Invariant regularization of anomaly-free chiral theories

We present a generalization of the Frolov-Slavnov invariant regularization scheme for chiral fermion theories in curved spacetimes. The Lagrangian level regularization is explicitly invariant under all the local gauge symmetries of the theory, including local Lorentz invariance. The perturbative scheme works for arbitrary representations which satisfy the chiral gauge anomaly and the mixed Lorentz-gauge anomaly cancellation conditions. Anomalous theories on the other hand manifest themselves by having divergent fermion loops which remain unregularized by the scheme. Since the invariant scheme is promoted to include also local Lorentz invariance, spectator fields which do not couple to gravity cannot be, and are not, introduced. Furthermore, the scheme is truly chiral (Weyl) in that all fields, including the regulators, are left handed; and only the left-handed spin connection is needed. The scheme is, therefore, well suited for the study of the interaction of matter with all four known forces in a completely chiral fashion. In contrast with the vectorlike formulation, the degeneracy between the Adler-Bell-Jackiw current and the fermion number current in the bare action is preserved by the chiral regularization scheme.