In differential geometry, given a metaplectic structure
the symplectic spinor bundle is the Hilbert space bundle
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
is called a symplectic spinor field.
be a metaplectic structure on a symplectic manifold
that is, an equivariant lift of the symplectic frame bundle
with respect to the double covering
{\displaystyle \rho \colon {\mathrm {Mp} }(n,{\mathbb {R} })\to {\mathrm {Sp} }(n,{\mathbb {R} }).\,}
The symplectic spinor bundle
is defined [2] to be the Hilbert space bundle associated to the metaplectic structure
via the metaplectic representation
also called the Segal–Shale–Weil [3][4][5] representation of
denotes the group of unitary operators acting on a Hilbert space
The Segal–Shale–Weil representation [6] 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.