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| In [[differential geometry]], a '''spin structure''' on an [[orientable]] [[Riemannian manifold]] ''(M,g)'' allows one to define associated [[spinor bundle]]s, giving rise to the notion of a [[spinor]] in differential geometry.
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| Spin structures have wide applications to [[mathematical physics]], in particular to [[quantum field theory]] where they are an essential ingredient in the definition of any theory with uncharged [[fermion]]s. They are also of purely mathematical interest in [[differential geometry]], [[algebraic topology]], and [[K theory]]. They form the foundation for [[spin geometry]].
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| ==Introduction==
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| In [[geometry]] and in [[Field_theory_(physics)#Field_theory|field theory]], mathematicians ask whether or not a given oriented Riemannian manifold ''(M,g)'' admits [[spinor]]s. One method for dealing with this problem is to require that ''M'' has a '''spin structure'''.<ref name="autogenerated558">{{cite journal|title=Sur l’extension du groupe structural d’un espace fibré|author=A. Haefliger|journal=C. R. Acad. Sci. Paris|volume=243|year=1956|pages=558–560}}</ref><ref>{{cite journal|title=Spin structures on manifolds|author=J. Milnor|journal=L'Enseignement Math.|volume=9|year=1963|pages=198–203}}</ref><ref>{{cite journal|title=Champs spinoriels et propagateurs en rélativité générale|author=A. Lichnerowicz|journal=Bull. Soc. Math. Fr.|volume=92|year=1964|pages=11–100}}</ref><ref>{{cite journal|title=Algèbres de Clifford et K-théorie|author=M. Karoubi|journal=Ann. Sci. Éc. Norm. Sup.|volume=1|year=1968|pages=161–270|issue=2}}</ref> This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second [[Stiefel-Whitney class]] ''w''<sub>2</sub>(''M'') ∈ H<sup>2</sup>(''M'', '''Z'''<sub>2</sub>) of ''M'' vanishes. Furthermore, if ''w''<sub>2</sub>(''M'') = 0, then the set of the isomorphism classes of spin structures on ''M'' is acted upon freely and transitively by H<sup>1</sup>(''M'', '''Z'''<sub>2</sub>) . As the manifold ''M'' is assumed to be oriented, the first Stiefel-Whitney class ''w''<sub>1</sub>(''M'') ∈ H<sup>1</sup>(''M'', '''Z'''<sub>2</sub>) of ''M'' vanishes too. (The Stiefel-Whitney classes ''w<sub>i</sub>''(''M'') ∈ H<sup>''i''</sup>(''M'', '''Z'''<sub>2</sub>) of a manifold ''M'' are defined to be the Stiefel-Whitney classes of its [[tangent bundle]] ''TM''.)
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| The bundle of spinors π<sub>''S''</sub>: ''S'' → ''M'' over ''M'' is then the [[complex vector bundle]] associated to the corresponding [[principal bundle]] π<sub>'''P'''</sub>: '''P''' → ''M'' of spin frames over ''M'' and the spin representation of its structure group Spin(''n'') on the space of spinors Δ<sub>''n''</sub>. The bundle ''S'' is called the '''[[spinor bundle]]''' for a given '''spin structure''' on ''M''.
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| A precise definition of '''spin structure''' on manifold was possible only after the notion of [[fiber bundle]] had been introduced; [[André Haefliger]] (1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and [[Max Karoubi]] (1968) extended this result to the non-orientable pseudo-Riemannian case.
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| ==Spin structures on Riemannian manifolds==
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| ===Definition===
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| A '''spin structure''' on an [[orientable]] [[Riemannian manifold]] ''(M,g)'' is an [[equivariant]] lift of the oriented orthonormal frame bundle ''F''<sub>SO</sub>(''M'') → ''M'' with respect to the double covering ρ: Spin(''n'') → SO(''n''). In other words, a pair ('''P''',''F''<sub>'''P'''</sub>) is a '''spin structure on the principal bundle''' π: ''F''<sub>SO</sub>(''M'') → ''M'' when
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| :a) π<sub>'''P'''</sub>: '''P''' → ''M'' is a principal Spin(''n'')-bundle over ''M'',
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| :b) ''F''<sub>'''P'''</sub>: '''P''' → ''F''<sub>SO</sub>(''M'') is an [[equivariant]] 2-fold [[covering map]] such that
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| ::<math>\pi\circ F_{\mathbf P}=\pi_{\mathbf P}</math> and ''F''<sub>'''P'''</sub>('''p''' ''q'') = ''F''<sub>'''P'''</sub>('''p''')ρ(''q'') for all '''p''' ∈ '''P''' and ''q'' ∈ Spin(''n'').
