Pointlike Hopf defects in Abelian projections

TL;DR

This paper introduces a new type of pointlike topological defect in Abelian projections characterized by the Hopf invariant, expanding the understanding of topological structures in gauge theories.

Contribution

It identifies and visualizes Hopf defects with second-order zeros in Abelian projections, linking them to instanton sectors and the Hopf invariant pi_3(S^2).

Findings

Hopf defects occur in the Laplacian Abelian gauge within instanton sectors.

Ensemble of Hopf defects accounts for the instanton number.

Visualisation of the Hopf invariant in gauge configurations.

Abstract

We present a new kind of defect in Abelian Projections, stemming from pointlike zeros of second order. The corresponding topological quantity is the Hopf invariant pi_3(S^2) (rather than the winding number pi_2(S^2) for magnetic monopoles). We give a visualisation of this quantity and discuss the simplest non-trivial example, the Hopf map. Such defects occur in the Laplacian Abelian gauge in a non-trivial instanton sector. For general Abelian projections we show how an ensemble of Hopf defects accounts for the instanton number.