Topology of $SU(N)$ lattice gauge theories coupled with ${\mathbb{Z}}_N$ $2$-form gauge fields

We extend the definition of Lüscher’s lattice topological charge to the case of $4$d $SU(N)$ gauge fields coupled with ${\mathbb{Z}}_N$ $2$-form gauge fields. This result is achieved while maintaining the locality, the $SU(N)$ gauge invariance, and ${\mathbb{Z}}_N$ $1$-form gauge invariance, and we find that the manifest $1$-form gauge invariance plays the central role in our construction. This result gives the lattice regularized derivation of the mixed 't Hooft anomaly in pure $SU(N)$ Yang-Mills theory between its ${\mathbb{Z}}_N$ $1$-form symmetry and the $\theta$ periodicity.