# 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