Symplectic spinor bundle

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

Formal definition

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

The symplectic spinor bundle $𝐐$ is defined to be the Hilbert space bundle

$𝐐=𝐏{\times }_{𝔪}{L}^2({ℝ}^n)$

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

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

Further reading

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