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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2013
   • (edited May 13th 2013)
   • PermaLink
   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
   • PermaLink
   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 -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
   • PermaLink
   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 -topos theoretic formulation of double dimensional reduction, in the following form:

   Let be any (∞,1)-topos and let be an ∞-group in . There is a pair of adjoint ∞-functors of the form

   where

   • denotes the internal hom in ,

   • denotes the homotopy quotient by the conjugation ∞-action for equipped with its canonical ∞-action by left multiplication and the argument regarded as equipped with its trivial --action, hence for this is the cyclic loop space construction.

   Hence for

   • a principal ∞-bundle

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

   then there is a natural equivalence

   given by

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

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexAdded the remark that the [[differential geometry]] of the double dimensional reduction of the [[M2-brane]]- and [[M5-brane]]-charges was maybe first clearly written out in: * [[Varghese Mathai]], [[Hisham Sati]], §4 in: *Some Relations between Twisted K-theory and $E_8$ Gauge Theory*, J. High Energ. Phys. **2004** 03 (2004) 016 &lbrack;[arXiv:hep-th/0312033](https://arxiv.org/abs/hep-th/0312033), [doi:10.1088/1126-6708/2004/03/016](https://doi.org/10.1088/1126-6708/2004/03/016)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/double+dimensional+reduction/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/double+dimensional+reduction/29">v29</a>, <a href="https://ncatlab.org/nlab/show/double+dimensional+reduction">current</a>

    Added the remark that the differential geometry of the double dimensional reduction of the M2-brane- and M5-brane-charges was maybe first clearly written out in:

    • Varghese Mathai, Hisham Sati, §4 in: Some Relations between Twisted K-theory and Gauge Theory, J. High Energ. Phys. 2004 03 (2004) 016 [arXiv:hep-th/0312033, doi:10.1088/1126-6708/2004/03/016]

    diff, v29, current

11. • CommentRowNumber11.
    • CommentAuthorperezl.alonso
    • CommentTimeOct 6th 2023
    • (edited Oct 6th 2023)
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    Author: perezl.alonso
    Format: MarkdownItexOne can regard Proposition 2.3 to be a statement about the double-dimensional reduction of $Fields$. But there should be a more general statement concerning the reduction of theories. For instance, if $Fields$ carries a [[prequantum circle n-bundle]] (e.g. take $Fields =BG_{conn}$ equipped with a [[characteristic class]], say the Chern-Simons 3-bundle to $B^3U(1)$), then one should be able to do double-dimensional reduction not only of the fields but also of the corresponding theory. This I guess will involve some statement similar to Proposition 2.3 involving a more general [[comma category]]. Is this sorted out anywhere? I'm particularly asking because there have been some recent papers arguing one can get [[generalized symmetry|Symmetry TFT's]] from doing dimensional reduction "of" [[M-theory]], so it would be nice to make this precise.

    One can regard Proposition 2.3 to be a statement about the double-dimensional reduction of . But there should be a more general statement concerning the reduction of theories. For instance, if carries a prequantum circle n-bundle (e.g. take equipped with a characteristic class, say the Chern-Simons 3-bundle to ), then one should be able to do double-dimensional reduction not only of the fields but also of the corresponding theory. This I guess will involve some statement similar to Proposition 2.3 involving a more general comma category. Is this sorted out anywhere? I’m particularly asking because there have been some recent papers arguing one can get Symmetry TFT’s from doing dimensional reduction “of” M-theory, so it would be nice to make this precise.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2023
    • (edited Oct 7th 2023)
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    Author: Urs
    Format: MarkdownItexJust to note that the double dimensional reduction by cyclification does not involve truncation: The space/stack of dd-reduced fields is still equivalent to that of the original fields (the "reduction/oxidation" equivalence). In that sense there is no necessary cause to change the prequantum $\infty$-bundle. And if one does want to truncate some of the KK-modes, that would typically amount to restricting to a sub-stack of the dd-reduced fields, for which the appropriate prequantum bundle would just be the restriction (pullack along the inclusion) of the original one.

    Just to note that the double dimensional reduction by cyclification does not involve truncation:

    The space/stack of dd-reduced fields is still equivalent to that of the original fields (the “reduction/oxidation” equivalence).

    In that sense there is no necessary cause to change the prequantum -bundle.

    And if one does want to truncate some of the KK-modes, that would typically amount to restricting to a sub-stack of the dd-reduced fields, for which the appropriate prequantum bundle would just be the restriction (pullack along the inclusion) of the original one.

13. • CommentRowNumber13.
    • CommentAuthorperezl.alonso
    • CommentTimeApr 4th 2024
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    Author: perezl.alonso
    Format: MarkdownItexIs it described anywhere how this process leads to getting WZW from CS on a $\Sigma\times S^1$ worldvolume? That is, how one uses this to go from $\Sigma\times S^1\to BG \to B^3 U(1)_{conn}$ to $\Sigma\to G\to B^2 U(1)_{conn}$ where in the first case one has the CS 2-gerbe and in the second case it is the WZW 1-gerbe?

    Is it described anywhere how this process leads to getting WZW from CS on a worldvolume? That is, how one uses this to go from to where in the first case one has the CS 2-gerbe and in the second case it is the WZW 1-gerbe?

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 5th 2024
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    Author: Urs
    Format: MarkdownItexYes, this is pp. 40 in [arXiv:1304.0236](https://arxiv.org/abs/1304.0236), following [arXiv:1207.5449](https://arxiv.org/abs/1207.5449) (pp. 3) and [arXiv:0410013](https://arxiv.org/abs/math/0410013).

