Abelian vortices from Sinh--Gordon and Tzitzeica equations

TL;DR

This paper links sinh--Gordon and Tzitzeica equations to Abelian vortices on specific surfaces, constructing vortex fields from surface data and analyzing radially symmetric solutions with explicit vortex strengths.

Contribution

It introduces a geometric interpretation of these equations as vortex equations on special surfaces and derives explicit vortex solutions using Painlevé transcendents.

Findings

Radially symmetric solutions correspond to vortices with topological charge one.

Explicit vortex strengths are computed using third Painlevé transcendents.

The equations are connected to surfaces embedded in ,1 and hyperbolic affine spheres.

Abstract

It is shown that both the sinh--Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in $\R^{2, 1}$, or on a surface conformally related to a hyperbolic affine sphere in $\R^3$. In both cases the Higgs field and the U(1) vortex connection are constructed directly from the Riemannian data of the surface corresponding to the sinh--Gordon or the Tzitzeica equation. Radially symmetric solutions lead to vortices with a topological charge equal to one, and the connection formulae for the resulting third Painlev\'e transcendents are used to compute explicit values for the strength of the vortices.