A topological perspective on bulk boundary thermodynamic equivalence

TL;DR

The paper develops an exact bulk–boundary thermodynamic duality for a five-dimensional charged Gauss–Bonnet AdS black hole and its four-dimensional boundary CFT, where the boundary theory naturally hosts two central charges $C$ and $A$ from trace anomalies. It constructs a holographic first law linking bulk variables to boundary quantities via a conformal boundary metric and a dictionary that identifies a boundary enthalpy-like energy $\tilde{E}$, temperature $\tilde{T}$, entropy $\tilde{S}$, and central charges, with the bulk mass $M$, temperature $T$, entropy $S$, and Gauss–Bonnet parameter $\alpha$; the boundary theory exhibits phase structure and critical behavior identical to the bulk, including van der Waals-like transitions. The authors then develop a thermodynamic topology analysis using Duan’s phi-mapping, defining topological charges for both the first-order phase transition and the critical point, and show that these charges match across bulk and boundary. This bulk–boundary topological equivalence reinforces holographic connections and suggests robust, universal topological features of holographic thermodynamics that may extend to more general higher-curvature gravities and dimensions.

Abstract

We establish an exact duality between the extended thermodynamics of five-dimensional charged Gauss-Bonnet AdS black holes and the thermodynamic framework of the dual boundary conformal field theory (CFT). The thermodynamics of the dual CFT involves two central charges originating from the trace anomaly. We demonstrate a precise correspondence between the extended first laws on the bulk and boundary sides. Moreover, the topological charges of the CFT thermodynamics, associated with the phase transition and critical point, coincide with those of the corresponding bulk black hole.

Figures (7)

• Figure 1: The $\Tilde{T}-\Tilde{S}$ oscillatory behavior and the swallowtail behavior of the dual CFT. Here we have set the charge $\Tilde{Q}=0$, the boundary radius $R=10000$, and the central charge $A=27$.
• Figure 2: The $\Tilde{T}-\Tilde{S}$ oscillatory behavior and the swallowtail behavior of the dual CFT. Here we have set the charge $\Tilde{Q}=0.0866855$, the boundary radius $R=10000$, and the central charge $A=27$.
• Figure 3: Zero points of $\phi^{\tilde{S}}$ in the $\tau$--$\tilde{S}$ plane for the dual CFT. The black solid, blue dashed, and red solid curves correspond to the low-entropy (LE), intermediate-entropy (IE), and high-entropy (HE) branches, respectively. Black and blue dots indicate the annihilation and generation points. We have taken the boundary radius $R=10,000$, the central charge $A=27$. (a) Zero points of $\phi^{\tilde{S}}$ in the $\tau$--$\tilde{S}$ plane for the neutral dual CFT ($\tilde{Q}=0$) with central charge $C=29$. (b) Zero points of $\phi^{\tilde{S}}$ in the $\tau$--$\tilde{S}$ plane for the charged dual CFT with electric charge $\tilde{Q}=0.0866855$ and central charge $C=30$.
• Figure 4: The arrows depict the unit vector field $\mathbf{n}$ on a portion of the $x-\Theta$ plane, with zero points indicated by red dots. (a) For the dual CFT with $\Tilde{Q}=0$, the zero points of the vector field are located at $(x,\Theta)=(0.148427,\pi/2)$, $(0.335925,\pi/2)$, and $(0.929365,\pi/2)$, corresponding to ZP$_1$, ZP$_2$, and ZP$_3$, respectively. These points are associated with the low-, intermediate-, and high-entropy states of the dual CFT. (b) For the dual CFT with $\Tilde{Q}\neq 0$, the zero points of the vector field are located at $(x,\Theta)=(0.250441,\pi/2)$, $(0.353396,\pi/2)$, and $(0.687455,\pi/2)$ for the charge value $\Tilde{Q}=0.0866855$, corresponding to ZP$'_1$, ZP$'_2$, and ZP$'_3$, respectively. These points again represent the low-, intermediate-, and high-entropy states of the dual CFT. Throughout the analysis, we set the boundary radius to $R=10000$ and the central charge to $A=27$.
• Figure 5: $\Omega$ vs. $\vartheta$ for the contours $C_{1}, C_{2}, C_{3}, C_{4}$ in subfigure (a) (with $\tilde{Q} = 0$) and $C'_{1}, C'_{2}, C'_{3}, C'_{4}$ in subfigure (b) (with $\tilde{Q} \neq 0$).