Completeness of Bethe ansatz for Gaudin models associated with gl(1|1)

TL;DR

This paper proves the completeness of the Bethe ansatz for Gaudin models related to gl(1|1), showing a bijection between eigenvectors and divisors of a polynomial, and confirming a prior conjecture.

Contribution

It provides an explicit description of the algebra of Hamiltonians for gl(1|1) Gaudin models and confirms the conjecture that each eigenspace is one-dimensional.

Findings

Each eigenspace of the Hamiltonian algebra is one-dimensional.

There is a bijection between eigenvectors and polynomial divisors.

The paper expresses transfer matrices in terms of the quadratic transfer matrix and the center of the universal enveloping algebra.

Abstract

We study the Gaudin models associated with $\mathfrak{gl}(1|1)$. We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}(1|1)[t]$-modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. Therefore, we confirm Conjecture 8.3 from arXiv:1809.01279. We also give dimensions of the generalized eigenspaces. Moreover, we express the generating pseudo-differential operator of Gaudin transfer matrices associated to antisymmetrizers in terms of the quadratic Gaudin transfer matrix and the center of $\mathrm{U}(\mathfrak{gl}(1|1)[t])$.