Spinor structure

on an $ n $- dimensional manifold $ M $, fibration of spin-frames

A principal fibre bundle $ \widetilde \pi : \widetilde{P} \rightarrow M $ over $ M $ with structure group $ \mathop{\rm Spin} _ {n} $( see Spinor group), covering some principal fibre bundle $ \pi : P \rightarrow M $ of co-frames with structure group $ \mathop{\rm SO} _ {n} $. The latter condition means that there is given a surjective homomorphism $ \kappa : \widetilde{P} \rightarrow P $ of principal fibre bundles, which is the identity on the base and is compatible with the natural homomorphism $ \rho : \mathop{\rm Spin} _ {n} \rightarrow \mathop{\rm SO} _ {n} $. One says that the spinor structure $ ( \widetilde \pi , \kappa ) $ is subordinate to the Riemannian metric $ g $ on $ M $ defined by $ \pi $. From the point of view of the theory of $ G $- structures, a spinor structure is a generalized $ G $- structure with structure group $ G = \mathop{\rm Spin} _ {n} $ together with a non-faithful representation $ \rho : \mathop{\rm Spin} _ {n} \rightarrow \mathop{\rm SO} _ {n} $( cf. $ G $- 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 $ M $ consist of the orientability of $ M $ and the vanishing of the Stiefel–Whitney class $ W _ {2} ( M) $. When these conditions hold, the number of non-isomorphic spinor structures on $ M $ subordinate to a given Riemannian metric coincides with the order of the group $ H ^ {1} ( M, \mathbf Z ) $( see [6]).

Let $ C $ be the complexification of the Clifford algebra of $ \mathbf R ^ {n} $ with quadratic form $ q= \sum _ {i=} 1 ^ {n} x _ {i} ^ {2} $. Then $ C $ has an irreducible representation in a space $ S $ of dimension $ 2 ^ {[ n/2] } $, which defines a representation of $ \mathop{\rm Spin} _ {n} \subset C $ in $ S $. Every spinor structure $ \widetilde \pi $ on $ M $ yields an associated vector bundle $ \pi _ {S} : S( M) \rightarrow M $ with typical fibre $ S $, called a spinor bundle. The Riemannian connection on $ M $ determines in a canonical way a connection on $ \pi _ {S} $. On the space $ \Gamma ( S) $ of smooth sections of $ \pi _ {S} $( spinor fields) there acts a linear differential operator $ D $ of order $ 1 $, the Dirac operator, which is locally defined by the formula

$$ Du = \sum _ { i= } 1 ^ { n } s _ {i} \cdot \nabla _ {s _ {i} } u ,\ u \in \Gamma ( S) , $$

where $ \nabla _ {s _ {i} } $( $ i= 1 \dots n $) are the covariant derivatives in the directions of the system of orthonormal vector fields $ s _ {i} $ and the dot denotes multiplication of spinor fields by vector fields which correspond to the above $ C $- module structure on $ S $.

Spinor fields in the kernel of $ D $ are sometimes called harmonic spinor fields. If $ M $ is compact, then $ \mathop{\rm dim} \mathop{\rm ker} D < \infty $, and this dimension does not change under conformal deformation of the metric [4]. If the Riemannian metric on $ M $ has positive scalar curvature, then $ \mathop{\rm ker} D = 0 $( see [4], [5]).

A spinor structure on a space-time manifold $ ( M, g) $( that is, on a $ 4 $- dimensional Lorentz manifold) is defined as a spinor structure subordinate to the Lorentz metric $ g $. The existence of a spinor structure on a non-compact space-time $ M $ is equivalent to the total parallelizability of $ M $( see [3]). As a module over the spinor group $ \mathop{\rm Spin} ( 1, 3) \approx \mathop{\rm SL} ( 2, G) $, the spinor space decomposes into the direct sum of two complex $ 2 $- dimensional complexly-conjugate $ \mathop{\rm SL} ( 2, G) $- modules $ {\mathcal C} ^ {2} $ and $ {\mathcal C} dot {} ^ {2} $. This corresponds to the decomposition $ S( M)= {\mathcal C} ^ {2} ( M) \oplus {\mathcal C} dot {} ^ {2} ( M) $ of the spinor bundle, where the tensor product $ {\mathcal C} ^ {2} ( M) \oplus {\mathcal C} dot {} ^ {2} ( M) $ is identified with the complexification of the tangent bundle $ TM $. Spinor fields in space-time that are eigenfunctions of the Dirac operator characterize free fields of particles with spin $ 1/2 $, 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
[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 Zbl 0661.53016