  By studying already known extrema of non-semi-simple Inonu-Wigner contraction
CSO(p, q)^{+} and non-compact SO(p, q)^{+}(p+q=8) gauged N=8 supergravity in
4-dimensions developed by Hull sometime ago, one expects there exists
nontrivial flow in the 3-dimensional boundary field theory. We find that these
gaugings provide first-order domain-wall solutions from direct extremization of
energy-density. We also consider the most general CSO(p, q, r)^{+} with p+q+r=8
gauging of N=8 supergravity by two successive SL(8,R) transformations of the de
Wit-Nicolai theory, that is, compact SO(8) gauged supergravity. The theory
found earlier has local SU(8)x CSO(p, q, r)^{+} gauge symmetry as well as local
N=8 supersymmetry. The gauge group CSO(p, q, r)^{+} is spontaneously reduced to
its maximal compact subgroup SO(p)^{+} x SO(q)^{+} x U(1)^{+r(r-1)/2}. The
T-tensor we obtain describes a two-parameter family of gauged N=8 supergravity
from which one can construct A_1 and A_2 tensors. The effective nontrivial
scalar potential can be written as the difference of positive definite terms.
We examine the scalar potential for critical points at which the expectation
value of the scalar field is SO(p)^{+} x SO(q)^{+} x SO(r)^{+} invariant. It
turns out that there is no new extra critical point. However, we do have flow
equations and domain-wall solutions for the scalar fields are the gradient flow
equations of the superpotential that is one of the eigenvalues of A_1 tensor.

High Energy Physics - Theory