Mathematicians Explore Mirror Link Between Two Geometric Worlds

Torus fibrations are useful in a few ways. One is that they give mathematicians a simpler way to think of complicated spaces. Just like you can construct a torus fibration of a two-dimensional sphere, you can construct a torus fibration of the six-dimensional symplectic and complex spaces that feature in mirror symmetry. Instead of circles, the fibers of those spaces are three-dimensional tori. And while a six-dimensional symplectic manifold is impossible to visualize, a three-dimensional torus is almost tangible. “That’s already a big help,” said Sheridan.
A torus fibration is useful in another way: It reduces one mirror space to a set of building blocks that you could use to build the other. In other words, you can’t necessarily understand a dog by looking at a duck, but if you break each animal into its raw genetic code, you can look for similarities that might make it seem less surprising that both organisms have eyes.


Here, in a simplified view, is how to convert a symplectic space into its complex mirror. First, perform a torus fibration on the symplectic space. You’ll get a lot of tori. Each torus has a radius (just like a circle — a one-dimensional torus — has a radius). Next, take the reciprocal of the radius of each torus. (So, a torus of radius 4 in your symplectic space becomes a torus of radius ¼ in the complex mirror.) Then use these new tori, with reciprocal radii, to build a new space.



In 1996, Andrew Strominger, Shing-Tung Yau and Eric Zaslow proposed this method as a general approach for converting any symplectic space into its complex mirror. The proposal that it’s always possible to use a torus fibration to move from one side of the mirror to the other is called the SYZ conjecture, after its originators. Proving it has become one of the foundational questions in mirror symmetry (along with the homological mirror symmetry conjecture, proposed by Maxim Kontsevich in 1994).
The SYZ conjecture is hard to prove because, in practice, this procedure of creating a torus fibration and then taking reciprocals of the radii is not easy to do. To see why, return to the example of the surface of the earth. At first it seems easy to stripe it with circles, but at the poles, your circles will have a radius of zero. And the reciprocal of zero is infinity. “If your radius equals zero, you’ve got a bit of a problem,” said Sheridan.


This same difficulty crops up in a more pronounced way when you’re trying to create a torus fibration of a six-dimensional symplectic space. There, you might have infinitely many torus fibers where part of the fiber is pinched down to a point — points with a radius of zero. Mathematicians are still trying to figure out how to work with such fibers. “This torus fibration is really the great difficulty of mirror symmetry,” said Tony Pantev, a mathematician at the University of Pennsylvania.
Put another way: The SYZ conjecture says a torus fibration is the key link between symplectic and complex spaces, but in many cases, mathematicians don’t know how to perform the translation procedure that the conjecture prescribes.


Long-Hidden Connections
Over the past 27 years, mathematicians have found hundreds of millions of examples of mirror pairs: This symplectic manifold is in a mirror relationship with that complex manifold. But when it comes to understanding why a phenomenon occurs, quantity doesn’t matter. You could assemble an ark’s worth of mammals without coming any closer to understanding where hair comes from.
“We have huge numbers of examples, like 400 million examples. It’s not that there’s a lack of examples, but nevertheless it’s still specific cases that don’t give much of a hint as to why the whole story works,” said Gross.
Mathematicians would like to find a general method of construction — a process by which you could hand them any symplectic manifold and they could hand you back its mirror. And now they believe that they’re getting close to having it. “We’re moving past the case-by-case understanding of the phenomenon,” said Auroux. “We’re trying to prove that it works in as much generality as we can.”


Mathematicians are progressing along several interrelated fronts. After decades building up the field of mirror symmetry, they’re close to understanding the main reasons the field works at all.
I think it will be done in a reasonable time,” said Kontsevich, a mathematician at the Institute of Advanced Scientific Studies (IHES) in France and a leader in the field. “I think it will be proven really soon.”
One active area of research creates an end run around the SYZ conjecture. It attempts to port geometric information from the symplectic side to the complex side without a complete torus fibration. In 2016, Gross and his longtime collaborator Bernd Siebert of the University of Hamburg posted a general-purpose method for doing so. They are now finishing a proof to establish that the method works for all mirror spaces. “The proof has now been completely written down, but it’s a mess,” said Gross, who said that he and Siebert hope to complete it by the end of the year.


Another major open line of research seeks to establish that, assuming you have a torus fibration, which gives you mirror spaces, then all the most important relationships of mirror symmetry fall out from there. The research program is called “family Floer theory” and is being developed by Mohammed Abouzaid, a mathematician at Columbia University. In March 2017 Abouzaid posted a paper that proved this chain of logic holds for certain types of mirror pairs, but not yet all of them.
And, finally, there is work that circles back to where the field began. A trio of mathematicians — Sheridan, Sheel Ganatra and Timothy Perutz — is building on seminal ideas introduced in 1990s by Kontsevich related to his homological mirror symmetry conjecture.


Cumulatively, these three initiatives would provide a potentially complete encapsulation of the mirror phenomenon. “I think we’re getting to the point where all the big ‘why’ questions are close to being understood,” said Auroux.