Loop Variables and the Virasoro Group

TL;DR

This paper introduces a novel method using a first order formalism to compute the action of finite Virasoro group elements on vertex operators, simplifying calculations by thickening loops to bands.

Contribution

It provides a closed-form expression for finite Virasoro group actions on vertex operators using a loop variable approach, enhancing computational simplicity.

Findings

Derived a closed-form expression for finite Virasoro group actions

Introduced a loop thickening technique to simplify calculations

Complemented previous algorithms with a new formalism

Abstract

We derive an expression in closed form for the action of a finite element of the Virasoro Group on generalized vertex operators. This complements earlier results giving an algorithm to compute the action of a finite string of generators of the Virasoro Algebra on generalized vertex operators. The main new idea is to use a first order formalism to represent the infinitesimal group element as a loop variable. To obtain a finite group element it is necessary to thicken the loop to a band of finite thickness. This technique makes the calculation very simple.