Contents

# Contents

## 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 $T X \oplus T^* X$ has as structure group the Narain group.

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto 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)$

### Maximal compact subgroup

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.

groupsymboluniversal coversymbolhigher coversymbol
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

## References

Last revised on March 30, 2019 at 10:03:36. See the history of this page for a list of all contributions to it.