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Narain group

Contents

Contents

Idea

The orthogonal group O(n,n)O(n,n) for signature (n,n)(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 XX a smooth manifold, the generalized tangent bundle TXT *XT 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)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)SU(3) \times SU(3)

Maximal compact subgroup

The maximal compact subgroup of the Narain group is the product group O(n)×O(n)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 groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)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.