Abstract
We classify the necessary and sufficient conditions to obtain the near-horizon geometry of extremal supersymmetric rotating black holes embedded in 11d supergravity which are associated to rotating M2-branes. Such rotating black holes admit an AdS2 near-horizon geometry which is fibered by the transverse spacetime directions. In this paper we allow for the most general fibration over AdS2 with a flux configuration permitting rotating M2-branes. Using G-structure techniques we rewrite the conditions for supersymmetry in terms of differential equations on an eight-dimensional balanced space. The 9d compact internal space is a U(1)-fibration over this 8d base. The geometry is constrained by a master equation reminiscent of the one found in the non-rotating case. We give a Lagrangian from which the equations of motion may be derived, and show how the asymptotically AdS4 electrically charged Kerr-Newman black hole in 4d mathcal{N} = 2 supergravity is embedded in the classification. In addition, we present the conditions for the near-horizon geometry of rotating black strings in Type IIB by using dualities with the 11d setup.
Highlights
Type IIB and AdS2 solutions in 11d supergravity
We classify the necessary and sufficient conditions to obtain the near-horizon geometry of extremal supersymmetric rotating black holes embedded in 11d supergravity which are associated to rotating M2-branes
In particular we show that when supersymmetry is imposed on the action it reduces to a simple form which computes the entropy of the black hole/string
Summary
In [33] the conditions for a solution of 11d supergravity to admit a timelike Killing spinor were derived. It would be interesting in the future to relax these assumptions To engineer such solutions one should place the rotating M2-branes in an asymptotic geometry of the form Rt CY5 and wrap the M2-brane on a Riemann surface inside the Calabi-Yau five-fold. Since the asymptotic geometry is Calabi-Yau it is natural to expect that our 10d base space is complex, which requires that w1 = w2 = 0 This is how the rotating M2-brane solution is embedded in the classification of [33] we have not been able to prove that restricting to just M2branes implies the complex condition. In addition to requiring the complex condition we want to eliminate the possibility of having flux sourcing M5-branes For this reason we will remove any terms appearing in the flux which are of Hodge type (4, 0)+(0, 4), since these would not come from M2-branes wrapped on a Riemann surface..
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