# nLab Spin(6)

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

The spin group in dimension 6.

## Properties

### Exceptional isomorphism

###### Proposition

There is an exceptional isomorphism

between Spin(6) and SU(4), reflecting, under the classification of simple Lie groups, the coincidence of the Dynkin diagramsD3” and A3.

One way to see the isomorphism $\mathrm{Spin}(6) \cong \mathrm{SU}(4)$ is as follows. Let $V$ be a 4-dimensional complex vector space with an inner product and a compatible complex volume form, meaning an element of the exterior product $\Lambda^4 V$ whose norm is 1 in the norm coming from the inner product on $V$. The inner product defines a conjugate-linear isomorphism $V \cong V^\ast$ (with the complex dual vector space) that together with the complex volume form can be used to define a conjugate-linear Hodge star operator on $\Lambda^2 V$. This Hodge star operator squares to the identity, and its $+1$ and $-1$ eigenspaces, say $\Lambda_{\pm}^2 V$, each become 6-dimensional real inner product spaces in a natural way. Thus, the group $\mathrm{SU}(V)$, consisting of all complex-linear transformations of $V$ that preserve the inner product and complex volume form, acts as linear transformations of $\Lambda_+^2 V$ that preserve the inner product, giving a homomorphism $\rho: \mathrm{SU}(V) \to \mathrm{O}(\Lambda_+^2 V)$. Since $\mathrm{SU}(V)$ is connected we in fact have $\rho: \mathrm{SU}(V) \to \mathrm{SO}(\Lambda_+^2 V)$.

Specializing to the case $V = \mathbb{C}^4$ we get a Lie group homomorphism $\rho: \mathrm{SU}(4) \to \mathrm{SO}(6)$. Since $d\rho$ is nonzero and $\mathrm{SU}(4)$ is simple, $d\rho$ must be injective. Since

$\mathrm{dim}(\mathrm{SU}(4)) = 15 = \mathrm{dim}(\mathrm{SO}(6)),$

$d\rho$ must also be surjective. Since $\mathrm{SO}(6)$ is connected and $d\rho$ is a bijection, $\rho$ must be a covering map. Since $\rho(\pm 1) = 1$, $\rho$ exhibits $\mathrm{SU}(4)$ as a connected cover of $\mathrm{SO}(6)$ that is at least a double cover. But the universal cover of $\mathrm{SO}(6)$, namely $\mathrm{Spin}(6)$, is only a double cover. Thus $\mathrm{SU}(4)$ is a double cover of $\mathrm{SO}(6)$, and $\mathrm{SU}(4) \cong \mathrm{Spin}(6)$.

### Coset spaces

coset space-structures on n-spheres:

standard:
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$this Prop.
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$this Prop.
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$this Prop.
exceptional:
$S^7 \simeq_{diff} Spin(7)/G_2$Spin(7)/G2 is the 7-sphere
$S^7 \simeq_{diff} Spin(6)/SU(3)$since Spin(6) $\simeq$ SU(4)
$S^7 \simeq_{diff} Spin(5)/SU(2)$since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
$S^6 \simeq_{diff} G_2/SU(3)$G2/SU(3) is the 6-sphere
$S^15 \simeq_{diff} Spin(9)/Spin(7)$Spin(9)/Spin(7) is the 15-sphere

(from FSS 19, 3.4)

### $G$-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structure$Spin(8,8)$$Spin(7) \times Spin(7)$
generalized G2-structure$Spin(7,7)$$G_2 \times G_2$
generalized CY3$Spin(6,6)$$SU(3) \times SU(3)$

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)