Indifferential geometry, aspin structure on anorientableRiemannian manifold(M,g) allows one to define associatedspinor bundles, giving rise to the notion of aspinor in differential geometry.

Spin structures have wide applications tomathematical physics, in particular toquantum field theory where they are an essential ingredient in the definition of any theory with unchargedfermions. They are also of purely mathematical interest indifferential geometry,algebraic topology, andK theory. They form the foundation forspin geometry.

Ingeometry and infield theory, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admitsspinors. One method for dealing with this problem is to require thatM have a spin structure.^[1][2][3] 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 secondStiefel–Whitney classw₂(M) ∈ H²(M,Z₂) ofM vanishes. Furthermore, ifw₂(M) = 0, then the set of the isomorphism classes of spin structures onM is acted upon freely and transitively by H¹(M,Z₂) . As the manifoldM is assumed to be oriented, the first Stiefel–Whitney classw₁(M) ∈ H¹(M,Z₂) ofM vanishes too. (The Stiefel–Whitney classesw_i(M) ∈ Hⁱ(M,Z₂) of a manifoldM are defined to be the Stiefel–Whitney classes of itstangent bundleTM.)

The bundle of spinors π_S:S →M overM is then thecomplex vector bundle associated with the correspondingprincipal bundle π_P:P →M ofspin frames overM and the spin representation of its structure group Spin(n) on the space of spinors Δ_n. The bundleS is called the spinor bundle for a given spin structure onM.

A precise definition of spin structure on manifold was possible only after the notion offiber bundle had been introduced;André Haefliger (1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold andMax Karoubi (1968) extended this result to the non-orientable pseudo-Riemannian case.^[4][5]

A spin structure on anorientableRiemannian manifold${\displaystyle (M,g)}$ with an oriented vector bundle${\displaystyle E}$ is anequivariantlift of the orthonormal frame bundle${\displaystyle P_{\mathrm{SO}}(E)\to M}$ with respect to the double covering${\displaystyle \rho :\mathrm{Spin}(n)\to \mathrm{SO}(n)}$. In other words, a pair${\displaystyle (P_{\mathrm{Spin}},\varphi )}$ is a spin structure on the SO(n)-principal bundle${\displaystyle \pi :P_{\mathrm{SO}}(E)\to M}$ when

a)${\displaystyle {\pi }_P:P_{\mathrm{Spin}}\to M}$ is a principal Spin(n)-bundle over${\displaystyle M}$, and

b)${\displaystyle \varphi :P_{\mathrm{Spin}}\to P_{\mathrm{SO}}(E)}$ is anequivariant 2-foldcovering map such that

${\displaystyle \pi \circ \varphi ={\pi }_P}$and${\displaystyle \varphi (pq)=\varphi (p)\rho (q)}$for all${\displaystyle p\in P_{\mathrm{Spin}}}$ and${\displaystyle q\in \mathrm{Spin}(n)}$.

Two spin structures${\displaystyle (P_1,{\varphi }_1)}$ and${\displaystyle (P_2,{\varphi }_2)}$ on the same orientedRiemannian manifold are called "equivalent" if there exists a Spin(n)-equivariant map${\displaystyle f:P_1\to P_2}$ such that

${\displaystyle {\varphi }_2\circ f={\varphi }_1}$ and${\displaystyle f(pq)=f(p)q}$ for all${\displaystyle p\in P_1}$ and${\displaystyle q\in \mathrm{Spin}(n)}$.

In this case${\displaystyle {\varphi }_1}$ and${\displaystyle {\varphi }_2}$ are two equivalent double coverings.

The definition of spin structure on${\displaystyle (M,g)}$ as a spin structure on the principal bundle${\displaystyle P_{\mathrm{SO}}(E)\to M}$ is due toAndré Haefliger (1956).

Haefliger^[1] 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 a certain element [k] of H²(M,Z₂) . For a spin structure the class [k] is the secondStiefel–Whitney classw₂(M) ∈ H²(M,Z₂) ofM. Hence, a spin structure exists if and only if the second Stiefel–Whitney classw₂(M) ∈ H²(M,Z₂) ofM vanishes.

