Two-dimensional Anomaly, Orbifolding, and Boundary States
Ken Kikuchi, Yang Zhou
TL;DR
The paper develops a modular-S transformation criterion to detect anomalies of discrete internal symmetries G in two-dimensional diagonal RCFTs via twisted torus partition functions Z_{(h,h')}. By comparing Z_{(h,h)} with its S-transform SZ_{(h,h)} and employing a truncation to the topological defect sector, the authors derive explicit anomaly-free conditions across a wide class of WZW models (A_r, B_r, C_r, D_r, E_6, E_7) and minimal models, and they connect these to orbifoldability and invariant boundary states. They show that invariant Cardy states imply anomaly decoupling for the corresponding subgroup, establishing a strong link between boundary-consistency and anomaly structure; conversely, the absence of such boundary states signals anomalies or mixed anomalies with outer automorphisms. Through detailed examples (SU(2)_k, SU(3)_k, Ising-type and Potts models), the work clarifies when discrete symmetries are anomaly-free or anomalous and interprets the anomalies as mixed with the S-dual of the symmetry. Overall, the approach provides a concrete, modular-dynamical toolkit for diagnosing 2D symmetry anomalies, orbifoldability, and boundary-state consistency in RCFTs and their WZW realizations.
Abstract
We study anomalies of discrete internal global symmetry $G$ in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of $G$ can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular $S$-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between $G$ and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.