Dirac monopoles embedded in $\mathbf{SU(N)}$ gauge theory with the $\mathbf{\theta}$ term

Dirac monopoles embedded in $SU(N)$ gauge theory with the $\theta$ term are considered. For $\theta=4\pi M$ (where $M$ is half-integer and integer for $N=2$ and $N>2$, respectively), these monopoles acquire an $SU(N)$ charge due to the $\theta$ term and become dyons. They belong to various (but not any) irreducible representations of the $SU(N)$ group. The admissible representations are listed. Their minimum dimension increases with $N$. The basic result of the study is the representation of the partition function of any $SU(N)$ model involving the $\theta$ term and complemented by singular gauge fields corresponding to the indicated monopoles in the form of a vacuum average of the product of Wilson loops viewed along the world lines of the monopoles. This vacuum average must be calculated in the corresponding model without the $\theta$ term.