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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 23rd 2014
   • (edited May 23rd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHere is a simple observation, which seems to be pointing to something deeper. I am trying to extract more of that "something deeper", and it feels like part of it must be well known in some circles, if maybe in different words. This here is to ask if the following resonates with anyone, and what useful pointers might be. So essentially the story is that a hefty number of steps done in differential geometry to geometrically quantize Chern-Simons type gauge theories may be collapsed to one single algebro-geometric step, as follows: for $\Sigma_{4k+2}$ a compact real $4k+2$-dimensional manifold, then any choice of conformal structure on it is supposed to induce on the moduli space of flat $(k+1)$-line bundles over it a complex structure whose holomorphic coordinates are connections with Hodge-self-dual curvature on $\Sigma$. Now by the [[Koszul-Malgrange theorem]] we may turn the construction of this complex moduli space into the following simple operation: realize $\Sigma$ as the manifold underlying the analytication of a scheme $S$ over $\mathbb{C}$. Then form the mapping stack $[S, \mathbf{B}^{k+1}\mathbb{G}_m]$. That, or its would-be analytification, is the desired moduli "space". It seems this state of affairs should have a neat expression in terms of differential refinements of $\mathbf{B}^{k+1}\mathbb{G}_m$ regarded as a $\flat$-modal object in the $\infty$-topos of "[[smooth E-infinity-groupoids]]", namely of $\infty$-stacks on the site of smooth manifolds with coefficients in $\infty$-stacks on a site of (arithmetic) schemes (or better spectral schemes, but that's maybe not so important for what I am after here at the moment). I am tinkering with the thought of considering "manifolds equipped with complex structure" here are objects in the product site, being the product of a smooth (arithmetic?) scheme with the manifold underlying the analytification of its complex points. While that models some of what one wants in the above context, it seems a strangely ad-hoc thing to consider in itself. I am wondering if there is some nicer general abstract something here which I should be paying attention to. Is there some nice general-abstract way to characterize analytification in the context of the above infinity-topos. Or something? I know that this is a vague question, sorry. But often these are the hardest.

   Here is a simple observation, which seems to be pointing to something deeper. I am trying to extract more of that “something deeper”, and it feels like part of it must be well known in some circles, if maybe in different words. This here is to ask if the following resonates with anyone, and what useful pointers might be.

   So essentially the story is that a hefty number of steps done in differential geometry to geometrically quantize Chern-Simons type gauge theories may be collapsed to one single algebro-geometric step, as follows:

   for $\Sigma_{4k+2}$ a compact real $4k+2$-dimensional manifold, then any choice of conformal structure on it is supposed to induce on the moduli space of flat $(k+1)$-line bundles over it a complex structure whose holomorphic coordinates are connections with Hodge-self-dual curvature on $\Sigma$.

   Now by the Koszul-Malgrange theorem we may turn the construction of this complex moduli space into the following simple operation: realize $\Sigma$ as the manifold underlying the analytication of a scheme $S$ over $\mathbb{C}$. Then form the mapping stack $[S, \mathbf{B}^{k+1}\mathbb{G}_m]$. That, or its would-be analytification, is the desired moduli “space”.

   It seems this state of affairs should have a neat expression in terms of differential refinements of $\mathbf{B}^{k+1}\mathbb{G}_m$ regarded as a $\flat$-modal object in the $\infty$-topos of “smooth E-infinity-groupoids”, namely of $\infty$-stacks on the site of smooth manifolds with coefficients in $\infty$-stacks on a site of (arithmetic) schemes (or better spectral schemes, but that’s maybe not so important for what I am after here at the moment).

   I am tinkering with the thought of considering “manifolds equipped with complex structure” here are objects in the product site, being the product of a smooth (arithmetic?) scheme with the manifold underlying the analytification of its complex points. While that models some of what one wants in the above context, it seems a strangely ad-hoc thing to consider in itself. I am wondering if there is some nicer general abstract something here which I should be paying attention to. Is there some nice general-abstract way to characterize analytification in the context of the above infinity-topos. Or something?

   I know that this is a vague question, sorry. But often these are the hardest.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 24th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex>...$\infty$-stacks on the site of smooth manifolds with coefficients in $\infty$-stacks on a site of (arithmetic) schemes. So that's the setting you mentioned back [here](http://nforum.mathforge.org/discussion/5169/what-points-do-we-have/?Focus=42802#Comment_42802). Is this part of some large story of blending algebraic and analytic information.

   …$\infty$-stacks on the site of smooth manifolds with coefficients in $\infty$-stacks on a site of (arithmetic) schemes.

