Thermodynamics of AdS Black Holes: Critical Behavior of the Central Charge

We reconsider the thermodynamics of anti-de Sitter black holes in the context of gauge-gravity duality. In this new setting, where both the cosmological constant Λ and the gravitational Newton's constant G are varied in the bulk, we rewrite the first law in a new form containing both Λ (associated with thermodynamic pressure) and the central charge C of the dual conformal field theory and their conjugate variables. We obtain a novel thermodynamic volume, in turn leading to a new understanding of the Van der Waals behavior of charged anti-de Sitter black holes in which phase changes are governed by the degrees of freedom in the conformal field theory. Compared to the "old" P-V criticality, this new criticality is "universal" (independent of the bulk pressure) and directly relates to the thermodynamics of the dual field theory and its central charge.