s almost impossible to say which compactification is more likely to describe reality than the rest. We can’t possibly study the physical properties of all of them. So you look for a toy model. And if you like having exact results instead of approximated results, which I like, then you often end up with a K3 compactification, which is a middle ground for compactifications between too simple and too complicated. It also captures the key properties of Calabi-Yau manifolds [the most highly studied class of compactifications] and how string theory behaves when it’s compactified on them. K3 also has the feature that you can often do direct and exact computations with it.

What does K3 actually look like?

You can think of a flat torus, then you fold it so that there’s a line or corner of sharp edges. Mathematicians have a way to smooth it, and the result of smoothing a folded flat torus is a K3 surface.

So you can figure out what the physics is in this setup, with strings moving through this space-time geometry?

Yes. In the context of my Ph.D., I explored how black holes behave in this theory. Once you have the curled-up dimensions being K3-related Calabi-Yaus, black holes can form. How do these black holes behave—especially their quantum properties?

So you could try to solve the information paradox—the long-standing puzzle of what happens to quantum information when it falls inside a black hole.

Absolutely. You can ask about the information paradox or properties of various types of black holes, like realistic astrophysical black holes or supersymmetric black holes that come out of string theory. Studying the second type can shed light on your realistic problems because they share the same paradox. That’s why trying to understand string theory in K3 and the black holes that arise in that compactification should also shed light on other problems. At least, that’s the hope, and I think it’s a reasonable hope.

Do you think string theory definitely describes reality? Or is it something you study purely for its own sake?

I personally always have the real world at the back of my mind—but really, really, really back. I use it as sort of an inspiration for determining roughly the big directions I’m going in. But my day-to-day research is not aimed at solving the real world. I see it as differences in taste and style and personal capabilities. New ideas are needed in fundamental high-energy physics, and it’s hard to say where those new ideas will come from. Understanding the basic, fundamental structures of string theory is needed and helpful. You’ve got to start somewhere where you can compute things, and that leads, often, to very mathematical corners. The payoff to understanding the real world might be really long term, but that’s necessary at this stage.

Have you always had a knack for physics and math?

As a child in Taiwan I was more into literature—that was my big thing. And then I got into music when I was 12 or so—pop music, rock, punk. I was always very good at math and physics, but I wasn’t really interested in it. And I always found school insufferable and was always trying to find a way around it. I tried to make a deal with the teacher that I wouldn’t need to go into the class. Or I had months of sick leave while I wasn’t sick at all. Or I skipped a year here and there. I just don’t know how to deal with authority, I guess.

And the material was probably too easy. I skipped two years, but that didn’t help. So then they moved me to a special class and that made it even worse, because everybody was very competitive, and I just couldn’t deal with the competition at all. Eventually I was super depressed, and I decided either I would kill myself or not go to school. So I stopped going to school when I was 16, and I also left home because I was convinced that my parents would ask me to go back to school and I really didn’t want to do that. So I started working in a record shop, and by that time I also played in a band, and I loved it.

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