Nonsingular solutions of Hitchin's equations for noncompact gauge groups

TL;DR

This paper finds smooth, localized solutions to Hitchin's equations for noncompact gauge groups, specifically SO(2,1), revealing integrability and connections to classical equations like Liouville and sine-Gordon.

Contribution

It introduces a new ansatz for solving Hitchin's equations with noncompact gauge groups, leading to explicit nonsingular solutions and linking to well-known integrable equations.

Findings

Smooth solutions with localized action densities for SO(2,1)

Reduction of equations to Liouville and sine-Gordon forms

Solutions on R^2, S^2, and T^2 surfaces

Abstract

We consider a general ansatz for solving the 2-dimensional Hitchin's equations, which arise as dimensional reduction of the 4-dimensional self-dual Yang-Mills equations, with remarkable integrability properties. We focus on the case when the gauge group G is given by a real form of SL(2,C). For G=SO(2,1), the resulting field equations are shown to reduce to either the Liouville, elliptic sinh-Gordon or elliptic sine-Gordon equations. As opposed to the compact case, given by G=SU(2), the field equations associated with the noncompact group SO(2,1) are shown to have smooth real solutions with nonsingular action densities, which are furthermore localized in some sense. We conclude by discussing some particular solutions, defined on R^2, S^2 and T^2, that come out of this ansatz.