Note on lattice description of generalized symmetries in $SU(N)/\mathbb{Z}_N$ gauge theories

Understand

Topology and generalized symmetries in the $SU(N)/\mathbb{Z}_N$ gauge theory are considered in the continuum and the lattice.

• Starting from the $SU(N)$ gauge theory with the 't~Hooft twisted boundary condition, we give a simpler explanation of the van~Baal's proof on the fractionality of the topological charge.
• This description is applicable to both continuum and lattice by using the generalized L\"uscher's construction of topology on the lattice.
• Thus we can recover the $SU(N)/\mathbb{Z}_N$ principal bundle from lattice $SU(N)$ gauge fields being subject to the $\mathbb{Z}_N$-relaxed cocycle condition.