In differential geometry, given a metaplectic structure on a -dimensional symplectic manifold the symplectic spinor bundle is the Hilbert space bundle 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 is called a symplectic spinor field. Formal definition Let be a metaplectic structure on a symplectic manifold that is, an equivariant lift of the symplectic frame bundle with respect to the double covering The symplectic spinor bundle is defined to be the Hilbert space bundle associated to the metaplectic structure via the metaplectic representation also called the Segal–Shale–Weil representation of Here, the notation denotes the group of unitary operators acting on a Hilbert space The Segal–Shale–Weil representation is an infinite dimensional unitary representation of the metaplectic group on the space of all complex valued square Lebesgue integrable square-integrable functions Because of the infinite dimension, the Segal–Shale–Weil representation is not so easy to handle. Notes Further reading Symplectic geometry Structures on manifolds Algebraic topology