View source for Spinor structure

''on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867802.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867803.png" />, fibration of spin-frames''

A principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867804.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867805.png" /> with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867806.png" /> (see [[Spinor group|Spinor group]]), covering some principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867807.png" /> of co-frames with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867808.png" />. The latter condition means that there is given a surjective homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s0867809.png" /> of principal fibre bundles, which is the identity on the base and is compatible with the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678010.png" />. One says that the spinor structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678011.png" /> is subordinate to the Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678013.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678014.png" />. From the point of view of the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678015.png" />-structures, a spinor structure is a generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678016.png" />-structure with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678017.png" /> together with a non-faithful representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678018.png" /> (cf. [[G-structure|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678019.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678020.png" /> consist of the orientability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678021.png" /> and the vanishing of the Stiefel–Whitney class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678022.png" />. When these conditions hold, the number of non-isomorphic spinor structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678023.png" /> subordinate to a given Riemannian metric coincides with the order of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678024.png" /> (see [[#References|[6]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678025.png" /> be the complexification of the [[Clifford algebra|Clifford algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678026.png" /> with quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678027.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678028.png" /> has an irreducible representation in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678029.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678030.png" />, which defines a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678032.png" />. Every spinor structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678034.png" /> yields an associated vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678035.png" /> with typical fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678036.png" />, called a spinor bundle. The Riemannian connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678037.png" /> determines in a canonical way a connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678038.png" />. On the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678039.png" /> of smooth sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678040.png" /> (spinor fields) there acts a linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678041.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678042.png" />, the Dirac operator, which is locally defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678043.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678044.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678045.png" />) are the covariant derivatives in the directions of the system of orthonormal vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678046.png" /> and the dot denotes multiplication of spinor fields by vector fields which correspond to the above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678047.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678048.png" />.

Spinor fields in the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678049.png" /> are sometimes called harmonic spinor fields. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678050.png" /> is compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678051.png" />, and this dimension does not change under conformal deformation of the metric [[#References|[4]]]. If the Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678052.png" /> has positive scalar curvature, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678053.png" /> (see [[#References|[4]]], [[#References|[5]]]).

A spinor structure on a [[Space-time|space-time]] manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678054.png" /> (that is, on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678055.png" />-dimensional Lorentz manifold) is defined as a spinor structure subordinate to the Lorentz metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678056.png" />. The existence of a spinor structure on a non-compact space-time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678057.png" /> is equivalent to the total parallelizability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678058.png" /> (see [[#References|[3]]]). As a module over the spinor group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678059.png" />, the spinor space decomposes into the direct sum of two complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678060.png" />-dimensional complexly-conjugate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678061.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678063.png" />. This corresponds to the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678064.png" /> of the spinor bundle, where the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678065.png" /> is identified with the complexification of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678066.png" />. Spinor fields in space-time that are eigenfunctions of the Dirac operator characterize free fields of particles with spin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086780/s08678067.png" />, such as electrons.

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Baum,   "Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten" , Teubner  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.J. Dodson,   "Categories, bundles, and spacetime topology" , Kluwer  (1988)  pp. Chapt. V, §3</TD></TR></table>