There’s a very intriguing new article out today by Kevin Hartnett at Quanta magazine, entitled Secret Link Uncovered Between Pure Math and Physics (also a video here). It’s about ideas relating number theory and physics from arithmetic geometer Minhyong Kim. He’s evidently on tour talking about them, with two talks on Gauge theory in arithmetic and a colloquium talk on “Gauge theory in geometry and number theory” in Heidelberg, and a talk on Gauge theory in arithmetic geometry in Paris.
In recent years Kim has been working on what he calls “arithmetic Chern-Simons” theory. For details about this, there are papers here, here, here and here, a workshop here, talks here and here. These ideas grew out of a beautiful and well-known analogy between topology and number theory that goes under the name “Arithmetic Topology”. For more about this, see the book Knots and Primes by Morishita, or the course notes by Chao Li and Charmaine Sia.
While these ideas look quite interesting and I have some idea what they’re about, the Quanta story seems to indicate that Kim has something new, an idea about “Diophantine gauge theory” going beyond the arithmetic Chern-Simons business, and with potential applications to deep problems in arithmetic geometry. Unfortunately the mathematical background here is beyond me (you can try to look at Jordan Ellenberg here, and this earlier paper of Kim’s), and as far as I can tell, the only source for details on the conjectured relations to gauge theory is Kim’s recent talks, which aren’t documented anywhere I can see.
I’m sure we’ll be hearing more about this as time goes on. It joins a host of other ideas relating gauge theory and number theory (in the context for instance of the Langlands program), and promises deeper links to come between fundamental ideas about physics and about mathematics.
Update: Some personal background on this story from John Baez.
Update: Reddit has a report of the Heidelberg talk here, unfortunately giving just enough information about the talk to make it sound very interesting, too little to figure out what the ideas discussed actually were. For yet more frustration on this front, Kim gave a general talk on the subject last year here with video supposedly available, but no browser I’ve tried can access it. I do hope we’ll soon see slides, notes, a paper, something, anything, so we can figure out what the Quanta article actually was about.
Update: Kim now has a paper out explaining these new ideas: Arithmetic Gauge Theory: A Brief Introduction.