nLab
n-plectic form

Context

Symplectic geometry

Higher geometry

Contents

Idea

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.

Definition

Definition

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.

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-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.

Remark

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.

References

See the references at n-plectic geometry and at multisymplectic geometry.

For instance definition 2.1 in

Revised on January 29, 2014 01:44:55 by Urs Schreiber (89.204.138.164)