Integrating the Chirally Split Diffeomorphism Anomaly on a Compact Riemann Surface

TL;DR

This paper constructs a well-defined functional for the 2D chirally split diffeomorphism anomaly on compact Riemann surfaces, using Beltrami parametrization and background metrics, generalizing the Polyakov action.

Contribution

It introduces a new functional that integrates the chirally split diffeomorphism anomaly on arbitrary compact Riemann surfaces, extending previous flat case results.

Findings

Reproduces the Polyakov action in the flat case

Works on the torus (genus 1) surface

Provides a new approach to anomaly integration on complex surfaces

Abstract

A well-defined chirally split functional integrating the 2D chirally split diffeomorphism anomaly 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. Although it works on the torus (genus 1), the proposed functional still remains to be related to a Wess-Zumino action for diffeomorphisms.