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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.
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
$\langle D q \wedge D p\rangle$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 $G$-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.
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 $\mathbf{d}q^i \wedge \mathbf{d}p_i$, 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 $2n$ vector bundle with affine connection $D$?
In the latter case, if you have a hermitean rank-$2n$ vector bundle with connection $D$, then for $\phi$ a section, the expression $\langle D \phi \wedge D \phi\rangle$ is the Lagrangian for a “topological Higgs-like field” $\phi$. It would be the actual kinetic Lagrangian if there is also a (pseudo-)Riemannian structure on the base space and you’d consider $\langle D \phi \wedge \star D \phi\rangle$, whith $\star$ the corresponding Hodge-operator.
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