Symplectic spinor bundle

In differential geometry, given a metaplectic structure ${\pi }_{\mathbf{𝐏}}:\mathbf{𝐏}\to M$ on a $2n$-dimensional symplectic manifold $(M,\omega ),$ the symplectic spinor bundle is the Hilbert space bundle ${\pi }_{\mathbf{𝐐}}:\mathbf{𝐐}\to M$ 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 ${\pi }_{\mathbf{𝐑}}:\mathbf{𝐑}\to M$ with respect to the double covering $\rho :\mathrm{M}\mathrm{p}(n,\mathbb{ℝ})\to \mathrm{S}\mathrm{p}(n,\mathbb{ℝ}).$

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 $\mathfrak{𝔪}:\mathrm{M}\mathrm{p}(n,\mathbb{ℝ})\to \mathrm{U}(L^2({\mathbb{ℝ}}^n)),$ also called the Segal–Shale–Weil representation of $\mathrm{M}\mathrm{p}(n,\mathbb{ℝ}).$ 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 $\mathrm{M}\mathrm{p}(n,\mathbb{ℝ})$ 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