I’m at a workshop this week. It’s part of a series of “Bethe Forums”, cozy little conferences run by the Bethe Center for Theoretical Physics in Bonn.
The workshop’s title, “Geometries and Special Functions for Physics and Mathematics”, covers a wide range of topics. There are talks on Calabi-Yau manifolds, elliptic (and hyper-elliptic) polylogarithms, and cluster algebras and cluster polylogarithms. Some of the talks are by mathematicians, others by physicists.
In addition to the talks, this conference added a fun innovative element, “my favorite problem sessions”. The idea is that a speaker spends fifteen minutes introducing their “favorite problem”, then the audience spends fifteen minutes discussing it. Some treated these sessions roughly like short talks describing their work, with the open directions at the end framed as their favorite problem. Others aimed broader, trying to describe a general problem and motivate interest in people of other sub-fields.
This was a particularly fun conference for me, because the seemingly distinct topics all connect in one way or another to my own favorite problem. In our “favorite theory” of N=4 super Yang-Mills, we can describe our calculations in terms of an “alphabet” of pieces that let us figure out predictions almost “by guesswork”. These alphabets, at least in the cases we know how to handle, turn out to correspond to mathematical structures called cluster algebras. If we look at interactions of six or seven particles, these cluster algebras are a powerful guide. For eight or nine, they still seem to matter, but are much harder to use.
For ten particles, though, things get stranger. That’s because ten particles is precisely where elliptic curves, and their related elliptic polylogarithms, show up. Things then get yet more strange, and with twelve particles or more we start seeing Calabi-Yau manifolds magically show up in our calculations.
We don’t know what an “alphabet” should look like for these Calabi-Yau manifolds (but I’m working on it). Because of that, we don’t know how these cluster algebras should appear.
In my view, any explanation for the role of cluster algebras in our calculations has to extend to these cases, to elliptic polylogarithms and Calabi-Yau manifolds. Without knowing how to frame an alphabet for these things, we won’t be able to solve the lingering mysteries that fill our field.
Because of that, “my favorite problem” is one of my biggest motivations, the question that drives a large chunk of what I do. It’s what’s made this conference so much fun, and so stimulating: almost every talk had something I wanted to learn.