Spinor bundle
In differential geometry, given a spin structure on an [math]\displaystyle{ n }[/math]-dimensional orientable Riemannian manifold [math]\displaystyle{ (M, g),\, }[/math] one defines the spinor bundle to be the complex vector bundle [math]\displaystyle{ \pi_{\mathbf S}\colon{\mathbf S}\to M\, }[/math] associated to the corresponding principal bundle [math]\displaystyle{ \pi_{\mathbf P}\colon{\mathbf P}\to M\, }[/math] of spin frames over [math]\displaystyle{ M }[/math] and the spin representation of its structure group [math]\displaystyle{ {\mathrm {Spin}}(n)\, }[/math] on the space of spinors [math]\displaystyle{ \Delta_n }[/math].
A section of the spinor bundle [math]\displaystyle{ {\mathbf S}\, }[/math] is called a spinor field.
Formal definition
Let [math]\displaystyle{ ({\mathbf P},F_{\mathbf P}) }[/math] be a spin structure on a Riemannian manifold [math]\displaystyle{ (M, g),\, }[/math]that is, an equivariant lift of the oriented orthonormal frame bundle [math]\displaystyle{ \mathrm F_{SO}(M)\to M }[/math] with respect to the double covering [math]\displaystyle{ \rho\colon {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n) }[/math] of the special orthogonal group by the spin group.
The spinor bundle [math]\displaystyle{ {\mathbf S}\, }[/math] is defined [1] to be the complex vector bundle [math]\displaystyle{ {\mathbf S}={\mathbf P}\times_{\kappa}\Delta_n\, }[/math] associated to the spin structure [math]\displaystyle{ {\mathbf P} }[/math] via the spin representation [math]\displaystyle{ \kappa\colon {\mathrm {Spin}}(n)\to {\mathrm U}(\Delta_n),\, }[/math] where [math]\displaystyle{ {\mathrm U}({\mathbf W})\, }[/math] denotes the group of unitary operators acting on a Hilbert space [math]\displaystyle{ {\mathbf W}.\, }[/math] It is worth noting that the spin representation [math]\displaystyle{ \kappa }[/math] is a faithful and unitary representation of the group [math]\displaystyle{ {\mathrm {Spin}}(n). }[/math][2]
See also
- Clifford bundle
- Clifford module bundle
- Orthonormal frame bundle
- Spin geometry
- Spinor
- Spinor representation
Notes
- ↑ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 page 53
- ↑ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 pages 20 and 24
Further reading
- Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
- Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1
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