Analytic Bethe Ansatz for Fundamental Representations of Yangians

TL;DR

This paper develops an analytic Bethe ansatz framework for fundamental representations of Yangians and their quantum affine analogues, providing explicit eigenvalue formulas and proving key properties like pole-freeness.

Contribution

It introduces explicit eigenvalue formulas for transfer matrices of all fundamental representations of Yangians and proves their pole-freeness under Bethe ansatz equations, extending to higher representations.

Findings

Eigenvalue formulas for all fundamental representations

Proof of pole-freeness under Bethe equations

Conjectures on higher representations using T-system

Abstract

We study the analytic Bethe ansatz in solvable vertex models associated with the Yangian $Y(X_r)$ or its quantum affine analogue $U_q(X^{(1)}_r)$ for $X_r = B_r, C_r$ and $D_r$. Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations of $Y(X_r)$. Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying the $T$-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.