Geometry of Solutions of Hitchin Equations on R^2

TL;DR

This paper analyzes the structure and geometry of solutions to the Hitchin equations on R^2, focusing on moduli spaces, symmetric solutions, and asymptotic behaviors for different polynomial degrees of the Higgs field.

Contribution

It provides a detailed study of the moduli space structure and geometric properties of solutions to Hitchin equations on R^2, especially for degree 3 Higgs fields.

Findings

Existence of moduli spaces for solutions with degree n≥3

Explicit description of symmetric solutions for n=1 and n=2

Analysis of asymptotic geometry and geodesic surfaces for n=3

Abstract

We study smooth SU(2) solutions of the Hitchin equations on R^2, with the determinant of the complex Higgs field being a polynomial of degree n. When n>=3, there are moduli spaces of solutions, in the sense that the natural L^2 metric is well-defined on a subset of the parameter space. We examine rotationally-symmetric solutions for n=1 and n=2, and then focus on the n=3 case, elucidating the moduli and describing the asymptotic geometry as well as the geometry of two totally-geodesic surfaces.