Supersymmetry of Bianchi attractors in gauged supergravity

Bianchi attractors are near horizon geometries with homogeneous symmetries in spatial directions. We construct supersymmetric Bianchi attractors in $\mathcal{N}=2,d=4$, 5 gauged supergravity. In $d=4$, we consider gauged supergravity coupled to vector and hypermultiplets. In $d=5$, we consider gauged supergravity coupled to vector multiplets with a generic gauging of symmetries of the scalar manifold and the $U(1{)}_R$ gauging of the $R$-symmetry. Analyzing the gaugino conditions, we show that when the fermionic shifts do not vanish, there are no supersymmetric Bianchi attractors. This is analogous to the known condition that for maximally supersymmetric solutions, all of the fermionic shifts must vanish. When the central charge satisfies an extremization condition, some of the fermionic shifts vanish and supersymmetry requires that the symmetries of the scalar manifold are not gauged. This allows supersymmetric Bianchi attractors sourced by massless gauge fields and a cosmological constant. In five dimensions in the Bianchi I class, we show that the anisotropic ${\mathrm{AdS}}_3\times {\mathbb{R}}^2$ solution is $1/2$ BPS (Bogomol’nyi-Prasad-Sommerfield). We also construct a new class of $1/2$ BPS Bianchi III geometries labeled by the central charge. When the central charge takes a special value, the Bianchi III geometry reduces to the known ${\mathrm{AdS}}_3\times {\mathbb{H}}^2$ solution. For the Bianchi V and VII classes, the radial spinor breaks all of supersymmetry. We briefly discuss the conditions for possible massive supersymmetric Bianchi solutions by generalizing the matter content to include tensor, hypermultiplets, and a generic gauging on the $R$-symmetry.