Dyonic black holes at arbitrary locations

Abstract: We construct and study stationary, asymptotically flat multicenter solutions describing regular black holes with non-Abelian hair (colored magnetic-monopole and dyon fields) in two models of $\mathcal{N}=2,d=4$ Super-Einstein-Yang-Mills theories: the quadratic model $\overline{\mathbb{CP}}^{3}$ and the cubic model ST$[2,6]$, which can be embedded in 10-dimensional Heterotic Supergravity. These solutions are based on the multicenter dyon recently discovered by one of us, which solves the SU$(2)$ Bogomol'nyi and dyon equations on $\mathbb{E}^{3}$. In contrast to the well-known Abelian multicenter solutions, the relative positions of the non-Abelian black hole centers are unconstrained. We study necessary conditions on the parameters of the solutions that ensure the regularity of the metric. In the case of the $\overline{\mathbb{CP}}^{3}$ model we show that it is enough to require the positivity of the "masses" of the individual black holes, the finiteness of each of their entropies and their superadditivity. In the case of the $ST[2,6]$ model we have not been able to show that analogous conditions are sufficient, but we give an explicit example of a regular solution describing thousands of non-Abelian dyonic black holes in equilibrium at arbitrary relative positions. We also construct non-Abelian solutions that interpolate smoothly between just two aDS$_{2}\times$S$^{2}$ vacua with different radii (dumbbell solutions).