Octonionic black holes

Abstract

Abstract Using algebraic tools inspired by the study of nilpotent orbits in simple Lie algebras, we obtain a large class of solutions describing interacting non-BPS black holes in $ \mathcal{N} = 8 $ supergravity, which depend on 44 harmonic functions. For this purpose, we consider a truncation $ {E_{{{6}({6})}}}/S{p_{\text{c}}}\left( {8,\mathbb{R}} \right) \subset {E_{{{8}({8})}}}/{\text{Spin}}_{\text{c}}^{ * }\left( {16} \right) $ of the non-linear sigma model describing stationary solutions of the theory, which permits a reduction of algebraic computations to the multiplication of 27 by 27 matrices. The lift to $ \mathcal{N} = 8 $ supergravity is then carried out without loss of information by using a pertinent representation of the moduli parametrizing E7(7)/SUc (8) in terms of complex valued Hermitian matrices over the split octonions, which generalise the projective coordinates of exceptional special K¨ahler manifolds. We extract the electromagnetic charges, mass and angular momenta of the solutions, and exhibit the duality invariance of the black holes distance separations. We discuss in particular a new type of interaction which appears when interacting non-BPS black holes are not aligned. Finally we will explain the possible generalisations toward the description of the most general stationary black hole solutions of $ \mathcal{N} = 8 $ supergravity.