Abelian Projection on the Torus for general Gauge Groups

TL;DR

This paper develops a generalized Abelian projection method for Yang-Mills theories on a four torus with various gauge groups, analyzing boundary conditions, gauge fixing, and magnetic monopole charges in relation to instanton sectors.

Contribution

It introduces a consistent Abelian projection framework for arbitrary gauge groups on the torus, including boundary conditions and monopole charge characterization.

Findings

Constructed gauge fixing domains for all instanton sectors.

Identified singularities as Dirac strings connecting monopoles.

Linked magnetic charges and Higgs windings to instanton numbers.

Abstract

We consider Yang-Mills theories with general gauge groups $G$ and twists on the four torus. We find consistent boundary conditions for gauge fields in all instanton sectors. An extended Abelian projection with respect to the Polyakov loop operator is presented, where $A_0$ is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed $A_0$ are constructed for arbitrary gauge groups. In the sectors with non-vanishing instanton number such gauge fixings are necessarily singular. The singularities can be restricted to Dirac strings joining magnetically charged defects. The magnetic charges of these monopoles take their values in the co-root lattice of the gauge group. We relate the magnetic charges of the defects and the windings of suitable Higgs fields about these defects to the instanton number.