n-plectic form


Symplectic geometry

Higher geometry



The notion of nn-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 XX a smooth manifold and nn \in \mathbb{N}, n1n \geq 1, a differential form ω\omega on XX is nn-plectic if

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

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

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

    ι ()ω:Γ(TX)Ω n(X) \iota_{(-)}\omega \;\colon\; \Gamma(T X) \to \Omega^{n}(X)

    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-nn-plectic form. Of course this is just a closed (n+1)(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 XX 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

Revised on February 23, 2015 22:06:09 by Urs Schreiber (