   So that’s the setting you mentioned back here. Is this part of some large story of blending algebraic and analytic information.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 24th 2014
   • (edited May 24th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat's what I am after at least. Of course Bunke-Tamme already showed that this is the context in which Beilinson's regulators "really" live ([[differential algebraic K-theory]]) but I am trying to see how to further fit that with the geometric quantization of self-dual higher gauge fields, which is very much about blending differential cohomological data with arithmetic geometry data. For instance it seems to me that Lurie's "[2-equivariant elliptic cohomology](http://ncatlab.org/nlab/show/equivariant+elliptic+cohomology#2EquivariantEllipticCohomology)" is (as briefly indicated at that link) essentially the story of the tower of transgressions of the extended 3d Chern-Simons functional $\mathbf{B}G_{conn}\longrightarrow \mathbf{B}^3U(1)_{conn}$ that I have been writing about a bit, but suitably extended from bare smooth to higher arithmetic geometry. I'd like to nail that down a bit more.

   That’s what I am after at least. Of course Bunke-Tamme already showed that this is the context in which Beilinson’s regulators “really” live (differential algebraic K-theory) but I am trying to see how to further fit that with the geometric quantization of self-dual higher gauge fields, which is very much about blending differential cohomological data with arithmetic geometry data.

   For instance it seems to me that Lurie’s “2-equivariant elliptic cohomology” is (as briefly indicated at that link) essentially the story of the tower of transgressions of the extended 3d Chern-Simons functional $\mathbf{B}G_{conn}\longrightarrow \mathbf{B}^3U(1)_{conn}$ that I have been writing about a bit, but suitably extended from bare smooth to higher arithmetic geometry. I’d like to nail that down a bit more.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 27th 2014
   • PermaLink
   Author: zskoda
   Format: MarkdownItexArithmetic and complex geometry are "mixed" in [[Arakelov geometry]] where even the original paper by Arakelov was in terms of Green functions tailored to arithmetic. [[Nikolai Durov]] incorporated a new version of Arakelov geometry into generalized algebraic scheme theory and has some homotopical algebra in such generalized geometry, based on model stacks rather than model categories.

   Arithmetic and complex geometry are “mixed” in Arakelov geometry where even the original paper by Arakelov was in terms of Green functions tailored to arithmetic. Nikolai Durov incorporated a new version of Arakelov geometry into generalized algebraic scheme theory and has some homotopical algebra in such generalized geometry, based on model stacks rather than model categories.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 27th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI wonder if Arakelov motivic cohomology provides some answers ([I](http://arxiv.org/abs/1012.2523), [II](http://arxiv.org/abs/1205.3890)). If #3 >Bunke-Tamme already showed that this is the context in which Beilinson’s regulators “really” live (differential algebraic K-theory), how does their construction relate to Scholbach's >We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from BGL to the Deligne cohomology spectrum.

   I wonder if Arakelov motivic cohomology provides some answers (I, II). If #3

   Bunke-Tamme already showed that this is the context in which Beilinson’s regulators “really” live (differential algebraic K-theory),

   how does their construction relate to Scholbach’s

   We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from BGL to the Deligne cohomology spectrum.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2014
   • (edited May 28th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry for the slow reply, was a busy day today. Thanks for highlighting Arakelov geometry once more! I still need to really absorb it. Before I do so, here is something I might say: currently I think I find what I am after in the above sketch when working with schemes/analytic spaces over $\mathbb{C}$.The key ingredient for defining the differential cohomology object here such that it is ordinary differential cohomology in the manifold-direction and such as to produce intermediate Jacobians as above in the "flat" algebraic direction is (entirely unsurprisingly) the [[exponential exact sequence]]. I was trying to see if it is possible to pass from working over $\mathbb{C}$ to working arithmetically by replacing the complex exponential sequence by the _[[Kummer-Artin-Schreier-Witt exact sequence]]_. Morally that seems to be just the right thing to do, but for that to work I may have to replace manifolds by something more adic. I am really not sure here for the moment. Maybe looking at Arakelov theory will help.Superficially at least it sounds just like the right connective tissue.

   Sorry for the slow reply, was a busy day today.

   Thanks for highlighting Arakelov geometry once more! I still need to really absorb it.

   Before I do so, here is something I might say: currently I think I find what I am after in the above sketch when working with schemes/analytic spaces over $\mathbb{C}$.The key ingredient for defining the differential cohomology object here such that it is ordinary differential cohomology in the manifold-direction and such as to produce intermediate Jacobians as above in the “flat” algebraic direction is (entirely unsurprisingly) the exponential exact sequence.

