X iv : h ep - t h / 96 02 13 7 v 1 2 6 Fe b 19 96 UCONN - 96 - 03 Anomalous Chiral Action from the Path - Integral

By generalizing the Fujikawa approach, we show in the path-integral formalism: (1) how the infinitesimal variation of the fermion measure can be integrated to obtain the full anomalous chiral action ; (2) how the action derived in this way can be identified as the Chern-Simons term in five dimensions, if the anomaly is consistent; (3) how the regularization can be carried out, so as to lead to the consistent anomaly and not to the covariant anomaly. Our method uses Schwinger's " proper-time " representation of the Green's function and the gauge invariant point-splitting technique. We find that the consistency requirement and the point-splitting technique allow both an anomalous and a non-anomalous action. In the end, the nature of the vacuum determines whether we have an anomalous theory, or, a non-anomalous theory.