Aspects of Symmetry in Sine-Gordon Theory

TL;DR

This paper explores a novel non-abelian symmetry, specifically a Witt algebra, in the sine-Gordon integrable system, highlighting its unique properties and potential implications for quantum theory and soliton solutions.

Contribution

It identifies and describes a Witt algebra of vector fields in sine-Gordon theory, distinct from the Virasoro algebra, and details its local action on soliton variables with geometric significance.

Findings

Witt algebra appears as a symmetry in sine-Gordon theory

The symmetry acts locally on N-soliton solutions

Preliminary insights into quantisation are discussed

Abstract

As a prototype of powerful non-abelian symmetry in an Integrable System, I will show the appearance of a Witt algebra of vector fields in the SG theory. This symmetry does not share anything with the well-known Virasoro algebra of the conformal $c=1$ unperturbed limit. Although it is quasi-local in the SG field theory, nevertheless it gives rise to a local action on $N$-soliton solution variables. I will explicitly write the action on special variables, which possess a beautiful geometrical meaning and enter the Form Factor expressions of quantum theory. At the end, I will also give some preliminary hints about the quantisation.