We study supersymmetric domain walls of four dimensional $SU(N)$ SQCD with $N$ and $N+1$ flavors. In $4d$ we analyze the BPS differential equations numerically. In $3d$ we propose the $mathcal{N}=1$ Chern-Simons-Matter gauge theories living on the wa
lls. Compared with the previously studied regime of $F<N$ flavors, we encounter a couple of novelties: with $N$ flavors, there are solutions/vacua breaking the $U(1)$ baryonic symmetry; with $N+1$ flavors, our $3d$ proposal includes a linear monopole operator in the superpotential.
We consider supersymmetric domain walls of four-dimensional $mathcal{N}!=!1$ $Sp(N)$ SQCD with $F!=!N+1$ and $F!=!N+2$ flavors. First, we study numerically the differential equations defining the walls, classifying the solutions. When $F!=!N+2$, in
the special case of the parity-invariant walls, the naive analysis does not provide all the expected solutions. We show that an infinitesimal deformation of the differential equations sheds some light on this issue. Second, we discuss the $3d$ $mathcal{N}!=!1$ Chern-Simons-matter theories that should describe the effective dynamics on the walls. These proposals pass various tests, including dualities and matching of the vacua of the massive $3d$ theory with the $4d$ analysis. However, for $F!=!N+2$, the semiclassical analysis of the vacua is only partially successful, suggesting that yet-to-be-understood strong coupling phenomena are into play in our $3d$ $mathcal{N}!=!1$ gauge theories.
We describe a new procedure to obtain consistent backgrounds that uplift vacua and deformations of various maximal gauged supergravities by taking a known solution and performing singular limits along the moduli space of the corresponding 4-dimension
al theory. We then apply this procedure to the S^3 x H^{2,2} background that provides the uplift of 4-dimensional Minkowski vacua of maximal supergravity with gauge group [SO(4) x SO(2,2)] $ltimes$ R^{16}. We find that the newly generated vacua are generally only locally geometric and correspond to asymmetric orbifolds, Q-flux backgrounds or combinations thereof. We also provide the uplift to eleven dimensions of all the four-parameter Cremmer-Scherk-Schwarz gaugings.
