nLab
metaplectic structure

Context

Symplectic geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

For (X,ω)(X, \omega) a symplectic manifold a metaplectic structure on XX is a G-structure for GG 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,)Sp(2n,)Mp(2n, \mathbb{R}) \to Sp(2n, \mathbb{R}):

BMp(2n,) metaplecticstructure X TX BSp(2n,). \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 cMp^c-structures.

Properties

Relation to metalinear structure

Theorem

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

(Bates-Weinstein, theorem 7.16)

Existence of Mp cMp^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.

References

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