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 $SU(N{)}_K$ Wess-Zumino-Witten (WZW) model and $SU(N{)}_{K^{\prime }}$ WZW model, enforcing the ${\mathbb{Z}}_N$ 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, we obtain the generalized pairing condition $K=\pm R^2{K}^{\prime }(\text{mod}.N)$ with $\text{gcd}(R,N)=1$. On the other hand, the anomaly matching condition results in the weaker condition $K=\pm RU{K}^{\prime }(\text{mod}.N)$ with $\text{gcd}(R,N)=\text{gcd}(U,N)=1$. The preservation law of the ${\mathbb{Z}}_N$ defects and symmetries is compatible only with the pairing condition.