LetM be aparacompacttopological manifold andE anoriented vector bundle onM of dimensionn equipped with afibre metric. This means that at each point ofM, the fibre ofE is aninner product space. A spinor bundle ofE is a prescription for consistently associating aspin representation to every point ofM. There are topological obstructions to being able to do it, and consequently, a given bundleE may not admit any spinor bundle. In case it does, one says that the bundleE isspin.

This may be made rigorous through the language ofprincipal bundles. The collection of orientedorthonormal frames of a vector bundle form aframe bundleP_SO(E), which is a principal bundle under the action of thespecial orthogonal group SO(n). A spin structure forP_SO(E) is alift ofP_SO(E) to a principal bundleP_Spin(E) under the action of thespin group Spin(n), by which we mean that there exists a bundle map${\displaystyle \varphi }$ :P_Spin(E) →P_SO(E) such that

${\displaystyle \varphi (pg)=\varphi (p)\rho (g)}$, for allp ∈P_Spin(E) andg ∈ Spin(n),

whereρ : Spin(n) → SO(n) is the mapping of groups presenting the spin group as a double-cover of SO(n).

In the special case in whichE is thetangent bundleTM over the base manifoldM, if a spin structure exists then one says thatM is aspin manifold. EquivalentlyM isspin if the SO(n) principal bundle oforthonormal bases of the tangent fibers ofM is aZ₂ quotient of a principal spin bundle.

If the manifold has acell decomposition or atriangulation, a spin structure can equivalently be thought of as ahomotopy class of a trivialization of thetangent 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.

For anorientablevector bundle${\displaystyle {\pi }_E:E\to M}$ a spin structure exists on${\displaystyle E}$ if and only if the secondStiefel–Whitney class${\displaystyle w_2(E)}$ vanishes. This is a result ofArmand Borel andFriedrich Hirzebruch.^[6] Furthermore, in the case${\displaystyle E\to M}$ is spin, the number of spin structures are in bijection with${\displaystyle H^1(M,\mathbb{Z}/2)}$. These results can be easily proven^{[7]pg 110-111} using a spectral sequence argument for the associated principal${\displaystyle \mathrm{SO}(n)}$-bundle${\displaystyle P_E\to M}$. Notice this gives afibration

${\displaystyle \mathrm{SO}(n)\to P_E\to M}$

hence theSerre spectral sequence can be applied. From general theory of spectral sequences, there is an exact sequence

${\displaystyle 0\to E_3^{0,1}\to E_2^{0,1}\stackrel{d_2}{\to }{E}_2^{2,0}\to E_3^{2,0}\to 0}$

where

In addition,${\displaystyle E_{\mathrm{\infty }}^{0,1}=E_3^{0,1}}$ and${\displaystyle E_{\mathrm{\infty }}^{0,1}=H^1(P_E,\mathbb{Z}/2)/F^1(H^1(P_E,\mathbb{Z}/2))}$ for some filtration on${\displaystyle H^1(P_E,\mathbb{Z}/2)}$, hence we get a map

${\displaystyle H^1(P_E,\mathbb{Z}/2)\to E_3^{0,1}}$

giving an exact sequence

${\displaystyle H^1(P_E,\mathbb{Z}/2)\to H^1(\mathrm{SO}(n),\mathbb{Z}/2)\to H^2(M,\mathbb{Z}/2)}$

