Symplectic spinor bundle

In differential geometry, given a metaplectic structure $\pi _{{{\mathbf P}}}\colon {{\mathbf P}}\to M\,$ on a $2n$ -dimensional symplectic manifold $(M,\omega ),\,$ one defines the symplectic spinor bundle to be 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

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.

Notes

↑ Kostant, B. (1974). "Symplectic Spinors". Symposia Mathematica. Academic Press. XIV: 139–152.
↑ Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0 page 37
↑ Segal, I.E (1962), Lectures at the 1960 Boulder Summer Seminar, AMS, Providence, RI
↑ Shale, D. (1962). "Linear symmetries of free boson fields". Trans. Amer. Math. Soc. 103: 149–167. doi:10.1090/s0002-9947-1962-0137504-6.
↑ Weil, A. (1964). "Sur certains groupes d’opérateurs unitaires". Acta Math. 111: 143–211. doi:10.1007/BF02391012.
↑ Kashiwara, M; Vergne, M. (1978). "On the Segal-Shale-Weil representation and harmonic polynomials". Inventiones Mathematicae. 44: 1–47. doi:10.1007/BF01389900.

Books

Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0

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Symplectic spinor bundle

Formal definition

See also

Notes

Books