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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2013
   • (edited May 13th 2013)
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   Author: Urs
   Format: MarkdownItexam 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 13th 2013
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   Author: David_Corfield
   Format: MarkdownItexWhat 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?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2013
   • (edited May 13th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 16th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to _[[double dimensional reduction]]_ a [formal definition](https://ncatlab.org/nlab/show/double+dimensional+reduction#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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 19th 2017
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   Author: Urs
   Format: MarkdownItexMeanwhile 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](https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#DoubleDimensionalReduction).

   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.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 21st 2017
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   Author: David_Corfield
   Format: MarkdownItexExpressed in HoTT, I imagine that could look beautifully simple.

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

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 14th 2017
   • (edited Feb 14th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have moved over ([here](https://ncatlab.org/nlab/show/double+dimensional+reduction#GeneralReduction)) 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|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 * $\hat X \to X$ a $G$ [[principal ∞-bundle]] * $A$ a [[coefficient]] object, such as for some [[differential cohomology|differential]] [[generalized cohomology theory]] 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 $$ \left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right) $$

   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

   • $\hat X \to X$ a $G$ principal ∞-bundle

   • $A$ a coefficient object, such as for some differential generalized cohomology theory

   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

   $$\left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right)$$
8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 11th 2019
   • (edited Jan 11th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 * [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_ ATMP Volume 22 (2018) Number 5 ([arXiv:1611.06536](https://arxiv.org/abs/1611.06536)) and in full [[homotopy theory]] in * [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], Section 2.2 of _[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory]]_ ([arXiv:1806.01115](https://arxiv.org/abs/1806.01115)) Exposition is in * [[Urs Schreiber]], [Section 4](https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#DoubleDimensionalReduction) of _[[schreiber:Super Lie n-algebra of Super p-branes]]_, talks at [Fields, Strings, and Geometry Seminar](http://www.surrey.ac.uk/maths/news/seminars/fsg/), Surrey, 5th-9th Dec. 2016 <a href="https://ncatlab.org/nlab/revision/diff/double+dimensional+reduction/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/double+dimensional+reduction/17">v17</a>, <a href="https://ncatlab.org/nlab/show/double+dimensional+reduction">current</a>

   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

   • Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 3 of T-Duality from super Lie n-algebra cocycles for super p-branes (schreiber) ATMP Volume 22 (2018) Number 5 (arXiv:1611.06536)

   and in full homotopy theory in

   • Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber, Section 2.2 of Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory (arXiv:1806.01115)

   Exposition is in

   • Urs Schreiber, Section 4 of Super Lie n-algebra of Super p-branes (schreiber), talks at Fields, Strings, and Geometry Seminar, Surrey, 5th-9th Dec. 2016

   diff, v17, current

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 11th 2019
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   Author: David_Corfield
   Format: MarkdownItexTypo fixed in link. <a href="https://ncatlab.org/nlab/revision/diff/double+dimensional+reduction/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/double+dimensional+reduction/18">v18</a>, <a href="https://ncatlab.org/nlab/show/double+dimensional+reduction">current</a>

   Typo fixed in link.

   diff, v18, current