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| The principal bundle π<sub>'''P'''</sub>: '''P''' → ''M'' is also called the bundle of '''spin frames''' over ''M''.
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| Two spin structures ('''P'''<sub>1</sub>, ''F''<sub>'''P'''<sub>1</sub></sub>) and ('''P'''<sub>2</sub>, ''F''<sub>'''P'''<sub>2</sub></sub>) on the same oriented [[Riemannian manifold]] ''(M,g)'' are called '''equivalent''' if there exists a Spin(''n'')-equivariant map ''f'': '''P'''<sub>1</sub> → '''P'''<sub>2</sub> such that
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| :<math>F_{\mathbf P_2}\circ f=F_{\mathbf P_1}</math> and ''f''('''p''' ''q'') = ''f''('''p''')''q'' for all <math>{\mathbf p}\in {\mathbf P_1}</math> and ''q'' ∈ Spin(''n'').
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| Of course, in this case <math>F_{\mathbf P_1}</math> and <math>F_{\mathbf P_2}</math> are two equivalent double coverings of the oriented orthonormal frame SO(''n'')-bundle ''F''<sub>SO</sub>(''M'') → ''M'' of the given Riemannian manifold ''(M,g)''.
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| This definition of '''spin structure''' on ''(M,g)'' as a spin structure on the principal bundle ''F''<sub>SO</sub>(''M'') → ''M'' is due to [[André Haefliger]] (1956).
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| ===Obstruction===
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| [[André Haefliger]] <ref name="autogenerated558"/> found necessary and sufficient conditions for the existence of a spin structure on an oriented Riemannian manifold ''(M,g)''. The obstruction to having a '''spin structure''' is certain element ''[k]'' of H<sup>2</sup>(''M'', '''Z'''<sub>2</sub>) . For a spin structure the class ''[k]'' is the second [[Stiefel-Whitney class]] ''w''<sub>2</sub>(''M'') ∈ H<sup>2</sup>(''M'', '''Z'''<sub>2</sub>) of ''M''. Hence, a spin structure exists if and only if the second Stiefel-Whitney class ''w''<sub>2</sub>(''M'') ∈ H<sup>2</sup>(''M'', '''Z'''<sub>2</sub>) of ''M'' vanishes.
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| ==Spin structures on vector bundles==
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| Let ''M'' be a [[paracompact]] [[topological manifold]] and ''E'' an [[Oriented#Orientation of vector bundles|oriented]] vector bundle on ''M'' of dimension ''n'' equipped with a [[fibre metric]]. This means that at each point of ''M'', the fibre of ''E'' is an [[inner product space]]. A '''spinor bundle''' of ''E'' is a prescription for consistently associating a [[spin representation]] to every point of ''M''. There are topological obstructions to being able to do it, and consequently, a given bundle ''E'' may not admit any spinor bundle. In case it does, one says that the bundle ''E'' is '''spin'''.
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| This may be made rigorous through the language of [[principal bundle]]s. The collection of oriented [[orthonormal frame]]s of a vector bundle form a [[frame bundle]] ''P''<sub>SO</sub>(''E''), which is a principal bundle under the action of the [[special orthogonal group]] SO(''n''). A spin structure for ''P''<sub>SO</sub>(''E'') is a ''lift'' of ''P''<sub>SO</sub>(''E'') to a principal bundle ''P''<sub>Spin</sub>(''E'') under the action of the [[spin group]] Spin(''n''), by which we mean that there exists a bundle map φ : ''P''<sub>Spin</sub>(''E'') → ''P''<sub>SO</sub>(''E'') such that
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| :<math>\phi(pg) = \phi(p)\rho(g)</math>, for all ''p'' ∈ ''P''<sub>Spin</sub>(''E'') and ''g'' ∈ Spin(''n''),
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| where ρ: Spin(''n'') → SO(''n'') is the mapping of groups presenting the spin group as a double-cover of SO(''n'').
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| In the special case in which ''E'' is the [[tangent bundle]] ''TM'' over the base manifold ''M'', if a spin structure exists then one says that ''M'' is a '''spin manifold'''. Equivalently ''M'' is ''spin'' if the SO(''n'') principal bundle of [[orthonormal basis|orthonormal bases]] of the tangent fibers of ''M'' is a '''Z'''<sub>2</sub> quotient of a principal spin bundle.
