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Reductions of 11-dimensional supergravity on a 4-dimensional space to a 7-dimensional theory.
The AdS7/CFT6-correspondence relates 7-dimensional supergravity on asymptocially 7d anti de Sitter spacetimes to the 6d (2,0)-superconformal QFT.
10-dimensional type II supergravity, heterotic supergravity
7-dimensional supergravity
The maximally $N=4$ supersymmetric $SO(5)$-gauged supergravity in 7-dimensions was first constructed in
There are also gaugings for non-compact groups
The $N=2$ $Sp(1)$-gauged theory was described in
See also
E. Bergshoeff, D C Jong and Ergin Sezgin, Noncompact gaugings, chiral reduction and dual sigma models in supergravity Class. Quantum Grav. 23 (2006) 2803–2831 (pdf)
Der-Chyn Jong, Aspects of 7D and 6D gauged supergravities PhD thesis (2007) (web)
The following references discuss construction of full 7d sugra theories or of their solutions by KK-reduction from 11-dimensional supergravity or 10d type II supergravity.
The construction of 7d supergravity by reduction from 11-dimensional supergravity was first indicated in
That their Ansatz is indeed consistent has been shown in
Horatiu Nastase, Diana Vaman, Peter van Nieuwenhuizen, Consistent nonlinear KK reduction of 11d supergravity on $AdS_7\times S_4$ and self-duality in odd dimensions Phys.Lett. B469 (1999) 96-102 (arXiv:hep-th/9905075)
Ali Chamseddine, W.A. Sabra, $D=7$ $SU(2)$ gauged supergravity from $D=10$ supergravity Physics Letters B Volume 476, Issues 3-4, 16 (2000), Pages 415-419
P. van Nieuwenhuizen, N. P. Warner, New Compactificafions of Ten and Eleven Dimensional Supergravity on Manifolds Which Are Not Direct Products Commun. Math. Phys. 99, 141-151 (1985)
Jerome P. Gauntlett, Nakwoo Kim, Stathis Pakis, Daniel Waldram, M-Theory solutions with AdS factors (arXiv:hep-th/0202184)
Discussion of quantum anomaly cancellation is in