# Confinement and Topological Charge in the Abelian Gauge of QCD

## TL;DR

This paper explores the relationship between instantons and monopoles in abelian gauge of QCD, revealing how monopole trajectories become complex at high instanton density and demonstrating monopole dominance in confinement phases.

## Contribution

It provides a detailed analysis of monopole-instanton interactions in both continuum and lattice QCD, highlighting the role of monopoles in confinement and topological charge.

## Key findings

- Monopole trajectories become complex at high instanton density.
- Monopole dominance observed in the confinement phase at finite temperature.
- Linear correlation between monopole-loop length and topological density.

## Abstract

We study the relation between instantons and monopoles in the abelian gauge. First, we investigate the monopole in the multi-instanton solution in the continuum Yang-Mills theory using the Polyakov gauge. At a large instanton density, the monopole trajectory becomes highly complicated, which can be regarded as a signal of monopole condensation. Second, we study instantons and monopoles in the SU(2) lattice gauge theory both in the maximally abelian (MA) gauge and in the Polyakov gauge. Using the $16^3 \times 4$ lattice, we find monopole dominance for instantons in the confinement phase even at finite temperatures. A linear-type correlation is found between the total monopole-loop length and the integral of the absolute value of the topological density (the total number of instantons and anti-instantons) in the MA gauge. We conjecture that instantons enhance the monopole-loop length and promote monopole condensation.