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spin group

Idea

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

Properties

General

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

$${\mathbb{Z}}_2\to \mathrm{Spin}\to \mathrm{SO}.$$\mathbb{Z}_2 \to Spin \to SO \,.

Homotopy theory

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

$$\cdots \to \mathrm{Fivebrane}(n)\to \mathrm{String}(n)\to \mathrm{Spin}(n)\to \mathrm{SO}(n)\to \mathrm{O}(n).$$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

The homotopy groups of $O(n)$ are for $k\in \mathbb{N}$ and for sufficiently large $n$

$$\begin{array}{cc}{\pi }_{8k+0}(O)& ={\mathbb{Z}}_2\\ {\pi }_{8k+1}(O)& ={\mathbb{Z}}_2\\ {\pi }_{8k+2}(O)& =0\\ {\pi }_{8k+3}(O)& =\mathbb{Z}\\ {\pi }_{8k+4}(O)& =0\\ {\pi }_{8k+5}(O)& =0\\ {\pi }_{8k+6}(O)& =0\\ {\pi }_{8k+7}(O)& =\mathbb{Z}\end{array}.$$\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

$$\begin{array}{ccccc}\mathrm{cokill}\mathrm{this}& & & & \mathrm{to}\mathrm{get}\\ \\ {\pi }_0(O)& ={\mathbb{Z}}_2& & & \mathrm{SO}\\ {\pi }_1(O)& ={\mathbb{Z}}_2& & & \mathrm{Spin}\\ {\pi }_2(O)& =0\\ {\pi }_3(O)& =\mathbb{Z}& & & \mathrm{String}\\ {\pi }_4(O)& =0\\ {\pi }_5(O)& =0\\ {\pi }_6(O)& =0\\ {\pi }_7(O)& =\mathbb{Z}& & & \mathrm{Fivebrane}\end{array}.$$\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 } \,.

Exceptional isomorphisms

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

We have

• in Riemannian signature

  • $\mathrm{Spin}(1)\simeq O(1)$

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

  • $\mathrm{Spin}(3)\simeq \mathrm{Sp}(1)\simeq \mathrm{SU}(2)$

  • $\mathrm{Spin}(4)\simeq \mathrm{Sp}(1)\times \mathrm{Sp}(1)$

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

  • $\mathrm{Spin}(6)\simeq \mathrm{SU}(4)$

• in Lorentzian signature

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

  • $\mathrm{Spin}(2,1)\simeq \mathrm{SL}(2,ℝ)$

  • $\mathrm{Spin}(3,1)\simeq \mathrm{SL}(2,ℂ)$

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

  • $\mathrm{Spin}(5,1)\simeq \mathrm{SL}(2,ℍ)$

• in anti de Sitter signature

  • $\mathrm{Spin}(2,2)\simeq \mathrm{SL}(2,ℝ)\times \mathrm{SL}(2,ℝ)$

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

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

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

Applications

Spin geometry

See spin geometry

In physics

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

See spin structure.

Related concepts

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.

group	symbol	universal cover	symbol	higher cover	symbol
orthogonal group	$\mathrm{O}(n)$	Pin group	$\mathrm{Pin}(n)$	Tring group	$\mathrm{Tring}(n)$
special orthogonal group	$\mathrm{SO}(n)$	Spin group	$\mathrm{Spin}(n)$	String group	$\mathrm{String}(n)$
Lorentz group	$\mathrm{O}(n,1)$	$$	$\mathrm{Spin}(n,1)$	$$	$$
anti de Sitter group	$\mathrm{O}(n,2)$	$$	$\mathrm{Spin}(n,2)$	$$	$$
Poincaré group	$\mathrm{ISO}(n,1)$	$$	$$	$$	$$
super Poincaré group	$\mathrm{sISO}(n,1)$	$$	$$	$$	$$

References

A standard textbook reference is

• H.B. Lawson and M.-L. Michelson, Spin Geometry , Princeton University Press, Princeton, NJ, (1989)

See also

• wikipedia Spin group