Almost thirty years ago, a group of physicists noticed some of the most important numbers in mathematics appearing where they didn’t seem to belong. A new proof finally explains why they’re there.
The work, which is still unpublished, is by four leading mathematicians in the field of mirror symmetry. It explains why the so-called zeta values — numbers that have preoccupied mathematicians since the mid-18th century — are implicated in the numerical mystery at the heart of one of the most active fields in contemporary mathematics.
“Our work gives some kind of geometric explanation for the origin of these strange numbers,” said Nick Sheridan, a research fellow at the University of Cambridge and co-author of the work.
The zeta values are numbers generated by taking an infinite sum. For example, zeta of 2 is equal to 1 + 1/22 + 1/32 + 1/42 + … , while zeta of 3 is equal to 1 + 1/23 + 1/33 + 1/43 + …. The zeta values appear in many areas of mathematics, including, most famously, the Riemann hypothesis, which relates to the distribution of prime numbers.
As I explained in my recent feature story on the state of mirror symmetry, “Mathematicians Explore Mirror Link Between Two Geometric Worlds,” the field was discovered by accident. In the early 1990s, physicists were trying to figure out the details of string theory. They wanted to explain the physical world as a product of tiny, vibrating strings woven through an additional six dimensions of space. They tried to understand what the geometry of those six dimensions might be. The first option came from the mathematical field of algebraic geometry; a second one came from the mathematical field of symplectic geometry. To the trained mathematical eye, the two could hardly have seemed more different.
And yet, the physicists noticed some strange similarities between them. In particular, when they performed a calculation on one space, they generated numbers that matched the numbers they generated when they performed a very different type of calculation on the other side. “Two things that looked, in principle, unrelated, magically were equal,” said Denis Auroux, a mathematician at the University of California, Berkeley.
Mathematicians and physicists began excavating the mirror relationship. They soon built increasingly abstracted mathematical entities atop the foundations of the underlying symplectic and complex geometric spaces. You can think of these more abstract mathematical entities as houses whose architecture reflects the foundations they’ve been built on.
In the setting of mirror symmetry, points on the house whose coordinate values previously were integers become points whose coordinate values are now multiples of different zeta values. The process effectively rotates the space. “It’s been rotated, and the amount it’s been rotated by maybe will involve some of these zeta values,” said Sheridan.
Mathematicians have noticed the presence of zeta values almost since the beginning of the study of mirror symmetry. “The arithmetic is fascinating and has been explored in a lot of examples, but it’s missing a conceptual explanation. We’ve been trying to get more of a conceptual explanation,” said Sheel Ganatra, a mathematician at the University of Southern California and co-author of the new work along with Sheridan, Mohammed Abouzaid of Columbia University and Hiroshi Iritani of Kyoto University. This new work explains why the zeta values are there.
The explanation has to do with intrinsic geometric features of the two sides of mirror symmetry. A foundational question in mirror symmetry, called the SYZ conjecture, says it should be possible to take one mirror space, break it down into pieces, manipulate those pieces, and then use them to construct the second mirror space. It’s as though you had a big shape made of Legos, took it apart, and used the pieces to build something new.
When you break down the first space into Lego-like pieces, most of the pieces will be the same, but there will also be some special blocks — the odd green or yellow piece in a sea of reds and blues.
In their new work, the mathematicians prove that each kind of special piece is associated to a zeta value. Maybe the green blocks are associated to zeta of 2, and therefore if you have five greens when you take apart your first mirror space, the structure atop your rebuilt mirror space will have its coordinates offset by a factor of five times zeta of 2.
In this new work, the four mathematicians use techniques from a field called tropical geometry. Using those techniques, they prove that these “special pieces” explain why numbers on opposite sides of the mirror differ by exactly a factor of zeta values. So far, their proof holds for many cases of mirror symmetry. The authors are waiting until they’ve been able to prove even more cases before they make the proof public.
The work is also emblematic of an overall trend in mirror symmetry. The field began in revelation and is advancing rapidly toward understanding. Instead of simply cataloging mysterious phenomena — these sets of numbers match! — mathematicians are beginning to really explain why mirror phenomena occur at all.