Spin(6)

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

**spin geometry**, **string geometry**, **fivebrane geometry** …

**rotation groups in low dimensions**:

see also

The spin group in dimension 6.

There is an exceptional isomorphism

between Spin(6) and SU(4), reflecting, under the classification of simple Lie groups, the coincidence of Dynkin diagram of “D3” with A3.

(e.g. Figueroa-O’Farrill 10, Lemma 8.1)

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 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 |

see also *Spin(8)-subgroups and reductions*

homotopy fibers of homotopy pullbacks of classifying spaces:

(from FSS 19, 3.4)

**Spin(8)-subgroups and reductions to exceptional geometry**

reduction | from spin group | to maximal subgroup |
---|---|---|

Spin(7)-structure | Spin(8) | Spin(7) |

G2-structure | Spin(7) | G2 |

CY3-structure | Spin(6) | SU(3) |

SU(2)-structure | Spin(5) | SU(2) |

generalized reduction | from Narain group | to 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 | $Spin(6,6)$ | $SU(3) \times SU(3)$ |

see also: *coset space structure on n-spheres*

**rotation groups in low dimensions**:

see also

- José Figueroa-O'Farrill,
*PG course on Spin Geometry*lecture 8:*Parallel and Killing spinors*, 2010 (pdf)

Last revised on January 16, 2021 at 01:50:32. See the history of this page for a list of all contributions to it.