Spin structure

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

a) ${\displaystyle \pi _{P}:P_{\operatorname {Spin} }\rightarrow M}$ is a principal Spin(n)-bundle over ${\displaystyle M}$, and

b) ${\displaystyle \phi :P_{\operatorname {Spin} }\rightarrow P_{\operatorname {SO} }(E)}$ is an equivariant 2-fold covering map such that

${\displaystyle \pi \circ \phi =\pi _{P}\quad }$and${\displaystyle \quad \phi (pq)=\phi (p)\rho (q)\quad }$for all ${\displaystyle p\in P_{\operatorname {Spin} }}$ and ${\displaystyle q\in \operatorname {Spin} (n)}$.

Two spin structures ${\displaystyle (P_{1},\phi _{1})}$ and ${\displaystyle (P_{2},\phi _{2})}$ on the same oriented Riemannian manifold are called "equivalent" if there exists a Spin(n)-equivariant map ${\displaystyle f:P_{1}\rightarrow P_{2}}$ such that

${\displaystyle \phi _{2}\circ f=\phi _{1}\quad }$ and ${\displaystyle \quad f(pq)=f(p)q\quad }$ for all ${\displaystyle p\in P_{1}}$ and ${\displaystyle q\in \operatorname {Spin} (n)}$.

In this case ${\displaystyle \phi _{1}}$ and ${\displaystyle \phi _{2}}$ are two equivalent double coverings.

The definition of spin structure on ${\displaystyle (M,g)}$ as a spin structure on the principal bundle ${\displaystyle P_{\operatorname {SO} }(E)\rightarrow M}$ is due to André 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 second Stiefel–Whitney class w₂(M) ∈ H²(M, Z₂) of M. Hence, a spin structure exists if and only if the second Stiefel–Whitney class w₂(M) ∈ H²(M, Z₂) of M vanishes.

For an orientable vector bundle ${\displaystyle \pi _{E}:E\to M}$ a spin structure exists on ${\displaystyle E}$ if and only if the second Stiefel–Whitney class ${\displaystyle w_{2}(E)}$ vanishes. This is a result of Armand Borel and Friedrich 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 ${\displaystyle \operatorname {SO} (n)}$-bundle ${\displaystyle P_{E}\to M}$. Notice this gives a fibration

${\displaystyle \operatorname {SO} (n)\to P_{E}\to M}$

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

${\displaystyle 0\to E_{3}^{0,1}\to E_{2}^{0,1}\xrightarrow {d_{2}} E_{2}^{2,0}\to E_{3}^{2,0}\to 0}$

where

${\displaystyle {\begin{aligned}E_{2}^{0,1}&=H^{0}(M,H^{1}(\operatorname {SO} (n),\mathbb {Z} /2))=H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)\\E_{2}^{2,0}&=H^{2}(M,H^{0}(\operatorname {SO} (n),\mathbb {Z} /2))=H^{2}(M,\mathbb {Z} /2)\end{aligned}}}$

In addition, ${\displaystyle E_{\infty }^{0,1}=E_{3}^{0,1}}$ and ${\displaystyle E_{\infty }^{0,1}=H^{1}(P_{E},\mathbb {Z} /2)/F^{1}(H^{1}(P_{E},\mathbb {Z} /2))}$ 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

${\displaystyle H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)\to H^{2}(M,\mathbb {Z} /2)}$

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

${\displaystyle {\begin{matrix}\operatorname {Spin} (n)&\to &{\tilde {P}}_{E}&\to &M\\\downarrow &&\downarrow &&\downarrow \\\operatorname {SO} (n)&\to &P_{E}&\to &M\end{matrix}}}$

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, from Hurewicz theorem and change of coefficients, this is exactly the cohomology group ${\displaystyle H^{1}(P_{E},\mathbb {Z} /2)}$. Applying the same argument to ${\displaystyle \operatorname {SO} (n)}$, the non-trivial covering ${\displaystyle \operatorname {Spin} (n)\to \operatorname {SO} (n)}$ corresponds to ${\displaystyle 1\in H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)=\mathbb {Z} /2}$, 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