Now, a spin structure is exactly a double covering of${\displaystyle P_E}$ fitting into a commutative diagram

where the two left vertical maps are the double covering maps. Now, double coverings of${\displaystyle P_E}$ are in bijection with index${\displaystyle 2}$ subgroups of${\displaystyle {\pi }_1(P_E)}$, which is in bijection with the set of group morphisms${\displaystyle \text{Hom}({\pi }_1(E),\mathbb{Z}/2)}$. But, fromHurewicz theorem and change of coefficients, this is exactly the cohomology group${\displaystyle H^1(P_E,\mathbb{Z}/2)}$. Applying the same argument to${\displaystyle \mathrm{SO}(n)}$, the non-trivial covering${\displaystyle \mathrm{Spin}(n)\to \mathrm{SO}(n)}$ corresponds to${\displaystyle 1\in H^1(\mathrm{SO}(n),\mathbb{Z}/2)=\mathbb{Z}/2}$, and the map to${\displaystyle H^2(M,\mathbb{Z}/2)}$ is precisely the${\displaystyle w_2}$ of the second Stiefel–Whitney class, hence${\displaystyle w_2(1)=w_2(E)}$. If it vanishes, then the inverse image of${\displaystyle 1}$ under the map

${\displaystyle H^1(P_E,\mathbb{Z}/2)\to H^1(\mathrm{SO}(n),\mathbb{Z}/2)}$

is the set of double coverings giving spin structures. Now, this subset of${\displaystyle H^1(P_E,\mathbb{Z}/2)}$ can be identified with${\displaystyle H^1(M,\mathbb{Z}/2)}$, showing this latter cohomology group classifies the various spin structures on the vector bundle${\displaystyle E\to M}$. This can be done by looking at the long exact sequence of homotopy groups of the fibration

${\displaystyle {\pi }_1(\mathrm{SO}(n))\to {\pi }_1(P_E)\to {\pi }_1(M)\to 1}$

and applying${\displaystyle \text{Hom}(-,\mathbb{Z}/2)}$, giving the sequence of cohomology groups

${\displaystyle 0\to H^1(M,\mathbb{Z}/2)\to H^1(P_E,\mathbb{Z}/2)\to H^1(\mathrm{SO}(n),\mathbb{Z}/2)}$

Because${\displaystyle H^1(M,\mathbb{Z}/2)}$ is the kernel, and the inverse image of${\displaystyle 1\in H^1(\mathrm{SO}(n),\mathbb{Z}/2)}$ is in bijection with the kernel, we have the desired result.

When spin structures exist, the inequivalent spin structures on a manifold have a one-to-one correspondence (not canonical) with the elements of H¹(M,Z₂), which by theuniversal coefficient theorem is isomorphic to H₁(M,Z₂). More precisely, the space of the isomorphism classes of spin structures is anaffine space over H¹(M,Z₂).

Intuitively, for each nontrivial cycle onM a spin structure corresponds to a binary choice of whether a section of the SO(N) bundle switches sheets when one encircles the loop. Ifw₂^[8] vanishes then these choices may be extended over the two-skeleton, then (byobstruction theory) they may automatically be extended over all ofM. Inparticle physics this corresponds to a choice of periodic or antiperiodicboundary conditions forfermions going around each loop. Note that on a complex manifold${\displaystyle X}$ the second Stiefel-Whitney class can be computed as the firstchern class${\displaystyle \text{mod }2}$.

1. AgenusgRiemann surface admits 2^2g inequivalent spin structures; seetheta characteristic.
2. IfH²(M,Z₂) vanishes,M isspin. For example,Sⁿ isspin for all${\displaystyle n\ne 2}$. (Note thatS² is alsospin, but for different reasons; see below.)
3. Thecomplex projective planeCP² is notspin.
4. More generally, all even-dimensionalcomplex projective spacesCP²ⁿ are notspin.
5. All odd-dimensionalcomplex projective spacesCP²ⁿ⁺¹ arespin.
6. All compact,orientable manifolds ofdimension 3 or less arespin.
7. AllCalabi–Yau manifolds arespin.

• The genus of a spin manifold is an integer, and is an even integer if in addition the dimension is 4 mod 8.

  In general the genus is a rational invariant, defined for any manifold, but it is not in general an integer.

  This was originally proven byHirzebruch andBorel, and can be proven by theAtiyah–Singer index theorem, by realizing the genus as the index of aDirac 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.

