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),

the double cover 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).

See also associated bundle.