Spinor bundle
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Given a differentiable manifold M with a tetrad of signature (p,q) over it, a spinor bundle over M is a vector SO(p,q)-bundle over M such that its fiber is a spinor representation of
- Spin(p,q),
a double cover of the identity component of the special orthogonal group SO(p,q).
Spinor bundles inherit a connection from a connection on the vector bundle V (see tetrad).
When
- p + q ≤ 3
there are some further possibilities for covering groups of the orthogonal group, so other bundles (anyonic bundles).
Spinor bundles
The language of associated bundles is helpful in expressing the meaning of spinor bundles.
Here the two groups SO and Spin are involved (for a fixed choice of signature <math>(p,\ q)</math>), the former having a faithful matrix representation of dimension <math>n\ =\ p\ +\ q</math>, but the latter acting (in general) only faithfully in a higher dimension, on a space of spinors. Spin is a double cover of the identity component of SO, so that the latter is a quotient of the former. (If p and q are both non-zero, then the special orthogonal group has 2 components, while the spin group has only one.) That does mean that transition data with values in Spin give rise to transition data for SO, automatically: passing to a quotient group simply loses information.
Therefore a Spin-bundle always gives rise to an associated bundle with fibers <math>\mathbb{R}^n</math>, since Spin acts on <math>\mathbb{R}^n</math>, via its quotient SO. Conversely, there is a lifting problem for SO-bundles: there is a consistency question on the transition data, in passing to a Spin-bundle. The existence of such a spin structure is extra information on a real vector bundle.