BRST treatment of zero modes for the worldline formalism in curved space

TL;DR

This paper develops a systematic BRST-based method for handling zero modes in the worldline formalism of quantum field theory in curved space, ensuring reparametrization invariance and analyzing anomalies.

Contribution

It introduces a novel BRST approach to treat zero modes in 1D path integrals for nonlinear sigma models, emphasizing reparametrization invariance and supersymmetry.

Findings

Verified the method with multiple examples

Confirmed the chiral anomaly does not get higher order contributions

Ensured reparametrization invariance in the treatment of zero modes

Abstract

One-loop quantities in QFT can be computed in an efficient way using the worldline formalism. The latter rests on the ability of calculating 1D path integrals on the circle. In this paper we give a systematic discussion for treating zero modes on the circle of 1D path integrals for both bosonic and supersymmetric nonlinear sigma models, following an approach originally introduced by Friedan. We use BRST techniques and place a special emphasis on the issue of reparametrization invariance. Various examples are extensively analyzed to verify and test the general set-up. In particular, we explicitly check that the chiral anomaly, which can be obtained by the semiclassical approximation of a supersymmetric 1D path integral, does not receive higher order worldline contributions, as implied by supersymmetry.