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recently there were some questions about it here on the nForum: now there is an entry on the Kaluza-Klein mechanism
inspired by this Physics.SE question I went through the entry Kaluza-Klein mechanism and polished and slightly expanded throughout.
Then I added another Examples-subsection: Cascades of KK-reductions from holographic boundaries.
Probably a wrong thought, but when I see geometric Langlands result from such a cascade of reductions and dualities, it sounds like it’s quite a way along the direction of concrete particularity. But if geometric Langlands is one Rosetta stone dialectic version of some theory, another of whose versions is arithmetic Langlands, shouldn’t we expect the latter to be similarly concretely particular? And then, wouldn’t that suggest an arithmetic analogue to the larger, less particular space of quantum field theories being related in the cascade?
Yes, that seems to be an evident big-scale conjecture. One term that comes to mind as a potential part of a potential answer is: p-adic physics.
While an evident conjecture, I am not aware that anyone has tried to make the connection. Or maybe someone has? I’d be interested in seeing references.
Hm, googling for this does yield some results. Many seem to be a bit dubious, though. Not sure about the following preprint here, but at least it does seem to try to connect $p$-adic string theory with number theoretic Langlands duality:
I see Kapustin has a student – Kevin Setter – who has fairly recently finished his thesis extending Kapustin & Witten on geometric Langlands. The introductory first 8 pages gives a clear account. He finds an equivalence of 2-categories of boundary conditions for two 3d gauged topological sigma models.
Thanks, I have now added that citation here.
But I thought you were after the non-geometric number-theoretic Langlands duality realized in physics. This thesis doesn’t mention that, or does it?
Sure, it doesn’t mention that. I just came across the thesis and hadn’t heard of it, so thought to mention it. On the other hand, if there is a great Rosetta analogy to be had, understanding Kapustin-Witten by generalising in this way may help pick up the analogues on the arithmetic side.
From my small search into the original issue, I should think Tamas Hausel’s work might give some pointers. E.g., Global topology of the Hitchin system
Abstract: Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory,
where he writes
Our studies will lead us into a circle of ideas relating arithmetic and the Langlands program to the physical ideas from gauge theory, S-duality and mirror symmetry in the study of the global topology of the Hitchin system. This could be considered the hyperkahler analogue of the fascinating parallels between the arithmetic approach of Harder-Narasimhan and the gauge theoretical approach of Atiyah-Bott in the study of H(N).
Also a talk Arithmetic and physics in discrete algebraic geometry.
Thanks. I have just glanced over it. Looks good. But this is still just on the geometric Langlands duality, isn’t it? Hausel’s main point is that Kapustin-Witten’s S-duality realization may also be thought of in terms of T-duality and mirror symmetry. But it’s still the geometric version, not the number-theoretic version. Or maybe I am missing something, of course.
To take the Hegelian path, if something like your four axiom account of synthetic QFT (and I guess you might be able to update axiom 4 now) is right, then we should expect any arithmetic Rosetta analogue of ordinary QFT to work via arithmetic forms of cohesion.
And if such a thing exists, shouldn’t we be stumbling over 1000s of traces of such arithmetic analogues? Maybe we are, e.g.,
Yes, probably.
added a section KK-compactification – Formalization with the following comments:
KK-compactification along trivial fibrations is closely related to forming mapping stacks: if $\mathbf{Fields}_n$ is the moduli stack of fields] for an $n$-dimensional field theory (see at prequantum field theory for more on this), then for $\Sigma_{k}$ a $k$-dimensional manifold with $k \lt n$ the mapping stack
$\mathbf{Fields}_{n-k} \coloneqq [\Sigma_k, \mathbf{Fields}]$may be thought of as the moduli stack of fields for an $(n-k)$-dimensional field theory. By the definition universal property of the mapping stack, this lower dimensional field theory is then such that a field configiuration over an $(n-k)$-dimensional spacetime $X_{n-k}$
$\phi \colon X_{n-k} \longrightarrow \mathbf{Fields}_{n-k}$is equivalently a field configuration of the $n$-dimensional field theory
$X_{n-k} \times \Sigma_k \longrightarrow \mathbf{Fields}_n$on the product space $X_{n-k}\times \Sigma_k$ (the trivial $\Sigma_{k}$-fiber bundle over $X_{n-k}$).
Traditionally KK-reduction is understood as retaining only parts of $\mathbf{Fields}_{n-k}$ (the “0-modes” of fields on $\Sigma_k$ only) but of course one may consider arbitrary corrections to this picture and eventially retain the full information.
One example of KK-reduction where the full mapping stack appears is the reduction of topologically twisted N=4 D=4 super Yang-Mills theory on a complex curve $C$ as it appears in the explanation of geometric Langlands duality as a special case of S-duality (Witten 08, section 6). Here $\mathbf{Fields}_4 = \mathbf{B}G_{\mathrm{conn}}$ is the universal moduli stack of $G$-principal connections (or rther that of $G$-Higgs bundles).
I have added two paragraphs to the Idea-section at KK-compactification, mentioning relation to geometrodynamics and the issue of moduli stabilization. Also cross-linked.
Edit to: Kaluza-Klein mechanism by Urs Schreiber at 2018-04-01 01:23:26 UTC.
Author comments:
added pointer to textbook by Ibanez-Uranga
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