Black Rings in Taub-NUT and D0-D6 interactions

TL;DR

This paper analyzes non-supersymmetric D0–D6 bound states in Taub-NUT via neutral black rings, using both perturbative thin-ring methods and exact solution-generating techniques. The authors show that, in the absence of worldvolume B-flux on the D6, D0 and D6 interact repulsively at finite separation, but can form bound states when the D0 gas is thermally excited or when a sufficiently large B-field is present, yielding black rings in Taub-NUT and, in extremal limits, a singular KK black hole. The perturbative analysis provides equilibrium conditions, interaction energies, and phase-space boundaries, while the exact constructions reveal the full bound-state spectrum, including extremal and non-extremal rings and their phase diagrams, along with conical-defect energetics and stability considerations. The results illuminate how bound states emerge from D0–D6 dynamics, connecting horizon physics, higher-dimensional solution-generating techniques, and four-dimensional KK physics with potential extensions to more general Taub-NUT backgrounds and multi-ring configurations.

Abstract

We analyze the dynamics of neutral black rings in Taub-NUT spaces and their relation to systems of D0 and D6 branes in the supergravity approximation. We employ several recent techniques, both perturbative and exact, to construct solutions in which thermal excitations of the D0-branes can be turned on or off, and the D6-brane can have $B$-fluxes turned on or off in its worldvolume. By explicit calculation of the interaction energy between the D0 and D6 branes, we can study equilibrium configurations and their stability. We find that although D0 and D6 branes (in the absence of $B$ fields, and at zero temperature) repeal each other at non-zero separation, as they get together they go over continuosly to an unstable bound state of an extremal singular Kaluza-Klein black hole. We also find that, for $B$-fields larger than a critical value, or sufficiently large thermal excitation, the D0 and D6 branes form stable bound states. The bound states with thermally excited D0 branes are black rings in Taub-NUT, and we provide an analysis of their phase diagram.

Figures (8)

• Figure 1: D0-D6 interaction energy $E_\text{int}$ (with no $B$-field and at zero temperature) as a function of separation $R$, for $Q/P=.1$ (thin), $1.$ (thick) and $10$ (thicker). The interaction energy is normalized relative to the one of the (singular) extremal black hole with the same electric and magnetic charges, and angular momentum $G_4J=PQ$. Here $E_\text{int}$ is computed using the exact solutions of sec. \ref{['sec:exactextr']}.
• Figure 2: D0-D6 interaction potential with flux $B=bL/2$ versus $R$, for $b=1/9$ (dashed), $b_c=1/\sqrt{3}$ (thin solid) and $.8$ (thick solid). The interaction energy is normalized with the electric charge so it corresponds to a fixed net number of D0 branes. The potential is obtained from the perturbative extremal solutions of sec. \ref{['sec:pertd0d6']}.
• Figure 3: Thermal D0-D6 interaction potential versus $R$, for fixed values of the entropy and charge, $G_4 S/(4\pi Q^2)=.1$ (dashed), critical $1/\sqrt{2}$ (thin solid) and $3.$ (thick solid). The interaction energy is normalized with the electric charge. The potential is obtained from the perturbative non-extremal solutions of sec. \ref{['sec:pertd0d6']} with $b=0$.
• Figure 4: Rod structure of the initial solution \ref{['eqn:seed']}. This is the same rod structure of the seed solution of the standard $S^1$--spinning ring.
• Figure 5: Conical defect versus radius $R$ in units of $P$ for $Q/P=.1$ (thin), $1$ (thick) and $10$ (thicker). Note that the conical defect is never zero for finite values of $R$. This means that the solution is never balanced. In the $R\rightarrow 0$ limit the conical defect is maximal: $2\pi$. The force $F_\text{def}$ is porportional to $\Delta$ (\ref{['epressure']}).