nLab metaplectic structure

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.

(Bates-Weinstein, theorem 7.16)

Existence of $Mp^c$-structures

Theorem

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

(Robinson-Rawnsley 89, theorem (6.2))

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

Related concepts

• metaplectic group

• spin structure

• metaplectic correction (in geometric quantization)

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)