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In [[differential geometry]], given a [[spin structure]] on a <math>n</math>-dimensional [[Riemannian manifold]] <math>(M, g),\,</math> one defines the '''spinor bundle''' to be the [[complex vector bundle]] <math>\pi_{\mathbf S}\colon{\mathbf S}\to M\,</math> associated to the corresponding [[principal bundle]] <math>\pi_{\mathbf P}\colon{\mathbf P}\to M\,</math> of spin frames over <math>M</math> and the [[spin representation]] of its structure group <math>{\mathrm {Spin}}(n)\,</math> on the space of [[spinor]]s <math>\Delta_n.\,</math>.
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A section of the '''spinor bundle''' <math>{\mathbf S}\,</math> is called a '''spinor field'''.
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==Formal definition==
Let <math>({\mathbf P},F_{\mathbf P})</math> be a [[spin structure]] on a [[Riemannian manifold]] <math>(M, g),\,</math> that is, an [[equivariant]] lift of the oriented [[orthonormal frame bundle]] <math>\mathrm F_{SO}(M)\to M</math> with respect to the double covering <math>\rho\colon {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n).\,</math>
 
The '''spinor bundle''' <math>{\mathbf S}\,</math> is defined <ref>{{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}} page 53
</ref> to be the [[complex vector bundle]]
: <math>{\mathbf S}={\mathbf P}\times_{\kappa}\Delta_n\,</math>
associated to the [[spin structure]] <math>{\mathbf P}</math> via the [[spin representation]] <math>\kappa\colon {\mathrm {Spin}}(n)\to {\mathrm U}(\Delta_n),\,</math> where <math>{\mathrm U}({\mathbf W})\,</math> denotes the [[Group (mathematics)|group]] of [[unitary operator]]s acting on a [[Hilbert space]] <math>{\mathbf W}.\,</math> It is worth noting that the spin representation <math>\kappa</math> is a faithful and [[unitary representation]] of the group <math>{\mathrm {Spin}}(n)</math>.<ref>{{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}} pages 20 and 24</ref>
 
==See also==
* [[Orthonormal frame bundle]]
* [[Spinor]]
* [[Spin manifold]]
* [[Spinor representation]]
* [[Spin geometry]]
 
==Notes==
{{Reflist}}
 
==Further reading==
* {{Cite book | last1=Lawson | first1=H. Blaine | last2=Michelsohn |author1-link=H. Blaine Lawson| first2=Marie-Louise |author2-link=Marie-Louise Michelsohn| title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 | postscript=<!--None-->}}
* {{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}}
 
[[Category:Riemannian geometry]]
[[Category:Structures on manifolds]]
[[Category:Algebraic topology]]
 
 
{{differential-geometry-stub}}

Latest revision as of 18:46, 9 March 2014

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