# Algebraic structure of classical integrability for complex sine-Gordon

### Submission summary

 As Contributors: Luc FRAPPAT Arxiv Link: https://arxiv.org/abs/1911.06720v2 (pdf) Date accepted: 2020-02-11 Date submitted: 2020-01-30 Submitted by: FRAPPAT, Luc Submitted to: SciPost Physics Discipline: Physics Subject area: Mathematical Physics Approach: Theoretical

### Abstract

The algebraic structure underlying the classical $r$-matrix formulation of the complex sine-Gordon model is fully elucidated. It is characterized by two matrices $a$ and $s$, components of the $r$ matrix as $r=a-s$. They obey a modified classical reflection/Yang--Baxter set of equations, further deformed by non-abelian dynamical shift terms along the dual Lie algebra $su(2)^*$. The sign shift pattern of this deformation has the signature of the twisted boundary dynamical algebra. Issues related to the quantization of this algebraic structure and the formulation of quantum complex sine-Gordon on those lines are introduced and discussed.

Published as SciPost Phys. 8, 033 (2020)

### List of changes

Dear Editor,

We have revised our manuscript according to the requested changes of the referee. Here are the major modifications:
- page 2, the sentence has been modified to clarify the point.
- we added three paragraphs in section 3 :
a/ page 8, a comment on an alternative procedure of quantizing the complex sine-Gordon model (with refs 23 and 24 added);
b/ page 8, a comment on the potential existence of discrete quantum systems related to CSG (with ref. 26).
c/ page 9, a final paragraph about the feasibility of the quantization itself.

In addition, the grammar and the punctuation have been revised.
Best regards,
The authors

### Submission & Refereeing History

Resubmission 1911.06720v2 on 30 January 2020
Submission 1911.06720v1 on 19 November 2019