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.