Spinor bundle

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 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).

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.

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Spinor bundle

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