Symplectic spinor bundle

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

Formal definition

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

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

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

associated to the metaplectic structure P {\displaystyle {\mathbf {P} }} via the metaplectic representation m : M p ( n , R ) → U ( L 2 ( R n ) ) , {\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 M p ( n , R ) . {\displaystyle {\mathrm {Mp} }(n,{\mathbb {R} }).\,} Here, the notation U ( W ) {\displaystyle {\mathrm {U} }({\mathbf {W} })\,} denotes the group of unitary operators acting on a Hilbert space W . {\displaystyle {\mathbf {W} }.\,}

The Segal–Shale–Weil representation [6] is an infinite dimensional unitary representation of the metaplectic group M p ( n , R ) {\displaystyle {\mathrm {Mp} }(n,{\mathbb {R} })} on the space of all complex valued square Lebesgue integrable square-integrable functions L 2 ( R n ) . {\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. XIV. Academic Press: 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