Thermal superpotential and thermodynamics of neutral hairy black holes in extended SUGRA

TL;DR

This work constructs an exact family of neutral hairy black holes in extended $\mathcal{N}=2$ gauged supergravity with Fayet–Iliopoulos terms, focusing on the pure electric sector and a consistent dilaton truncation. It derives a closed-form thermal superpotential that serves as a counterterm, showing that no additional finite counterterms are needed to regularize the Euclidean action and quasi-local stress tensor, and computes the holographic stress tensor, which matches that of a thermal conformal fluid under mixed dilaton boundary conditions. Thermodynamically, the hairy solutions exhibit Hawking–Page transitions between the hairy black hole and thermal AdS, with the hair parameter $\nu$ controlling the minimum and transition temperatures. The results provide a coherent holographic interpretation of the hairy black holes, including a triple-trace boundary deformation with $\Delta=1$ and a well-defined holographic renormalization framework for nonconformal RG flows in curved boundaries.

Abstract

We present a family of exact neutral hairy black-hole solutions with spherical horizon topology in extended supergravity with Fayet--Iliopoulos terms. We consider a consistent dilaton truncation and analyze in detail a sector where the magnetic part of the FI terms vanishes. Using appropriate dilaton counterterms, we compute the thermodynamic quantities and show the existence of Hawking--Page phase transitions. As a holographic application, we derive the thermal superpotential in closed form and use it as a counterterm, explicitly demonstrating that no additional finite counterterms are required to regularize the Euclidean action and the quasi-local stress tensor. The dual stress tensor matches that of a thermal gas of massless particles and is consistent with mixed dilaton boundary conditions that preserve conformal symmetry.

Figures (5)

• Figure 1: The left panel shows the free energy as a function of temperature, exhibiting a first-order phase transition at $F = 0$, corresponding to the Hawking--Page temperature $T_\textsc{hp} = 1/(\pi L)$. The right panel displays the mass-temperature relation, which features two branches corresponding to small (unstable) and large (stable) black holes.
• Figure 2: The plots show the free energy as a function of temperature, illustrating the presence of a first-order phase transition at $F=0$. On the left, for $2.1\leq\nu\leq3$, the minimum temperature remains nearly constant. On the right, for $\nu>3$, the scalar hair significantly affects both the minimum temperature and the Hawking--Page temperature.
• Figure 3: The left panel ($2.1\leq\nu\leq 2.5$) shows the mass–temperature relation, where a discontinuity in the heat capacity at $T_\text{min}$ confirms the existence of two distinct branches, corresponding to small and large black holes. In the right panel ($\nu\geq3.1$) the temperature at which hairy black holes exist increases consistently with the hair parameter $\nu$.
• Figure 4: The left panel ($2.1 \leq \nu \leq 2.5$) shows that the minimum temperature $T_\text{min}$ does not vary significantly. In contrast, in the right panel ($\nu \geq 3.5$) $T_\text{min}$ grows more rapidly as the scalar hair increases.
• Figure 5: In the left panel ($\nu=2.2$), the plots of $g_{tt}$ versus $\Upsilon$ for various values of $M$ reveal the presence of the event horizon. The dashed curve represents $M_\text{min}\,G/L=0.447$, indicating that an event horizon forms only when $M>M_\text{min}$. The right panel shows the dependence of $M_\text{min}$ on $\nu$, exhibiting a minimum at $\nu=3$ with $M_\text{min}=L/(3G)$.