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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2015
   • (edited Jun 17th 2015)
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   Author: Urs
   Format: MarkdownItexA 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](https://plus.google.com/+UrsSchreiber/posts/EjuY7gc6XYW)) 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](http://ncatlab.org/nlab/show/hyperbolic+manifold#Zickert07)). (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...

   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 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 the contribution of the supergravity C-field.

   Not sure how it all hangs together, but there might be some interesting relation here…

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 17th 2015
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   Author: David_Corfield
   Format: MarkdownItexDoes the volume of the complement of a hyperbolic knot come into the picture here?

   Does the volume of the complement of a hyperbolic knot come into the picture here?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2015
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   Author: Urs
   Format: MarkdownItexI suppose so, but I don't understand this yet.

   I suppose so, but I don’t understand this yet.