# nLab SemiSpin(16)

Contents

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

group theory

# Contents

## Idea

The semi-spin group in dimension 16.

## Properties

### As a subgroup of $E_8$

The subgroup of the exceptional Lie group E8 which corresponds to the Lie algebra-inclusion $\mathfrak{so}(16) \hookrightarrow \mathfrak{e}_8$ is the semi-spin group in that dimension

$SemiSpin(16) \;\subset\; E_8$

On the other hand, the special orthogonal group $SO(16)$ is not a subgroup of $E_8$ (e.g. McInnes 99a, p. 11).

### In heterotic string theory

In heterotic string theory with gauge group the direct product group $E_8 \times E_8$ it is usually only this subgroup $Semispin(16) \times SemiSpin(16)$ which is considered (but typically denoted $Spin(16)/\mathbb{Z}_2$, see also Distler-Sharpe 10, Sec. 1).

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)