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| If the manifold has a [[CW complex|cell decomposition]] or a [[Triangulation (topology)|triangulation]], a spin structure can equivalently be thought of as a homotopy-class of trivialization of the [[tangent bundle]] over the 1-[[skeleton]] that extends over the 2-skeleton. If the dimension is lower than 3, one first takes a Whitney sum with a trivial line bundle.
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| ===Obstruction===
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| A spin structure on a vector bundle ''E'' exists if and only if the second [[Stiefel-Whitney class]] ''w''<sub>2</sub> of ''E'' vanishes. This is a result of [[Armand Borel]] and [[Friedrich Hirzebruch]].<ref>{{cite journal|author=A. Borel|coauthors=F. Hirzebruch|title=Characteristic classes and homogeneous spaces I|journal=[[American Journal of Mathematics]]|volume=80|year=1958|pages=97–136|doi=10.2307/2372795|jstor=2372795|issue=2}}</ref> Note, we have assumed π<sub>''E''</sub>: ''E'' → ''M'' is an [[orientable]] [[vector bundle]].
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| ===Classification===
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| When spin structures exist, the inequivalent spin structures on a manifold have a one-to-one correspondence (not canonical) with the elements of H<sup>1</sup>(''M'','''Z'''<sub>2</sub>), which by the [[universal coefficient theorem]] is isomorphic to H<sub>1</sub>(''M'','''Z'''<sub>2</sub>). More precisely, the space of the isomorphism classes of spin structures is an [[affine space]] over H<sup>1</sup>(''M'','''Z'''<sub>2</sub>).
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| Intuitively, for each nontrivial cycle on ''M'' a spin structure corresponds to a binary choice of whether a section of the SO(''N'') bundle switches sheets when one encircles the loop. If ''w''<sub>2</sub> vanishes then these choices may be extended over the two-[[skeleton (topology)|skeleton]], then (by [[obstruction theory]]) they may automatically be extended over all of ''M''. In [[particle physics]] this corresponds to a choice of periodic or antiperiodic [[boundary condition]]s for [[fermions]] going around each loop.
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| ===Application to particle physics===
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| In [[particle physics]] the [[spin statistics theorem]] implies that the [[wavefunction]] of an uncharged [[fermion]] is a section of the [[associated vector bundle]] to the ''spin'' lift of an SO(''N'') bundle ''E''. Therefore the choice of spin structure is part of the data needed to define the wavefunction, and one often needs to sum over these choices in the [[partition function (quantum field theory)|partition function]]. In many physical theories ''E'' is the [[tangent bundle]], but for the fermions on the worldvolumes of [[D-branes]] in [[string theory]] it is a [[normal bundle]].
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| ===Examples===
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| # A [[genus (mathematics)|genus]] ''g'' [[Riemann surface]] admits 2<sup>2''g''</sup> inequivalent spin structures; see [[theta characteristic]].
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| # If ''H''<sup>2</sup>(''M'','''Z'''<sub>2</sub>) vanishes, ''M'' is ''spin''. For example, ''S''<sup>n</sup> is ''spin'' for all ''n'' except 2. (''S''<sup>2</sup> is also ''spin'' for different reasons; see below.)
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| # The complex [[projective plane]] '''CP'''<sup>2</sup> is not ''spin''.
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| # More generally, all even-dimensional [[complex projective space]]s '''CP'''<sup>2n</sup> are not ''spin''.
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| # All odd-dimensional [[complex projective space]]s '''CP'''<sup>2n+1</sup> are ''spin''.
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| # All compact, [[orientable manifold]]s of dimension 3 or less are ''spin''.
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| # All [[Calabi-Yau manifold]]s are ''spin''.
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| === Properties ===
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| * The [[Â genus]] of a spin manifold is an integer, and is an even integer if in addition the dimension is 4 mod 8.
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| *:In general the [[Â genus]] is a rational invariant, defined for any manifold, but it is not in general an integer.
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| *:This was originally proven by [[Friedrich Hirzebruch|Hirzebruch]] and [[Armand Borel|Borel]], and can be proven by the [[Atiyah–Singer index theorem]], by realizing the [[Â genus]] as the index of a [[Dirac operator]] – a Dirac operator is a square root of a second order operator, and exists due to the spin structure being a "square root". This was a motivating example for the index theorem.
