The Z2 anomaly in some chiral gauge theories

We revisit the simplest Bars-Yankielowicz (BY) model (the ?? model), starting from a model with an additional Dirac pair of fermions in the fundamental representation, together with a complex color-singlet scalar 0 coupled to them through a Yukawa interaction. This model possesses a color-flavor-locked 1-form Z(N) symmetry, due to the intersection of the color SU(N) and two nonanomalous U(1) groups. In the bulk, the model reduces to the ?? model studied earlier when 0 acquires a nonzero vacuum expectation value and the extra fermions pair up, get massive and decouple (thus we will call our extended theory as the "X-ray model"), while it provides a regularization of the Z(2) fluxes needed to study the Z(2 )anomaly. The anomalies involving the 1-form Z(N) symmetry reduce, for N even, exactly to the mixed Z(2) anomaly found earlier in the ?? model. The present work is a first significant step to clarify the meaning of the mixed Z(2) - [Z(N)((1)) ](2) anomaly found in the ?? and in other BY and Georgi-Glashow type SU(N) models with even N.