Magnetic monopoles vs. Hopf defects in the Laplacian (Abelian) gauge

TL;DR

This paper explores how the Laplacian Abelian gauge reveals magnetic monopoles and Hopf defects on S^4 in the presence of a 't Hooft instanton, linking topological defects to confinement mechanisms.

Contribution

It demonstrates the emergence of monopoles and Hopf defects from the eigenfunctions of the covariant Laplacian in the Laplacian Abelian gauge, providing new insights into topological structures in gauge theories.

Findings

First-order zeros produce magnetic monopoles.

Second-order zeros lead to pointlike Hopf defects.

Similar defects are found in the fundamental representation.

Abstract

We investigate the Laplacian Abelian gauge on the sphere S^4 in the background of a single `t Hooft instanton. To this end we solve the eigenvalue problem of the covariant Laplace operator in the adjoint representation. The ground state wave function serves as an auxiliary Higgs field. We find that the ground state is always degenerate and has nodes. Upon diagonalisation, these zeros induce toplogical defects in the gauge potentials. The nature of the defects crucially depends on the order of the zeros. For first-order zeros one obtains magnetic monopoles. The generic defects, however, arise from zeros of second order and are pointlike. Their topological invariant is the Hopf index S^3 -> S^2. These findings are corroborated by an analysis of the Laplacian gauge in the fundamental representation where similar defects occur. Possible implications for the confinement scenario are discussed.