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nLab > Latest Changes: D=5 Maxwell theory

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2020
    • (edited May 4th 2020)

    starting something, for the moment mainly to give a home to relation between 5d Maxwell theory and self-dual 3-forms in 6d – section

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2020

    added a section “Relation between 5d Maxwell theory and massive vector mesons in 4d” (here) spelling out a key step in

    (but not the full construction there, and also using some other assumptions, as I have highlighted in the text – maybe more later)

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2020
    • (edited May 6th 2020)

    I am starting a subsection “Relation to massive vector mesosns in 4d – The Sakai-Sugimoto model” (here), following Sutcliffe.

    So far I have added a (somewhat lengthy) proof (here) that the 5d vacuum Maxwell equations on the Sakai-Sugimoto spacetime

    g≔H +1η μνdx μ⊗dx ν+H −1dz⊗dz g \;\coloneqq\; H^{+1} \, \eta_{\mu \nu} d x^\mu \otimes d x^\nu + H^{-1} d z \otimes d z

    with

    H({x μ},z)≔H(z)≔(1+z 2L 2) 2/3 H(\{x^\mu\}, z) \;\coloneqq\; H(z) \;\coloneqq\; \left( 1 + \frac{z^2}{L^2} \right)^{2/3}

    are equivalent to the following massive 4d wave equation:

    ⋆d⋆dA=0 ⇔ η μν∂ μ∂ νA+H 1/2∂ z(H 3/2∂ zA)=0 \begin{aligned} & \star d \star d A \;=\; 0 \\ \Leftrightarrow \;\;\; & \eta^{\mu \nu}\partial_\mu \partial_\nu A \;+\; H^{1/2} \partial_z \left( H^{3/2} \partial_z A \right) \;=\; 0 \end{aligned}

    This statement is sort of in the references cited, but in a somewhat roundabout manner. I wanted to write out an explicit statement and proof.

    diff, v7, current

1 to 3 of 3

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