Embedding the SU(3) sector of SO(8) supergravity in $D=11$

Understand

The SU(3)--invariant sector of maximal supergravity in four dimensions with an SO(8) gauging is uplifted to $D=11$ supergravity.

• In order to do this, the SU(3)--neutral sector of the tensor and duality hierarchies of the $D=4$ ${\cal N}=8$ supergravity is first worked out.
• The consistent $D=11$ embedding of the full, dynamical SU(3) sector is then expressed at the level of the $D=11$ metric and three-form gauge field in terms of these $D=4$ tensors.
• The redundancies introduced by this approach are eliminated at the level of the $D=11$ four-form field strength by making use of the $D=4$ duality hierarchy.