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Extremal solutions of the S3 model and nilpotent orbits of G2(2)

Sung-Soo Kim, Josef Lindman Hörnlund, Jakob Palmkvist, Amitabh Virmani

TL;DR

This work develops a group-theoretic framework to classify extremal, static black holes in the S^3 model by analyzing nilpotent tilde{K}-orbits of the duality group G_{2(2)}. The authors identify six orbits (three supersymmetric, one non-supersymmetric, two unphysical) and show that all supersymmetric solutions uplift to five-dimensional minimal supergravity with a Gibbons-Hawking base, while also constructing novel extremal pressureless black strings and exploring connections to black rings. They provide explicit charge data, GH representations, and orbit-generating procedures, tying the four-dimensional solutions to five-dimensional geometry and to the Einstein-Maxwell truncation. The results offer a concrete map from orbit structure to physical solution spaces, clarifying attractor flows and the role of nilpotency in extremal black hole classification, with potential implications for the broader landscape of BPS and non-BPS configurations. The work thus advances the understanding of extremal solutions in symmetric moduli spaces and demonstrates the power of group-theoretical methods in supergravity.

Abstract

We study extremal black hole solutions of the S3 model (obtained by setting S=T=U in the STU model) using group theoretical methods. Upon dimensional reduction over time, the S3 model exhibits the pseudo-Riemannian coset structure G/K with G=G2(2) and K=SO(2,2). We study nilpotent K-orbits of G2(2) corresponding to non-rotating single-center extremal solutions. We find six such distinct K-orbits. Three of these orbits are supersymmetric, one is non-supersymmetric, and two are unphysical. We write general solutions and discuss examples in all four physical orbits. We show that all solutions in supersymmetric orbits when uplifted to five-dimensional minimal supergravity have single-center Gibbons-Hawking space as their four-dimensional Euclidean hyper-Kähler base space. We construct hitherto unknown extremal (supersymmetric as well as non-supersymmetric) pressureless black strings of minimal five-dimensional supergravity and briefly discuss their relation to black rings.

Extremal solutions of the S3 model and nilpotent orbits of G2(2)

TL;DR

This work develops a group-theoretic framework to classify extremal, static black holes in the S^3 model by analyzing nilpotent tilde{K}-orbits of the duality group G_{2(2)}. The authors identify six orbits (three supersymmetric, one non-supersymmetric, two unphysical) and show that all supersymmetric solutions uplift to five-dimensional minimal supergravity with a Gibbons-Hawking base, while also constructing novel extremal pressureless black strings and exploring connections to black rings. They provide explicit charge data, GH representations, and orbit-generating procedures, tying the four-dimensional solutions to five-dimensional geometry and to the Einstein-Maxwell truncation. The results offer a concrete map from orbit structure to physical solution spaces, clarifying attractor flows and the role of nilpotency in extremal black hole classification, with potential implications for the broader landscape of BPS and non-BPS configurations. The work thus advances the understanding of extremal solutions in symmetric moduli spaces and demonstrates the power of group-theoretical methods in supergravity.

Abstract

We study extremal black hole solutions of the S3 model (obtained by setting S=T=U in the STU model) using group theoretical methods. Upon dimensional reduction over time, the S3 model exhibits the pseudo-Riemannian coset structure G/K with G=G2(2) and K=SO(2,2). We study nilpotent K-orbits of G2(2) corresponding to non-rotating single-center extremal solutions. We find six such distinct K-orbits. Three of these orbits are supersymmetric, one is non-supersymmetric, and two are unphysical. We write general solutions and discuss examples in all four physical orbits. We show that all solutions in supersymmetric orbits when uplifted to five-dimensional minimal supergravity have single-center Gibbons-Hawking space as their four-dimensional Euclidean hyper-Kähler base space. We construct hitherto unknown extremal (supersymmetric as well as non-supersymmetric) pressureless black strings of minimal five-dimensional supergravity and briefly discuss their relation to black rings.

Paper Structure

This paper contains 33 sections, 207 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and summary
  2. Nilpotent orbits
  3. Single modulus S$^3$ $N=2, D=4$ supergravity
  4. The theory
  5. Reduction on time
  6. Five-dimensional lift and hyper-Kähler base space
  7. Asymptotic frame
  8. Orbit structure of G$_{2(2)}$
  9. Generalities on G$_{2(2)}$
  10. $\mathrm{G}_{2}$-orbits
  11. $\mathrm{G}_{2(2)}$-orbits
  12. $\tilde{K}$-orbits
  13. Generating orbits in practice
  14. Supersymmetric orbits
  15. The $\mathcal{O}_1$ orbit
  16. ...and 18 more sections

Figures (2)

  • Figure 1: The roots of $\mathfrak{g}_{2}$ given as vectors in the two-dimensional root space, dual to the Cartan subalgebra $\mathfrak{h}$. The positive (negative) roots $\pm\alpha_1,\,\ldots,\,\pm\alpha_6$ correspond to the root vectors $e_1,\,\ldots,\,e_6$ ($f_1,\,\ldots,\,f_6$), which under the automorphism $\varphi$ are mapped into $E_1,\,\ldots,\,E_6$ ($F_1,\,\ldots,\,F_6$). Applying this automorphism, we can associate the horizontal and vertical axes with the subalgebra $\tilde{\mathfrak{k}}$, and the rectangle with the representation space $\tilde{\mathfrak{p}}$.
  • Figure 2: Hasse diagram for the partial ordering of the nilpotent orbits in $\mathfrak{g}_{2(2)}$.