nLab exceptional geometry




The classification of Riemannian manifolds with special holonomy contains two “exceptional” cases: G2-holonomy in dimension 7, and Spin(7)-holonomy in dimension 8. Their study is the topic of exceptional geometry.

Sometimes more generally, exceptional geometry is understood to study spaces controled by exceptional Lie groups in some way.


In terms of twisted Cohomotopy

coset space-structures on n-spheres:

S n1 diffSO(n)/SO(n1)S^{n-1} \simeq_{diff} SO(n)/SO(n-1)this Prop.
S 2n1 diffSU(n)/SU(n1)S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)this Prop.
S 4n1 diffSp(n)/Sp(n1)S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)this Prop.
S 7 diffSpin(7)/G 2S^7 \simeq_{diff} Spin(7)/G_2Spin(7)/G2 is the 7-sphere
S 7 diffSpin(6)/SU(3)S^7 \simeq_{diff} Spin(6)/SU(3)since Spin(6) \simeq SU(4)
S 7 diffSpin(5)/SU(2)S^7 \simeq_{diff} Spin(5)/SU(2)since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
S 6 diffG 2/SU(3)S^6 \simeq_{diff} G_2/SU(3)G2/SU(3) is the 6-sphere
S 15 diffSpin(9)/Spin(7)S^15 \simeq_{diff} Spin(9)/Spin(7)Spin(9)/Spin(7) is the 15-sphere

see also Spin(8)-subgroups and reductions

homotopy fibers of homotopy pullbacks of classifying spaces:

(from FSS 19, 3.4)



General discussion is in

Discussion of G2 manifolds is in

  • Spiro Karigiannis, G 2G_2-manifolds – Exceptional structures in geometry arising from exceptional algebra (pdf)

In supergravity

Applications to KK-compactification of 11d supergravity (see also at M-theory on G2-manifolds) is discussed in

For more along these lines see the references at exceptional generalized geometry.

As M-brane target space

Discusssion of M-brane sigma-models on exceptional geometry target spaces:

Last revised on April 11, 2024 at 12:41:01. See the history of this page for a list of all contributions to it.