# nLab Pin(2)

Contents

group theory

## 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

# Contents

## Idea

A Pin group in dimension 2.

## Properties

Let $\mathbb{H} \simeq \mathbb{C} \oplus j \mathbb{C}$ be the quaternions realized as the Cayley-Dickson double of the complex numbers, and identify the circle group

$SO(2) \simeq S\big( \mathbb{C}\big) \hookrightarrow \mathbb{H}$

with the unit circle in $\mathbb{C} \hookrightarrow \mathbb{H}$ this way, with group structure now thought of as given by multiplication of quaternions. Then the Pin group $Pin_-(2)$ is isomorphic to the subgroup of the group of units $\mathbb{H}^\times$ of the quaternions which consists of this copy of SO(2), together with the multiplies of the imaginary quaternion $j$ with this copy:

$Pin_-(2) \;\simeq\; S\big( \mathbb{C}\big) \;\cup\; j \cdot S\big( \mathbb{C}\big) \;\subset\; \mathbb{H}^\times \,.$

This is in fact inside the unit sphere $S(\mathbb{H}) \simeq$ Spin(3).

By the classification of the finite subgroups of Spin(3) this means that the restriction of $Pin_-(2)$ to the inclusion of the dihedral group $D_{2n}$ into the orthogonal group $O(2)$ is the binary dihedral group $2 D_{2n}$:

$\array{ 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow \\ D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }$

### Appearance in Seiberg-Witten theory and Floer homology

On a spin 4-manifold, the $U(1)$-symmetry of the Seiberg-Witten equations? enhances to a $Pin(2)$-symmetry (Furuta 01). Because of this, $Pin(2)$-equivariance appears in Seiberg-Witten theory and Floer homology. See the references below.

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)

$Pin(2)$-equivariant homotopy theory/equivariant cohomology theory
$Pin(2)$-equivariant KO-theory: