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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 11th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI see Minhyong Kim has a paper out on this [Arithmetic Chern-Simons Theory I](http://arxiv.org/abs/1510.05818). This differs from [your approach](http://nforum.ncatlab.org/discussion/6592/is-the-emtopos-of-the-jetcomonad-cohesive-if-the-base-is/?Focus=53750#Comment_53750), Urs? >... based on two observations: >a) (with Domenico Fiorenza): the extended TQFT with coefficients in higher spans "phased" over a moduli stack $\mathbf{B}^n U(1)$ which to the point assings a given higher Chern-Simons bundle $\mathbf{B}G \to \mathbf{B}^n U(1)$ will in codimension-1 assign the corresponding theta-line, and the functoriality in codimension-1 expresses this as a higher modular functor; >b) (with suggestions from Minhyong Kim): The system of sections of that theta-line, something like a corrected exponentiated eta-invariant, is the by far best differential geometric analog of the Artin L-functions

   I see Minhyong Kim has a paper out on this Arithmetic Chern-Simons Theory I. This differs from your approach, Urs?

   … based on two observations:

   a) (with Domenico Fiorenza): the extended TQFT with coefficients in higher spans “phased” over a moduli stack $\mathbf{B}^n U(1)$ which to the point assings a given higher Chern-Simons bundle $\mathbf{B}G \to \mathbf{B}^n U(1)$ will in codimension-1 assign the corresponding theta-line, and the functoriality in codimension-1 expresses this as a higher modular functor;

   b) (with suggestions from Minhyong Kim): The system of sections of that theta-line, something like a corrected exponentiated eta-invariant, is the by far best differential geometric analog of the Artin L-functions

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexAt one point he quotes Bruce Bartlett. I checked with Bruce, the statement he mentioned to Minhyong Kim was taken from the $n$Lab, and it was one of the notes I had made a while back. Regarding the close [analogy](http://ncatlab.org/nlab/show/Artin%20L-function#AnalogyWithSelbergZeta) between Artin L-functions and the Selberg/Ruelle zeta functions, as well as the role of the latter as factors in the perturbative Chern-Simons invariant (Reidemeister torsion, eta invariant).

   At one point he quotes Bruce Bartlett. I checked with Bruce, the statement he mentioned to Minhyong Kim was taken from the $n$Lab, and it was one of the notes I had made a while back. Regarding the close analogy between Artin L-functions and the Selberg/Ruelle zeta functions, as well as the role of the latter as factors in the perturbative Chern-Simons invariant (Reidemeister torsion, eta invariant).

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 14th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI wonder if there could be a [[complex volume]] of the spec of a number field.

   I wonder if there could be a complex volume of the spec of a number field.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 20th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThe sequel has appeared * Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo, _Arithmetic Chern-Simons Theory II_, ([arXiv:1609.03012](http://arxiv.org/abs/1609.03012)) >We apply ideas of Dijkgraaf and Witten on three-dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications.

   The sequel has appeared

   • Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo, Arithmetic Chern-Simons Theory II, (arXiv:1609.03012)

   We apply ideas of Dijkgraaf and Witten on three-dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 20th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexYou should record this in some $n$Lab entry!

   You should record this in some $n$Lab entry!

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 20th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI just did at [[arithmetic Chern-Simons theory]].

   I just did at arithmetic Chern-Simons theory.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 20th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks!

   Thanks!

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 17th 2017
   • (edited Dec 17th 2017)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI added Minhyong's nice talk * [[Minhyong Kim]], _Arithmetic topological quantum field theory?_, ([slides](ttps://simonsfoundation.s3.amazonaws.com/share/mps/conferences/2017_Conference_on_Number_Theory_Geometry_Moonshine_and_Strings/Kim.pdf)) Slide 22 (or 64/114) >Moduli Spaces >Given an arithmetic scheme $Y$ and a $p$-adic Lie group $A$, we will be interested in >$$M(Y,A) := Hom(\pi_1(Y),A)//A,$$ >the moduli space of continuous homomorphisms >$$\rho: \pi_1(Y) \to A$$ >up to conjugation. >Key point of lecture: Even though an arithmetic scheme is quite different from a manifold, $M(Y,A)$ is structurally similar to moduli spaces of bundles in geometry and physics. Ripe for HoTT expression as dependent sum over $B A$ of a hom-space?

   I added Minhyong’s nice talk

   • Minhyong Kim, Arithmetic topological quantum field theory?, (slides)

   Slide 22 (or 64/114)

   Moduli Spaces

   Given an arithmetic scheme $Y$ and a $p$-adic Lie group $A$, we will be interested in

   $$M(Y,A) := Hom(\pi_1(Y),A)//A,$$

   the moduli space of continuous homomorphisms

   $$\rho: \pi_1(Y) \to A$$

   up to conjugation.

   Key point of lecture: Even though an arithmetic scheme is quite different from a manifold, $M(Y,A)$ is structurally similar to moduli spaces of bundles in geometry and physics.

   Ripe for HoTT expression as dependent sum over $B A$ of a hom-space?

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 17th 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexUrs may remember we discussed this very object in Edinburgh at the conference dinner. I have the sketchy notes in my notebook.

   Urs may remember we discussed this very object in Edinburgh at the conference dinner. I have the sketchy notes in my notebook.

