Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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
    • 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

    • CommentRowNumber11.
    • CommentAuthorperezl.alonso
    • CommentTimeOct 6th 2023
    • (edited Oct 6th 2023)

    One can regard Proposition 2.3 to be a statement about the double-dimensional reduction of FieldsFields. But there should be a more general statement concerning the reduction of theories. For instance, if FieldsFields carries a prequantum circle n-bundle (e.g. take Fields=BG connFields =BG_{conn} equipped with a characteristic class, say the Chern-Simons 3-bundle to B 3U(1)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 Symmetry TFT’s from doing dimensional reduction “of” M-theory, so it would be nice to make this precise.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2023
    • (edited Oct 7th 2023)

    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 \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.