The Bogomolnyi-Prasad-Sommerfield (BPS) baby Skyrme model coupled to gravity is considered. We show that in an asymptotically flat space-time the model still possesses the BPS property, i.e., admits a BPS reduction to first order Bogomolnyi equations, which guarantees that the corresponding proper energy is a linear function of the topological charge. We also find the mass-radius relation as well as the maximal mass and radius. All these results are obtained in an analytical manner, which implies the complete solvability of this selfgravitating matter system.If a cosmological constant is added, then the BPS property is lost. In de Sitter (dS ) spacetime both extremal and non-extremal solutions are found, where the former correspond to finite positive pressure solutions of the flat space-time model. For the asymptotic anti-de Sitter (AdS ) case, extremal solutions do not exist as there are no negative pressure BPS baby Skyrmions in flat space-time. Non-extremal solutions with AdS asymptotics do exist and may be constructed numerically. The impact of the negative cosmological constant on the mass-radius relation is studied. We also found two potentials for which exact multisoliton solutions in the external AdS space can be obtained. Finally, we elaborate on the implications of these findings for certain three-dimensional models of holographic QCD.
PACS numbers:
arXiv:1802.07278v2 [hep-th] 4 Apr 2018Both models have exact solutions, for any value of the topological charge, which solve the pertinent Bogomolny equations and, therefore, saturate the corresponding topological bounds. Hence, the full baby Skyrme model can be viewed as a certain point in parameter space {c 0 , c 2 , c 4 } with two distinguished limits -the BPS models [6]. As a consequence, properties of baby Skyrmions in a given model (given values of the coupling constants) can be interpreted as resulting from a competition between these two BPS limits. In any case, understanding these two limiting models provides a solid starting point for an analytical treatment of the full model.Recently, the baby Skyrme model has been used as a toy model of holographic QCD [7] (see also [8]-[10]) in the spirit of the Sakai-Sugimoto framework [11] (for recent results on holographic/AdS Skyrmions we refer to [12], [13], [14]). In the simplest set-up it just means putting the baby Skyrme model in an AdS background. Because of this relation, properties of gravitating baby Skyrmions may teach us something about phases of dense nuclear matter (in particular baryonic popcorn [15]