Characteristic determinant and Manakov triple for the double elliptic integrable system

TL;DR

This paper introduces a determinant representation for the generating function of commuting Hamiltonians in the double elliptic integrable system, connecting it to spectral curves, Lax matrices, and the Manakov triple framework.

Contribution

It provides a novel determinant formula for the Hamiltonians, reproduces eigenvalues for the dual elliptic Ruijsenaars model, and constructs an explicit L-matrix satisfying the Manakov triple.

Findings

Determinant representation for Hamiltonians derived

Eigenvalues for dual elliptic Ruijsenaars model reproduced

Explicit L-matrix satisfying Manakov triple constructed

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.