# nLab gauge reduction

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

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 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

Around page 11 of

• A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy, Reduction of Presymplectic Manifolds with Symmetry (arXiv:math-ph/9911008)
Revised on September 16, 2013 01:56:04 by Urs Schreiber (89.204.139.122)