A Master of Umbral Moonshine Toys With String Theory
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?”
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.
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.
How did you get from there to string theory?
Long story short, I got a little bit discouraged or bored. I wanted to do something else aside from music. So I tried to go back to university, but then I had the problem that I hadn’t graduated from high school. But before I quit school I was in a special class for kids who are really good in science. I could get in the university with this. So I thought, OK, great, I’ll just get into university first by majoring in physics or math, and then I can switch to literature. So I enrolled in the physics department, having a very on- and off-again relationship to it, going to class every now and then, and then trying to study literature, while still playing in the band. Then I realized I’m not good enough in literature. And also there was a very good teacher teaching quantum mechanics. Just once I went to his class and thought, that’s actually pretty cool. I started paying a bit more attention to my studies of math and physics, and I started to find peace in it. That’s what started to attract me about math and physics, because my other life in the band playing music was more chaotic somehow. It sucks a lot of emotions out of you. You’re always working with people, and the music is too much about life, about emotions—you have to give a lot of yourself to it. Math and physics seems to have this peaceful quiet beauty. This space of serenity.
Then at the end of university I thought, well, let me just have one more year to study physics, then I’m really done with it and can move on with my life. So I decided to go to Holland to see the world and study some physics, and I got really into it there.
You got your master’s at Utrecht under Nobel Prize-winning physicist Gerard ’t Hooft, and then you did your PhD in Amsterdam. What drew you in?
Working with [’t Hooft] was a big factor. But just learning more is also a big factor—to realize that there are so many interesting questions. That’s the big-picture part. But for me the day-to-day part is also important. The learning process, the thinking process, really the beauty of it. Every day you encounter some equations or some way of thinking, or this fact leads to that fact—I thought, well, this is pretty. Gerard is not a string theorist—he’s very open-minded about what the correct area of quantum gravity should be—so I got exposed to a few different options. I got attracted by string theory because it’s mathematically rigorous, and pretty.
With the work you’re doing now, aside from the beauty, are you also drawn to the mystery of these connections between seemingly different parts of math and physics?
The mystery part connects to the bad side of my character, which is the obsessive side. That’s one of the driving forces that I would call slightly negative from the human point of view, though not the scientist point of view. But there’s also the positive driving force, which is that I really enjoy learning different stuff and feeling how ignorant I am. I enjoy that frustration, like, “I know nothing about this subject; I really want to learn!” So that’s one motivation—to be at this boundary place between math and physics. Moonshine is a puzzle that might require inspirations from everywhere and knowledge from everywhere. And the beauty, certainly—it’s a beautiful story. It’s kind of hard to say why it is beautiful. It’s beautiful not the same way as a song is beautiful or a picture is beautiful.
What’s the difference?
Typically a song is beautiful because it triggers certain emotions. It resonates with part of your life. Mathematical beauty is not that. It’s something much more structured. It gives you a feeling of something much more permanent, and independent of you. It makes me feel small, and I like that.
What is a moonshine, exactly?
A moonshine relates representations of a finite symmetry group to a function with special symmetries [ways that you can transform the function without affecting its output]. Underlying this relationship, at least in the case of monstrous moonshine, is a string theory. String theory has two geometries. One is the “worldsheet” geometry. If you have a string—essentially a circle—moving in time, then you get a cylinder. That’s what we call the worldsheet geometry; it’s the geometry of the string itself. If you roll the cylinder and connect the two ends, you get a torus. The torus gives you the symmetry of the j-function. The other geometry in string theory is space-time itself, and its symmetry gives you the monster group.
If or when you find the K3 string theory underlying the 23 umbral moonshines, what would the moonshines buy you in terms of new ways that you can study K3 string theory?
We don’t know yet, but these are educated guesses: To have a moonshine tells you that this theory has to have an algebraic structure [you have to be able to do algebra with its elements]. If you look at a theory and you ask what kind of particles you have at a certain energy level, this question is infinite, because you can go to higher and higher energies, and then this question goes on and on. In monstrous moonshine, this is manifested in the fact that if you look at the j-function, there are infinitely many terms that basically capture the energy of the particles. But we know there’s an algebraic structure underlying it—there’s a mechanism for how the lower energy states can be related to higher energy states. So this infinite question has a structure; it’s not just random.
As you can imagine, having an algebraic structure helps you understand what the structure is that captures a theory—how, if you look at the lower energy states, they will tell you something about the higher energy states. And then it also gives you more tools to do computations. If you want to understand something at a high-energy level [such as inside black holes], then I have more information about it. I can compute what I want to compute for high-energy states using this low-energy data I already have in hand. That’s the hope.
Umbral moonshine tells you that there should be a structure like this that we don’t understand yet. Understanding it more generally will force us to understand this algebraic structure. And that will lead to a much deeper understanding of the theory. That’s the hope.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication 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.