Domain Wall from Gauged d=4, N=8 Supergravity: Part II
  The scalar potentials of the non-semi-simple CSO(p,8-p)(p=7,6,5) gaugings of N=8 supergravity are studied for critical points. The CSO(7,1) gauging has no G_2-invariant critical points, the CSO(6,2) gauging has three new SU(3)-invariant AdS critical points and the CSO(5,3) gauging has no SO(5)-invariant critical points. The scalar potential of CSO(6,2) gauging in four dimensions we discovered provides the SU(3) invariant scalar potential of five dimensional SO(6) gauged supergravity. The nontrivial effective scalar potential can be written in terms of the superpotential which can be read off from A_1 tensor of the theory. We discuss first-order domain wall solutions by analyzing the supergravity scalar-gravity action and using some algebraic relations in a complex eigenvalue of A_1 tensor. We examine domain wall solutions of G_2 sectors of noncompact SO(7,1) and CSO(7,1) gaugings and SU(3) sectors of SO(6,2) and CSO(6,2) gaugings. They share common features with each sector of compact SO(8) gauged N=8 supergravity in four dimensions. We analyze the scalar potentials of the CSO(p,q,8-p-q) gauged supergravity we have found before. The CSO(p,6-p,2) gauge theory in four dimensions can be reduced from the SO(p,6-p) gauge theory in five dimensions. Moreover, the SO(p,5-p) gauge theory in seven dimensions reduces to CSO(p,5-p,3) gauge theory in four dimensions. Similarly, CSO(p,q-p,8-q) gauge theories in four dimensions are related to SO(p,q-p)(q=2,3,4,7) gauge theories in other dimensions. 
Changhyun Ahn and Kyungsung Woo