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| ==Spin<sup>c</sup> structures==
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| A '''spin<sup>''c''</sup> structure''' is analogous to a spin structure on an oriented [[Riemannian manifold]],<ref>{{Cite book | last1=Lawson | first1=H. Blaine | last2=Michelsohn | first2=Marie-Louise|author2-link=Marie-Louise Michelsohn | title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 | postscript=<!--None-->}} page 391
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| </ref> but uses the spin<sup>''c''</sup> group, which is defined instead by the [[exact sequence]]
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| :<math>1 \to \mathbb{Z}_2 \to {\mathrm {Spin}}^{\mathbb C}(n) \to {\mathrm {SO}}(n)\times {\mathrm U}(1) \to 1.</math>
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| To motivate this, suppose that κ: Spin(''n'') → U(''N'') is a complex spinor representation. The center of U(''N'') consists of the diagonal elements coming from the inclusion i: U(1) → U(''N''), i.e., the scalar multiples of the identity. Thus there is a [[homomorphism]]
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| :<math>\kappa\times i\colon {\mathrm {Spin}}(n)\times {\mathrm U}(1)\to {\mathrm U}(N).</math>
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| This will always have the element (-1,-1) in the kernel. Taking the quotient modulo this element gives the group Spin<sup>'''C'''</sup>(''n''). This is the twisted product
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| :<math>{\mathrm {Spin}}^{\mathbb C}(n) = {\mathrm {Spin}}(n)\times_{\Bbb Z_2} {\mathrm U}(1)\, ,</math>
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| where U(1) = SO(2) = '''S'''<sup>1</sup>. In other words, the group Spin<sup>''c''</sup>(''n'') is a [[central extension (mathematics)|central extension]] of SO(''n'') by '''S'''<sup>1</sup>. | |
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| Viewed another way, Spin<sup>''c''</sup>(''n'') is the quotient group obtained from Spin(''n'') × Spin(2) with respect to the normal '''Z'''<sub>2</sub> which is generated by the pair of covering transformations for the bundles Spin(''n'') → SO(''n'') and Spin(2) → SO(2) respectively. This makes the '''spin<sup>''c''</sup>''' group both a bundle over the circle with fibre Spin(''n''), and a bundle over SO(''n'') with fibre a circle.<ref>{{cite journal|title=''Spin<sup>c</sup>–structures and homotopy equivalences''|author=R. Gompf|doi=10.2140/gt.1997.1.41|journal=[[Geometry & Topology]]|volume=1|year=1997|pages=41–50}}</ref><ref>{{Cite book | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| page 26</ref>
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| The fundamental group π<sub>1</sub>(Spin<sup>'''C'''</sup>(''n'')) is isomorphic to '''Z'''.
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| If the manifold has a [[CW complex|cell decomposition]] or a [[Triangulation (topology)|triangulation]], a '''spin<sup>''c''</sup> structure''' can be equivalently thought of as a homotopy class of [[complex structure]]{{disambiguation needed|date=September 2012}} over the 2-[[skeleton]] that extends over the 3-skeleton. Similarly to the case of spin structures, one takes a Whitney sum with a trivial line bundle if the manifold is odd dimensional.
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| Yet another definition is that a '''spin<sup>''c''</sup> structure''' on a manifold ''N'' is a complex line bundle ''L'' over ''N'' together with a spin structure on ''TN'' ⊕ ''L''.
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| ===Obstruction===
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| A spin<sup>''c''</sup> structure exists when the bundle is orientable and the second [[Stiefel-Whitney class]] of the bundle ''E'' is in the image of the map ''H''<sup>2</sup>(''M'', '''Z''') → ''H''<sup>2</sup>(''M'', '''Z'''/2'''Z''') (in other words, the third '''integral''' Stiefel-Whitney class vanishes). In this case one says that ''E'' is spin<sup>''c''</sup>. Intuitively, the lift gives the [[Chern class]] of the square of the U(1) part of any obtained ''spin''<sup>''c''</sup> bundle.
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| By a theorem of Hopf and Hirzebruch, closed orientable 4-manifolds always admit a spin<sup>c</sup> structure.
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| ===Classification===
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| When a manifold carries a spin<sup>''c''</sup> structure at all, the set of spin<sup>''c''</sup> structures forms an affine space. Moreover, the set of spin<sup>''c''</sup> structures has a free transitive action of ''H''<sup>2</sup>(''M'', '''Z'''). Thus, spin<sup>''c''</sup>-structures correspond to elements of ''H''<sup>2</sup>(''M'', '''Z''') although not in a natural way.
