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

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 $\mathrm{U}(\mathfrak{gl}(1|1)[t])$.

arXiv commentLaTeX, 22 pages. This is a continuation of arXiv:1910.13360

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Cited by in corpus (2)

• On Bethe eigenvectors and higher transfer matrices for supersymmetric spin chains
• Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras