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Exact solutions for supersymmetric stationary black hole composites

Brandon Bates, Frederik Denef

TL;DR

The paper develops a general method to construct explicit analytic multicentered BPS black hole solutions in four-dimensional N=2 supergravity by leveraging the entropy function of a single black hole. The full solution is determined by a single function Σ(H) proportional to the horizon entropy, enabling explicit expressions for the warp factor U, moduli z, the angular-momentum one-form ω, and the gauge fields once S_BH(Q) is known. Through the diagonal T^6 example and the two-centered bound-state analysis, the authors illustrate how horizon entropy is tied to Σ and how moduli control the existence and properties of multi-centered configurations, including walls of marginal stability and entropy enhancement. This work links microscopic entropy Calculations to macroscopic supergravity solutions, clarifying moduli dependence and offering a concrete framework for understanding D-brane bound states in Calabi–Yau compactifications.

Abstract

Four dimensional N=2 supergravity has regular, stationary, asymptotically flat BPS solutions with intrinsic angular momentum, describing bound states of separate extremal black holes with mutually nonlocal charges. Though the existence and some properties of these solutions were established some time ago, fully explicit analytic solutions were lacking thus far. In this note, we fill this gap. We show in general that explicit solutions can be constructed whenever an explicit formula is known in the theory at hand for the Bekenstein-Hawking entropy of a single black hole as a function of its charges, and illustrate this with some simple examples. We also give an example of moduli-dependent black hole entropy.

Exact solutions for supersymmetric stationary black hole composites

TL;DR

The paper develops a general method to construct explicit analytic multicentered BPS black hole solutions in four-dimensional N=2 supergravity by leveraging the entropy function of a single black hole. The full solution is determined by a single function Σ(H) proportional to the horizon entropy, enabling explicit expressions for the warp factor U, moduli z, the angular-momentum one-form ω, and the gauge fields once S_BH(Q) is known. Through the diagonal T^6 example and the two-centered bound-state analysis, the authors illustrate how horizon entropy is tied to Σ and how moduli control the existence and properties of multi-centered configurations, including walls of marginal stability and entropy enhancement. This work links microscopic entropy Calculations to macroscopic supergravity solutions, clarifying moduli dependence and offering a concrete framework for understanding D-brane bound states in Calabi–Yau compactifications.

Abstract

Four dimensional N=2 supergravity has regular, stationary, asymptotically flat BPS solutions with intrinsic angular momentum, describing bound states of separate extremal black holes with mutually nonlocal charges. Though the existence and some properties of these solutions were established some time ago, fully explicit analytic solutions were lacking thus far. In this note, we fill this gap. We show in general that explicit solutions can be constructed whenever an explicit formula is known in the theory at hand for the Bekenstein-Hawking entropy of a single black hole as a function of its charges, and illustrate this with some simple examples. We also give an example of moduli-dependent black hole entropy.

Paper Structure

This paper contains 12 sections, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and setup
  3. General formalism
  4. Example: diagonal $T^6$
  5. BPS equations of motion
  6. Solutions
  7. Solutions for $U$ and $z$: general case
  8. Solutions for $U$ and $z$: diagonal $T^6$ example
  9. Solutions for $\omega$
  10. Solution for ${\cal F}$: general case
  11. Dependence of entropy on moduli
  12. Conclusions

Figures (1)

  • Figure 1: Left: Profile of $\tau({\bf x})$ in the upper half plane for the 2-centered example described in the text. The shaded region is the domain of allowed values of $\tau_\infty$. The red curve is the line of marginal stability. Right:$e^U$ as a function of $x$ with the two charges located on the $x$-axis.