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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 $CS(A) + i vol$ 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…

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.