### Arithmetic Gauge Theory

#### Posted by David Corfield

Around 2008-9 we had several exchanges with Minhyong Kim here at the Café, in particular on his views of approaching number theory from a homotopic perspective, in particular in the post Kim on Fundamental Groups in Number Theory. (See also the threads Afternoon Fishing and The Elusive Proteus.)

I even recall proposing a polymath project based on his ideas in Galois Theory in Two Variables. Something physics-like was in the air, and this seemed a good location with two mathematical physicists as hosts, John having extensively written on number theory in This Week’s Finds.

Nothing came of that, but it’s interesting to see Minhyong is very much in the news these days, including in a popular article in Quanta magazine, Secret Link Uncovered Between Pure Math and Physics.

The Quanta article has Minhyong saying:

“I was hiding it because for many years I was somewhat embarrassed by the physics connection,” he said. “Number theorists are a pretty tough-minded group of people, and influences from physics sometimes make them more skeptical of the mathematics.”

Café readers had an earlier alert from an interview I conducted with Minhyong, reported in Minhyong Kim in The Reasoner. There he was prepared to announce

The work that occupies me most right now, arithmetic homotopy theory, concerns itself very much with arithmetic moduli spaces that are similar in nature and construction to moduli spaces of solutions to the Yang-Mills equation.

Now his articles are appearing bearing explicit names such as ‘Arithmetic Chern-Simons theory’ (I and II), and today, we have Arithmetic Gauge Theory: A Brief Introduction.

What’s moved on in the intervening years from our side (‘our’ largely in the sense of the nLab) is an approach (very much due to Urs) to gauge field theory which looks to extract its abstract essence, and even to express this in the language of cohesive homotopy type theory, see nLab: geometry of physics. What I would love to know is how best to think of the deepest level of commonality between constructions deserving of the name ‘gauge theory’.

On the apparently non-physics side, who knows what depths might be reached if topological Langlands is ever worked out in stable homotopy theory, there being a gauge theoretic connection to geometric Langlands and even to the arithmetic version, as Minhyong remarks in his latest article:

We note also that the Langlands reciprocity conjecture … has as its goal the rewriting of arithmetic L-functions quite generally in terms of automorphic L-functions… it seems reasonable to expect the geometry of arithmetic gauge fields to play a key role in importing quantum field theoretic dualities to arithmetic geometry.

Perhaps the deepest idea would have to reflect the lack of uniformity in arithmetic. Minhyong writes in his latest paper about the action of $G_K = Gal(\bar K, K)$

The $G_K$-action is usually highly non-trivial, and this is a main difference from geometric gauge theory, where the gauge group tends to be constant over spacetime.

Even if orbifolds and singularities appear in the latter, maybe there’s still a difference. From a dilettantish wish to make sense of Buium and Borger’s nLab: arithmetic jet spaces, I had hoped that the geometric jet space constructions as beautifully captured by the nlab: jet comonad, might help. But arithmetic always seems a little obstructive, and one can’t quite reach the adjoint quadruples of the cohesive world: nlab: Borger’s absolute geometry. James Borger explained this to me as follows:

the usual, differential jet space of X can be constructed by gluing together jet spaces on open subsets, essentially because an infinitesimal arc can never leave an open subset. However, the analogous thing is not true for arithmetic jet spaces, because a Frobenius lift can jump outside an open set. So you can’t construct them by gluing local jet spaces together!

So plenty to explore. Where Minhyong speaks of arithmetic Euler-Lagrange equations, how does this compare with the jet comonadic version of Urs, outlined in Higher Prequantum Geometry II: The Principle of Extremal Action - Comonadically?

## Re: Arithmetic Gauge Theory

Also of interest to nCat cafe denizens: Kim says in footnote 36 at the start of the last section of Arithmetic Gauge Theory that he is working on the (presumably derived analytic) foundations of the material in that section with Kremnitzer and Behrend.