In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the **spinor bundle** to be the complex vector bundle associated to the corresponding principal bundle of spin frames over and the spin representation of its structure group on the space of spinors .

A section of the **spinor bundle** is called a **spinor field**.

## Formal definition

Let be a spin structure on a Riemannian manifold that is, an equivariant lift of the oriented orthonormal frame bundle with respect to the double covering of the special orthogonal group by the spin group.

The **spinor bundle** is defined ^{[1]} to be the complex vector bundle

associated to the spin structure via the spin representation where denotes the group of unitary operators acting on a Hilbert space It is worth noting that the spin representation is a faithful and unitary representation of the group .^{[2]}

## See also

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

## 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