exceptional structures, exceptional isomorphisms
exceptional finite rotation groups:
and Kac-Moody groups:
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.
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 |
see also Spin(8)-subgroups and reductions
homotopy fibers of homotopy pullbacks of classifying spaces:
(from FSS 19, 3.4)
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
Applications to KK-compactification of 11d supergravity (see also at M-theory on G2-manifolds) is discussed in
George Papadopoulos, 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.
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 May 14, 2019 at 00:53:35. See the history of this page for a list of all contributions to it.