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Non-Supersymmetric Attractor Flow in Symmetric Spaces

Davide Gaiotto, Wei Li, Megha Padi

TL;DR

This work develops a unified framework to construct extremal black hole attractor flows in theories with symmetric 3D moduli spaces by reducing to a geodesic problem on ${\cal M}_{3D}=\mathbf{G}/\mathbf{H}$ and exponentiating nilpotent generators $k\in\mathbf{k}$. It applies the method to torus-reduced pure gravity and to ${\cal N}=2$ supergravity with one vector multiplet, deriving explicit BPS and non-BPS single-centered flows, and extends to multi-centered configurations. A key finding is that BPS flows arise from specific nilpotent coset elements, while non-BPS flows require different nilpotent structures, and multi-centered non-BPS solutions typically have mutually local charges and no intrinsic angular momentum under the flat 3D-slice assumption. The results clarify when the harmonic-function replacement for charges extends to non-BPS cases and provide exact 4D mappings from 3D geodesics to 4D black-hole metrics, with potential applicability to other symmetric spaces and broader ${\cal N}=2$ systems.

Abstract

We derive extremal black hole solutions for a variety of four dimensional models which, after Kaluza-Klein reduction, admit a description in terms of 3D gravity coupled to a sigma model with symmetric target space. The solutions are in correspondence with certain nilpotent generators of the isometry group. In particular, we provide the exact solution for a non-BPS black hole with generic charges and asymptotic moduli in N=2 supergravity coupled to one vector multiplet. Multi-centered solutions can also be generated with this technique. It is shown that the non-supersymmetric solutions lack the intricate moduli space of bound configurations that are typical of the supersymmetric case.

Non-Supersymmetric Attractor Flow in Symmetric Spaces

TL;DR

This work develops a unified framework to construct extremal black hole attractor flows in theories with symmetric 3D moduli spaces by reducing to a geodesic problem on and exponentiating nilpotent generators . It applies the method to torus-reduced pure gravity and to supergravity with one vector multiplet, deriving explicit BPS and non-BPS single-centered flows, and extends to multi-centered configurations. A key finding is that BPS flows arise from specific nilpotent coset elements, while non-BPS flows require different nilpotent structures, and multi-centered non-BPS solutions typically have mutually local charges and no intrinsic angular momentum under the flat 3D-slice assumption. The results clarify when the harmonic-function replacement for charges extends to non-BPS cases and provide exact 4D mappings from 3D geodesics to 4D black-hole metrics, with potential applicability to other symmetric spaces and broader systems.

Abstract

We derive extremal black hole solutions for a variety of four dimensional models which, after Kaluza-Klein reduction, admit a description in terms of 3D gravity coupled to a sigma model with symmetric target space. The solutions are in correspondence with certain nilpotent generators of the isometry group. In particular, we provide the exact solution for a non-BPS black hole with generic charges and asymptotic moduli in N=2 supergravity coupled to one vector multiplet. Multi-centered solutions can also be generated with this technique. It is shown that the non-supersymmetric solutions lack the intricate moduli space of bound configurations that are typical of the supersymmetric case.

Paper Structure

This paper contains 38 sections, 244 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. 3D Moduli Space
  4. Attractor Flow Equation
  5. Models with Symmetric Moduli Space
  6. Torus Reduction of $D$-dimensional Pure Gravity
  7. Kaluza-Klein Reduction
  8. Nilpotency
  9. A Toy Example: Hyperkähler Euclidean Metrics in Four Dimensions
  10. Single-centered Black Holes in Pure Gravity
  11. Constructing the flow generator $k$
  12. Full flow
  13. Example: 5D pure gravity compactified on a circle.
  14. Multi-centered Solutions in Pure Gravity
  15. Attractor Flows in $G_{2(2)}/(SL(2,\mathbb{R})\times SL(2,\mathbb{R}))$
  16. ...and 23 more sections

Figures (4)

  • Figure 1: Root Diagram of Cartan Decomposition of $G_{2(2)}$
  • Figure 2: Root Diagram of Isometry of $M_{3D}=G_{2(2)}/(SL(2,\mathbb{R})\times SL(2,\mathbb{R}))$. $\{\mathbf{u},\mathbf{y},\mathbf{x}, \bm{\sigma},\mathbf{A}^{0},\mathbf{A}^{1},\mathbf{B}_{1},\mathbf{B}_{0}\}$ generates the solvable subgroup.
  • Figure 3: Sample BPS flow. The attractor point is labeled $(x^*,y^*)$. The initial points of each flow are given by $(x_1=1.5, y_1=0.5), (x_2=2, y_2=4), (x_3=-0.2, y_3=0.1), (x_4=-1, y_4=3)$
  • Figure 4: Sample non-BPS flow. The attractor point is labeled $(x^*,y^*)$. The initial points of each flow are given by: $(x_1=0.539624, y_1= 5.461135), (x_2=1.67984, y_2=0.518725), (x_3=-0.432811,y_3=0.289493),(x_4=1.28447, y_4=1.49815), (x_5=-0.499491, y_5=0.181744)$