Supersymmetric black rings and three-charge supertubes

TL;DR

The paper constructs a seven-parameter family of 1/8-BPS black ring solutions in five dimensions by uplifting a three-charge, three-dipole M-theory configuration; upon reduction to IIB, these become D1-D5-P black supertubes with rich horizon structure and KK-dipole quantization. It demonstrates a finite, but continuous, non-uniqueness of supersymmetric black holes due to dipole charges, and analyzes a decoupling limit that connects to AdS$_3$/CFT physics. A parallel worldvolume description via calibrated M5-branes reveals a three-charge, three-dipole calibrated supertube, elucidating how cross-sections and fluxes encode M2 charges but with limitations in capturing the full angular momentum structure. The work maps between supergravity and worldvolume viewpoints, clarifying when each description aligns and highlighting remaining puzzles, notably the origin of the second angular momentum $J_ ext{φ}$ and a precise microscopic entropy counting for three-charge rings.

Abstract

We present supergravity solutions for 1/8-supersymmetric black supertubes with three charges and three dipoles. Their reduction to five dimensions yields supersymmetric black rings with regular horizons and two independent angular momenta. The general solution contains seven independent parameters and provides the first example of non-uniqueness of supersymmetric black holes. In ten dimensions, the solutions can be realized as D1-D5-P black supertubes. We also present a worldvolume construction of a supertube that exhibits three dipoles explicitly. This description allows an arbitrary cross-section but captures only one of the angular momenta.

Figures (6)

• Figure 1: Coordinate system for black ring metrics (from ER2RE). The diagram sketches a section at constant $t$ and $\phi$. Surfaces of constant $y$ are ring-shaped, while $x$ is a polar coordinate on the $S^2$ (roughly $x\sim\cos\theta$). $x=\pm 1$ and $y=-1$ are fixed-point sets ( i.e., axes) of $\partial_\phi$ and $\partial_\psi$, respectively. Asymptotic infinity lies at $x=y=-1$.
• Figure 2: Coordinates $(\rho,\Theta)$, in a section at constant $t$, $\phi$, $\psi$ (the four quadrants are obtained by including also constant $\phi+\pi$ and $\psi+\pi$). Solid lines are surfaces of constant $\rho$, dashed lines are at constant $\Theta$. The ring lies at $\rho=R$, $\Theta=\pi/2$.
• Figure 3: Coordinates $(r,\theta)$, in a section at constant $t$, $\phi$, $\psi$ (and $\phi+\pi$, $\psi+\pi$). Solid lines are surfaces of constant $r$, dashed lines are at constant $\theta$. The axis of $\phi$ consists of the segments $r=0$ and $\theta=\pi/2$.
• Figure 4: The dimensionless area $a_H$ as a function of $j_\psi$ and $j_\phi$ for fixed $\eta=0.4$. Note that the third dipole charge is not held constant, but is determined by the other parameters. Darker regions correspond to smaller area. On the left, the triangular region is bounded by the line $j_\phi=j_\psi$. The lower bound is a consequence of $j_\phi$ being bounded from below for non-vanishing dipole moment: $2 \sqrt{2}\,j_\phi \ge \eta$. The region is bounded on the right by the line determined by $j_\psi^{\mathrm{max}}$ in (\ref{['jpsimax']}). This is a consequence of requiring that there are no naked CCCs. The area $a_H$ vanishes at the bottom and right boundaries of the triangular region. For fixed $j_\phi$, $a_H$ is maximized when $j_\psi \to j_\phi$.
• Figure 5: Contour plots of the dimensionless area $a_H$ versus $\eta_1$ and $\eta_2$, for fixed values of $j_\phi$ and $j_\psi$. For all three plots $4\sqrt{2}\,j_\psi = 2.1$, but $j_\phi$ varies for each case: from left to right $4\sqrt{2}\,j_\phi = 0.7$, $1.$, $1.7$. The plots are symmetric in $\eta_1$ and $\eta_2$, and the darker regions correspond to smaller area. The regions for which the black rings exist are bounded by the condition that there be no naked CCCs. At this boundary, the area $a_H$ vanishes.