Contents

group theory

spin geometry

string geometry

# Contents

## Idea

The group $Spin(2)$ is the spin group in 2 dimensions, hence the double cover of SO(2)

$\array{ \mathbb{Z}/2 &\hookrightarrow& Spin(2) \\ && \downarrow \\ && SO(2) }$

In fact:

###### Proposition

There is an isomorphism $Spin(2) \simeq SO(2) \simeq U(1)$ with the circle group which exhibits the double cover of SO(2) by Spin(2) as the real Hopf fibration

$\array{ \mathbb{Z}/2 &\hookrightarrow& S^1 &\simeq& Spin(2) \\ && \downarrow^{\mathrlap{\cdot 2}} && \downarrow^{\mathrlap{}} \\ && S^1 &\simeq& SO(2) }$
sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
$\vdots$$\vdots$
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)