7.1 Summary

Let us summarize the key results that have been derived so far. Any extremal black hole containing a compact U (1) axial symmetry admits a Virasoro algebra in its near-horizon geometry with a non-trivial central charge. The black-hole entropy is reproduced by a chiral half of Cardy’s formula. This result is robust for any diffeomorphism-invariant theory and holds even including scalar and gauge field couplings and higher-derivative corrections. Moreover, if a U (1) gauge field can be geometrized into a Kaluza–Klein vector in a higher-dimensional spacetime, a Virasoro algebra can be defined along the Kaluza–Klein compact U (1) direction and all analysis goes through in a similar fashion as for the axial U (1) symmetry. The deep similarity between the effects of rotation and electric charge can be understood from the fact that these charges are on a similar footing in the higher-dimensional geometry. When two U (1) symmetries are present, one can mix up the compact directions using a modular transformation and the construction of Virasoro algebras can still be made.

Independent of these constructions, the scattering probabilities of probes around the near-extremal Kerr–Newman black hole can be reproduced near the superradiant bound by manipulating near-chiral thermal two-point functions of a two-dimensional CFT. The result extends straightforwardly to other asymptotically-flat or AdS black holes in various gravity theories. Finally away from extremality, hidden SL (2,ℝ ) × SL (2,ℝ ) symmetries are present in some scalar probes around the Kerr–Newman black hole close enough to the horizon. We showed that several CFTs are required to account for the entire probe dynamics in the near region in the regime of small mass, small energy and small charge. This analysis does not extend straightforwardly to AdS black holes.

These results – obtained in gravity coupled to matter – are naturally accounted for by assuming that the microstates of asymptotically-flat black holes, at extremality and away from extremality, can be described by 2d CFTs and that the microstates of asymptotically-AdS black holes at extremality can be described by chiral halves of 2d CFTs. Scattering amplitudes and hidden symmetries are also accounted for by assuming that part of the dynamics of black holes can be mapped to the dynamics of these CFTs once they are suitably coupled to the exterior black-hole region. By consistency with the gravitational analysis, several CFT descriptions are available when several compact U (1) symmetries are present. The existence of such CFTs is conjectural and only future research will tell how far these Kerr/CFT correspondences and their extensions can be made more precise.

A fair concluding remark would be that our understanding of the Kerr, Reissner–Nordström and Kerr–Newman black hole has increased over the last four years, but there is still a long road ahead of us to comprehend what these CFTs really are and what they are telling us about the nature of quantum black holes.

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