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A little bit of moonshine-style observations:
My last round of arithmetic geometry back then, after visiting Minhyong Kim last year, culminated in the observation (here) that the true differential-geometric analog of the Artin L-function for a given Galois representation is the perturbative Chern-Simons invariant of a flat connection on a hyperbolic manifold expressed as a combination of Selberg/Ruelle zeta functions.
Now the combination of Chern-Simons invariants and hyperbolic manifolds appears prominently in analytically continued Chern-Simons theory, where the imaginary part of the CS-action is given by volumes of hyperbolic manifolds (e.g. Zickert 07). (Thanks to Hisham Sati for highlighting this.)
Next, this formal complex combination $CS(A) + i vol$ also appears as the contribution of membrane instantons in M-theory, it’s the contribution of the Polyakov action functional of a membrane wrapped on a 3-cycle, if we read $CS(A) = C$ the contribution of the supergravity C-field.
Not sure how it all hangs together, but there might be some interesting relation here…
Does the volume of the complement of a hyperbolic knot come into the picture here?
I suppose so, but I don’t understand this yet.
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