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 $SU(N{)}_{K^{\mathrm{\prime }}}$ 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 matching conditions in conformal field theories. For the (half-)integer spin simple current condition of the folded models $SU(N{)}_K{⊠}_{{\mathbb{Z}}_N}\stackrel{‾}{SU(N{)}_{K^{\mathrm{\prime }}}}$ or $SU(N{)}_K{⊠}_{{\mathbb{Z}}_N}SU(N{)}_{K^{\mathrm{\prime }}}$, we obtain the generalized pairing condition $K=\pm R^2{K}^{\mathrm{\prime }}(\text{mod}\mathrm{.}N)$ with $\text{gcd}(R,N)=1$. On the other hand, the anomaly matching condition results in the weaker condition $K=\pm RU{K}^{\mathrm{\prime }}(\text{mod}\mathrm{.}N)$ with $\text{gcd}(R,N)=\text{gcd}(U,N)=1$. Surprisingly, the anomaly matching without the pairing condition results in the intrinsic mismatch of the preservation law of the ${\mathbb{Z}}_N$ defects and symmetries. The pairing condition canonically relates symmetries and defects, as in category theory, whereas it is unclear whether it can be a guiding principle in prohibiting the massless renormalization group flows.