Resolution of Gauss' law in Yang-Mills theory by Gauge Invariant Projection: Topology and Magnetic Monopoles

TL;DR

This paper introduces a gauge-invariant method to resolve Gauss' law in Yang-Mills theory, revealing how magnetic monopoles and topological structures emerge naturally in the gauge-invariant formulation.

Contribution

It presents a novel gauge-invariant projection approach that simplifies the description of Yang-Mills theory and elucidates the role of magnetic monopoles in its topology.

Findings

Magnetic monopoles arise in the gauge-invariant formulation.

The Pontryagin index relates to magnetic charges.

Monopoles account for the topological structure.

Abstract

An efficient way of resolving Gauss' law in Yang-Mills theory is presented by starting from the projected gauge invariant partition function and integrating out one spatial field variable. In this way one obtains immediately the description in terms of unconstrained gauge invariant variables which was previously obtained by explicitly resolving Gauss' law in a modified axial gauge. In this gauge, which is a variant of 't Hooft's Abelian gauges, magnetic monopoles occur. It is shown how the Pontryagin index of the gauge field is related to the magnetic charges. It turns out that the magnetic monopoles are sufficient to account for the non-trivial topological structure of the theory.