Bethe ansatz conjecture for quantum toroidal $\mathcal E_2$ transfer-matrix spectra

Let

$$W=\mathcal F_0(v_{0,1})\otimes\cdots\otimes\mathcal F_0(v_{0,M_0})\otimes\mathcal F_1(v_{1,1})\otimes\cdots\otimes\mathcal F_1(v_{1,M_1})$$

be a tensor product of Fock spaces, and let $T_\nu(u)$ be the transfer matrix acting on $W$. For an eigenvector $w$ of principal degree $N_0+N_1$ and weight $(N_0-N_1)\bar\alpha_1$, define

$$Q_0(u)=\prod_{i=1}^{N_0}(1-s_i/u),\qquad Q_1(u)=\prod_{i=1}^{N_1}(1-t_i/u).$$

The functions $\mathbf{\mathfrak a}_0(u)$ and $\mathbf{\mathfrak a}_1(u)$ are the rational functions specified by the products of the Fock parameters, the $Q_i$, and the parameters $p_0=\bar p\bar p_1^{-1}q^{-M_0}$ and $p_1=\bar p_1q^{-M_1}$ in the statement.

Quantum toroidal $\mathcal E_2$ Bethe ansatz conjecture. For each such eigenvector, there exist polynomials $Q_0$ and $Q_1$ whose roots satisfy the two displayed Bethe ansatz equations, and the corresponding eigenvalue of $T_\nu(u)$ is the displayed partition sum involving $\varphi_\nu(u)$, $Q_\nu(q_2^{-1}u)/Q_\nu(u)$, and $\mathbf{\mathfrak a}_{c_\nu(\square)}(q^{-\square}u)$.

This is the proposed spectral description of the $\mathcal E_2$ transfer matrices by Bethe roots. It is formulated as a conjecture in the paper and is used later to motivate the affine Gaudin description of qKdV non-local integrals of motion.