A spin^C structure is analogous to a spin structure on an orientedRiemannian manifold,^[9] but uses the Spin^C group, which is defined instead by theexact sequence

${\displaystyle 1\to {\mathbb{Z}}_2\to {\mathrm{Spin}}^{\mathbf{C}}(n)\to \mathrm{SO}(n)\times \mathrm{U}(1)\to 1.}$

To motivate this, suppose thatκ : Spin(n) → U(N) is a complex spinor representation. The center of U(N) consists of the diagonal elements coming from the inclusioni : U(1) → U(N), i.e., the scalar multiples of the identity. Thus there is ahomomorphism

${\displaystyle \kappa \times i:\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\times \mathrm{U}(1)\to \mathrm{U}(N).}$

This will always have the element (−1,−1) in the kernel. Taking the quotient modulo this element gives the group Spin^C(n). This is the twisted product

${\displaystyle {\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{\mathbb{C}}(n)=\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n){\times }_{{\mathbb{Z}}_2}\mathrm{U}(1),}$

where U(1) = SO(2) =S¹. In other words, the group Spin^C(n) is acentral extension of SO(n) byS¹.

Viewed another way, Spin^C(n) is the quotient group obtained fromSpin(n) × Spin(2) with respect to the normalZ₂ which is generated by the pair of covering transformations for the bundlesSpin(n) → SO(n) andSpin(2) → SO(2) respectively. This makes the Spin^C group both a bundle over the circle with fibre Spin(n), and a bundle over SO(n) with fibre a circle.^[10][11]

The fundamental group π₁(Spin^C(n)) is isomorphic toZ ifn ≠ 2, and toZ ⊕Z ifn = 2.

If the manifold has acell decomposition or atriangulation, a spin^C structure can be equivalently thought of as a homotopy class ofcomplex structure 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.

Yet another definition is that a spin^C structure on a manifoldN is a complex line bundleL overN together with a spin structure onTN ⊕L.

A spin^C structure exists when the bundle is orientable and the secondStiefel–Whitney class of the bundleE is in the image of the mapH²(M,Z) →H²(M,Z/2Z) (in other words, the third integral Stiefel–Whitney class vanishes). In this case one says thatE is spin^C. Intuitively, the lift gives theChern class of the square of the U(1) part of any obtained spin^C bundle.By a theorem of Hopf and Hirzebruch, closed orientable 4-manifolds always admit a spin^C structure.

When a manifold carries a spin^C structure at all, the set of spin^C structures forms an affine space. Moreover, the set of spin^C structures has a free transitive action ofH²(M,Z). Thus, spin^C-structures correspond to elements ofH²(M,Z) although not in a natural way.

This has the following geometric interpretation, which is due toEdward Witten. When the spin^C structure is nonzero this square root bundle has a non-integral Chern class, which means that it fails thetriple overlap condition. In particular, the product of transition functions on a three-way intersection is not always equal to one, as is required for aprincipal bundle. Instead it is sometimes −1.

This failure occurs at precisely the same intersections as an identical failure in the triple products of transition functions of the obstructedspin bundle. Therefore, the triple products of transition functions of the fullspin^c bundle, which are the products of the triple product of thespin and U(1) component bundles, are either1² = 1 or(−1)² = 1 and so the spin^C bundle satisfies the triple overlap condition and is therefore a legitimate bundle.

The above intuitive geometric picture may be made concrete as follows. Consider theshort exact sequence0 →Z →Z →Z₂ → 0, where the secondarrow ismultiplication by 2 and the third is reduction modulo 2. This induces along exact sequence on cohomology, which contains

${\displaystyle \cdots ⟶{\text{H}}^2(M;\mathbf{Z})\stackrel{2}{⟶}{\text{H}}^2(M;\mathbf{Z})⟶{\text{H}}^2(M;{\mathbf{Z}}_2)\stackrel{\beta }{⟶}{\text{H}}^3(M;\mathbf{Z})⟶\dots ,}$

where the secondarrow is induced by multiplication by 2, the third is induced by restriction modulo 2 and the fourth is the associatedBockstein homomorphismβ.

