Mathematicians Explore Mirror Link Between Two Geometric Worlds

by Kevin Hartnett

Decades after physicists happened upon a stunning mathematical coincidence, researchers are getting close to understanding the link between two seemingly unrelated geometric universes.


Twenty-seven years ago, a group of physicists made an accidental discovery that flipped mathematics on its head. The physicists were trying to work out the details of string theory when they observed a strange correspondence: Numbers emerging from one kind of geometric world matched exactly with very different kinds of numbers from a very different kind of geometric world.
To physicists, the correspondence was interesting. To mathematicians, it was preposterous. They’d been studying these two geometric settings in isolation from each other for decades. To claim that they were intimately related seemed as unlikely as asserting that at the moment an astronaut jumps on the moon, some hidden connection causes his sister to jump back on earth.
It looked totally outrageous,” said David Morrison, a mathematician at the University of California, Santa Barbara, and one of the first mathematicians to investigate the matching numbers.


Nearly three decades later, incredulity has long since given way to revelation. The geometric relationship that the physicists first observed is the subject of one of the most flourishing fields in contemporary mathematics. The field is called mirror symmetry, in reference to the fact that these two seemingly distant mathematical universes appear somehow to reflect each other exactly. And since the observation of that first correspondence — a set of numbers on one side that matched a set of numbers on the other — mathematicians have found many more instances of an elaborate mirroring relationship: Not only do the astronaut and his sister jump together, they wave their hands and dream in unison, too.
Recently, the study of mirror symmetry has taken a new turn. After years of discovering more examples of the same underlying phenomenon, mathematicians are closing in on an explanation for why the phenomenon happens at all.
We’re getting to the point where we’ve found the ground. There’s a landing in sight,” said Denis Auroux, a mathematician at the University of California, Berkeley.

The effort to come up with a fundamental explanation for mirror symmetry is being advanced by several groups of mathematicians. They are closing in on proofs of the central conjectures in the field. Their work is like uncovering a form of geometric DNA — a shared code that explains how two radically different geometric worlds could possibly hold traits in common.


Discovering the Mirror
What would eventually become the field of mirror symmetry began when physicists went looking for some extra dimensions. As far back as the late 1960s, physicists had tried to explain the existence of fundamental particles — electrons, photons, quarks — in terms of minuscule vibrating strings. By the 1980s, physicists understood that in order to make “string theory” work, the strings would have to exist in 10 dimensions — six more than the four-dimensional space-time we can observe. They proposed that what went on in those six unseen dimensions determined the observable properties of our physical world.
“You might have this small space that you can’t see or measure directly, but some aspects of the geometry of that space might influence real-world physics,” said Mark Gross, a mathematician at the University of Cambridge.


Mark Gross, a mathematician at the University of Cambridge, and a colleague are putting the finishing touches on a proof that establishes a universal method for constructing one mirror space from another.