# nLab metaplectic structure

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

cohomology

# 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

### Relation to metalinear structure

###### Theorem

Let $(X,\omega)$ be a symplectic manifold and $L \subset T X$ a subbundle of Lagrangian subspaces of the tangent bundle. Then $T X$ admits a metaplectic structure precisely if $L$ admits a metalinear structure.

### Existence of $Mp^c$-structures

###### Theorem

Every Sp-principal bundle has a lift to an Mp^c-principal bundle.

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

## References

Revised on January 21, 2015 23:35:06 by Urs Schreiber (88.100.66.95)