# Characteristic determinant and Manakov triple for the double elliptic integrable system

### Submission summary

 Authors (as Contributors): Andrei Zotov
Submission information
Date accepted: 2021-02-24
Date submitted: 2021-02-04 21:40
Submitted by: Zotov, Andrei
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Mathematical Physics
Approach: Theoretical

### Abstract

Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding $L$-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the $L$-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the $L$-matrix.

Published as SciPost Phys. 10, 055 (2021)

This version of the paper is very close to original one.

### List of changes

We added Appendix E, which contains alternative proofs of main statements. Also some typos were corrected and some comments added. We tried to make all corrections from the lists of recommendations by Reviewers.

### Submission & Refereeing History

Resubmission 2010.08077v3 on 4 February 2021
Submission 2010.08077v1 on 1 November 2020

## Reports on this Submission

### Report 1 by Yegor Zenkevich on 2021-2-20 (Invited Report)

• Cite as: Yegor Zenkevich, Report on arXiv:2010.08077v3, delivered 2021-02-20, doi: 10.21468/SciPost.Report.2589

### Report

Two typos still need to be corrected (see "Requested changes"). Otherwise, the paper is ready for publication.

### Requested changes

(All page, equation and section numbers are according to the new version of the manuscript)

1) p.6, fourth sentence of sec.2: "paring" should read "pairing".

2) p.10, Eq.(2.32): the sign in front of q_0 hasn't been changed and still looks wrong to me. This can also be noticed by comparing Eq.(2.28) to (2.33): the two equations are compatible if there is z - N q_0 instead of z + N q_0 in the last equality of Eq.(2.32).

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -