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BPS Black Hole Horizons in N=2 Gauged Supergravity

Nick Halmagyi

TL;DR

This work analyzes static BPS black hole horizons in four-dimensional N=2 gauged supergravity with a general cubic prepotential, focusing on AdS_2 × Σ_g horizons and including both electric and magnetic gaugings as well as dyonic charges. By adopting a symplectically covariant formalism, the authors derive horizon equations that reduce to holomorphic quadratic relations for variables rak{p}^\Lambda and rak{q}_\Lambda, enabling explicit solutions for symmetric special Kähler manifolds and implicit holomorphic solutions in general cases. The entropy is shown to be governed by a refined invariant related to the quartic 𝓘_4, with explicit results for cases where p^0=0 or for symmetric spaces; an STU-model embedding in M-theory is discussed, including a numerical dyonic spinning M2-brane on Σ_g. The paper extends attractor-like horizon analyses from ungauged to gauged N=2 supergravity, provides a unified framework to connect horizon data to charges and gaugings, and points to UV completions and M-theory embeddings as important directions for future work.

Abstract

We study static BPS black hole horizons in four dimensional N=2 gauged supergravity coupled to $n_v$-vector multiplets and with an arbitrary cubic prepotential. We work in a symplectically covariant formalism which allows for both electric and magnetic gauging parameters as well as dyonic background charges and obtain the general solution to the BPS equations for horizons of the form $AdS_2\times Σ_g$. In particular this means we solve for the scalar fields as well as the metric of these black holes as a function of the gauging parameters and background charges. When the special Kahler manifold is a symmetric space, our solution is completely explicit and the entropy is related to the familiar quartic invariant. For more general models our solution is implicit up to a set of holomorphic quadratic equations. For particular models which have known embeddings in M-theory, we derive new horizon geometries with dyonic charges and numerically construct black hole solutions. These correspond to M2-branes wrapped on a Riemann surface in a local Calabi-Yau five-fold with internal spin.

BPS Black Hole Horizons in N=2 Gauged Supergravity

TL;DR

This work analyzes static BPS black hole horizons in four-dimensional N=2 gauged supergravity with a general cubic prepotential, focusing on AdS_2 × Σ_g horizons and including both electric and magnetic gaugings as well as dyonic charges. By adopting a symplectically covariant formalism, the authors derive horizon equations that reduce to holomorphic quadratic relations for variables rak{p}^\Lambda and rak{q}_\Lambda, enabling explicit solutions for symmetric special Kähler manifolds and implicit holomorphic solutions in general cases. The entropy is shown to be governed by a refined invariant related to the quartic 𝓘_4, with explicit results for cases where p^0=0 or for symmetric spaces; an STU-model embedding in M-theory is discussed, including a numerical dyonic spinning M2-brane on Σ_g. The paper extends attractor-like horizon analyses from ungauged to gauged N=2 supergravity, provides a unified framework to connect horizon data to charges and gaugings, and points to UV completions and M-theory embeddings as important directions for future work.

Abstract

We study static BPS black hole horizons in four dimensional N=2 gauged supergravity coupled to -vector multiplets and with an arbitrary cubic prepotential. We work in a symplectically covariant formalism which allows for both electric and magnetic gauging parameters as well as dyonic background charges and obtain the general solution to the BPS equations for horizons of the form . In particular this means we solve for the scalar fields as well as the metric of these black holes as a function of the gauging parameters and background charges. When the special Kahler manifold is a symmetric space, our solution is completely explicit and the entropy is related to the familiar quartic invariant. For more general models our solution is implicit up to a set of holomorphic quadratic equations. For particular models which have known embeddings in M-theory, we derive new horizon geometries with dyonic charges and numerically construct black hole solutions. These correspond to M2-branes wrapped on a Riemann surface in a local Calabi-Yau five-fold with internal spin.

Paper Structure

This paper contains 12 sections, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Algebraic Fixed Point Equations
  3. $AdS_2\times \Sigma_g$ Fixed Point Equations
  4. Symplectic Invariant Equations
  5. Solving the Horizon Equations
  6. $\frak{p}^0= 0$
  7. $\frak{p}^0\neq 0$
  8. Symmetric Spaces
  9. Revisiting Black Hole Horizons in Ungauged ${\cal N}=2$ Supergravity
  10. Embedding Dyonic Black Holes in M-theory
  11. Conclusions
  12. Some Special Geometry

Figures (2)

  • Figure 1: The metric functions $U'(\rho)$ and $V'(\rho)$
  • Figure 2: The scalar functions $y^1=y^2$, $y^3$ and $x^1$