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Using a cohomological characterization of the consistent and the covariantLorentz and gauge anomalies, derived from the complexification of the relevantalgebras, we study in $d=2$ the definition of the Weyl determinant for anon-abelian theory with Riemannian background. We obtain two second orderoperators that produce, by means of $\zeta$-function regularization,respectively the consistent and the covariant Lorentz and gauge anomalies,preserving diffeomorphism invariance. We compute exactly their functionaldeterminants and the W-Z-W terms: the ``consistent'' determinant agrees withthe non-abelian generalization of the classical Leutwyler's result, while the``covariant'' one gives rise to a covariant version of the usualWess-Zumino-Witten action.Comment: Latex File, 19 page

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