Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model

TL;DR

This paper constructs the quantum spectral curve for the elliptic Gaudin model using Manin matrices, linking quantum groups, integrable systems, and the Langlands program.

Contribution

It introduces a novel approach to generating commuting elements in elliptic Gaudin models via Manin matrices and quantum elliptic groups.

Findings

Construction of the quantum spectral curve for elliptic Gaudin model

Realization of commutative families via Felder's elliptic quantum group

Application of Manin matrices to integrable systems

Abstract

In this paper we construct the quantum spectral curve for the quantum dynamical elliptic gl(n) Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group and taking the appropriate limit. The approach of Manin matrices here suits well to the problem of constructing the generation function of commuting elements which plays an important role in SoV and Langlands concept.