Difference between pair creation and anomaly matching: Nontrivial aspects of invertible symmetries and defects in $SU(N)$ models

We study the connection between the $SU(N)_{K}$ Wess-Zumino-Witten (WZW) model and $SU(N)_{K'}$ WZW models, enforcing the unbroken $\mathbb{Z}_{N}$ symmetries. The fundamental underlying structures are the (half-)integer spin simple currents in the topological quantum field theories and anomaly matc