Instanton Solutions from Abelian Sinh-Gordon and Tzitzeica Vortices

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 Poincaré 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 Kähler manifold with non-vanishing scalar curvature. The instantons we construct in this way cannot be obtained via a twistorial approach.

24 pages, 6 figures; corrected typos, added section 5, published version

References in corpus (5)

• Integrability of Vortex Equations on Riemann Surfaces
• Fractional Vortices and Lumps
• Vortex Motion on Surfaces of Small Curvature
• Vortices as degenerate metrics
• A ladder of topologically non-trivial non-BPS states

Cited by in corpus (3)

• Manton's five vortex equations from self-duality
• Integrable Abelian Vortex-like solitons
• Elizabethan vortices