# nLab n-plectic form

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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 $n \in \mathbb{N}$, $n \geq 1$, 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: $d_{dR} \omega = 0$;

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

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

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

Created on September 2, 2011 11:55:33 by Urs Schreiber (131.211.238.38)