Circular symmetry in the Hitchin system

TL;DR

This paper investigates circularly symmetric solutions in the SU(2) Hitchin system, deriving differential equations and presenting an exact pure gauge solution with a nontrivial Higgs scalar.

Contribution

It introduces a symmetry-based reduction of the Hitchin system and provides an explicit exact solution with nontrivial Higgs field configurations.

Findings

Derived differential equations for symmetric configurations

Obtained an exact pure gauge solution with nonzero Higgs scalar

Identified vacuum expectation values for the Higgs field

Abstract

To study circularly symmetric field configurations in the SU(2) Hitchin system an SO(2) symmetry, [J_3, \phi]=0 and [J_3, A_{\pm}]=\pm A_{\pm}, is imposed on the Higgs scalar \phi and the gauge fields A_{\pm} of the system, respectively, where J_3 is a sum of the third components of the orbital angular momenta and the generators of the SU(2). The circular symmetry and the equation \bar{D}\phi=0 yield onstant, generally nonzero, vacuum expectation values for {\rm Tr}(\phi^{2}). The equation 4F_{z\bar{z}}=[\phi, \phi^{*}] yields a system of differential equations which govern the circularly symmetric field configurations and an exact solution to these equations in a pure gauge form with nontrivial Higgs scalar is obtained.