spin geometry

string geometry

∞-Lie theory

# Contents

## Idea

The spin group $\mathrm{Spin}\left(n\right)$ is the universal covering space of the special orthogonal group $\mathrm{SO}\left(n\right)$. By the usual arguments it inherits a group structure for which the operations are smooth and so is a Lie group like $\mathrm{SO}\left(n\right)$.

## Properties

### General

By definition the spin group sits in a short exact sequence of groups

${ℤ}_{2}\to \mathrm{Spin}\to \mathrm{SO}\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{Z}_2 \to Spin \to SO \,.

### Relation to Whitehead tower of orthogonal group

The spin group is one element in the Whitehead tower of $O\left(n\right)$, which starts out like

$\cdots \to \mathrm{Fivebrane}\left(n\right)\to \mathrm{String}\left(n\right)\to \mathrm{Spin}\left(n\right)\to \mathrm{SO}\left(n\right)\to \mathrm{O}\left(n\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

The homotopy groups of $O\left(n\right)$ are for $k\in ℕ$ and for sufficiently large $n$

$\begin{array}{cc}{\pi }_{8k+0}\left(O\right)& ={ℤ}_{2}\\ {\pi }_{8k+1}\left(O\right)& ={ℤ}_{2}\\ {\pi }_{8k+2}\left(O\right)& =0\\ {\pi }_{8k+3}\left(O\right)& =ℤ\\ {\pi }_{8k+4}\left(O\right)& =0\\ {\pi }_{8k+5}\left(O\right)& =0\\ {\pi }_{8k+6}\left(O\right)& =0\\ {\pi }_{8k+7}\left(O\right)& =ℤ\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.

By co-killing these groups step by step one gets

$\begin{array}{ccccc}\mathrm{cokill}\phantom{\rule{thinmathspace}{0ex}}\mathrm{this}& & & & \mathrm{to}\phantom{\rule{thinmathspace}{0ex}}\mathrm{get}\\ \\ {\pi }_{0}\left(O\right)& ={ℤ}_{2}& & & \mathrm{SO}\\ {\pi }_{1}\left(O\right)& ={ℤ}_{2}& & & \mathrm{Spin}\\ {\pi }_{2}\left(O\right)& =0\\ {\pi }_{3}\left(O\right)& =ℤ& & & \mathrm{String}\\ {\pi }_{4}\left(O\right)& =0\\ {\pi }_{5}\left(O\right)& =0\\ {\pi }_{6}\left(O\right)& =0\\ {\pi }_{7}\left(O\right)& =ℤ& & & \mathrm{Fivebrane}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ cokill\, this &&&& to\,get \\ \\ \pi_{0}(O) & = \mathbb{Z}_2 &&& SO \\ \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin \\ \pi_{2}(O) & = 0 \\ \pi_{3}(O) & = \mathbb{Z} &&& String \\ \pi_{4}(O) & = 0 \\ \pi_{5}(O) & = 0 \\ \pi_{6}(O) & = 0 \\ \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane } \,.

Via the J-homomorphism this is related to the stable homotopy groups of spheres:

$n$012345678910111213141516
homotopy groups of stable orthogonal group${\pi }_{n}\left(O\right)$${ℤ}_{2}$${ℤ}_{2}$0$ℤ$000$ℤ$${ℤ}_{2}$${ℤ}_{2}$0$ℤ$000$ℤ$${ℤ}_{2}$
stable homotopy groups of spheres${\pi }_{n}\left(𝕊\right)$$ℤ$${ℤ}_{2}$${ℤ}_{2}$${ℤ}_{24}$00${ℤ}_{2}$${ℤ}_{240}$${ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{2}\oplus {ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{6}$${ℤ}_{504}$0${ℤ}_{3}$${ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{480}\oplus {ℤ}_{2}$${ℤ}_{2}\oplus {ℤ}_{2}$
image of J-homomorphism$\mathrm{im}\left({\pi }_{n}\left(J\right)\right)$0${ℤ}_{2}$0${ℤ}_{24}$000${ℤ}_{240}$${ℤ}_{2}$${ℤ}_{2}$0${ℤ}_{504}$000${ℤ}_{480}$${ℤ}_{2}$

### Exceptional isomorphisms

In low dimensions the spin group happens to be isomorphic (“sporadic isomorphisms”) to various other classical group (among them the general linear group $\mathrm{GL}\left(p,V\right)$ for $V$ the real numbers $ℝ$, the complex numbers $ℂ$ and the quaternions $ℚ$, the orthogonal group $O\left(p,q\right)$, the unitary group $U\left(p,q\right)$ and the symplectic group $\mathrm{Sp}\left(p,q\right)$).

