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 coicidence of Dynkin diagram of “D3” with A3.

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

**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 August 29, 2019 at 10:03:17. See the history of this page for a list of all contributions to it.