Integrating the chirally split diffeomorphism anomaly on a compact Riemann surface

A well-defined chirally split functional integrating the 2D chirally split diffeomorphism is exhibited on an arbitrary compact Riemann surface without boundary. The construction requires both the use of the Beltrami parametrisation of complex structures and the introduction of a background metric possibly subject to a Liouville equation. This formula reproduces in the flat case the so-called Polyakov action. Althrough it works on the torus (genus 1), the proposed functional still remains to be related to a Wess-Zumino action for diffeomorphisms.