Spin connection - Imported Knowlpedia Content

Definition

Let $(M,g)$ be an oriented Riemannian $n$ -manifold with spin structure $\Phi:P_{\mathrm{Spin}}\to P_{\mathrm{SO}}$ , and let $\lambda:\mathrm{Spin}(n)\to\mathrm{SO}(n)$ be the double covering. The spin connection is the unique principal connection $\widetilde\omega$ on $P_{\mathrm{Spin}}$ satisfying

d\lambda\circ\widetilde\omega=\Phi^*\omega_{\mathrm{LC}},

where $\omega_{\mathrm{LC}}$ is the Levi–Civita connection on $P_{\mathrm{SO}}$ . Since $d\lambda:\mathfrak{spin}(n)\to\mathfrak{so}(n)$ is an isomorphism, this equation both defines the lift and proves its uniqueness. Thus the spin connection depends on the metric and the chosen spin structure.

Associated covariant derivative

For a complex spin representation $\Delta_n$ , the spin connection induces a covariant derivative $\nabla^S$ on the associated spinor bundle $S=P_{\mathrm{Spin}}\times_{\mathrm{Spin}(n)}\Delta_n$ . It is compatible with the spinor metric and Clifford multiplication:

\nabla^S_X(Y\mathbin{\cdot}\psi) =(\nabla_XY)\mathbin{\cdot}\psi+Y\mathbin{\cdot}\nabla^S_X\psi.

This compatibility is the bridge from Riemannian parallel transport to the Dirac operator; see Lawson–Michelsohn, Chapter II, §§4–5.

Curvature and local form

The curvature of $\widetilde\omega$ maps under $d\lambda$ to the pullback of the Riemannian curvature form. In a local oriented orthonormal frame, if $\omega_{ij}$ are the Levi–Civita connection one-forms, then the induced spinor derivative has the familiar form

\nabla^S=d+\frac14\sum_{i,j}\omega_{ij}\,e_i\mathbin{\cdot}e_j.

The factor depends on summation conventions, but the invariant lifting equation in the core does not Friedrich, Chapters 2–3.

Examples and conventions

On Euclidean space with its standard spin structure and constant orthonormal frame, the spin connection form vanishes and $\nabla^S$ is ordinary differentiation. An arbitrary principal connection on $P_{\mathrm{Spin}}$ is not the Riemannian spin connection unless it projects to $\omega_{\mathrm{LC}}$ . In some physics texts “spin connection” denotes a connection built from a more general metric connection, possibly with torsion; the present knowl uses the canonical Levi–Civita convention.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record. Relevant: Chapter II, especially §§4–5 on spinors, connections, and Dirac operators.
Thomas Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics 25, American Mathematical Society, 2000. Publisher record. Relevant: Chapters 2–3 and Appendix B.