Metaplectic structure

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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[edit]

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[edit]

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[edit]

Manifolds admitting a metaplectic structure[edit]

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

See also[edit]

Notes[edit]

  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". Commun. Math. Phys. 64: 269–278. doi:10.1007/bf01221734. 

References[edit]