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 ${\displaystyle (M,\omega ),\,}$ the symplectic spinor bundle is 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.^[1]

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