Symplectic spinor bundle

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

Formal definition

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

The symplectic spinor bundle ${\displaystyle {\mathbf {Q} }\,}$ is defined [2] to be the Hilbert space bundle

${\displaystyle {\mathbf {Q} }={\mathbf {P} }\times _{\mathfrak {m}}L^{2}({\mathbb {R} }^{n})\,}$

associated to the metaplectic structure ${\displaystyle {\mathbf {P} }}$ via the metaplectic representation ${\displaystyle {\mathfrak {m}}\colon {\mathrm {Mp} }(n,{\mathbb {R} })\to {\mathrm {U} }(L^{2}({\mathbb {R} }^{n})),\,}$ also called the Segal–Shale–Weil [3][4][5] representation of ${\displaystyle {\mathrm {Mp} }(n,{\mathbb {R} }).\,}$ Here, the notation ${\displaystyle {\mathrm {U} }({\mathbf {W} })\,}$ denotes the group of unitary operators acting on a Hilbert space ${\displaystyle {\mathbf {W} }.\,}$

The Segal–Shale–Weil representation [6] is an infinite dimensional unitary representation of the metaplectic group ${\displaystyle {\mathrm {Mp} }(n,{\mathbb {R} })}$ on the space of all complex valued square Lebesgue integrable square-integrable functions ${\displaystyle L^{2}({\mathbb {R} }^{n}).\,}$ Because of the infinite dimension, the Segal–Shale–Weil representation is not so easy to handle.

Notes

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

Further reading

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