Conjecture 4.1: Bethe ansatz solutions for evaluation Verma modules of quantum toroidal gl2

Establish that for generic parameters q, q1, µ, p0, and p1, every joint eigenvector of the Bethe algebra A(p0, p1) acting on the evaluation Verma module Gµ of the quantum toroidal gl2 algebra (in the perpendicular realization) of degree (l, d) = (l − l0, −l0) admits polynomials y0(x) = ∏_{i=1}^{l0} (x − s_i) and y1(x) = ∏_{i=1}^{l1} (x − t_i) with non-zero distinct roots that satisfy the explicit Bethe ansatz equations provided in the paper, while ensuring none of the factors appearing in those equations vanish.

Background

Section 4 introduces evaluation Verma modules Gµ for the perpendicular realization E⊥ of the quantum toroidal gl2 algebra via the Miki evaluation map. The authors describe the Bethe algebra action and formulate explicit Bethe ansatz equations in terms of polynomials y0 and y1 with roots s_i and t_i. They conjecture that for generic parameters, each eigenvector produces polynomials satisfying these equations and a non-vanishing condition on all factors.

This conjecture provides a precise spectral characterization for non-highest weight modules in the quantum toroidal setting, extending earlier results known for highest weight modules and connecting eigenvectors to q-difference opers via Bethe ansatz relations.