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. 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. A metaplectic structure on a symplectic manifold ( M , ω ) {displaystyle (M,omega )} is an equivariant lift of the symplectic frame bundle π R : R → M {displaystyle pi _{mathbf {R} }colon {mathbf {R} } o M,} with respect to the double covering ρ : M p ( n , R ) → S p ( n , R ) . {displaystyle ho colon {mathrm {Mp} }(n,{mathbb {R} }) o {mathrm {Sp} }(n,{mathbb {R} }).,} In other words, a pair ( P , F P ) {displaystyle ({mathbf {P} },F_{mathbf {P} })} is a metaplectic structure on the principal bundle π R : R → M {displaystyle pi _{mathbf {R} }colon {mathbf {R} } o M,} when The principal bundle π P : P → M {displaystyle pi _{mathbf {P} }colon {mathbf {P} } o M,} is also called the bundle of metaplectic frames over M {displaystyle M} . Two metaplectic structures ( P 1 , F P 1 ) {displaystyle ({mathbf {P} _{1}},F_{mathbf {P} _{1}})} and ( P 2 , F P 2 ) {displaystyle ({mathbf {P} _{2}},F_{mathbf {P} _{2}})} on the same symplectic manifold ( M , ω ) {displaystyle (M,omega )} are called equivalent if there exists a M p ( n , R ) {displaystyle {mathrm {Mp} }(n,{mathbb {R} })} -equivariant map f : P 1 → P 2 {displaystyle fcolon {mathbf {P} _{1}} o {mathbf {P} _{2}}} such that Of course, in this case F P 1 {displaystyle F_{mathbf {P} _{1}}} and F P 2 {displaystyle F_{mathbf {P} _{2}}} are two equivalent double coverings of the symplectic frame S p ( n , R ) {displaystyle {mathrm {Sp} }(n,{mathbb {R} })} -bundle π R : R → M {displaystyle pi _{mathbf {R} }colon {mathbf {R} } o M,} of the given symplectic manifold ( M , ω ) {displaystyle (M,omega )} . Since every symplectic manifold M {displaystyle M} 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 ( M , ω ) {displaystyle (M,omega )} admits a metaplectic structures if and only if the second Stiefel-Whitney class w 2 ( M ) ∈ H 2 ( M , Z 2 ) {displaystyle w_{2}(M)in H^{2}(M,{mathbb {Z} _{2}})} of M {displaystyle M} vanishes. In fact, the modulo 2 {displaystyle _{2}} reduction of the first Chern class c 1 ( M ) ∈ H 2 ( M , Z ) {displaystyle c_{1}(M)in H^{2}(M,{mathbb {Z} })} is the second Stiefel-Whitney class w 2 ( M ) {displaystyle w_{2}(M)} . Hence, ( M , ω ) {displaystyle (M,omega )} admits metaplectic structures if and only if c 1 ( M ) {displaystyle c_{1}(M)} is even, i.e., if and only if w 2 ( M ) {displaystyle w_{2}(M)} is zero. If this is the case, the isomorphy classes of metaplectic structures on ( M , ω ) {displaystyle (M,omega )} are classified by the first cohomology group H 1 ( M , Z 2 ) {displaystyle H^{1}(M,{mathbb {Z} _{2}})} of M {displaystyle M} with Z 2 {displaystyle {mathbb {Z} _{2}}} -coefficients. As the manifold M {displaystyle M} is assumed to be oriented, the first Stiefel-Whitney class w 1 ( M ) ∈ H 1 ( M , Z 2 ) {displaystyle w_{1}(M)in H^{1}(M,{mathbb {Z} _{2}})} of M {displaystyle M} vanishes too.