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N=8 non-BPS Attractors, Fixed Scalars and Magic Supergravities

S. Ferrara, A. Marrani

TL;DR

This work analyzes the Hessian of the black hole potential $V_{BH}$ in $ ext{N}=8$, d=4 supergravity, focusing on non-BPS critical points and their stability, and connects these results to the $ ext{N}=2$ magic supergravities obtained via consistent truncations. The authors derive attractor equations and characterize the Hessian spectra for the $ rac{1}{8}$-BPS and non-BPS branches, showing a rank of $30$ for the $ rac{1}{8}$-BPS Hessian and a model-dependent, representation-driven splitting for non-BPS points, governed by embeddings into $ ext{USp}(8)$ and its subgroups. They map the $ ext{N}=8$ spectra to $ ext{N}=2$ descendants, clarifying how vector- and hypermultiplet scalars reorganize into $ ext{N}=2$ multiplets and how the well-known $n_V+1$ / $n_V-1$ degeneracy pattern for generic special Kahler geometries arises in this higher-dimensional context. Through explicit analyses of the magic models $ ext{J}_{3}^{ ext{H}}, ext{J}_{3}^{ ext{C}}, ext{J}_{3}^{ ext{R}}, ext{stu}, ext{J}_{3,M}^{ ext{R}}, ext{J}_{3,M}^{ ext{C}}$, the paper demonstrates how the Hessian eigenstructure and stability properties are inherited under $ ext{N}=8 ightarrow ext{N}=2$ reductions, including cases with and without flat directions. The results offer a coherent picture of non-BPS attractors across extended supergravities and highlight practical routes for extending the analysis to non-maximal theories.

Abstract

We analyze the Hessian matrix of the black hole potential of N=8, d=4 supergravity, and determine its rank at non-BPS critical points, relating the resulting spectrum to non-BPS solutions (with non-vanishing central charge) of N=2, d=4 magic supergravities and their ``mirror'' duals. We find agreement with the known degeneracy splitting of N=2 non-BPS spectrum of generic special Kahler geometries with cubic holomorphic prepotential. We also relate non-BPS critical points with vanishing central charge in N=2 magic supergravities to a particular reduction of the N=8, 1/8-BPS critical points.

N=8 non-BPS Attractors, Fixed Scalars and Magic Supergravities

TL;DR

This work analyzes the Hessian of the black hole potential in , d=4 supergravity, focusing on non-BPS critical points and their stability, and connects these results to the magic supergravities obtained via consistent truncations. The authors derive attractor equations and characterize the Hessian spectra for the -BPS and non-BPS branches, showing a rank of for the -BPS Hessian and a model-dependent, representation-driven splitting for non-BPS points, governed by embeddings into and its subgroups. They map the spectra to descendants, clarifying how vector- and hypermultiplet scalars reorganize into multiplets and how the well-known / degeneracy pattern for generic special Kahler geometries arises in this higher-dimensional context. Through explicit analyses of the magic models , the paper demonstrates how the Hessian eigenstructure and stability properties are inherited under reductions, including cases with and without flat directions. The results offer a coherent picture of non-BPS attractors across extended supergravities and highlight practical routes for extending the analysis to non-maximal theories.

Abstract

We analyze the Hessian matrix of the black hole potential of N=8, d=4 supergravity, and determine its rank at non-BPS critical points, relating the resulting spectrum to non-BPS solutions (with non-vanishing central charge) of N=2, d=4 magic supergravities and their ``mirror'' duals. We find agreement with the known degeneracy splitting of N=2 non-BPS spectrum of generic special Kahler geometries with cubic holomorphic prepotential. We also relate non-BPS critical points with vanishing central charge in N=2 magic supergravities to a particular reduction of the N=8, 1/8-BPS critical points.

Paper Structure

This paper contains 15 sections, 70 equations, 4 tables.

Table of Contents

  1. Introduction
  2. $\mathcal{N}=8$ and $\mathcal{N}=2$ Magic Supergravities
  3. $\mathcal{N}=8$, $d=4$ Critical Points and Hessian
  4. Solutions to Attractor Equations
  5. Critical Spectra
  6. $\mathcal{N}=8$, $\frac{1}{8}$-BPS Critical Points and their $\mathcal{N}=2$ Descendants
  7. $\mathcal{N}=2$, $\frac{1}{2}$-BPS
  8. $\mathcal{N}=2$ non-BPS $Z=0$
  9. $\mathcal{N}=8$ non-BPS Critical Points and $\mathcal{N}=2$ non-BPS $Z\neq 0$ Critical Points
  10. $J_{3}^{\mathbb{H}}$
  11. $J_{3}^{\mathbb{C}}$
  12. $J_{3}^{\mathbb{R}}$
  13. $stu$
  14. $J_{3,M}^{\mathbb{R}}$
  15. Conclusion