nLab
metalinear structure
Context
Symplectic geometry
symplectic geometry

higher symplectic geometry

Background
geometry

differential geometry

Basic concepts
almost symplectic structure , metaplectic structure , metalinear structure

symplectic form , n-plectic form

symplectic Lie n-algebroid

symplectic infinity-groupoid

symplectomorphism , symplectomorphism group

Hamiltonian action , moment map

symplectic reduction , BRST-BV formalism

isotropic submanifold , Lagrangian submanifold , polarization

Classical mechanics and quantization
Hamiltonian mechanics

quantization

deformation quantization ,

geometric quantization , higher geometric quantization

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Contents
Definition
A metalinear structure on a smooth manifold of dimension $n$ is a lift of the structure group of the tangent bundle along the group extension $Ml(n) \to GL(n)$ of the general linear group by the metalinear group .

Properties
Obstruction and existence
A metalinear structure on a manifold $Q$ of dimension $n$ exists precisely if the Chern class of the canonical bundle $\wedge^n T^*Q$ is divisible by 2. So a metalinear structure is equivalent to the existence of a square root line bundle $\sqrt{\wedge^n T^* Q}$ ( Theta characteristic ).

This means that for $E \to Q$ any hermitean line bundle , sections of the tensor product $E \otimes \sqrt{\wedge^n T^* Q}$ have a canonical inner product (if $Q$ is compact and orientable). This is the use of metalinear structure in metaplectic correction .

(Bates-Weinstein, theorem 7.16 )

The following table lists classes of examples of square roots of line bundles

References
Lecture notes include

Discussion with an eye towards Theta characteristics is in

Last revised on January 2, 2015 at 19:47:59.
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