Chiral fermions on 2D curved spacetimes

TL;DR

This paper explores the quantization of chiral fermions on 2D curved spacetimes, addressing anomalies and singularities, and identifies geometric conditions for consistent conformal symmetry.

Contribution

It introduces a quantization scheme based on the Green's function of the classical field equation and analyzes conditions for well-defined short-distance behavior.

Findings

Explicit computation of the Green's function singular term

Identification of geometric constraints for anomaly resolution

Conditions for local conformal symmetry on special backgrounds

Abstract

The theory of free Majorana-Weyl spinors is the prototype of conformal field theory in two dimensions in which the gravitational anomaly and the Weyl anomaly obstruct extending the flat spacetime results to curved backgrounds. In this paper, we investigate a quantization scheme in which the short distance singularity in the two-point function of chiral fermions on a two dimensional curved spacetime is given by the Green's function corresponding to the classical field equation. We compute the singular term in the Green's function explicitly and observe that the short distance limit is not well-defined in general. We identify constraints on the geometry which are necessary to resolve this problem. On such special backgrounds the theory has locally $c=\frac{1}{2}$ conformal symmetry.