Diffeomorphism Cohomology in Beltrami Parametrization II : The 1-Forms

TL;DR

This paper investigates the cohomology of 1-forms in a 2D conformal field theory using Beltrami parametrization, analyzing classical and quantum anomalies related to currents and holomorphy regions.

Contribution

It provides a detailed analysis of 1-form diffeomorphism cohomologies in a Beltrami parametrized model, including quantum anomaly extensions and their relation to external field holomorphy.

Findings

Anomalies originate from the holomorphy region of external fields.

Only differential spin 1 and 2 currents can be anomalous.

Quantum extension clarifies current conservation violations.

Abstract

We study the 1-form diffeomorphism cohomologies within a local conformal Lagrangian Field Theory model built on a two dimensional Riemann surface with no boundary. We consider the case of scalar matter fields and the complex structure is parametrized by Beltrami differential. The analysis is first performed at the Classical level, and then we improve the quantum extension, introducing the current in the Lagrangian dynamics, coupled to external source fields. We show that the anomalies which spoil the current conservations take origin from the holomorphy region of the external fields, and only the differential spin 1 and 2 currents (as well their c.c) could be anomalous.