higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
The notion of $n$-plectic form is a generalization of the notion of symplectic form to differential forms of more than two arguments. This is considered in higher symplectic geometry, specifically: in n-plectic geometry/multisymplectic geometry.
For $X$ a smooth manifold and $n \in \mathbb{N}$, $n \geq 1$, a differential form $\omega$ on $X$ is $n$-plectic if
it is an $(n+1)$-form, $\omega \in \Omega^{n+1}(X)$;
it is closed: $d_{dR} \omega = 0$;
it is non-degenerate in that the contraction map
has trivial kernel.
An 1-plectic form is equivalently a symplectic form.
If the last condition is dropped, then by analogy with presymplectic forms one may speak of a pre-$n$-plectic form. Of course this is just a closed $(n+1)$-form, but as in the presymplectic case, the plectic-terminology indicates that one wants to regard it as input datum for higher geometric quantization.
This definition has an evident generalization to the case where also $X$ is allowed to be a “higher” generalization of a smooth manifold, namely a smooth ∞-groupoid or L-∞ algebroid. See higher symplectic geometry for more on this case.
See the references at n-plectic geometry and at multisymplectic geometry.
For instance definition 2.1 in
See also