Conformal Holographic Extended Thermodynamics

• Conformal holographic extended thermodynamics is an approach in gauge/gravity duality that treats the boundary conformal factor as an independent thermodynamic variable.
• It explores the role of central charge variations in driving van der Waals-like phase transitions and critical phenomena in higher-dimensional AdS black holes.
• The framework links microscopic CFT properties with macroscopic bulk observables, offering new insights into black hole stability and quantum gravitational effects.

Conformal holographic extended thermodynamics is an advanced framework in gauge/gravity duality that systematically explores the interplay between extended black hole thermodynamics in anti-de Sitter (AdS) spacetimes and the thermodynamic properties of boundary conformal field theories (CFTs). By treating the conformal rescaling factor of the CFT boundary as a thermodynamic variable and analyzing the effects of varying the central charge and spacetime dimensionality, this formalism yields new insight into phase transitions, stability, and the microscopic origins of gravitational thermodynamics in higher-dimensional black holes (Ladghami et al., 7 Oct 2025).

1. Conformal Rescaling Factor as an Independent Thermodynamic Variable

The foundation of conformal holographic extended thermodynamics (abbreviated as CHET) is the promotion of the boundary CFT’s conformal factor, ω, to a thermodynamic parameter. The boundary metric is rescaled as

$$ds^2_{\text{CFT}} = \omega^2 (-dt^2 + L^2 d\Omega^2_{d-2}),$$

where $L$ is the AdS radius and $d\Omega^2_{d-2}$ is the metric on the unit $(d-2)$-sphere. The spatial volume becomes $V \propto (\omega L)^{d-2}$ and the CFT energy is rescaled as $Ẽ = E / \omega$. This formalism allows for independent variations in the CFT size (through $\omega$ or $V$) and the number of degrees of freedom (encoded in the central charge $C$). This approach resolves degeneracies present in earlier holographic dictionaries where variations of $L$ simultaneously changed the CFT volume and central charge, and it ensures a non-degenerate extended first law of thermodynamics for the boundary CFT.

2. Role of Central Charge and Its Variation

The central charge $C$ quantifies the effective number of degrees of freedom in the boundary CFT, scaling as $C = L^{d-2}/G$ with fixed Newton constant $G$. Varying $C$ is equivalent to exploring families of CFTs (e.g., $C \propto N^2$ in $SU(N)$ gauge theory) and, via the bulk/boundary correspondence, directly influences the thermodynamic landscape of the dual gravitational theory. For charged higher-dimensional AdS black holes (such as five- and six-dimensional Reissner–Nordström–AdS solutions), all thermodynamic quantities—mass, temperature, free energy, and the critical parameters for phase transitions—depend explicitly on $C$. Criticality in these systems is determined by equations such as

$$\left( \frac{\partial \hat{T}}{\partial S} \right)_{Q̃, C} = 0, \quad \left( \frac{\partial^2 \hat{T}}{\partial S^2} \right)_{Q̃, C} = 0,$$

yielding critical entropy $S_c$, critical central charge $C_c$, and critical temperature $\hat{T}_c$, with transitions classified by the value of $C$ relative to $C_c$.

3. Dimensionality Dependence and Comparison with BTZ Black Holes

The emergence and structure of thermodynamic phase transitions are strongly dimension-dependent. In $d \geq 5$, the T–S and F–T phase diagrams of charged AdS black holes display classic van der Waals-like behavior:

• First-order phase transitions occur for $C > C_c$, signaled by swallowtail structures in the Helmholtz free energy and multiple T–S branches.
• At $C = C_c$, a second-order critical point is found, with critical exponents matching mean-field universality.

In lower dimensions (notably in $d=3$, for BTZ black holes), such phase transitions are absent; T(S) is monotonic, the system remains thermodynamically stable for all $C$, and no swallowtail appears in free energy plots. This contrast underlines the essential role of spacetime dimensionality and boundary degrees of freedom in the gravitational phase structure.

4. Van der Waals-Like Phase Transitions and Critical Phenomena

Higher-dimensional charged AdS black holes exhibit thermodynamic behavior closely paralleling that of a van der Waals fluid. For fixed $Q̃$ and $C$:

• The T–S diagram shows stable “small” and “large” black holes (phases with positive heat capacity) separated by an unstable “medium” branch with negative heat capacity.
• The Helmholtz free energy $F̂ = M̂ - T̂ S$ develops swallowtail features for $C > C_c$, indicating first-order phase coexistence.
• The loci of second-order criticality are defined by stationary points of $T̂(S)$ as above.
• The coexistence of phases, and the amplitude of phase separation, are governed by the central charge.

This correspondence further cements the analogy between black hole thermodynamics and classical liquid–gas transitions, now controlled by holographic data.

5. Extended First Law and Smarr Relation in CHET

Treating the conformal factor ω as a variable produces a non-degenerate, scale-covariant first law and Smarr formula for the boundary dual, e.g.,

$$d\hat{M} = \hat{T}\,dS + \hat{\Phi}\,dQ̃ + \hat{\mu}\,dC,$$

$\hat{M} = \hat{T}S + \hat{\Phi} Q̃ + \hat{\mu} C.$

Here, all “hatted” variables are rescaled by ω and L for conformity with the boundary geometry. Notably, the chemical potential term $\hat{\mu}\,dC$ encodes the energetic cost of changing the number of degrees of freedom, and its sign or vanishing delineates quantum, classical, and non-holographic regimes for the black hole system.

6. Microscopic Implications and Holographic Insights

The conformal rescaling parameter ω, combined with variations in central charge C, permits a fine-grained exploration of how microscopic CFT properties shape the bulk gravitational thermodynamics. The boundary degrees of freedom, as indexed by $C$, modulate the stability, criticality, and phase behavior of AdS black holes. For $C < C_c$, only a single stable phase is present. For $C = C_c$ or $C > C_c$, rich critical and first-order transition phenomena—analogous to those in condensed matter systems—are found. The absence of such structure in three-dimensional BTZ black holes further illustrates the unique interplay of boundary theory characteristics with bulk gravity, mediated by holography.

7. Conclusions and Broader Significance

The formalism of conformal holographic extended thermodynamics, with a variable boundary conformal factor and explicit central charge dependence, provides a robust tool for dissecting the thermodynamics of black holes in higher-dimensional AdS spacetimes. This approach:

• Recovers and clarifies van der Waals-like phase transitions and their critical exponents,
• Highlights the necessity of sufficient boundary degrees of freedom for non-trivial bulk phase structures,
• Demonstrates how dimensionality shapes gravitational thermodynamics,
• Establishes precise connections between boundary CFT properties and macroscopic bulk observables,
• Suggests new avenues for probing quantum origins of black hole thermodynamics and for studying non-equilibrium phase structure in gravitational systems.

These connections elucidate the fundamental role of boundary CFT data in governing the thermodynamics of gravitational systems and set the stage for further studies into quantum gravitational phenomena, quantum criticality, and phase transitions in the broader holographic context (Ladghami et al., 7 Oct 2025).