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

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.