Spinor bundle

A spinor bundle is a vector bundle over a Riemannian manifold whose fibers carry representations of the Clifford algebra of the tangent space. It is the geometric object on which the Dirac operator acts, and its existence requires that the manifold admit a spin structure — a topological condition equivalent to the vanishing of the second Stiefel-Whitney class. Without a spinor bundle, there is no Dirac operator; without the Dirac operator, much of modern index theory and quantum field theory on curved spacetime would not exist.

The sections of a spinor bundle are called spinor fields, and they generalize the notion of scalar and vector fields to objects that transform under the double cover of the rotation group rather than the rotation group itself. This seemingly minor distinction — replacing SO(n) by Spin(n) — is responsible for the Pauli exclusion principle in physics and for the existence of harmonic spinors in geometry. The spinor bundle is not an optional refinement of differential geometry; it is the natural bundle for first-order elliptic operators, and its absence from a manifold is a topological fact with analytic consequences.