Symplectic spinor bundle
In differential geometry, given a metaplectic structure π P : P → M {\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to M\,} on a 2 n {\displaystyle 2n} -dimensional symplectic manifold ( M , ω ) , {\displaystyle (M,\omega ),\,} the symplectic spinor bundle is the Hilbert space bundle π Q : Q → M {\displaystyle \pi _{\mathbf {Q} }\colon {\mathbf {Q} }\to M\,} associated to the metaplectic structure via the metaplectic representation. The metaplectic representation of the metaplectic group — the two-fold covering of the symplectic group — gives rise to an infinite rank vector bundle; this is the symplectic spinor construction due to Bertram Kostant.