Towards a quadratic Poisson algebra for the subtracted classical monodromy of Symmetric Space Sine-Gordon theories

TL;DR

This paper develops a regularisation method to compute the Poisson algebra of the subtracted monodromy in Symmetric Space Sine-Gordon theories, demonstrating their integrability through conserved quantities.

Contribution

It introduces a regularisation approach for the non-ultralocal Poisson algebra of the subtracted monodromy in these theories, establishing their integrability structure.

Findings

Poisson algebra satisfies the Jacobi identity

Existence of infinite conserved quantities in involution

Regularisation enables analysis of non-ultralocal theories

Abstract

Symmetric Space Sine-Gordon theories are two-dimensional massive integrable field theories, generalising the Sine-Gordon and Complex Sine-Gordon theories. To study their integrability properties on the real line, it is necessary to introduce a subtracted monodromy matrix. Moreover, since the theories are not ultralocal, a regularisation is required to compute the Poisson algebra for the subtracted monodromy. In this article, we regularise and compute this Poisson algebra for certain configurations, and show that it can both satisfy the Jacobi identity and imply the existence of an infinite number of conserved quantities in involution.