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 n-plectic geometry/multisymplectic geometry.

Definition

Definition

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

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

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

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

   $$\iota_{(-)}\omega \;\colon\; \Gamma(T X) \to \Omega^{n}(X)$$

   has trivial kernel.

Remark

An 1-plectic form is equivalently a symplectic form.

Remark

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.

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 n-plectic geometry and at multisymplectic geometry.

For instance definition 2.1 in

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

See also

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)