All superalgebras for warped AdS ₂ and black hole near horizon geometries

Abstract

We identify all symmetry superalgebras 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 11dimensional supergravities. If the index of a particular Dirac operator vanishes, we find that the even symmetry subalgebra decomposes as g 0 = sl(2, R)⊕ t 0 , where t 0 /c is the Lie algebra of a group that acts transitively and effectively on spheres, and c is the center of 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.

N = 4 near-horizon geometries in D = 11 supergravity

“…The geometric structure of near-horizon solutions admitting N = 2 JHEP07(2021)155 supersymmetry was classified in [25,26]. Following on from this, analysis of the superalgebras associated with supersymmetric near-horizon geometries [27,28] was used to further elucidate the nature of the symmetry algebras of solutions with N > 2 supersymmetry. In conjunction with the homogeneity theorem in [29], together with a classification of all compact homogeneous 9-dimensional Riemannian manifolds in [30], it was shown that there are no supersymmetric near horizon geometries with 16 < N < 32 supersymmetries.…”

“…In this section, we collate and summarize the key results obtained from [25][26][27][28]37], which we shall utilize in considering the analysis of near-horizon geometries.…”

“…and in particular, it has been shown that if φ r − = 0 then φ r + defined as above is nonvanishing. A further algebraic relation, based on a superalgebra computation [27,28], has also been constructed which relates φ r − to φ r + via…”

Classification, geometry and applications of supersymmetric backgrounds

“…One uses a similar methodology to establish the classification statement for AdS 4 backgrounds in Section 12.7 for (massive) IIA and IIB supergravities. This method can also be extended to prove a non-existence result for AdS 2 and AdS 3 backgrounds that preserve N > 16 supersymmetries [261,265].…”

A non-existence theorem for N > 16 supersymmetric AdS3 backgrounds

“…It has also been shown in [18] that the only smooth AdS 4 solution with compact without boundary internal space that preserves 16 < N < 32 supersymmetries is locally isometric to the N = 24 IIA solution AdS 4 × CP 3 of [19]. Moreover it has been shown in [20] that there are no smooth AdS 2 backgrounds in 10-and 11-dimensional supergravities with compact without boundary internal space that preserve N > 16 supersymmetries. Product solutions AdS n × M D−n with M D−n a symmetric space have been classified in [21]- [24].…”