A ${\bf Z_2}$ Structure in the Configuration Space of Yang-Mills Theories

TL;DR

This paper demonstrates a ${f Z}_2$ topological structure in the configuration space of certain Yang-Mills theories, revealing new disconnected sectors and saddle point solutions related to symmetry breaking.

Contribution

It rigorously proves the existence of a ${f Z}_2$ homotopy group in the configuration space of $SU(2n)$ and $SO(2n)$ Yang-Mills theories, uncovering novel topological features.

Findings

Existence of a ${f Z}_2$ homotopy group in the configuration space.

Presence of infinite surfaces of odd-parity static gauge configurations.

Identification of $CS=1/2$ saddle point solutions related to symmetry breaking.

Abstract

We argue for the presence of a ${\bf Z}_2$ topological structure in the space of static gauge-Higgs field configurations of $SU(2n)$ and $SO(2n)$ Yang-Mills theories. We rigorously prove the existence of a ${\bf Z}_2$ homotopy group of mappings from the 2-dim. projective sphere ${\bf R}P^2$ into $SU(2n)/{\bf Z}_2$ and $SO(2n)/{\bf Z}_2$ Lie groups respectively. Consequently the symmetric phase of these theories admits infinite surfaces of odd-parity static and unstable gauge field configurations which divide into two disconnected sectors with integer Chern-Simons numbers $n$ and $n+1/2$ respectively. Such a ${\bf Z}_2$ structure persists in the Higgs phase of the above theories and accounts for the existence of $CS=1/2$ odd-parity saddle point solutions to the field equations which correspond to spontaneous symmetry breaking mass scales.