Metaplectic structure

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.
A metaplectic structure on a symplectic manifold formula_1 is an equivariant lift of the symplectic frame bundle formula_2 with respect to the double covering formula_3 In other words, a pair formula_4 is a metaplectic structure on the principal bundle formula_2 when
The principal bundle formula_6 is also called the bundle of metaplectic frames over formula_8.
Two metaplectic structures formula_17 and formula_18 on the same symplectic manifold formula_1 are called equivalent if there exists a formula_7-equivariant map formula_21 such that
Of course, in this case formula_26 and formula_27 are two equivalent double coverings of the symplectic frame formula_28-bundle formula_2 of the given symplectic manifold formula_1.
Obstruction.
Since every symplectic manifold formula_8 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. In other words, a symplectic manifold formula_1 admits a metaplectic structures if and only if the second Stiefel-Whitney class formula_33 of formula_8 vanishes. In fact, the modulo formula_35 reduction of the first Chern class formula_36 is the second Stiefel-Whitney class formula_37. Hence, formula_1 admits metaplectic structures if and only if formula_39 is even, i.e., if and only if formula_37 is zero.
If this is the case, the isomorphy classes of metaplectic structures on formula_1 are classified by the first cohomology group formula_42 of formula_8 with formula_44-coefficients.
As the manifold formula_8 is assumed to be oriented, the first Stiefel-Whitney class formula_46 of formula_8 vanishes too.