# Spinor structure

on an -dimensional manifold , fibration of spin-frames

A principal fibre bundle over with structure group (see Spinor group), covering some principal fibre bundle of co-frames with structure group . The latter condition means that there is given a surjective homomorphism of principal fibre bundles, which is the identity on the base and is compatible with the natural homomorphism . One says that the spinor structure is subordinate to the Riemannian metric on defined by . From the point of view of the theory of -structures, a spinor structure is a generalized -structure with structure group together with a non-faithful representation (cf. -structure).

In a similar way one defines spinor structures subordinate to pseudo-Riemannian metrics, and spinor structures on complex manifolds subordinate to complex metrics. Necessary and sufficient conditions for the existence of a spinor structure on consist of the orientability of and the vanishing of the Stiefel–Whitney class . When these conditions hold, the number of non-isomorphic spinor structures on subordinate to a given Riemannian metric coincides with the order of the group (see ).

Let be the complexification of the Clifford algebra of with quadratic form . Then has an irreducible representation in a space of dimension , which defines a representation of in . Every spinor structure on yields an associated vector bundle with typical fibre , called a spinor bundle. The Riemannian connection on determines in a canonical way a connection on . On the space of smooth sections of (spinor fields) there acts a linear differential operator of order , the Dirac operator, which is locally defined by the formula where ( ) are the covariant derivatives in the directions of the system of orthonormal vector fields and the dot denotes multiplication of spinor fields by vector fields which correspond to the above -module structure on .

Spinor fields in the kernel of are sometimes called harmonic spinor fields. If is compact, then , and this dimension does not change under conformal deformation of the metric . If the Riemannian metric on has positive scalar curvature, then (see , ).

A spinor structure on a space-time manifold (that is, on a -dimensional Lorentz manifold) is defined as a spinor structure subordinate to the Lorentz metric . The existence of a spinor structure on a non-compact space-time is equivalent to the total parallelizability of (see ). As a module over the spinor group , the spinor space decomposes into the direct sum of two complex -dimensional complexly-conjugate -modules and . This corresponds to the decomposition of the spinor bundle, where the tensor product is identified with the complexification of the tangent bundle . Spinor fields in space-time that are eigenfunctions of the Dirac operator characterize free fields of particles with spin , such as electrons.