String-inspired Methods and the Worldline Formalism in Curved Space

TL;DR

This paper advances the worldline formalism in curved space by deriving Friedan's map in Riemann Normal Coordinates for maximally symmetric spaces, enabling accurate anomaly calculations.

Contribution

It provides a closed-form expression for Friedan's map in RNC coordinates and demonstrates its effectiveness in computing trace anomalies in curved space.

Findings

Derived Friedan's map explicitly for RNC coordinates in maximally symmetric spaces

Validated the approach by successfully computing trace anomalies matching known results

Enhanced the string-inspired worldline formalism for curved space applications

Abstract

The worldline approach to Quantum Field Theory (QFT) allows to efficiently compute several quantities, such as one-loop effective actions, scattering amplitudes and anomalies, which are linked to particle path integrals on the circle. A helpful tool in the worldline formalism on the circle, are string-inspired (SI) Feynman rules, which correspond to a specific way of factoring out a zero mode. In flat space this is known to generate no difficulties. In curved space, it was shown how to correctly achieve the zero mode factorization by applying BRST techniques to fix a shift symmetry. Using special coordinate systems, such as Riemann Normal Coordinates, implies the appearance of a non-linear map---originally introduced by Friedan---which must be taken care of in order to obtain the correct results. In particular, employing SI Feynman rules, the map introduces further interactions in the worldline path integrals. In the present paper, we compute in closed form Friedan's map for RNC coordinates in maximally symmetric spaces, and test the path integral model by computing trace anomalies. Our findings match known results.