Loop Variables and Gauge Invariance in Closed Bosonic String Theory

TL;DR

This paper extends a gauge-invariant loop variable approach from open to closed bosonic strings at tree level, enabling off-shell descriptions that match on-shell amplitudes and preserve gauge invariance.

Contribution

It introduces a novel method to describe off-shell closed string amplitudes using loop variables, building on open string techniques and ensuring gauge invariance.

Findings

Off-shell closed string amplitudes are constructed as products of open string amplitudes.

The approach maintains gauge invariance off-shell.

On-shell amplitudes match traditional closed string results.

Abstract

We extend an earlier proposal for a gauge invariant description of off-shell open strings (at tree level), using loop variables, to off-shell closed strings (at tree level). The basic idea is to describe the closed string amplitudes as a product of two open string amplitudes (using the technique of kawai, Lewellen and Tye). The loop variable techniques that were used earlier for open strings can be applied here {\it mutatis mutandis}. It is a proposal for a theory whose on-shell amplitudes coincide with those of the closed bosonic string in 26 dimensions. It is also gauge invariant off-shell. As was the case with the open string, the interacting closed string looks like a free closed string thickened to a band.