Difference between revisions of "Spinor structure"
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''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'' | ''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 | + | 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]]]). | 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]]]). |
Revision as of 08:32, 19 October 2014
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 |
Comments
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 |
Spinor structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spinor_structure&oldid=16824