Family of anomalies in two dimensions in path-integral formulation. I

We study the regularization of the path integral for a two-dimensional fermionic system in terms of the eigenfunctions and eigenvalues of the operator Da=∂+ieaA where a is a real continuous parameter. We derive the associated Ward-Takahashi identities for the local chiral and vector transformations and obtain expressions for ∂μJμA and ∂μJμV. We propose a straightforward regularization that ultimately leads to the family of anomalies in which the parameter appearing in the family of anomalies is related to a. A comparison with the work of Alfaro, Urrutia, and Vergara is given