nLab
metaplectic structure
Context
Symplectic geometry
symplectic geometry

higher symplectic geometry

Background
geometry

differential geometry

Basic concepts
almost symplectic structure , metaplectic structure , metalinear structure

symplectic form , n-plectic form

symplectic Lie n-algebroid

symplectic infinity-groupoid

symplectomorphism , symplectomorphism group

Hamiltonian action , moment map

symplectic reduction , BRST-BV formalism

isotropic submanifold , Lagrangian submanifold , polarization

Classical mechanics and quantization
Hamiltonian mechanics

quantization

deformation quantization ,

geometric quantization , higher geometric quantization

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Cohomology
cohomology

Special and general types Special notions Variants Operations Theorems
Contents
Idea
For $(X, \omega)$ a symplectic manifold a metaplectic structure on $X$ is a G-structure for $G$ the metaplectic group , hence a lift of structure groups of the tangent bundle from the symplectic group to the metaplectic group through the double cover map $Mp(2n, \mathbb{R}) \to Sp(2n, \mathbb{R})$ :

$\array{
&& \mathbf{B}Mp(2n, \mathbb{R})
\\
& {}^{\mathllap{metaplectic \atop structure}}\nearrow & \downarrow
\\
X &\stackrel{T X}{\to}& \mathbf{B} Sp(2n, \mathbb{R})
}
\,.$

Analogously for the Mp^c -group one considers $Mp^c$ -structures .

Properties
(Bates-Weinstein, theorem 7.16 )

Existence of $Mp^c$ -structures
(Robinson-Rawnsley 89, theorem (6.2) )

For more details, see at metaplectic group – (Non-)Triviality of Extensions .

References
Michael Forger , Harald Hess, Universal metaplectic structures and geometric quantization , Comm. Math. Phys. Volume 64, Number 3 (1979), 269-278. (EUCLID )

R. J. Plymen, The Weyl bundle , Journal of Functional Analysis 49, 186-197 (1982) (journal )

P. L. Robinson, John Rawnsley , The metaplectic representation, $Mp^c$ -structures and geometric quantization , 1989

Michel Cahen, Simone Gutt , John Rawnsley , Symplectic Dirac Operators and $Mp^c$ -structures (arXiv:1106.0588 )

Sean Bates, Alan Weinstein , Lectures on the geometry of quantization , (pdf )

Last revised on January 21, 2015 at 23:35:06.
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