Three-Charge Black Holes on a Circle

TL;DR

This work extends the phase analysis of Kaluza-Klein black holes by generating non-extremal and near-extremal three-charge (F1-D4-D0) black holes on a transverse circle from neutral seeds via boosts and U-dualities. It derives exact mappings of physical quantities, analyzes uniform, non-uniform, and localized phases, and provides analytic corrections to the metric and thermodynamics in the small-mass and near-extremal limits. A key result is that the near-extremal three-charge system on a circle maintains a constant relative tension r=2, and the localized phase yields finite extremal entropy with first-order corrections matching microscopic brane-counting once circle interactions are accounted for. The paper also explores partial extremal limits, numerical phase diagrams, and the microstate counting for localized configurations, offering insights into brane dynamics on compact spaces and the AdS/CFT implications for these systems.

Abstract

We study phases of five-dimensional three-charge black holes with a circle in their transverse space. In particular, when the black hole is localized on the circle we compute the corrections to the metric and corresponding thermodynamics in the limit of small mass. When taking the near-extremal limit, this gives the corrections to the constant entropy of the extremal three-charge black hole as a function of the energy above extremality. For the partial extremal limit with two charges sent to infinity and one finite we show that the first correction to the entropy is in agreement with the microscopic entropy by taking into account that the number of branes shift as a consequence of the interactions across the transverse circle. Beyond these analytical results, we also numerically obtain the entire phase of non- and near-extremal three- and two-charge black holes localized on a circle. More generally, we find in this paper a rich phase structure, including a new phase of three-charge black holes that are non-uniformly distributed on the circle. All these three-charge black hole phases are found via a map that relates them to the phases of five-dimensional neutral Kaluza-Klein black holes.

Figures (6)

• Figure 1: Diagram with $\mu$ versus $n$ for the uniform black string (red), non-uniform black string (blue) and localized black hole phase (magenta) for five-dimensional Kaluza-Klein black holes, using numerical results of Kudoh:2004hsKleihaus:2006ee.
• Figure 2: The non-extremal $(\epsilon,r)$ phase diagram for four different values of the charges. The three charges are all taken to be equal and have the value $q=1$, $q=10$, $q=100$, and $q=1000$. Notice how all the phases collapse to the line $r=2$ as the charges go to infinity. The curves were found from Equation (\ref{['eq:goestor2']}) using $n=1/2$ for the uniform phase (red curve), numerical data from Kudoh:2004hs for the localized phase (magenta curve) and numerical data from Kleihaus:2006ee for the non-uniform phase (blue curve).
• Figure 3: The entropy $\hat{\mathfrak s}$ as a function of the energy above extremality $\epsilon$ for the localized phase (magenta), the uniform phase (red) and the non-uniform phase (blue). The curves are based on numerical data from Kudoh:2004hsKleihaus:2006ee.
• Figure 4: The free energy $\hat{\mathfrak f}$ as a function of the temperature $\hat{\mathfrak t}$ for the localized phase (magenta), the uniform phase (red) and the non-uniform phase (blue). The curves are based on numerical data from Kudoh:2004hsKleihaus:2006ee.
• Figure 5: $(\epsilon,r)$ phase diagram for near-extremal two-charge black holes on a circle. Shown are the localized phase (magenta), the uniform phase (red) and the non-uniform phase (blue). The curves are based on numerical data from Kudoh:2004hsKleihaus:2006ee.