Symplectic spinor bundle

In differential geometry, given a metaplectic structure ${\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to M\,}$ on a ${\displaystyle 2n}$-dimensional symplectic manifold ${\displaystyle (M,\omega ),\,}$ one defines the symplectic spinor bundle to be the Hilbert space bundle ${\displaystyle \pi _{\mathbf {Q} }\colon {\mathbf {Q} }\to M\,}$ 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.

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