nForum - Discussion Feed (double dimensional reduction) 2021-06-17T20:25:11-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher David_Corfield comments on "double dimensional reduction" (75197) https://nforum.ncatlab.org/discussion/4923/?Focus=75197#Comment_75197 2019-01-11T05:14:28-05:00 2021-06-17T20:25:11-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ Typo fixed in link. diff, v18, current

Typo fixed in link.

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Urs comments on "double dimensional reduction" (75195) https://nforum.ncatlab.org/discussion/4923/?Focus=75195#Comment_75195 2019-01-11T04:36:05-05:00 2021-06-17T20:25:11-04:00 Urs https://nforum.ncatlab.org/account/4/ I have taken the liberty of adding pointers to our formalization of double dimensional reduction: Formalization of double dimensional reduction is discussed in rational homotolpy theory ...

I have taken the liberty of adding pointers to our formalization of double dimensional reduction:

Formalization of double dimensional reduction is discussed in rational homotolpy theory in

and in full homotopy theory in

Exposition is in

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Urs comments on "double dimensional reduction" (61314) https://nforum.ncatlab.org/discussion/4923/?Focus=61314#Comment_61314 2017-02-14T05:15:15-05:00 2021-06-17T20:25:11-04:00 Urs https://nforum.ncatlab.org/account/4/ I have moved over (here) at least statement and proof of the abstract &infin;\infty-topos theoretic formulation of double dimensional reduction, in the following form: Let H\mathbf{H} be any ...

I have moved over (here) at least statement and proof of the abstract $\infty$-topos theoretic formulation of double dimensional reduction, in the following form:

Let $\mathbf{H}$ be any (∞,1)-topos and let $G$ be an ∞-group in $\mathbf{H}$. There is a pair of adjoint ∞-functors of the form

$\mathbf{H} \underoverset {\underset{[G,-]/G}{\longrightarrow}} {\overset{hofib}{\longleftarrow}} {\bot} \mathbf{H}_{/\mathbf{B}G} \,,$

where

• $[G,-]$ denotes the internal hom in $\mathbf{H}$,

• $[G,-]/G$ denotes the homotopy quotient by the conjugation ∞-action for $G$ equipped with its canonical ∞-action by left multiplication and the argument regarded as equipped with its trivial $G$-$\infty$-action, hence for $G = S^1$ this is the cyclic loop space construction.

Hence for

then there is a natural equivalence

$\underset{ \text{original} \atop \text{fluxes} }{ \underbrace{ \mathbf{H}(\hat X\;,\; A) } } \;\; \underoverset {\underset{oxidation}{\longleftarrow}} {\overset{reduction}{\longrightarrow}} {\simeq} \;\; \underset{ \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} } }{ \underbrace{ \mathbf{H}(X \;,\; [G,A]/G) } }$

given by

$\left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right)$ ]]>
David_Corfield comments on "double dimensional reduction" (61033) https://nforum.ncatlab.org/discussion/4923/?Focus=61033#Comment_61033 2017-01-21T03:37:35-05:00 2021-06-17T20:25:11-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ Expressed in HoTT, I imagine that could look beautifully simple.

Expressed in HoTT, I imagine that could look beautifully simple.

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Urs comments on "double dimensional reduction" (61015) https://nforum.ncatlab.org/discussion/4923/?Focus=61015#Comment_61015 2017-01-19T17:54:56-05:00 2021-06-17T20:25:11-04:00 Urs https://nforum.ncatlab.org/account/4/ Meanwhile we have a much more sophisticated formulation of double dimensional reduction. It’s not reflected in the entry yet. But I am writing an exposition as talk notes here.

Meanwhile we have a much more sophisticated formulation of double dimensional reduction. It’s not reflected in the entry yet. But I am writing an exposition as talk notes here.

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Urs comments on "double dimensional reduction" (55471) https://nforum.ncatlab.org/discussion/4923/?Focus=55471#Comment_55471 2015-12-16T07:28:39-05:00 2021-06-17T20:25:11-04:00 Urs https://nforum.ncatlab.org/account/4/ added to double dimensional reduction a formal definition for double dimensional reduction of cocycles in differential cohomology.

added to double dimensional reduction a formal definition for double dimensional reduction of cocycles in differential cohomology.

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Urs comments on "double dimensional reduction" (39690) https://nforum.ncatlab.org/discussion/4923/?Focus=39690#Comment_39690 2013-05-13T10:45:34-04:00 2021-06-17T20:25:11-04:00 Urs https://nforum.ncatlab.org/account/4/ This is I think part of the original Witten and Kapustin-Witten story on geometric Langlands. Roughly like this: the 6d (2,0)(2,0)-superconformal QFT on the worldvolume of the M5-brane in 11-d ...

This is I think part of the original Witten and Kapustin-Witten story on geometric Langlands. Roughly like this:

• the 6d $(2,0)$-superconformal QFT on the worldvolume of the M5-brane in 11-d SuGra has a conformal invariance, specifically Moebius transformations when taken to be a product of a 4d space with a torus

• double dimensional reduction makes this the 5d super-Yang-Mills theory on the worldvolume of the D4-brane in 10-d SuGra

• further ordinary dimensional reduction of the 5d worldvolume theory to a 4d theory yields 4d Yang-Mills and its topological twists. Now the Montonen-Olive S-duality of that theory is supposed to be the shadow of the original conformal invariance of the (2,0)-theory on the torus which was “dimensionally reduced”.

• further compactifying down to d=2 turns this into geometric Langlands duality.

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David_Corfield comments on "double dimensional reduction" (39688) https://nforum.ncatlab.org/discussion/4923/?Focus=39688#Comment_39688 2013-05-13T10:24:20-04:00 2021-06-17T20:25:11-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ What happens to the S-duality connected to 6d (2,0)-superconformal QFT when undergoing this reduction? What happens to holographic duals when one is reduced?

What happens to the S-duality connected to 6d (2,0)-superconformal QFT when undergoing this reduction? What happens to holographic duals when one is reduced?

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Urs comments on "double dimensional reduction" (39686) https://nforum.ncatlab.org/discussion/4923/?Focus=39686#Comment_39686 2013-05-13T01:14:27-04:00 2021-06-17T20:25:11-04:00 Urs https://nforum.ncatlab.org/account/4/ am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the ...

am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.

started a stubby double dimensional reduction in this context and added some first further pointers and references to M5-brane, to D=5 super Yang-Mills theory and maybe elsewhere.

But this still needs more details to be satisfactory, clearly.

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