Topology of SU(N) lattice gauge theories coupled with &Z;N 2-form gauge fields

We extend the definition of Lüscher's lattice topological charge to the case of 4d SU(N) gauge fields coupled with &Z;_N 2-form gauge fields. This result is achieved while maintaining the locality, the SU(N) gauge invariance, and &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 &Z;_N 1-form symmetry and the θ periodicity.