Reflections on Topology - Viewpoints on Abelian Projections

TL;DR

This paper explores the topological features of non-abelian gauge theories, focusing on abelian projection, the Hopf invariant, and monopole structures, with implications for understanding gauge vacua and topological charge.

Contribution

It establishes a connection between the Hopf invariant, abelian projection, and monopole configurations, providing a detailed analysis of their interplay in gauge theories.

Findings

Hopf invariant relates to monopole world lines and topological charge.

Degenerate eigenvalues of Polyakov loop identify monopole constituents.

Correlation between defect locations and monopole structure demonstrated in SU(3) example.

Abstract

This talk discusses two topological features in non-abelian gauge theories, related by the notion of abelian projection and the Hopf invariant. Minimising the energy of the non-linear sigma model with a Skyrme-like term (the Faddeev-Niemi model), can be identified with a non-linear maximal abelian gauge fixing of the SU(2) gauge vacua with a winding number equal to the Hopf invariant. In the context of abelian projection the Hopf invariant can also be associated to a monopole world line, through the Taubes winding, measuring its contribution to topological charge. Calorons with non-trivial holonomy provide an explicit realisation. We discuss the identification of its constituent monopoles through degenerate eigenvalues of the Polyakov loop (the singularities or defects of the abelian projection). It allows us to study the correlation between the defect locations and the explicit constituent monopole structure, through a specific SU(3) example.