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

TL;DR

This paper calculates the Weyl determinant for two-dimensional non-abelian gauge theories in curved space using $$-function regularization, revealing new forms of anomalies and Wess-Zumino-Witten terms.

Contribution

It introduces a cohomological approach to define and compute the Weyl determinant, deriving both consistent and covariant anomalies and their associated W-Z-W terms in curved backgrounds.

Findings

Exact functional determinants for anomalies computed

Consistent determinant aligns with Leutwyler's classical result

Covariant determinant yields a covariant Wess-Zumino-Witten action

Abstract

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 $\zeta$-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.