Symplectic spinor bundle

In differential geometry, given a metaplectic structure [math]\displaystyle{ \pi_{\mathbf P}\colon{\mathbf P}\to M\, }[/math] on a [math]\displaystyle{ 2n }[/math]-dimensional symplectic manifold [math]\displaystyle{ (M, \omega),\, }[/math] the symplectic spinor bundle is the Hilbert space bundle [math]\displaystyle{ \pi_{\mathbf Q}\colon{\mathbf Q}\to M\, }[/math] 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 [math]\displaystyle{ {\mathbf Q}\, }[/math] is called a symplectic spinor field.

Formal definition

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

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

[math]\displaystyle{ {\mathbf Q}={\mathbf P}\times_{\mathfrak m}L^2({\mathbb R}^n)\, }[/math]

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

The Segal–Shale–Weil representation is an infinite dimensional unitary representation of the metaplectic group [math]\displaystyle{ {\mathrm {Mp}}(n,{\mathbb R}) }[/math] on the space of all complex valued square Lebesgue integrable square-integrable functions [math]\displaystyle{ L^2({\mathbb R}^n).\, }[/math] 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