RG flows from (1,0) 6D SCFTs to N=1 SCFTs in four and three dimensions

We study $Ad{S}_5\times {\mathrm{\Sigma }}_2$ and $Ad{S}_4\times {\mathrm{\Sigma }}_3$ solutions of $N=2$, $SO(4)$ gauged supergravity in seven dimensions with ${\mathrm{\Sigma }}_{2,3}$ being $S^{2,3}$ or $H^{2,3}$. The $SO(4)$ gauged supergravity is obtained from coupling three vector multiplets to the pure $N=2$, $SU(2)$ gauged supergravity. With a topological mass term for the 3-form field, the $SO(4)\sim SU(2)\times SU(2)$gaugedsupergravityadmitstwosupersymmetric$AdS_7$criticalpoints,with$SO(4)$and$SO(3)$symmetries,providedthatthetwo$SU(2)$gaugecouplingsaredifferent\mathrm{.}Thesevacuacorrespondto$N=(1,0)$superconformalfieldtheories(SCFTs)insixdimensions\mathrm{.}Inthecaseof$\Sigma_2$,wefindaclassof$AdS_5\times S^2$and$AdS_5\times H^2$solutionspreservingeightsuperchargesand$SO(2)\times SO(2)$ symmetry, but only $Ad{S}_5\times H^2$ solutions exist for $SO(2)$ symmetry. These should correspond to some $N=1$ four-dimensional SCFTs. We also give RG flow solutions from the $N=(1,0)$ SCFTs in six dimensions to these four-dimensional fixed points including a two-step flow from the $SO(4)$ $N=(1,0)$ SCFT to the $SO(3)$ $N=(1,0)$ SCFT that eventually flows to the $N=1$ SCFT in four dimensions. For $Ad{S}_4\times {\mathrm{\Sigma }}_3$, we find a class of $Ad{S}_4\times S^3$ and $AdS_4\times H^3$solutionswithfoursupercharges,correspondingto$N=1$ SCFTs in three dimensions. When the two $SU(2)$ gauge couplings are equal, only $AdS_4\times H^3$arepossible\mathrm{.}Theupliftedsolutionsforequal$SU(2)$ gauge couplings to eleven dimensions are also given.