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am writing stuff about "covariant field theory" into multisymplectic geometry, following this very useful reference.
at multisymplectic geometry and covariant phase space I have added pointers to the the recent article by Frédéric Hélein here that around its equations (21)-(25) has a discussion of how the ordinary canonical symplectic structure on covariant phase space is induced by what should be the right version of transgression of the canonical multisymplectic form.
I went through the first two subsections (up to and excluding the examples) of the section Extended phase spaces in covariant field theory and tried to polish, prettify, and clarify the discussion a bit more. I didn’t change any actual content, but I think it is better readable now.
(The rest of the entry can do woth cleanup, too, maybe I find time for more…)
started a new section
which states (and shows) that the canonical multisymplectic form is equal to the “covariant symplectic potential current density” that comes out of the covariant phase space formalism – according to this remark there – for a free field theory Lagrangian (and hence for any standard Lagrangian with standard kinetic and potential term).
(If anyone can give me a pointer to the literature of this precise statement, I’ll add it. Right now I have trouble finding such…, basic as the statement is.)
concerning the pointer: thanks to Igor Khavkine for reminding me (once again): it’s in math-ph/0408008 around p. 25 (somewhat hidden maybe, but it’s there).
added a section on the field equations
with emphasis on the way that Forger-Romero neatly relate the multisymplectic de Donder-Weyl field equations to the Euler-Lagrange equations.
Special thanks to Igor Khavkine for discussion of this point, and for persistent pointers to the literature.
Chris Rogers, $L_\infty$-algebras from multisymplectic geometry, Lett. Math. Phys., 100(1):29–50 (2012); 2-plectic geometry, Courant algebroids, and categorified prequantization, J. Symplectic Geom. 11(1):53–91 (2013)
Antonio Michele Miti, Marco Zambon, Observables on multisymplectic manifolds and higher Courant algebroids, arXiv:2209.05836
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