7.2 Set of open problems

We close this review with a list of open problems. We hope that the interested reader will tackle them with the aim of shedding more light on the Kerr/CFT correspondence. We tried to order the problems with increasing difficulty but the evaluation is rough and highly subjective.

  1. Hidden symmetries have been discussed so far for spin 0 probes. Discuss hidden symmetries for a probe gauge field or a probe graviton on Kerr or Kerr–Newman. Does one obtain the same temperatures TL and TR as in the scalar probe case?
  2. A black hole in de Sitter spacetime can be extremal in the sense that its outer radius coincides with the cosmological horizon. The resulting geometry, called the rotating Narirai geometry, has many similarities with the near-horizon geometries of extremal black holes in flat spacetime or in AdS spacetime. The main difference is that the near-horizon geometry is a warped product of dS2 with S2 instead of AdS 2 with S2. It has been conjectured that these extremal black holes are dual to the chiral half of a Euclidean CFT [7]. Test the conjecture by generalizing all arguments of the Kerr/CFT correspondence to this cosmological setting.
  3. Away from extremality, it is curious that the right-moving temperature is given by TR = TH ∕ΩJ for the Kerr–Newman black hole. Account for this fact. Also, for all known asymptotically-flat extremal black holes in Einstein gravity coupled to matter, the product of the horizon areas of the inner and outer horizon can be expressed in terms of quantized charges (J, Q, …) and fundamental constants only [195, 103, 106]. Explain this feature from a fundamental perspective.
  4. In the analysis of near-extremal superradiant scattering for any spin, we discarded the unstable modes that are below the Breitenlohner–Freedman bound. Such modes have imaginary β; see (177View Equation). Clarify the match between these modes and CFT expectations for the Kerr–Newman black hole.
  5. The probe scalar wave equation in Kerr–Newman–AdS has two complex poles in addition to poles corresponding to the inner and outer horizon and infinity. This prevented a straightforward generalization of the hidden SL (2,ℝ ) × SL (2,ℝ) symmetry. Clarify the role of these additional poles. Also explain why the product of all horizon areas (inner, outer and complex horizons) seems in general not to depend on the mass of the black hole [102].
  6. Near-horizon geometries of black-hole solutions of (1View Equation) have been classified. Classify the four-dimensional near-horizon geometries of extremal black holes for gravity coupled to charged scalars, massive vectors, p-forms and non-abelian gauge fields.
  7. Compute the central charges cL and cR away from extremality. Also, compute the quantum corrections to the central charge c L and investigate the matching between the quantum-corrected entropy of extremal black holes derived in [241] and the asymptotic growth of states in the dual CFT.
  8. Understand how the extension of the Kerr/CFT correspondence to extremal AdS black holes fits within the AdS/CFT correspondence. As discussed in [204], the extremal AdS–Kerr/CFT correspondence suggests that one can identify a non-trivial Virasoro algebra acting on the low-energy states of strongly coupled large N super-Yang–Mills theory in an extremal thermal ensemble. Try to make this picture more precise.
  9. From the point of view of 2d CFTs, study if a SL (2,ℤ) action exists that transforms a CFT into another CFT. This would clarify the existence of an SL (2,ℤ) family of CFTs dual to the Kerr–Newman black hole. Note that this can be done for three-dimensional CFTs with a U(1) current [267].
  10. Compute the superradiant scattering amplitude of probe scalar fields on the Kerr–Newman geometry with first-order backreaction. Compare the result with the scattering amplitude defined in the CFT at one loop order (using two and three-point correlation functions).
  11. Formulate a general scattering theory around near-extremal black-hole solutions of (1View Equation). This would require one to classify the geometries admiting a Killing–Yano tensor so that the wave equation could be separated. A long-standing problem already consists in separating and decoupling the wave e quation of a probe spin 1 or spin 2 field in the Kerr–Newman geometry.
  12. Construct one example in string theory of an exact quantum field theory dual to (an embedding in string theory of) the Kerr black hole. Characterize whether that field theory is a CFT, a limit of a CFT, or a deformation thereof.

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