nLab
spin structure

Idea

A spin structure on a manifold $X$ with an orientation is a lift $\hat{g}$ of the classifying map $g:X\to ℬSO(n)$ of the tangent bundle through the second step $ℬ\mathrm{Spin}(n)\to ℬSO(n)$ in the Whitehead tower of $O(n)$.

$$\begin{array}{ccc}& & ℬ\mathrm{Spin}(n)\\ & {}^{\hat{g}}\nearrow & \downarrow \\ X& \stackrel{g}{\to }& ℬSO(n)\end{array}$$\array{ && \mathcal{B}Spin(n) \\ & {}^{\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathcal{B}S O(n) }

Spin structures derive their name from the fact that their existence on a space $X$ make the quantum anomaly for spinning particles propagating on $X$ vanish. See there.

The next steps correspond to

• String structure

• Fivebrane structure.

Definition

For $X$ a manifold, the groupoid/homotopy 1-type $\mathrm{Spin}(X)$ of spin structures over $X$ is the homotopy fiber in ∞Grpd $\simeq$ Top of the second Stiefel-Whitney class

$$\begin{array}{ccc}\mathrm{Spin}(X)& \to & *\\ {}^{\eta }\downarrow & & \downarrow \\ \mathrm{Top}(X,B\mathrm{SO})& \stackrel{(w_2{)}_{*}}{\to }& \mathrm{Top}(X,B^2{\mathbb{Z}}_2)\end{array}.$$\array{ Spin(X) &\to& * \\ {}^{\mathllap{\eta}}\downarrow && \downarrow \\ Top(X,B SO) &\stackrel{(w_2)_*}{\to}& Top(X, B^2 \mathbb{Z}_2) } \,.

Here an object $s\in \mathrm{Spin}(X)$ over an $\mathrm{SO}$-principal bundle $\eta (s)$ on $X$ is called a spin structure on $\eta (s)$ ($\mathrm{SO}$ is the special orthogonal group).

For $\eta (s)$ the $\mathrm{SO}$-principal bundle for which the tangent bundle $TX$ is the canonically associated bundle, one says that a spin-structure on $\eta (s)$ is a spin structure on the manifold $X$.

As quantum anomaly cancellation condition

In the context of quantum field theory the existence of a spin structure on a Riemannian manifold $X$ arises notably as the condition for quantum anomaly cancellation of the sigma-model for the spinning particle – the superparticle – propagating on $X$.

Discussions of spin structures along these lines date back to

• Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

• L. Alvarez-Gaumé Communications in Mathematical Physics 90 (1983) 161

• D. Friedan, P. Windey, Nucl. Phys. B235 (1984) 395

It is the generalization of this anomaly computation from the worldlines of superparticles to superstrings that leads to string structure, and then further the generalizaton to the worldvolume anomaly of fivebranes that leads to fivebrane structure.

Related concepts

• (twisted differential) c-structure

  • orientation

  • spin structure, twisted spin structure, differential spin structure

    spin^c structure, twisted spin^c structure

  • 2-framing

  • string structure, differential string structure

  • fivebrane structure, differential fivebrane structure

References

A disccussion of the full groupoid of spin structures is in

• Johannes Ebert, Characteristic classes of spin surface bundles: Applications of the Madsen-Weiss theory Phd thesis (2006) (pdf)