nLab
n-plectic form

Idea

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 multisymplectic geometry.

Definition

Definition

For $X$ a smooth manifold and $n\in \mathbb{N}$, $n\ge 1$, a differential form $\omega$ on $X$ is $n$-plectic if

1. it is an $n$-forms, $\omega \in {\Omega }^n(X)$;

2. it is closed: $d_{\mathrm{dR}}\omega =0$;

3. it is non-degenerate in that the contraction map

   $${\iota }_{(-)}\omega :\Gamma (TX)\to {\Omega }^{n-1}(X)$$\iota_{(-)}\omega : \Gamma(T X) \to \Omega^{n-1}(X)

   has trivial kernel.

Remark

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.

References

See the references at multisymplectic geometry.

For instance definition 2.1 in

• Chris Rogers, Higher symplectic geometry PhD thesis (arXiv).