Generating Series for Nested Bethe Vectors

TL;DR

This paper introduces a new framework for expressing nested relations among off-shell Bethe vectors using generating series, leading to hierarchical relations that facilitate deriving Bethe equations for quantum affine algebras.

Contribution

It reformulates nested Bethe vector relations as generating series equations and derives hierarchical relations, advancing the analytical Bethe ansatz methodology.

Findings

Derived hierarchical relations between Bethe vectors.

Applied relations to obtain Bethe equations for U_q(gl_2).

Provided a new approach to analyze off-shell Bethe vectors.

Abstract

We reformulate nested relations between off-shell $U_q(\widehat{\mathfrak{gl}}_N)$ Bethe vectors as a certain equation on generating series of strings of the composed $U_q(\widehat{\mathfrak{gl}}_N)$ currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincar\'e 7 (2006), 1217-1268, math-ph/0512037] for the parameters of universal Bethe vectors of the algebra $U_q(\widehat{\mathfrak{gl}}_2)$.