   I was trying to see if it is possible to pass from working over $\mathbb{C}$ to working arithmetically by replacing the complex exponential sequence by the Kummer-Artin-Schreier-Witt exact sequence. Morally that seems to be just the right thing to do, but for that to work I may have to replace manifolds by something more adic. I am really not sure here for the moment. Maybe looking at Arakelov theory will help.Superficially at least it sounds just like the right connective tissue.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeMay 28th 2014
   • (edited May 28th 2014)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexYour mentioning Chern-Simons theory in this context tells me that you should talk to [[Maxim Kontsevich]] about this at some point. I do not understand what is going on here, but he was explaining in 2004 how the Chern-Simons works in number theoretic setup, if I remember right. This discussion was few days after his talk on his unfinished work with Gangl in Spring 2004 on algebraic K-theory of fields. In rough outline he was using some sort of double filtration (one of the gradings was from Adams operations, if I recall right) and he said that the stuff is zero above the diagonal, known below and questionable on the diagonal and that is where they looked for some rational information. Then he went on introducing Chern-Simons, from one side there is hyperbolic geometry in odd dim, then the boundary, the moduli space of flat connections...he also said that it looks like on the boundary you are working with the solution of Euler-Lagrange equations for the Morse theoretic problem, but the thing is that the "action functional" concerning the connection is complex so one can not say that literally. He introduced some Azumaya algebras as well, subject to transformations which close the commutative pentagon and so on; cluster algebras lurking in the picture (the talk first started with some new number theoretical formulas over some huge field which had also to do with this identity). His talk had Soule regulator in the title, as he produced some new viewpoint on this which Soule liked and Goncharov said it was obvious; the same year M.K. gave another inspiring talk with some similar stuff in Stockholm, related to quantization of Teichmueller (and higher Teichmuller) theory and work of Fock, Goncharov and others. Sorry for totally incoherent impressions from the memory, but maybe it is better first to mention the key words and then we can try to dig out related stuff. Gangl's side in project was working with concrete identities and calculations in K-theory of fields; pentagon identity mentioned is reflecting in the pentagon for the [[dilogarithm]] and [[quantum dilogarithm]] and has higher analogues (see more recent work fo Gangl and earlier work of [[Werner Nahm]] on CFT, Bloch-Wigner function, K-theory and dilogarithms). Partly incorporated in recent work of Bloch, Goncharov and others related to motives and their appearance in QFT. Some people cite a related joint preprint of Goncharov and Kontsevich which seem not to be publically available.

   Your mentioning Chern-Simons theory in this context tells me that you should talk to Maxim Kontsevich about this at some point. I do not understand what is going on here, but he was explaining in 2004 how the Chern-Simons works in number theoretic setup, if I remember right. This discussion was few days after his talk on his unfinished work with Gangl in Spring 2004 on algebraic K-theory of fields. In rough outline he was using some sort of double filtration (one of the gradings was from Adams operations, if I recall right) and he said that the stuff is zero above the diagonal, known below and questionable on the diagonal and that is where they looked for some rational information. Then he went on introducing Chern-Simons, from one side there is hyperbolic geometry in odd dim, then the boundary, the moduli space of flat connections…he also said that it looks like on the boundary you are working with the solution of Euler-Lagrange equations for the Morse theoretic problem, but the thing is that the “action functional” concerning the connection is complex so one can not say that literally.

   He introduced some Azumaya algebras as well, subject to transformations which close the commutative pentagon and so on; cluster algebras lurking in the picture (the talk first started with some new number theoretical formulas over some huge field which had also to do with this identity). His talk had Soule regulator in the title, as he produced some new viewpoint on this which Soule liked and Goncharov said it was obvious; the same year M.K. gave another inspiring talk with some similar stuff in Stockholm, related to quantization of Teichmueller (and higher Teichmuller) theory and work of Fock, Goncharov and others. Sorry for totally incoherent impressions from the memory, but maybe it is better first to mention the key words and then we can try to dig out related stuff.

   Gangl’s side in project was working with concrete identities and calculations in K-theory of fields; pentagon identity mentioned is reflecting in the pentagon for the dilogarithm and quantum dilogarithm and has higher analogues (see more recent work fo Gangl and earlier work of Werner Nahm on CFT, Bloch-Wigner function, K-theory and dilogarithms). Partly incorporated in recent work of Bloch, Goncharov and others related to motives and their appearance in QFT. Some people cite a related joint preprint of Goncharov and Kontsevich which seem not to be publically available.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for all the keywords. Would be nice to have some kind of something for where to start. Is there any talk notes or anything? I'll try to google...

   Thanks for all the keywords. Would be nice to have some kind of something for where to start. Is there any talk notes or anything? I’ll try to google…

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexDimofte and others speak about "arithmetic TQFT" if in perturbation theory amplitudes are over number fields. Not sure yet how that might connect...

   Dimofte and others speak about “arithmetic TQFT” if in perturbation theory amplitudes are over number fields. Not sure yet how that might connect…