# nLab metaplectic structure

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

For $\left(X,\omega \right)$ a symplectic manifold a metaplectic structure on $X$ is a lift of structure groups of the tangent bundle from the symplectic group to the metaplectic group through the double cover map $\mathrm{Mp}\left(2n,ℝ\right)\to \mathrm{Sp}\left(2n,ℝ\right)$:

$\begin{array}{ccc}& & B\mathrm{Mp}\left(2n,ℝ\right)\\ & {}^{\genfrac{}{}{0}{}{\mathrm{metaplectic}}{\mathrm{structure}}}↗& ↓\\ X& \stackrel{TX}{\to }& B\mathrm{Sp}\left(2n,ℝ\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && \mathbf{B}Mp(2n, \mathbb{R}) \\ & {}^{\mathllap{metaplectic \atop structure}}\nearrow & \downarrow \\ X &\stackrel{T X}{\to}& \mathbf{B} Sp(2n, \mathbb{R}) } \,.

Revised on July 10, 2012 18:35:58 by David Corfield (129.12.18.29)