A thought on Langlands etc.

Stream: maths

[Mr Proof]

I heard the news about the (unramified) geometric Langlands in 2024.

Now, being someone more interesting in physics and string theory, this seems to have some correspondence with closed string amplitudes which involve sums over integrals on Riemann surfaces. The "unramified" part would correspond it them having no external legs. i.e. being a pure "partition function" in the physics sense of the word. (a vacuum amplitude).

Witten, a few years back now (https://arxiv.org/pdf/1506.04293) suggested the full Geometric Langlands could be explained by the conjectured S-Duality of N=4 Super Yang Mills. Where S-Duality is generalisation of the duality between electric charges and magnetic monopoles, specifically where the coupling constant goes from g->1/g. The Riemann surfaces appear again by compactifying two dimensions.

Now, here's the thing: Physicists have at the moment become a bit "stuck" with trying to formulate a non-perturbative formulation of string theory or M-Theory. By which I mean a formulation which doesn't rely on an asymptotic series expansion. (There is the AdS/CFT correspondence but since our universe is dS not AdS this is only partly helpful).

So my view is that in fact the path to M-Theory, may come from solving the full geometric Langlands conjecture. Such a proof may point the way to the existence and construction of the full non-perturbative M-Theory. (In a similar way the the proof of the Moonshine conjecture pointed to the existence of the Monster CFT).

So in that sense a formalisation of the geometric Langlands could be a path to solving the ultimate questions in physics.

It may involve formalising path integrals which is notoriously hard to do. Although in certain cases they can be shown to give finite answers such as topological quantum field theory on a sphere or torus. (The result often being a modular form in terms of the parameters. The duality of g->1/g appearing as the symmetry of the modular form).

What do you think? Anyone who works in this area have any thoughts on the matter?

[Kevin Buzzard]

My thoughts on the matter are "what does this have to do with Lean?"