Moonshine Master Toys With String Theory

The physicist-mathematician Miranda Cheng is working to harness a mysterious connection between string theory, algebra and number 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 evidencethat 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.

https://www.quantamagazine.org/20160804-miranda-cheng-moonshine-string-theory/