Gauge-invariant signatures of spontaneous gauge symmetry breaking by the Hosotani mechanism

The Hosotani mechanism claims to achieve gauge-symmetry breaking, for instance $SU(3)\to SU(2)\times U(1)$. To verify this claim, we propose to monitor the stability of a topological defect stable under a gauge subgroup but not under the whole gauge group, like a $U(1)$ flux state or monopole in the case above. We use gauge invariant operators to probe the presence of the topological defect to avoid any ambiguity introduced by gauge fixing. Our method also applies to an ordinary gauge-Higgs system.