Topology and geometry of 6-dimensional (1,0) supergravity black hole horizons

Abstract

We show that the supersymmetric near horizon black hole geometries of 6-dimensional supergravity coupled to any number of scalar and tensor multiplets are either locally $AdS_3\times \Sigma^3$, where \Sigma^3 is a homology 3-sphere, or $\bR^{1,1}\times {\cal S}^4$, where ${\cal S}^4$ is a 4-manifold whose geometry depends on the hypermultiplet scalars. In both cases, we find that the tensorini multiplet scalars are constant and the associated 3-form field strengths vanish. We also demonstrate that the $AdS_3\times \Sigma^3$ horizons preserve 2, 4 and 8 supersymmetries. For horizons with 4 supersymmetries, \Sigma^3 is in addition a non-trivial circle fibration over a topological 2-sphere. The near horizon geometries preserving 8 supersymmetries are locally isometric to either $AdS_3\times S^3$ or $\bR^{1,1}\times T^4$. Moreover, we show that the $\bR^{1,1}\times {\cal S}$ horizons preserve 1, 2 and 4 supersymmetries and the geometry of ${\cal S}$ is Riemann, K\"ahler and hyper-K\"ahler, respectively.