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| ====Geometric picture====
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| This has the following geometric interpretation, which is due to [[Edward Witten]]. When the ''spin''<sup>''c''</sup> structure is nonzero this square root bundle has a non-integral Chern class, which means that it fails the [[triple overlap condition]]. In particular, the product of transition functions on a three-way intersection is not always equal to one, as is required for a [[principal bundle]]. Instead it is sometimes −1.
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| This failure occurs at precisely the same intersections as an identical failure in the triple products of transition functions of the obstructed [[spinor bundle|spin bundle]]. Therefore the triple products of transition functions of the full ''spin''<sup>''c''</sup> bundle, which are the products of the triple product of the ''spin'' and U(1) component bundles, are either 1<sup>2</sup>=1 or -1<sup>2</sup>=1 and so the ''spin''<sup>''c''</sup> bundle satisfies the triple overlap condition and is therefore a legitimate bundle.
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| ====The details====
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| The above intuitive geometric picture may be made concrete as follows. Consider the [[short exact sequence]] 0 → '''Z''' → '''Z''' → '''Z'''<sub>2</sub> → 0 where the second [[arrow]] is [[multiplication]] by 2 and the third is reduction modulo 2. This induces a [[long exact sequence]] on cohomology, which contains
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| ::<math>\dots \longrightarrow \textrm H^2(M;\mathbf Z) \stackrel {2} {\longrightarrow} \textrm H^2(M;\mathbf Z) \longrightarrow \textrm H^2(M;\mathbf Z_2) \stackrel {\beta}\longrightarrow \textrm H^3(M;\mathbf Z) \longrightarrow \dots</math>
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| where the second [[arrow]] is induced by multiplication by 2, the third is induced by restriction modulo 2 and the fourth is the associated [[Bockstein homomorphism]] β.
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| The obstruction to the existence of a ''spin'' bundle is an element ''w''<sub>2</sub> of H<sup>2</sup>(''M'','''Z'''<sub>2</sub>). It reflects the fact that one may always locally lift an SO(N) bundle to a ''spin'' bundle, but one needs to choose a '''Z'''<sub>2</sub> lift of each transition function, which is a choice of sign. The lift does not exist when the product of these three signs on a triple overlap is -1, which yields the [[Čech cohomology]] picture of ''w''<sub>2</sub>.
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| To cancel this obstruction, one tensors this ''spin'' bundle with a U(1) bundle with the same obstruction ''w''<sub>2</sub>. Notice that this is an abuse of the word ''bundle'', as neither the ''spin'' bundle nor the U(1) bundle satisfies the triple overlap condition and so neither is actually a bundle.
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| A legitimate U(1) bundle is classified by its [[Chern class]], which is an element of H<sup>2</sup>(''M'','''Z'''). Identify this class with the first element in the above exact sequence. The next arrow doubles this Chern class, and so legitimate bundles will correspond to even elements in the second H<sup>2</sup>(''M'','''Z'''), while odd elements will correspond to bundles that fail the triple overlap condition. The obstruction then is classified by the failure of an element in the second H<sup>2</sup>(''M'','''Z''') to be in the image of the arrow, which, by exactness, is classified by its image in H<sup>2</sup>(''M'','''Z'''<sub>2</sub>) under the next arrow.
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| To cancel the corresponding obstruction in the ''spin'' bundle, this image needs to be ''w''<sub>2</sub>. In particular, if ''w''<sub>2</sub> is not in the image of the arrow, then there does not exist any U(1) bundle with obstruction equal to ''w''<sub>2</sub> and so the obstruction cannot be cancelled. By exactness, ''w''<sub>2</sub> is in the image of the preceding arrow only if it is in the kernel of the next arrow, which we recall is the [[Bockstein homomorphism]] β. That is, the condition for the cancellation of the obstruction is
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| :::<math>W_3=\beta w_2=0</math>
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| where we have used the fact that the third '''integral''' Stiefel-Whitney class ''W''<sub>3</sub> is the Bockstein of the second Stiefel-Whitney class ''w''<sub>2</sub> (this can be taken as a definition of ''W''<sub>3</sub>).
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| ====Integral lifts of Stiefel-Whitney classes====
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| This argument also demonstrates that second Stiefel-Whitney class defines elements not only of '''Z'''<sub>2</sub> cohomology but also of integral cohomology in one higher degree. In fact this is the case for all even Stiefel-Whitney classes. It is traditional to use an uppercase ''W'' for the resulting classes in odd degree, which are called the integral Stiefel-Whitney classes, and are labeled by their degree (which is always odd).