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 17th 2017
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexTypo in the url: ([slides](https://simonsfoundation.s3.amazonaws.com/share/mps/conferences/2017_Conference_on_Number_Theory_Geometry_Moonshine_and_Strings/Kim.pdf))

    Typo in the url: (slides)

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2017
    • (edited Dec 18th 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexAt the moment I don't have time for this, and I lost motivation a little when _<a href="https://arxiv.org/abs/1510.05818">Arithmetic Chern-Simons theory I</a>_ was put on the arXiv, after we had chatted about this. On p. 17 in that article Bruce Bartlett is credited for pointing out the relation to Reidemeister torsion. I happened to be around Bruce at ESI Vienna when this came out and I checked with him: He had been recounting material that he had seen me write on the nLab. To point this out I had sent the following by email to Minhyong Kim: > Date: Thu, 22 Oct 2015 10:02:27 +0200 > Dear Minhyong Kim, > how are you doing? I saw your nice article on arithmetic Chern-Simons appear. That reminded me of some of the things that we were discussing back then. > I happen to be around with Bruce Bartlett in Vienna at the moment. Bruce tells me that the suggestion which you cite, about the relation to Reidemeister torsion, he got from the nLab, where I had been pointing out these relations. > Here is how I see it: > 1) the differential geometric analog of Artin L-functions is clearly the Ruelle/Selberg zeta function > [http://ncatlab.org/nlab/show/Ruelle+zeta+function#AnalogyToTheArtinLFunction](http://ncatlab.org/nlab/show/Ruelle+zeta+function#AnalogyToTheArtinLFunction) > 2) the Ruelle/Selberg zeta function are known to serve to express both Reidemeister torsion as well as the eta invariant > [http://ncatlab.org/nlab/show/Selberg+zeta+function#RelationToTheEtaFunction](http://ncatlab.org/nlab/show/Selberg+zeta+function#RelationToTheEtaFunction) > [http://ncatlab.org/nlab/show/Selberg+zeta+function#RelationToAnalyticTorsion](http://ncatlab.org/nlab/show/Selberg+zeta+function#RelationToAnalyticTorsion) > 3) these are of course the factors in the perturbative Chern-Simons quantum invariant > [http://ncatlab.org/nlab/show/Chern-Simons+theory#PerturbativePathIntegralQuantization](http://ncatlab.org/nlab/show/Chern-Simons+theory#PerturbativePathIntegralQuantization) I see that the article is now at version number 4 on the arXiv. But this attribution has not been modified.

    At the moment I don’t have time for this, and I lost motivation a little when Arithmetic Chern-Simons theory I was put on the arXiv, after we had chatted about this.

    On p. 17 in that article Bruce Bartlett is credited for pointing out the relation to Reidemeister torsion. I happened to be around Bruce at ESI Vienna when this came out and I checked with him: He had been recounting material that he had seen me write on the nLab. To point this out I had sent the following by email to Minhyong Kim:

    Date: Thu, 22 Oct 2015 10:02:27 +0200

    Dear Minhyong Kim,

    how are you doing? I saw your nice article on arithmetic Chern-Simons appear. That reminded me of some of the things that we were discussing back then.

    I happen to be around with Bruce Bartlett in Vienna at the moment. Bruce tells me that the suggestion which you cite, about the relation to Reidemeister torsion, he got from the nLab, where I had been pointing out these relations.

    Here is how I see it:

    1) the differential geometric analog of Artin L-functions is clearly the Ruelle/Selberg zeta function

    http://ncatlab.org/nlab/show/Ruelle+zeta+function#AnalogyToTheArtinLFunction

    2) the Ruelle/Selberg zeta function are known to serve to express both Reidemeister torsion as well as the eta invariant

    http://ncatlab.org/nlab/show/Selberg+zeta+function#RelationToTheEtaFunction http://ncatlab.org/nlab/show/Selberg+zeta+function#RelationToAnalyticTorsion

    3) these are of course the factors in the perturbative Chern-Simons quantum invariant

    http://ncatlab.org/nlab/show/Chern-Simons+theory#PerturbativePathIntegralQuantization

    I see that the article is now at version number 4 on the arXiv. But this attribution has not been modified.

12. • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 18th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThat's a shame. Minhyong's program is receiving plenty of interest at the moment, even a popular [article](https://www.quantamagazine.org/secret-link-uncovered-between-pure-math-and-physics-20171201/).

    That’s a shame. Minhyong’s program is receiving plenty of interest at the moment, even a popular article.

13. • CommentRowNumber13.
    • CommentAuthorRichard Williamson
    • CommentTimeDec 18th 2017
    • PermaLink
    Author: Richard Williamson
    Format: MarkdownItexThis is indeed very unfortunate. I get the impression that you know Minhyong a little, David; would you be able to find a way to remind him to try to put an attribution in place? He may have simply have accidentally omitted it, but such things can be deflating. Perhaps also if the nLab work on this were to continue to be developed, it would make it clearer that much of the original thoughts about it occurred with Urs (and others?) here.

    This is indeed very unfortunate. I get the impression that you know Minhyong a little, David; would you be able to find a way to remind him to try to put an attribution in place? He may have simply have accidentally omitted it, but such things can be deflating.

    Perhaps also if the nLab work on this were to continue to be developed, it would make it clearer that much of the original thoughts about it occurred with Urs (and others?) here.

14. • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 18th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThanks, Richard. I was intending to write to Minhyong anyway, so will mention this. And, yes, I think there is an issue with correct attribution for information on a wiki. For one thing, some people imagine the name at the bottom of the page is the author, when it might be generated by the correction of a single letter. Second, the other day I saw an article referring to some nLab entries and listing all contributing authors. This is better, but still may list typo-correctors.

    Thanks, Richard. I was intending to write to Minhyong anyway, so will mention this.

    And, yes, I think there is an issue with correct attribution for information on a wiki. For one thing, some people imagine the name at the bottom of the page is the author, when it might be generated by the correction of a single letter. Second, the other day I saw an article referring to some nLab entries and listing all contributing authors. This is better, but still may list typo-correctors.