Holographic CFT thermodynamics of charged, rotating black holes in $D=4$ dimension

We study the holographic thermodynamics of $4-D$ Kerr-Newman AdS black holes. We consider the conformal thermal states dual to KN AdS black holes and work out the corresponding thermodynamics in 10 ensembles. These ensembles are: fixed (Q,J,V,C), fixed (Q,Ω,V,C), fixed (φ,Ω,V,C), fixed (φ,J,V,C), fixed (Q,J,p,C), fixed (Q,Ω,p,C),fixed (φ,J,p,C),fixed (Q,J,p,μ) and fixed (φ,Ω,p,μ) ensembles. Here φ, Q, Ω, J, p, V, C and μ denotes the electric potential, electric charge, angular velocity, angular momentum, CFT pressure, CFT volume, central charge, and chemical potential respectively. In the fixed (Q,J,V,C) ensemble, we observe a first order phase transition for Q<Qcrit, J<Jcrit and C>Ccrit. In the fixed (Q,Ω,V, C) ensemble, we again find a first-order phase transition for Q<Qcrit, Ω<Ωcrit and C>Ccrit. The fixed (φ,Ω,V, C) ensemble is characterized by a confinement/de-confinement phase transition. In the fixed (φ,J,V,C) ensemble, we see a first order phase transition for J<Jcrit, φ<φcrit and C>Ccrit. Finally, in the fixed (Q,J,p, C),(Q,Ω,p, C),(φ,J,p, C),(Q,J,p,μ),(φ,Ω,p,μ) ensembles, we do not observe any critical behavior or phase transition.