We have

• in Riemannian signature

• $\mathrm{Spin}\left(1\right)\simeq O\left(1\right)$

• $\mathrm{Spin}\left(2\right)\simeq U\left(1\right)\simeq \mathrm{SO}\left(2\right)$

• $\mathrm{Spin}\left(3\right)\simeq \mathrm{Sp}\left(1\right)\simeq \mathrm{SU}\left(2\right)$ (the special unitary group SU(2))

• $\mathrm{Spin}\left(4\right)\simeq \mathrm{Sp}\left(1\right)×\mathrm{Sp}\left(1\right)$

• $\mathrm{Spin}\left(5\right)\simeq \mathrm{Sp}\left(2\right)$

• $\mathrm{Spin}\left(6\right)\simeq \mathrm{SU}\left(4\right)$ (the special unitary group SU(4))

• in Lorentzian signature

• $\mathrm{Spin}\left(1,1\right)\simeq \mathrm{GL}\left(1,ℝ\right)$

• $\mathrm{Spin}\left(2,1\right)\simeq \mathrm{SL}\left(2,ℝ\right)$ – 2d special linear group of real numbers

• $\mathrm{Spin}\left(3,1\right)\simeq \mathrm{SL}\left(2,ℂ\right)$ – 2d special linear group of complex numbers

• $\mathrm{Spin}\left(4,1\right)\simeq \mathrm{Sp}\left(1,1\right)$

• $\mathrm{Spin}\left(5,1\right)\simeq \mathrm{SL}\left(2,ℍ\right)$ – 2d special linear group of quaternions

• $\mathrm{Spin}\left(9,1\right){\simeq }_{\mathrm{in}\phantom{\rule{thickmathspace}{0ex}}\mathrm{some}\phantom{\rule{thickmathspace}{0ex}}\mathrm{sense}}\mathrm{SL}\left(2,𝕆\right)$ – 2d special linear group of octonions

• in anti de Sitter signature

• $\mathrm{Spin}\left(2,2\right)\simeq \mathrm{SL}\left(2,ℝ\right)×\mathrm{SL}\left(2,ℝ\right)$

• $\mathrm{Spin}\left(3,2\right)\simeq \mathrm{Sp}\left(4,ℝ\right)$

• $\mathrm{Spin}\left(4,2\right)\simeq \mathrm{SU}\left(2,2\right)$

Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
$3=2+1$$\mathrm{Spin}\left(2,1\right)\simeq \mathrm{SL}\left(2,ℝ\right)$$ℝ$ the real numbers
$4=3+1$$\mathrm{Spin}\left(3,1\right)\simeq \mathrm{SL}\left(2,ℂ\right)$$ℂ$ the complex numbers
$6=5+1$$\mathrm{Spin}\left(5,1\right)\simeq \mathrm{SL}\left(2,ℍ\right)$$ℍ$ the quaternionslittle string
$10=9+1$$\mathrm{Spin}\left(9,1\right){\simeq }_{\mathrm{some}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sense}}\mathrm{SL}\left(2,𝕆\right)$$𝕆$ the octonionsheterotic/type II string

## Applications

### In physics

The name arises due to the requirement that the structure group of the tangent bundle of spacetime lifts to $\mathrm{Spin}\left(n\right)$ so as to ‘define particles with spin’… (Someone more awake and focused please put this into proper words!)

See spin structure.

The Whitehead tower of the orthogonal group looks like

$\cdots \to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group.

Another extension of $\mathrm{SO}$ is the spin^c group.

groupsymboluniversal coversymbolhigher coversymbol
orthogonal group$\mathrm{O}\left(n\right)$Pin group$\mathrm{Pin}\left(n\right)$Tring group$\mathrm{Tring}\left(n\right)$
special orthogonal group$\mathrm{SO}\left(n\right)$Spin group$\mathrm{Spin}\left(n\right)$String group$\mathrm{String}\left(n\right)$
Lorentz group$\mathrm{O}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\mathrm{Spin}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
anti de Sitter group$\mathrm{O}\left(n,2\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\mathrm{Spin}\left(n,2\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
Narain group$O\left(n,n\right)$
Poincaré group$\mathrm{ISO}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
super Poincaré group$\mathrm{sISO}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$

## References

A standard textbook reference is