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An Infinite-Dimensional Family of Black-Hole Microstate Geometries

Iosif Bena, Nikolay Bobev, Stefano Giusto, Clement Ruef, Nicholas P. Warner

TL;DR

This work constructs an explicit, horizonless family of three-charge microstate geometries whose moduli space is infinite dimensional, by embedding a round two-charge supertube with fluctuating charge densities into a two-center ambipolar Gibbons-Hawking base and solving the fully back-reacted supergravity equations. A central technical advance is the scalar Green function on ambipolar GH spaces, obtained by dimensional reduction from a Green function on AdS_3 × S^2, which enables explicit back-reaction with density modes parameterized by a single continuous function. The authors derive bubble (integrability) equations that govern center locations, and show these functional equations agree with those obtained from the Born-Infeld action, signaling a non-renormalization property in moduli space. They demonstrate an entropy-enhancement mechanism in which a small wobbling tube in a strongly magnetized background stores more entropy than in flat space, with the leading entropy set by background charges and fluxes rather than the tube’s own electric charges, while still bounded by global consistency constraints. Collectively, the results substantiate the fuzzball program’s premise that smooth microstate geometries can account for substantial black-hole entropy and illuminate how density fluctuations extend the microstate landscape beyond previously known finite-dimensional families.

Abstract

We construct the first explicit, smooth, horizonless black-hole microstate geometry whose moduli space is described by an arbitrary function of one variable and is thus infinite-dimensional. This is achieved by constructing the scalar Green function on a simple D6 anti-D6 background, and using this Green function to obtain the fully back-reacted solution for a supertube with varying charge density in this background. We show that this supertube can store parametrically more entropy than in flat space, confirming the entropy enhancement mechanism that was predicted using brane probes. We also show that all the local properties of the fully back-reacted solution can, in fact, be obtained using the DBI action of an appropriate brane probe. In particular, the supergravity and the DBI analysis yield identical functional bubble equations that govern the relative locations of the centers. This indicates that there is a non-renormalization theorem that protects these functional equations as one moves in moduli space. Our construction creates configurations that are beyond the scope of recent arguments that appear to put strong limits on the entropy that can be found in smooth supergravity solutions.

An Infinite-Dimensional Family of Black-Hole Microstate Geometries

TL;DR

This work constructs an explicit, horizonless family of three-charge microstate geometries whose moduli space is infinite dimensional, by embedding a round two-charge supertube with fluctuating charge densities into a two-center ambipolar Gibbons-Hawking base and solving the fully back-reacted supergravity equations. A central technical advance is the scalar Green function on ambipolar GH spaces, obtained by dimensional reduction from a Green function on AdS_3 × S^2, which enables explicit back-reaction with density modes parameterized by a single continuous function. The authors derive bubble (integrability) equations that govern center locations, and show these functional equations agree with those obtained from the Born-Infeld action, signaling a non-renormalization property in moduli space. They demonstrate an entropy-enhancement mechanism in which a small wobbling tube in a strongly magnetized background stores more entropy than in flat space, with the leading entropy set by background charges and fluxes rather than the tube’s own electric charges, while still bounded by global consistency constraints. Collectively, the results substantiate the fuzzball program’s premise that smooth microstate geometries can account for substantial black-hole entropy and illuminate how density fluctuations extend the microstate landscape beyond previously known finite-dimensional families.

Abstract

We construct the first explicit, smooth, horizonless black-hole microstate geometry whose moduli space is described by an arbitrary function of one variable and is thus infinite-dimensional. This is achieved by constructing the scalar Green function on a simple D6 anti-D6 background, and using this Green function to obtain the fully back-reacted solution for a supertube with varying charge density in this background. We show that this supertube can store parametrically more entropy than in flat space, confirming the entropy enhancement mechanism that was predicted using brane probes. We also show that all the local properties of the fully back-reacted solution can, in fact, be obtained using the DBI action of an appropriate brane probe. In particular, the supergravity and the DBI analysis yield identical functional bubble equations that govern the relative locations of the centers. This indicates that there is a non-renormalization theorem that protects these functional equations as one moves in moduli space. Our construction creates configurations that are beyond the scope of recent arguments that appear to put strong limits on the entropy that can be found in smooth supergravity solutions.

Paper Structure

This paper contains 28 sections, 181 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The general back-reacted solution with fluctuating electric charge densities
  3. Scalar Green functions on a GH space
  4. Review of known Green functions on regular GH spaces
  5. Preliminaries: the simplest metric with an ambi-polar base, global $AdS_3 \times S^2$
  6. The strategy: reducing a Green function from five to four dimensions
  7. The scalar Green function on $AdS_3 \times S^2$
  8. Reducing the scalar Green function to the GH base
  9. Properties of the Green function
  10. Returning to the GH geometry
  11. Residues and pole structure
  12. Solutions and their asymptotic behavior
  13. Short-distance behaviour
  14. Asymptotics at the Gibbons-Hawking points
  15. Behaviour at infinity
  16. ...and 13 more sections

Figures (1)

  • Figure 1: A round supertube can retain its round shape but concentrate or dilute charges in different regions as it expands or contracts. Some large sectors of the original tube can be mapped into smaller sectors, concentrating charge, while some sectors can become expanded resulting in a charge dilution. Even though the shape of the supertube remains circular, the charge density modes induce a radius change and can thus be seen as special class of shape mode.