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Four Dimensional Black Hole Microstates: From D-branes to Spacetime Foam

Vijay Balasubramanian, Eric G. Gimon, Thomas S. Levi

TL;DR

The paper proposes that every supersymmetric 4D black hole of finite area consists of microstates formed from mutually nonlocal 1/2-BPS atoms, whose binding grows as the string coupling increases, yielding a spacetime foam in M-theory at strong coupling. It constructs a broad class of smooth, horizonless 4D solutions by dimensional reduction from five-dimensional microstate geometries and shows how these states flow—from D-brane bound states at weak coupling to a foam-like spacetime at strong coupling—via a scaling relation that links the Coulomb and Higgs branches of a corresponding quiver gauge theory. The authors demonstrate a precise correspondence between the gravitational bubble equations and the Coulomb-branch vacua of the open-string gauge theory, and argue that the Higgs branch vacua provide a complementary description, with their counts matching under suitable conditions. They further discuss how E7(7) dualities organize the charge configurations and how coarse-graining of microstates can reproduce finite-area black hole horizons, offering a unified picture that links D-brane microstate counting with spacetime foam pictures of entropy. The work lays out a program to establish a detailed, quantitative map between microstate counting, flow between gauge-theory branches, and the emergence of macroscopic horizons, with significant implications for understanding black hole entropy from multiple ultraviolet/infrared perspectives.

Abstract

We propose that every supersymmetric four dimensional black hole of finite area can be split up into microstates made up of primitive half-BPS "atoms''. The mutual non-locality of the charges of these "atoms'' binds the state together. In support of this proposal, we display a class of smooth, horizon-free, four dimensional supergravity solutions carrying the charges of black holes, with multiple centers each carrying the charge of a half-BPS state. At vanishing string coupling the solutions collapse to a bound system of intersecting D-branes. At weak coupling the system expands into the non-compact directions forming a topologically complex geometry. At strong coupling, a new dimension opens up, and the solutions form a "foam'' of spheres threaded by flux in M-theory. We propose that this transverse growth of the underlying bound state of constitutent branes is responsible for the emergence of black hole horizons for coarse-grained observables. As such, it suggests the link between the D-brane and "spacetime foam'' approaches to black hole entropy.

Four Dimensional Black Hole Microstates: From D-branes to Spacetime Foam

TL;DR

The paper proposes that every supersymmetric 4D black hole of finite area consists of microstates formed from mutually nonlocal 1/2-BPS atoms, whose binding grows as the string coupling increases, yielding a spacetime foam in M-theory at strong coupling. It constructs a broad class of smooth, horizonless 4D solutions by dimensional reduction from five-dimensional microstate geometries and shows how these states flow—from D-brane bound states at weak coupling to a foam-like spacetime at strong coupling—via a scaling relation that links the Coulomb and Higgs branches of a corresponding quiver gauge theory. The authors demonstrate a precise correspondence between the gravitational bubble equations and the Coulomb-branch vacua of the open-string gauge theory, and argue that the Higgs branch vacua provide a complementary description, with their counts matching under suitable conditions. They further discuss how E7(7) dualities organize the charge configurations and how coarse-graining of microstates can reproduce finite-area black hole horizons, offering a unified picture that links D-brane microstate counting with spacetime foam pictures of entropy. The work lays out a program to establish a detailed, quantitative map between microstate counting, flow between gauge-theory branches, and the emergence of macroscopic horizons, with significant implications for understanding black hole entropy from multiple ultraviolet/infrared perspectives.

Abstract

We propose that every supersymmetric four dimensional black hole of finite area can be split up into microstates made up of primitive half-BPS "atoms''. The mutual non-locality of the charges of these "atoms'' binds the state together. In support of this proposal, we display a class of smooth, horizon-free, four dimensional supergravity solutions carrying the charges of black holes, with multiple centers each carrying the charge of a half-BPS state. At vanishing string coupling the solutions collapse to a bound system of intersecting D-branes. At weak coupling the system expands into the non-compact directions forming a topologically complex geometry. At strong coupling, a new dimension opens up, and the solutions form a "foam'' of spheres threaded by flux in M-theory. We propose that this transverse growth of the underlying bound state of constitutent branes is responsible for the emergence of black hole horizons for coarse-grained observables. As such, it suggests the link between the D-brane and "spacetime foam'' approaches to black hole entropy.

Paper Structure

This paper contains 34 sections, 101 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of five dimensional solutions
  3. Basic setup:
  4. Solution:
  5. Constraints:
  6. Charges:
  7. Reducing to Four Dimensions
  8. Introducing Taub-NUT
  9. Reduction to IIA
  10. Reduction to four dimensions and special geometry
  11. $E_7$ structure:
  12. Explicit solution and special geometry:
  13. $E_{7(7)}$ and more general solutions:
  14. Generalized constraint and coarse graining:
  15. Relation to finite area black holes
  16. ...and 19 more sections

Figures (3)

  • Figure 1: A sample quiver with two nodes.
  • Figure 2: A quiver with a closed loop.
  • Figure 3: The different phases as we increase $g_s$.