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nLab > Latest Changes: double dimensional reduction

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  1.  
    • 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,−]/G⟵hofibH /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)⏟originalfluxes≃⟵oxidation⟶reductionH(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
    • CommentTimeJan 11th 2019
    • (edited Jan 11th 2019)

    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

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 11th 2019

    Typo fixed in link.

    diff, v18, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2022

    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:

    diff, v29, current

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