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Atrium > Mathematics, Physics & Philosophy: twisted symplectic structures

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    • CommentRowNumber1.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 21st 2013
    Where is there written a twisted simplectic form like
    pDq whee D is a covariant rather than exterior derivative?
    • CommentRowNumber2.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 21st 2013

    Hi - I’d imagine something like this arises when thinking about electromagnetism (more generally, the Yang-Mills equations) from the symplectic point of view? I’m thinking of chapter III “Motion in a Yang-Mills field and the principle of general covariance” in Guillemin and Sternberg, Symplectic techniques in physics.

    • CommentRowNumber3.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 22nd 2013
    That sounds quite likely, but I don't have G&S here. Anyone with a copy who would check it out or
    a later arXiv reference I could follow?
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2013
    • (edited Sep 24th 2013)

    Jim,

    what you write down in #1 is a 1-form. I suspect you want that to be the analog of a symplectic potential, not of a symplectic 2-form.

    So what’s the form of a “twisted symplectic 2-form” that you have in mind? I gather you are imaginging an affine connection and a covariant derivative. Now you need some invariant bilinear pairing ⟨−,−⟩\langle -,-\rangle to produce a 2-form of the schematic form

    ⟨Dq∧Dp⟩ \langle D q \wedge D p\rangle

    I suppose. In this form this immediately reminds one of the symplectic form on the phase space of Chern-Simons theory (e.g. here). That de-transgresses to a 3-plectic form on the moduli of GG-principal connections, and that does involve a covariant derivative of the above form.

    But I can’t tell if this is going in the direction that you are after. Maybe you could give more details on what you have in mind.

    • CommentRowNumber5.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 23rd 2013
    Yes, of the schematic form

    ⟨Dq∧Dp⟩

    where does such appear in the litt?
    sorry for confusing the potential with the form
    I still don't speak this lingo fluently
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2013
    • (edited Sep 24th 2013)

    Hard to say where this appears, since we might still have to say better what “this” is.

    By Darboux’s theorem, an ordinary symplectic form is guaraneteed to locally look like dq i∧dp i\mathbf{d}q^i \wedge \mathbf{d}p_i, so it doesn’t appear there.

    The example of Chern-Simons theory which I mentioned is not an example of the kind you want?

    So I am not sure. But can you maybe say more in detail what you are after? Ar you looking at a hermitean rank 2n2n vector bundle with affine connection DD?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2013
    • (edited Sep 24th 2013)

    In the latter case, if you have a hermitean rank-2n2n vector bundle with connection DD, then for ϕ\phi a section, the expression ⟨Dϕ∧Dϕ⟩\langle D \phi \wedge D \phi\rangle is the Lagrangian for a “topological Higgs-like field” ϕ\phi. It would be the actual kinetic Lagrangian if there is also a (pseudo-)Riemannian structure on the base space and you’d consider ⟨Dϕ∧⋆Dϕ⟩\langle D \phi \wedge \star D \phi\rangle, whith ⋆\star the corresponding Hodge-operator.

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