nLab presymplectic structure

Definition

A presymplectic structure on a smooth manifold or more generally on a smooth space $X$ is simply a closed differential 2-form $\omega \in \Omega^2(X)$.

(In parts of the literature a pre-symplectic 2-form is required to have constant rank. However, other parts of the literature do not require this, e.g. Bottacin, def. 3.1. )

If the 2-form is happens to be non-degenerate (have maximal rank) then it is a symplectic structure on $X$.

Given a presymplectic structure, the quotient of $X$ by the flow of the vector fields in the kernel of $\omega$ is, if it exists in a reasonable way, a symplectic manifold.

One speaks of closed 2-forms as presymplectic structures if one is interested in eventually forming this quotient and obtaining a symplectic structure.

The central application of this appears in the theory of quantization of action functionals. The covariant phase space of a local action functional is canonically presymplectic, and one is interested in its quotientient by symmetries to obtain a symplectic structure. This quotient generically is very ill behaved, though, when taken in the naive way. The BV-BRST formalism is all about forming this quotient “up to homotopy”, such that it exists in a reasonable way. See derived critical locus for more on this.

The notion of presymplectic structure is a weakening of the notion of symplectic structure roughly orthogonal to the notion of Poisson structure.

Properties

Ambient symplectic manifold

Under mild technical conditions, presymplectic manifolds arise as submanifolds of ambient symplectic manifolds. See (EMR, theorem 3).

Examples

• covariant phase space

Related concepts

• symplectic structure, almost symplectic structure

• geometric quantization, geometric quantization of non-integral forms

Reductions of (pre-)symplectic manifolds:

symplectic geometry	physics
presymplectic manifold	covariant phase space
$\downarrow$ gauge reduction	$\downarrow$ quotient by gauge symmetry
symplectic manifold	reduced phase space
$\downarrow$ symplectic reduction	$\downarrow$ quotient by global symmetry
symplectic manifold	reduced phase space

References

The generalization of symplectic reduction for presymplectic manifolds, presymplectic reduction is discussed in

• Francesco Bottacin, A Marsden-Weinstein reduction theorem for presymplectic manifold (pdf)

• A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy, Reduction of Presymplectic Manifolds with Symmetry (arXiv:math-ph/9911008)

The geometric quantization of presymplectic manifolds by geometric quantization by push-forward is discussed in

• Ana Canas da Silva, Yael Karshon, Susan Tolman, Quantization of Presymplectic Manifolds and Circle Actions, Trans. Amer. Math. Soc. 352 (2000), 525-552 (arXiv:dg-ga/9705008)