Symplectic spinor bundle

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

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

Formal definition

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

The symplectic spinor bundle ${\mathbf {Q} }\,$ is defined to be the Hilbert space bundle

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

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

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

Further reading

• Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0