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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2013
    • (edited May 13th 2013)

    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.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2013

    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?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2013
    • (edited May 13th 2013)

    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 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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2015

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

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2017

    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.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 21st 2017

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

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2017
    • (edited Feb 14th 2017)

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

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

    where

    Hence for

    then there is a natural equivalence

    H(X^,A)originalfluxesoxidationreductionH(X,[G,A]/G)doublydimensionally reducedfluxes \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

    (X^A)(X [G,A]/G BG) \left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right)
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTime5 days ago
    • (edited 5 days ago)

    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

    diff, v17, current

  1. Typo fixed in link.

    diff, v18, current

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