Metaplectic structure

Definition

A metaplectic structure on a symplectic manifold is a lifting of its gluing cocycle to the symplectic group $Sp_{2}n(\mathbb {R} )$, to a gluing cocycle to the metaplectic group $Mp_{2n}(\mathbb {R} )$. The metaplectic group is the double cover of the symplectic group.

Equivalently, it can be viewed as realizing a symplectic manifold as a principal $Mp_{2n}(\mathbb {R} )$-bundle that gives rise to its given realization as a $Sp_{2n}(\mathbb {R} )$-bundle.

Related notions

• Spin structure to an oriented Riemannian manifold is very similar: it lifts from the special orthogonal group to its double cover, the spin group.