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

• Published in: Nuclear physics. B 2022-07, Vol.980, p.115790, Article 115790
• Main Author: Lu, Kang
• Format: Article
• Language: English
• ISSN: 0550-3213; 1873-1562
• DOI: 10.1016/j.nuclphysb.2022.115790

We study the Gaudin models associated with gl(1|1). We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation 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 [7, Conjecture 8.3]. 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 U(gl(1|1)[t]).