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.

Formal definition

Let ${\displaystyle ({\mathbf {P} },F_{\mathbf {P} })}$ be a metaplectic structure on a symplectic manifold ${\displaystyle (M,\omega ),\,}$ that is, an equivariant lift of the symplectic frame bundle ${\displaystyle \pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,}$ with respect to the double covering ${\displaystyle \rho \colon {\mathrm {Mp} }(n,{\mathbb {R} })\to {\mathrm {Sp} }(n,{\mathbb {R} }).\,}$

The symplectic spinor bundle ${\displaystyle {\mathbf {Q} }\,}$ is defined ^[2] to be the Hilbert space bundle

${\displaystyle {\mathbf {Q} }={\mathbf {P} }\times _{\mathfrak {m}}L^{2}({\mathbb {R} }^{n})\,}$

associated to the metaplectic structure ${\displaystyle {\mathbf {P} }}$ via the metaplectic representation ${\displaystyle {\mathfrak {m}}\colon {\mathrm {Mp} }(n,{\mathbb {R} })\to {\mathrm {U} }(L^{2}({\mathbb {R} }^{n})),\,}$ also called the Segal–Shale–Weil ^[3][4][5] representation of ${\displaystyle {\mathrm {Mp} }(n,{\mathbb {R} }).\,}$ Here, the notation ${\displaystyle {\mathrm {U} }({\mathbf {W} })\,}$ denotes the group of unitary operators acting on a Hilbert space ${\displaystyle {\mathbf {W} }.\,}$

The Segal–Shale–Weil representation ^[6] is an infinite dimensional unitary representation of the metaplectic group ${\displaystyle {\mathrm {Mp} }(n,{\mathbb {R} })}$ on the space of all complex valued square Lebesgue integrable square-integrable functions ${\displaystyle L^{2}({\mathbb {R} }^{n}).\,}$ Because of the infinite dimension, the Segal–Shale–Weil representation is not so easy to handle.