Fractional topological charge in $SU(N)$ gauge theories without dynamical quarks

In $SU(N)$ gauge theories without dynamical quarks, we discuss how configurations with fractional topological charge, $\sim 1/N$, can arise in the vacuum and dominate in the confining phase. They are not solutions of the classical equations of motion, but arise as quantum solutions of the effective Lagrangian. Their size is essentially fixed, on the order of the confinement scale. We give both a general mathematical analysis and illustrative solutions. Their presence can be measured through numerical simulations on the lattice using known methods. As they carry $\mathbb{Z}_N$ magnetic charge, the introduction of dynamical quarks significantly complicates their dynamics.