${\displaystyle H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)}$

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

${\displaystyle \pi _{1}(\operatorname {SO} (n))\to \pi _{1}(P_{E})\to \pi _{1}(M)\to 1}$

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

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

Because ${\displaystyle H^{1}(M,\mathbb {Z} /2)}$ is the kernel, and the inverse image of ${\displaystyle 1\in H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)}$ 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 the universal coefficient theorem is isomorphic to H₁(M,Z₂). More precisely, the space of the isomorphism classes of spin structures is an affine space over H¹(M,Z₂).

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 w₂[8] vanishes then these choices may be extended over the two-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 conditions for fermions going around each loop. Note that on a complex manifold ${\displaystyle X}$ the second Stiefel-Whitney class can be computed as the first chern class ${\displaystyle {\text{mod }}2}$.

1. A genus g Riemann surface admits 2^2g inequivalent spin structures; see theta characteristic.
2. If H²(M,Z₂) vanishes, M is spin. For example, Sⁿ is spin for all ${\displaystyle n\neq 2}$. (Note that S² is also spin, but for different reasons; see below.)
3. The complex projective plane CP² is not spin.
4. More generally, all even-dimensional complex projective spaces CP²ⁿ are not spin.
5. All odd-dimensional complex projective spaces CP²ⁿ⁺¹ are spin.
6. All compact, orientable manifolds of dimension 3 or less are spin.
7. All Calabi–Yau manifolds are spin.

• 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 by Hirzebruch and Borel, and can be proven by the Atiyah–Singer index theorem, by realizing the  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.

A spin^C structure exists when the bundle is orientable and the second Stiefel–Whitney class of the bundle E is in the image of the map H²(M, Z) → H²(M, Z/2Z) (in other words, the third integral Stiefel–Whitney class vanishes). In this case one says that E is spin^C. Intuitively, the lift gives the Chern 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 of H²(M, Z). Thus, spin^C-structures correspond to elements of H²(M, Z) although not in a natural way.

This has the following geometric interpretation, which is due to Edward Witten. When the spin^C 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.

This failure occurs at precisely the same intersections as an identical failure in the triple products of transition functions of the obstructed spin bundle. Therefore, the triple products of transition functions of the full spin^c bundle, which are the products of the triple product of the spin and U(1) component bundles, are either 1² = 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 the short exact sequence 0 → Z → Z → Z₂ → 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

${\displaystyle \dots \longrightarrow {\textrm {H}}^{2}(M;\mathbf {Z} ){\stackrel {2}{\longrightarrow }}{\textrm {H}}^{2}(M;\mathbf {Z} )\longrightarrow {\textrm {H}}^{2}(M;\mathbf {Z} _{2}){\stackrel {\beta }{\longrightarrow }}{\textrm {H}}^{3}(M;\mathbf {Z} )\longrightarrow \dots ,}$

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.

The obstruction to the existence of a spin bundle is an element w₂ of H²(M,Z₂). It reflects the fact that one may always locally lift an SO(n) bundle to a spin bundle, but one needs to choose a Z₂ 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 w₂.

To cancel this obstruction, one tensors this spin bundle with a U(1) bundle with the same obstruction w₂. 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.

A legitimate U(1) bundle is classified by its Chern 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 second H²(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 the spin bundle, this image needs to be w₂. In particular, if w₂ is not in the image of the arrow, then there does not exist any U(1) bundle with obstruction equal to w₂ 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 the Bockstein 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 third integral Stiefel–Whitney class W₃ is the Bockstein of the second Stiefel–Whitney class w₂ (this can be taken as a definition of W₃).

This argument also demonstrates that second Stiefel–Whitney class defines elements not only of Z₂ 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).

1. All oriented smooth manifolds of dimension 4 or less are spin^C.[12]
2. All almost complex manifolds are spin^C.
3. All spin manifolds are spin^C.