Warped CFT duals of the Pleba\'nski-Demia\'nski family of solutions

In this paper, we analyze the symmetry properties of the complete family of type D spacetimes generalized form the Pleba\'nski-Demia\'nski solution in four dimensions holographically in terms of a warped CFT. The generalized Pleba\'nski-Demia\'nski solutions are black hole-like spacetimes characterized by seven physical parameters. Most of the black holes in four dimensions are included within this family. Generically consider a solution with horizon in this family, we figure out the possible warped conformal symmetry attached to the horizon. The horizon can be either extremal or non-extremal. In the extremal case, the near horizon region can be mapped to an infinite spacetime with geometry given by a warped and twist product of AdS$_2$ and S$^2$. The new boundary conditions for AdS$_2$ as well as their higher dimensional uplifts are applied here to manifest the asymptotic symmetry as the warped conformal symmetry. In the non-extremal case, the global warped conformal symmetry is singled out by analyzing the scalar wave equation with constant frequency. The local warped conformal symmetries are represented by the charge algebra associated to the vector fields which preserve the scalar wave equation as well as its frequency. In defining the variation of the covariant charges, a proper counterterm is introduced for consistency conditions which is supposed to be suitable for all the solutions within the family. As a consistency check, the horizon entropy is reproduced by the entropy formula of the warped CFT by using its modular covariance and the central terms derived in the bulk spacetimes.