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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 , {\displaystyle ,\,} 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.
A section of the symplectic spinor bundle Q {\displaystyle {\mathbf {Q} }\,} is called a symplectic spinor field.