# nLab presymplectic structure

Contents

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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

Reductions of (pre-)symplectic manifolds:

symplectic geometryphysics
presymplectic manifoldcovariant phase space
$\downarrow$ gauge reduction$\downarrow$ quotient by gauge symmetry
symplectic manifoldreduced phase space
$\downarrow$ symplectic reduction$\downarrow$ quotient by global symmetry
symplectic manifoldreduced 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)

Last revised on September 21, 2017 at 17:58:05. See the history of this page for a list of all contributions to it.