Spinor bundle explained

In differential geometry, given a spin structure on an

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

A section of the spinor bundle

{ S} is called a spinor field.

Formal definition

Let

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

The spinor bundle

{ S} is defined ^[1] to be the complex vector bundle$$=\backslash times\_\backslash Delta\_n\backslash ,$$associated to the spin structure { P} via the spin representation \kappa\colon { Spin }(n)\to (\Delta_n),\, where { U}({ W}) denotes the group of unitary operators acting on a Hilbert space { W}. It is worth noting that the spin representation \kappa is a faithful and unitary representation of the group { Spin }(n).^[2]

See also

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

Further reading

• Book: Lawson . H. Blaine . Michelsohn . H. Blaine Lawson. Marie-Louise . Marie-Louise Michelsohn. Spin Geometry . . 978-0-691-08542-5 . 1989 .

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Notes and References

1. page 53
2. pages 20 and 24