# Algebraic structure of classical integrability for complex sine-Gordon

### Submission summary

 As Contributors: Luc FRAPPAT Arxiv Link: https://arxiv.org/abs/1911.06720v1 (pdf) Date submitted: 2019-11-19 01:00 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.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 1911.06720v2 on 30 January 2020

Submission 1911.06720v1 on 19 November 2019

## Reports on this Submission

### Anonymous Report 1 on 2019-12-20 Invited Report

• Cite as: Anonymous, Report on arXiv:1911.06720v1, delivered 2019-12-20, doi: 10.21468/SciPost.Report.1406

### Strengths

1. Comprehensive analysis of the complex sine-Gordon model in the Hamiltonian formalism.
2. Uncovering of the novel Poisson bracket structure of the model.
3. Interesting discussion of how the classical structure could be quantized.

### Weaknesses

1. Not clear whether this approach will lead to a successful quantization of the model, are there many similar model where that is known (e.g. sine Gordon theory is related to a spin chain, can we expect the same for the complex sine Gordon?)

### Report

The paper analyses the classical integrability of the complex sine-gordon theory in the Hamiltonian formalism as a precursor to a quantum treatment. There is a preliminary discussion of the quantum version of the model. The structure uncovered in rather non-standard and this makes the investigation interesting and novel and worthy of publication.

### Requested changes

1. bookkeeping -> book-keeping
2. page 2 "We answer this last question in this paper". What question is this referring to?
3. generally the grammar used in the paper is awkward. There are instances where an article should be used (the or a), for example. The use (or lack) of commas is another issue. I won't go through the paper line by line pointing out the language deficiencies, but suggest a careful edit to make it read better.

• validity: high
• significance: good
• originality: high
• clarity: good
• formatting: good
• grammar: reasonable