Hidden conformal symmetry of the Kerr black hole

Extreme and very-near-extreme spin $J$ Kerr black holes have been conjectured to be holographically dual to two-dimensional (2D) conformal field theories (CFTs) with left and right central charges $c_L=c_R=12J$. In this paper it is observed that the 2D conformal symmetry of the scalar wave equation at low frequencies persists for generic nonextreme values of the mass $M\ne \sqrt{J}$. Interestingly, this conformal symmetry is not derived from a conformal symmetry of the spacetime geometry except in the extreme limit. The $2\pi$ periodic identification of the azimuthal angle $\varphi$ is shown to correspond to a spontaneous breaking of the conformal symmetry by left and right temperatures $T_L=M^2/2\pi J$ and $T_R=\sqrt{M^4-J^2}/2\pi J$. The well-known low-frequency scalar-Kerr scattering amplitudes coincide with correlators of a 2D CFT at these temperatures. Moreover, the CFT microstate degeneracy inferred from the Cardy formula agrees exactly with the Bekenstein-Hawking area law for all $M$ and $J$. These observations provide evidence for the conjecture that the Kerr black hole is dual to a $c_L=c_R=12J$ 2D CFT at temperatures ($T_L,T_R$) for every value of $M$ and $J$.