Spin structure - Knowlpedia

Let $(M,g)$ be an oriented Riemannian $n$ -manifold with $n\ge1$ , let $P_{\mathrm{SO}}(M)$ be its oriented orthonormal frame bundle, and let $\lambda:\mathrm{Spin}(n)\to\mathrm{SO}(n)$ be the double covering. A spin structure on $M$ is a principal $\mathrm{Spin}(n)$ -bundle $P_{\mathrm{Spin}}(M)\to M$ together with a bundle map

\Phi:P_{\mathrm{Spin}}(M)\longrightarrow P_{\mathrm{SO}}(M)

over $M$ satisfying $\Phi(pg)=\Phi(p)\lambda(g)$ . Thus $\Phi$ lifts each oriented orthonormal frame through the spin group in a way compatible with the group actions.

Existence and classification

An oriented manifold admits a spin structure exactly when the second Stiefel–Whitney class of its tangent bundle vanishes:

w_2(TM)=0.

When spin structures exist, their isomorphism classes form a torsor for $H^1(M;\mathbb Z/2)$ ; there is generally no preferred origin in this torsor.

The Riemannian definition uses orthonormal frames, but existence and equivalence do not depend on the chosen metric: the space of Riemannian metrics is contractible, so changing the metric does not change the lifting problem up to isomorphism class.

Associated geometry

A spin structure lifts the Levi-Civita connection to $P_{\mathrm{Spin}}(M)$ . Associating a spin representation produces the spinor bundle, and composing its covariant derivative with Clifford multiplication gives the Dirac operator. The spin structure is additional global data: an orientation and Riemannian metric do not by themselves choose one.

Examples and non-examples

The standard oriented $\mathbb R^n$ has a spin structure obtained from its trivial oriented orthonormal frame bundle. Every oriented surface is spin; its spin structures form a torsor for $H^1(M;\mathbb Z/2)$ . Complex projective space $\mathbb{CP}^2$ is oriented but not spin because $w_2(T\mathbb{CP}^2)\neq0$ . The latter computation follows from the relation between Chern and Stiefel–Whitney classes.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record. Relevant: Chapter II, especially §2, spin structures and their obstruction.
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. Publisher record. Relevant: Chapter 14, spin characteristic classes and examples.