# nLab exceptional geometry

Contents

## Philosophy

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

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.

## Properties

### In terms of twisted Cohomotopy

coset space-structures on n-spheres:

standard:
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$this Prop.
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$this Prop.
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$this Prop.
exceptional:
$S^7 \simeq_{diff} Spin(7)/G_2$Spin(7)/G2 is the 7-sphere
$S^7 \simeq_{diff} Spin(6)/SU(3)$since Spin(6) $\simeq$ SU(4)
$S^7 \simeq_{diff} Spin(5)/SU(2)$since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
$S^6 \simeq_{diff} G_2/SU(3)$G2/SU(3) is the 6-sphere
$S^15 \simeq_{diff} Spin(9)/Spin(7)$Spin(9)/Spin(7) is the 15-sphere

(from FSS 19, 3.4)

## References

### General

General discussion is in

Discussion of G2 manifolds is in

• Spiro Karigiannis, $G_2$-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 is in

• Yuho Sakatani, Shozo Uehara, Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry, Phys. Rev. Lett. 117, 191601 (2016) (arXiv:1607.04265, talk slides)

• Yuho Sakatani, Shozo Uehara, Exceptional M-brane sigma models and $\eta$-symbols (arXiv:1712.10316)

Last revised on July 10, 2019 at 02:50:36. See the history of this page for a list of all contributions to it.