The orthogonal group $O(n,n)$ for signature $(n,n)$ is sometimes called the *Narain group* or *generalized T-duality group* for the role that it plays in T-duality of type II string theory. See also at *type II geometry*.

For $X$ a smooth manifold, the generalized tangent bundle $T X \oplus T^* X$ has as structure group the Narain group.

**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*

The maximal compact subgroup of the Narain group is the product group $O(n) \times O(n)$ (e.g. Ooguri-Yin 96, p. 44).

A reduction of the structure group of the generalized tangent bundle along the inclusion defines a type II geometry.

group | symbol | universal cover | symbol | higher cover | symbol |
---|---|---|---|---|---|

orthogonal group | $\mathrm{O}(n)$ | Pin group | $Pin(n)$ | Tring group | $Tring(n)$ |

special orthogonal group | $SO(n)$ | Spin group | $Spin(n)$ | String group | $String(n)$ |

Lorentz group | $\mathrm{O}(n,1)$ | $\,$ | $Spin(n,1)$ | $\,$ | $\,$ |

anti de Sitter group | $\mathrm{O}(n,2)$ | $\,$ | $Spin(n,2)$ | $\,$ | $\,$ |

conformal group | $\mathrm{O}(n+1,t+1)$ | $\,$ | |||

Narain group | $O(n,n)$ | ||||

Poincaré group | $ISO(n,1)$ | Poincaré spin group | $\widehat {ISO}(n,1)$ | $\,$ | $\,$ |

super Poincaré group | $sISO(n,1)$ | $\,$ | $\,$ | $\,$ | $\,$ |

superconformal group |

- Hirosi Ooguri, Zheng Yin,
*TASI Lectures on Perturbative String Theories*(arXiv:hep-th/9612254)

Last revised on March 30, 2019 at 14:03:36. See the history of this page for a list of all contributions to it.