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We have had our share of the debate of whether $Spec(\mathbb{Z})$ is really usefully analogous to a 3-manifold, and of how the $Spec(\mathbb{F}_p)$-s inside it then are analogous to knots in a 3-manifold.
Here is a thought (maybe this was voiced before and I am just being really slow, please bear with me):
things would seem to fall into place much better if we thought of the $Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z})$ not as analogous to knots, but as analogous to the prime geodesics inside a hyperbolic 3-manifold.
With this and its generalization to function fields, then the analogy between the Selberg zeta function for 3-manifolds and the Artin L-function (pointed out here) would become even better: in both cases we’d have the infinite product over all prime geodesics of, essentially, the determinant of the monodromy of the given flat connection over that geodesic.
Also, thinking of the $Spec(\mathbb{F}_p)$ not as knots but as prime geodesics removes all the awkward aspects of the former interpretation, such as why on earth one would be required to consider all these knots at once (which does not fit the analogy with knots in CS theory). Of course the prime geodesics would also be knots, technically, but I am talking here about the difference between thinking of them playing the conceptual role of the knots in CS theory (which are things we choose at will to build observables) and the prime geodesics, which are given to us by the gods as a means to compute the perturbative CS path integral.
Finally, there is of course much support from other directions of an analogy between prime geodesics and prime numbers (asymptotics etc.).
So it would seem to make much sense.
I think you ought to post your thoughts to the Café. It’s just not clear to me how many relevantly knowledgeable people read the nForum, people like James Borger and Minhyong Kim.
I’ll maybe post it to MO a little later. Right now I am in Oxford visiting Minhyong Kim. We are tossing some ideas around…
Is this at all related to Mirzakhani’s work?
David Corfield kindly points me to this question recently on MO, where somebody asks what the arithmetic analog of Mirzakhani’s result would be, under the translation prime geodesics $\leftrightarrow$ prime ideals. And this reply to the question does mention in passing the kind of reasoning that I am after.
So from this it seems that Mirzakhani’s work in itself does not shed light on this, but it is the kind of result that one would want to consider in view of the analogy, yes.
On the other hand, these results mentioned in #5 concern surfaces, not 3-folds, it seems.
Not quite the statement about the “curve in 3d”-aspect, but otherwise all of the analogy that I am talking about here has been highlighted in
Okay, to sum up, altogether this gives now a very detailed analogy that says that the map that takes a Galois representation to its Artin L-function is in fact analogous to the map that takes a flat connection on a 3-manifold to the perturbative (semiclassical) Chern-Simons path integral perturbing around that classical solution.
Good. Now next the question is: in view of this, what is it that the Langlands conjecture really says?
It’s something to do with extracting from the perturbative path integral, being a section of the determinant line, the partition theta function which itself is that section, up to isomorphism. Maybe Langlands’ conjecture 1 is really the arithmetic version of this isomorphism, the one on p.31 of Freed’s “On determinant line bundles” (pdf).
But enough for today. More tomorrow…
Morishita talks of on p, 156 of Knots and Primes of complete hyperbolic 3-manifolds providing a “closer analogy”. I can’t see whose work he’s referring to: Fr, Sg1, Sg2, Sg3. Presumably the last three are by Selberg.
Perhaps also worth a look through Watkin’s list.
Thanks! This remark 12.7 on that page is just excellent, just the kind of thing I am after. I need to chase the references he gives there.
the map that takes a Galois representation to its Artin L-function is in fact analogous to the map that takes a flat connection on a 3-manifold to the perturbative (semiclassical) Chern-Simons path integral perturbing around that classical solution.
Should we be looking for a non-perturbative number theory then?
Hmm, did you see Bruce’s addition at Chern-Simons theory:
Trying to interest your number theory friends with Chern–Simons theory? How about this: the Chern–Simons path integral Z(k) above is (in a certain precise sense) a modular form. This correspondence between the Chern–Simons quantum invariants and modular forms sheds light in both directions, and is a fascinating idea to me. The key words here (which I don’t understand) are “Eichler integral” and “mock theta function?”. See:
Lawrence and Zagier, Modular forms and quantum invariants of 3-manifolds, Asian Journal of Mathematics vol 3 no 1 (1999).
Hikami, Quantum invariant, modular forms, and lattice points, arXiv. See also the follow ups to this paper.
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