  We study a renormalization group flow of ABJM theory embedded into the warped
A_8 geometry and explore the dependence of the vacuum structure on the
parameters of the theory. This model has a product group gauge structure
U(N)xU(n+l) and comes equipped with discrete parameters N, l and k, a
continuous parameter b related to the ratio of the Yang-Mills coupling for the
two gauge groups, and one dimensionful parameter gYM^2 setting the overall
scale. A supersymmetric supergravity solution exists when Q=N-l(l-k)/2k-k/24 is
positive and is interpretable as a RG flow from a Yang-Mills like UV fixed
point to a superconformal IR fixed point with free energy of order Q^3/2. The
fate of the theory when Q is taken to be negative is less clear. We explore the
structure of the possible gravity solution for small negative Q by considering
the linearized gravitational back reaction from adding a small number of
anti-branes on the Q=0 background. Following the work of Bena, et.al., we find
that a sensible solution satisfying appropriate boundary conditions does not
appear to exist. This leaves the status of the RG flow for the Q<0 theories a
mystery. We offer the following speculative resolution to the puzzle: the -k/24
unit of charge induced by the curvature correction to supergravity should be
considered an allowed physical object, and one should be adding an anti brane
not to the Q=0 background but rather the Q=-k/24 background. Such a solution
has a repulson singularity, and gives rise to a picture of the vacuum
configuration where a cluster of anti-branes are floating around the repulson
singularity, but are stabilized from being pushed off to infinity by other
fluxes. Such a state is non-supersymmetric and appears to describe a vacuum
with dynamical breaking of supersymmetry. Based on these considerations, we
construct a phase diagram for this theory exhibiting various interesting
regions.
