Classical defects in higher-dimensional Einstein gravity coupled to nonlinear $$\sigma $$-models

We construct solutions of higher-dimensional Einstein gravity coupled to nonlinear \(\sigma \)-model with cosmological constant. The \(\sigma \)-model can be perceived as exterior configuration of a spontaneously-broken \(SO(D-1)\) global higher-codimensional “monopole”. Here we allow the kinetic term of the \(\sigma \)-model to be noncanonical; in particular we specifically study a quadratic-power-law type. This is some possible higher-dimensional generalization of the Bariola–Vilenkin (BV) solutions with k-global monopole studied recently. The solutions can be perceived as the exterior solution of a black hole swallowing up noncanonical global defects. Even in the absence of comological constant its surrounding spacetime is asymptotically non-flat; it suffers from deficit solid angle. We discuss the corresponding horizons. For \(\Lambda >0\) in 4d there can exist three extremal conditions (the cold, ultracold, and Nariai black holes), while in higher-than-four dimensions the extremal black hole is only Nariai. For \(\Lambda <0\) we only have black hole solutions with one horizon, save for the 4d case where there can exist two horizons. We give constraints on the mass and the symmetry-breaking scale for the existence of all the extremal cases. In addition, we also obtain factorized solutions, whose topology is the direct product of two-dimensional spaces of constant curvature (\(M_2\), \(dS_2\), or \(AdS_2\)) with (D-2)-sphere. We study all possible factorized channels.