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Flow Equations for Non-BPS Extremal Black Holes

Anna Ceresole, Gianguido Dall'Agata

TL;DR

The paper demonstrates that stable non-BPS extremal black holes in four-dimensional N=2 supergravity can be described by first-order flow equations using a fake superpotential W that replaces the central charge in VBH. It develops a general framework based on symplectic structure to construct multiple W's yielding the same VBH and validates this with explicit one-modulus and STU models, including truncated and field-dependent W realizations. It also develops algebraic attractor equations for non-BPS points and discusses potential links to entropy functionals and pseudo-supersymmetry, suggesting a broader applicability of first-order formalisms beyond BPS solutions. The results illuminate how duality and symplectic rotations generate alternative first-order descriptions, offering a path to extend attractor-like behavior to non-BPS extremal black holes and potentially connect to microscopic entropy counts.

Abstract

We exploit some common features of black hole and domain wall solutions of (super)gravity theories coupled to scalar fields and construct a class of stable extremal black holes that are non-BPS, but still can be described by first-order differential equations. These are driven by a "superpotential'', which replaces the central charge Z in the usual black hole potential. We provide a general procedure for finding this class and deriving the associated "superpotential''. We also identify some other cases which do not belong to this class, but show a similar behaviour.

Flow Equations for Non-BPS Extremal Black Holes

TL;DR

The paper demonstrates that stable non-BPS extremal black holes in four-dimensional N=2 supergravity can be described by first-order flow equations using a fake superpotential W that replaces the central charge in VBH. It develops a general framework based on symplectic structure to construct multiple W's yielding the same VBH and validates this with explicit one-modulus and STU models, including truncated and field-dependent W realizations. It also develops algebraic attractor equations for non-BPS points and discusses potential links to entropy functionals and pseudo-supersymmetry, suggesting a broader applicability of first-order formalisms beyond BPS solutions. The results illuminate how duality and symplectic rotations generate alternative first-order descriptions, offering a path to extend attractor-like behavior to non-BPS extremal black holes and potentially connect to microscopic entropy counts.

Abstract

We exploit some common features of black hole and domain wall solutions of (super)gravity theories coupled to scalar fields and construct a class of stable extremal black holes that are non-BPS, but still can be described by first-order differential equations. These are driven by a "superpotential'', which replaces the central charge Z in the usual black hole potential. We provide a general procedure for finding this class and deriving the associated "superpotential''. We also identify some other cases which do not belong to this class, but show a similar behaviour.

Paper Structure

This paper contains 10 sections, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Black Holes and Domain Walls
  3. First order equations
  4. Multiple $W$ for the same $V_{BH}$
  5. One class of $V_{BH}$ with multiple $W$
  6. Examples
  7. One-modulus case
  8. STU black hole
  9. More general solutions
  10. Outlook

Figures (1)

  • Figure 1: Plots of the sections of $W$ and $|{\cal Z}|$ at Im $z=0$, for unit charges. Where the central charge shows a minimum, the "fake superpotentials" crosses zero. Changing the signs of the $q_0$ and $p^0$ charges exchanges the two pictures.