Loop Variables and Gauge Invariant Interactions - I

TL;DR

This paper extends the loop variable approach to formulate gauge-invariant interacting equations for all modes of bosonic open strings, emphasizing off-shell invariance and a higher-dimensional framework.

Contribution

It generalizes the loop variable method to include interactions with gauge invariance for all string modes, introducing a band-like interaction structure.

Findings

Interaction equations are exactly gauge invariant off shell.

The formulation involves infinite gauge coordinates for space-time fields.

The theory is set in a higher dimension where modes are massless.

Abstract

We describe a method of writing down interacting equations for all the modes of the bosonic open string. It is a generalization of the loop variable approach that was used earlier for the free, and lowest order interacting cases. The generalization involves, as before, the introduction of a parameter to label the different strings involved in an interaction. The interacting string has thus becomes a ``band'' of finite width. The interaction equations expressed in terms of loop variables, has a simple invariance that is exact even off shell. A consistent definition of space-time fields requires the fields to be functions of all the infinite number of gauge coordinates (in addition to space time coordinates). The theory is formulated in one higher dimension, where the modes appear massless. The dimensional reduction that is needed to make contact with string theory (which has been discussed earlier for the free case) is not discussed here.