Loop Variables and the Interacting Open String in a Curved Background

TL;DR

This paper develops a systematic, gauge-invariant method to derive interacting equations of motion for open string modes in curved backgrounds using loop variables, extending previous free-space approaches.

Contribution

It introduces a covariantization procedure for interacting open string equations in curved space, accounting for mode mixing and coupling among all string modes.

Findings

Derived gauge-invariant equations for open strings in curved backgrounds.

Showed that loop variables are deformed to include mode mixing.

Established that all string modes are necessary for gauge invariance in interactions.

Abstract

Applying the loop variable proposal to a sigma model (with boundary) in a curved target space, we give a systematic method for writing the gauge and generally covariant interacting equations of motion for the modes of the open string in a curved background. As in the free case described in an earlier paper, the equations are obtained by covariantizing the flat space (gauge invariant) interacting equations and then demanding gauge invariance in the curved background. The resulting equation has the form of a sum of terms that would individually be gauge invariant in flat space or at zero interaction strength, but mix amongst themselves in curved space when interactions are turned on. The new feature is that the loop variables are deformed so that there is a mixing of modes. Unlike the free case, the equations are coupled, and all the modes of the open string are required for gauge invariance.