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Global Structure of Five-dimensional BPS Fuzzballs

G. W. Gibbons, N. P. Warner

TL;DR

Five-dimensional BPS fuzzballs provide smooth, horizonless geometries that mimic black holes through deep AdS throats supported by nontrivial topology and Chern–Simons interactions. The authors show that the Smarr relation acquires bulk, topological terms in the presence of nontrivial $H^2$ and CS terms, enabling nontrivial mass and charges without horizons. They develop a linear system on a hyper-Kähler base, construct explicit bubbling solutions, derive bubble equations to avoid closed timelike curves, and analyze both BPS and non-BPS examples, including a soliton that Violates the BPS bound due to spin-structure obstructions. The work clarifies how topology, fluxes, and global structure underpin the microstate geometry program and informs the semiclassical understanding of black-hole entropy in higher-dimensional supergravity.

Abstract

We describe and study families of BPS microstate geometries, namely, smooth, horizonless asymptotically-flat solutions to supergravity. We examine these solutions from the perspective of earlier attempts to find solitonic solutions in gravity and show how the microstate geometries circumvent the earlier "No-Go" theorems. In particular, we re-analyse the Smarr formula and show how it must be modified in the presence of non-trivial second homology. This, combined with the supergravity Chern-Simons terms, allows the existence of rich classes of BPS, globally hyperbolic, asymptotically flat, microstate geometries whose spatial topology is the connected sum of N copies of S^2 x S^2 with a "point at infinity" removed. These solutions also exhibit "evanescent ergo-regions," that is, the non-space-like Killing vector guaranteed by supersymmetry is time-like everywhere except on time-like hypersurfaces (ergo-surfaces) where the Killing vector becomes null. As a by-product of our work, we are able to resolve the puzzle of why some regular soliton solutions violate the BPS bound: their spactimes do not admit a spin structure.

Global Structure of Five-dimensional BPS Fuzzballs

TL;DR

Five-dimensional BPS fuzzballs provide smooth, horizonless geometries that mimic black holes through deep AdS throats supported by nontrivial topology and Chern–Simons interactions. The authors show that the Smarr relation acquires bulk, topological terms in the presence of nontrivial and CS terms, enabling nontrivial mass and charges without horizons. They develop a linear system on a hyper-Kähler base, construct explicit bubbling solutions, derive bubble equations to avoid closed timelike curves, and analyze both BPS and non-BPS examples, including a soliton that Violates the BPS bound due to spin-structure obstructions. The work clarifies how topology, fluxes, and global structure underpin the microstate geometry program and informs the semiclassical understanding of black-hole entropy in higher-dimensional supergravity.

Abstract

We describe and study families of BPS microstate geometries, namely, smooth, horizonless asymptotically-flat solutions to supergravity. We examine these solutions from the perspective of earlier attempts to find solitonic solutions in gravity and show how the microstate geometries circumvent the earlier "No-Go" theorems. In particular, we re-analyse the Smarr formula and show how it must be modified in the presence of non-trivial second homology. This, combined with the supergravity Chern-Simons terms, allows the existence of rich classes of BPS, globally hyperbolic, asymptotically flat, microstate geometries whose spatial topology is the connected sum of N copies of S^2 x S^2 with a "point at infinity" removed. These solutions also exhibit "evanescent ergo-regions," that is, the non-space-like Killing vector guaranteed by supersymmetry is time-like everywhere except on time-like hypersurfaces (ergo-surfaces) where the Killing vector becomes null. As a by-product of our work, we are able to resolve the puzzle of why some regular soliton solutions violate the BPS bound: their spactimes do not admit a spin structure.

Paper Structure

This paper contains 37 sections, 192 equations, 2 figures.

Table of Contents

  1. Introduction: Six Impossible things before breakfast
  2. A brief history of solitons versus particles
  3. Global structure
  4. Some Microstate geometries
  5. The ${\cal N}\!=\!{2}$ supergravity theory
  6. BPS solutions
  7. Smarr Formula in $4+1$ spacetime dimensions
  8. Equations of Motion
  9. Invariances
  10. Mass and Charge
  11. Expansions at infinity
  12. Normalizing the Komar Integrals
  13. Komar Integrals in five dimensional supergravity
  14. A class of examples
  15. The hyper-Kähler base
  16. ...and 22 more sections

Figures (2)

  • Figure 1: Graph of ${{\cal Q} \over Z^2 V^2}$ for $-2 \le z \le 2$, $0 \le \rho \le 0.5$. Observe that it is globally non-negative.
  • Figure 2: Graph of $-g^{tt}$ given in (\ref{['stabcausal']}) for $-2 \le z \le 2$, $0 \le \rho \le 0.5$. Observe that $-g^{tt}>0$ everywhere and so the space-time is stably causal. The cusps at $\rho=0$, $z=\pm1$ and $\rho=0$, $z=0$ are a coordinate artifacts arising from the fact that $\rho$ and $z$ are not good coordinates at these points.