# Contents

## Idea

The orthogonal group $O\left(n,n\right)$ for signature $\left(n,n\right)$ 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.

## Properties

### Structure group of generalized tangent bundle

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

### Maximal compact subgroup

The maximal compact subgroup of the Narain group is the product group $O\left(n\right)×O\left(n\right)$. A reduction of the structure group of the generalized tangent bundle along the inclusion defines a type II geometry.

groupsymboluniversal coversymbolhigher coversymbol
orthogonal group$\mathrm{O}\left(n\right)$Pin group$\mathrm{Pin}\left(n\right)$Tring group$\mathrm{Tring}\left(n\right)$
special orthogonal group$\mathrm{SO}\left(n\right)$Spin group$\mathrm{Spin}\left(n\right)$String group$\mathrm{String}\left(n\right)$
Lorentz group$\mathrm{O}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\mathrm{Spin}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
anti de Sitter group$\mathrm{O}\left(n,2\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\mathrm{Spin}\left(n,2\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
Narain group$O\left(n,n\right)$
Poincaré group$\mathrm{ISO}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
super Poincaré group$\mathrm{sISO}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
Created on May 28, 2012 01:47:29 by Urs Schreiber (82.113.99.74)