Abstract:We study the effective dynamics of black hole horizons in Einstein-Maxwell theory in a large number of spacetime dimensions D. We demonstrate that horizon dynamics may be recast as a well posed initial value problem for the motion of a codimension one non gravitational membrane moving in flat space. The dynamical degrees of freedom of this membrane are its shape, charge density and a divergence free velocity field. We determine the equations that govern membrane dynamics at leading order in the large D expansion. Our derivation of the membrane equations assumes that the solution preserves an SO(D − p − 2) isometry with p held fixed as D is taken to infinity. However we are able to cast our final membrane equations into a completely geometric form that makes no reference to this symmetry algebra.

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We develop new techniques to efficiently evaluate heat kernel coefficients for the Laplacian in the short-time expansion on spheres and hyperboloids with conical singularities. We then apply these techniques to explicitly compute the logarithmic contribution to black hole entropy from an N = 4 vector multiplet about a Z N orbifold of the nearhorizon geometry of quarter-BPS black holes in N = 4 supergravity. We find that this vanishes, matching perfectly with the prediction from the microstate counting. We also discuss possible generalisations of our heat kernel results to higher-spin fields over Z N orbifolds of higher-dimensional spheres and hyperboloids.

Abstract:We compute the logarithmic correction to black hole entropy about exponentially suppressed saddle points of the Quantum Entropy Function corresponding to Z N orbifolds of the near horizon geometry of the extremal black hole under study. By carefully accounting for zero mode contributions we show that the logarithmic contributions for quarter-BPS black holes in N = 4 supergravity and one-eighth BPS black holes in N = 8 supergravity perfectly match with the prediction from the microstate counting. We also find that the logarithmic contribution for half-BPS black holes in N = 2 supergravity depends non-trivially on the Z N orbifold. Our analysis draws heavily on the results we had previously obtained for heat kernel coefficients on Z N orbifolds of spheres and hyperboloids in arXiv:1311.6286 and we also propose a generalization of the Plancherel formula to Z N orbifolds of hyperboloids to an expression involving the Harish-Chandra character of sl (2, R), a result which is of possible mathematical interest.

It has recently been shown that the dynamics of black holes in large number of dimensions D can be recast as the dynamics of a probe membrane propagating in the background spacetime which solves Einstein equations without matter. The equations of motion of this membrane are simply the statement of conservation of the stress tensor and charge current defined on this membrane. In this paper we obtain the effective equations of motion for stationary membranes in any empty background both in presence and absence of charge. It turns out that the thermodynamic quantities associated with the stationary membranes that satisfy these effective equations also satisfy the first law of black hole thermodynamics. These stationary membrane equations have some interesting solutions such as charged rotating black holes in flat and AdS backgrounds as well as black ring solutions in large D. arXiv:1806.04637v2 [hep-th] 4 Jul 2018 ContentsRecently the authors of [1][2][3][4] made an observation that in the limit where D 1, the degrees of freedom of the black hole spacetime separate into light degrees with length scale r 0 and heavy degrees with length scale r 0 D , where r 0 is a characteristic length scale associated with black hole horizon. They also showed in [5,6] that the quasinormal modes about a static, spherically symmetric black hole in such a large number of dimensions show a similar separation of scales. So it should be possible 1 See [7-11] for related discussion. 2 See [19-21] for more work in this membrane paradigm. 3 See [22-33] for other work that uses large D expansion.