Clifford Algebras and Spin Structures

Clifford multiplication is defined pointwise by the metric, but a globally defined spinor field requires additional topology: the oriented, time-oriented orthonormal frame bundle must lift through the double cover $\mathrm{Spin}^+(1,3)\to SO^+(1,3)$. After orientation and time orientation are secured, the obstruction is the second Stiefel–Whitney class $w_2(TM)$; when the obstruction vanishes, inequivalent lifts form a torsor for $H^1(M;\mathbb Z_2)$.

Required background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian inner products and signature; Orthonormal Tetrads and Measured Components supplies orthonormal frames and coframes. Affine Connections and Covariant Derivatives supplies the general covariant-derivative framework later lifted to spinors.

Helpful background. Calculus and Linear Algebra for Relativity supplies linear maps and bilinear forms; the associative algebra and representation language is introduced here. Exterior Algebra and Differential Forms supplies alternating forms, wedge products, and their grading. Killing Fields and Isometry Groups supplies the spacetime-isometry side of the comparison; this page introduces the internal spin action.

Clifford multiplication and sign

For a vector space with Lorentzian metric $\eta$, this chapter uses the Clifford relation

$$\gamma^a\gamma^b+\gamma^b\gamma^a=2\eta^{ab}\mathbf 1.$$

Thus $(\gamma^0)^2=-\mathbf1$ and $(\gamma^i)^2=+\mathbf1$ in an orthonormal frame with signature $(-,+,+,+)$. Define

$$\gamma^{ab}=\frac12[\gamma^a,\gamma^b].$$

The antisymmetric matrices $\gamma^{ab}$ generate the infinitesimal spin representation up to the conventional factor used in the connection.

Wernli adopts the opposite Clifford sign, $c(v)c(w)+c(w)c(v)=-2g(v,w)$. After complexification, the translation $\gamma(v)=i\,c(v)$ gives the plus-sign relation used here. Statements and formulas drawn from that source are translated as a whole, not by changing isolated signs. The algebraic construction and its representations are developed in Wernli 2019, §§1.2–1.3, pp. 12–21.

The Clifford algebra is generated by tangent vectors subject to this relation. Its antisymmetrized products resemble exterior forms as vector spaces, but Clifford multiplication also contains metric contractions; it is not the wedge product.

Spin group and the double cover

The spin group is built from suitable products of unit vectors in the even Clifford algebra. Conjugation on vectors induces an orthogonal transformation, producing a two-to-one homomorphism

$$\mathrm{Spin}^+(1,3)\longrightarrow SO^+(1,3).$$

The two elements $S$ and $-S$ determine the same Lorentz transformation. The half-angle behavior follows directly from the Clifford relation. For two orthonormal spatial directions, $(\gamma^{12})^2=-\mathbf1$, so one choice of oriented rotation lift is

$$S(\alpha)=\exp\!\left(\frac\alpha2\gamma^{12}\right) =\cos\frac\alpha2\,\mathbf1+\sin\frac\alpha2\,\gamma^{12}.$$

Thus $S(2\pi)=-\mathbf1$ and $S(4\pi)=\mathbf1$. The associated vector rotation is already the identity at $2\pi$, because conjugation by $-\mathbf1$ acts trivially on vectors. Spinors and vectors transform in different representations of the covering group; the sign is an algebraic consequence, not a failure of ordinary rotation geometry.

On a contractible tetrad patch, one can choose gamma matrices and a local spin representation without encountering topology. Such a local construction does not prove that the choices glue consistently over every overlap of a manifold.

Global spin structures

Assume the spacetime is oriented and time-oriented, so its orthonormal frame bundle has structure group $SO^+(1,3)$. A spin structure consists of a principal $\mathrm{Spin}^+(1,3)$ bundle together with an equivariant two-to-one map to that frame bundle whose restriction on each fiber is the standard covering map. The associated spinor bundle is then formed from a chosen spin representation.

Three global words carry most of the idea. A lift chooses spin-valued transition functions that project to the orthonormal-frame transitions. An obstruction is the failure of those choices to satisfy the consistency condition on triple overlaps; for spin structures that failure is encoded by $w_2$. A torsor is a set on which a group acts freely and transitively, but with no preferred origin: once one spin structure is chosen, every other one is obtained by adding a class in $H^1(M;\mathbb Z_2)$, although topology does not single out which structure should be called zero.

The obstruction to this lift is

$$w_2(TM)\in H^2(M;\mathbb Z_2).$$

A spin structure exists exactly when $w_2(TM)=0$, after the required orientation reduction. Orientability itself is controlled by $w_1(TM)$ and must not be skipped by quoting only $w_2$. Time orientability is an additional Lorentzian reduction. Geroch analyzes these global spacetime requirements in Geroch 1968, §I, pp. 1739–1742.

When a spin structure exists on a connected manifold, its isomorphism classes form an affine space—a torsor—for $H^1(M;\mathbb Z_2)$. There is no preferred zero structure in general, so saying “equal to $H^1$” would hide the absence of a canonical choice.

Existence versus choice

Minkowski space is contractible, so $w_2=0$ and $H^1=0$; its spin structure is unique up to isomorphism. By contrast,

$$M=\mathbb R\times S^1\times\mathbb R^2$$

is parallelizable and hence spin, but $H^1(M;\mathbb Z_2)\cong\mathbb Z_2$. For a concrete comparison give it the flat product metric $ds^2=-dt^2+a^2d\phi^2+dx^2+dy^2$, where $a\gt0$ and $\phi\sim\phi+2\pi$. The product coframe is parallel. In that trivialization the two spin structures impose periodic or antiperiodic spinor components around the circle, and the lifted flat connection has holonomy $+\mathbf1$ or $-\mathbf1$. Both have the same local Clifford algebra and metric; they differ globally.

The Clifford bundle associated directly with the tangent bundle and metric can be defined without first choosing a spin lift. What can fail is a globally defined bundle carrying the desired spin representation with Clifford multiplication. Other structures, such as spin-c, may exist when spin does not, but they require additional data and lie outside this page.

Wernli gives the principal-bundle lift, obstruction, classification, and associated spinor construction in Wernli 2019, §§2.3.1–2.3.3, pp. 60–70.

Exercises

For the flat product metric and parallel product coframe above, explain why the two spin structures cannot be distinguished inside a contractible coordinate patch, but can be distinguished by parallel transport once around $S^1$.

Solution

Every contractible patch has trivial first cohomology, so either global lift restricts to the same trivial local spin bundle up to isomorphism. The parallel product coframe has zero local connection, so only the spin transition across the circle’s seam remains. It is $+\mathbf1$ or $-\mathbf1$ in the kernel of $\mathrm{Spin}^+(1,3)\to SO^+(1,3)$. A parallel spinor therefore returns with the same sign in the periodic structure and the opposite sign in the antiperiodic structure. Ordinary vectors cannot distinguish the two signs. For a general curved metric, connection holonomy also contributes, so topology alone would not fix the total transport matrix to $\pm\mathbf1$.

References

• Geroch, Robert. “Spinor Structure of Space-Times in General Relativity. I.” Journal of Mathematical Physics 9, 1739–1744 (1968), DOI.
• Wernli, Konstantin. Lecture Notes on Spin Geometry. arXiv:1911.09766 (2019), arXiv.