All Vacuum Near-Horizon Geometries in $D$-dimensions with $(D-3)$ Commuting Rotational Symmetries

We explicitly construct all stationary, non-static, extremal near horizon geometries in $D$ dimensions that satisfy the vacuum Einstein equations, and that have $D-3$ commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in $D=4,5$. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology $S^2\times T^{D-4}$, or $S^3\times T^{D-5}$, or quotients thereof. Our metrics depend on two discrete parameters specifying the topology type, as well as $(D-2)(D-3)\mathrm{/}2$ continuous parameters. Not all of our metrics in $D\ge 6$ seem to arise as the near horizon limits of known black hole solutions.