Nikiya Anton Bettey 2021: 09-28-2021-1814 - symplectic spinor bundle field

Wednesday, September 29, 2021

09-28-2021-1814 - symplectic spinor bundle field

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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 bundleis 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.^[1]

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

Formal definition[edit]

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 ^[2] 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 ^[3]^[4]^[5]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 ^[6] 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.

https://en.wikipedia.org/wiki/Symplectic_spinor_bundle

Nikiya Anton Bettey 2021

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Wednesday, September 29, 2021

09-28-2021-1814 - symplectic spinor bundle field

Formal definition[edit]

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