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.^[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. XIV. Academic Press: 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.

Formal definition

Notes

Further reading