nLab
metaplectic structure

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Context

Symplectic geometry

symplectic geometry

higher symplectic geometry

Background

Basic concepts

Classical mechanics and quantization

Contents

Idea

For (X,ω) a symplectic manifold a metaplectic structure on X is a lift of structure groups of the tangent bundle from the symplectic group to the metaplectic group through the double cover map Mp(2n,ℝ)→Sp(2n,ℝ):

BMp(2n,ℝ) metaplecticstructure↗ ↓ X →TX BSp(2n,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}) } \,.

Revised on July 10, 2012 18:35:58 by David Corfield (129.12.18.29)
Edit | Back in time (1 revision) | See changes | History | Views: Print | TeX | Source | Linked from: BV-BRST formalism, Courant algebroid, AKSZ sigma-model, symplectic geometry, geometric quantization, symplectic vector space, symplectic manifold, Poisson manifold, spin structure, symplectic groupoid, symplectic Lie n-algebroid, Poisson Lie algebroid, multisymplectic geometry, Maslov index, Poisson Lie group, symplectomorphism, lagrangian submanifold, deformation quantization, n-plectic geometry, Heisenberg group, Poisson n-algebra, Lagrangian subspace, isotropic subspace, presymplectic structure, isotropic submanifold, moment map, polarization, symplectic field theory, symplectic geometry - contents, higher symplectic geometry, symplectic infinity-groupoid, geometric quantization of symplectic groupoids, Hamiltonian action, Hamiltonian vector field, Bohr-Sommerfeld leaf, quantum observable, reduction and lift of structure groups, 2-plectic geometry, metaplectic structure, canonical momentum, metaplectic representation, contact manifold, symplectic leaf, symplectomorphism group, symplectic reduction, metaplectic group, metaplectic correction (in geometric quantization), C-star algebraic deformation quantization, metalinear group, metalinear structure, Heisenberg Lie n-algebra, Weinstein symplectic category, Heisenberg n-group, Hamiltonian form, geometry of physics, Dirac manifold, Liouville integrable system, Poisson bracket Lie n-algebra, Lie-Poisson structure, geometric quantization of non-integral forms, Guillemin-Sternberg geometric quantization conjecture, Moyal quantization, symplectic connection, Lagrangian Grassmannian, Poisson geometry, symplectic realization, prequantum geometry, Kostant-Souriau extension, Poisson tensor, Poisson cohomology