Spin Connections and Spinor Curvature

On a spacetime with a chosen spin structure, the Levi-Civita connection lifts from orthonormal frames to spinors. In the convention $\{\gamma^a,\gamma^b\}=2\eta^{ab}$ and $\gamma^{ab}=\tfrac12[\gamma^a,\gamma^b]$, the lifted derivative is $\nabla_\mu\psi=\partial_\mu\psi+\tfrac14\omega_{\mu ab}\gamma^{ab}\psi$. Its commutator represents Riemann curvature in the spin representation.

Required background. Pseudo-Riemannian Metrics and Signature supplies the Lorentzian inner product and signature; Affine Connections and Covariant Derivatives supplies the general covariant-derivative framework, The Levi-Civita Connection supplies its metric-compatible torsion-free specialization, and The Riemann Curvature Tensor fixes the curvature commutator. Connection Forms and Cartan Structure Equations supplies the frame connection and curvature forms lifted below.

Helpful background. Calculus and Linear Algebra for Relativity supplies linear maps and changes of basis; matrix commutators are used explicitly below. Exterior Algebra and Differential Forms supplies forms and wedge products, while The Exterior Derivative and the Poincare Lemma supplies $d$. Killing Fields and Isometry Groups supplies the spacetime-isometry comparison; local Lorentz-frame freedom is developed in this chapter.

Lifting the frame connection

Let $\theta^a$ be an orthonormal coframe and let $\omega^a{}_b$ be its Levi-Civita connection forms. Metric compatibility gives $\omega_{ab}=-\omega_{ba}$. In a local spin frame, define the matrix-valued connection

$$\Gamma=\frac14\omega_{ab}\gamma^{ab}.$$

The spinor covariant derivative is

$$\nabla\psi=d\psi+\Gamma\psi, \qquad \nabla_\mu\psi =\partial_\mu\psi +\frac14\omega_{\mu ab}\gamma^{ab}\psi.$$

Repeated antisymmetric indices are summed over both orders; the factor $1/4$ is consistent with $\gamma^{ab}=\tfrac12[\gamma^a,\gamma^b]$. Changing either definition changes the visible factor, so both must always be stated together.

The factor can be checked without memorization. The Clifford relation gives

$$[\gamma^{ab},\gamma^c] =2\left(\eta^{bc}\gamma^a-\eta^{ac}\gamma^b\right), \qquad [\Gamma,\gamma^c]=-\omega^c{}_b\gamma^b.$$

Thus the spin commutator cancels exactly the vector connection on an upper frame index. Equivalently, the map $A^a{}_b\mapsto\tfrac14A_{ab}\gamma^{ab}$ is the spin representation of the Lorentz Lie algebra. This checks the coefficient, the index lowering, and the sign together.

Wernli develops the spin connection and Clifford-compatible derivative in Wernli 2019, §2.3.4, pp. 70–71. That source uses the Clifford relation with the opposite sign; the formulas here have been translated to $\{\gamma^a,\gamma^b\}=2\eta^{ab}$ rather than copied term by term.

Local Lorentz covariance

Use the same coframe convention as the frame-gauge page: $\theta'=\Lambda\theta$, so vector components obey $V'^a=\Lambda^a{}_bV^b$. Choose its spin lift $S(x)$ by

$$S^{-1}\gamma^aS=\Lambda^a{}_b\gamma^b.$$

Equivalently, $S\gamma^aS^{-1}=(\Lambda^{-1})^a{}_b\gamma^b$. The inverse matters because an upper gamma label follows the dual basis convention. With $\gamma(V)=V^a\gamma_a$, this gives $\gamma(V')=S\gamma(V)S^{-1}$. Therefore $\psi'=S\psi$ makes Clifford multiplication and the derivative transform together. Covariance $\nabla'\psi'=S\nabla\psi$ requires

$$\Gamma'=S\Gamma S^{-1}-dS\,S^{-1}.$$

This is the spin-representation version of the inhomogeneous frame-connection law. The derivative term cancels the derivative of the local basis change. A spinor component is therefore frame-dependent, while equations formed with the covariant derivative transform consistently.

Compatibility can also be expressed by saying that Clifford multiplication is parallel. If a vector and a spinor are differentiated together, then

$$\nabla_X\bigl(\gamma(V)\psi\bigr) =\gamma(\nabla_XV)\psi+\gamma(V)\nabla_X\psi.$$

The displayed spin connection satisfies this identity and is the unique lift of the Levi-Civita principal spin connection. Clifford compatibility alone is a weaker requirement on a general spinor-bundle connection: one can add a scalar one-form times the identity, which commutes with every gamma matrix. Such an extra term describes additional twisting data, not the pure spin lift considered here.

