Supersymmetric AdS${}_2\times {\mathrm{\Sigma }}_2$ solutions from tri-sasakian truncation

A class of $Ad{S}_2\times {\mathrm{\Sigma }}_2$, with ${\mathrm{\Sigma }}_2$ being a two-sphere or a hyperbolic space, solutions within four-dimensional $N=4$ gauged supergravity coupled to three-vector multiplets with dyonic gauging is identified. The gauged supergravity has non-semisimple $SO(3)\ltimes ($\mathbf{T}$^3,\hat{$\mathbf{T}$}^3)$ gauge group and can be obtained from a consistent truncation of eleven-dimensional supergravity on a tri-sasakian manifold. The maximally symmetric vacua contain $Ad{S}_4$ geometries with $N=1,3$ supersymmetry corresponding to $N=1$ and $N=3$ superconformal field theories (SCFTs) in three dimensions. We find supersymmetric solutions of the form $Ad{S}_2\times {\mathrm{\Sigma }}_2$ preserving two supercharges. These solutions describe twisted compactifications of the dual $N=1$ and $N=3$ SCFTs and should arise as near horizon geometries of dyonic black holes in asymptotically $Ad{S}_4$ space-time. Most solutions have hyperbolic horizons although some of them exhibit spherical horizons. These provide a new class of $Ad{S}_2\times {\mathrm{\Sigma }}_2$ geometries with known M-theory origin.