Spinor bundle

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In mathematics and theoretical physics, spinors are certain geometric entities bound up with physical theories of 'spin', and the mathematics of Clifford algebras, that in a sense are kinds of twisted tensors. From a geometric point of view, spinors are organised into spinor bundles.

Given a differentiable manifold M with a metric 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 spin connection).

When

p + q ≤ 3

there are some further possibilities for covering groups of the orthogonal group, so other bundles (anyonic bundles).

[edit] From associated bundles

The language of associated bundles is helpful in expressing the meaning of spinor bundles. The existence of a spin structure is extra information on a real vector bundle.

Here the two groups SO and Spin are involved (for a fixed choice of signature $(p, q)$ ), the former having a faithful matrix representation of dimension $n = p + q$ , 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 $\mathbb{R}^n$ , since Spin acts on $\mathbb{R}^n$ , 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 obstruction to the lifting is known to be the second Stiefel-Whitney class.