Curvature on spinors

The spin curvature is

$$F=d\Gamma+\Gamma\wedge\Gamma.$$

The map from a Lorentz-algebra element to its spin generator is a Lie-algebra representation. It therefore preserves brackets, and applying it to Cartan’s second structure equation gives

$$F=\frac14\Omega_{ab}\gamma^{ab}.$$

If $\Omega_{ab}=\tfrac12R_{ab\mu\nu}dx^\mu\wedge dx^\nu$, then

$$[\nabla_\mu,\nabla_\nu]\psi =\frac14R_{ab\mu\nu}\gamma^{ab}\psi.$$

The curvature components use the site’s convention $[\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma$, converted to the orthonormal pair $a,b$. A convention-explicit Riemannian derivation appears in Derdzinski n.d., §31, Lemma 31.1, pp. 179–180. That source uses the opposite Clifford sign and a different curvature-index order; both are translated together here, rather than changing the visible sign of a single formula in isolation.

The equation says that the same geometric curvature acts in two representations: on vectors through Lorentz generators and on spinors through $\gamma^{ab}/2$. It does not introduce a new independent curvature tensor.

If a spinor is parallel on an open region, then applying the commutator gives the integrability condition

$$R_{ab\mu\nu}\gamma^{ab}\psi=0.$$

This need not force the entire Riemann tensor to vanish because a particular spinor can have a nontrivial stabilizer. Conversely, satisfying the algebraic condition at one point does not by itself construct a parallel spinor on a neighborhood; the differential equation and holonomy must also be controlled.

Flat polar tetrad check

In Minkowski spacetime with cylindrical coordinates,

$$ds^2=-dt^2+dr^2+r^2d\phi^2+dz^2,$$

choose $\theta^0=dt$, $\theta^1=dr$, $\theta^2=r\,d\phi$, and $\theta^3=dz$ on $r>0$. The only nonzero spatial connection pair is

$$\omega^1{}_2=-d\phi, \qquad \omega^2{}_1=d\phi.$$

Consequently,

$$\Gamma=-\frac12d\phi\,\gamma^{12}.$$

The spin connection is nonzero because the polar tetrad rotates. Nevertheless,

$$F=d\Gamma+\Gamma\wedge\Gamma=0$$

on the patch: $d(d\phi)=0$ locally and a one-form wedges with itself to zero. The commutator of spinor covariant derivatives therefore vanishes, agreeing with the Riemann tensor of flat spacetime. This calculation checks the factor in $\Gamma$ and separates frame rotation from curvature.

Around a noncontractible loop, a spin lift additionally depends on the chosen spin structure. Local vanishing of $F$ does not by itself select global periodicity conditions. In particular, integrating $\Gamma_\phi$ around $2\pi$ within polar components must be combined with the spin-frame transition at the angular seam. A component sign from that rotating frame alone is not yet the invariant spin holonomy.

One can also check the flat result by returning to the Cartesian tetrad, where $\Gamma=0$. The polar and Cartesian expressions are related by a local spin rotation on a simply connected angular patch. The inhomogeneous derivative of that rotation produces the nonzero polar $\Gamma$, while conjugation leaves $F=0$. Agreement of these two gauges is stronger than merely observing that the coordinate Riemann components vanish.

Scope and limitations

The lifted connection requires a spin structure, not merely gamma matrices written on one patch. Conversely, the existence of a spin structure does not choose a preferred tetrad, gamma-matrix representation, or spin frame. All are related by appropriate local changes of basis.

Coordinate covariance and spin-frame covariance operate simultaneously. The spacetime index on $\nabla_\mu$ transforms as a covector under coordinate changes, while the spinor index transforms through $S(x)$. Keeping those two actions separate is essential when converting between $\omega_{\mu ab}$, the one-form $\omega_{ab}$, and frame components $\omega_{cab}$.

This page stops at differential geometry. Dirac dynamics, field quantization, anomalies, Newman–Penrose equations, and gravitational field equations require additional structures and belong elsewhere.

Exercises

For the flat polar tetrad above, compute $[\nabla_r,\nabla_\phi]\psi$ directly from $\Gamma_r=0$ and $\Gamma_\phi=-\tfrac12\gamma^{12}$.

Solution

The matrix $\Gamma_\phi$ is independent of $r$ and $\phi$ on the patch, while $\Gamma_r=0$. Therefore

$$[\nabla_r,\nabla_\phi]\psi =\left(\partial_r\Gamma_\phi-\partial_\phi\Gamma_r +[\Gamma_r,\Gamma_\phi]\right)\psi=0.$$

The nonzero angular connection changes components under transport, but its curvature vanishes. Any nonzero result would signal an inconsistent connection coefficient, an omitted frame term, or an incorrect commutator convention.

References

• Derdzinski, Andrzej. Einstein Metrics in Dimension Four. Expository course notes, The Ohio State University, n.d. Course PDF.
• Wernli, Konstantin. Lecture Notes on Spin Geometry. arXiv:1911.09766 (2019), arXiv.