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| In [[differential geometry]], given a [[metaplectic structure]] <math>\pi_{\mathbf P}\colon{\mathbf P}\to M\,</math> on a <math>2n</math>-dimensional [[symplectic manifold]] <math>(M, \omega),\,</math> one defines the '''symplectic spinor bundle''' to be the [[Hilbert space]] bundle <math>\pi_{\mathbf Q}\colon{\mathbf Q}\to M\,</math> 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]].<ref>{{cite journal|title=Symplectic Spinors|last=Kostant |first=B. |journal=Symposia Mathematica|volume= XIV|year=1974|publisher=Academic Press|pages=139–152}}</ref>
| | Chemistry Technician Leibold from Halifax, has many hobbies and interests which include playing music, health and fitness and riddles. Finds plenty of encouragement from life by touring locales like Thatta.<br><br>Visit my site ... [http://www.le-gratos.fr/petites-annonces-gratuites/author/kaaddison/ Http://www.le-gratos.Fr] |
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| A section of the '''symplectic spinor bundle''' <math>{\mathbf Q}\,</math> is called a '''symplectic spinor field'''.
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| ==Formal definition==
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| Let <math>({\mathbf P},F_{\mathbf P})</math> be a [[metaplectic structure]] on a [[symplectic manifold]] <math>(M, \omega),\,</math> that is, an [[equivariant]] lift of the [[symplectic frame bundle]] <math>\pi_{\mathbf R}\colon{\mathbf R}\to M\,</math> with respect to the double covering <math>\rho\colon {\mathrm {Mp}}(n,{\mathbb R})\to {\mathrm {Sp}}(n,{\mathbb R}).\,</math>
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| The '''symplectic spinor bundle''' <math>{\mathbf Q}\,</math> is defined <ref>{{citation | last1=Habermann|first1=Katharina|last2=Habermann|first2=Lutz |title = Introduction to Symplectic Dirac Operators| publisher=[[Springer-Verlag]] | year=2006|isbn=978-3-540-33420-0}} page 37
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| </ref> to be the Hilbert space [[bundle (mathematics)|bundle]]
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| : <math>{\mathbf Q}={\mathbf P}\times_{\mathfrak m}L^2({\mathbb R}^n)\,</math> | |
| associated to the metaplectic structure <math>{\mathbf P}</math> via the metaplectic representation <math>{\mathfrak m}\colon {\mathrm {Mp}}(n,{\mathbb R})\to {\mathrm U}(L^2({\mathbb R}^n)),\,</math> also called the '''Segal-Shale-Weil''' <ref>{{citation | last1=Segal|first1=I.E|title = Lectures at the 1960 Boulder Summer Seminar| publisher=AMS, Providence, RI| year=1962}}
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| </ref><ref>{{cite journal |first=D. |last=Shale |title=Linear symmetries of free boson fields |journal=Trans. Amer. Math. Soc. |volume=103 |year=1962 |pages=149–167}}</ref><ref>{{cite journal |first=A. |last=Weil |title={{lang|fr|Sur certains groupes d’opérateurs unitaires}} |journal=Acta Math. |volume=111 |year=1964 |pages=143–211 |doi=10.1007/BF02391012}}</ref> representation of <math>{\mathrm {Mp}}(n,{\mathbb R}).\,</math> Here, the notation <math>{\mathrm U}({\mathbf W})\,</math> denotes the [[group (mathematics)|group]] of [[unitary operator]]s acting on a [[Hilbert space]] <math>{\mathbf W}.\,</math>
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| The Segal-Shale-Weil representation <ref>{{cite journal | last1=Kashiwara |first1=M |last2=Vergne|first2=M. | title = On the Segal-Shale-Weil representation and harmonic polynomials| journal=Inventiones Mathematicae |volume=44 |year=1978 |pages=1–47|doi=10.1007/BF01389900}}
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| </ref> is an infinite dimensional [[unitary representation]]
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| of the metaplectic group <math>{\mathrm {Mp}}(n,{\mathbb R})</math> on the space of all complex
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| valued square [[Lebesgue integrable]] functions <math>L^2({\mathbb R}^n).\,</math> Because of the infinite dimension,
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| the Segal-Shale-Weil representation is not so easy to handle.
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| ==See also==
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| * [[Metaplectic group]]
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| * [[Metaplectic structure]]
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| * [[Symplectic frame bundle]]
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| * [[Symplectic group]]
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| ==Notes==
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| {{Reflist}}
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| ==Books==
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| * {{citation | last1=Habermann|first1=Katharina|last2=Habermann|first2=Lutz |title = Introduction to Symplectic Dirac Operators| publisher=[[Springer-Verlag]] | year=2006|isbn=978-3-540-33420-0}}
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| {{DEFAULTSORT:Symplectic Spinor Bundle}}
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| [[Category:Symplectic geometry]]
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| [[Category:Structures on manifolds]]
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| [[Category:Algebraic topology]]
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| {{differential-geometry-stub}}
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Chemistry Technician Leibold from Halifax, has many hobbies and interests which include playing music, health and fitness and riddles. Finds plenty of encouragement from life by touring locales like Thatta.
Visit my site ... Http://www.le-gratos.Fr