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

Abstract: We study the connection between the $SU(N)<em>{K}$ Wess-Zumino-Witten (WZW) model and $SU(N)</em>{K&#39;}$ WZW models, enforcing the unbroken $\mathbb{Z}<em>{N}$ 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 $SU(N)</em>{K}\boxtimes_{\mathbb{Z}<em>{N}} \overline{SU(N)</em>{K&#39;}}$ or $SU(N)<em>{K}\boxtimes</em>{\mathbb{Z}<em>{N}} SU(N)</em>{K&#39;}$, we obtain the generalized pairing condition $K= \pm R^{2}K&#39; (\text{mod}.N)$ with $\text{gcd}(R,N)=1$. On the other hand, the anomaly matching condition results in the weaker condition $K= \pm RUK&#39; (\text{mod}.N)$ 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.