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    • CommentRowNumber1.
    • CommentAuthorcfgauss
    • CommentTimeAug 2nd 2010
    • (edited Aug 2nd 2010)

    I’m a physics grad student and I’ve been looking through Daniel Freed’s lectures in QFT and strings: a course for mathematicians, and had some notation questions. Trying to google to find help seems to only lead me to an n-lab page (http://ncatlab.org/nlab/show/curvature+characteristic+form), and Daniel Freeds web page and papers, which didn’t really help me, so I thought this would be a good place to ask!

    In his lectures on classical field theories, he gives the example of a lagrangian of classical mechanics, and writes the symplectic 2-form we get out of the action as ω=δγ=δxδx where γ=langlex,δx and .,. is the inner product.

    But I do not understand what the notation AA could mean? If I vary gamma I would think I should get: δx,δx+x,δ2x=δx,δx which looks almost like the \omega above except for a wedge instead of a comma!

    But this notation is used again later several times, e.g., in defining the chern-simons action as S=AF16A[AA] when the usual form looks similar but with a +23 and a trace instead of the brackets. It also seems that 1/2[AA]=[A,A]? Where does this come from?

    I don’t see any explanation of either of these notations and don’t recall seeing it in any of my differential geometry books.

    The link above claims <.> is used for the “curvature characteristic form” but that’s not something I’ve ever heard of, and it doesn’t seem to really define it… So does anyone know of any texts / resources where this is defined? Or is there a simple definition in terms of the inner product as the notation would indicate?

    • CommentRowNumber2.
    • CommentAuthorEric
    • CommentTimeAug 2nd 2010

    Hi cfgauss,

    I don’t have an immediate answer to your question, just a note about formatting.

    If you edit your comment, throw some dollar signs around your tex commands and click “Markdown+Itex”, it should render as math.

    Gotta run for now!