The obstruction to the existence of aspin bundle is an elementw₂ ofH²(M,Z₂). It reflects the fact that one may always locally lift an SO(n) bundle to aspin bundle, but one needs to choose aZ₂ 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 theČech cohomology picture ofw₂.

To cancel this obstruction, one tensors thisspin bundle with a U(1) bundle with the same obstructionw₂. Notice that this is an abuse of the wordbundle, as neither thespin bundle nor the U(1) bundle satisfies the triple overlap condition and so neither is actually a bundle.

A legitimate U(1) bundle is classified by itsChern class, which is an element of H²(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 secondH²(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 H²(M,Z) to be in the image of the arrow, which, by exactness, is classified by its image in H²(M,Z₂) under the next arrow.

To cancel the corresponding obstruction in thespin bundle, this image needs to bew₂. In particular, ifw₂ is not in the image of the arrow, then there does not exist any U(1) bundle with obstruction equal tow₂ and so the obstruction cannot be cancelled. By exactness,w₂ is in the image of the preceding arrow only if it is in the kernel of the next arrow, which we recall is theBockstein homomorphism β. That is, the condition for the cancellation of the obstruction is

${\displaystyle W_3=\beta w_2=0}$

where we have used the fact that the thirdintegral Stiefel–Whitney classW₃ is the Bockstein of the second Stiefel–Whitney classw₂ (this can be taken as a definition ofW₃).

This argument also demonstrates that second Stiefel–Whitney class defines elements not only ofZ₂ 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 uppercaseW 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).

1. Allorientedsmooth manifolds of dimension 4 or less are spin^C.^[12]
2. Allalmost complex manifolds are spin^C.
3. Allspin manifolds are spin^C.

Inparticle physics thespin–statistics theorem implies that thewavefunction of an unchargedfermion is a section of theassociated vector bundle to thespin lift of an SO(N) bundleE. 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 thepartition function. In many physical theoriesE is thetangent bundle, but for the fermions on the worldvolumes ofD-branes instring theory it is anormal bundle.

Inquantum field theory charged spinors are sections of associatedspin^c bundles, and in particular no charged spinors can exist on a space that is notspin^c. An exception arises in somesupergravity theories where additional interactions imply that other fields may cancel the third Stiefel–Whitney class. The mathematical description of spinors in supergravity and string theory is a particularly subtle open problem, which was recently addressed in references.^[13][14] It turns out that the standard notion of spin structure is too restrictive for applications to supergravity and string theory, and that the correct notion of spinorial structure for the mathematical formulation of these theories is a "Lipschitz structure".^[13][15]

• Metaplectic structure
• Orthonormal frame bundle
• Spinor

• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989).Spin Geometry.Princeton University Press.ISBN 978-0-691-08542-5.
• Friedrich, Thomas (2000).Dirac Operators in Riemannian Geometry.American Mathematical Society.ISBN 978-0-8218-2055-1.
• Karoubi, Max (2008).K-Theory. Springer. pp. 212–214.ISBN 978-3-540-79889-7.
• Greub, Werner; Petry, Herbert-Rainer (2006) [1978]."On the lifting of structure groups".Differential Geometrical Methods in Mathematical Physics II. Lecture Notes in Mathematics. Vol. 676. Springer-Verlag. pp. 217–246.doi:10.1007/BFb0063673.ISBN 9783540357216.
• Scorpan, Alexandru (2005)."4.5 Notes Spin structures, the structure group definition; Equivalence of the definitions of".The wild world of 4-manifolds. American Mathematical Society. pp. 174–189.ISBN 9780821837498.

• Something on Spin Structures by Sven-S. Porst is a short introduction toorientation and spin structures for mathematics students.

• Riemannian manifolds
• Structures on manifolds
• Algebraic topology
• K-theory
• Mathematical physics

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