Loop Variables and the (Free) Open String in a Curved Background

TL;DR

This paper develops a systematic, gauge-invariant method for deriving covariant equations of motion for free open string modes in curved backgrounds using the loop variable formalism, including explicit examples.

Contribution

It introduces a covariantization procedure for string equations in curved space that maintains gauge invariance, extending previous flat space approaches.

Findings

Derived gauge and covariant equations for open string modes in curved backgrounds.

Explicitly worked out the spin two case demonstrating the method.

Showed how curvature couplings naturally emerge from gauge invariance requirements.

Abstract

Using the loop variable formalism as applied to a sigma model in curved target space, we give a systematic method for writing down gauge and generally covariant equations of motion for the modes of the free open string in curved space. The equations are obtained by covariantizing the flat space equation and then demanding gauge invariance, which introduces additional curvature couplings. As an illustration of the procedure, the spin two case is worked out explicitly.