ζ-function calculation of the Weyl determinant for two-dimensional non-abelian gauge theories in a curved background and its W-Z-W terms

Using a cohomological characterization of the consistent and the covariant Lorentz and gauge anomalies, derived from the complexification of the relevant algebras, we study in d = 2 the definition of the Weyl determinant for a non-abelian theory with Riemannian background. We obtain two second order operators that produce, by means of ζ-function regularization, respectively the consistent and the covariant Lorentz and gauge anomalies, preserving diffeomorphism invariance. We compute exactly their functional determinants and the W-Z-W terms: the “consistent” determinant agrees with the non-abelian generalization of the classical Leutwyler’s result, while the “covariant” one gives rise to a covariant version of the usual Wess-Zumino-Witten action.