Symplectic spinor bundle

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In differential geometry, given a metaplectic structure on a -dimensional symplectic manifold one defines the symplectic spinor bundle to be the Hilbert space bundle 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 is called a symplectic spinor field.

Formal definition[edit]

Let be a metaplectic structure on a symplectic manifold that is, an equivariant lift of the symplectic frame bundle with respect to the double covering

The symplectic spinor bundle is defined [2] to be the Hilbert space bundle

associated to the metaplectic structure via the metaplectic representation also called the Segal-Shale-Weil [3][4][5] representation of Here, the notation denotes the group of unitary operators acting on a Hilbert space

The Segal-Shale-Weil representation [6] is an infinite dimensional unitary representation of the metaplectic group on the space of all complex valued square Lebesgue integrable square-integrable functions Because of the infinite dimension, the Segal-Shale-Weil representation is not so easy to handle.

See also[edit]

Notes[edit]

  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. 

Books[edit]