nLab
Narain group

Idea

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.

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 $O(n)\times O(n)$. A reduction of the structure group of the generalized tangent bundle along the inclusion defines a type II geometry.

Related concepts

group	symbol	universal cover	symbol	higher cover	symbol
orthogonal group	$\mathrm{O}(n)$	Pin group	$\mathrm{Pin}(n)$	Tring group	$\mathrm{Tring}(n)$
special orthogonal group	$\mathrm{SO}(n)$	Spin group	$\mathrm{Spin}(n)$	String group	$\mathrm{String}(n)$
Lorentz group	$\mathrm{O}(n,1)$	$$	$\mathrm{Spin}(n,1)$	$$	$$
anti de Sitter group	$\mathrm{O}(n,2)$	$$	$\mathrm{Spin}(n,2)$	$$	$$
Narain group	$O(n,n)$
Poincaré group	$\mathrm{ISO}(n,1)$	$$	$$	$$	$$
super Poincaré group	$\mathrm{sISO}(n,1)$	$$	$$	$$	$$