Witten anomaly in 4d heterotic compactificaitons with N $$ \mathcal{N} $$ = 2 supersymmetry

Abstract We showed that there is no SU(2) Witten anomaly in a large class of 4d N $$ \mathcal{N} $$ = 2 supersymmetric Heterotic string compactifications. The consistency conditions we consider are the modularity of the new supersymmetric index, the integrality of BPS indices, and the discrete Peccei-Quinn shift symmetries. We also found an example where these conditions are not sufficient to show that the theory is anomaly free. This suggests that there are more conditions in the worldsheet SCFT essential for consistent string compactifications.