Magic Supergravities, N= 8 and Black Hole Composites

TL;DR

The paper provides explicit U-duality invariants and entropy functions for magic N=2 and N=8 supergravities, linking four-, five-, and six-dimensional invariants via cubic and quartic forms. It constructs general multicenter BPS solutions, including 1/2 BPS octonionic magic and 1/8 BPS N=8 configurations obtained by truncation to quaternionic magic, and extends attractor analysis to non-BPS cases in STU-like truncations. By presenting universal d=4 ⇄ d=5 relations and manifest symmetry forms, it unifies the entropy and stabilization framework across magic models and clarifies how BPS and non-BPS states relate under dimensional and supersymmetry reductions. The results have implications for the structure of extremal black holes in extended supergravities and provide a practical toolkit for exploring duality-invariant multicenter configurations.

Abstract

We present explicit U-duality invariants for the R, C, Q, O$ (real, complex, quaternionic and octonionic) magic supergravities in four and five dimensions using complex forms with a reality condition. From these invariants we derive an explicit entropy function and corresponding stabilization equations which we use to exhibit stationary multi-center 1/2 BPS solutions of these N=2 d=4 theories, starting with the octonionic one with E_{7(-25)} duality symmetry. We generalize to stationary 1/8 BPS multicenter solutions of N=8, d=4 supergravity, using the consistent truncation to the quaternionic magic N=2 supergravity. We present a general solution of non-BPS attractor equations of the STU truncation of magic models. We finish with a discussion of the BPS-non-BPS relations and attractors in N=2 versus N= 5, 6, 8.