Black Holes with Varying Flux: A Numerical Approach

TL;DR

The paper develops a numerical framework to construct black-hole solutions in type IIB supergravity with varying flux, capturing horizons with nontrivial $F_3$ and $F_5$ that holographically describe the deconfined phase of confining gauge theories. It derives a radial, one-dimensional effective action yielding coupled ODEs for metric and flux functions, and validates the approach by reproducing analytic backgrounds such as the nonextremal D3-brane and Klebanov-Tseytlin. It then demonstrates the existence of regular-horizon solutions with finite area and KT-like asymptotics, maps their thermodynamics, and shows rich parameter-dependent behavior including islands with no black-hole solutions. The results offer a robust numerical method to explore flux-varying backgrounds in IIB supergravity and illuminate the finite-temperature phase structure of holographic confining theories, with possible implications for Hawking-Page-type transitions and related phenomena.

Abstract

We present a numerical study of type IIB supergravity solutions with varying Ramond-Ramond flux. We construct solutions that have a regular horizon and contain nontrivial five- and three-form fluxes. These solutions are holographically dual to the deconfined phase of confining field theories at finite temperature. As a calibration of the numerical method we first numerically reproduce various analytically known solutions including singular and regular nonextremal D3 branes, the Klebanov-Tseytlin solution and its singular nonextremal generalization. The horizon of the solutions we construct is of the precise form of nonextremal D3 branes. In the asymptotic region far away from the horizon we observe a logarithmic behavior similar to that of the Klebanov-Tseytlin solution.

Figures (15)

• Figure 1: Plot of the difference between analytical solutions and the corresponding numerical outputs: $\mathbb{R}^{3,1}\times \mathbb{CY}$ (red), Klebanov-Tseytlin (black), standard nonextremal D3 (green) and singular nonextremal generalization of Klebanov-Tseytlin (blue).
• Figure 2: Plot of the numerical output of the constrain for four study cases: $\mathbb{R}^{3,1}\times \mathbb{CY}$ (red), Klebanov-Tseytlin (black), standard nonextremal D3 (green) and singular nonextremal generalization of Klebanov-Tseytlin (blue).
• Figure 3: Plot of the $g_{00}$ component of the metrics (\ref{['eq:metric']}). The solid black curve represents the numerical solution with $P=a=1000$ and $Q=1$, and the dashed red curve the analytical solution with $P=0$, $a=1000$ and $Q=1$. Here $u_{90\%}\approx 0.0001$.
• Figure 4: Plot of the $g_{ii}$ component of the metrics (\ref{['eq:metric']}). The solid black curve represents the numerical solution with $P=a=1000$ and $Q=1$, and the dashed red curve the analytical solution with $P=0$, $a=1000$ and $Q=1$. Here $u_{90\%}\approx 0.0001$.
• Figure 5: Plot of the $g_{uu}$ component of the metrics (\ref{['eq:metric']}). The solid black curve represents the numerical solution with $P=a=1000$ and $Q=1$, and the dashed red curve the analytical solution with $P=0$, $a=1000$ and $Q=1$. Here $u_{90\%}\approx 0.0001$.