Near-Extremal Kerr $AdS_2\times S^2$ Solution and Black-Hole/Near-Horizion-CFT Duality

We study the thermodynamics of the near horizon of near extremal Kerr geometry ($near-NHEK$) within an $AdS_2/CFT_1$ correspondence. We do this by shifting the horizon by a general finite mass, which does not alter the geometry and the resulting solution is still diffeomorphic to $NHEK$, however it allows for a Robertson Wilczek two dimensional Kaluza-Klein reduction and the introduction of a finite regulator on the $AdS_2$ boundary. The resulting asymptotic symmetry group of the two dimensional Kaluza-Klein reduction leads to a non-vanishing quantum conformal field theory ($CFT$) on the respective $AdS_2$ boundary. The $s$-wave contribution of the energy-momentum-tensor of the $CFT$, together with the asymptotic symmetries, generate a Virasoro algebra with calculable center and non-vanishing lowest Virasoro eigen-mode. The central charge and lowest eigen-mode reproduce the $near-NHEK$ Bekenstein-Hawking entropy via the statistical Cardy Formula and our derived central charge agrees with the standard Kerr/$CFT$ Correspondence. We also compute the Hawking temperature of the shifted $near-NHEK$ by analyzing quantum holomorphic fluxes of the Robinson and Wilczek two dimensional analogue fields.