presymplectic structure



A presymplectic structure on a smooth manifold or more generally on a smooth space XX is simply a closed differential 2-form ωΩ 2(X)\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 XX.

Given a presymplectic structure, the quotient of XX by the flow of the vector fields in the kernel of ω\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.


Ambient symplectic manifold

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


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


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)

Revised on September 21, 2017 17:58:05 by Urs Schreiber (