cohomology

# Contents

## Idea

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

$\begin{array}{ccc}& & B\mathrm{Spin}\left(n\right)\\ & {}^{\stackrel{^}{g}}↗& ↓\\ X& \stackrel{g}{\to }& BSO\left(n\right)\end{array}$\array{ && B Spin(n) \\ & {}^{\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& 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.

## Definition

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

$\begin{array}{ccc}\mathrm{Spin}\left(X\right)& \to & *\\ {}^{\eta }↓& & ↓\\ \mathrm{Top}\left(X,B\mathrm{SO}\right)& \stackrel{\left({w}_{2}{\right)}_{*}}{\to }& \mathrm{Top}\left(X,{B}^{2}{ℤ}_{2}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $s\in \mathrm{Spin}\left(X\right)$ over an $\mathrm{SO}$-principal bundle $\eta \left(s\right)$ on $X$ is called a spin structure on $\eta \left(s\right)$ ($\mathrm{SO}$ is the special orthogonal group).

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

## Properties

### Over a Riemann surface

Over a Riemann surface spin structures correspond to square roots of the canonical bundle. See at Theta characteristic.

### 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 $X$.

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.

## Higher spin structures

Spin structures are one step in a tower of conditions that are related to the quantum anomaly cancellation of higher dimensional spinning/super branes.

This is controled by the Whitehead tower of the classifying space/delooping of the orthogonal group $O\left(n\right)$, which starts out as

$\begin{array}{cc}& \mathrm{Whitehead}\mathrm{tower}\\ & ⋮\\ & B\mathrm{Fivebrane}& \to & \cdots & \to & *\\ & ↓& & & & ↓\\ \mathrm{second}\mathrm{frac}\mathrm{Pontr}.\mathrm{class}& B\mathrm{String}& \to & \cdots & \stackrel{\frac{1}{6}{p}_{2}}{\to }& {B}^{8}ℤ& \to & *\\ & ↓& & & & ↓& & ↓\\ \mathrm{first}\mathrm{frac}\mathrm{Pontr}.\mathrm{class}& B\mathrm{Spin}& & & & & \stackrel{\frac{1}{2}{p}_{1}}{\to }& {B}^{4}ℤ& \to & *\\ & ↓& & & & ↓& & ↓& & ↓\\ \mathrm{second}\mathrm{SW}\mathrm{class}& BSO& \to & \cdots & \to & & \to & & \stackrel{{w}_{2}}{\to }& {B}^{2}{ℤ}_{2}& \to & *\\ & ↓& & & & ↓& & ↓& & ↓& & ↓\\ \mathrm{first}\mathrm{SW}\mathrm{class}& BO& \to & \cdots & \to & {\tau }_{\le 8}BO& \to & {\tau }_{\le 4}BO& \to & {\tau }_{\le 2}BO& \stackrel{{w}_{1}}{\to }& {\tau }_{\le 1}BO\simeq B{ℤ}_{2}& \mathrm{Postnikov}\mathrm{tower}\end{array}$\array{ & Whitehead tower \\ &\vdots \\ & B Fivebrane &\to& \cdots &\to& * \\ & \downarrow && && \downarrow \\ second frac Pontr. class & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z} &\to& * \\ & \downarrow && && \downarrow && \downarrow \\ first frac Pontr. class & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & * \\ & \downarrow && && \downarrow && \downarrow && \downarrow \\ second SW class & B S O &\to& \cdots &\to& &\to& & \stackrel{w_2}{\to} & B^2 \mathbb{Z}_2 &\to& * \\ & \downarrow && && \downarrow && \downarrow && \downarrow && \downarrow \\ first SW class & B O &\to& \cdots &\to& \tau_{\leq 8 } B O &\to& \tau_{\leq 4 } B O &\to& \tau_{\leq 2 } B O &\stackrel{w_1}{\to}& \tau_{\leq 1 } B O \simeq B \mathbb{Z}_2 & Postnikov tower }

where the stages are the deloopings of

$\to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group,

where lifts through the stages correspond to

and where the obstruction classes are the universal characteristic classes

and where every possible square in the above is a homotopy pullback square (using the pasting law).

Notice that for instance ${w}_{2}$ is identified as such by using that $\left[{S}^{2},-\right]$ preserves homotopy pullbacks and sends $BO\to {\tau }_{\le 2}BO$ to a equivalence, so that $B\mathrm{SO}\to {B}^{2}ℤ$ is an isomorphism on the second homotopy group and hence by the Hurewicz theorem is also an isomorphism on the cohomology group ${H}^{2}\left(-,{ℤ}_{2}\right)$. Analogously for the other characteristic maps.

In summary, more concisely, the tower is

$\begin{array}{c}⋮\\ ↓\\ B\mathrm{Fivebrane}\\ ↓\\ B\mathrm{String}& \stackrel{\frac{1}{6}{p}_{2}}{\to }& {B}^{7}U\left(1\right)& \simeq {B}^{8}ℤ\\ ↓\\ B\mathrm{Spin}& \stackrel{\frac{1}{2}{p}_{1}}{\to }& {B}^{3}U\left(1\right)& \simeq {B}^{4}ℤ\\ ↓\\ B\mathrm{SO}& \stackrel{{w}_{2}}{\to }& {B}^{2}{ℤ}_{2}\\ ↓\\ BO& \stackrel{{w}_{1}}{\to }& B{ℤ}_{2}\\ {↓}^{\simeq }\\ B\mathrm{GL}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \vdots \\ \downarrow \\ B Fivebrane \\ \downarrow \\ B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) & \simeq B^8 \mathbb{Z} \\ \downarrow \\ B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& B^3 U(1) & \simeq B^4 \mathbb{Z} \\ \downarrow \\ B SO &\stackrel{w_2}{\to}& B^2 \mathbb{Z}_2 \\ \downarrow \\ B O &\stackrel{w_1}{\to}& B \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ B GL } \,,

where each “hook” is a fiber sequence.

smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
$⋮$
$↓$
fivebrane 6-group$B\mathrm{Fivebrane}$fivebrane structuresecond fractional Pontryagin class
$↓$
string 2-group$B\mathrm{String}\stackrel{\frac{1}{6}{p}_{2}}{\to }{B}^{7}U\left(1\right)$string structurefirst fractional Pontryagin class
$↓$
spin group$B\mathrm{Spin}\stackrel{\frac{1}{2}{p}_{1}}{\to }{B}^{3}U\left(1\right)$spin structuresecond Stiefel-Whitney class
$↓$
special orthogonal group$B\mathrm{SO}\stackrel{{w}_{2}}{\to }{B}^{2}{ℤ}_{2}$orientation structurefirst Stiefel-Whitney class
$↓$
orthogonal group$BO\stackrel{{w}_{1}}{\to }B{ℤ}_{2}$orthogonal structure/vielbein/Riemannian metric
$↓$
general linear group$B\mathrm{GL}$smooth manifold

(all hooks are homotopy fiber sequences)

## References

### General

A discussion 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)

### In quantum anomaly cancellation

Discussions of spin structures in the context of quantum anomaly cancellation for the spinning particle date back to

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

• Luis Alvarez-Gaumé, Communications in Mathematical Physics 90 (1983) 161

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

Revised on January 10, 2013 16:29:22 by Urs Schreiber (89.204.153.52)