Spinor bundle

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

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

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

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