In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define the symplectic spinor bundle, which is the Hilbert space bundle associated to the metaplectic structure via the metaplectic representation, giving rise to the notion of a symplectic spinor field in differential geometry.

Symplectic spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in establishing the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for symplectic spin geometry.

Formal definition

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A metaplectic structure [1] on a symplectic manifold   is an equivariant lift of the symplectic frame bundle   with respect to the double covering   In other words, a pair   is a metaplectic structure on the principal bundle   when

a)   is a principal  -bundle over  ,
b)   is an equivariant  -fold covering map such that
  and   for all   and  

The principal bundle   is also called the bundle of metaplectic frames over  .

Two metaplectic structures   and   on the same symplectic manifold   are called equivalent if there exists a  -equivariant map   such that

  and   for all   and  

Of course, in this case   and   are two equivalent double coverings of the symplectic frame  -bundle   of the given symplectic manifold  .

Obstruction

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Since every symplectic manifold   is necessarily of even dimension and orientable, one can prove that the topological obstruction to the existence of metaplectic structures is precisely the same as in Riemannian spin geometry.[2] In other words, a symplectic manifold   admits a metaplectic structures if and only if the second Stiefel-Whitney class   of   vanishes. In fact, the modulo   reduction of the first Chern class   is the second Stiefel-Whitney class  . Hence,   admits metaplectic structures if and only if   is even, i.e., if and only if   is zero.

If this is the case, the isomorphy classes of metaplectic structures on   are classified by the first cohomology group   of   with  -coefficients.

As the manifold   is assumed to be oriented, the first Stiefel-Whitney class   of   vanishes too.

Examples

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Manifolds admitting a metaplectic structure

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  • Phase spaces     any orientable manifold.
  • Complex projective spaces     Since   is simply connected, such a structure has to be unique.
  • Grassmannian   etc.

See also

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Notes

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  1. ^ Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0 page 35
  2. ^ M. Forger, H. Hess (1979). "Universal metaplectic structures and geometric quantization" (PDF). Commun. Math. Phys. 64: 269–278. doi:10.1007/bf01221734.

References

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