Instanton Solutions from Abelian Sinh-Gordon and Tzitzeica Vortices

TL;DR

This paper constructs vortex solutions on conical surfaces from Abelian Higgs vortices related to sinh-Gordon and Tzitzeica equations, and uplifts these to generate novel four-dimensional Yang-Mills instantons on curved Kähler manifolds.

Contribution

It introduces a method to derive vortex solutions on conifolds from simpler surfaces and constructs new instantons on non-self-dual Kähler manifolds, expanding the understanding of gauge theories on curved spaces.

Findings

Explicit vortex solutions on the hyperbolic cone were obtained.

A new class of Yang-Mills instantons on curved Kähler manifolds was constructed.

The method links vortices on surfaces of revolution to those on conifolds.

Abstract

We study the Abelian Higgs vortex solutions to the sinh-Gordon equation and the elliptic Tzitzeica equation. Starting from these particular vortices, we construct solutions to the Taubes equation with higher vortex number, on surfaces with conical singularities. We then, analyse more general properties of vortices on such singular surfaces and propose a method to obtain vortices on conifolds from vortices on surfaces of revolution. We apply our method to construct explicit vortex solutions on the Poincar\'e disk with a conical singularity in the centre, to which we refer as the "hyperbolic cone". We uplift the Abelian sinh-Gordon and Tzitzeica vortex solutions to four dimensions and construct cylindrically symmetric, self-dual Yang-Mills instantons on a non-self-dual (nor anti-self-dual) $4$-dimensional K\"ahler manifold with non-vanishing scalar curvature. The instantons we construct in this way cannot be obtained via a twistorial approach.