    Yes, this is pp. 40 in arXiv:1304.0236, following arXiv:1207.5449 (pp. 3) and arXiv:0410013.

15. • CommentRowNumber15.
    • CommentAuthorperezl.alonso
    • CommentTimeApr 5th 2024
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    Author: perezl.alonso
    Format: MarkdownItexThanks. A couple of questions. If you start with an action $G\to \text{Aut}(\nabla_{CS})$ of a higher group $G$ on Chern-Simons, what happens to this action when passing to WZW? And also, in p.41 it is mentioned that the sections of the WZW gerbe are twisted unitary bundles, referencing Example 2.5.4 for this, which does not talk about gerbes with connection. But isn't the fact that the sections of a line bundle gerbe with connection $\mathcal{G}$ can be expressed as bundle gerbe morphisms from a trivial gerbe $\mathcal{I}_{\omega}$ to $\mathcal{G}$, that is as twisted vector bundles with connection, only valid for torsion Dixmier-Douady classes?

    Thanks. A couple of questions. If you start with an action of a higher group on Chern-Simons, what happens to this action when passing to WZW? And also, in p.41 it is mentioned that the sections of the WZW gerbe are twisted unitary bundles, referencing Example 2.5.4 for this, which does not talk about gerbes with connection. But isn’t the fact that the sections of a line bundle gerbe with connection can be expressed as bundle gerbe morphisms from a trivial gerbe to , that is as twisted vector bundles with connection, only valid for torsion Dixmier-Douady classes?

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 5th 2024
    • PermaLink
    Author: Urs
    Format: MarkdownItexRight, the twisted bundles discussed there are finite-rank, hence the corresponding gerbe must have torsion DD-class. But this is the case after pullback to D-brane worldvolume submanifolds $Q \hookrightarrow G$ as considered there, if by Freed-Witten anomaly cancellation and assuming $\mathrm{Spin}^c$-structure, the restriction of the class of the gerbe to the brane worldvolume $Q$ is torsion. On the other hand, in a more sophisticated perspective the quantization is anyways described directly by pushforward in K-theory twisted by that gerbe, as in section 5.2.4 of Joost Nuiten's thesis ([pp. 115](https://ncatlab.org/schreiber/files/thesisNuiten.pdf#page=115)).

    Right, the twisted bundles discussed there are finite-rank, hence the corresponding gerbe must have torsion DD-class.

    But this is the case after pullback to D-brane worldvolume submanifolds as considered there, if by Freed-Witten anomaly cancellation and assuming -structure, the restriction of the class of the gerbe to the brane worldvolume is torsion.

    On the other hand, in a more sophisticated perspective the quantization is anyways described directly by pushforward in K-theory twisted by that gerbe, as in section 5.2.4 of Joost Nuiten’s thesis (pp. 115).

17. • CommentRowNumber17.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 19th 2025
    • (edited Feb 19th 2025)
    • PermaLink
    Author: perezl.alonso
    Format: MarkdownItexIf one has a cocycle $X \times S^1 \to \mathbf{B} G$ for $G$ a finite group, how is this reduced to a cocycle on $X$? I understand topologically the free loop space of $BG$ is $G//_{ad} G$ but what about the cyclic one, now that I am working with cocycles?

    If one has a cocycle for a finite group, how is this reduced to a cocycle on ? I understand topologically the free loop space of is but what about the cyclic one, now that I am working with cocycles?

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2025
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    Author: Urs
    Format: MarkdownItexGood question. This is answered in *[[schreiber:Cyclification of Orbifolds|Cyclification of Orbifolds]]*: The object you are after is there called $Cyc_{S^1_{coh}}(\mathbf{B}G)$ and the main theorem says that this is equivalent to *[[Huan's inertia orbifold]]* of $\mathbf{B}G$.

    Good question. This is answered in Cyclification of Orbifolds: The object you are after is there called and the main theorem says that this is equivalent to Huan’s inertia orbifold of .

19. • CommentRowNumber19.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 19th 2025
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    Author: perezl.alonso
    Format: MarkdownItexSo if $X$ in #17 is connected, how does the cocycle $X\times S^1\to \mathbf{B}G$ determine to which connected component of $\Lambda_{S^1}\mathbf{B}G = \coprod_{[g]\in G/G} \mathbf{B}\Lambda_g$ $X$ maps to?

    So if in #17 is connected, how does the cocycle determine to which connected component of maps to?

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeFeb 20th 2025
    • PermaLink
    Author: Urs
    Format: MarkdownItexBy adjointness, see the review [here](https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#eq:GeneralAdjointnessForDoubleDimReduction). In your special case of the trivial $S^1$-fibration you can in fact just use just the ordinary hom-adjunction, see [here](https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#via_free_looping_no_0brane_effect): At each point of $X$ you already manifestly have a map $S^1 \to \mathbf{B}G$ which is a map $\mathbf{B}\mathbb{Z} \to \ \mathbf{B}G$, which is an object of $\Lambda \mathbf{B}G$, which gives an object in $\Lambda_{S^1} \mathbf{B}G$.

    By adjointness, see the review here. In your special case of the trivial -fibration you can in fact just use just the ordinary hom-adjunction, see here: At each point of you already manifestly have a map which is a map , which is an object of , which gives an object in .