Twisted and Singular Gravitating Vortices

Abstract

We introduce the notion of twisted gravitating vortices on a compact Riemann surface. If the genus of the Riemann surface is greater than 1 and the twisting forms have suitable signs, we prove an existence and uniqueness result for suitable range of the coupling constant generalizing the result of Álvarez-Cónsul et al. in (Math Ann 1–34, 2020) in the nontwisted setting. It is proved via solving a continuity path deforming the coupling constant from 0 for which the system decouples as twisted Kähler–Einstein metric and twisted vortices. Moreover, specializing to a family of twisting forms smoothing delta distribution terms, we prove the existence of singular gravitating vortices whose Kähler metric has conical singularities and Hermitian metric has parabolic singularities. In the Bogomol’nyi phase, we establish an existence result for singular Einstein–Bogomol’nyi equations, which represents cosmic strings with singularities.