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| ===Application to particle physics===
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| In [[quantum field theory]] charged spinors are sections of associated ''spin''<sup>''c''</sup> bundles, and in particular no charged spinors can exist on a space that is not ''spin''<sup>''c''</sup>. An exception arises in some [[supergravity]] theories where additional interactions imply that other fields may cancel the third Stiefel-Whitney class.
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| ===Examples===
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| # All [[oriented]] [[smooth manifold]]s of dimension 4 or less are ''spin''<sup>''c''</sup>.<ref>{{Cite book | last1=Gompf | first1=Robert E. | last2=Stipsicz | first2=Andras I. | title=4-Manifolds and Kirby Calculus | publisher=[[American Mathematical Society]] | isbn=0-8218-0994-6 | year=1999 | pages=55–58, 186–187}}</ref>
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| # All [[almost complex manifold]]s are ''spin''<sup>''c''</sup>.
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| # All ''spin'' manifolds are ''spin''<sup>''c''</sup>.
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| ==Vector structures==
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| While spin structures are lifts of [[vector bundle]]s to associated spin bundles, vector structures are lifts of other bundles to associated vector bundles.
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| ===Obstruction===
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| For example, consider an SO(8) bundle. The group SO(8) has three 8-dimensional representations, two of which are spinorial and one of which is the vector representation. These three representations are exchanged by an isomorphism known as triality. Given an SO(8) vector bundle E, the obstruction to the construction of an associated spin bundle is the second Stiefel-Whitney class ''w''<sub>2</sub>(''E''), which is an element of the second cohomology group with '''Z'''<sub>2</sub> coefficients. By triality, given an SO(8) spin bundle F, the obstruction to the existence of an associated vector bundle is another element of the same cohomology group, which is often denoted <math>\hat{w}_2(F)</math>.
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| ===Application to particle physics===
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| Vector structures were first considered in physics, in the paper [http://www.arxiv.org/abs/hep-th/9605184 Anomalies, Dualities and Topology of ''D''=6, ''N''=1 Superstring Vacua] by [[Micha Berkooz]], [[Robert Leigh]], [[Joseph Polchinski]], [[John Henry Schwarz|John Schwarz]], [[Nathan Seiberg]] and [[Edward Witten]]. They were considering [[type I string theory]], whose configurations consist of a 10-manifold with a Spin(32)/'''Z'''<sub>2</sub> principle bundle over it. Such a bundle has a vector structure, and so lifts to an SO(32) bundle, when the triple product of the transition functions on all triple intersection is the trivial element of the '''Z'''<sub>2</sub> quotient. This happens precisely when <math>\hat{w}_2</math>, the characteristic 2-cocycle with '''Z'''<sub>2</sub> coefficients, vanishes.
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| The following year, in
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| [http://www.arxiv.org/abs/hep-th/9703157 The Mirror Transform of Type I Vacua in Six Dimensions], [[Ashoke Sen]] and [[Savdeep Sethi]] demonstrated that type I superstring theory is only consistent, in the absence of fluxes, when this characteristic class is trivial. More generally, in type I string theory the [[B-field]] is also a class in the second cohomology with Z<sub>2</sub> coefficients and they demonstrated that it must be equal to <math>\hat{w}_2</math>. | |
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| ==See also==
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| * [[Orthonormal frame bundle]]
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| * [[Spinor]]
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| * [[Spinor bundle]]
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| * [[Spin manifold]]
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| ==References==
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| <references/>
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| ==Further reading==
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| * {{Cite book | last1=Lawson | first1=H. Blaine | last2=Michelsohn | first2=Marie-Louise | title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 | postscript=<!--None-->}}
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| * {{Cite book | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| * {{Cite book | last1=Karoubi | first1=Max|title=K-Theory | publisher=Springer | isbn=978-3-540-79889-7 | year=2008 |pages=212-214| postscript=<!--None-->}}
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| ==External links==
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| *[http://earthlingsoft.net/ssp/studium/2001spring/Spin.pdf Something on Spin Structures] by Sven-S. Porst is a short introduction to [[orientability|orientation]] and spin structures for mathematics students.
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| [[Category:Riemannian manifolds|Structures on Riemannian manifolds]]
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| [[Category:Structures on manifolds]]
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| [[Category:Algebraic topology]]
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| [[Category:K-theory]]
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| [[Category:Mathematical physics]]
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