Given a presymplectic manifold $(X,\omega)$, the quotient (if it exists suitably) by the flow of the kernel of the presymplectic form $\omega : T X \to T^* X$ is the symplectic manifold obtained by *gauge reduction*. Also called *presymplectic reduction*

In the interpretation in physics this takes one from a covariant phase space to a *reduced phase space*.

In constrast, in *symplectic reduction* one takes the quotient of a pre-specified Hamiltonian action *and* restricts to the 0-locus of the corresponding Hamiltonians (the momentum map).

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 |

Around page 11 of

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

Last revised on September 16, 2013 at 01:56:04. See the history of this page for a list of all contributions to it.