Symplectic spinor bundle

In differential geometry, given a metaplectic structure on a -dimensional symplectic manifold the symplectic spinor bundle is 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

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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.

Notes

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  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

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