Spin structure - Imported Knowlpedia Content

Definition

Let $(M,g)$ be an oriented Riemannian $n$ -manifold, 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. These obstruction and classification statements are proved in Lawson–Michelsohn, Chapter II, §2.

The Riemannian definition uses orthonormal frames, but existence and equivalence do not depend on the chosen metric: changing the metric canonically transports the lifting problem to another oriented frame reduction.

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 Milnor–Stasheff, Chapter 14.

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.