Spinor bundle

In differential geometry, given a spin structure on an ${\displaystyle n}${\displaystyle n}-dimensional orientable Riemannian manifold ${\displaystyle (M,g),,円}${\displaystyle (M,g),,円} one defines the spinor bundle to be the complex vector bundle ${\displaystyle \pi _{\mathbf {S} }\colon {\mathbf {S} }\to M,円}${\displaystyle \pi _{\mathbf {S} }\colon {\mathbf {S} }\to M,円} associated to the corresponding principal bundle ${\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to M,円}${\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to M,円} of spin frames over ${\displaystyle M}${\displaystyle M} and the spin representation of its structure group ${\displaystyle {\mathrm {Spin} }(n),円}${\displaystyle {\mathrm {Spin} }(n),円} on the space of spinors ${\displaystyle \Delta _{n}}${\displaystyle \Delta _{n}}.

A section of the spinor bundle ${\displaystyle {\mathbf {S} },円}${\displaystyle {\mathbf {S} },円} is called a spinor field.

Let ${\displaystyle ({\mathbf {P} },F_{\mathbf {P} })}${\displaystyle ({\mathbf {P} },F_{\mathbf {P} })} be a spin structure on a Riemannian manifold ${\displaystyle (M,g),,円}${\displaystyle (M,g),,円}that is, an equivariant lift of the oriented orthonormal frame bundle ${\displaystyle \mathrm {F} _{SO}(M)\to M}${\displaystyle \mathrm {F} _{SO}(M)\to M} with respect to the double covering ${\displaystyle \rho \colon {\mathrm {Spin} }(n)\to {\mathrm {SO} }(n)}${\displaystyle \rho \colon {\mathrm {Spin} }(n)\to {\mathrm {SO} }(n)} of the special orthogonal group by the spin group.

The spinor bundle ${\displaystyle {\mathbf {S} },円}${\displaystyle {\mathbf {S} },円} is defined ^[1] to be the complex vector bundle $${\displaystyle {\mathbf {S} }={\mathbf {P} }\times _{\kappa }\Delta _{n},円}$${\displaystyle {\mathbf {S} }={\mathbf {P} }\times _{\kappa }\Delta _{n},円} associated to the spin structure ${\displaystyle {\mathbf {P} }}${\displaystyle {\mathbf {P} }} via the spin representation ${\displaystyle \kappa \colon {\mathrm {Spin} }(n)\to {\mathrm {U} }(\Delta _{n}),,円}${\displaystyle \kappa \colon {\mathrm {Spin} }(n)\to {\mathrm {U} }(\Delta _{n}),,円} where ${\displaystyle {\mathrm {U} }({\mathbf {W} }),円}${\displaystyle {\mathrm {U} }({\mathbf {W} }),円} denotes the group of unitary operators acting on a Hilbert space ${\displaystyle {\mathbf {W} }.,円}${\displaystyle {\mathbf {W} }.,円} The spin representation ${\displaystyle \kappa }${\displaystyle \kappa } is a faithful and unitary representation of the group ${\displaystyle {\mathrm {Spin} }(n).}${\displaystyle {\mathrm {Spin} }(n).}^[2]

• Clifford bundle
• Clifford module bundle
• Orthonormal frame bundle
• Spin geometry
• Spinor
• Spinor representation

• 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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