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:** More about this story here, including a discussion of the use of Kim’s methods to deal with the “Cursed Curve”.

**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.

“What I started out trying to find” was a least-action principle for the mathematical setting, he [Minhyong Kim] wrote in an email. “I still don’t quite have it. But I am pretty confident it’s there.”

Then shouldn’t the title be “Secret Link Conjectured Between Pure Math and Physics” or even “Secret Analogy Conjectured Between Pure Math and Physics”?

Or probably the title refers to the fact that Minhyong Kim is finally revealing/uncovering a link between pure math and physics he had in his head for a while. Then the title is a clickbait.

I don’t know anything about the math, but I love the headline.

I thought the article was extremely interesting and clear. For whatever reason, this one grabbed me like few others of its kind. The notion that somehow there’s something analogous to finding rational solutions to Diophantine equations in the way nature “finds” the classical path of a light beam is kind of mind-blowing.

“Principle of Least Action now considered for use in Number Theory” perchance?

Thanks for blogging about this!

One question I’ve wanted to know about arithmetic Chern-Simons theory is: in topology, Chern-Simons theory manifests in many different ways (e.g. path integrals, 3-manifold invariants, functors out of a cobordism category), so arithmetic Chern-Simons must carry over some subset of those things into number theory. Which parts have been brought over, and how much of the rest of the story is expected to apply to the number-theoretic side?

I know very little number theory, but looking at these papers, it looks like thus far, Kim and his collaborators have constructed the analogue of the classical Dijkgraaf-Witten action. Has there been discussion of quantizing this action? What could the analogue of functorial TQFT for this action look like?

Arun Debray,

Good questions. From looking at the papers a bit, there’s no quantization or TQFT here that I can see, this is quite a different story than the Witten-CS QFT one. What appears is just the analog of the classical action of Chern-Simons (Abelian case), with the novelty coming from the arithmetic context,

Pingback: Arithmetic topology in Quanta – neverendingbooks

Ah, ok. Thanks!

To me, this has much the same look as the “hype” you complain about.

I would be much more sceptical about mathematical PR: here and elsewhere, you seem surprisingly willing to accept it at face value.

Baez made a nice google plus post expanding on this

Mathematician passing by,

I’d agree that the title of the piece is over the top, but the Quanta article on the whole I think gives a non-hyped take on a complicated story. The last section of the article both presents Kim’s hopes and skepticism from others, explaining well the current situation. It is difficult for others to evaluate this right now, since Kim has not (at least in print that I know of) explained either how gauge theory ideas motivated his earlier solid mathematical results, or what the conjectured variational problem he has in mind is.

I don’t think this is at all like the typical physics hype that I complain about, which generally deals with bogus claims being made about failed ideas that have been around for over 30 years, and studied in detail by thousands of scientists. Whether Kim really has something or not, we’ll see, but he has a significant amount of credibility (unlike other common cases of physicists or mathematicians making non-credible personal claims for a new idea, which I tend to ignore and not bring up here).

As a mathematician not working in number theory or geometry, my impression is that this is exactly the kind of hype we are used to in geometry and number theory, which has elevated these subjects to an over-inflated status, much like string theory, and has resulted in other branches of mathematics being starved of funds, and of bright minds, and going out of fashion. It is not a theorem, not even a conjecture, just a vague idea. It might lead somewhere spectacular, of course, but mathematicians are only impressed by proofs, not by speculation.

I wholeheartedly concur with the contributions from the two mathematicians. I think Minhyong Kim is an awesome mathematician, but even before his coming-out as a physics-minded number theorist, his work tended to gear towards grand programmes and sweeping conjectures. There is nothing inherently wrong with that, but we can’t all be Grothendieck, dreaming up a bunch of mysterious connections and slowly working towards a grand materialization 10, 20 years down the road. And even in Grothendieck’s case it didn’t all quite work out as planned. The whole field of number theory, and arithmetic geometry more generally, is absolutely littered with unproven conjectures and half-fulfilled dreams, and in my humble view no-one should be encouraged to add to this castle in the sky.

As for me, I am practicing what I preach, and now working in a field where I see concrete results everyday, and where moreover I don’t have to engage in mental gymnastics to justify the fact that people are giving me money for my labours.

I can access the cgp video on safari and also on my iPhone. (There are tons of interesting videos on that site too btw, so it is very much worth figuring out how to access it.)

Trent,

Thanks. With Safari I at least got a frozen screen of video with the title slide, the Kim talk looks like it was about the arithmetic Chern-Simons stuff, not the more recent ideas. I can’t stand the idea of trying to watch a math talk on my phone, I’m too old for that.

Tried Safari again on the Kim talk, and managed to get a few screens showing later slides, which looked quite interesting. But staring at the frozen screen got old after a while. Got so desperate I tried the phone, but that doesn’t work at all. This is really maddening.

On Safari worked fine for me: https://goo.gl/JtyBxu

Peter wrote:

People have looked at the quantization too; a good reference is Analogies between knots and primes, 3-manifolds and number rings by Masanori Morishita.

The basic idea is that the Galois group of Q is like the fundamental group of Spec(Z), since it acts as “deck transformations” of the “universal cover” of Spec(Z), which is the spectrum of the algebraic integers in the algebraic completion of Q. Going further with this analogy, representations of the Galois group of Q are like flat vector bundles over Spec(Z), since flat vector bundles over a space correspond to representations of its fundamental group. And since Spec(Z) is 3-dimensional from the viewpoint of etale cohomology, while Spec(Z/p) is 1-dimensional for any prime p, and we have a map Spec(Z/p) -> Spec(Z), we should think of Spec(Z) as being a bit like a 3-manifold, and each prime as giving a knot in this 3-manifold.

(All this generalizes to arbitrary algebraic number fields, too.)

Since Chern-Simons theory is a topological quantum field theory involving flat vector bundles on a 3-manifold, and it gives invariants of knots and links in 3-manifolds, it’s irresistibly tempting to create an “arithmetic Chern-Simons theory” that gives invariants for primes or collections of primes.

The first success of this viewpoint is that the “Gauss sum” formula for the Legendre symbol of a pair of primes can be seen as analogous to the path integral for the vacum expectation of the product of Wilson loops for a pair of knots in U(1) Chern-Simons theory, which is a Gaussian integral. Yes, the Gauss sum looks like a discretized Gaussian integral! And this vacuum expectation is basically just the linking number!

Thus, quadratic reciprocity, which says how the Legendre symbol changes when you switch the two primes, becomes the usual symmetry of the linking number under switching the two knots.

MK has a new paper out, on this: “Arithmetic Gauge Theory: A Brief Introduction”, arXiv:1712.07602

There are more such analogies:

1) the differential geometric analog of Artin L-functions is clearly the Ruelle/Selberg zeta functions;

2) the Ruelle/Selberg zeta functions are known (I, II) to serve to express both Reidemeister torsion as well as the eta invariant;

3) these are of course the factors in the perturbative Chern-Simons quantum invariant (here).

Under this identification, all the zeta-. eta-. theta-, and L-functions in physics and number theory fit neatly into a dictionary.

> 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.

.

This link does work if you open it from a media player like vlc but keeps buffering a lot:

http://cgpvod.postech.ac.kr:1935/vod/voddata/IBS_CGP_2016_NT&QFT/1-1.Minhyong_Kim(20160822).MP4/playlist.m3u8