Thermodynamics and topological classifications of static non-extremal four-charge AdS black hole in the five-dimensional $\mathcal{N} = 2$, $STU-W^2U$ gauged supergravity

TL;DR

The paper constructs a new static non-extremal four-charge AdS_5 black hole in five-dimensional $\mathcal{N}=2$ gauged supergravity with prepotential $\mathcal{V} = STU - W^2U \equiv 1$ and analyzes its thermodynamics in depth. It derives the full set of thermodynamic quantities, verifies that they satisfy both the differential form $dM = T dS + \Phi_1 dQ_1 + \Phi_2 dQ_2 + \Phi_3 dQ_3 - 2\Phi_4 dQ_4 + V dP$ and the integral Smarr relation $M = \frac{3}{2} T S + \Phi_1 Q_1 + \Phi_2 Q_2 + \Phi_3 Q_3 - 2\Phi_4 Q_4 - V P$, with a characteristic $-2$ weighting for the fourth charge arising from the model's prepotential. A thermodynamic topology analysis, using the off-shell free energy $\mathcal{F} = M - \frac{S}{\tau}$ and the Duan vector $\phi$, reveals a new subclass $\bar{W}^{0-}$ for $\bar{w}=0$ and assigns the known class $W^{1+}$ to $\bar{w}=1,2$, indicating a charge-driven transition in the state space. The work also recovers the three-charge AdS_5 solution in the appropriate limit and points to future avenues, including rotating generalizations and implications for AdS_5/CFT_4 correspondences.

Abstract

In this paper, we investigate thermodynamical aspects of a novel static non-extremal black hole solution in the five-dimensional anti-de Sitter (AdS) space carrying four independent electric charge parameters and with a negative cosmological constant, within the framework of $D=5$, $\mathcal{N} = 2$ gauged supergravity coupled to three vector multiplets specified by the pre-potential $\mathcal{V} = STU -W^2U \equiv 1$. A central outcome of our work is the demonstration that the thermodynamics of this solution consistently satisfies both the differential and integral forms of the first law of black hole thermodynamics. When the fourth charge is set to zero, our configuration reduces to the known three-charge static non-extremal AdS$_5$ black hole within the gauged $STU$ model, and its thermodynamic behavior connects smoothly to that limit. Finally, we also provide a brief discussion about topological classifications of its thermodynamics.

Figures (1)

• Figure 1: Zero points of $\phi^{r_h}$ in the $\tau-r_h$ plane for the four-charge static non-extremal AdS$_5$ black hole in the $\mathcal{N} = 2$, $STU-W^2U$ gauged supergravity with $q_1 = r_0$, $q_2 = 2r_0$, $q_3 = 3r_0$, $q_4 = 0.5r_0$, $l = 5r_0$, and (a) $\bar{w} = 0$; (b) $\bar{w} = 1$; (c) $\bar{w} = 2$, respectively. The blue line corresponds to a thermodynamically unstable black hole branch with the winding number $w = -1$, while the red line corresponds to a thermodynamically stable black hole branch with the winding number $w = 1$.