nLab
exceptional geometry

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 understod to study spaces controled by exceptional Lie groups in some way.

Related concepts

• exceptional Lie group, exceptional generalized geometry

References

General

General discussion is in

• Dominic Joyce, The exceptional holonomy groups and calibrated geometry (pdf)

• Simon Salamon, A tour of exceptional geometry (pdf)

• Simon Salamon, Self-duality and exceptional geometry (pdf)

Discussion of G2 manifolds is in

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

In supergravity

Applications to supergravity are discussed in

• G. Papadopoulos and Paul Townsend, Compactifications of $D=11$ supergravity on spaces of exceptional holonomy (arXiv:hep-th/9506150)

• K. Koepsell, Hermann Nicolai, Henning Samtleben, An exceptional geometry for d=11 supergravity? (arXiv:hep-th/0006034)

• Chris Hull, Generalised Geometry for M-Theory (arXiv:hep-th/0701203)

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