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Register a new userAll superalgebras for warped AdS$_2$ and black hole near horizon geometries
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We identify all symmetry superalgebras $mathfrak{g}$ of near horizon geometries of black holes with a Killing horizon, assuming the solution is smooth and that the spatial cross section of the event horizon is compact without boundary. This includes all warped AdS$_2$ backgrounds with the most general allowed fluxes in 10- and 11-dimensional supergravities. If the index of a particular Dirac operator vanishes, we find that the even symmetry subalgebra decomposes as $mathfrak{g}_0=mathfrak{sl}(2,mathbb{R})oplus mathfrak{t}_0$, where $mathfrak{t}_0/mathfrak{c}$ is the Lie algebra of a group that acts transitively and effectively on spheres, and $mathfrak{c}$ is the center of $mathfrak{g}$. If the Dirac operator index does not vanish, then the symmetry superalgebra is nilpotent with one even generator. We also demonstrate that there are no near horizon geometries, and also therefore no warped AdS$_2$ backgrounds, in 10- and 11-dimensions that preserve more than 16 supersymmetries.
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We present all the symmetry superalgebras $mathfrak{g}$ of all warped AdS$_ktimes_w M^{d-k}$, $k>2$, flux backgrounds in $d=10, 11$ dimensions preserving any number of supersymmetries. First we give the conditions for $mathfrak{g}$ to decompose into a direct sum of the isometry algebra of AdS$_k$ and that of the internal space $M^{d-k}$. Assuming this decomposition, we identify all symmetry superalgebras of AdS$_3$ backgrounds by showing that the isometry groups of internal spaces act transitively on spheres. We demonstrate that in type II and $d=11$ theories the AdS$_3$ symmetry superalgebras may not be simple and also present all symmetry superalgebras of heterotic AdS$_3$ backgrounds. Furthermore, we explicitly give the symmetry superalgebras of AdS$_k$, $k>3$, backgrounds and prove that they are all classical.
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