The 11-dimensional metric for AdS/CFT RG flows with common SU (3) invariance

The compact 7-manifold arising in the compactification of 11-dimensional supergravity is described by the metric encoded in the vacuum expectation values (vevs) in d = 4, N = 8 gauged supergravity. Especially, the space of SU(3)-singlet vevs contains various critical points and RG flows (domain walls) developing along AdS₄ radial coordinate. Based on the nonlinear metric ansatz of de Wit-Nicolai-Warner, we show the geometric construction of the compact 7-manifold metric and find the local frames (siebenbeins) by decoding the SU(3)-singlet vevs into squashing and stretching parameters of the 7-manifold. Then the 11-dimensional metric for the whole SU(3)-invariant sector is obtained as a warped product of an asymptotically AdS₄ space with a squashed and stretched 7-sphere. We also discuss the difference in the 7-manifold between two sectors, namely (SU(3) x U(1))-invariant sector and G₂-invariant sector. In spite of the difference in base 6-sphere, both sectors share the 4-sphere of CP² associated with the common SU(3)-invariance of various 7-manifolds.