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

$Spin(6) \simeq SU(4)$

with the special unitary group of complex dimension 4.

(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 May 14, 2019 at 00:36:24. See the history of this page for a list of all contributions to it.