Given a differentiable manifold M with a Cartan connection applications of metric signature ( p , q ) over it, a spinor bundle over M is a vector bundle over M such that its fiber bundle is a spinor representations of Lie groups/algebras of

: Spin ( p , q ),

the double cover of the special orthogonal group SO ( p , q ).

Spinor bundles inherit a connection (vector bundle) from a connection on the vector bundle V (see Cartan connection applications).

When

: p + q ≤ 3

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