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.