Bubbles on Manifolds with a U(1) Isometry

TL;DR

This paper develops five-dimensional, three-charge BPS solutions with only a rotational $U(1)$ isometry by employing ambi-polar hyper-Kähler bases. It derives the BPS equations in a Toda-frame and analyzes regularity on critical surfaces where the base signature changes, showing that fluxes can yield fully regular five-dimensional solutions despite base ambi-polarity. Explicit bubbling constructions are presented on the Atiyah-Hitchin base and its ambi-polar generalization, as well as on ambi-polar Eguchi-Hanson, including a global $AdS_2\times S^3$ realization and discussions of wormhole vs. pinch-off geometries. The work highlights how bubbling geometries may form the microstate structure of black holes by assembling horizonless, flux-supported bases and points to future directions for counting moduli and extending to more general ambi-polar hyper-Kähler spaces. Overall, it strengthens the program that black holes are ensembles of smooth supergravity configurations and provides concrete, highly symmetric examples that illuminate the role of the base geometry and fluxes in microstate geometries.

Abstract

We investigate the construction of five-dimensional, three-charge supergravity solutions that only have a rotational U(1) isometry. We show that such solutions can be obtained as warped compactifications with a singular ambi-polar hyper-Kahler base space and singular warp factors. We show that the complete solution is regular around the critical surface of the ambi-polar base. We illustrate this by presenting the explicit form of the most general supersymmetric solutions that can be obtained from an Atiyah-Hitchin base space and its ambi-polar generalizations. We make a parallel analysis using an ambi-polar generalization of the Eguchi-Hanson base space metric. We also show how the bubbling procedure applied to the ambi-polar Eguchi-Hanson metric can convert it to a global AdS_2xS^3 compactification.

Figures (4)

• Figure 1: This shows the three functions, $w_j$, as a function of $x = \sin^2{ {\theta \over 2}}$ when $u$ is given by (\ref{['simpgenu']}). One has $w_1 < 0$ and $w_3 > 0$ and $w_2$ has a simple zero at $\theta = \pi/2$. All three functions diverge at both ends of the interval.
• Figure 2: This shows plots of $Z$ as a function of $x = \sin^2{ {\theta \over 2}}$. We have taken $\delta =1$, fixed $\gamma$ in terms of $\alpha_1$ using (\ref{['gamres']}) and then we have chosen three values of $\alpha_1$ that ensure that $\gamma$ is negative: $\alpha_1= 0.6$, $\alpha_1= 1.0$ and $\alpha_1=2.0$. The steeper graphs at $x=0.5$ correspond to larger values of $\alpha_1$. Note that $Z \to 1$ as $x \to 1$, but that $Z$ is generically negative for $x>0.6$.
• Figure 3: These are graphs of $Z$ (on the left) and $\mu$ (on the right) as functions of $x = \sin^2{ {\theta \over 2}}$ for $\gamma =0$, $\delta =1$ and $\alpha_2 \approx 0.4890$. Both functions are singular at $x=0.5$ and both vanish, $\mu$ with a double root, at $x \approx 0.1837$.
• Figure 4: This graph shows the three metric coefficients in the angular directions, $Z a^2$, $Z^{-2} {\cal Q}$ and $Z c^2$ (in this order from top to bottom on the right-hand side of the graph), as functions of $x = \sin^2{ {\theta \over 2}}$ for $\gamma =0$, $\delta =1$ and $\alpha_2 \approx 0.4890$. All of these functions vanish at the pinching-off point, $x \approx 0.1837$, and are positive-definite to the right of it.