Twisted and Singular gravitating vortices

TL;DR

This paper introduces twisted gravitating vortices on compact Riemann surfaces, proving existence, uniqueness, and singular solutions with conical and parabolic singularities, generalizing previous results in the field.

Contribution

It extends the theory of gravitating vortices by incorporating twisting forms, establishing existence and uniqueness results, and constructing singular solutions with conical and parabolic singularities.

Findings

Proved existence and uniqueness of twisted gravitating vortices for certain parameters.

Constructed singular solutions with conical and parabolic singularities.

Extended the framework to include cosmic string solutions in the Bogomol'nyi phase.

Abstract

We introduce the notion of twisted gravitating vortex 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 arXiv:1510.03810v2 in the non twisted setting. It is proved via solving a continuity path deforming the coupling constant from 0 for which the system decouples as twisted K\"ahler-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 K\"ahler 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.