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 [6]).

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 [4]. If the Riemannian metric on has positive scalar curvature, then (see [4], [5]).

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 [3]). 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.

References

[1]	G. Casanova, "L'algèbre vectorielle" , Presses Univ. France (1976)
[2]	R. Penrose, "The structure of space-time" C. deWitt (ed.) , Batelle Rencontres 1967 Lectures in Math. Physics , Benjamin (1968) pp. 121–235 (Chapt. VII)
[3]	R. Geroch, "Spinor structure of space-times in general relativity" J. Math. Phys. , 9 (1968) pp. 1739–1744
[4]	N. Hitchin, "Harmonic spinors" Adv. in Math. , 14 (1974) pp. 1–55
[5]	A. Lichnerowicz, "Champs spinoriels et propagateurs en rélativité génerale" Bull. Soc. Math. France , 92 (1964) pp. 11–100
[6]	J. Milnor, "Spin structure on manifolds" Enseign. Math. , 9 (1963) pp. 198–203
[7]	R. Penrose, "The twistor programme" Reports Math. Phys. , 12 (1977) pp. 65–76
[8]	R.O., jr. Wells, "Complex manifolds and mathematical physics" Bull. Amer. Math. Soc. , 1 (1979) pp. 296–336

References

[a1]	H. Baum, "Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten" , Teubner (1981)
[a2]	C.T.J. Dodson, "Categories, bundles, and spacetime topology" , Kluwer (1988) pp. Chapt. V, §3